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9-22-17
Holocaust Denier’s Sentence: Visit 5 Ex-Nazi Camps, and Write About ItBreaking News
tags: Holocaust, Belgium, Holocaust Denier. Mr. Louis left Parliament in 2014.
Mr. Louis was given a six-month suspended jail sentence and fined over $20,000 at his 2015 trial, which centered on online statements he made that questioned the number of Jews killed in gas chambers during the Holocaust. After that sentence was changed on Wednesday, he celebrated on Facebook and apologized “to anyone who may have been hurt by my remarks.”
| 52,337
|
section "Tabulating the Balanced Predicate"
theory Root_Balanced_Tree_Tab
imports
Root_Balanced_Tree
"HOL-Decision_Procs.Approximation"
"HOL-Library.IArray"
begin
locale Min_tab =
fixes p :: "nat \<Rightarrow> nat \<Rightarrow> bool"
fixes tab :: "nat list"
assumes mono_p: "n \<le> n' \<Longrightarrow> p n h \<Longrightarrow> p n' h"
assumes p: "\<exists>n. p n h"
assumes tab_LEAST: "h < length tab \<Longrightarrow> tab!h = (LEAST n. p n h)"
begin
lemma tab_correct: "h < length tab \<Longrightarrow> p n h = (n \<ge> tab ! h)"
apply auto
using not_le_imp_less not_less_Least tab_LEAST apply auto[1]
by (metis LeastI mono_p p tab_LEAST)
end
definition bal_tab :: "nat list" where
"bal_tab = [0, 1, 1, 2, 4, 6, 10, 16, 25, 40, 64, 101, 161, 256, 406, 645, 1024,
1625, 2580, 4096, 6501, 10321, 16384, 26007, 41285, 65536, 104031, 165140,
262144, 416127, 660561, 1048576, 1664510, 2642245, 4194304, 6658042, 10568983,
16777216, 26632170, 42275935, 67108864, 106528681, 169103740, 268435456,
426114725, 676414963, 1073741824, 1704458900, 2705659852, 4294967296\<^cancel>\<open>,
6817835603\<close>]"
(*ML\<open>floor (Math.pow(2.0,5.0/1.5))\<close>*)
axiomatization where c_def: "c = 3/2"
fun is_floor :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
"is_floor n h = (let m = floor((2::real) powr ((real(h)-1)/c)) in n \<le> m \<and> m \<le> n)"
text\<open>Note that @{prop"n \<le> m \<and> m \<le> n"} avoids the technical restriction of the
\<open>approximation\<close> method which does not support \<open>=\<close>, even on integers.\<close>
lemma bal_tab_correct:
"\<forall>i < length bal_tab. is_floor (bal_tab!i) i"
apply(simp add: bal_tab_def c_def All_less_Suc)
apply (approximation 50)
done
(* FIXME mv *)
lemma ceiling_least_real: "ceiling(r::real) = (LEAST i. r \<le> i)"
by (metis Least_equality ceiling_le le_of_int_ceiling)
lemma floor_greatest_real: "floor(r::real) = (GREATEST i. i \<le> r)"
by (metis Greatest_equality le_floor_iff of_int_floor_le)
lemma LEAST_eq_floor:
"(LEAST n. int h \<le> \<lceil>c * log 2 (real n + 1)\<rceil>) = floor((2::real) powr ((real(h)-1)/c))"
proof -
have "int h \<le> \<lceil>c * log 2 (real n + 1)\<rceil>
\<longleftrightarrow> 2 powr ((real(h)-1)/c) < real(n)+1" (is "?L = ?R") for n
proof -
have "?L \<longleftrightarrow> h < c * log 2 (real n + 1) + 1" by linarith
also have "\<dots> \<longleftrightarrow> (real h-1)/c < log 2 (real n + 1)"
using c1 by(simp add: field_simps)
also have "\<dots> \<longleftrightarrow> 2 powr ((real h-1)/c) < 2 powr (log 2 (real n + 1))"
by(simp del: powr_log_cancel)
also have "\<dots> \<longleftrightarrow> ?R"
by(simp)
finally show ?thesis .
qed
moreover have "((LEAST n::nat. r < n+1) = nat(floor r))" for r :: real
by(rule Least_equality) linarith+
ultimately show ?thesis by simp
qed
interpretation Min_tab
where p = bal_i and tab = bal_tab
proof(unfold bal_i_def, standard, goal_cases)
case (1 n n' h)
have "int h \<le> ceiling(c * log 2 (real n + 1))" by(rule 1[unfolded bal_i_def])
also have "\<dots> \<le> ceiling(c * log 2 (real n' + 1))"
using c1 "1"(1) by (simp add: ceiling_mono)
finally show ?case .
next
case (2 h)
show ?case
proof
show "int h \<le> \<lceil>c * log 2 (real (2 ^ h - 1) + 1)\<rceil>"
apply(simp add: of_nat_diff log_nat_power) using c1
by (metis ceiling_mono ceiling_of_nat order.order_iff_strict mult.left_neutral mult_eq_0_iff of_nat_0_le_iff mult_le_cancel_iff1)
qed
next
case 3
thus ?case using bal_tab_correct LEAST_eq_floor
by (simp add: eq_iff[symmetric]) (metis nat_int)
qed
text\<open>Now we replace the list by an immutable array:\<close>
definition bal_array :: "nat iarray" where
"bal_array = IArray bal_tab"
text\<open>A trick for code generation: how to get rid of the precondition:\<close>
lemma bal_i_code:
"bal_i n h =
(if h < IArray.length bal_array then IArray.sub bal_array h \<le> n else bal_i n h)"
by (simp add: bal_array_def tab_correct)
end
| 194,933
|
Our view: Anger over MNsure Paul Bunyan ads misplacedFolks.
Folks.
There’s Paul Bunyan water-skiing into a tree, Paul Bunyan flying off a treadmill into a wall and Paul Bunyan sledding into a tree. Minnesota, the “land of 10,000 reasons to get health insurance,” goes the catchy catch phrase at the end.
“Not impressed,” Bemidji Mayor Rita Albrecht tweeted.
“To have him be a doofus, it doesn’t make sense,” she told the St. Paul Pioneer Press.
“Paul Bunyan is a Minnesota icon and should not be used as an actor,” chimed in state Sen. Carrie Ruud, R-Breezy Point, in an op-ed she sent around the state, including to the News Tribune.
“He has been made into a punch line,” she wrote, calling the campaign and its ads “offensive,” “inappropriate” and “wrong.”
Lighten up.
Whether you championed or fought bitterly against it, the president’s health-care overhaul is reality. It’s kicking in. And there’s a definite need to make the best of what some predict will be a taxing and expensive, bad situation. There’s a definite need to educate Minnesotans about MNsure and how to sign up for care. MNsure’s Paul Bunyan ads are light-hearted, humorous, memorable and very Minnesotan, much like the Minnesota Lottery ads or Minnesota State Fair ads. They’re effectively doing the job.
If anyone wants to get mad they can balk at the advertising campaign’s hefty price tag: anywhere from $9 million, according to numerous news reports, to $22 million, according to Sen. Ruud. Advertising and marketing campaigns aren’t cheap. But can taxpayers be assured they’re getting good value here? Multimillions is a whole lot of our tax dollars, no matter what the specific number.
Plenty of Minnesotans already aren’t so sure. In a Pioneer Press online poll last week, nearly 80 percent of 1,228 respondents said even $9 million was too much. While not scientific, the poll was quite telling.
Anyone skeptical of the spending couldn’t have felt much better finding out that MNsure officials tried to exploit “a gap in the law” (MNsure spokesman John Reich’s words in a Star Tribune article) to keep secret details of their marketing campaign. The state Department of Administration rightly denied a July 8 MNsure request to keep public information about the public spending of public money out of the public eye. As a result of the ruling, the marketing data will become public Sept. 9.
To their credit, Bemidji officials changed their tune about the way their statue is being depicted.
“Ultimately, we’re flattered,” chamber President Lori Paris later acknowledged.
“We’ve decided to make lemonade out of lemons,” Mayor Albrecht said, also later, according to the Bemidji Pioneer.
Cynics, critics of the Affordable Care Act and others can do likewise. But none of us should stop keeping a close eye on how our tax dollars are being spent.
Tags: our view, opinion, editorials, minnesota, health, politics, money
| 264,625
|
TITLE: Left adjoint of evaluation functor
QUESTION [3 upvotes]: This is Theorem 1.10, pg5
(Kan) Let $A$ be a small category, together with a locally small category $C$ with has small colimits. For any functor $u:A \rightarrow C$, the evaluation at $u$,
$$u^*:C \rightarrow \hat{A}, \quad Y \mapsto u^*(Y):(a \mapsto Hom(u(a),Y).$$
has a left adjoint $u_{!}:\hat{A} \rightarrow C$. Moreover, there is a unique natural isomorphism
$$u(a) \simeq u_{!}(h_a), a \in ob(A). $$
It begins with
For each presheaf $X$ over $A$, we choose a colimit of the functor
$$A/X \rightarrow C, (a,s) \mapsto u(a). $$
which we denote by $u_{!}(X)$.
$A/X$ is the category of elements of $X$ defined on pg4. I do not understand the proof where it says
We have a canonical isomorphism $u(a) \simeq u_{!}(h_a)$ since $(a,1_a)$ is a final object of $(A,h_a)$.
REPLY [4 votes]: It's just an application of the fact that if a category $\mathcal{J}$ has a terminal object $1$, then the colimit of any functor $F:\mathcal{J}\to \mathcal{C}$ is just $F(1)$, with the cocone defined by the maps $F(\tau_j):F(j)\to F(1)$. For a proof of this fact, see this question; alternatively, if you know about final functors, you can show that the inclusion functor $\mathbf{1}\to\mathcal{J}$ that takes the unique object of $\mathbf{1}$ to the terminal object of $\mathcal{J}$ is final.
| 150,306
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\begin{document}
\author{\normalsize Rory B.B. Lucyshyn-Wright\thanks{Partial financial assistance by the Ontario Graduate Scholarship program is gratefully acknowledged.}\let\thefootnote\relax\footnote{2010 Mathematics Subject Classification: 18A35, 18B25, 18B30, 18B35}\let\thefootnote\relax\footnote{Keywords: totally distributive category; topos; injective topos; essential subtopos; essential localization; continuous category}
\\
\small York University, 4700 Keele St., Toronto, ON, Canada M3J 1P3}
\title{\large \textbf{Totally distributive toposes}}
\date{}
\maketitle
\abstract{A locally small category $\E$ is \textit{totally distributive} (as defined by Rosebrugh-Wood) if there exists a string of adjoint functors $t \dashv c \dashv \y$, where $\y:\E \rightarrow \h{\E}$ is the Yoneda embedding. Saying that $\E$ is \textit{lex totally distributive} if, moreover, the left adjoint $t$ preserves finite limits, we show that the lex totally distributive categories with a small set of generators are exactly the \textit{injective Grothendieck toposes}, studied by Johnstone and Joyal. We characterize the totally distributive categories with a small set of generators as exactly the \textit{essential} subtoposes of presheaf toposes, studied by Kelly-Lawvere and Kennett-Riehl-Roy-Zaks.
}
\section{Introduction} \label{sec:intro}
The aim of this paper is to establish certain connections between the work of Marmolejo, Rosebrugh, and Wood \cite{RWsets,MRW} on \textit{totally distributive categories} and two other bodies of work on distinct topics: Firstly, that of Johnstone and Joyal \cite{Jinj,JJ} on \textit{injective toposes} and \textit{continuous categories}, and secondly, that of Kelly-Lawvere \cite{KL} and Kennett-Riehl-Roy-Zaks \cite{KRRZ} on \textit{essential localizations} and \textit{essential subtoposes}. One of our observations, \ref{thm:main} (2), when taken together with a theorem of Kelly-Lawvere which we recall in \ref{rem:KL_thm}, yields a concrete combinatorial description of all totally distributive categories with a small set of generators.
We adopt the foundational conventions of $\cite{Jele}$ (and \cite{Jinj,JJ}), since our only use of the stronger foundational assumptions of $\cite{SW,St,W,RWsets,MRW}$ is made in finally deducing our main results (\ref{thm:main}) as strengthened variants of propositions which precede them. We let $\CCAATT$ represent the meta-2-category of categories, functors, and natural transformations (see \cite{Jele}, 1.1.1), and we let $\CAT$ be its full sub-(meta)-2-category consisting of locally small categories.
\renewcommand*\theparagraph{\thesection.\arabic{subsection}}
\addtocounter{subsection}{1}
\paragraph{Completely distributive lattices, totally distributive categories.} \label{sec:intro_cd_lats_td_cats}
A poset $\E$ is a \textit{constructively completely distributive lattice} \cite{FW}, or \textit{ccd lattice}, if there exist adjunctions
$$
\xymatrix {
\E \ar@/_1.2pc/[rr]_\threeheaddownarrow^{\top} \ar@/^1.2pc/[rr]^\downarrow_{\top} & & \Dn(\E) \ar[ll]|\vee
}
$$
where $\Dn(\E)$ is the poset of down-closed subsets of $\E$, ordered by inclusion, and $\downarrow:\E \rightarrow \Dn(\E)$ is the embedding given by $v \mapsto \;\downarrow v := \{u \in \E \mid u \lt v\}$. The existence of the left adjoint $\vee$ of $\downarrow$ is equivalent to the cocompleteness of $\E$, i.e. the condition that $\E$ be a complete lattice, and if such a map $\vee$ exists, it necessarily sends each down-closed subset to its join in $\E$. In the presence of the axiom of choice, a poset is a ccd lattice iff it is a completely distributive lattice in the usual sense \cite{FW}.
Rosebrugh and Wood \cite{RWsets} have defined an analogue of this notion for arbitrary categories rather than just posets\footnote{Marmolejo, Rosebrugh, and Wood \cite{MRW} have also studied an apparently distinct analogue --- the notion of \textit{completely distributive category}.}. A locally small category $\E$ is \textit{totally distributive} if there exist adjunctions
$$
\xymatrix {
\E \ar@/_1.1pc/[rr]_t^{\top} \ar@/^1.1pc/[rr]^\y_{\top} & & \h{\E} \ar[ll]|c
}
$$
where $\h{\E}$ is the presheaf category $[\E^\op, \Set]$ and $\y$ is the Yoneda embedding, given by $v \mapsto \h{v} := \E(-,v)$. We say that a totally distributive category $\E$ is \textit{lex totally distributive} if the associated functor $t:\E \rightarrow \h{\E}$ preserves finite limits.
The existence of the left adjoint $c$ of $\y$ is the requirement that $\E$ be \textit{total} \cite{SW}, or \textit{totally cocomplete}. This left adjoint $c$ of $\y$ is characterized by the property that
\begin{equation}\label{eq:col_of_presh}cE \cong \cl{\h{u} \rightarrow E} u = \col((\E \downarrow E) \rightarrow \E) \cong \int^{u \in \E}Eu \cdot u\end{equation}
naturally in $E \in \h{\E}$, so that totality is equivalent to the existence of a colimit in $\E$ of the (possibly large) canonical diagram of each presheaf $E$ on $\E$.
Note that any totally distributive category $\E$ is in particular \textit{lex total}, meaning that $\E$ is total and the associated functor $c:\h{\E} \rightarrow \E$ preserves finite limits. Wood \cite{W} attributes to Walters the theorem that those lex total categories with a small set of generators are exactly the Grothendieck toposes; the paper \cite{St} of Street includes a proof of this result.
\addtocounter{subsection}{1}
\paragraph{Continuous dcpos, continuous categories.}
A poset $\X$ is a \textit{continuous dcpo} if there exist adjunctions
$$
\xymatrix {
\X \ar@/_1.2pc/[rr]_\twoheaddownarrow^{\top} \ar@/^1.2pc/[rr]^\downarrow_{\top} & & \Idl(\X) \ar[ll]|\vee
}
$$
where $\Idl(\X)$ is the poset of ideals of $\X$ (i.e. upward-directed down-closed subsets of $\X$), ordered by inclusion, and $\downarrow:\X \rightarrow \Idl(\X)$ is the embedding given by $y \mapsto \;\downarrow y := \{x \in \X \mid x \lt y\}$. The existence of the left adjoint $\vee$ of $\downarrow$ is equivalent to the existence of all directed joins in $\X$, i.e. the condition that $\X$ be a \textit{dcpo}, or \textit{directed complete partial order}, and if such a map $\vee$ exists, it necessarily sends each ideal to its join in $\E$.
Johnstone and Joyal \cite{JJ} have defined a generalization of this notion to arbitrary categories, rather than just posets, as follows. We say that a locally small category $\X$ is \textit{continuous} if there exist adjunctions
$$
\xymatrix {
\X \ar@/_1.1pc/[rr]_w^{\top} \ar@/^1.1pc/[rr]^\m_{\top} & & \Ind{\X} \ar[ll]|\col
},
$$
where $\Ind{\X}$ is the \textit{ind-completion} of $\X$, whose objects are all small filtered diagrams in $\X$, and $\m$ is the canonical full embedding sending each object $x \in \X$ to the diagram $1 \rightarrow \X$, indexed by the terminal category $1$, with constant value $x$.
The existence of the left adjoint $\col$ of $\m:\X \rightarrow \Ind{X}$ is equivalent to the requirement that $\X$ be equipped with colimits for all small filtered diagrams, and $\col$ necessarily sends each $D \in \Ind{\X}$ to a colimit of $D$ in $\X$.
\addtocounter{subsection}{1}
\paragraph{Stone duality for continuous dcpos.} \label{sec:stone_duality_for_cts_dcpos}
It was shown by Hoffmann \cite{H} and Lawson \cite{L} that the category of continuous dcpos and directed-meet-preserving maps is dually equivalent to the category of completely distributive lattices and maps preserving finite meets and arbitrary joins.
The category of continuous dcpos is isomorphic to the full subcategory of topological spaces consisting of continuous dcpos endowed with the \textit{Scott topology}, and the given dual equivalence of this category of spaces with the category of completely distributive lattices is a restriction of the dual equivalence between sober spaces and spatial frames (see, e.g., \cite{Jsto}), associating to a space its frame of open sets.
Subsequent work of Banaschewski \cite{B} entails that this dual equivalence restricts further to a dual equivalence between \textit{continuous lattices} (i.e. those continuous dcpos which are also complete lattices) and \textit{stably supercontinuous lattices}, also known as \textit{lex ccd lattices} \cite{MRW} or \textit{lex completely distributive lattices}, which are those ccd lattices for which the left adjoint $\threeheaddownarrow$ preserves finite meets. Scott \cite{Sc} had shown earlier that the continuous lattices, when endowed with their Scott topologies, are exactly the \textit{injective} $T_0$ spaces.
\addtocounter{subsection}{1}
\paragraph{Continuous categories and injective toposes.}
Scott's isomorphism between injective $T_0$ spaces and continuous lattices \cite{Sc} has a topos-theoretic analogue, given by Johnstone-Joyal \cite{JJ}, which we now recall.
First let us record the following earlier result of Johnstone \cite{Jinj}:
\begin{ThmSubSub} \label{thm:injtop_as_retracts}
\emph{(Johnstone \cite{Jinj}).}
A Grothendieck topos $\E$ is injective (with respect to geometric inclusions) if and only if $\E$ is a retract, by geometric morphisms, of a presheaf topos $\h{\C}$ with $\C$ a small finitely-complete category.
\end{ThmSubSub}
We call such Grothendieck toposes \textit{injective toposes}. A \textit{quasi-injective topos} \cite{JJ} is defined as a Grothendieck topos which is a retract, by geometric morphisms, of an arbitrary presheaf topos $\h{\C}$ (with $\C$ a small category). A continuous category $\X$ is \textit{ind-small} if there exists a small \textit{ind-dense} subcategory $\A$ of $\X$, by which we mean a small, full, dense subcategory $\A$ of $\X$ for which each comma category $(\A \downarrow x)$, with $x \in \X$, is filtered\footnote{The term \textit{ind-small} was introduced not in \cite{JJ} but later in \cite{Jele}, where it is defined in terms of a different criterion, which, by 2.17 of \cite{JJ} and C4.2.18 of \cite{Jele}, is equivalent to the given condition, employed in \cite{JJ}. Chapter C4 of \cite{Jele} includes an alternate exposition of much of the content of \cite{JJ}.}.
\begin{ThmSubSub} \label{thm:injcts}
\emph{(Johnstone-Joyal \cite{JJ}).}
\begin{enumerate}
\item There is an equivalence of 2-categories between the 2-category of quasi-injective toposes, with geometric morphisms, and the 2-category of ind-small continuous categories, with morphisms all filtered-colimit-preserving functors. This equivalence sends a quasi-injective topos $\E$ to its category of points $\pt(\E)$.
\item This equivalence restricts to an equivalence between the full sub-2-categories of injective toposes and cocomplete ind-small continuous categories.
\end{enumerate}
\end{ThmSubSub}
\addtocounter{subsection}{1}
\paragraph{Totally distributive toposes.}\label{sec:qinjtop}
Having seen that Scott's isomorphism between injective $T_0$ spaces and continuous lattices has a topos-theoretic analogue relating injective toposes and cocomplete ind-small continuous categories, we are led to seek a topos-theoretic analogue of the dual equivalence between continuous lattices and lex completely distributive lattices. We prove the following, where by a \textit{small dense generator} for a category $\E$ we mean a small dense full subcategory $\G$ of $\E$. Recall that every Grothendieck topos has a small dense generator.
\begin{ThmSubSub} \label{thm:lex_td_sdg_equals_inj_top}
The lex totally distributive categories with a small dense generator are exactly the injective toposes. Hence, the 2-category of cocomplete ind-small continuous categories (\ref{thm:injcts}) is equivalent to the the 2-category of lex totally distributive categories with a small dense generator (with geometric morphisms).
\end{ThmSubSub}
One may also ask whether there is a similar analogue of the broader dual equivalence between continuous dcpos and completely distributive lattices, and in this regard we provide a partial result, as follows:
\begin{PropSubSub} \label{thm:qinj_implies_td}
Every quasi-injective topos is totally distributive.
\end{PropSubSub}
In proving these theorems, we come upon a further result of independent interest. An \textit{essential subtopos} of a topos $\F$ is a topos $\E$ for which there is a geometric inclusion $i:\E \rightarrow \F$ whose inverse image functor $i^*:\F \rightarrow \E$ has a left adjoint.
\begin{ThmSubSub} \label{thm:charn_of_td_sdg}
Those totally distributive categories having a small dense generator are exactly the essential subtoposes of presheaf toposes $\h{\C} = [\C^\op,\Set]$ (with $\C$ a small category).
\end{ThmSubSub}
\begin{RemSubSub} \label{rem:KL_thm}
It was shown by Kelly-Lawvere \cite{KL} that the essential subtoposes of a presheaf topos $\h{\C}$ correspond bijectively to \textit{idempotent ideals} of arrows in the small category $\C$.
\end{RemSubSub}
\begin{ExaSubSub}
The cases in which $\h{\C}$ is the topos $\h{\Delta}$ of \textit{simplicial sets}, the topos $\h{\mathbb{I}}$ of \textit{cubical sets}, or the topos $\h{\mathbb{G}}$ of \textit{reflexive globular sets} are of interest in homotopy theory and higher category theory. It is shown in \cite{KRRZ} that the essential subtoposes of these toposes are classified by the dimensions $n \in \mathbb{N}$. In general, the essential subtoposes of a topos $\F$ (or rather, their associated equivalent full replete subcategories of $\F$) form a complete lattice \cite{KL}.
\end{ExaSubSub}
\begin{RemSubSub} \label{rem:small_set_gens}
As noted in \ref{sec:intro_cd_lats_td_cats}, it was proved in \cite{St} that any lex total category $\E$ with a small set of generators is a Grothendieck topos. Using this result, whose proof in \cite{St} appears to make use of the foundational assumption that there is a category of sets $S'$ such that both $\E$ and the category $\Set$ of small sets are categories internal to $S'$, we obtain the following corollaries to Theorems \ref{thm:lex_td_sdg_equals_inj_top} and \ref{thm:charn_of_td_sdg}:
\end{RemSubSub}
\begin{ThmSubSub} \label{thm:main}
\emptybox
\begin{enumerate}
\item Those lex totally distributive categories having a small set of generators are exactly the injective toposes.
\item Those totally distributive categories having a small set of generators are exactly the essential subtoposes of presheaf toposes $\h{\C} = [\C^\op,\Set]$ (with $\C$ small).
\end{enumerate}
\end{ThmSubSub}
\renewcommand*\theparagraph{\thesubsubsection.\arabic{paragraph}}
\section{Preliminaries on totally distributive categories}
It is shown in \cite{RWsets}, by means of a result of \cite{SW}, that every presheaf category $\h{\C}$ on a small category $\C$ is totally distributive. In order to clearly establish this in the absence of the foundational assumptions of \cite{RWsets}, we give a self-contained elementary proof, by means of the following lemma (cf. Corollary 14 of \cite{SW}). We prove also that if $\C$ is finitely complete, then $\h{\C}$ is lex totally distributive.
\begin{LemSub} \label{thm:presh_lemma}
Let $\C$ be a small category. Then there is an adjunction
$$
\xymatrix {
\h{\C} \ar@/^1.2pc/[rr]^{\y_{\h{\C}}}_{\top} & & \h{\h{\C}} \ar[ll]|{\widehat{\y_\C}}
}\;,
$$
where $\y_\C:\C \rightarrow \h{\C}$ and $\y_{\h{\C}}:\h{\C} \rightarrow \h{\h{\C}}$ are the Yoneda embeddings.
\end{LemSub}
\begin{proof}
Each $\mathbb{C} \in \h{\h{\C}}$ is a coend $\mathbb{C} \cong \int^{C \in \h{\C}}\mathbb{C}(C) \cdot \h{C}$, and these isomorphisms are natural in $\mathbb{C}$. Using this and the Yoneda Lemma, we obtain isomorphisms
$$(\widehat{\y_{\C}}(\mathbb{C}))(c) = \mathbb{C}(\h{c}) \cong \int^{C \in \h{\C}}\mathbb{C}(C) \times \h{C}(\h{c}) \cong \int^{C \in \h{\C}}\mathbb{C}(C) \times C(c)$$
natural in $\mathbb{C} \in \h{\h{\C}}$ and $c \in \C$. Hence we have an isomorphism
$$\widehat{\y_{\C}}(\mathbb{C}) \cong \int^{C \in \h{\C}}\mathbb{C}(C) \cdot C$$
natural in $\mathbb{C} \in \h{\h{\C}}$, so with reference to \eqref{eq:col_of_presh}, $\widehat{\y_{\C}} \dashv \y_{\h{\C}}$.
\end{proof}
\begin{PropSub} \label{thm:preshtd}
Let $\C$ be a small category. Then $\h{\C}$ is totally distributive. Moreover, if $\C$ has finite limits, then $\h{\C}$ is lex totally distributive.
\end{PropSub}
\begin{proof}
We have an adjunction as in Lemma \ref{thm:presh_lemma}, and the left adjoint $\widehat{\y_\C}:\h{\h{\C}} \rightarrow \h{\C}$ has a further left adjoint $\exists_{\y_\C}:[\C^\op,\Set] \rightarrow [{\h{\C}\,}^\op,\Set]$, which is given by left Kan-extension along $\y_\C^\op:\C^\op \rightarrow {\h{\C}\,}^\op$. Hence $\h{\C}$ is totally distributive. If $\C$ has finite limits, then $\y_\C:\C \rightarrow \h{\C}$ is a cartesian functor between cartesian categories, and it follows that the associated functor $\exists_{\y_\C}$ is also cartesian.
\end{proof}
The following lemma, based on Lemma 3.5 of Marmolejo-Rosebrugh-Wood \cite{MRW}, provides a means of deducing that a category is totally distributive. We have augmented the lemma slightly in order to handle lex totally distributive categories as well.
\begin{LemSub} \label{thm:mrwlemma}
Let $\D$ and $\E$ be locally small categories. Suppose we are given adjunctions
$$
\xymatrix {
\D \ar@/_1.1pc/[rr]_q^{\top} \ar@/^1.1pc/[rr]^s_{\top} & & \E \ar[ll]|r
}
$$
with $q,s$ fully faithful and $\E$ totally distributive. Then $\D$ is totally distributive.
Moreover, if $\E$ is lex totally distributive and $q$ preserves finite limits, then $\D$ is lex totally distributive.
\end{LemSub}
\begin{proof}
There is a 2-functor $\h{(-)} := \CAT((-)^\op,\Set) : \CAT^\coop \rightarrow \CCAATT$, where $\CAT^\coop$ is the (meta)-2-category gotten by reversing both the 1-cells and 2-cells in $\CAT$. This 2-functor sends the adjunctions $q \dashv r \dashv s : \D \rightarrow \E$ in $\CAT$ to adjunctions $\h{q} \dashv \h{r} \dashv \h{s}$, so we have a diagram
$$
\xymatrix @+1pc {
\D \ar@/_1.0pc/[rr]_q^{\top} \ar@/^1.0pc/[rr]^s_{\top} \ar@/^1.0pc/[d]^{\y'} & & \E \ar[ll]|r \ar@{->}@/_1.0pc/[d]_t^{\dashv} \ar@{->}@/^1.0pc/[d]^{\y}_{\dashv}\\
\h{\D} \ar@{<-}@/_1.1pc/[rr]_{\h{q}}^{\top} \ar@{<-}@/^1.1pc/[rr]^{\h{s}}_{\top} & & \h{\E} \ar@{<-}[ll]|{\h{r}} \ar[u]|c
}
$$
where $\y'$ is the Yoneda embedding. Observe that $\y' \cong \h{s} \cdot \y \cdot s$, since we have
$$(\h{s} \cdot \y \cdot s)(d) = \h{s}(\E(-,sd)) = \E(\Op{s}-,sd) \cong \D(-,d) = \y'(d) \;\;\;\;$$
naturally in $d \in \D$, as $s$ is fully faithful. Therefore, letting $c' := r \cdot c \cdot \h{r}$ and $t' := \h{q} \cdot t \cdot q$ we find that
$$
\xymatrix {
\D \ar@/_1.1pc/[rr]_{t'}^{\top} \ar@/^1.1pc/[rr]^{\y'}_{\top} & & \h{\D} \ar[ll]|{c'}
}
$$
so $\D$ is totally distributive.
If $t$ and $q$ are cartesian, then since $\h{q}$ is also cartesian, $t' = \h{q} \cdot t \cdot q$ is cartesian and hence $\D$ is lex totally distributive.
\end{proof}
\section{A construction of Johnstone-Joyal} \label{sec:const_of_jj}
Let $\X$ be an ind-small continuous category, and let $\A$ be a small ind-dense subcategory of $\X$. We now recall from \cite{JJ} an explicit manner of constructing a quasi-injective topos $\F$ such that $\X$ is equivalent to the category of points of $\F$.
Firstly, there is an associated functor $W:\X^\op \times \X \rightarrow \Set$, given by
$$W(x,y) := \Ind{\X}(\m x,wy),\;\;\;\;x,y \in \X\;.$$
The elements of $W(x,y)$ are called \textit{wavy arrows} from $x$ to $y$ in $\X$. Johnstone and Joyal \cite{JJ} show that this functor $W$, when viewed as a profunctor $W:\X \mrelarrow \X$, underlies an \textit{idempotent profunctor comonad} on $\X$, and that the restriction $V:\A^\op \times \A \rightarrow \Set$ of $W$ is again an idempotent profunctor comonad on $\A$. In the latter case, since $\A$ is small, this means precisely that $V:\A \mrelarrow \A$ is an idempotent comonad on $\A$ in the bicategory $\Prof$ of small categories, profunctors, and morphisms of profunctors. Further, $V$ is \textit{left-flat}, meaning that for each $y \in \A$, $V(-,y):\A^\op \rightarrow \Set$ is a flat presheaf.
Recall that for small categories $\C,\D$, each profunctor $M:\C \mrelarrow \D$ (by which we mean a functor $M:\C^\op \times \D \rightarrow \Set$) gives rise to a cocontinuous functor $\widetilde{M}:[\C,\Set] \rightarrow [\D,\Set]$. Indeed, $\widetilde{M}$ is the left Kan extension along the Yoneda embedding $\C^\op \rightarrow [\C,\Set]$ of the transpose $\C^\op \rightarrow [\D,\Set]$ of $M$. This passage defines an equivalence of the bicategory $\Prof$ with another bicategory, in fact a 2-category, whose objects are again all small categories, but whose 1-cells $\C \rightarrow \D$ are all cocontinuous functors $[\C,\Set] \rightarrow [\D,\Set]$, and whose 2-cells are all natural transformations.
Hence our idempotent comonad $V:\A \mrelarrow \A$ in $\Prof$ determines an idempotent comonad $\widetilde{V}:[\A,\Set] \rightarrow [\A,\Set]$. Moreover, since $V(-,y):\A^\op \rightarrow \Set$ is flat for each $y \in \A$, it follows that $\widetilde{V}$ preserves finite limits and so is said to be a \textit{cartesian comonad}. Further, since $\widetilde{V}$ is also cocontinuous, $\widetilde{V}$ is the inverse-image part of a geometric morphism:
\begin{DefSub}
Given an ind-small continuous category $\X$ with a small ind-dense subcategory $\A$, the \textit{associated geometric endomorphism} is defined to be the geometric morphism $m_{\A,\X}:[\A,\Set] \rightarrow [\A,\Set]$ whose inverse-image part is \textit{the associated idempotent comonad} $m_{\A,\X}^* =\widetilde{V}$.
\end{DefSub}
\begin{PropSub} \label{thm:jjassoctopos}
\emph{(Johnstone-Joyal \cite{JJ}).}
Let $\X$ be an ind-small continuous category, and let $\A$ be a small ind-dense subcategory of $\X$. Let $[\A,\Set] \rightarrow \F \rightarrow [\A,\Set]$ be a factorization of the associated geometric endomorphism $m_{\A,\X}$ into a geometric surjection followed by a geometric inclusion. Then $\F$ is a quasi-injective topos whose category of points of is equivalent to $\X$. Further, if $\X$ is cocomplete, then we may take $\A$ to be finitely cocomplete, and it follows that $\F$ is an injective topos.
\end{PropSub}
\section{Totally distributive toposes from continuous categories}
We now show that the toposes corresponding to continuous categories under the equivalence of Theorem \ref{thm:injcts} are totally distributive, so that every quasi-injective topos is totally distributive.
\begin{LemSub} \label{thm:left_adjoint_idempotent_comonad}
Let $i:\C \rightarrow \D$ be a fully faithful functor with a right adjoint $r$, and suppose that the induced comonad $i \cdot r$ on $\D$ has a right adjoint $n$. Then $r$ has a right adjoint $s := n \cdot i$, so that $i \dashv r \dashv s$.
\end{LemSub}
\begin{proof}
$$\C(r(d),c) \cong \D(i \cdot r(d),i(c)) \cong \D(d,n \cdot i(c)) = \D(d,s(c))\;,$$
naturally in $d \in \D, c \in \C$.
\end{proof}
\begin{LemSub} \label{thm:assoctopos}
Let $\X$ be an ind-small continuous category, let $\A$ be a small ind-dense subcategory of $\X$, and let $i:\F \hookrightarrow [\A,\Set]$ be the coreflective embedding induced by the associated idempotent comonad $m_{\A,\X}^*$ on $[\A,\Set]$ (so that $\F$ is the category of fixed points of $m_{\A,\X}^*$). Then
\begin{enumerate}
\item i preserves finite limits;
\item The right adjoint $r:[\A,\Set] \rightarrow \F$ to $i$ has a further right adjoint $s$, so that
$$
\xymatrix {
\F \ar@/_1.1pc/[rr]_i^{\top} \ar@/^1.1pc/[rr]^s_{\top} & & {[\A,\Set]\;;} \ar[ll]|r
}
$$
\item $\F$ is a quasi-injective topos whose category of points is equivalent to $\X$;
\item If $\X$ is cocomplete, we may take $\A$ to be finitely cocomplete, and $\F$ is then an injective topos.
\end{enumerate}
\end{LemSub}
\begin{proof}
Since $\F$ is isomorphic to the category of coalgebras of the cartesian comonad $m_{\A,\X}^*$, $\F$ is an elementary topos, and the forgetful functor $i:\F \hookrightarrow [\A,\Set]$ is the inverse-image part of a geometric surjection $p:[\A,\Set] \twoheadrightarrow \F$; see, e.g., \cite{Jele}, A4.2.2. Further, the idempotent comonad $i \cdot r = m_{\A,\X}^*$ has a right adjoint ${m_{\A,\X}}_*$, so we deduce by Lemma \ref{thm:left_adjoint_idempotent_comonad} that $r$ has a right adjoint $s$, so that $i \dashv r \dashv s$. In particular, $r$ is left adjoint and cartesian, so we obtain a geometric morphism $q:\F \rightarrow [\A,\Set]$ with $q^* = r$ and $q_* = s$. Since $i \dashv r \dashv s$ and $i$ is fully faithful, it follows that $s = q_*$ is also fully faithful, so $q:\F \rightarrow [\A,\Set]$ is a geometric inclusion. Further, the composite $[\A,\Set] \xrightarrow{p} \F \xrightarrow{q} [\A,\Set]$ is $m_{\A,\X}$, or, more precisely, has inverse-image part $(q \cdot p)^* = p^* \cdot q^* = i \cdot r = m_{\A,\X}^*$. Hence 3 and 4 follow from Proposition \ref{thm:jjassoctopos}.
\end{proof}
\begin{DefSub}
For an ind-small continuous category $\X$ and a small ind-dense subcategory $\A$ of $\X$, we call the topos $\F$ of Lemma \ref{thm:assoctopos} the \textit{associated topos}.
\end{DefSub}
\begin{LemSub} \label{thm:assoctopostd}
Let $\X$ be an ind-small continuous category, so that $\X$ has some small ind-dense subcategory $\A$. Then the the associated topos $\F$ is totally distributive. If $\X$ is also cocomplete, then we may take $\A$ to be finitely cocomplete, and it follows that $\F$ is lex totally distributive.
\end{LemSub}
\begin{proof}
By Lemma \ref{thm:assoctopos}, we have adjunctions
$$
\xymatrix {
\F \ar@/_1.1pc/[rr]_i^{\top} \ar@/^1.1pc/[rr]^s_{\top} & & [\A,\Set] \ar[ll]|r
}
$$
with $i,s$ fully faithful and $i$ cartesian. By Proposition \ref{thm:preshtd}, $[\A,\Set]$ is totally distributive, so we deduce by Lemma \ref{thm:mrwlemma} that $\F$ is totally distributive. If $\X$ is also cocomplete, then we can take $\A$ to be finitely cocomplete, so $\A^\op$ is finitely complete and hence, by \ref{thm:preshtd}, $\h{\A^\op} = [\A,\Set]$ is lex totally distributive, so we deduce by \ref{thm:mrwlemma} that $\F$ is lex totally distributive.
\end{proof}
\begin{ThmSub} \label{thm:injimpliesltd}
Every quasi-injective topos is totally distributive, and every injective topos is lex totally distributive.
\end{ThmSub}
\begin{proof}
Given a quasi-injective topos $\E$, Theorem \ref{thm:injcts} entails that the category of points $\X := \pt(\E)$ of $\E$ is an ind-small continuous category. Taking any small ind-dense subcategory $\A$ of $\X$, the associated topos $\F$ is a quasi-injective topos whose category of points is equivalent to $\X$, so by Theorem \ref{thm:injcts} we deduce that $\E$ is equivalent to $\F$. But the latter topos is totally distributive by Lemma \ref{thm:assoctopostd}, and total distributivity is clearly invariant under equivalences, so $\E$ is totally distributive. The second statement may be deduced analogously.
\end{proof}
\section{Totally distributive categories as essential localizations}
\begin{PropSub} \label{thm:resttogen}
Let $\E$ be a totally distributive category with a small dense generator $i:\G \hookrightarrow \E$. We then conclude the following:
\begin{enumerate}
\item There are adjunctions
$$
\xymatrix {
\E \ar@/_1.1pc/[rr]_{t'}^{\top} \ar@/^1.1pc/[rr]^{\y'}_{\top} & & {\h{\G}} \ar[ll]|{c'}
}
$$
with $\y'$ and $t'$ fully faithful, where $\y'$ is the composite $\E \xrightarrow{\y} \h{\E} \xrightarrow{\h{i}} \h{\G}$.
\item $\E$ is an essential subtopos of $\h{\G}$ and, in particular, a Grothendieck topos.
\item If $\E$ is lex totally distributive, then $t':\E \rightarrow \h{\G}$ preserves finite limits.
\end{enumerate}
\end{PropSub}
\begin{proof}
We let
$$c' := c \cdot \forall_i = (\h{\G} \xrightarrow{\forall_i} \h{\E} \xrightarrow{c} \E)\;,$$
$$t' := \h{i} \cdot t = (\E \xrightarrow{t} \h{\E} \xrightarrow{\h{i}} \h{\G})\;,$$
where $\forall_i:\h{\G} \rightarrow \h{\E}$ is the functor given by right Kan extension along $i^\op:\G^\op \hookrightarrow \E^\op$. Since $\h{i} \dashv \forall_i$ and $t \dashv c$, we have that $t' = \h{i} \cdot t \dashv c \cdot \forall_i = c'$. Since $i:\G \hookrightarrow \E$ is fully faithful, the counit of the adjunction $\h{i} \dashv \forall_i$ is an isomorphism (e.g., by \cite{Ma}, X.3.3), so $\forall_i$ is fully faithful.
Observe that the diagram
$$
\xymatrix {
\E \ar[d]_\y \ar[r]^\y & \h{\E} \\
\h{\E} \ar[r]_{\h{i}} & \h{\G} \ar[u]_{\forall_i}
}
$$
commutes up to isomorphism, since the density of $\G$ in $\E$ gives us exactly that
$u \cong \int^{g \in \G} \E(g,u) \cdot g$
naturally in $u \in \E$, so
$$(\y v)u = \E(u,v) \cong \E(\int^{g \in \G} \E(g,u) \cdot g,v) \cong \int_{g \in \G} [\E(g,u),\E(g,v)] = ((\forall_i \cdot \h{i} \cdot \y)v)u$$
naturally in $u,v \in \E$.
We find that $c' = c \cdot \forall_i \dashv \h{i} \cdot \y = \y'$, since by using the adjointness $c \dashv \y$, the commutativity of the above diagram, and the fact that $\forall_i$ is fully faithful, we deduce that
$$\E(c \cdot \forall_i(G),v) \cong \h{\E}(\forall_i(G),\y v) \cong \h{\E}(\forall_i(G),\forall_i \cdot \h{i} \cdot \y(v)) \cong \h{\G}(G,\h{i} \cdot \y(v))$$
naturally in $G \in \h{\G}, v \in \E$.
Since $\G$ is a dense generator for $\E$ we have that $\y'$ is fully faithful, and since $t' \dashv c' \dashv \y'$ it follows that $t'$ is fully faithful as well.
If $\E$ is lex totally distributive, then $t$ preserves finite limits, so since $\h{i}$ preserves all limits, $t' = \h{i} \cdot t$ preserves finite limits.
\end{proof}
\begin{ThmSub} \label{thm:lextdinjective}
Let $\E$ be a lex totally distributive category with a small dense generator. Then $\E$ is an injective Grothendieck topos.
\end{ThmSub}
\begin{proof}
By \ref{thm:resttogen} we know that $\E$ is a Grothendieck topos, and it follows from Giraud's Theorem that there exists a \textit{finitely complete} small dense full subcategory $\G$ of $\E$. (Indeed, this follows readily from 4.1 and 4.2 in the Appendix of \cite{MM}, for example). We have adjunctions $t' \dashv c' \dashv \y'$ as in Proposition \ref{thm:resttogen}, with $\y'$ fully faithful and $t'$ cartesian. Hence we obtain geometric morphisms $s:\E \rightarrow \h{\G}$ and $r:\h{\G} \rightarrow \E$ with
$s_* = \y'$, $s^* = c'$, $r_* = c'$, $r^* = t'$, since $c'$ is right adjoint and hence cartesian. Further, since $\y'$ is fully faithful and $c' \dashv \y'$, we have that
$$(r \cdot s)_* = r_* \cdot s_* = c' \cdot \y' \cong 1_\E \;,$$
so $\E$ is a (pseudo-)retract of the presheaf topos $\h{\G}$ by geometric morphisms, and the result follows by \ref{thm:injtop_as_retracts}.
\end{proof}
Hence Theorem \ref{thm:lex_td_sdg_equals_inj_top} is proved. To prove Theorem \ref{thm:charn_of_td_sdg}, it remains only to show the following:
\begin{PropSub}
Let $\E$ be an essential subtopos of a presheaf topos $\h{\C}$ (with $\C$ small). Then $\E$ is totally distributive and has a small dense generator.
\end{PropSub}
\begin{proof}
There is a geometric inclusion $s:\E \rightarrow \h{\C}$ whose inverse-image functor $s^*:\h{\C} \rightarrow \E$ has a left adjoint $s_!$. Hence we have $s_! \dashv s^* \dashv s_*$ with $s_!$ and $s_*$ fully faithful, so $\E$ is totally distributive, by Lemma \ref{thm:mrwlemma}.
\end{proof}
\bibliographystyle{amsplain}
\bibliography{td1}
\end{document}
| 7,949
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Digital Displays
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tag:blogger.com,1999:blog-19323171.post8803371874758456823..comments2020-06-15T12:25:38.674-04:00Comments on All Things Anderson: Anderson Cooper 360 on Tuesday, January 12, 2015 Unknownnoreply@blogger.comBlogger4125tag:blogger.com,1999:blog-19323171.post-79007384114074921992016-01-14T23:20:47.461-05:002016-01-14T23:20:47.461-05:00@JULIE MOORE: SOUNDS TO ME THAT YOU ARE BITTER AND...@JULIE MOORE: SOUNDS TO ME THAT YOU ARE BITTER AND BITTER IS BETTER THAN FEELING NOTHING AT ALL.<br />HOWEVER, VAN JONES AND DAVID AXELROD ARE BOTH "CREDIBLE COMMENTATORS," IN MY BOOK. AXELROD RAN<br />I BELIEVE, THE OBAMA CAMPAIGN FOR A LONG TIME AND EVEN WROTE A BOOK ABOUT IT, SO YOU MIGHT<br />SAY HE KNOWS WHAT HE'S TALKING ABOUT AND NOT JUST GIVING HIS OPINION.<br />AND VAN JONES OFTEN FINDS HIMSELF HAVING TO DEFEND THE OBAMA ADMINISTRATION BECAUSE CNN, THE NETWORK YOU CARE NEVER TO WATCH AGAIN, IS SO BIAS AGAINST THIS ADMINISTRATION, JONES IS CONSTANTLY <br />DEFENDING EVERY LITTLE ACTION THE PRESIDENT MAKES.<br />THIS PAST TUESDAY THE HUGE QUESTION WAS "WHY DIDN'T THE POTUS MENTION THE CAPTIVE SAILORS IN HIS SOTU<br />ADDRESS? COULD IT BE FOR SECURITY REASONS FOLKS?<br />AND DANA BASH WAS BUSY BADGERING JOHN KERRY FROM THE MOMENT SHE SAW HIM ENTER THE CAPITOL BUILDING, ABOUT THE SPECIFICS OF THE IRAN NEGOTIATIONS AND TRANSPARENCY, AND THE CAPTIVE SAILOR CONTROVERSY.<br />YOU SEEM TO HAVE MISSED ALL THE IMPLIED OBAMA TRANSGRESSIONS FROM ANDERSON COOPER, WHEN IT COMES<br />TO PRESIDENT OBAMA....NOW THAT HAS TO BE A PLUS FOR CNN AND SURELY A REASON TO WATCH MORE OF CNN'S TOTAL 'FICTIONAL REPORTAGE.'<br />BUT FOX NEWS STILL BEATS CNN AND GETS AND A++ FOR JUST "MAKING STUFF UP."Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-19323171.post-698065937814151122016-01-14T12:46:26.457-05:002016-01-14T12:46:26.457-05:00I was so frustrated and disappointed at the disres...I was so frustrated and disappointed at the disrespect from Van Jones and David Axelrod toward Mike Rogers--and so were the 30 others in my home. We are mostly independents that like to watch both CNN and Fox news just to get all sides but I will likely never watch CNN again. David Axelrod and Van Jones are not credible commentators on anything--why would you expose them to the American public that are searching for answers---only to get such a slanted and really mislead point of view. I think of Andersen Cooper as a journalist and just expect more honesty and truth. Obama is a nice man and he means well but he has been neither a Commander in Chief nor a highly performing CEO of our country, its not his fault, we threw him in there as a figure head but he had a real job to do and he really has never held one and been exposed to folks that aren't such good mentors. The reality is that our representatives in congress and in the White House should be real people, working in the private sector that leave this for a while to serve the country----that is how our founding fathers meant for this to be---not a life of favors both monetary and otherwise to stay in office and continue the constant cycle of unethical behavior and favors. We are under attack in this country--I have watched terrorist attacks all over this country not be covered thoroughly, the hostages left in Tehran just left there---no one covering that yet we are buddying up to them and releasing sanctions without our hostages being released---does anyone care??? We are scorned as Christians in this country while we have to cater to other radicalized religions. We can't afford to feed, clothe, and shelter current citizens of the US yet we want to bring immigrants in that we can't determine their intentions and know that they have no means to support themselves ---where does the madness end? We are spending money we don't have on things and people that have nothing to do with good for our country. Can CNN start to focus on that or do we have to have a massive attack or a major revolt to get those things covered---please drop your political bias and just report honestly and ethically---there are a lot of hard working, well educated folks out here with nothing but honorable intentions and we want the right things for our country--we are mothers, veterans, students and we are American and want someone to stand up for us---all of us....Black, white, Asian, Latino, Christian, Gay, lesbian, male,female but we don't want career politicians looking for a promotion and donors---just honesty.Anonymous MARCO RUBIO IS UNABLE TO FULFILL HIS COMMITMENT...IF MARCO RUBIO IS UNABLE TO FULFILL HIS COMMITMENT AS A SENATOR, HOW THEN CAN HE FULFILL HIS COMMITMENT AS PRESIDENT?<br />I ALWAYS GOT THE IMPRESSION HE WAS TRYING TO IMITATE OBAMA'S METEORIC RISE TO THE PRESIDENCY. ALTHOUGH<br />RUBIO HAS MASTERED A RHETORICAL SMOOTHNESS OF DELIVERY, HE NEVER QUITE MEASURES UP....THERE'S SOMETHING LACKING AND I JUST CAN'T PUT MY FINGER ON IT, AND NEITHER CAN ANYONE ELSE.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-19323171.post-15310528905073899542016-01-13T01:01:59.371-05:002016-01-13T01:01:59.371-05:00Tuesday's show can't really be called &quo...Tuesday's show can't really be called "360". It was instead the "State of the Union Preview Show". Wolf Blitzer was the co-host of this preview show with Anderson. I watched about 3/4 of it.<br /><br />It started with splashy opening credits that made clear the GOP 2016 candidates would get some time. But the lead story was the 10 sailors who drifted into Iranian waters and were arrested. Barbara Starr provided more info and then Blitzer, Jake Tapper and Jim Sciutto discussed it further. Anderson popped up to give a couple of details on the SOTU before handing it off to Jim Acosta. Dana Bash was at the Capitol and was able to interview Sen. Marco Rubio. I liked her question about being there for SOTU but not for many of the Senate votes as his competitors in the GOP have pointed out. Rubio babbled something about the importance of running for President and I wished Bash asked which Senate votes he thought were particularly un-important. The first segment ended with a panel of David Axelrod, Michael Smerconish, Gloria Borger and John King.<br /><br />Next Acosta and Tapper talked about Obama's plans for his last year in office. Anderson showed new polls for the Democratic candidates and Hillary's comment on Biden's praise for Sanders in that interview with Borger yesterday. A new panel discussed all of that - Van Jones, Amanda Carpenter, Paul Begala and Mike Rogers. The middle of the panel was interrupted to show video footage of Bash asking Sect. of State Kerry about the 10 sailors. It looked pathetic. Bash, in the hallways of the Capitol, saw Kerry. Kerry said hello to her and she asked him about the 10 sailors. Kerry had a short answer and kept going. Bash tried and tried to get him to stay and talk more But Kerry left as if he had an important event to get to somewhere else in the building.<br /><br />Bash was also there for the arrival of the SCOTUS Justices. It looked like a B-Grade bizarro-world Red Carpet parody. The preview show continued with footage of Biden and various senators arriving. while various CNN folk babbled about varous SOTU-related topics. Jeff Zeleny appeared for a short report predicting Obama's SOTU content; best guess was comments on gun control. Then it was talk about Biden and Speaker of the House Rep. Paul Ryan, blah blah blah.<br /><br />It just started sounding redundant. I channel surfed and missed a chunk from 20 minutes before SOTU to 7 minutes before SOTU. The CNN chatter sounded about the same. My attention picked up when they mentioned Rep. Keith Eliison (a Minnesotan!) because he's Muslim. Other than that it really wasn't interesting. <br /><br />I didn't watch the post-SOTU show but imagine it was mostly the panels of pundits pontificating opinions profusely.Jaanza
| 282,501
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\begin{document}
\maketitle
\begin{abstract}
Le Roux and Ziegler asked whether every simply connected compact nonempty planar $\Pi^0_1$ set always contains a computable point.
In this paper, we solve the problem of le Roux and Ziegler by showing that there exists a planar $\Pi^0_1$ dendroid without computable points.
We also provide several pathological examples of tree-like $\Pi^0_1$ continua fulfilling certain global incomputability properties:
there is a computable dendrite which does not $\ast$-include a $\Pi^0_1$ tree;
there is a $\Pi^0_1$ dendrite which does not $\ast$-include a computable dendrite;
there is a computable dendroid which does not $\ast$-include a $\Pi^0_1$ dendrite.
Here, a continuum $A$ {\em $\ast$-includes} a member of a class $\mathcal{P}$ of continua if, for every positive real $\varepsilon$, $A$ includes a continuum $B\in\mathcal{P}$ such that the Hausdorff distance between $A$ and $B$ is smaller than $\varepsilon$.
\end{abstract}
\section{Background}
Every nonempty open set in a computable metric space (such as Euclidean $n$-space $\mathbb{R}^n$) contains a computable point.
In contrast, the Non-Basis Theorem asserts that a nonempty {\em co-c.e.\ closed} set (also called a $\Pi^0_1$ set) in Cantor space (hence, even in Euclidean $1$-space) can avoid any computable points.
Non-Basis Theorems can shed new light on connections between {\em local} and {\em global} properties by incorporating the notions of {\em measure} and {\em category}.
For instance, Kreisel-Lacombe \cite{KrLa} and Tanaka \cite{Tan} showed that there is a $\Pi^0_1$ set with positive measure that contains no computable point.
Recent exciting progress in {\em Computable Analysis} \cite{Wei} naturally raises the question whether Non-Basis Theorems exist for {\em connected} $\Pi^0_1$ sets.
However, we observe that, if a nonempty $\Pi^0_1$ subset of $\mathbb{R}^1$ contains no computable points, then it must be totally disconnected.
Then, in higher dimensional Euclidean space, can there exist a connected $\Pi^0_1$ set containing no computable points?
It is easy to construct a nonempty connected $\Pi^0_1$ subset of $[0,1]^2$ without computable points, and a nonempty simply connected $\Pi^0_1$ subset of $[0,1]^3$ without computable points.
An open problem, formulated by Le Roux and Ziegler \cite{RZ} was whether every nonempty simply connected compact planar $\Pi^0_1$ set contains a computable point.
As mentioned in Penrose's book ``{\em Emperor's New Mind}'' \cite{Pen}, {\em the Mandelbrot set} is an example of a simply connected compact planar $\Pi^0_1$ set which contains a computable point, and he conjectured that the Mandelbrot set is not computable {\em as a closed set}.
Hertling \cite{Her} observed that the Penrose conjecture has an implication for a famous open problem on local connectivity of the Mandelbrot set.
Our interest is which topological assumption (especially, connectivity assumption) on a $\Pi^0_1$ set can force it to possess a given computability property.
Miller \cite{Mil} showed that every $\Pi^0_1$ sphere in $\mathbb{R}^n$ is computable, and so it contains a dense c.e.\ subset of computable points.
He also showed that every $\Pi^0_1$ ball in $\mathbb{R}^n$ contains a dense subset of computable points.
Iljazovi\'c \cite{Ilj} showed that chainable continua (e.g., arcs) in certain metric spaces are almost computable, and hence there always is a dense subset of computable points.
In this paper, we show that {\em not} every $\Pi^0_1$ dendrite is almost computable, by using a tree-immune $\Pi^0_1$ class in Cantor space.
This notion of immunity was introduced by Cenzer, Weber Wu, and the author \cite{CKWW}.
We also provide pathological examples of tree-like $\Pi^0_1$ continua fulfilling certain global incomputability properties:
there is a computable dendrite which does not $\ast$-include a $\Pi^0_1$ tree;
there is a computable dendroid which does not $\ast$-include a $\Pi^0_1$ dendrite.
Finally, we solve the problem of Le Roux and Ziegler \cite{RZ} by showing that there exists a planar $\Pi^0_1$ dendroid without computable points.
Indeed, our planar dendroid is contractible.
Hence, our dendroid is also the first example of a contractible Euclidean $\Pi^0_1$ set without computable points.
\section{Preliminaries}
\noindent
{\bf Basic Notation:}
$2^{<\nn}$ denotes the set of all finite binary strings.
Let $X$ be a topological space.
For a subset $Y\subseteq X$, $cl(Y)$ ($int(Y)$, resp.) denotes the closure (the interior, resp.) of $Y$.
Let $(X;d)$ be a metric space.
For any $x\in X$ and $r\in\mathbb{R}$, $B(x;r)$ denotes the open ball $B(x;r)=\{y\in X:d(x,y)<r\}$.
Then $x$ is called {\em the center of $B(x;r)$}, and $r$ is called {\em the radius of $B(x;r)$}.
For a given open ball $B=B(x;r)$, $\hat{B}$ denotes the corresponding closed ball $\hat{B}=\{y\in X:d(x,y)\leq r\}$.
For $a,b\in\mathbb{R}$, $[a,b]$ denotes the closed interval $[a,b]=\{x\in\mathbb{R}:a\leq x\leq b\}$, $(a,b)$ denotes the open interval $(a,b)=\{x\in\mathbb{R}:a<x<b\}$, and $\lrangle{a,b}$ denotes a point of Euclidean plane $\mathbb{R}^2$.
For $X\subseteq \mathbb{R}^n$, ${\rm diam}(X)$ denotes $\max\{d(x,y):x,y\in X\}$.
\medskip
\noindent
{\bf Continuum Theory:}
{\em A continuum} is a compact connected metric space.
For basic terminology concerning {\em Continuum Theory}, see Nadler \cite{Nad} and Illanes-Nadler \cite{IlNa}.
Let $X$ be a topological space.
The set $X$ is {\em a Peano continuum} if it is a locally connected continuum.
The set $X$ is {\em a dendrite} if it is a Peano continuum which contains no Jordan curve.
The set $X$ is {\em unicoherent} if $A\cap B$ is connected for every connected closed subsets $A,B\subseteq X$ with $A\cup B=X$.
The set $X$ is {\em hereditarily unicoherent} if every subcontinuum of $X$ is unicoherent.
The set $X$ is {\em a dendroid} if it is an arcwise connected hereditary unicoherent continuum.
For a point $x$ of a dendroid $X$, $r_X(x)$ denotes the cardinality of the set of arc-components of $X\setminus\{x\}$.
If $r_X(x)\geq 3$ then $x$ is said to be {\em a ramification point of $X$}.
The set $X$ is {\em a tree} if it is dendrite with finitely many ramification points.
Note that a topological space $X$ is a dendrite if and only if it is a locally connected dendroid.
Hahn-Mazurkiewicz's Theorem states that a Hausdorff space $X$ is a Peano continuum if and only if $X$ is an image of a continuous curve.
\begin{example}[Planar Dendroids]\label{exa:dend:dend}~
\begin{enumerate}
\item Put $\mathcal{B}_t=\{2^{-t}\}\times[0,2^{-t}]$.
Then the following set $\mathcal{B}\subseteq\mathbb{R}^2$ is dendrite.
\[\mathcal{B}=\bigcup_{t\in\nn}\mathcal{B}_t\cup([-1,1]\times\{0\}).\]
We call $\mathcal{B}$ {\em the basic dendrite}.
The set $\mathcal{B}_{t}$ is called {\em the $t$-th rising of $\mathcal{B}$}.
See Fig.\ \ref{fig:dendrite}.
\item The set $\mathcal{H}=cl((\{1/n:n\in\nn\}\times[0,1])\cup([0,1]\times\{0\}))$ is called {\em a harmonic comb}.
Then $\mathcal{H}$ is a dendroid, but not a dendrite.
The set $\{1/n\}\times[0,1]$ is called {\em the $n$-th rising of the comb $\mathcal{H}$}, and the set $[0,1]\times\{0\}$ is called {\em the grip of $\mathcal{H}$}.
See Fig.\ \ref{fig:Harmonic_comb}.
\item Let $C\subseteq\mathbb{R}^1$ be the middle third Cantor set.
Then the one-point compactification of $C\times(0,1]$ is called {\em the Cantor fan}.
(Equivalently, it is the quotient space ${\rm Cone}(C)=(C\times[0,1])/(C\times\{0\})$.)
The Cantor fan is a dendroid, but not a dendrite.
See Fig.\ \ref{fig:Cantor_fan}.
\end{enumerate}
\end{example}
\begin{figure}[t]\centering
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\vspace{-0.5em}
\caption{The basic dendrite}
\label{fig:dendrite}
\end{minipage}
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\begin{center}
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\end{center}
\caption{The harmonic comb}
\label{fig:Harmonic_comb}
\end{minipage}
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\begin{center}
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\end{center}
\vspace{-1.5em}
\caption{The Cantor fan}
\label{fig:Cantor_fan}
\end{minipage}
\end{figure}
Let $X$ be a topological space.
$X$ is {\em $n$-connected} if it is path-connected and $\pi_i(X)\equiv 0$ for any $1\leq i\leq n$, where $\pi_i(X)$ is the $i$-th homotopy group of $X$.
$X$ is {\em simply connected} if $X$ is 1-connected.
$X$ is {\em contractible} if the identity map on $X$ is null-homotopic.
Note that, if $X$ is contractible, then $X$ is $n$-connected for each $n\geq 1$.
It is easy to see that the dendroids in Example \ref{exa:dend:dend} are contractible.
\medskip
\noindent
{\bf Computability Theory:}
We assume that the reader is familiar with Computability Theory on the natural numbers $\mathbb{N}$, Cantor space $2^\mathbb{N}$, and Baire space $\mathbb{N}^\mathbb{N}$ (see also Soare \cite{Soa}).
For basic terminology concerning {\em Computable Analysis}, see Weihrauch \cite{Wei}, Brattka-Weihrauch \cite{BW}, and Brattka-Presser \cite{BP}.
Hereafter, we fix a countable base for the Euclidean $n$-space $\mathbb{R}^n$ by $\rho=\{B(x;r):x\in\mathbb{Q}^n\;\&\;r\in\mathbb{Q}^+\}$, where $\mathbb{Q}^+$ denotes the set of all positive rationals.
Let $\{\rho_n\}_{n\in\nn}$ be an effective enumeration of $\rho$.
We say that a point $x\in\mathbb{R}^n$ is {\em computable} if the code of its principal filter $\mathcal{F}(x)=\{i\in\mathbb{N}:x\in\rho_i\}$ is computably enumerable (hereafter c.e.)
A closed subset $F\subseteq\mathbb{R}^n$ is $\Pi^0_1$ if there is a c.e.\ set $W\subseteq\nn$ such that $F=X\setminus\bigcup_{e\in W}\rho_e$.
A closed subset $F\subseteq\mathbb{R}^n$ is {\em computably enumerable} (hereafter {\em c.e.}) if $\{e\in\nn:F\cap\rho_e\not=\emptyset\}$ is c.e.
A closed subset $F\subseteq\mathbb{R}^n$ is {\em computable} if it is $\Pi^0_1$ and c.e.\ on $\mathbb{R}^n$.
\medskip
\noindent
{\bf Almost Computability:}
Let $A_0,A_1$ be nonempty closed subsets of a metric space $(X,d)$.
Then {\em the Hausdorff distance} between $A_0$ and $A_1$ is defined by
\[d_H(A_0,A_1)=\max_{i<2}\sup_{x\in A_i}\inf_{y\in A_{1-i}}d(x,y).\]
Let $\mathcal{P}$ be a class of continua.
We say that a continuum {\em $A$ $\ast$-includes a member of $\mathcal{P}$} if $\inf\{d_H(A,B):A\supseteq B\in\mathcal{P}\}=0$.
\begin{prop}\label{prop:astinclude}
Every Euclidean dendroid $\ast$-includes a tree.
\end{prop}
\begin{proof}\upshape
Fix a Euclidean dendroid $D\subseteq\mathbb{R}^n$, and a positive rational $\varepsilon\in\mathbb{Q}$.
Then $D$ is covered by finitely many open rational balls $\{B_i\}_{i<n}$ of radius $\varepsilon/2$.
Choose $d_i\in D\cap B_i$ for each $i<n$ if $B_i$ intersects with $D$.
Note that $\{B(d_i;\varepsilon)\}_{i<n}$ covers $D$.
Since $D$ is dendroid, there is a unique arc $\gamma_{i,j}\subseteq D$ connecting $d_i$ and $d_j$ for each $i,j<n$.
Then, $E=\bigcup_{\{i,j\}\subseteq n}\gamma_{i,j}$ is connected and locally connected, since $E$ is a union of finitely many arcs (i.e., it is a graph, in the sense of Continuum Theory; see also Nadler \cite{Nad}).
It is easy to see that $E$ has no Jordan curve, since $E$ is a subset of the dendroid $D$.
Consequently, $E$ is a tree.
Moreover, clearly $d_H(E,D)<\varepsilon$, since $d_i\in E$ for each $i<n$.
\end{proof}
The class $\mathcal{P}$ has {\em the almost computability property} if every $A\in\mathcal{P}$ $\ast$-includes a computable member of $\mathcal{P}$ as a closed set.
In this case, we simply say that {\em every $A\in\mathcal{P}$ is almost computable}.
Iljazovi\'c \cite{Ilj} showed that every $\Pi^0_1$ chainable continuum is almost computable, hence every $\Pi^0_1$ arc is almost computable.
\section{Incomputability of Dendrites}
By Proposition \ref{prop:astinclude}, topologically, every planar dendrite $\ast$-includes a tree.
However, if we try to effectivize this fact, we will find a counterexample.
\begin{theorem}\label{thm:main:dendrite1}
Not every computable planar dendrite $\ast$-includes a $\Pi^0_1$ tree.
\end{theorem}
\begin{proof}\upshape
Let $A\subseteq\mathbb{N}$ be an incomputable c.e.\ set.
Thus, there is a total computable function $f_A:\mathbb{N}\to\mathbb{N}$ such that ${\rm range}(f_A)=A$.
We may assume $f_A(s)\leq s$ for every $s\in\mathbb{N}$.
Let $A_s$ denote the finite set $\{f_A(u):u\leq s\}$.
Then ${\rm st}^A:\nn\to\nn$ is defined as ${\rm st}^A(n)=\min\{s\in\nn:n\in A_s\}$.
Note that ${\rm st}^A(n)\geq n$ by our assumption $f_A(s)\leq s$.
\begin{construction}\upshape
Recall the definition of the basic dendrite from Example \ref{exa:dend:dend}.
We construct a computable dendrite by modifying the basic dendrite $\mathcal{B}$.
For every $t\in\mathbb{N}$, we introduce {\em the width of the $t$-rising} $w(t)$ as follows:
\[
w(t)=
\begin{cases}
2^{-(2+{\rm st}^A(t))}, & \mbox{ if } t\in A,\\
0, & \mbox{ otherwise.}
\end{cases}
\]
Let $I_t$ be the closed interval $[2^{-t}-w(t),2^t+w(t)]$.
Since ${\rm st}^A(n)\geq n$, we have $I_t\cap I_s=\emptyset$ whenever $t\not=s$.
We observe that $\{w(t)\}_{t\in\nn}$ is a uniformly computable sequence of real numbers.
Now we define a computable dendrite $D\subseteq\mathbb{R}^2$ by:
\begin{align*}
D^0_t&=(\{2^{-t}-w(t)\}\cup\{2^{-t}+w(t)\})\times [0,2^{-t}]\\
D^1_t&=[2^{-t}-w(t),2^{-t}+w(t)]\times\{2^{-t}\}\\
D^2_t&=(2^{-t}-w(t),2^{-t}+w(t))\times(-1,2^{-t})\\
D&=\Big(\bigcup_{t\in\mathbb{N}}(D^0_{t}\cup D^1_{t})\Big)\cup\Big(([-1,1]\times \{0\})\setminus\bigcup_{t\in\mathbb{N}}D^2_{t,m}\Big).
\end{align*}
We call $D_t=D^0_t\cup D^1_t$ {\em the $t$-th rising of $D$}.
See Fig.\ \ref{fig:Basic_dendrite2}.
\end{construction}
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\caption{The dendrite $D$ for $0,2,4\not\in A$ and $1,3\in A$.}
\label{fig:Basic_dendrite2}
\end{figure}
\begin{claim}
The set $D$ is a dendrite.
\end{claim}
To prove $D$ is a Peano continuum, by the Hahn-Mazurkiewicz Theorem, it suffices to show that $D={\rm Im}(h)$ for some continuous curve $h:[-1,1]\to\mathbb{R}^2$.
We divide the unit interval $[0,1]$ into infinitely many parts $I_t=[2^{-(t+1)},2^{-t}]$.
Furthermore, we also divide each interval $I_{2t}$ into three parts $I^0_{2t}$, $I^1_{2t}$, and $I^2_{2t}$, where $I^i_{2t}=[(5-i)\cdot 3^{-1}\cdot 2^{-(2t+1)},(6-i)\cdot 3^{-1}\cdot 2^{-(2t+1)}]$ for each $i<3$.
Then we define a desired curve $h$ as follows.
\[
h(x)\mbox{ moves in }
\begin{cases}
\{2^{-t}+w(t)\}\times[0,2^{-t}]&\mbox{if }x\in I^0_{2t},\\
[2^{-t}-w(t),2^{-t}+w(t)]\times\{2^{-t}\}&\mbox{if }x\in I^1_{2t},\\
\{2^{-t}-w(t)\}\times[0,2^{-t}]&\mbox{if }x\in I^2_{2t},\\
[2^{-(t+1)}+w(t+1),2^{-t}-w(t)]\times\{0\}&\mbox{if }x\in I_{2t+1},\\
[-1,0]\times\{0\}&\mbox{if }x\in [-1,0].
\end{cases}
\]
Clearly, $h$ can be continuous, and indeed computable, since the map $w:\mathbb{R}\to\mathbb{R}$ is computable.
It is easy to see that $D={\rm Im}(h)$.
Moreover, ${\rm Im}(h)$ contains no Jordan curve since $\pi_0(h(x))\leq\pi_0(h(y))$ whenever $x\leq y$, where $\pi_0(p)$ denotes the first coordinate of $p\in\mathbb{R}^2$.
Consequently, $D$ is a dendrite.
\medskip
Moreover, by construction, it is easy to see that $D$ is computable.
\begin{claim}
The computable dendrite $D$ does not $\ast$-include a $\Pi^0_1$ tree.
\end{claim}
Suppose that $D$ contains a $\Pi^0_1$ subtree $T\subseteq D$.
We consider a rational open ball $B_t$ with center $\lrangle{2^{-t},2^{-t}}$ and radius $2^{-(t+2)}$, for each $t\in\nn$.
Note that $B_t\cap D\subseteq D_t$ for every $t\in\nn$.
Since $T$ is $\Pi^0_1$ in $\mathbb{R}^2$, $B=\{t\in\mathbb{N}:\hat{B}_{t}\cap T=\emptyset\}$ is c.e.
If $w(t)>0$ (i.e., $t\in A$) then $D\setminus(D_t\cap B_t)$ is disconnected.
Therefore, either $T\subseteq [-1,2^{-t}]\times\mathbb{R}$ or $T\subseteq[2^{-t},1]\times\mathbb{R}$ holds whenever $\hat{B}_{t}\cap T=\emptyset$ (i.e., $t\in B$), since $T$ is connected.
Thus, if the condition $\#(A\cap B)=\aleph_0$ is satisfied, then either $T\subseteq [-1,0]\times\mathbb{R}$ or $T\subseteq [0,1]\times\mathbb{R}$ holds.
Consequently, we must have $d_H(T,D)\geq 1$.
Therefore, we may assume $\#A\cap B<\aleph_0$.
Since $A$ is coinfinite, $D$ has infinitely many ramification points $\lrangle{2^{-t},0}$ for $t\not\in A$.
However, by the definition of tree, $T$ has only finitely many ramification points.
Thus we must have $(D^0_{t}\cap T)\setminus \{\lrangle{2^{-t},0}\}=\emptyset$ for almost all $t\not\in A$.
Since $\hat{B}_{t}\cap T\subseteq (D^0_{t}\cap T)\setminus \{\lrangle{2^{-t},0}\}$, we have $t\in B$ for almost all $t\in\nn\setminus A$.
Consequently, we have $\#((\mathbb{N}\setminus A)\triangle B)<\aleph_0$.
This implies that $\mathbb{N}\setminus A$ is also c.e., since $B$ is c.e.
This contradicts that $A$ is incomputable.
\end{proof}
Note that a Hausdorff space (hence every metric space) is (locally) arcwise connected if and only if it is (locally) pathwise connected.
However, Miller \cite{Mil} pointed out that the effective versions of arcwise connectivity and pathwise connectivity do {\em not} coincide.
Theorem \ref{thm:main:dendrite1} could give a result on effective connectivity properties.
Note that {\em effectively pathwise connectivity} is defined by Brattka \cite{Bra08}.
Clearly, the dendrite $D$ is effectively pathwise connected.
We now introduce a new effective version of arcwise connectivity property by thinking arcs as closed sets.
Let $\mathcal{A}_-(X)$ denote the hyperspace of closed subsets of $X$ with negative information (see also Brattka \cite{Bra08}).
\begin{definition}
A computable metric space $(X,d,\alpha)$ is {\em semi-effectively arcwise connected} if there exists a total computable multi-valued function $P:X^2\rightrightarrows\mathcal{A}_-(X)$ such that $P(x,y)$ is the set of all arcs $A$ whose two end points are $x$ and $y$, for any $x,y\in X$.
\end{definition}
Obviously $D$ is not semi-effectively arcwise connected.
Indeed, for every $\varepsilon>0$ there exists $x_0,x_1\in [0,1]$ with $d(x_0,x_1)<\varepsilon$ such that $\lrangle{x_0,0},\lrangle{x_1,0}\in D$ cannot be connected by any $\Pi^0_1$ arc.
Thus, we have the following corollary.
\begin{cor}
There exists an effectively pathwise connected Euclidean continuum $D$ such that $D$ is not semi-effectively arcwise connected.
\end{cor}
\begin{comment}
Let $\mathcal{P}$ be a class of continua.
For continua $A,B\in\mathcal{P}$ let $\mathcal{F}(\mathcal{P};A,B)$ be the set of all computable sequence $F:\mathbb{N}\to\mathcal{A}_\Pi$ of pairwise disjoint computable closed sets such that, for each $i$, $F(i)\cap A$ and $F(i)\cap B$ are nonempty and satisfy $\mathcal{P}$ whenever $F(i)$ is nonempty.
Then we say that a continuum $A$ {\em strongly $\ast$-includes} a member of $\mathcal{P}$ if
\[\inf\{\sum_id_H(A\cap F(i), B\cap F(i)):A\supseteq B\in\mathcal{P}\;\&\;F\in\mathcal{F}(\mathcal{P};A,B)\}=0.\]
The class $\mathcal{P}$ has strongly almost computability property if every $A\in\mathcal{P}$ strongly $\ast$-includes a computable member of $\mathcal{P}$ as a closed set.
In this case, we simply say that {\em every $A\in\mathcal{P}$ is strongly almost computable}.
It is easy to see that an arc (or a tree) $\ast$-includes $\mathcal{P}$ if and only if it strongly $\ast$-includes $\mathcal{P}$.
However, this does not hold for $\Pi^0_1$ dendrites.
\begin{theorem}
Not every almost computable $\Pi^0_1$ planar dendrite is strongly almost computable.
\end{theorem}
\begin{proof}\upshape
The construction is similar to Theorem \ref{thm:main:dendrite1}.
Let $L\subseteq\mathbb{N}$ be a simple non-hypersimple set.
Then there exists a computable function $m_L:\mathbb{N}\to\mathbb{N}$ which majorizes $\mathbb{N}\setminus L$.
Without loss of generality, we can assume that $m_L$ is strictly increasing, and we put $m_L(-1)=-1$.
We define a function $\hat{m}_L:\mathbb{N}\to\mathbb{N}$ by $\hat{m}_L(t)=1/(n+1)$, where $n$ is the unique number such that $m_L(n-1)<t\leq m_L(n)$, for each $t$.
For $t\in\mathbb{N}$, put $v(t)=\hat{m}_L(t)$ if $t\not\in L$; $v(t)=0$ otherwise.
Note that $\sum_tv(t)=\sum_{t\not\in L}\hat{m}_L(t)\geq\sum_n1/n=\infty$.
Obviously $v(t)$ is not computable, however it is upper semi-computable.
Then, we modify the definition of $D$ in Theorem \ref{thm:main:dendrite1} as follows:
\begin{align*}
E^0_{t,m}&=(\{c(t,m)-w(t)\}\cup\{c(t,m)+w(t)\})\times [0,2^{-t}+v(t)]\\
E&=\Big(\bigcup_{t\in\mathbb{N}}\bigcup_{m<2^t}(E^0_{t,m}\cup D^1_{t,m})\Big)\cup\Big(([0,1]\times \{0\})\setminus\bigcup_{t\in\mathbb{N}}\bigcup_{m<2^t}D^2_{t,m}\Big).
\end{align*}
We can prove that $E$ is a dendrite in the same way as in the proof of Theorem \ref{thm:main:dendrite1} since $\lim_t(2^{-t}+v(t))=0$ holds, and clearly $E$ is $\Pi^0_1$.
To prove that $E$ is almost computable, fix a real number $\varepsilon>0$, and pick $u$ for which $1/u<\varepsilon$.
Then the set
\begin{align*}
F=\Big(\bigcup_{t<u}\bigcup_{m<2^t}(E^0_{t,m}\cup D^1_{t,m})\cup\bigcup_{t\geq u}\bigcup_{m<2^t}(D^0_{t,m}\cup D^1_{t,m})\Big)\\
\cup\Big(([0,1]\times \{0\})\setminus\bigcup_{t\in\mathbb{N}}\bigcup_{m<2^t}D^2_{t,m}\Big)
\end{align*}
is a computable dendrite such that $d_H(F,E)\leq 1/u<\varepsilon$.
Now we consider a rational open ball $C_{t,m}$ with center $\lrangle{c(t,m),2^{-s}+r(t,m)}$ and radius $r(t,m)$ for any $\lrangle{t,m}\in X$.
Note that $t\in L$ if and only if $C_{t,m}$ does not intersect with $E^0_{t,m}\cup D^1_{t,m}$ for each $m<2^t$.
Let $G\subseteq E$ be a computable dendrite.
We can assume $G$ contains at least three points in $D\cap([0,1]\times\{0\})$ otherwise $d_H(G,F)\geq 1/4$ as in the proof of Theorem \ref{thm:main:dendrite1}.
Then $G$ is c.e.\ closed, so $M=\{t\in\mathbb{N}:(\forall m<2^t)\;C_{t,m}\cap G\not=\emptyset\}$ is c.e., and clearly $M\subseteq\mathbb{N}\setminus L$.
Since $L$ has no infinite c.e.\ subset, $M$ is finite.
Put $F_{t,m}=\{c(t,m)\}\times[0,2^{-t}+v(t)]$ if $\lrangle{c(t,m),0}\in G$ for $\max M<t\not\in L$ and $m<2^t$; $F_{t,m}=\emptyset$ otherwise.
Since $G$ and $\mathbb{N}\setminus L$ are $\Pi^0_1$, the sequence $\lrangle{t,m}\mapsto F_{t,m}$ is computable as a function $\mathbb{N}^2\to\mathcal{A}_\Pi$.
By our assumption, since $\#(G\cap D\cap([0,1]\times\{0\}))\geq 3$, there is $u\geq\max M$ such that $G$ must contain $\lrangle{c(t,m),0}$ for all $t>u$ and some $m<2^t$, otherwise $G$ is disconnected.
This implies $\sum_{t,m}d_H(G\cap F_{t,m}, E\cap F_{t,m})\geq\sum_{u<t\not\in L}v(t)=\infty$ since $\sum_{u<t\not\in L}v(t)\geq\sum_{n\geq j}1/n$.
\end{proof}
\end{comment}
\begin{theorem}\label{thm:rite_alcom}
Not every $\Pi^0_1$ planar dendrite is almost computable.
\end{theorem}
To prove Theorem \ref{thm:rite_alcom}, we need to prepare some tools.
For a string $\sigma\in 2^{<\mathbb{N}}$, let $lh(\sigma)$ denote the length of $\sigma$.
Then
\[\psi(\sigma)=\left\lrangle{2^{-1}\cdot 3^{-i}+2\sum_{i<lh(\sigma)\;\&\; \sigma(i)=1}3^{-(i+1)},2^{-lh(\sigma)}\right}\in\mathbb{R}^2.\]
For two points $\vec{x},\vec{y}\in\mathbb{R}^2$, the closed line segment $L(\vec{x},\vec{y})$ from $\vec{x}$ to $\vec{y}$ is defined by $L(\vec{x},\vec{y})=\{(1-t)\vec{x}+t\vec{y}:t\in [0,1]\}$.
For a (possibly infinite) tree $T\subseteq 2^{<\mathbb{N}}$, we plot an embedded tree $\Psi(T)\subseteq\mathbb{R}^2$ by
\[\Psi(T)=cl\left(\bigcup\{L(\psi(\sigma),\psi(\tau)):\sigma,\tau\in T\;\&\;lh(\sigma)=lh(\tau)+1\}\right).\]
Then $\Psi(T)$ is a dendrite (but not necessarily a tree, in the sense of Continuum Theory), for any (possibly infinite) tree $T\subseteq 2^\mathbb{N}$.
See Fig.\ \ref{fig:Tree}.
\begin{figure}[t]\centering
\begin{center}
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\end{picture}
\end{center}
\vspace{-0.5em}
\caption{The plotted tree $\Psi(2^{<\nn})$.}
\label{fig:Tree}
\end{figure}
We can easily prove the following lemmata.
\begin{lemma}\label{lem:rite:1}
Let $T$ be a subtree of $2^{<\mathbb{N}}$, and $D$ be a planar subset such that $\psi(\lrangle{})\in D\subseteq \Psi(T)$ for the root $\lrangle{}\in 2^{<\mathbb{N}}$.
Then $D$ is a dendrite if and only if $D$ is homeomorphic to $\Psi(S)$ for a subtree $S\subseteq T$.
\end{lemma}
\begin{proof}\upshape
The ``if'' part is obvious.
Let $D$ be a dendrite.
For a binary string $\sigma$ which is not a root, let $\sigma^-$ be an immediate predecessor of $\sigma$.
We consider the set $S=\{\lrangle{}\}\cup\{\sigma\in 2^{<\mathbb{N}}:\sigma\not=\lrangle{}\;\&\;D\cap(L(\psi(\sigma^-),\psi(\sigma))\setminus\{\psi(\sigma^-)\})\not=\emptyset\}$.
Since $D$ is connected, $S$ is a subtree of $T$.
It is easy to see that $D$ is homeomorphic to $\Psi(S)$.
\end{proof}
\begin{lemma}\label{lem:rite:2}
Let $T$ be a subtree of $2^{<\mathbb{N}}$.
Then $T$ is $\Pi^0_1$ (c.e., computable, resp.) if and only if $\Psi(T)$ is a $\Pi^0_1$ (c.e., computable, resp.) dendrite in $\mathbb{R}^2$.
\end{lemma}
\begin{proof}\upshape
With our definition of $\Psi$, the dendrite $\Psi(2^{<\mathbb{N}})$ is clearly a computable closed subset of $\mathbb{R}^2$.
So, if $T$ is $\Pi^0_1$, then it is easy to prove that $\Psi(T)$ is also $\Pi^0_1$.
Assume that $T$ is a c.e.\ tree.
At stage $s$, we compute whether $L(\psi(\sigma^-),\psi(\sigma))$ intersects with the $e$-th open rational ball $\rho_e$, for any $e<s$ and any string $\sigma$ which is already enumerated into $T$ by stage $s$.
If so, we enumerate $e$ into $W_T$ at stage $s$.
Then $\{e\in\nn:\Psi(T)\cap\rho_e\not=\emptyset\}=W_T$ is c.e.
Assume that $\Psi(T)$ is $\Pi^0_1$.
We consider an open rational ball $B_-(\sigma)=B(\psi(\sigma);2^{-(lh(\sigma)+2)})$ for each $\sigma\in 2^{<\mathbb{N}}$.
Note that $\hat{B}_-(\sigma)\cap\hat{B}_-(\tau)=\emptyset$ for $\sigma\not=\tau$.
Since $\Psi(T)$ is $\Pi^0_1$, $T^*=\{\sigma\in 2^{<\mathbb{N}}:\Psi(T)\cap\hat{B}_-(\sigma)=\emptyset\}$ is c.e., and it is easy to see that $T=2^{<\mathbb{N}}\setminus T^*$.
Thus, $T$ is a $\Pi^0_1$ tree of $2^{<\mathbb{N}}$.
We next assume that $\Psi(T)$ is c.e.
We can assume that $\Psi(T)$ contains the root $\psi(\lrangle{})$, otherwise $T=\emptyset$, and clearly it is c.e.
For a binary string $\sigma$ which is not a root, let $\sigma^-$ be an immediate predecessor of $\sigma$.
Pick an open rational ball $B_+(\sigma)$ such that $\Psi(2^{<\mathbb{N}})\cap B_+(\sigma)\subseteq L(\psi(\sigma^-),\psi(\sigma))$ for each $\sigma$.
Since $\Psi(T)$ is c.e., $T^*=\{\sigma\in 2^{<\mathbb{N}}:\Psi(T)\cap B_+(\sigma)\not=\emptyset\}$ is c.e.
If $\sigma$ is not a root and $\sigma\in T$ then $L(\psi(\sigma^-),\psi(\sigma))\subseteq\Psi(T)$, so $\Psi(T)\cap B_+(\sigma)\not=\emptyset$.
We observe that if $\sigma\not\in T$ then $L(\psi(\sigma^-),\psi(\sigma))\cap\Psi(T)=\emptyset$, so $\Psi(T)\cap B_+(\sigma)=\emptyset$.
Thus, we have $T=T^*$.
In the case that $\Psi(T)$ is computable, $\Psi(T)$ is c.e.\ and $\Pi^0_1$, hence $T$ is c.e.\ and $\Pi^0_1$, i.e., $T$ is computable.
\end{proof}
\begin{lemma}\label{lem:rite:3}
Let $D$ be a computable subdendrite of $\Psi(2^{<\mathbb{N}})$.
Then there exists a computable subtree $T^+\subseteq 2^{<\mathbb{N}}$ such that $D\subseteq \Psi(T^+)$ and $([0,1]\times\{0\})\cap D=([0,1]\times\{0\})\cap\Psi(T^+)$.
\end{lemma}
\begin{proof}\upshape
We can assume $\psi(\lrangle{})\in D$, otherwise we connect $\psi(\lrangle{})$ and the root of $D$ by a subarc of $\Psi(2^{<\mathbb{N}})$.
Again we consider an open rational ball $B_-(\sigma)=B(\psi(\sigma);2^{-(lh(\sigma)+2)})$, and an open rational ball $B_+(\sigma)$ such that $\Psi(2^{<\mathbb{N}})\cap B_+(\sigma)\subseteq L(\psi(\sigma^-),\psi(\sigma))$ for each $\sigma\in 2^{<\nn}$.
Since $D$ is $\Pi^0_1$, $U^*=\{\sigma\in 2^{<\mathbb{N}}:D\cap\hat{B}_-(\sigma)=\emptyset\}$ is c.e.
Since $D$ is c.e., $T^*=\{\sigma\in 2^{<\mathbb{N}}:D\cap B_+(\sigma)\not=\emptyset\}$ is c.e., and it is a tree by Lemma \ref{lem:rite:1}.
For every $\sigma\in 2^{<\mathbb{N}}$, either $D\cap\hat{B}_-(\sigma)=\emptyset$ or $D\cap B_+(\sigma)\not=\emptyset$ holds.
Therefore, we have $T^*\cup U^*=2^{<\nn}$.
Moreover, for the set of {\em leaves of $T^*$}, $L_T^*=\{\rho\in T^*:(\forall i<2)\;\rho\fr\lrangle{i}\not\in T^*\}$, we observe that $T^*\cap U^*\subseteq L_T^*$.
Recall that the pointclass $\Sigma^0_1$ has the reduction property, that is, for two c.e.\ sets $T^*$ and $U^*$, there exist c.e.\ subsets $T\subseteq T^*$ and $U\subseteq U^*$ such that $T\cup U=T^*\cup U^*$ and $T\cap U=\emptyset$.
This is because, for $\sigma\in T^*\cap U^*$, $\sigma$ is enumerated into $T$ when $\sigma$ is enumerated into $T^*$ before it is enumerated into $U^*$; $\sigma$ is enumerated into $U$ otherwise.
Since $T^*\cap U^*\subseteq L_T^*$, $T$ must be tree.
Furthermore, $T$ is c.e., and $U=2^{<\mathbb{N}}\setminus T$ is also c.e.
Thus, $T$ is a computable tree.
Therefore, $T^+=\{\sigma\fr\lrangle{i}:\sigma\in T\;\&\;i<2\}$ is also a computable tree.
Then, $D\subseteq\Psi(T^+)$, and we have $([0,1]\times\{0\})\cap D=([0,1]\times\{0\})\cap\Psi(T^+)$ since the set of all infinite paths of $T$ and that of $T^+$ coincide.
\end{proof}
Cenzer, Weber and Wu, and the author \cite{CKWW} introduced the notion of {\em tree-immunity} for closed sets in Cantor space $2^\mathbb{N}$.
For $\sigma\in 2^{<\mathbb{N}}$, define $I_\sigma$ as $\{f\in 2^\mathbb{N}:(\forall n<lh(\sigma))\;f(n)=\sigma(n)\}$.
Note that $\{I_\sigma:\sigma\in 2^{<\mathbb{N}}\}$ is a countable base for Cantor space.
\begin{definition}[Cenzer-Kihara-Weber-Wu \cite{CKWW}]
A nonempty closed set $F\subseteq 2^\mathbb{N}$ is said to be {\em tree-immune} if the tree $T_F=\{\sigma\in 2^{<\mathbb{N}}:F\cap I_\sigma\not=\emptyset\}\subseteq 2^{<\mathbb{N}}$ contains no infinite computable subtree.
\end{definition}
For a nonempty $\Pi^0_1$ subset $P\subseteq 2^\mathbb{N}$, the corresponding tree $T_P$ is $\Pi^0_1$, and it has no dead ends.
The set of {\em all complete consistent extensions of Peano Arithmetic} is an example of a tree-immune $\Pi^0_1$ subset of $2^\mathbb{N}$.
Tree-immune $\Pi^0_1$ sets have the following remarkable property.
\begin{lemma}\label{lem:rite:4}
Let $P$ be a tree-immune $\Pi^0_1$ subset of $2^\mathbb{N}$ and let $D\subseteq\Psi(T_P)$ be any computable subdendrite.
Then $([0,1]\times\{0\})\cap D=\emptyset$ holds.
\end{lemma}
\begin{proof}\upshape
By Lemma \ref{lem:rite:3}, there exists a computable subtree $T\subseteq 2^{<\mathbb{N}}$ such that $D\subseteq\Psi(T)$ and $\Psi(T)$ agrees with $D$ on $[0,1]\times\{0\}$.
Since $D\subseteq\Psi(T_P)$, and since $T_P$ has no dead ends, $T\subseteq T_P$ holds.
Since $P$ is tree-immune, $T$ must be finite.
By using weak K\"onig's lemma (or, equivalently, compactness of Cantor space), $T\subseteq 2^l$ holds for some $l\in\mathbb{N}$.
Thus, $D\subseteq\Psi(T)\subseteq [0,1]\times [2^{-l},1]$ as desired.
\end{proof}
Note that if $P$ is a nonempty $\Pi^0_1$ set in Cantor space $2^\mathbb{N}$, then for every $\delta>0$ it holds that $((0,1)\times(0,\delta))\cap\Psi(T_P)\not=\emptyset$.
Finally, we are ready to prove Theorem \ref{thm:rite_alcom}.
\begin{proof}[Proof of Theorem \ref{thm:rite_alcom}]\upshape
Again, we adapt the construction in the proof of Theorem \ref{thm:main:dendrite1}.
We fix a nonempty tree-immune $\Pi^0_1$ set $P\subseteq 2^\mathbb{N}$.
For $\sigma\in 2^{<\mathbb{N}}$, put $E(\sigma)=\{\tau\in 2^{<\mathbb{N}}:\tau\supseteq\sigma\}$.
For a $\Pi^0_1$ tree $T_P\subseteq 2^{<\mathbb{N}}$, there exists a computable function $f_P:\mathbb{N}\to 2^{<\mathbb{N}}$ such that $T_P=2^{<\mathbb{N}}\setminus\bigcup_nE(f_P(n))$ and such that for each $\sigma\in 2^{<\mathbb{N}}$ and $s\in\mathbb{N}$ we have $\sigma\in\bigcup_{t<s}E(f_P(t))$ whenever $\sigma\fr 0,\sigma\fr 1\in\bigcup_{t<s}E(f_P(t))$.
For such a computable function $f_P:\mathbb{N}\to 2^{<\mathbb{N}}$, we let $T_{P,s}$ denote $2^{<\mathbb{N}}\setminus\bigcup_{t<s}E(f_P(t))$.
Then $T_{P,s}$ is a tree without dead ends, and $\{T_{P,s}:s\in\nn\}$ is computable uniformly in $s$.
\begin{construction}\upshape
Let $\vec{e}_1$ denote $\lrangle{1,0}\in\mathbb{R}^2$.
For a tree $T\subseteq 2^{<\mathbb{N}}$ and $w\in\mathbb{Q}$, we define $\Psi(T;w)$, {\em the edge of the fat approximation of the tree $T$ of width $w$}, by
\begin{align*}
\Psi(T;w)=cl\biggl(\bigcup\Big\{L&\left(\psi(\sigma)\pm(3^{-|\sigma|}\cdot w)\vec{e}_1,\psi(\tau)\pm(3^{-|\tau|}\cdot w)\vec{e}_1\right)\\
&:\pm\in\{-,+\}\;\&\;\sigma,\tau\in T\;\&\;lh(\sigma)=lh(\tau)+1\Big\}\biggr).
\end{align*}
If $\lim_s w_s=0$ then we have $\lim_s\Psi(T;w_s)=\Psi(T)$.
Moreover, if $\{w_s:s\in\mathbb{N}\}$ is a uniformly computable sequence of rational numbers, then $\{\Psi(T;w_s):s\in\mathbb{N}\}$ is also a uniformly computable sequence of computable closed sets.
Additionally, the set $\Psi(T;w,c,t,q)$, for a tree $T\subseteq 2^{<\mathbb{N}}$, for $w,c,q\in\mathbb{Q}$, and for $t\in\mathbb{N}$, is defined by
\[\Psi(T;w,c,t,q)=\left\{\left\lrangle{c+q\cdot\left(x-\frac{1}{2}\right),\frac{2-y}{2^{t+1}}\right}\in\mathbb{R}^2:\lrangle{x,y}\in\Psi(T;w)\right\}.\]
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\end{center}
\vspace{-0.5em}
\caption{The fat approximation $\Psi(T;w)$.}
\label{fig:Treec}
\end{minipage}
\begin{minipage}{0.48\hsize}
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\end{center}
\vspace{-0.5em}
\caption{The basic object $\Psi(T;w,c,t,q)$.}
\label{fig:Tree2}
\end{minipage}
\end{figure}
Note that $\Psi(T;w,c,t,q)\subseteq [c-q/2,c+q/2]\times [2^{-(t+1)},2^{-t}]$ as in Fig. \ref{fig:Tree2}.
For $t\in\mathbb{N}$, and for ${\rm st}^A(t)=\min\{s:t\in A_s\}$ in the proof of Theorem \ref{thm:main:dendrite1}, let $l(t)\in 2^\mathbb{N}$ be the leftmost path of $T_{P,{\rm st}^A(t)}$.
If ${\rm st}^A(t)$ is undefined (i.e., $t\not\in A$) then $l(t)$ is also undefined.
For each $t\in\mathbb{N}$ we define $F(t)=\{\sigma\in 2^{<\mathbb{N}}:\sigma\subseteq l(t)\}$ if $l(t)$ is defined; $F(t)=T_P$ otherwise.
Then $\{F(t):t\in\mathbb{N}\}$ is a computable sequence of $\Pi^0_1$ subsets of $2^{<\mathbb{N}}$.
Furthermore, we have $\Psi(F(t))\cap([0,1]\times\{0\})\not=\emptyset$, since $F(t)$ has a path for every $t\in\mathbb{N}$.
For each $t\in\nn$, $w(t)$ is defined again as in the proof of Theorem \ref{thm:main:dendrite1}.
Now we define a $\Pi^0_1$ dendrite $H\subseteq\mathbb{R}^2$ as follows:
\begin{align*}
H^*_{t}&=\Psi(F(t);w(t),2^{-t},t,2^{-(t+2)})\\
H^0_{t}&=(\{2^{-t}-w(t)\}\cup\{2^{-t}+w(t)\})\times [0,2^{-(t+1)}]\\
H^{**}_{t}&=(2^{-t}-w(t),2^{-t}+w(t))\times\{2^{-(t+1)}\}\\
H^2_{t}&=(2^{-t}-w(t),2^{-t}+w(t))\times(-1,2^{-(t+1)})\\
H&=\Big(\bigcup_{t\in\mathbb{N}}\left(H^*_{t}\cup H^0_{t}\setminus(H^{**}_t\cup int H^*_t)\right)\Big)\cup\Big(([-1,1]\times \{0\})\setminus\bigcup_{t\in\mathbb{N}}H^2_{t}\Big).
\end{align*}
\end{construction}
Put $H_t=H^*_{t}\setminus(H^{**}_t\cup int H^*_t)$ (see Fig.\ \ref{fig:Tree_dendrite}).
We can also show that $H$ is a $\Pi^0_1$ dendrite in the same way as for Theorem \ref{thm:main:dendrite1}.
\begin{claim}
The $\Pi^0_1$ dendrite $H$ does not $\ast$-include a computable dendrite.
\end{claim}
Let $J$ be a computable subdendrite of $H$.
Put $S(t)=[3\cdot 2^{-(t+2)},5\cdot 2^{-(t+2)}]\times[2^{-(t+1)},2^{-t}]$.
Then, we note that $J(t)=J\cap S(t)$ is also a computable dendrite, since $H_t\subseteq S(t)$ and it is a dendrite.
However, by Lemma \ref{lem:rite:4}, if $t\not\in A$ then we have $J(t)\cap(\mathbb{R}\times\{2^{-t}\})=\emptyset$.
So we consider the following set:
\begin{align*}
C=\{t\in\mathbb{N}:J(t)\cap\left([3\cdot 2^{-(t+2)},5\cdot 2^{-(t+2)}]\times[2^{-t},1]\right)=\emptyset\}.
\end{align*}
Since $J(t)$ is uniformly computable in $t$, the set $C$ is clearly c.e., and we have $\mathbb{N}\setminus A\subseteq C$.
However, if $\mathbb{N}\setminus A=C$, then this contradicts the incomputability of $A$.
Thus, there must be infinitely many $t\in A$ such that $t$ is enumerated into $C$.
However, if $t\in A$ is enumerated into $C$, it {\em cuts} the dendrite $H$.
In other words, since $J\subseteq H$ is connected, either $J\subseteq [-1,5\cdot 2^{-(t+2)}]\times\mathbb{R}$ or $J\subseteq [3\cdot 2^{-(t+2)},1]\times\mathbb{R}$.
Hence we must have $d_H(J,H)\geq 1$.
\end{proof}
\begin{figure}[t]\centering
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\end{center}
\vspace{-0.5em}
\caption{The dendrite $H$ for $0,2,4\not\in A$ and $1,3\in A$.}
\label{fig:Tree_dendrite}
\end{figure}
\begin{cor}
There exists a nonempty $\Pi^0_1$ subset of $[0,1]^2$ which is contractible, locally contractible, and $*$-includes no connected computable closed subset.
\end{cor}
\section{Incomputability of Dendroids}
\begin{theorem}\label{thm:roid:alinc}
Not every computable planar dendroid $\ast$-includes a $\Pi^0_1$ dendrite.
\end{theorem}
\begin{lemma}\label{lem:comp_roid}
There exists a limit computable function $f$ such that, for every uniformly c.e.\ sequence $\{U_n:n\in\mathbb{N}\}$ of cofinite c.e.\ sets, we have $f(n)\in U_n$ for almost all $n\in\mathbb{N}$.
\end{lemma}
\begin{proof}\upshape
Let $\{V_e:e\in\mathbb{N}\}$ be an effective enumeration of all uniformly c.e.\ non-increasing sequences $\{U_n:n\in\mathbb{N}\}$ of c.e.\ sets such that $\min U_n\geq n$, where $(V_e)_n=U_n=\{x\in\mathbb{N}:(n,x)\in V_e\}$.
{\em The $e$-state of $y$} is defined by $\sigma(e,y)=\{i\leq e:y\in(V_i)_e\}$, and {\em the maximal $e$-state} is defined by $\sigma(e)=\max_z\sigma(e,z)$.
The construction of $f:\nn\to\nn$ is to maximize the $e$-state.
For each $e\in\nn$, $f(e)$ chooses the least $y\in\nn$ having the maximal $e$-state $\sigma(e,y)=\sigma(e)$.
Since $\{\sigma(e,y):e,y\in\mathbb{N}\}$ is uniformly c.e., and $\sigma(e,y)\in 2^e$, the function $e\mapsto\sigma(e)=\max_z\sigma(e,z)$ is total limit computable.
Thus, $f$ is limit computable.
It is easy to see that $\lim_e\sigma(e)(n)$ exists for each $n\in\mathbb{N}$.
Let $U=\{U_n:n\in\mathbb{N}\}$ be a uniformly c.e.\ sequence of cofinite c.e.\ sets.
Then $V=\{\bigcap_{m\leq n}U_m:n\in\mathbb{N}\}$ is a uniformly c.e.\ non-increasing sequence of cofinite c.e.\ sets.
Thus, $V_i=V$ for some index $i$.
Then $i\in\sigma(e,y)$ for almost all $e,y\in\mathbb{N}$.
This ensures that $i\in\sigma(e)$ for almost all $e\in\mathbb{N}$ by our assumption $\min U_n\geq n$.
Hence we have $f(n)\in U_n$ for almost all $n\in\mathbb{N}$.
\end{proof}
\begin{remark}
The proof of Lemma \ref{lem:comp_roid} is similar to the standard construction of a maximal c.e.\ set (see Soare \cite{Soa}).
Recall that the principal function of the complement of a maximal c.e.\ set is {\em dominant}, i.e., it dominates all total computable functions.
The limit computable function $f$ in Lemma \ref{lem:comp_roid} is also dominant.
Indeed, for any total computable function $g$, if we set $U^g_n=\{y\in\mathbb{N}:y\geq g(n)\}$ then $\{U^g_n:n\in\mathbb{N}\}$ is a uniformly c.e.\ sequence of cofinite c.e.\ sets, and if $f(n)\in U^g_n$ holds then we have $f(n)\geq g(n)$.
\end{remark}
\begin{proof}[Proof of Theorem \ref{thm:roid:alinc}]\upshape
Pick a limit computable function $f=\lim_sf_s$ in Lemma \ref{lem:comp_roid}.
For every $t,u\in\mathbb{N}$, put $v(t,u)=2^{-s}$ for the least $s$ such that $f_s(t)=u$ if such $s$ exists; $v(t,u)=0$ otherwise.
Since $\{f_s:s\in\mathbb{N}\}$ is uniformly computable, $v:\mathbb{N}^2\to\mathbb{R}$ is computable.
\begin{construction}\upshape
For each $t\in\mathbb{N}$, {\em the center position of the $u$-th rising of the $t$-th comb} is defined as $c_*(t,u)=2^{-(2t+1)}+2^{-(2t+u+1)}$, and {\em the width of the $u$-th rising of the $t$-th comb} is defined as $v_*(t,u)=v(t,u)\cdot 2^{-(2t+u+3)}$.
Then, we define {\em the $t$-th harmonic comb} $K_{t}$ for each $t\in\mathbb{N}$ as follows:
\begin{align*}
K^*_{t}&=\{2^{-(2t+1)}\}\times [0,2^{-t}]\\
K^0_{t,u}&=\{c_*(t,u)-v_*(t,u),c_*(t,u)+v_*(t,u)\}\times [0,2^{-t}]\\
K^1_{t,u}&=[c_*(t,u)-v_*(t,u),c_*(t,u)+v_*(t,u)]\times\{2^{-t}\}\\
K^2_{t,u}&=(c_*(t,u)-v_*(t,u),c_*(t,u)+v_*(t,u))\times(-1,2^{-t})\\
K_{t}&=\left(K^*_{t}\cup\bigcup_{i<2}\bigcup_{u\in\mathbb{N}}K^i_{t,u}\right)\cup\left(([2^{-(2t+1)},2^{-2t}]\times \{0\})\setminus\bigcup_{u\in\mathbb{N}}K^2_{t,u}\right).
\end{align*}
Note that $K_{t}$ is homeomorphic to the harmonic comb $\mathcal{H}$ for each $t\in\mathbb{N}$.
This is because, for fixed $t\in\mathbb{N}$, since $\lim_sf_s(t)$ exists we have $v(t,u)=0$ for almost all $u\in\mathbb{N}$.
Then the desired computable dendroid is defined as follows.
\[K=([-1,0]\times\{0\})\cup\bigcup_{t\in\mathbb{N}}\left(\big([2^{-(2t+2)},2^{-(2t+1)}]\times\{0\}\big)\cup K_{t}\right).\]
\end{construction}
\begin{figure}[t]\centering
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\begin{center}
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\put(7.8000,-8.8000){\makebox(0,0)[lb]{$K_2$}}
\end{picture}
\end{center}
\vspace{-0.5em}
\caption{The dendroid $K$.}
\label{fig:Dendroid1}
\end{minipage}
\begin{minipage}{0.48\hsize}
\begin{center}
\unitlength 0.1in
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\end{center}
\vspace{-0.5em}
\caption{The harmonic comb $K_t$ for $f_0(t)=0$, $f_1(t)=0$, $f_2(t)=2$, $\dots$}
\label{fig:Dendroid2}
\end{minipage}
\end{figure}
\begin{claim}
The set $K$ is a computable dendroid.
\end{claim}
Clearly $K$ is a computable closed set.
To show that $K$ is pathwise connected, we consider the following subcontinuum $K_{t}^-$, {\em the grip of the comb $K_{t,m}$}, for each $t\in\mathbb{N}$.
\begin{align*}
K^-_{t}&=(\bigcup_{i<2}\bigcup_{v(t,u)>0}K^i_{t,u})\cup\big(([2^{-(2t+1)},2^{-2t}]\times \{0\})\setminus\bigcup_{v(t,u)>0}K^2_{t,u}\big).
\end{align*}
Then $K^-=([-1,0]\times\{0\})\cup\bigcup_{t\in\mathbb{N}}\left(([2^{-(2t+2)},2^{-(2t+1)}]\times\{0\})\cup K^-_{t}\right)$ has no ramification points.
We claim that $K^-$ is connected and $K^-$ is even an arc.
To show this claim, we first observe that $K^-_{t}$ is an arc for any $t\in\mathbb{N}$, since $v(t,u)>0$ occurs for finitely many $u\in\mathbb{N}$.
Moreover $K^-_{t}\subseteq S(t)$, and $\lim_t{\rm diam}(S(t))=0$ holds.
Therefore, we see that $K^-$ is locally connected and, hence, an arc.
For points $p,q\in K$, if $p,q\in K^-$ then $p$ and $q$ are connected by a subarc of $K^-$.
In the case $p\in K\setminus K^-$, the point $p$ lies on $K^0_{t,u}$ for some $t,u$ such that $v(t,u)=0$.
If $q\in K^-$ then there is a subarc $A\subseteq K^-$ (one of whose endpoints must be $\lrangle{c_*(t,u),0}$) such that $A\cup K^0_{t,u}$ is an arc containing $p$ and $q$.
In the case $q\in K\setminus K^-$, similarly we can connect $p$ and $q$ by an arc in $K$.
Hence, $K$ is pathwise connected.
$K$ is hereditarily unicoherent, since the harmonic comb is hereditarily unicoherent.
Thus, $K$ is a dendroid.
\begin{claim}
The computable dendroid $K$ does not $\ast$-include a $\Pi^0_1$ dendrite.
\end{claim}
What remains to show is that every $\Pi^0_1$ subdendrite $R\subseteq K$ satisfies $d_H(R,K)$
$\geq 1$.
Let $R\subseteq K$ be a $\Pi^0_1$ dendrite.
Put $S(t)=[2^{-(2t+1)},2^{-2t}]\times[0,2^{-t}]$.
Since $R$ is locally connected, $R\cap S(t)=R\cap K_t$ is also locally connected for each $t\in\mathbb{N}$ and $m<2^t$.
Thus, for fixed $t\in\mathbb{N}$, let $K^{1*}_{t,u}=[c_*(t,u)-2^{-(2t+u+3)},c_*(t,u)+2^{-(2t+u+3)}]\times\{2^{-t}\}$.
For any continuum $R^*\subset K_{t}$, if $R^*\cap K^{1*}_{t,u}\not=\emptyset$ for infinitely many $u\in\nn$, then $R^*$ must be homeomorphic to the harmonic comb $\mathcal{H}$.
Hence, $R^*$ is not locally connected.
Therefore, we have $R\cap K^{1*}_{t,u}=\emptyset$ for almost all $u\in\mathbb{N}$.
Since $K^{1*}_{t,u}$ and $K^{1*}_{s,v}$ is disjoint whenever $\lrangle{t,u}\not=\lrangle{s,v}$, and since $R$ is $\Pi^0_1$, we can effectively enumerate $U_t=\{u\in\mathbb{N}:R\cap K^{1*}_{t,u}=\emptyset\}$, i.e., $\{U_t:t\in\mathbb{N}\}$ is uniformly c.e.
Moreover, $U_t$ is cofinite for every $t\in\mathbb{N}$.
Then, by our definition of $f=\lim_sf_s$ in Lemma \ref{lem:comp_roid}, there exists $t^*\in\mathbb{N}$ such that $f(t)\in U_t$ for all $t\geq t^*$.
Note that $v(t,f(t))>0$ and thus the condition $f(t)\in U_t$ (i.e., $R\cap K^{1*}_{t,f(t)}=\emptyset$) implies that, for every $t\geq t^*$, either $R\subseteq [-1,c_*(t,u)+v_*(t,u)]\times[0,1]$ or $R\subseteq [c_*(t,u)-v_*(t,u),1]\times[0,1]$ holds.
Thus we obtain the desired condition $d_H(R,K)\geq 1$.
\end{proof}
\begin{remark}
It is easy to see that the dendroid constructed in the proof of Theorem \ref{thm:roid:alinc} is contractible.
\end{remark}
\begin{cor}
There exists a nonempty contractible planar computable closed subset of $[0,1]^2$ which $\ast$-includes no $\Pi^0_1$ subset which is connected and locally connected.
\end{cor}
\begin{theorem}\label{thm:special_roid}
Not every nonempty $\Pi^0_1$ planar dendroid contains a computable point.
\end{theorem}
\begin{proof}\upshape
One can easily construct a $\Pi^0_1$ Cantor fan $F$ containing at most one computable point $p\in F$, and such $p$ is the unique ramification point of $F$.
Our basic idea is to construct a topological copy of the Cantor fan $F$ along a pathological located arc $A$ constructed by Miller \cite[Example 4.1]{Mil}.
We can guarantee that moving the fan $F$ along the arc $A$ cannot introduce new computable points.
Additionally, this moving will make a ramification point $p^*$ in a copy of $F$ incomputable.
\medskip
\noindent
{\bf Fat Approximation.}
To archive this construction, we consider a fat approximation of a subset $P$ of the middle third Cantor set $C\subseteq\mathbb{R}^1$, by modifying the standard construction of $C$.
For a tree $T\subseteq 2^{<\mathbb{N}}$, put $\pi(\sigma)=3^{-1}+2\sum_{i<lh(\sigma)\;\&\; \sigma(i)=1}3^{-(i+2)}$ for $\sigma\in T$, and $J(\sigma)=[\pi(\sigma)-3^{-(lh(\sigma)+1)},\pi(\sigma)+2\cdot 3^{-(lh(\sigma)+1)}]$.
If a binary string $\sigma$ is incomparable with a binary string $\tau$ then $J(\sigma)\cap J(\tau)=\emptyset$.
We extend $\pi$ to a homeomorphism $\pi_*$ between Cantor space $2^\mathbb{N}$ and $C\cap[1/3,2/3]$ by defining $\pi_*(f)=3^{-1}+2\sum_{f(i)=1}3^{-(i+2)}$ for $f\in 2^\mathbb{N}$.
Let $P^*\subseteq 2^\mathbb{N}$ be a nonempty $\Pi^0_1$ set without computable elements.
Then there exists a computable tree $T_P$ such that $P^*$ is the set of all paths of $T_P$, since $P^*$ is $\Pi^0_1$.
{\em A fat approximation $\{P_s:s\in\mathbb{N}\}$ of $P=\pi_*(P^*)$} is defined as $P_s=\bigcup\{J(\sigma):lh(\sigma)=s\;\&\;\sigma\in T_P\}$.
Then $\{P_s:s\in\mathbb{N}\}$ is a computable decreasing sequence of computable closed sets, and we have $P=\bigcap_sP_s$.
Since $P$ is a nonempty bounded closed subset of a real line $\mathbb{R}^1$, both $\min P$ and $\max P$ exist.
By the same reason, both $l_s^-=\min P_s$ and $r_s^+=\max P_s$ also exist, for each $s\in\mathbb{N}$, and $\lim_sl_s^-=\min P$ and $\lim_sr_s^+=\max P$, where $\{l_s:s\in\mathbb{N}\}$ is increasing, and $\{r_s:s\in\mathbb{N}\}$ is decreasing.
Let $l_s=l_s^-+3^{-(s+1)}$ and $r_s=r_s^+-3^{-(s+1)}$.
We also set $l_s^*=l_s^-+3^{-(s+2)}$ and $r_s=r_s^+-3^{-(s+2)}$.
Note that $l_s<r_s$, $\lim_sl_s=\min P$, and $\lim_sr_s=\max P$.
Since $\min P,\max P\in P$ and $P$ contains no computable points, $\min P$ and $\max P$ are non-computable, and so $l_s<\min P<\max P<r_s$ holds for any $s\in\mathbb{N}$.
The fat approximation of $P$ has the following remarkable property:
\[[l_s^-,l_s]\subseteq P_s,\ [l_s^-,l_s]\cap P=\emptyset,\ [r_s,r_s^+]\subseteq P_s,\ \mbox{and }[r_s,r_s^+]\cap P=\emptyset.\]
To simplify the construction, we may also assume that $P$ has the following property:
\[P=\{1-x\in\mathbb{R}:x\in P\}\]
Because, for any $\Pi^0_1$ subset $A\subseteq C$, the $\Pi^0_1$ set $A^*=\{x/3:x\in A\}\cup\{1-x/3:x\in A\}\subseteq C$ has that property.
\medskip
\noindent
{\bf Basic Notation.}
For each $i,j<2$, for each $a,b\in\mathbb{R}^2$, and for each $q,r\in\mathbb{R}$, {\em the $2$-cube} $\Delta_{ij}(a,b;q,r)\subseteq [a,a+q]\times [b,b+r]$ is defined as the smallest convex set containing the three points $\{(a,b),(a+q,b),(a,b+r),(a+q,b+r)\}\setminus\{(a+(1-i)q,b+(1-j)r)\}$.
Namely,
\begin{align*}
\Delta_{ij}(a,b;q,r)=\{&\lrangle{(-1)^ix+a+iq,(-1)^jy+b+jr}\in\mathbb{R}^2\\
&:x,y\geq 0\;\&\;rx+qy\leq qr\}.
\end{align*}
For a set $R\subseteq\mathbb{R}^1$ and real numbers $r,y\in\mathbb{R}$, put $\Theta(R;r,y)=\{rx+y\in\mathbb{R}:x\in R\}$.
Clearly $\Theta(R;r,y)$ is computably homeomorphic to $R$.
Let four symbols $\llcorner$, $\urcorner$, $\lrcorner$, and $\ulcorner$ denote $\lrangle{10,01}$, $\lrangle{01,10}$, $\lrangle{00,11}$, and $\lrangle{11,00}$, respectively.
For $v\in\{\llcorner,\urcorner,\lrcorner,\ulcorner\}$ and for any $R\subseteq[0,1]$, $a,b\in\mathbb{R}^2$, and $q,r\in\mathbb{R}$, we define $[v](R;a,b;q,r)\subseteq[a,a+q]\times[b,b+r]$ as follows:
\begin{align*}
[v](R;a,b;q,r)=\big(([a,a+q]\times\Theta(R;r,b))\cap\Delta_{v(0)}(a,b;q,r)\big)\\
\cup\big((\Theta(R;q,a)\times [b,b+r])\cap\Delta_{v(1)}(a,b;q,r)\big).
\end{align*}
\begin{figure}[t]\centering
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\put(1.0000,-4.7000){\makebox(0,0)[lb]{$b+r$}}
\end{picture}
\end{center}
\vspace{-0.5em}
\caption{The cubes $\Delta_{ij}(a,b,q,r)$.}
\label{fig:Cube1}
\end{minipage}
\begin{minipage}{0.48\hsize}
\begin{center}
\unitlength 0.1in
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\end{picture}
\end{center}
\vspace{-0.5em}
\caption{$[\llcorner](R;a,b;q,r)$ for $R=[1/6,1/2]$}
\label{fig:Cube2}
\end{minipage}
\end{figure}
\begin{sublem}
$[v](P;a,b;q,r)$ is computably homeomorphic to $P\times [0,1]$.
In particular, $[v](P;a,b;q,r)$ contains no computable points.
\end{sublem}
To simplify our argument, we use the normalization $\tilde{P}_t^s$ of $P_t$ for $t\geq s$, that is defined by $\tilde{P}_t^s=\{(x-l_s^-)/(r_s^+-l_s^-)\in\mathbb{R}:x\in P_t\}$, for each $s\in\mathbb{N}$.
Note that $\tilde{P}^s_t\subseteq [0,1]$ for $t\geq s$, and $0,1\in\tilde{P}^s_s$ holds for each $s\in\mathbb{N}$.
Put $[v]^s_t([a,a+q]\times[b,b+r])=[v](\tilde{P}_t^s;a,b;q,r)$ for $t\geq s$.
We also introduce the following two notions:
\begin{align*}
[-]^s_t([a,a+q]\times[b,b+r])&=[a,a+q]\times\Theta(\tilde{P}_t^s;r,b);\\
[\;\mid\;]^s_t([a,a+q]\times[b,b+r])&=\Theta(\tilde{P}_t^s;q,a)\times[b,b+r].
\end{align*}
Here we code two symbols $-$ and $\mid$ as $0$ and $1$ respectively.
\begin{sublem}
$[v]^s_t([a,a+q]\times[b,b+r])\subseteq[a,a+q]\times[b,b+r]$, and $[v]^s_t([a,a+q]\times[b,b+r])$ intersects with the boundary of $[a,a+q]\times[b,b+r]$.
\end{sublem}
\begin{sublem}
There is a computable homeomorphism between $[v]^s_t(a,b;q,r)$ and $P_t\times [0,1]$ for any $t\in\nn$.
Therefore, $\bigcap_t[v]^s_t(a,b;q,r)$ is computably homeomorphic to $P\times [0,1]$.
\end{sublem}
\noindent
{\bf Blocks.}
A {\em block} is a set $Z\subseteq\mathbb{R}^2$ with {\em a bounding box ${\rm Box}(Z)=[a,a+q]\times[b,b+r]$}.
Each $\delta\in 2^2$ is called a {\em direction}, and directions $\lrangle{00}$, $\lrangle{01}$, $\lrangle{10}$, and $\lrangle{11}$ are also denoted by $[\leftarrow]$, $[\rightarrow]$, $[\downarrow]$, and $[\uparrow]$, respectively.
For $\delta\in 2^2$, $\delta^\circ=\lrangle{\delta(0),1-\delta(0)}$ is called {\em the reverse direction of $\delta$}.
Put ${\rm Line}(Z;[\leftarrow])=\{a\}\times[b,b+r]$; ${\rm Line}(Z;[\rightarrow])=\{a+q\}\times[b,b+r]$; ${\rm Line}(Z;[\downarrow])=[a,a+q]\times\{b\}$; ${\rm Line}(Z;[\uparrow])=[a,a+q]\times\{b+r\}$.
Assume that a class $\mathcal{Z}$ of blocks is given.
We introduce a relation $\touch{\delta}$ on $\mathcal{Z}$ for each direction $\delta$.
Fix a block $Z_{\rm first}\in\mathcal{Z}$, and we call it {\em the first block}.
Then we declare that $\touch{[\leftarrow]}Z_{\rm first}$ holds.
We inductively define the relation $\touch{\delta}$ on $\mathcal{Z}$.
If $Z\touch{\delta}Z_0$ (resp.\ $Z_0\touch{\delta}Z$) for some $Z$ and $\delta$, then we also write it as $\touch{\delta}Z_0$ (resp.\ $Z_0\touch{\delta}$).
For any two blocks $Z_0$ and $Z_1$, the relation $Z_0\touch{\delta}Z_1$ holds if the following three conditions are satisfied:
\begin{enumerate}
\item $Z_0\cap Z_1={\rm Line}(Z_0;\delta)\cap Z_0={\rm Line}(Z_1;\delta^\circ)\cap Z_1\not=\emptyset$.
\item $\touch{\varepsilon}Z_0$ has been already satisfied for some direction $\varepsilon$.
\item $Z_1\touch{\varepsilon}Z_0$ does not satisfied for any direction $\varepsilon$
\end{enumerate}
If $Z_0\touch{\delta}Z_1$ for some $\delta$, then we say that {\em $Z_1$ is a successor of $Z_0$} ($Z_0$ is a predecessor of $Z_1$), and we also write it as $Z_0\touch{}Z_1$.
We will construct a partial computable function $\mathcal{Z}:\nn^3\to\mathcal{A}(\mathbb{R}^2)$ with a computable function $k:\nn\to\nn$ and ${\rm dom}(\mathcal{Z})=\{(u,i,t)\in\nn^3:u\leq t\;\&\;i<k(u)\}$ such that $\mathcal{Z}(u,i,t)$ is a block with a bounding box for any $(u,i,t)\in{\rm dom}(\mathcal{Z})$, and the block $\mathcal{Z}(u,i,t)$ is computably homeomorphic to $P_t\times[0,1]$ uniformly in $(u,i,t)$.
Here $\mathcal{A}(\mathbb{R}^2)$ is the hyperspace of all closed subsets in $\mathbb{R}^2$ with positive and negative information.
For each stage $t$, $\mathcal{Z}_t(u)=\{\mathcal{Z}(t,u,i):i<k(u)\}$ for each $u\leq t$ is defined.
Let $\mathcal{Z}(u)$ denote the finite set $\{\lambda t.\mathcal{Z}(t,u,i):i<k(u)\}$ of functions, for each $u\in\nn$.
The relation $\touch{}$ induces a pre-ordering $\prec$ on $\bigcup_{u\in\nn}\mathcal{Z}(u)$ as follows:
$Z_0\prec Z_1$ if there is a finite path from $Z_0(t)$ to $Z_1(t)$ on the finite directed graph $(\bigcup_{u\leq t}\mathcal{Z}_t(u),\touch{})$ at some stage $t\in\nn$.
We will assure that $\prec$ is a well-ordering of order type $\omega$, and $Z_0\prec Z_1$ whenever $Z_0\in\mathcal{Z}(u)$, $Z_1\in\mathcal{Z}(v)$, and $u<v$.
In particular, for every $Z\in\bigcup_{u\in\nn}\mathcal{Z}(u)$, the predecessor $Z_{\rm pre}$ of $Z$ and the successor $Z_{\rm suc}$ of $Z$ under $\prec$ are uniquely determined.
If $Z_{\rm pre}(t)\touch{\delta}Z(t)\touch{\varepsilon}Z_{\rm suc}(t)$, then we say that {\em $Z$ moves from $\delta$ to $\varepsilon$}, and that $\lrangle{\delta,\varepsilon}$ is {\em the direction of $Z$}.
\begin{example}\label{example}
Fig. \ref{fig:mthmblock} is an example satisfying $\touch{[\leftarrow]}Z_{\rm first}\touch{[\leftarrow]}Z_0\touch{[\downarrow]}Z_1\touch{[\rightarrow]}Z_2$.
\end{example}
\begin{figure}[t]\centering
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\vspace{-0.5em}
\caption{Example \ref{example}.}
\label{fig:mthmblock}
\end{figure}
\medskip
\noindent
{\bf Destination Point.}
Basically, our construction is similar as the construction by Miller \cite{Mil}.
Pick the standard homeomorphism $\rho$ between $2^\mathbb{N}$ and the middle third Cantor set, i.e., $\rho(M)=2\sum_{i\in M}(1/3)^{i+1}$ for $M\subseteq\mathbb{N}$, and pick a non-computable c.e. set $B\subseteq\mathbb{N}$ and put $\gamma=\rho(B)$.
We will construct a Cantor fan so that the first coordinate of the unique ramification point is $\gamma$, hence the fan will have a non-computable ramification point.
Let $\{B_s:s\in\mathbb{N}\}$ be a computable enumeration of $B$, and let $n_s$ denote the element enumerated into $B$ at stage $s$, where we may assume just one element is enumerated into $B$ at each stage.
Put $\gamma^{\min}_s=\rho(B_s)$ and $\gamma^{\max}_s=\rho(B_s\cup\{i\in\mathbb{N}:i\geq n_s\})$.
Note that $\gamma$ is not computable, and so $\gamma^{\min}_s\not=\gamma$ and $\gamma^{\max}_s\not=\gamma$ for any $s\in\mathbb{N}$.
This means that for every $s\in\mathbb{N}$ there exists $t>s$ such that $\gamma^{\min}_s\not=\gamma^{\min}_t$ and $\gamma^{\max}_s\not=\gamma^{\max}_t$.
By this observation, without loss of generality, we can assume that $\gamma^{\min}_s\not=\gamma^{\min}_t$ and $\gamma^{\max}_s\not=\gamma^{\max}_t$ whenever $s\not=t$.
We can also assume $1/3\leq\gamma^{\min}_s\leq\gamma^{\max}_s\leq 2/3$ for any $s\in\mathbb{N}$.
\medskip
\noindent
{\bf Stage $0$.}
We now start to construct a $\Pi^0_1$ Cantor fan $Q=\bigcap_{s\in\mathbb{N}}Q_s$.
At the first stage $0$, and for each $t\geq 0$, we define the following sets:
\[Z_{0,t}^{\rm st}=[-]^s_t([\gamma_0^{\min},\gamma_0^{\max}]\times[l_0^-,r_0^+]);\ Z_0^{\rm end}=[\gamma_0^{\min}-1/3,\gamma_0^{\min}]\times[l_0^-,r_0^+].\]
Moreover, we set $Q_0=Z_{0,0}^{\rm st}\cup Z_0^{\rm end}$.
By our choice of $P_0$, actually $Q_0=[\gamma_0^{\min}-1/3,\gamma_0^{\max}]\times[l_0^-,r_0^+]$.
$Z_{0,0}^{\rm st}$ is called {\em the straight block from $2/3$ to $1/3$ at stage $0$}, and $Z_0^{\rm end}$ is called {\em the end box at stage $0$}.
The {\em bounding box} of the block $Z_0^{\rm st}$ is defined by $[\gamma_0^{\min},\gamma_0^{\max}]\times[l_0^-,r_0^+]$.
{\em The collection of $0$-blocks at stage $t$} is $\mathcal{Z}_t(0)=\{Z_{0,t}^{\rm st}\}$.
We declare that $Z_0^{\rm st}$ is the first block, and that $\touch{[\leftarrow]}Z_0^{\rm st}$.
\medskip
\noindent
{\bf Stage $s+1$.}
Inductively assume that we have already constructed the collection of $u$-blocks $\mathcal{Z}_t(u)$ at stage $t\geq u$ is already defined for every $u\leq s$.
For any $u$, we think of the collection $\mathcal{Z}(u)=\{\mathcal{Z}_t(u):t\geq u\}$ as a finite set $\{Z^u_i\}_{i<\#\mathcal{Z}_u(u)}$ of computable functions $Z_i^u:\{t\in\nn:t\geq u\}\to\bigcup_{t}\mathcal{Z}_t(u)$ such that $\mathcal{Z}_t(u)=\{Z^u_i(t):i<\#\mathcal{Z}_u(u)\}$ for each $t\geq u$.
We inductively assume that the collection $\mathcal{Z}(u)=\{\mathcal{Z}_t(u):t\geq u\}$ satisfies the following conditions:
\begin{enumerate}
\item[(IH1)] For each $Z\in\mathcal{Z}(u)$ and for each $t\geq v\geq u$, $Z(t)\subseteq Z(v)$.
\item[(IH2)] There is a computable function $f:\mathbb{R}^2\to\mathbb{R}^2$ such that $f\res\bigcup\bigcup_{u\leq s}\mathcal{Z}_t(u)$ is a homeomorphism between $\bigcup\bigcup_{u\leq s}\mathcal{Z}_t(u)$ and $P_t\times [0,1]$ for any $t\geq s$.
\item[(IH3)] There are $y,z,\zeta\in\mathbb{Q}$ such that the blocks $Z_{s,t}^{\rm st}$ and $Z_s^{\rm end}$ are constructed as follows:
\begin{align*}
Z_{s,t}^{\rm st}&=[-]^s_t([\gamma_s^{\min},\gamma_s^{\max}]\times[y+zl_s^-,y+zr_s^+]);\\
Z_s^{\rm end}&=[\gamma_s^{\min}-\zeta,\gamma_s^{\min}]\times[y+zl_s^-,y+zr_s^+].
\end{align*}
Here, a computable closed set $Q_s$, {\em an approximation of our $\Pi^0_1$ Cantor fan $Q$ at stage $s$}, is defined by $Q_s=Z_s^{\rm end}\cup\bigcup\bigcup_{u\leq s}\mathcal{Z}_{s}(u)$.
\end{enumerate}
\begin{figure}[t]\centering
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\vspace{-0.5em}
\caption{The active block $Z_s^{\rm st}\cup Z_s^{\rm end}$ at stage $s$.}
\label{fig:mthm2}
\end{figure}
\noindent
{\bf Non-injured Case.}
First we consider the case $[\gamma^{\min}_{s+1},\gamma^{\max}_{s+1}]\subseteq[\gamma^{\min}_s,\gamma^{\max}_s]$, i.e., this is the case that our construction is {\em not injured} at stage $s+1$.
In this case, we construct $(s+1)$-blocks in the active block $Z_s^{\rm st}\cup Z_s^{\rm end}$.
We will define $Z_t(s,i,j)$ and ${\rm Box}(s,i,j)={\rm Box}(Z_t(s,i,j))$ for each $j<6$.
{\em The first two corner blocks} at stage $t\geq s+1$ are defined by:
\begin{align*}
{\rm Box}(s,0)=&[\gamma_s^{\min}-\zeta,\gamma_s^{\min}]\times[y+zl_s^-,y+zr_s^*],\\
Z_t(s,0)=&[\llcorner]^s_t([\gamma_s^{\min}-\zeta,\gamma_s^{\min}]\times[y+zl_s^-,y+zr_s^+])\cap{\rm Box}(s,0),\\
{\rm Box}(s,1)=&[\gamma_s^{\min}-\zeta,\gamma_s^{\min}]\times[y+zr_s^*,y+zr_s^+],\\
Z_t(s,1)=&[\ulcorner]^s_t({\rm Box}(s,1)).
\end{align*}
\begin{sublem}\label{sublem:1}
$Z_t(s,0)\cup Z_t(s,1)\subseteq Z_s^{\rm end}$ for any $t\geq s+1$.
\end{sublem}
\begin{sublem}\label{sublem:2}
$Z_{s,t}^{\rm st}\touch{[\leftarrow]}Z_t(s,0)\touch{[\uparrow]}Z_t(s,1)$, for any $t\geq s+1$.
\end{sublem}
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\end{picture}
\end{center}
\vspace{-0.5em}
\caption{The first two corner blocks $Z_s(s,0)$ and $Z_s(s,1)$.}
\label{fig:mthm3}
\end{figure}
The next block is {\em a straight block from $\gamma_s^{\min}$ to $\gamma_{s+1}^{\max}$} which is defined as follows:
\begin{align*}
{\rm Box}(s,2)=&[\gamma_s^{\min},\gamma_s^{\max}]\times[y+zr_s^*,y+zr_s^+].\\
Z_t(s,2)=&[-]({\rm Box}_t(s,2)).
\end{align*}
For given $a,b,\alpha,\beta\in\mathbb{Q}$, we can calculate $N_{0,s}(a,b;\alpha,\beta)$ and $N_{1,s}(a,b;\alpha,\beta)$ satisfying $N_{0,s}(a,b;\alpha,\beta)+N_{1,s}(a,b;\alpha,\beta)\cdot l_s^-=a+b\alpha$, and $N_{0,s}(a,b;\alpha,\beta)+N_{1,s}(a,b;\alpha,\beta)\cdot r_s^+=a+b\beta$.
Put $y^{\star}=N_{0,s}(y,z;r_s^*,r_s^+)$, and $z^{\star}=N_{1,s}(y,z;r_s^*,r_s^+)$.
\begin{sublem}\label{sublem:b1}
${\rm Box}(s,2)=[\gamma_s^{\min},\gamma_s^{\max}]\times[y^{\star}+z^{\star}l_s^-,y^{\star}+z^{\star}r_s^+]$.
\end{sublem}
Put $\zeta^{\star}=(\gamma_s^{\max}-\gamma_{s+1}^{\max})/3^s$.
Note that $\zeta^{\star}>0$ since $\gamma_s^{\max}>\gamma_{s+1}^{\max}$.
We then again define {\em corner blocks}.
\begin{align*}
{\rm Box}(s,3)=&[\gamma_{s+1}^{\max},\gamma_{s+1}^{\max}+\zeta^{\star}]\times[y^{\star}+z^{\star}l_s^-,y^{\star}+z^{\star}r_s^*],\\
Z_t(s,3)=&[\lrcorner]^s_t([\gamma_{s+1}^{\max},\gamma_{s+1}^{\max}+\zeta^{\star}]\times[y^{\star}+z^{\star}l_s^-,y^{\star}+z^{\star}r_s^+])\cap{\rm Box}(s,3),\\
{\rm Box}(s,4)=&[\gamma_{s+1}^{\max},\gamma_{s+1}^{\max}+\zeta^{\star}]\times[y^{\star}+z^{\star}r_s^*,y^{\star}+z^{\star}r_s^+],\\
Z_t(s,4)=&[\urcorner]^s_t({\rm Box}(s,4)).
\end{align*}
Next, {\em a straight block from $\gamma_s^{\min}$ to $\gamma_{s+1}^{\max}$} is defined as follows:
\begin{align*}
{\rm Box}(s,5)&=[\gamma_{s+1}^{\min},\gamma_{s+1}^{\max}]\times[y^{\star}+z^{\star}r_s^*,y^{\star}+z^{\star}r_s^+],\\
Z_t(s,5)&=[-]^s_t[{\rm Box}(s,5)].
\end{align*}
Put $y^{\star\star}=N_{0,s}(y^{\star},z^{\star};r_s^*,r_s^+)$, and $z^{\star\star}=N_{1,s}(y^{\star},z^{\star};r_s^*,r_s^+)$.
\begin{sublem}\label{sublem:b2}
${\rm Box}(s,5)=[\gamma_s^{\min},\gamma_s^{\max}]\times[y^{\star\star}+z^{\star\star}l_s^-,y^{\star\star}+z^{\star\star}r_s^+]$.
\end{sublem}
Put $\zeta^{\star\star}=(\gamma_{s+1}^{\min}-\gamma_{s}^{\min})/3^s$.
Note that $\zeta^{\star\star}>0$ since $\gamma_{s+1}^{\min}>\gamma_{s}^{\max}$.
{\em The end box at stage $s+1$} is:
\[Z(s,6)=[\gamma_{s+1}^{\min}-\zeta^{\star\star},\gamma_{s+1}^{\min}]\times[y^{\star\star}+z^{\star\star}l_s^-,y^{\star\star}+z^{\star\star}r_s^+].\]
Then put $Z_{s+1,t}^{\rm st}=Z_{t}(s,5)$, $Z_{s+1}^{\rm st}=Z_{s+1,s+1}^{\rm st}$, and $Z_{s+1}^{\rm end}=Z(s,6)$.
{\em The active block at stage $s+1$} is the set $Z_{s+1,s+1}^{\rm st}\cup Z_{s+1}^{\rm end}$, and {\em the collection of $(s+1)$-blocks at stage $t$} is defined by $\mathcal{Z}_t(s+1)=\{Z_t(s,i):i\leq 5\}$.
Clearly, our definition satisfies the induction hypothesis (IH3) at stage $s+1$.
\begin{figure}[t]\centering
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\end{center}
\vspace{-0.5em}
\caption{$Z_s(s-1,5)\cup\bigcup\mathcal{Z}_s(s+1)$.}
\label{fig:mthm4}
\end{minipage}
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\end{picture}
\end{center}
\vspace{-0.5em}
\caption{$Z_{s+1}(s-1,5)\cup\bigcup\mathcal{Z}_{s+1}(s+1)$.}
\label{fig:mthm5}
\end{minipage}
\end{figure}
\begin{sublem}\label{sublem:3}
$Z_t(s,i)\subseteq Z_v(s,i)$ for each $t\geq v\geq s+1$ and $i\leq 5$.
\end{sublem}
\begin{sublem}\label{sublem:4}
For any $t\geq s+1$,
\[Z_{s,t}^{\rm st}\touch{[\leftarrow]}Z_t(s,0)\touch{[\uparrow]}Z_t(s,1)\touch{[\rightarrow]}Z_t(s,2)\touch{[\rightarrow]}Z_t(s,3)\touch{[\uparrow]}Z_t(s,4)\touch{[\leftarrow]}Z_t(s,5).\]
\end{sublem}
\begin{proof}\upshape
It follows straightforwardly from the definition of these blocks $Z_t(s,i)$, and Sublemma \ref{sublem:b1} and \ref{sublem:b2}.
\end{proof}
\begin{sublem}\label{sublem:5}
$\bigcup_{2\leq i\leq 6}Z_t(s,i)\subseteq Z_s^{\rm st}\cap [\gamma_s^{\min},\gamma_s^{\max}]\times (y+zr_s,y+zr_s^+]$.
Hence, $\left(\bigcup_{2\leq i\leq 6}Z_t(s,i)\right)\cap Z^{\rm st}_{s,s+1}=\emptyset$
\end{sublem}
Consequently, we can show the following extension property.
\begin{sublem}\label{sublem:6}
Assume that we have a computable function $f_s:\mathbb{R}^2\to\mathbb{R}^2$ such that $f_s\res\bigcup\bigcup_{u\leq s}\mathcal{Z}_{t}(u)$ is a computable homeomorphism between $\bigcup\bigcup_{u\leq s}\mathcal{Z}_{t}(u)$ and $P_t\times[1/(s+2),1]$ for any $t\geq s$.
Then we can effectively find a computable function $f_{s+1}:\mathbb{R}^2\to\mathbb{R}^2$ extending $f_s\res\bigcup\bigcup_{u\leq s}\mathcal{Z}_{s+1}(u)$ such that $f_{s+1}\res\bigcup\bigcup_{u\leq s+1}\mathcal{Z}_{t}(u)$ is a computable homeomorphism between $\bigcup\bigcup_{u\leq s+1}\mathcal{Z}_{t}(u)$ and $P_t\times[1/(s+3),1]$ for any $t\geq s+1$.
\end{sublem}
\begin{proof}\upshape
By Sublemma \ref{sublem:2}, \ref{sublem:4}, and \ref{sublem:5}.
\end{proof}
By Sublemma \ref{sublem:3} and \ref{sublem:6}, induction hypothesis (IH1) and (IH2) are satisfied.
Since $Z_{s+1}^{\rm end}\cup\bigcup\mathcal{Z}_{s+1}(s+1)\subseteq Z_s^{\rm st}\cup Z_s^{\rm end}$ by Sublemma \ref{sublem:1} and \ref{sublem:5}, and $\bigcup\mathcal{Z}_{s+1}(u)\subseteq\bigcup\mathcal{Z}_s(u)$ for each $u\leq s$, by induction hypothesis (IH1), we have the following:
\[Q_{s+1}=Z_{s+1}^{\rm end}\cup\bigcup\bigcup_{u\leq s+1}\mathcal{Z}_{s+1}(u)\subseteq Z_s^{\rm st}\cup Z_s^{\rm end}\cup\bigcup\bigcup_{u\leq s}\mathcal{Z}_{s}(u)\subseteq Q_s.\]
\noindent
{\bf Injured Case.}
Secondly we consider the case that our construction {\em is injured}.
This means that $[\gamma_{s+1}^{\min},\gamma_{s+1}^{\max}]\not\subseteq[\gamma_s^{\min},\gamma_s^{\max}]$.
In this case, indeed, we have $[\gamma_{s+1}^{\min},\gamma_{s+1}^{\max}]\cap[\gamma_s^{\min},\gamma_s^{\max}]=\emptyset$.
Fix the greatest stage $p\leq s$ such that $[\gamma_{s+1}^{\min},\gamma_{s+1}^{\max}]\subseteq[\gamma_p^{\min},\gamma_p^{\max}]$ occurs.
We again, inside the end box $Z_s^{\rm end}$ at stage $s$, define corner blocks $Z_t(s,0)$ and $Z_t(s,1)$ as non-injuring stage, whereas the construction of $Z_t(s,i)$ for $i\geq 2$ differs from non-injuring stage.
The end box of our construction at stage $s+1$ will turn back along all blocks belonging $\mathcal{Z}_s(u)$ for $p<u\leq s$ in the reverse ordering of $\prec$.
Let $\{Z_i:i<k_{s+1}\}$ be an enumeration of all blocks in $\mathcal{Z}_s(u)$ for $p<u\leq s$, under the reverse ordering of $\prec$.
In other words, $Z_i$ is the successor block of $Z_{i+1}$ under $\touch{}$, for each $i<k_{s+1}-1$.
There are two kind of blocks; one is {\em a straight block}, and another is {\em a corner block}.
We will define blocks $Z_t(s,i,j)$ for $i<k_{s+1}$ and $j<3$.
Now we check the direction $\lrangle{\delta_i,\varepsilon_i}$ of $Z_i$.
Here, we may consistently assume that the condition $Z_0\touch{[\leftarrow]}$ holds.
\medskip
\noindent
{\bf Subcase 1.}
If $\delta_i(0)=\varepsilon_i(0)$ then $Z_i$ is a straight block.
In this case, we only construct $Z_t(s,i,0)$.
Since $Z_i$ is straight, there are $y_i,z_i,\alpha,\beta\in\mathbb{Q}$ and $u\leq s$ such that, for $B_i(0)=[\alpha,\beta]$ and $B_i(1)=[y_i+z_il_u^-,y_i+z_ir_u^+]$ such that ${\rm Box}(Z_i)=B_i(\delta_i(0))\times B_i(1-\delta_i(0))$.
If $\delta_i(1)=0$, then set $y_i^{\star}=N_{0,s}(y_i,z_i;l_s^-,l_s^*)$ and $z_i^{\star}=N_{1,s}(y_i,z_i;l_s^-,l_s^*)$.
If $\delta_i(1)=1$, then set $y_i^{\star}=N_{0,s}(y_i,z_i;r_s^-,r_s^+)$ and $z_i^{\star}=N_{1,s}(y_i,z_i;r_s^*,r_s^+)$.
Then, we define $Z_t(s,i,0)$ as the following straight block:
\begin{align*}
B_i^{\star}(0)=B_i(0);\quad B_i^{\star}(1)=[y_i^{\star}+z_i^{\star}l_s^-,y_i^{\star}+z_i^{\star}r_s^+];\\
Z_t(s,i,0)=[\delta_i(0)]^s_t(B_i^\star(\delta_i(0))\times B_i^\star(1-\delta_i(0))).
\end{align*}
Here, ${\rm Box}(Z_t(s,i,0))$ is defined by $B_i^\star(\delta_i(0))\times B_i^\star(1-\delta_i(0))$.
\begin{figure}[t]\centering
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\vspace{-0.5em}
\caption{The block $Z_i$.}
\label{fig:inj1}
\end{minipage}
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\end{center}
\vspace{-0.5em}
\caption{The block $Z_t(s,i,0)$.}
\label{fig:inj2}
\end{minipage}
\end{figure}
\begin{sublem}
$Z_t(s,i,0)\subseteq Z_i$.
\end{sublem}
\begin{proof}\upshape
By our definition of $N_{0,s}$ and $N_{1,s}$, we have $B_i^{\star}(1)=[y_i+z_il_s^-,y_i+z_il_s^*]$ or $B_i^{\star}(1)=[y_i+z_ir_s^*,y_i+z_ir_s^+]$.
\end{proof}
\noindent
{\bf Subcase 2.}
If $\delta_i(0)\not=\delta_i(2)$ then $Z_i$ is a corner block.
We will construct three blocks; one corner block $Z_t(s,i,0)$, and two straight blocks $Z_t(s,i,1)$ and $Z_t(s,i,2)$.
We may assume that $Z_i$ is of the following form:
\begin{align*}
Z_i=[e]^u_s&([x_i+\zeta_il_u^-,x_i+\zeta_ir_u^+]\times[y_i+z_il_u^-,y_i+z_ir_u^+]),\\
\text{or }Z_i=[e]^u_s&([x_i+\zeta_il_u^-,x_i+\zeta_ir_u^+]\times[y_i+z_il_u^-,y_i+z_ir_u^+])\\
&\cap([x_i+\zeta_il_u^-,x_i+\zeta_ir_u^+]\times[y_i+z_il_u^-,y_i+z_ir_u^*])
\end{align*}
Set $z=0$ if the former case occurs; otherwise, set $z=1$.
Let $\{p_n:n<6\}$ be an enumeration of $\{l_u^-,l_s^-,l_s^*,r_s^*,r_s^+,r_u^+\}$ in increasing order, and let $p_6$ be $r_u^*$.
First we compute the value ${\tt rot}=2|\varepsilon_i(0)-|\delta_i(1)-\varepsilon_i(1)||+1$.
Note that ${\tt rot}\in\{1,3\}$, and, if $Z_i$ rotates clockwise then ${\tt rot}=1$; and if $Z_i$ rotates counterclockwise then ${\tt rot}=3$.
If $\touch{[\rightarrow]}Z_i$ or $Z_i\touch{[\rightarrow]}$, then put $D(0)=1$; otherwise put $D(0)=3$.
If $\touch{[\downarrow]}Z_i$ or $Z_i\touch{[\downarrow]}$, then put $D(1)=1$; otherwise put $D(1)=3$.
If $\touch{[\rightarrow]}Z_i$ or $Z_i\touch{[\leftarrow]}$, then put $E(0)=0$; otherwise put $E(0)=5-{\rm rot}$.
If $\touch{[\uparrow]}Z_i$ or $Z_i\touch{[\downarrow]}$, then put $E(1)=0$; otherwise put $E(1)=5-{\rm rot}$.
Then we now define $Z_t(s,i,j)$ for $j<3$ as follows:
\begin{align*}
{\rm Box}(s,i,0)&=[x_i+\zeta_ip_{D(0)},x_i+\zeta_ip_{D(0)+2}]\times[y_i+z_ip_{D(1)},y_i+z_ip_{D(1)+2}],\\
{\rm Box}(s,i,1)&=[x_i+\zeta_ip_{E(0)},x_i+\zeta_ip_{E(0)+{\tt rot}}]\times[y_i+z_ip_{D(1)},y_i+z_ip_{D(1)+2}],\\
{\rm Box}(s,i,2)&=[x_i+\zeta_ip_{D(0)},x_i+\zeta_ip_{D(0)+2}]\times[y_i+z_ip_{E(1)},y_i+z_ip_{E(1)+{\tt rot}+z}],\\
Z_t(s,i,0)&=[e]^s_t({\rm Box}(s,i,0)),\\
Z_t(s,i,1)&=[-]^s_t({\rm Box}(s,i,1)),\\
Z_t(s,i,2)&=[\;\mid\;]^s_t({\rm Box}(s,i,2)).
\end{align*}
Intuitively, $D(0)=1$ (resp.\ $D(0)=3$) indicates that $Z_t(s,i,0)$ passes the west (resp.\ the east) of $Z_i$; $D(1)=1$ (resp.\ $D(1)=3$) indicates that $Z_t(s,i,0)$ passes the south (resp.\ the north) of $Z_i$; $E(0)=0$ (resp.\ $E(0)=5-{\tt rot}$) indicates that $Z_t(s,i,1)$ passes the west (resp.\ the east) border of the bounding box of $Z_i$; and $E(1)=0$ (resp.\ $E(1)=5-{\tt rot}$) indicates that $Z_t(s,i,2)$ passes the south (resp.\ the north) border of the bounding box of $Z_i$.
Note that the corner block $Z_t(s,i,0)$ leaves $Z_i$ on his right, and $Z_t(s,i,0)$ has the reverse direction to $Z_i$.
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\vspace{-0.5em}
\caption{${\rm rot}=1$.}
\label{fig:inj4}
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\put(4.0000,-18.2000){\makebox(0,0)[lb]{$Z_s(s,i,1)\touch{\leftarrow}Z_s(s,i,0)\touch{\uparrow}Z_s(s,i,2)$}}
\end{picture}
\end{center}
\vspace{-0.5em}
\caption{${\rm rot}=3$.}
\label{fig:inj5}
\end{minipage}
\end{figure}
\begin{sublem}
$Z_t(s,i,2-\delta_i(0))\touch{\varepsilon^{\circ}}Z_t(s,i,0)\touch{\delta^{\circ}}Z_t(s,t,1+\delta_i(0))$.
\end{sublem}
\begin{sublem}
$Z_t(s,i,j)\subseteq Z_i$.
\end{sublem}
For each $i<k_{s+1}$, we have already constructed $\mathcal{Z}_t(s+1;i)=\{Z_t(s,i,j):j<3\}$.
All of these blocks constructed at the current stage are included in $Z_s^{\rm end}\cup\bigcup\bigcup_{p<u\leq s}\mathcal{Z}_s(u)$.
Let $Z^{0}[i]$ (resp.\ $Z^{1}[i]$) be the $\prec$-least (resp. the $\prec$-greatest) element of $\{\lambda t.Z_t(s,i,j):j<3\}$.
It is not hard to see that our construction ensures the following condition.
\begin{sublem}
$Z_t^1[i]\touch{}Z_t^0[i+1]$.
\end{sublem}
Thus, $\bigcup_{i<k_{s+1}}\mathcal{Z}_t(s+1;i)$ is computably homeomorphic to $P_t\times[0,1]$, uniformly in $t\geq s+1$.
Therefore, we can connect blocks $Z_s(s,i)$ for $i<k_{s+1}$, and we succeed to return back on the current approximation of the $\prec$-greatest $p$-block $Z_s(p)=Z_{p,s}^{\rm st}\in\mathcal{Z}_s(p)$.
Then we construct blocks $Z_t(s,k)$ for $2\leq k\leq 6$ on the block $Z_s(p)$.
The construction is essentially similar as the non-injuring case.
By induction hypothesis (IH3), we note that $Z_s(p)$ must be of the following form for some $y_p,z_p\in\mathbb{Q}$:
\[Z_s(p)=[-]^p_s([\gamma_p^{\rm min},\gamma_p^{\max}]\times[y_p+z_pl_p^-,y_p+z_pr_p^+]).\]
On $Z_s(p)$, we define {\em a straight block from $\gamma_p^{\min}$ to $\gamma_{s+1}^{\max}$} as follows:
\[Z_t(s,2)=[-]^p_s([\gamma_p^{\rm min},\gamma_{s+1}^{\max}]\times[y_p+z_pr_s^*,y_p+z_pr_s^+]).\]
Here, by our assumption, $\gamma^{\max}_{s+1}<\gamma^{\max}_p$ holds since $\gamma^{\max}_{s+1}\leq\gamma^{\max}_p$.
The blocks $Z_t(s,k)$ for $3\leq k\leq 6$ are defined as in the same method as non-injuring case.
The active block at stage $s+1$ is $Z_{s+1}(s,5)$, and the end box at stage $s+1$ is $Z_{s+1}(s,6)$.
{\em $(s+1)$-blocks at stage $t$} are $Z_t(s,i)$ for $i<6$, and $Z_t(s,i,j)$ for $i<k_{s+1}$ and $j<3$ if it is constructed.
$\mathcal{Z}_t(s+1)$ denotes {\em the collection of $(s+1)$-blocks at stage $t$}.
\begin{sublem}
$Z_{s+1}^{\rm end}\cup\bigcup\mathcal{Z}_{s+1}(s+1)\subseteq Z_s^{\rm end}\cup\bigcup\bigcup_{p\leq u\leq s}\mathcal{Z}_s(u)$.
\end{sublem}
Thus we again have the following:
\[Q_{s+1}=Z_{s+1}^{\rm end}\cup\bigcup\bigcup_{u\leq s+1}\mathcal{Z}_{s+1}(u)\subseteq Z_s^{\rm st}\cup Z_s^{\rm end}\cup\bigcup\bigcup_{u\leq s}\mathcal{Z}_{s}(u)\subseteq Q_s.\]
\begin{sublem}\label{sublem:15}
Assume that we have a computable function $f_s:\mathbb{R}^2\to\mathbb{R}^2$ such that $f_s\res\bigcup\bigcup_{u\leq s}\mathcal{Z}_{t}(u)$ is a computable homeomorphism between $\bigcup\bigcup_{u\leq s}\mathcal{Z}_{t}(u)$ and $P_t\times[1/(s+2),1]$ for any $t\geq s$.
Then we can effectively find a computable function $f_{s+1}:\mathbb{R}^2\to\mathbb{R}^2$ extending $f_s\res\bigcup\bigcup_{u\leq s}\mathcal{Z}_{s+1}(u)$ such that $f_{s+1}\res\bigcup\bigcup_{u\leq s+1}\mathcal{Z}_{t}(u)$ is a computable homeomorphism between $\bigcup\bigcup_{u\leq s+1}\mathcal{Z}_{t}(u)$ and $P_t\times[1/(s+3),1]$ for any $t\geq s+1$.
\end{sublem}
\begin{figure}[t]\centering
\begin{center}
\unitlength 0.1in
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\end{picture}
\end{center}
\vspace{-0.5em}
\caption{Outline of our construction of the injured case.}
\label{fig:mthm10}
\end{figure}
Finally we put $Q=\bigcap_{s\in\mathbb{N}}Q_s$ and $\mathcal{Z}^*=\bigcup_{u\in\mathbb{N}}\mathcal{Z}(u)$.
The construction is completed.
\medskip
\noindent
{\bf Verification.}
Now we start to verify our construction.
\begin{lemma}\label{lem:verify:6}
$Q$ is $\Pi^0_1$.
\end{lemma}
\begin{sublem}\label{sublem:16}
$\bigcap_{t\in\mathbb{N}}\bigcup_{Z\in\mathcal{Z}^*}Z_t=\bigcup_{Z\in\mathcal{Z}^*}\bigcap_{t\in\mathbb{N}}Z_t$.
\end{sublem}
\begin{proof}\upshape
The intersection $Z_s(p)\cap Z^i_s$ for $i<2$ is included in some line segment $L_i\in\{[0,1]\times\{b\},\{b\}\times[0,1]:b\in\mathbb{R}\}$, and $Z_s(p)\cap L_i=Z_s(p)\cap Z^i_s$ holds.
\end{proof}
\begin{sublem}\label{sublem:18}
$\bigcup_{Z\in\mathcal{Z}(u)}\bigcap_{t\in\mathbb{N}}Z_t$ is computably homeomorphic to $[0,1]\times P$, for each $u\in\nn$.
\end{sublem}
\begin{proof}\upshape
By the induction hypothesis (IH2).
\end{proof}
\begin{sublem}\label{sublem:17}
$\bigcup_{Z\in\mathcal{Z}^*}\bigcap_{t\in\mathbb{N}}Z_t$ is homeomorphic to $(0,1]\times P$.
\end{sublem}
\begin{proof}\upshape
By Sublemma \ref{sublem:6} and \ref{sublem:15}.
\end{proof}
\begin{lemma}\label{lem:verify:7}
$Q$ is homeomorphic to a Cantor fan.
\end{lemma}
\begin{proof}\upshape
By Sublemma \ref{sublem:16}, there exists a real $y_0\in\mathbb{R}$ such that the following holds:
\[Q=\left(\bigcup_{Z\in\mathcal{Z}^*}\bigcap_{t\in\mathbb{N}}Z_t\right)\cup\{\lrangle{\gamma,y_0}\}.\]
Therefore, by Sublemma \ref{sublem:17}, $Q$ is homeomorphic to the one-point compactification of $(0,1]\times P$.
\end{proof}
\begin{lemma}\label{lem:verify:8}
$Q$ contains no computable point.
\end{lemma}
\begin{proof}\upshape
By Sublemma \ref{sublem:18}, $\bigcup_{Z\in\mathcal{Z}^*}\bigcap_{t\in\mathbb{N}}Z_t$ contains no computable point.
\end{proof}
By Lemmata \ref{lem:verify:6}, \ref{lem:verify:7}, and \ref{lem:verify:8}, $Q$ is the desired dendroid.
\end{proof}
\begin{remark}
Since dendroids are compact and simply connected, Theorem \ref{thm:special_roid} is the solution to the question of Le Roux and Ziegler \cite{RZ}.
Indeed, the dendroid constructed in the proof of Theorem \ref{thm:special_roid} is contractible.
\end{remark}
\begin{cor}
Not every nonempty contractible $\Pi^0_1$ subset of $[0,1]^2$ contains a computable point.
\end{cor}
\begin{question}
Does every locally connected planar $\Pi^0_1$ set contain a computable point?
\end{question}
\section{Immediate Consequences}
\subsection{Effective Hausdorff Dimension}
For basic definition and properties of the the effective Hausdorff dimension of a point of Euclidean plane, see Lutz-Weihrauch \cite{LW}.
For any $I\subseteq[0,2]$, let ${\rm DIM}^I$ denote the set of all points in $\mathbb{R}^2$ whose effective Hausdorff dimensions lie in $I$.
Lutz-Weihrauch \cite{LW} showed that ${\rm DIM}^{[1,2]}$ is path-connected, but ${\rm DIM}^{(1,2]}$ is totally disconnected.
In particular, ${\rm DIM}^{(1,2]}$ has no nondegenerate connected subset.
It is easy to see that ${\rm DIM}^{(0,2]}$ has no nonempty $\Pi^0_1$ simple curve, since every $\Pi^0_1$ simple curve contains a computable point, and the effective Hausdorff dimension of each computable point is zero.
\begin{theorem}
${\rm DIM}^{[1,2]}$ has a nondegenerate contractible $\Pi^0_1$ subset.
\end{theorem}
\begin{proof}\upshape
For any strictly increasing computable function $f:\nn\to\nn$ with $f(0)=0$ and any infinite binary sequence $\alpha\in 2^\nn$, put $\iota_f(\alpha)=\prod_{i\in\nn}\lrangle{\alpha(i),\alpha(f(i)),\alpha(f(i)+1),\dots,\alpha(f(i+1)-1)}$, where $\sigma\times\tau$ denotes the concatenation of binary strings $\sigma$ and $\tau$.
Then, $r:2^\nn\to\mathbb{R}$ is defined as $r(\alpha)=\sum_{i\in\nn}(\alpha(i)\cdot 2^{-(i+1)})$.
Note that $\alpha\not=\beta$ and $r(\alpha)=r(\beta)$ hold if and only if there is a common initial segment $\sigma\in 2^{<\nn}$ of $\alpha$ and $\beta$ such that $\sigma 0$ and $\sigma 1$ are initial segments of $\alpha$ and $\beta$ respectively, and that $\alpha(m)=1$ and $\beta(m)=0$ for any $m>lh(\sigma)$, where $lh(\sigma)$ denotes the length of $\sigma$.
In this case, we say that $\alpha$ {\em sticks to $\beta$ on $\sigma$}.
If $r(\alpha)\not=r(\beta)$, then clearly $r\circ\iota_f(\alpha)\not=r\circ\iota_f(\beta)$.
Assume that $\alpha$ sticks to $\beta$ on $\sigma$.
Then there are $m_0<m_1$ such that $\iota_f(\alpha)(m_0)=\iota_f(\alpha)(m_1)=\alpha(lh(\sigma))=0$ and $\iota_f(\beta)(m_0)=\iota_f(\beta)(m_1)=\beta(lh(\sigma))=1$ by our definition of $\iota_f$.
Therefore, $\iota_f(\alpha)$ does not stick to $\iota_f(\beta)$.
Hence, $r\circ\iota_f(\alpha)\not=r\circ\iota_f(\beta)$ whenever $\alpha\not=\beta$.
Actually, $r\circ\iota_f:2^\nn\to\mathbb{R}$ is a computable embedding.
For each $n\in\nn$, put $k_f(n)=\#\{s:f(s)<n\}$.
Then, there is a constant $c\in\nn$ such that, for any $\alpha\in 2^\nn$ and $n\in\nn$, we have $K(\iota_f(\alpha)\res n+k_f(n)+1)\geq K(\alpha\res n)-c$, where $K$ denotes the prefix-free Kolmogorov complexity.
Therefore, for any sufficiently fast-growing function $f:\nn\to\nn$ and any Martin-L\"of random sequence $\alpha\in 2^\nn$, the effective Hausdorff dimension of $r\circ\iota_f(\alpha)$ must be $1$.
Thus, for any nonempty $\Pi^0_1$ set $R\subseteq 2^\nn$ consisting of Martin-L\"of random sequences, $\{0\}\times(r\circ\iota_f(R))$ is a $\Pi^0_1$ subset of ${\rm DIM}^{\{1\}}$.
Let $Q$ be the dendroid constructed from $P=r\circ\iota_f(R)$ as in the proof of Theorem \ref{thm:special_roid}, where we choose $\gamma=\rho(B)$ as Chaitin's halting probability $\Omega$.
For every point $\lrangle{x_0,x_1}\in Q$, the effective Hausdorff dimension of $x_i$ for some $i<2$ is equivalent to that of an element of $P$ or that of $\Omega$.
Consequently, $Q\subseteq{\rm DIM}^{[1,2]}$.
\end{proof}
\subsection{Reverse Mathematics}
\begin{theorem}
For every $\Pi^0_1$ set $P\subseteq 2^\nn$, there is a contractible planar $\Pi^0_1$ set $Q$ such that $Q$ is Turing-degree-isomorphic to $P$, i.e., $\{\deg_T(x):x\in P\}=\{\deg_T(x):x\in D\}$.
\end{theorem}
\begin{proof}\upshape
We choose $B$ as a c.e.\ set of the same degree with the leftmost path of $P$.
Then, the dendroid $Q$ constructed from $P$ and $B$ as in the proof of Theorem \ref{thm:special_roid} is the desired one.
\end{proof}
A compact $\Pi^0_1$ subset $P$ of a computable topological space is {\em Muchnik complete} if every element of $P$ computes the set of all theorems of $T$ for some consistent complete theory $T$ containing Peano arithmetic.
By Scott Basis Theorem (see Simpson \cite{Sim}), $P$ is Muchnik complete if and only if $P$ is nonempty and every element of $P$ computes an element of any nonempty $\Pi^0_1$ set $Q\subseteq 2^\nn$.
\begin{cor}
There is a Muchnik complete contractible planar $\Pi^0_1$ set.
\end{cor}
A compact $\Pi^0_1$ subset $P$ of a computable topological space is {\em Medvedev complete} (see also Simpson \cite{Sim}) if there is a uniform computable procedure $\Phi$ such that, for any name $x\in\nn^\nn$ of an element of $P$, $\Phi(x)$ is the set of all theorems of $T$ for some consistent complete theory $T$ containing Peano arithmetic.
\begin{question}
Does there exist a Medvedev complete simply connected planar $\Pi^0_1$ set?
Does there exist a Medvedev complete contractible Euclidean $\Pi^0_1$ set?
\end{question}
Our Theorem \ref{thm:special_roid} also provides a reverse mathematical consequence.
For basic notation for Reverse Mathematics, see Simpson \cite{SimRM}.
Let ${\sf RCA}_0$ denote the subsystem of second order arithmetic consisting of $I\Sigma^0_1$ (Robinson arithmetic with induction for $\Sigma^0_1$ formulas) and $\Delta^0_1$-${\sf CA}$ (comprehension for $\Delta^0_1$ formulas).
Over ${\sf RCA}_0$, we say that a sequence $(B_i)_{i\in\nn}$ of open rational balls is {\em flat} if there is a homeomorphism between $\bigcup_{i<n}B_i$ and the open square $(0,1)^2$ for any $n\in\nn$.
It is easy to see that ${\sf RCA}_0$ proves that every flat cover of $[0,1]$ has a finite subcover.
\begin{theorem}
The following are equivalent over ${\sf RCA}_0$.
\begin{enumerate}
\item Weak K\"onig's Lemma: every infinite binary tree has an infinite path.
\item Every open cover of $[0,1]$ has a finite subcover.
\item Every flat open cover of $[0,1]^2$ has a finite subcover.
\end{enumerate}
\end{theorem}
\begin{proof}\upshape
The equivalence of the item (1) and (2) is well-known.
It is not hard to see that ${\sf RCA}_0$ proves the existence of the sequence $\{Q_s\}_{s\in\nn}$ as in our construction of the dendroid $Q$ in Theorem \ref{thm:special_roid}, by formalizing our proof in Theorem \ref{thm:special_roid} in ${\sf RCA}_0$.
Here we may assume that $\{Q_s\}_{s\in\nn}$ is constructed from the set of all infinite paths of a given infinite binary tree $T\subseteq 2^{<\nn}$, and a c.e.\ complete set $B\subseteq\nn$.
Note that $\bigcup_{s<t}([0,1]^2\setminus Q_s)$ does not cover $[0,1]^2$ for every $t\in\nn$.
Over ${\sf RCA}_0$, there is a flat sequence $\{[0,1]^2\setminus Q^*_s\}_{s\in\nn}$ of open rational balls such that $\bigcap_{s<t}Q^*_s\supseteq\bigcap_{s<t}Q_s$ for any $t\in\nn$, and that an open rational ball $U$ is removed from some $Q^*_s$ if and only if an open rational ball $U$ is removed from some $Q_u$.
However, if $T$ has no infinite path, then $Q$ has no element.
In other words, $\{[0,1]^2\setminus Q^*_s\}_{s\in\nn}$ covers $[0,1]^2$.
\end{proof}
\begin{ack}\upshape
The author thank Douglas Cenzer, Kojiro Higuchi and Sam Sandars for valuable comments and helpful discussion.
\end{ack}
| 60,845
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TITLE: A basic question on omega limit sets equilibrium points
QUESTION [4 upvotes]: Consider the following O.D.E $$\dot{x}(t)=h(x(t))$$ with $h$ being lipschitz. consider a trajectory of it. Assume that its omega limit sets are finite. Then I have read in a paper that its omega limit set necessarily consists of equilibrium points of the O.D.E. Why ?
REPLY [2 votes]: Suppose the $\omega$ limit set of a trajectory has an isolated point $p$. I claim $p$ is an equilibrium point.
Proof by contradiction:
Suppose $p$ is not an equilibrium point. Take some $q \ne p$ such that the solution with $x(0) = p$ has $x(T) = q$, where $T > 0$. Take $0 < \epsilon < |p - q|/2$, and $0 < \delta < \epsilon/2$ such that if $|x(0) - p| < \delta$ then $|x(T) - q| < \epsilon$.
Thus every solution starting (at time $0$) inside the circle $C$ of radius $\delta$ centred at $p$ is outside $C$ at time $T$. By the Intermediate Value
Theorem, it must be on the circle at some time in the interval $(0,T)$.
Now since $p$ is an $\omega$ limit point of your trajectory $x(t)$, there are arbitrarily large times at which the trajectory is inside the circle.
By the previous paragraph, there are arbitrarily large times $t_n$ at which
the trajectory is on the circle. A limit point of $x(t_n)$, which exists
by compactness of $C$, is then an $\omega$ limit point of the trajectory.
So the trajectory contains other $\omega$ limit points arbitrarily close to $p$, contradicting the assumption that $p$ is isolated.
| 129,366
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TITLE: How does Einstein's relativity explain the results of Ruyong Wang's Sagnac experiment that reveals that the Sagnac effect does not depend on rotation?
QUESTION [1 upvotes]: How does special relativity explain the results of Ruyong Wang's Sagnac experiment that reveals that the Sagnac effect does not depend on rotation?
Refer to this paper:
https://arxiv.org/abs/physics/0609222
Journal reference: Physics Letters A, Volume 312, Issues 1-2, 2 June 2003, Pages 7-10
REPLY [2 votes]: The following is an expansion of a comment I wrote to the answer by contributor 'Dale'
The fiber optic conveyer is discussed in 'Sagnac effect, twin paradox and space-time topology — Time and length in rotating systems and closed Minkowski space-times', Olaf Wucknitz, 2007.
If memory serves me: Olaf Wucknitz argues the case that there is an overarching category of which Sagnac setup is a sub-category. The overarching category is then 'loop closing setup'. The general loop closing setup then has two sub-categories: make the loop enclose an area, or make the loop not enclose an area.
I think is is not helpful to cast the distinction in terms of 'involves rotation' versus 'does not involve rotation'.
A fiber conveyer setup can be readily changed between two configurations: (A) the loop encloses an area; (B) the loop does not enclose an area. It would not make sense to suggest: with setup (A): rotation; with setup (B): no rotation.
I agree of course that historically the Sagnac effect has been associated with rotation
However, the very purpose of the fiber optic conveyer experiment is to challenge the notion that the Sagnac effect is in essence a rotation effect.
For the overarching category the crucial factor is that a loop is closed.
A Sagnac setup has the following characteristics: (1) a loop is closed, (2) the loop encloses an area. (3) the detector is not stationary with respect to the loop; the detector is in motion along the perimeter of the loop.
A state of rotation is in most cases characterized by motion along a circle. But in the case of a Sagnac setup what is actually necessary to obtain interference effect is motion along the perimeter of the loop. That perimeter can be any shape, as long as it closes a loop.
That is why Olaf Wucknitz sets his discussion in the wider context of topology.
The historical association between Sagnac effect and rotation is not necessarily correct.
It's just that when the setup involves rotation you get automatically that the detector is in motion along the perimeter of the loop.
[Later addition]
Your question is the relation between relativistic physics and the optic fiber conveyor experiment.
The following is things I learned from the article by Olaf Wucknitz. These are my own words; I strongly recommend studying the Wucknitz article.
The length of the optic fiber conveyor can be treated as a Minkowski spacetime. Treating the diameter of the optic fiber as negligable: the spacetime has one spatial dimension and of course time dimension.
The optic fiber is of course moving through space.
It is of course the case that in order for an object to move along a loop through space acceleration will be involved. However, the experimental setup can always be arranged such that the acceleration occurs at right angles to the direction of motion that is probed by the experiment.
Wucknitz points out: consider what happens if you reverse the direction of motion of the optic fiber conveyor. The direction of the acceleration doesn't change, but the direction the loop-effect does reverse.
Treating the length of the optic fiber as a Minkowski spacetime:
Here is the crucial bit: not all properties of Minkowski spacetime in general carry over to the special case of a Minkowski spacetime that loops back on itself.
The general concept of Minkowski spacetime is one of a spacetime without curvature of any type. In the general concept looping back on itself is excluded. In the absence of looping back on itself there is unrestricted relativity of simultaneity.
In a Minkowski spacetime that loops back on itself, however, there is a global procedure that identifies a globally preferred frame.
In the animation the red and blue dots represent propagating light. The four grey dots represent detectors that are in motion along the perimeter of the closed loop.
As a matter of principle the speed of light in the clockwise and counterclockwise directions is the same. (In the animation the dots keep crossing each other at the same 4 points.)
If you perform a clock synchronization along a sub-section of the perimeter then you apply Einstein synchronization convention.
However, if you would proceed to apply Einstein synchronization convention on adjacent sub-sections all the way around the loop, then you would end up with a time gap; it doesn't go all the way around.
So we see that in the case of a spacetime that loops back on itself there is only one self-consistent synchronization all the way around. You must take into account that the space loops back to itself.
This synchronization effect has been experimentally verified, it is documented in an article by Neil Ashby titled The Sagnac effect in the Global Positioning System. In normal operation the clocks onboard the GPS satellites are maintained to be in sync with Earth based clocks, but for this experiment synchronizing signals were relayed from satellite to satellite, in both directions.
(The GPS base stations on Earth maintain a global synchronized time.)
(The loop-closing effect is also the operating principle of a ring laser gyroscope. Light is generated, and it propagates in both directions along a loop. An interference effect is obtained, and from that measurement the rate of rotation can be inferred. Crucially, a ring laser gyroscope does not need calibration to establish the point of zero rotation rate; the fundamental reference for the rotation rate sensing is that in both directions the speed of light is the same.)
The key points
It is possible to define a Minkowski spacetime for the case of the optic fiber conveyor setup.
When a loop is closed there is no longer unrestricted applicability of relativity of simultaneity. Relativity of simultaneity is still applicable for synchronization along arbitrary subsections of the perimeter. However, for self-consistent synchronization all the way around there is no relativity of simultaneity. All the way around there is only a single synchronization.
| 69,186
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RCRD LBL’s Elliot Aronow’s Our Show vid/podcast is well worth a damn viewing. With a bunch of seasons under their belt, the show features featurettes and live performances all tailored around music and shit. We love it.
Now, the show attacks printed matter with the release of their first zine under the same name. 300 handprinted copies featuring wrtitings and words from Diplo, Nick Zinner, chef Eddie Huang, Ruvan, Justine D, Brooklyn Tailors, and others fill the issue. As with all things print, we are more than happy to add it to our bookshelves. Find it online for $5 shipped; a “nice price.” Let’s take a look inside….
Photography: Selectism.com
More looks in our gallery…
| 272,381
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WWE star Bianca Belair recently spoke with Sporf and Inside The Ropes to discuss all things pro-wrestling, including how her Royal Rumble appearance this year was a surprise to her, and how she wants a high profile matchup with either Sasha Banks or Charlotte Flair at WrestleMania 37. Highlights are below.
On wanting to face Sasha Banks or Charlotte:
I would love, at WrestleMania, to be facing Sasha Banks. Or I would love to also get my revenge and face Charlotte. One of those two.
On the Royal Rumble:
Last year, the Royal Rumble kind of came out of nowhere. I found out the night before that I was going to be a part of it, and I went and had this great moment there. So I’m very excited about this year because I’m fully expecting to be in the Royal Rumble, and I can prepare for it and so I’m just super excited about this being an even bigger Royal Rumble. And now I can be prepared to actually win it and that be my path to WrestleMania.
Full interview can be seen here. (Transcribed by Inside The Ropes)
👀 Watch out @SashaBanksWWE & @MsCharlotteWWE…
👊 …@BiancaBelairWWE is coming for you at @WrestleMania! pic.twitter.com/9JoiEHKyfN
— SPORF (@Sporf) December 7, 2020
| 88,460
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\begin{document}
\thispagestyle{empty}
\
\medskip
\medskip
\vskip2cm
\begin{center}
\medskip
\medskip
\noindent {\LARGE{\bf {Poisson--Hopf algebra deformations\\[6pt] of Lie--Hamilton systems}}}
\medskip
\medskip
\medskip
\medskip
\medskip
{\sc \'Angel Ballesteros$^{1}$, Rutwig Campoamor-Stursberg$^{2,3}$, Eduardo Fern\'andez-Saiz$^{3}$,\\[4pt] Francisco J.~Herranz$^{1}$ and Javier de Lucas$^{4}$}
\end{center}
\medskip
\noindent
$^1$ Departamento de F\1sica, Universidad de Burgos, E-09001 Burgos, Spain
\noindent
$^2$ Instituto de Matem\'atica Interdisciplinar I.M.I-U.C.M., E-28040 Madrid, Spain
\noindent
$^3$ Departamento de Geometr\1a y Topolog\1a, Universidad Complutense de Madrid, Plaza de Ciencias 3, E-28040 Madrid, Spain
\noindent
$^4$ Department of Mathematical Methods in Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland
\medskip
\noindent
{\small
E-mail: {\rm angelb@ubu.es, rutwig@ucm.es, eduardfe@ucm.es, fjherranz@ubu.es, javier.de.lucas@fuw.edu.pl
}}
\medskip
\begin{abstract}
\noindent
Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie--Hamilton systems, to devise a novel formalism: the Poisson--Hopf algebra deformations of Lie--Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie--Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie--Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie--Hamilton system associated with a Poisson--Hopf algebra of functions that allows for the explicit description of its $t$-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson--Hopf algebra analogue of the non-standard quantum deformation of $\mathfrak{sl}(2)$ and its applications to deform well-known Lie--Hamilton systems describing oscillator systems, Milne--Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency.
\end{abstract}
\bigskip
\noindent
MSC: 16T05, 17B66, 34A26
\medskip
\noindent
PACS: {02.20.Uw, 02.20.Sv, 02.60.Lj}
\medskip
\noindent{KEYWORDS}: Lie system, Vessiot--Guldberg Lie algebra, Hopf algebra, Poisson coalgebra, oscillator system, position-dependent mass, Riccati equation
\sect{Introduction}
A {\it Lie system} is a nonautonomous
system of first-order ordinary differential equations whose general solution can be written as a function, a so-called {\it superposition rule}, of a family of particular solutions and some constants \cite{LS,VES,DAV}. Superposition rules constitute a structural
property that emerges naturally from the group-theoretical
approach to differential equations initiated by Lie, Vessiot, and Guldberg, within the context of
the development of the geometric program based on transformation groups, as well as from the analytic classification
of differential equations developed by Painlev\'e and Gambier,
among others. Indeed, Lie proved that every Lie system can be described by a finite-dimensional Lie algebra of vector fields, a {\it Vessiot--Guldberg Lie algebra} \cite{LS}, and Vessiot used Lie groups to derive superposition rules \cite{VES}.
In the frame
of physical applications, it was not until the 1980s that the power
of superposition rules and Lie systems was fully recognized \cite{PW}, motivating a systematic
analysis of their applications in classical dynamics and their potential generalization to quantum systems (see
\cite{PW,CGM00,CGM07,Dissertations} and references therein).
Although Lie systems, as well as their refinements and generalizations, represent a valuable
auxiliary tool in the integrability study of physical systems, it seems surprising that the methods
employed have always remained within the limitations of Lie group and distribution theory, without considering
other
frameworks that have turned out to be a very successful
approach to integrability, such as quantum groups and Poisson--Hopf algebras ~\cite{Abe,Chari,Majid,coalgebra1,coalgebra2}. We recall that, beyond superintegrable systems~\cite{coalgebra1,coalgebra2}, Poisson coalgebras have been recently applied to integrable bi-Hamiltonian deformations of Lie--Poisson systems~\cite{Ballesteros1} and to integrable deformations of R\"ossler and Lorenz systems~\cite{Ballesteros2}.
This paper presents a novel generic procedure for the Poisson--Hopf algebra deformations of {\it Lie--Hamilton (LH) systems}, namely Lie systems endowed with a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a Poisson structure \cite{CLS13}. LH systems posses also a finite-dimensional Lie algebra of functions, a so-called {\it LH algebra}, governing their dynamics \cite{CLS13}. Then, our approach is based on the Poisson coalgebra formalism extensively used in the context of superintegrable systems together with the notion of involutive distributions in the sense of Stefan--Sussman (see \cite{Va94,Pa57,WA} for details). The crux will be to consider a Poisson--Hopf algebra structure
that replaces the LH algebra of the non-deformed LH system, thus allowing for an explicit construction of $t$-independent constants of the
motion, that will be expressed in terms of the deformed Casimir invariants. Moreover, the deformation will generally transform the Vessiot--Guldberg Lie algebra of the LH system into a mere set of vector fields generating an integrable distribution in the sense of Stefan--Sussman. Consequently, the deformed LH systems are not, in general, Lie systems anymore.
Our novel approach is presented in the next section, where the basics of LH systems and Poisson--Hopf algebras are recalled (for details on the general theory of Lie and LH systems, the reader is referred to~\cite{LS,PW,CGM00,CGM07,Dissertations,CLS13, CGL10,CGL11,BCHLS13Ham,BBHLS,BHLS,C132,C135, Ibragimov16,Ibragimov17,HLT}).
To illustrate this construction, we consider in section 3 the Poisson--Hopf algebra analogue of the so-called non-standard quantum deformation of $\mathfrak{sl}(2)$~\cite{Ohn,beyond, non, Shariati} together with its deformed Casimir invariant.
Afterwards, relevant examples of deformed LH systems that can be extracted from this deformation are given. Firstly, the non-standard deformation of the Milney--Pinney equation is presented in section 4, where this deformation is shown to give rise to a new oscillator system with a position-dependent mass and a time-dependent frequency, whose (time-independent) constants of the motion are also explicitly deduced.
In sections 5 and 6 several deformed (complex and coupled) Riccati equations are obtained as a straightforward application of the formalism here presented. We would like to stress that, albeit these applications are carried out on the plane, thus allowing a deeper insight in the proposed formalism, the method here presented is by no means constrained dimensionally, and its range of applicability goes far beyond the particular cases here considered.
Finally, some remarks and open problems are addressed in the concluding section.
\sect{Formalism}
For the sake of simplicity we will develop our formalism and its corresponding applications on $\mathbb{R}^2$, but we stress that this approach can be applied, mutatis mutandis, to construct Poisson--Hopf algebra deformations of LH systems defined on any manifold.
\subsect{Lie--Hamilton systems}
Let us consider the global coordinates $\{x,y\}$ on the Euclidean plane $\mathbb{R}^2$. Geometrically, every nonautonomous system of first-order differential equations on $\mathbb{R}^2$ of the form
\begin{equation}
\frac{{\rm d} x}{{\rm d} t }=f(t,x,y), \qquad \frac{{\rm d} y}{{\rm d} t }=g(t,x,y),
\label{system}
\end{equation}
where $f,g:\mathbb{R}^3\rightarrow \mathbb{R}$ are arbitrary functions, amounts to a $t$-dependent vector field ${\bf X}:\mathbb{R}\times \mathbb{R}^2\rightarrow {\rm T}\mathbb{R}^2$ given by
\begin{equation}\label{Vect}
{\bf X}:\mathbb{R}\times\mathbb{R}^2\ni (t,x,y)\mapsto f(t,x,y)\frac{\partial}{\partial x}+g(t,x,y)\frac{\partial}{\partial y}
\in {\rm T}\mathbb{R}^2 .
\end{equation}
This justifies to represent (\ref{Vect}) and its related system of differential equations (\ref{system}) by ${\bf X}$ (cf. \cite{Dissertations}). Let us assume ${\bf X}$ to be a {\it Lie system} on $\mathbb{R}^2$, namely it admits a superposition rule (see \cite{LS,PW,CGM00,CGM07,Dissertations,CGL10} for details). Since the general solution to a Lie system is not generally known, the use of a superposition rule enables one to unveil its general properties or to simplify the use of numerical methods \cite{PW,Dissertations}. Lie systems are, for instance, several Riccati, Kummer--Schwarz and Milne--Pinney equations when written as first-order systems of differential equations \cite{Dissertations,BCHLS13Ham, BBHLS,BHLS}.
According to the {Lie--Scheffers Theorem}~\cite{LS,CGM00,CGM07}, a system $\bf X$ is a Lie system if and only if
\be
{\bf X}_t(x,y):= {\bf X}(t,x,y)=\sum_{i=1}^l b_i(t){\bf X}_i(x,y) ,
\label{aabb}
\ee
for some $t$-dependent functions $b_1(t),\ldots,b_l(t)$ and vector fields ${\bf X}_1,\ldots,{\bf X}_l$ on $\mathbb{R}^2$ that span an $l$-dimensional real Lie algebra $V$ of vector fields, i.e. the Vessiot--Guldberg Lie algebra of ${\bf X}$.
A Lie system ${\bf X}$ is, furthermore, a LH one~\cite{Dissertations,CLS13,BCHLS13Ham,BBHLS,BHLS, HLT} if it admits a Vessiot--Guldberg Lie algebra $V$ of Hamiltonian vector fields relative to a Poisson structure. This amounts to the existence, around each generic point of $\mathbb{R}^2$, of a symplectic form, $\omega$, such that:
\be
\mathcal{L}_{{\bf X}_i}\omega=0 ,
\label{der}
\ee
for a basis ${\bf X}_1,\ldots,{\bf X}_l$ of $V$ (cf.~Lemma 4.1 in \cite{BBHLS}). To avoid minor technical details and to highlight our main ideas, hereafter it will be assumed, unless otherwise stated, that the symplectic form and remaining structures are defined globally. More accurately, a local description around a generic point in $\mathbb{R}^2$ could easily be carried out.
Each vector field ${\bf X}_i$ admits a Hamiltonian function $h_i$ given by the rule:
\be
\iota_{{\bf X}_i}\omega={\rm d}h_i,
\label{contract}
\ee
where $\iota_{{\bf X}_i}\omega$ stands for the contraction of the vector field ${\bf X}_i$ with the symplectic form $\omega$. Since $\omega$ is non-degenerate, every function $h$ induces a unique associated Hamiltonian vector field ${\bf X}_h$. This fact gives rise to a Poisson bracket on $C^\infty(\mathbb{R}^2)$ given by
\begin{equation}\label{LB}
\{\cdot,\cdot\}_\omega\ :\ C^\infty\left(\mathbb{R}^2\right)\times C^\infty\left(\mathbb{R}^2\right)\ni (f_1,f_2)\mapsto X_{f_2} f_1\in C^\infty\left(\mathbb{R}^2\right),
\end{equation}
turning $(C^\infty(\mathbb{R}^2),\{\cdot,\cdot\}_\omega)$ into a Poisson algebra \cite{Va94}. The space ${\rm Ham}(\omega)$ of Hamiltonian vector fields on $\mathbb{R}^2$ relative to $\omega$ is also a Lie algebra relative to the commutator of vector fields. Moreover, we have the following exact sequence of Lie algebra morphisms (see \cite{Va94})
\begin{equation}\label{seq}
0\hookrightarrow \mathbb{R}\hookrightarrow (C^\infty(\mathbb{R}^2),\{\cdot,\cdot\}_\omega)\stackrel{\varphi}{\longrightarrow} ({\rm Ham}(\omega),[\cdot,\cdot])\stackrel{\pi}{\longrightarrow} 0,
\end{equation}
where $\pi$ is the projection onto $0$ and $\varphi$ maps each $f\in C^\infty(\mathbb{R}^2)$ onto the Hamiltonian vector field ${\bf X}_{-f}$.
In view of the sequence (\ref{seq}), the Hamiltonian functions $ h_1,\ldots,h_l$ and their successive Lie brackets with respect to (\ref{LB}) span a finite-dimensional Lie algebra of functions contained in $\varphi^{-1}(V)$. This Lie algebra is called a
{\em LH algebra} $\lh$ of $X$. We recall that LH algebras play a relevant role in the derivation of constants of motion and superposition rules for LH systems~\cite{BCHLS13Ham, BHLS,HLT}.
\subsect{Poisson--Hopf algebras}
The core in what follows is the fact that the space $C^\infty\left(\lh^* \right)$ can be endowed with a {\it Poisson--Hopf algebra} structure.
We recall that an associative algebra $A$ with a {\it product} $m$ and a {\it unit} $\eta$ is said to be a {\em Hopf algebra} over $\Bbb R$
\cite{Abe,Chari, Majid} if there exist two homomorphisms called {\em coproduct} $(\Delta :
A\longrightarrow A\otimes A )$ and {\em counit} $(\epsilon : A\longrightarrow
\Bbb R)$, along with an antihomomorphism, the {\em antipode} $\gamma :
A\longrightarrow A$, such that for every $ a\ \! \in A$ one gets:
$$
\begin{gathered}
({\rm Id}\otimes\Delta)\Delta (a)=(\Delta\otimes {\rm Id})\Delta (a),
\nonumber \\
({\rm Id}\otimes\epsilon)\Delta (a)=(\epsilon\otimes {\rm Id})\Delta (a)= a,
\nonumber\\
m(( {\rm Id} \otimes \gamma)\Delta (a))=m((\gamma \otimes {\rm Id})\Delta (a))=
\epsilon (a) \eta,\nonumber
\end{gathered}
$$
where $m$ is the usual multiplication $m(a\otimes b)=ab$. Hence the following diagram is commutative:
\\
\centerline{ {\xymatrix@C=0.4em{&A\otimes A\ar[rr]^{{\rm Id}\,\otimes\, \gamma}&&A\otimes A\ar[rd]^m&&\\
A\ar[rd]^\Delta\ar[rr]^{\epsilon}\ar[ur]^\Delta&&\mathbb{R}\ar[rr]^{\eta}&&A\\
&A\otimes A\ar[rr]^{\gamma\,\otimes\, {\rm Id}}&&A\otimes A\ar[ru]^m&} }}
\medskip
\noindent
If $A$ is a commutative Poisson algebra and $\Delta$ is a Poisson algebra morphism, then $(A,m, \eta,\Delta,\epsilon,\gamma)$ is a {\it Poisson--Hopf algebra} over $\Bbb R$.
We recall that the Poisson bracket on $A\otimes A$ reads
$$
\{ a\otimes b, c\otimes d\}_{A\otimes A}=\{ a, c\}\otimes b d + a c\otimes \{ b, d\} ,\qquad \forall a,b,c,d\in A .
$$
In our particular case, $C^\infty\left(\lh^* \right)$ becomes a Hopf algebra relative to its natural associative algebra with unit provided that
$$
\begin{gathered}
\Delta (f)(x_1,x_2):=f(x_1+x_2),\qquad m(h\otimes g)(x):=h(x)g(x),\\
\epsilon (f):=f(0),\qquad \eta(1)(x):=1,\qquad \gamma(f)(x):=f(-x),
\end{gathered}
$$
for every $x,x_1,x_2\in \lh$ and $f,g,h\in C^\infty(\lh^*)$. Therefore, the space $C^\infty\left(\lh^* \right)$ becomes a Poisson--Hopf algebra by endowing it with the Poisson structure defined by the Kirillov--Kostant--Souriau bracket related to a Lie algebra structure on $\lh$.
\subsect{Deformations of Lie--Hamilton systems and generalized distributions}
The aim of this paper is to provide a systematic procedure to obtain deformations of LH systems by using LH algebras and deformed Poisson--Hopf algebras that lead to appropriate extensions of the theory of LH systems. Explicitly, the construction is based upon the following {four} steps:
\begin{enumerate}
\item Consider a LH system ${\bf X}$ (\ref{aabb}) on $\mathbb{R}^2$ with respect to a symplectic form $\omega$ and admitting a LH algebra $\lh$ spanned by a basis of functions $h_1,\ldots, h_l\in C^\infty(\mathbb{R}^2)$ with structure constants $c_{ij}^k$, i.e.
\be
\{h_i,h_j\}_{\omega}= \sum_{{k=1}}^{l} c_{ij}^{k}h_{k},\qquad i,j=1,\ldots,l.
\nonumber
\ee
\item Introduce a Poisson--Hopf algebra deformation $\qlhzd$ of $C^\infty(\lh^*)$ with deformation parameter $z\in\mathbb R$ (in a quantum group setting we would have $q:={\rm e}^z$) as the space of smooth functions $F(h_{z,1},\ldots,h_{z,l})$ with fundamental Poisson bracket given by
\be
\{h_{z,i},h_{z,j}\}_{\omega}= F_{z,ij}(h_{z,1},\dots,h_{z,l }),
\label{zab}
\ee
where $F_{z,ij}$ are certain smooth functions also depending smoothly on the deformation parameter $z$ and such that
\be
\lim_{z\to 0} h_{z,i}=h_i ,\qquad \lim_{z\to 0}\nabla h_{z,i}=\nabla h_i, \qquad \lim_{z\to 0} F_{z,ij}(h_{z,1},\dots,h_{z,l }) =\sum_{k=1}^l c_{ij}^k h_k,
\label{zac}
\ee
where $\nabla$ stands for the gradient relative to the Euclidean metric on $\mathbb{R}^2$.
Hence,
\be
\lim_{z\to 0} \{h_{z,i},h_{z,j}\}_{\omega}=\{h_i,h_j\}_\omega .
\label{zad}
\ee
\item Define the deformed vector fields ${\bf X}_{z,i}$ by the rule
\be
\iota_{{\bf X}_{z,i}}\omega :={\rm d}h_{z,i},
\label{contract2}
\ee
so that
\be
\lim_{z\to 0} {\bf X}_{z,i}= {\bf X}_i.
\label{zae}
\ee
\item Define the deformed LH system of the initial system ${\bf X}$ (\ref{aabb}) by
\be
{\bf X}_z:=\sum_{i=1}^lb_i(t){\bf X}_{z,i} .
\label{aabbc}
\ee
\end{enumerate}
Now some remarks are in order. First, note that for a given LH algebra $\lh$ there exist as many Poisson--Hopf algebra deformations as non-equivalent Lie bialgebra structures $\delta$ on $\lh$~\cite{Chari}, where the 1-cocycle $\delta$ essentially provides the first-order deformation in $z$ of the coproduct map $\Delta$. For three-dimensional real Lie algebras the full classification of Lie bialgebra structures is known~\cite{Gomez}, and some classification results are also known for certain higher-dimensional Lie algebras (see~\cite{Gomez, dualPL, BBM3d} and references therein). Once a specific Lie bialgebra $(\lh,\delta)$ is chosen, the full Poisson--Hopf algebra deformation can be systematically obtained by making use of the Poisson version of the `quantum duality principle'\ for Hopf algebras, as we will explicitly see in the next section for an $(\mathfrak{sl}(2),\delta)$ Lie bialgebra.
Second, the deformed vector fields ${\bf X}_{z,i}$ (\ref{contract2}) will not, in general, span a finite-dimensional Lie algebra, which implies that (\ref{aabbc}) is not a Lie system. In fact, the
sequence of Lie algebra morphisms (\ref{seq}) and the properties of Hamiltonian vector fields \cite{Va94} lead to
\be
[{\bf X}_{z,i},{\bf X}_{z,j}]= [
\varphi(h_{z,i}),\varphi(h_{z,j})]= \varphi(
\{h_{z,i},h_{z,j}\}_{\omega})= \varphi( F_{z,ij}(h_{z,1},\dots,h_{z,l }))=-\sum_{k=1}^l\frac{\partial F_{z,ij}}{\partial h_{z,k}}{\bf X}_{z,k}.
\nonumber
\ee
In other words,
\be
\left[{\bf X}_{z,i},{\bf X}_{z,j}\right]= \sum_{k=1}^{l} G_{z,ij}^{k} (x,y ) {\bf X}_{z,k}, \label{FRO}
\ee
where the $G_{z,ij}^{k} (x,y)$ are smooth functions relative to the coordinates $x,y$ and the deformation parameter $z$.
Despite this, the relations (\ref{zad}) and the continuity of $\varphi$ imply that
\be
[{\bf X}_i,{\bf X}_j]=\varphi(\{h_i,h_j\})_\omega=\varphi\left(\lim_{z\rightarrow 0}\{h_{z,i},h_{z,j}\}_\omega\right)=\lim_{z\to 0}\varphi\{h_{z,i},h_{z,j}\}_\omega=\lim_{z\to 0}[{\bf X}_{z,i},{\bf X}_{z,j}] .
\nonumber
\ee
Hence
\be \lim_{z\to 0}G_{z,ij}^{k} (x,y ) ={\rm constant}
\nonumber
\ee
holds for all indices.
Geometrically, the conditions (\ref{FRO}) establish that the vector fields ${\bf X}_{z,i}$ span an involutive smooth generalized distribution $\mathcal{D}_z$. In particular, the distribution $\mathcal{D}_0$ is spanned by the Vessiot--Guldberg Lie algebra $\langle {\bf X}_1,\dots,{\bf X}_l\rangle$. This causes $\mathcal{D}_0$ to be integrable on the whole $\mathbb{R}^2$ in the sense of Stefan--Sussman~\cite{Va94,Pa57,WA}. The integrability of $\mathcal{D}_z$, for $z\neq 0$, can only be ensured on open connected subsets of $\mathbb{R}^2$ where $\mathcal{D}_z$ has constant rank~\cite{Va94}.
Third, although the vector fields ${\bf X}_{z,i}$ depend smoothly on $z$, the distribution $\mathcal{D}_z$ may change abruptly. For instance, consider the case given by the LH system ${\bf X}=\partial_x+ty\partial_x$ relative to the symplectic form $\omega=\dd x\wedge \dd y$ and admitting a LH algebra $\lh=\langle h_1:=y,\ h_2:=y^2/2\rangle$. Let us define $h_{z,1}:=y$ and $h_{z,2}:=y^2/2+zx$. Then ${\bf X}_z=\partial_x+t(y\partial_x-z\partial_y)$ and $\dim \mathcal{D}_0(x,y)=1$, but $\dim \mathcal{D}_z(x,y)=2$ for $z\neq 0$. Hence, the deformation of LH systems may change in an abrupt way the dynamical and geometrical properties of the systems ${\bf X}_z$ (cycles, periodic solutions, etc).
Fourth, the deformation parameter $z$ provides an additional degree of freedom that enables the control or modification of the deformed system ${\bf X}_z$. In fact, as $z$ can be taken small, perturbations of the initial Lie system ${\bf X}$ can be obtained from the deformed one ${\bf X}_z$ in a natural way.
And, finally,
we stress that, by construction, the very same procedure can be applied to other two-dimensional manifolds different to $\mathbb{R}^2$, to higher dimensions as well as to multiparameter Poisson--Hopf algebra deformations of Lie algebras endowed with two or more deformation parameters.
\subsect{Constants of the motion}
The fact that $ \qlhzd$ is a Poisson--Hopf algebra allows us to apply the coalgebra formalism established in~\cite{BCHLS13Ham} in order to obtain $t$-independent constants of the motion for ${\bf X}_z$.
Let $S\left(\lh\right)$ be the {\it symmetric algebra} of $\lh$, i.e. the associative unital algebra of polynomials on the elements of $\lh$. The Lie algebra structure on $\lh$ can be extended to a Poisson algebra structure in $S\left(\lh\right)$ by requiring $[v,\cdot ]$ to be a derivation on the second entry for every $v\in \lh$. Then, $S\left(\lh\right)$ can be endowed with a Hopf algebra structure with a non-deformed (trivial) coproduct map $\Delta$ defined by
\begin{equation}
{\Delta} :S\left(\lh \right)\rightarrow
S\left(\lh\right) \otimes S\left(\lh\right) ,\qquad {\Delta}(v_i):=v_i\otimes 1+1\otimes v_i, \qquad i=1,\dots, l,
\label{baa}
\end{equation}
which is a Poisson algebra homomorphism relative to the Poisson structure on $S(\lh)$ and the one induced in $S(\lh)\otimes S(\lh)$. Recall that every element of $S(\lh)$ can be understood as a function on $\lh^*$. Moreover, as $S(\mathcal{H}_\omega)$ is dense in the space
$C^\infty(\mathcal{H}^*_\omega)$ of smooth functions on the dual $\mathcal{H}^*_\omega$ of the LH algebra $\mathcal{H}_\omega$, the coproduct in $S(\mathcal{H}_\omega)$ can be extended in a unique way to
\begin{equation}
{\Delta} :C^\infty\left(\lh^*\right)\rightarrow
C^\infty\left(\lh^*\right) \otimes C^\infty\left(\lh^*\right).
\nonumber
\end{equation}
Similarly, all structures on $S(\lh)$ can be extended turning $C^\infty(\mathcal{H}^*_\omega)$ into a Poisson--Hopf algebra. Indeed, the resulting structure is the natural one in $C^\infty(\mathcal{H}^*_\omega)$ given in section 2.2.
Let us assume now that $C^\infty\left(\lh^*\right)$
has a Casimir invariant
\be
C=C(v_1,\dots,v_l),
\nonumber
\ee
where $v_1,\ldots,v_l$ is a basis for $\lh$. The initial LH system allows us to define a Lie algebra morphism $\phi:\lh\rightarrow C^\infty(M)$, where $M$ is a submanifold of $\mathbb R^2$ where all functions $h_i:=\phi(v_i)$, for $i=1,\ldots,l$, are well defined.
Then, the Poisson algebra morphisms
\be
D: C^\infty\left( \lh^* \right) \rightarrow C^\infty(M),\qquad D^{(2)} : C^\infty\left( \lh^* \right)\otimes C^\infty\left( \lh^* \right)\rightarrow C^\infty(M)\otimes C^\infty(M),
\label{morphisms}
\ee
defined respectively by
\be
D( v_i):= h_i(x_1,y_1), \qquad
D^{(2)} \left( {\Delta}(v_i) \right):= h_i(x_1,y_1)+h_i(x_2,y_2) ,\qquad i=1,\dots, l,
\label{bb}
\ee
lead to the $t$-independent constants of motion $F^{(1)}:= F$ and $F^{(2)}$ for the Lie system ${\bf X}$ given in (\ref{aabb}) where
\be
F:= D(C),\qquad F^{(2)}:= D^{(2)} \left( {\Delta}(C) \right).
\label{bc}
\ee
The very same procedure can also be applied to any Poisson--Hopf algebra $ \qlhzd$ with deformed coproduct $\Delta_z$ and Casimir invariant
$C_z=C_z(v_{1},\dots,v_{l})$, where $\{ v_{1},\dots,v_{l} \}$ fulfill the same Poisson brackets ({\ref{zab}), and such that
\be
\lim_{z\to 0} \Delta_{z}=\Delta , \qquad \lim_{z\to 0} C_{z}= C.
\nonumber
\ee
Following~\cite{BCHLS13Ham}, the element $C_z$ turns out to be the cornerstone in the construction of the deformed constants of the motion for the `generalized' LH system ${\bf X}_z $.
\sect{A Poisson--Hopf algebra deformation of $\mathfrak{sl}(2)$}
Once the general description of our approach has been introduced, we present in this section the general properties of the Poisson analogue of the so-called non-standard quantum deformation of the simple real Lie algebra $\mathfrak{sl}(2)$. This deformation will be applied in the sequel to get deformations of the Milne--Pinney equation or Ermakov system and of some Riccati equations, since all these systems are known to be endowed with a LH algebra $\lh$ isomorphic to $\mathfrak{sl}(2)$~\cite{BCHLS13Ham, BBHLS, BHLS}.
Let us consider the basis $\{J_3, J_+,J_-\}$ for $\mathfrak{sl}(2)$ with Lie brackets and Casimir operator given by
\be
[ J_3,J_\pm ] = \pm 2J_\pm ,\qquad [J_+ , J_- ] = J_3,\qquad {\cal C}=\tfrac 12 J_3^2+(J_+ J_- + J_- J_+).
\label{crules}
\ee
Amongst the three possible quantum deformations of $\mathfrak{sl}(2)$~\cite{Tmatrix}, we shall hereafter consider the non-standard (triangular or Jordanian) quantum deformation, $U_z(\mathfrak{sl}(2))$ (see~\cite{Ohn,beyond, non, Shariati} for further details). The Hopf algebra structure of $U_{z}(\mathfrak{sl}(2))$ has the following
deformed coproduct and compatible deformed commutation rules
\be
\Delta_z(J_+)=J_+\otimes 1 + 1 \otimes J_+,\qquad
\Delta_z(J_j)=J_j\otimes {\rm e}^{2z J_+} + {\rm e}^{- 2z J_+} \otimes J_j ,\qquad j \in \{-, 3\},
\label{codef}\nonumber
\ee
\be
[ J_3,J_+ ] _z= \frac{\sinh ( 2z J_+)}{ z} ,\qquad [ J_3, J_-]_z=- J_- \cosh(2zJ_+) - \cosh(2zJ_+) J_- ,\qquad [J_+ , J_- ]_z= J_3.
\label{corudef}\nonumber
\ee
The counit and antipode can be explicitly found in~\cite{Ohn,non}, and the deformed Casimir reads~\cite{beyond}
\be
{\cal C}_z=\frac 12 J_3^2+\frac{\sinh(2z J_+) }{2z} \, J_- + J_- \,\frac{\sinh(2z J_+) }{2z} + \frac 12 \cosh^2( 2z J_+) .
\label{bf}\nonumber
\ee
Let $\mathfrak{g}$ be the Lie algebra of $G$.
It is well known (see~\cite{Chari,Majid}) that quantum algebras $U_z(\mathfrak{g})$ are Hopf algebra duals of quantum groups $G_z$. On the other hand, quantum groups $G_z$ are just quantizations of Poisson--Lie groups, which are Lie groups endowed with a multiplicative Poisson structure, {i.e.}~a Poisson structure for which the Lie group multiplication is a Poisson map. In the case of $U_z(\mathfrak{sl}(2))$, such Poisson structure on $SL(2)$ is explicitly given by the Sklyanin bracket coming from the classical $r$-matrix
\be
r=z J_3\wedge J_+,
\label{rmns}
\ee
which is a solution of the (constant) classical Yang--Baxter equation.
Moreover, the `quantum duality principle`~\cite{Dri,STS} states that quantum algebras can be thought of as `quantum dual groups' $G_z^\ast$, which means that any quantum algebra can be obtained as the Hopf algebra quantization of the dual Poisson--Lie group $G^\ast$. The usefulness of this approach to construct explicitly the Poisson analogue of quantum algebras was developed in~\cite{dualPL}.
In the case of $U_z(\mathfrak{sl}(2))$, the Lie algebra $\mathfrak{g}^\ast$ of the dual Lie group $G^\ast$ is given by the dual of the cocommutator map $\delta$ that is obtained from the classical $r$-matrix as
\begin{equation}
\delta(x)=[ x\otimes 1+1\otimes x , r],\qquad \forall x\in \mathfrak{g}.
\label{rmatrix}
\end{equation}
In our case, from~\eqref{crules} and~\eqref{rmns} we explicitly obtain
\be
\delta(J_3)=2z \, J_3 \wedge J_+ ,\qquad
\delta(J_+)= 0,\qquad
\delta(J_-)= 2z \, J_- \wedge J_+ ,
\nonumber
\ee
and the dual Lie algebra $\mathfrak{g}^\ast$ reads
\begin{equation}
[j^+,j^3]=-2 z \, j^3, \qquad [j^+,j^-]=-2 z \, j^-, \qquad [j^3,j^-]=0,
\label{book}
\end{equation}
where $\{j^3,j^+,j^-\}$ is the basis of $\mathfrak{g}^\ast$, and $\{J_3,J_+,J_-\}$ can now be interpreted as local coordinates on the dual Lie group $G^\ast$. The dual Lie algebra~\eqref{book} is the so-called `book' Lie algebra, and the complete set of its Poisson--Lie structures was explicitly obtained in~\cite{BBM3d} (see also~\cite{LV}, where book Poisson--Hopf algebras were used to construct integrable deformations of Lotka--Volterra systems). In particular, if we consider the coordinates on $G^\ast$ given by
\be
v_1= J_+,\qquad v_2 = \tfrac 12 J_3,\qquad v_3= - J_-,
\nonumber
\ee
the Poisson--Lie structure on the book group whose Hopf algebra quantization gives rise to the quantum algebra $U_z(\mathfrak{sl}(2))$ is given by the fundamental Poisson brackets~\cite{BBM3d}
\be
\{v_1,v_2\}_z=-\shc (2z v_1)v_1,\qquad
\{v_1,v_3\}_z=-2 v_2,\qquad
\{v_2,v_3\}_z= - \cosh(2 z v_1) v_3,
\label{gb}
\ee
together with the coproduct map
\begin{equation}
\Delta_z(v_1)= v_1 \otimes 1+
1\otimes v_1 , \qquad
\Delta_z(v_k)=v_k \otimes \eee^{2 z v_1} + \eee^{-2 z v_1} \otimes
v_k ,\qquad k=2,3,
\label{ga}
\end{equation}
which is nothing but the group law for the book Lie group $G^\ast$ in the chosen coordinates (see~\cite{dualPL,BBM3d,LV} for a detailed explanation). Therefore, ~\eqref{gb} and~\eqref{ga} define a Poisson--Hopf algebra structure on $C^\infty(G^\ast)$, which can be thought of as a Poisson--Hopf algebra deformation of the Poisson algebra $C^\infty(\mathfrak{sl}(2)^\ast)$, since we have identified the local coordinates on $C^\infty(G^\ast)$ with the generators of the Lie--Poisson algebra $\mathfrak{sl}(2)^\ast$.
Notice that we have introduced in (\ref{gb}) the hereafter called {\it cardinal hyperbolic sinus function} defined by
\be
\shc(x):=\frac {\sinh (x)}{x}.
\label{shc}
\ee
Some properties of this function along with its relationship with Lie systems are given in the Appendix.
Summarizing, the Poisson--Hopf algebra given by~\eqref{gb} and~\eqref{ga}, together with its Casimir function
\be
{C}_z=\shc( 2z v_1)\, v_1v_3-v_2^2 ,
\label{gc}
\ee
will be the deformed Poisson--Hopf algebra that we will use in the sequel in order to construct deformations of LH systems based on $\mathfrak{sl}(2)$. Note that the usual Poisson--Hopf algebra $C^\infty(\mathfrak{sl}(2)^\ast)$ is smoothly recovered under the $z\to 0$ limit leading to the non-deformed Lie--Poisson coalgebra
\be
\{ v_1,v_2\}=- v_1,\qquad \{ v_1,v_3\} =- 2v_2,\qquad \{ v_2,v_3\} =-v_3,
\label{brack2}
\ee
with undeformed coproduct (\ref{baa}) and Casimir
\be
C=v_1 v_3 - v_2^2 .
\label{ai}
\ee
We stress that this application of the `quantum duality principle' would allow one to obtain the Poisson analogue of any quantum algebra $U_z(\mathfrak{g})$, which by following the method here presented could be further applied in order to construct the corresponding deformation of the LH systems associated to the Lie--Poisson algebra $\mathfrak{g}$. In particular, the Poisson versions of the other quantum algebra deformations of $\mathfrak{sl}(2)$ can be obtained in the same manner with no technical obstructions (for instance, see~\cite{dualPL} for the explicit construction of the `standard' or Drinfel'd--Jimbo deformation).
\sect{Deformed Milne--Pinney equation and oscillator systems}
As a first application of our approach, we will construct the non-standard deformation of the well-known Milne--Pinney (MP) equation \cite{Mi30,Pi50}, which is known to be a LH system~\cite{BBHLS, BHLS}. Recall that the MP equation corresponds to the equation of motion of the isotropic oscillator with a time-dependent frequency and a `centrifugal' or Rosochatius--Winternitz term. As we will show in the sequel, the main feature of this deformation is that the new oscillator system has both a position-dependent mass and a time-dependent frequency.
\subsect{Non-deformed system}
The MP equation \cite{Mi30,Pi50} has the following expression
\begin{equation}\label{mp}
\frac{\dd^2x}{\dd t^2}=-\Omega^2(t)x+\frac{c}{x^3},
\end{equation}
where $\Omega(t)$ is any $t$-dependent function and $c\in \mathbb{R}$.
By introducing a new variable $y:= \dd x/\dd t$, the system \eqref{mp} becomes a first-order system of differential equations on ${\rm T}\mathbb{R}_0$, where $\mathbb{R}_0:=\mathbb{R}\backslash\{0\}$, of the form
\be
\frac{\dd x}{\dd t}=y,\qquad \frac{\dd y}{\dd t}=-\Omega^2(t)x+\frac{c}{x^3}.
\label{FirstLie}
\ee
This system is indeed part of the one-dimensional Ermakov system~\cite{Dissertations,Er08,Le91,LA08} and diffeomorphic to the one-dimensional $t$-dependent frequency counterpart~\cite{BCHLS13Ham, BBHLS, BHLS} of the Smorodinsky--Winternitz oscillator \cite{WSUF65}.
The system (\ref{FirstLie}) determines a Lie system with associated $t$-dependent vector field~\cite{BHLS}
\be
{\bf X} ={\bf X}_3+\Omega^2(t){\bf X}_1,
\label{MP}
\ee
where
\begin{equation}\label{FirstLieA}
{\bf X}_1:=-x\frac{\partial}{\partial y},\qquad {\bf X}_2:=\frac 12 \left(y\frac{\partial}{\partial y}-x\frac{\partial}{\partial x}\right),\qquad {\bf X}_3:=y\frac{\partial}{\partial x}+\frac{c}{x^3}\frac{\partial}{\partial y},
\end{equation}
span a Vessiot--Guldberg Lie algebra $V^{\rm MP}$ of vector fields isomorphic to $\mathfrak{sl}(2)$ (for any value of $c$) with commutation relations given by
\begin{equation}\label{aa}
[{\bf X}_1,{\bf X}_2]={\bf X}_1,\qquad [{\bf X}_1,{\bf X}_3]=2{\bf X}_2,\qquad [{\bf X}_2,{\bf X}_3]={\bf X}_3 .
\end{equation}
The vector fields of $V^{\rm MP}$ are defined on $\mathbb R^2_{x\ne 0}$, where they span a regular distribution of order two.
Furthermore, ${\bf X} $ is a LH system with respect to the symplectic form $\omega={\rm d}x\wedge {\rm d}y$ and the vector fields (\ref{FirstLieA}) admit Hamiltonian functions given by
\be
h_1=\frac 12 x^2 ,\qquad h_2=-\frac 12 xy ,\qquad h_3=\frac 12 \left(y^2 +\frac{c}{x^2} \right),
\label{ham2}
\ee
that fulfill the following commutation relations with respect to the Poisson bracket induced by $\omega$:
\be
\{ h_1,h_2\}_\omega=- h_1,\qquad \{ h_1,h_3\}_\omega =- 2h_2,\qquad \{ h_2,h_3\}_\omega =-h_3 .
\label{brack}
\ee
Then, the functions $h_1,h_2,h_3$ span a LH algebra ${\cal H}_{\omega}^{\rm {MP}} \simeq \mathfrak{sl}(2)$ of functions on $\mathbb R^2_{x\ne 0}$; the $t$-dependent Hamiltonian associated with the $t$-dependent vector field (\ref{MP}) reads
\be
h =h_3+\Omega^2(t)h_1 .
\label{hMP}
\ee
We recall that this Hamiltonian is a natural one, that is, it can be written in terms of a kinetic energy $T$ and potential $U$
by identifying the variable $y$ as the conjugate momentum $p$ of the coordinate $x$:
\be
h =T+U= \frac 12\, p^2 + \frac 12 \Omega^2(t) x^2 + \frac{c}{2x^2}.
\label{ham}
\ee
Hence $h$ determines the composition of a one-dimensional oscillator with a time-dependent frequency $ \Omega(t)$ and unit mass with a Rosochatius or Winternitz potential; the latter is just a centrifugal barrier whenever $c>0$ (see~\cite{nonlinear} and references therein). The LH system (\ref{FirstLie}) thus comes from the Hamilton equations of $h$ and, obviously, when $c$ vanishes, these reduce to the equations of motion of a harmonic oscillator with a
time-dependent frequency.
We stress that it has been already proved in~\cite{BBHLS,BHLS} that the MP equations (\ref{FirstLie}) comprise the {\em three} different types of possible $\mathfrak{sl}(2)$-LH systems according to the value of the constant $c$: class {\rm P}$_2$ for $c>0$; class {\rm I}$_4$ for $c<0$; and class {\rm I}$_5$ for $c=0$. This means that any other LH system related to a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields isomorphic to $\mathfrak{sl}(2)$ must be, up to a $t$-independent change of variables, of the form (\ref{FirstLie}) for a positive, zero or negative value of $c$.
This implies that the second-order Kummer--Schwarz equations~\cite{CGL11, LS12} and several types of Riccati equations~\cite{Mariton, Eg07, Wi08, SSVG11, SSVG14,CGLS, pilar} are comprised within ${\cal H}_{\omega}^{\rm {MP}}$ (depending on the sign of $c$). The relationships amongst all of these systems are ensured by construction and these can be explicitly obtained through either diffeomorphisms or changes of variables (see~\cite{BBHLS,BHLS} for details).
The constants of motion for the MP equations can be obtained by applying the coalgebra formalism introduced in~\cite{BCHLS13Ham} and briefly summarized in section 2.4. Explicitly,
let us consider the Poisson--Hopf algebra $C^\infty({\cal H}_{\omega}^{\rm {MP}*} )$ with basis $\{ v_1,v_2,v_3\}$, coproduct (\ref{baa}), fundamental Poisson brackets (\ref{brack2}) and Casimir (\ref{ai}).
The Poisson algebra morphisms (\ref{morphisms})
\be
D: C^\infty({\cal H}_{\omega}^{\rm {MP}*}) \rightarrow C^\infty(\mathbb R^2_{x\ne 0}) ,\quad D^{(2)} : C^\infty( {\cal H}_{\omega}^{\rm {MP}*} ) \otimes C^\infty ( {\cal H}_{\omega}^{ \rm {MP}*} )\rightarrow C^\infty(\mathbb R^2_{x\ne 0})\otimes C^\infty(\mathbb R^2_{x\ne 0}) ,
\nonumber
\ee
defined by (\ref{bb}), where $h_i$ are the Hamiltonian functions (\ref{ham2}), lead to the $t$-independent constants of the motion $F^{(1)}:=F$ and $F^{(2)}$ given by (\ref{bc}), through the Casimir (\ref{ai}), for the Lie system ${\bf X}$ (\ref{FirstLie}); namely~\cite{BCHLS13Ham}
\bea
&& F= h_1(x_1,y_1) h_3(x_1,y_1)- h_2^2(x_1,y_1)=\frac c 4 ,\nonumber\\[2pt]
&&F^{(2)}=\bigl( \left[ h_1(x_1,y_1)+h_1(x_2,y_2)\right] \left[ h_3(x_1,y_1)+h_3(x_2,y_2)\right] \bigr) -\bigl( h_2(x_1,y_1)+h_2(x_2,y_2) \bigl)^2\nonumber\\[2pt]
&&\qquad\, = \frac 14 ({x_1}{y_2} -{x_2} {y_1})^2
+\frac c 4\, \frac{(x_1^2+x_2^2)^2}{x_1^2 x_2^2} .
\label{am}
\eea
We observe that $F^{(2)}$ is just a Ray--Reid invariant for generalized Ermakov systems \cite{Le91,RR79} and that it is related to the one obtained in~\cite{coalgebra2,letterBH} from a coalgebra approach applied to superintegrable systems.
By permutation of the indices corresponding to the variables of the non-trivial invariant $F^{(2)}$, we find two other constants of the motion:
\be
F_{13}^{(2)}=S_{13} ( F^{(2)} ) ,\qquad F_{23}^{(2)}=S_{23} ( F^{(2)} ) ,
\label{an}
\ee
where $S_{ij}$ is the permutation of variables $(x_{i},y_i)\leftrightarrow
(x_j,y_j)$. Since $\partial(F^{(2)},F^{(2)}_{23})/\partial(x_1,y_1)\neq 0$, both constants of motion are functionally independent (note that the pair $(F^{(2)},F^{(2)}_{13})$ is functionally independent as well). From these two invariants, the corresponding superposition rule can be derived in a straightforward manner. Its explicit expression can be found in~\cite{BCHLS13Ham}.
\subsect{Deformed Milne--Pinney equation}
In order to apply the non-standard deformation of $\mathfrak{sl}(2)$ described in section 3 to the MP equation, we need to find the deformed counterpart $h_{z,i}$ $(i=1,2,3)$ of the Hamiltonian functions $h_i$ (\ref{ham2}), so fulfilling the Poisson brackets (\ref{gb}), by keeping the canonical symplectic form $\omega$.
This problem can be rephrased as the one consistent in finding symplectic realizations of a given Poisson algebra, which can be solved once a particular symplectic leave is fixed as a level set for the Casimir functions of the algebra, where the generators of the algebra can be expressed in terms of the corresponding Darboux coordinates.
In the particular case of the $U_z(\mathfrak{sl}(2))$ algebra, the explicit solution (modulo canonical transformations) was obtained in~\cite{chains} where
the algebra~\eqref{gb} was found to be generated by the functions
\bea
&&v_1(q,p)=\frac 12\,q^2,\cr
&&v_2(q,p)=-\frac 12\frac {\sinh z q^2}{z q^2} \, q p , \cr
&&v_3(q,p)=\frac 12\frac {\sinh z q^2}{z q^2}\, p^2 +
\frac 12\frac{z c}{\sinh z q^2},
\nonumber
\eea
where $\omega={\rm d} q\wedge {\rm d} p$, and the Casimir function~\eqref{gc} reads ${C}_z=c/4$. In practical terms, such a solution can easily be found by solving firstly the non-deformed case $z\to 0$ and, afterwards, by deforming the $v_i(q,p)$ functions under the constraint that the Casimir ${C}_z$ has to take a constant value.
With this result at hand, the corresponding deformed vector fields ${\bf X}_{z,i}$ can be computed by imposing the relationship (\ref{contract2}) and the final result is summarized in the following statement.
\begin{proposition}
\label{proposition1} (i) The Hamiltonian functions defined by
\be
h_{z,1}:=\frac 12 x^2 , \qquad
h_{z,2}:= - \frac 12\shc (z x^2)\, x y , \qquad
h_{z,3}:= \frac 12 \left(\! \shc (z x^2) \, y^2 + \frac 1{ \shc (zx^2)}\,
\frac{c}{x^2} \right) ,
\label{gd}
\ee
close the Poisson brackets (\ref{gb}) with respect to the symplectic form $\omega={\rm d}x\wedge {\rm d}y$ on $\mathbb R^2_{x\ne 0}$, namely
\begin{equation}\label{gb2}
\begin{gathered}
\{h_{z,1},h_{z,2}\}_\omega=-\shc (2z h_{z,1} )h_{z,1},\qquad
\{h_{z,1},h_{z,3}\}_\omega=-2 h_{z,2},\\[2pt]
\{h_{z,2},h_{z,3}\}_\omega= - \cosh(2 z h_{z,1}) h_{z,3},
\end{gathered}
\end{equation}
where $\shc(x)$ is defined in (\ref{shc}).
Relations~\eqref{gb2} define the deformed Poisson algebra $C^\infty({\cal H}_{z,\omega}^{\rm {MP}*})$.
\noindent
(ii) The vector fields ${\bf X}_{z,i}$ corresponding to $h_{z,i}$ read
\begin{equation}
\begin{gathered}
{\bf X}_{z,1} = -x\frac{\partial}{\partial y},\qquad {\bf X}_{z,2}=\left(\cosh(zx^{2})-\frac 12 \shc(zx^{2})\right)y\frac{\partial}{\partial y}-\frac 12 \shc(zx^{2})\,x\frac{\partial}{\partial x}, \\
{\bf X}_{z,3} = \shc(zx^{2})\,y\frac{\partial}{\partial x}+\left[\frac{c}{x^{3}}\, \frac{\cosh(zx^{2})}{\shc^{2}(zx^{2})} +\frac{\shc(zx^{2})-\cosh(zx^{2})}{x}\,y^{2}\right]\frac{\partial}{\partial y},\nonumber
\end{gathered}
\end{equation}
which satisfy
\begin{equation}
\begin{gathered}
\left[{\bf X}_{z,1},{\bf X}_{z,2}\right]=\cosh (z x^2) \, {\bf X}_{z,1},\qquad [{\bf X}_{z,1},{\bf X}_{z,3}]=2 {\bf X}_{z,2}, \\[2pt]
[{\bf X}_{z,2},{\bf X}_{z,3}]=\cosh (z x^2) \, {\bf X}_{z,3}+ z^2\left( c + x^2 y^2\, \shc^2 (z x^2) \right) {\bf X}_{z,1}.
\label{com2}
\end{gathered}
\end{equation}
\end{proposition}
\smallskip
Since $\lim_{z\to 0}\shc(z x^2)=1$ and $\lim_{z\to 0}\cosh(z x^2)=1$, it can directly be checked that
all the classical limits (\ref{zac}), (\ref{zad}) and (\ref{zae}) are fulfilled. As expected, the Lie derivative of $\omega$ with respect to each ${\bf X}_{z,i}$ vanishes.
At this stage, it is important to realize that, albeit (\ref{gb2}) are genuine Poisson brackets defining the Poisson algebra $C^\infty({\cal H}_{z,\omega}^{\rm {MP}*})$, the commutators (\ref{com2}) show that ${\bf X}_{z,i}$ do not span a new Vessiot--Guldberg Lie algebra; in fact, the commutators give rise to linear combinations of the vector fields ${\bf X}_{z,i}$ with coefficients that are functions depending on the coordinates and the deformation parameter.
Consequently, proposition~\ref{proposition1} leads to a deformation of the initial Lie system (\ref{MP}) and of the LH one (\ref{hMP}) defined by
\be
{\bf X}_z:={\bf X}_{z,3}+\Omega^2(t){\bf X}_{z,1},\qquad h_z:=h_{z,3}+\Omega^2(t)h_{z,1}.
\label{MPz}
\ee
Thus we obtain the following $z$-parametric system of differential equations that generalizes (\ref{FirstLie}):
\bea
&&\frac{\dd x}{\dd t}=\shc (z x^2)\, y,\nonumber\\[2pt]
&& \frac{\dd y}{\dd t}=-\Omega^2(t)x+
\frac{c}{x^3} \, \frac{\cosh (z x^2) }{ \shc^2(z x^2) }+ \frac{\shc (z x^2)- \cosh (z x^2)}{x} \, y^2 .
\label{FirstLie2}
\eea
From the first equation, we can write $$y=\frac 1{ \shc (z x^2) }\, \frac{\dd x }{\dd t},
$$
and by substituting this expression into the second equation in (\ref{FirstLie2}), we obtain
a deformation of the MP equation~\eqref{mp} in the form
\be
\frac{\dd^2 x}{\dd t^2} + \left(\frac{1}{x}- \frac{z x}{ \tanh (z x^2)} \right) \biggl(\frac{\dd x}{\dd t} \biggr)^2 =-\Omega^2(t) \, x \shc (z x^2)+ \,\frac{c\,z }{x \tanh (z x^2)} .
\nonumber
\ee
Note that this really is a deformation of the MP equation in the sense that the limit $z\to 0$ recovers the standard one (\ref{mp}).
\subsect{Constants of motion for the deformed Milne--Pinney system}
An essential feature of the formalism here presented is the fact that $t$-independent constants of motion for the deformed system ${\bf X}_z$ (\ref{MPz}) can be deduced by using the coalgebra structure of $C^\infty( {\cal H}_{z,\omega}^{\rm {MP}*})$. Thus we start with the Poisson--Hopf algebra $C^\infty( {\cal H}_{z,\omega}^{\rm {MP}*} )$ with deformed coproduct $\Delta_z$ given by (\ref{ga}) and, following section 2.4~\cite{BCHLS13Ham}, we consider the Poisson algebra morphisms
\be
D_z: C^\infty( {\cal H}_{z,\omega}^{\rm {MP}*} )\rightarrow C^\infty( \mathbb R^2_{x\ne 0}),\quad D_z^{(2)} : C^\infty( {\cal H}_{z,\omega}^{\rm {MP}*} )\otimes C^\infty( {\cal H}_{z,\omega}^{\rm {MP}*} )\rightarrow C^\infty(\mathbb R^2_{x\ne 0})\otimes C^\infty(\mathbb R^2_{x\ne 0}),
\nonumber
\ee
which are defined by
\bea
&& D_z( v_i)= h_{z,i}(x_1,y_1):= h_{z,i}^{(1)} , \quad i=1,2,3, \nonumber\\
&& D_z^{(2)} \left( {\Delta}_z(v_1) \right) = h_{z,1}(x_1,y_1)+h_{z,1}(x_2,y_2):= h_{z,1}^{(2)} \, , \nonumber \\
&&
D_z^{(2)} \left( {\Delta}_z(v_j) \right) = h_{z,j}(x_1,y_1) {\rm e}^{2 z h_{z,1}(x_2,y_2)} + {\rm e}^{-2 z h_{z,1}(x_1,y_1)} h_{z,j}(x_2,y_2):= h_{z,j}^{(2)} \, ,\quad j= 2,3,
\nonumber
\eea
where $h_{z,i}$ are the Hamiltonian functions (\ref{gd}), so fulfilling (\ref{gb2}). Hence (see \cite{chains})
\bea
&& h_{z,1}^{(2)} = \frac 12(x_1^2+x_2^2) ,\nonumber\\[2pt]
&& h_{z,2}^{(2)} = - \frac 12 \left( \! {\shc (z x_1^2)} \, x_1 y_1 {\rm e}^{z x_2^2} + {\rm e}^{- z x_1^2} {\shc (z x_2^2)} \, x_2 y_2 \right) , \nonumber\\[2pt]
&& h_{z,3}^{(2)} =\frac 12 \left( \! {\shc (z x_1^2)}\, y_1^2 +
\frac{c}{ x_1^2 \shc (zx_1^2)} \right) {\rm e}^{z x_2^2} + \frac 12\, {\rm e}^{- z x_1^2} \left( \! {\shc (z x_2^2)} \, y_2^2 +
\frac{c}{ x_2^2\, \shc (zx_2^2)} \right).
\nonumber
\eea
Recall that, by construction, the functions $h_{z,i}^{(2)}$ fulfill the Poisson brackets (\ref{gb2}).
The
$t$-independent constants of motion are then obtained through
$$
F_z= D_z(C_z),\qquad F_z^{(2)}= D_z^{(2)} \left( {\Delta_z}(C_z) \right),
$$
where $C_z$ is the Casimir (\ref{gc}); these are
\bea
&& \!\!\!\! \!\!\!\! \!\!\!\! \! F_z= \shc\bigl(2 z h_{z,1}^{(1)} \bigr) h_{z,1}^{(1)} h_{z,3}^{(1)} - \bigl( h_{z,2}^{(1)} \bigr)^2 = \frac c 4 \, ,\nonumber\\[2pt]
&& \!\!\!\! \!\!\!\! \!\!\!\! \! F_z^{(2)}= \shc\bigl(2 z h_{z,1}^{(2)} \bigr) h_{z,1}^{(2)} h_{z,3}^{(2)} - \bigl( h_{z,2}^{(2)} \bigr)^2 \label{amz}\\[2pt]
&& =
\frac 14 \left[ \shc(z x_1^2) \shc(z x_2^2) \, ({x_1}{y_2} -{x_2} {y_1})^2 +c\, \frac{ \shc^2\bigl( z(x_1^2+x_2^2) \bigr)}{\shc( z x_1^2) \shc( z x_2^2)}\, \frac{(x_1^2+x_2^2)^2}{x_1^2 x_2^2} \right] {\rm e}^{- z x_1^2} {\rm e}^{z x_2^2} ,
\nonumber
\eea
so providing the corresponding deformed Ray--Reid invariant, being (\ref{am}) its non-deformed counterpart with $z=0$. Notice that this invariant is related to the so-called `universal constant of the motion' coming from $U_z(\mathfrak{sl}(2))$ and given in~\cite{letterBH}.
As in (\ref{an}), other equivalent constants of motion can be deduced from $ F_z^{(2)}$ by permutation of the variables.
\subsect{A new oscillator system with position-dependent mass}
If we set $p:= y$, the $t$-dependent Hamiltonian $h_z$ in (\ref{MPz}) can be written, through (\ref{gd}), as:
\be
h_z= T_z+U_z= \frac 12 {\shc (z x^2)}\, p^2+ \frac 12\Omega^2(t) x^2 +
\frac{ c}{2 x^2 {\shc (z x^2)} } \, ,
\nonumber
\ee
so deforming $h$ given in (\ref{ham}). The corresponding Hamilton equations are just (\ref{FirstLie2}).
It is worth mentioning that $h_z$ can be interpreted naturally within the framework of
position-dependent mass oscillators (see~\cite{CrNN07, Quesne07, CrR09, BurgosAnnPh11, Ran14Jmp, GhoshRoy15, MustJpa15, Quesne15Jmp} and references therein). The above Hamiltonian naturally suggests the definition of a position-dependent mass function in the form
\be
m_z(x) :=\frac 1{ \shc (z x^2) }= \frac{z x^2}{ \sinh (z x^2)} \, ,\qquad \lim_{z\to 0} m_z(x) =1, \qquad \lim_{x\to \pm\infty } m_z(x) =0.
\label{masa}
\ee
Then $h_z$ can be rewritten as
\be
h_z= \frac {p^2}{2m_z(x)} + \frac 12m_z(x) \Omega^2(t)\left[ x^2 \shc (z x^2) \right] +
\frac{ c}{2m_z(x)} \left[ \frac{1}{x ^2\shc^2(z x^2)} \right] .
\nonumber
\ee
Thus the Hamiltonian $h_z$ can be regarded as a system corresponding to a particle with position-dependent mass $m_z(x)$ under a deformed oscillator potential $U_{z,{\rm osc}}(x)$ with time-dependent frequency $\Omega(t)$ and a deformed Rosochatius--Winternitz potential $U_{z,{\rm RW}}(x)$ given by
\bea
&&
U_{z,{\rm osc}}(x): = x^2 \shc (z x^2)= \frac{\sinh (z x^2)}{z} \, ,\label{oscz}\\
&&
U_{z,{\rm RW}}(x):=\frac{1}{x ^2\shc^2(z x^2)}= \left( \frac{z x}{\sinh (z x^2)}\right)^2,
\nonumber
\eea
such that
\bea
&& \lim_{z\to 0} U_{z,{\rm osc}}(x) =x^2, \qquad\ \lim_{x\to \pm\infty }U_{z,{\rm osc}}(x) =+\infty, \nonumber\\[2pt]
&& \lim_{z\to 0} U_{z,{\rm RW}}(x)=\frac 1{x^2}, \qquad\ \lim_{x\to \pm\infty } U_{z,{\rm RW}}(x) =0 .
\nonumber
\eea
The deformed mass and the oscillator potential functions are represented in figures 1 and 2.}
The Hamilton equations (\ref{FirstLie2}) can easily be expressed in terms of $m_z(x)$ as
\bea
&&\!\!\!\!\!\!\!\!\!\! \dot x = \frac{\partial h^{\rm MP}_z}{\partial p}= \frac p{m_z(x)} ,\nonumber\\
&&\!\!\!\!\!\!\!\!\!\! \dot p= - \frac{\partial h^{\rm MP}_z}{\partial x} = -m_z(x) \Omega^2(t) \, x \shc (z x^2) + \frac{c}{m_z(x)} \, \frac {\cosh (z x^2)} {x^3 \shc^3 (z x^2)} +p^2 \frac{ m^\prime_z(x)}{2 m^2_z(x)},
\nonumber
\eea
and the constant of the motion (\ref{amz}) turns out to be
\be
F_z^{(2)}= \frac 14 \left[ \frac {({x_1}{p_2} -{x_2} {p_1})^2 }{m_z(x_1)m_z(x_2) } +c\, m_z(x_1)m_z(x_2) { \shc^2\bigl( z(x_1^2+x_2^2) \bigr)} \, \frac{(x_1^2+x_2^2)^2}{x_1^2 x_2^2} \right] {\rm e}^{- z x_1^2} {\rm e}^{z x_2^2}.
\nonumber
\ee
\begin{figure}[t]
\begin{center}
\includegraphics[height=6.0cm]{fig1.pdf}
\caption{\small The position-dependent mass (\ref{masa}) for different values of the deformation parameter $z$.}
\label{pdm}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[height=6.0cm]{fig2.pdf}
\caption{\small The deformed oscillator potential (\ref{oscz}) for different values of the deformation parameter $z$.}
\label{osc}
\end{center}
\end{figure}
\sect{Deformed complex Riccati equation}
In this section we consider the complex Riccati equation given by
\begin{equation}
\frac{{\rm d} z}{{\rm d} t}=b_1(t)+b_2(t)z+b_3(t)z^2,\qquad z\in\mathbb{C},
\label{da}
\end{equation}
where $b_i(t)$ are arbitrary $t$-dependent real coefficients. We recall that (\ref{da}) is related to certain planar Riccati equations~\cite{Eg07,Wi08} and that several mathematical and physical applications can be found in~\cite{Ju97,FMR10,Or12,Sc12}.
By writing $z= \xx+i \yy$, we find that (\ref{da}) gives rise to a system of the type (\ref{system}), namely
\begin{equation}
\frac{{\rm d} \xx}{{\rm d} t}=b_1(t)+b_2(t)\xx+b_3(t)(\xx^2- \yy^2),\qquad \frac{{\rm d} \yy}{{\rm d} t}=b_2(t)\yy+2b_3(t)\xx \yy .
\label{db}
\end{equation}
Thus the associated $t$-dependent vector field reads
\be
{\bf X}=b_1(t){\bf X}_1+b_2(t){\bf X}_2+b_3(t){\bf X}_3,
\label{dc}
\ee
where
\begin{equation}
{\bf X}_1= \frac{\partial}{\partial \xx},\qquad {\bf X}_2= \xx\frac{\partial}{\partial \xx}+\yy\frac{\partial}{\partial \yy} ,\qquad {\bf X}_3= (\xx^2- \yy^2)\frac{\partial}{\partial \xx}+2\xx \yy\frac{\partial}{\partial \yy} ,
\label{vectRiccati2}
\end{equation}
span a Vessiot--Guldberg Lie algebra $V^{\rm CR}\simeq \mathfrak{sl}(2)$ with the same commutation relations (\ref{aa}). It has already be proven that the system ${\bf X}$ is a LH one belonging to the class {\rm P}$_2$~\cite{ BBHLS, BHLS} and that their vector fields span a regular distribution on $\mathbb{R}^2_{\yy\neq 0}$. The symplectic form, coming from (\ref{der}), and the corresponding Hamiltonian functions (\ref{contract}) turn out to be
\be
\omega=\frac{\dd \xx\wedge \dd \yy}{\yy^2},\qquad h_1= -\frac 1{\yy},\qquad h_2= -\frac \xx \yy , \qquad h_3=- \frac{\xx^2+\yy^2}{\yy} ,
\label{de}
\ee
which fulfill the commutation rules (\ref{brack}) so defining a LH algebra ${\cal H}_{\omega}^{\rm {CR}}$.
A $t$-dependent Hamiltonian associated with ${\bf X}$ reads
\be
h=b_1(t)h_1+b_2(t)h_2+b_3(t)h_3 .
\label{de2}
\ee
In this case, the constants of the motion (\ref{bc}) are found to be $F=1$ and~\cite{BHLS}
\be
F^{(2)}= \frac{(\xx_1-\xx_2)^2+(\yy_1+ \yy_2)^2}{\yy_1 \yy_2} \, .
\label{df}
\ee
As commented above, the Riccati system (\ref{db}) is locally diffeomorphic to the MP equations (\ref{FirstLie}) with $c>0$, both belonging to the same class {\rm P}$_2$~\cite{BBHLS}. Explicitly, the change of variables
\be
x=\pm \frac{c^{1/4}}{ \sqrt{|v| }},\qquad y=\mp \frac{c^{1/4} \, u}{ \sqrt{|v|}},\qquad u=-\frac{y}x,\qquad |v|= \frac{c^{1/2}}{x^2},\qquad c>0,
\label{dg}
\ee
map, in this order, the vector fields (\ref{FirstLieA}) on $\mathbb{R}^2_{x\neq 0}$, the symplectic form $\omega={\rm d}x\wedge {\rm d}y$, Hamiltonian functions (\ref{ham2}) and the constant of motion (\ref{am})
onto the vector fields (\ref{vectRiccati2}) on $\mathbb{R}^2_{\yy\neq 0}$, (\ref{de}) and (\ref{df}) (up to a multiplicative constant $\pm\frac 12 c^{1/2}$).
To obtain the corresponding (non-standard) deformation of the complex Riccati system (\ref{db}), the very same change of variables (\ref{dg}) can be considered since, in our approach, the symplectic form (\ref{de}) is kept non-deformed. Thus, by starting from proposition~\ref{proposition1} and applying (\ref{dg}) (with $c=4$ for simplicity), we get the following result.
\begin{proposition}
\label{proposition2}
(i) The Hamiltonian functions given by
\be
h_{z,1}=- \frac 1 {\yy} \, , \qquad
h_{z,2}= - \shc (2z /\yy) \, \frac \xx\yy \, , \qquad
h_{z,3}=- \frac{ \shc^2 (2z /\yy) \, \xx^2 + \yy^2 }{\shc (2z /\yy)\, \yy} \, ,
\nonumber
\ee
fulfill the commutation rules (\ref{gb2}) with respect to the Poisson bracket induced by the symplectic form $\omega$ (\ref{de}) defining the deformed Poisson algebra $C^\infty({\cal H}_{z,\omega}^{\rm {CR}*})$.
\noindent
(ii) The corresponding vector fields ${\bf X}_{z,i}$ read
\begin{eqnarray}
&& {\bf X}_{z,1} = \frac{\partial}{\partial \xx},\qquad {\bf X}_{z,2}=\xx \cosh(2z/ \yy)\frac{\partial}{\partial \xx} + \yy \shc(2z/ \yy) \frac{\partial}{\partial \yy} , \nonumber\\
&& {\bf X}_{z,3} = \left(\xx^2 -\frac{\yy^2}{ \shc^2(2z/ \yy)} \right) \cosh(2z /\yy)\frac{\partial}{\partial \xx}+2\xx\yy \shc(2z/ \yy)\frac{\partial}{\partial \yy} ,\nonumber
\end{eqnarray}
which satisfy
\bea
&& [{\bf X}_{z,1},{\bf X}_{z,2}]=\cosh (2z/ \yy) \, {\bf X}_{z,1},\qquad [{\bf X}_{z,1},{\bf X}_{z,3}]=2 {\bf X}_{z,2},\nonumber\\[2pt]
&& [{\bf X}_{z,2},{\bf X}_{z,3}]=\cosh (2z/ \yy) \, {\bf X}_{z,3}+ 4 z^2\left( 1 + \frac{\xx^2}{\yy^2} \shc^2 (2z/ \yy) \right) {\bf X}_{z,1}.
\nonumber
\eea
\end{proposition}
Next the deformed counterpart of the Riccati Lie system (\ref{dc}) and of the LH one (\ref{de2}) is defined by
\be
{\bf X}_z:=b_1(t){\bf X}_{z,1}+b_2(t){\bf X}_{z,2}+b_3(t){\bf X}_{z,3},\qquad h_z:=b_1(t)h_{z,1}+b_2(t)h_{z,2}+b_3(t)h_{z,3} .
\label{dl}
\ee
And the $t$-independent constants of motion turn out to be $F_z=1$ and
\be
F_z^{(2)}=\left( \shc (2z/ \yy_1) \shc (2z/ \yy_2) \frac{(\xx_1-\xx_2)^2 }{\yy_1 \yy_2} + \frac{\shc^2 (2z/ \yy_1+ 2z/ \yy_2) }{ \shc (2z/ \yy_1) \shc (2z/ \yy_2) } \frac{ (\yy_1+ \yy_2)^2}{\yy_1 \yy_2} \right) \eee^{2z/\yy_1} \eee^{-2z/\yy_2} \, .
\nonumber
\ee
Therefore the deformation of the system (\ref{db}), defined by ${\bf X}_z$ (\ref{dl}), reads
\bea
&& \frac{{\rm d} \xx}{{\rm d} t}=b_1(t)+b_2(t) \xx \cosh(2z/ \yy) +b_3(t) \left(\xx^2 -\frac{\yy^2}{ \shc^2(2z/ \yy)} \right) \cosh(2z /\yy), \nonumber\\[2pt]
&& \frac{{\rm d} \yy}{{\rm d} t}=b_2(t) \yy \shc(2z/ \yy)+2b_3(t) \xx\yy \shc(2z/ \yy) .
\nonumber
\eea
\sect{Deformed coupled Riccati equations}
As a last application, let us consider two coupled Riccati equations given by
\cite{Mariton}
\begin{equation}
\frac{{\rm d}\xx}{{\rm d}t}=a_0(t)+a_1(t)\xx+a_2(t)\xx^2,\qquad \frac{{\rm d}\yy}{{\rm d}t}=a_0(t)+a_1(t)\yy+a_2(t)\yy^2,
\label{ea}
\end{equation}
constituting a particular case of the systems of Riccati equations studied in~\cite{BCHLS13Ham,CGLS}.
Clearly, the system (\ref{ea}) is a Lie system associated with a $t$-dependent vector field
\be
{\bf X}=a_0(t){\bf X}_1+a_1(t){\bf X}_2+a_2(t){\bf X}_3,
\label{eb}
\ee
where
\begin{equation}
{\bf X}_1= \frac{\partial}{\partial \xx}+ \frac{\partial}{\partial \yy}, \qquad {\bf X}_2= \xx\frac{\partial}{\partial \xx}+\yy\frac{\partial}{\partial \yy} ,\qquad {\bf X}_3= \xx^2\frac{\partial}{\partial \xx}+\yy^2\frac{\partial}{\partial \yy} ,
\label{ec}
\end{equation}
close on the commutation rules (\ref{aa}), so spanning a Vessiot--Guldberg Lie algebra $V^{\rm 2R}\simeq \mathfrak{sl}(2)$.
Furthermore, ${\bf X}$ is a LH system which belongs to the class {\rm I}$_4$~\cite{ BBHLS, BHLS} restricted to $\mathbb{R}^2_{\xx\neq \yy}$.
The symplectic form and Hamiltonian functions for ${\bf X}_1,{\bf X}_2, {\bf X}_3$ read
\be
\omega=\frac{\dd \xx\wedge \dd \yy}{(\xx-\yy)^2} ,\qquad h_1= \frac 1{\xx-\yy},\qquad h_2= \frac 12\left( \frac {\xx+\yy} {\xx-\yy}\right) , \qquad h_3= \frac{\xx \yy}{\xx-\yy}.
\label{ee}
\ee
The functions $h_1,h_2,h_3$ satisfy the commutation rules (\ref{brack}), thus spanning a LH algebra ${\cal H}_{\omega}^{\rm {2R}}$.
Hence, the $t$-dependent Hamiltonian associated with ${\bf X}$ is given by
\be
h=a_0(t)h_1+a_1(t)h_2+a_2(t)h_3 .
\label{ee2}
\ee
The constants of the motion (\ref{bc}) are now $F=-1/4$ and~\cite{BHLS}
\be
F^{(2)}=- \frac{ (\xx_2- \yy_1 ) (\xx_1- \yy_2 )} { (\xx_1- \yy_1 ) (\xx_2- \yy_2 )} \, .
\label{ef}
\ee
The LH system (\ref{ea}) is locally diffeomorphic to the MP equations (\ref{FirstLie}) but now with $c<0$~\cite{BBHLS}.
Such a diffeomorphism is achieved through the change of variables given by
\bea
&& x=\pm \frac{ (4|c|)^{1/4}}{ \sqrt{|\xx-\yy| }},\qquad y=\mp \frac { (4|c|)^{1/4} (\xx+\yy)}{2 \sqrt{|\xx-\yy| }},\qquad c<0, \nonumber\\[2pt]
&& \xx=\pm \frac{ |c|^{1/2}}{x^2}-\frac y x ,\qquad \yy=\mp \frac{ |c|^{1/2}}{x^2}-\frac y x ,
\label{eg}
\eea
which map the MP vector fields (\ref{FirstLieA}) with domain $\mathbb{R}^2_{x\neq 0}$, symplectic form $\omega={\rm d}x\wedge {\rm d}y$, Hamiltonian functions (\ref{ham2}) and constant of motion (\ref{am})
onto (\ref{ec}) with domain $\mathbb{R}^2_{\xx\neq \yy}$, (\ref{ee}) and (\ref{ef}) (up to a multiplicative constant $\pm |c|^{1/2}$), respectively.
As in the previous section, the (non-standard) deformation of the coupled Riccati system (\ref{ea}) is obtained by starting again from proposition~\ref{proposition1} and now applying the change of variables (\ref{eg}) with $c=-1$ (without loss of generality) finding the following result.
\begin{proposition}
\label{proposition3}
(i) The Hamiltonian functions given by
\bea
h_{z,1}= \frac 1 {\xx-\yy} \, , \qquad
h_{z,2}= \frac 12 \shc \bigl(\tfrac{2z} {\xx-\yy}\bigr) \biggl( \frac{ \xx+\yy}{ \xx-\yy} \biggr) , \qquad h_{z,3}= \frac{ \shc^2 \bigl(\frac{2z} {\xx-\yy}\bigr)( \xx+\yy)^2 -( \xx-\yy)^2 }{ 4 \shc \bigl(\frac{2z} {\xx-\yy}\bigr) (\xx-\yy) } \, ,
\nonumber
\eea
satisfy the commutation relations (\ref{gb2}) with respect to the symplectic form $\omega$ (\ref{ee}) and define the deformed Poisson algebra $C^\infty({\cal H}_{z,\omega}^{\rm {2R}*})$.\\
(ii) Their corresponding deformed vector fields turn out to be
\begin{eqnarray}
&& {\bf X}_{z,1} = \frac{\partial}{\partial \xx}+ \frac{\partial}{\partial \yy},\nonumber\\[2pt]
&& {\bf X}_{z,2}=\frac 12 (\xx+\yy) \cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) \left( \frac{\partial}{\partial \xx}+ \frac{\partial}{\partial \yy} \right)
+\frac 12 (\xx-\yy) \shc \bigl(\tfrac{2z} {\xx-\yy}\bigr) \left( \frac{\partial}{\partial \xx}- \frac{\partial}{\partial \yy} \right), \nonumber\\
&& {\bf X}_{z,3} = \frac 14 \left[ (\xx+\yy)^2+\frac{ (\xx-\yy)^2 }{ \shc^2 \bigl(\tfrac{2z} {\xx-\yy}\bigr)} \right] \cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) \left( \frac{\partial}{\partial \xx}+ \frac{\partial}{\partial \yy} \right)
+\frac 12 (\xx^2-\yy^2) \shc \bigl(\tfrac{2z} {\xx-\yy}\bigr) \left( \frac{\partial}{\partial \xx}- \frac{\partial}{\partial \yy} \right) ,\nonumber
\end{eqnarray}
which fulfill
\bea
&& [{\bf X}_{z,1},{\bf X}_{z,2}]=\cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) {\bf X}_{z,1},\qquad [{\bf X}_{z,1},{\bf X}_{z,3}]=2 {\bf X}_{z,2},\nonumber\\[2pt]
&& [{\bf X}_{z,2},{\bf X}_{z,3}]=\cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) {\bf X}_{z,3}- z^2\left[ 1 - \biggl(\frac{\xx+\yy}{\xx-\yy} \biggr)^2\! \shc^2 \bigl(\tfrac{2z} {\xx-\yy}\bigr) \right] {\bf X}_{z,1}.
\nonumber
\eea
\end{proposition}
The deformed counterpart of the coupled Ricatti Lie system (\ref{eb}) and of the LH one (\ref{ee2}) is defined by
\be
{\bf X}_z:=a_0(t){\bf X}_{z,1}+a_1(t){\bf X}_{z,2}+a_2(t){\bf X}_{z,3},\qquad h_z:=a_0(t)h_{z,1}+a_1(t)h_{z,2}+a_2(t)h_{z,3} .
\label{el}
\ee
And the $t$-independent constants of motion are $F_z=-1/4$ and
\bea
&& F_z^{(2)}=\frac{ \eee^{-\frac{2z}{\xx_1-\yy_1}} \eee^{\frac{2z}{\xx_2-\yy_2}} }{4(\xx_1-\yy_1)(\xx_2-\yy_2)} \left[ \shc \bigl(\tfrac{2z} {\xx_1-\yy_1}\bigr) \shc \bigl(\tfrac{2z} {\xx_2-\yy_2}\bigr) (\xx_1-\xx_2+\yy_1-\yy_2)^2 \right. \nonumber\\[2pt]
&& \qquad \left. - \left( \frac{ \eee^{\frac{2z}{\xx_1-\yy_1}} (\xx_1-\yy_1) }{ \shc \bigl(\tfrac{2z} {\xx_1-\yy_1}\bigr) } + \frac{ \eee^{-\frac{2z}{\xx_2-\yy_2}} (\xx_2-\yy_2) }{ \shc \bigl(\tfrac{2z} {\xx_2-\yy_2}\bigr) } \right) \shc \bigl(\tfrac{2z} {\xx_1-\yy_1}+\tfrac{2z} {\xx_2-\yy_2}\bigr) (\xx_1+\xx_2-\yy_1-\yy_2) \right] .
\label{em}\nonumber
\eea
Therefore, the deformation of the system (\ref{ea}) is determined by ${\bf X}_z$ (\ref{el}). Note that the resulting system presents a strong interaction amongst the variables $(u,v)$ through $z$,
which goes far beyond the initial (naive) coupling corresponding to set the same $t$-dependent parameters $a_i(t)$ in both one-dimensional Riccati equations; namely
\bea
&& \frac{{\rm d} \xx}{{\rm d} t}=a_0(t)+\frac{a_1(t)}{2}\left[ (\xx+\yy) \cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) + (\xx-\yy) \shc \bigl(\tfrac{2z} {\xx-\yy}\bigr) \right]
\nonumber\\[2pt]
&&\qquad\quad +\frac{a_2(t)}4 \left[ \left( (\xx+\yy)^2+\frac{ (\xx-\yy)^2 }{ \shc^2 \bigl(\tfrac{2z} {\xx-\yy}\bigr)} \right) \cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) + 2 (\xx^2-\yy^2) \shc \bigl(\tfrac{2z} {\xx-\yy}\bigr) \right] , \nonumber\\[2pt]
&& \frac{{\rm d} \yy}{{\rm d} t}=a_0(t)+\frac{a_1(t)}{2}\left[ (\xx+\yy) \cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) - (\xx-\yy) \shc \bigl(\tfrac{2z} {\xx-\yy}\bigr) \right]
\nonumber\\[2pt]
&&\qquad\quad +\frac{a_2(t)}4 \left[ \left( (\xx+\yy)^2+\frac{ (\xx-\yy)^2 }{ \shc^2 \bigl(\tfrac{2z} {\xx-\yy}\bigr)} \right) \cosh \bigl(\tfrac{2z} {\xx-\yy}\bigr) - 2 (\xx^2-\yy^2) \shc \bigl(\tfrac{2z} {\xx-\yy}\bigr) \right] .
\label{en}\nonumber
\eea
\section{Conclusions}
In this work, the notion of Poisson--Hopf deformation of LH systems
has been proposed. This framework differs radically from
other approaches to the LH systems theory~\cite{PW,Dissertations, CLS13, CGL10, BCHLS13Ham}, as our resulting deformations do not formally correspond to LH systems, but to an extended notion of them that requires a (non-trivial) Hopf structure and is related with the non-deformed LH system by means of a limiting process in which the deformation parameter $z$ vanishes. Moreover, the introduction of Poisson--Hopf structures allows for the generalization of the type of systems under inspection, since the finite-dimensional Vessiot--Guldberg Lie algebra is replaced by an involutive distribution in the Stefan--Sussman sense.
This framework has been illustrated via the Poisson analogue of the non-standard quantum deformation of
$\mathfrak{sl}(2)$, and deformations of physically relevant LH systems such as the oscillator system, as well as the complex and coupled Riccati equations have been presented. In the former case the deformation can be interpreted as the transformation of the initial system into a new one
possessing a position-dependent mass, hence suggesting an alternative approach to the latter type of systems that presents an ample potential of applications. In particular, the Schr\"odinger problem for position-dependent mass Hamiltonians is directly connected with the quantum dynamics of charge carriers in semiconductor heterostructures and nanostructures (see, for instance,~\cite{Roos, Bastard, qDWW}). In this respect, it is worth remarking that the standard or Drinfel'd--Jimbo deformation of
$\mathfrak{sl}(2)$ would not lead to an oscillator with a position-dependent mass since, in that case, the deformation function would be $\!\shc(z q p)$ instead of $\!\shc(z q^2)$; this can clearly be seen in the corresponding symplectic realization given in~\cite{AR}. This fact explains that, in order to illustrate our approach, we have chosen the non-standard deformation of $\mathfrak{sl}(2)$ due to its physical applications.
In spite of this, the Drinfel'd--Jimbo deformation would provide another deformation for the MP and Riccati equations which would be
non-equivalent to the ones here studied.
There are still many questions to be analyzed in detail. Since the formalism here presented is applicable in a more wide context, with other types of Hopf algebra deformations and dealing with higher-dimensional Vessiot--Guldberg Lie algebras, this would lead to a richer spectrum of properties for the deformed systems that deserve further investigation. For instance, the deformed LH systems studied in this work are such that the distribution spanned by the deformed
vector fields is the same as the initial one. As it has been observed previously, this constraint could not be preserved for generic Poisson--Hopf algebra deformations of LH systems defined
on more general manifolds.
An important question to be addressed is whether this approach can provide an effective procedure to derive a deformed analogue of superposition principles for deformed LH systems. Also, it would be interesting to know
whether such a description is simultaneously applicable to the various non-equivalent deformations, like an extrapolation of the notion of Lie algebra contraction to Lie systems. Another open problem worthy to be considered is the possibility of getting a unified description of such systems in terms of a certain amount of fixed `elementary' systems, thus implying a first rough systematization of LH-related systems from a more general perspective than that of finite-dimensional Lie algebras. Work in these directions is currently in progress.
\section*{Appendix. The hyperbolic sinc function}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
The hyperbolic counterpart of the well-known sinc function is defined by
\be
\shc( x):= \left\{
\begin{array}{ll}
\frac{\sinh (x)}{x}, &\mbox{for}\ x\ne 0, \\
1,&\mbox{for}\ x=0.
\end{array}
\right.
\nonumber
\ee
The power series around $x=0$ reads
\be
\shc( x) = \sum_{n=0}^\infty \frac{x^{2n}}{(2n+1)!} \, .
\nonumber
\ee
And its derivative is given by
\be
\frac{\rm d}{{\rm d}x} \shc(x)= \frac{\cosh (x)}{x}- \frac{\sinh (x)}{x^2} = \frac {\cosh( x) - \shc(x)}{x} \, .
\nonumber
\ee
Hence the behaviour of $\shc( x)$ and its derivative remind that of the hyperbolic cosine and sine functions, respectively.
We represent them in figure~\ref{fig3}.
\begin{figure}[t]
\begin{center}
\includegraphics[height=6.0cm]{fig3.pdf}
\caption{\small The hyperbolic sinc function versus the hyperbolic cosine function and the derivative of the former versus the hyperbolic sine function.}
\label{fig3}
\end{center}
\end{figure}
A novel relationship of the $\shc$ function (and also of the $\sinc$ one) with Lie systems can be established by considering the following second-order ordinary differential equation
\be
t\, \frac{\dd^2 x}{\dd t^2}+ 2 \, \frac{\dd x}{\dd t}-\eta^2 t\, x =0 ,
\label{diff}
\ee
where $\eta$ is a non-zero real parameter. Its general solution can be written as
\be
x(t)=A \shc(\eta t)+B\, \frac{\cosh(\eta t)}{t} \, ,\qquad A,B\in \mathbb{R}.
\nonumber
\ee
Notice that if we set $\eta=i \lambda$ with $\lambda\in \mathbb{R}^\ast$ we recover the known result for the sinc function:
\be
t\, \frac{\dd^2 x}{\dd t^2}+ 2 \, \frac{\dd x}{\dd t}+\lambda^2 t\, x =0 , \qquad x(t)=A \sinc(\lambda t)+B\, \frac{\cos(\lambda t)}{t} \, .
\label{Ad}
\ee
Next the differential equation (\ref{diff}) can be written as a system of two first-order differential equations by setting $y={\dd x}/{\dd t}$, namely
\be
\frac{\dd x}{\dd t}=y,\qquad \frac{\dd y}{\dd t}=- \frac{2}{t}\, y+ \eta^2 x.
\label{eqLie}
\nonumber
\ee
Remarkably enough, these equations determine a Lie system with associated $t$-dependent vector field
\be
{\bf X}=- \frac 2 t\, {\bf X}_1+ {\bf X}_2 + \eta^2 {\bf X}_3,
\label{Aa}
\ee
where
\begin{equation}
{\bf X}_1=y\frac{\partial}{\partial y},\qquad {\bf X}_2= y\frac{\partial}{\partial x} ,\qquad {\bf X}_3= x\frac{\partial}{\partial y} ,\qquad {\bf X}_4=x \frac{\partial}{\partial x}+ y\frac{\partial}{\partial y},
\nonumber
\end{equation}
fulfill the commutation relations
\begin{equation}
[{\bf X}_1,{\bf X}_2]={\bf X}_2,\qquad [{\bf X}_1,{\bf X}_3]=-{\bf X}_3,\qquad [{\bf X}_2,{\bf X}_3]=2 {\bf X}_1- {\bf X}_4 ,\qquad [{\bf X}_4, \, \cdot \, ]=0 .
\nonumber
\end{equation}
Hence, these vector fields span a Vessiot--Guldberg Lie algebra $V$ isomorphic to $\mathfrak{gl}(2)$ with domain $\mathbb R^2_{x\ne 0}$.
In fact, $V$ is diffeomorphic to the class ${\rm I}_7 \simeq \mathfrak{gl}(2)$ of the classification given in~\cite{BBHLS}. The diffemorphism can be explictly performed by means of the change of variables
$u=y/x$ and $v=1/x$, leading to the vector fields of class ${\rm I}_7$ with domain $\mathbb R^2_{v\ne 0}$ given in~\cite{BBHLS}
$$
{\bf X}_1= u\frac{\partial}{\partial u},\qquad {\bf X}_2= -u^2\frac{\partial}{\partial u}- u v \frac{\partial}{\partial v},\qquad
{\bf X}_3= \frac{\partial}{\partial u},\qquad {\bf X}_4= - v \frac{\partial}{\partial v} .
$$
Therefore ${\bf X}$ (\ref{Aa}) is a Lie system but not a LH one since there does not exist any compatible symplectic form satisfying (\ref{der}) for class ${\rm I}_7$ as shown in~\cite{BBHLS}.
Finally, we point out that the very same result follows by starting from the differential equation (\ref{Ad}) associated with the sinc function.
\section*{Acknowledgments}
\small
A.B.~and F.J.H.~have been partially supported by Ministerio de Econom\'{i}a y Competitividad (MINECO, Spain) under grants MTM2013-43820-P and MTM2016-79639-P (AEI/FEDER, UE), and by Junta de Castilla y Le\'on (Spain) under grants BU278U14 and VA057U16. The research of R.C.S.~was partially supported by grant MTM2016-79422-P (AEI/FEDER, EU).
E.F.S.~acknowledges a fellowship (grant CT45/15-CT46/15) supported by the Universidad Complutense de Madrid. J.~de L.~acknowledges funding from the Polish National Science Centre under grant HARMONIA 2016/22/M/ST1/00542.
| 39,561
|
If when you try to install the ServiceM8 MYOB Connector it fails with an error regarding unable to install the Microsoft .Net Framework 4.
You can fix this by manually installing the Microsoft .Net Framework 4 from this link:
Once you've installed the .Net Framework, re-try to install the ServiceM8 MYOB Connector and you should have success.
If not, get in touch with the helpdesk for additional support.
| 4,349
|
TITLE: Solving $\cos x+\sin x-1=0$
QUESTION [7 upvotes]: How does one solve this equation?
$$\cos {x}+\sin {x}-1=0$$
I have no idea how to start it.
Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?
Thanks in advance!
REPLY [3 votes]: I'll throw my hat in the ring to get a picture in edgewise. :-)
I'll use $\theta$ for the angle, rather than $x$ (to avoid confusion with the $x$-coordinate). If we then let $x = \cos \theta, y = \sin \theta$, then the allowable results are along the red circle with $x^2+y^2 = 1$ (since $\cos^2 \theta + \sin^2 \theta = 1$). The solutions of the given equation are at the intersections of the blue line $x+y = 1$ with that red circle, yielding $(\cos \theta, \sin \theta) = (1, 0)$ and $(0, 1)$.
This in turn yields
$$
\theta = 2k\pi, \qquad \theta = 2k\pi + \frac{\pi}{2}
$$
for $k$ an integer.
| 32,296
|
\begin{document}
\title [Injectivity theorems and algebraic closures]
{Injectivity theorems and algebraic closures of groups with
coefficients}
\author{Jae Choon Cha}
\address{Information and Communications University, Daejeon 305--732,
Korea}
\email{jccha@icu.ac.kr}
\def\subjclassname{\textup{2000} Mathematics Subject Classification}
\expandafter\let\csname subjclassname@1991\endcsname=\subjclassname
\expandafter\let\csname subjclassname@2000\endcsname=\subjclassname
\subjclass{Primary 20J05, 57M07, Secondary 55P60, 57M27}
\keywords{Algebraic closure, Localization, Injectivity, Torsion-free
derived series}
\begin{abstract}
Recently, Cochran and Harvey defined torsion-free derived series of
groups and proved an injectivity theorem on the associated
torsion-free quotients. We show that there is a universal
construction which extends such an injectivity theorem to an
isomorphism theorem. Our result relates injectivity theorems to a
certain homology localization of groups. In order to give a
concrete combinatorial description and existence proof of the
necessary homology localization, we introduce a new version of
algebraic closures of groups with coefficients by considering a
certain type of equations.
\end{abstract}
\maketitle
\section{Introduction}
\label{section:introduction}
Let $R$ be a subring of the rationals. A map $f\colon X \to Y$
between two spaces $X$ and $Y$ is called an \emph{$R$-homology
equivalence} if $f$ induces isomorphisms on $H_*(-;R)$. Homology
equivalences play an important role in the study of various problems
in geometric topology, including homology cobordism of manifolds and
concordance of embeddings, in particular knot and link concordance.
In this regard, a central question is how the fundamental groups of
homology equivalent spaces relate. As a preliminary observation, it
can be easily seen that an $R$-homology equivalence induces a
homomorphism on the fundamental groups which is 2-connected on
$R$-homology; we say that a group homomorphism $\phi$ is
\emph{2-connected on $R$-homology} if $\phi$ induces an isomorphism on
$H_1(-;R)$ and a surjection on $H_2(-;R)$. When $R=\Z$ (resp.\ $\Q$),
$\phi$ is called \emph{integrally (resp.\ rationally) 2-connected}.
We also remark that in most applications it suffices to consider
finite complexes (for example, compact manifolds) and so fundamental
groups can be assumed to be finitely presented.
Probably the first landmark result on the relationship between
homology equivalences and fundamental groups is an isomorphism theorem
of Stallings, which says that an integrally 2-connected homomorphism
$\pi\to G$ induces an isomorphism $\pi/\pi_q \to G/G_q$, where $G_q$
denotes the $q$-th lower central subgroup of $G$
\cite{Stallings:1965-1}. This has several well-known topological
applications: abelian invariants such as linking numbers can be viewed
as an application of the simplest nontrivial case $G/G_2=H_1(G)$ of
Stallings' theorem. More generally, the invariance of $G/G_q$ plays a
key role in understanding Milnor's $\bar\mu$-invariants of links
\cite{Milnor:1954-1,Milnor:1957-1} as shown in Casson's
work~\cite{Casson:1975-1}. Orr defined further homotopy invariants of
links using Stallings' theorem~\cite{Orr:1987-1,Orr:1989-1}. In some
cases homology cobordism invariants can be obtained by combining
Stallings' theorem and the index theorem in a similar way as Levine's
work on Atiyah--Patodi--Singer signatures of
links~\cite{Levine:1994-1}. In \cite{Friedl:2003-1}, Friedl applied
this method to reformulate and generalize a link concordance invariant
obtained from certain nonabelian and irregular covers due to the
author and Ko~\cite{Cha-Ko:1999-1}.
In 2004, Cochran and Harvey announced a remarkable discovery of an
injectivity theorem relating the rational homology of a group $G$ to a
certain type of derived series $\{G_H\^n\}$, which is called the
\emph{torsion-free derived series}~\cite{Cochran-Harvey:2004-1}. (The
series is due to Harvey~\cite{Harvey:2005-1}; for a precise definition
of $G_H\^n$, see \cite{Cochran-Harvey:2004-1,Harvey:2005-1}, or
Section~\ref{section:torsion-free-derived-series}.) The main result
of~\cite{Cochran-Harvey:2004-1} can be stated as follows: if $\pi \to
G$ is a rationally 2-connected homomorphism of a finitely generated
group $\pi$ into a finitely presented group $G$, then it induces an
injection $\pi/\pi_H\^n \to G/G_H\^n$. It has interesting
applications as illustrated in a recent result of Harvey; she obtained
invariants of homology cobordism by combining the injectivity with
rank invariants and $L^{(2)}$-signature invariants.
The advent of the injectivity theorem leads us to ask a natural
question: can one extend the torsion-free quotient $G/G_H\^n$ in such
a way that an isomorphism is induced instead of an injection? More
generally, when can such an injectivity theorem be extended to an
isomorphism theorem? Regarding topological applications, we remark
that the fundamental ideas of the applications of Stallings' theorem
could be reused when one has an isomorphism theorem.
In this paper we study injectivity theorems and their extensions to
isomorphism theorems in a general setting motivated from
Cochran--Harvey's result. To formalize injectivity theorems, we
introduce a notion of an \emph{I-functor}. For an arbitrary
coefficient ring $R\subset \Q$, we think of the collection $\Omega^R$
of homomorphisms of finitely generated groups into finitely presented
groups which are 2-connected on $R$-homology. Roughly speaking, we
define an \emph{I-functor} $\H$ to be a functorial association $G \to
\H(G)$ such that to each $\pi\to G$ in $\Omega^R$, an injection
$\H(\pi)\to \H(G)$ is associated. Of course the main example of an
I-functor is $\H(G)=G/G_H\^n$ where $R=\Q$. We also formalize an
isomorphism theorem extending the injectivity as follows: a
\emph{container} of an I-functor $\H$ is defined to be another
I-functor $\F$ such that $\H$ injects into $\F$, that is,
$\H(G)\subset \F(G)$, and $\F$ associates an isomorphism to each
morphism in $\Omega^R$. (Because there is some technical
sophistication, we postpone precise definitions to
Section~\ref{section:I-functors-and-containers}; here we just remark
that everything is required to have certain functorial properties
which are naturally expected.)
Then our question can be stated as whether there is a container of a
given I-functor. Note that if one has a container, then it is easy to
construct a larger container by extending it; for example, take the
direct sum with a constant functor. So the most essential one is a
minimal container. We prove the following result:
\begin{theorem}
\label{theorem:rough-version-of-main-theorem}
If an I-functor $\H$ commutes with limits, then there exists a
container $\hat\H$ of $\H$ which is universal (initial) in the
following sense: if $\F$ is another container, then $\hat\H$ injects
into $\F$ in a unique way.
\end{theorem}
In other words, $\hat\H$ provides an isomorphism theorem which extends
the injectivity of $\H$, and it is universal among such extensions.
For a more precise statement, see
Section~\ref{section:I-functors-and-containers}. We remark that from
its universality it follows that $\hat\H$ is a unique minimal
container of~$\H$.
As a corollary of Theorem~\ref{theorem:rough-version-of-main-theorem},
we show that the torsion-free derived quotient $G \to G/G_H\^n$ has a
universal container (see
Corollary~\ref{corollary:existence-of-container-of-C-H-quotient}). We
remark that this special case was partially addressed
in~\cite{Cochran-Harvey:2004-1}; they constructed a container of
$G/G_H\^n$ by using an iterated semidirect product of certain homology
groups. We also show that the container in
\cite{Cochran-Harvey:2004-1} fails to be universal in our sense (see
Theorem~\ref{theorem:non-universality-of-C-H-container}). An
interesting observation is that this is closely related to the use of
the \emph{Ore localization} of a group ring $\Q[G]$ of a
poly-torsion-free-abelian group $G$ in the construction of the
container in~\cite{Cochran-Harvey:2004-1}. In the Ore localization
all nonzero elements are inverted, but it turns out that this is too
excessive; in showing that our universal container is strictly
smaller, it is illustrated that the unnecessarily inverted elements
are ones in the kernel of the augmentation $\Q[G] \to \Q$. (See
Section~\ref{section:torsion-free-derived-series} for more details.)
This gives us a motivation for expecting a similar but more natural
theory using the \emph{Cohn localization}, instead of the Ore
localization.
On the other hand, in our results there is something beyond the
existence of a universal container. It relates injectivity theorems
to a certain localization functor of groups. In general, one can view
a localization functor as a \emph{universal} construction inverting a
given collection $\Omega$ of morphisms in a category. (Our definition
of a localization is given in
Section~\ref{section:localization-wrt-2-connected-morphisms}.) So, if
a localization with respect to $\Omega$ exists, it provides a natural
isomorphism theorem for morphisms in $\Omega$. That is, it associates
an isomorphism to each morphism in~$\Omega$. In order to prove
Theorem~\ref{theorem:rough-version-of-main-theorem}, we use a
particular localization functor $G \to \hat G$ with respect to the
collection $\Omega^R$ considered above. In fact, in the proof of
Theorem~\ref{theorem:rough-version-of-main-theorem}, we show that the
universal container $\hat\H$ of $\H$ is given by $\hat\H(G)=\H(\hat
G)$. This presents another point of view that the injective
homomorphism induced by $\H$ can be regarded as a restriction of a
natural isomorphism obtained from the localization; for any
homomorphism in $\Omega^R$, the induced isomorphism on the
localization gives rise to an isomorphism on $\hat\H(-)$, and the
induced homomorphism on $\H(-)$ is its restriction.
We remark that several homology localizations of groups have been
studied in the literature, including works of Bousfield
\cite{Bousfield:1975-1}, Vogel, Levine
\cite{Levine:1989-2,Levine:1989-1}, and
Farjoun--Orr--Shelah~\cite{Farjoun-Orr-Shelah:1989}. In particular,
in \cite{Levine:1989-2,Levine:1989-1} Levine introduced the notion of
an algebraic closure of a group, which turns out to be equivalent to
Vogel's localization with respect to integrally 2-connected
homomorphisms from finitely generated groups into finitely presented
groups which are normally surjective. The localization with respect
to our $\Omega^R$ which is used to prove
Theorem~\ref{theorem:rough-version-of-main-theorem} is similar to that
of Levine, but distinguished in two points: first the normal
surjectivity condition is not required, and second, an arbitrary
subring $R$ of $\Q$ is used as homology coefficients. Although it
may be regarded as folklore that there exists a localization with
respect to $\Omega^R$, we give an existence proof for concreteness
since we could not find any published one in the literature. In
addition our work provides a combinatorial description of the desired
$R$-homology localization. For this purpose, we introduce a new
version of algebraic closures of groups, modifying the idea of
Farjoun, Orr, and Shelah~\cite{Farjoun-Orr-Shelah:1989} and
Levine~\cite{Levine:1989-2,Levine:1989-1}. We think of a certain
class of systems of equations over a group $G$ which we call
\emph{$R$-nullhomologous}, and define an \emph{(algebraic)
$R$-closure} $\hat G$ of $G$ in terms of solubility of such systems
of equations. We show the existence of an $R$-closure $\hat G$ for
any group $G$, and show that it is equivalent to the desired
localization with respect to~$\Omega^R$.
The remaining part of this paper is organized as follows. In
Section~\ref{section:I-functors-and-containers} we prove the existence
of a universal container, assuming the existence of the $R$-closure of
a group. In Section~\ref{section:torsion-free-derived-series} we
apply the results of Section~\ref{section:I-functors-and-containers}
to the case of the torsion-free derived quotient, and show that our
universal container is strictly smaller than a container constructed
in~\cite{Cochran-Harvey:2004-1}. In the remaining sections, we prove
results on the $R$-closure which are used in previous sections. In
Section~\ref{section:nullhomologous-equations} we introduce the notion
of $R$-nullhomologous systems of equations, and in
Section~\ref{section:localization-wrt-2-connected-morphisms} it is
related to the localization of groups with respect to~$\Omega^R$. In
Section~\ref{section:existence-of-closures} we show the existence of
the $R$-closure and some properties of the $R$-closure of a finitely
presented group.
\subsection*{Acknowledgements}
The author would like to thank Stefan Friedl and Kent Orr for
discussions from which the fundamental idea of this paper is obtained.
The author also thanks Tim Cochran and Shelly Harvey who kindly
provided a copy of slides containing their results in
\cite{Cochran-Harvey:2004-1,Harvey:2005-1} before the manuscripts
became available. Finally, comments from an anonymous referee were
very helpful in improving this paper.
\section{I-functors and containers}
\label{section:I-functors-and-containers}
We start with a formalization of the notion of a container of the
torsion-free derived quotient $G/G_H\^n$. Here we have a technical
issue that the association $G \to G/G_H\^n$ is not a functor of the
category $\G$ of groups; not all group homomorphisms induce a morphism
on the quotients, although the result of Cochran--Harvey guarantees
that a rationally 2-connected homomorphism of a finitely generated
group into a finitely presented group gives rise to an induced
homomorphism.
This leads us to consider what follows. Recall that for any subring
$R$ of $\Q$, we denote by $\Omega^R$ the class of group homomorphisms
$\alpha\colon \pi \to G$ such that $\pi$ is finitely generated, $G$ is
finitely presented, and $\alpha$ is 2-connected on $R$-homology, that
is, $\alpha$ induces an isomorphism on $H_1(-;R)$ and a surjection on
$H_2(-;R)$. Denoting $\H(G) = G/G_H\^n$, in
\cite{Cochran-Harvey:2004-1} it was shown that $\H$ has the following
properties for $R=\Q$:
\begin{enumerate}
\item To each group $G$, a homomorphism $p_G \colon G \to \H(G)$ is
associated.
\item To each homomorphism $\alpha \colon \pi \to G$ in $\Omega^R$, an
injection $\H(\pi) \to \H(G)$ is associated.
\item The above associations have naturality, that is,
$\H(\beta\circ\alpha) = \H(\beta)\circ \H(\alpha)$, $\H(\alpha)\circ
p_\pi = p_G \circ \alpha$ (see the diagrams below), and
$\H(\id_G)=\id_{\H(G)}$ whenever the involved homomorphisms exist.
\[
\begin{diagram}
\node{\H(\pi)} \arrow[2]{e,t}{\H(\beta\circ\alpha)} \arrow{se,b}{\H(\alpha)}
\node[2]{\H(P)}
\\
\node[2]{\H(G)} \arrow{ne,b}{\H(\beta)}
\end{diagram}
\qquad
\begin{diagram}
\node{\pi}\arrow{e,t}{\alpha} \arrow{s,l}{p_\pi}
\node{G} \arrow{s,r}{p_G}
\\
\node{\H(\pi)} \arrow{e,b}{\H(\alpha)}
\node{\H(G)}
\end{diagram}
\]
\xdef\lastcount{\arabic{enumi}}
\end{enumerate}
While $\H$ is not a functor of the category $\G$ of groups, we can
view $\H$ as a functor of a subcategory of finitely presented groups,
which are of our main interest regarding topological applications: let
$\G^R$ be the category whose objects are finitely presented groups and
whose morphisms are homomorphisms between finitely presented groups
which are 2-connected on $R$-homology. Then from the above properties
it follows that $\H$ induces a functor $\G^R \to \G$. Also $p$
induces a natural transformation from the obvious inclusion functor
$\G^R \to \G$ to (the functor induced by)~$\H$. In particular,
homology equivalences between finite complexes give rise to morphisms
in $\G^R$ and then one can apply $\H$ to obtain injections.
One more obvious property of $\H$ which might be easily overlooked is
the following:
\begin{enumerate}
\setcounter{enumi}{\lastcount}
\item To any isomorphism $\alpha\colon \pi\to G$, an isomorphism
$\H(\alpha)\colon \H(\pi) \to \H(G)$ is associated.
\end{enumerate}
We remark that (4) does not follow from (1)--(3) since (1)--(3) do not
guarantee that $\H(\alpha)$ is defined for an isomorphism~$\alpha$ in
general.
Results of this section are not specific to the torsion-free
quotients; we consider any association $\H$ with the properties
above.
\begin{definition}
$(\H,p)$ is called an \emph{I-functor with respect to
$R$-coefficients} if the above (1)--(4) are satisfied.
\end{definition}
Note that if $R'$ is a subring of $R$, then an $I$-functor with
respect to $R$-coefficients is automatically an $I$-functor with
respect to $R'$-coefficients. When the coefficient ring $R$ is
clearly understood, we simply say that $\H$ is an I-functor.
I-functors form (objects of) a category; a morphism $\tau$ between two
I-functors $\H$ and $\H'$ is defined to be a natural transformation
$\tau \colon \H \to \H'$, where $\H$ and $\H'$ are viewed as functors
$\G^R \to \G$ as an abuse of notation, such that the diagram
\[
\begin{diagram}
\node{G}\arrow{s,l}{p_G}\arrow{se,t}{p'_G}\\
\node{\H(G)}\arrow{e,b}{\tau_G} \node{\H'(G)}
\end{diagram}
\]
commute for each object $G$ in $\G^R$ where $p_G$ and $p'_G$ are the
natural transformations that the I-functors $\H$ and $\H'$ are endowed
with, respectively. If each $\tau_G$ is injective, then we say that
$\tau$ is \emph{injective} and $\H'$ is an \emph{extension} of~$\H$.
\begin{definition}
For an I-functor $\H$, a morphism $\tau\colon \H \to \F$ into
another I-functor $\F$ is called a \emph{container} of $\H$ if
$\tau$ is injective and $\F(\alpha)\colon \F(\pi)\to \F(G)$ is an
isomorphism for any morphism $\alpha\colon \pi\to G$ in~$\Omega^R$.
\end{definition}
Sometimes we say that $\F$ is a container of $\H$ when we do not have
to specify $\tau\colon \H \to \F$ explicitly.
As mentioned in the introduction, we are interested in a universal
(initial) container of a given I-functor~$\H$. To give its
definition, we consider the category of containers and injective
morphisms; objects are containers $\F$ of $\H$, and morphisms from
$\F$ to $\F'$ are injective morphisms $\F \to \F'$ between the two
I-functors $\F$ and $\F'$ which makes the diagram
\[
\begin{diagram}
\node{\H}\arrow{s}\arrow{se}\\
\node{\F}\arrow{e}
\node{\F'}
\end{diagram}
\]
commute.
\begin{definition}
A universal (initial) object $\F$ in the category of containers of
$\H$ is called a \emph{universal} container of~$\H$, that is, for
any container $\F'$ of $\H$, there is a unique morphism from $\F$
to~$\F'$.
\end{definition}
Obviously a universal container is unique if it exists. Also, a
universal container is automatically minimal, in the sense that it is
not a proper extension of another container. So if a universal
container exists, it is a unique minimal container.
For our existence result of a universal container, we need to
formulate a relationship of an I-functor $\H$ and limits. In this
paper it suffices to consider the direct limit of a sequence
\[
G_0 \to G_1 \to G_2 \to \cdots.
\]
of group homomorphisms in $\G^R$.
Usually, if $\H$ were an ordinary functor $\G\to \G$, we would say
that $\H$ commutes with limits when $\varinjlim \H(G_k) \cong
\H(\varinjlim G_k)$; more precisely, the isomorphism is explicitly
specified in this case. Namely, $G_k \to \varinjlim G_k$ induces
$\H(G_k) \to \H(\varinjlim G_k)$, and then
\[
\varinjlim \H(G_k) \to \H(\varinjlim G_k)
\]
is induced. If it is an isomorphism, then we say that $\H$ commutes
with limits. However, in our case, because $\H$ is just an I-functor,
the homomorphism $G_k \to \varinjlim G_k$ does not necessarily induce
$\H(G_k) \to \H(\varinjlim G_k)$ in general. So we need to adopt the
existence of this induced homomorphism as a part of a definition:
\begin{definition}
An I-functor $\H$ is said to \emph{commute with limits} if for any
sequence
\[
G_0 \to G_1 \to G_2 \to \cdots
\]
of morphisms $G_k \to G_{k+1}$ in $\G^R$, $\H$ associates to $G_k
\to \varinjlim G_k$ a homomorphism $\H(G_k) \to \H(\varinjlim G_k)$,
and its limit
\[
\varinjlim \H(G_k) \to \H(\varinjlim G_k)
\]
is an isomorphism.
\end{definition}
We note that even though each $G_k \to G_{k+1}$ is in $\G^R$,
$\varinjlim G_k$ is not necessarily (an object) in~$\G^R$.
Now we can precisely state the main result of this section.
\begin{theorem}
\label{theorem:existence-of-container-of-I-functor}
Suppose that $R$ is a subring of $\Q$ and $\H$ is an I-functor with
respect to $R$-coefficients which commutes with limits. Then there
exists a universal container $\tau \colon \H \to \hat\H$ of $\H$,
that is, for any container $\sigma\colon \H \to \F$, there is a
unique injective morphism $\hat\sigma \colon \hat\H \to \F$ such
that the diagram
\[
\begin{diagram}
\node[2]{G}\arrow{ssw}\arrow{s}\arrow{sse}
\\
\node[2]{\hat \H(G)}\arrow{se,b,..}{\hat\sigma_G}
\\
\node{\H(G)} \arrow{ne,b}{\tau_G} \arrow[2]{e,b}{\sigma_G}
\node[2]{\F(G)}
\end{diagram}
\]
commutes.
\end{theorem}
To prove Theorem~\ref{theorem:existence-of-container-of-I-functor}, we
use a homology localization functor $E\colon \G \to \G$ with respect
to $R$-coefficients. At this moment we just need the following
properties of~$E$, which are analogues of Levine's results on
algebraic closures of groups \cite{Levine:1989-2,Levine:1989-1}; a
construction of our $E$ and proofs of the necessary properties are
postponed to later sections.
\begin{theorem}
\label{theorem:properties-of-closures}
For any subring $R$ of $\Q$, there is a pair $(E,i)$ of a functor
$E\colon \G \to \G$ and a natural transformation $i \colon \id_\G
\to E$ which has the following properties:
\begin{enumerate}
\item For any $\alpha\colon \pi \to G$ in $\Omega^R$, the induced
homomorphism $E(\alpha) \colon E(\pi) \to E(G)$ is an isomorphism.
\item For any object $G$ in $\G^R$, there is a sequence
\[
G=G_0 \to G_1 \to \cdots \to G_k \to \cdots
\]
of morphisms $G_k \to G_{k+1}$ in $\G^R$ such that $E(G)=
\varinjlim G_k$ and $i_G \colon G \to E(G)$ is the limit
homomorphism.
\end{enumerate}
\end{theorem}
We denote $E(G)$ by $\hat G$. For any I-functor $\H$ which commutes
with limits, we will prove that the composition of $E$ and $\H$ is a
universal container of $\H$.
\begin{proof}
[Proof of Theorem~\ref{theorem:existence-of-container-of-I-functor}]
We define $\hat\H(G) = \H(\hat G)$ and $\hat p_G \colon G \to
\hat\H(G)$ to be the composition
\[
G \xrightarrow{i_G} \hat G \xrightarrow{p_{\hat G}} \H(\hat G)=\hat\H(G).
\]
In other words, $\hat\H = \H \circ E$ and $\hat p = p \circ i$. We
will show that $(\hat\H,\hat p)$ is an I-functor. For any
$\alpha\colon \pi \to G$ which is in $\Omega^R$, $\hat\alpha\colon
\hat\pi \to \hat G$ is an isomorphism. Applying $\H$, we obtain an
induced isomorphism $\H(\hat\pi) \to \H(\hat G)$. We define
$\hat\H(\alpha)$ to be this isomorphism. Viewing $(\hat H, \hat p)$
as $(\H \circ E, p \circ i)$, the required naturality of $(\hat H,
\hat p)$ follows from that of $(\H,p)$ and $(E,i)$.
For a finitely presented group $G$, $(\hat\H, \hat p)$ can be
interpreted as follows. Choose a sequence
\[
G=G_0 \to G_1 \to G_2 \to \cdots
\]
of morphisms in $\G^R$ such that $i_G \colon G \to \hat G$ is the
limit homomorphism $G \to \varinjlim G_k \cong \hat G$. By the naturality
of $p$,
\[
\begin{diagram}
\node{G} \arrow{e} \arrow{s,l}{p_G}
\node{G_k} \arrow{e} \arrow{s,r}{p_{G_k}}
\node{\varinjlim G_k} \arrow{s,r}{p_{\varinjlim G_k}}
\\
\node{\H(G)} \arrow{e}
\node{\H(G_k)} \arrow{e}
\node{\H(\varinjlim G_k)}
\end{diagram}
\]
commutes. Taking the limit, we have a commutative diagram
\[
\begin{diagram}
\node{G} \arrow{e,t}{i_G} \arrow{s,l}{p_G}
\node{\varinjlim G_k} \arrow{e,=} \arrow{s,r}{\varinjlim p_{G_k}}
\node{\varinjlim G_k \hbox to 0mm{ $= \hat G$\hss}}
\arrow{s,r}{p_{\varinjlim G_k}}
\\
\node{\H(G)} \arrow{e,b}{\text{limit map}}
\node{\varinjlim \H(G_k)} \arrow{e,b}{\cong}
\node{\H(\varinjlim G_k)\hbox to 0mm{ $= \H(\hat G)$\hss}}
\end{diagram}
\hphantom{= \H(\hat G)}
\]
That is, $\hat\H(G) = \varinjlim \H(G_k)$ and $\H$ associates to
$i_G$ the limit homomorphism
\[
\H(i_G)\colon \H(G) \to \hat\H(G) = \varinjlim \H(G_k).
\]
Also, $\hat p_G\colon G \to \hat\H(G)$ is the composition
\[
G \xrightarrow{p_G} \H(G) \xrightarrow{\H(i_G)} \varinjlim \H(G_k).
\]
Now we construct an injective morphism $\tau\colon (\H,p) \to
(\hat\H, \hat p)$ between the I-functors $(\H,p)$ and $(\hat\H, \hat
p)$ as follows. For a finitely presented group $G$, there exists
$\H(i_G)\colon \H(G) \to \H(\hat G)$ as discussed above. We define
$\tau_G \colon \H(G) \to \hat\H(G)$ to be $\H(i_G)$, that is, $\tau
= \H \circ i$. Viewing $\tau$ as a transformation between functors
$\H, \H'\colon \G^R \to \G$, the naturality of $\tau$ follows from
that of $\H$ and~$i$. Furthermore,
\[
\begin{diagram}
\node{G} \arrow{s,l}{p_G} \arrow{se,t}{\hat p_G}
\\
\node{\H(G)} \arrow{e,b}{\tau_G}
\node{\hat\H(G)}
\end{diagram}
\]
commutes since $\hat p_G = \H(i_G) \circ p_G$. This shows that
$\tau$ is a morphism $(\H,p) \to (\hat\H,\hat p)$. To show the
injectivity, we consider a sequence $G=G_0 \to G_1 \to \cdots$ with
limit $\hat G$ as above. Since $\H$ is an I-functor and $G \to G_k$
is in $\Omega^R$, $\H(G) \to \H(G_k)$ is injective. Since $\tau_G =
\H(i_G)$ is the limit of $\H(G) \to \H(G_k)$, $\tau_G$ is injective
too.
We will show that $\tau\colon (\H,p) \to (\hat\H,\hat p)$ has the
universal property. Suppose that $\sigma\colon (\H,p) \to (\F,q)$
is a container. We define a morphism $\hat\sigma \colon
(\hat\H,\hat p) \to (\F,q)$ as follows: for a finitely presented
group $G$, choose $G=G_0 \to G_1 \to \cdots$ with limit $\hat G$ as
above. Taking the limit of
\[
\begin{diagram}\dgARROWLENGTH=1.9em
\node[2]{G} \arrow{e}\arrow{s,r}{p_G}\arrow{ssw,t}{q_G}
\node{G_k} \arrow{s,l}{p_{G_k}} \arrow{sse,t}{q_{G_k}}
\\
\node[2]{\H(G)} \arrow{e,J} \arrow{sw,b,J}{\sigma_G}
\node{\H(G_k)} \arrow{se,b,J}{\sigma_{G_k}}
\\
\node{\F(G)} \arrow[3]{e,b}{\cong}
\node[3]{\F(G_k)}
\end{diagram}
\]
we obtain a commutative diagram
\begin{equation}
\begin{diagram}\dgARROWLENGTH=1.9em
\node[2]{G} \arrow{e}\arrow{s,r}{p_G}\arrow{ssw,t}{q_G}
\node{\varinjlim G_k}
\arrow{s,l}{\varinjlim p_{G_k}} \arrow{sse,t}{\varinjlim q_{G_k}}
\\
\node[2]{\H(G)} \arrow{e,J} \arrow{sw,b,J}{\sigma_G}
\node{\varinjlim \H(G_k)} \arrow{se,b,J}{\varinjlim \sigma_{G_k}}
\\
\node{\F(G)} \arrow[3]{e,b}{\cong}
\node[3]{\varinjlim \F(G_k)}
\end{diagram}
\tag*{$(*)_{\{G_k\}}$}
\end{equation}
We define $\hat \sigma_G$ to be $\varinjlim \sigma_{G_k}$, that is,
\[
\hat\sigma_G \colon \hat\H(G) =\H(\hat G) = \varinjlim \H(G_k)
\xrightarrow{\varinjlim \sigma_{G_k}} \varinjlim \F(G_k) = \F(G).
\]
Since each $\sigma_{G_k}$ is injective, so is~$\hat\sigma_G$.
At the moment, our $\hat\sigma_G$ depends on the choice of
$\{G_k\}$. Before showing that it is well-defined, we prove the
naturality of $\hat\sigma$. Suppose $\alpha\colon \pi \to G$ is in
$\G^R$ and $\pi=\pi_0 \to \pi_1 \to \cdots$ and $G=G_0 \to G_1 \to
\cdots$ are sequences giving $\hat\pi$ and $\hat G$ as above. First
we consider a special case that $\{\pi_k\}$ and $\{G_k\}$ behave
nicely under $\alpha$, that is, we suppose that there are
homomorphisms $\pi_k \to G_k$ which fit into the following
commutative diagram:
\[
\begin{diagram}
\node{\hbox to 0mm{\hss$\pi=$ }\pi_0} \arrow{s,l}{\alpha}\arrow{e}
\node{\pi_1} \arrow{s}\arrow{e}
\node{\pi_2} \arrow{s}\arrow{e}
\node{\cdots}
\\
\node{\hbox to 0mm{\hss$G=$ }G_0} \arrow{e}
\node{G_1} \arrow{e}
\node{G_2} \arrow{e}
\node{\cdots}
\end{diagram}
\]
Then $\alpha$, together with $\pi_k \to G_k$, induces a ``morphism''
between the diagrams $(*)_{\{\pi_k\}}$ and $(*)_{\{G_k\}}$. In
particular, the morphisms $\hat\H(\alpha)$ and $\F(\alpha)$ give us
the commutative diagram below, which says that $\hat\sigma$ is
natural in this special case:
\[
\begin{diagram}
\node{\hat\H(\pi)}\arrow{e,t}{\hat\H(\alpha)}
\arrow{s,l}{\varinjlim\sigma_{\pi_k}=\hat\sigma_\pi}
\node{\hat\H(G)} \arrow{s,r}{\hat\sigma_G=\varinjlim\sigma_{G_k}}
\\
\node{\F(\pi)} \arrow{e,b}{\F(\alpha)}
\node{\F(G)}
\end{diagram}
\]
To reduce the general case to the special case above, we appeal to
the following result, which is a horseshoe-type lemma:
\begin{lemma}
\label{lemma:horseshoe-special-case}
Suppose that $\pi=\pi_0 \to \pi_1 \to \cdots$ and $G=G_0 \to G_1
\to \cdots$ are sequences of morphisms in $\G^R$ such that
$\hat\pi=\varinjlim \pi_k$ and $\hat G = \varinjlim G_k$, and
$\alpha\colon \pi \to G$ is in $\G^R$. Then there exists a
commutative diagram
\[
\hphantom{G=}
\begin{diagram}\dgARROWLENGTH=1em
\node{\hbox to 0mm{\hss$\pi=$ }\pi_0} \arrow{e} \arrow{s,l}{\alpha}
\node{\pi_1} \arrow{e} \arrow{s}
\node{\cdots} \arrow{e}
\node{\pi_k} \arrow{e} \arrow{s}
\node{\cdots} \arrow{e}
\node{\varinjlim \pi_k \hbox to 0mm{ $= \hat\pi$\hss}} \arrow{s}\\
\node{\hbox to 0mm{\hss$G=$ }P_0} \arrow{e}
\node{P_1} \arrow{e}
\node{\cdots} \arrow{e}
\node{P_k} \arrow{e}
\node{\cdots} \arrow{e}
\node{\varinjlim P_k} \\
\node{\hbox to 0mm{\hss$G=$ }G_0} \arrow{e} \arrow{n,=}
\node{G_1} \arrow{e} \arrow{n}
\node{\cdots} \arrow{e}
\node{G_k} \arrow{e} \arrow{n}
\node{\cdots} \arrow{e}
\node{\varinjlim G_k \hbox to 0mm{ $= \hat G$\hss}} \arrow{n}
\end{diagram}\hphantom{= \hat G}
\]
where $\pi_k \to P_k$, $G_k \to P_k$, and $P_k \to P_{k+1}$ are in
$\G^R$,
\[
\hat G =\varinjlim G_k \to \varinjlim P_k
\]
is an isomorphism, and the limit homomorphism
\[
G=P_0 \to \varinjlim P_k \cong \hat G
\]
is equal to $i_G \colon G \to \hat G$.
\end{lemma}
Applying the above special case to $(\{\pi_k\}, \{P_k\})$ and
$(\{P_k\},\{G_k\})$, we obtain a commutative diagram
\[
\begin{diagram}
\node{\hat\H(\pi)}\arrow{e,t}{\hat\H(\alpha)}\arrow{s,l}{\varinjlim\sigma_{\pi_k}}
\node{\hat\H(G)} \arrow{e,t}{\hat\H(\id)=\id}\arrow{s,r}{\varinjlim\sigma_{P_k}}
\node{\hat\H(G)} \arrow{s,r}{\varinjlim\sigma_{G_k}}
\\
\node{\F(\pi)} \arrow{e,b}{\F(\alpha)}
\node{\F(G)} \arrow{e,b}{\F(\id)=\id}
\node{\F(G)}
\end{diagram}
\tag{$**$}
\]
This shows that $\hat\sigma$ behaves naturally for $\pi \to G$ even
when $\hat\sigma_\pi$ and $\hat\sigma_G$ are defined using
arbitrarily chosen $\{\pi_k\}$ and $\{G_k\}$.
We can also use the same argument to show the well-definedness of
$\hat\sigma_G$, that is, $\hat\sigma_G$ is independent of the choice
of $\{G_k\}$. For this, we apply
Lemma~\ref{lemma:horseshoe-special-case} to a special case that
$\pi=G$ and $\alpha\colon \pi \to G$ is the identity. Then for any
$\{\pi_k\}$ and $\{G_k\}$ with limit $\hat G$, there is $\{P_k\}$
which gives the diagram~($**$). Since $\hat\H(\alpha)=\id$ in this
case, it follows that the homomorphisms
\[
\varinjlim \sigma_{\pi_k},\, \varinjlim
\sigma_{P_k},\, \varinjlim \sigma_{G_k}\colon \hat\H(G) \to \F(G)
\]
are all equal.
From the diagram~$(*)_{\{G_k\}}$, we obtain a commutative diagram
\[
\begin{diagram}
\node[2]{G} \arrow{ssw,t}{\hat p_G} \arrow{sse,t}{q_G} \arrow{s,l,3}{p_G}
\\
\node[2]{\H(G)} \arrow{sw,b}{\tau_G} \arrow{se,b}{\sigma_G}
\\
\node{\hat\H(G)} \arrow[2]{e,b}{\hat\sigma_G}
\node[2]{\F(G)}
\end{diagram}.
\]
From this it follows that $\hat\sigma$ can be viewed as a morphism
between containers.
Finally we show the uniqueness of $\hat\sigma$. Suppose that
$\hat\sigma' \colon (\hat\H,\hat p) \to (\F,q)$ is another morphism
between the containers $(\hat\H,\hat p)$ and $(\F,q)$. For a
sequence $G=G_0 \to G_1 \to \cdots$ giving $\hat G$, we have the
following commutative diagram:
\[
\begin{diagram}\dgARROWLENGTH=.7em
\node[3]{\H(G)}
\arrow[2]{sw,t}{\tau_G} \arrow[2]{s,r}{\sigma_G}
\arrow[2]{se}
\\
\\
\node{\hat\H(G)} \arrow[2]{e,t}{\hat\sigma'_G} \arrow[2]{se,b}{\cong}
\node[2]{\F(G)} \arrow[2]{se,t,1}{\cong}
\node[2]{\H(G_k)} \arrow{sw,-} \arrow[2]{s,r}{\sigma_{G_k}}
\\
\node[4]{}\arrow{sw,t}{\tau_{G_k}}
\\
\node[3]{\hbox to0mm{\hss$\varinjlim\H(G_k)=$ }\hat\H(G_k)}
\arrow[2]{e,b}{\hat\sigma'_{G_k}}
\node[2]{\F(G_k)}
\end{diagram}
\]
Since $\hat\sigma'_{G_k}\tau_{G_k} = \sigma_{G_k} = \tau_{G_k}
\hat\sigma_{G_k} $, $\hat\sigma'_{G}$ and $\hat\sigma_{G}\colon
\hat\H(G) \to \F(G)$ coincide on the image of $\tau_{G_k}\colon
\H(G_k) \to \hat\H(G_k)=\hat\H(G)$. From this it follows that
$\hat\sigma'_{G}=\hat\sigma_{G}$ since
$\hat\H(G)=\varinjlim\H(G_k)$.
\end{proof}
\section{Containers of torsion-free derived series}
\label{section:torsion-free-derived-series}
In this section we focus on a special case of an I-functor, namely the
torsion-free derived quotient $G \to G/G_H\^n$. We begin by recalling
the definition of $G_H\^n$ in~\cite{Cochran-Harvey:2004-1}. For a
group $G$, $G_H\^0$ is defined to be $G$ itself. Suppose $G_H\^n$ has
been defined to be a normal subgroup of $G$ such that that $G/G_H\^n$
is a poly-torsion-free-abelian (PTFA) group. Since the integral group
ring of a PTFA group is an Ore domain, there exists the skew field
$\K[G/G_H\^n]$ of (right) quotients of $\Z[G/G_H\^n]$, that is,
$\K[G/G_H\^n] = \Z[G/G_H\^n] (\Z[G/G_H\^n]-\{0\})^{-1}$. Note that
$\K[G/G_H\^n]$ is $\Z[G/G_H\^n]$-flat. $G_H\^{n+1}$ is defined to be
the kernel of the following composition:
\[
\begin{aligned}
G_H\^n & \to G_H\^n/[G_H\^n,G_H\^n] = H_1(G;\Z[G/G_H\^n]) \\
& \to H_1(G;\Z[G/G_H\^n])\mathbin{\mathop{\otimes}_{\Z[G/G_H\^n]}}
\K[G/G_H\^n] = H_1(G;\K[G/G_H\^n])
\end{aligned}
\]
Since $G/G_H\^{n+1}$ is an extension of $G_H\^n/G_H\^{n+1}$ by
$G/G_H\^n$ and $G_H\^n/G_H\^{n+1}$ is a subgroup of
$H_1(G;\K[G/G_H\^n])$ which is a torsion-free abelian group,
$G/G_H\^{n+1}$ is PTFA so that one can continue this process. For
$n=\omega$, the first infinite ordinal, $G_H\^n$ is defined to be the
intersection of $G_H\^k$ where $k$ runs over all integers. For
further details see~\cite{Cochran-Harvey:2004-1}.
We denote $\H_n(G) = G/G_H\^n$. It was shown in
\cite{Cochran-Harvey:2004-1} that $\H_n$, equipped with the projection
$G \to \H_n(G)$, is an I-functor with respect to $\Q$-coefficients.
(In this section the coefficient ring $R$ is always $\Q$.) From
Theorem~\ref{theorem:existence-of-container-of-I-functor}, it follows
that $\H_n$ has a universal container:
\begin{corollary}
\label{corollary:existence-of-container-of-C-H-quotient}
There exists a universal container of~$\H_n$ for any~$n \le \omega$.
\end{corollary}
\begin{proof}
By Theorem~\ref{theorem:existence-of-container-of-I-functor}, it
suffices to show that $\H_n$ commutes with limits. Suppose that
\[
G=G_0\to G_1 \to G_2 \to \cdots
\]
is a sequence of morphisms in $\G^R$. We use the following two
properties of~$\H_n$: first, $G \to \H_n(G)$ is obviously surjective
for any $G$, and second, we need Lemma 5.3
in~\cite{Cochran-Harvey:2004-1}: the natural map $G_k \to \varinjlim
G_k$ gives rise to an injection $\H_n(G_k) \to \H_n(\varinjlim G_k)$
for any $k$ and any $n\le \omega$.
Taking the limit of the injections $\H_n(G_k) \to
\H_n(\varinjlim G_k)$, we obtain an
injection
\[
\varinjlim \H_n(G_k) \longrightarrow \H_n(\varinjlim G_k).
\]
Since every $x$ in $\H_n(\varinjlim G_k)$ is represented by an
element in $\varinjlim G_k$, $x$ is in the image of some $G_k$, and
so in the image of some $\H_n(G_k)$. From this the surjectivity
follows.
\end{proof}
Recall that the universal container $\hat\H_n$ of $\H_n$ is given by
$\hat\H_n(G)=\H_n(\hat G)$, where $\hat G$ denotes our homology
localization with respect to $\Q$-coefficients given in
Theorem~\ref{theorem:properties-of-closures}.
\begin{remark}
For a $(4k-1)$-manifold $M$ with fundamental group $\pi$, Harvey
considered the $L^{(2)}$-signature of $M$ associated to $\pi \to
\H_n(\pi)$ as a homology cobordism invariant. Since $\H_n(\pi) \to
\hat \H_n(\pi) = \hat\pi/\hat\pi_H\^n$ is injective by
Corollary~\ref{corollary:existence-of-container-of-C-H-quotient},
from the induction property of the $L^{(2)}$-signature it follows
that Harvey's invariant coincides with the $L^{(2)}$-signature
associated to a characteristic quotient of the localization of
$\pi$, namely $\pi \to \hat \pi \to \hat\pi/\hat\pi_H\^n$.
\end{remark}
On the other hand, Cochran and Harvey defined a container $\F_n(G) =
\tilde G_n$ of $\H_n$ as follows. Let $\tilde G_0 = \{e\}$, a trivial
group. Suppose $\tilde G_n$ has been defined as a PTFA group. Then
$\tilde G_{n+1}$ is defined to be $\tilde G_{n+1} = H_1(G;\K\tilde
G_n) \rtimes \tilde G_n$, where the semidirect product is formed by
viewing $H_1(G;\K\tilde G_n)$ as a $\Z[\tilde G_n]$-module. Also,
injections $\sigma_{n,G} \colon \H_n(G)=G/G_H\^n \to \tilde
G_n=\F_n(G)$ are defined as follows. Initially $\sigma_{0,G}$ is the
trivial homomorphism. Suppose $\sigma_{n,G}$ has been defined.
Consider the composition
\[
\phi_{n,G}\colon G_H\^n/G_H\^{n+1} \xrightarrow{\Phi_{n,G}} H_1(G;\K\H_n(G))
\xrightarrow{f} H_1(G;\K\F_n(G))
\]
where $\Phi_{n,G}$ is the injection induced by the homomorphism used
above to define $G_H\^{n+1}$, and $f$ is induced by $\sigma_{n,G}
\colon \H_n(G) \to \F_n(G)$. In~\cite{Cochran-Harvey:2004-1} the
followings were shown: there is a derivation $G \to H_1(G;\K\F_n(G))$
which induces~$\phi_{n,G}$. This derivation, together with $G \to
\H_n(G) \to \F_n(G)$, gives rise to a homomorphism $G \to
H_1(G;\K\F_n(G)) \rtimes \F_n(G) = \F_{n+1}(G)$ with kernel
$G_H\^{n+1}$. So it induces an injection $\sigma_{n+1,G}\colon
\H_{n+1}(G) \to \F_{n+1}(G)$. For $n=\omega$, $\tilde G_\omega$
(which is denoted by $\tilde G$ and called the solvable completion of
$G$ in \cite{Cochran-Harvey:2004-1}) is defined to be $\tilde G_\omega
= \varprojlim \tilde G_k$ where $k<\omega$. For any $n\le \omega$,
$\sigma_n \colon \H_n \to \F_n$ is a container of $\H_n$. For further
details see~\cite{Cochran-Harvey:2004-1}.
We note that from the definitions it follows that there is a
commutative diagram
\[
\tag*{$(*\mathord{*}*)$}
\begin{diagram}\dgARROWLENGTH=1.8em \dgHORIZPAD=.2em
\node{\kern 2em 1} \arrow{e}
\node{G_H\^n/G_H\^{n+1}} \arrow{e} \arrow{s,r}{\phi_{n,G}}
\node{\H_{n+1}(G)} \arrow{e} \arrow{s,r}{\sigma_{n+1,G}}
\node{\H_n(G)} \arrow{e} \arrow{s,r}{\sigma_{n,G}}
\node{1\kern 2em}
\\
\node{\kern 2em 1} \arrow{e}
\node{H_1(G;\K\F_n(G))} \arrow{e}
\node{\F_{n+1}(G)} \arrow{e}
\node{\F_n(G)} \arrow{e}
\node{1\kern 2em}
\end{diagram}
\]
with exact rows.
In the remaining part of this section, we compare the container $\F_n$
with the universal container $\hat\H_n$ of $\H_n$. More precisely, by
Corollary~\ref{corollary:existence-of-container-of-C-H-quotient},
there is an injective morphism $\hat\sigma_{n,G}$ of our universal
container $\hat\H_n(G) = \hat G / \hat G_H\^n$ to $\F_n(G)=\tilde
G_n$. Then our question is whether $\hat\sigma_{n,G}$ is an
isomorphism. The following proposition says that this is closely
related to the structure of certain homology modules of $\hat G$ in an
inductive manner.
\begin{proposition}
\label{proposition:criterion-for-universality-of-C-H-container}
For a finitely presented group $G$, $\hat\sigma_{n+1,G}$ is an
isomorphism if and only if $\hat\sigma_{n,G}$ is an isomorphism and
the canonical homomorphism
\[
H_1(\hat G;\Z\H_n(\hat G)) \to H_1(\hat G;\K\H_n(\hat G))
\]
is surjective.
\end{proposition}
\begin{proof}
Before proving the proposition, we assert that if
$\hat\sigma_{n,G}\colon \H_n(\hat G) \to \F_n(G)$ is an isomorphism
then there is a commutative diagram
\[
\begin{diagram}\dgHORIZPAD=.2em
\node{\kern 2em 1} \arrow{e}\dgARROWLENGTH=2em
\node{\hat G_H\^n/\hat G_H\^{n+1}}
\arrow{e} \arrow{s,r}{\Phi_{n,\hat G}}
\node{\H_{n+1}(\hat G)} \arrow{e} \arrow{s,r}{\hat \sigma_{n+1,G}}
\node{\H_n(\hat G)} \arrow{e} \arrow{s,r}{\hat \sigma_{n,G}}
\node{1\kern 2em}
\\
\node{\kern 2em 1} \arrow{e}
\node{H_1(\hat G;\K\H_n(\hat G))} \arrow{e}
\node{\F_{n+1}(G)} \arrow{e}
\node{\F_n(G)} \arrow{e}
\node{1\kern 2em}
\end{diagram}
\]
with exact rows.
To prove this, we recall that the morphism $\hat\sigma_n$ can be
described as follows. Choose a sequence $G = G_0 \to G_1 \to
\cdots$ of morphisms in $\Omega^\Q$ whose limit is $\hat G$. Then
$\H_n(\hat G) \cong \varinjlim \H_n(G_k)$ and
$\hat\sigma_{n,G}\colon \H_n(\hat G) \to \F_n(G)$ is the limit of
\[
\sigma_{n,G_k}\colon \H_n(G_k) \to \F_n(G_k) \cong \F_n(G)
\]
as $k\to \infty$.
By the hypothesis of the assertion, $\H_n(\hat G) \cong \F_n(G)
\cong \F_n(G_k)$ where the latter isomorphism is induced by $G \to
G_k$. So from the diagram $(*\mathord{*}*)$ we obtain a commutative
diagram
{\footnotesize
\[\hbox to 0mm{\hss$
\begin{diagram}\dgARROWLENGTH=-6em\dgHORIZPAD=.4em
\node{\kern 1em 1} \arrow{e}
\node{\frac{G_H\^n}{G_H\^{n+1}}}
\arrow[2]{e} \arrow{sse} \arrow[4]{s,l,3}{\phi_{n,G}}
\node[2]{\H_{n+1}(G)} \arrow[2]{e} \arrow{sse} \arrow[2]{s,-}
\node[2]{\H_{n}(G)} \arrow{e} \arrow{sse} \arrow[2]{s,-}
\node{1 \kern 1em}
\\
\\
\node[2]{\kern 1em 1} \arrow{e}
\node{\frac{(G_k)_H\^n}{(G_k)_H\^{n+1}}}
\arrow[2]{e} \arrow[4]{s,r,3}{\phi_{n,G_k}}
\node{} \arrow[2]{s,r}{\sigma_{n+1,G}}
\node{\H_{n+1}(G_k)} \arrow[2]{e} \arrow[4]{s,r,3}{\sigma_{n+1,G_k}}
\node{} \arrow[2]{s,r}{\sigma_{n,G}}
\node{\H_{n}(G_k)} \arrow{e} \arrow[4]{s,r,3}{\sigma_{n,G_k}}
\node{1 \kern 1em}
\\
\\
\node{\kern 1em 1} \arrow{e}
\node{H_1(G;\K\H_n(\hat G))} \arrow{e,-} \arrow{sse}
\node{} \arrow{e}
\node{\F_{n+1}(G)} \arrow{e,-} \arrow{sse,b}{\cong}
\node{} \arrow{e}
\node{\F_{n}(G)} \arrow{e} \arrow{sse,b}{\cong}
\node{1\kern 1em}
\\
\\
\node[2]{\kern 1em 1} \arrow{e}
\node{H_1(G_k;\K\H_n(\hat G))} \arrow[2]{e}
\node[2]{\F_{n+1}(G_k)} \arrow[2]{e}
\node[2]{\F_n(G_k)} \arrow{e}
\node{1 \kern 1em}
\end{diagram}
$\hss}
\]
}
with exact rows. $H_1(G;\K\H_n(\hat G)) \to H_1(G_k;\K\H_n(\hat
G))$ is an isomorphism since $\F_{n+1}(G) \to \F_{n+1}(G_k)$ and
$\F_{n}(G) \to \F_{n}(G_k)$ are isomorphisms. Since $H_1$ commutes
with limits,
\[
H_1(G;\K\H_n(\hat G)) = \varinjlim H_1(G_k;\K\H_n(\hat G)) =
H_1(\hat G;\K\H_n(\hat G)).
\]
Moreover, from the limit of the second row it follows that
\[
\varinjlim\, (G_k)_H\^n/(G_k)_H\^{n+1} = \hat G_H\^n / \hat
G_H\^{n+1}.
\]
Also the limit homomorphism
\[
\varinjlim \phi_{n,G_k} \colon \hat G_H\^n / \hat G_H\^{n+1} \to
H_1(\hat G; \K\H_n(\hat G))
\]
is equal to the homomorphism $\Phi_{n,\hat G}$. So the commutative
diagram in our assertion is obtained by taking the limit. This
completes the proof of the assertion.
Now we prove the proposition. For the if part, note that it
suffices to investigate the surjectivity of $\hat\sigma_{n+1,G}$
since it is always injective. Since $\hat\sigma_{n,G}$ is an
isomorphism by the hypothesis, from our assertion it follows that
$\hat\sigma_{n+1,G}$ is surjective if and only if $\Phi_{n,\hat G}$
is surjective. Since $\Phi_{n,\hat G}$ is induced by the
composition
\[
\hat G_H\^n \xrightarrow{p} \hat G_H\^n/[\hat G_H\^n,\hat G_H\^n] =
H_1(\hat G;\Z\H_n(\hat G)) \xrightarrow{q} H_1(\hat G;\K\H_n(\hat
G))
\]
and $p$ is surjective, $\Phi_{n,\hat G}$ is surjective if and only
if $q$ is surjective. This proves the if part.
For the only if part, note that $\hat\sigma_{n+1,G}$ induces
$\hat\sigma_{n,G}$ on quotient groups. (To prove this, one may use
the argument of the proof of our assertion above; it shows that the
right square in the diagram of the assertion commutes even without
assuming that $\hat\sigma_{n,G}$ is an isomorphism.) Since
$\hat\sigma_{n+1,G}$ is surjective by the hypothesis, so is
$\hat\sigma_{n,G}$. Since $\hat\sigma_{n,G}$ is always injective,
it is an isomorphism. So from our assertion it follows that
$H_1(\hat G;\Z\H_n(\hat G)) \to H_1(\hat G;\K\H_n(\hat G))$ is
surjective as in the previous paragraph. This proves the only if
part.
\end{proof}
Now we investigate inductively whether $\hat\sigma_{n,G}$ is an
isomorphism.
\begin{proposition}
\label{proposition:C-H-container-for-low-index}
$\hat\sigma_{n,G}$ is an isomorphism between $\hat\H_n(G)$ and
$\F_n(G)$ for $n<2$.
\end{proposition}
\begin{proof}
For $n=0$, $\hat\sigma_{0,G}$ is obviously an
isomorphism, being a homomorphism between trivial groups.
For $n=1$, we consider
\[
\Phi_{0,\hat G}\colon H_1(\hat G;\Z) \to H_1(\hat G;\Q).
\]
By the lemma below, which is a special case of
Lemma~\ref{lemma:divisibility-of-H_1-of-R-closed-group} proved
later, it follows that $\Phi_{0,\hat G}$ is an isomorphism:
\begin{lemma}
\label{lemma:divisibility-of-H_1-of-hat-G}
For any group $G$, $H_1(\hat G;\Z)$ is divisible.
\end{lemma}
By
Proposition~\ref{proposition:criterion-for-universality-of-C-H-container},
the proof of
Proposition~\ref{proposition:C-H-container-for-low-index} is
completed.
\end{proof}
However, for $n=2$, the following result illustrates that $\hat
\sigma_n \colon \hat\H_n \to \F_n$ is not necessarily an isomorphism:
\begin{proposition}
\label{proposition:non-surjectivity-for-free-groups}
For any free group $F$ with rank $>1$, $\hat \sigma_{2,F} \colon
\hat\H_2(F) \to \F_2(F)$ is not surjective.
\end{proposition}
From Proposition \ref{proposition:non-surjectivity-for-free-groups},
it follows that $\hat \sigma_{n,F} \colon \hat\H_n(F) \to \F_n(F)$ is
not surjective for any $n > 1$ (including $n=\omega$), since $\hat
\sigma_{n,F}$ induces a non-surjective homomorphism, namely
$\hat\sigma_{2,F}$, on quotient groups. Therefore we obtain the
following result:
\begin{theorem}
\label{theorem:non-universality-of-C-H-container}
The container $\F_n$ of $\H_n$ is not universal for any $2 \le n \le
\omega$.
\end{theorem}
\begin{proof}[Proof of
Proposition~\ref{proposition:non-surjectivity-for-free-groups}]
We start with a general discussion about an arbitrary finitely
presented group~$G$. By
Lemma~\ref{lemma:divisibility-of-H_1-of-hat-G}, $H_1(\hat G;\Z)$ is
divisible. In fact, $H_1(\hat G;\Z)$ is a $\Q$-module (see
Lemma~\ref{lemma:divisibility-of-H_1-of-R-closed-group}). So, by
the definition, $\hat G_H\^1$ is the kernel of the surjection $\hat
G \to H_1(\hat G;\Q) = H_1(\hat G;\Z)\otimes \Q = H_1(\hat G;\Z)$.
It follows that $\hat G_H\^1$ is equal to the ordinary commutator
subgroup $\hat G\^1 = [\hat G,\hat G]$. Also,
\[
\H_1(\hat G)=\hat G/\hat G_H\^1 = H_1(\hat G;\Q)=H_1(G;\Q)=\Q^\mu,
\]
where $\mu$ is the first betti number of~$G$. (The third equality
is a well-known property of a homology localization. For
concreteness we remark that it can be shown by appealing to Theorem
\ref{theorem:properties-of-closures} (2): there is a sequence $G=G_0
\to G_1 \to \cdots$ of rationally 2-connected homomorphisms with
limit $\hat G$.)
We consider
\[
\Psi \colon H_1(\hat G; \Z H_1(\hat G;\Q)) \to H_1(\hat G;
\K H_1(\hat G;\Q)).
\]
Since $H_1(\hat G;\Q)$ is abelian, $\K H_1(\hat G;\Q)$ is the
ordinary localization $S^{-1} \cdot \Z H_1(\hat G;\Q)$ of the
commutative ring $\Z H_1(\hat G;\Q)$ where $S=\Z H_1(\hat
G;\Q)-\{0\}$. Moreover
\[
H_1(\hat G; \K H_1(\hat G;\Q)) = S^{-1} \cdot
H_1(\hat G; \Z H_1(\hat G;\Q))
\]
since $S^{-1} \cdot \Z H_1(\hat G;\Q)$ is a flat $\Z H_1(\hat
G;\Q)$-module. Therefore $\Psi$ is surjective if and only if every
element in $H_1(\hat G; \Z H_1(\hat G;\Q))$ is divisible by any
element in $S$, that is, for any $u \in H_1(\hat G; \Z H_1(\hat
G;\Q))$ and $s\in S$, there exists $v \in H_1(\hat G; \Z H_1(\hat
G;\Q))$ such that $s\cdot v = u$.
We will show that the divisibility criterion is not satisfied in
case that $G$ is a free group $F$ of rank $\mu > 1$. Let $x$ and
$y$ be two distinct generators of~$F$. As an abuse of notation, for
an element $g$ in $F$, we denote the image of $g$ under $F \to \hat
F$ by~$g$. (Indeed, it can be shown that $F \to \hat F$ is
injective for a free group~$F$, although we will not use it.)
Consider the element
\[
u \in \hat F\^1 / [\hat F\^1, \hat F\^1] = H_1(\hat F; \Z H_1(\hat
F;\Q))
\]
which is represented by $xyx^{-1}y^{-1} \in \hat F\^1$, and the
element
\[
s = [x]-1 \in \Z H_1(F;\Q)=\Z H_1(\hat F;\Q)
\]
where $[x]$ is the homology class of~$x$. Suppose that there exists
$v \in \hat F\^1$ such that $s\cdot v = u$ in $H_1(\hat F; \Z
H_1(\hat F;\Q))$. Since the action of $H_1(\hat F;\Q)$ on $H_1(\hat
F;\Z H_1(\hat F;\Q))$ is given by conjugation, we have
\[
xvx^{-1}v^{-1} \equiv u = xyx^{-1}y^{-1} \mod [\hat F\^1, \hat F\^1].
\]
From this it follows that $xyx^{-1}y^{-1}$ is in $\hat F_3 = [\hat
F, [\hat F, \hat F]]$, the third term of the lower central series
of~$\hat F$.
We will show that a contradiction is derived from this. Obviously,
$xyx^{-1}y^{-1}$ is not in $F_3$, say by Hall's basis theorem. To
generalize this to $\hat F$, we use the rational version of
Stallings' theorem: the rational derived series $G_q^\Q$ of a group
$G$ is defined inductively by
\[
G_1^\Q = G, \quad G_{q+1}^\Q = \text{kernel of } G_q^\Q \to
\frac{G_q^\Q}{[G,G_q^\Q]} \to \frac{G_q^\Q}{[G,G_q^\Q]}
\mathbin{\mathop{\otimes}_\Z} \Q.
\]
Then for any group homomorphism $\pi \to G$ which is rationally
2-connected, $\pi/\pi_q^\Q \to G/G_q^\Q$ is injective for
all~$q$~\cite{Stallings:1965-1}. We also need the following facts:
obviously $G_q \subset G_q^\Q$ for any group $G$, and for a free
group $F$, $F_q = F_q^\Q$ since $F_q/F_{q+1}$ is known to be torsion
free as an abelian group.
Now, applying the rational version of Stallings' theorem to our $F
\to \hat F$ which is rationally 2-connected, it follows that
$xyx^{-1}y^{-1} \in F_3^\Q = F_3$ since $xyx^{-1}y^{-1} \in \hat F_3
\subset \hat F_3^\Q$. This is a contradiction.
\end{proof}
\begin{remark}
In the proof of
Proposition~\ref{proposition:non-surjectivity-for-free-groups}, we
considered a particular element $s=[x]-1$ in $S$ to show that the
divisibility criterion is not satisfied. Our argument also works
for any $s$ contained in the kernel of the augmentation homomorphism
$\Z H_1(F;\Q) \to \Z$. Such an element $s$ can be used for this
purpose since it is invertible in the Ore localization; but it is
not in the Cohn localization. As mentioned in the introduction,
this fact motivates a study of a more natural series similar to the
torsion-free derived series but defined using the Cohn localization
instead of the Ore localization. (See also Remark 5.22 of
\cite{Cochran-Harvey:2004-1}.) We do not address this issue in
depth in the present paper.
\end{remark}
\section{Nullhomologous equations and $R$-closures}
\label{section:nullhomologous-equations}
Let $G$ be a group. We call an element $w$ in the free product
$G*F\langle x_1,\ldots,x_n\rangle$ a \emph{monomial over $G$ in
$x_1,\ldots,x_n$}, where $F\langle x_1,\ldots,x_n\rangle$ denotes
the free group generated by $x_1,\ldots, x_n$. Viewing a monomial $w$
as a word in elements of $G$ and $x_1,\ldots,x_n$, we sometimes write
$w=w(x_1,\ldots,x_n)$.
We consider systems of equations over $G$ of the following form:
\[
x_i^e = w_i(x_1,\ldots,x_n), \quad i=1,\ldots, n
\]
where $x_1,\ldots,x_n$ are considered as indeterminates, $e$ is a
nonzero integer, and $w_i(x_1,\ldots,x_n)$ is a monomial over~$G$. An
$n$-tuple $(g_1,\ldots, g_n)$ of elements in $G$ is called a
\emph{solution} of the system if $x_i = g_i$ satisfies the equations,
that is, $g_i^e$ is equal to $w_i(g_1,\ldots,g_n)$ in $G$ for all~$i$.
Henceforth we fix a subring $R$ of~$\Q$. We denote by $D_R$ the set
of denominators of reduced fractional expressions of elements in~$R$.
$D_R$~is a multiplicatively closed set.
\begin{definition}
A system $\{x_i^e = w_i(x_1,\ldots,x_n)\}$ over $G$ is called
\emph{$R$-nullhomologous} if $e$ is in $D_R$ and each
$w_i(x_1,\ldots,x_n)$ is sent to the trivial element by the
canonical projection
\[
G*F \to F \to F/[F,F] = H_1(F)
\]
where $F=F\langle x_1,\ldots,x_n\rangle$.
\end{definition}
\begin{definition}
\begin{enumerate}
\item A group $A$ is called \emph{$R$-closed} if every
$R$-nullhomologous system over $A$ has a unique solution in~$A$.
\item For a group $G$, an $R$-closed group $\hat G$ equipped with a
homomorphism $G \to \hat G$ is called an \emph{$R$-closure} of $G$
if for any homomorphism of $G$ into an $R$-closed group $A$, there
exists a unique homomorphism $\hat G \to A$ making the following
diagram commute:
\[
\begin{diagram}
\node{G}\arrow{s}\arrow{e}
\node{\hat G}\arrow{sw,..}\\
\node{A}
\end{diagram}
\]
That is, $G\to \hat G$ is the universal (initial) object in the
category of homomorphisms of $G$ into $R$-closed groups.
\end{enumerate}
\end{definition}
\begin{remark}
Although we do not need it in this paper, it can be seen that every
$R$-nullhomologous system has a unique solution if and only if so
does every system of the form $\{x_i^e = g_i u_i(x_1,\ldots,x_n)\}$
where $g_i \in G$, $u_i \in [F,F]$, and $e\in D_R$. This form is
more similar to the equations considered in work of
Farjoun--Orr--Shelah~\cite{Farjoun-Orr-Shelah:1989}. The only if
part is clear. For the if part, suppose an $R$-nullhomologous
system $\{x_i^e = w_i(x_1,\ldots,x_n)\}$ over $G$ is given. If the
variables $x_i$ commuted with elements of $G$ appearing in $w_i$,
then $w_i$ would be of the form $g_i \cdot u_i$ where $g_i \in G$
and $u_i\in [F,F]$. Therefore we can rewrite $w_i$ as $g_i \cdot
\big(\prod_j [h_{ij}, x_{ij}^{\pm1}]\big) \cdot u_i$ where $h_{ij}
\in G$. For each $h_{ij}$, we adjoin to the system an indeterminate
$y_{ij}$ and an equation $y_{ij} = h_{ij}$. Replacing each
occurrence of $h_{ij}$ in the original equation $x_i^e=w_i$ by the
new indeterminate $y_{ij}$, we obtain a system of the desired form.
From this the assertion follows.
\end{remark}
The following definition generalizes the notion of invisible subgroups
in Levine's work~\cite{Levine:1989-1}.
\begin{definition}
A normal subgroup $N$ in $G$ is called \emph{$R$-invisible} if
\begin{enumerate}
\item $N$ is normally finitely generated in $G$, and
\item The order of every element in $N/[G,N]$ is (finite and)
in~$D_R$.
\end{enumerate}
\end{definition}
We recall that $N$ is said to be \emph{normally finitely generated} in
$G$ if there exist finitely many elements $a_1,\ldots,a_n$ in $G$
such that $N$ is the smallest normal subgroup containing the~$a_i$.
In this case the $a_i$ are called normal generators of~$N$. Note that
a normally finitely generated subgroup $N$ in $G$ is $R$-invisible if
and only if $(N/[G,N])\otimes_\Z R =0$.
\begin{remark}
In his work on homology localizations, Bousfield called a normal
subgroup $N$ in $G$ \emph{$\pi$-perfect} if $N=[G,N]$
\cite{Bousfield:1975-1}. For our purpose, we need to modify it
regarding the coefficient $R$ and the finiteness assumption as in the
above definition. When $R=\Z$, our definition agrees with the
definition of an \emph{invisible} subgroup due to
Levine~\cite{Levine:1989-1}.
\end{remark}
In what follows we discuss some useful relationships between
$R$-invisible subgroups and $R$-nullhomologous systems.
\begin{lemma}\label{lemma:invisible-subgps-are-killed-in-closed-gps}
Suppose $\phi\colon G \to A$ is a homomorphism into an $R$-closed
group~$A$. Then every $R$-invisible subgroup $N$ in $G$ is
contained in the kernel of~$\phi$.
\end{lemma}
\begin{proof}
Choose normal generators $a_1,\ldots,a_n$ of $N$. Since the order
of $a_i$ is in $D_R$, there exist an element $e\in D_R$ such that
$a_i^e \in [G,N]$ for all~$i$. So we can write $a_i^e$ as a product
of commutators $[b_{ij},c_{ij}]$ where $b_{ij}\in N$, $c_{ij} \in
G$. Furthermore $b_{ij}$ can be written as a product of conjugates
of the~$a_k$. Replacing each occurrence of $a_k$ in this expression
of $b_{ij}$ by an indeterminate $x_k$ and plugging the result into
the above expression of $a_i^e$, we obtain a word
$w_i(x_1,\ldots,x_n)$ in $G*F$, $F$ is the free group generated by
the $x_i$, such that the system
\[
x_i^e = w_i(x_1,\ldots,x_n), \qquad i=1,\ldots,n
\]
has two sets of solutions, $\{x_i = 1\}$ and $\{x_i = a_i\}$. It is
easily seen that this system is $R$-nullhomologous.
Denote by $w_i^\phi$ the image of $w_i$ under $G*F \to A*F$. It
gives rise to an $R$-nullhomologous system
\[
x_i^e = w_i^\phi(x_1,\ldots,x_n), \qquad i=1,\ldots,n
\]
over $A$, which has two solution sets $\{x_i = 1\}$ and $\{x_i =
\phi(a_i)\}$. By the uniqueness of a solution over $A$, it follows
that $\phi(a_i)=1$. Thus $\phi(N)$ is trivial.
\end{proof}
\begin{lemma}\label{lemma:product-of-R-invisible-subgps}
If $N_1$ and $N_2$ are $R$-invisible subgroups in $G$, then
$N=N_1N_2$ is also $R$-invisible in~$G$.
\end{lemma}
\begin{proof}
Obviously $N$ is normally finitely generated. For any $n\in N$,
write $n=n_1n_2$ where $n_i \in N_i$. Since $N_i$ is $R$-invisible,
there exist $e\in D_R$ such that $n_i^e \in [G,N_i]$. Then
\[
(n_1n_2)^e \equiv n_1^e n_2^e \equiv 1 \mod [G,N].
\]
This shows that the order of $n[G,N]$ in $N/[G,N]$ is (a divisor
of)~$e$.
\end{proof}
\begin{lemma}\label{lemma:no-nontrivial-invisible-subgroup}
Suppose $G$ is a group and $N$ is the union of all $R$-invisible
subgroups in~$G$. Then $G/N$ has no nontrivial $R$-invisible
subgroup.
\end{lemma}
\begin{proof}
First of all, $N$ is a normal subgroup by the previous lemma.
Suppose $H/N$ is $R$-invisible in $G/N$ for some $H\subset G$.
Choose a finite normal generator set $\{h_iN \}$ of $H/N$, $h_i \in
H$. It suffices to show that $h_i \in N$ for each~$i$. For some
$e\in D_R$, $h_i^e \in [H/N,G/N]$. Therefore $h_i^e \in n_i[H,G]$
for some $n_i \in N$. By the previous lemma, there exists an
$R$-invisible subgroup $K$ in $G$ such that $n_i \in K$ for all~$i$.
Then the normal subgroup $K_1$ generated by $K$ and the $h_i$ is
$R$-invisible in~$G$. It follows that $h_i \in K_1 \subset N$.
\end{proof}
\begin{lemma}\label{lemma:uniqueness-of-solution}
If $G$ has no nontrivial $R$-invisible subgroup, then any
$R$-nullhomologous system over $G$ has at most one solution.
\end{lemma}
\begin{proof}
Suppose $S=\{x_i^e = w_i(x_1,\ldots,x_n)\}$ is $R$-nullhomologous
and $\{x_i=a_i\}$ and $\{x_i=b_i\}$ are solutions of $S$ over~$G$.
Let $N$ be the normal subgroup in $G$ generated by the
$a_i^{\mathstrut} b_i^{-1}$, and let
\[
u_i(x_1,\ldots,x_n)=w_i(x_1b_1,\ldots,x_nb_n)b_i^{-e}.
\]
Since $u_i(1,\ldots,1) = w_i(b_1,\ldots,b_n)b_i^{-e} = 1$,
we can write $u_i$ as a word of the form
\begin{align*}
u_i &= \prod_j g_{ij}^{\mathstrut} x_{ij}^{\pm1} g_{ij}^{-1} \\
&= \Big( \prod_j (x_{i1}^{\pm1} \cdots x_{i,j-1}^{\pm1})
[g_{ij}^{\mathstrut}, x_{ij}^{\pm1}] (x_{i1}^{\pm1} \cdots
x_{i,j-1}^{\pm1})^{-1} \Big) \prod_j x_{ij}^{\pm1}
\end{align*}
where $g_{ij} \in G$ and $x_{ij} = x_{k_{ij}}$ for some~$k_{ij}$.
Furthermore, each $u_i$ is killed by $G*F \to F/[F,F]$ where
$F=F\langle x_1,\ldots,x_n \rangle$, since so is~$w_i$. It follows
that $\prod_j x_{ij}^{\pm1}$, the last term of the above expression,
is contained in $[F,F]$. Therefore
\begin{align*}
a_i^{e} b_i^{-e} &= w_i(a_1,\ldots,a_n)b_i^{-e} \\
&= u_i(a_1^{\mathstrut} b_1^{-1},\ldots, a_n^{\mathstrut} b_n^{-1}) \in [G,N].
\end{align*}
Now we have
\begin{align*}
(a_i^{\mathstrut} b_i^{-1})^e &\equiv (a_i^{\mathstrut} b_i^{-1})^e \cdot a_i^{-e} b_i^{e} \\
& \equiv (a_i^{\mathstrut}b_i^{-1})^{e-1} \cdot a_i^{\mathstrut}b_i^{-1} \cdot a_i^{-e} b_i^{e} \\
& \equiv (a_i^{\mathstrut}b_i^{-1})^{e-1} \cdot a_i^{-e} \cdot a_ib_i^{-1} \cdot b_i^{e} \\
& \equiv (a_i^{\mathstrut}b_i^{-1})^{e-1} \cdot a_i^{-e+1} b_i^{e-1} \equiv
\cdots \equiv 1 \mod [G,N]
\end{align*}
This shows that $N$ is $R$-invisible. By the hypothesis, $N$ is
trivial and $a_i=b_i$.
\end{proof}
As an immediate consequence of the definition, we have the following
divisibility result:
\begin{lemma}
\label{lemma:divisibility-of-H_1-of-R-closed-group}
If $A$ is $R$-closed, then $H_1(A;\Z)$ is an $R$-module.
\end{lemma}
\begin{proof}
Let $g$ be an element in $A$, and let $e$ be an element in~$D_R$.
Consider the equation $x^e = g$. Since it is $R$-nullhomologous,
there is a solution $x=h$ in~$A$. It follows that the homology
class of $g$ is divisible by~$e$.
\end{proof}
\section{$R$-closures and localizations}
\label{section:localization-wrt-2-connected-morphisms}
We begin this section by recalling the definition of a localization.
In general, we think of a category $\mathcal{C}$ and a class of
morphisms~$\Omega$ in $\mathcal{C}$.
\begin{definition}
\begin{enumerate}
\item An object $A$ in $\mathcal{C}$ is called \emph{local} with
respect to $\Omega$ if for any morphism $\pi \to G$ in $\Omega$
and any morphism $\pi \to A$, there exists a unique morphism $G
\to A$ making
\[
\begin{diagram}
\node{\pi}\arrow{e}\arrow{s}
\node{G} \arrow{sw,..} \\
\node{A}
\end{diagram}
\]
commute.
\item A \emph{localization} with respect to $\Omega$ is a pair
$(E,p)$ of a functor $E\colon \mathcal{C} \to \mathcal{C}$ and a
natural transformation $p\colon \id_\mathcal{C} \to E$ (that is,
each object $G$ is equipped with a morphism $p_G\colon G \to
E(G)$) such that for any morphism $G\to A$ into a local object $A$,
there is a unique morphism $E(G) \to A$ making
\[
\begin{diagram}
\node{G}\arrow{e}\arrow{s}
\node{E(G)} \arrow{sw,..} \\
\node{A}
\end{diagram}
\]
commute.
\end{enumerate}
\end{definition}
Of course our main interest is the localization of groups with respect
to the class $\Omega^R$; recall that $\Omega^R$ is the class of group
homomorphisms $\phi\colon \pi \to G$ where $\pi$ is finitely
generated, $G$ is finitely presented, and $\phi$ is 2-connected on
$R$-homology. From now on local groups and localizations are always
with respect to our~$\Omega^R$.
\begin{theorem}\label{theorem:closed-iff-local}
A group $A$ is $R$-closed if and only if $A$ is local with respect
to~$\Omega^R$.
\end{theorem}
\begin{proof}
Suppose that $A$ is $R$-closed. To show that $A$ is local, suppose
that a morphism $\alpha\colon \pi \to G$ in $\Omega^R$ and a
morphism $\phi\colon \pi \to A$ are given. We will show that there
exists a unique morphism $\varphi\colon G \to A$ that $\phi\colon
\pi\to A$ factors through.
Choose generators $h_1,\ldots,h_n$ of~$G$. Since $\alpha$ induces
an isomorphism
\[
H_1(\pi;R)=\frac{\pi}{[\pi,\pi]}
\mathbin{\mathop{\otimes}\limits_\Z} R \to \frac{G}{[G,G]}
\mathbin{\mathop{\otimes}\limits_\Z} R,
\]
it follows that for any element $x$ in $G/[G,G]$, $x^r$ is contained
in the image of $\pi/[\pi,\pi]$ for some $r\in D_R$. Therefore
there exists $e\in D_R$ such that each $h_i^e$ can be written as
\[
h_i^e = \alpha(g_i) \cdot \prod_j [u_{ij}, v_{ij}]
\]
where $g_i\in \pi$ and $u_{ij}=u_{ij}(h_1,\ldots,h_n)$,
$v_{ij}=v_{ij}(h_1,\ldots,h_n)$ are words in $h_1,\ldots,h_n$.
Consider the system $S$ of equations $x_i^e = w_i(x_1,\ldots,x_n)$
where the element $w_i$ in $G*F\langle x_1,\ldots,x_n\rangle$ is
given by
\[
w_i(x_1,\ldots,x_n) = g_i \cdot \prod_j
[u_{ij}(x_1,\ldots,x_n),v_{ij}(x_1,\ldots,x_n)].
\]
Then $S$ is $R$-nullhomologous.
We associated to the system $S$ a new group $\pi_S$ obtained by
``adding'' to $\pi$ a solution $\{z_i\}$ to $S$; formally, it is
defined to be a amalgamated product of $\pi$ and $F\langle
z_1,\ldots,z_n\rangle$:
\[
\pi_S = \langle \pi, z_1,\ldots,z_n \mid z_i^e=w_i(z_1,\ldots,z_n),
\, i=1,\ldots,n\rangle.
\]
Note that, since $e\in D_R$, the canonical homomorphism $\pi \to
\pi_S$ is 2-connected on $R$-homology; it can be seen by computing
$H_1(\pi_S;R)$ and $H_2(\pi_S;R)$ using the complex obtained by
attaching to $K(\pi,1)$ 1-cells and 2-cells corresponding the new
generators and relations.
$\pi \to G$ and $z_i \to h_i \in G$ induce a surjection $\beta
\colon \pi_S \to G$. Since $A$ is $R$-closed, there is a unique
solution $\{x_i=a_i\}$ of the $R$-nullhomologous system $S^\phi =
\{x_i^e = w_i^\phi(x_1,\ldots,x_n)\}$ over~$A$, which is the image
of $S$ under~$\phi$. $\phi\colon \pi \to A$ and $z_i \to a_i \in A$
induce a homomorphism $\phi_S\colon \pi_S \to A$.
\[
\begin{diagram}
\node{\pi}\arrow{e}\arrow{s,l}{\phi}
\node{\pi_S} \arrow{sw,l}{\phi_S} \arrow{e,t}{\beta}
\node{G} \arrow{sww,b,..}{\varphi} \\
\node{A}
\end{diagram}
\]
We will show that $\phi_S$ factors through a homomorphism
$\varphi\colon G \to A$. Let $N$ be the kernel of~$\beta$.
Applying the Stallings exact sequence~\cite{Stallings:1965-1} to
$\beta$ which is surjective, we obtain a long exact sequence
\[
H_2(\pi_S;R) \to H_2(G;R) \to \frac{N}{[\pi_S,N]}\otimes_\Z R \to
H_1(\pi_S;R) \to H_1(G;R).
\]
Since $\alpha \colon \pi \to G$ and $\pi \to \pi_S$ are 2-connected
on $R$-homology, so is~$\beta$. Thus $(N/[\pi_S,N])\otimes R=0$.
Since $\pi_S$ is finitely generated and $G$ is finitely presented,
$N$ is finitely normally generated in~$\pi_S$. This shows that $N$
is $R$-invisible in~$\pi_S$. By
Lemma~\ref{lemma:invisible-subgps-are-killed-in-closed-gps},
$\phi_S(N)$ is trivial. It follows that $\phi_S$ induces a
homomorphism $\varphi\colon G\to A$ as desired.
If another homomorphism $\varphi' \colon G \to A$ satisfies $\phi =
\varphi'\circ \alpha$, then $\{x_i = \varphi'(h_i)\}$ is a solution of
the system $S^\phi$. By the uniqueness of a solution, we have
$\varphi(h_i) = a_i = \varphi'(h_i)$, that is, $\varphi = \varphi'$.
This completes the proof of the only if part.
For the converse, suppose that $A$ is local, and an
$R$-nullhomologous system $S=\{x_i^e = w_i(x_1,\ldots,x_n)\}$ over
$A$ is given. There are finitely many elements in $A$ which appears
in the words $w_i$. Let $G$ be the free group generated by (symbols
associated to) these elements, and $\phi \colon G \to A$ be the
natural homomorphism. Lifting $S$, we obtain a system $S' = \{x_i^e
= w_i(x_1,\ldots,x_n)\}$ over $G$ which is sent to $S$ by~$\phi$.
Consider the group $G_{S'}$ obtained by ``adding a solution
$\{z_i\}$ to the system $S'$'' as before. Then $G_{S'}$ is finitely
presented and the canonical homomorphism $\alpha \colon G \to
G_{S'}$ is 2-connected. Since $A$ is local, there exists a unique
homomorphism $\varphi\colon G_{S'} \to A$ making
\[
\begin{diagram}
\node{G}\arrow{e,t}{\alpha} \arrow{s,l}{\phi}
\node{G_{S'}} \arrow{sw,r,..}{\varphi} \\
\node{A}
\end{diagram}
\]
commute. Now $\{\varphi(z_i)\}$ is a solution of $S$ over $A$.
If there is another solution $\{x_i = a_i\}$ of $S$, then $x_i \to
a_i$ gives rise to another homomorphism $\varphi'\colon G_{S'} \to
A$ making the above diagram commute. By the uniqueness of
$\varphi$, $\varphi' =\varphi$, and therefore, $a_i = \varphi(z_i)$.
This completes the proof.
\end{proof}
\section{Existence of $R$-closures}
\label{section:existence-of-closures}
\begin{theorem}\label{theorem:existence-of-closure}
For any subring $R$ of $\Q$ and any group $G$, there is an
$R$-closure $G \to \hat G$.
\end{theorem}
\begin{proof} Basically the construction consists of two parts: adjoin
solutions repeatedly so that every system has at least one solution
eventually, and take a quotient of the resulting group to identify
different solutions if any.
This idea is formalized as follows. We construct a sequence $G_0,
G_1,\ldots$ of groups inductively. Let $G_0=G$. Suppose $G_k$ has
been defined. Let $\mathcal{S}_k$ be the set of all
$R$-nullhomologous systems over~$G_k$. We associate a symbol $z_i$
to an indeterminate $x_i$ of a system in~$\mathcal{S}_k$, and let
$F_k$ be the free group generated by all the symbols $z_i$. Let
$G_{k+1} = G_k*F_k$ modulo the relations corresponding the systems
in~$\mathcal{S}_k$, that is, each equation
$x_i^e=w_i(x_1,\ldots,x_n)$ gives rise to a defining relation
$z_i^e=w_i(z_1,\ldots,z_n)$ of~$G_{k+1}$. Let $\bar G=\varinjlim
G_k$. Let $N$ be the union of all $R$-invisible subgroups in $\bar
G$, and finally let $\hat G = \bar G/N$.
We will show that the canonical homomorphism $\Phi\colon G \to \hat
G$ is an $R$-closure of~$G$. First we claim that $\hat G$ is
$R$-closed. For the existence of a solution, suppose that
$S=\{x_i^e = w_i(x_1,\ldots,x_n)\}$ be an $R$-nullhomologous system
over~$\hat G$. Since the $w_i$ involve only finitely many elements
of $\hat G$, $S$ lifts to an $R$-nullhomologous system over some
$G_k$, that is, each $w_i$ is the image of an element $w_i'$ in $G_k
* F$ where $F=F\langle x_1,\ldots,x_n\rangle$. Sending $w_i'$ via
$G_k *F \to G_{k+1}*F$, we obtain an $R$-nullhomologous system over
$G_{k+1}$ which has a solution $\{x_i = a_i\}$ by our construction
of~$G_{k+1}$; recall that $G_{k+1}$ is obtained by adjoining
solutions of all $R$-nullhomologous systems over~$G_k$. Obviously
the image of the $a_i$ in $\hat G$ is a solution of the given
system~$S$. On the other hand, $\hat G$ has no nontrivial
$R$-invisible subgroup by
Lemma~\ref{lemma:no-nontrivial-invisible-subgroup}. From
Lemma~\ref{lemma:uniqueness-of-solution}, the uniqueness of a
solution follows. This proves the claim.
Now it remains to show that $\Phi\colon G\to \hat G$ is a universal
(initial) object. Suppose that $\phi\colon G \to A$ is a
homomorphism of $G$ into an $R$-closed group~$A$. Since $G_1$ is
obtained from $G=G_0$ by adjoining solutions of $R$-nullhomologous
systems, there exists a unique homomorphism $\phi_1\colon G_1 \to A$
making the below diagram commute; $\phi_1$ is defined by sending new
generators $z_i$ of $G_1$ associated to a system $S=\{x_i =
w_i(x_1,\ldots,x_n)\}$ over $G_0$ to the solution of $S^\phi$ over
$A$, and the uniqueness of $\phi_1$ follows from the uniqueness of a
solution over~$A$.
\[
\begin{diagram}
\node{G} \arrow{s,l}{\phi}\arrow{e}
\node{G_1} \arrow{sw,l,..}{\phi_1}\arrow{e}
\node{G_2} \arrow{sww,r,..}{\phi_2}\arrow{e}
\node{\cdots} \arrow{e}
\node{\bar G = \varinjlim G_k} \\
\node{A}
\end{diagram}
\]
Repeating the same argument, we can inductively construct a sequence
of homomorphisms $\phi_k\colon G_k \to A$ which make the diagram
commute. Passing to the limit, the $\phi_k$ induce a homomorphism
$\bar\phi\colon \bar G \to A$. By
Lemma~\ref{lemma:invisible-subgps-are-killed-in-closed-gps},
$\varphi'$ kills each $R$-invisible subgroup, and so $N$ is
contained in the kernel of~$\bar\phi$. It follows that $\bar\phi$
gives rise to a homomorphism $\varphi\colon \hat G = \bar G/N \to A$
such that $\varphi \circ \Phi = \phi$.
Suppose another homomorphism $\varphi'\colon \hat G \to A$ satisfies
$\varphi' \circ \Phi = \phi$. Consider the composition
$\phi_k'\colon G_k \to \hat G \xrightarrow{\varphi'} A$. Then the
$\phi_k$ make the above diagram commute as well as the $\phi_k$.
From the uniqueness of a solution over $A$, it follows that $\phi_k =
\phi_k'$ for every~$k$. Passing to the limit, it follows that
$\varphi=\varphi'$. This completes the proof.
\end{proof}
{\long\def\ignoreme{
\begin{remark}
Compared with the proofs of the existence of algebraic closures for
the Vogel--Levine localization and the Bousfield localization, the
most significant difference is that we need to iterate the process
of adjoining solutions; recall that we have constructed groups $G_k$
and showed that the algebraic closure is a quotient of the limit.
In the construction of Levine~\cite{Levine:1989-1} and
Farjoun--Orr--Shelah~\cite{Farjoun-Orr-Shelah:1989}, it is
sufficient to consider $G_1$ only, that is, the algebraic closure of
$G$ is a quotient of the group obtained by adjoining solutions of
all systems \emph{over $G$}. The main reason of this sophistication
in our case is that the exponent $e$ can be greater than~$1$. In
fact, if $e$ were always 1, then we could reduce a nullhomologous
system over $G_{k+1}$ into a system over $G_{k}$ by using a sort of
back-substitution argument as in \cite{Levine:1989-1}
and~\cite{Farjoun-Orr-Shelah:1989}, so that every element in the
limit $\varinjlim G_k$ was (a part of) a solution of a system
over~$G$. In particular, if the coefficient ring $R$ is $\Z$, then
our closure has the following property: any element in $\hat G$ is
(a part of) a solution of a nullhomologous system over~$G$.
\end{remark}
}}
\begin{remark}
There is an alternative construction of an $R$-closure: let $G_0=G$
as before, and assuming $G_k$ has been defined, consider the group
obtained from $G_k$ by adjoining solutions of all $R$-nullhomologous
systems, and let $G_{k+1}$ be its quotient by the union of all
$R$-invisible subgroups. Then it can be proved that $\varinjlim
G_k$ is an $R$-closure of~$G$.
\end{remark}
\begin{corollary}\label{corollary:existence-of-localization}
There is a localization $(E,p)$ with respect to $\Omega^R$.
\end{corollary}
\begin{proof}
For each group $G$, define $E(G)=\hat G$ and $p_G \colon G \to \hat
G$ to be the homomorphism $\Phi$ constructed above. For any
homomorphism $\phi\colon \pi \to G$, the composition $\pi\to G \to
E(G)=\hat G$ is a homomorphism of $\pi$ into an $R$-closed group.
By the universal property of the $R$-closure $\pi \to E(\pi)=\hat
\pi$, there is a unique homomorphism $E(\pi) \to E(G)$ that the
composition factors through; we denote this homomorphism by
$E(\phi)\colon E(\pi) \to E(G)$. One can check that $E$ is a
functor and $p$ is a natural transformation by using the universal
property of $R$-closures in a straightforward way. Finally, since
$G \to E(G)$ is an $R$-closure, it is initial among homomorphisms of
$G$ into local groups, by Theorem~\ref{theorem:closed-iff-local}.
\end{proof}
We sometimes denote the induced homomorphism $E(\phi)$ by~$\hat\phi$.
From the universal property of the $R$-closure, a natural isomorphism
theorem follows:
\begin{proposition}
\label{proposition:isomorphism-induced-by-R-closure}
If $\alpha\colon\pi \to G$ is in $\Omega^R$, then the induced
homomorphism $\hat \alpha \colon \hat \pi \to \hat G$ is an
isomorphism.
\end{proposition}
\begin{proof}
We consider the following diagram:
\[
\begin{diagram}
\node{\pi} \arrow{e,t}{\alpha} \arrow{s,l}{p_\pi}
\node{G} \arrow{s,r}{p_G} \arrow{sw,t,..}{\phi} \\
\node{\hat \pi} \arrow{e,b}{\hat\alpha}
\node{\hat G}
\end{diagram}
\]
Since $\alpha$ is in $\Omega^R$ and $\hat\pi$ is local, there is
$\phi\colon G\to \hat \pi$ such that $\phi\alpha=p_\pi$. Since
$\hat\pi$ is local and $\hat G$ is the localization, there is
$\psi\colon \hat G \to \hat\pi$ such that $\psi p_G = \phi$. We
will show that $\psi$ is an inverse of $\hat\alpha$. Observe that
$(\hat\alpha \psi p_G)\alpha = p_G\alpha$. Since $\hat G$ is local
and $\alpha\in \Omega^R$, we have $\hat\alpha \psi p_G = p_G$. It
follows that $\hat\alpha\psi=\mathrm{id}_{\hat G}$ by the uniqueness
of a map $\beta \colon \hat G \to \hat G$ such that $\beta p_G =
p_G$. On the other hand, since $\psi \hat\alpha p_\pi = \psi p_G
\alpha = \phi \alpha = p_\pi$, $\psi \hat\alpha = \mathrm{id}_{\hat
\pi}$ by a similar uniqueness argument.
\end{proof}
\begin{remark}
\begin{enumerate}
\item For subrings $R\subset S\subset \Q$, temporarily denote by
$\hat G^R$ and $\hat G^S$ the $R$- and $S$-closures of a group
$G$, respectively. Then, since $\Omega^R \subset \Omega^S$, there
is a natural transformation $\hat G^R \to \hat G^S$ making the
following diagram commute:
\[
\begin{diagram}
\node{G} \arrow[2]{e} \arrow{se}
\node[2]{\hat G^S} \\
\node[2]{\hat G^R} \arrow{ne}
\end{diagram}
\]
\item Similar conclusion holds for Levine's algebraic closure
defined in~\cite{Levine:1989-1}, which is a localization with
respect to the collection $\Omega^{Levine}$ of group homomorphisms
$\alpha\colon \pi\to G$ such that $\pi$, $G$ are finitely
presented, $\alpha$ is (integrally) 2-connected, and $G$ is
normally generated by the image of~$\alpha$. Namely, denoting
Levine's algebraic closure by $\hat G^{Levine}$, there is a
natural transformation $\hat G^{Levine} \to \hat G^\Z$ making a
similar diagram commute, since $\Omega^{Levine} \subset
\Omega^\Z$.
\end{enumerate}
\end{remark}
We finish this section with some results on the $R$-closure of a
finitely presented group.
\begin{proposition}\label{theorem:closure-as-limit}
If $G$ is finitely presented, then there is a sequence
\[
G=P_0 \to P_1 \to P_2 \to \cdots
\]
of homomorphisms in $\Omega^R$ (in particular, each $P_k$ is
finitely presented) such that the limit homomorphism $G = P_0 \to
\varinjlim P_k$ is an $R$-closure, that is, $\hat G\cong \varinjlim
P_k$.
\end{proposition}
\begin{proof}
We use the notations of the proof of
Theorem~\ref{theorem:existence-of-closure}. Recall that
$\mathcal{S}_k$ is the set of all $R$-nullhomologous systems over
$G_k$. We denote $\mathcal{S}=\bigcup \mathcal{S}_k$. Then an
element $S=\{x_i = w_i\}$ in $\mathcal{S}$ is a system over $G_p$
for some~$p$. By our construction of $G_p$, each $w_i$ can be
viewed as a word in indeterminates $x_1,x_2,\ldots$ and solutions of
other systems over some $G_q$ where $q<p$.
Since $G_0 = G$ is countable, an induction shows that $G_k$ and
$\mathcal{S}_k$ are always countable. Thus the union $\mathcal{S}$
of all $\mathcal{S}_k$ is countable. From this it can be seen that
we can enumerate elements of $\mathcal{S}$ as a sequence $T_1,
T_2,\ldots$ of systems which satisfies the following: suppose the
system $T_k = \{x_i=w_i\}$ is over~$G_p$. Then each $w_i$ involves
only elements in $G_p$ that can be expressed as a product of
solutions (and their inverses) of other systems $T_q$ such that
$q<k$. In other words, the system $T_{k}$ can be lifted to a system
over the group
\[
Q_{k-1} = (\cdots((G_{T_1})_{T_2})\cdots)_{T_{k-1}}.
\]
(Recall our notation that $Q_{k-1}$ is the group obtained from $G$
by adjoining the solutions of the systems $T_1, T_2, \ldots, T_k$.)
Note that there is a canonical map $Q_k \to Q_{k+1}$; it is in
$\Omega^R$ as we discussed before. Furthermore, it is obvious that
$\varinjlim Q_k \cong \bar G=\varinjlim G_k$.
We claim that for each $k$ there is an $R$-invisible subgroup $N_k$
in $Q_k$ such that
\begin{enumerate}
\item $Q_k \to Q_{k+1}$ sends $N_k$ into $N_{k+1}$, and
\item for any $R$-invisible subgroup $K$ in $\bar G$, there is $k$
such that $K$ is contained in the normal subgroup of $G$ generated
by the image of $N_k$ under $Q_k \to \bar G$.
\end{enumerate}
For, since $\bar G$ is countable, we can arrange all $R$-invisible
subgroups in $\bar G$ as a sequence, and by appealing to
Lemma~\ref{lemma:product-of-R-invisible-subgps}, we can produce an
increasing sequence $L_1 \subset L_2 \subset \cdots $ of
$R$-invisible subgroups in $\bar G$ such that every $R$-invisible
subgroup in $\bar G$ is contained in some~$L_k$. Since each $L_k$
$R$-invisible, it is normally generated by finitely many elements
$b_i$ which satisfy equations of the following form:
\[
b_i^e = \prod_j h_j[b_{i_j},g_j]h_j^{-1} \quad \text{where }h_j, g_j
\in \bar G,\, e\in D_R.
\]
All the $b_i,h_j,g_j \in \bar G$ which appear in this expression can
be lifted to $Q_n$ for some $n$ in such a way that the equations can
also be lifted, that is, replacing these elements in the equation by
the lifts, we again obtain an equality in~$Q_n$. Choosing a
subsequence of $\{Q_k\}$, we may assume that $n=k$. Then the lifts
of $b_i$ in $Q_k$ generate an $R$-invisible subgroup $N_k'$
in~$Q_k$. Let $N_k$ be the union of the image of $N_\ell'$ under
$Q_\ell \to Q_k$ where $\ell$ runs over $1,\ldots,k$. Now the claim
follows.
Let $P_k = Q_k/N_k$. By our claim (1), $Q_k \to Q_{k+1}$ induces
$P_k \to P_{k+1}$. We will show that these groups $P_k$ has the
desired properties. First, $P_k \to P_{k+1}$ is in $\Omega^R$ since
$Q_k \to Q_{k+1}$ is 2-connected and so is $Q_k \to P_k$ by the
Stallings exact sequence.
So it remains to show that $\varinjlim P_k \cong \hat G$. For each
$k$ the composition
\[
j_k \colon Q_k \to \bar G \to \hat G
\]
gives rise to $P_k \to \hat G$ since it kills the $R$-invisible
subgroup~$N_k$. These morphisms induces $\varinjlim P_k \to \hat
G$, which is obviously surjective. To show that it is injective,
suppose that $j_k$ sends an element $x\in Q_k$ into an $R$-invisible
subgroup $K$ in~$\bar G$. We may assume that $K$ is contained in
the normal closure of $j_k(N_k)$ by our claim (2) above. Then
$j_k(x)$ is of the form $\prod_j g_j y_j g_j^{-1}$, $g_j\in \bar G$,
$y_j \in j_k(N_k)$. By choosing a sufficiently large $k$, we may
assume that every $g_j$ is in the image of~$j_k$. So $j_k(x) \in
j_k(N_k)$. By choosing a larger $k$, we may assume that $x \in
N_k$. Now it follows that $x$ is sent to the identity in $P_k$. It
proves that $\varinjlim P_k \to \hat G$ is injective.
\end{proof}
\begin{proposition}
Suppose that $\pi_0 \to \pi_1 \to \pi_2 \to \cdots$ and $G_0 \to G_1
\to G_2 \to \cdots$ are sequences of homomorphisms in $\Omega^R$
such that $\pi_0 \to \varinjlim \pi_k$ and $G_0 \to \varinjlim G_k$
are the $R$-closures. Then for any $\phi \colon \pi_0 \to G_0$,
there exist a sequence $G_0=P_0 \to P_1 \to P_2 \to \cdots$ of
homomorphisms in $\Omega^R$ which fits into the commutative diagram
\[
\begin{diagram}\dgARROWLENGTH=1em
\node{\pi_0} \arrow{e} \arrow{s,l}{\phi}
\node{\pi_1} \arrow{e} \arrow{s}
\node{\cdots} \arrow{e}
\node{\pi_k} \arrow{e} \arrow{s}
\node{\pi_{k+1}} \arrow{e} \arrow{s}
\node{\cdots} \arrow{e}
\node{\varinjlim \pi_k \hbox to 0mm{ $= \hat \pi$\hss}} \arrow{s}\\
\node{P_0} \arrow{e}
\node{P_1} \arrow{e}
\node{\cdots} \arrow{e}
\node{P_k} \arrow{e}
\node{P_{k+1}} \arrow{e}
\node{\cdots} \arrow{e}
\node{\varinjlim P_k} \\
\node{G_0} \arrow{e} \arrow{n,=}
\node{G_1} \arrow{e} \arrow{n}
\node{\cdots} \arrow{e}
\node{G_k} \arrow{e} \arrow{n}
\node{G_{k+1}} \arrow{e} \arrow{n}
\node{\cdots} \arrow{e}
\node{\varinjlim G_k \hbox to 0mm{ $= \hat G$\hss}} \arrow{n}
\end{diagram}\hphantom{= \hat G}
\]
in such a way that each $G_k \to P_k$ is in $\Omega^R$ and $\hat G
\to \varinjlim P_k$ is an isomorphism, that is, $G_0 = P_0 \to
\varinjlim P_k$ is an $R$-closure.
\end{proposition}
\begin{proof}
We will construct inductively a sequence $n_0 < n_1 < \cdots$ such
that $P_k = G_{n_k}$ has the desired property. Let $n_0=0$. As our
induction hypothesis, suppose that $n_0,\ldots,n_k$ have been chosen
in such a way that the following diagram commutes, where $\hat \phi$
is the map induced by~$\phi$. (Note that $n_k \ge k$
automatically.)
\[
\begin{diagram}\dgARROWLENGTH=1.6em \dgHORIZPAD=.4em
\node{\pi_0} \arrow{e} \arrow{s,l}{\phi}
\node{\cdots} \arrow{e}
\node{\pi_k} \arrow{s} \arrow{e}
\node{\pi_{k+1}} \arrow{e} \arrow{sseee,..}
\node{\cdots} \arrow[3]{e}
\node[3]{\varinjlim \pi_k}
\arrow[2]{s,r}{\hat \phi}
\\
\node{G_{n_0}} \arrow{e} \arrow{s,=}
\node{\cdots} \arrow{e}
\node{G_{n_k}} \arrow{seee,=}
\\
\node{G_0} \arrow{e}
\node{\cdots} \arrow{e}
\node{G_k} \arrow{e} \arrow{n}
\node{G_{k+1}} \arrow{e}
\node{\cdots} \arrow{e}
\node{G_{n_k}} \arrow{e}
\node{\cdots} \arrow{e}
\node{\varinjlim G_k}
\end{diagram}
\]
Since $\pi_{k+1}$ is finitely generated, the composition $\pi_{k+1}
\to \varinjlim \pi_k \to \varinjlim G_k$ factors through $G_{r}$ for
some $r > n_k$. Note that the two compositions $\pi_k \to \pi_{k+1}
\to G_r$ and $\pi_k \to G_{n_k} \to G_r$ may not be identical.
However, composing $G_r \to \varinjlim G_k$ with them, we obtain the
same maps. Since $\pi_k$ is finitely generated, it follows that
$\pi_k \to \pi_{k+1} \to G_{s}$ and $\pi_k \to G_{n_k} \to G_s$ are
identical for some $s\ge r$. We choose $s$ as~$n_{k+1}$. Our
induction hypothesis is maintained so that the construction of
$\{n_k\}$ can be continued.
Now, letting $P_k = G_{n_k}$, obviously $P_k \to P_{k+1}$, $G_k \to
P_k$ are in $\Omega^R$, and $\varinjlim P_k \cong \varinjlim G_k$.
\end{proof}
\bibliographystyle{amsplainabbrv}
\bibliography{research}
\end{document}
| 169,020
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Foreign Spouse
[…] […]
More Weddings in 2010 […]
| 417,677
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Forums / Property Investing / Creative Investing / Seminar – Mark Rolton
- Pro investor wrote:hi everyone
there is a cd on options in the back of Australia property investor for $159.00 called options made simple i haven't bought it but plan on doin so
thanks Rob
We’re happy to pay for good quality information that helps to take us closer towards our goals. We haven’t done Mark’s seminar but the CD you are referring to is ‘Options Made Simple’ by Rob Balanda.
We found it helped us in the beginning to understand how Options work. It explains how to use property options to buy and speculate in real estate. It includes really good checklists plus it has a sample Call Option Agreement and Put and Call Agreement.
We personally think that Options are just another strategy that can be used to control or tie up a property. For example some developers may enter into an Option agreement with a Land Owner where, for a relatively small non-refundable fee, a property developer has the right, but not an obligation, to purchase the property by a pre-determined date, by which time it is hoped that a Development Permit has been approved.
Options allow property developers to “lay by” a property and buy it at a later time if they wish to do so.
Adrian and Amber Zenere
Yes I agree however … reading from books and CD's is not the same as dealing PERSONALLY with some one who is achieving success now ….
You need coaching and to learn from the real people … how much would you pay Kerry Packer, Frank Lowery, Harry Triggerbof and the lists go on …
D
bluesilverParticipant@bluesilverJoin Date: 2007Post Count: 2
dear mr fair go
hi my name is rocco
iv done the massland home study if you havnt brought the package yet i have a spear im will 2 sell 2 u for less then 5k
if your interestd plz get in contact wit me
Thanks
Rocco
Just went to a seminar from Barry Pickering, ADAYOUN Capital Markets. Anyone heard of this company before, I googled him and it came up with another company name onsite direct. Just wondering what people think of his practices and strategies. That way i can make an informed decision. His fee is $250 to be a wealth coach. not quiet sure what that entails but i had to leave the seminar before getting to talk to him.
If anyone can shed some light that would be good.
Damowangster wrote:I signed up and then the problems started. They quickly took my money and gave me nothing. It took weeks to get the home study pack and then to get some answers on attending a course date – well. There was a course in a few weeks time and I wanted to attend. The staff could not tell me where or what time, only that it was on the Gold Coast. I said I needed to book flights and accommodation, but still no answers. I finally got an attendance welcome email, 3 days before the event – Guess my answer.
There was another course 2 months later and once again, the response was slow, but I at least had 3 weeks notice.
Hi all,
I can relate to Wangster's experience regarding Mark's course. They are an ever-so-helpful…. until they receive your payment that is, after that you can forget it. It took a couple of weeks to receive any learning materials (half of it anyway), then another couple of weeks to receive the other half and without a covering letter or receipt. I finally received a receipt almost 2 months later but I had to jump up and down for it. After endless phone calls to their office, my calls not being returned, now they have twice postponed the Options event on the Gold Coast which has also left me out of pocket because of accommodation cancellation fees. Needless to say I'm hardly impressed especially when they did not even respond to my request for a refund – hey can you blame me? Apparently, seems the only way to get a refund is to attend the first day of the event and hand back your materials – if that's what I have to do then that's what I have to do.
Just wondering if anyone else has actually done this to get their refund, how difficult was the process and how much of the first day did you attend?
Appreciate your experiences/comments.
Hi
I got a refund but only after they suspected I knew a lot more about them than they were comfortable with. I did not have to hand back any course materials either.
Hope that helps.
Thanks Matt007. Have found your comments in (several) forums very helpful. I'm new to this forum thing – is it possible to PM you?
Cheers
Poss.. sure, PM me here if you like.
Hi Matt007,
I'm new to this forum but would like to know, did you get your refund after going to the seminar or did you talk to them over the phone? I have been reading the comments here and they have confirmed my early feelings that these people are not very professional at all.
Thankyou.
Hello all,
You can buy a Mercedes Benz and have problems, we have …
We have done Marks course and other stuff and yes it is expensive but his deals are real.
Within "options" are several opportunities to prosper, I attended (we) last year and connected with several intelligent people (solicitors and bankers) and out of that connection at "Marks" seminar have created a lot of business. Today I am flying to Brisbane to sit down with a major finance company who has lots and lots of liquidated projects that they want to off load and need our help … this would not have happened if we did not attend that seminar.
There is a 97% failure rate of people who attend Tony Robbins seminars who just don't apply the strategies and fall back into their hum drum life styles several days after attending the events.
You can read all the great books and they will all tell you that only 3% (possibly 5%) become wealthy because they learn to "do things in a certain way" (the science of getting rich, if you don't know).
Are you or do you want to be in the 3 – 5% and if so what do you have to do to get there?
Finally if you want to play in this league you need to have "balls" it's hard it's scary but the rewards can be outstanding IF "you learn to do things in a certain way"
D … good luck to all and never give up.
Hi wealth4life: interesting post saying Marks' deals are 'real' and all the other info. I'd be interested in speaking with you further, off-line and in person if possible, so please PM and let me know if you're open to that as I think we may be operating in a similar space.
Cheers
M007
For damo001
Would not touch this Barry Pickering or Adayoun as they are both involved
in Ondect tas onsite both ** UNDER EXTERNAL ADMINISTRATION and/or
CONTROLLER APPOINTED ** BUNKRUPT!!!, Also involved in Australia's largest
building crash of RPC ( Real Property Constructions). As my research
continues Adayoun is a new name for Onsite and the new builder is Bloomer.
All of these companies are from the Gold Coast, Qld. You work it out!!
Hey people,
I am new to this site and indeed new to blogging and forums so forgive me for being such a novice, but could someone please tell me what it means to PM someone.
To Matt007 if your reading this, I would be very interested in emailing you or you can give me a call to talk about your experiences with MassLand. After doing some further research including reading some of your comments I want to ensure I will get my money back when I arrive at the event in less than two weeks. Do not want to waste any more time here with this group as I have since found many better alternatives for learning about property development.
Any help would be much appreciated,
Thank you.
Hi Polski23: my experiences and those of others with Massland are well documented, if not too well documented…. and I can't really offer more than what's been said already, but especially I'd suggest going over the post by Inthegame on page 1 of this post. I suggest you take that particular post to heart.
My only suggestion is that you ask for proof of all claims made, in written form (at least something that will then hold them legally liable for the claims they make if they turn out to be false or not) including company value, projects and location, referees from past students (in person, not something posted on a website as they can be faked). If you know a good lawyer tell them (massland) you want to do a 10A (I think that's the associated number) due dilligence inquiry on Massland and its associated companies before entering into a JV with them (have your lawyer look at that too by the way and see if it compels them to pay you anything at all)…..see what reaction you get. Its' standard risk management practice by the way if anyone tells you otherwise….If they can't or won't provide any kind of proof, you can ask yourself why that is, make your own judgements….. I've been blissfully ignorant of that lot for a while now and i'm quite happy with that. Things may have changed there, but my own opinion is probably not based on what I keep hearing and reading.
Having heard others experiences with trying to get money back I'd suggest a lot of persistence is required and always remember that the media love a good story
…I'm not advocating or suggesting you do that, that's entirely up to you..just something I hear once…..but be persistent and have a good lawyer at your disposal just in case.. otherwise they'll just keep ignoring and trying to upsell you.
Good luck.
There's a few videos on youtube about him.
I watched the DVD that came with the last issue of Wealth Creator. All very inspiring, so I responded to the website and was called the very next day by a nice chap wanting to sign me up for the Gold Coast or Sydney seminars. I told him I was in WA and he assured me that there were graduates doing million dollar deals over here. 'Great', I said, 'before I sign up, put me in touch with a couple as I would love to talk to them and I'm sure they wouldn't mind as it could benefit everyone'
I'm still waiting…………..
attrill: i asked the same questions…. still waiting… 3 years later.. still waiting.. finding anyone who's been paid out over there is quite a task I find! i think some of the people who speak of successful students over there may mean one or two who have successfully managed to get an option over something. The confusion then arises as many think that's a ticket to a million bucks…when in reality it means very little… Its getting the DA, adding vlaue and then being able to sell it to someone or have an end buyer in place and people who'll deal with you, that's the key, this is when you can profit from it..not a lot of point in having the option if you don't have those other factors in place…at least in the development sphere…whether or not the profits occur to the tune Rolton claims is another matter.. especially in this market place. Loads of stock out there, overabundance of it in fact, not a lot of buyers for it.
Hi Matt and sorry 4 the delay,
Yes there are so many like you who are tall poppies but that is what what Australia is all about, right? …
Yes I attended the last seminar … oh my god I am sooo guilty.
Yes I am entering into some option deals …
You wouldn't believe it but we just optioned the last of an unsold development of 50 lots from a liquidated land deal and sold it to a major company for a 100k profit (thats 2k per site with NO money down) all within 30 days using Marks strategies … if you want to challenge me I'll do u double or no deal on a personal guarantee …
Sure it's not all for every body but just one strategy could be worth the risk after all are you any better … no I don't work for Mark I have been active on this forum way before he was here … yes I am qualified … any way what is your true gripe or are you like so many just jealous … opps what if you are wrong and he is right … or what is your strategy?
D
Hi Wealth4Life: thanks for your reply. To answer some of your…comments..
a – no i'm far from a tall poppy, still very much a learner learning but am going along just fine thanks for asking, long way to go yet but it's fun so far.
b – no one said attending a seminar made you guilty of anything.. and if you read my posts around the place, i've never said the strategy of using options doesn't work. Fine strategy indeed in amongst the many available if the right conditions exist….
c – i have no interest in 'challenging' you double or nothing for anything.. I don't care. Do what you do and good luck to you
d – my 'gripe' as you put it is not jealousy. Not by a long shot, certainly not of Mark.
My only real issue can be described in a general commentary on the whole spruiking industry, and it is seeing written, verifiable proof of claims made, by anyone in business,whether they're a seminar spruiker or otherwise. It's called due dilligence and it's a very common practice. So if you and I were to enter into a business arrangement and you hopped up and said "I'm worth half a billion and I'm this that and the other" I wouldn't hand over a cent to you or sign any binding paperwork like a JV until my accountant and solicitor had verified that… not too much to ask given the potential sums of money involved. At least not in my humble opinion. But hypothetically speaking.. if we did enter into an arrangement, and I said all those things to you, and you made business decisions based on those and it cost you time and money, and those things turned out to be either false or inflated or whatever.. what would your response be? I somehow doubt it'd be a position of trust.
Whichever. Best of luck in your deals. Merry Christmas to you and yours.
Matt you have to be one of the sanest people on here. I wouldn't call myself a seminar junkie, but I will go to a free info evening and watch a free DVD and it's amazing what can be picked up for nothing.
There are only so many legal ways to make money out of property in Australia and everything else is in the packaging. What I have learnt at no cost to myself is that I can use Google Earth, a zoning map and RP Data to find potential development sites. I then negotiate an option on these, add value and on sell it. No problem, so why not develop it myself?
Of course the access to finance is the key. Most potential seminar candidates are looking for the golden egg that will make them a million dollars with "no money down" If you have been in the property market for a few years and have access to equity then it becomes much, much easier.
Wealth4life, good on you for doing what you are doing. It's good to hear the other side of any argument..
You must be logged in to reply to this topic.
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The over 400,000 clients of housing banks, which operate on the basis of saving – lending systems, shall receive state bonuses amounting to RON 42.55 M (EUR 9.45 M), which represent the last installments corresponding to the savings in 2010 and one third for those corresponding to 2011. The state pays a guaranteed bonus up to 25 per cent of the annual saved amounts but not more than EUR 250 per year to each client. The housing banks pay fix interest loans, 4.5 – 6 per cent per year, throughout the lending period, the clients having also access to anticipated.
| 380,246
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ROBERT “BOB” KRAUTH, 75, of Massena (Svcs. May 4th, 2013)
April 22nd, 2013 by Ric Hanson
ROBERT “BOB” KRAUTH, 75, of Massena, died Feb. 27th, at the Banner Heart Hospital in Mesa, AZ. Celebration of Life services for BOB KRAUTH will be held 2:30-p.m.May 4th at the Massena United Methodist Church. Steen Funeral Home in Massena has the arrangements.
The family will greet friends one-hour prior to the service on May 4th, and a luncheon will be held immediately after the service. Online condolences may be left to the family at.
Memorials may be directed to the ROBERT “BOB” KRAUTH Memorial Fund, to be established by the family.
| 209,682
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TITLE: Probability problem
QUESTION [0 upvotes]: Question:
Let's say that I have $23$ people in my workplace. What is the probability that $2$ people in my workplace have the same birthday? Please answer in exact form with all decimal points that are implied.
I am sort of confused and would like to find the solution. If I could get a step-by-step solution, that would be great.
Thanks!
REPLY [1 votes]: Quoting Wikipedia
$p(n)$ is the probability of at least two of the n people sharing a
birthday. It is easier to first calculate the probability $\bar p(n)$
that all $n$ birthdays are different. According to the pigeonhole
principle, $p(n)$ is zero when $n > 365$. When $n ≤ 365$:
$$\begin{align*} \bar p(n) &= 1 \times \left(1-\frac{1}{365}\right)
\times \left(1-\frac{2}{365}\right) \times \cdots \times
\left(1-\frac{n-1}{365}\right)=\\\\ &= { 365 \times 364 \times \cdots
\times (365-n+1) \over 365^n }= { 365! \over 365^n (365-n)!} =
\frac{n!\cdot{365 \choose
> n}}{365^n}=\\\\&=\frac{_{365}P_n}{365^n}\end{align*}$$where '!' is the
factorial operator, $\textstyle {365 \choose n}$ is the binomial
coefficient and ${_{k}P_r}$ denotes permutation. The equation
expresses the fact that the first person has no one to share a
birthday, the second person cannot have the same birthday as the first
$(364/365)$, the third cannot have the same birthday as the first two
$(363/365)$, and in general the nth birthday cannot be the same as any
of the $n − 1$ preceding birthdays. The event of at least two of the
$n$ persons having the same birthday is complementary to all n
birthdays being different. Therefore, its probability $p(n)$ is $$p(n)= 1 - \bar p(n)$$ This probability surpasses $1/2$ for $n = 23$ (with value about $50.7%$).
More precisely $$p(23)=1-\frac{23!\cdot \dbinom{365}{23}}{365^{23}}=0.507297234323986$$
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TITLE: Finding closed form of exponential generating function
QUESTION [0 upvotes]: Let $S(n, k)$ be the Stirling number of the second kind. For a fixed
positive integer $k$, find a closed form for the exponential
generating function $B(x) = \sum_{n\ge0}S(n,k)\frac{x^n}{n!}$.
I believe the closed form of
$$\sum_{n\ge0}n!\frac{x^n}{n!}$$
is $\frac{1}{1-x}$ but the inclusion of $S(n,k)$ confuses me.
REPLY [1 votes]: Stirling numbers of the second kind, $\genfrac{\{}{\}}{0pt}{}{n}{k}$, count the number of ways to divide a set of $n$ elements into $k$ classes. This gives, considering element $n$:
If $n$ is in a class by itself: This leaves $n - 1$ elements to group into $k - 1$ classes, $\genfrac{\{}{\}}{0pt}{}{n - 1}{k - 1}$ ways to do this
If $n$ is in a class with other elements: There are $n - 1$ other elements to divide into $k$ classes, $n$ can be in any one of them, $k \genfrac{\{}{\}}{0pt}{}{n - 1}{k}$ ways
Pulling this together gives the recurrence:
$\begin{align*}
\genfrac{\{}{\}}{0pt}{}{n}{k}
&= \genfrac{\{}{\}}{0pt}{}{n - 1}{k - 1} + k \genfrac{\{}{\}}{0pt}{}{n - 1}{k}
\end{align*}$
As boundary conditions we have $\genfrac{\{}{\}}{0pt}{}{n}{0} = 0$ if $n > 0$, $\genfrac{\{}{\}}{0pt}{}{n}{n} = 1$.
To get an exponential generating function in $n$ for $\genfrac{\{}{\}}{0pt}{}{n}{k}$,
call it $S_k(z) = \sum_{n \ge 0} \genfrac{\{}{\}}{0pt}{}{n}{k} \frac{z^n}{n!}$,
multiply the shifted recurrence by $z^n / n!$ and sum over $n \ge 0$ and recognize resulting sums:
$\begin{align*}
\sum_{n \ge 0} \genfrac{\{}{\}}{0pt}{}{n + 1}{k} \frac{z^n}{n!}
&= \sum_{n \ge 0} \genfrac{\{}{\}}{0pt}{}{n}{k - 1} \frac{z^n}{n!}
+ k \sum_{n \ge 0} \genfrac{\{}{\}}{0pt}{}{n}{k} \frac{z^n}{n!} \\
S_k'(z)
&= S_{k - 1}(z) + k S_k(z)
\end{align*}$
From the boundary conditions we know $S_0(z) = 1$ and $S_k(0) = 0$ if $k > 0$.
The solution to the differential equation is:
$\begin{align*}
S_k(z)
&= e^{k z} \int_0^z e^{-k t} S_{k - 1}(t) d t
\end{align*}$
This gives:
$\begin{align*}
S_0(z)
&= 1 \\
S_1(z)
&= e^z - 1 \\
S_2(z)
&= \frac{(e^z - 1)^2}{2} \\
S_3(z)
&= \frac{(e^z - 1)^3}{3!} \\
S_4(z)
&= \frac{(e^z - 1)^4}{4!}
\end{align*}$
Induction confirms that:
$\begin{align*}
S_k(z)
&= \frac{(e^z - 1)^k}{k!}
\end{align*}$
| 65,624
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A last-minute grant from the state will cut costs for Norwell taxpayers on the $200,000 renovation to the Jacobs Farmhouse property.
A last-minute grant from the state will cut costs for Norwell taxpayers on the $200,000 renovation to the Jacobs Farmhouse property.
Time has taken its toll on the 288-year-old property, which is in need of new roofs on the carriage house, barns and farmhouse along with window and shutter repairs.
In May, Town Meeting voters approved a $206,775 transfer from Community Preservation Committee (CPC) funds to help pay for repairs. An emergency-matching grant from the Massachusetts Preservation Projects Fund through the Massachusetts Historical Commission for $35,000 will lower the town’s total costs to about $170,000, according to Wendy Bawabe, Norwell Historical Commission secretary.
Using the state and CPC funds, the construction will replace the asphalt roofs on the East and West barns with more historically accurate wooden shingles. The money will also be used to replace the carriage house roof, pay for repairs to windows and shutters and cover spot-roof fixes to the farmhouse.
“We prioritized the projects at the farmhouse and these are the projects that were considered the first priority,” Bawabe said. “We’re just trying to restore farmhouse so it will last another 300 years.”
The state grant is a 50 percent reimbursement grant, meaning the town will cover the costs until the work is completed in accordance with the grant application.
Bawabe estimated that more than 100 hours were spent in securing the grant, which will allow for the expensive, yet more historically accurate, repairs.
"The commission felt that the use of traditional materials was vital to this highly visible and important site,” Nancy McBride, chairman of the Norwell Historical Commission, said in a statement to the Mariner. “Hence, we applied for a grant to ensure that the roofing can be completed in cedar."
The estimated lifetime of the cedar-shingled roofs is 30-40 years, with proper care. Bawabe said the wood would need to be oiled every five to 10 years.
“No matter what the maintenance or lifespan is, everybody agreed it’s the most appropriate and that it will look fabulous,” Bawabe said.
Century-old photos of the farmhouse clearly show the original roof was constructed using the wood shingles. The commission is currently researching when the switch to asphalt tiles occurred.
Construction will begin in the late summer, or early fall.
“The shutters will be coming off soon to be stored for restoration and people are going to start noticing a lot of work going on there,” Bawabe said.
The town’s facilities manager, David Sutton will be instrumental in ensuring the proper maintenance of the Jacobs Farmhouse property once renovations are completed.
“The farmhouse and barns will be protected from a harsh New England winter by new cedar roofs, as it had been previously for hundreds of years," he said.
Follow Erin Tiernan on Twitter @ErinTiernan.
| 132,833
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Did You Hear About The Affair New York’s Democrat Governor Had?
No, not the old one. The new.
Spitzer. McGreevey. Paterson.
One gets the idea that not a lot of parents in the New York area are letting their kids watch the news these days. I mean, you thought Hollywood was bad. East coast politics has Paris Hilton beat handily these days. The way things are going there will be a sex tape emerging any day now.
| 359,063
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- ‘My Lips Are Sealed’ The Impact Of Resealable Packages On Consumption Self-Regulation
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\begin{document}
\title{Presentations of linear monoidal categories and their endomorphism algebras}
\author{Bingyan Liu}
\address{
Department of Mathematics and Statistics \\
University of Ottawa
}
\address{
School of Mathematical Science \\
Tongji University
}
\email{bingyanliu21@gmail.com}
\begin{abstract}
We give the definition of presentations of linear monoidal categories. Our main result is that given a presentation of a linear monoidal category, we can produce a presentation of the same category as a linear category. We apply this result to endomorphism algebras of certain important linear monoidal categories.
\end{abstract}
\subjclass[2010]{18D10}
\keywords{Monoidal category, linear category, free category, presentation of category, endomorphism algebra}
\maketitle
\thispagestyle{empty}
\tableofcontents
\section{Introduction}
Categorification is an exciting and relatively new area of research. It is the process of replacing set-theoretical objects by their categorical analogues. Strict linear monoidal categories have been playing an important role in the study of categorification, since they are categorification of classic algebraic objects---rings, where the tensor product is the categorification of the product in the ring.
In practice, (strict) linear monoidal categories are often defined by some generating objects and generating morphisms where objects and morphisms are formal tensor products of generators, with certain relations imposed. This leads to the concept of presentation of linear monoidal categories. Although common, the definition of presentation of linear monoidal categories and certain prerequisite concepts are not well represented in the literature. To make these notations precise is one of the goals of the current article.
We first review the notion of linear categories and define the notion of presentation of linear categories in \cref{sec:LinCat}. We denote such a presentation by $\left<X,E,R\right>$ where $X$ is a set of generating objects, $E$ is a set of generating morphisms, and $R$ is a set of relations on morphisms.
Then we build the monoidal analogue of these notions in \cref{sec: LinMonCat}. Concretely speaking, we define free linear monoidal categories, tensor ideals and presentations of linear monoidal categories, denoted by $\left<X,E,R\right>_\otimes$ if the category is generated by objects in $X$ and morphisms in $E$ with relations $R$ imposed. In particular, we show the existence of the free linear monoidal categories by concrete constructions in \cref{sec:LinMonCatExist}.
In a linear monoidal category $\rC$, the endomorphism sets $\End_\rC(a)$ form algebras, where the product is the composition of morphisms. The tensor product induces an extra structure on the endomorphism algebras. Thus to study the structure of the endomorphism algebras, it proves useful to forget the tensor product and view the category as only a linear category. Therefore, for a monoidal category given by presentation $\left<X,E,R\right>_\otimes$, we are motivated to give a presentation $\left<X^\prime,E^\prime,R^\prime\right>$ of it as a linear category and study the structure of its endomorphism algebras according to the presentation $\left<X^\prime,E^\prime,R^\prime\right>.$
In \cref{sec:MainTheorem}, we introduce our main result \cref{MainTheorem}, which is stated without proof in \cite[\S 2.6]{Bru14}. Namely, given a presentation of a linear monoidal category $\rC$ with presentation $\left<X,E,R\right>_\otimes$, we produce a presentation $\left<M(X),\bar{E},\bar{R}\right>$ of $\rC$ as a linear category. Roughly speaking, $M(X)$ is the free monoid over $X$, the morphisms $\bar{E}$ are morphisms of the form $1_v \otimes e \otimes 1_w$, and the relations are those obtained from $R$ by replacing each $r$ with $1_v \otimes r \otimes 1_w$ together with the interchange law for morphisms in $\bar{E}$.
For the structure of the endomorphism algebras, we consider a special case where all generating morphisms are endomorphisms. Given the presentation of the linear monoidal category $\left<X,E,R\right>_\otimes$, we describe a presentation of the endomorphism algebras, with the generators and relations of that algebra in those of the presentation $\left<M(X), \bar{E}, \bar{R}\right>$. This result is stated in \cref{thm:presentationofalgebra}.
As important categories in the study of categorification, we introduce several examples of linear monoidal categories and visualize these categories with the help of string diagrams. As applications of our main results, we identify their endomorphism algebras with certain well-studied associative algebras. These identifications are stated in \cite[\S 3]{Savage18} without proof.
\color{black}
Fix a commutative ground ring $\kk$. In this article, all linear categories are $\kk$-linear categories and monoidal categories are strict monoidal categories.
\subsection*{Acknowledgements.} I would like to express my sincere gratitude to Professor Alistair Savage for his patience and guidance throughout the whole project. I would like to thank the Mitacs Organization, the China Scholarship Council and the University of Ottawa for providing me this great opportunity.
\section{Generators, relations and presentations of linear categories}
In this section, the main goal is to give a precise definition of presentations of ($\kk$-)linear categories. We will first review the definition of quivers, (small) linear categories and the free linear category over a quiver. Afterwards, we recall the definition of ideals of linear categories, the quotient category by an ideal and presentations of linear categories.
\subsection{Quivers and linear categories}\label{sec:LinCat}
A \emph{quiver} $Q=(V,E,\partial_0,\partial_1)$ consists of a set of vertices $V$ and a set of edges $E$, together with two functions $\partial_0, \partial_1 \colon E \to V$, which are called the start point function and end point function respectively. We write $e \colon a \to b$ if $\partial_0 e= a $ and $\partial_1 e=b$ and often write $Q=(V,E)$ for conciseness leaving the maps $\delta_0$ and $\delta_1$ implied. A \emph{quiver morphism} $\phi$ from a quiver $Q=(V,E)$ to a quiver $Q^\prime=(V^\prime,E^\prime)$ is a pair of functions $\phi=(\phi_0,\phi_1)$ where $\phi_0$ is a function from $V$ to $V^\prime$ and $\phi_1$ is a function from $E$ to $E^\prime$ such that the following two diagrams commute:
\begin{equation}\label{QuivDiag}
\begin{tikzcd}
E\arrow{r}{\phi_1} \arrow{d}{\partial_0} & E^\prime\arrow{d}{\partial_0}\\
V\arrow{r}{\phi_0}&V^\prime
\end{tikzcd},
\qquad
\begin{tikzcd}
E\arrow{r}{\phi_1}\arrow{d}{\partial_1}&E^\prime\arrow{d}{\partial_1}\\
V\arrow{r}{\phi_0}&V^\prime
\end{tikzcd}.
\end{equation}
Quivers and quiver morphisms constitute the category of quivers $\Quiv$, where quivers serve as objects, quiver morphisms serve as morphisms and the composition of morphisms is the composition of functions. The composition of morphisms is clearly associative. The pair of identity maps $id_Q=(id_V,id_E)$ serves as the identity morphism over a object $Q=(V,E)$.
A (small) category is called a \emph{linear category} if its hom-sets are $\kk$-modules and the composition of morphisms is bilinear, namely
\begin{itemize}
\item $(k_1\ f_1+k_2\ f_2) \circ g = k_1 (f_1 \circ g) + k_2 (f_2 \circ g),$
\item $f \circ (k_1\ g_1+k_2\ g_2) = k_1 (f \circ g_1) + k_2 (f \circ g_2),$
\end{itemize}
for $k_1,k_2 \in \kk$ and $f,f_1,f_2,g,g_1,g_2$ morphisms such that the above operations are defined.
A \emph{linear functor} $F$ between linear categories is a functor satisfying \[F ( k_1\ f_1 + k_2\ f_2) = k_1 F(f_1)+k_2 F(f_2).\]
The category of linear categories $\LCat$ has linear categories as objects and linear functors as morphisms.
We can define the \emph{forgetful functor} $U$ from the category of (small) linear categories $\LCat$ to the category of quivers $\Quiv$. It is based on the fact that any (small) linear category $\rC$ is manifestly a quiver $U\rC$ where the set of objects $Ob(\rC)$ forms the set of vertices and the edges consist of all morphisms $\Hom(\rC)$.
\begin{defin}\label{def:freelincat} The \emph{free linear category} over a quiver $Q$ is a linear category $\tL Q$ together with a quiver morphism $\iota$ from $Q$ to $U\tL Q$ that satisfies the following universal property: for any linear category $\rC$ and quiver morphism $\phi$ from $Q$ to $U\rC$, there exists a unique linear functor $\phi^\prime \colon \tL Q \to \rC$, such that $\phi$ factors through $\iota$ as $U\phi^\prime \circ \iota = \phi$, as the diagram below indicates:
\begin{equation}
\begin{tikzcd}
& &&Q\arrow{rd}{\phi}\arrow{d}{\iota}&\\
\tL Q\arrow{r}{\exists! \phi^\prime}&\rC, &&U \tL Q\arrow[swap,dashrightarrow]{r}{U \phi^\prime}&U\rC.
\end{tikzcd}
\end{equation}
\end{defin}
The free linear category $\tL Q$ over a quiver $Q$ is unique up to isomorphism. This follows from the fact that $(\tL Q,\iota)$ is an initial object in the comma category $(Q \downarrow U)$, and the initial object in a category is unique up to isomorphism. One can find a detailed discussion in \cite[II.8]{MacLaneCat}.
The existence of $\tL Q$ also holds. Before we show this, we need the concept of free category. The \emph{free category} $\rF$ over a quiver $Q=(V,E)$ is the category whose objects are $V$ and morphisms are formal finite compositions of composable edges, where two edges $e_1, e_2$ are composable if $\partial_0 e_1 = \partial_1 e_2$. We may also call the morphisms in the free category \emph{paths} on $Q$ and call an identity morphism $1_v$ a trivial path.
Let $U_0$ be the forgetful functor from $\Cat$ to $\Quiv$ and $\iota_0$ be the evident embedding quiver morphism from $Q$ to $U_0\rF$. The free category $\rF$ together with $\iota_0$ satisfies the following universal property: given any category $\rC$ and quiver morphism $\phi \colon Q \to U_0\rC$, there exists a unique functor $\psi$ from $\rF$ to $\rC$, such that $U_0\psi \circ \iota_0 = \phi$. One can find a detailed discussion of free category in \cite[II.7]{MacLaneCat}.
\begin{theo}
Let $\rF$ be the free category over a quiver $Q$ and $\rN$ be the category whose objects are the same as those of $\rF$ and whose hom-sets are the free $\kk$-modules on the hom-sets of $\rF$. Then $\rN$ is the free linear category $\tL Q$.
\end{theo}
\begin{proof}
Let $\Cat$ be the category of (small) categories. Consider the natural forgetful functors in the following diagram:
\[\begin{tikzcd}
& \Cat \arrow{dr}{U_0} &\\
\LCat \arrow{ur}{U_1} \arrow {rr}{U}& & \Quiv.
\end{tikzcd}\]
It is clear that this diagram commutes. Let $\iota_0$ be the natural embedding quiver morphism from $Q$ to $U_0\rF$ and $\iota_1$ be the natural embedding functor from $\rF$ to $U_1\rN$. We define $\iota \colon Q \to U\rN$ to be $\iota = U_0\iota_1 \circ \iota_0 $.
Now, for any linear category $\rC$ and quiver morphism $\phi$ from $Q$ to $U\rC$, by the universal property of $\rF$, there exists a unique functor $\psi \colon \rF \to U_1 \rC$, such that $ \phi = U_0 \psi \circ \iota_0 $, as the diagram below indicates. For $\psi$, since the hom-sets of $\rN$ are free $\kk$-modules, it's easy to show there exists a unique linear functor $\phi^\prime \colon \rN \to \rC$, such that $\psi =U_1\phi^\prime \circ \iota_1$, as the diagram below indicates.
\begin{equation}\label{diag:freelincatexsitprove}
\begin{tikzcd}
Q\arrow{r}{\iota_0}\arrow[swap]{rd}{\phi} &
U_0 \rF\arrow[dashrightarrow]{d}{U_0\psi} & \rF\arrow[dashrightarrow]{d}{\psi}
\\
& UL=U_0U_1\rC & U_1\rC
\end{tikzcd},\quad
\begin{tikzcd}
\rF\arrow{r}{\iota_1}\arrow[swap]{rd}{\psi}
& U_1\rN \arrow[dashrightarrow]{d}{U_1\phi^\prime}& \rN \arrow[dashrightarrow]{d}{\phi^\prime}& \\
& U_1\rC & \rC
\end{tikzcd}.
\end{equation}
Thus, applying $U_0$ to the diagram of $\phi^\prime$ on the right in \cref{diag:freelincatexsitprove}, and connecting it with the other diagram, we have that the diagram below commutes.
\begin{equation}\label{diag:freelincatexistprove2}
\begin{tikzcd}
Q \arrow[swap]{rd}{\phi}\arrow{r}{\iota_0}
& U_0\rF \arrow[dashrightarrow]{d}{U_0\psi} \arrow{r}{U_0\iota_1}
& U_0U_1\rN \arrow[dashrightarrow]{ld}{U_0U_1\phi^\prime}\\
& U_0U_1\rC &
\end{tikzcd}
\end{equation}
Thus we have $\phi= U_0U_1 \phi^\prime \circ U_0 \iota_1 \circ \iota_0 = U\phi^\prime \circ \iota.$
Suppose there exists another linear functor $\psi^\prime$ which satisfies $\phi = U \psi^\prime \circ \iota$. We have $ \phi = U_0 ( U_1 \psi^\prime \circ \iota_1 )\circ \iota_0$. By the uniqueness of $\psi$, we have $ U_1 \psi^\prime \circ \iota_1 = \psi$. By the uniqueness of $\phi^\prime$, we have $\psi^\prime=\phi^\prime$. Thus $\rN=\tL Q$.
\end{proof}
As constructed above, the hom-set ${\tL Q}(a,b)$ for any objects $a$, $b$ are free $\kk$-modules over the hom-sets ${\rF}(a,b)$ of $\rF$. Therefore morphisms in $\tL Q$ can always be uniquely written as linear combinations of morphisms in $\rF$, namely linear combinations of paths on $Q$.
\subsection {Ideals, relations and presentations} \label{sec:LinCatPre}
\begin{defin}
An \emph{ideal} $\rI$ of a linear category $\rL$ is a linear subcategory $\rI$ such that
\begin{itemize}
\item for all $f \in \rI(a,b)$, $g \in \rL(b,c)$, we have $g \circ f \in \rI(a,c)$,
\item for all $f \in \rL(a,b)$, $g \in \rI(b,c)$, we have $g \circ f \in \rI(a,c)$.
\end{itemize}
Let $R$ be a set of morphisms of a linear category $\rL$, not necessary in a single hom-set.
The \emph{ideal generated by} $R$ is the minimal ideal that contains $R$, denoted as $\left<R\right>$, which can also be described as the intersection of all ideals that contain $R$.
\end{defin}
To give a explicit description of the generated ideal generated by a set of morphisms, we have the following proposition.
\begin{prop}\label{idealstruc}
Let $R$ be a set of morphisms of a linear category $\rC$. Then the ideal $\left<R\right>$ consists of all morphisms of the form
\begin{equation} \label{equ:idealstruc}
\sum_{i=1}^n k_i\ (f^\prime_i\circ r_i \circ f_i),
\end{equation}
where $n \in \N$, $k_i \in \kk$, $r_i \in R$ and $f_i$, $f^\prime_i$ are morphisms in $\rC$ such that the above operations are defined.
\end{prop}
\begin{proof}
Let $\rI$ denote the category with all morphisms of the form \cref{equ:idealstruc}. Then $\rI$ forms a linear subcategory.
Since the composition of morphisms is bilinear, we have
\begin{align*}
g^\prime \circ \left(\sum_{i=1}^n k_i (f^\prime_i\circ r_i \circ f_i)\right) \circ g
=\sum_{i=1}^n k_i (g^\prime \circ f^\prime_i) \circ r_i \circ (f_i \circ f^\prime_i) \in \Hom(\rI),
\end{align*}
for any morphisms $g$, $g^\prime$ such that the operations above are defined. Thus $\rI$ forms an ideal and $\rI$ contains $R$. Hence $\rI$ contains $\left<R\right>$. Conversely, since $R$ is contained in $\left<R\right>$ and $\left<R\right>$ is an ideal, we have $f^\prime_i\circ r_i \circ f_i \in \left<R\right>$ for any $f$,$f^\prime$ such that these compositions are defined. Since $\left<R\right>$ is closed under linear operations, all morphisms in \cref{idealstruc} are in $\left<R\right>$, namely $\rI \subseteq \left<R\right>$. Thus $\rI=\left<R\right>$.
\end{proof}
\begin{defin}
The \emph{quotient category} of a linear category $\rC$ by an ideal $\rI$ is the linear category $\rC/\rI$ whose objects are the same as those of $\rC$ and whose hom-sets are the quotient $\kk$-modules $\rC(a,b)/\rI(a,b)$. For $f \in \rC(a,b)$, let $[f]$ denote the coset of $f$. Then the linear operation and composition of morphisms are defined as follows:
\begin{align*}
[f]+[g]=[f+g], \quad k[f]=[kf], \quad [g] \circ [f]=[g \circ f],
\end{align*}
for $k\in \kk$ and $f,g$ morphisms in $\rC$, such that the above operations are defined.
\end{defin}
The linear operations are exactly the linear operations of the quotient module. For the composition of morphisms, the definition of ideals guarantees that it is well-defined: Let $[f]=[f^\prime]$, $[g]=[g^\prime]$, then $f-f^\prime, g-g^\prime \in \Hom(\rI)$, we have
\begin{align*}
g \circ f - g^\prime \circ f^\prime
= g \circ f - g \circ f^\prime + g \circ f^\prime - g^\prime \circ f^\prime
= g \circ (f - f^\prime) + (g - g^\prime) \circ f \in \Hom(\rI),
\end{align*}
therefore $[g \circ f] = [g^\prime \circ f^\prime]$.
Now we define presentations of linear categories.
\begin{defin}
A \emph{presentation} of a linear category $\rC$ is a triple $(V,E,R)$, where $R$ is a set of morpihsms of the free linear category $\tL Q$ over the quiver $Q=(V,E)$, such that $\rC$ is isomorphic to the quotient category of $\tL Q$ by the ideal generated by $R$, namely the linear category $\tL Q / \left<R\right>$.
\end{defin}
\section{Presentations of linear monoidal categories}
In this section, our main goal is to give a precise definition of presentations of (strict) linear monoidal categories. This section is developed by making a monoidal analogue of the previous section. We will first define monoidal quivers, small linear monoidal categories and the free linear monoidal category over a quiver. The existence of the free linear monoidal category is demonstrated in \cref{sec:LinMonCatExist}. Afterwards, we will define tensor ideals of linear monoidal categories, the quotient category by a tensor ideal and presentations of linear monoidal categories.
\subsection{Monoidal quivers and linear monoidal categories} \label{sec: LinMonCat}
We make a monoidal analogue of quivers, linear categories, the forgetful functor and the free linear category.
A \emph{monoidal quiver} $Q=(V,E)$ is defined to be a quiver whose vertex set $V$ is a monoid. A \emph{monoidal quiver morphism} is a quiver morphism such that the map on vertices is also a monoidal morphism. They constitute the category $\MQuiv$ of monoidal quivers.
A \emph{(strict) linear monoidal category} $\rD$ is a linear category equipped with a tensor product (bifunctor) $\otimes$ such that
\begin{itemize}
\item $\otimes$ is associative on both objects and morphisms, and
\item there exists $\one \in Ob(\rD)$ such that $\one \otimes a = a = a \otimes \one$ for any object $a \in Ob(\rD)$
and $1_\one \otimes f = f = f \otimes 1_\one$ for any morphism $f \in \Hom(\rD)$, and
\item the bifunctor $\otimes$ is bilinear, namely
\begin{align*}
(k_1\ f_1+k_2\ f_2) \otimes g = k_1 (f_1 \otimes g) + k_2 (f_2 \otimes g),\\
f \otimes (k_1\ g_1+k_2\ g_2) = k_1 (f \otimes g_2) + k_2 (f \otimes g_2),
\end{align*}
where $k_1,k_2 \in \kk$ and $f,f_1,f_2,g,g_1,g_2$ are morphisms such that the above operations are defined.
\end{itemize}
A \emph{linear monoidal functor} $F$ between linear monoidal categories is a linear functor $F$ such that $F$ also commutes with $\otimes$, namely $F(a\otimes b) = F(a) \otimes F(b)$ for any objects $a,b$ and $F(f \otimes g) = F(f) \otimes F(g)$ for any morphisms $f,g$.
The category of small linear monoidal categories $\LMCat$ consists of linear monoidal categories as objects and linear monoidal functors as morphisms.
We can define the \emph{forgetful functor} $U$ from the category of small linear monoidal categories $\LMCat$ to the category of monoidal quivers $\MQuiv$. It is based on the fact that any (small) linear monoidal category $\rD$ is manifestly a monoidal quiver $U\rD$ where the vertice are $Ob(\rD)$ and edges consists of all morphisms $\Hom(\rD)$.
\begin{defin}\label{def:freelinmoncat} The \emph{free linear monoidal category} over a monoidal quiver $Q$ is a category $\tM Q$ together with a monoidal quiver morphism $\tau$ from $Q$ to $U\tM Q$, where $U$ is the forgetful functor defined above, that satisfies the following universal property: for any linear monoidal category $\rD$ and monoidal quiver morphism $\phi$ from $Q$ to $U\rD$, there exists a unique linear functor $\phi^\prime \colon \tM Q \to \rD$, such that $\phi$ factors through $\tau$ as $U\phi^\prime \circ \tau = \phi$, as the diagram below indicates:
\begin{equation}
\begin{tikzcd}
& &&Q\arrow{rd}{\phi}\arrow{d}{\tau}&\\
\tM Q\arrow{r}{\phi^\prime}&\rD, &&U \tM Q\arrow[swap,dashrightarrow]{r}{U \phi^\prime}&U\rD.
\end{tikzcd}
\end{equation}
\end{defin}
The free linear monoidal category $\tM Q$ is unique up to isomorphism by a conventional proof. The question is does the free linear monoidal category always exist? Fortunately, the answer is \emph{yes}. We will give a detailed construction in \cref{sec:LinMonCatExist}.
\subsection{Tensor ideals, relations and presentations}
\begin{defin}
A \emph{tensor ideal} of a linear monoidal category $\rD$ is a linear monoidal subcategory $\rI$ such that
\begin{itemize}
\item $\rI$ is an ideal of $\rD$ when $\rI$ and $\rD$ are viewed as merely linear categories, and
\item it satisfy the ideal property, namely
\begin{align*}
g \otimes f \in \rI(a \otimes c,b \otimes d),\ \text{for all } f \in \rI(a,b),\ g \in \rD(c,d)\\
g \otimes f \in \rI(a \otimes c,b \otimes d),\ \text{for all } f \in \rD(a,b),\ g \in \rI(c,d)
\end{align*}
\end{itemize}
Let $R$ be a set of morphisms of a linear monoidal category $\rD$, not necessary in a single hom-set.
The \emph{tensor ideal generated by} $R$ is the minimal tensor ideal that contains $R$, denoted as $\left<R\right>_\otimes$, which can also be described as the intersection of all tensor ideals that contain $R$.
\end{defin}
The following proposition proves useful.
\begin{prop} \label{prop:tenequivdef}
Let $\rD$ be a linear monoidal category. Suppose a linear monoidal subcategory $\rI$ satisfies the following conditions:
\begin{itemize}
\item $\rI$ is an ideal of $\rD$ when $\rI$ and $\rD$ are viewed as merely linear categories.
\item $\rI$ satisfies the ideal property for identity morphisms; namely for any objects $c$ and morphisms $f \in \rI(a,b)$,
we have
\[1_c \otimes f \in \rI(c\otimes a, c \otimes b)\quad \text{ and } \quad f \otimes 1_c \in \rI( a \otimes c, b \otimes c). \]
\end{itemize}
Then $\rI$ is a tensor ideal of $\rD$.
\end{prop}
\begin{proof}
Notice that $f\otimes g=(f\otimes 1_{\partial_1 g}) \circ (1_{\partial_0 f} \otimes g)$. Since $\rI$ satisfies the ideal property for identity morphisms, it satisfies the ideal property for any morphisms.
\end{proof}
To give an explicit description of the tensor ideal generated by a set of morphisms, we have the following proposition:
\begin{prop}\label{tenidealstruc}
Let $R$ be a set of morphisms of a linear monoidal category $\rD$. Then $\left<R\right>_\otimes$ consists of all morphisms of the form:
\begin{equation}\label{equ:tenidealstruc}
\sum_{i=1}^n k_i\ f^\prime_i\circ (g^\prime_i \otimes r_i \otimes g_i) \circ f_i,
\end{equation}
where $n \in \N$, $k_i \in \kk$, $r_i \in R$ and $f_i,f^\prime_i,g_i,g^\prime_i$ are morphisms in $\rD$ such that the above operations are defined.
\end{prop}
\begin{proof}
Let $\rI$ denote the subcategory of all morphisms of the form in \cref{equ:tenidealstruc}. It forms an ideal by the same argument as in the proof of \cref{idealstruc}. For any two objects $a,a^\prime$ in $\rD$, since the tensor product is bilinear, we have
\begin{align*}
&1_a \otimes \left(\sum_{i=1}^n k_i\ f^\prime_i\circ \left(g^\prime_i \otimes r_i \otimes g_i\right) \circ f_i \right) \otimes 1_a^\prime \\
=& \sum_{i=1}^n k_i\ 1_a\otimes \left(f^\prime_i\circ \left(g^\prime_i \otimes r_i \otimes g_i\right) \circ f_i \right) \otimes 1_{a^\prime}\\
=& \sum_{i=1}^n k_i\
\left( 1_{a} \otimes f^\prime_i \otimes 1_{a^\prime} \right)
\circ
\left(1_a \otimes \left(g^\prime_i \otimes r_i \otimes g_i\right) \otimes 1_{a^\prime} \right)
\circ
\left(1_{a} \otimes f_i \otimes 1_{a^\prime} \right).
\end{align*}
Since $\rI$ is a linear subcategory and
\begin{align*}
1_a \otimes \left(g^\prime_i \otimes r_i \otimes g_i \right) \otimes 1_{a^\prime} = \left( 1_a \otimes g^\prime_i\right) \otimes r_i \otimes \left( g_i \otimes 1_{a^\prime} \right) \in \Hom(\rI),
\end{align*}
we have that $\rI$ satisfies the ideal property for identity morphisms. By \cref{prop:tenequivdef}, $\rI$ is a tensor ideal. Since $\rI$ contains $R$, $\rI$ contains $\left<R\right>_\otimes$. On the other hand, since $r_i$ is in $\left<R\right>_\otimes$ and $\left<R\right>_\otimes$ is a tensor ideal, morphisms of the form \cref{equ:tenidealstruc} are in $R$. Thus we have $\rI=R$.
\end{proof}
Since linear monoidal categories are linear categories and any tensor ideal is an ideal by definition, combining \cref{idealstruc} with \cref{tenidealstruc}, we have the following corollary.
\begin{prop}\label{tenidealandidealstruc}
Let $R$ be a set of morphisms of a linear monoidal category $\rD$. Then viewing $\rD$ as a linear category, we have that $\left<R\right>_\otimes$, as an ideal of $\rD$, is generated by
\[
\tilde{R}= \left\lbrace 1_a \otimes r \otimes 1_b \mid r \in R,\ a,b \in \mathrm{Ob}(\rD) \right\rbrace ,
\]
namely $\left<R\right>_\otimes=\left<\tilde{R}\right>.$
\end{prop}
\begin{proof}
By the structure theorem of ideals and tensor ideals, namely \cref{tenidealstruc} and \cref{idealstruc} , we have $\left<R\right>_\otimes$, as an ideal, is generated by
\[
\left\lbrace g^\prime \otimes r \otimes g \mid r \in R,\ g,g^\prime \in \Hom(\rD) \right\rbrace .
\]
At the same time we have
\[
g^\prime \otimes r \otimes g = (g^\prime \otimes 1_{x^\prime} \otimes 1_{b^\prime}) \circ (1_a \otimes r \otimes 1_b) \circ ( 1_a \otimes 1_x \otimes g) \in \left<\tilde{R}\right>,
\]
where $r \colon x \to x^\prime$, $g \colon a \to a^\prime$ and $g^\prime \colon b \to b^\prime.$ Thus we have $\left<R\right>_\otimes \subseteq \left<\tilde{R}\right>$. Conversely, $1_a \otimes r \otimes 1_b \in \left<R\right>_\otimes$ implies $\left<\tilde{R}\right> \subseteq \left<R\right>_\otimes$, and so we have $\left<R\right>_\otimes = \left<\tilde{R}\right>$.
\end{proof}
\color{black}
\begin{defin}
The \emph{quotient category} $\rD/\rI$ of a linear monoidal category $\rD$ by a tensor ideal $\rI$ is the quotient category of $\rD$ by $\rI$ viewed respectively as a linear category and an ideal, equipped with a tensor product defined by
\begin{align*}
[f] \otimes [g] = [f \otimes g]
\end{align*}
for $f,g$ morphisms in $\rD$.
\end{defin}
The definition of ideals of linear categories guarantees that the tensor product of cosets are well-defined: let $[f]=[f^\prime]$, $[g]=[g^\prime]$. Then we have $f - f^\prime \in \Hom(\rI)$, $g- g ^\prime \in \Hom(\rI)$. Thus
\begin{align*}
f \otimes g - f^\prime \otimes g^\prime
= f \otimes g - f^\prime \otimes g + f^\prime \otimes g - f^\prime \otimes g^\prime
= (f - f^\prime) \otimes g - f^\prime \otimes (g -g^\prime) \in \Hom(\rI).
\end{align*}
Thus we have $ [f \otimes g] = [f ^\prime \otimes g^\prime]$.
For a set $X$, let $M(X)$ denote the free monoid generated by $X$. Now we define presentations of linear monoidal categories.
\begin{defin}
A \emph{presentation} of a linear monoidal category $\rD$ is a triple $(X,E,R)_\otimes$, where $R$ is a set of morphisms of the free linear monoidal category $\tM Q$ over the monoidal quiver $Q=(M(X),E)$ such that $\rD$ is isomorphic to the quotient category of $\tM Q$ by the tensor ideal generated by $R$, namely the linear monoidal category $\tM Q / \left<R\right>_\otimes$.
\end{defin}
\section{Construction of the free linear monoidal category} \label{sec:LinMonCatExist}
In this section, we prove the existence of the free (strict) linear monoidal category over a monoidal quiver by giving an explicit construction. We follow the work of \cite[\S 5]{Pie17}.
\subsection{Step one}
Let $V$ be a monoid where the product is denoted by concatenation and let $Q=(V,E)$ be a quiver on $V$. We define a new quiver $\bar{Q}=(V,\bar{E})$, where the set of edges $\bar{E}$ consists of all triples
\[
(v,e,w), \qquad v,w \in V,\ e \in E.
\]
For an edge $e \colon x \to y$, we define
\[
\partial_0(v,e,w)=vxw,\qquad \partial_1 (v,e,w) = vyw.
\]
Now take $\rM^\prime$ to be the free linear category over $\bar{Q}$, namely $\rM^\prime = \tL \bar{Q}$. Then all morphisms in $\rM^\prime$ are linear combinations of paths on $\bar{Q}$.
We now define a product $\otimes$ on both objects and morphisms of $\rM^\prime$ inductively.
\begin{itemize}
\item For two objects $a$ and $b$ of $\rM^\prime$, we define
\[
a \otimes b = ab, \quad 1_a \otimes 1_b = 1_{ab}.
\]
\item For a triple $(v,e,w)$ and a path $f=\alpha_1 \circ \dotsb \circ \alpha_n$ where $\alpha_i=(v_i,e_i,w_i),i=1,...,n$ are edges, we define
\begin{align}
\begin{split}
1_a \otimes (v,e,w) = (av,e,w)\quad \text{and}\quad
1_a \otimes f= (1_a \otimes \alpha_1) \circ \dotsb \circ (1_a \otimes \alpha_n),\\
(v,e,w) \otimes 1_a = (v,e,w a)\quad \text{and}\quad
f \otimes 1_a = ( \alpha_1 \otimes 1_a) \circ \dotsb \circ ( \alpha_n \otimes 1_a).\\
\end{split}
\end{align}
\item For two paths $f \colon a \to b$ and $g \colon c \to d$, we define
\begin{align} \label{productreduce}
\begin{split}
f \otimes g = (f \otimes 1_d) \circ (1_a \otimes g)
\end{split}
\end{align}
(From now on we adopt the convention that the product $\otimes$ will be perfomed before the composition $\circ$ to avoid unnecessary parentheses.) In particular, for two edges $(v,e,w)$ and $(v^\prime,e^\prime,w^\prime)$, where $e \colon x \to y$ and $e^\prime \colon x^\prime \to y^\prime$, we have that
\begin{align}
\begin{split}
(v,e,w) \otimes (v^\prime,e^\prime,w^\prime)
=&(v,e,w) \otimes 1_{v^\prime y^\prime w^\prime} \circ 1_{vxw} \otimes (v^\prime,e^\prime,w^\prime)\\
=&(v,e,wv^\prime y^\prime w^\prime) \circ (vxwv^\prime,e^\prime,w^\prime).
\end{split}
\end{align}
\item We extend $\otimes$ to all morphisms of $\rM^\prime$ by bilinearity.
\end{itemize}
It's easy to check that $\otimes$ is associative and the identity $\one$ of the monoid $M(X)$ and the identity morphisms $1_{\one}$ on it are the identities of objects and morphisms of the product, respectively. We may be attempted to think $(M^\prime,\otimes)$ is the free linear monoidal category. However, it is not, since the interchange law doesn't hold. In particular, the interchange law for triples fails by definition:
\begin{align}
(v,e,wv^\prime y^\prime w^\prime) \circ (vxwv^\prime,e^\prime,w^\prime) \neq
(vywv^\prime,e^\prime,w^\prime) \circ (v,e,wv^\prime x^\prime w^\prime).
\end{align}
Thus we are motivated to mod out the differences of the two sides of the above expression.
\subsection{Step two} \label{sec:freelinmoncatrel}
Let $C$ be
all morphisms of the following form
\begin{equation}\label{interchangelawoftriples}
(v,e,wv^\prime y^\prime w^\prime) \circ (vxwv^\prime,e^\prime,w^\prime) -
(vywv^\prime,e^\prime,w^\prime) \circ (v,e,wv^\prime x^\prime w^\prime),
\end{equation}
where $v,w,v^\prime, w^\prime \in \mathrm{Ob}(M)$ and $e \colon x \to y, e^\prime \colon x^\prime \to y^\prime \in E$.
Let $\rM$ be the quotient category of $\rM^\prime$ by the ideal generated by $C$, namely $\rM=\rM^\prime/\left<C\right>$. Then $\otimes$ induces a product on $\rM$, still denoted by $\otimes$.
Before we show that $(\rM,\otimes)$ actually forms a linear monoidal category, we need to show that $\otimes$ is well defined on $\rM$. Namely, for a morphism $f$ in $\left<C\right>$ and an arbitrary morphism $g$ we have that $f \otimes g \in \left<C\right>$ and $g \otimes f \in \left<C\right>$.
By \cref{idealstruc}, morphisms in $\left<C\right>$ are of the form
\[
f=\sum_{i=1}^n k_i\ p_i^\prime \circ r_i \circ p_i,
\]
where $k_i \in \kk$, $r_i \in C$ and $p_i,p_i^\prime$ are paths in $\rM^\prime$. Let $a$ and $b$ be the domain and codomain of $f$, respectively. Since for an arbitrary path $g\colon c \to d$ in $\rM^\prime$ we have
\begin{align}
\begin{split}
f \otimes g &= f \otimes 1_d \circ 1_a \otimes g\\
&= \sum_{i=1}^n k_i\ (p_i^\prime \circ r_i \circ p_i) \otimes 1_d \circ 1_a \otimes g\\
&= \sum_{i=1}^n k_i\ p_i^\prime \otimes 1_d \circ r_i \otimes 1_d \circ p_i \otimes 1_d \circ 1_a \otimes g,
\end{split}
\end{align}
to show $f\otimes g$ is in $\left<C\right>$, it suffices to show that morphisms of the form $r \otimes 1_d$ are in $\left<C\right>$.
By the definition of $C$, we have $$r=\alpha \otimes 1_{y^\prime} \circ 1_{x} \otimes \alpha^\prime -
1_{y} \otimes \alpha^\prime \circ \alpha \otimes 1_{x^\prime} ,$$ for some triples $\alpha \colon x \to y$, $\alpha^\prime \colon x^\prime \to y^\prime$. Let $\beta^\prime=\alpha^\prime \otimes 1_d \colon x^\prime d \to y^\prime d$. Then we have
\begin{align}
\begin{split}
r \otimes 1_d &=
\alpha \otimes 1_{y^\prime} \otimes 1_d \circ 1_{x} \otimes \alpha^\prime \otimes 1_d -
1_{y} \otimes \alpha^\prime \otimes 1_d \circ \alpha \otimes 1_{x^\prime} \otimes 1_d\\
&= \alpha \otimes 1_{y^\prime d} \circ 1_x \otimes \beta^\prime - 1_y \otimes \beta^\prime \circ \alpha \otimes 1_{x^\prime d} \in C.
\end{split}
\end{align}
Thus we have $f \otimes g \in \left<C\right>$, and similar arguements yield that $g \otimes f$ is also in $\left<C\right>$.
Thus $\otimes$ is well-defined on $\rM$. It follows directly that $\otimes$ on $\rM$ is bilinear and associative since $\otimes$ on $\rM^\prime$ is bilinear and associative.
Morphisms in $\rM$ are cosets of morphisms in $\rM^\prime$, denoted by $[f]$, where $f \in \Hom(\rM^\prime)$. We will omit the brackets for identity morphisms and triples to simplify our notation.
Since we define $\rM$ by modding out morphisms of the form \cref{interchangelawoftriples}, in $\rM$ we already have the interchange law for triples:
\begin{align}
(v,e,wv^\prime y^\prime w^\prime) \circ (vxwv^\prime,e^\prime,w^\prime) =
(vywv^\prime,e^\prime,w^\prime) \circ (v,e,wv^\prime x^\prime w^\prime).
\end{align}
It quickly follows that the interchange law for arbitrary morphisms also holds, as we now explain. Since $\otimes$ is bilinear, it suffices to check the interchange law for arbitrary paths. For two paths $f=\alpha_1 \circ \dotsb \alpha_n$ and $g=\beta_1 \circ \dotsb
\circ \beta_m$ where $\alpha_i: x_i \to y_i $ and $\beta_j: z_i \to w_i$ are triples, we have that
\[
[f] \otimes 1_{w_1} \circ 1_{x_n} \otimes [g] = (\alpha_1 \otimes 1_{w_1}) \circ \dotsb \circ (\alpha_n \otimes 1_{w_1}) \circ (1_{x_n} \otimes \beta_1) \circ \dotsb \circ (1_{x_n} \otimes \beta_m).
\]
By using the interchange law for triples repeatedly we can move all morphisms of the form $1_{x_i} \otimes \beta_j$ to the left. After that, the equation above reads exactly $1_{y_1} \otimes [g] \circ [f] \otimes 1_{z_m}$.
Thus the interchange law for arbitrary tensor products holds. It follows that $\otimes$ is a bifunctor since for $f_i \colon a_i \to b_i,i=1,2,$ and $g_i \colon c_i \to d_i,i=1,2$, we have
\begin{align}
\begin{split}
([f_1] \circ [f_2])\otimes( [g_1] \circ [g_2])
&= ([f_1] \circ [f_2]) \otimes 1_{d_1} \circ 1_{a_2} \otimes([g_1] \circ [g_2])\\
&= [f_1] \otimes 1_{d_1} \circ [f_2] \otimes 1_{d_1} \circ 1_{a_2} \otimes [g_1] \circ 1_{a_2} \otimes [g_2]\\
&\stackrel{\star}{=} [f_1] \otimes 1_{d_1} \circ 1_{b_2} \otimes [g_1] \circ [f_2] \otimes 1_{z_1} \circ 1_{a_2} \otimes [g_2] \\
&= [f_1] \otimes 1_{d_1} \circ 1_{a_1} \otimes [g_1] \circ [f_2] \otimes 1_{w_2} \circ 1_{a_2} \otimes [g_2] \\
&= [f_1] \otimes [g_1] \circ [f_2] \otimes [g_2],
\end{split}
\end{align}
where $\star$ is due to the interchange law.
Now $(\rM,\otimes)$ does form a linear monoidal category.
\color{black}
\subsection{Step three: universal property}
Now we claim that $(\rM,\otimes)$ satisfies the universal property in \cref{def:freelinmoncat} and therefore it is the free linear monoidal category. We denote by $\tau=(\tau_0,\tau_1)$ the natural inclusion from the base monoidal quiver $Q=(V,E)$ to $U\rM$, where $U$ is the natural forgetful functor, $\tau_0$ is identity on $V$ and $\tau_1$ maps $e$ to $(\one,e,\one)$.
\begin{theo}[Universal property]
The category $\rM$ constructed above is the free linear monoidal category over $Q$, as defined in \cref{def:freelinmoncat}.
\end{theo}
\begin{proof}
To prove that $\rM$ is the free linear monoidal category, we should show that for any linear monoidal category $\rD$ and monoidal quiver morphism $\phi$ from $Q$ to $U\rD$, where U is the forgetful functor from $\LMCat$ to $\MQuiv$, there exists a unique linear monoidal functor $\phi^\prime \colon \rM \to \rD$ such that $\phi=U\phi^\prime \circ \tau$, as the diagram below indicates:
\begin{equation}
\begin{tikzcd}
& &&Q\arrow{rd}{\phi}\arrow{d}{\tau}&\\
\rM\arrow{r}{\exists! \phi^\prime}&\rD, &&U\rM\arrow[swap,dashrightarrow]{r}{U\phi^\prime}&U\rD.
\end{tikzcd}
\end{equation}
Define a linear monoidal functor $\phi^\prime$ in the following way: for $[f] = \sum_{i=1}^n k_i [f_i]$, where $f_i$ are paths and $[f_i] = \alpha^i_{1} \circ \dotsb \circ \alpha^i_{n_i}$, with $\alpha^i_{j}=(v^i_j,e^i_j,w^i_j)=1_{v^i_j} \otimes (\one,e^i_j,\one) \otimes 1_{w^i_j},$ we define $\phi^\prime([f])$ as follows:
\begin{align*}
\begin{split}
\phi^\prime([f]) &= \sum_{i=1}^n k_i\ \phi^\prime([f_i]),\\
\phi^\prime([f_i]) &=\phi^\prime(\alpha^i_1) \circ \dotsb \circ \phi^\prime(\alpha^i_{n_i}),\\
\phi^\prime(\alpha^i_j) &=\phi^\prime(1_{v^i_j}) \otimes \phi^\prime(\one,e^i_j,\one) \otimes \phi^\prime(1_{w^i_j}),\\
\phi^\prime(1_{v^i_j})&=1_{\phi(v^i_j)},\quad \phi^\prime (\one,e^i_j,\one) = \phi(e^i_j).
\end{split}
\end{align*}
To show $\phi^\prime$ is well-defined, it suffices to show for any morphism $[f]=[0]$, we have $\phi^\prime([f])=0$.
By the choice of $f$ we have $f \in \left<C\right>$, where $C$ is defined in \cref{interchangelawoftriples}, and thus there exists some $g_i,g^\prime_i,r_i \in \rM$ and $k_i \in \kk$ such that $[f]=\sum_{i=1}^n k_i(g^\prime_i \circ r_i \circ g_i)$. By the definition of $\phi^\prime$, we have
\begin{equation}\label{phiwelldefined}
\phi^\prime([f])=\sum_{i=1}^n k_i \phi^\prime([g^\prime_i]) \circ \phi^\prime([r_i]) \circ \phi^\prime([g_i]).
\end{equation}
Consider $\phi^\prime(r)$ for any $r \in C$. There exists $(v,e,w)$ and $(v^\prime,e^\prime,w^\prime)$, where $e \colon x \to y$ and $e^\prime \colon x^\prime \to y^\prime$ such that
\[
r=(v,e,w v^\prime y^\prime w^\prime) \circ (vxw v^\prime,e,w^\prime) - (vyw v^\prime,e,w^\prime) \circ (v,e,w v^\prime x^\prime w^\prime).
\]
By the definition of $\phi^\prime$, we have
\begin{align*}
\phi^\prime([r])=&\phi^\prime(v,e,w v^\prime y^\prime w^\prime) \circ \phi^\prime(vxw v^\prime,e,w^\prime) -
\phi^\prime(vyw v^\prime,e,w^\prime) \circ \phi^\prime(v,e,w v^\prime x^\prime w^\prime)\\
=& 1_{\phi(v)} \otimes \phi(e) \otimes 1_{\phi(w v^\prime y^\prime w^\prime)} \circ 1_{\phi(vxw v^\prime)} \otimes \phi(e^\prime) \otimes 1_{\phi(w^\prime)} \\
&- 1_{\phi(vyw v^\prime)} \otimes \phi(e^\prime) \otimes 1_{\phi(w^\prime)} \circ 1_{\phi(v)} \otimes \phi(e) \otimes 1_{\phi(w v^\prime x^\prime w^\prime)}
\end{align*}
Thus we have $\phi^\prime([r])=0$ by the interchange law in $\rD$.
With the well-definedness established, it's easy to show that $\phi^\prime$ is indeed a linear monoidal functor.
\end{proof}
Therefore, $\rM= \tM Q$, the free linear monoidal category over a monoidal quiver, always exists. It follows that one can always talk about presentations of a linear monoidal category.
\section{Presentation of linear monoidal categories as linear categories}\label{sec:MainTheorem}
In this section, we will give our main results: given a presentation of a linear monoidal category, we can produce a presentation of it as a linear category, and when all generating morphisms are endomorphisms, we can produce presentations of its endomorphism algebras. These results are in \cref{thm:presentationofalgebra,MainTheorem} and the former one is stated in \cite[\S 2.6]{BCNR17} without proof.
Before we get to the main theorem, we need the following lemma.
\begin{lem}[The third isomorphism theorem]\label{ThirdIso} Let $\rL$ be a linear category and $\rI$, $\rJ$ be two ideals of $\rL$ such that $\rI$ is a subcategory of $\rJ$. Then $\rJ/\rI$ is also an ideal of $\rL/\rI$ and we have
\begin{equation*}
\rL/\rJ \cong (\rL/\rI)/(\rJ/\rI),
\end{equation*}
where $\cong$ means isomorphic as linear categories.
\end{lem}
\begin{proof}
Consider a family of module morphisms $P_{a,b}$, $a,b \in \mathrm{Ob}(\rL)$: for each $f \in \rL(a,b)$, $P_{a,b}$ sends $[f]=f + \rI(a,b)$ to $f+ \rJ(a,b)$. It is well defined since $\rI$ is contained in $\rJ$. By the third isomorphism theorem of modules, we have there exists module isomorphism $\bar{P}_{a,b}$ from
\[ {( \rL/ \rI)/( \rJ/ \rI)}(a,b)=({ \rL/ \rI}(a,b))/({ \rJ/ \rI}(a,b))\]
to $\rL/\rJ(a,b)$. In particular, $\bar{P}_{a,b}$, sending each $[f]+\rJ/\rI(a,b)$ to $f + \rJ(a,b)$, is well-defined. Thus we can define a linear functor $\bar{P}$ induced by $\bar{P}_{a,b},\ a,b \in \mathrm{Ob}(\rL)$. The functor $\bar{P}$ does form a functor since we have
\begin{align*}
P ([g] + \rJ/\rI(b,c)) ([f] +\rJ/\rI(a,b))= P([g \circ f] + \rJ/\rI(a,c))
= g \circ f + \rJ(a,c) \\= (g + \rJ(b,c))(f+\rJ(a,b)) = P([g] + \rJ(b,c)) \circ P( [f] +\rJ(a,b)).
\end{align*}
Since $\bar{P}_{a,b}$ are linear isomorphisms, $\bar{P}$ is an isomorphism of linear categories.
\end{proof}
Now we reformulate and prove the theorem stated in \cite[\S 2.6]{Bru14} without proof.
\begin{theo}\label{MainTheorem}
If a linear monoidal category $\rM$ has a presentation $(X,E,R)_\otimes$ as a linear monoidal category, then it has a presentation $(M(X),\bar{E},\bar{R})$ as a linear category, where $M(X),\bar{E},\bar{R}$ are defined as follows.
\begin{itemize}
\item $M(X)$ is the free monoid generated by $X$, where the product is denoted by concatenation.
\item The set of generating morphisms $\bar{E}$ is
\begin{align*}
\bar{E}=\left\lbrace (v,e,w) \mid v,w \in M(X),\ e \in E \right\rbrace.
\end{align*}
\item The set of relations $\bar{R}$ is $C \cup R^\prime$, where $C$ and $R^\prime$ are defined as follows.
\begin{itemize}
\item The set $C$ is the relation of the interchange law. To be precise, $C$ consists of all morphisms of the form
\[
(v,e,wv^\prime y^\prime w^\prime) \circ (vxwv^\prime,e^\prime,w^\prime) -
(vywv^\prime,e^\prime,w^\prime) \circ (v,e,wv^\prime x^\prime w^\prime),
\]
where $v,v^\prime,w,w^\prime$ are objects and $e \colon x \to y, e^\prime \colon x^\prime \to y^\prime$ are edges in $E$.
\item The set $R^\prime$ is constructed in the following way: for each relation $r$ in $R$ written as
\begin{align*}
[r]=\sum_{i=1}^n k_i [f_i],\quad
[f_i]=\alpha^i_1 \circ \dotsb \circ \alpha^i_{n_i},\quad
\alpha^i_j=(v^i_j, e^i_j,w^i_j),
\end{align*}
where $n$, $n_i \in \N$, $e^i_j \in E$,
let $R^\prime$ include $r_{a,b}$, for $a$, $b \in M(X)$, where
\begin{align*}
r_{a,b}&= \sum_{i=1}^n k_i (f_i)_{a,b},\\
(f_i)_{a,b}&=(\alpha^i_1)_{a,b} \circ \dotsb \circ (\alpha^i_{n_i})_{a,b},\\
(\alpha^i_j)_{a,b}&=(a v^i_j, e^i_j, w^i_j b).
\end{align*}
\end{itemize}
\end{itemize}
\end{theo}
\begin{rem}
In the construction of $R^\prime$, we choose some representatives of the cosets $[r]$, which might contradict well-definedness. However, we can see from the proof below that only the coset itself matters, namely the resulting presentation is independent with the choice of the representatives.
\end{rem}
\begin{proof}
Let $Q=(X,E)$ and $\bar{Q}=(M(X),\bar{E})$ where $M(X)$ and $\bar{E}$ are defined as above. We shall prove that $\rM = \tM Q/ \left<R\right>_\otimes$, as a linear category, is isomorphic to $\tL \bar{Q} / \left<\bar{R}\right>$.
Let $\rL= \tL \bar{Q}$, $\rI=\left<C\right>$ and $\rJ=\left<\bar{R}\right>$. Now $\rD = \rL/\rJ$ and by \cref{ThirdIso}, $\rD =(\rL/\rI)/(\rJ/\rI)$.
Notice $\rI$ is exactly the interchange law for triples, so we can equip $\rL/\rI$ with the natural tensor product as in \cref{sec:LinMonCatExist}. Therefore $\rL/\rI$ coincides with $\tM Q$. Thus it suffices to show $\rJ/\rI$ is the same ideal as $\left<R\right>_\otimes$.
On the one hand, by \cref{tenidealandidealstruc}, we have $\left<R\right>_\otimes$, as an ideal, is generated by $\tilde{R}$, where
\[
\tilde{R}=\left\lbrace 1_a \otimes [r] \otimes 1_b \mid a,b \in M(X), r \in R \right\rbrace .
\]
One the other hand, since $\rJ = \left<\bar{R}\right>= \left<C\cup R^\prime\right>$, the ideal $\rJ/\rI$ of $\rL/\rI=\tM Q$ is generated by
\[
[s], \quad s \in C \cup R^\prime.
\]
If $s \in C$, we have $[s]=[0]$. For $s \in R^\prime$, there exists $r \in R$ such that $s =r_{a,b}$. Namely
\begin{align*}
[s] &= [r_{a,b}] = \sum_{i=1}^n k_i [(f_i)_{a,b}],\\
[(f_i)_{a,b}] &= (\alpha^i_1)_{a,b} \circ \dotsb \circ (\alpha^i_{n_i})_{a,b},\\
(\alpha^i_j)_{a,b}&=(a v^i_j, e^i_j, w^i_j b)= 1_{a} \otimes (v^i_j,e^i_j,w^i_j) \otimes 1_{b},
\end{align*}
for some $r$ such that
\begin{align*}
[r]=\sum_{i=1}^n k_i [f_i],\quad
[f_i]=\alpha^i_1 \circ \dotsb \circ \alpha^i_{n_i},\quad
\alpha^i_j=(v^i_j, e^i_j,w^i_j).
\end{align*}
Since $(\alpha^i_j)_{a,b} = 1_{a} \otimes (v^i_j,e^i_j,w^i_j) \otimes 1_{b}$, we have $$[r_{a,b}]=1_a \otimes [r] \otimes 1_b.$$ (In particular, this implies that the choice of representatives doesn't matter.)
Thus $\rJ/\rI$ is also generated by $\tilde{R}$.
So we have $\rJ/\rI= \left<R\right>_\otimes.$
\end{proof}
Now we focus on the endomorphisms algebras. We consider a special case where all generators are endomorphisms.
\begin{theo}\label{thm:presentationofalgebra}
Let $\rD$ be a linear monoidal category with a presentation $\left<X,E,R\right>_\otimes$, where all generating morphisms in $E$ are endomorphisms and let $\left<M(X),\bar{E},\bar{R}\right>$ be its presentation as a linear category in \cref{MainTheorem}. For an object $a$, let $\bar{E}_a$ ($\bar{R}_a$) be the set of all generating morphisms in $\bar{E}$ (all the relations in $\bar{R}$) that are also in the endomorphism algebra $\End(a)$. Then the endomorphism algebra $\End(a)$ has an algebra presentation
\[
\End(a) \cong \left< \bar{E}_a \mid \bar{R}_a \right>.
\]
\end{theo}
\begin{proof}
Since all morphisms in $E$ are endomorphisms, all morphisms in $\bar{E}$, namely
\[
(1_v,e,1_w), \quad v,w \in M(X) \text{ and } e \in E,
\]
are endomorphisms. Therefore $\End_\rD(a)$, as a algebra, is generated by generators of $\rD$ which are also in $\End_\rD(a)$, denoted by $E_a$.
On the other hand, the ideal generated by $\bar{R}$, by \cref{idealstruc}, consists of morphisms of the form
\[
\sum_{i=1}^n k_i (f^\prime_i\circ r_i \circ f_i),
\]
where $k_i \in \kk$, $r_i \in R_a$ and $f_i, f^\prime_i \in \Hom(\rD)$.
Since all morphisms in $M$ are endomorphisms, its endomorphism algebra $\End_{\left<R\right>}(a)$ consists of morphisms of the above form with $f, f^\prime \in \End_\rD(a)$. Thus $\End_{\left<R\right>}(a)$ is exactly the ideal of the algebra $\End(a)$ generated by $\bar{R}_a$. Hence $\End(a)$ has the presentation $\left< \bar{E}_a \mid \bar{R}_a \right>$.
\end{proof}
\color{black}
\section{Monoidally generated algebras} \label{sec:MonGenAlg}
In this section, we will introduce some important (strict) linear monoidal categories by following the presentation in \cite[\S 3]{Savage18}, where all generating morphisms are endomorphisms, and apply \cref{thm:presentationofalgebra,MainTheorem} to produce presentations of their endomorphism algebras.
\subsection{String diagrams}\label{sec:StringDiag}
To help visualize these linear monoidal categories, we will use the notation of string diagrams. We follow the presentation of \cite[\S 2.2]{Savage18}.
A morphism $f \colon a \to b$ is denoted by a vertical strand with a coupon labeled $f$, read from the bottom to the top. In particular, identity morphisms are denoted by empty strands.
\[
f \colon a \to b \ \text{is denoted by}\ \
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{a}} to (0,1) node[anchor=south] {\regionlabel{b}};
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{f}};
\end{tikzpicture}, \qquad
1_a \colon a \to a \ \text{is denoted by}\ \
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{a}} to (0,1) node[anchor=south] {\regionlabel{a}};
\end{tikzpicture}.
\]
Composition of morphisms is denoted by vertical stacking and tensor product is denoted by horizontal juxtaposition:
\[
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{b}} to (0,1) node[anchor=south] {\regionlabel{c}};
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{f}};
\end{tikzpicture}
\ \circ \
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{a}} to (0,1) node[anchor=south] {\regionlabel{b}};
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{g}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{a}} to (0,1.4) node[anchor=south] {\regionlabel{c}};
\filldraw[black,fill=white] (0,1.2) arc(90:450:0.2);
\node at (0,1) {\tokenlabel{f}};
\filldraw[black,fill=white] (0,0.6) arc(90:450:0.2);
\node at (0,0.4) {\tokenlabel{g}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{a}} to (0,1.4) node[anchor=south] {\regionlabel{c}};
\filldraw[black,fill=white] (0,1) arc(90:450:0.3);
\node at (0,0.7) {\tokenlabel{f \circ g}};
\end{tikzpicture},
\qquad
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{a}} to (0,1) node[anchor=south] {\regionlabel{b}};
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{f}};
\end{tikzpicture}
\ \otimes \
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{c}} to (0,1) node[anchor=south] {\regionlabel{d}};
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{g}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{a}} to (0,1) node[anchor=south] {\regionlabel{b}};
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{f}};
\draw (0.5,0) node[anchor=north] {\regionlabel{c}} to (0.5,1) node[anchor=south] {\regionlabel{d}};
\filldraw[black,fill=white] (0.5,0.7) arc(90:450:0.2);
\node at (0.5,0.5) {\tokenlabel{g}};
\end{tikzpicture}.
\]
The \emph{interchange law} for $f \colon a \to b$ and $g \colon c \to d$
\[
(f \otimes 1_{d}) \circ (1_{a} \otimes g)
\ =\
f \otimes g
\ =\
(1_{b} \otimes g) \circ (f \otimes 1_{c})
\]
then becomes the following, where we omit the object labels:
\[
\begin{tikzpicture}[anchorbase]
\draw (0,0) to (0,1.6);
\filldraw[black,fill=white] (0,1.3) arc(90:450:0.2);
\node at (0,1.1) {\tokenlabel{f}};
\draw (0.5,0) to (0.5,1.6);
\filldraw[black,fill=white] (0.5,0.7) arc(90:450:0.2);
\node at (0.5,0.5) {\tokenlabel{g}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw (0,0) to (0,1.6);
\filldraw[black,fill=white] (0,1) arc(90:450:0.2);
\node at (0,0.8) {\tokenlabel{f}};
\draw (0.5,0) to (0.5,1.6);
\filldraw[black,fill=white] (0.5,1) arc(90:450:0.2);
\node at (0.5,0.8) {\tokenlabel{g}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw (0,0) to (0,1.6);
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{f}};
\draw (0.5,0) to (0.5,1.6);
\filldraw[black,fill=white] (0.5,1.3) arc(90:450:0.2);
\node at (0.5,1.1) {\tokenlabel{g}};
\end{tikzpicture}.
\]
In the following sections we will use string diagrams to visualize the generators and relations of several linear monoidal categories. We may generalize the horizontal juxaposition by using it to denote tensor products of objects and triples $(v,e,w)$, namely
\[
1_{v \otimes w} =
\begin{tikzpicture}[anchorbase]
\draw (0,0) node[anchor=north] {\regionlabel{v}}
to (0,1) node[anchor=south] {\regionlabel{v}} ;
\draw (0.4,0) node[anchor=north] {\regionlabel{w}}
to (0.4,1) node[anchor=south] {\regionlabel{w}} ;
\end{tikzpicture}
\quad \text{and} \quad
(v,e,w)
\ =\
\begin{tikzpicture}[anchorbase]
\draw (-0.5,0) node[anchor=north] {\regionlabel{v}} to (-0.5,1)
node[anchor=south] {\regionlabel{v}};
\draw (-0,-0) node[anchor=north] {\regionlabel{x}} to (0,1) node[anchor=south] {\regionlabel{y}};
\filldraw[black,fill=white] (0,0.7) arc(90:450:0.2);
\node at (0,0.5) {\tokenlabel{e}};
\draw (0.5,0) node[anchor=north] {\regionlabel{w}} to (0.5,1) node[anchor=south] {\regionlabel{w}};
\end{tikzpicture}
\quad \text{for } e\colon x \to y.
\]
\color{black}
\subsection{The Symmetric group} \label{sec:Sym}
Let $\cS$ be the linear monoidal category with the presentation $\left<X_\cS,E_\cS,R_\cS\right>_\otimes$, where
\[
X_\cS=\left\lbrace a\right\rbrace ,\ E_\cS=\left\lbrace e \colon a\otimes a \to a \otimes a\right\rbrace \text{ and }R_\cS=\left\lbrace r_1,r_2\right\rbrace ,
\]
where
\[
r_1 = e^2-1_a,\quad
r_2 = (e\otimes 1_a) \circ (1_a \otimes e) \circ (e \otimes 1_a) - (1_a \otimes e) \circ (e \otimes 1_a) \circ (1_a \otimes e).
\]
Using string diagrams, we denote
\begin{itemize}
\item the generating object $a$ as $\uparrow$ and,
\item the generating morphism $e$ as
$
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.25,-0.25) to (0.25,0.25);
\draw[->] (0.25,-0.25) to (-0.25,0.25);
\end{tikzpicture},
$
\end{itemize}
then $R_\cS$ becomes
\begin{equation} \label{Sn-strings}
\begin{tikzpicture}[anchorbase]
\draw[->] (0.3,0) to[out=up,in=down] (-0.1,0.5) to[out=up,in=down] (0.3,1);
\draw[->] (-0.1,0) to[out=up,in=down] (0.3,0.5) to[out=up,in=down] (-0.1,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.2,0) -- (-0.2,1);
\draw[->] (0.2,0) -- (0.2,1);
\end{tikzpicture}
\qquad \text{and} \qquad
\begin{tikzpicture}[anchorbase]
\draw[->] (0.4,0) -- (-0.4,1);
\draw[->] (0,0) to[out=up, in=down] (-0.4,0.5) to[out=up,in=down] (0,1);
\draw[->] (-0.4,0) -- (0.4,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (0.4,0) to (-0.4,1);
\draw[->] (0,0) to[out=up, in=down] (0.4,0.5) to[out=up,in=down] (0,1);
\draw[->] (-0.4,0) to (0.4,1);
\end{tikzpicture}
\ .
\end{equation}
By \cref{MainTheorem}
, we have that $\cS$ has the presentation $\left<M(X_\cS),\overline{E_\cS},\overline{R_\cS}\right>$ as a linear category.
\begin{itemize}
\item The objects are $M(X_\cS)=\left\lbrace \uparrow^{\otimes n} \mid n \in \N\right\rbrace $. To simplify our notation, we also denote the tensor product on objects $\otimes$ as concatenation. Thus $M(X_\cS)=\left\lbrace \uparrow^{ n} \mid n \in \N\right\rbrace $
\item The generating morphisms in $\overline{E_\cS}$ are $(v,e,w)$, where $e \colon \uparrow^2 \to \uparrow^2$ and $v,w \in M(X_\cS)$, namely
\[
\bar{E}_\cS = \left\lbrace \
\left(\uparrow^m,
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.25,-0.25) to (0.25,0.25);
\draw[->] (0.25,-0.25) to (-0.25,0.25);
\end{tikzpicture},
\uparrow^n\right)
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture}\
\mid
m,n \in \N \right\rbrace .
\]
\item The relations are $\overline{R_\cS} = C_\cS \cup R_\cS^\prime$. The relations in $C_\cS$ are the interchange laws of triples, namely
\begin{align} \label{SymInt}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,0.5) to [out=90, in=270] (0.4,1);
\draw[->] (0.4,0) to (0.4,0.5) to [out=90, in=270] (0,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{l}};
\draw[->] (1.6,0) to (1.6,1);
\draw[->] (2,0) to [out=90, in=270] (2.4,0.5) to (2.4,1);
\draw[->] (2.4,0) to [out=90, in=270] (2,0.5) to (2,1);
\draw[->] (2.8,0) to (2.8,1);
\node at (3.2,0.5) {$\cdots$};
\node at (3.2,0) {\regionlabel{n}};
\draw[->] (3.6,0) to (3.6,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to (0.4,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{l}};
\draw[->] (1.6,0) to (1.6,1);
\draw[->] (2,0) to (2.0,0.5) to [out=90, in=270] (2.4,1);
\draw[->] (2.4,0) to (2.4 ,0.5) to [out=90, in=270] (2,1);
\draw[->] (2.8,0) to (2.8,1);
\node at (3.2,0.5) {$\cdots$};
\node at (3.2,0) {\regionlabel{n}};
\draw[->] (3.6,0) to (3.6,1);
\end{tikzpicture},
\end{align}
for all $m,n,l \in \N$.
The relations in $R_\cS^\prime$ are $(r_i)_{a,b}$ as defined in \cref{MainTheorem}, where $r_i \in R$, $a,b \in M(X)$, namely
\begin{align} \label{SymRel}
\begin{split}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to [out=up, in=down] (0,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to [out=up,in=down] (0.4,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ &=\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\draw[->] (0.4,0) to (0.4,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
,\\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0.8,0) -- (0,1);
\draw[->] (0.4,0) to[out=up, in=down] (0,0.5) to[out=up,in=down] (0.4,1);
\draw[->] (0,0) -- (0.8,1);
\draw[->] (1.2,0) to (1.2,1);
\node at (1.6,0.5) {$\cdots$};
\node at (1.6,0) {\regionlabel{n}};
\draw[->] (2,0) to (2,1);
\end{tikzpicture}
\ &=\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0.8,0) to (0,1);
\draw[->] (0.4,0) to[out=up, in=down] (0.8,0.5) to[out=up,in=down] (0.4,1);
\draw[->] (0,0) to (0.8,1);
\draw[->] (1.2,0) to (1.2,1);
\node at (1.6,0.5) {$\cdots$};
\node at (1.6,0) {\regionlabel{n}};
\draw[->] (2,0) to (2,1);
\end{tikzpicture},
\end{split}
\end{align}
for all $m,n \in \N$.
\end{itemize}
Fix $d \in \N$, and consider the endomorphism algebra $\End(\uparrow^d)$.
Since the only generating morphism is an endomorphism, by \cref{thm:presentationofalgebra}, we have that
generators of $\End(\uparrow^d)$ are
\begin{align}\label{sym-end-gen}
s_i =
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture},
\quad i=1,...,d-1.
\end{align}
Then relations that the generators $s_i,i=1,...,d-1$ satisfy, denoted by $\overline{R_\cS}^d$, are:
\begin{align}\label{sym-end-rel}
\begin{split}
s_i s_j = s_j s_i, \quad& \left|i-j\right|>1,\\
s_i^2 = 1, \quad& i=1,\dots,d-1,\\
s_{i+1} s_i s_{i+1} = s_i s_{i+1} s_i, \quad & i=1,\dots,d-2,
\end{split}
\end{align}
where the first one is from $C_\cS$
and the last two are from $R^\prime_\cS$
By \cref{thm:presentationofalgebra}, we have that $\End_\cS(\uparrow^d)$, as an algebra, has the presentation $\left< s_1,\dots,s_{d-1} \mid \overline{R_\cS}^d \right>$. Notice this is exactly the presentation of the group algebra of symmetric group $S_d$. Thus we have
\begin{align*}
\End_\cS(\uparrow^d) \cong \kk S_d.
\end{align*}
\subsection{Degenerate affine Hecke algebras\label{sec:dAHA}}
Let $\AH^\dg$ be the strict $\kk$-linear monoidal category $\cS$ defined in \cref{sec:Sym}, but with an additional generating morphism
$
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,0.6);
\redcircle{(0,0.3)};
\end{tikzpicture}
\ \colon \uparrow\ \to\ \uparrow
$
and one additional relation:
\[
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) -- (0.6,0.6);
\draw[->] (0.6,0) -- (0,0.6);
\redcircle{(0.15,.45)};
\end{tikzpicture}
\ -\
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) -- (0.6,0.6);
\draw[->] (0.6,0) -- (0,0.6);
\redcircle{(.45,.15)};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) -- (0,0.6);
\draw[->] (0.3,0) -- (0.3,0.6);
\end{tikzpicture}\ .
\]
It follows that all morphisms in $\AH^\dg$ are endomoprhisms, and we have that \cref{thm:presentationofalgebra} applies.
Thus if we fix $d$, we have that the endormorphism algebra $\End(\uparrow^d)$ is generated by
\[
s_i =
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture},
\quad \text{and} \quad
t_j =
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\redcircle{(0,0.25)};
\draw[->] (0.4,0) to (0.4,0.5);
\node at (0.8,0.25) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,0.5);
\end{tikzpicture},
\]
where $i=1,\dots,d-1$ and $j=1,\dots,d$. The generators $s_i$ satisfy the relations in \cref{SymRel} and \cref{SymInt}. Besides, the generators $s_i$ and $t_j$ satisfy that:
\begin{align}\label{ahdeg-end-rel}
\begin{split}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to (0.4,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
\redcircle{(0,0.75)};
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ -\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,0.5) to [out=90, in=270] (0.4,1);
\draw[->] (0.4,0) to (0.4,0.5) to [out=90, in=270] (0,1);
\redcircle{(0.4,0.25)};
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\draw[->] (0.4,0) to (0.4,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture},
\end{split}
\end{align}
\begin{align}\label{ahdeg-end-interchange}
\begin{split}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\redcircle{(0,0.75)};
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\redcircle{(1.6,0.25)};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture}
\ =&\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\redcircle{(0,0.25)};
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\redcircle{(1.6,0.75)};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture},
\\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.6,0) to (-1.6,1);
\node at (-1.2,0.5) {$\cdots$};
\node[below] at (-1.2,0) {\regionlabel{d-1-j}};
\draw[->] (-0.8,0) to (-0.8,1);
\draw[->] (-0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
\draw[->] (0,0) to [out=90, in=270] (-0.4,0.5) to (-0.4,1);
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\redcircle{(1.6,0.75)};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture}
\ =&\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.6,0) to (-1.6,1);
\node at (-1.2,0.5) {$\cdots$};
\node[below] at (-1.2,0) {\regionlabel{d-1-j}};
\draw[->] (-0.8,0) to (-0.8,1);
\draw[->] (-0.4,0) to (-0.4,0.5) to [out=90, in=270] (0,1);
\draw[->] (0,0) to (0,0.5) to [out=90, in=270] (-0.4,1);
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\redcircle{(1.6,0.25)};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture},
\end{split}
\end{align}
where $i,j=1,\dots,d-1,$ and in the last relation $i\neq j, i\neq j+1$.
The generators and relations form the presentation of the \emph{degenerate affine Hecke algebra} of type $A_{d-1}$. One can refer to \cite[5.55]{Mol00} for more details about these algebras.
\color{black}
\subsection{The braid group} \label{braid-cat}
Let $\cB$ be a linear monoidal category with the presentation
$ \left<
\uparrow,E_\cB,R_\cB
\right>_\otimes,$
where the generating morphisms in $E_\cB$ are
\begin{equation} \label{poscross}
\begin{tikzpicture}[anchorbase]
\draw[->] (0.25, -0.25) to (-0.25,0.25);
\draw[wipe] (-0.25,-0.25) to (0.25,0.25);
\draw[->] (-0.25,-0.25) to (0.25,0.25);
\end{tikzpicture}
\quad \text{and} \quad
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.25, -0.25) to (0.25,0.25);
\draw[wipe] (0.25,-0.25) to (-0.25,0.25);
\draw[->] (0.25,-0.25) to (-0.25,0.25);
\end{tikzpicture},
\end{equation}
and relations in $R_\cB$ are
\begin{equation} \label{braid-rel}
\begin{tikzpicture}[anchorbase]
\draw[->] (0.3,0) to[out=up,in=down] (-0.1,0.5) to[out=up,in=down] (0.3,1);
\draw[wipe] (-0.1,0) to[out=up,in=down] (0.3,0.5) to[out=up,in=down] (-0.3,1);
\draw[->] (-0.1,0) to[out=up,in=down] (0.3,0.5) to[out=up,in=down] (-0.1,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.2,0) -- (-0.2,1);
\draw[->] (0.2,0) -- (0.2,1);
\end{tikzpicture},
\quad
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.1,0) to[out=up,in=down] (0.3,0.5) to[out=up,in=down] (-0.1,1);
\draw[wipe] (0.3,0) to[out=up,in=down] (-0.1,0.5) to[out=up,in=down] (0.3,1);
\draw[->] (0.3,0) to[out=up,in=down] (-0.1,0.5) to[out=up,in=down] (0.3,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.2,0) -- (-0.2,1);
\draw[->] (0.2,0) -- (0.2,1);
\end{tikzpicture}
\quad \text{and} \quad
\begin{tikzpicture}[anchorbase]
\draw[->] (0.4,0) -- (-0.4,1);
\draw[wipe] (0,0) to[out=up, in=down] (-0.4,0.5) to[out=up,in=down] (0,1);
\draw[->] (0,0) to[out=up, in=down] (-0.4,0.5) to[out=up,in=down] (0,1);
\draw[wipe] (-0.4,0) -- (0.4,1);
\draw[->] (-0.4,0) -- (0.4,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (0.4,0) -- (-0.4,1);
\draw[wipe] (0,0) to[out=up, in=down] (0.4,0.5) to[out=up,in=down] (0,1);
\draw[->] (0,0) to[out=up, in=down] (0.4,0.5) to[out=up,in=down] (0,1);
\draw[wipe] (-0.4,0) -- (0.4,1);
\draw[->] (-0.4,0) -- (0.4,1);
\end{tikzpicture}
\ .
\end{equation}
The first two relations in \cref{braid-rel} actually mean that the two generating morphisms are inverses of each other, so it suffices to give only the first one generating morphism and assume it is invertible.
By \cref{MainTheorem}, we obtain that $\cB$ has a presentation $\left< M(\uparrow), \overline{E_\cB}, \overline{R_\cB}\right>$ as a linear category, where:
\begin{itemize}
\item $M(\uparrow)=\left\lbrace \ \uparrow^n \ \mid n \in \N\ \right\rbrace $,
\item Morphisms in $\overline{E_\cB}$ are
\begin{equation}\label{braid-generator}
b_{n,m}=
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[wipe] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture}
\quad \text{and its inverse} \quad
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[wipe] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture},
\end{equation}
where $m,n \in \N$
\item Relations in $\overline{R_\cB}$ are $C_\cB$ and ${R_\cB}^\prime$, where ${R_\cB}^\prime$ consists of morphisms
\begin{align}
\begin{split}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to [out=up,in=down] (0.4,1);
\draw[wipe] (0,0) to [out=90, in=270] (0.4,0.5) to [out=up, in=down] (0,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to [out=up, in=down] (0,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ &=\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\draw[->] (0.4,0) to (0.4,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
,\\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to [out=up, in=down] (0,1);
\draw[wipe] (0.4,0) to [out=90, in=270] (0,0.5) to [out=up,in=down] (0.4,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to [out=up,in=down] (0.4,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ &=\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\draw[->] (0.4,0) to (0.4,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{n}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
,\\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0.8,0) -- (0,1);
\draw[wipe] (0.4,0) to[out=up, in=down] (0,0.5) to[out=up,in=down] (0.4,1);
\draw[->] (0.4,0) to[out=up, in=down] (0,0.5) to[out=up,in=down] (0.4,1);
\draw[wipe] (0,0) -- (0.8,1);
\draw[->] (0,0) -- (0.8,1);
\draw[->] (1.2,0) to (1.2,1);
\node at (1.6,0.5) {$\cdots$};
\node at (1.6,0) {\regionlabel{n}};
\draw[->] (2,0) to (2,1);
\end{tikzpicture}
\ &=\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0.8,0) to (0,1);
\draw[wipe] (0.4,0) to[out=up, in=down] (0.8,0.5) to[out=up,in=down] (0.4,1);
\draw[->] (0.4,0) to[out=up, in=down] (0.8,0.5) to[out=up,in=down] (0.4,1);
\draw[wipe] (0,0) to (0.8,1);
\draw[->] (0,0) to (0.8,1);
\draw[->] (1.2,0) to (1.2,1);
\node at (1.6,0.5) {$\cdots$};
\node at (1.6,0) {\regionlabel{n}};
\draw[->] (2,0) to (2,1);
\end{tikzpicture},
\end{split}
\end{align}
for all $m,n \in \N$, and $C_\cB$ consists of morphisms
\begin{align} \label{Briad-Int}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,0.5) to [out=90, in=270] (0.4,1);
\draw[wipe] (0.4,0) to (0.4,0.5) to [out=90, in=270] (0,1);
\draw[->] (0.4,0) to (0.4,0.5) to [out=90, in=270] (0,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{l}};
\draw[->] (1.6,0) to (1.6,1);
\draw[->] (2,0) to [out=90, in=270] (2.4,0.5) to (2.4,1);
\draw[wipe] (2.4,0) to [out=90, in=270] (2,0.5) to (2,1);
\draw[->] (2.4,0) to [out=90, in=270] (2,0.5) to (2,1);
\draw[->] (2.8,0) to (2.8,1);
\node at (3.2,0.5) {$\cdots$};
\node at (3.2,0) {\regionlabel{n}};
\draw[->] (3.6,0) to (3.6,1);
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{m}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to (0.4,1);
\draw[wipe] (0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node at (1.2,0) {\regionlabel{l}};
\draw[->] (1.6,0) to (1.6,1);
\draw[->] (2,0) to (2.0,0.5) to [out=90, in=270] (2.4,1);
\draw[wipe] (2.4,0) to (2.4 ,0.5) to [out=90, in=270] (2,1);
\draw[->] (2.4,0) to (2.4 ,0.5) to [out=90, in=270] (2,1);
\draw[->] (2.8,0) to (2.8,1);
\node at (3.2,0.5) {$\cdots$};
\node at (3.2,0) {\regionlabel{n}};
\draw[->] (3.6,0) to (3.6,1);
\end{tikzpicture},
\end{align}
where $m,n,l\in \N$.
\end{itemize}
Fix a natural number $d$, we have the endomoprhism algebra $\End(\uparrow^d)$ is generated by
\begin{equation}\label{braid-end-generator}
s_i=
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[wipe] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture},
\quad \text{and} \quad
s_i^{-1}=
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[wipe] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture},
\end{equation}
where $i=1,\dots,d-1$, and they satisfy the relation
\begin{align}\label{braid-end-rel}
\begin{split}
s_{i+1} s_i s_{i+1} &= s_i s_{i+1} s_i, \quad i=1,\dots,d_1\\
s_is_j &= s_js_i, \qquad \left|i-j\right|>1.
\end{split}
\end{align}
This presentation is the presentation of the group algebra of the briad group on $d$ strands.
\subsection{Hecke algebras}
Fix $z \in \kk^\times$. Let $\cH(z)$ be the strict $\kk$-linear monoidal category $\cB$ defined in \cref{braid-cat}, but with one more relation:
\begin{equation} \label{skein}
\begin{tikzpicture}[anchorbase]
\draw[->] (0.25,-0.25) to (-0.25,0.25);
\draw[wipe] (-0.25,-0.25) to (0.25,0.25);
\draw[->] (-0.25,-0.25) to (0.25,0.25);
\end{tikzpicture}
\ -\
\begin{tikzpicture}[anchorbase]
\draw[->] (-0.25,-0.25) to (0.25,0.25);
\draw[wipe] (0.25,-0.25) to (-0.25,0.25);
\draw[->] (0.25,-0.25) to (-0.25,0.25);
\end{tikzpicture}
\ = z\
\begin{tikzpicture}[anchorbase]
\draw[->] (0.2,-0.25) to (0.2,0.25);
\draw[->] (-0.2,-0.25) to (-0.2,0.25);
\end{tikzpicture}
\ .
\end{equation}
We take $\kk = \C(q)$ and $z = q - q^{-1}$ and fix $d\in \N$. Since all morphisms in $\cH$ are endomorphisms, \cref{thm:presentationofalgebra} applies. We have that all generators in $\End_{\cH(z)}(\uparrow^d)$ are those in \cref{braid-end-generator} and all relations are those in \cref{braid-end-rel} together with
\begin{align}
\begin{tikzpicture}[anchorbase, yscale=1.2]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[wipe] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture}
\ -\
\begin{tikzpicture}[anchorbase, yscale=1.2]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0)to [out=90, in=270] (0.4,0.5);
\draw[wipe] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture}
\ =\
(q - q^{-1})
\begin{tikzpicture}[anchorbase, yscale=1.2]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\draw[->] (0.4,0) to (0.4,0.5);
\draw[->] (0.8,0) to (0.8,0.5);
\node at (1.2,0.25) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,0.5);
\end{tikzpicture}.
\end{align}
then $\End_{\cH(z)} (\uparrow^{\otimes d})$ is the \emph{Iwahori--Hecke algebra} of type $A_{d-1}$.
\subsection{Wreath product algebras}
In this section we give the definition of wreath product categories by following the presentation in \cite[\S 3.6]{Savage18}.
Let $A$ be an associative $\kk$-algebra. Define the \emph{wreath product category} $\cW(A)$ to be the strict $\kk$-linear monoidal category obtained from $\cS$ by adding morphisms such that we have an algebra homomorphipsm
\[
A \to \End(\uparrow),
\qquad
a \mapsto
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,0.6);
\bluedot{(0,0.3)} node[anchor=west,color=black] {\dotlabel{a}};
\end{tikzpicture}
\ .
\]
In particular, this means that
\begin{equation} \label{dotlin}
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,0.6);
\bluedot{(0,0.3)} node[anchor=west,color=black] {\dotlabel{(\alpha a+ \alpha b)}};
\end{tikzpicture}
= \alpha\
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,0.6);
\bluedot{(0,0.3)} node[anchor=west,color=black] {\dotlabel{a}};
\end{tikzpicture}
+ \beta\
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,0.6);
\bluedot{(0,0.3)} node[anchor=west,color=black] {\dotlabel{b}};
\end{tikzpicture}
\qquad \text{and} \qquad
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,1);
\bluedot{(0,0.3)} node[anchor=west,color=black] {\dotlabel{b}};
\bluedot{(0,0.6)} node[anchor=west,color=black] {\dotlabel{a}};
\end{tikzpicture}
=\
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,1);
\bluedot{(0,0.5)} node[anchor=west,color=black] {\dotlabel{ab}};
\end{tikzpicture}
\qquad \text{for all } \alpha,\beta \in \kk,\ a,b \in A.
\end{equation}
We call the closed circles appearing in the above diagrams \emph{tokens}. We then impose the additional relation
\begin{equation} \label{tokslide}
\begin{tikzpicture}[anchorbase, scale=0.75]
\draw[->] (0,0) -- (1,1);
\draw[->] (1,0) -- (0,1);
\bluedot{(.25,.25)} node [anchor=east, color=black] {\dotlabel{a}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase, scale=0.75]
\draw[->](0,0) -- (1,1);
\draw[->](1,0) -- (0,1);
\bluedot{(0.75,.75)} node [anchor=east, color=black] {\dotlabel{a}};
\end{tikzpicture}
\ ,\quad a \in A.
\end{equation}
Note that we can compose \cref{tokslide} on the top and bottom with a crossing to obtain
\[
\begin{tikzpicture}[anchorbase,scale=0.75]
\draw[->] (0,0) -- (1,1);
\draw[->] (1,0) -- (0,1);
\bluedot{(.25,.25)} node [anchor=east, color=black] {\dotlabel{a}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase,scale=0.75]
\draw[->](0,0) -- (1,1);
\draw[->](1,0) -- (0,1);
\bluedot{(0.75,.75)} node [anchor=east, color=black] {\dotlabel{a}};
\end{tikzpicture}
\ \implies\
\begin{tikzpicture}[anchorbase]
\draw[->] (0.25,-0.5) to[out=up,in=down] (-0.25,0) to[out=up,in=down] (0.25,0.5) to[out=up,in=down] (-0.25,1);
\draw[->] (-0.25,-0.5) to[out=up,in=down] (0.25,0) to[out=up,in=down] (-0.25,0.5) to[out=up,in=down] (0.25,1);
\bluedot{(-0.25,0)} node[anchor=east,color=black] {\dotlabel{a}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase]
\draw[->] (0.25,-0.5) to[out=up,in=down] (-0.25,0) to[out=up,in=down] (0.25,0.5) to[out=up,in=down] (-0.25,1);
\draw[->] (-0.25,-0.5) to[out=up,in=down] (0.25,0) to[out=up,in=down] (-0.25,0.5) to[out=up,in=down] (0.25,1);
\bluedot{(0.25,0.5)} node[anchor=west,color=black] {\dotlabel{a}};
\end{tikzpicture}
\ \stackrel{\cref{Sn-strings}}{\implies} \
\begin{tikzpicture}[anchorbase,scale=0.75]
\draw[->] (0,0) -- (1,1);
\draw[->] (1,0) -- (0,1);
\bluedot{(0.25,.75)} node [anchor=north east, color=black] {\dotlabel{a}};
\end{tikzpicture}
\ =\
\begin{tikzpicture}[anchorbase,scale=0.75]
\draw[->] (0,0) -- (1,1);
\draw[->] (1,0) -- (0,1);
\bluedot{(.75,.25)} node [anchor=south west, color=black] {\dotlabel{a}};
\end{tikzpicture}
\ .
\]
So tokens also slide up-left through crossings.
We have the following theorem in \cite[\S 3.6]{Savage18} stated without proof.
\begin{prop}
The endomoprhism algebras of the wreath product algebra satisfy
\[
\End_{\cW(A)} (\uparrow^{\otimes d}) \cong A^{\otimes d} \rtimes S_d,
\]
which is called the $d$-th \emph{wreath product algebra} associated to $A$, where the multiplication is determined by
\[
(\baone \otimes \pi_1) (\batwo \otimes \pi_2)
= (\baone (\pi_1\cdot \ba_{2})\otimes \pi_1 \pi_2),
\quad \baone, \batwo \in A^{\otimes d},\ \pi_1, \pi_2 \in S_d,
\]
where $\pi_1 \cdot a_2$ denotes the natural action of $\pi_1 \in S_d$ on $a_2 \in A^{\otimes d}$ by permutation of the factors.
\end{prop}
\begin{proof}
All generators of $\End_{\cW(A)} (\uparrow^{\otimes d})$ are endomorphisms; thus \cref{thm:presentationofalgebra} applies. The generators are $s_i$ defined in \cref{sym-end-gen} together with
\[
u_j(a) =
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\bluedot{(0,0.25)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (0.4,0) to (0.4,0.5);
\node at (0.8,0.25) {$\cdots$};
\node at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,0.5);
\end{tikzpicture},
\quad a \in A.
\]
By \cref{thm:presentationofalgebra}, the generators $s_i$ satisfy the relations of $\cS$ as in \cref{sym-end-rel}, and the generators $u_j(a)$ satisfy the relations
\begin{align}\label{prod-end-rel}
\begin{split}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\bluedot{(0,0.25)} node[anchor=west,color=black] {\dotlabel{(\alpha a+ \beta b)}};
\draw[->] (1.6,0) to (1.6,0.5);
\node at (2,0.25) {$\cdots$};
\node at (2,0) {\regionlabel{j-1}};
\draw[->] (2.4,0) to (2.4,0.5);
\end{tikzpicture}
=& \alpha\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\bluedot{(0,0.25)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (0.4,0) to (0.4,0.5);
\node at (0.8,0.25) {$\cdots$};
\node at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,0.5);
\end{tikzpicture}
+ \beta\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\bluedot{(0,0.25)} node[anchor=west,color=black] {\dotlabel{b}};
\draw[->] (0.4,0) to (0.4,0.5);
\node at (0.8,0.25) {$\cdots$};
\node at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,0.5);
\end{tikzpicture},\\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\bluedot{(0,0.75)} node[anchor=west,color=black] {\dotlabel{a}};
\bluedot{(0,0.25)} node[anchor=west,color=black] {\dotlabel{b}};
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,1);
\end{tikzpicture}
=&\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\bluedot{(0,0.25)} node[anchor=west,color=black] {\dotlabel{ab}};
\draw[->] (0.6,0) to (0.6,0.5);
\node at (1,0.25) {$\cdots$};
\node at (1,0) {\regionlabel{j-1}};
\draw[->] (1.4,0) to (1.4,0.5);
\end{tikzpicture},\\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\bluedot{(0,0.75)} node[anchor=west,color=black] {\dotlabel{b}};
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\bluedot{(1.6,0.25)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture}
\ =&\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\bluedot{(0,0.25)} node[anchor=west,color=black] {\dotlabel{b}};
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\bluedot{(1.6,0.75)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture},
\end{split}
\end{align}
where $\alpha,\beta \in \kk$ and $a,b \in A$.
Most importantly, by \cref{tokslide}, these two kinds of generators together satisfy the following relations:
\begin{align} \label{prod-end-rel-int}
\begin{split}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,0.5) to [out=90, in=270] (0.4,1);
\draw[->] (0.4,0) to (0.4,0.5) to [out=90, in=270] (0,1);
\bluedot{(0.4,0.25)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ =&\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to (0.4,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
{(0,0.75)};
\bluedot{(0,0.75)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture},
\\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.6,0) to (-1.6,1);
\node at (-1.2,0.5) {$\cdots$};
\node[below] at (-1.2,0) {\regionlabel{d-1-j}};
\draw[->] (-0.8,0) to (-0.8,1);
\draw[->] (-0.4,0) to (-0.4,0.5) to [out=90, in=270] (0,1);
\draw[->] (0,0) to (0,0.5) to [out=90, in=270] (-0.4,1);
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\bluedot{(1.6,0.25)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture}
\ =&\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.6,0) to (-1.6,1);
\node at (-1.2,0.5) {$\cdots$};
\node[below] at (-1.2,0) {\regionlabel{d-1-j}};
\draw[->] (-0.8,0) to (-0.8,1);
\draw[->] (-0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
\draw[->] (0,0) to [out=90, in=270] (-0.4,0.5) to (-0.4,1);
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{(j-i)-1}};
\draw[->] (1.2,0) to (1.2,1);
\draw[->] (1.6,0) to (1.6,1);
\bluedot{(1.6,0.75)} node[anchor=west,color=black] {\dotlabel{a}};
\draw[->] (2,0) to (2, 1);
\node at (2.4,0.5) {$\cdots$};
\node[below] at (2.4,0) {\regionlabel{i-1}};
\draw[->] (2.8,0) to (2.8,1);
\end{tikzpicture},
\end{split}
\end{align}
where $a \in A$. Algebraically, \cref{prod-end-rel-int} reads
\[
s_i u_j(b) = u_{s_i(j)}(b) s_i = (s_i \cdot u_j)(b) s_i, \quad \text{for all } b \in A,
\]
where we identify the generator $s_i$ with the transiposition $(i,i+1)$ and thus $s_i(j)$ denote the image of $j$ under $(i,i+1).$
Let $B$ be the free algebra generated by $u_j(b),\ j=1,\dots,n,\ b \in A$ and $s_i,\ i=1,\dots ,d-1$, we define a $\kk$-algebra morphism $\phi$ from $B$ to $A^{\otimes d} \rtimes S_d$ determined by the following:
\begin{align*}
\phi(s_i) =& 1^{\otimes d} \otimes s_i, \\
\phi(u_j(b))=& 1^{\otimes {d-j}}
\otimes {b} \otimes 1^{\otimes j} \otimes \id_d,
\end{align*}
where $s_i$ denotes the transposition $(i,i+1)$ and $\id_d$ is the identity of $S_d$.
It is clear that $\phi$ is surjective and the kernel of $\phi$ contains all the relations on the generators of $\End_{\cW(A)}\left(\uparrow^d\right)$. In particular, \cref{prod-end-rel-int} is satisfied due to the product
\begin{align}
\begin{split}
(\baone \otimes \pi_1)(\batwo \otimes \pi_2)
=& (\baone(\pi_1 \cdot \batwo) \otimes \pi_1\pi_2).
\end{split}
\end{align}
Thus $\phi$ induces a surjective algebra morphism $\bar{\phi}$ from $\End_{\cW(A)}\left(\uparrow^d\right)$ to $A^{\otimes d} \rtimes S_d$. Conversely, consider the following multi-linear map
\begin{align*}
\psi: A^{d}\times S_d
\ \to& \
\End_{\cW(A)}(\uparrow^{\otimes d}), \\
(a_d\times \dotsb\times a_1 \times \pi) \ \mapsto&\ u_d(a_d)\dotsm u_1(a_1) \pi .
\end{align*}
It induces a natural $\kk$-module morphism as follows:
\begin{align*}
\bar{\psi}: A^{\otimes d} \rtimes S_d= A^{\otimes d} \otimes \kk S_d
\ \to& \
\End_{\cW(A)}(\uparrow^{\otimes d}), \\
(a_d\otimes \dotsb\otimes a_1)\otimes \pi \ \mapsto&\ u_d(a_d)\dotsm u_1(a_1) \pi .
\end{align*}
The $\kk$-module morphism $\bar{\psi}$ actually forms an algebra morphism: for $\baone=(a_i)_{i=1}^d,\batwo=(b_i)_{i=1}^d \in A^{\otimes d}$ and $\pi_1,\pi_2 \in S_d$,
\begin{align}
\begin{split}
\bar{\psi}((\baone \otimes \pi_1)(\batwo \otimes \pi_2))
=& \bar{\psi} (\baone(\pi_1 \cdot \batwo) \otimes \pi_1\pi_2)\\
=& u_d(a_d)\dotsm u_1(a_1)u_d(b_{\pi_1(d)}) ...u_1(b_{\pi_1(1)})\pi_1\pi_2\\
\stackrel{\star}{=}& u_d(a_d)\dotsm u_1(a_1)\pi_1u_d(b_d) ...u_1(b_1)\pi_2\\
=& \bar{\psi}(\baone \otimes \pi_1)\bar{\psi}(\batwo \otimes \pi_2),
\end{split}
\end{align}
where one can check that $\star$ holds by writting $\pi_1$ as a product of transpositions.
Since $\bar{\psi}$ is $\kk$-algebra morphism, it is clear that it is the inverse of $\bar{\phi}$.
This shows that $\End_\cW(A)\left(\uparrow^d\right)$ is isomorphic to $A^{\otimes d} \rtimes S_d$.
\end{proof}
\subsection{Affine wreath product algebras}
The wreath product category $\cW(A)$ is a generalization of the symmetric group category $\cS$ that depends on a choice of associative $\kk$-algebra $A$. We now suppose that we have a $\kk$-linear \emph{trace map}
$ \tr \colon A \to \kk $
and dual bases $B$ and $\left\lbrace \chk{b} : b \in B\right\rbrace $ of $A$ such that
\[
\tr(ab) = \tr(ba) \quad \text{and} \quad
\tr(\chk{a}b) = \delta_{a,b}
\quad \text{for all } a,b \in B.
\]
It can be shown that
$ \sum_{b \in B} b \otimes \chk{b} \in A \otimes A$
is independent of the choice of basis $B$. Also, for all $x \in A$, we have
\begin{equation} \label{teleport}
\begin{split}
\sum_{b \in B} bx \otimes \chk{b}
= \sum_{a \in B} a \otimes x\chk{a}.
\end{split}
\end{equation}
We define the \emph{affine wreath product category} $\AW(A)$ to be the strict $\kk$-linear monoidal category obtained from $\cW(A)$ by adding a generating morphism
$
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,0.6);
\redcircle{(0,0.3)};
\end{tikzpicture}
\ \colon \uparrow\ \to\ \uparrow
$
and the additional relations
\begin{equation} \label{AWPA}
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) -- (0.6,0.6);
\draw[->] (0.6,0) -- (0,0.6);
\redcircle{(0.15,.45)};
\end{tikzpicture}
\ -\
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) -- (0.6,0.6);
\draw[->] (0.6,0) -- (0,0.6);
\redcircle{(.45,.15)};
\end{tikzpicture}
\ = \sum_{b \in B}
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) -- (0,0.6);
\draw[->] (0.3,0) -- (0.3,0.6);
\bluedot{(0,0.3)} node[anchor=east,color=black] {\dotlabel{b}};
\bluedot{(0.3,0.3)} node[anchor=west,color=black] {\dotlabel{\chk{b}}};
\end{tikzpicture}
\qquad \text{and} \qquad
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,1);
\redcircle{(0,0.3)};
\bluedot{(0,0.6)} node[anchor=east,color=black] {\dotlabel{a}};
\end{tikzpicture}
\ =
\begin{tikzpicture}[anchorbase]
\draw[->] (0,0) to (0,1);
\redcircle{(0,0.6)};
\bluedot{(0,0.3)} node[anchor=west,color=black] {\dotlabel{a}};
\end{tikzpicture}
\ ,\quad \text{for all } a \in A.
\end{equation}
\begin{prop} The endomorphism algebra
$ \End_{\AW(A)}(\uparrow^{\otimes d}) $ in the affine wreath product category is an affine wreath product algebra as defined in \cite[Definition 3.1]{Sav17}. Namely, as an $\kk$-algebra, $ \End_{\AW(A)}(\uparrow^{\otimes d}) $ is generated by
\[
t_i, \ i=1,\dots,d, \quad u_i(a),\ i=1,\dots,d,\ a\in A, \quad \text{and} \quad s_i,\ i=1,\dots,d-1,
\]
where they satisfy \cref{sym-end-rel,ahdeg-end-interchange,prod-end-rel,prod-end-rel-int} and the following relations:
\begin{align}
\begin{split}
t_iu_j(a) = u_j(a)t_i, \quad &\text{for all } a \in A\\
t_{i+1}s_i - s_it_i = m_i, \quad &\text{for }i=1,\dots,d-1,
\end{split}
\end{align}
where $m_i=\sum_{b \in B} u_{i+1}(b)u_i(\chk{b})$.
\end{prop}
\begin{proof}
As a linear monoidal category, all generators of $\AW(A)$ are endormophisms, thus \cref{thm:presentationofalgebra} applies. Thus for fixed $d$, we have that the generators of
$ \End_{\AW(A)}(\uparrow^{\otimes d}) $
are the generators of $\End_{\cW(A)}(\uparrow^{\otimes d})$ together with
\begin{equation}
t_i =
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,0.5);
\node at (-0.8,0.25) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,0.5);
\draw[->] (0,0) to (0,0.5);
\redcircle{(0,0.25)};
\draw[->] (0.4,0) to (0.4,0.5);
\node at (0.8,0.25) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,0.5);
\end{tikzpicture}.
\end{equation}
The relations they satisfy in $\AW(A)$ should be the relations they satisfy in $\cW(A)$, namely \cref{sym-end-rel,prod-end-rel,prod-end-rel-int}, together with the interchange law involving $t_i$ and the new relation induced by the first relation in \cref{AWPA}. To be precise, the interchange law involving $t_i$ corresponds to \cref{ahdeg-end-interchange} and
\[
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\redcircle{(0,0.25)};
\bluedot{(0,0.75)} node[anchor=west, color=black] {\dotlabel{a}};
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,1);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-j}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\redcircle{(0,0.75)};
\bluedot{(0,0.25)} node[anchor=west, color=black] {\dotlabel{a}};
\draw[->] (0.4,0) to (0.4,1);
\node at (0.8,0.5) {$\cdots$};
\node[below] at (0.8,0) {\regionlabel{j-1}};
\draw[->] (1.2,0) to (1.2,1);
\end{tikzpicture}.
\]
The induced relations, by \cref{MainTheorem}, are
\[
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to [out=90, in=270] (0.4,0.5) to (0.4,1);
\draw[->] (0.4,0) to [out=90, in=270] (0,0.5) to (0,1);
\redcircle{(0,0.75)};
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ -\
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,0.5) to [out=90, in=270] (0.4,1);
\draw[->] (0.4,0) to (0.4,0.5) to [out=90, in=270] (0,1);
\redcircle{(0.4,0.25)};
\draw[->] (0.8,0) to (0.8,1);
\node at (1.2,0.5) {$\cdots$};
\node[below] at (1.2,0) {\regionlabel{i-1}};
\draw[->] (1.6,0) to (1.6,1);
\end{tikzpicture}
\ =\
\sum_{b \in B}
\begin{tikzpicture}[anchorbase]
\draw[->] (-1.2,0) to (-1.2,1);
\node at (-0.8,0.5) {$\cdots$};
\node[below] at (-0.8,0) {\regionlabel{d-1-i}};
\draw[->] (-0.4,0) to (-0.4,1);
\draw[->] (0,0) to (0,1);
\bluedot{(0,0.5)} node[anchor=east, color=black] {\dotlabel{b}};
\draw[->] (0.4,0) to (0.4,1);
\bluedot{(0.4,0.5)} node[anchor=west, color=black] {\dotlabel{\chk{b}}};
\draw[->] (1,0) to (1,1);
\node at (1.4,0.5) {$\cdots$};
\node[below] at (1.4,0) {\regionlabel{i-1}};
\draw[->] (1.8,0) to (1.8,1);
\end{tikzpicture}.
\]
Thus the endomorphism algebra is an affine wreath product algebra.
\end{proof}
It worth mentioning that $\End_{\AW(A)}(\uparrow^{\otimes d})$, as a $\kk$-module, is isomorphic to
\[
\kk[x_1,\dots,x_d] \otimes A^{\otimes d} \otimes \kk S_d.
\]
This is because each morphism in the basis, by the relations, is a linear combination of morphisms of the form
\[
t_d^{l_d} \dots t_1^{l_1} u_d(a_d) \dots u_1(a_1) \pi, \quad l_i \in \N,\ a_i \in A,\ \pi \in S_d.
\]
Thus there is a natural surjective linear map determined by
\begin{align}
\begin{split}
\kk[x_1,\dots,x_d] \otimes A^{\otimes d} \otimes \kk S_d\ \to\ & \End_{\AW(A)}(\uparrow^{\otimes d}), \\
x_d^{l_d} \dotsm x_1^{l_1} \otimes a_d \otimes \dotsb \otimes a_1 \otimes \pi \ \mapsto\ & t_d^{l_d} \dotsm t_1^{l_1} u_d(a_d) \dotsm u_1(a_1) \pi.
\end{split}
\end{align}
Hence it suffices to check the injectivity. One can refer to \cite[Theorem 4.6]{Sav17} for a detailed proof.
\color{black}
\subsection{Quantum affine wreath product algebras}
One can also define affine versions of the Hecke category and, more generally, quantum versions of the affine wreath product category. We refer the reader to \cite{BS18a,BS18b} for further details.
\color {black}
\bibliographystyle{alphaurl}
\bibliography{biblist-2}
\par
\end{document}
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TITLE: If 3 people put their hat in a box, but the hats are mixed up. How likely is it that AT LEAST one person getting their hat back.
QUESTION [0 upvotes]: If 3 people put their hat in a box, but the hats are mixed up. How likely is it that AT LEAST one person gets their hat back. Consider all possibilities. Then what about 4 people. Please use simplicity (I know similar questions have been posted before, but none make sense to me.)
I believe there are 24 possible outcomes, 15 being favorable that at least 1 person get their hat back, which comes out to 62.5% is this correct?
REPLY [0 votes]: For three people there are six possibilities. Simply number the people 1, 2, 3. Then suppose they pick a hat out of the box in that order, then the possible hat id's are:
123 (everyone gets their own hat back)
132 (only person 1 gets their hat back)
213 (only person 3 gets their hat back)
231 (nobody gets their hat back)
312 (nobody gets their hat back)
321 (only person 2 gets their hat back)
These are the possibilities. There are 6 of them (corresponding to $3! = 6$ permutations of three distinct objects, and in 3 cases someone gets their hat back (ie, the permutation has at least one fixed point)
For four people, there are $4! = 24$ permutations. In this case you could simply list the permutations and count the number of cases where at least 1 person got their own hat back), or you could argue as follows:
How many permutations have exactly 1 fixed point?
There are 4 possibilities for the 1 element fixed, and for each such choice there are 2 arrangements for the other 3 elements which do not introduce additional fixed points (using the result for 3 people), for a total of $4\cdot 2 = 8$.
How many permutations have exactly 2 fixed points?
There are (4 choose 2) = 6 possible choices for the pair of elements fixed. The remaining elements must be fixed, so there are 6 permutations with exactly 2 fixed points.
How many permutations have exactly 3 fixed points?
Note that if three elements are fixed, the fourth must also be fixed, so there are no permutations in this category.
How many permutations have exactly 4 fixed points?
Of course there is just 1, namely 1234.
Thus, the total is $8 + 6 + 1 = 15$ permutations with at least one fixed point.
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TITLE: Why is the fundamental group of the Volodin space $X(R)$ the Steinberg group $St(R)$?
QUESTION [3 upvotes]: The Volodin space $X(R)$ is defined in A.A. Suslin's "On the Equivalence of K-Theories" (https://www.tandfonline.com/doi/abs/10.1080/00927878108822666) as the union of classifying spaces $\bigcup_{\sigma,n} BT^\sigma_n(R) \subset BE(R)$. He states that the fundamental group of this space is $\pi_1 X(R) \cong St(R)$; the Steinberg group. Is there a proof of this? Suslin's argument is that $X(R)$ can be written as a quotient space of $V(St(R), \{T^\sigma_n(R)\})$; the simplicial complex with vertices $St(R)$ (as a set) where elements $(x_0,...,x_p)$ form a $p$-simplex if each $x_i x_j^{-1}$ lies in the same $T^\sigma_n(R)$.
In particular, Suslin states that it can be easily seen that $V(St(R), \{T^\sigma_n(R)\})$ is simply connected and one can quotient out by the natural, free action of $St(R)$ on vertices to yield $X(R)$. This makes sense to me but I can't find a reason why $V(St(R), \{T^\sigma_n(R)\})$ is simply connected.
Weibel in his K-book notes that the map $\iota_*: \pi_1 X(R) \to E(R)$ induced by the inclusion $X(R) \hookrightarrow BE(R)$ is surjective since the $T^\sigma_n(R)$ generate $E(R)$. In light of this an alternative approach I've tried is to show that the kernel of $\iota_*$ is contained in the centre of $\pi_1 X(R)$ (i.e. $\iota_*$ is a central extension of $E(R)$). Since $X(R)$ is acyclic we have $H_n(\pi_1 X(R), \mathbb{Z}) = 0$ for $n =1,2$ (see Weibel's K-book, IV, Lemma 1.3.1 for this result). Altogether this would show that $\pi_1 X(R)$ is a universal central extension of $E(R)$, giving us that $St(R) \cong \pi_1 X(R)$. Only trouble is I can't show that $\iota_*: \pi_1 X(R) \to E(R)$ is a central extension.
Any ideas to the end of showing $\pi_1 X(R) \cong St(R)$ would be much appreciated!
REPLY [1 votes]: Since thinking about this problem some more, I've figured out a solution so I'm answering my own question. By the remarks in my question, all we have to do is show that $\pi_1 X(R) \to E(R)$ is a central extension.
We first bear three things in mind;
For subgroups $H,H' \leq G$ of some group $G$, we have continuous inclusions $BH, BH' \hookrightarrow BG$ that allow us to make the identification $BH \cap BH' = B(H \cap H')$ (I can include details on this if anyone asks). This also shows that
$BH \cap BH'$ is path connected.
Since CW complexes are locally contractible, we can find open sets $BT^\sigma_n(R)' \supset BT^\sigma_n(R)$ that deformation retract onto the $BT^\sigma_n(R)$ and hence $X(R) = \bigcup_{n, \sigma} BT^\sigma_n(R)'$ and in particular Van-Kampen's theorem holds.
The proof that $St(R) \to E(R): x_{ij}(r) \mapsto e_{ij}(r)$ is a central extension only relies on the generators $x_{ij}(r)$ of $St(R)$ satisfying the Steinberg relations (see Srinivas' "Algebraic K-Theory" and/or Weibel's K-book for this proof), so we can generalize this proof to show that for any group $\Theta(R)$ generated by symbols $\theta_{ij}(r)$ that satisfy the Steinberg relations, the homomorphism $\Theta(R) \to E(R): \theta_{ij}(r) \to e_{ij}(r)$ (provided it is well-defined) is a central extension.
Now for the proof we were after:
By Van-Kampen's theorem we have an isomorphism
$$ \rho: (*_{n, \sigma} \pi_1 BT^\sigma_n(R)')/N = (*_{n, \sigma} \pi_1 BT^\sigma_n(R))/N \overset{\sim}{\to} \pi_1 X(R) $$
where $*_{n, \sigma}$ denotes the free product of the $\pi_1 BT^\sigma_n(R) = T^\sigma_n(R)$ and $N$ is the normal subgroup generated by elements of the form $\iota_1(\alpha) \iota_2(\alpha)^{-1}$ where $\iota_1$ denotes an inclusion
$T^{\sigma_1}_{n_1}(R) \cap T^{\sigma_2}_{n_2}(R) \hookrightarrow T^{\sigma_1}_{n_1}(R)$ and $\iota_2$ is defined similarly. In particular,
$\theta_{ij}(r) := \rho(e_{ij}(r))$ is well defined and the symbols $\theta_{ij}(r)$ generate $\pi_1 X(R)$ and satisfy the
Steinberg relations since each $T^\sigma(n,R)$
has the Steinberg relations.
As an aside, one might suppose that the $\theta_{ij}(r)$ satisfy any relation that the $e_{ij}(r)$ satisfy in $E(R)$. However this is not
necessarily the case as, for example, relations in $E(R)$ written in the form $ e_{ij}(r_1) ... e_{ji}(r_2) ... = 1$ will not hold on
any of the $T^\sigma(n,R)$ since $e_{ij}(r_1), e_{ji}(r_2)$ cannot simultaneously exist in any one of these subgroups. By what follows it turns out that no further relations are actually satisfied by the $\theta_{ij}(r)$.
Now the composite map $\pi_1 BT^\sigma(n,R) \to \pi_1 X(R) \to \pi_1 BE(R)$ induced from inclusions is an inclusion of groups,
so the map $\pi_1 X(R) \to E(R)$ sends $\theta_{ij}(r)$ to $e_{ij}(r)$ and by our third remark above this map is in fact a central extension.
As I've already said, the isomorphism $\pi_1 X(R) \cong St(R)$ follows from the remark in the original question.
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March 5, 2008
#5
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| 149,744
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\begin{document}
\begin{abstract}
Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an~unknown change has occurred or not. For instance, detecting a~change in trend for a~randomly spaced time series is a~special case of the investigated framework. The designed changepoint tests are shown to be consistent and involve neither nuisance parameters nor tuning constants, which makes the testing procedures effortlessly applicable. A~changepoint estimator is also introduced and its consistency is proved. A~boundary issue is avoided, meaning that the changepoint can be detected when being close to the extremities of the observation regime. As a~theoretical basis for the developed methods, a~weak invariance principle for the smallest singular value of the data matrix is provided, assuming weakly dependent and non-stationary errors. The results are presented in a~simulation study, which demonstrates computational efficiency of the techniques. The completely data-driven tests are illustrated through a~calibration problem, however, the methodology can be applied to other areas such as clinical measurements, dietary assessment, computational psychometrics, or environmental toxicology as manifested in the paper.
\end{abstract}
\keywords{changepoint, errors-in-variables, hypothesis testing, non-stationarity, nuisance-parameter-free, singular value, weak invariance principle}
\section{Introduction and main aims}
If measured input and output data are supposed to be in some linear relations, then it is of particular interest to detect whether impact of the input characteristics has changed over time on the output observables. Moreover, only \emph{error-prone surrogates} of the unobservable input-output characteristics are in hand instead of a~precise measurement. Despite the fact that the relations and, consequently, suitable underlying stochastic models are linearly defined, the possible estimates and the corresponding inference may be highly non-linear~\citep{gleser}. It becomes even more challenging to handle measurement errors in input and output data simultaneously, when the linear relations are subject to change at some unknown time point---\emph{changepoint}.
There is a~vast literature aimed at linear relations modeled through so-called \emph{measurement error models} or \emph{errors-in-variables models} (for an~over\-view, see \cite{Fuller1987}, \cite{huffel}, \cite{CRSC2006}, \cite{Buonaccorsi2010}, or \cite{Yi2017}), but very little has been explored in the changepoint analysis for these models yet. A~change in regression has been explored thoroughly, cf.~\cite{H1995} or~\cite{AHHK2008}. However, such a~framework does not cover the case of measurement error models. Maximum likelihood approach~\citep{ChH1997,SS2002} and Bayesian approach~\citep{CRW1999,GK2001} to the changepoint estimation in the measurement error models were applied, both requiring parametric distributional assumptions on the errors. \cite{KMV2007} estimated the changepoint in the input data only. A~change in the variance parameter of the normally distributed errors within the measurement error models was investigated by~\cite{DTJM2016}. All of these mentioned contributions dealt with the changepoint estimation solely. Our \emph{main goal} is to test for a~possible change in the parameters relating the input and output data, both encumbered by some errors. Consequently, if a~change is detected, we aim to estimate it. By our best knowledge, we are not aware of any similar results even for the independent and identically distributed errors. Additionally to that, our changepoint tests are supposed to be \emph{nuisance-parameter-free}, \emph{distributional-free}, and to allow for very general error structures.
\subsection{Outline}
The paper is organized as follows: In the next section, our data model for the changepoint in errors-in-variables is introduced and several practical motivations for such a~model are given. Section~\ref{sec:SWIP} contains a~spectral weak invariance principle for weakly dependent and non-stationary random variables. It serves as the main theoretical tool for the consequent inference. The technical assumptions are discussed as well. Two test statistics for the changepoint detection are proposed in Section~\ref{sec:test}. Consequently, their asymptotic behavior is derived under the null as well as under the alternative hypothesis. Moreover, a~consistent changepoint estimator is introduced. Section~\ref{sec:simulations} contains a~simulation study that compares finite sample performance of the investigated tests. It numerically emphasizes the advantages of the proposed detection procedures. A~practical application of the developed approach to a~calibration problem is presented in Subsection~\ref{subsec:Papplication}. On the other hand, a~theoretical application to randomly spaced time series is performed in Subsection~\ref{subsec:Tapplication}. Afterwards, our conclusion follows. Proofs are given in the Appendix~\ref{sec:proofs}.
\section{Changepoint in errors-in-variables}\label{sec:CPTinEIV}
\emph{Errors-in-variables} (EIV) or also called \emph{measurement error} model
\begin{equation}\label{eq:M}\tag{$\mathcal{M}$}
\bX=\bZ+\btheta
\end{equation}
and
\begin{equation}\label{eq:H0}\tag{$\mathcal{H}_0$}
\bY=\bZ\bbeta+\beps
\end{equation}
is considered, where $\bbeta\in\mathbb{R}^p$ is a~vector of unknown \emph{regression parameters} possibly subject to change, $\bX\in\mathbb{R}^{n\times p}$ and $\bY\in\mathbb{R}^{n\times 1}$ consist of \emph{observable random} variables ($\bX$ are covariates and $\bY$ is a~response), $\bZ\in\mathbb{R}^{n\times p}$ consists of \emph{unknown constants} and has full rank, $\beps\in\mathbb{R}^{n\times 1}$ and $\btheta\in\mathbb{R}^{n\times p}$ are \emph{random errors}. This setup can be extended to a~\emph{multivariate case}, where $\bbeta\in\mathbb{R}^{p\times q}$, $\bY\in\mathbb{R}^{n\times q}$, and $\beps\in\mathbb{R}^{n\times q}$, $q\geq 1$, see Subsection~\ref{subsec:extension}.
The EIV model~\eqref{eq:M}--\eqref{eq:H0} with non-random unknown constants $\bZ$ is sometimes called \emph{functional EIV} model~\citep{Fuller1987,hall}. On the other hand, a~different approach may handle $\bZ$ as random covariates, which is called \emph{structural EIV} model~\citep{ChH1997}. \cite{S2000} stated: `However, functional models played an~important role in the study of measurement error models and in statistics more generally.' And here, we will concentrate on the functional EIV model not because of this matter-of-fact quote, but because we wish to demonstrate a~distributional-free approach, where `no, or only minimal, assumptions are made about the distribution of the $\bX$s' \citep{CRSC2006}, as challenged in the introduction. Nevertheless with respect to derivation of the forthcoming theory for the functional EIV model, changing some technical assumptions would allow to prove suitable results for the structural case as well.
To estimate the unknown parameter~$\bbeta$, one usually minimizes the \emph{Frobenius matrix norm} of the errors $[\btheta,\beps]$, see~\cite{golub}. This approach leads to a~\emph{total least squares} (TLS) estimate $\hat{\bbeta}=(\bX^{\top}\bX-\lambda_{min}([\bX,\bY]^{\top}[\bX,\bY])\bI_p)^{-1}\bX^{\top}\bY$, where $\lambda_{min}(\bM)$ is the smallest eigenvalue of the matrix~$\bM$ and $\bI_p$ is a~$(p\times p)$ identity matrix. Geometrically speaking, the Frobenius norm tries to minimize the \emph{orthogonal distance} between the observations and the fitted hyperplane. Therefore, the TLS are usually known as \emph{orthogonal regression}. One can generalize this method by replacing the Frobenius norm by \emph{any unitary invariance matrix norm}, which surprisingly yields the same TLS estimate, having interesting invariance and equivariance properties~\citep{P2016}. The TLS estimate is shown to be strongly and weakly consistent~\citep{gleser,gallo,P2011} as well as to be asymptotically normal~\citep{gallophd,P2013,P2017} under various conditions.
We aim to detect a~possible change in the linear relation parameter~$\bbeta$. The interest lies in testing the \emph{null hypothesis}~\eqref{eq:H0} of all observations~$Y_i$'s being random variables having expectations~$\bZ_{i,\bullet}\bbeta$'s. Our goal is to test against the alternative of the first $\tau$ outcome observations have expectations~$\bZ_{i,\bullet}\bbeta$'s and the remaining $n-\tau$ observations come from distributions with expectations $\bZ_{i,\bullet}(\bbeta+\bdelta)$'s, where $\bdelta\neq\zero$. A~`row-column' notation for a~matrix $\bM$ is used in this manner: $\bM_{i,\bullet}$ denotes the $i$th row of~$\bM$ and $\bM_{\bullet,j}$ corresponds to the $j$th column of~$\bM$. Furthermore, if $i\in\mathbb{N}_0$, then $\bM_{i}$ stays for the first~$i$ rows of~$\bM$ and $\bM_{-i}$ represents the remaining $n-i$ rows of~$\bM$, when the first~$i$ rows are deleted. Now more precisely, our \emph{alternative hypothesis} is
\begin{equation}\label{eq:HA}\tag{$\mathcal{H}_A$}
\bY_{\tau}=\bZ_{\tau}\bbeta+\beps_{\tau}\quad\mbox{and}\quad\bY_{-\tau}=\bZ_{-\tau}(\bbeta+\bdelta)+\beps_{-\tau}.
\end{equation}
Here, $\bdelta\equiv\bdelta(n)\neq\zero$ is an~unknown vector parameter representing the size of change and is possibly depending on~$n$. The \emph{changepoint} $\tau\equiv\tau(n)<n$ is also an~unknown scalar parameter, which depends on~$n$ as well. Although, $\bbeta$ is considered to be independent of~$n$. One may also think of the changepoint in errors-in-variables framework as \emph{segmented regression with measurement errors}, cf.~\cite{SS2002}.
\subsection{Intercept and fixed regressors}
Note that the EIV model~\eqref{eq:M}--\eqref{eq:H0} has no intercept and all the covariates are encumbered by some errors. To overcome such a~restriction, one can think of an~extended regression model, where some explanatory variables are \emph{subject to error} and some are measured \emph{precisely}. I.e., $\bY=\bW\bgamma+\bZ\bbeta+\beps$, where $\bW$ are \emph{observable true} and $\bZ$ are \emph{unobservable true} constants, both having full rank. Regression parameters $\bgamma$ and $\bbeta$ remain unknown. Then, the non-random (fixed) \emph{intercept} can be incorporated into the regression model by setting one column of the matrix~$\bW$ equal to $[1,\ldots,1]^{\top}$. Consequently, we may \emph{project out} exact observations using projection matrix $\bR:=\bI_n-\bW(\bW^{\top}\bW)^{-1}\bW^{\top}$. Notice that $\bR$ is symmetric and idempotent. Finally, one may work with $\bR\bY=\bR\bZ\bbeta+\bR\beps$ instead of~\eqref{eq:H0}.
\subsection{Motivations}
The proposed class of models---\emph{errors-in-variables with changepoint}---is very rich and general. Our approach and results are motivated in the context of several applications taken from chemistry, biological sciences, medicine, and epidemiological studies.
\vspace{-0.3cm}
\paragraph{Case~1: Assessing agreement in clinical measurement.}
Direct measurement of cardiac stroke volume or blood pressure without adverse effects is difficult or even impossible. The true values remain unknown. Indirect methods are, therefore, used instead. When a~new measurement technique is developed, it has to be evaluated by comparison with an~established technique rather than with the true quantity~\citep{BA1986}. Clinicians need to test whether both measurement techniques agree sufficiently. Thereafter, the old technique may be replaced by the new one.
\vspace{-0.3cm}
\paragraph{Case~2: Nutritional epidemiology.}
\cite{SS2002} analyzed data from a~nutritional study that investigates the relation between dietary folate intake (calories adjusted $\mu$g/day) on plasma homocysteine concentration ($\mu$mol/liter of blood). There exists a~suspicion that serum homocysteine is significantly elevated when ingested folate is below a~certain changepoint. Moreover, the analysis used estimates of folate that were developed with a~food frequency questionnaire, which is recognized to be imperfect.
\vspace{-0.3cm}
\paragraph{Case~3: Psychometric testing.}
Let us think of two psychometric instruments: unspeeded 15-item vocabulary tests and highly speeded 75-item vocabulary tests, cf.~\cite{L1973}. The results of both tests are error-prone. Within a~group of people, there is a~speculation that individuals with an~unspeeded test's result exceeding some unknown level should perform dramatically better in the highly speeded test.
\vspace{-0.3cm}
\paragraph{Case~4: Environmental toxicology.}
A~threshold limiting value in toxicology is the dose of a~toxin or a~substance under which there is harmless or insignificant influence on some response. In a~dose-response relationship, both of them are measured with errors. And the goal is to set the threshold limiting value. Such a~problem was dealt by~\cite{GK2001} using fully Bayesian approach. Moreover, a~similar task regarding the NO$_2$ concentration is discussed by~\cite{S2000}.
\vspace{-0.3cm}
\paragraph{Case~5: Device calibration.}
Later on in Subsection~\ref{subsec:Papplication}, we concentrate in more details on the \emph{calibration} task and \emph{exemplify the proposed methodology} through analysis of data from a~calibrated device and a~casual device (needs to be calibrated) in order to demonstrate practical efficiency of our detection method.
\bigskip
Besides that, there are many other applications of the changepoint within the linear relations framework in, for instance, glaciology~\citep{watson}, empirical economics~\citep{ChH1997}, dietary assessment~\citep{CRW1999}, or image forensics~\citep{RL2014}.
\section{Spectral weak invariance principle}\label{sec:SWIP}
A~theoretical device is going to be developed in order to construct the changepoint tests. The \emph{smallest eigenvalue} of~$\bSigma^{-1}[\bX,\bY]^{\top}[\bX,\bY]$---the squared smallest singular value of~$[\bX,\bY]\bSigma^{-1/2}$, i.e., the data matrix~$[\bX,\bY]$ multiplied by the inverse of a~matrix square root from the error variance structure (cf.~subsequent Assumption~\ref{assump:EIVerrors})---plays a~key role. We proceed to the assumptions that are needed for deriving forthcoming asymptotic results. Henceforth, $\xrightarrow{\prob}$ denotes convergence in probability, $\xrightarrow{\dist}$ convergence in distribution, $\xrightarrow[n\to\infty]{\weak}$ weak convergence in the Skorokhod topology $\weak$ of c\`adl\`ag functions on $[0,1]$, and $[x]$ denotes the integer part of the real number~$x$.
\subsection{Assumptions}\label{subsec:assump}
Firstly, a~\emph{design assumption} on the unobservable regressors is needed.
\begin{assumpD}\label{assump:EIVdesign}
$\bDelta_{t}:=\lim_{n\to\infty}n^{-1}\bZ_{[nt]}^{\top}\bZ_{[nt]}$, $\bDelta_{-t}:=\lim_{n\to\infty}n^{-1}\bZ_{-[nt]}^{\top}\bZ_{-[nt]}$ for every $t\in(0,1)$, and $\bDelta:=\lim_{n\to\infty}n^{-1}\bZ^{\top}\bZ$ are positive definite.
\end{assumpD}
It basically says that the error-free design points do not concentrate to close to each other (i.e., strict positive definiteness) and, simultaneously, they do not spread-out too far (i.e., existence of limits). For example in one-dimensional case (i.e., $p=1$), a~simple equidistant design, where $Z_{i,1}=i/(n+1)$, provides $\bDelta_{t}=t^3/3$ and $\bDelta=1/3$.
Prior to postulating an~\emph{errors' assumption}, we summarize the notion of \emph{strong mixing} ($\alpha$-mixing) dependence in more detail, which will be imposed on the model's errors. Suppose that $\{\xi_n\}_{n=1}^{\infty}$ is a~sequence of random elements on a~probability space $(\Omega,\mathcal{F},\P)$. For sub-$\sigma$-fields $\mathcal{A},\mathcal{B}\subseteq\mathcal{F}$, let $\alpha(\mathcal{A}|\mathcal{B}):=\sup_{A\in\mathcal{A},B\in\mathcal{B}}\left|\P(A\cap B)-\P(A)\P(B)\right|$. Intuitively, $\alpha(\cdot|\cdot)$ measures the dependence of the events in $\mathcal{B}$ on those in $\mathcal{A}$.
There are many ways in which one can describe weak dependence or, in other words, \emph{asymptotic independence} of random variables, see~\cite{Bradley2005}. Considering a~filtration $\mathcal{F}_m^n=\sigma\{\xi_i\in\mathcal{F},m\leq i\leq n\}$, sequence $\{\xi_n\}_{n=1}^{\infty}$ of random variables is said to be \emph{strong mixing} ($\alpha$-mixing) if $\alpha(\xi_{\circ},n)=\sup_{k\in\mathbb{N}}\alpha(\mathcal{F}_{1}^k|\mathcal{F}_{k+n}^{\infty})\to 0$ as $n\to\infty$. \cite{Anderson1958} comprehensively analyzed a~class of $m$-dependent processes. They are $\alpha$-mixing, since they are finite order ARMA processes with innovations satisfying \emph{Doeblin's condition} \citep[p.~168]{Billingsley1968}. Finite order processes, which do not satisfy Doeblin's condition, can be shown to be $\alpha$-mixing \citep[pp.~312--313]{IL1971}. \cite{Rosenblatt1971} provides general conditions under which stationary Markov processes are $\alpha$-mixing. Since functions of mixing processes are themselves mixing \citep{Bradley2005}, time-varying functions of any of the processes just mentioned are mixing as well. This means that the class of the $\alpha$-mixing processes is sufficiently large for the further practical applications and that is why we chose such a~mixing condition.
\begin{assumpE}\label{assump:EIVerrors}
$\{[\btheta_{n,\bullet},\eps_n]\}_{n=1}^{\infty}$ is a~sequence of $\alpha$-mixing absolutely continuous random vectors having zero mean and a~variance matrix~$\sigma^2\bSigma$ with an~unknown $\sigma^2>0$ and a~known positive definite~$\bSigma=\begin{bmatrix}
\bSigma_{\btheta} & \bSigma_{\btheta,\beps}\\
\bSigma_{\btheta,\beps}^{\top} & 1
\end{bmatrix}$ such that $\alpha([\btheta_{\circ,\bullet},\eps_{\circ}],n)=\Oo(n^{-1-\varpi})$ as $n\to\infty$ for some $\varpi>0$, $\sup_{n\in\mathbb{N}} Z_{n,j}^2<\infty$, $\sup_{n\in\mathbb{N}}\E|\Theta_{n,j}|^{4+\omega}<\infty$, $j\in\{1,\ldots,p\}$, and $\sup_{n\in\mathbb{N}}\E|\eps_n|^{4+\omega}<\infty$ for some $\omega>0$ such that $\omega\varpi>2$.
\end{assumpE}
Let us emphasize that the sequence of the errors \emph{do not have to be stationary}. The assumption of an~unknown~$\sigma^2$ and a~known~$\bSigma$ implies that we know the ratio of any pair of covariances in advance. In the simplest situation, a~homoscedastic covariance structure of the within-individual errors $[\btheta_{n,\bullet},\eps_n]$ can be assumed (i.e., $\bSigma=\bI_{p+1}$), if prior experience or essence of the analyzed problem allow for that. On the other hand, if the covariance matrix $\bSigma$ is unknown, it can be estimated when possessing \emph{replicate measurements} or \emph{validation data} as commented by~\cite{S2000}. There are various approaches proposed to serve this purpose. In order ot mention at least some of them, we refer to \cite{ChR2006}, \cite{GL2011}, \cite{P2013}, or \cite{LML2019}. On the top of that, we have to bear in mind that~$\bSigma$ cannot be completely unspecified. \cite{N1977} showed that if~$\bSigma$ is unrestricted, no strongly consistent estimator for~$\bbeta$ can exist even under normally distributed errors.
Furthermore, a~\emph{variance assumption} for the misfit disturbances is stated. It can be considered as a~typical assumption for the \emph{long-run variance} of residuals. Let us denote $\bSigma^{-1/2}=\begin{bmatrix}
\bar{\bSigma}_{\btheta} & \bar{\bSigma}_{\btheta,\beps}\\
\bar{\bSigma}_{\btheta,\beps}^{\top} & \bar{\Sigma}_{\beps}
\end{bmatrix}$ a~symmetric square root of~$\bSigma^{-1}$, where $\bar{\Sigma}_{\beps}\in\mathbb{R}$ is a~scalar.
\begin{assumpV}\label{assump:EIVmisfit}
There exist
$\phi:=\bar{\Sigma}_{\beps}-\bar{\bSigma}_{\btheta,\beps}^{\top}(\bar{\bSigma}_{\btheta}+\bbeta\bar{\bSigma}_{\btheta,\beps}^{\top})^{-1}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})\neq 0$ and $\upsilon:=\lim_{n\to\infty}n^{-1}\var\|\bY-\bX\bbeta\|_2^2>0$.
\end{assumpV}
Let us remark that $\bar{\bSigma}_{\btheta,\beps}=\zero$ for the uncorrelated error structure and, then, $\phi=\bar{\Sigma}_{\beps}$.
\subsection{SWIP}
Finally, the spectral weak invariance principle for the smallest eigenvalues is provided. Let us denote $\lambda_{i}:=\lambda_{min}(\bSigma^{-1}[\bX_{i},\bY_{i}]^{\top}[\bX_{i},\bY_{i}])$ for $2\leq i\leq n$, $\lambda_0:=\lambda_1:=0$ and $\widetilde{\lambda}_{i}:=\lambda_{min}(\bSigma^{-1}[\bX_{-i},\bY_{-i}]^{\top}[\bX_{-i},\bY_{-i}])$ for $0\leq i\leq n-2$, $\widetilde{\lambda}_n:=\widetilde{\lambda}_{n-1}:=0$. Note that $\lambda_n\equiv\widetilde{\lambda}_0$.
\begin{proposition}[SWIP]\label{prop:SWIP}
Let~\ref{eq:M} and~\ref{eq:H0} hold. Under Assumptions~\ref{assump:EIVdesign}, \ref{assump:EIVerrors}, and \ref{assump:EIVmisfit},
\begin{equation*}
\left\{\frac{1}{\sqrt{n}}\left(\lambda_{[nt]}-[nt]\sigma^2\right)\right\}_{t\in[0,1]}\xrightarrow[n\to\infty]{\dist[0,1]}\left\{\frac{\phi^2\upsilon}{1+\|\balpha\|_2^2}\mathcal{W}(t)\right\}_{t\in[0,1]}
\end{equation*}
and
\begin{equation*}
\left\{\frac{1}{\sqrt{n}}\left(\widetilde{\lambda}_{[n(1-t)]}-[n(1-t)]\sigma^2\right)\right\}_{t\in[0,1]}\xrightarrow[n\to\infty]{\dist[0,1]}\left\{\frac{\phi^2\upsilon}{1+\|\balpha\|_2^2}\widetilde{\mathcal{W}}(t)\right\}_{t\in[0,1]},
\end{equation*}
where $\{\mathcal{W}(t)\}_{t\in[0,1]}$ is a~standard Wiener process , $\widetilde{\mathcal{W}}(t)=\mathcal{W}(1)-\mathcal{W}(t)$, and $\balpha=(\bar{\bSigma}_{\btheta}+\bbeta\bar{\bSigma}_{\btheta,\beps}^{\top})^{-1}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})$.
\end{proposition}
\subsection{Extension to multivariate case}\label{subsec:extension}
Suppose that $\bbeta\in\mathbb{R}^{p\times q}$, $\bY\in\mathbb{R}^{n\times q}$, and $\beps\in\mathbb{R}^{n\times q}$, $q\geq 1$. Let the singular value decomposition (SVD) of the partial transformed data be
\[
[\bX_{[nt]},\bY_{[nt]}]\bSigma^{-1/2}=\bU(t)\bGamma(t)\bV^{\top}(t)=\sum_{i=1}^{p+q}\varsigma(t)^{(i)}\bu(t)^{(i)}\bv(t)^{(i)\top},
\]
where $\bu(t)^{(i)}$'s are the left-singular vectors, $\bv(t)^{(i)}$'s are the right-singular vectors, and $\varsigma(t)^{(i)}$'s are the singular values in the non-increasing order. One may replace $\lambda_{[nt]}$ by
\[
\Lambda_{[nt]}:=\sum_{j=1}^{q}\Big(\varsigma(t)^{(p+j)}\Big)^2
\]
in Proposition~\ref{prop:SWIP} (and analogously for~$\widetilde{\lambda}_{[n(1-t)]}$). Then, the SWIP can be derived again (see the proof of Proposition~\ref{prop:SWIP}), provided adequately extended assumptions on the errors $\{\beps_{n,1}\}_{n=1}^{\infty},\ldots,\{\beps_{n,q}\}_{n=1}^{\infty}$ instead of the original ones $\{\eps_n\}_{n=1}^{\infty}$. However, the consequent proofs would become more technical.
\section{Nuisance-parameter-free detection}\label{sec:test}
Consistent estimation of~$\bbeta$ can be performed via the generalized TLS approach~\citep{gallo,HV1989}. The optimizing problem
\begin{equation*}
[\bb,\hat{\btheta},\hat{\beps}]:=\mathop{\arg\min}_{\left[\btheta,\beps\right]\in\mathbb{R}^{n\times (p+1)},{\bbeta}\in\mathbb{R}^p}\left\|\left[\btheta,\beps\right]\bSigma^{-1/2}\right\|_F\quad\mbox{s.t.}\quad \bY-{\beps}=({\bX}-{\btheta}){\bbeta},
\end{equation*}
where $\|\cdot\|_F$ stands for the Frobenius matrix norm, has a~solution consisting of the estimator
\begin{equation}\label{eq:TLS}
\bb=(\bX^{\top}\bX-\lambda_{n}\bSigma_{\btheta})^{-1}(\bX^{\top}\bY-\lambda_{n}\bSigma_{\btheta,\beps})
\end{equation}
and the fitted errors $[\hat{\btheta},\hat{\beps}]$ such that
\begin{equation}\label{eq:Fnorm}
\big\|[\hat{\btheta},\hat{\beps}]\bSigma^{-1/2}\big\|_F^2=\lambda_n.
\end{equation}
We construct the changepoint test statistics based on property~\eqref{eq:Fnorm}.
\subsection{Changepoint test statistics}
Let us think of two TLS estimates of~$\bbeta$: The first one based on the first~$i$ data lines $[\bX_i,\bY_i]$ and the second one based on the first~$k$ data lines $[\bX_k,\bY_k]$ such that $1\leq i\leq k\leq n$. Under the null~\ref{eq:H0}, these two TLS estimates should be close to each other. On the other hand, under the alternative~\ref{eq:HA} such that $\tau\in\{i,\ldots,k\}$, they should be somehow different. A~similar conclusion can be made for the \emph{goodness-of-fit} statistics coming from~\eqref{eq:Fnorm}. It means that
\begin{equation*}
\lambda_i-\frac{i}{k}\lambda_k
\end{equation*}
should be reasonably small under the null~\ref{eq:H0}. Under the alternative~\ref{eq:HA} such that $\tau\in\{i,\ldots,k\}$, it should be relatively large. For the multivariate case described in previous Subsection~\ref{subsec:extension}, one has to replace~$\lambda_k$ by $\Lambda_k=\sum_{j=1}^{q}\big(\varsigma(k/n)^{(p+j)}\big)^2$.
We rely on \emph{self-normalized test statistics} introduced by~\cite{shao2010testing}, because the unknown quantity $\phi^2\upsilon/(1+\|\balpha\|_2^2)$ from Proposition~\ref{prop:SWIP} cancels out in the test statistics. Our \emph{supremum-type self-normalized test statistic} based on the \emph{goodness-of-fit} is defined as
\begin{equation}\label{eq:statisticS}
\mathscr{S}_n:=\max_{1\leq k< n}\frac{\big|\lambda_k-\frac{k}{n}\lambda_n\big|}{\max_{1\leq i< k}\big|\lambda_i-\frac{i}{k}\lambda_k\big|+\max_{k< i\leq n}\big|\widetilde{\lambda}_{i}-\frac{n-i}{n-k}\widetilde{\lambda}_{k}\big|}
\end{equation}
and the \emph{integral-type self-normalized test statistic} is defined as
\begin{equation}\label{eq:statisticT}
\mathscr{T}_n:=\sum_{k=1}^{n-1}\frac{\big(\lambda_k-\frac{k}{n}\lambda_n\big)^2}{\sum_{i=1}^{k-1}\big(\lambda_i-\frac{i}{k}\lambda_k\big)^2+\sum_{i=k+1}^{n}\big(\widetilde{\lambda}_{i}-\frac{n-i}{n-k}\widetilde{\lambda}_{k}\big)^2}.
\end{equation}
Let us note that evaluations of the above defined test statistics require just several singular value decompositions, which is reasonably \emph{quick}. Our new test statistics involve \emph{neither nuisance parameters nor tuning constants} and will work for non-stationary and weakly dependent data. On the top of that, \emph{no boundary issue} is present meaning that the tests can detect the change close to the beginning or to the end of the studied regime.
Under the null hypothesis and the technical assumptions from Subsection~\ref{subsec:assump}, the test statistics defined in~\eqref{eq:statisticS} and~\eqref{eq:statisticT} converge to \emph{non-degenerate limit distributions} (their quantiles can be found in Subsection~\ref{subsec:crit}).
\begin{thm}[Under the null]\label{thm:H0}
Let~\ref{eq:M} and~\ref{eq:H0} hold. Under Assumptions~\ref{assump:EIVdesign}, \ref{assump:EIVerrors}, and \ref{assump:EIVmisfit},
\begin{equation}\label{eq:limit_distS}
\mathscr{S}_n\xrightarrow[n\to\infty]{\dist}\sup_{t\in[0,1]}\frac{\big|\mathcal{W}(t)-t\mathcal{W}(1)\big|}{\sup_{s\in[0,t]}\big|\mathcal{W}(s)-\frac{s}{t}\mathcal{W}(t)\big|+\sup_{s\in[t,1]}\big|\widetilde{\mathcal{W}}(s)-\frac{1-s}{1-t}\widetilde{\mathcal{W}}(t)\big|}
\end{equation}
and
\begin{equation}\label{eq:limit_distT}
\mathscr{T}_n\xrightarrow[n\to\infty]{\dist}\int_0^1\frac{\big\{\mathcal{W}(t)-t\mathcal{W}(1)\big\}^2}{\int_0^t\big\{\mathcal{W}(s)-\frac{s}{t}\mathcal{W}(t)\big\}^2\ud s+\int_t^1\big\{\widetilde{\mathcal{W}}(s)-\frac{1-s}{1-t}\widetilde{\mathcal{W}}(t)\big\}^2\ud s}\ud t,
\end{equation}
where $\{\mathcal{W}(t)\}_{t\in[0,1]}$ is a~standard Wiener process and $\widetilde{\mathcal{W}}(t)=\mathcal{W}(1)-\mathcal{W}(t)$.
\end{thm}
The null hypothesis is rejected at significance level $\alpha$ for large values of $\mathscr{S}_n$ and $\mathscr{T}_n$. The critical values can be obtained as the $(1-\alpha)$-quantiles of the asymptotic distributions from~\eqref{eq:limit_distS} and~\eqref{eq:limit_distT}. In order to describe limit behavior of the test statistics under the alternative, an~additional \emph{changepoint assumption} is required.
\begin{assumpC}\label{assump:EIVchange}
For some $\zeta\in(0,1)$, as $n\to\infty$,
\begin{equation}\label{eq:C}
\|\bdelta\|_2\to 0\quad\mbox{and}\quad (\eta\kappa-\bvarphi^{\top}\bvarphi)\sqrt{n}\to\infty,
\end{equation}
where $\kappa:=(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_{\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}(\bbeta+\bdelta)^{\top})\bDelta_{-\zeta}(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})$, $\bvarphi:=(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}\bbeta^{\top})\bDelta_{\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}(\bbeta+\bdelta)^{\top})\bDelta_{-\zeta}(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})$, and $\eta:=\lambda_{min}((\bar{\bSigma}_{\btheta}+\bar{\bSigma}_{\btheta,\beps}\bbeta^{\top})\bDelta(\bar{\bSigma}_{\btheta}+\bbeta\bar{\bSigma}_{\btheta,\beps}^{\top})+\sigma^2\bI_p)-\sigma^2$.
\end{assumpC}
This assumption may be considered as a~changepoint \emph{detectability requirement} for local alternatives, because it manages the relationship between the size of the change, the location of the change, and the noisiness of the data in order to be able to detect the changepoint. In case of uncorrelated error structure, the previous formulae become simpler due to $\bar{\bSigma}_{\btheta,\beps}=\zero$. Assumption~\ref{assump:EIVchange} is automatically fulfilled, for instance, for an~arbitrary $\delta\to 0$ and the one-dimensional equidistant design points $Z_i$'s on $(0,1)$ with homoscedastic error structure, because then $\eta\kappa-\bvarphi^{\top}\bvarphi=\beta^2\{\zeta^3+(1-\zeta)^3\}\{1-\zeta^3-(1-\zeta)^3\}/9+O(\delta)$ as $\delta\to 0$. Furthermore, let us remark that $\bvartheta:=\bar{\bSigma}_{\btheta}+\bbeta\bar{\bSigma}_{\btheta,\beps}^{\top}$ has full rank under Assumption~\ref{assump:EIVmisfit}.
Now, the tests based on $\mathscr{S}_n$ and $\mathscr{T}_n$ are shown to be \emph{consistent}, as the test statistics converge to infinity under some local alternatives, provided that the size of the change does not convergence to zero too fast, cf.~Assumption~\ref{assump:EIVchange} where $\kappa$ and $\bvarphi$ depend on~$\bdelta$.
\begin{thm}[Under local alternatives]\label{thm:H1}
Let~\ref{eq:M} and~\ref{eq:HA} hold such that $\tau=[n\zeta]$ for some $\zeta\in(0,1)$. Under Assumptions~\ref{assump:EIVchange}, \ref{assump:EIVdesign}, \ref{assump:EIVerrors}, and \ref{assump:EIVmisfit},
\begin{equation}\label{eq:alt}
\mathscr{S}_n\xrightarrow[n\to\infty]{\prob}\infty\xleftarrow[n\to\infty]{\prob}\mathscr{T}_n.
\end{equation}
\end{thm}
Assumption~\ref{assump:EIVchange} can be sharpened as remarked below with the corresponding proof in the Appendix~\ref{sec:proofs}.
\begin{remark}\label{rmk:sharp}
The second part of relation~\eqref{eq:C} can be replaced by
\begin{equation}\label{eq:Csharp}
\{\kappa+\eta-\sqrt{(\kappa+2\sigma^2+\eta)^2-4(\kappa+\sigma^2-\bvarphi^{\top}(\bvartheta^{\top}\bDelta\bvartheta+\sigma^2\bI_p)^{-1}\bvarphi)(\sigma^2+\eta)}\}\sqrt{n}\to\infty
\end{equation}
and the assertion of Theorem~\ref{thm:H1} still holds.
\end{remark}
Basically, Theorem~\ref{thm:H1} discloses that in presence of the structural change in linear relations, the test statistics \emph{explode above all bounds}. Hence, the asymptotic distributions from Theorem~\ref{thm:H0} can be used to construct the tests. Although, explicit forms of those distributions stated in~\eqref{eq:limit_distS} and~\eqref{eq:limit_distT} are unknown.
\subsection{Asymptotic critical values}\label{subsec:crit}
The critical values may be determined by simulations from the limit distributions $\mathscr{S}_n$ and $\mathscr{T}_n$ from Theorem~\ref{thm:H0}. Theorem~\ref{thm:H1} ensures that we reject the null hypothesis for large values of the test statistics. We have simulated the asymptotic distributions~\eqref{eq:limit_distS} and~\eqref{eq:limit_distT} by \emph{discretizing} the standard Wiener process and using the relationship of a~random walk to the standard Wiener process. We considered $1000$ as the number of discretization points within $[0,1]$ interval and the number of simulation runs equals to $100000$. In Table~\ref{tab:crit_val}, we present several critical values for the test statistics~$\mathscr{S}_n$ and~$\mathscr{T}_n$.
\begin{table}
\caption{\label{tab:crit_val}Simulated asymptotic critical values for~$\mathscr{S}_n$ and~$\mathscr{T}_n$}
\centering
\begin{tabular}{rrrrrr}
\toprule
$100(1-\alpha)\%$ & $90\%$ & $95\%$ & $97.5\%$ & $99\%$ & $99.5\%$ \\
\midrule
$\mathscr{S}$-based & $1.209008$ & $1.393566$ & $1.571462$ & $1.782524$ & $1.966223$\\
$\mathscr{T}$-based & $5.700222$ & $7.165705$ & $8.807070$ & $10.597625$ & $11.755233$\\
\bottomrule
\end{tabular}
\end{table}
\subsection{Changepoint estimator}
If a~change is \emph{detected}, it is of interest to estimate the changepoint. It is sensible to use
\begin{equation*}
\hat{\tau}_n:=\mathop{\operatorname{argmax}}_{1\leq k \leq n-1}\frac{\big|\lambda_k-\frac{k}{n}\lambda_n\big|+\big|\widetilde{\lambda}_k-\frac{n-k}{n}\widetilde{\lambda}_0\big|}{\max_{1\leq i< k}\big|\lambda_i-\frac{i}{k}\lambda_k\big|+\max_{k< i\leq n}\big|\widetilde{\lambda}_{i}-\frac{n-i}{n-k}\widetilde{\lambda}_{k}\big|}
\end{equation*}
as a~\emph{changepoint estimator}. Our next theorem shows that under the alternative, the changepoint $\tau$ is consistently estimated by the estimator $\hat{\tau}_n$.
\begin{corollary}[Consistency]\label{cor:est}
Let the assumptions of Theorem~\ref{thm:H1} hold. If
\begin{align}
&\forall t\in(\zeta,1):\,\{\eta(t)\kappa(t)-\bvarphi(t)^{\top}\bvarphi(t)\}\sqrt{n}\xrightarrow{n\to\infty}\infty;\label{eq:Cpt}\\
&\forall t\in(0,\zeta):\,\{\tilde{\eta}(t)\tilde{\kappa}(t)-\tilde{\bvarphi}(t)^{\top}\tilde{\bvarphi}(t)\}\sqrt{n}\xrightarrow{n\to\infty}\infty,\label{eq:Cpt2}
\end{align}
where $\kappa(t):=(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_{\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}(\bbeta+\bdelta)^{\top})(\bDelta_{t}-\bDelta_{\zeta})(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})$, $\bvarphi(t):=(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}\bbeta^{\top})\bDelta_{\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}(\bbeta+\bdelta)^{\top})(\bDelta_{t}-\bDelta_{\zeta})(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})$, $\tilde{\kappa}(t):=(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_{-\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}(\bbeta+\bdelta)^{\top})(\bDelta_{-t}-\bDelta_{-\zeta})(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})$, $\tilde{\bvarphi}(t):=(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}\bbeta^{\top})\bDelta_{-\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}(\bbeta+\bdelta)^{\top})(\bDelta_{-t}-\bDelta_{-\zeta})(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})$, $\eta(t):=\lambda_{min}((\bar{\bSigma}_{\btheta}+\bar{\bSigma}_{\btheta,\beps}\bbeta^{\top})\bDelta_t(\bar{\bSigma}_{\btheta}+\bbeta\bar{\bSigma}_{\btheta,\beps}^{\top})+t\sigma^2\bI_p)-t\sigma^2$, and $\tilde{\eta}(t):=\lambda_{min}((\bar{\bSigma}_{\btheta}+\bar{\bSigma}_{\btheta,\beps}\bbeta^{\top})\bDelta_{-t}(\bar{\bSigma}_{\btheta}+\bbeta\bar{\bSigma}_{\btheta,\beps}^{\top})+(1-t)\sigma^2\bI_p)-(1-t)\sigma^2$, then
\[
\frac{\hat{\tau}_n}{n}\xrightarrow[n\to\infty]{\prob}\zeta.
\]
\end{corollary}
Conditions~\eqref{eq:Cpt} and~\eqref{eq:Cpt2} serve as a~uniform intermediary between the size of the change, the location of the change, the sample size, and the heteroscedasticity of the disturbances for assuring changepoint estimator's consistency. These assumptions are again automatically fulfilled for the case discussed below Assumption~\ref{assump:EIVchange}.
In order to estimate more than one changepoint, it is possible to use an~arbitrary `divide-and-estimate' \emph{multiple changepoints} method relying on our changepoint estimator, for instance, wild binary segmentation by~\cite{F2014}.
\section{Simulation study}\label{sec:simulations}
We are interested in the performance of the tests based on the self-normalized test statistics $\mathscr{S}_n$ and $\mathscr{T}_n$ that are completely nuisance-parameter-free. We focused on the comparison of the \emph{accuracy of critical values} obtained by the simulation from the limit distributions.
In Figures~\ref{fig:H0}--\ref{fig:H12D}, one may see \emph{size-power plots} considering the test statistics $\mathscr{S}_n$ and $\mathscr{T}_n$ under the null hypothesis and under the alternative. Figures~\ref{fig:H0} and~\ref{fig:H1} correspond to one input covariate (i.e., $p=1$) with choices of $\beta=1$ and $Z_{i,1}=100i/(n+1)$. A~case with two error-prone regressors (i.e., $p=2$) is illustrated in Figures~\ref{fig:H02D} and~\ref{fig:H12D} for choices of $\bbeta=[1,1]^{\top}$ and $Z_{i,\bullet}=100\times[i/(n+1),(i/(n+1))^{3/2}]$. Next, $n\in\{200,1000\}$ and $\tau\in\{n/4,n/2\}$. The size of change is $\delta\in\{0.1,0.5\}$ for $p=1$ and $\bdelta\in\{[0.1,0.1]^{\top},[0.5,0.5]^{\top}\}$ for $p=2$. Especially smaller values of the break should represent the situations under the local alternatives. In Figures~\ref{fig:H0} and~\ref{fig:H02D}, the empirical rejection frequency under the null hypothesis (actual $\alpha$-errors) is plotted against the theoretical size (theoretical $\alpha$-errors with $\alpha\in\{1\%,5\%,10\%\}$), illustrating the size of the tests. The ideal situation under the null hypothesis is depicted by the straight diagonal dotted line. The empirical rejection frequencies ($1-$errors of the second type) under the alternative (with different changepoints and values of the change) are shown in Figures~\ref{fig:H1} and~\ref{fig:H12D}, illustrating the power of the tests. Under the alternative, the desired situation would be a~steep function with values close to~1. For more details on the size-power plots we may refer, e.g., to~\cite{kir2006}. The standard deviation of the random disturbances was set to $\sigma\in\{0.5,1.0\}$ and the random error terms $\{\Theta_{n,1}\}_{n=1}^{\infty},\ldots,\{\Theta_{n,p}\}_{n=1}^{\infty}$, and $\{\eps_n\}_{n=1}^{\infty}$ were independently simulated as three time series:
\begin{itemize}
\setlength\itemsep{-0.1cm}
\item \textsf{IID} \ldots~independent and identically distributed random variables;
\item \textsf{AR(1)} \ldots~autoregressive (AR) process of order one having a~coefficient of autoregression equal $0.5$;
\item \textsf{ARCH(1)} \ldots~autoregressive conditional heteroscedasticity (ARCH) process with the second coefficient equal $0.5$.
\end{itemize}
The standard normal distribution and the Student $t$-distribution with 3~degrees of freedom are used for generating the innovations of the models' errors. All of the time series are standardized such that they have variance equal~$\sigma^2$. Let us remark that the setup of Student $t_3$-distribution does not satisfy Assumption~\ref{assump:EIVerrors}. However, it can be considered as a~misspecified model and one would like to inspect performance of our procedures on such a~model that violates our assumptions. In the simulations of the rejection rates, we used $10000$ repetitions.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\textwidth]{FigH0-LR.pdf}
\caption{\label{fig:H0}Size-power plots for $\mathscr{S}_n$ and $\mathscr{T}_n$ under~\ref{eq:H0} ($p=1$)}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\textwidth]{FigH1-LR.pdf}
\caption{\label{fig:H1}Size-power plots for $\mathscr{S}_n$ and $\mathscr{T}_n$ under~\ref{eq:HA} ($p=1$)}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\textwidth]{FigH0-LR-2D.pdf}
\caption{\label{fig:H02D}Size-power plots for $\mathscr{S}_n$ and $\mathscr{T}_n$ under~\ref{eq:H0} ($p=2$)}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\textwidth]{FigH1-LR-2D.pdf}
\caption{\label{fig:H12D}Size-power plots for $\mathscr{S}_n$ and $\mathscr{T}_n$ under~\ref{eq:HA} ($p=2$)}
\end{figure}
In all of the subfigures of Figures~\ref{fig:H0} and~\ref{fig:H02D} depicting a~situation under the null hypothesis, we may see that comparing the accuracy of $\alpha$-levels (sizes) for different self-normalized test statistics, the integral-type ($\mathscr{T}$-based) method seems to keep the theoretical significance level more firmly than the supre\-mum-type ($\mathscr{S}$-based) method. Comparing the case of $\mathsf{N}(0,1)$ innovations with the case of~$t_3$ innovations, the rejection rates under the null tend to be slightly higher for the~$t_3$-distribution. In spite of the fact that the $t_3$-distributed errors violate Assumption~\ref{assump:EIVerrors}, the performance of our tests is still surprisingly satisfactory in such case. As expected, the accuracy of the critical values tends to be better for larger~$n$. The more complicated dependence structure of errors is assumed, the worse performance of the tests is obtained. Furthermore, the less volatile errors are set, the better tests' sizes are attained.
The $\mathscr{T}$-method performs better under the null. However under the alternative, the $\mathscr{S}$-method has a~tendency to have slightly higher power than the $\mathscr{T}$-method (see Figures~\ref{fig:H1} and~\ref{fig:H12D}). We may also conclude that under~\ref{eq:HA} with less volatile errors, the power of the test increases. The power decreases when the changepoint is closer to the beginning or the end of the input-output data. The heavier tails ($t_3$ against $\mathsf{N}(0,1)$) give worse results in general for both test statistics. Moreover, `more dependent' scenarios reveal worsening of the test statistics' performance. Furthermore, the smaller size of the change is considered, the lower power of the test is achieved. And again, the power gets higher for larger~$n$.
Afterwards, a~simulation experiment is performed to study the \emph{finite sample} properties of the changepoint estimator for a~change in the linear relations' parameter. We numerically present only the case of $p=1$. In particular, the interest lies in the \emph{empirical distributions} of the proposed estimator visualized via boxplots, see Figure~\ref{fig:Estimator}. The simulation setup is kept the same as described above.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\textwidth]{FigBoxplotLR.pdf}
\caption{\label{fig:Estimator}Boxplots of the estimated changepoint $\hat{\tau}_n$ ($p=1$)}
\end{figure}
It can be concluded that the precision of our changepoint estimate is satisfactory even for relatively small sample sizes regardless of the errors' structure. Less volatile model errors provide more precise changepoint estimate. The less complicated dependence structure is assumed, the higher accuracy of the estimator is obtained. Furthermore, the disturbances with heavier tails yield less precise estimates than innovations with light tails. One may notice that higher precision is obtained when the changepoint is closer to the middle of the data. It is also clear that the precision of $\hat{\tau}_n$ improves markedly as the size of change increases.
\section{Applications}\label{sec:applications}
\subsection{Practical application: Calibration}\label{subsec:Papplication}
A~company has two industrial devices, where the first one is calibrated according to some institute of standards and the second one is just a~casual device. We want to test whether the second device is calibrated according to the first one. In this \emph{calibration problem}, it means to know whether the second device has approximately the \emph{same performance up to some unknown multiplication constant} as the first one. Consequently, other devices of the same type are needed to be calibrated as well. For some reasons, e.g., economic or logistic, it is only possible to calibrate one device by the official authorities.
Our data set, provided by a~Czech steelmaker, contains 100 couples of speed values of two hammer rams (see Figure~\ref{fig:Presser}), where the first forging hammer is calibrated. We set the same power level on both hammers and measure the speed of each hammer ram repeatedly changing only the power level. Our measurements of the speed are encumbered with errors of the same variability in both cases, because we use the same device for measuring the speed and both forging hammers are of the same type. Since the power set for the forging hammer is directly proportional to the speed of the hammer ram, our goal is to test whether the \emph{ratio of two hammer rams' speeds is kept constant} over changing the power level or not. Therefore, our changepoint in the EIV model is very suitable for this setup---a~linear dependence and errors in both measured speeds (with the same variance).
Both our changepoint tests---$\mathscr{S}_n=83.2$ and $\mathscr{T}_n=861.4$---reject the null hypothesis of a~constant linear coefficient between two hammer rams' speed values at the significance level of $\alpha=0.5\%$ (cf.~Table~\ref{tab:crit_val}; the significance level for technical fields is usually smaller than the standard $5\%$), indicating a~changed performance of the second non-calibrated hammer ram.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{Presser.pdf}
\caption{Speeds of two hammer rams, where the first one displayed on the x-axis is calibrated. The changepoint estimate corresponding to the technical issues of the second hammer ram after the 60th measurement is depicted by the vertical line}
\label{fig:Presser}
\end{center}
\end{figure}
As an~estimate for our change, we obtain $\hat{\tau}_n=60$ (depicted by a~vertical line in Figure~\ref{fig:Presser}), which corresponds to the 60th measurement of pair of speeds. After this particular measurement, we have background information that a~technical issue appeared to the second hammer ram---one of its oil tubes started to leak. Our procedure is indeed capable to detect and, consequently, to estimate the changepoint in the ratio of the hammer rams' speeds. And this is done fully automatically without expert knowledge about the oil tube issue and also without setting tuning parameters. Moreover, the estimated ratio via the TLS approach before the change is $1.000891$ (the slope of the green line in Figure~\ref{fig:Presser}), which basically says that the hammer rams work approximately in the same way. However, the estimated ratio via the TLS approach after the change is $0.9892154$ (the slope of the red line in Figure~\ref{fig:Presser}), which is significantly different from constant~1 (see a~formal statistical test by~\cite{P2013}).
Other calibration examples, where our methodology is applicable, can be found in, e.g., \cite{ChR2006} or~\cite{GL2011}.
\subsection{Theoretical application: Randomly spaced time series}\label{subsec:Tapplication}
A~motivation for the changepoint problem in randomly spaced time series comes from the changepoint in the polynomial \emph{trending regression}~\citep{AHH2009}. Let us think of a~single regressor measured precisely such that $X_{i,1}\equiv Z_{i,1}=i/(n+1)$. This indeed corresponds to a~situation of a~one-dimensional equally (regularly) spaced time series, where the original time points $\{i\}_{i=1}^n$ are `squeezed' into the interval~$[0,1]$ by dividing of~$n+1$.
Now, let us assume that our outcome observations $Y_i$'s are supposed to be measured at some unknown time points~$Z_{i,1}$'s. However, due to some measurement imprecision or some outer random influence, the actual observation~$Y_i$, which should correspond to~$Z_{i,1}$, is not recorded at time point~$Z_{i,1}$, but at time point~$X_{i,1}$. One can imagine a~long-distance time trial against the chronometer (e.g., an~individual competition in cross-country skiing). There are $n$ intermediate spots on the track, where the athlete's time is recorded. If we think of one particular athlete, we measure at the intermediate spot~$i$ her/his error-prone competition time $X_{i,1}$, which was encumbered by some randomness, instead of the true unobservable time is $Z_{i,1}$. This is because each race is specific and every athlete has a~unique performance during that particular race. We also observe a~time lag~$Y_{i}$ between her/him and the current leader at that spot. Now, one is interested whether there is a~change in linear trend. This would help to analyze whether the particular athlete tried to improve or not during the time trial. One can argue that a~distance of the intermediate spot should be taken into account instead of the athlete's intermediate time. However, the intermediate distance is also measured with error, for instance, a~rounded value of the true unobserved distance is provided. Another example of randomly spaced time series is a~case when the observation times are driven by the series itself. For instance, cumulative counts of occurrences of a~disease in a~given area~\citep{W1986}.
The unobservable sequence $\{Z_{i,1}\}_{i=1}^n$ can be regularly or irregularly spaced. The key issue is to have satisfiable Assumption~\ref{assump:EIVdesign}. Since the developed detection procedures rely on the orthogonal regression, it is sufficient to transform the original randomly spaced time series~$\{X_{i,1},Y_i\}_{i=1}^n$ into, e.g., $\{X_{i,1}/(\max_i\{|X_{i,1}|\}+\epsilon),Y_i/(\max_i\{|X_{i,1}|\}+\epsilon)\}_{i=1}^n$, where a~constant~$\epsilon$ is reasonably large. Afterwards, the proposed tests remain valid when applied to the transformed randomly spaced time series, because~$\beta$ stays unchanged after such a~transformation. Hence, one can test whether the \emph{linear trend} has or has not changed over time.
\section{Conclusions}
Our changepoint problem in linear relations is linearly defined, but comes with a~highly non-linear solution and inference. We have proposed two tests for changepoints with desirable theoretical properties: The asymptotic size of the tests is guaranteed by a~limit theorem even under non-stationarity and weak dependency, the tests and the related changepoint estimator are consistent. We are not aware of any similar results even for independent and identically distributed errors. By combining self-normalization and the proposed spectral weak invariance principle, there are neither tuning constants nor nuisance parameters involved in the whole testing procedure. Therefore, the detection methods are completely data-driven, which makes this framework effortlessly applicable as demonstrated. In our simulations, the tests show reliable performance.
\appendix
\section{Proofs}\label{sec:proofs}
\begin{proof}[Proof of Proposition~\ref{prop:SWIP}]
Let the singular value decomposition of the transformed `partial' data matrix be
\[
[\bX_{[nt]},\bY_{[nt]}]\bSigma^{-1/2}=\bU(t)\bGamma(t)\bV(t)^{\top}=\sum_{i=1}^{p+1}\varsigma(t)^{(i)}\bu(t)^{(i)}\bv(t)^{(i)\top}
\]
for some $t\in(0,1]$. Note that we are in a~situation of no change in the parameter~$\bbeta$. Bearing in mind Assumptions~\ref{assump:EIVdesign} and~\ref{assump:EIVerrors}, \citet[Lemma~2.1]{gleser} and~\citet[Theorem~3.1]{P2011} provide that $0\neq v_{p+1}(t)^{(p+1)}$ (i.e., the last element of the last right-singular vector $\bv(t)^{(p+1)}$ corresponding to the smallest singular value) with probability tending to one as~$n$ increases.
According to~\citet[proof of Lemma~4.2]{gleser}, one gets
\begin{align}
&\frac{1}{\sqrt{n}}\left(\lambda_{[nt]}-[nt]\sigma^2\right)=\Big(v_{p+1}(t)^{(p+1)}\Big)^2[\ba_{t}^{\top},-1]\left\{\frac{1}{\sqrt{n}}\left(\bD_{t}-\E\bD_{t}\right)\right\}\begin{bmatrix}
\ba_{t}\\
-1
\end{bmatrix}\label{eq:swip1}\\
&\quad+\Big(v_{p+1}(t)^{(p+1)}\Big)^2\sqrt{n}
[\ba_{t}^{\top},-1]\bSigma^{-1/2}\begin{bmatrix}
\bI_{p}\\
\bbeta^{\top}
\end{bmatrix}\frac{1}{n}\bZ_{[nt]}^{\top}\bZ_{[nt]}[\bI_{p},\bbeta]\bSigma^{-1/2}\begin{bmatrix}
\ba_{t}\\
-1
\end{bmatrix},\label{eq:swip2}
\end{align}
where $\ba_{t}:=(\tilde{\bX}_{[nt]}^{\top}\tilde{\bX}_{[nt]}-\lambda_{[nt]}\bI_{p})^{-1}\tilde{\bX}_{[nt]}^{\top}\tilde{\bY}_{[nt]}$ is the TLS estimator for the transformed data $[\tilde{\bX}_{[nt]},\tilde{\bY}_{[nt]}]:=[\bX_{[nt]},\bY_{[nt]}]\bSigma^{-1/2}$ and $\bD_{t}:=\bSigma^{-1/2}[\bX_{[nt]},\bY_{[nt]}]^{\top}[\bX_{[nt]},\bY_{[nt]}]\bSigma^{-1/2}$. With respect to~\cite{P2011}, we have
\begin{equation*}
\Big(v_{p+1}(t)^{(p+1)}\Big)^2=1-\left\|[v_{1}(t)^{(p+1)},\ldots,v_{p}(t)^{(p+1)}]^{\top}\right\|_2^2\to\frac{1}{1+\|\balpha\|_2^2}
\end{equation*}
almost surely as $n\to\infty$. Moreover, $\sqrt{n}(\ba_{t}-\balpha)=\Op(1)$ as $n\to\infty$ by~\cite{Pesta2013}. The strong law of large numbers for $\alpha$-mixing by~\cite{chenwu1989} together with Theorem~3.1 by~\cite{P2011} lead to $\ba_{t}-\balpha=o(1)$ almost surely. Since Assumption~\ref{assump:EIVdesign} holds, the expression in~\eqref{eq:swip2} is $\op(1)$. Furthermore, the expression on the right hand side of~\eqref{eq:swip1} is $o(1)$ away from
\begin{equation}\label{eq:approx}
\frac{1}{1+\|\balpha\|_2^2}[\balpha^{\top},-1]\left\{\frac{1}{\sqrt{n}}\left(\bD_{t}-\E\bD_{t}\right)\right\}
\begin{bmatrix}
\balpha\\
-1
\end{bmatrix}
\end{equation}
as $n\to\infty$. Hence, the process from the left hand side of~\eqref{eq:swip1} in $\dist[0,1]$ has approximately the same distribution as the process~\eqref{eq:approx}.
Note that
\[
[\balpha^{\top},-1]\bD_{t}\begin{bmatrix}
\balpha\\
-1
\end{bmatrix}=(\bar{\Sigma}_{\beps}-\bar{\bSigma}_{\btheta,\beps}^{\top}\balpha)^2\big\|\bY_{[nt]}-\bX_{[nt]}\bbeta\big\|_2^2.
\]
Using the functional central limit theorem for $\alpha$-mixing by~\cite{Herrndorf1983} or~\citet[Corollary 3.2.1]{linlu1997} in an~analogous fashion as in the proof of Theorem~2.3 by~\cite{Pesta2013}, one gets
\[
\left\{[\balpha^{\top},-1]\left\{\frac{1}{\sqrt{n}}\left(\bD_{t}-\E\bD_{t}\right)\right\}\begin{bmatrix}
\balpha\\
-1
\end{bmatrix}\right\}_{t\in[0,1]}\xrightarrow[n\to\infty]{\dist[0,1]}\{\phi^2\upsilon\mathcal{W}(t)\}_{t\in[0,1]}
\]
due to Assumption~\ref{assump:EIVmisfit}.
Similarly for $\left\{\frac{1}{\sqrt{n}}\left(\widetilde{\lambda}_{[n(1-t)]}-[n(1-t)]\sigma^2\right)\right\}_{t\in[0,1]}$ and $\{\widetilde{\mathcal{W}}(t)\}_{t\in[0,1]}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:H0}]
The spectral weak invariance principle from Proposition~\ref{prop:SWIP} and Lemma~1 by~\cite{PW2019} in combination with the continuous mapping device complete the proof.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:H1}]
Under~\ref{eq:HA}, let us find a~lower bound for the smallest eigenvalue of the positive semi-definite matrix
\begin{equation}\label{eq:ABCmatrix}
\frac{1}{n}\bSigma^{-1/2}[\bX,\bY]^{\top}[\bX,\bY]\bSigma^{-1/2}=\frac{1}{n}\begin{bmatrix}
\tilde{\bX}^{\top}\tilde{\bX} & \tilde{\bX}^{\top}\tilde{\bY}\\
\tilde{\bY}^{\top}\tilde{\bX} & \tilde{\bY}^{\top}\tilde{\bY}
\end{bmatrix}=:\begin{bmatrix}
\bA & \bc\\
\bc^{\top} & d
\end{bmatrix},
\end{equation}
where $[\tilde{\bX},\tilde{\bY}]:=[\bX,\bY]\bSigma^{-1/2}$. With respect to~\citet[Theorem~1]{D1988}, we get
\begin{equation}\label{eq:lower}
\lambda_{min}\left(\begin{bmatrix}
\bA & \bc\\
\bc^{\top} & d
\end{bmatrix}\right)\geq \frac{d+\ell}{2}-\sqrt{\frac{(d-\ell)^2}{4}+\bc^{\top}\bc},
\end{equation}
where $\ell$ is any lower bound on the smallest eigenvalue of the matrix~$\bA$. Recall that Assumption~\ref{assump:EIVerrors} and the proof of Theorem~3.1 by~\cite{P2011} provide
\begin{equation}\label{eq:limits}
\frac{1}{n}\tilde{\beps}^{\top}\tilde{\beps}\to\sigma^2,\,\frac{1}{n}\tilde{\btheta}^{\top}\tilde{\beps}\to\zero,\,\frac{1}{n}\tilde{\btheta}^{\top}\tilde{\btheta}\to\sigma^2\bI_{p},\,\frac{1}{n}\tilde{\bZ}^{\top}\tilde{\beps}\to 0,\,\frac{1}{n}\tilde{\bZ}^{\top}\tilde{\btheta}\to\zero
\end{equation}
almost surely as $n\to\infty$, where $[\tilde{\btheta},\tilde{\beps}]:=[\btheta,\beps]\bSigma^{-1/2}$ and $\tilde{\bZ}:=\bZ [\bI_{p},\bbeta]\begin{bmatrix}
\bar{\bSigma}_{\btheta}\\
\bar{\bSigma}_{\btheta,\beps}^{\top}
\end{bmatrix}$. By Assumptions~\ref{assump:EIVchange} and~\ref{assump:EIVdesign}, one can obtain
\begin{multline}\label{eq:lower2}
\lambda(\bA)_{min}=\lambda_{min}\left(\frac{1}{n}(\tilde{\bZ}+\tilde{\btheta})^{\top}(\tilde{\bZ}+\tilde{\btheta})\right)\\
\to\lambda_{min}\left([\bar{\bSigma}_{\btheta},\bar{\bSigma}_{\btheta,\beps}]\begin{bmatrix}
\bI_{p}\\
\bbeta^{\top}
\end{bmatrix}\bDelta[\bI_{p},\bbeta]\begin{bmatrix}
\bar{\bSigma}_{\btheta}\\
\bar{\bSigma}_{\btheta,\beps}^{\top}
\end{bmatrix}+\sigma^2\bI_p\right)=\sigma^2+\eta
\end{multline}
almost surely as $n\to\infty$. Relation~\eqref{eq:lower2} immediately provides a~limit of a~candidate for~$\ell$. Now, \eqref{eq:ABCmatrix} and \eqref{eq:lower} lead to
\begin{multline}\label{eq:ineqLimInf}
\liminf_{n\to\infty}\lambda_{min}\left(\frac{1}{n}\bSigma^{-1/2}[\bX,\bY]^{\top}[\bX,\bY]\bSigma^{-1/2}\right)\\
\geq\frac{\displaystyle\lim_{n\to\infty}\frac{1}{n}\tilde{\bY}^{\top}\tilde{\bY}+\sigma^2+\eta}{2}-\sqrt{\frac{\Big(\displaystyle\lim_{n\to\infty}\frac{1}{n}\tilde{\bY}^{\top}\tilde{\bY}-\sigma^2-\eta\Big)^2}{4}+\lim_{n\to\infty}\left\|\frac{1}{n}\tilde{\bX}^{\top}\tilde{\bY}\right\|_2^2}.
\end{multline}
Assumptions~\ref{assump:EIVchange}, \ref{assump:EIVdesign}, and relations~\eqref{eq:limits} yield
\begin{multline*}
\frac{1}{n}\tilde{\bY}^{\top}\tilde{\bY}=\frac{1}{n}\tilde{\bY}_{\tau}^{\top}\tilde{\bY}_{\tau}+\frac{1}{n}\tilde{\bY}_{-\tau}^{\top}\tilde{\bY}_{-\tau}=(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_{\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})\\
+\sigma^2+(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}(\bbeta+\bdelta)^{\top})\bDelta_{-\zeta}(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})+o(1)=\kappa+\sigma^2+o(1)
\end{multline*}
and
\begin{multline*}
\frac{1}{n}\tilde{\bX}^{\top}\tilde{\bY}=\frac{1}{n}\tilde{\bX}_{\tau}^{\top}\tilde{\bY}_{\tau}+\frac{1}{n}\tilde{\bX}_{-\tau}^{\top}\tilde{\bY}_{-\tau}=(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}\bbeta^{\top})\bDelta_{\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})\\
+(\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}(\bbeta+\bdelta)^{\top})\bDelta_{-\zeta}(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})+o(1)=\bvarphi+o(1)
\end{multline*}
almost surely as $n\to\infty$. Thus,
\begin{multline}\label{eq:ineqLimInf2}
\frac{\frac{1}{n}\tilde{\bY}^{\top}\tilde{\bY}+\sigma^2+\eta}{2}-\sqrt{\frac{\big(\frac{1}{n}\tilde{\bY}^{\top}\tilde{\bY}-\sigma^2-\eta\big)^2}{4}+\left\|\frac{1}{n}\tilde{\bX}^{\top}\tilde{\bY}\right\|_2^2}\\
=\frac{\kappa+\eta-\sqrt{(\kappa-\eta)^2+4\bvarphi_{2}^{\top}\bvarphi_{2}}}{2}+\sigma^2+o(1)
\end{multline}
almost surely as $n\to\infty$. Hence, combining~\eqref{eq:ineqLimInf} and~\eqref{eq:ineqLimInf2} ends up with
\begin{equation*}
\liminf_{n\to\infty}\lambda_{min}\left(\frac{1}{n}[\tilde{\bX},\tilde{\bY}]^{\top}[\tilde{\bX},\tilde{\bY}]\right)-\sigma^2\geq\lim_{n\to\infty}\frac{2\{\eta\kappa-\bvarphi^{\top}\bvarphi\}}{\kappa+\eta+\sqrt{(\kappa-\eta)^2+4\bvarphi^{\top}\bvarphi}}.
\end{equation*}
Then,
\begin{equation}\label{eq:delta}
\frac{1}{\sqrt{n}}|\lambda_n-n\sigma^2|\xrightarrow[n\to\infty]{\mbox{a.s.}}\infty
\end{equation}
by Assumption~\ref{assump:EIVchange}.
With respect to Assumptions~\ref{assump:EIVdesign}, \ref{assump:EIVerrors}, \ref{assump:EIVmisfit} and according to the underlying proof of Theorem~\ref{thm:H0}, $\frac{1}{\sqrt{n}}\max_{1\leq i<\tau}\big|\lambda_i-\frac{i}{\tau}\lambda_{\tau}\big|$ and $\frac{1}{\sqrt{n}}\max_{\tau< i\leq n}\big|\widetilde{\lambda}_{i}-\frac{n-i}{n-\tau}\widetilde{\lambda}_{\tau}\big|$ are $\Op(1)$ as $n\to\infty$.
Moreover, $\frac{1}{\sqrt{n}}\big|\lambda_{\tau}-\tau\sigma^2\big|=\Op(1)$ as $n\to\infty$ due to Proposition~\ref{prop:SWIP}.
Note that there are no changes in the linear parameter corresponding to the first~$\tau$ observations as well as to the last (remaining)~$n-\tau$ observations. Let $k=\tau$. Thus, under~\ref{eq:HA},
\begin{align*}
\mathscr{S}_n&\geq\frac{\big|\lambda_{\tau}-\frac{\tau}{n}\lambda_n\big|}{\max_{1\leq i<\tau}\big|\lambda_i-\frac{i}{\tau}\lambda_{\tau}\big|+\max_{\tau< i\leq n}\big|\widetilde{\lambda}_{i}-\frac{n-i}{n-\tau}\widetilde{\lambda}_{\tau}\big|}\\
&\geq\frac{\frac{1}{\sqrt{n}}\Big|\big|\lambda_{\tau}-\tau\sigma^2\big|-\frac{\tau}{n}\big|n\sigma^2-\lambda_n\big|\Big|}{\frac{1}{\sqrt{n}}\max_{1\leq i<\tau}\big|\lambda_i-\frac{i}{\tau}\lambda_{\tau}\big|+\frac{1}{\sqrt{n}}\max_{\tau< i\leq n}\big|\widetilde{\lambda}_{i}-\frac{n-i}{n-\tau}\widetilde{\lambda}_{\tau}\big|}\xrightarrow[n\to\infty]{\prob}\infty,
\end{align*}
because of~\eqref{eq:delta}.
Furthermore, again under~\ref{eq:HA},
\begin{align*}
\mathscr{T}_n&\geq\frac{\big(\lambda_{\tau}-\frac{\tau}{n}\lambda_n\big)^2}{\sum_{i=1}^{\tau-1}\big(\lambda_i-\frac{i}{\tau}\lambda_{\tau}\big)^2+\sum_{i=\tau+1}^{n}\big(\widetilde{\lambda}_{i}-\frac{n-i}{n-\tau}\widetilde{\lambda}_{\tau}\big)^2}\\
&\geq\frac{\frac{1}{n}\Big(\big|\lambda_{\tau}-\tau\sigma^2\big|-\frac{\tau}{n}\big|n\sigma^2-\lambda_n\big|\Big)^2}{\frac{1}{n}\sum_{i=1}^{\tau-1}\big(\lambda_i-\frac{i}{\tau}\lambda_{\tau}\big)^2+\frac{1}{n}\sum_{i=\tau+1}^{n}\big(\widetilde{\lambda}_{i}-\frac{n-i}{n-\tau}\widetilde{\lambda}_{\tau}\big)^2}\xrightarrow[n\to\infty]{\prob}\infty,
\end{align*}
because of similar arguments as in the case of $\mathscr{S}_n$.
\end{proof}
\begin{proof}[Proof of Remark~\ref{rmk:sharp}]
It is sufficient to replace Theorem~1 by~\cite{D1988} with Theorem~3.1 by~\cite{MZ1995} in the proof of Theorem~\ref{thm:H1}.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:est}]
The estimator can be rewritten as
\begin{equation}\label{eq:numdenom}
\hat{\tau}_n=\mathop{\operatorname{argmax}}_{1\leq k \leq n-1}\frac{\frac{1}{n}\big|\lambda_k-\frac{k}{n}\lambda_n\big|+\frac{1}{n}\big|\widetilde{\lambda}_k-\frac{n-k}{n}\widetilde{\lambda}_0\big|}{\max_{1\leq i< k}\frac{1}{\sqrt{n}}\big|\lambda_i-\frac{i}{k}\lambda_k\big|+\max_{k< i\leq n}\frac{1}{\sqrt{n}}\big|\widetilde{\lambda}_{i}-\frac{n-i}{n-k}\widetilde{\lambda}_{k}\big|}.
\end{equation}
We will treat the numerator~$N_n(k)$ and the denominator~$D_n(k)$ of the above stated ratio separately. Let us use notations from the previous proofs and let us recall Assumption~\ref{assump:EIVchange}, \ref{assump:EIVdesign}, and relations~\eqref{eq:limits}. If $[nt]\leq\tau$, then
\begin{equation*}
\frac{1}{n}\tilde{\bY}_{[nt]}^{\top}\tilde{\bY}_{[nt]}=(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_t(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+t\sigma^2+o(1)
\end{equation*}
almost surely as $n\to\infty$. Otherwise, if $[nt]>\tau$, then
\begin{multline*}
\frac{1}{n}\tilde{\bY}_{[nt]}^{\top}\tilde{\bY}_{[nt]}=(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_{\zeta}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+t\sigma^2\\
+(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}(\bbeta+\bdelta)^{\top})(\bDelta_{t}-\bDelta_{\zeta})(\bar{\bSigma}_{\btheta,\beps}+(\bbeta+\bdelta)\bar{\Sigma}_{\beps})+o(1)
\end{multline*}
almost surely as $n\to\infty$. In both cases, we have
\begin{multline*}
\frac{1}{n}[\tilde{\bX}_{[nt]},\tilde{\bY}_{[nt]}]^{\top}[\tilde{\bX}_{[nt]},\tilde{\bY}_{[nt]}]\\
\xrightarrow[n\to\infty]{\mbox{a.s.}}\begin{bmatrix}
\bvartheta^{\top}\bDelta_{t}\bvartheta+t\sigma^2\bI_p & (\bar{\bSigma}_{\btheta}+\bar{\Sigma}_{\btheta,\beps}\bbeta^{\top})\bDelta_{t}(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})\\
(\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_{t}(\bar{\bSigma}_{\btheta}+\bbeta\bar{\Sigma}_{\btheta,\beps}^{\top}) & (\bar{\bSigma}_{\btheta,\beps}^{\top}+\bar{\Sigma}_{\beps}\bbeta^{\top})\bDelta_t(\bar{\bSigma}_{\btheta,\beps}+\bbeta\bar{\Sigma}_{\beps})+t\sigma^2
\end{bmatrix}\\
=t\sigma^2\bI_{p+1}+\bSigma^{-1/2}\begin{bmatrix}
\bI_p\\
\bbeta^{\top}
\end{bmatrix}\bDelta_{t}[\bI_p,\bbeta]\bSigma^{-1/2}.
\end{multline*}
Therefore, for the Frobenius matrix norm $\|\cdot\|_F$,
\begin{multline*}
\lim_{n\to\infty}\Bigg|\lambda_{min}\left(\frac{1}{n}[\tilde{\bX}_{[nt]},\tilde{\bY}_{[nt]}]^{\top}[\tilde{\bX}_{[nt]},\tilde{\bY}_{[nt]}]\right)-\frac{[nt]}{n}\lambda_{min}\left(\frac{1}{n}[\tilde{\bX},\tilde{\bY}]^{\top}[\tilde{\bX},\tilde{\bY}]\right)\Bigg|\\
=:\lambda_{dif}(t)\leq\left\|\bSigma^{-1/2}\begin{bmatrix}
\bI_p\\
\bbeta^{\top}
\end{bmatrix}\bDelta_{-t}[\bI_p,\bbeta]\bSigma^{-1/2}\right\|_F
\end{multline*}
uniformly in~$t$ almost surely, because $|\lambda_{min}(\bA)-\lambda_{min}(\bB)|\leq\|\bA-\bB\|_F$ due to~\citet[proof of Lemma~2.3]{gallophd}.
For $k=\tau$, Proposition~\ref{prop:SWIP} together with the continuous mapping theorem yield that the denominator from~\eqref{eq:numdenom}
\begin{equation*}
D_n(\tau)\xrightarrow[n\to\infty]{\dist}\frac{\phi^2\upsilon}{1+\|\balpha\|_2^2}\bigg\{\sup_{0\leq t\leq \zeta}\bigg|\mathcal{W}(t)-\frac{t}{\zeta}\mathcal{W}(\zeta)\bigg|+\sup_{\zeta<t\leq 1}\bigg|\widetilde{\mathcal{W}}(t)-\frac{1-t}{1-\zeta}\widetilde{\mathcal{W}}(\zeta)\bigg|\bigg\}=:W,
\end{equation*}
where the limit~$W$ is strictly positive almost surely. We conclude that $|N_n(\tau)/D_n(\tau)|$ converge in distribution to the random variable $\lambda_{dif}(\zeta)+\widetilde{\lambda}_{dif}(\zeta)/W$ such that $\widetilde{\lambda}_{dif}(t):=\lim_{n\to\infty}|\widetilde{\lambda}_{[nt]}-\frac{n-[nt]}{n}\widetilde{\lambda}_{0}|$. For $k=[nt]$ with $t>\zeta$, we obtain
\begin{align*}
&\max_{1\leq i< [nt]}\frac{1}{\sqrt{n}}\bigg|\lambda_i-\frac{i}{[nt]}\lambda_{[nt]}\bigg|+\max_{[nt]< i\leq n}\frac{1}{\sqrt{n}}\bigg|\widetilde{\lambda}_{i}-\frac{n-i}{n-[nt]}\widetilde{\lambda}_{[nt]}\bigg|\\
&\geq\frac{1}{\sqrt{n}}\bigg|\lambda_{[n\zeta]}-\frac{[n\zeta]}{[nt]}\lambda_{[nt]}\bigg|\geq\frac{1}{\sqrt{n}}\Bigg|\bigg|\lambda_{[n\zeta]}-[n\zeta]\sigma^2\bigg|-\frac{[n\zeta]}{[nt]}\bigg|\lambda_{[nt]}-[nt]\sigma^2\bigg|\Bigg|\\
&\approx\Bigg|\Op(1)-\sqrt{n}\frac{\zeta}{t}\bigg|\frac{\lambda_{[nt]}}{n}-t\sigma^2\bigg|\Bigg|\\
&\approx\Bigg|\Op(1)-2\sqrt{n}\frac{\zeta}{t}\frac{\big|\eta(t)\kappa(t)-\bvarphi(t)^{\top}\bvarphi(t)\big|}{\kappa(t)+\eta(t)+\sqrt{(\kappa(t)-\eta(t))^2+4\bvarphi(t)^{\top}\bvarphi(t)}}\Bigg|\xrightarrow[n\to\infty]{\prob}\infty
\end{align*}
according to the proof of Theorem~\ref{thm:H1} and assumption~\eqref{eq:Cpt}. Similar arguments can be applied in the case~$t<\zeta$ and the convergence holds uniformly for all~$t$ outside any $\epsilon$-neighborhood of~$\zeta$. It follows that for an arbitrary $\epsilon>0$,
\[
\max_{k: |k-\tau|\geq n\epsilon}\frac{|N_n(k)|}{D_n(k)}=\Op\left(\frac{1}{|\eta(t)\kappa(t)-\bvarphi(t)^{\top}\bvarphi(t)|\sqrt{n}}\right).
\]
Now, let us chose a~sequence $d_n\to 0$ with $d_n|\eta(t)\kappa(t)-\bvarphi(t)^{\top}\bvarphi(t)|\sqrt{n}\to\infty$. Then, for any $\epsilon>0$,
\begin{equation*}
\P[|\hat{\tau}/n-\zeta|>\epsilon]\leq \P[|N_n(\tau)/D_n(\tau)|<d_n]+\P\bigg[\max_{k: |k-\tau|\geq n\epsilon}|N_n(k)/D_n(k)|>d_n\bigg]\xrightarrow{n\to\infty}0.
\end{equation*}
\end{proof}
\subsection*{Acknowledgements}
The research of Michal Pe\v{s}ta was supported by the Czech Science Foundation project GA\v{C}R No.~18-01781Y.
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\begin{document}
\theoremstyle{definition}
\newtheorem{Definition}{Definition}[section]
\newtheorem*{Definitionx}{Definition}
\newtheorem{Convention}{Convention}[section]
\newtheorem{Notation}[Definition]{Notation}
\newtheorem{Construction}{Construction}[section]
\newtheorem{Example}[Definition]{Example}
\newtheorem{Examples}[Definition]{Examples}
\newtheorem{Remark}[Definition]{Remark}
\newtheorem*{Remarkx}{Remark}
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\author{Gebhard Martin}
\address{\hfill \newline
Mathematisches Institut \newline
Universit\"at Bonn \newline
Endenicher Allee 60 \newline
53115 Bonn \newline
Germany}
\email{gmartin@math.uni-bonn.de}
\title{Infinitesimal automorphisms of algebraic varieties\\ \hspace{-3mm}and vector fields on elliptic Surfaces}
\date{\today}
\subjclass[2010]{14J27, 14J50, 14G17}
\begin{abstract}
We give several results concerning the connected component $\Aut_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behaviour with respect to birational modifications, normalization, restrictions to closed subschemes and deformations. Then, we apply our results to study the automorphism scheme of not necessarily Jacobian elliptic surfaces $f: X \to C$ over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension $h^0(X,T_X)$ of the space of global vector fields on an elliptic surface $X$ if the generic fiber of $f$ is ordinary or if $f$ admits no multiple fibers, and show that, without these assumptions, the number $h^0(X,T_X)$ can be arbitrarily large for any base curve $C$ and any field of positive characteristic. If $f$ is not isotrivial, we prove that $\Aut_X^0 \cong \mu_{p^n}$ and give a bound on $n$ in terms of the genus of $C$ and the multiplicity of multiple fibers of $f$. As a corollary, we re-prove the non-existence of global vector fields on K3 surfaces and calculate the connected component of the automorphism scheme of a generic supersingular Enriques surface in characteristic $2$. Finally, we present additional results on horizontal and vertical group scheme actions on elliptic surfaces which can be applied to determine $\Aut_X^0$ explicitly in many concrete cases.
\end{abstract}
\maketitle
\vspace{-5mm}
{\bf Contents}
\begin{itemize}
\item[\S \ref{introduction}]
Introduction \hfill \pageref{introduction}
\item[\S \ref{preliminaries}]
Generalities \hfill \pageref{preliminaries}
\item[\S \ref{vertical}]
Vertical automorphisms \hfill \pageref{vertical}
\item[\S \ref{horizontal}]
Horizontal automorphisms \hfill \pageref{horizontal}
\item[\S \ref{examples}]
Examples \hfill \pageref{examples}
\item[\S \ref{proofs}]
Proofs of the main theorems \hfill \pageref{proofs}
\item[\S \ref{appendix}]
Appendix: Some quadratic twists \hfill \pageref{appendix}
\end{itemize}
\section{Introduction} \label{introduction}
Let $X$ be a scheme which is proper over a field $k$. In \cite{MatsumuraOort} Matsumura and Oort proved that the automorphism functor $\Aut_X$ of $X$ over $k$ is representable by a group scheme that is locally of finite type over $k$. Its connected component of the identity $\Aut_X^0$ together with its tangent space at the identity $H^0(X,T_X)$, consisting of global vector fields, play a central r\^ole in the deformation theory of $X$. Indeed, if $h^0(X,T_X) = 0$, then the deformation funtor $\Def_X$ of $X$ is prorepresentable and, conversely, if $\Aut_X^0$ is not smooth, then $\Def_X$ can never be prorepresentable. Similarly, if $X$ is a proper variety with $h^0(X,T_X) = 0$ and if a moduli stack $\cal{M}$ parametrizing objects of the same type as $X$ exists, this stack is Deligne--Mumford at the point corresponding to $X$, since the stabilizer of $\cal{M}$ at $X$ is reduced. This leads to the following geometric question.
\begin{Questionx}[{\bf A}]\label{QuestionA}
Let $X$ be a proper scheme over $k$. What is the dimension of $H^0(X,T_X)$?
\end{Questionx}
On the other hand, it was observed in \cite{AbramovichOlssonVistoli} and \cite{Alper} that Artin stacks with finite linearly reductive stabilizers behave better than general Deligne--Mumford stacks in many ways; for example, they are \'etale locally quotient stacks by finite and linearly reductive group schemes (see \cite[Theorem 3.2]{AbramovichOlssonVistoli}). Since linearly reductive group schemes may very well be connected, a first step towards checking whether $\Aut_X$ is linearly reductive is to check it for $\Aut_X^0$.
\begin{Questionx}[{\bf B}]\label{QuestionB}
When is $\Aut_X^0$ linearly reductive?
\end{Questionx}
Finally, to better understand how close $\cal{M}$ is to being a scheme \'etale locally at $X$, we can ask for the size of $\Aut_X^0$.
\begin{Questionx}[{\bf C}]\label{QuestionC}
If $\Aut_X^0$ is finite, what is its length?
\end{Questionx}
If $X$ is a smooth projective curve, answers to all three of the above questions are known, since $\Aut_X^0$ is always smooth in this case and the automorphism groups of $X$ are well-known. More precisely, we have $\Aut_{\PP^1} \cong {\rm PGL}_2$ and $\Aut_E^0 \cong E$ for an elliptic curve $E$ and if $X$ has higher genus, then $\Aut_X^0$ is trivial. However, already for singular curves or smooth projective surfaces, the three Questions (A), (B), and (C) are wide open. In recent years, however, some structural results in the case of surfaces of general type have been obtained by Tziolas in \cite{Tziolas1} and \cite{Tziolas2}.
The purpose of this paper is to give answers to Questions (A), (B) and (C) for elliptic surfaces over an algebraically closed field $k$ of arbitrary characteristic. Before we start explaining our setup, let us remark that Question (A) for elliptic surfaces without multiple fibers has been studied by Rudakov and Shafarevich \cite{RudakovShafarevich} and the first proof of the non-existence of global vector fields on K3 surfaces is a corollary of their work. Unfortunately, it turns out that Lemma $3$, Lemma $4$, and, as a result, also Theorem $6$ in \cite{RudakovShafarevich} are false as they are stated there. Some of these issues were also addressed in \cite{RudaShafaVector}, but the classification of counterexamples stated there is incomplete. In Section \ref{examples}, we give counterexamples to these claims and complete their classification of counterexamples begun in \cite{RudaShafaVector}. As a special case of our analysis, we will recover a modified version of \cite[Theorem 6]{RudakovShafarevich} in Theorem (D), which gives a characterization of elliptic surfaces with vector fields and without multiple fibers.
Let $k$ be an algebraically closed field of characteristic $\Char(k) = p \geq 0$. Let $f: X \to C$ be an elliptic surface, that is, $X$ is a smooth projective surface, $C$ is a smooth projective curve and $f$ is a proper morphism such that $f_*\OO_X = \OO_C$, almost all fibers of $f$ are smooth curves of genus one and there are no $(-1)$-curves in the fibers of $f$. By Blanchard's Lemma (see \cite[Theorem 7.2.1]{Brion}), there is a natural morphism of group schemes $f_*: \Aut_X^0 \to \Aut_C^0$. We say that $\Ker(f_*)$ is the group scheme of \emph{vertical automorphisms} and $\im(f_*)$ is the group scheme of \emph{horizontal automorphisms}.
In characteristic $0$, the structure of elliptic surfaces with non-trivial $\Aut_X^0$ is simple and well-known: If $\Aut_X^0$ is non-trivial, then either $X$ is ruled or, after a finite base change $C' \to C$, it becomes isomorphic to the trivial elliptic surface $F \times C'$ where $F$ is a general fiber of $f$. We leave it to the reader to check that the same conclusion follows from our results in arbitrary characteristic under the stronger assumption $\dim \Aut_X^0 > 0$ (see also the very recent preprint by Fong \cite{Fong}).
Recall that an elliptic surface is called \emph{isotrivial} if all smooth fibers of $f$ are isomorphic, or equivalently, if the $j$-map of $f$ is constant.
In the following, in Theorems (A), (B), and (C), we will give a summary of our answers to Questions (A), (B), and (C). We refer the reader to Section \ref{vertical} and Section \ref{horizontal} for more refined and more general results on the structure of $\Ker(f_*)$ and $\im(f_*)$, such as possible fiber types, further information on multiplicities of fibers, as well as geometric restrictions on multisections for elliptic surfaces with non-trivial $\Aut_X^0$. The proofs of the following results can be found in Section \ref{proofs}, where we combine our results on vertical and horizontal automorphisms.
\begin{Theoremx}[{\bf A}]
Let $f: X \to C$ be an elliptic surface. Then, the following hold:
\begin{enumerate}[(i)]
\item If $f$ is not isotrivial, then $h^0(X,T_X) \leq 1$.
\item If the generic fiber of $f$ is ordinary or $f$ admits no multiple fibers, then $h^0(X,T_X) \leq 4$. If additionally $h^0(X,T_X) \geq 2$, then one of the following holds:
\begin{enumerate}[(1)]
\item $X$ is ruled over an elliptic curve.
\item $X$ is an Abelian surface isogeneous to a product of elliptic curves.
\item $X$ is a bielliptic surface with $\omega_X \cong \OO_X$. These surfaces exist if and only if $p \in \{2,3\}$.
\end{enumerate}
\item For every field $K$ of characteristic $\Char(K) > 0$, for every smooth projective curve $C$ over $K$ and for every $n \geq 0$, there is an elliptic surface $f: X \to C$ with $h^0(X,T_X) \geq n$.
\end{enumerate}
\end{Theoremx}
In particular, the elliptic surfaces appearing in Theorem (A) (iii) are isotrivial with supersingular generic fiber and they admit multiple fibers. The relevant examples can be found in Example \ref{importantexample}. In the non-isotrivial cases, the following theorems give a description of $\Aut_X^0$ as well as a bound on its length that depends on the number $h_p$, which is defined in the discussion before Corollary \ref{Igusainequality} and which coincides with the number of supersingular $j$-invariants over $k$ if $p \neq 2,3$.
\begin{Theoremx}[{\bf B}]
Let $f: X \to C$ be a non-isotrivial elliptic surface. Then, $\Aut_X^0 \cong \mu_{p^n}$ for some $n \geq 0$. In particular, $\Aut_X^0$ is linearly reductive.
\end{Theoremx}
\begin{Theoremx}[{\bf C}]
Let $f: X \to C$ be a non-isotrivial elliptic surface with $\Aut_X^0 \cong \mu_{p^n}$. Then,
$$
\frac{1}{48}(p-1)(p^{2n-1} - 12 p^{n-1} + 1) + 1 - \frac{h_p}{2} \leq g(C).
$$
If, additionally,
\begin{enumerate}[(a)]
\item $p^n \geq 4$, or
\item $C \not \cong \PP^1$, or
\item $p^n = 3$ and the singular fibers of $f$ are not of type $({\rm II},{\rm I}_{3^{2k}})$ or $({\rm II},{\rm I}^*_{3^{2k-1}})$ for any $k \geq 1$, or
\item $p^n = 2$ and the singular fibers of $f$ are not of type $({\rm II},{\rm I}_{2^{2k+1}})$ or $({\rm III},{\rm I}_{2^{2k+1}})$ for any $k \geq 1$,
\end{enumerate}
then $\Ker(f_*) \cong \Aut_X^0$ and all additive or supersingular fibers of $f$ are multiple fibers with multiplicity divisible by $p^n$.
\end{Theoremx}
The exceptions in Theorem (C) (c) and (d) occur for every $k \geq 1$ (see Example \ref{caseb3} and Example \ref{caseb2}). Our Examples \ref{caseb3} and Example \ref{caseb2} are elliptic surfaces with a section and therefore they are counterexamples to \cite[Theorem 6]{RudakovShafarevich} in characteristic $2$ and $3$. Three of these four families of counterexamples were already exhibited in \cite{RudaShafaVector}.
In the other characteristics, our analysis recovers \cite[Theorem 6]{RudakovShafarevich}. More precisely, we obtain the following theorem.
\begin{Theoremx}[{\bf D}]
Let $f: X \to C$ be an elliptic surface without multiple fibers and such that $h^0(X,T_X) \neq 0$. Then, one of the following holds:
\begin{enumerate}[(i)]
\item $f$ is isotrivial and $c_2(X) = 0$.
\item $f$ is Jacobian and isotrivial with two singular fibers, $X$ is rational, and $C \cong \PP^1$.
\item $p = 3$, $C \cong \PP^1$, and the singular fibers of $f$ are of type $({\rm II},{\rm I}_{3^{2k}})$ or $({\rm II},{\rm I}^*_{3^{2k-1}})$ for some $k \geq 1$.
\item $p = 2$, $C \cong \PP^1$, and the singular fibers of $f$ are of type $({\rm II},{\rm I}_{2^{2k+1}})$ or $({\rm III},{\rm I}_{2^{2k+1}})$ for some $k \geq 1$.
\item $p \in \{2,3\}$, $C \cong \PP^1$, and $f$ is isotrivial with a unique singular fiber.
\end{enumerate}
\end{Theoremx}
One can classify the possible singular fibers of $f$ in Theorem (D) (ii). They are precisely the types described in Lemma \ref{twofibers} and Lemma \ref{twofibers23}.
All cases in Theorem (D) actually occur and we give the corresponding examples in Section \ref{sectionhorizontal}. Theorem (D) has the following well-known consequence (compare \cite[Theorem 7]{RudakovShafarevich}).
\begin{Corollary}\label{K3corollary}
There are no global regular vector fields on K3 surfaces.
\end{Corollary}
Our proof follows the strategy of Rudakov and Shafarevich and by using Theorem (D) we can circumvent the issues with Lemma 3 and Lemma 4 in \cite{RudakovShafarevich}. In characteristic $2$ and $3$, the proof builds on the fact that supersingular K3 surfaces admit an elliptic fibration with at least two singular fibers (see \cite[p.1502]{RudakovShafarevich2}). Let us also remark that there is an independent proof of Corollary \ref{K3corollary} due to Nygaard in \cite{Nygaard}.
Since our analysis of elliptic surfaces does not assume the existence of a section, we can also apply it to study the automorphism group scheme of Enriques surfaces in characteristic $2$. For example, we prove the following result in Example \ref{multiplicativeEnriques}.
\begin{Corollary}\label{Enriquescorollary}
Let $X$ be a generic supersingular Enriques surface in characteristic $2$. Then, $\Aut_X^0 \cong \mu_2$.
\end{Corollary}
Using the more refined results we give in Section \ref{vertical} and Section \ref{horizontal}, it is possible to determine the group scheme $\Aut_X^0$ in many concrete cases. For example, as an extension of Corollary \ref{Enriquescorollary}, we will use our results to calculate the connected components of the identity of the automorphism schemes of elliptic Enriques surfaces in characteristic $2$ in an upcoming article.
The outline of this article is as follows: In Section \ref{preliminaries}, we will give several general results on automorphism schemes of proper schemes, such as the behaviour under birational modifications, the relation to deformation theory, and a fixed point formula for actions of connected linearly reductive group schemes. This part of the article applies to arbitrary proper schemes over arbitrary fields and we hope that our results will help to answer Questions (A), (B), and (C) for more general classes of proper varieties. Then, we give some background on elliptic surfaces and recall the structure of the automorphism scheme of curves of genus one. In Section \ref{vertical}, we study the group scheme $\Ker(f_*)$ of vertical automorphisms of an elliptic surface and in Section \ref{horizontal}, we describe the group scheme $\im(f_*)$ of horizontal automorphisms. Before deducing our main results from this in Section \ref{proofs}, we give several examples in Section \ref{examples}, illustrating the different phenomena that occur in the context of automorphism schemes of elliptic surfaces.
\vspace{3mm}
\noindent
\textbf{Acknowledgements:}
The author gratefully acknowledges funding from the DFG under research grant MA 8510/1-1.
Also, the author would like to thank the Department of Mathematics at the University of Utah for its hospitality while this article was written. I would like to thank Michel Brion for pointing me towards Blanchard's Lemma and Claudia Stadlmayr for fruitful discussions. Finally, I would like to thank Fabio Bernasconi, Michel Brion, Leo Herr, and Claudia Stadlmayr for helpful remarks on an earlier version of this article.
\section{Generalities} \label{preliminaries}
\subsection{Generalities on group scheme actions}
In this section, we will be working over an arbitrary field $K$ and all schemes we consider will be of finite type over $K$. The following theorem of Matsumura and Oort \cite{MatsumuraOort} was mentioned in the introduction and proves the existence of our main object of interest.
\begin{Theorem} \label{MatsumuraOort}
Let $X$ be a proper scheme over $K$. Then, the functor
\begin{eqnarray*}
\Aut_{X/K}: &(Sch/K)^{op} &\to (Sets) \\
&S &\mapsto \Aut(X \times S \to S),
\end{eqnarray*}
where $\Aut(X \times S \to S)$ is the group of automorphisms of $X \times S$ over $S$, is representable by a group scheme $\Aut_{X/K}$ which is locally of finite type over $K$. In particular, its connected component of the identity $\Aut_X^0$ is of finite type over $K$.
\end{Theorem}
\begin{Remark}
More generally, one can prove the existence of a scheme of automorphisms $\Aut_{X/C}$ for a proper and flat morphism $f: X \to C$ of schemes, where $C$ is a normal and locally Noetherian scheme of dimension at most $1$, as follows: The relative Hilbert functor for $X \times_C X \to C$ is representable by a separated algebraic space $\mathcal{H}$ which is locally of finite presentation over $C$ (by \cite[Section 6]{ArtinI}), the functor $\Aut_{X/C}$ is an open subfunctor of $\mathcal{H}$ (by \cite[Proposition 4.6.7 (ii)]{Ega3}), and every separated algebraic group space that is locally of finite type over $C$ is in fact a group scheme over $C$ (by \cite[Th\'eor\`eme (3.3.1).]{Raynaud}). This also shows that $\Aut_{X/C}$ exists as an algebraic group space under much weaker assumptions on $C$.
\end{Remark}
If the base scheme $C$ is clear from the context, we will simply write $\Aut_{X}$ for the functor of automorphisms of $X$ over $C$. For a scheme $X$ and a closed subscheme $Z \subseteq X$, we let $\Aut_{(Z,X)} \subseteq \Aut_X$ be the subgroup functor of automorphisms of $X$ preserving $Z$. Its $S$-valued points are given by
\begin{eqnarray*}
\Aut_{(Z,X)}(S) = \{ \alpha \in \Aut_X(S) \mid Z \times S = (X \times S) \times_{\alpha,(X \times S)} (Z \times S) \hookrightarrow X \times S\}.
\end{eqnarray*}
That is, we want $Z \times S = (X \times S) \times_{\alpha,(X \times S)} (Z \times S)$ as closed subschemes of $X \times S$. Equivalently, $\Aut_{(Z,X)}$ is the stabilizer of the $K$-valued point corresponding to $Z$ in the Hilbert functor of $X$ over $K$. This second interpretation shows the following.
\begin{Corollary}
If $X$ is proper, then $\Aut_{(Z,X)}$ is a closed subgroup scheme of $\Aut_X$.
\end{Corollary}
\begin{Remark}\label{autzxremark}
If a group scheme $G$ acts on $X$, then the condition that $G \to \Aut_X$ factors through $\Aut_{(Z,X)}$ can be rephrased as $\rho^{-1}I_Z = {\rm pr}_2^{-1}I_Z$, where $\rho: G \times X \to X$ is the action, ${\rm pr}_2$ is the second projection, $I_Z$ is the ideal sheaf of $Z$ in $X$, and ${\rm pr}_2^{-1}I_Z$ and $\rho^{-1}I_Z$ denote the corresponding inverse image ideal sheaves. For more details, see \cite[Section 2]{Fogarty}.
\end{Remark}
\vspace{1mm}
\subsubsection{Equivariant morphisms and birational modifications}
In this section, we will study how group scheme actions behave with respect to (birational) morphisms. First, we note that the construction of $\Aut_{(Z,X)}$ is compatible with scheme theoretic images in the following sense.
\begin{Lemma}\label{image}
Let $Z \subseteq X$ be a closed subscheme of a scheme $X$ and $G$ a subgroup functor of $\Aut_{(Z,X)}$. Let $f: X \to Y$ be a $G$-equivariant morphism. Then, the induced morphism $G \to \Aut_Y$ factors through $\Aut_{(f(Z),Y)}$, where $f(Z)$ is the scheme theoretic image of $Z$ under $f$.
\end{Lemma}
\prf
It suffices to observe that for every $K$-scheme $S$ and $f_S: X \times S \to Y \times S$, the scheme theoretic images satisfy $f_S(Z \times S) = f(Z) \times S$. This is true by \cite[Lemma 14.6]{GoertzWedhorn}.
\qed
\vspace{3mm}
In general, not every $G$-action on $X$ descends to $Y$. An important case where we get an induced $G$-action is given by Blanchard's Lemma (see e.g. \cite[Theorem 7.2.1]{Brion}):
\begin{Theorem}[Blanchard's Lemma]\label{Blanchard}
Let $f: X \to Y$ be a proper morphism of schemes with $f_* \OO_X \cong \OO_Y$ and let $G$ be a connected group scheme acting on $X$. Then, the following hold:
\begin{enumerate}[(i)]
\item There is a unique $G$-action on $Y$ such that $f$ is $G$-equivariant.
\item If $X$ and $Y$ are proper, there is a natural homomorphism $f_*: \Aut_X^0 \to \Aut_Y^0$.
\item If, additionally, $f$ is birational, then $f_*: \Aut_X^0 \to \Aut_Y^0$ is a closed immersion.
\end{enumerate}
\end{Theorem}
Alternatively, we can start with a group scheme action on $Y$ and ask whether it lifts along a birational modification $f: X \to Y$ to a compatible action on $X$.
\begin{Proposition} \label{blowup}
Let $Z \subseteq X$ be a closed subscheme of a scheme $X$ and let $\pi: {\rm Bl}_Z(X) \to X$ be the blow-up of $X$ in $Z$. Let $G$ be a group scheme acting on $X$. If $G \to \Aut_X$ factors through $\Aut_{(Z,X)}$, then the $G$-action lifts to ${\rm Bl}_Z(X)$. The converse holds if $G \to \Aut_{{\rm Bl}_Z(X)}$ factors through $\Aut_{(\pi^{-1}(Z),{\rm Bl}_Z(X))}$.
\end{Proposition}
\prf
Since the action map $\rho: G \times X \to X$ is flat and blow-up commutes with flat base-change, we have the following diagram of solid arrows with cartesian square
$$
\xymatrix{
G \times {\rm Bl}_Z(X) \ar[dr]^{{\rm id} \times \pi} \ar@{.>}[r]^(.4){\iota} & Y := {\rm Proj} \bigoplus\limits_{i = 0}^{\infty} (\rho^{-1}I_Z)^i \ar[r]^(.7){\rho'} \ar[d]^{\pi'} & {\rm Bl}_Z(X) \ar[d]^\pi \\
& G \times X \ar[r]^\rho & X
}
$$
\noindent and we are asking for the existence of the map $\iota$ such that $\rho' \circ \iota$ is an action of $G$ and such that the above diagram commutes. If $G \to \Aut_X$ factors through $\Aut_{(Z,X)}$, then $Y \cong G \times {\rm Bl}_Z(X)$ over $G \times X$ by Remark \ref{autzxremark} and we get the desired map $\iota$.
The converse statement follows from Lemma \ref{image}, since $Z$ is the scheme-theoretic image of $\pi^{-1}(Z)$.
\qed
\vspace{1mm}
\begin{Remark}
In Corollary \ref{criteria}, we will see that if $G$ is connected and the normal bundle $N_{E/{\rm Bl}_Z(X)}$ of $E := \pi^{-1}(Z)$ in ${\rm Bl}_Z(X)$ satisfies $h^0(E,N_{E/{\rm Bl}_Z(X)}) = 0$, then $G$ preserves $E$. In particular, this holds if $\pi$ is the contraction of a negative definite configuration of curves on a smooth surface.
\end{Remark}
Let $\nu: \widetilde{X} \to X$ be a finite and birational morphism. Then, the \emph{conductor ideal} $I_\nu$ of $\nu$ is defined as
$$
I_\nu = \Hom_{\OO_X}(\nu_* \OO_{\widetilde X}, \OO_X) = {\rm Ann}_{\OO_X}(\nu_* \OO_{\widetilde X}/\OO_X)\subseteq \OO_X.
$$
We let $C_\nu \subseteq X$ be the closed subscheme defined by $I_\nu$ and call it the \emph{conductor} of $\nu$. Then, the locus where $\nu$ is not an isomorphism is precisely $C_\nu$.
If both $\widetilde{X}$ and $X$ are Gorenstein, it follows from relative duality \cite{Kleiman} that $I_\nu \cdot \omega_X = \nu_* (\omega_{\widetilde{X}})$. In particular, $I_\nu$ is reflexive and thus $C_\nu$ is a generalized divisor in the sense of \cite{Hartshornegeneralized}. The fact that $\nu^{-1} I_\nu$ is locally principal can be used to show that $\nu$ is the blow-up of $I_\nu$ (see \cite[Proposition 2.9]{Piene}). Using Proposition \ref{blowup}, we obtain the following proposition.
\begin{Proposition}\label{normalization}
Let $\nu: \widetilde{X} \to X$ be a finite and birational morphism between Gorenstein schemes. Then, $\widetilde{X} = {\rm Bl}_{C_\nu}(X)$ and thus a $G$-action on $X$ lifts to $\widetilde{X}$ if and only if $G \to \Aut_X$ factors through $\Aut_{(C_\nu,X)}$.
\end{Proposition}
\prf
By Proposition \ref{blowup}, the $G$-action on $X$ lifts to $\widetilde{X}$ if $G \to \Aut_X$ factors through $\Aut_{(C_\nu,X)}$.
For the converse, note that, because $\rho$ and ${\rm pr_2}: G \times X \to X$ are flat and ${\rm id} \times \nu$ is the base change of $\nu$ along both $\rho$ and ${\rm pr_2}$, we can use the fact that cohomology, annihilators, and quotients commute with flat base change to obtain
\begin{eqnarray*}
{\rm pr_2}^{-1}(I_{C_\nu}) &=& {\rm pr_2}^{-1}({\rm Ann}_{\OO_X}(\nu_* \OO_{\widetilde X}/\OO_X))
= {\rm Ann}_{\OO_{G \times X}}((({\rm id} \times \nu)_*\OO_{G \times {\widetilde X}})/\OO_{G \times X}) \\
&=& \rho^{-1}({\rm Ann}_{\OO_X}(\nu_* \OO_{\widetilde X}/\OO_X)) = \rho^{-1}(I_{C_\nu}).
\end{eqnarray*}
Hence, by Remark \ref{autzxremark}, the $G$-action on $X$ preserves $C_{\nu}$.
\qed
\begin{Remark}
In particular, if $X$ is a reduced proper scheme over $K$ such that $X$ and its normalization $\widetilde{X}$ are Gorenstein, the scheme of automorphisms of $X$ that lift to $\widetilde{X}$ is precisely the stabilizer of the conductor. This seems to be the "general principle" mentioned in the calculation of the automorphism scheme of a cuspidal plane cubic curve in \cite[p. 213]{BombieriMumford3}.
\end{Remark}
\vspace{0.5mm}
\subsubsection{Fixed points}
Recall that if a group scheme $G$ acts on a scheme $X$, then the subfunctor of fixed points for this action is defined as
$$
X^G(S) := \{ x \in X(S) \mid g(x_T) = x_T \text{ for all }S\text{-schemes } T \text{ and } g \in G(T)\}
$$
By \cite[Theorem 2.3]{Fogarty}, $X^G$ is representable by a closed subscheme of $X$. We have the following lemma, whose proof is the same as the one of the corresponding statement for actions of abstract groups and thus left to the reader.
\begin{Lemma} \label{fixedofsubscheme}
Let $X$ be a scheme and let $H \subseteq G$ be a normal subgroup scheme of a group scheme $G$. Assume that $G$ acts on $X$. Then, $G$ acts on the fixed locus of the induced $H$-action on $X$, that is, $G \to \Aut_X$ factors through $\Aut_{(X^H,X)}$.
\end{Lemma}
This simple observation can sometimes be used to obtain information about fixed points of $G$ via the following corollary.
\begin{Corollary} \label{isolatedefixedpoints}
Let $X$ be a scheme and let $H \subseteq G$ be a normal subgroup scheme of a connected group scheme $G$ that acts on $X$. Assume that $X^H$ admits a connected component $P$ isomorphic to $\Spec K$. Then, $P \in X^G$
\end{Corollary}
\prf
By Lemma \ref{fixedofsubscheme}, $G$ acts on $X^H$ and since $G$ is connected, this action preserves the connected components of $X^H$. By our assumption, the connected component of $X^H$ containing $P$ is isomorphic to $\Spec K$. Therefore, the induced $G$-action on $P$ is trivial, hence $P \in X^G$.
\qed
\vspace{5mm}
\subsubsection{Some deformation theory}
In this section, we will use the deformation theory of a closed subscheme $Z$ of a scheme $X$ to obtain information about the functor ${\rm Aut}_{(Z,X)}$.
For the necessary background on deformation theory, we refer the reader to \cite{Sernesi}.
We fix the following notation.
\begin{Notation}\label{Deformationnotation}
Let $Z \subseteq X$ be a closed subscheme of a scheme $X$.
\begin{itemize}
\item $\Def_Z$ is the functor of deformations of $Z$.
\item ${\rm Def}_{Z/X}$ is the functor of deformations of $Z$ in $X$.
\item ${\rm Def}'_{Z/X}$ is the subfunctor of deformations of $Z$ in $X$ mapping to the trivial deformation of $Z$ via the forgetful map $F: {\rm Def}_{Z/X} \to \Def_Z$.
\item $\widehat{\Aut}_{X}$ is the restriction of $\Aut_X$ to the category ${\rm Art}_K^{\rm op}$ of Artinian local $K$-schemes with residue field $K$ whose closed points map to ${\rm id}_X$. For every such $S \in {\rm Art}_K^{\rm op}$, there is a natural map $\widehat{\Aut}_{X}(S) \to {\rm Def}'_{Z/X}(S)$ given by $\alpha \mapsto (X \times S) \times_{\alpha,(X \times S)} (Z \times S)$.
\end{itemize}
\end{Notation}
We warn the reader that, even if $X$ is smooth and proper, the functor $\Def'_{Z/X}$ does not have a hull in general. The reason is that if $H$ is a hull for $\Def_Z$ and $\{\ast\} \to \Def_Z$ is the morphism that maps $S$ to the trivial deformation, then $H' := \{\ast\} \times_{\Def_Z} H \to H$ is a monomorphism of functors of Artin rings but $H'$ may not be prorepesentable. If $H'$ is prorepresentable, we say that $H$ is a \emph{good} hull for $\Def_Z$.
Note that if $\Def_Z$ is prorepresentable, then it admits a good hull.
\begin{Lemma}\label{prorep}
Let $Z \subseteq X$ be a closed subscheme of a proper scheme $X$. If $\Def_Z$ admits a good hull, then $\Def_{Z/X}'$ is prorepresentable.
\end{Lemma}
\prf
By definition, $\Def_{Z/X}'$ is the fiber product $\{\ast\} \times_{\Def_Z} \Def_{Z/X}$. If $\Def_Z$ admits a good hull $H$ with $H' := \{\ast\} \times_{\Def_Z} H$, then $\Def_{Z/X}' = H' \times_H \Def_{Z/X}$ and thus $\Def_{Z/X}'$ is prorepresentable.
\qed
\begin{Example} \label{cuspexample}
If $Z$ is a cuspidal plane cubic over an algebraically closed field $k$ of characteristic $p \geq 0$, then $\Def_Z$ admits a good hull if and only if $p \not\in \{2,3\}$ (see \cite[p.202]{BombieriMumford3}).
\end{Example}
\begin{Remark}
In fact, using Schlessinger's criteria \cite{Schlessinger}, one can prove that $\Def'_{Z/X}$ is prorepresentable if and only if it has a hull. Note, however, that $\Def'_{Z/X}$ may be prorepresentable even if $\Def_Z$ does not admit a good hull, for example if $\Def'_{Z/X} = \Def_{Z/X}$.
\end{Remark}
The reason why we care about the functor $\Def_{Z/X}'$ is that we can use it to check whether the inclusion $\Aut_{(Z,X)}^0 \subseteq \Aut_X^0$ is an equality.
\begin{Lemma} \label{restriction}
Let $X$ be a proper scheme and $Z \subseteq X$ a closed subscheme. The natural map $\widehat{\Aut}_{X} \to {\rm Def}'_{Z/X}$ is constant if and only if $\Aut_{(Z,X)}^0 = \Aut_X^0$.
\end{Lemma}
\prf
Since $\Aut_{(Z,X)}^0 \to \Aut_X^0$ is a closed immersion and both sides are connected, the equality $\Aut_{(Z,X)}^0 = \Aut_X^0$ holds if and only if $\widehat{\Aut}_{X} = \widehat{\Aut}_{(Z,X)}$.
From the definitions, we see that $\widehat{\Aut}_{(Z,X)}$ is the fiber of $\widehat{\Aut}_{X} \to {\rm Def}'_{Z/X}$ over the trivial deformation of $Z$ in $X$. Thus, $\widehat{\Aut}_{X} = \widehat{\Aut}_{(Z,X)}$ if and only if $\widehat{\Aut}_{X} \to {\rm Def}'_{Z/X}$ is constant.
\qed
\vspace{5mm}
Now, we want to understand the tangent spaces of the functors recalled in Notation \ref{Deformationnotation}. To this end, we define a subsheaf $T_X\langle Z \rangle \subset T_X$ of the tangent sheaf $T_X$ of $X$ via
$$
T_X\langle Z \rangle(U) = \{ D \in T_X(U) \mid D(I_Z(U)) \subseteq I_Z(U) \}.
$$
We recall that $N_{Z/X}$ denotes the normal sheaf of $Z$ in $X$ and that $K[\epsilon] := K[x]/x^2$ is the ring of dual numbers. Then, with a slight abuse of notation, we get the following well-known identifications of the relevant tangent spaces:
\begin{itemize}
\item $\widehat{\Aut}_{(Z,X)}(K[\epsilon]) = H^0(X,T_X\langle Z \rangle)$.
\item $\widehat{\Aut}_{X}(K[\epsilon]) = H^0(X,T_X)$.
\item $\Def_{Z/X}(K[\epsilon]) = H^0(Z,N_{Z/X})$.
\item $\Def'_{Z/X}(K[\epsilon]) = \ker(H^0(Z,N_{Z/X}) \to \Def_Z(k[\epsilon]))$.
\item If $Z$ is reduced, then $\Def_Z(K[\epsilon]) = {\rm Ext}^1(\Omega_Z,\OO_Z)$ and if $Z$ is smooth, then ${\rm Ext}^1(\Omega_Z,\OO_Z) = H^1(Z,T_Z)$. In these cases, the differential of the forgetful map $F: \Def_{Z/X} \to \Def_Z$ is induced by the conormal sequence.
\end{itemize}
\begin{Corollary} \label{criteria}
If $\Def'_{Z/X}$ is trivial and $X$ is proper, then $\Aut_{(Z,X)}^0 = \Aut_X^0$. This holds in each of the following cases:
\begin{enumerate}[(a)]
\item $H^0(Z,N_{Z/X}) = 0$.
\item $Z$ is reduced, $\Def_{Z/X}'$ is prorepresentable and $H^0(Z,N_{Z/X}) \to {\rm Ext}^1(\Omega_Z,\OO_Z)$ is injective.
\item $X$ is smooth in a neighborhood of $Z$ and $Z$ is a reduced, connected and singular effective divisor on $X$ with $N_{Z/X} = \OO_Z$ such that $\Def_{Z/X}'$ is prorepresentable.
\end{enumerate}
\end{Corollary}
\prf
First, observe that Lemma \ref{restriction} shows that $\Aut_{(Z,X)}^0 = \Aut_X^0$ holds if $\Def'_{Z/X}$ is trivial, so we have to check that $\Def'_{Z/X}$ is trivial under any of the stated conditions.
Since $\Def_{Z/X}$ is prorepresentable, it is trivial as soon as $H^0(Z,N_{Z/X}) = 0$. As $\Def'_{Z/X}$ is a subfunctor of $\Def_{Z/X}$, it is also trivial in this case. This is Claim (a).
As for Claim (b), since $\Def'_{Z/X}$ is prorepresentable, it suffices to check that $\Def'_{Z/X}(K[\epsilon])$ is trivial. But by the facts recalled above and since $Z$ is reduced, we have $\Def'_{Z/X}(K[\epsilon]) = \ker(H^0(Z,N_{Z/X}) \to {\rm Ext}^1(\Omega_Z,\OO_Z))$.
To prove Claim (c), we thus have to prove that $H^0(Z,N_{Z/X}) \to {\rm Ext}^1(\Omega_Z,\OO_Z)$ is injective. Since $Z$ is a reduced effective Cartier divisor, we have the short exact conormal sequence
$$
0 \to \OO_Z(Z) \to \Omega_X|_Z \to \Omega_Z \to 0.
$$
Applying $\Hom(-,\OO_Z)$, we obtain
$$
H^0(Z,T_X|_Z) \to H^0(Z,N_{Z/X}) = \Hom(\OO_Z(Z),\OO_Z) \overset{f}{\to} \Ext^1(\Omega_Z,\OO_Z).
$$
The map $f$ associates to a morphism $\varphi: \OO_Z(Z) \to \OO_Z$ the pushout of the conormal sequence along $\varphi$. Since $Z$ is connected, the space $H^0(Z,N_{Z/X}) = H^0(Z,\OO_Z)$ is $1$-dimensional, so that $f$ is either trivial or injective.
Suppose that $f$ is trivial, that is, that $f({\rm id}) = 0$. This means that the conormal sequence splits. Thus, $\Omega_Z$ is locally free, being a direct summand of the locally free sheaf $\Omega_X|_Z$ and the rank of $\Omega_Z$ is $\dim Z = \dim X - 1$. Therefore, $Z$ is smooth, contradicting our assumption that $Z$ is singular. Hence, $f$ is injective and Claim (c) follows from Claim (b).
\qed
\vspace{3mm}
\begin{Remark} \label{criteriaremark}
Without the assumption on the prorepresentabilty of $\Def_{Z/X}'$, the proof of Corollary \ref{criteria} (b) and (c) shows that $H^0(X,T_X\langle Z \rangle) = H^0(X,T_X)$. Indeed, the map $\varphi: H^0(X,T_X) \to H^0(Z,N_{Z/X})$ factors through $H^0(Z,T_X|_Z)$, so the above proof shows that $\varphi$ is trivial, hence $H^0(X,T_X\langle Z \rangle) = \Ker(\varphi) = H^0(X,T_X)$. In particular, even if $\Def_{Z/X}'$ is not prorepresentable, the functors $\Aut_{(Z,X)}$ and $\Aut_{X}$ have the same tangent space at the identity in case (b) and (c). This implies, for example, that for every connected subgroup scheme $G$ of $\Aut_X$ the intersection $G \cap \Aut_{(Z,X)}$ is non-trivial.
\end{Remark}
Another case where $\Aut_{(Z,X)}$ and $\Aut_X$ have the same tangent space is if $Z$ is given by a Frobenius power of an ideal. Recall that if $I \subseteq \OO_X$ is an ideal sheaf, then its \emph{Frobenius power} $I^{[p]}$ is the ideal sheaf which is locally generated by the $p$-th powers of generators of $I$. If $I$ is locally principal, then $I^{[p]} = I^p$.
\begin{Lemma} \label{Frobeniuspower}
Let $Z \subseteq X$ be a closed subscheme of a scheme $X$ and let $Z^{[p]} \subseteq X$ be the closed subscheme defined by $I_Z^{[p]}$. Then, $T_X\langle Z^{[p]} \rangle = T_X$. In particular, $H^0(X,T_X\langle Z^{[p]} \rangle) = H^0(X,T_X)$ holds.
\end{Lemma}
\prf
Let $U \subseteq X$ be an open subset with $I_Z(U) = \langle f_1,\hdots,f_n \rangle$ and $D \in T_X(U)$. Then, using the Leibniz rule, we deduce for arbitrary $a_i \in \OO_X(U)$ that
$$
D(\sum_{i=1}^n a_i f_i^p) = \sum_{i=1}^n f_i^p D(a_i) \in I_{Z^{[p]}}(U).
$$
\vspace{-5mm}
\qed
\vspace{5mm}
\subsubsection{Examples of group schemes and some structure theory}
If $\Char(K) > 0$ and $X$ is a scheme over $K$, we write $X^{(p)}$ for the pullback of $X$ along the $K$-linear Frobenius. For a group scheme $G$ over a field of positive characteristic $K$, the notation $G[F^n]$ denotes the kernel of the $n$-fold $K$-linear Frobenius $F^n: G \to G^{(p^n)}$. If $G$ is finite and connected, then $G[F^n] = G$ for $n \gg 0$. If $k \subseteq K$ is a field extension, we write $G_K$ for $G \times_{\Spec k} \Spec K$. Let us recall $G[F^n]$ for some common group schemes over an algebraically closed field $k$:
\begin{itemize}
\item If $G = \GG_m$, then $G[F^n] =: \mu_{p^n}$.
\item If $G = \GG_a$, then $G[F^n] =: \alpha_{p^n}$.
\item If $E$ is an \emph{ordinary} elliptic curve, then $E[F^n] \cong \mu_{p^n}$.
\item If $E$ a \emph{supersingular} elliptic curve over $k$, then $E[F^n] =: M_n$ and $M_n$ is an $n$-fold non-split extension of $\alpha_p$ by itself.
\end{itemize}
In each case, ${\rm length}(G[F^n]/G[F^{n-1}]) = p$, so the above list is a complete enumeration of all finite connected subgroup schemes of these four group schemes. Next, let us recall some structural results for a group scheme $G$ of finite type over a field $K$.
\begin{Lemma}\label{generalgroupscheme}
Let $G$ be a group scheme of finite type over a field $K$. Then,
\begin{enumerate}[(i)]
\item \emph{(Cartier's Theorem)} If $\Char(K) = 0$, then $G$ is reduced.
\item The connected component of the identity $G^0 \subseteq G$ is a closed subgroup scheme.
\item There is a smallest normal subgroup scheme $H \subseteq G$ such that $G/H$ is affine. This $H$ is smooth, connected, commutative and contained in the center of $G^0$.
\item There is a smallest normal subgroup scheme $H \subseteq G$ such that $G/H$ is proper. This $H$ is affine and connected.
\item There is a smallest normal subgroup scheme $H \subseteq G$ such that $G^{ab} := G/H$ is commutative.
\item If $K$ is perfect, then $G_{red} \subseteq G$ is a closed and smooth subgroup scheme.
\end{enumerate}
\end{Lemma}
\prf
Claim (i) is \cite[VIB.1.6.1.]{SGA3}, claims (ii), (iii), and (iv) can be found in \cite[Theorem 1, Theorem 2]{Brion}, for claim (v) see \cite[Proposition 6.17]{Milne}, and for claim (vi) see \cite[Proposition 2.5.2.]{Brion}.
\qed
\vspace{3mm}
Recall that the \emph{Cartier dual} $G^\vee := \Hom(G,\GG_m)$ of a finite commutative group scheme $G$ is also a finite and commutative group scheme and we have $(G^\vee)^\vee \cong G$. The homomorphism $V: G^{(p)} \to G$ induced by the Frobenius on $G^\vee$ is called \emph{Verschiebung} and denoted by $V$. For the following well-known lemma, we refer the reader to the chapters on finite group schemes in \cite{Waterhouse} or \cite{Milne}.
\begin{Lemma}\label{finitecommgroupscheme}
Let $G$ be a finite and commutative group scheme over a field $K$. Then, the following hold:
\begin{enumerate}[(i)]
\item If $K$ is perfect, then there is a functorial decomposition
$$
G \cong G_{rr} \times G_{rl} \times G_{lr} \times G_{ll}
$$
where $G_{xy}$ is reduced if $x = r$ and connected if $x = l$, and $G_{xy}^\vee$ is reduced if $y = r$ and connected if $y = l$. We say that $G$ is of type $xy$ if $G \cong G_{xy}$ for $x,y \in \{r,l\}$.
\item If $K = k $ is algebraically closed, then
\begin{enumerate}[(1)]
\item $G_{rr}$ is the constant group scheme associated to an abelian group of order prime to $p$,
\item $G_{rl}$ is the constant group scheme associated to an abelian group of $p$-power order,
\item $G_{lr} \cong \prod_{i=1}^m \mu_{p^{n_i}}$ for some $n_i,m \geq 0$,
\item $G_{ll}$ is an iterated extension of $\alpha_p$ by itself. Moreover, $G_{ll} \cong \alpha_p^r$ for some $r \geq 0$ if and only if both $F$ and $V$ are trivial on $G_{ll}$.
\end{enumerate}
\end{enumerate}
\end{Lemma}
\begin{Lemma}\label{homsandexts}
Let $G$ and $H$ be finite and commutative group schemes over an algebraically closed field $k$. Then, the following hold:
\begin{enumerate}[(i)]
\item If $G$ is of type $xy$ and $H$ is of type $x'y'$ and if there is a non-trivial homomorphism $G \to H$, then $(x,y) = (x',y')$.
\item If $M$ is an extension of $G$ by $H$, then we have the following:
\begin{enumerate}[(1)]
\item If $G$ and $H$ are of type $lr$, then $M$ is commutative of type $lr$.
\item If $G$ is of type $ll$ and $H$ is of type $lr$, then $M \cong H \times G$.
\item If $G$ is of type $lr$ and $H$ is of type $ll$, then $M \cong H \rtimes G$.
\end{enumerate}
\end{enumerate}
\end{Lemma}
\prf
Claim (i) follows from functoriality of the canonical decomposition of a finite commutative group scheme. Claim (ii) (1) is \cite[Theorem 15.39.]{Milne}, the splitting in Claim (ii) (2) is \cite[Theorem 15.37.]{Milne} and that $M$ is in fact a direct product follows from the fact that $\Aut_H$ is \'etale. Finally, Claim (ii) (3) is proved in the same way as \cite[Theorem 15.34.(b)]{Milne}.
\qed
\vspace{5mm}
\subsection{Linearization of $\mu_p^n$-actions and a fixed point formula}\label{Linearization} From now on, we will work over an algebraically closed field $k$ of characteristic $p > 0$.
Let $X$ be a smooth variety over $k$ with a faithful $\mu_{p^n}$-action. Let $P \in X$ be a fixed point of this action. Since $\mu_{p^n}$ is linearly reductive, it is well-known (see e.g. \cite[Proof of Corollary 1.8]{Satriano}) that the action of $\mu_{p^n}$ on $X$ can be linearized in a formal neighborhood of $P$ in $X$. If $X$ is a surface, "linearizability" means that there is a $\mu_{p^n}$-equivariant isomorphism
$$
\widehat{\OO}_{X,P} \to k[[x,y]],
$$
where the action of $\mu_{p^n}$ on $k[[x,y]]$ is linear. After possibly conjugating this action, we can assume that $\mu_{p^n}$ acts diagonally on $k[[x,y]]$, that is, via the coaction
\begin{eqnarray*}
k[[x,y]] &\to& k[[x,y]] \otimes_k k[\lambda]/(\lambda^p -1) \\
x & \mapsto & x \otimes \lambda \\
y & \mapsto & y \otimes \lambda^a
\end{eqnarray*}
for some $0 \leq a \leq p^n-1$. We say that the $\mu_{p^n}$-action on $X$ is of type $\frac{1}{p^n}(1,a)$ at $P$. The fact that one can linearize $\mu_{p^n}$-actions has the following consequences for the fixed locus $X^{\mu_{p^n}}$ and the quotient $q: X \to Y:= X/\mu_{p^n}$.
\begin{itemize}
\item $X^{\mu_{p^n}}$ is representable by a smooth closed subscheme of $X$ (see \cite[Theorem 5.4]{Fogarty}).
\item If $X$ is a surface, then $q(P)$ is a singular point of $Y$ if and only if $P$ is an isolated fixed point of the $\mu_{p^n}$-action (see \cite[Theorem 1.3]{Satriano}).
\item If $X$ is a surface and $p \nmid a$, then $q(P)$ is a cyclic quotient singularity of type $\frac{1}{p^n}(1,a)$ (see e.g. \cite[Theorem 2.3]{Hirokado} for $n = 1$; the general case is similar).
\end{itemize}
\begin{Remark}\label{vectorfield}
Actions of $\mu_p$ (and $\alpha_p$) on $X$ correspond bijectively to vector fields $D$ on $X$ with $D^p = D$ (resp. $D^p = 0$). An explicit description of this correspondence can be found for example in \cite[Section 3]{Tziolas3}. We remark that the fixed locus of the action is identified with the zero locus of the vector field via this correspondence. Vector fields with $D^p = \lambda D$ for some $\lambda \in k(X)$ are called \emph{$p$-closed}, and $D$ is called \emph{multiplicative} (resp. \emph{additive}) if $D^p = D$ (resp. $D^p = 0$).
\end{Remark}
We will now prove a fixed point formula for $\mu_{p^n}$-actions on smooth projective varieties. It may be possible to give a proof similar to the proof of the fixed point formula for torus actions on smooth varieties by Iversen \cite{Iversen}, but we were not able to find a suitable reference.
\begin{Theorem} \label{fixedpointformula}
Let $X$ be a smooth proper variety with an action of a finite commutative group scheme $G$ of type $lr$. Then, the $\ell$-adic Euler characteristics of $X$ and $X^{G}$ coincide, that is,
$$
e(X) = e(X^{G})
$$
\end{Theorem}
\prf
Since $X^{\mu_{p^m}}$ is a smooth proper variety for all $\mu_{p^m} \subseteq G$ and $G$ is commutative, we can use Lemma \ref{fixedofsubscheme}, Lemma \ref{finitecommgroupscheme} (ii) (3), and induction on the length of $G$ to assume without loss of generality that $G \cong \mu_p$.
Then, by Remark \ref{vectorfield}, the $G$-action corresponds to the action of a multiplicative vector field $D$, and the fixed locus $X^{\mu_p}$ coincides with the zero locus of $D$. Phrased differently, the fixed locus $X^{\mu_p}$ is the zero locus of a section of $T_X$ and therefore $e(X^{\mu_p}) = c_{\dim(X)}(T_X) = e(X)$.
\qed
\begin{Remark}
The fixed point formula for torus actions given in \cite{Iversen} can be deduced from Theorem \ref{fixedpointformula} by considering the $\mu_{p^n}$-actions induced by a given $\GG_m$-action and letting $n \to \infty$.
\end{Remark}
\begin{Remark}\label{alphapfixedpoint}
Note that if $X$ is a smooth proper variety with an action of $\alpha_p$, then also $e(X) = e(X^{\alpha_p})$, by the same proof as in the $\mu_p$-case. However, as $X^{\alpha_p}$ is not necessarily smooth, it is unclear how to extend this to actions of, say, $M_n$.
\end{Remark}
\vspace{5mm}
\subsection{Elliptic surfaces}
In this section, we will recall the necessary background on elliptic surfaces over the algebraically closed field $k$.
Following \cite{Mumford}, we say that a non-zero effective divisor $F = \sum_{i=1}^n a_iF_i$ on a smooth surface $X$ is \emph{of canonical type} if $F.F_i = K_X.F_i = 0$ for all $i$. We say that $F$ is \emph{indecomposable}, if it is not a non-trivial sum of divisors of canonical type. Every fiber of an elliptic fibration is a curve of canonical type and, conversely, for many surfaces, curves of canonical type can be used to prove the existence of elliptic fibrations.
Let us recall the Kodaira--N\'eron classification of indecomposable divisors $F$ of canonical type (see e.g. \cite{Kodaira}). If $F$ is irreducible, it is either an elliptic curve (Type ${\rm I}_0$), a nodal rational curve (Type ${\rm I}_1$) or a cuspidal rational curve (Type ${\rm II}$). If $F$ is not irreducible, its components are $(-2)$-curves. If the components of $F$ do not intersect transversally, then $F$ consists either of two $(-2)$-curves which meet with multiplicity $2$ at one point (Type ${\rm III}$) or of three $(-2)$-curves meeting transversally in a single point (Type ${\rm IV}$). In all other cases, all curves intersect transversally in distinct points and the resulting dual graphs are given in the following table. We call $F$ \emph{additive} if it is not of type ${\rm I}_n$ and \emph{multiplicative} if it is of type ${\rm I}_n$ with $n \geq 1$.
\vspace{-5mm}
$$
\begin{array}[t]{|l*{10}{|c}|}
\hline
\text{Type of $F$}
& {\rm I}_0 & {\rm I}_1 & {\rm I}_m & \p{\rm I}_m^* & {\rm II} & {\rm III} & {\rm IV} & \p{\rm IV}^* & \p{\rm III}^* & \p{\rm II}^* \\
\hline&&&&&&&&&&\\[-2ex]
\text{Dual Graph} & - & - & \widetilde{A}_{m-1} & \widetilde{D}_{m+4} & - & \widetilde{A}_1 & \widetilde{A}_2 & \widetilde{E}_6 & \widetilde{E}_7 & \widetilde{E}_8 \\
\hline
\end{array}
$$
\vspace{2mm}
Recall that by a result of Lichtenbaum and Shafarevich, the minimal proper regular model of a curve of positive genus over a Dedekind scheme exists and is unique (see \cite[Theorem 9.3.21]{Liu}). In the setting of elliptic surfaces, this can be rephrased as follows.
\begin{Lemma}\label{uniquemodel}
Let $C$ be a smooth projective curve over $k$ and let $F_\eta \to \Spec k(C)$ be a smooth projective curve of genus $1$ over $k(C)$. Then, there exists a unique elliptic surface $f: X \to C$ with generic fiber $F_\eta$.
\end{Lemma}
Using this, the \emph{Jacobian} $J(f):J(X) \to C$ of an elliptic surface $f:X \to C$ is simply the minimal proper regular model of the Jacobian $\Pic^0_{F_\eta}$ of the generic fiber $F_\eta$ of $f$. Note that the line bundle $\OO_{F_\eta}$ induces a canonical section of $J(f)$ and, away from the multiple fibers of $f$, the smooth locus of $f$ is a torsor under the smooth locus of its Jacobian. We call an elliptic surface $f$ \emph{Jacobian} if $f$ admits a section.
Now, let $f: X \to C$ be an elliptic surface and choose integers $m_i$ and indecomposable divisors $F_i$ of canonical type for $i=1,\hdots,n$ such that the $m_iF_i$ are precisely the multiple fibers of $f$. Then, $m_i$ is called \emph{multiplicity} of $F_i$ and whenever we say that $mF$ is a fiber of $f$, we implicitly assume that $m$ is the multiplicity of the fiber.
Set $P_i := f(F_i)$.
Let $\cal{L} \oplus \cal{T}$ be the decomposition of $R^1f_* \OO_X$ into its locally free part $\cal{L}$ and its torsion part $\cal{T}$. A multiple fiber $m_iF_i$ of $f$ is called \emph{wild} if $\cal{T}_{P_i} \neq 0$ and \emph{tame} otherwise. Equivalently, $m_iF_i$ is tame if and only if $\nu_i = m_i$, where $\nu_i$ is the order of the normal bundle of $F_i$ in $X$.
Recall the following formulas:
\begin {itemize}
\item (\emph{Application of Riemann--Roch})
$$\chi(X,\OO_X) = \chi(X, \omega_X^{\otimes n}) \geq 0$$ for all $n \in \mathbb{Z}$.
\item (\emph{Noether formula})
$$12 \chi(X,\OO_X) = c_2(X).$$
\item (\emph{Igusa inequality})
$${\rm rk}(\Num(X)) \leq b_2(X)$$.
\item (\emph{Canonical bundle formula} (see \cite[Theorem 2]{BombieriMumford2}))
There are integers $0 \leq a_i \leq m_i - 1$ and $\gamma_i$ such that
\begin{eqnarray*}
\omega_X & \cong& f^*(\omega_C \otimes \cal{L}^{-1}) \otimes \OO_X(\sum_{i=1}^n a_iF_i) \\
m_i & = & p^{\gamma_i} \nu_i
\end{eqnarray*}
where $\deg(\omega_C \otimes \cal{L}^{-1}) = 2g(C) - 2 + \chi(X,\OO_X) + {\rm length}(\cal{T})$ and $\nu_i$ is the order of the normal bundle of $F_i$ in $X$.
\item (\emph{Ogg's formula} (see \cite{Ogg}))
Let $\Delta_f$ be the discriminant of $f$ and $v_P(\Delta_f)$ the order of vanishing of $\Delta_f$ at $P \in C$. Then,
$$
c_2(X) = \sum_{P \in C} v_P(\Delta_f).
$$
Moreover, if $F_P$ denotes the fiber over $P$, then $v_P(\Delta_f) = e(F_P) + \delta_{F_P}$, where $e(F_P)$ is the topological Euler characteristic of $F_P$ and $\delta_{F_P}$ is the Swan conductor of $F_P$. If $m$ is the number of components of $F_P$, then
$$
e(F_P) = \begin{cases}
0 & \text{ if } (F_P)_{red}\text{ is smooth }, \\
m & \text{ if } F_P \text{ is multiplicative}, \\
m+1 & \text{ if } F_P \text{ is additive}.
\end{cases}
$$
and
$$
\delta_{F_P} = \begin{cases}
0 & \text{ if } p \neq 2,3 \text{ or } F_P \text{ multiplicative}, \\
0 & \text{ if } p = 3 \text{ and } F_P \text{ of type } {\rm III},{\rm III}^* \text{ or } {\rm I}_n^*, \\
0 & \text{ if } p = 2 \text{ and } F_P \text{ of type } {\rm IV} \text{ or } {\rm IV}^*, \\
\geq 2 & \text{ if } p = 2 \text{ and } F_P \text{ of type } {\rm II} \text{ or } {\rm I}_n^* \text{ with } n \neq 1, \\
\geq 1 & \text{ else }.
\end{cases}.
$$
For the list of Swan conductors, see e.g. \cite[p. 67]{schuettShioda}.
\item (\emph{Comparison of $f$ and $J(f)$ }(see \cite{LiuLorenziniRaynaud}))
Let $P \in C$ and $F_P$ resp. $F'_P$ be the fibers of $f$ resp. $J(f)$ over $P$. Then,
\begin{enumerate}[(i)]
\item $f$ and $J(f)$ have the same $j$-map,
\item $F_P$ and $F'_P$ are of the same type,
\item $v_P(\Delta_f) = v_P(\Delta_{J(f)})$,
\item $\delta_{F_P} = \delta_{F'_P}$,
\item $c_2(X) = c_2(J(X))$.
\end{enumerate}
\end {itemize}
Finally, we introduce the notion of \emph{movable} fiber, which will play an important r\^ole throughout this article. The letters $a,m,\gamma,$ and $\nu$ will have the same meaning as the corresponding letters with indices in the canonical bundle formula recalled above.
\begin{Definition}\label{movabledefn}
A fiber $mF$ of an elliptic surface $f:X \to C$ is called \emph{$n$-movable}, if $\Aut_{(nF,X)}^0 \neq \Aut_X^0$. An $m$-movable fiber is simply called \emph{movable}. We say that $mF$ is \emph{$n$-movable by vector fields}, if $H^0(X,T_X\langle nF \rangle) \neq H^0(X,T_X)$.
\end{Definition}
Clearly, an $n$-movable fiber is $k$-movable for all $k \leq n$, and if $mF$ is $n$-movable by vector fields, it is $n$-movable. The following lemma shows that movable fibers satisfy very special properties.
\begin{Lemma} \label{movable}
Let $mF$ be an $n$-movable fiber of $f$ with $n \geq 1$. Then,
\begin{enumerate}[(i)]
\item $\nu = 1$, $m = p^{\gamma}$ with $\gamma \geq 0$ and either $a > n$ with $p \mid a$ or $a = 0$. In particular, if $mF$ is movable, then $a = 0$.
\item $F$ is smooth, or $p \in \{2,3\}$ and $F$ is of type ${\rm II}$. In the latter case, $mF$ is not $1$-movable by vector fields.
\end{enumerate}
\end{Lemma}
\prf
By Corollary \ref{criteria} (a), we have $H^0(F,N_{F/X}) \neq 0$. Since $N_{F/X}$ has degree $0$ on every component of $F$, we deduce $N_{F/X} = \OO_{F}$ and therefore $\nu = 1$ and $m = p^\gamma$ for some $\gamma \geq 0$.
Suppose that $a$ is prime to $p$. Then, there exists $l \geq 0$ such that $F$ appears as a reduced irreducible component of the scheme-theoretic base locus of $|lK_X|$. Since $\Aut_X^0$ acts naturally on this base locus and preserves its connected components, we obtain $\Aut_{(F,X)}^0 = \Aut_X^0$. Hence, $p \mid a$. Moreover, if $a \neq 0$, then $aF$ is an irreducible component of the scheme-theoretic fixed locus of $|K_X|$, so that $a > n$. Since $a$ is bounded above by the multiplicity of $F$, we deduce that $a = 0$ if $F$ is movable.
Since $(-2)$-curves are infinitesimally rigid in $X$, Corollary \ref{criteria} (a) shows that $F$ is integral. Next, if $F$ is of type ${\rm I}_1$, then $\Def_{F}$ is prorepresentable and thus so is $\Def_{F/X}'$ by Lemma \ref{prorep}. Then, Corollary \ref{criteria} (c) shows that $mF$ is not $1$-movable.
If $F$ is of type ${\rm II}$, then $mF$ is not $1$-movable by vector fields by Remark \ref{criteriaremark}. Moreover, if $p \neq 2,3$, then $\Def_{F}$ admits a good hull (see Remark \ref{cuspexample}), so that $F$ is not $1$-movable by Lemma \ref{prorep} and Corollary \ref{criteria} (c).
\qed
\begin{Remark}
In Example \ref{movableexample}, we will give examples of elliptic surfaces over an affine curve with a movable fiber of type ${\rm II}$ in characteristic $2$ and $3$.
\end{Remark}
\vspace{2mm}
\subsection{Automorphism schemes of genus $1$ curves}
In this section, we recall the structure of the automorphism scheme of a curve $C$ of genus $0$ or $1$ over an algebraically closed field $k$. This is well-known if $\Char(k) = p \neq 2,3$ and we refer the reader to \cite[Proposition 6]{BombieriMumford3} for proofs in the case of the cuspidal cubic if $p = 2,3$.
\begin{Lemma} \label{autoschemes}
Let $C$ be a reduced, irreducible curve of arithmetic genus $0$ or $1$ over $k$. Then, the following hold:
\begin{enumerate}[(i)]
\item If $C \cong \PP^1$, then $\Aut_C \cong \rm{PGL}_2$.
\item If $C$ is an elliptic curve, then $\Aut_C^0 \cong C$.
\item If $C$ is a nodal cubic curve, then $\Aut_C^0 \cong \GG_m$.
\item If $C$ is a cuspidal cubic curve, then $\Aut_C^0 \cong (\GG_a \rtimes A_p) \rtimes \GG_m$, where
$$
A_p = \begin{cases}
\{1\} & \text{ if } p \neq 2,3, \\
\alpha_3 & \text{ if } p = 3, \\
(\alpha_2 \times \alpha_2) \cdot \alpha_2 & \text{ if } p=2.
\end{cases}
$$
\end{enumerate}
\end{Lemma}
We will need further information on the fixed loci of some finite subgroup schemes of $\Aut_C^0$ in the above cases.
\begin{Lemma} \label{fixedloci}
Let $C$ be a reduced, irreducible curve of arithmetic genus $0$ or $1$ over $k$ and let $G \subseteq \Aut_C$ be a non-trivial connected subgroup scheme.
\begin{enumerate}[(i)]
\item If $C = \PP^1$, then $G$ has at most $2$ fixed points on $C$. Moreover, $G$ has precisely $2$ fixed points if and only if $G \subseteq \GG_m$.
\item If $C$ is an elliptic curve, then $G$ admits no fixed points on $C$.
\item If $C$ is a nodal cubic curve, then $G$ has exactly $2$ fixed points on $C$ and one of them is the node of $C$.
\item If $C$ is a cuspidal cubic curve and $G \cong \mu_{p^n}$, then one of the following holds
\begin{enumerate}[(1)]
\item $G$ has exactly $2$ fixed points on $C$ and one of them is the cusp of $C$,
\item $p^n = 2$, $G$ has exactly $3$ fixed points on $C$ and one of them is the cusp of $C$.
\item $p^n = 2$, $G$ has exactly $4$ fixed points on $C$ and all of them are smooth points of $C$,
\item $p^n = 3$, $G$ has exactly $3$ fixed points on $C$ and all of them are smooth points of $C$,
\item $p^n = 4$, $G$ has exactly $2$ fixed points on $C$ and both of them are smooth points of $C$. In this case, the induced $\mu_2$-action is as in case (2).
\end{enumerate}
\end{enumerate}
\end{Lemma}
\prf
Claims (ii) and (iii) and the first part of Claim (i) are well-known. Let us prove the second part of Claim (i). If $G$ fixes two points on $\PP^1$, then $G \subseteq \GG_m$. Conversely, if $G \subseteq \GG_m$, then we can conjugate $G \subseteq {\rm PGL}_2$ so that it lies in the diagonal torus. Then, $G$ fixes $0$ and $\infty$ on $\PP^1$.
To prove Claim (iv), we recall that by \cite[Proposition 6]{BombieriMumford3}, one can identify the smooth locus of $C$ with $\mathbb{A}^1 = \Spec k[t]$ such that automorphisms of $\mathbb{A}^1$ induced by automorphisms of $C$ are of the following form:
\begin{eqnarray*}
t \mapsto& at + b, &a \in \GG_m, b \in \GG_a \text{ if } p \neq 2,3, \\
t \mapsto& at + b + ct^3, & a \in \GG_m, b \in \GG_a, c^3 = 0 \text{ if } p = 3, \\
t \mapsto& at + b + ct^2 + dt^4, & a \in \GG_m, b \in \GG_a, c^4 = d^2 = 0 \text{ if } p = 2.
\end{eqnarray*}
Moreover, we refer the reader to \cite[p. 212]{BombieriMumford3} for the calculation of the stabilizer of the cusp of $C$, which is given by all substitutions if $p \neq 2,3$, by the substitutions with $c = 0$ if $p = 3$, and by the substitutions with $c^2 = d = 0$ if $p = 2$.
If $p \neq 2,3$, then $G$ is conjugate to the $\mu_{p^n}$ of maps $t \mapsto at$, $a \in \mu_{p^n}$. Its fixed points are $t = 0$ and the cusp of $C$.
If $p = 3$, then we can conjugate $G$ such that either $G$ acts as above or as $t \mapsto at + (1-a)t^3$ with $a^3 = 1$. In the latter case, the fixed points are given by $t^3 = t$. This $\mu_3$-action does not fix the cusp of $C$.
If $p = 2$, we can conjugate $G$ such that it acts in one of the following ways with $\lambda,\mu \in k$:
\begin{eqnarray*}
t \mapsto & at, & a \in \mu_{2^n}, \\
t \mapsto & at + \lambda(1+a)t^2 + \mu(1+a)t^4, & a \in \mu_2, \\
t \mapsto & at + (a+a^2)t^2 + (1+a^2)t^4,& a \in \mu_4.
\end{eqnarray*}
In the first case, $G$ fixes $t = 0$ and the cusp of $C$.
In the second case, $G$ fixes the points where $\mu t^4 + \lambda t^2 + t = 0$. If $\mu \neq 0$, this shows that $G$ fixes $4$ smooth points on $C$ and does not fix the cusp, whereas if $\mu = 0$, the action of $G$ has $2$ smooth fixed points on $C$ and fixes the cusp.
In the third case, $G$ fixes $t \in \{0,1\}$ and does not fix the cusp of $C$. Moreover, $G[F] = \mu_2$ acts as in the second case with $\mu = 0$. This proves Claim (iv).
\qed
\vspace{3mm}
Using Lemma \ref{fixedloci}, we can determine how $\mu_{p^n}$-actions on an elliptic surface $f: X \to C$ can restrict to reducible fibers of $f$ and determine the possible fixed loci.
\begin{Corollary}\label{fixedlocireducible}
Let $f:X \to C$ be an elliptic surface with $\mu_{p^n} \subseteq \Aut_X^0$. Let $mF$ be a singular fiber of $f$. Then, the following hold:
\begin{enumerate}[(i)]
\item If $F$ is not of type ${\rm II}$ or ${\rm III}$, then $e(F) = e(F^{\mu_{p^n}})$.
\item If $F$ is of type ${\rm III}$, then $e(F) = e(F^{\mu_{p^n}}) = 3$, or $p^n = 2$ and $e(F^{\mu_2}) = 4$.
\item If $F$ is of type ${\rm II}$, then $F$ is preserved by the $\mu_{p^n}$-action, and
\begin{enumerate}[(1)]
\item $e(F) = e(F^{\mu_{p^n}}) = 2$, or
\item $p^n = 3$ and $e(F^{\mu_3}) = 3$, or
\item $p^n = 2$ and $e(F^{\mu_2}) \in \{3,4\}$.
\end{enumerate}
\end{enumerate}
\end{Corollary}
\prf
If $F$ is not of type ${\rm II}$, then $mF$ is not $1$-movable by Lemma \ref{movable}, so the $\mu_{p^n}$-action on $X$ restricts to a $\mu_{p^n}$-action on $F$.
In the first case, all intersections of components of $F$ are transversal. Since $\mu_{p^n}$ preserves all components, it fixes all their intersections. Now, the statement can be checked case by case and the proof is the same as in \cite[Lemma 2]{DolgachevNumerical}.
In the second case, the intersection of the two components $E_1,E_2$ of $F$ is not transversal. If $\mu_{p^n}$ fixes $(E_1 \cap E_2)_{red}$, then $e(F^{\mu_{p^n}}) = 3$ by Lemma \ref{fixedloci} (i) and, since $e(F) = 3$, this gives the desired equality of Euler characteristics. If $\mu_{p^n}$ does not fix $(E_1 \cap E_2)_{red}$, then it fixes two points on each of the $E_i$ by Lemma \ref{fixedloci} (i). Consider the contraction $\pi: X \to X'$ of $E_1$. Then, $\pi(E_2)$ is a cuspidal curve on $X'$, the $\mu_{p^n}$-action on $X'$ induced via Theorem \ref{Blanchard} has three fixed points on $\pi(E_2)$ and one of them is the cusp of $\pi(E_2)$ by Proposition \ref{blowup}. Hence, $p^n = 2$ by Lemma \ref{fixedloci}.
If $F$ is of type ${\rm II}$, the only statement that is not already included in Lemma \ref{fixedloci} is the fact that $F$ is preserved by $\mu_{p^n}$. To prove this, note that $F$ is not $1$-movable by vector fields by Lemma \ref{movable} and hence $\mu_{p^n} \cap \Aut_{(F,X)}^0$ is non-trivial. In particular, the induced $\mu_p$-action preserves $F$. By Lemma \ref{fixedloci}, this $\mu_p$-action has an isolated fixed point $Q$ on $F$. Since $X^{\mu_p}$ is smooth at $Q$, the point $Q$ is also a fixed point of the $\mu_{p^n}$-action by Corollary \ref{isolatedefixedpoints}. Therefore, by Lemma \ref{blowup}, the $\mu_{p^n}$-action lifts to the blowup $\widetilde{X}$ of $X$ at $Q$. Since the strict transform $\widetilde{F}$ of $F$ in $\widetilde{X}$ is a negative curve, we have $\Aut_{(\widetilde{F},\widetilde{X})}^0 = \Aut_{\widetilde{X}}^0$ by Corollary \ref{criteria} and therefore the $\mu_{p^n}$-action on $X$ preserves $F$ by Lemma \ref{image}.
\qed
\begin{Remark}
It was claimed in \cite[Lemma 3]{RudakovShafarevich} that the exceptional case in Corollary \ref{fixedlocireducible} (ii) does not occur. We will give a counterexample to this statement in Example \ref{caseb2}. The proof of \cite[Lemma 3]{RudakovShafarevich} seems to be correct up until the last sentence, where it is claimed that the configuration described in \cite[p. 1224]{RudakovShafarevich} is not of Kodaira type. In fact, the configuration described there is of type ${\rm I}_1^*$.
\end{Remark}
\begin{Remark} \label{fixedlocireducibleremark}
The proof of Corollary \ref{fixedlocireducible} (iii) shows more generally that an irreducible fiber $F$ of an elliptic fibration $f: X \to C$ is preserved by a group scheme action as soon as the action has a fixed point on $F$.
\end{Remark}
\begin{Example} \label{movableexample}
The following examples show that, at least locally, there may be group scheme actions on elliptic surfaces that actually move fibers of type ${\rm II}$ in characteristic $2$ and $3$: Let $p = 2$ and let $X \subseteq \PP^2_{k[t]}$ be the smooth surface defined by
$$
y^2z + t^4yz^2 = x^3 + tz^3.
$$
The generic fiber of $X \to \Spec k[t]$ is an elliptic curve and the fiber $F$ at $t = 0$ is a cuspidal cubic.
There is an $\alpha_4$-action on $X$ defined by
\begin{eqnarray*}
(x,z) &\mapsto& (x,z) \\
y &\mapsto& y + az\\
t &\mapsto& t + a^2 + at^4
\end{eqnarray*}
where $a^4 = 0$. Note that the induced $\alpha_2$-action preserves $F$, but the $\alpha_4$-action itself does not.
Moreover, the $\alpha_4$-action has no fixed point on $f$, since the induced $\alpha_4$-action on $\Spec k[t]$ has no fixed point.
A similar example of an $\alpha_9$-action in characteristic $3$ exists on the surface defined by
$$y^2z = x^3 + t^9xz^2 + tz^3.$$
\end{Example}
\section{Vertical automorphisms} \label{vertical}
In this section, $f: X \to C$ is an elliptic surface over an algebraically closed field $k$. The purpose of this section is to study the group scheme of vertical automorphisms of elliptic surfaces.
Recall that because of Blanchard's Lemma (see Theorem \ref{Blanchard}) there is a natural map $f_*: \Aut_X^0 \to \Aut_C^0$.
\begin{Definition}
Let $f: X \to C$ be an elliptic surface. The group scheme of \emph{vertical automorphisms} of $X$ is defined as $\Ker(f_*)$, where $f_*: \Aut_X^0 \to \Aut_C^0$ is the natural map.
\end{Definition}
After recalling the notion of Weil restrictions of group schemes along the field extension $h: \Spec k(C) \to \Spec k$, we will first study the action of $\Ker(f_*)$ on the generic fiber $F_\eta$ of $f$ and then determine obstructions to extending such actions to the surface $X$.
\subsection{Automorphisms of the generic fiber}
Recall the following results on Weil restrictions from \cite[Section 7.6]{NeronModels}.
\begin{Definition}
The \emph{Weil restriction} of a scheme $G$ over $k(C)$ along $h: \Spec k(C) \to \Spec k$ is defined as the presheaf
\begin{eqnarray*}
h_*G: & (Sch/k)^{op} &\to (Sets) \\
& T &\mapsto G(T \times_{\Spec k} \Spec k(C)).
\end{eqnarray*}
\end{Definition}
\begin{Lemma}\label{Weilrestrictionlemma}
There is a bijection of sets of homomorphisms of presheaves
$$
\Hom_{\Spec k}(T,h_*G) \to \Hom_{\Spec k(C)}(T \times_{\Spec k} \Spec k(C), G)
$$
which is functorial in the $k$-scheme $T$ and the $k(C)$-scheme $G$.
\end{Lemma}
\begin{Lemma} \label{unexpectedkernel}
Let $G$ be a group scheme over $k(C)$, let $G'$ be a group scheme over $k$ and let $g: G'_{k(C)} \to G$ be a morphism of group schemes such that the induced map $h_*g: G' \to h_* G$ is a monomorphism of presheaves. Then, the only subscheme of $\Ker(g)$ that can be defined over $k$ is the trivial subgroup scheme. In particular, $\Ker(g)_{red}$ is trivial.
\end{Lemma}
\prf
Let $H \subseteq \Ker(g)$ be a subscheme which can be defined over $k$. Then, by definition, there exists a scheme $H'$ over $k$ and a morphism $H' \to G'$ whose base change along $h$ agrees with $H \to G'_{k(C)}$. Since the induced map $H \to G$ is constant, it follows from the adjunction in Lemma \ref{Weilrestrictionlemma} that the map $H' \to h_*G$ is constant. But $h_*g$ is a monomorphism, hence $H'$ is trivial and thus so is $H$. In particular, $\Ker(g)$ contains only one point and hence $\Ker(g)_{red}$ is trivial.
\qed
\vspace{3mm}
We will now apply the Weil restriction to automorphisms of elliptic surfaces. For every $k$-scheme $T$, we have a natural injective map
$$
\Ker(f_*)(T) \to \Aut_{F_\eta/k(C)}(T \times_{\Spec k} \Spec k(C)),
$$
where $F_\eta$ is the generic fiber of $f$. This defines a monomorphism of presheaves of groups $\Ker(f_*) \to h_*\Aut_{F_\eta/k(C)}$ and hence we obtain a morphism of group schemes $\varphi: \Ker(f_*)_{k(C)} \to \Aut_{F_\eta/k(C)}$ from Lemma \ref{Weilrestrictionlemma}. The connected component of the identity of the latter group scheme is isomorphic to the generic fiber $J_\eta$ of the Jacobian $J(f)$ of $f$ and we denote the induced map of identity components by $\varphi^0: \Ker(f_*)_{k(C)}^0 \to J_\eta$. The following lemma shows that $\varphi^0$ is injective as long as $F_\eta$ is ordinary.
\newpage
\begin{Lemma}\label{connsub}
Let $f: X \to C$ be an elliptic surface. Let $G = \Ker(f_*)^0$ and let $\varphi^0$ as above.
\begin{enumerate}[(i)]
\item The group scheme $G$ is commutative and $\dim(G) \leq 1$,
\item If $\dim(G) = 1$, then the Jacobian $J(f): J(X) \to C$ of $f$ is trivial.
\item If $J_\eta$ is ordinary, then $\varphi^0$ is injective. In this case, either $\dim(G) = 1$ and $G_{k(C)} \cong J_\eta$, or $\dim(G) = 0$ and $G \cong \mu_{p^n}$ for some $n \geq 0$.
\item If $J_\eta$ is supersingular, then we have $G[F] \cong \alpha_p^r$ for some $r \geq 0$. If $r = 1$, then either $\dim(G) = 1$ and $G_{k(C)} \cong J_\eta$, or $\dim(G) = 0$ and $G \cong M_n$ for some $n \geq 0$.\end{enumerate}
\end{Lemma}
\prf
First, note that the action of $G$ on $X$ factors through $G^{ab}$ on a dense open subset of $X$, because the automorphism scheme of a smooth curve of genus one is commutative. Therefore, the action of $G$ on all of $X$ factors through $G^{ab}$ and thus $G$ is commutative, proving the first part of (i).
To prove Claim (i) and (ii), let $G_r := G_{red}$ be the reduction of $G$. Since $k$ is perfect, Lemma \ref{generalgroupscheme} (vi) shows that this is a closed and smooth subgroup scheme of $G$. Assume that $\dim(G) \geq 1$. Then, $G_r$ is non-trivial. Consider the morphism $\varphi_r: (G_r)_{k(C)} \to J_\eta$ obtained by restricting $\varphi^0$ to $(G_r)_{k(C)}$. By Lemma \ref{unexpectedkernel}, the group scheme $\Ker(\varphi_r)$ is zero-dimensional and connected, hence $\varphi_r$ is a purely inseparable isogeny of elliptic curves over $k(C)$. But all finite connected subgroup schemes of $(G_r)_{k(C)}$ are of the form $(G_r)_{k(C)}[F^n]$ for some $n \geq 0$. Since $(G_r)_{k(C)}[F^n] = (G_r[F^n])_{k(C)}$, these subschemes can be defined over $k$. Hence, by Lemma \ref{unexpectedkernel}, the map $\varphi_r$ is an isomorphism. Then, $G \times C \to C$ is a minimal proper regular model for $J_\eta$ over $C$ and hence coincides with $J(f)$ by Lemma \ref{uniquemodel}. In particular, $J(f)$ is trivial. This yields Claim (i) and (ii).
To prove Claim (iii) and (iv), we use Lemma \ref{finitecommgroupscheme} (ii) to write $G[F^n] \cong G[F^n]_{lr} \times G[F^n]_{ll}$ and consider the action of $G[F^n]$ on $X$.
For Claim (iii), assume that $J_\eta$ is ordinary. Then, almost all fibers of $f$ are ordinary. By Lemma \ref{homsandexts} (i), every action of $G[F^n]_{ll}$ on an ordinary elliptic curve is trivial, hence $G[F^n]$ is of type $lr$ and thus isomorphic to $\prod_{i=1}^m \mu_{p^{n_i}}$ for some $m,n_i \geq 0$ by Lemma \ref{finitecommgroupscheme} (ii) (3). Therefore, we have $(G[F^n])_{k(C)} \cong \prod_{i=1}^m (\mu_{p^{n_i}})_{k(C)}$. Subgroup schemes of this group scheme correspond to quotients of its reduced Cartier dual, hence all of them are defined over $k$. Thus, by Lemma \ref{unexpectedkernel} the intersection $\Ker(\varphi^0) \cap G[F^n]$ is trivial and we have $(G[F^n])_{k(C)} \cong J_\eta[F^n] \cong (\mu_{p^{n}})_{k(C)}$ and thus $G[F^n] \cong \mu_{p^n}$. Since $\Ker(\varphi^0)$ is finite and connected, we have $\Ker(\varphi^0) \subseteq G[F^n]$ for $n \gg 0$, so we can in fact deduce that $\varphi^0$ is injective. If $\dim(G) = 0$, then $G \cong G[F^n]$ for $n \gg 0$ and if $\dim(G) = 1$, then $G = G_{red}$ and, as in the second paragraph of the proof, $\varphi^0$ induces an isomorphism $G_{k(C)} \to J_\eta$. This yields Claim (iii).
As for Claim (iv), we assume that $J_\eta$ is supersingular. Then, the group $G[F]$ is of type $ll$ by Lemma \ref{homsandexts} (i). Moreover, the action of $G[F]$ on a general fiber $E$ of $f$ factors through $E[F]$. Since Verschiebung is trivial on $E[F]$, the action of $G[F]$ on $X$ factors through $G[F]/(VG[F])$, hence $VG[F] = 0$. Therefore, by Lemma \ref{finitecommgroupscheme} (ii), we have $G[F] \cong \alpha_p^r$ for some $r \geq 0$. Now, if $r = 1$, then $\alpha_p$ is the unique simple closed subgroup scheme of $G[F^n]$ for every $n \geq 1$. Therefore, if the morphism $G[F^n] \to E[F^n]$ is not injective for a general fiber $E$, then $G[F] = \alpha_p$ is in its kernel and therefore $(G[F])_{k(C)} \subseteq \Ker(\varphi)$, which is impossible by Lemma \ref{unexpectedkernel}. Hence, $G[F^n]$ is isomorphic to its image in $E[F^n] \cong M_n$. This yields Claim (iv).
\qed
\vspace{2mm}
\begin{Remark}
In Example \ref{importantexample}, we will show that the integer $r$ appearing in Lemma \ref{connsub} (iv) can be arbitrarily large in every positive characteristic.
\end{Remark}
If $J_\eta$ is ordinary, the existence of a subscheme isomorphic to $(\mu_{p^n})_{k(C)}$ with $n \geq 1$ in $J_\eta$ already gives strong restrictions on the geometry of $J(f)$. This is closely related to the Igusa moduli problem, which is defined as follows.
\begin{Definition} \label{Igusadefn}
The ordinary part ${\rm Ig}(p^n)^{\rm ord}$ of the \emph{Igusa stack} is the stack over the category of $k$-schemes whose objects over a $k$-scheme $T$ are families $E \to T$ of ordinary elliptic curves over $T$ together with a generator of $E^{(p^n)}[V^n] := {\rm Ker}(V^n)(E^{(p^n)} \to E)$, where $V: E^{(p)} \to E$ denotes Verschiebung on $E$.
\end{Definition}
This moduli problem has been first studied by Igusa in \cite{Igusa}. If $p^n \geq 3$, then ${\rm Ig}(p^n)^{\rm ord}$ is representable by a smooth curve defined over $\FF_p$ (see \cite[Corollary 12.6.3]{KatzMazur}). We denote its smooth projective compactification by ${\rm Ig}(p^n)$.
Now, the following lemma is a straightforward consequence of the definition of ${\rm Ig}(p^n)$.
\begin{Lemma} \label{Igusa}
Let $J(f): J(X) \to C$ be a Jacobian elliptic surface with generic fiber $J_\eta$. Assume that $p^n > 2$ and $(\mu_{p^n})_{k(C)} \subseteq J_\eta$. Then, $J_\eta$ is the pullback of the universal elliptic curve over ${\rm Ig}(p^n)$ along a morphism $\Spec k(C) \to {\rm Ig}(p^n)$.
\end{Lemma}
\prf
Since $(\mu_{p^n})_{k(C)} \subseteq J_\eta$ has length $p^n$, we have $(\mu_{p^n})_{k(C)} = J_\eta[F^n]$. Therefore, we have $(\mu_{p^n})_{k(C)} = J_\eta[F^n] \cong J_\eta^{(p^n)}[V^n]^\vee$, so that $J_\eta^{(p^n)}[V^n] \cong \ZZ/p^n\ZZ$. Choosing a generator of the latter group, we obtain a morphism $\Spec k(C) \to {\rm Ig}(p^n)$ inducing $J_\eta$ via pullback of the universal elliptic curve over ${\rm Ig}(p^n)$.
\qed
\vspace{3mm}
In \cite{LiedtkeSchroer}, Liedtke and Schr\"oer studied the singular fibers of the universal elliptic surfaces over the Igusa curves ${\rm Ig}(p)$. Using their results and Lemma \ref{Igusa}, we obtain the following corollary.
\begin{Corollary}\label{singularfibers}
Assume $p > 3$. Let $f: X \to C$ be an elliptic surface with $(\mu_p)_{k(C)} \subseteq \Aut_{F_\eta/k(C)}^0$. Let $mF$ be an additive fiber of $f$. Then, the following hold:
\begin{enumerate}[(i)]
\item $F$ is not of type ${\rm I}_n^*$ with $n \geq 1$.
\item If $p \equiv 1$ mod $12$, then $F$ is of type ${\rm I}_0^*$.
\item If $p \equiv 7$ mod $12$, then $F$ is of type ${\rm III},{\rm III}^*$ or ${\rm I}_0^*$.
\item If $p \equiv 5$ mod $12$, then $F$ is of type ${\rm II},{\rm IV},{\rm IV}^*,{\rm II}^*$ or ${\rm I}_0^*$.
\end{enumerate}
\end{Corollary}
\prf
We have $(\mu_p)_{k(C)} \subseteq \Aut_{F_\eta/k(C)}^0 \cong J_\eta$, where $J_\eta$ is the generic fiber of the Jacobian of $f$. Therefore, the curve $J_\eta$ is a pullback of the universal elliptic curve over ${\rm Ig}(p^n)$ along a morphism $\Spec k(C) \to {\rm Ig}(p^n)$ by Lemma \ref{Igusa}. Since $f$ and $J(f)$ have the same types of singular fibers, the claim now follows by comparing the reduction types of the universal elliptic curve given in \cite[Theorem 10.1 and Theorem 10.3]{LiedtkeSchroer} with the tables in \cite[Section 5.2.]{schuettShioda}.
\qed
\begin{Remark}
There is no analogue of Lemma \ref{Igusa} if $J_\eta$ is supersingular. For $\alpha_p$ and $M_2$, this follows immediately from \cite[Theorem 6.1]{Liedtkeptors}, and for $M_n$ with $n \geq 3$ one can simply iterate the argument given in the proof there.
\end{Remark}
The genus $g({\rm Ig}(p^n))$ of the Igusa curve has been computed by Igusa in \cite{Igusa}. We have
$$
g({\rm Ig}(p^n)) = \frac{1}{48}(p-1)(p^{2n-1}-12p^{n-1} + 1) + 1 - \frac{h_p}{2},
$$
where
$$
h_p = \text{number of supersingular $j$-invariants in } k +
\begin{cases}
0 & \text{ if } p \neq 2,3 \\
\frac{1}{3} &\text{ if } p = 3 \\
\frac{3}{8} & \text{ if } p = 2
\end{cases}.
$$ In particular,
$$
g({\rm Ig}(p^n)) =
\begin{cases}
0 &\text{ if } p^n \leq 12 \\
1 &\text{ if } p^n \in \{13,16\} \\
\geq 2 &\text{ else}
\end{cases}
$$
\begin{Corollary}\label{Igusainequality}
Let $f: X \to C$ be an elliptic surface with $c_2(X) \neq 0$. If $\mu_{p^n} \subseteq \Ker(f_*)$, then
$$
g(C) \geq \frac{1}{48}(p-1)(p^{2n-1}-12p^{n-1} + 1) + 1 - \frac{h_p}{2}
$$
\end{Corollary}
\prf
We can assume that $p^n > 2$, for otherwise the right hand side of the inequality is negative. Since the genus of smooth curves does not go down under taking finite covers, it suffices to show that $C$ admits a dominant rational map to ${\rm Ig}(p^n)$.
By Lemma \ref{connsub} (iv), the map $\varphi: (\mu_{p^n})_{k(C)} \to J_\eta$ induced by the inclusion $\mu_{p^n} \subseteq \Ker(f_*)$ is injective. Hence, Lemma \ref{Igusa} shows that there is a morphism $\Spec k(C) \to {\rm Ig}(p^n)$. Seeking a contradiction, we assume that this map is constant. Then, we have $J_\eta = E \times_{\Spec k} \Spec k(C)$ for some ordinary elliptic curve $E$ over $k$. Thus, $E \times C$ is a minimal proper regular model of $J_\eta$ over $C$ and therefore it coincides with $J(X)$ by Lemma \ref{uniquemodel}. Then, $0 = c_2(E \times C) = c_2(J(X)) = c_2(X)$, contradicting our assumption. Hence, $\Spec k(C) \to {\rm Ig}(p^n)$ is dominant, which is what we had to prove.
\qed
\vspace{5mm}
\subsection{Extending the action to $X$}
In the previous subsection, we have seen how the existence of a $\mu_{p^n}$-action on the generic fiber $F_\eta$ of an elliptic surface $f:X \to C$ gives restrictions on $f$. In this section, we gather several criteria for a connected group scheme action on $F_\eta$ to extend to an action on $X$. Using these criteria, we give geometric restrictions that have to be satisfied by elliptic surfaces with non-trivial $\Ker(f_*)$.
\begin{Definition}
Let $f: X \to C$ be an elliptic surface with generic fiber $F_\eta$. Let $G$ be a group scheme and let $\rho_\eta: G \times F_\eta \to F_\eta$ an action of $G$ on $F_\eta$ such that $\rho_\eta$ is a morphism of $C$-schemes. We say that $\rho_\eta$ \emph{extends over} $p \in C$ if there is a commutative diagram
$$
\xymatrix{
G \times F_\eta \ar[r] \ar[d] & F_\eta \ar[d] \\
G \times X_p \ar[r]^{\rho_p} & X_p,
}
$$
where $X_p := (X \times \Spec \OO_{C,p})$ and the vertical arrows are induced by the inclusion $\OO_{C,p} \subseteq k(C)$.
We say that $\rho_\eta$ \emph{extends to} $X$ if there is a similar diagram with a morphism $\rho: G \times X \to X$ in the second row.
\end{Definition}
\begin{Remark}
Note that $\rho$ and $\rho_p$ are automatically actions of $G$, as associativity can be checked on the schematically dense subscheme $G \times G \times F_\eta$ of $G \times G \times X$.
\end{Remark}
Recall from the previous subsection that an action $\rho_\eta: G \times F_\eta \to F_\eta$ as above gives rise to a translation action $\rho'_\eta: G \times J_\eta \to J_\eta$, where $J_\eta$ is the generic fiber of the Jacobian $J(f)$ of $f$. In the following proposition, we relate extendability of $\rho_\eta$ to extendability of $\rho'_\eta$.
\begin{Proposition} \label{extending}
Let $f: X \to C$ be an elliptic surface with generic fiber $F_\eta$ and let $\rho_\eta: G \times F_\eta \to F_\eta$ be an action of a connected group scheme $G$ such that $\rho_\eta$ is a morphism of $C$-schemes. Then, the following hold:
\begin{enumerate}[(i)]
\item The action $\rho_\eta$ extends to $X$ if and only if it extends over every $p \in C$.
\item If $p \in C$ is a point such that the fiber $F_p$ of $f$ over $p$ is simple, then $\rho_\eta$ extends over $p$ if and only if the corresponding action $\rho'_\eta$ on the generic fiber $J_\eta$ of the Jacobian $J(f)$ of $f$ extends over $p$.
\end{enumerate}
\end{Proposition}
\prf
The action $\rho_\eta$ gives rise to a rational map $C \dashrightarrow {\rm Hom}_C(G \times F_\eta, F_\eta)$. Since the latter scheme is separated and $C$ is a smooth curve, this rational map extends to a morphism if and only if it extends over every closed point of $C$. This shows Claim (i).
Next, let $p \in C$ be a point such that $F_p$ is simple and let $A := \Spec \OO_{C,p}$. Since the smooth locus of $X_p \to A$ is a torsor under its Jacobian and $\rho_\eta$ is induced by restricting the action of $J_\eta$ on $F_\eta$ to $G$, there is an \'etale cover $B \to A$, which we may assume to be Galois with covering group $H$, and a $G$-equivariant isomorphism of $B$-schemes
$$
\varphi: J(X)_p \times_A B \cong X_p \times_A B.
$$
Both sides of the isomorphism are equipped with the natural action of $H$ on the second factor and the $G$-action on $J(X)_p \times_A B$ (resp. $X_p \times_A B$) descends to $J(X)_p$ (resp. $X_p$) if and only if it is normalized by this action of $H$. Thus, we obtain two actions of $H$ on both sides of the above isomorphism and one can check that these two actions differ by translation by a $B$-valued section of $J(X)_p \times_A B$ (see \cite[p.1233]{RudakovShafarevich}). By construction, $\rho_\eta$ and $\rho'_\eta$ commute with translations, so if one of them, say $\rho_\eta$, extends over $p$, then the induced $G$-action on $X_p \times_A B$ is normalized by both $H$-actions, hence this $G$-action also descends to $J(X)_p$. The induced action agrees with $\rho'_\eta$ on $J_\eta$, hence it extends $\rho'_\eta$ over $p$. By the same argument, $\rho_\eta$ extends over $p$ if $\rho_\eta'$ does. This proves Claim (ii).
\qed
\vspace{5mm}
Hence, if $f: X \to C $ admits no multiple fibers, then $G$-actions on $X$ which are trivial on $C$ correspond naturally to $G$-actions on $J(X)$ which are trivial on $C$, that is, we have the following corollary.
\begin{Corollary}\label{ExtendabilityCorollary}
Let $f: X \to C$ be an elliptic surface without multiple fibers. Then, $\Ker(f_*)^0 \cong \Ker(J(f)_*)^0$.
\end{Corollary}
The situation becomes more complicated if $f$ admits multiple fibers. Nevertheless, it turns out that an elliptic fibration $f$ with non-trivial $\Ker(f_*)^0$ must satisfy severe geometric constraints.
\begin{Theorem}\label{Main}
Let $f:X \to C$ be an elliptic surface with $\Ker(f_*)^0$ non-trivial. Then, the following hold:
\begin{enumerate}[(i)]
\item Every separable multisection $\Sigma$ of $f$ satisfies $\Sigma^2 \geq 0$.
\item Either $\chi(X,\OO_X) = 0$ or $f$ admits a multiple fiber.
\item One of the following two cases holds:
\begin{enumerate}[(1)]
\item $\alpha_p^r \subseteq \Ker(f_*)$ for some $r \geq 1$, every separable multisection $\Sigma$ of $f$ satisfies the inequality $h^0(\Sigma,N_{\Sigma/X}) \geq r$ and $f$ is isotrivial with supersingular generic fiber, or
\item $\mu_{p^n} \subseteq \Ker(f_*)$ for some $n \geq 1$ and all additive or supersingular fibers of $f$ are multiple fibers with multiplicity divisible by $p^n$.
\end{enumerate}
\end{enumerate}
\end{Theorem}
\prf
Let $\Sigma \subseteq X$ be an irreducible curve such that $f|_\Sigma: \Sigma \to C$ is finite and separable. Then, the curve $\Sigma$ intersects a general fiber of $f$ transversally, say in $n$ points. Since $\Ker(f_*)^0$ acts without fixed point on a general fiber of $f$ by Lemma \ref{fixedloci} (ii), this implies that $\Sigma$ is not preserved by $\Ker(f_*)^0$. Thus, by Corollary \ref{criteria}, we have $\Sigma^2 \geq 0$. This is Claim (i).
To prove Claim (ii), we need to show that if $f$ admits no multiple fibers, then $\chi(X,\OO_X) = 0$. By Corollary \ref{ExtendabilityCorollary} and since $\chi(X,\OO_X) = \chi(J(X),\OO_{J(X)})$, we may assume that $f$ admits a section $\Sigma$. Applying adjunction and the canonical bundle formula, we obtain
$$
2g(\Sigma) - 2 = 2g(C) - 2 + \chi(X,\OO_{X}) + \Sigma^2.
$$
Since $\chi(X,\OO_{X})$ is always non-negative and $\Sigma^2$ is non-negative by Claim (i), we deduce from $g(\Sigma) = g(C)$ that $\chi(X,\OO_X) = \Sigma^2 = 0$, as claimed.
For the proof of Claim (iii), let us first assume that $(\alpha_p)^r \subseteq \Ker(f_*)$ for some $r \geq 1$. If $h^0(\Sigma,N_{\Sigma/X}) \leq r-1$, then $\Aut_{(\Sigma,X)}^0 \cap \Ker(f_*)$ is non-trivial, which is impossible by the same argument as in the first paragraph, since $\Sigma$ intersects a general fiber of $f$ transversally. By Lemma \ref{connsub}, the existence of $\alpha_p \subseteq \Ker(f_*)$ implies that the generic fiber of the Jacobian of $f$ is supersingular. Since a supersingular elliptic curve can be defined over a finite field, this implies that $J(f)$ is isotrivial and therefore the same holds for $f$.
If $\alpha_p \not \subseteq \Ker(f_*)$, then $\mu_{p^n} \subseteq \Ker(f_*)$ for some $n \geq 1$ by Lemma \ref{connsub}. Let $mF$ be an additive or supersingular fiber of $f$, where $m$ is the multiplicity of $F$ and let $P = f(F)$. To finish the proof, we have to show that $p^n \mid m$.
First, assume that $\mu_{p^n}$ preserves $F$. Since $F$ is additive or smooth, there is a reduced component $F_1$ of $F$ which meets at most one other component of $F$. Then, Lemma \ref{autoschemes} shows that $\mu_{p^n}$ has a fixed point $Q$ on $F_1$ that does not lie on any other component of $F$.
By Section \ref{Linearization}, we can linearize the $\mu_{p^n}$-action in a formal neighborhood of $Q$, i.e. there is a $\mu_{p^n}$-equivariant isomorphism
$$
\widehat{\OO}_{X,Q} \cong k[[x,y]]
$$
such that $\mu_{p^n}$ acts via $x \mapsto \lambda x, y \mapsto \lambda^a y$ for some $0 \leq a \leq p^n-1$.
Since $F_1$ is preserved by the $\mu_{p^n}$-action and the fixed locus of the $\mu_{p^n}$-action is contained in fibers of $f$, we can assume without loss of generality that $F_1$ is defined by $x = 0$. Let $t$ be a parameter on $C$ at $P$. The morphism $\varphi^{\#}: \widehat{\OO}_{C,P} \cong k[[t]] \to k[[x,y]]$ is then given by $\varphi^{\#}(t) = ux^m$, where $u \in k[[x,y]]$ is a unit and $m$ is the multiplicity of $F_1$.
Now, since the $\mu_{p^n}$-action on $C$ is trivial, we must have $ux^m \in k[[x,y]]^{\mu_{p^n}}$. In particular, the leading monomial of $ux^m$, which is of the form $c x^m$ for some $c \in k^{\times}$, has to be $\mu_{p^n}$-invariant. Thus, $p^n \mid m$.
If $\mu_{p^n}$ does not preserve $F$, then $F$ is smooth by Lemma \ref{movable} and Corollary \ref{fixedlocireducible}. The induced $\mu_p$-action does not preserve $F$ either, because otherwise it would fix $F$ pointwise by Lemma \ref{fixedloci} and then $\mu_{p^n}$ would preserve $F$ by Lemma \ref{fixedofsubscheme}. Hence, the quotient $X/\mu_{p^n}$ is smooth in a neighborhood of the image $F'$ of $F$ and the inverse image of $F'$ under the quotient map $X \to X/\mu_{p^n}$ is $p^nF$. Therefore, the multiplicity $m$ of $mF$ is divisible by $p^n$.
\qed
\vspace{5mm}
In the simpler case where $f: X \to C$ admits no multiple fibers, Theorem \ref{Main} specializes to the following corollary.
\begin{Corollary}\label{corollarymain}
Let $f: X \to C$ be an elliptic surface without multiple fibers and with $\Ker(f_*)^0$ non-trivial. Then, $\chi(X,\OO_X) = 0$ and $\Ker(f_*)^0 \in \{\mu_{p^n}, M_n, E\}$, where $n \geq 0$ and $E$ is an elliptic curve.
\end{Corollary}
\prf
Since $\Ker(f_*)^0 \cong \Ker(J(f)_*)^0$ by Corollary \ref{ExtendabilityCorollary}, we may assume that $f$ admits a section $\Sigma$. As in the proof of Theorem \ref{Main}, we have $\Sigma^2 = \chi(X,\OO_X) = 0$. In particular, $\Sigma$ satisfies $h^0(\Sigma,N_{\Sigma/X}) = 1$ and therefore $\Ker(f_*)[F] \in \{\mu_p,\alpha_p\}$ by Theorem \ref{Main}. Then, the result follows from Lemma \ref{connsub}.
\qed
\vspace{3mm}
\section{Horizontal automorphisms} \label{horizontal}
Recall that the group scheme of horizontal automorphisms of an elliptic surface is defined as follows.
\begin{Definition}
Let $f: X \to C$ be an elliptic surface. The group scheme of \emph{horizontal automorphisms} of $X$ is defined as $\im(f_*)$, where $f_*: \Aut_X^0 \to \Aut_C^0$ is the natural map.
\end{Definition}
If $f: X \to C$ is an elliptic surface such that $\im(f_*)$ is non-trivial, then certainly $\Aut_C^0$ is non-trivial and therefore $H^0(C,T_C) \neq 0$. In particular, either $C = \PP^1$ or $g(C) = 1$
Let us first treat the simpler case where the base curve $C$ satisfies $g(C) = 1$.
\begin{Proposition}\label{Propelliptic}
Let $f: X \to C$ be an elliptic surface with $g(C) = 1$. Assume that $\im(f_*)$ is non-trivial. Then, the following hold:
\begin{enumerate}[(i)]
\item All fibers of $f$ are movable and $f$ is isotrivial.
\item We have $\chi(X,\OO_X) = 0$, unless possibly if $p \in \{2,3\}$, $f$ admits a multiple fiber and both the generic fiber of $f$ and $C$ are supersingular.
\item If additionally $h^0(X,T_X) \geq 2$, then one of the following holds:
\begin{enumerate}[(1)]
\item $X$ is an Abelian surface and $h^0(X,T_X) = 2$.
\item $X$ is a bielliptic surface with $\omega_X = \OO_X$ and $h^0(X,T_X) = 2$.
\item The generic fiber of $f$ is supersingular and $f$ admits a multiple fiber.
\end{enumerate}
\end{enumerate}
\end{Proposition}
\prf
First, note that if $Z \subseteq X$ is any closed subscheme contained in a fiber of $f$, then $\Aut_{(Z,X)}^0 \subseteq \Ker(f_*)$. Indeed, the action of $\Aut_{(Z,X)}^0$ on $C$ preserves the reduced point $f(Z)$ by Lemma \ref{image} and is therefore trivial by Lemma \ref{fixedloci} (ii). Hence, all fibers of $f$ are movable and thus, by Lemma \ref{movable}, the fibration $f$ is isotrivial because all fibers of $f$ are either of type ${\rm II}$ or smooth and therefore the $j$-map has no poles. This proves Claim (i).
For Claim (ii), note that, by Ogg's formula, we have $\chi(X,\OO_X) = 0$ if and only if $f$ admits no fiber of type ${\rm II}$. Assume that $f$ admits a fiber $F$ of type ${\rm II}$. Then, $p \in \{2,3\}$ and the $j$-map of $f$ is identically $0$ so that the generic fiber of $f$ is supersingular and $\Aut_{(F,X)}^0 \subseteq \Ker(f_*)$ is non-trivial by Remark \ref{criteriaremark}. In particular, $f$ admits a multiple fiber by Theorem \ref{Main} (ii). Now, if $C$ is ordinary, then $\mu_p \subseteq \im(f_*)$ and since the induced extension of $\mu_p$ by $\Ker(f_*)$ splits by Lemma \ref{homsandexts} (ii) (3), there is a $\mu_p$-action on $X$. But then we can use Remark \ref{criteriaremark} again to deduce that $\mu_p \subseteq \Ker(f_*)$, which is impossible since the generic fiber of $f$ is supersingular. Thus, $C$ has to be supersingular, too.
For Claim (iii), assume that $h^0(X,T_X) \geq 2$. Let $D \in H^0(X,T_X)$ be a $p$-closed vector field. Then, we have the following short exact sequence obtained by saturating the inclusion $\OO_X \to T_X$ induced by $D$, where $Z$ (resp. $W$) is the divisorial (resp. codimension $2$) part of the zero locus of $D$:
$$
0 \to \OO_X(Z) \to T_X \to I_W(-Z-K_X) \to 0.
$$
Note that $K_X$ is effective by the canonical bundle formula and $Z$ is effective by definition. Now, we get two cases according to whether $-Z-K_X$ is effective or not.
If $-Z-K_X$ is not effective, then $h^0(X,T_X) = h^0(X,\OO_X(Z)) \geq 2$. Therefore, the zero locus of every $p$-closed vector field contains a divisor linearly equivalent to $Z$, hence all these vector fields induce the trivial vector field on $C$. In particular, the tangent space of $\Ker(f_*)$ is at least $2$-dimensional. By Lemma \ref{connsub} and Corollary \ref{corollarymain}, this implies that the generic fiber of $f$ is supersingular and $f$ admits a multiple fiber, that is, $X$ is as in Case (3).
If $-Z-K_X$ is effective, then both $Z$ and $K_X$ are trivial. Since $h^0(X,T_X) \geq 2$, we must have $h^0(X,I_W) \geq 1$ and hence $W$ is trivial. Thus, $h^0(X,T_X) = 2$ and $X$ is Abelian or bielliptic with $\omega_X \cong \OO_X$ by the classification of surfaces. In particular, $X$ is as in Case (1) or (2).
\qed
\vspace{2mm}
Now that we understand the case where the base curve $C$ has genus $1$, it remains to treat the case where $C = \PP^1$. If $C$ is rational, then $b_1(X) \in \{0,2\}$ by \cite[Lemma 3.4]{KatsuraUeno}. In the second case we can argue in a similar fashion as in Proposition \ref{Propelliptic} because of the following lemma.
\begin{Lemma}\label{b1=2}
Let $f: X \to \PP^1$ be an elliptic surface with $b_1(X) = 2$. Then, the following hold:
\begin{enumerate}[(i)]
\item $\chi(X,\OO_X) = 0$.
\item There is an isomorphism $J(X) \cong \PP^1 \times E$, where $E$ is a general fiber of $f$.
\item The Albanese map $a_X: X \to {\rm Alb}(X)$ is a fibration over an elliptic curve and all fibers of $a_X$ are irreducible and reduced.
\item There is an action of $E$ on $X$ that induces a transitive action on ${\rm Alb}(X)$.
\end{enumerate}
\end{Lemma}
\prf
By Ogg's formula, to prove that $c_2(X) = \chi(X,\OO_X) = 0$, it suffices to show that $f$ admits no singular fibers. But this follows immediately from the criterion in\cite[Lemma 3.4]{KatsuraUeno}, since singular fibers consist of rational curves and are thus contracted by the Albanese morphism of $X$. This proves Claim (i).
Next, we prove Claim (ii) and show that $E$ acts on $X$. Since $c_2(X) = 0$, the fibration $f$ is isotrivial and $J(f)$ is a smooth elliptic fibration over $\PP^1$ by Ogg's formula. As there are no non-trivial finite \'etale covers of $\PP^1$, this implies that $J(X) \cong \PP^1 \times E$ for a general fiber $E$ of $f$. Hence, there is a finite Galois cover $C \to \PP^1$ with group $G$ such that the normalization of $X \times_{\PP^1} C$ is isomorphic to $E \times C$. Moreover, the quotient of $E \times C$ by the induced action of $G$ maps via a finite and birational map to $X$ and hence coincides with $X$. Since $X$ is smooth, the group $G$ acts via translations on the first factor of $E \times C$, for otherwise it would have an isolated fixed point and then $X$ would be singular. In particular, the translation action of $E$ on the first factor of $E \times C$ commutes with the $G$-action and thus descends to $X$.
To finish the proof, note that, by Igusa's formula, we have ${\rm rk} (\Pic(X)) \leq b_2(X) = c_2(X) + 2b_2(X) - 2 = 2$, so that all fibers of $f$ and $a_X$ have to be irreducible. Since $b_1(X) = 2$, the Albanese variety ${\rm Alb}(X)$ is an elliptic curve. Moreover, a general fiber $E$ of $f$ maps surjectively onto ${\rm Alb}(X)$, so that the target of the Stein factorization of $a_X$ is an elliptic curve, which then has to coincide with ${\rm Alb}(X)$ by the universal property of $a_X$. This also shows that the action of $E$ on ${\rm Alb}(X)$ is transitive and, because a general fiber of $a_X$ is reduced, this implies that in fact all fibers of $a_X$ are reduced.
\qed
\begin{Proposition}\label{Proprational}
Let $f: X \to \PP^1$ be an elliptic surface with $b_1(X) = 2$. If $\im(f_*)$ is non-trivial, then the following hold:
\begin{enumerate}[(i)]
\item At most two fibers of $f$ are non-movable and $\chi(X,\OO_X) = 0$.
\item If additionally $h^0(X,T_X) \geq 2$, then one of the following holds:
\begin{enumerate}[(1)]
\item $X$ is ruled over an elliptic curve.
\item $X$ is bielliptic, $\omega_X \cong \OO_X$ and $h^0(X,T_X) = 2$.
\item The generic fiber of $f$ is supersingular, $f$ admits a multiple fiber, and all fibers of $a_X$ are rational curves.
\end{enumerate}
\end{enumerate}
\end{Proposition}
\prf
The argument for Claim (i) is the same as in the proof of Proposition \ref{Propelliptic} (i), with the only difference that a subgroup scheme of $\Aut_{\PP^1}$ can have up to two fixed points by Lemma \ref{fixedloci} (i).
Note that by Lemma \ref{b1=2} the equality $\chi(X,\OO_X) = 0$ holds even if $\im(f_*)$ is trivial.
Before we start proving Claim (ii), observe that if $f$ does not admit a multiple fiber, then $X$ is ruled by the canonical bundle formula and the base curve of the ruling must be an elliptic curve, since $X$ admits an elliptic fibration. Hence, we are in Case (1) if $f$ admits a section.
Now, let us prove Claim (ii). Assume that $h^0(X,T_X) \geq 2$. Then, $\Ker((a_X)_*)$ is non-trivial. Therefore, if a general fiber $C$ of $a_X$ is smooth, then $C$ is either $\PP^1$ or an elliptic curve. Thus, in the first case, the Albanese map $a_X$ yields a ruling of $X$ over ${\rm Alb}(X)$. In the latter case, the Albanese map $a_X$ is an elliptic fibration over an elliptic curve and $a_X$ has no multiple fibers by Lemma \ref{b1=2}. Then, Corollary \ref{corollarymain} shows that the tangent space of $\Ker((a_X)_*)$ is $1$-dimensional, hence $\im((a_X)_*)$ is non-trivial. Thus, we can apply Proposition \ref{Propelliptic} (iii). It shows that $X$ is bielliptic with $\omega_X \cong \OO_X$, since an Abelian surface does not admit an elliptic fibration over $\PP^1$ and $a_X$ admits no multiple fibers by Lemma \ref{b1=2} (iii).
So, we may assume that the general fiber $C$ of $a_X$ is singular.
Let $E$ be a general fiber of $f$. The map $E \to {\rm Alb}(X)$ factors through an \'etale morphism $A \to {\rm Alb}(X)$. Pulling back $X$ along this map, we obtain a smooth surface $X'$ with a fibration $g': X' \to A$ and an elliptic fibration $f': X' \to D$ obtained as the Stein factorization of $X' \to \PP^1$.
Now, both the action of $\Ker((a_X)_*)$ on $X$ and the action of $E$ on $X$ constructed in Lemma \ref{b1=2} lift to $X'$ and these two actions generate $\Aut_X^0$. Therefore, we have $h^0(X',T_X') \geq 2$ and $\im(f'_*)$ is non-trivial. Thus, $D$ is either an elliptic curve or $\PP^1$. If $D$ is an elliptic curve, then ${\rm Alb}(X')$ is a surface by \cite[Lemma 3.4]{KatsuraUeno} and the morphism $g': X' \to A$ factors through ${\rm Alb}(X')$. This is impossible, since the fibers of $g'$ are singular whereas the fibers of ${\rm Alb}(X')$ are (unions of) elliptic curves. Hence, we must have $D \cong \PP^1$ and we may replace $X$ by $X'$ to assume that a general fiber of $f$ maps purely inseparably to ${\rm Alb}(X')$ and in particular all multiple fibers of $f$ have multiplicity $p^n$ for some $n \geq 1$. The remainder of the proof splits into two cases according to whether the generic fiber of $f$ is ordinary or supersingular.
If the generic fiber of $f$ is ordinary then so is ${\rm Alb}(X)$. Let $m_iF_i$ be a multiple fiber of $f$. Since the map $F_i \to X \to {\rm Alb}(X)$ is purely inseparable, the dual map $\Pic^0_{{\rm Alb}(X)} \to \Pic^0_X \to \Pic^0_{F_i}$ is \'etale. In particular, the map $H^1({\rm Alb}(X), \OO_{{\rm Alb}(X)}) \to H^1(X,\OO_X) \to H^1(F_i,\OO_{F_i})$ is an isomorphism. By \cite[Section 6]{KatsuraUeno}, one can use cocycles $\rho \in H^1(X,\OO_X)$ that are fixed by Frobenius and map to a non-trivial element in $H^1(F_i,\OO_{F_i})$ to construct an \'etale cover $\widetilde{X}$ of $X$ with an elliptic fibration $\widetilde{f}: \widetilde{X} \to D$ without multiple fibers. Choosing the cocycles $\rho$ in the image of $H^1({\rm Alb}(X), \OO_{{\rm Alb}(X)}) \to H^1(X,\OO_X)$, we can assume that $\widetilde{X}$ arises as pullback of $a_X$ along an \'etale isogeny $A \to {\rm Alb}(X)$. Then, as in the previous paragraph, the group scheme $\Aut_X^0$ acts on $\widetilde{X}$ and $\im(\widetilde{f}_*)$ is non-trivial and the image $D$ of the Stein factorization of $\widetilde{X} \to \PP^1$ satisfies $D \cong \PP^1$. Since $\widetilde{f}$ admits no multiple fibers and $\PP^1$ admits no \'etale covers, the equality $c_2(\widetilde{X}) = c_2(X) = 0$ implies $\widetilde{X} \cong \PP^1 \times A$, contradicting our assumption that the fibers of $a_X$ are singular.
Hence, the generic fiber of $f$ is supersingular. Then, it is shown in \cite[Proposition 3.1]{Kawazoe} that there is a purely inseparable cover $\widetilde{\pi}: \widetilde{X} \to X$ such that the Stein factorization of $f \circ \widetilde{\pi}$ is an elliptic fibration $\widetilde{f}: \widetilde{X} \to D$ without multiple fibers. Since $\widetilde{\pi}$ is purely inseparable, so is $D \to \PP^1$. Hence, $D \cong \PP^1$ and $\widetilde{f}$ admits a section $\Sigma$. The image $\widetilde{\pi}(\Sigma)$ is a rational curve and is therefore contracted by $a_X$. The fibers of $a_X$ are integral, so that $\widetilde{\pi}(\Sigma)$ coincides with a fiber of $a_X$. Hence, all fibers of $a_X$ are rational curves.
\qed
\vspace{5mm}
Thus, the last remaining case are elliptic surfaces $f: X \to \PP^1$ with $b_1(X) = 0$. We will use the following lemma, which is well-known in characteristic $0$.
\begin{Lemma}\label{twofibers}
Let $f: X \to \PP^1$ be an elliptic surface with $b_1(X) = 0$ and at most two singular fibers $F_1,F_2$ with Swan conductors $\delta_{F_1} = \delta_{F_2} = 0$. Then, $\chi(X,\OO_X) = 1$ and the possible types of $F_1$ and $F_2$ are as follows:
\begin{enumerate}[(i)]
\item $({\rm II},{\rm II}^*)$ and $p \not\in \{ 2,3 \}$.
\item $({\rm III},{\rm III}^*)$ and $p \neq 2$.
\item $({\rm IV},{\rm IV}^*)$ and $p \neq 3$.
\item $({\rm I}_0^*,{\rm I}_0^*)$ and $p \neq 2$.
\end{enumerate}
\end{Lemma}
\prf
Since $f$ and $J(f)$ have the same types of singular fibers and $\chi(X,\OO_X) = \chi(J(X),\OO_{J(X)})$, we may assume that $f$ admits a section. Let $F_1,F_2$ be the singular fibers of $f$.
Let $T = U \oplus T_1 \oplus T_2$, where $U$ is the unimodular lattice generated by a section of $f$ and the class of a fiber of $f$, and $T_i$ is the lattice generated by the components of $F_i$ disjoint from the zero section of $J(f)$. Then, Igusa's inequality yields
$$
{\rm rk}(T) = 2 + {\rm rk}(T_1) + {\rm rk}(T_2) \leq {\rm rk}(\Num(X)) \leq b_2(X) = c_2(X) - 2.
$$
On the other hand, by Ogg's formula and our assumption that the Swan conductor of every fiber is trivial, we have
$$
c_2(X) - 2 = e(F_1) + e(F_2) - 2 \leq 2 + {\rm rk}(T_1) + {\rm rk}(T_2)
$$
with equality if and only if $F_1$ and $F_2$ are additive. Thus, both $F_1$ and $F_2$ are additive fibers and $T \subseteq \Num(X)$ is of finite index. Moreover, $T_1$ and $T_2$ are either trivial or root lattices of type $A_1,A_2,D_n,E_6,E_7$ or $E_8$. Their discriminants are $2,3,4,3,2$ and $1$, respectively.
Since ${\rm rk} (\Num(X)) = b_2(X)$, we can use $\ell$-adic Poincar\'e duality for all $\ell \neq p$ to deduce that the discriminant ${\rm disc}(\Num(X))$ is a power of $p$. Moreover, since $T \subseteq \Num(X)$ is of finite index, the discriminants of these two lattices differ by a square. Taking into account ${\rm rk}(T) = b_2(X) = c_2(X) - 2 = 10 +12k$ for some $k \geq 0$, we thus have the following cases, where in each case we have ${\rm disc}(\Num(X)) = 1$:
\begin{enumerate}[(i)]
\item $T_1 = 0$, $T_2 = E_8$ and $p \neq 2,3$,
\item $T_1 = A_1$, $T_2 = E_7$ and $p \neq 2$,
\item $T_1 = A_2$, $T_2 = E_6$ and $p \neq 3$,
\item $T_1 = D_m$, $T_2 = D_n$ for some $m,n$ and $p \neq 2$,
\item $T_1 = D_m$, $T_2 \in \{E_8,0\}$ for some $m$ and $p \neq 2$.
\end{enumerate}
Now, if $p \neq 2$, we can apply a quadratic twist to $f$ that only changes the fibers $F_1$ and $F_2$. Then, either all fibers of the twisted fibration are smooth or the fibration satisfies the assumptions of the lemma and then its singular fibers have to appear in the above list. Hence, Lemma \ref{twist} (i) shows that $m = n = 4$ in Case $(iv)$ and that Case $(v)$ does not exist.
\qed
\vspace{2mm}
\begin{Remark} \label{twofibersremark}
The Jacobian $J(X)$ of each of these four types of surfaces in the above Lemma \ref{twofibers} is a rational surface that can be defined over $\ZZ[j(F_\eta)]$, where $F_\eta$ is the generic fiber of $f$. The reductions of these fibrations modulo the excluded characteristics in the respective cases in Lemma \ref{twofibers} become quasi-elliptic (see e.g. \cite{Lang3}). It is straightforward, e.g. from the equations given in \cite{Lang3}, to check that $J(X)$ is the minimal resolution of $(\PP^1 \times E)/(\ZZ/n\ZZ)$ with $n \in \{2,3,4,6\}$ and $\ZZ/n\ZZ$ acting diagonally with a fixed point on $E$.
\end{Remark}
If we allow the Swan conductors $\delta_{F_i}$ to be non-trivial in Lemma \ref{twofibers}, there are many examples of elliptic surfaces with only one or two singular fibers. In the following lemma, we will treat a very special case that will appear in Theorem \ref{mainrational}.
\begin{Lemma} \label{twofibers23}
Let $f: X \to \PP^1$ be an elliptic surface with at most two singular fibers $F_1,F_2$. Assume that $\delta_{F_2} = 0$, and either $p = 3$ and $F_1$ is of type ${\rm II}$ with $\delta_{F_1} = 1$ or $p = 2$ and $F_1$ is of type ${\rm II}$ or ${\rm III}$ with $\delta_{F_1} = 2$ or $\delta_{F_1} = 1$, respectively. Then, the following hold:
\begin{enumerate}[(i)]
\item If $p = 3$, the possible types of $F_1$ and $F_2$ are $({\rm II},{\rm III}^*),({\rm II},{\rm I}_{3^{2k}}),({\rm II},{\rm I}_{3^{2k-1}}^*)$. In particular, $c_2(X) = 3^{2k} + 3$ or $c_2(X) = 3^{2k-1} + 9$ for some $k \geq 1$.
\item If $p = 2$, the possible types of $F_1$ and $F_2$ are $({\rm II},{\rm IV}^*),({\rm III},{\rm IV}^*),({\rm II},{\rm I}_{2^{2k+1}}),({\rm III},{\rm I}_{2^{2k+1}})$. In particular, $c_2(X) = 2^{2k+1} + 4$ for some $k \geq 1$.
\end{enumerate}
\end{Lemma}
\prf
As in the proof of Lemma \ref{twofibers}, we may assume that $f$ admits a section. We will split the proof in two cases according to whether $f$ is isotrivial or not.
Assume first that $f$ is isotrivial. Then, since the $j$-map of $f$ has a zero at $f(F_1)$, the generic fiber of $f$ is supersingular with $j$-invariant $0$. If $p = 3$, then $v_{f(F_1)}(\Delta_f) = 3$ by assumption so that $v_{f(F_2)}(\Delta_f) \equiv 9$ mod $12$ by Ogg's formula. Moreover, we have $\delta_{F_2} = 0$ and the $j$-map has no pole at $f(F_2)$, so that $F_2$ is additive with $8$ components. This implies that $F_2$ is of type ${\rm III}^*$. If $p = 2$, then $v_{f(F_1)}(\Delta_f) = 4$ and thus, by the same argument as before, $F_2$ is additive with $7$ components. Since $\delta_{F_2} = 0$, this implies that $F_2$ is of type ${\rm IV}^*$.
Next, assume that $f$ is not isotrivial. Then, the $j$-map of $f$ has a pole. If $p = 3$, this implies that $F_2$ is of type ${\rm I}_n^*$ or ${\rm I}_n$. Then, by Lemma \ref{twist}, we can replace $f$ by a quadratic twist and assume that $F_2$ is of type ${\rm I}_n$ and $F_1$ is of type ${\rm II}$ or ${\rm II}^*$ with $\delta_{F_1} = 1$.
If $p = 2$, the assumption $\delta_{F_2} = 0$ forces $F_2$ to be of type ${\rm I}_n$. Moreover, by Lemma \ref{twist}, we can replace $f$ by a quadratic twist to assume that $F_1$ is of type ${\rm III}$ with $\delta_{F_1} = 1$.
Now, we let $T = U \oplus T_1 \oplus T_2$, where $T_i$ is spanned by non-identity components of $F_i$, and $U$ is generated by the class of a fiber and a section of $f$. Note that $T_1$ is unimodular by the previous paragraph. Since $F_2$ is multiplicative and $F_1$ satisfies $\delta_{F_1} = 1$, we obtain ${\rm rk} (T) = b_2(X)$ from Ogg's formula. Hence, $T$ has finite index in $\Num(X)$. As in the proof of Lemma \ref{twofibers}, $\ell$-adic Poincar\'e duality shows that ${\rm disc}(\Num(X))$ is a power of $p$. By \cite[Section 11.10]{schuettShioda} this implies that $n = p^i m^2$, where $i$ is some integer and $m$ is the order of the group of torsion sections of $f$ of order prime to $p$. Since, on the one hand, a torsion section of order prime to $p$ is disjoint from the zero section \cite[Proposition 3.5 (iv)]{OguisoShioda} and, on the other hand, $f$ admits the fiber $F_1$ whose underlying group is $\GG_a$, we have $m = 1$. Thus, we have $v_{f(F_1)}(\Delta_f) + p^i \equiv 0$ mod $12$ by Ogg's formula.
If $p = 3$ and $F_1$ is of type ${\rm II}$, this implies that $i = 2k$ for some $k \geq 1$, and if $F_1$ is of type ${\rm II}^*$, this implies that $i = 2k-1$ for some $k \geq 1$. Undoing the quadratic twist we applied in the second paragraph of the proof, we obtain the stated types of singular fibers.
If $p = 2$, then this implies $i = 2k+1$ for some $k \geq 1$. Again, undoing the quadratic twist, we obtain the stated types of singular fibers.
\qed
\vspace{2mm}
After having prepared the necessary technical lemmas, we are now ready to prove the main result of this section on elliptic surfaces $f: X \to \PP^1$ with $b_1(X) = 0$.
\begin{Theorem} \label{mainrational}
Let $f: X \to \PP^1$ be an elliptic surface with $b_1(X) = 0$. Assume that $\im(f_*)$ is non-trivial. Then, $f$ has at most two non-movable fibers and one of following holds:
\begin{enumerate}[(i)]
\item $f$ is isotrivial with precisely two singular fibers of the types given in Lemma \ref{twofibers}. Moreover, $\Aut_X^0 \subseteq \GG_m$.
\item $p \in \{2,3\}$ and $f$ admits precisely two singular fibers $F_1,F_2$ of the types given in Lemma \ref{twofibers23}. Moreover, $\Aut_X^0 \cong \im(f_*) \cong \mu_p$ and there are no multiple fibers except possibly $F_1$ and $F_2$.
\item $p = 2$, the generic fiber of $f$ is ordinary, $f$ admits a fiber $F$ of type ${\rm I}_{8k + 4}^*$ with $\delta_{F} = 4k+8$ for some $k \geq 0$ and all other fibers of $f$ are smooth and non-multiple. Moreover, we have $\Aut_X^0 \cong \im(f_*) \subseteq \GG_a$.
\item $p \in \{2,3\}$ and $f$ is isotrivial with supersingular generic fiber and at most one non-movable fiber. Moreover, the group scheme $\Aut_X^0$ does not contain $\mu_p$.
\end{enumerate}
\end{Theorem}
\prf
Since $b_1(X) = 0$, we have $c_2(X) > 0$ and thus $f$ admits at least one singular fiber $m_1F_1$, say over $\infty \in \PP^1$, where $m_1$ is the multiplicity of the fiber. On the other hand, by the same argument as in Proposition \ref{Proprational}, $f$ admits at most two non-movable fibers. To prove the remaining claims, we will make use of fact that we understand the fixed loci of $\mu_p$-actions on $X$ by Theorem \ref{fixedpointformula}. To do this, we will split the proof into three cases according to whether $\mu_p \subseteq \Ker(f_*)$, $\mu_p \subseteq \im(f_*)$ or $\Aut_X^0$ does not contain any $\mu_p$ at all.
\vspace{2mm}
\underline{Case $\mu_p \subseteq \Ker(f_*)$:}
Assume that $\mu_p \subseteq \Ker(f_*)$. Then, by Lemma \ref{connsub}, we have $\mu_p = \Ker(f_*)[F]$ and in particular $\mu_p$ is preserved by every automorphism of $\Ker(f_*)$. Since $\Ker(f_*)$ is normal in $\Aut_X^0$, this implies that $\mu_p$ is normal in $\Aut_X^0$. Therefore, Lemma \ref{fixedofsubscheme} implies that the action of $\Aut_X^0$ on $X$ preserves $X^{\mu_p}$. Since $\im(f_*)$ is non-trivial by assumption, we can apply Lemma \ref{fixedloci} to deduce that there is a fiber $m_2F_2$ of $f$, say over $0 \in \PP^1$, such that $X^{\mu_p} \subseteq F_1 \cup F_2$, for otherwise the action of $\Aut_X^0$ on $\PP^1$ would have more than two fixed points, which is impossible. By Remark \ref{criteriaremark} and Lemma \ref{fixedloci}, the $\mu_p$-action on $X$ preserves every singular fiber of $f$ and has at least one fixed point on each such fiber. In particular, $m_1F_1$ and $m_2F_2$ are the only possibly singular fibers of $f$. The fixed point formula given in Theorem \ref{fixedpointformula} then yields
\begin{equation}\label{fixedpointeqn}
c_2(X) = e(F_1^{\mu_p}) + e(F_2^{\mu_p}) \tag{$\ast$}
\end{equation}
Assume that $p \not \in \{2,3\}$. Then, Lemma \ref{twofibers} implies that both $F_1$ and $F_2$ are singular of the types given in Lemma \ref{twofibers}.
If $p \in \{2,3\}$, we can compare Equation \eqref{fixedpointeqn} with Ogg's formula to obtain
$$
e(F_1^{\mu_p}) + e(F_2^{\mu_p}) = c_2(X) = e(F_1) + e(F_2) + \delta_{F_1} + \delta_{F_2}.
$$
By Corollary \ref{fixedlocireducible}, we know that $e(F_i^{\mu_{p}}) = e(F_i)$ holds unless $F_i$ is of type ${\rm II}$, or $F_i$ is of type ${\rm III}$ and $p = 2$. If neither $F_1$ nor $F_2$ are of these types, then $\delta_{F_1} = \delta_{F_2} = 0$ and we conclude as in the case $p \not \in \{2,3\}$. Note also that if $F_i$ is of type ${\rm II}$ or ${\rm III}$, then $e(F_i^{\mu_p}) \leq 4$ by Lemma \ref{fixedloci} and Corollary \ref{fixedlocireducible}, so that $c_2(X) = 12 \chi(X,\OO_X) \geq 12$ implies that not both $F_1$ and $F_2$ are of these exceptional types. Hence, we may assume that $e(F_2) = e(F_2^{\mu_p})$.
Assume that $p = 3$ and $F_1$ is of type ${\rm II}$. Then, we have $e(F_1^{\mu_3}) = 3 = e(F_1) + 1$ by Corollary \ref{fixedlocireducible}, and hence Equation \eqref{fixedpointeqn} shows that $\delta_{F_1} + \delta_{F_2} = 1$. Since $\delta_{F_1} \geq 1$, this implies $\delta_{F_2} = 0$ and thus we can apply Lemma \ref{twofibers23} to determine the types of $F_1$ and $F_2$.
If $p = 2$ and $F_1$ is of type ${\rm III}$, then $e(F_1^{\mu_2}) = 4 = e(F_1) + 1$ and Equation \eqref{fixedpointeqn} shows that $\delta_{F_1} + \delta_{F_2} = 1$. The rest of the argument is as in the case $p = 3$. Similarly, if $F_1$ is of type ${\rm II}$, then $\delta_{F_1} \geq 2$, so that again $\delta_{F_2} = 0$ and Lemma \ref{twofibers23} applies.
We have shown that the singular fibers of $f$ are as claimed in (i), (ii), or (iii) and it remains to prove the assertions on the structure of $\Aut_X^0$ and the multiple fibers. For this, we will first show that $h^0(X,T_X) \leq 1$ holds.
Denote the divisorial part of $X^{\mu_p}$ by $Z$ and the isolated part by $W$. Then, the saturation of the section of $T_X$ given by the $\mu_p$-action yields an exact sequence
$$
0 \to \OO_X(Z) \to T_X \to I_W(- K_X - Z) \to 0.
$$
Since $F_1$ and $F_2$ are singular and $Z$ is smooth, the $F_i$ cannot be contained in $Z$. Hence, we have $h^0(X,\OO_X(Z)) \leq 1$ and the above sequence shows that $h^0(X,T_X) \leq 1$, unless possibly if $- K_X$ is effective. If $-K_X$ is effective, then the canonical bundle formula shows that $f$ admits no wild fibers and at most one multiple fiber. In this case, if $f$ admits no multiple fiber, then $\Ker(f_*)$ is trivial by Theorem \ref{Main} so that $h^0(X,T_X) \leq 1$. If $f$ admits a multiple fiber $F$, then $h^0(X,\OO_X(-K_X)) = 1$ and $-K_X \sim F$. In this case, we also have $h^0(X,T_X) \leq 1$, unless $X^{\mu_p} \subseteq F$. But Lemma \ref{fixedloci} shows that $\mu_p$ has fixed points on both $F_1$ and $F_2$, so $X^{\mu_p} \subseteq F$ is impossible. Therefore, we have $h^0(X,T_X) \leq 1$ in all cases.
Now, since $\mu_p$ has fixed points on $F_1$ and $F_2$ and $\Aut_X^0$ acts on $X^{\mu_p}$, we have $\im(f_*) \subseteq \Aut^0_{0 \cup \infty, \PP^1} \cong \GG_m$. By Theorem \ref{Main}, we also have $\Ker(f_*) \cong \mu_{p^n}$ for some $n \geq 0$. Thus, the group scheme $\Aut_X^0[F^n]$, being an extension of finite commutative group schemes of type $lr$, is also commutative of type $lr$ by Lemma \ref{homsandexts}. Since $h^0(X,T_X) \leq 1$, Lemma \ref{finitecommgroupscheme} implies $\Aut_X^0[F^n] \cong \mu_{p^n}$ and therefore either $\Aut_X^0 \cong \GG_m$ or $\Aut_X^0 \cong \mu_{p^n}$ for some $n \geq 1$. In the cases where $p \in \{2,3\}$ and $F_1$ is of type ${\rm II}$ or ${\rm III}$, Lemma \ref{fixedloci} and Corollary \ref{fixedlocireducible} imply that $n = 1$. But then $\Aut_X^0$ acts trivially on the base, contradicting $\mu_p \subseteq \Ker(f_*)$. Putting everything together, we see that $\mu_p \subseteq \Ker(f_*)$ implies that we are in Case (i).
\vspace{2mm}
\underline{Case $\mu_p \not \subseteq \Ker(f_*)$ and $\mu_p \subseteq \im(f_*)$:}
If $\mu_p \subseteq \im(f_*)$ and $\mu_p \not \subseteq \Ker(f_*)$, then by Lemma \ref{connsub} $\Ker(f_*)$ is either trivial or finite and commutative of type $ll$. Thus, the extension of $\mu_p$ by $\Ker(f_*)$ splits by Lemma \ref{homsandexts} and we get a $\mu_p$-action on $X$ whose fixed locus is contained in two fibers, which are then necessarily the only singular or multiple fibers of $f$. Then, the arguments where we compute the types of $F_1$ and $F_2$ and deduce $h^0(X,T_X) \leq 1$ are the same as in the previous case. But this shows that $\Ker(f_*)$ has to be trivial, for otherwise $\Aut_X^0$ would contain $\alpha_p \rtimes \mu_p$ and thus its tangent space would be too big. Moreover, the fiber $F_2$ is not movable by Lemma \ref{movable}, so $\Aut_X^0 \cong \im(f_*) \subseteq \Aut^0_{(0,\PP^1)} \cong \GG_a \rtimes \GG_m$. The only subgroup schemes of $\GG_a \rtimes \GG_m$ which have $1$-dimensional tangent space and contain $\mu_p$ are $\mu_{p^n}$ and $\GG_m$, so $\Aut_X^0$ has to be one of those two group schemes. Moreover, in the cases where $p \in \{2,3\}$ and $F_1$ is of type ${\rm II}$ or ${\rm III}$, Lemma \ref{fixedloci} and Corollary \ref{fixedlocireducible} imply that $\Aut_X^0 \cong \mu_p$. Thus, $\mu_p \not \subseteq \Ker(f_*)$ and $\mu_p \subseteq \im(f_*)$ imply that we are in Case (i) or (ii).
\vspace{2mm}
\underline{Case $\mu_p \not \subseteq \Ker(f_*)$ and $\mu_p \not \subseteq \im(f_*)$:}
Since $\mu_p \not \subseteq \im(f_*)$, Lemma \ref{fixedloci} shows that the action of $\Aut_X^0$ on $\PP^1$ has at most one fixed point and thus $f$ has at most one non-movable fiber. Hence, by Lemma \ref{movable} and Lemma \ref{twofibers}, we have $p \in \{2,3\}$.
If the generic fiber of $f$ is supersingular, then we are in Case (iv), so we may assume that the generic fiber of $f$ is ordinary.
Assume that $f$ has ordinary generic fiber. Since $\mu_p \not \subseteq \Ker(f_*)$, we have $\Ker(f_*) = \{ {\rm id}\}$. The $j$-map is not identically $0$, so $F_1$ is not of type ${\rm II}$ and in particular not movable by Lemma \ref{movable}. Hence, $\Aut_X^0$ acts on $\PP^1$ with a fixed point and thus $\Aut_X^0 \subseteq \Aut_{(\infty,\PP^1)}^0 \cong \GG_a \rtimes \GG_m$. In fact, by our assumption that $\mu_p \not \subseteq \im(f_*)$, we have $\Aut_X^0 \subseteq \GG_a$.
In particular, there is an $\alpha_p \subseteq \Aut_X^0$ that acts non-trivially on $\PP^1$. By Lemma \ref{movable} singular fibers are preserved by $\alpha_p$, hence $F_1$ is the only singular fiber of $f$. Similarly, if $f$ admits a multiple fiber $mF$ different from $F_1$, then $mF$ is movable so that $pF \subseteq mF$ by Lemma \ref{movable}. But then $\alpha_p$ preserves $pF$ by Lemma \ref{Frobeniuspower}, contradicting the fact that $\alpha_p$ acts with only one fixed point on $\PP^1$. Therefore, all multiple or singular fibers of $f$ are equal to $F_1$.
By Lemma \ref{twist} (iii), this implies that $f$ is isotrivial and $F_1$ is of type ${\rm I}_{8k+4}^*$ with $\delta_{F_1} = 4k + 8$ for some $k \geq 0$. Hence, we are in Case (iii). This finishes the proof.
\qed
\vspace{15mm}
\section{Examples}\label{examples}
The purpose of this section is to give several examples illustrating the different phenomena discussed in the previous two sections.
\subsection{Examples with many global vector fields} \label{Vectorfieldssection}
In this section, we show that all types of surfaces $X$ with $h^0(X,T_X) \geq 2$ listed in Theorem (A) actually occur. Moreover, we give a series of examples proving Theorem (A) (iii). Throughout, $E$ denotes an elliptic curve.
\begin{Example}[\emph{Elliptic ruled surfaces}]\label{ruled}
If $X$ is ruled over $E$, let $a_X: X \to E$ be the ruling and assume that $X$ admits an elliptic fibration $f: X \to \PP^1$. Being a ruled surface, $X$ can be written as $X = \PP(\cal{E})$ for some normalized (in the sense of \cite[Chapter V, Proposition 2.8]{Hartshorne}) locally free sheaf $\cal{E}$ of rank $2$ on $E$. Let $e := -\deg(\cal{E})$. Using the results of \cite[Chapter V, Corollary 2.18]{Hartshorne}, it is straightforward to check that $e \in \{0,-1\}$. Therefore, either $\cal{E}$ is the unique indecomposable vector bundle of rank $2$ on $E$ with $e \in \{0,-1\}$ or $\cal{E} \cong \OO_E \oplus \cal{L}$ for a torsion line bundle $\cal{L}$ on $E$ of order $n \geq 0$. Finally, it follows from \cite[Theorem 9]{Atiyah} that if $p = 0$ and $\cal{E}$ is indecomposable with $e = 0$, then $X$ does not admit an elliptic fibration while \cite[Proposition, p.336]{Mumford} implies that the corresponding ruled surface admits an elliptic fibration in positive characteristic.
Alternatively, these surfaces can be described as $X = (E \times \PP^1)/G$, where $G \subseteq E$ is a finite subgroup scheme acting faithfully on $\PP^1$. With this description, it is clear that if $N$ is the normalizer of $G$ in $\Aut_{E \times \PP^1}$, then $\Aut_X = N/G$. Since $\Aut_{E \times \PP^1}^0 \cong E \times {\rm PGL}_2$ and $E$ is commutative, we can calculate $N^0$ as the product of the centralizers of $G$ in $E$ and ${\rm PGL}_2$, respectively. Putting all of this together, the connected component of the automorphism scheme of an elliptic surface which is also ruled over an elliptic curve $E$ is as in the following table:
\begin{table}[h!]
\resizebox{\textwidth}{!}{$\displaystyle
\begin{array}{|l|l|l|l|} \hline
\cal{E} & G & \Aut_X^0/E & h^0(X,T_X) \\ \hline
\OO_E \oplus \OO_E & \{1\} & {\rm PGL}_2 & 4 \\ \hline
\OO_E \oplus \cal{L} & \mu_n & \GG_m & 2 \\ \hline
indec., e = -1 &
E[2] & \{1\} & 1 \\ \hline
indec., e = 0 & \begin{cases}
\ZZ/p\ZZ &\text{ if } E \text{ is ordinary} \\
\alpha_p &\text{ if } E \text{ is supersingular}
\end{cases} &
\begin{cases}
\GG_a &\text{ if } p \neq 2\\
\GG_a \rtimes \mu_2 & \text{ if } p = 2 \text{ and } E \text{ is ordinary} \\
\GG_a \times \alpha_2 & \text{ if } p = 2 \text{ and } E \text{ is supersingular}
\end{cases} & \begin{cases}
2 &\text{ if } p \neq 2\\
3 &\text{ if } p = 2
\end{cases} \\ \hline
\end{array} $
}
\end{table}
\noindent
The calculation of $(\Aut_X^0)_{red}$ and $h^0(X,T_X)$ for all ruled surfaces can be found in \cite{Maruyama}. Therefore, the only thing in the above table that still needs to be checked is the case $p = 2$ and $e = 0$ and we leave this case to the reader.
\end{Example}
\begin{Example}[\emph{Abelian and bielliptic surfaces}]\label{abelianbielliptic}
If $X$ is Abelian, then $\Aut_X^0 \cong X$ and in particular $h^0(X,T_X) = 2$. If $X$ is bielliptic, then, by \cite{BombieriMumford3}, the canonical sheaf $\omega_X$ can be trivial if and only if $p \in \{2,3\}$. In these cases, one can prove that $\Aut_X^0$ is not reduced. We refer the reader to the upcoming article \cite{Martin} of the author, where the group scheme $\Aut_X$ is calculated for all (quasi-)bielliptic surfaces in all characteristics.
\end{Example}
\begin{Example}[\emph{Examples with supersingular generic fiber}]\label{supersingularexample}
This example will serve as the basic example of isotrivial elliptic surfaces with supersingular generic fiber and many vector fields from which we will derive a whole series of examples in Example \ref{importantexample}.
Consider the rational curve $C \subseteq \PP^2$ of degree $p+1$ given by the homogeneous equation
$$
y^pz = x^{p+1}.
$$
Then, $p_a(C) = \frac{p(p-1)}{2}$ and $C$ has a single isolated singularity at $P = [0:0:1]$. Consider the $\alpha_p$-action defined by
$$
[x:y:z] \mapsto [x: y+az:z] \hspace{1cm} a^p = 0
$$
and note that $P$ is not a fixed point of this action. More precisely, the reduced fixed locus of $\alpha_p$ on $C$ consists of the single smooth point $Q = [0:1:0]$.
Now, let $E$ be a supersingular elliptic curve and let $X := (E \times C)/\alpha_p$, where $\alpha_p \subseteq E$ acts on $C$ via the action defined above. By the same argument as in the proof of \cite[Proposition 7]{BombieriMumford3}, the surface $X$ is smooth, since $\alpha_p$ does not fix $P$. Moreover, $X$ comes with two fibrations $a_X: X \to E/\alpha_p$ and $f: X \to \PP^1$, where the latter is obtained by taking the normalization of $C/\alpha_p$. By construction, the morphism $f$ is an elliptic fibration with general fiber isomorphic to $E$ and $f$ admits a unique multiple fiber of multiplicity $p$, namely the image of $E \times Q$ on $X$.
Finally, note that there is an $(\alpha_p)^2$-action on $C$ given by
$$
[x:y:z] \mapsto [x^2 + bxy: xy + cx^2 + bcxy: xz + byz] \hspace{1cm} b^p = c^p = 0
$$
and this action commutes with the $\alpha_p$-action used to construct $X$. Thus, we get an induced action of $E \times (\alpha_p)^2$ on $X$. In particular, we have $h^0(X,T_X) \geq 3$.
\begin{Remark}
Alternatively, and analogously to the construction of Raynaud's counterexamples to Kodaira vanishing on surfaces in positive characteristic \cite{Raynaud}, one can describe the above example as follows: Let $E$ be a supersingular elliptic curve and let $\cal{E}$ be the indecomposable vector bundle of rank $2$ on $E$ with $e = 0$. Since $E$ is supersingular, the Frobenius map $F$ is trivial on $H^1(E,\OO_E) \cong {\rm Ext}^1(\OO_X,\OO_X)$, so the pullback of $\cal{E}$ along $F$ splits and this splitting yields an inseparable multisection of the ruling $\PP(\cal{E}) \to E$. Then, $X$ can be defined as the degree $(p+1)$ cover of $\PP(\cal{E})$ branched over the inseparable multisection and a disjoint section.
\end{Remark}
\end{Example}
\begin{Example}[\emph{Examples with unbounded vector fields}]\label{importantexample}
Here, we will use Example \ref{supersingularexample} to construct the elliptic surfaces announced in Theorem (A) (iii). More precisely, for every field $K$ of positive characteristic, for every smooth projective curve $\widetilde{C}$ over $K$ and for every $n \geq 1$, we will use $f: X \to C$ to construct an elliptic surface $\widetilde{f}:\widetilde{X} \to \widetilde{C}$ with $h^0(\widetilde{X},T_{\widetilde{X}}) \geq n$.
Let $E$ be a supersingular elliptic curve over $\FF_p$. Then, the surface $X := (E \times C)/\alpha_p$ constructed in Example \ref{supersingularexample} is also defined over $\FF_p$. Moreover, the elliptic fibration $f: X \to \PP^1$ has exactly one multiple fiber, corresponding to the unique fixed point of the $\alpha_p$-action on $C$.
Now, let $n \geq 1$ be arbitrary, let $K$ be some field extension of $\FF_p$ and let $\widetilde{C}$ be a smooth projective curve over $K$. If $K$ is finite, choose a finite separable morphism $g': \widetilde{C} \to \PP^1$ which is ramified over only one point (this is possible by the "wild Belyi Theorem" \cite[Theorem 1]{AnbarTutdere}) and let $g$ be the composition of $g'$ with a tame finite map $\PP^1 \to \PP^1$ of degree at least $n$. If $K$ is infinite, let $g$ be any finite and separable map $g: \widetilde{C} \to \PP^1$. In both cases, we can modify $g$ by an automorphism of $\PP^1$ such that the multiple fiber of $f$ does not map to a branch point of $g$.
Now, let $\widetilde{f}: \widetilde{X} \to \widetilde{C}$ be the base change of $f$ along $g$. The branch locus of $\widetilde{X} \to X$ consists of a disjoint union of smooth fibers, so $\widetilde{X}$ is smooth. We claim that $h^0({\widetilde{X}},T_{\widetilde{X}}) \geq n$. By flat base change, we may assume that $K$ is algebraically closed. Since $\Ker(f_*)$ preserves the fibers of $f$, it acts naturally on the fiber product $\widetilde{X} = \widetilde{X} \times_{\PP^1} \widetilde{C}$ and we obtain an inclusion $\Ker(f_*) \subseteq \Ker(\widetilde{f}_*)$. Next, consider the short exact sequence associated to any $D \in H^0(X,T_X)$, where $Z$ is the divisorial part and $W$ is the isolated part of the zero locus of $D$
$$
0 \to \OO_X(Z) \to T_X \to I_W(-Z-K_X) \to 0.
$$
Since $H^0(X,T_X)$ contains a $3$-dimensional subspace generated by the additive vector fields corresponding to the $\alpha_p^3$-action constructed in the previous example, we must have $h^0(X,\OO_X(Z)) \geq 2$ and there is an $\alpha_p$-action $\rho$ on $X$ that fixes a simple fiber $F$ of $f$ such that $f(F)$ is not a branch point of $g$. By construction, the preimage of $F$ in $\widetilde{X}$ consists of at least $n$ disjoint simple fibers $\widetilde{F}_1,\hdots,\widetilde{F}_n$, all of which must be fixed pointwise by the $\alpha_p$-action $\widetilde{\rho}$ on $\widetilde{X}$ that induces the action $\rho$. Then, we consider the short exact sequence induced by the action $\widetilde{\rho}$, where $\widetilde{Z}$ and $\widetilde{W}$ are the divisorial and isolated part of the fixed locus, respectively:
$$
0 \to \OO_{\widetilde{X}}(\widetilde{Z}) \to T_{\widetilde{X}} \to I_{\widetilde{W}}(-\widetilde{Z}-K_{\widetilde{X}}) \to 0.
$$
Since $\bigcup_{i=1}^n F_i \subseteq \widetilde{Z}$, we have $n \leq h^0(X,\OO_X(\sum_{i=1}^n F_i)) \leq h^0(X,T_X)$ by a Clifford argument. In particular, for every curve $C$ over every field $K$ of positive characteristic, the set of numbers
$$
\{ h^0(X,T_X) \mid X \text{ admits an elliptic fibration } f:X \to C \}
$$
is unbounded.
\end{Example}
\subsection{Non-isotrivial examples with vertical automorphisms and movable multiple fibers}
In this section, we let $k$ be an algebraically closed field of characteristic $2$. We give examples of non-isotrivial elliptic surfaces with vertical automorphisms and also show that $1$-movable multiple fibers exist over $k$, even for non-isotrivial elliptic surfaces.
Recall that a supersingular Enriques surface $X$ over $k$ is a smooth projective surface with $\omega_X \cong \OO_X$, $b_2(X) = 10$ and $\Pic^\tau_X \cong \alpha_2$. The associated $\alpha_2$-torsor induces a global $1$-form on $X$ and hence $h^0(X,T_X) = h^0(X,\Omega_X) = 1$. The next example proves Corollary \ref{Enriquescorollary}.
\begin{Example}[\emph{The automorphism scheme of generic supersingular Enriques surfaces}]\label{multiplicativeEnriques}
Assume that $X$ is generic. Then, it is known that $X$ contains no $(-2)$-curves (see e.g. \cite[Proposition 5.2]{Martinunnodal}) and that $X$ admits a multiplicative $p$-closed global vector field (see \cite[Theorem 8.16]{EkedahlHylShep}). By \cite[Theorem 5.7.1.]{CossecDolgachev}, there is an elliptic fibration $f: X \to \PP^1$, which, by \cite[Theorem 5.7.2.]{CossecDolgachev}, admits a unique multiple fiber $2F$, which is either additive or supersingular. By Remark \ref{vectorfield}, the existence of a multiplicative vector field implies $\mu_2 \subseteq \Aut_X^0$. Since $X$ contains no $(-2)$-curves, the fibration $f$ admits no reducible fibers, so Theorem \ref{mainrational} shows that $\im(f_*)$ is trivial. Hence, Lemma \ref{connsub} implies that $\Aut_X^0 \cong \Ker(f_*) \cong \mu_{2^n}$ and finally Theorem \ref{Main} shows that $n = 1$.
\end{Example}
\begin{Example}[\emph{$1$-movable fibers exist on non-isotrivial surfaces}]
Again, let $X$ be a generic supersingular Enriques surface. In particular, we have $\Aut_X^0 = \mu_2$. By \cite[Theorem 3.4.1.]{CossecDolgachev}, the surface $X$ admits two elliptic fibrations $f_1,f_2: X \to \PP^1$ with unique double fibers $2F_i$ satisfying $F_1.F_2 = 1$. We claim that $F_1$ is movable if it is smooth. In fact, one can show that this condition is automatically satisfied for generic $X$, but for the sake of brevity we will not prove this here.
Seeking a contradiction, we assume that $F_1$ is not $1$-movable. Since $F_1$ is smooth, it is supersingular, and thus it is fixed pointwise by $\Aut_X^0 = \mu_2$. But then $\mu_2$ fixes a point on a general fiber of $f_2$ and hence it fixes a general fiber of $f_2$ pointwise by Lemma \ref{fixedloci}. This is a contradiction and therefore $F_1$ is $1$-movable.
\end{Example}
\begin{Remark}
Taking base changes of Example \ref{multiplicativeEnriques} along suitable finite and separable covers $C \to \PP^1$, one can construct non-isotrivial surfaces with non-trivial $\Ker(f_*)$ over every curve $C$ in characteristic $2$. We do not know how to construct similar examples if $p^n$ is bigger than $2$. This has essentially two reasons: First, the bounds given in Theorem (C) become very strong for $p^n \gg 0$ and second, it seems to be a very hard problem to construct elliptic surfaces with multiple supersingular and additive fibers whose multiplicity is a big power of $p$ (see for example \cite{Kawazoe} where this problem is studied in a very special case).
\end{Remark}
\subsection{Examples with horizontal automorphisms} \label{sectionhorizontal}
In this section, we give examples of elliptic surfaces $f: X \to \PP^1$ where $\im(f_*)$ is non-trivial. More precisely, we will realize all cases described in Theorem \ref{mainrational} and the numbering of the examples will refer to the numbering in Theorem \ref{mainrational}. Since our examples admit a section, they will also show that all cases described in Theorem (D) occur.
We will use the following technical Lemma, which allows us to construct some $\alpha_p$- or $\mu_p$-actions on $X$ by describing them on an affine Weierstrass equation.
\begin{Lemma} \label{extendvectorfield}
Let $f: X \to \PP^1$ be a Jacobian elliptic surface and let $D$ be a rational $p$-closed vector field on $X$.
Assume that $D$ is regular away from a fiber $F$ of $f$ and such that the induced rational vector field on $\PP^1$ is regular everywhere and has a zero at $f(F)$. Let $W$ be the isolated part of the zero locus of $D$, let $Z$ be the divisorial part, and let $S$ be a non-empty set of disjoint sections of $f$ to which $D$ is tangent. Then, $D$ is regular everywhere in each of the following cases:
\begin{enumerate}[(i)]
\item $F$ is of type ${\rm II}$ and ${\rm length}(W|_{X-F}) - (Z|_{X-F})^2 > c_2(X) - 4|S|$.
\item $p = 2$, $|S| \geq 2$, $F$ is of type ${\rm III}$, ${\rm length}(W|_{X-F}) - (Z|_{X-F})^2 > c_2(X) - 6$, and $D^2 = D$.
\end{enumerate}
\end{Lemma}
\prf
Let $t = f^{\#}(s)$, where $s$ is a parameter at $f(F)$. Let $S = \{\Sigma_1,\hdots,\Sigma_n\}$ be disjoint sections of $f$ such that $D$ is tangent to $\Sigma_i$ and let $x_i$ be a local equation for $\Sigma_i$ in a neighborhood of $P_i := F \cap \Sigma_i$. Then, in the completion $\widehat{\OO}_{X,P_i} \cong k[[x_i,t]]$, we can write $D$ as
$$
D = t^{-{m_i}}(t^{m_i+l} \frac{\partial}{\partial t} + g_i \frac{\partial}{\partial x_i}),
$$
where $m_i$ is the pole order of $D$ along the component of $F$ meeting $\Sigma_i$, $l \geq 1$ is the zero order of the induced vector field on $\PP^1$ at $f(F)$ and $g_i \in k[[x_i,t]]$ is a power series with $t \nmid g$ and $x_i \mid g$, since $D$ is tangent to $\Sigma_i$.
In particular, $W$ has multiplicity ${\rm mult}_{P_i}(W) = (m_i+l) \cdot {\rm mult}_{P_i}(g_i)$ at $P_i$.
Since $Z$ is contained in fibers of $f$, we can apply \cite[Proposition 2.1]{KatsuraTakeda} to the part $W'$ of $W$ with support in $F$ to obtain
\begin{equation}\label{eqnvectorfield}
c_2(X) - \sum_{i=1}^n {\rm mult}_{P_i}(W) \geq c_2(X) - {\rm length}(W') = {\rm length}(W|_{X-F}) - Z^2. \tag{$\ast$}
\end{equation}
Assume first that $F$ is of type ${\rm II}$ and $D$ has a pole along $F$. Then, all the $m_i$ are equal and $m := m_1 > 0$. Moreover, $t^mD$ is a regular $p$-closed vector field near $F$. We have $(t^mD)^p(t) = 0$, hence $t^mD$ is additive and thus ${\rm mult}_{P_i}(g_i) \geq 2$. But then ${\rm mult}_{P_i}(W) = (m+l) \cdot {\rm mult}_{P_i}(g_i) \geq 4$. Plugging into equation \eqref{eqnvectorfield}, this proves Claim (i).
Next, assume that $F$ is of type ${\rm III}$. If $D$ has poles along both components $F_1$ and $F_2$ of $F$, then Claim (ii) follows by the same argument as in the previous paragraph. If $D$ has a pole along $F_1$ but not along $F_2$, then we consider the contraction $\pi:X \to X'$ of $F_1$. Then, $D$ induces a $\mu_2$-action on $X'$ that preserves the image $F'$ of $F$. Note that $F'$ is a cuspidal rational curve. Moreover, by Proposition \ref{blowup}, the $\mu_2$-action does not fix the cusp of $F'$, for otherwise it would lift to $X$. Hence, by Lemma \ref{fixedloci}, the $\mu_2$-action on $F'$ has four isolated fixed points. In particular, we have ${\rm length}(W') \geq 4$. On the other hand, the pole of $D$ along $F_1$ contributes at least $(-2)$ to the right hand side of Equation \eqref{eqnvectorfield}. Hence,
$$
{\rm length}(W|_{X-F}) - (Z|_{X-F})^2 \leq {\rm length}(W|_{X-F}) - Z^2 - 2 \leq c_2(X) - {\rm length}(W') - 2 \leq c_2(X) - 6
$$
contradicting our assumption. This proves Claim (ii).
\qed
\vspace{3mm}
\begin{Example}[\emph{Case $(i)$}]\label{casea}
Consider the following four affine Weierstrass equations, where $u,v \in k$ are parameters and $t$ is a coordinate on $C = \PP^1$:
\begin{eqnarray*}
y^2 &=& x^3 + t \\
y^2 &=& x^3 + tx \\
y^2 &=& x^3 + t^2 \\
y^2 &=& x^3 + ut^2x + vt^3
\end{eqnarray*}
The induced minimal proper regular models $f: X \to \PP^1$ are precisely the four types of surfaces described in Lemma \ref{twofibers} (see \cite{MirandaPersson}). Now, note that each of these Weierstrass models admits a $\GG_m$-action given by $t \mapsto \lambda^a t, x \mapsto \lambda^2 x, y \mapsto \lambda^3 y$, where $a = 6,4,3,$ and $2$, respectively. Moreover, since $\GG_m$ is smooth and $X$ is the minimal resolution of the corresponding Weierstrass model, we obtain a $\GG_m$-action on $X$. Since $\Ker(f_*) \cap \GG_m$ is finite in every case, we have $\GG_m \subseteq \im(f_*)$.
\end{Example}
\begin{Remark}
Note that for the first surface, the $\GG_m$-action on $X$ induces the vector field $D = 6t\frac{\partial}{\partial_t} + 2x\frac{\partial}{\partial_x}$ in a neighborhood of the fiber of type ${\rm II}$ at $t = 0$. This $D$ is a counterexample to \cite[Lemma 4]{RudakovShafarevich} for all $p > 3$. The problem with the proof of \cite[Lemma 4]{RudakovShafarevich} is that not every vector field on a Weierstrass model is of the form claimed there.
\end{Remark}
\begin{Example}[\emph{Case $(ii)$ with $p = 3$}]\label{caseb3}
Let $p =3$ and consider the following three affine Weierstrass equations, where $k \geq 1$ is an integer and $t$ is a coordinate on $C = \PP^1$:
\begin{eqnarray*}
y^2 &=& x^3 + tx + t \\
y^2 &=& x^3 + x^2 + t^{3^{2k}}\\
y^2 &=& x^3 + tx^2 + t^{3^{2k-1} + 3}
\end{eqnarray*}
We claim that that the corresponding elliptic surface $f: X \to \PP^1$ has precisely two fibers of type $({\rm II},{\rm III}), ({\rm II},{\rm I}_{3^{2k}}),$ and $({\rm II},{\rm I}_{3^{2k-1}}^*)$, respectively and that $X$ admits a $\mu_3$-action which is non-trivial on $\PP^1$ in each of these cases.
The first Weierstrass model $X'$ can be embedded in $\PP(1,1,2,3)$ as
$$
y^2 = x^3 + ts^3 x + ts^5
$$
and it follows immediately from Tate's algorithm that $f$ admits a fiber of type ${\rm II}$ over $t = 0$ and a fiber of type ${\rm III}^*$ over $s = 0$. Note that the surface admits a unique singularity at the point $P$ given by $[s:t:x:y] = [0:1:0:0]$ and this singularity is a rational double point of type $E_7$. There is a $\mu_3$-action on the Weierstrass model given by
$$
[s:t:x:y] \mapsto [s:at:a^2x + (1-a)s^2:y] \hspace{1cm} a^3 = 1.
$$
This action fixes $P$, so it lifts to the blow-up of $X'$ at $P$ by Proposition \ref{blowup}. By \cite[Theorem 4.1 (iii)]{Hirokado}, this already implies that the $\mu_3$-action lifts to $X$.
The second Weierstrass model $X'$ is an affine chart of the pullback along the $(2k)$-fold Frobenius on $\PP^1$ of the surface $Y' \subseteq \PP(1,1,2,3)$ given by
$$
y^2 = x^3 + s^2x^2 + s^5t.
$$
By Tate's algorithm, the minimal proper regular model $g: Y \to \PP^1$ of $Y'$ has a fiber $F'$ of type ${\rm II}^*$ with $\delta_{F'} = 1$ over $s = 0$ and a fiber of type ${\rm I}_1$ over $t = 0$. Since the Swan conductor does not change if we pull back along Frobenius and the vanishing order of $\Delta_g$ gets multiplied by $3$, the elliptic surface $f: X \to \PP^1$ admits a fiber $F$ over $s = 0$ with $\delta_{F} = 1$ and $v_{f(F)}(\Delta_f)= 11 \cdot 3^{2k} = 3$ mod $12$. This shows that $F$ is of type ${\rm II}$. Moreover, the fiber of $f$ over $t = 0$ is of type ${\rm I}_{3^{2k}}$.
The affine Weierstrass equation for $X$ admits a $\mu_3$-action given by
$$
(t,x,y) \mapsto (at,x,y) \hspace{1cm} a^3 = 1.
$$
As in the previous case, this $\mu_3$-action preserves the singular point $(0,0,0)$ of the affine Weierstrass equation and lifts to the minimal resolution. Moreover, the $\mu_3$-action corresponds to the rational vector field $D = t\frac{\partial}{\partial_t}$ on $X$ which is regular away from $F$ and tangent to the zero section of $f$. A straightforward local computation shows that ${\rm length}(\langle D \rangle|_{X-F}) - ((D)|_{X-F})^2 = 3^{2k} = c_2(X) - 3$, hence $D$ is regular on $X$ by Lemma \ref{extendvectorfield}, giving the desired $\mu_3$-action on $X$.
The third Weierstrass model $X'$ is an affine chart of the quadratic twist by $t$ of the pullback along the $(2k-1)$-fold Frobenius on $\PP^1$ of the surface $Y' \subseteq \PP(1,1,2,3)$ given by
$$
y^2 = x^3 + s^2x^2 + s^5t.
$$
By a similar argument as in the previous case, the minimal proper regular model of the pulled back surface admits a singular fiber of type ${\rm II}^*$ and a fiber of type ${\rm I}_{3^{2k-1}}$. Therefore, by Lemma \ref{twist} (i), the singular fibers of the minimal proper regular model $f: X \to \PP^1$ of $X'$ are of the stated types. There is a $\mu_3$-action on the affine chart $X'$ given by
$$
(t,x,y) \mapsto (at,ax,y) \hspace{1cm} a^3 = 1.
$$
The rest of the argument is similar to the previous case.
\end{Example}
\begin{Remark}
We remark that all three of the above surfaces are counterexamples to \cite[Lemma 4]{RudakovShafarevich} in characteristic $3$. Moreover, the second and third example are counterexamples to \cite[Theorem 6]{RudakovShafarevich}. The proof of this Theorem fails in Case (6), where \cite[Lemma 4]{RudakovShafarevich} is applied. Moreover, note that our equation for the fibration with fibers of type $({\rm II},{\rm I}_{3^{2k-1}}^*)$ differs from Equation (3) given in \cite{RudaShafaVector} and the corresponding equation given in \cite[p.1503]{RudakovShafarevich2}. Using Tate's algorithm, one can check that, at least for general $k$, these two equations do not admit a fiber $F$ of type ${\rm II}$ with $v_{f(F)}(\Delta_f) = 3$, hence they do not admit global vector fields.
\end{Remark}
\begin{Example}[\emph{Case $(ii)$ with $p = 2$}]\label{caseb2} Let $p =2$ and consider the following four affine Weierstrass equations, where $k \geq 1$ is an integer and $t$ is a coordinate on $C = \PP^1$:
\begin{eqnarray*}
y^2 + ty &=& x^3 + t \\
y^2 + ty &=& x^3 \\
y^2 + xy &=& x^3 + t^{2^{2k}}x \\
y^2 + xy &=& x^3 + t^{2^{2k-1}}x^2 + t^{2^{2k}}x
\end{eqnarray*}
We claim that that the corresponding elliptic surface $f: X \to \PP^1$ has precisely two fibers of type $({\rm II},{\rm IV}^*),({\rm III},{\rm IV}^*), ({\rm III},{\rm I}_{2^{2k+1}}),$ and $({\rm II},{\rm I}_{2^{2k+1}})$, respectively, and that $X$ admits a $\mu_2$-action which is non-trivial on $\PP^1$ in every case.
The first Weierstrass model $X'$ can be embedded in $\PP(1,1,2,3)$ as
$$
y^2 + s^2ty = x^3 + s^5t
$$
and it follows from Tate's algorithm that $f$ admits a fiber of type ${\rm II}$ over $t = 0$ and a fiber of type ${\rm IV}^*$ over $s = 0$. There is a $\mu_2$-action on the Weierstrass model given by
$$
[s:t:x:y] \mapsto [as:t:x:y + (1+a)s^3] \hspace{1cm} a^2 = 1.
$$
This action fixes the unique singular point $P = [0:1:0:0]$ of $X'$, hence it lifts to the blow-up of $X'$ at $P$ by Lemma \ref{blowup}. Since $P$ is of type $E_6$, it follows from \cite[Theorem 5.1 (iii)]{Hirokado} that the action lifts to $X$.
Similarly, the second Weierstrass model $X'$ can be embedded in $\PP(1,1,2,3)$ as
$$
y^2 + s^2ty = x^3.
$$
This time, Tate's algorithm shows that $f$ admits a fiber of type ${\rm III}$ over $t = 0$ and a fiber of type ${\rm IV}^*$ over $s = 0$. There is a $\mu_2$-action on the Weierstrass model given by
$$
[s:t:x:y] \mapsto [as:t:x:y] \hspace{1cm} a^2 = 1.
$$
This action fixes the two singular points $[0:1:0:0]$ and $[1:0:0:0]$ and hence, as in the previous case, it lifts to $X$.
The third Weierstrass model $X'$ is an affine chart of the pullback along the $(2k-2)$-fold Frobenius on $\PP^1$ of the surface $Y' \subseteq \PP(1,1,2,3)$ given by
$$
y^2 + sxy = x^3 + t^4x.
$$
By Tate's algorithm, the minimal proper regular model $g: Y \to \PP^1$ of $Y'$ has a fiber $F'$ of type ${\rm III}$ with $\delta_{F'} = 1$ over $s = 0$ and a fiber of type ${\rm I}_8$ over $t = 0$. Similarly to the analogous case if $p = 3$, it is easy to check that $f:X \to \PP^1$ admits a fiber $F$ of type ${\rm III}$ with $\delta_F = 1$ over $s = 0$ and a fiber of type ${\rm I}_{2^{2k+1}}$ over $t = 0$. There is a $\mu_2$-action on $X'$ given by
$$
(t,x,y) \mapsto (at,x,y) \hspace{1cm} a^2 = 1.
$$
This action corresponds to the vector field $D = t \frac{\partial}{\partial t}$ on $X$, which satisfies $D^2 = D$ and is tangent to the zero section $\Sigma_1$ and to the $2$-torsion section $\Sigma_2$ given by $x = y = 0$. Using the height pairing (see \cite[p.110]{schuettShioda}), one can check that $\Sigma_2$ is disjoint from $\Sigma_1$. Since $D$ fixes $(0,0,0)$, it lifts to the minimal resolution of this singularity. Moreover, a straightforward local computation shows that ${\rm length}(\langle D \rangle|_{X-F}) - ((D)|_{X-F})^2 = 2^{2k+1} = c_2(X) - 4$, so that $D$ is regular on all of $X$ by Lemma \ref{extendvectorfield}. This yields the desired $\mu_2$-action on $X$.
The fourth Weierstrass model $X'$ is an affine chart of the pullback along the $(2k-2)$-fold Frobenius on $\PP^1$ of the surface $Y' \subseteq \PP(1,1,2,3)$ given by
$$
y^2 + sxy = x^3 + t^2x^2 + t^4x.
$$
By Tate's algorithm, the minimal proper regular model $g: Y \to \PP^1$ of $Y'$ has a fiber $F'$ of type ${\rm II}$ with $\delta_{F'} = 2$ over $s = 0$ and a fiber of type ${\rm I}_8$ over $t = 0$. Using Tate's algorithm, one can check that a fiber of type ${\rm II}$ with $\delta_{F'} = 2$ remains of the same type when pulled back along an even power of the Frobenius, hence $f: X \to \PP^1$ admits a fiber of type ${\rm II}$ with $\delta_{F} = 2$ over $s = 0$ and a fiber of type ${\rm I}_{2^{2k+1}}$ over $t = 0$. The $\mu_2$-action on $X'$ given by
$$
(t,x,y) \mapsto (at,x,y) \hspace{1cm} a^2 = 1
$$
extends to a $\mu_2$-action on $X$ by the same argument as in the previous case.
\end{Example}
\begin{Remark}
We remark that the first and the fourth of the above surfaces are counterexamples to \cite[Lemma 4]{RudakovShafarevich} in characteristic $2$ and the second and the third are counterexamples to \cite[Lemma 3]{RudakovShafarevich}. Moreover, if we choose $k$ such that $2^{2k+1}+4 = 12$ mod $24$, we obtain counterexamples to \cite[Theorem 6]{RudakovShafarevich}. Again, the proof of the latter fails in Case (6), where the erroneous Lemmas 3 and 4 are applied. Moreover, we remark that the surfaces with fibers of type $({\rm III},{\rm I}_{2^{2k+1}})$ are missing from the classification in \cite{RudaShafaVector}.
\end{Remark}
\begin{Example}[\emph{Cases $(iii)$ and $(iv)$}]\label{casecd}
Consider the following affine Weierstrass equations, where $t$ is a coordinate on $C = \PP^1$ and $u \in k^\ast$:
\begin{eqnarray*}
p=3: & y^2 &= x^3 + x + t \\
p=2: & y^2 + y &= x^3 + t \\
& y^2 + uxy &= x^3 + tx^2 + x
\end{eqnarray*}
Each of these surfaces admits a $\GG_a$-action given by
\begin{eqnarray*}
(t,x,y) &\mapsto& (t + a^3 + a, x - a,y) \\
(t,x,y) &\mapsto& (t + a^2 + a, x,y + a) \\
(t,x,y) &\mapsto& (t + a^2 + ua, x, y + ax) \hspace{1cm} a \in k
\end{eqnarray*}
which lifts to the respective minimal proper regular model $f: X \to \PP^1$. The first two surfaces admit a unique singular fiber of type ${\rm II}^*$ over $t = \infty$ and the generic fiber of $f$ is supersingular. The third surface admits a unique singular fiber of type ${\rm I}_4^*$ over $t = \infty$ and the generic fiber of $f$ is ordinary with $j$-invariant $u^8$.
\end{Example}
\section{Proofs of the Main Theorems}\label{proofs}
In this section, we combine our study of horizontal and vertical automorphisms in order to prove Theorem (A), (B), (C), and (D) of the introduction. Moreover, we recall how the non-existence of global vector fields on K3 surfaces follows from Theorem (D).
\vspace{3mm}
{\sc Proof of Theorem (A)}
\vspace{1mm}
Let us prove Claim (i). Since $f$ is not isotrivial, we have $\Ker(f_*) \cong \mu_{p^n}$ for some $n \geq 0$ by Lemma \ref{connsub}. Moreover, by Proposition \ref{Propelliptic}, Proposition \ref{Proprational} and Theorem \ref{mainrational}, the group of horizontal automorphisms $\im(f_*)$ is trivial unless possibly in the cases described in Theorem \ref{mainrational} (ii). In these latter cases, we have $\Aut_X^0 \cong \im(f_*) \subseteq \mu_p$, so $h^0(X,T_X) \leq 1$ holds in every case.
As for Claim (ii), assume that the generic fiber of $f$ is ordinary or that $f$ admits no multiple fibers, and that $h^0(X,T_X) \geq 2$. Then, by Lemma \ref{connsub} and Corollary \ref{corollarymain}, we have $\Ker(f_*)^0 \in \{\mu_{p^n},M_n,E\}$ where $n \geq 0$ and $E$ is an elliptic curve, so $\im(f_*)$ has to be non-trivial. Now, Proposition \ref{Propelliptic}, Proposition \ref{Proprational} and Theorem \ref{mainrational} imply that $X$ is either ruled over an elliptic curve, an Abelian surface isogeneous to a product of elliptic curves, bielliptic with $\omega_X \cong \OO_X$, or an elliptic surface $f: X \to \PP^1$ with a unique singular fiber, without multiple fibers, and with supersingular generic fiber. In the first case, we have described the automorphism scheme in Example \ref{ruled}. In particular, we have seen that $h^0(X,T_X) \leq 4$ holds. In the second and third case, we have $h^0(X,T_X) = 2$ by Proposition \ref{Propelliptic}. In the fourth case, the group of vertical automorphism $\Ker(f_*)^0$ is trivial by Theorem \ref{Main} and the group of horizontal automorphisms $\im(f_*)$ fixes a point on $\PP^1$ and thus $\im(f_*) \subseteq \GG_a \rtimes \GG_m$. In particular, we have $\im(f_*)[F] \subseteq \GG_a \rtimes \GG_m[F] = \alpha_p \rtimes \mu_p$. Now, Theorem \ref{mainrational} shows that $\im(f_*)[F] = \alpha_p$, hence $h^0(X,T_X) \leq 1$, so this case does not occur.
Finally, Claim (iii) is Example \ref{importantexample}.
\qed
\vspace{3mm}
{\sc Proof of Theorem (B)}
\vspace{1mm}
Assume first that $\im(f_*)$ is non-trivial. Then, by Proposition \ref{Propelliptic}, Proposition \ref{Proprational} and Theorem \ref{mainrational}, we have $\Aut_X^0 \cong \im(f_*) \cong \mu_p$ with $p \in \{2,3\}$. If $\im(f_*)$ is trivial, then $\Aut_X^0 \cong \Ker(f_*) \cong \mu_{p^n}$ for some $n \geq 0$ by Lemma \ref{connsub}. This proves Theorem (B).
\qed
\vspace{3mm}
{\sc Proof of Theorem (C)}
\vspace{1mm}
The inequality is trivial if $p^n \in \{2,3\}$, so we may assume $p^n \geq 4$. Then, by Proposition \ref{Propelliptic}, Proposition \ref{Proprational} and Theorem \ref{mainrational}, we have $\Aut_X^0 \cong \Ker(f_*) \cong \mu_{p^n}$ and then the inequality is proved in Lemma \ref{Igusainequality}. Next, note that by the results of Section \ref{horizontal}, the conditions given in Theorem (C) guarantee that $\Aut_X^0 \cong \Ker(f_*)$, so the statement about the multiplicities of additive and supersingular fibers of $f$ is exactly Theorem \ref{Main} (iii) (2).
\qed
\vspace{3mm}
{\sc Proof of Theorem (D)}
\vspace{1mm}
Assume that $c_2(X) \neq 0$. Since $f$ admits no multiple fibers, Theorem \ref{Main} shows that $\ker(f_*)^0$ is trivial. Moreover, by Proposition \ref{Propelliptic} and Proposition \ref{Proprational}, we have $C = \PP^1$ and $b_1(X) = 0$. In particular, $f$ admits a singular fiber and thus we have an inclusion $\Aut_X^0 \cong \im(f_*) \subseteq \GG_a \rtimes \GG_m$, since $\GG_a \rtimes \GG_m$ is the stabilizer of a point on $\PP^1$. Thus, either $\alpha_p \subseteq \im(f_*)$ or $\mu_p \subseteq \im(f_*)$.
If $\alpha_p \subseteq \im(f_*)$, then the $\alpha_p$-action on $X$ preserves every singular fiber of $f$ by Remark \ref{criteriaremark} and only one point on $\PP^1$ by Lemma \ref{fixedloci}, hence $f$ is isotrivial with a unique singular fiber. In particular, by Theorem \ref{mainrational} (iv), we have $p \in \{2,3\}$. Thus, we are in Case (v) of Theorem (D).
If $\mu_p \subseteq \im(f_*)$, then Theorem \ref{mainrational} shows that either the singular fibers of $f$ are as in Lemma \ref{twofibers} or $p \in \{2,3\}$ and the singular fibers of $f$ are as in Lemma \ref{twofibers23}. In particular, if the fibers are not of the types described in Theorem (D) (iii) and (iv), then $f$ is isotrivial and $X$ satisfies $c_2(X) = 12$
by Ogg's formula. Hence $\chi(X,\OO_X) = 1$ and therefore $\omega_X \cong \OO_X(-F)$, where $F$ is the class of a fiber of $f$. In particular, we have $h^1(X,\OO_X) = h^2(X, \omega_X^{\otimes 2}) = 0$ and thus $X$ is rational and $f$ admits a section. This is Case (ii) of Theorem (D).
\qed
\vspace{3mm}
{\sc Proof of Corollary \ref{K3corollary}}
\vspace{1mm}
Let $X$ be a K3 surface and assume by contradiction that $h^0(X,T_X) \neq 0$. By \cite[p. 1502]{RudakovShafarevich2}, this implies that the surface $X$ admits an elliptic fibration $f: X \to \PP^1$ with at least two singular fibers. This contradicts Theorem (D), because no elliptic surface listed in Theorem (D) (i)-(iv) satisfies the equality $c_2(X) = 24$, which holds for the K3 surface $X$. Therefore, we must have $h^0(X,T_X) = 0$.
\qed
\newpage
\section{Appendix: Some quadratic twists} \label{appendix}
In this section, we give some background on quadratic twists, which we needed for example in the proof of Lemma \ref{twofibers}. Let $f: X \to C$ be a Jacobian elliptic surface. Then, a \emph{quadratic twist} of $f$ is a Jacobian elliptic surface $f': X' \to C$ that becomes isomorphic to $f$ after passing to a degree two cover of $C$. If the generic fiber of $f$ is ordinary, then all its twists are quadratic. To make this more explicit, let $d \in k(C)$ be a rational function. Then, the quadratic twist $f_d: X_d \to C$ of $f$ by $d$ is defined as follows: If $p \neq 2$ and the generic fiber of $f$ is given by
$$
y^2 = x^3 + a_2x^2 + a_4x + a_6
$$
with $a_i \in k$, then $f_d$ is given by
$$
y^2 = x^3 + da_2x^2 + d^2a_4x + d^3a_6.
$$
If $p = 2$ and the generic fiber of $f$ is given by
$$
y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6
$$
with $a_i \in k$, then $f'$ is given by
$$
y^2 + a_1xy + a_3y = x^3 + (a_2 + da_1^2)x^2 + a_4x + a_6 + da_3^2.
$$
The fibers of $f$ and $f_d$ are isomorphic except possibly over the set $S$ of poles and zeroes of $d$ (resp. the set of poles if $p = 2$) and we say that $f_d$ is a \emph{quadratic twist of $f$ at $S$}. Quadratic twists by $d_1$ and $d_2$ are isomorphic if and only if $d_1/d_2$ is a square if $p \neq 2$ (resp. if and only if $d_1 + d_2 = c^2 + c$ for some $c \in k(C)$ if $p = 2$).
In the following lemma, we summarize the facts about quadratic twists that we used in this article.
\begin{Lemma} \label{twist}
Let $f: X \to C$ be an elliptic surface and let $d \in k(C)$. Let $F$ be a fiber of $f$ and $F_d$ the corresponding fiber of $f_d$. Then, the following hold:
\begin{enumerate}[(i)]
\item If $p \neq 2$ and $d$ has a zero or pole at $f(F)$, then the types of $F$ and $F_d$ are related as follows: ${\rm I}_n \leftrightarrow {\rm I}_n^*$, ${\rm II} \leftrightarrow {\rm IV}^*, {\rm III} \leftrightarrow {\rm III}^*, {\rm IV} \leftrightarrow {\rm II}^*$.
\item If $p = 2$, $C = \PP^1$ and $F$ is of type ${\rm II}$ with $\delta_{F} = 2$, then we can choose $d \in k(t)$ with a single simple pole such that $F_d$ is of type ${\rm III}$ with $\delta_{F} = 1$.
\item If $p = 2$ , $C = \PP^1$ and $f$ has ordinary generic fiber and a unique singular fiber $F$, then $f$ is isotrivial and $F$ is of type ${\rm I}_{8k + 4}^*$ with $\delta_{F} = 4k + 2$ for some $k \geq 0$.
\end{enumerate}
\end{Lemma}
\prf
Claim (i) is well-known, see for example \cite[Section 5.4.]{schuettShioda}.
As for Claim (ii), choose a Weierstrass equation
$$
y^2 + a_1xy + a_3y = x^3 + a_2 x^2 + a_4 x + a_6
$$
with coefficients $a_i \in k[t]$, where $t$ is a parameter at $f(F)$. Then, $F$ being of type ${\rm II}$ with $\delta_F = 2$ means that we can choose the $a_i$ such that $t \mid a_1,a_3,a_4,a_6$ but $t^2 \nmid a_3,a_6$. Let $c_3$ resp. $c_6$ be the linear terms of $a_3$ resp. $a_6$. If we set $d = c_6/c_3^2$, the quadratic twist
$$
y^2 + a_1xy + a_3y = x^3 + (a_2 + da_1^2)x^2 + a_4 x + a_6 + da_3^2
$$
still has coefficients in $k[t]$ and we have $t^2 \mid a_6 + da_3^2$. Note that $t^3 \nmid b_8 := (a_1^2a_6 + a_1a_3a_4 + a_2a_3^2 + a_4^2)$ and the quadratic twist does not change $b_8$. Thus, Tate's algorithm shows that $F_d$ is of type ${\rm III}$ with $\delta_{F_d} = 1$. Moreover, the twist parameter $d$ has a simple pole at $f(F)$ and no other poles.
Next, let us prove Claim (iii). First, we prove that $f$ is isotrivial. For this, choose a parameter $t$ on $\PP^1$ such that $F$ is located at $t = 0$. By \cite[Appendix A]{Silverman}, the assumption that the generic fiber of $f$ is ordinary allows us to find a Weierstrass equation of the form
$$
y^2 + xy = x^3 + \frac{a}{b} x^2 + a_6
$$
with $a,b \in k[t]$ and $a_6 \in k(t)$. Write $a/b = \sum_{i = -n}^\infty d_it^i \in k((t))$ and twist the above equation by $d = \sum_{i = -n}^{-1} d_it_i \in k(t)$. This quadratic twist only changes the fiber over $t = 0$, so we may assume that $t \nmid b$.
We have $\Delta = a_6$ and $j = 1/a_6$. Since $f$ has no singular fibers away from $t = 0$, the $j$-map has no poles away from $t = 0$ and $\Delta$ is constant up to $12$-th powers. Therefore, we can write $a_6 = t^{12n}/c^{12}$ for some $n \geq 0$ and $c \in k[t]$ with $t \nmid c$ and $\deg(c) \leq n$. Then, we can rescale the Weierstrass equation to an integral Weierstrass equation of the following form
$$
y^2 + bc^{2}xy = x^3 + abc^{4} x^2 + t^{12n}b^6.
$$
If $n > 0$, then Tate's algorithm shows that $F$ is of type ${\rm I}_{12n}$, because $t \nmid b,c$. Then, Igusa's inequality shows $12n \leq b_2(X) = c_2(X) - 2$, which contradicts Ogg's formula $12n = c_2(X)$. Hence, we must have $n = 0$ and thus $j$ is constant.
This implies that the generic fiber of $f$ is a quadratic twist of the ordinary elliptic curve with $j$-invariant $j$ given by
$$
y^2 + xy = x^3 + j
$$
by a twist parameter $d \in k(t)$ whose only poles are at $t = 0$.
Every non-trivial such twist can be written as
$$
y^2 + xy = x^3 + \frac{a}{t^{2k+1}}x^2 + j
$$
for some $a \in k[t]$ of degree at most $2k+1$ with $t \nmid a$, where $k \geq 0$ is an integer. Clearing denominators and applying $y \mapsto \sqrt{j} t^{3k + 3}$, we obtain the equation
$$
y^2 + t^{k+1}xy = x^3 + atx^2 + \sqrt{j}t^{4k+4}x.
$$
By Tate's algorithm, this equation is minimal and $F$ is of type ${\rm I}_{8k+4}^*$. Moreover, $\Delta = t^{12k+12}$, so that $\delta_{F} = 4k + 2$.
\qed
\bibliographystyle{alpha}
\bibliography{AutomorphismsEllipticSurfaces}
\end{document}
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\begin{document}
\begin{center}
{\large\bf Homogeneous linear matrix difference equations of higher order: Singular case}
\vskip.20in
Charalambos P. Kontzalis$^{1}$ and\ Grigoris Kalogeropoulos$^{2}$\\[2mm]
{\footnotesize
$^{1}$Department of Informatics, Ionian University, Corfu, Greece\\
$^{2}$Department of Mathematics, University of Athens, Greece}
\end{center}
{\footnotesize
\noindent
\textbf{Abstract:} In this article, we study the singular case of an homogeneous generalized discrete time system with given initial conditions. We consider the matrix pencil singular and provide necessary and sufficient conditions for existence and uniqueness of solutions of the initial value problem.\\
\\[3pt]
{\bf Keywords} : linear difference equations, matrix, singular, system.
\\[3pt]
\vskip.2in
\section{Introduction}
Many authors have studied generalized discrete \& continouus time systems, see [1-28], and their applications, see [29-35]. Many of these results have already been extended to systems of differential \& difference equations with fractional operators, see [36-45]. In this article, our purpose is to study the solutions of a generalized initial value problem of linear matrix difference equations into the mainstream of matrix pencil theory. Thus, we consider
\[
A_nX_{k+n}+A_{n-1}X_{k+n-1}+...+A_1X_{k+1}+A_0X_k=0_{m_1,1},
\]
with known initial conditions
\[
X_{k_0},X_{k_0+1},...,X_{k_0+n-1},
\]
where $A_i, i=0,1,...,n \in \mathcal{M}({m_1 \times r_1;\mathcal{F}})$, (i.e. the
algebra of square matrices with elements in the field
$\mathcal{F}$) with $X_k \in\mathcal{M}({m_1 \times
1;\mathcal{F}})$ and det$A_n$=0 if $A_n$ is square. In the sequel we adopt the following notations
\[
\begin{array}{c}
Y_{k,1}=X_k,\\
Y_{k,2}=X_{k+1},\\
\dots \\
Y_{k,n-1}=X_{k+n-2},\\
Y_{k,n}=X_{k+n-1}.
\end{array}
\]
and
\[
\begin{array}{c}
Y_{k+1,1}=X_{k+1}=Y_{k,2},\\
Y_{k+1,2}=X_{k+2}=Y_{k,3},\\
\dots \\
Y_{k+1,n-1}=X_{k+n-1}=Y_{k,n},\\
A_nY_{k+1,n}=A_nX_{k+n}=-A_{n-1}Y_{k,n}-...-A_1Y_{k,2}-A_0Y_{k,1}.
\end{array}
\]
Let $m_1n$=$r$ and $m_1n+r_1-m_1$=$m$. Then the above system can be written in Matrix form in the following way
\begin{equation}
FY_{k+1}=GY_k,
\end{equation}
with known initial conditions
\begin{equation}
Y_{k_0}.
\end{equation}
Where
\[
F = \left[
\begin{array}{ccccc}
I_{m_1}&0_{m_1, m_1}&...&0_{m_1, m_1}&0_{m_1, m_1}\\
0_{m_1, m_1}&I_{m_1}&...&0_{m_1, m_1}&0_{m_1, m_1}\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0_{m_1, m_1}&0_{m_1, m_1}&...&I_{m_1}&0_{m_1, m_1}\\
0_{m_1, r_1}&0_{m_1, r_1}&...&0_{m_1, r_1}&A_n
\end{array}
\right],
\]
\[
G = \left[\begin{array}{cccc} 0_{m_1, m_1}&I_{m_1}&\ldots&0_{m_1, m_1}\\0_{m_1, m_1}&0_{m_1, m_1}&\ldots&0_{m_1, m_1}\\\vdots&\vdots&\ddots&\vdots\\0_{m_1, m_1}&0_{m_1, m_1}&\ldots&I_{m_1}\\-A_0&-A_1&\ldots&-A_{n-1}\end{array}\right].
\]
and
\[
Y_k=\left[\begin{array}{cccc} Y_{k,1}^T Y_{k,2}^T \dots Y_{k,m_1}^T\end{array}\right]^T.
\]
With dimensions $F,G \in \mathcal{M}({r \times m;\mathcal{F}})$, (i.e. the
algebra of matrices with elements in the field
$\mathcal{F}$) with $Y_k \in \mathcal{M}({m \times
1;\mathcal{F}})$. For the sake of simplicity we set
${\mathcal{M}}_m = {\mathcal{M}}({m \times m;\mathcal{F}})$ and
${\mathcal{M}}_{rm} = {\mathcal{M}}({r \times m;\mathcal{F}} )$. The matrices $F$ and $G$ can be non-square (when $r\neq m$) or square ($r = m$) and $F$ singular (det$F$=0).
\section{Singular matrix pencils: Mathematical background and notation}
In this section we will give the mathematical background and the notation that is used throughout the paper
\\\\
\textbf{Definition 2.1} Given $F,G\in \mathcal{M}_{rm}$ and an indeterminate $s\in\mathcal{F}$, the matrix
pencil $sF-G$ is called regular when $r=n$ and $\det(sF-G)\neq 0$.
In any other case, the pencil will be called singular.
\\\\
\textbf{Definition 2.2}
The pencil $sF-G$ is said to be \emph{strictly equivalent} to the
pencil $s\tilde F - \tilde G$ if and only if there exist nonsingular
$P\in\mathcal{M}_n$ and $Q\in\mathcal{M}_m$ such as
\[
P({sF - G})Q = s\tilde F - \tilde G.
\]
In this article, we consider the case that the pencil is \emph{singular}. Unlike the case of the regular pencils, the characterization of a singular matrix pencil, apart from the set of the determinantal divisors requires the definition of additional sets of invariants, the minimal indices. Let $\mathcal{N}_r$, $\mathcal{N}_l$ be right, left null space of a matrix respectively. Then the equations
\[
(sF-G)U(s)=0_{m, 1}
\]
and
\[
V^T(s)(sF-G)=0_{1, m}
\]
have solutions in $U(s), V(s)$, which are vectors in the rational vector spaces $\mathcal{N}_r(sF-G)$ and $\mathcal{N}_l(sF-G)$ respectively. The binary vectors X(s) and $Y^T(s)$ express dependence relationships among the colums or rows of $sF-G$ respectively. $U(s), V(s)$ are polynomial vectors. Let $d$=dim$\mathcal{N}_r(sF-G)$ and $t$=$\mathcal{N}_l(sF-G)$. It is known [46-53] that $\mathcal{N}_r(sF-G)$, $\mathcal{N}_l(sF-G)$, as rational vector spaces, are spanned by minimal polynomial bases of minimal degrees
\[
\epsilon_1=\epsilon_2=...=\epsilon_g=0<\epsilon_{g+1}\leq...\leq\epsilon_d
\]
and
\[
\zeta_1=\zeta_2=...=\zeta_h=0<\zeta_{h+1}\leq...\leq\zeta_t
\]
respectively. The set of minimal indices $\epsilon_i$ and $\zeta_j$ are known [46-53] as \textit{column minimal indices} (c.m.i.) and \textit{row minimal indices} (r.m.i) of sF-G respectively.
To sum up in the case of a singular matrix pencil, we have invariants, a set of \emph{elementary divisors} (e.d.) and \emph{minimal indices}, of the following type:
\begin{itemize}
\item e.d. of the type $(s-a)^{p_j}$, \emph{finite elementary
divisors} (nz. f.e.d.)
\item e.d. of the type $\hat{s}^q=\frac{1}{s^q}$, \emph{infinite elementary divisors}
(i.e.d.).
\item m.c.i. of the type $\epsilon_1=\epsilon_2=...=\epsilon_g=0<\epsilon_{g+1}\leq...\leq\epsilon_d$, \emph{minimal column indices}
\item m.r.i. of the type $\zeta_1=\zeta_2=...=\zeta_h=0<\zeta_{h+1}\leq...\leq\zeta_t$, \emph{minimal row indices}
\end{itemize}
\textbf{Definition 2.3.} Let $B_1 ,B_2 ,\dots, B_n $ be elements of $\mathcal{M}_n$. The direct sum
of them denoted by $B_1 \oplus B_2 \oplus \dots \oplus B_n$ is
the blockdiag$\left[\begin{array}{cccc} B_1& B_2& \dots& B_n\end{array}\right]$.
\\\\
The existence of a complete set of invariants for singular pencils implies the existence of canonical form, known as Kronecker canonical form [46-53] defined by
\[
sF_K - Q_K :=
sI_p - J_p \oplus sH_q - I_q \oplus sF_{\epsilon}-G_{\epsilon}\oplus sF_{\zeta}-G_{\zeta}\oplus 0_{h, g}
\]
where $sI_p - J_p$ is uniquely defined by the set of f.e.d.
\[
({s - a_1 })^{p_1 } , \dots ,({s - a_\nu }
)^{p_\nu },\quad \sum_{j = 1}^\nu {p_j = p}
\]
of $sF-G$ and has the form
\[
sI_p - J_p := sI_{p_1 } - J_{p_1 } (
{a_1 }) \oplus \dots \oplus sI_{p_\nu } - J_{p_\nu }
({a_\nu })
\]
The $q$ blocks of the second uniquely defined block $sH_q -I_q$ correspond to the i.e.d.
\[
\hat s^{q_1} , \dots ,\hat s^{q_\sigma}, \quad \sum_{j =
1}^\sigma {q_j = q}
\]
of $sF-G$ and has the form
\[
sH_q - I_q := sH_{q_1 } - I_{q_1 } \oplus
\dots \oplus sH_{q_\sigma } - I_{q_\sigma}
\]
Thus, $H_q$ is a nilpotent element of $\mathcal{M}_n$ with index
$\tilde q = \max \{ {q_j :j = 1,2, \ldots ,\sigma } \}$, where
\[
H^{\tilde q}_q=0_{q, q},
\]
and $I_{p_j } ,J_{p_j } ({a_j }),H_{q_j }$ are defined as
\[
I_{p_j } = \left[\begin{array}{ccccc}
1&0& \ldots & 0&0\\
0& 1 & \ldots&0 &0 \\
\vdots & \vdots & \ddots & \vdots &\vdots \\
0 & 0 & \ldots & 0 &1
\end{array}\right]
\in {\mathcal{M}}_{p_j } ,
\]
\[
J_{p_j } ({a_j }) = \left[\begin{array}{ccccc}
a_j & 1 & \dots&0 & 0 \\
0 & a_j & \dots&0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \ldots& a_j& 1\\
0 & 0 & \ldots& 0& a_j
\end{array}\right] \in {\mathcal{M}}_{p_j }
\]
\[
H_{q_j } = \left[
\begin{array}{ccccc} 0&1&\ldots&0&0\\0&0&\ldots&0&0\\\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\ldots&0&1\\0&0&\ldots&0&0
\end{array}
\right] \in {\mathcal{M}}_{q_j }.
\]
For algorithms about the computations of the Jordan matrices see [46-53]. For the rest of the diagonal blocks of $F_K$ and $G_K$, s$F_{\epsilon}-G_{\epsilon}$ and s$F_{\zeta}-G_{\zeta}$, the matrices $F_{\epsilon}$, $G_{\epsilon}$ are defined as
\[
F_\epsilon=blockdiag\left\{L_{\epsilon_{g+1}}, L_{\epsilon_{g+2}}, ..., L_{\epsilon_d}\right\}
\]
Where $L_\epsilon= \left[
\begin{array}{ccc} I_\epsilon & \vdots & 0_{\epsilon, 1}
\end{array}
\right]$, for $\epsilon=\epsilon_{g+1}, ..., \epsilon_d$
\[
G_\epsilon=blockdiag\left\{\bar L_{\epsilon_{g+1}}, \bar L_{ \epsilon_{g+2}}, ..., \bar L_{\epsilon_d}\right\}
\]
Where $\bar L_\epsilon=\left[
\begin{array}{ccc} 0_{\epsilon, 1} & \vdots & I_\epsilon
\end{array}
\right]$, for $\epsilon=\epsilon_{g+1}, ..., \epsilon_d$. The matrices $F_{\zeta}$, $G_{\zeta}$ are defined as
\[
F_\zeta=blockdiag\left\{L_{\zeta_{h+1}}, L_{\zeta_{h+2}}, ..., L_{\zeta_t}\right\}
\]
Where $L_\zeta= \left[
\begin{array}{c} I_\zeta \\ 0_{1, \zeta}
\end{array}
\right]$, for $\zeta=\zeta_{h+1}, ..., \zeta_t$
\[
G_\zeta=blockdiag\left\{\bar L_{\zeta_{h+1}}, \bar L_{ \zeta_{h+2}}, ..., \bar L_{\zeta_t}\right\}
\]
Where $\bar L_\zeta=\left[
\begin{array}{c} 0_{1, \zeta}\\I_\zeta
\end{array}
\right]$, for $\zeta=\zeta_{h+1}, ..., \zeta_t$.
\section{Main results}
Following the given analysis in section 2, there exist non-singular matrices $P, Q$ such that
\begin{equation}
PFQ=F_K, \quad PGQ=G_K.
\end{equation}
Let
\begin{equation}
Q=\left[\begin{array}{ccccc}Q_p & Q_q &Q_\epsilon & Q_\zeta & Q_g\end{array}\right]
\end{equation}
where $Q_p\in \mathcal{M}_{rp}$, $Q_q\in \mathcal{M}_{rq}$, $Q_\epsilon\in \mathcal{M}_{r\epsilon}$, $Q_\zeta\in \mathcal{M}_{r\zeta}$ and $Q_g\in \mathcal{M}_{rg}$
\\\\
\textbf{Lemma 3.1.} System (1) is divided into five subsystems:
\begin{equation}
Z_{k+1}^p = J_p Z_k^p
\end{equation}
the subsystem
\begin{equation}
H_q Z^q_{k+1} = Z_k^q
\end{equation}
the subsystem
\begin{equation}
F_\epsilon Z^\epsilon_{k+1}=G_\epsilon Z^\epsilon_k
\end{equation}
the subsystem
\begin{equation}
F_\zeta Z^\zeta_{k+1}=G_\zeta Z^\zeta_k
\end{equation}
and the subsystem
\begin{equation}
0_{h, g}\cdot Z^g_{k+1}=0_{h, g}\cdot Z^g_k
\end{equation}
\textbf{Proof.} Consider the transformation
\[
Y_k=QZ_k
\]
By substituting this transformation into (1) we obtain
\[
FQZ_{k+1}=GQZ_k.
\]
Whereby, multiplying by $P$ and using (3), we arrive at
\[
F_KZ_{k+1}=G_K Z_k+PV_k.
\]
Moreover, we can write $Z_k$ as
\[
Z_k=\left[
\begin{array}{c} Z_k^p\\Z_k^q\\Z_k^\epsilon\\Z_k^\zeta\\Z_k^g
\end{array}
\right]
\]
where $Z_p^k\in \mathcal{M}_{p1}$, $Z_q^k \in \mathcal{M}_{q1}$, $Z_\epsilon^k\in \mathcal{M}_{\epsilon1}$, $Z_\zeta^k \in \mathcal{M}_{\zeta1}$ and $Z_g^k\in \mathcal{M}_{h1}$. Taking into account the above expressions, we arrive easily at the subsystems (5), (6), (7), (8), and (9).
\\\\
Solving the system (1) is equivalent to solving subsystems (5), (6), (7), (8), and (9).
\\\\
\textbf{Remark 3.1.} The subsystem (5) is a regular type system and its solution is given from, see [1-28].
\begin{equation}
Z^p_k=J_p^{k-k_0}Z^p_{k_0}.
\end{equation}
\textbf{Remark 3.2.} The subsystem (6) is a singular type system but its solution is very easy to compute, see [8-18].
\begin{equation}
Z^q_k=0_{q,1}.
\end{equation}
\textbf{Proposition 3.1.} The subsystem (7) has infinite solutions and can be taken arbitrary
\begin{equation}
Z_k^\epsilon=C_{k,1}.
\end{equation}
\textbf{Proof.}
If we set
\[
Z_k^\epsilon=\left[\begin{array}{c} Z_k^{\epsilon_{g+1}}\\Z_k^{\epsilon_{g+2}}\\\vdots\\Z_k^{\epsilon_d}\end{array}\right],
\]
by using the analysis in section 2, system (22) can be written as:
\[
blockdiag\left\{L_{\epsilon_{g+1}}, ..., L_{\epsilon_d}\right\}\left[\begin{array}{c} Z_{k+1}^{\epsilon_{g+1}}\\Z_{k+1}^{\epsilon_{g+2}}\\\vdots\\Z_{k+1}^{\epsilon_d}\end{array}\right]=blockdiag\left\{\bar L_{\epsilon_{g+1}}, ..., \bar L_{\epsilon_d}\right\}\left[\begin{array}{c} Z_k^{\epsilon_{g+1}}\\Z_k^{\epsilon_{g+2}}\\\vdots\\Z_k^{\epsilon_d}\end{array}\right].
\]
Then for the non-zero blocks a typical equation can be written as
\[
\begin{array}{ccc} L_{\epsilon_i} Z_{k+1}^{\epsilon_i}=\bar L_{\epsilon_i} Z_k^{\epsilon_i} & , &i =g+1, g+2, ..., d, \end{array}
\]
or
\[
\left[\begin{array}{ccc} I_{\epsilon_i} & \vdots & 0_{{\epsilon_i}, 1}\end{array}\right]Z_{k+1}^{\epsilon_i}=\left[
\begin{array}{ccc} 0_{{\epsilon_i}, 1} & \vdots & I_{\epsilon_i}
\end{array}
\right]Z_k^{\epsilon_i},
\]
or
\[
\left[\begin{array}{ccccc} 1 &0&\ldots&0&0\\0&1&\ldots&0&0\\ \vdots &\vdots &\ldots&\vdots&\vdots\\0&0&\ldots&1&0\end{array}\right]\left[\begin{array}{c} z_{k+1}^{{\epsilon_i},1}\\ z_{k+1}^{{\epsilon_i},2}\\ \vdots \\ z_{k+1}^{{\epsilon_i},{\epsilon_i}} \\ z_{k+1}^{{\epsilon_i},{\epsilon_i}+1}\end{array}\right]=\left[\begin{array}{ccccc} 0 &1&\ldots&0&0\\0&0&\ldots&0&0\\ \vdots &\vdots &\ldots&\vdots&\vdots\\0&0&\ldots&0&1\end{array}\right]\left[\begin{array}{c} z_k^{{\epsilon_i},1}\\ z_k^{{\epsilon_i},2}\\ \vdots \\ z_k^{{\epsilon_i},{\epsilon_i}} \\ z_k^{{\epsilon_i},{\epsilon_i}+1}\end{array}\right],
\]
or
\[
\begin{array}{ccccc} z_{k+1}^{{\epsilon_i},1}=z_k^{{\epsilon_i},2}\\ z_{k+1}^{{\epsilon_i},2}=z_k^{{\epsilon_i},3}\\\vdots\\ z_{k+1}^{{\epsilon_i},{\epsilon_i}}=z_{k+1}^{{\epsilon_i},{\epsilon_i}+1}.
\end{array}
\]
This is of a regular type system of difference equations with ${\epsilon_i}$ equations and ${\epsilon_i}+1$ unknowns. It is clear from the above analysis that in every one of the $d-g$ subsystems one of the coordinates of the solution has to be arbitrary by assigned total. The solution of the system can be assigned arbitrary
\[
Z_k^\epsilon=C_{k,1}
\]
\textbf{Proposition 3.2.} The subsystem (8) has the unique solution
\begin{equation}
Z_k^\zeta=0_{\zeta,1}.
\end{equation}
\\\\
\textbf{Proof.} If we set
\[
Z_k^\zeta=\left[\begin{array}{c} Z_k^{\zeta_{h+1}}\\Z_k^{\zeta_{h+2}}\\\vdots\\Z_k^{\zeta_t}\end{array}\right],
\]
then the subsystem (8) can be written as:
\[
blockdiag\left\{L_{\zeta_{h+1}}, ..., L_{\zeta_t}\right\}\left[\begin{array}{c} Z_{k+1}^{\zeta_{h+1}}\\Z_{k+1}^{\zeta_{h+2}}\\\vdots\\Z_{k+1}^{\zeta_t}\end{array}\right]=blockdiag\left\{\bar L_{\zeta_{h+1}}, ..., \bar L_{\zeta_t}\right\}\left[\begin{array}{c} Z_k^{\zeta_{h+1}}\\Z_k^{\zeta_{h+2}}\\\vdots\\Z_k^{\zeta_t}\end{array}\right].
\]
Then for the non-zero blocks, a typical equation can be written as
\[
\begin{array}{ccc} L_{\zeta_j} Z_{k+1}^{\zeta_j}=\bar L_{\zeta_j} Z_k^{\zeta_j} & , & j=h+1, h+2, ..., t, \end{array}
\]
or
\[
\left[\begin{array}{c} I_{\zeta_j} \\ \cdots \\ 0_{1, {\zeta_j}}\end{array}\right]Z_{k+1}^{\zeta_j}=\left[
\begin{array}{c} 0_{1,{\zeta_j}} \\ \cdots \\ I_{\zeta_j}
\end{array}
\right]Z_k^{\zeta_j},
\]
or
\[
\left[\begin{array}{cccc} 1 &0&\ldots&0\\0&1&\ldots&0\\ \vdots &\vdots &\ldots&\vdots\\0&0&\ldots&1\\0&0&\ldots&0\end{array}\right]\left[\begin{array}{c} z_{k+1}^{{\zeta_j},1}\\ z_{k+1}^{{\zeta_j},2}\\ \vdots \\ z_{k+1}^{{\zeta_j},{\zeta_j}}\end{array}\right]=\left[\begin{array}{cccc} 0 &0&\ldots&0\\1&0&\ldots&0\\ \vdots &\vdots &\ldots&\vdots\\0&0&\ldots&0\\0&0&\ldots&1\end{array}\right]\left[\begin{array}{c} z_k^{{\zeta_j},1}\\ z_k^{{\zeta_j},2}\\ \vdots \\ z_k^{{\zeta_j},{\zeta_j}}\end{array}\right],
\]
or
\[
\begin{array}{c} z_{k+1}^{{\zeta_j},1}=0\\ z_{k+1}^{{\zeta_j},2}=z_k^{{\zeta_j},1}\\\vdots\\ z_{k+1}^{{\zeta_j},{\zeta_j}}=z_k^{{\zeta_j},{\zeta_j}-1}\\0=z_k^{{\zeta_j},{\zeta_j}}\end{array}.
\]
We have a system of ${\zeta_j}$+1 difference equations and ${\zeta_j}$ unknowns. Starting from the last equation we get the solutions
\[
\begin{array}{c} z_k^{{\zeta_j},{\zeta_j}}=0\\ z_k^{{\zeta_j},{\zeta_j}-1}=0\\z_k^{{\zeta_j},{\zeta_j}-2}=0\\\vdots\\ z_k^{{\zeta_j},1}=0\end{array}
\]
Hence, the system (8) has the following unique solution
\[
Z_k^\zeta=0_{\zeta, 1}
\]
\textbf{Remark 3.3.}The subsystem (9) has an infinite number of solutions that can be taken arbitrary
\begin{equation}
Z_k^g=C_{k,2}
\end{equation}
We can state the following Theorem
\\\\
\textbf{Theorem 3.1.} Consider the system (1), with known initial conditions (2) and a singular matrix pencil $sF-G$. Then its solution is unique if and only if the c.m.i. are zero
\begin{equation}
dim\mathcal{N}_r(sF-G)=0
\end{equation}
and
\begin{equation}
Y_{k_0}\in colspan Q_p
\end{equation}
The unique solution is then given from the formula
\begin{equation}
Y_k=Q_pJ_p^{k-k_0}Z_{k_0}^p
\end{equation}
where $Z_{k_0}^p$ is the unique solution of the algebraic system $Y_{k_0}=Q_pZ_{k_0}^p$. In any other case the system has infinite solutions.
\\\\
\textbf{Proof.} First we consider that the system has non zero c.m.i and non zero r.m.i. By using transformation
\[
Y_k=QZ_k
\]
then from (10), (11), (12), (13) and (14) the solutions of the subsystems (5), (6), (7), (8) and (9) respectively are
\[
Z_k=\left[\begin{array}{c}
Z_k^p \\
Z_k^q\\Z_k^\epsilon
\\Z_k^\delta
\\Z_k^g
\end{array}\right] =\left[\begin{array}{c}
J_p^{k-k_0}Z_{k_0}^p \\
0_{q, 1}\\C_{k,1}
\\0_{t-h, 1}
\\C_{k,2}
\end{array}\right].
\]
Then by using (4)
\[
Y_k = QZ_k =
\left[\begin{array}{ccccc}Q_p & Q_q &Q_\epsilon & Q_\zeta & Q_g \end{array}\right]
\left[\begin{array}{c}
J_p^{k-k_0}Z_{k_0}^p \\
0_{q, 1}\\C_{k,1}
\\0_{t-h, 1}
\\C_{k,2}\end{array}\right]
\]
and
\[
Y_k =
Q_pJ_p^{k-k_0}Z_{k_0}^p+Q_\epsilon C_{k,1}+Q_g C_{k,2}
\]
Since $C_{k,1}$ and $C_{k,2}$ can be taken arbitrary, it is clear that the general singular discrete time system for every suitable defined initial condition has an infinite number of solutions. It is clear that the existence of c.m.i. is the reason that the systems (7) and consequently (9) exist. These systems have always infinite solutions. Thus a necessary condition for the system to have unique solution is not to have any c.m.i. which is equal to
\[
dim\mathcal{N}_r(sF-G)=0.
\]
In this case the Kronecker canonical form of the pencil $sF-G$ has the following form
\[
sF_K - Q_K :=
sI_p - J_p \oplus sH_q - I_q \oplus sF_{\zeta}-G_{\zeta}
\]
and then the system (1) is divided into the three subsystems (5), (6), (8) with solutions (10), (11), (13) respectively. Thus
\[
Y_k = QZ_k =
\left[\begin{array}{ccc}Q_p & Q_q & Q_\zeta\end{array}\right]
\left[\begin{array}{c}
J_p^{k-k_0}Z_{k_0}^p \\
0_{q, 1}
\\0_{t-h, 1}
\end{array}\right]
\]
and
\[
Y_k =
Q_pJ_p^{k-k_0}Z_{k_0}^p.
\]
The solution that exists if and only if
\[
Y_{k_0}=Q_pZ_{k_0}^p,
\]
or
\[
Y_{k_0}\in colspan Q_p.
\]
In this case the system has the unique solution
\[
Y_k=Q_pJ_p^{k-k_0}Z_{k_0}^p.
\]
| 196,806
|
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TITLE: Parabolas and axis of symmetry?
QUESTION [0 upvotes]: I have the parabola $$(x+y)^2 = 8(x−y)$$ and know that the axis of symmetry is $$x+y=0$$ but I know when this is the case the left hand side equals 0 but apart from that I can't see how this equation was found. Can someone please help and explain a $(x+y)^2 = 8(x−y)$ way to find it (if there is one)?
REPLY [0 votes]: Think of "regular" parabolas, with equation $y = ax^2 + bx + c$. These always have an axis of symmetry given by $x = -b/2a$. We could replace the standard coordinates of the plane to the coordinate $w,z$, with $w = x + y$ and $z = x - y$ (turning the coordinate axes of the plane 45 degrees counterclockwise). Then the equation of $(x+y)^2 = 8(x-y)$ becomes $w^2 = 8z$, so we have the parabola $z = 1/8 \cdot w^2$ with axis of symmetry $w = 0$, which is precisely $x + y = 0$.
| 179,886
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Listed:264 days ago
125 Tuckerman Road
Ashburnham, MA 01430
Beds:3
Baths:3
Type:Single Family
Year Built:2001
Sq. Feet:1,664 sq. ft.
Source ID:72075491
Listed:264 days ago
Published with REALTOR(r) Association of Pioneer Valley on 10/01.
Updated 06/22.
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Christina Sargent
Foster-Healey Real Estate
Foster-Healey Real Estate
Phone: 978-537-8301
Home Description
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Know more about this business than we do? Cool! Please submit any corrections or missing details you may have.Help us make it right
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Vinny Guadagnino: Stand-Up Comic? Check Him Out in Action! (PHOTO)
Vinny Guadagnino has always had a lot to say on Jersey Shore, and now that he’s got his own talk show, The Show With Vinny, our favorite Staten Island smartie has another platform to express his views and let fans get to know more about him.
Recently Vinny tried his hand at something similar to hosting, but in front of a live audience instead of a film crew: stand-up comedy! Vinny Instagrammed this cute pic of him wearing his glasses and a cool shirt trying his hand at comedy “for the first time” at the famous Laugh Factory in Los Angeles.
As we can imagine, Vinny hashtagged the experience as “#nervewracking but amazing!” One of the scariest parts of stand-up is the unpredictable crowd, as some people definitely heckle! We have a feeling Vinny did great though, and we wish we could watch his routine for ourselves. He’s always witty one-liners on Jersey Shore, so we think he might make for an excellent stand-up comedian.
What do you think about Vinny trying his hand at stand-up comedy? Would you go see him perform? Sound off below!
| 217,950
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TITLE: Does there exist some probability space $(\Omega,\mathcal F,\mathbb P)$ that admits random variables with all possible laws on $\mathbb R^n$?
QUESTION [6 upvotes]: I have a question about a statement which intuitively seems like it should be a canonical fact, but which I cannot find in any common textbook on probability. Namely, does there exist a probability space $(\Omega,\mathcal F,\mathbb P)$ such that for any probability measure $\mu$ on $\mathbb R^d$, there exists a random variable $X:\Omega\to\mathbb R^d$ such that $X$ has law $\mu$, i.e. $X_\#\mathbb P=\mu$? Can we do this on a single probability space as $d$ ranges over the natural numbers?
I suspect that the space $$([0,1]^\omega,\mathcal B_{[0,1]}^{\otimes\omega},\mathcal L^\omega)$$ should do the trick, where $\mathcal L$ is the Lebesgue measure on $[0,1]$, which I see can simply reduce to proving that $$([0,1]^d,\mathcal B_{[0,1]}^{\otimes d},\mathcal L^d)$$ works for $\mathbb R^d$, but the details of this last part are a bit beyond me.
REPLY [1 votes]: Suppose we have a (valid) joint distribution $F_{X_1, ..., X_d}(x_1, ..., x_d)=P[(X_1, ..., X_d)\leq (x_1,..., x_d)]$. We want to generate a random vector $(X_1, ..., X_d)$ with that joint CDF.
Method 1 (assuming we can compute conditional distributions):
Start with a single random variable $U$ that is uniform over $[0,1]$.
From that, generate i.i.d. uniform variables $Z_1, ..., Z_d$ as I mentioned in comments above.
Using $Z_1$, generate $X_1$ from the marginal $F_{X_1}(x_1)$ using the 1-d method in my comments above. Call $u_1$ the particular value of $X_1$ that is chosen.
Using $Z_2$, generate $X_2$ from the conditional CDF $P[X_2\leq x_2 |X_1=u_1]$ via the same 1-d method. Call $u_2$ the particular value of $X_2$ that is chosen.
Using $Z_3$, generate $X_3$ from the conditional CDF $P[X_3 \leq x_3 | X_1=u_1, X_2=u_2]$.
and so on.
Method 2:
Start with $U$ uniform over $[0,1]$.
From $U$, generate the infinite i.i.d. sequence $\{Z_i\}_{i=1}^{\infty}$ of uniform variables.
Chop up $\mathbb{R}^d$ into a countable collection of disjoint unit hypercubes $\{S_i\}$. Let $p_i = P[(X_1, ..., X_d) \in S_i$]. Use $Z_1$ to randomly choose a hypercube $i \in \{1, 2, 3, ...\}$ with prob $p_i$. This is like determining the integer parts of the random vector $(X_1, ..., X_d)$. To get the remaining fractional parts do the next steps.
Refine the chosen (hyper)cube by chopping into $2^d$ subcubes, use $Z_2$ to randomly choose a subcube (according to the appropriate conditional probabilities given you are already in the larger cube).
Refine the chosen subcube by chopping into $2^d$ sub-subcubes, use $Z_3$ to randomly choose a sub-subcube.
And so on, so each $Z_i$ refines our (binary) expansion of the random vector $(X_1, ..., X_d)$.
One disadvantage of this method is that it involves an infinite number of steps. The nice thing is that the conditional probabilities needed are always well defined as the denominators are always nonzero. Indeed, if at any step the prob 0 event occurs that you end up in a cube with mass 0, just stop the process and define $(X_1, ..., X_d)=(0,0,...,0)$ (or, just start again if you like). This will not affect the distribution as it happens with prob 0. That is, with prob 1, at each of the infinite number of steps, we choose a subcube with positive probability mass.
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“Break things down into smaller, more manageable pieces.”
There are times where a concept, idea, or task, completely overwhelms us such that we can’t make any progress on it. It can be building a cupboard, writing an essay, drawing a comic, learning a new skill etc. The task itself appears so complicated that we simply don’t know where to begin. Part of the problem stems from the fact that we are staying with the bigger picture, instead of seeing that as complicated as the bigger picture seems, there are smaller parts that compose the greater whole.
Breaking something down into smaller chunks allows you to see more clearly how the pieces fit, and how to tackle each piece individually. Seeing its smaller components can often help get you started because now you have something you can actually do. This is a creative exercise that forces you to momentarily push out something that you are used to, and enter a slightly more unfamiliar territory (only in the sense that it isn’t the actual problem before you). But what you will find is that although not the original problem, the new smaller chunks are easier to deal with.
One potential pitfall of breaking things down into smaller chunks is not knowing where to start because instead of ONE subject, now you’ve got multiple. If there is a starting point that is obvious, start there. If not, then it may not matter. Some people prefer to start with the bit that is easiest, and progress to the harder bits…others do the reverse. It depends entirely on how you function best.
While this post was more about managing apparently difficult tasks, it is also a wonderful exercise in combating writer’s block, or stalled creativity. These tips and many more found in the link below!
Source: How to Overcome Creative Blocks – Lifehacker
| 202,717
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TITLE: Immersion between spheres.
QUESTION [1 upvotes]: Let $F:\mathbb{S}^{2}\to \mathbb{S}^{5}$ be given by $F(x,y,z)=(x^{2},y^{2},z^{2},\sqrt{2}xy,\sqrt{2}yz,\sqrt{2}xz)$. Prove that is an immersion. I have a minor with order three with determinant equal to $xyz$ then $Jac F(x,y,z)$ has rank $3$ except when $x=0$ or $y=0$ or $z=0$, now i have a problem for to be continue.Any tips?
REPLY [1 votes]: You're right that the rank of ${\rm Jac \ } F(x,y,z)$ (viewed as a linear map from $\mathbb R^3 $ to $\mathbb R^6$) is less than $3$ when $x = 0$, $y = 0$ or $z = 0$. But this is not a problem! Your real goal is to prove that ${\rm Jac \ } F(x,y,z)$ is injective on the space of tangent vectors for any point $\vec x\in \mathbb S^2$. Any such tangent space is only two-dimensional. So you could approach this by finding two basis vectors for the tangent space, and verify that the images of these two tangent vectors under ${\rm Jac \ } F(x,y,z)$ are linearly independent.
For example, let's look at the $z = 0$ case. A typical point on $\mathbb S^2$ with $z = 0$ looks like this:
$$ \vec x = (\cos \phi, \sin \phi, 0).$$
The tangent space at this point is generated by the basis vectors,
$$ \vec v_1 =(-\sin \phi, \cos \phi, 0), \ \ \ \vec v_2 = (0, 0, 1).$$
Your task is to show that the images ${\rm Jac }F(\vec x)(\vec v_1)$ and ${\rm Jac }F(\vec x)(\vec v_2)$ are linearly independent. Hopefully this should be too difficult to verify!
| 154,290
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Possible reasons:
2000-2009 DpcAuto.com - (718)297-4404 144-24 Hillside Ave Jamaica, NY 11435 USA
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| 396,781
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Sighting
Ken Harris 29 Oct 2015
Pyrgotid Fly - Osa sp.
Identifications
- Ken Harris 29 October 2015Diptera(order)Taxonomy: Animalia: Arthropoda: Insecta: Diptera0 votes0
- Tony D. 30 October 2015Osa(genus)Taxonomy: Animalia: Arthropoda: Insecta: Diptera: Pyrgotidae: Osa
This one is a different genus to your recent upload [ ] - this one lacking the right-angle bend of Sc vein, thus here meeting the costa at an acute angle. The long ovipositor is found in Facilina and Osa, but the face not being depressed from antennae to mouth rules out the former. Good chance it is O. commoni by what I can see but not fully confident.0 votes0
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\section{Query enumeration}\label{sec:enumeration}
In this section we prove Theorem~\ref{thm:intro-main-qe}, which we recall below for convenience.
\newenvironment{red}
{\color{red}
\ignorespaces}
{}
\Enumeration*
First, we need to define our notion of enumerators and prepare some tools for them.
\paragraph{Enumerators.}
Let $x_1,\ldots,x_n$ be a sequence of elements.
An \emph{enumerator} of the sequence $x_1,\ldots,x_n$
is a data structure that implements a single method,
such that at the $i$th invocation of the method,
it outputs the element $x_j$ of the sequence,
where $j={i\mod n}$,
and reports an `end of sequence' message if $j=0$.
We say that the enumerator has \emph{delay} $t$ if
each invocation takes at most $t$ computation steps,
including the steps needed to output the element $x_j$
(assuming each element has a fixed representation).
An enumerator for a set $X$ is an enumerator for
any sequence $x_1,\ldots,x_n$ with $\set{x_1,\ldots,x_n}=X$
and $n=|X|$.
Enumerators
for Cartesian products and disjoint unions of sets
can be obtained in an obvious way:
\begin{lemma}\label{lem:enum-prod}
Suppose we are given an enumerator for a set $X$
with delay $t$ and an enumerator for a set $Y$ with delay $t'$, where $t,t'\ge 1$.
Then we can construct in time $\Oof(1)$ an enumerator
with delay $t+t'+\Oof(1)$
for the set $X\times Y$
and
-- if $X$ and $Y$ are disjoint --
for the set $X\uplus Y$.
\end{lemma}
We will also construct enumerators
for disjoint unions of families of sets, as follows.
\begin{lemma}\label{lem:enum-sum}
Suppose $X_1,\ldots,X_n$ are pairwise disjoint, nonempty sets,
such that $X_i$ has an enumerator $\cal E_i$ with delay $t$.
Suppose furthermore we have an enumerator
for the sequence $\cal E_1,\ldots,\cal E_n$
with delay $t'$.
Then one can construct, in time $\Oof(1)$
an enumerator for the set $\bigcup_{1\le i\le n} X_i$
with delay $t+t'+\Oof(1)$.
\end{lemma}
Finally, we will use the following lemma, proved in \cite[Lemma 7.15]{PilipczukSSTV21}.
\begin{lemma}
Fix a finite set $Q$ of size $q$ and a set of functions $\cal F\subset Q^Q$. There is a constant $c$ computable from $q$ and an algorithm that, given a rooted tree $T$, in which each edge $vw$ ($v$ child of $w$) is labelled by a function $f_{vw}\from Q\to Q$, computes in time $c\cdot |T|$ a collection $(\cal E_w)_{w\in V(T)}$ of enumerators, where each
$\cal E_w$ is an enumerator with delay $c$ that enumerates all descendants $v$ of $w$ such that the composition of the functions labeling the edges of the path from $v$ to $w$, belongs to $\Ff$.
\end{lemma}
This immediately yields the following.
\begin{corollary}\label{cor:composer-enum}
Fix a number $q$. There is a constant $c$ computable from $q$ and an algorithm that, given a rooted tree $T$,
in which each node $v$ is labeled by a set $X_v$ with $|X_v|\le q$ and a set $Y_v\subset X_v$, and each
edge $vw$ ($v$ child of $w$) is labelled by a function $f_{wv}\from X_v\to X_w$, computes in time $c\cdot |T|$ a collection $(\cal E_w^\tau)_{w\in V(T),\tau\in X_w}$ of enumerators, where each
$\cal E_w^\tau$ is an enumerator with delay $c$ that enumerates all descendants $v$ of $w$ such that
there is some $\sigma\in Y_v$ that is mapped to $\tau$ by the composition $f\from X_v\to X_w$ of the functions labeling the edges of the path from $v$ to $w$.
\end{corollary}
\paragraph{Proof of Theorem~\ref{thm:intro-main-qe}.}
We now proceed to the proof of Theorem~\ref{thm:intro-main-qe}.
Fix a number $k$, a set of variables $\tup x$,
an $n$-vertex graph $G$, together with its contraction sequence $\Pp_1,\ldots,\Pp_n$.
Denote $r:=2^k$.
For every $s\in [n]$ and $\tup x$-tuple $\tup u\in \Pp_s^{\tup x}$, and local type $\tau\in \Types_{\tup u,s}^k$,
denote
\[S_{\tup u,s}^\tau\coloneqq \setof{\tup w\in V(G)^{\tup x}}{\iwarp{\tup w}{s}=\tup u
\text{ and } \ltp^k_s(\tup w)=\tau}.\]
Recall that the root of $T_{r,\tup x}$ is the pair $(\tup r,n)$,
where $\tup r$ is the constant $\tup x$-tuple with all components equal to the unique
part of $\Pp_n$. Then $S_{\tup r,n}^\tau$ is the set of all $\tup x$-tuples $\tup w\in V(G)^{\tup x}$
with $\ltp^k_n(\tup w)=\tau$.
From Lemma~\ref{lem:ltp_type_as_local_type} and Lemma~\ref{prop:types_mc}
we get:
\begin{lemma}\label{lem:enum wrap up}
Fix a formula $\phi(\tup x)$ of quantifier-rank $k$.
Then there is a set $\Gamma\subset \Types_{\tup r,n}^n$
such that
$\phi(G):=\setof{\tup w\in V(G)^{\tup x}}{G\models \phi(\tup w)}$ is the disjoint union of
the family of sets
$\setof{S_{\tup r,n}^\tau}{\tau\in \Gamma}.$
\end{lemma}
Therefore, an enumerator for $\phi(G)$ can be obtained by concatenating
enumerators for the sets $S_{\tup r,n}^\tau$, for $\tau\in \Gamma$.
Note that here we are concatenating only $\Oof_{k,d,\tup x}(1)$
enumerators, by Lemma~\ref{lem:ltp_number_of_types},
so, by applying Lemma~\ref{lem:enum-prod} repeatedly,
the resulting enumerator can be obtained in time
$\Oof_{k,d,\tup x}(1)$ and has delay $\Oof_{k,d,\tup x}(1)$.
So to prove Theorem~\ref{thm:intro-main-qe},
it suffices to prove that we can efficiently compute an enumerator
for each of the sets $S_{\tup r,n}^\tau$.
Recall that $T_{r,\tup x}$ is the tree of $r$-close $\tup x$-tuples (see Def. \ref{def:close-tree}), and can be computed in time $\Oof_{d,k,\tup x}(n)$, by Lemma~\ref{lem:construct-tree}.
In the following proposition,
we will show how to compute enumerators for all of the sets $S_{\tup u,s}^\tau$,
for $(\tup u,s)\in T_{r,\tup x}$.
All the enumerators jointly will be computed in time $\Oof_{d,k,\tup x}(n)$.
\begin{proposition}\label{prop:enum}
Fix a nonempty set $\tup x$ of variables and $k\in\N$.
Assume $G$ is a graph on $n$ vertices provided on input through a contraction sequence
$\Pp_1,\ldots, \Pp_n$ of width $d$.
Then one can in time $\Oof_{d,k,\tup x}(n)$ construct a data structure that
associates, to every node $(\tup u,s)$ of $T_{r,\tup x}$
and every local type $\tau\in \Types_{\tup u, s}^k$,
an enumerator for all tuples in $S_{\tup u,s}^\tau$
with delay $\Oof_{d,k,\tup x}(1)$.
\end{proposition}
As noted above, Theorem~\ref{thm:intro-main-qe} follows from
Proposition~\ref{prop:enum}, using Lemma~\ref{lem:enum wrap up}.
The rest of Section~\ref{sec:enumeration} is devoted to proving
Proposition~\ref{prop:enum}.
We prove Proposition~\ref{prop:enum} by induction on $|\tup x|$.
So suppose the statement holds for all strict subsets of $\tup x$.
Recall that we may construct the tree $T_{r,\tup x}$, in time $\Oof_{d,k,\tup x}(n)$, using Lemma~\ref{lem:construct-tree}.
Let $v,u$ be two nodes of $T_{r,\tup x}$
with $v=(\tup v,t)$ and $u=(\tup u,s)$ and
$u\preceq v$.
By Lemma~\ref{lem:warp-ds}, there is a function
$f_{vu}\from \Types_{\tup v,t}^k\to \Types_{\tup u,s}^k$
such that for every $\tup w\in V(G)^{\tup x}$,
with $\tup u=\iwarp{\tup w}{t}$ we have
$f_{vu}(\ltp^k_t(\tup w))=\ltp^k_s(\tup w).$
\medskip
For a tuple $\tup w\in V(G)^{\tup x}$,
let $s\in [n]$
be the first time such that $\iwarp{\tup w}{s}$ is $r$-close at time $s$, where $r=2^k$.
We then say that $\tup w$ \emph{registers}
at $(\tup u,s)$, where $\tup u=\iwarp{\tup w}{s}$.
By Claim~\ref{cl:pinned}, in this case, the pair $(\tup u,s)$ is a node of $T_{r,\tup x}$.
For each node $(\tup u,s)$ of $T_{r,\tup x}$
and type $\tau\in \Types_{\tup u,s}^k$,
denote:
\[R_{\tup u,s}^\tau=\setof{\tup w\in V(G)^{\tup x}}{\tup w \text{ registers at $(\tup u,s)$
and $\ltp^k_s(\tup w)=\tau$}}.\]
Fix a node $(\tup u,s)\in T_{r,\tup x}$ and a type
$\tau\in \Types_{\tup u,s}^k$.
Clearly, every tuple $\tup w\in S_{\tup u,s}^\tau$
registers at exactly one descendant $v=(\tup v,t)$ of $u=(\tup u,s)$
(possibly, $v=u$),
and moreover,
${f_{vu}(\ltp^k_t(\tup w))=\tau}$.
This proves the following.
\begin{lemma}\label{l0}
For every node $u\in T_{r,\tup x}$ and type
$\tau\in \Types_u^k$,
the set $S_u^\tau$
is the disjoint union of
all the sets $R_v^\sigma$, for
$v\in T_{r,\tup x}$ with $v\succcurlyeq u$ and $\sigma\in\Types_v^k$ such that
$f_{vu}(\sigma)=\tau$.
\end{lemma}
We shall prove the following two lemmas.
\begin{lemma}\label{l1}
For every given node $u=(\tup u,s)\in T_{r,\tup x}$ and type
$\tau\in \Types_{\tup u,s}^k$,
an enumerator for the set $R_{\tup u,s}^\tau$
with delay $\Oof_{d,k,\tup x}(1)$ can be constructed in time $\Oof_{d,k,\tup x}(1)$.
\end{lemma}
\begin{lemma}\label{l2}
One can construct in time $\Oof_{d,k,\tup x}(n)$
a collection of enumerators $\cal E_u^\tau$,
one per each node $u=(\tup u,s)\in T_{r,\tup x}$
and type $\tau\in \Types_{\tup u,s}^k$,
where $\cal E_u^\tau$ has delay $\Oof_{d,k,\tup x}(1)$ and enumerates all descendants
$v=(\tup v,t)$ of $u$ in $T_{r,\tup x}$
such that there is some $\sigma\in \Types_{\tup v,t}^k$
with
$f_{vu}(\sigma)=\tau$ and
$R_{\tup v,t}^\sigma\neq\emptyset$.
\end{lemma}
Observe that combining Lemma~\ref{l0}, Lemma~\ref{l1}, Lemma~\ref{l2} and Lemma~\ref{lem:enum-sum} yields the required collection of enumerators
for each of the sets $S_{\tup u,s}^k$,
thus proving Proposition~\ref{prop:enum} and Theorem~\ref{thm:intro-main-qe}.
Thus, we are left with proving Lemmas~\ref{l1} and~\ref{l2}, which we do in order.
\begin{proof}[Proof of Lemma~\ref{l1}]
Let $u=(\tup u,s)\in T_{r,\tup x}$ be a node.
We consider two cases: either $s=1$ or $s>1$.
\subparagraph{Leaves.}
Consider the case that $s=1$.
As $\tup u\in \cal P_1^{\tup x}$ is $r$-close at the time $1$,
there is some $v\in V(G)$ such that
all the components $\tup u$ are equal to the part $\set{v}$.
Then a tuple $\tup w\in V(G)^{\tup x}$ satisfies $\iwarp{\tup w}{1}=\tup u$ if and only if $\tup w$ is the constant $\tup x$-tuple $\vec v$ consisting of the vertex $v$.
Therefore,
\[R_{\tup u,s}^\tau=
\begin{cases}\set{\vec v}& \text{if }\ltp^k_1(\vec v)=\tau,\\
\emptyset &\text{if }\ltp^k_1(\vec v)\neq \tau.
\end{cases}
\]
In either case, it is trivial to construct an enumerator for the (empty or singleton) set $R_{\tup u,s}^\tau$
and we can distinguish which case occurs by computing $\ltp^k_1(\vec v)$ in time $\Oof_{d,k,\tup x}(1)$.
The delay of the enumerator is bounded by $\Oof_{|\tup x|}(1)$,
as this is the size of the representation of the tuple $\vec v$.
\subparagraph{Inner nodes.}
Suppose now that $s>1$, that is, $(\tup u,s)$ is an inner node.
Denote
\[\cal V\coloneqq \setof{\tup v\in \cal P_{s-1}^{\tup x}} {\iwarp{\tup v}{s-1\to s}=\tup u}.\]
\begin{claim}
The set $\cal V$ has size $\Oof_{k,\tup x,d}(1)$ and can be computed in this time, given $(\tup u,s)\in T_{r,\tup x}$.
\end{claim}
\begin{proof}
The tuples $\tup v$ in $\cal V$ are precisely those tuples that can be obtained from the tuple $\tup u$ by replacing each occurrence of $B_s$ in $\tup u$ (which might occur zero or more times) by one of the two parts in $\Pp_{s-1}$ that are contained in $B_s$.
Since $B_s$ may occur in $\tup u$ at most $|\tup x|$ many times,
we have at most $2^{|\tup x|}$ many possibilities for $\tup v$,
and all of them can be computed in the required time.
\cqed\end{proof}
Fix $\tup v\in \cal V$, and consider the graph $H_{\tup v}$ with vertices $\tup x$
where any two distinct $y,y'\in \tup x$ are adjacent whenever $\dist_s(\tup v(y),\tup v(y'))\le r$.
Let $C_{\tup v}$ denote the set of connected components of $H_{\tup v}$,
where each connected component is viewed as a set $\tup y\subset \tup x$ of vertices of $H_{\tup v}$. Then each of the sets $C_{\tup v}$, can be computed in time $\Oof_{k,\tup x,d}(1)$, given $(\tup u,s)\in T_{r,\tup x}$
and $\tup v\in \cal V$.
Call a tuple
$\tup v\in \cal V$ \emph{disconnected} if $|C_{\tup v}|>1$, that is, $H_{\tup v}$ has more than one connected component.
Note that if $\tup v$ is disconnected and $\tup y\in C_{\tup v}$, then $|\tup y|<|\tup x|$.
Let $\cal V'\subset \cal V$ be the set of disconnected tuples.
\medskip
Fix an disconnected tuple $\tup v\in \cal V'$ and $\tup y\in C_{\tup v}$.
Denote by $\tup v_{\tup y}$ the restriction of $\tup v$ to $\tup y$.
Note that the pair $(\tup v_{\tup y},s-1)$ is a node of $T_{r,\tup y}$.
Indeed, by assumption, $\dist_{s}(B_s,\tup u)\le r$ holds, so
$\dist_{s}(B_s,\iwarp{\tup v}{s\to s+1})\le r$,
and in particular $\dist_{s}(B_s,\iwarp{\tup v_{\tup y}}{s\to s+1})\le r$.
Moreover, $\tup v_{\tup y}$ is $r$-close, since $\tup y$ is a connected component of
$H_{\tup v}$.
As $|\tup y|<|\tup x|$, by inductive assumption, we have already computed enumerators for each of the sets
$S_{\tup v_{\tup y},s-1}^\sigma$, for all adequate local types $\sigma$.
The following claim is obtained by repeatedly applying Lemma~\ref{lem:tp-merge}.
\begin{claim}\label{cl:step}
For every $\tup w\in V(G)^{\tup x}$ with $\iwarp{\tup w}s=\tup u$,
the local type $\ltp^k_s(\tup w)$ can be computed in time
$\Oof_{k,d,\tup x}(1)$ from the following data:
\begin{itemize}
\item the tuple $\tup v:=\iwarp{\tup w}{s-1}\in \cal V$,
\item the family of local types $\bar \tau:=(\ltp^k_{s-1}(\tup v_{\tup y}): \tup y\in C_{\tup v})$.
\end{itemize}
\end{claim}
More precisely, there is a function $\Gamma$
such that for each pair
$(\tup v,\bar \tau)$, where $\tup v\in \cal V$ and $\bar \tau =(\tau_{\tup y}:\tup y\in C_\tup v)$
is a family with $\tau_{\tup y}\in\Types_{\tup v_{\tup y},s-1}^k$, we have that
\[\Gamma(\tup v,\bar \tau)=\ltp^k_s(\tup w)\]
holds for every $\tup w\in V(G)^{\tup x}$ such that $\iwarp{\tup w}{s-1}=\tup v$
and $\ltp^k_{s-1}(\tup v_{\tup y})=\tau_{\tup y}$ for all $\tup y\in C_\tup v$.
For $\tup v\in \cal V$ and
$\bar \tau =(\tau_{\tup y}\colon \tup y\in C_\tup v)$
a family with $\tau_{\tup y}\in\Types_{\tup v_{\tup y},s-1}^k$,
define the set
\[S_{\tup v,s-1}^{\bar \tau}\coloneqq
\setof{\tup w\in V(G)^{\tup x}}
{\ltp^k_{s-1}(\tup w_{\tup y})=\tau_{\tup y} \text{ for all }\tup y\in C_{\tup v}}
.\]
Recall that the tuples $\tup y\in C_{\tup v}$ form a partition of $\tup x$.
Hence, we have the following.
\begin{claim}\label{cl:prod}
Fix an disconnected tuple $\tup v\in \cal V'$ and $\bar \tau =(\tau_{\tup y}:\tup y\in C_\tup v)$
a family with $\tau_{\tup y}\in\Types_{\tup v_{\tup y},s-1}^k$.
One can compute in time $\Oof_{k,d,\tup x}(1)$ an enumerator for the set $S_{\tup v,s-1}^{\bar \tau}$,
with delay $\Oof_{k,d,\tup x}(1)$.
\end{claim}
\begin{proof}
The enumerator is the Cartesian product (see Lemma~\ref{lem:enum-prod}) of the enumerators for the sets $S_{\tup v_{\tup y},s-1}^{\tau_{\tup y}}$, which have been computed by inductive assumption. Each of those enumerators has delay $\Oof_{k,d,\tup x}(1)$, and their total number is $\Oof_{k,d,\tup x}(1)$, by Lemma~\ref{lem:ltp_number_of_types}.
\cqed\end{proof}
\begin{claim}
The set $R_{\tup u,s}^\tau$
is equal to the disjoint union of the family of sets
\[\setof{S_{\tup v,s-1}^{\bar \tau}}{(\tup v,\bar \tau)\in \Gamma^{-1}(\tau), \tup v\in \cal V'}.\]
\end{claim}
\begin{proof}
We first show the right-to-left inclusion.
Let
$\tup w\in S^{\bar \tau}_{\tup v,s-1}$.
Then registers at $(\tup u,s)$ as $\iwarp{\tup w}{s-1}$
is not $r$-close at time $s-1$ (because $\tup v\in \cal V'$), and $\iwarp{\tup w}{s}=\tup u$.
Moreover, $\ltp_k^s(\tup w)=\tau$ since $(\tup v,\bar \tau)\in\Gamma^{-1}(\tau)$. This proves that $\tup w\in R_{\tup u,s}^\tau$.
Conversely, let $\tup w\in R_{\tup u,s}$.
Define $\tup v$ as $\iwarp{\tup w}{s-1}$.
Since $\tup w$ registers at $(\tup u,s)$,
it follows that $H_{\tup v}$ is disconnected.
Hence, $\tup v\in \cal V'$. For each connected component $\tup y\subset \tup x$ of $H_{\tup v}$, let $\tau_\tup y=\ltp_k^s(\iwarp{\tup w_\tup y}{s-1})$, and let
$\bar \tau=(\tau_{\tup y}\colon \tup y\in C_\tup v)$.
By construction, $\tup w\in S_{\tup v,s-1}^{\bar \tau}$.
This proves the left-to-right inclusion.
Moreover, it is easy to see that the union is disjoint.
\cqed\end{proof}
Therefore, an enumerator for the set $R_{\tup u,s}^\tau$ above can be obtained by concatenating the enumerators for the sets $S_{\tup v,s-1}^{\bar \tau}$, for $(\tup v,\bar \tau)\in \Gamma^{-1}(\tau)$ with $\tup v\in \cal V'$, and those can be computed in time $\Oof_{d,k,\tup x}(1)$ by Claim~\ref{cl:prod}.
Note that we are taking a disjoint union of at most $\Oof_{d,k,\tup x}(1)$ sets, by Lemma~\ref{lem:ltp_number_of_types},
so the concatenation can be computed by repeatedly applying Lemma~\ref{lem:enum-prod}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{l2}]
Label each node $u=(\tup u,s)$ of $T_{r,\tup x}$ by the
set $X_u:=\Types^k_{\tup u,s}$
and its subset $Y_u\subset X_u$
of all $\tau\in \Types^k_{\tup u,s}$
such that $R_{\tup u,s}^{\tau}$ is nonempty.
This can be computed in time $\Oof_{k,d,\tup x}(n)$,
by testing emptiness of the enumerators produced in Lemma~\ref{l1}.
Moreover, $|X_u|$ is bounded by $\Oof_{k,d,\tup x}$, by Lemma~\ref{lem:ltp_number_of_types}.
Label each child-parent edge $e=((\tup v,t),(\tup u,s))$ of $T_{r,\tup x}$ by the function $f_e\colon \Types^k_{\tup v,t}\to \Types^k_{\tup u,s}$ such that for every tuple $\tup w\in V(G)^{\tup x}$ with $\tup v=\iwarp{\tup w}{s}$, we have
\begin{equation}
\ltp_t^k(\tup w)=f_e(\ltp_s^k(\tup w)),
\end{equation}
where we denote $\ltp^k_t(\cdot)\coloneqq \ltp^k_{\Pp_t}(\cdot)$ for brevity. Such a function exists by Lemma~\ref{lem:tp-warp}.
Note that if $v:=(\tup v,t)$ is a descendant of $u:=(\tup u,s)$,
then the composition of the functions $f_e$ along the edges $e$ of the path from $v$ to $w$
is a function $f_{vu}\from \Types^k_{\tup v,t}\to \Types^k_{\tup u,s}$
such that for every tuple $\tup w\in V(G)^{\tup x}$ with $\tup v=\iwarp{\tup w}{s}$, we have
\begin{equation}
\ltp_t^k(\tup w)=f(\ltp_s^k(\tup w)).
\end{equation}
We are now in the setting of Corollary~\ref{cor:composer-enum}.
Hence we can compute in time $\Oof_{k,d,\tup x}(n)$
a collection of enumerators $\cal E_u^\tau$,
where for each node $u=(\tup u,s)$ of $T_{r,\tup x}$
and type $\tau\in \Types^k_{\tup u,s}$,
the enumerator $\cal E_w^\tau$ enumerates
all descendants $v=(\tup v,t)$ of $u$ in $T_{r,\tup x}$
such that $f_{vu}^{-1}(\tau)\cap Y_v$ is nonempty.
Unravelling the definitions, this means
that there is some $\sigma\in \Types^k_{\tup v,t}$
such that $f_{vu}(\sigma)=\tau$ and
$R_{\tup v,t}^\sigma$ is nonempty.
\end{proof}
| 65,040
|
TITLE: convert hyperbola in rectangular form to polar form
QUESTION [1 upvotes]: i am trying to convert the rectangular equation of a conic (hyperbola) to a polar form.
the rectangular equation is:
$$3y^2 - 16y -x^2 + 16 = 0.$$
i substituted $r\sin\theta$ for y and $r\cos\theta$ for x, and tried to simplify, but I am stuck.
i tried substituting $1-\sin^2\theta$ for the resulting $\cos^2\theta$ term, and ended up with the following expression after some manipulation:
$$r^2(2\sin\theta-1)(2\sin\theta+1)-16r\sin\theta + 16 = 0.$$
the solution to the question is supposed to be:
$$r = \frac{4}{1 + 2\sin\theta}.$$
I just can't figure out how to get from where I am to the expected solution. Perhaps I made the wrong substitution? I just need to figure out the intermediate steps to get from the original rectangular equation to the final equation in polar form given above.
Hope someone can help me -- thanks!
REPLY [2 votes]: Staring with $3 y^2 - 16 y - x^2 + 16 = 0 $
Put it first in the standard format. Complete the square in $y$
$ 3 \left(y - \dfrac{8}{3} \right)^2 - \dfrac{64}{3} - x^2 + 16 = 0 $
$ 3 \left(y - \dfrac{8}{3} \right)^2 - x^2 = \dfrac{16}{3} $
$ \dfrac{9}{16} \left( y - \dfrac{8}{3} \right)^2 - \dfrac{3}{16} x^2 = 1 $
Thus the center is at $\left(0, \dfrac{8}{3} \right) $ , $ a = \dfrac{4}{3} $ and $b = \dfrac{4}{\sqrt{3}} $
The focal distance $ c = \sqrt{a^2 + b^2} = 4 \sqrt{ \dfrac{1}{9} + \dfrac{1}{3} } = \dfrac{8}{3} $
Hence, the foci are at $ (0, 0) $ and $ \left(0, \dfrac{16}{3} \right) $
Taking the focus that is at the origin, then $ x = r \cos \theta, y = r \sin \theta $
Back to the original equation
$3 y^2 - 16 y - x^2 + 16 = 0 $
Substituting the polar expressions,
$ 3 (r \sin \theta)^2 - 16 r \sin \theta - r^2 \cos^2 \theta + 16 = 0 $
Using $\cos^2 \theta = 1 - \sin^2 \theta$ and collecting terms
$ r^2 ( 4 \sin^2 \theta - 1 ) - 16 r \sin \theta + 16 = 0$
From the quadratic formula, and taking the positive root
$ r = \dfrac{1}{ 2(4 \sin^2 \theta - 1 ) } \left ( 16 \sin \theta - \sqrt{ 256 \sin^2 \theta - (256 \sin^2 \theta - 64 ) } \right) $
And this simplifies to
$ r = \dfrac{1}{2 (4 \sin^2 \theta - 1) } \left( 16 \sin \theta - 8 \right) $
Factoring the denominator and cancelling equal terms
$ r = \dfrac{ 4 }{ 2 \sin \theta + 1 } $
| 210,022
|
TITLE: Why does $\text{arctanh}(2^{-k})$ approach powers of $2$?
QUESTION [5 upvotes]: This is from a piece of verilog code I generated for $\text{arctanh}(2^{-k})$:
localparam bit [31:0][47:0] arctanhTable = {
48'b100011001001111101010011110101010110100000011000,
48'b010000010110001010111011111010100000010001010001,
48'b001000000010101100010010001110010011110101011101,
48'b000100000000010101011000100010101101001101110101,
48'b000010000000000010101010110001000100100011010111,
48'b000001000000000000010101010101100010001000101011,
48'b000000100000000000000010101010101011000100010001,
48'b000000010000000000000000010101010101010110001000,
48'b000000001000000000000000000010101010101010101100,
48'b000000000100000000000000000000010101010101010101,
48'b000000000010000000000000000000000010101010101010,
48'b000000000001000000000000000000000000010101010101,
48'b000000000000100000000000000000000000000010101010,
48'b000000000000010000000000000000000000000000010101,
48'b000000000000001000000000000000000000000000000010,
48'b000000000000000100000000000000000000000000000000,
48'b000000000000000010000000000000000000000000000000,
48'b000000000000000001000000000000000000000000000000,
48'b000000000000000000100000000000000000000000000000,
48'b000000000000000000010000000000000000000000000000,
48'b000000000000000000001000000000000000000000000000,
48'b000000000000000000000100000000000000000000000000,
48'b000000000000000000000010000000000000000000000000,
48'b000000000000000000000001000000000000000000000000,
48'b000000000000000000000000100000000000000000000000,
48'b000000000000000000000000010000000000000000000000,
48'b000000000000000000000000001000000000000000000000,
48'b000000000000000000000000000100000000000000000000,
48'b000000000000000000000000000010000000000000000000,
48'b000000000000000000000000000001000000000000000000,
48'b000000000000000000000000000000100000000000000000
};
The $\text{arctanh}(2^{-k})$ values start converging to successive powers of $2$. It's apparent that the number of zeroes between the first $1$ and the second $1$ keeps increasing, so the form $\text{arctanh}(2^{-k}) = 2^{-1} + q(k)$ for values of $q(k)$ decreasing much more quickly than values of $2^{-k}$. Wolfram Alpha tells me $q(k) = -2^{-k} - \dfrac{\ln(1-2^{-k}) + \ln(1+2^{-k})}{2}$
Why does this divergence happen? I don't think I can leverage this to cheaply calculate $\text{arctanh}$, although it does give me an interesting way to compress the table by storing a floating-point $q(k)$, not that that's actually useful either.
REPLY [8 votes]: We know that
$$ \DeclareMathOperator{\arctanh}{artanh}
\arctanh(x) = \frac12\left(\log(1 + x) - \log(1 - x)\right). \tag1$$
The Taylor series for $\log(1+x)$ centered at $x=0$ is
$$ \log(1 + x) = x - \frac12 x^2 + r_1(x) $$
where $r_1(x)$ is $\mathcal O(x^3)$ as $x \to 0$.
Similarly, the Taylor series for $\log(1 - x)$ centered at $x=0$ is
$$ \log(1 - x) = -x - \frac12 x^2 - r_2(x) $$
where $r_2(x)$ is $\mathcal O(x^3)$ as $x \to 0$.
Substituting these two Taylor series into Equation $(1)$ gives
\begin{align}
\arctanh(x)
&= \frac12\left(\left(x - \frac12 x^2 + r_1(x)\right)
- \left(-x - \frac12 x^2 + r_2(x)\right)\right) \\
&= x + r_1(x) + r_2(x) \\
&\subseteq x + \mathcal O(x^3), \quad x \to 0. \\
\end{align}
Let $x = 2^{-k}$ for a positive integer $k$ and you have
$$ \arctanh(2^{-k}) \subseteq 2^{-k} + \mathcal O(2^{-3k}), \quad k \to \infty. $$
In other words, $q(k)$ is a function of order $2^{-3k}.$
So its first non-zero digit is about three times as many places to the right as the $1$ digit of $2^{-k}$, leaving about $2k$ zeros between the digits.
For example, consider the following two rows of your table.
48'b000000000100000000000000000000010101010101010101,
48'b000000000010000000000000000000000010101010101010,
The first of these two rows has nine zeros followed by a $1$,
which is the $2^{-k}$ digit of $\arctanh(2^{-k})$,
then we have $21$ zeros followed by some non-zero digits;
if this is the row for $k = 10$ then indeed we have about $2k$ zeros
($2k+1$ to be exact) between the non-zero digits.
In the next row the $2^{-k}$ digit has moved one place to the right,
but the rest of the digits have moved three places, adding two zeros to the gap between non-zero digits.
And we will just keep on adding two more zeros on every row,
which is a simple consequence of the fact that $q(k)$ is of order
$2^{-3k},$ or in other words, when $k$ is large,
$q(k)$ is approximately $\lambda 2^{-3k}$ for some constant $\lambda$.
| 211,313
|
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| 276,100
|
NOT QUITE HAL 9000, BUT IT VACUUMS [1]
News From:
MIT robotics professor Rodney Brooks helped bring about a paradigm shift in robotics in the late 1980s when he advocated a move away from top-down programming (which required complete control of the robot's environment) toward a biologically inspired model that helped robots navigate dynamic, constantly changing surroundings on their own. His breakthroughs paved the way for Roomba, the vacuuming robot disc that uses multiple sensors to adapt to different floor types and avoid obstalces in its path. (Brooks is chief technology officer and cofounder of Roomba's parent company, iRobot.) Brooks talked to NEWSWEEK's Katie Baker about the challenges involved in creating robots that can interact in social settings. Excerpts:
NEWSWEEK: Sociologists talk about the importance of culture and sociability in humans, and why [it should be equally important] in robots. Do roboticists consider things such as culture when thinking about how to integrate robots into human lives?
Rodney Brooks: Some of us certainly do, absolutely. My lab has been working on gaze direction. This is the one thing that you and I don't have right now [over the telephone], but if we were doing some task together, working in the same workspace, we would continuously be looking up at each other's eyes, to see what the other one was paying attention to. Certainly that level of integration with a robot has been of great interest to me. And if you're going to have a robot doing really high-level tasks with a person, I think you will want to know where its eyes are pointing, what it's paying attention to. Dogs do that with us and we do that with dogs, it happens all the time. Somehow cats don't seem to bother.
So are there ethical implications involved when you think about developing sociable robots, in terms of how they might change human behavior?
Well, every technology that we build changes us. There's a great piece on Edge.org by Kevin Kelly, I think it was, talking about how printing changed us, reading changed us. Computers have changed us, and robots will change us, in some way. It doesn't necessarily mean it's bad.
What are some of the more interesting robots that you've seen, or that you're developing or have developed?
I think what gets interesting is when robots have an ongoing existence. And Roomba has an ongoing existence, [though] less than [that of] an insect. But the ones that I have in my lab here, that I've scheduled to come out and clean every night, they do it day after day and recharge themselves. And they just sort of live as other entities in the lab and we never see them doing anything, except every so often we go and empty their dirt bins. So they've got an ongoing existence, but at a very, very primitive level. All the robots that you see from Honda and all those places don't even have that level of ongoing-ness. They're demo robots. But up until now, people haven't been building robots to have an ongoing existence, so they're sitting in the world, ready to do their thing when the situation is appropriate. So I think that's where the really interesting things will start to happen.
When you don't have to completely control the robot's environment?
When it becomes [something that] can have an ongoing existence with people … that is where things get interesting. We've done a few things like that here, starting back with [MIT professor] Cynthia Brezeal and [her sociable robot] Kismet, and in her new lab she's got some fantastic new robots where she's pushing towards that. We've had other robots here in my lab—Mertz, which was trying to recognize people from day to day as their looks change, and know about them. And some of our robots, like Domo, will interact with a person for 10 minutes or so, and [it] has face detectors and things like that. There are other projects in Europe—the RoboCub, which is focused in Genoa [Italy], is building these robots that many labs in Europe now have, which are all about emulating child development.
Obviously, we can tell when something's a robot and when something's a human. But when a robot is too humanlike, do we get concerned?
Our robots that we've built in my lab and the European robots, [if you] show a picture of that robot—just a static picture—to a kid, they'll tell you, "That's a robot." There's no question in their mind. And nevertheless, when the robots are acting and interacting, they can draw even skeptics into interacting with them—for a few seconds at least—where the person responds in a social way. And some people will go on responding for minutes and minutes. Then there are these super-realistic robots that a couple of different groups, one in Asia and one in the U.S., are building. One of them looks like Albert Einstein, and [the other] looks like this television reporter. And there it gets a little weirder. Because very quickly, you realize that they're not Albert Einstein or they're not the television reporter. But they look so much like it, you get this--some of the researchers in Japan call it the Uncanny Valley, I think. There's this dissonance in your head, because the expectations go so far up when it looks so realistic.
What else is important to understand about the robotics field today?
There are two typical reporter questions that you haven't asked me, and I'm glad you haven't. [The first] is: but a robot can't do anything it's not programmed to do anyway, so it's totally different from us. And my answer to that is that it's an artificial distinction, I think. Because my belief is that we are machines. And I think modern science treats us as machines. When you have a course in molecular biology, it's all mechanistic, and likewise in neuroscience. So if we are machines, then at least it seems to me, in principle, there could exist machines built out of other stuff, silicon and steel maybe, which are lawfully following the laws of physics and chemistry, just as we are, but could in principle have as much free will and as much soul as we have. Whether we humans are smart enough to build such machines is a different question. Maybe we're just not smart enough. That pisses off the scientists when I say that.
Well don't physicists say that, in a way? That there may be things that our brains are just not configured to understand about the universe?
Yes. Actually, Patrick Winston, who is a professor here—I used to co-teach his AI [artificial intelligence] course many years ago—he'd always start the first lecture on artificial intelligence with the undergrads here, talking about his pet raccoon he'd had as a kid, growing up in the Midwest. And it was very dexterous with its hands. But, he said, it never occurred to him that that raccoon was going to build a robot replica of itself. The raccoon just isn't smart enough. And maybe there are flying saucers up there, with little green men or green entities from somewhere and they're looking down at my lab and saying "What, he's trying to build robot replicas of himself? Isn't that funny! He'll never make it!"
And you said there was one other [typical reporter] question.…
When? When are we going to have them in our homes? When, when, when?
| 417,514
|
Resources library Clinical research participation among adolescent and young adults at an NCI-designated Comprehensive Cancer Center and affiliated pediatric hospital Minimal clinical trial participation among adolescents and young adults (AYAs) with cancer limits scientific progress and ultimately their clinical care and outcomes. These analyses examine the current state of AYA clinical research participation at a Midwestern comprehensive cancer center and affiliated pediatric hospital to advise program development and increase availability of trials and AYA participation. Enrollment is examined across all diagnoses, the entire AYA age spectrum (15–39), and both cancer therapeutic and supportive care protocols. Stacy D. Sanford, Jennifer L. Beaumont, Mallory A. Snyder, Jennifer Reichek, John M. Salsman Read more
| 102,154
|
TITLE: If the order of a number (mod n) equals n-1 then n is prime?
QUESTION [0 upvotes]: I have trouble in understanding the last part of the sufficiency proof of Pépin´s Test (https://en.wikipedia.org/wiki/Pépin%27s_test).
"In particular, there are at least least F_{n}-1 numbers below F_{n} coprime to F_{n}, and this can happen only if F_{n} is prime".
Can anybody explain me that? Is it true that if the order of a number (mod n) equals n-1 then n is prime?
REPLY [1 votes]: It is essentially saying that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic of order $n-1$ iff $n$ is prime. Which is true since if $n$ is not prime, then $|(\mathbb{Z}/n\mathbb{Z})^\times|=\varphi(n)<n-1$.
Edit: In other words, by Euler's theorem we have the order of any element divides $\varphi(n)$. So, if the order of an element is $n-1$, we would have $n-1|\varphi(n)$. But $\varphi(n)<n-1$ if $n$ is not prime. We conclude that $n$ has to be prime.
| 47,048
|
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| 85,170
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Last week, GFM was in the house for TAU Chapter of Omega Psi Phi Fraternity, fraternity member Shaquille O’Neal and High Aspirations Foundation, Inc., a 501c3 foundation that focuses on efforts to retrieve grants and other monetary contributions to provide scholarships for graduating high school seniors and undergraduate students, for their annual Spring Scholarship Extravaganza with special guests Avery Sunshine and Tank. First on the stage with a spirited smile and equally radiant performance was our beloved soultress, singer/songwriter, Avery *Sunshine. From her apologetic love song, “Ugly Part of Me” to her sassy rendition of Aretha Franklin’s “Jump to It”, the audience was left amazed by her “sistah girl” charm and incredible vocal talent. But as the night drew closer, women swarmed the front of the stage just to get close to their R&B headliner, Tank. With extended hands and excited screams, Tank made sure to deliver his customary form of female stimulation with songs like “Slowly” and “Celebration”, yet by the same token, an intimate feel with songs like “I Can’t Make You Love Me” and “Please Don’t Go” played tenderly on the piano. If you didn’t catch this show, we made sure to capture some of the pictorial highlights from the event. Hope WestHope is a music enthusiast and continual student of photography. Also known to rock out a quiet church or elevator with her current ringtone, Tom & Jerry’s Uncle Paco, “Crambone”. You might also like:Avery Sunshine-“Avery Sunshine” Album SamplerElle Varner – Only Wanna Give It To You feat.…Who Is Avery Sunshine???Avery*Sunshine – “SUMMER”#NowPlaying: Avery*Sunshine: “Come Do Nothing”GFM Spotlight Interview – Avery*SunshineAvery Sunshine Omega Psi Phi Shaquille O'Neal Tank RELATED POSTS
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\begin{document}
\begin{frontmatter}
\title{{\large{Correct energy evolution of stabilized formulations:
The relation between VMS, SUPG and GLS via dynamic orthogonal
small-scales and isogeometric analysis.}}\\ {\large{II: The incompressible Navier--Stokes equations}}}
\author{M.F.P. ten Eikelder\corref{cor1}}
\ead{m.f.p.teneikelder@tudelft.nl}
\cortext[cor1]{Corresponding author}
\author{I. Akkerman}
\ead{i.akkerman@tudelft.nl}
\address{Delft University of Technology, Department of Mechanical,
Maritime and Materials Engineering, P.O. Box 5, 2600 AA Delft, The Netherlands}
\date{}
\begin{abstract}
This paper presents the construction of a correct-energy stabilized finite
element method for the incompressible Navier-Stokes equations.
The framework of the methodology and the correct-energy concept have
been developed in the convective--diffusive context in the preceding
paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of
stabilized formulations: The relation between VMS, SUPG and GLS
via dynamic orthogonal small-scales and isogeometric analysis.
I: The convective--diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259--280].
The current work extends ideas of the preceding paper to build a stabilized method within the
variational multiscale (VMS) setting which displays correct-energy behavior.
Similar to the convection--diffusion case, a key ingredient is the proper
dynamic and orthogonal behavior of the small-scales. This is demanded
for correct energy behavior and links the VMS framework to the
streamline-upwind Petrov-Galerkin (SUPG) and the
Galerkin/least-squares method (GLS).
The presented method is a Galerkin/least-squares formulation with
dynamic divergence-free small-scales (GLSDD). It is locally
mass-conservative for both the large- and small-scales separately.
In addition, it locally conserves linear and angular momentum.
The computations require and employ NURBS-based isogeometric
analysis for the spatial discretization. The resulting formulation
numerically shows improved energy behavior for turbulent flows
comparing with the original VMS method.
\end{abstract}
\begin{keyword}
Stabilized methods \sep
Energy decay \sep
Residual-based variational multiscale method \sep
Orthogonal small-scales \sep
Incompressible flow \sep
Isogeometric analysis
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:Introduction}
The creation of artificial energy in numerical methods is undesirable from
both a physical and a numerical stability point of view. Therefore methods
precluding this deficiency are often sought after. This work continues the
construction of the correct-energy displaying stabilized finite element methods.
The first episode \cite{EAk17} exposes the developed methodology in the
convective--diffusive context. The current study deals with the incompressible
Navier--Stokes equations and is the second piece of work within the framework.
The setup of this paper is closely related to that of \cite{EAk17}. In particular,
the \textit{correct-energy} demand is the same, thus it represents that the
method (i) does not create artificial energy and (ii) closely resembles the
energy evolution of the continuous setting. The precise definition is stated
in Section \ref{sec:Towards correct energy behavior}.
What sets the Navier--Stokes problem apart from convection--diffusion case
is the inclusion of the incompressibility constraint.
In this work we use a divergence-conforming basis which allows exact
pointwise satisfaction of this constraint.
This is considered a beneficial property. Therefore it is added as a design criterion.
\added[id=Rev.1]{In a two-phase context this property is essential for correct energy behavior \cite{akkerman2018toward}.}
\subsection{Contributions of this work}
This paper derives a novel VMS formulation which exhibits the correct
energy behavior and to this purpose combines several ingredients.
The final formulation is summarized in \ref{Appendix: Galerkin/least-squares
formulation with dynamic divergence-free small-scales}. The new method
is a residual-based approach that employs (i) dynamic behavior of the
small-scales, (ii) solenoidal NURBS basis functions and (iii) a Lagrange-multiplier
construction to ensure the incompressibility of the small-scale velocities.
The formulation is of skew-symmetric type, rather than conservative, which
is motivated by both the correct-energy demand and its improved behavior
in the single scale setting (i.e. the Galerkin method) \cite{HuWe05}.
Moreover, \added[id=Rev.1]{the formulation reduces to a Galerkin formulation in case of a vanishing Reynolds number due to a Stokes-projector.}
The use of dynamic small-scales, firstly proposed in \cite{Cod02}, is also driven
from an energy point of view. In addition, it leads to global momentum
conservation and \added[id=Rev.1]{the numerical results of \cite{CodPriGuaBad07} show improved behavior of the dynamic small-scales} with respect
to their static counterpart.
\subsection{Context}
\added[id=Rev.1]{This work falls within the variational multiscale framework \cite{Hug95, Hug98}.
The basic idea of this method is to split solution into the large/resolved-scales and small/unresolved-scales.
The small-scales are modeled in terms of (the residual of) the large scales and substituted into the equation for the large-scales.
This approach was first applied in a residual-based LES context to incompressible
turbulence computations in \cite{BaCaCoHu07}.
The VMS methodology has enjoyed a lot of progress since then. For an overview of the development
consult the review paper \cite{codina2017variational}.
Our work is not the first to analyze the energy behavior of the VMS method.
A spectral analysis of the VMS method can be found in \cite{wang2010spectral}.
That paper proves dissipation of the model terms under restrictive conditions.
Additional to the optimality projector, they require $L_2$-orthogonality of the large- and small-scales.
This condition naturally leads to the use of spectral methods.
Principe et al. \cite{principe2010dissipative} provide a precise definition
of the numerical dissipation within the variational multiscale context for
incompressible flows. Equally important, they numerically show that the
concept of dynamic small-scales, which we apply in this work, is able
to model turbulence.
Colom\'{e}s et al. \cite{colomes2015assessment} assess the performance of several VMS methods for turbulent flow problems and provide an energy analysis of these methods.
They conclude that algebraic subgrid scales (ASGS) and orthogonal subscales (OSS) yield similar results, whereas the latter one is more convenient in terms of numerical performance.
We build onto \cite{wang2010spectral,principe2010dissipative,colomes2015assessment} without requiring $L_2$-orthogonality.
Therefore we are not restricted to the use of spectral methods, while retaining a strict energy relation.
Other} recent related work includes the IGA divergence-conforming VMS method
of Opstal et al. in \cite{OpYanCoEvKvBa17}. They also employ an
$H_0^1$-orthogonality between the velocity large- and small-scales \added[id=Rev.1]{on a local level}.
Our work deviates from
\cite{OpYanCoEvKvBa17} in that we motivate the required orthogonalities
with the correct energy demand. Furthermore, our work distinguishes
itself by enforcing the divergence-free velocity small-scales with a
Lagrange-multiplier construction. We believe the Stokes orthogonality
between the large- and small-scales is a natural path to take, since it
reduces the scheme to the Galerkin method in the vanishing Reynolds number limit.
The discretizations throughout this work are based on the isogeometric
analysis (IGA) concept, proposed by Hughes et al. in \cite{HuCoBa04}.
This idea integrates the historically distinct fields of computer aided
design (CAD) and finite element analysis. Isogeometric analysis rapidly
became a valuable tool in computational fluid dynamics, in particular in
turbulence computations. It provides several advantages over standard
finite element analysis, including an exact description of CAD geometries,
increased robustness and superior approximation properties
\cite{HuCoBa04, LEBEH09, BACHH07}. \added[id=Both Rev]{This work requires in particular
inf--sup stable discretizations for which we use \cite{Evans13steadyNS, Evans13unsteadyNS}.
Moreover these spaces allow the pointwise satisfaction of the incompressibility constraint.}
The smooth NURBS basis functions are convenient for the computation of second derivatives.
\subsection{Outline}
The organization of this paper in Section \ref{sec:GE} and \ref{sec:EESSM}
is very comparable with that of the convective--diffusive context \cite{EAk17},
and at some points mirrors it. The purpose thereof is (i) to indicate the great
similarities of the methodologies and (ii) to clarify the approach.
The remainder of this paper presents the actual construction of a stabilized
variational formulation for the incompressible Navier--Stokes equations
which displays correct-energy behavior. We summarize it as follows.
Section \ref{sec:GE} states the continuous form of the governing
incompressible flow equations, both in the strong formulation and the
standard weak formulation. It additionally provides the energy evolution
of the continuous equation, in both global and local form.
Section \ref{sec:EESSM} discusses the energy evolution of the variational
multiscale approach with dynamic small-scales. The path toward correct
energy behavior actually starts in Section \ref{sec:Towards correct energy behavior}.
This Section presents the required orthogonality of the large-scales and small-scales.
This converts the residual-based variational multiscale method into the
Galerkin/least-squares method with the correct energy behavior.
Section \ref{sec:CP} presents conservation properties of the method.
Section \ref{sec:ns_case} provides a computational test case, namely a
three-dimensional Taylor--Green vortical flow.
In particular it examines the energy behavior and compares the novel
method with the standard VMS method with static small-scales \cite{BaCaCoHu07}.
The calculations employ the generalized-$\alpha$ method with favorable
energy behavior which is also discussed in \cite{EAk17}.
In Section \ref{sec:CONC}, we wrap up and present avenues for future research.
\section{The continuous incompressible Navier--Stokes equations}
\label{sec:GE}
\subsection{Strong formulation}
\label{sec:GE, subsec:SF}
Let $\Omega \in \mathbb{R}^d$, \added[id=Rev.2]{$d=2,3$}, denote the spatial domain
and $\partial \Omega = \Gamma=\Gamma_g \cup \Gamma_h$ its
boundary, see Figure \ref{fig:domain1}.
\\
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\draw[line width=0.3mm, black ] (2,0.4) .. controls (3.5,1) .. (5.1, -0.25) .. controls (6.5,-1.5) .. (5, -2.5) .. controls (4.5,-2.75) .. (3.5, -2).. controls (3.0,-1.5) .. (2.2, -2) .. controls (1.1,-2.5) .. (1.2, -1.0) .. controls (1.3,0.1) .. (2.0,0.4);
\node[text width=3cm] at (5.0,-0.5) {$\Omega$};
\node[text width=3cm] at (6.0,-3.0) {$\Gamma_h$};
\node[text width=3cm] at (3.8,1.0) {$\Gamma_g$};
\node[text width=3cm] at (7.0,-2.1) {$\boldsymbol{\backslash}$};
\node[text width=3cm] at (3.0,0.25) {$\boldsymbol{\backslash}$};
\end{tikzpicture}
\end{center}
\caption{Spatial domain $\Omega$ with its boundaries
$\Gamma = \Gamma_g \cup \Gamma_h$. This is the same figure as in \cite{EAk17}.}
\label{fig:domain1}
\end{figure}\\
The problem consists of solving the incompressible Navier--Stokes equations
governing the fluid flow, which read in strong form
\begin{subequations}
\label{sec:GE, subsec:SF, NS Strong}
\begin{alignat}{1}
\pd_t \mathbf{u} + \nabla \cdot \left( \mathbf{u} \otimes \mathbf{u}\right)
+ \nabla p- \nabla \cdot \left( 2 \nu \nabla^s \mathbf{u} \right)= \mathbf{f} & \quad \text{in} \quad \Omega\times \mathcal{I},
\label{sec:GE, subsec:SF, NS Strong, mom eq}\\
\nabla \cdot \mathbf{u} = 0 & \quad \text{in} \quad \Omega \times \mathcal{I}, \label{sec:GE, subsec:SF, NS Strong, div free eq}\\
\mathbf{u} =\mathbf{g} & \quad \text{in} \quad \Gamma_g \times \mathcal{I}, \label{sec:GE, subsec:SF, NS Strong, dir bc}\\
-u_n^-\mathbf{u}-p\mathbf{n}+\nu\pd_n \mathbf{u} =\mathbf{h} & \quad \text{in} \quad \Gamma_h \times \mathcal{I},
\label{sec:GE, subsec:SF, NS Strong, neum bc}\\
\mathbf{u}(\mathbf{x},0) = \mathbf{u}_0(\mathbf{x}) & \quad \text{in} \quad \Omega, \label{sec:GE, subsec:SF, NS Strong, IC}
\end{alignat}
\end{subequations}
for the velocity $\mathbf{u}: \Omega \times \mathcal{I} \rightarrow \mathbb{R}^d$
and the pressure divided by the density $p: \Omega \times \mathcal{I} \rightarrow \mathbb{R}$.
A constant density is assumed. Eqs. (\ref{sec:GE, subsec:SF, NS Strong, mom eq})-(\ref{sec:GE, subsec:SF, NS Strong, IC})
describe the balance of linear momentum, the conservation of mass, the inhomogeneous Dirichlet
boundary condition, the traction boundary condition and the initial conditions, respectively. The
spatial coordinate denotes $\mathbf{x} \in \Omega$ and the time denotes $t \in \mathcal{I}=(0,T)$
with end time $T>0$. The given dynamic viscosity is $\nu: \Omega \rightarrow \mathbb{R}^+$, the
body force is $\mathbf{f}: \Omega \times \mathcal{I} \rightarrow \mathbb{R}^d$, the initial velocity
is $\mathbf{u}_0: \Omega \rightarrow \mathbb{R}^d$ and the boundary data are
$\mathbf{g}: \Gamma_g \times \mathcal{I} \rightarrow \mathbb{R}^d$ and
$\mathbf{h}: \Gamma_h \times \mathcal{I} \rightarrow \mathbb{R}^d$.
We assume a zero-average pressure for all $t \in \mathcal{I}$ \added[id=Both Rev]{in case of an empty Neumann boundary}. The normal velocity
denotes $u_n=\mathbf{u}\cdot\mathbf{n}$ with positive and negative parts
$u_n^{\pm}=\tfrac{1}{2}(u_n\pm|u_n|)$. The various derivative operators are the
temporal one $\pd_t$, the symmetric gradient $\nabla^s\cdot=\tfrac{1}{2}\left(\nabla\cdot + \nabla^T\cdot\right)$
and the normal gradient $\pd_n =\mathbf{n}\cdot \nabla $, with $\mathbf{n}$ the outward unit normal.
\subsection{Weak formulation}
\label{sec:GE, subsec:SWF}
Let $\WW^0$ denote the trial weighting function space satisfying the homogeneous
Dirichlet conditions on $\mathbf{u}$ and $\WW^g$ the trial solution space with
non-homogeneous Dirichlet conditions on $\mathbf{u}$. The standard variational formulation writes:\\
\textit{Find $\left\{\mathbf{u}, p\right\} \in \WW^g$ such that for all $\left\{\mathbf{w}, q\right\} \in \WW^0$,}
\begin{subequations}
\label{sec:GE, subsec:SWF, standard weak form1}
\begin{alignat}{2}
B_{\Omega,\Gamma_h}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)=& \added[id=Rev.1]{L_{\Omega,\Gamma_h}}\left(\left\{\mathbf{w}, q\right\}\right),
\end{alignat}
\indent \textit{where}
\begin{alignat}{2}
B_{D,\Gamma_h}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)=&B_{D}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)+\left(\mathbf{w},u_n^+\mathbf{u}\right)_{\Gamma_h\left(D\right)},\\
\added[id=Rev.1]{L_{D,\Gamma_h}}\left(\left\{\mathbf{w}, q\right\}\right)=&\added[id=Rev.1]{L_{D}}\left(\left\{\mathbf{w}, q\right\}\right) +\left(\mathbf{w},\mathbf{h}\right)_{\Gamma_h\left(D\right)},\\
B_D\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)=&\left( \mathbf{w}, \pd_t \mathbf{u} \right)_{D}
-(\nabla \mathbf{w}, \mathbf{u} \otimes \mathbf{u})_{D}+(\nabla \mathbf{w}, 2 \nu \nabla^s \mathbf{u} )_{D}\nonumber\\
&+(q,\nabla \cdot \mathbf{u})_{D}- (\nabla \cdot \mathbf{w}, p )_{D},\\
\added[id=Rev.1]{L_D}\left(\left\{\mathbf{w}, q\right\}\right)=&(\mathbf{w},\mathbf{f})_{D}.
\end{alignat}
\end{subequations}
Here $B_D$ is the bilinear form and $\left(\cdot,\cdot\right)_D$ is the
$L^2\left(D\right)$ inner product over $D$. The Dirichlet and traction
boundary of domain $D$ denote
$\Gamma_g(D):=\Gamma_g\cap\partial D$
and $\Gamma_h(D):=\Gamma_h\cap\partial D$
respectively. The strong (\ref{sec:GE, subsec:SF, NS Strong}) and the
weak formulation (\ref{sec:GE, subsec:SWF, standard weak form1})
are equivalent for smooth solutions.
\subsubsection*{Remark}
The variational form (\ref{sec:GE, subsec:SWF, standard weak form1})
is of conservative type: the incompressibility constraint
(\ref{sec:GE, subsec:SF, NS Strong, div free eq}) is not directly
employed in the convective terms. A discretization of the conservative
form may lead to spurious oscillations caused by the error in the
incompressibility constraint acting as a distribution of sinks and sources.
Employing (\ref{sec:GE, subsec:SF, NS Strong, div free eq}) can be
used to generate a \textit{convective form} which is sometimes
preferred and often adopted in Galerkin computations \cite{HuWe05}.
Here we write the variational formulation of \textit{skew-symmetric}
type which will be used in Section \ref{sec:Towards correct energy behavior}:\\
\textit{Find $\left\{\mathbf{u}, p\right\} \in \WW^g$
such that for all $\left\{\mathbf{w}, q\right\} \in \WW^0$,}
\begin{subequations}
\label{sec:GE, subsec:SWF, standard weak form1: convective}
\begin{alignat}{2}
C_{\Omega,\Gamma_h}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)=& \added[id=Rev.1]{L_{\Omega,\Gamma_h}}\left(\left\{\mathbf{w}, q\right\}\right),
\end{alignat}
\indent \textit{where}
\begin{alignat}{2}
C_{D,\Gamma_h}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)=&C_{D}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)+\tfrac{1}{2}\left(\mathbf{w},|u_n|\mathbf{u}\right)_{\Gamma_h\left(D\right)},\\
C_{D}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)=&\left( \mathbf{w}, \pd_t \mathbf{u} \right)_{D}
+\tfrac{1}{2}(\mathbf{w}, \mathbf{u} \cdot \nabla \mathbf{u})_{D}-\tfrac{1}{2}(\mathbf{u}\cdot\nabla\mathbf{w}, \mathbf{u})_{D}
+(\nabla \mathbf{w}, 2 \nu \nabla^s \mathbf{u} )_{D}\nonumber\\
&+(q,\nabla \cdot \mathbf{u})_{D}- (\nabla \cdot \mathbf{w}, p )_{D}.
\end{alignat}
\end{subequations}
Again, this form is equivalent to the strong form (\ref{sec:GE, subsec:SF, NS Strong}).
Form (\ref{sec:GE, subsec:SWF, standard weak form1: convective}) does not
possess all conservation properties when discretized in a standard way.
However, this can be restored using a multiscale split, see \cite{HuWe05}
for details. In the following we continue with the conservative form
(\ref{sec:GE, subsec:SWF, standard weak form1}).\\
To obtain the energy evolution linked to (\ref{sec:GE, subsec:SF, NS Strong})
we want to substitute $\mathbf{w}=\mathbf{u}$. This is not possible in
(\ref{sec:GE, subsec:SWF, standard weak form1}) due to the different
boundary conditions of the solution and test function spaces.
The enforcement of the Dirichlet boundary conditions in the spaces
bypasses when employing a Lagrange multiplier construction.
This converts the variational formulation into a \textit{mixed formulation}:\\
\textit{Find $\left(\left\{\mathbf{u}, p\right\}, \boldsymbol{\lambda}_\Omega\right) \in \WW \times \VV$
such that for all $\left(\left\{\mathbf{w}, q\right\}, \boldsymbol{\vartheta} \right) \in \WW \times \VV$,}
\begin{align}\label{sec:GE, subsec:SWF, standard weak form1 mixed}
\left(\boldsymbol{\lambda}_\Omega,\mathbf{w}\right)_{\Gamma_g} = B_{\Omega,\Gamma_h}\left(\left\{\mathbf{u}, p \right\},\left\{\mathbf{w}, q \right\}\right)
-\added[id=Rev.1]{L_{\Omega,\Gamma_h}}(\left\{\mathbf{w},q\right\})+\left(\boldsymbol{\vartheta},\mathbf{u}-\mathbf{g}\right)_{\Gamma_g}.
\end{align}
Here $\WW$ is the unrestricted space used for the solution and test
functions and $\VV$ is a suitable Lagrange multiplier space.
Section \ref{sec:GE, subsec:GEE} employs formulation
(\ref{sec:GE, subsec:SWF, standard weak form1 mixed}) to derive the
corresponding global energy statement. The equivalence of this form
with the strong form (\ref{sec:GE, subsec:SF, NS Strong}) follows from
Green's formula and an appropriate choice of the weighting functions.
The expression of the Lagrange multiplier is a by-product of this execution and yields
\begin{align}\label{sec:GE, subsec:SWF, LM}
\boldsymbol{\lambda}_\Omega =-\added[id=Authors]{\tfrac{1}{2}}u_n \mathbf{u} - p \mathbf{n} + \nu \pd_n \mathbf{u}.
\end{align}
The multiplier can be interpreted as an auxiliary flux with a convective,
a pressure and a viscous contribution. Consult \cite{HEML00} for details
about auxiliary fluxes in weak formulations.
\subsubsection*{Remark}
\added[id=Authors]{Note that we get the same expression when employing the skew-symmetric form (\ref{sec:GE, subsec:SWF, standard weak form1: convective}).}
\subsection{Global energy evolution}\label{sec:GE, subsec:GEE}
The evolution of the global energy follows when substituting
$\left(\left\{\mathbf{w},q\right\},\boldsymbol{\vartheta}\right)=\left(\left\{\mathbf{u},p\right\},\boldsymbol{\lambda}_{\Omega}\right)$
in (\ref{sec:GE, subsec:SWF, standard weak form1 mixed}).
Employing Green's formula and the strong incompressibility constraint
(\ref{sec:GE, subsec:SF, NS Strong, div free eq}) we see that the
convective term only contributes to the energy evolution via a boundary term.
The global energy, which is defined as
$E_{\Omega}:=\tfrac{1}{2}(\mathbf{u}, \mathbf{u})_{\Omega}$, evolves as
\begin{align}\label{sec:GE, subsec:EE, energy1}
\dfrac{{\rm d}}{{\rm d}t} E_{\Omega}=-\|\nu^{1/2}\nabla \mathbf{u}\|^2_{\Omega}+ \left(\mathbf{u},\mathbf{f}\right)_{\Omega}-(1,F_\Omega)_{\Gamma},
\end{align}
where $ \dfrac{{\rm d}}{{\rm d}t}$ is the time derivative and
$\|\cdot\|^2_{D}$ defines the standard $L^2$-norm over $D$.
The flux reads:
\begin{align}
\label{eq:glb_flux}
F_\Omega =
\left \{ \begin{array}{lc}
- \mathbf{g}\cdot \boldsymbol{\lambda}_\Omega &\text{on}~~~ \Gamma_g, \\
|u_n|e - \mathbf{u}\cdot\mathbf{h}&\text{on} ~~~ \Gamma_h,
\end{array} \right .
\end{align}
with $e:=\frac{1}{2}\mathbf{u}\cdot\mathbf{u}$ the pointwise energy.
The terms of (\ref{sec:GE, subsec:EE, energy1}) represent from
left to right: (i) the energy loss due to viscous molecular dissipation,
(ii) the power exerted by the body force and (iii) the energy change
due to the boundary conditions. Substitution of the Lagrange
multiplier and the boundary conditions leads to the expected
expression of the flux
\begin{align}
\label{eq:glb_flux alt}
F_\Omega = u_n (e+p) - \nu \pd_n e \quad \text{on} ~~~ \Gamma.
\end{align}
These terms represent the convective and viscous flux as well as
the rate of work due to the pressure. We emphasize that the
continuous convective--diffusive equation displays very similar
energy behavior (obviously the pressure term is absent there)
\cite{EAk17}. This provides an additional indication of the
similarity in the discrete setting.
\subsubsection*{Remark}
The transition from expression (\ref{eq:glb_flux}) to (\ref{eq:glb_flux alt})
is only possible in the continuous setting. In the discrete setting no
closed-form expression for the Lagrange multiplier exists. This also
applies to the localized version in Section \ref{sec:GE, subsec:LEE}.
\subsection{Local energy evolution}\label{sec:GE, subsec:LEE}
The procedure to find the local energy evolution is very similar to that
of the global energy. Let $\omega \subset \Omega$ be an arbitrary
subdomain with boundary $\pd \omega$, let $\Omega-\omega$
denote its complement and let their shared boundary denote
$\chi_\omega = \pd \omega \cap \pd (\Omega-\omega)$.
Figure \ref{fig:domain2} shows the subdomains and their boundaries.
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}
\draw[line width=0.3mm, black ] (2,0.4) .. controls (3.5,1) .. (5.1, -0.25) .. controls (6.5,-1.5) .. (5, -2.5) .. controls (4.5,-2.75) .. (3.5, -2).. controls (3.0,-1.5) .. (2.2, -2) .. controls (1.1,-2.5) .. (1.2, -1.0) .. controls (1.3,0.1) .. (2.0,0.4);
\draw[line width=0.3mm, black ] (5.1, -0.25) .. controls (3.6,-0.2) .. (3.5, -2);
\node[text width=3cm] at (3.5,-0.5) {$\Omega-\omega$};
\node[text width=3cm] at (6.0,-1.5) {$\omega$};
\node[text width=3cm] at (6.0,-3.0) {$\Gamma_h(\omega)$};
\node[text width=3cm] at (7.7,-1.0) {$\Gamma_g(\omega)$};
\node[text width=3cm] at (3.8,1.0) {$\Gamma_g(\Omega-\omega)$};
\node[text width=3cm] at (2.6,-2.8) {$\Gamma_h(\Omega-\omega)$};
\node[text width=3cm] at (5.3,-0.1) {$\chi_\omega$};
\node at (5cm,-1) {\pgfuseplotmark{cross*}};
\node[text width=3cm] at (7.0,-2.1) {$\boldsymbol{\backslash}$};
\node[text width=3cm] at (3.0,0.25) {$\boldsymbol{\backslash}$};
\node[text width=3cm] at (6.5, -0.25) {$\boldsymbol{/}$};
\node[text width=3cm] at (4.9, -2) {$\boldsymbol{/}$};
\end{tikzpicture}
\end{center}
\caption{Spatial domain $\Omega$ with a subdomain
$\omega\subset \Omega$. The shared boundary of $\omega$
and its complement is $\chi_\omega$. The boundaries
$\Gamma_g$ and $\Gamma_h$ split according to $\omega$.
This is the same figure as in \cite{EAk17}.}
\label{fig:domain2}
\end{figure}\\
The continuity across the interface is enforced with a Lagrange multiplier
in the appropriate space $\VV_\omega$. The discontinuous test function
space writes $\WW_\omega$. The weak statement enforced on $\omega$
is again a mixed formulation and reads:\\
\textit{Find $\left(\left\{\mathbf{u}, p\right\}, \boldsymbol{\lambda}_{\omega}
\right) \in \WW \times \VV$ such that for all $\left(\left\{\mathbf{w}, q\right\}, \boldsymbol{\vartheta}\right) \in \WW \times \VV$,}
\begin{align}\label{sec:GE, subsec:LEE, standard weak form1}
\left(\mathbf{w},\boldsymbol{\lambda}_\omega\right)_{\chi_\omega}
+\left(\mathbf{w},\boldsymbol{\lambda}_\omega\right)_{\Gamma_g \left(\omega\right)}
&= B_{\omega,\Gamma_h}\left(\left\{\mathbf{u}, p\right\},\left\{\mathbf{w}, q\right\}\right)
- L_{\omega,\Gamma_h}(\left\{\mathbf{w},q\right\}), \nonumber \\
\left(\boldsymbol{\vartheta},[\![ \mathbf{u} ]\!] \right)_{\chi_\omega}
+\left(\boldsymbol{\vartheta},\mathbf{u}-\mathbf{g}\right)_{\Gamma_g\left(\omega\right)}&=0.
\end{align}
We have here employed the jump term $[\![ \mathbf{u} ]\!]$ given by
\begin{equation}
[\![ \mathbf{u} ]\!]:=\mathbf{u}|_\omega -\mathbf{u}|_{\Omega-\omega},
\end{equation}
where the terms are defined on $\omega$ and $\Omega-\omega$, respectively.
Furthermore, $\mathbf{n}_\omega$ is the outward normal of domain
$\omega$, $u_{n_\omega}$ is the outward velocity in direction
$\mathbf{n}_\omega$ and $\partial_{n_\omega}$ the direction derivative
outward of $\omega$. The equivalence of this form with the strong form
(\ref{sec:GE, subsec:SF, NS Strong}) leads to the expression of the Lagrange multiplier:
\begin{align}\label{}
\boldsymbol{\lambda}_\omega= - u_{n_\omega} \mathbf{u} - p \mathbf{n}_\omega+\nu \pd_{n_\omega} \mathbf{u} ,
\end{align}
which is clearly the localized version of (\ref{sec:GE, subsec:SWF, LM}).
A direct consequence is the symmetry of the Lagrange multipliers
(these are also called auxiliary fluxes in this setting, see \cite{HEML00}):
\begin{align}
\boldsymbol{\lambda}_\omega+\boldsymbol{\lambda}_{\Omega-\omega}=\B{0},
\end{align}
i.e. that what flows out $\omega$ through $\chi_\omega$ enters its complement.
The energy evolution linked to each of the domains is a natural split of the global energy evolution:
\begin{subequations}
\label{gal: localized energy evolution}
\begin{alignat}{1}
\dfrac{{\rm d}}{{\rm d}t} E_\omega=&-\|\nu^{1/2}\nabla \mathbf{u}\|^2_{\omega}+ \left(\mathbf{u},\mathbf{f}\right)_{\omega}-\left(1,F_{\omega}\right)_{\partial \omega},
\end{alignat}
\end{subequations}
with energy fluxes
\begin{align}
\label{eq:glb_flux2}
F_{\omega} =
\left \{ \begin{array}{ll}
- \mathbf{g}\cdot \boldsymbol{\lambda}_\omega &\text{on}~~~ \Gamma_g\left(\omega\right), \\
|u_{n_\omega}|e - \mathbf{u}\cdot\mathbf{h}&\text{on} ~~~ \Gamma_h\left(\omega\right),\\
- \mathbf{u}\cdot \boldsymbol{\lambda}_\omega &\text{on} ~~~\chi_\omega.
\end{array} \right .
\end{align}
The last term of (\ref{eq:glb_flux2}) redistributes energy over the domain.
It represents an energy flux across the subdomain interface $\chi_\omega$
with a convective, a pressure and a viscous component. Similarly as before,
substitution of the terms in the energy flux leads to
\begin{align}
\label{eq:glb_flux alt2}
F_\omega = u_{n_\omega} (e+p) - \nu \pd_{n_\omega} e \quad \text{on} ~~~ \partial \omega.
\end{align}
This is obviously the localized version of (\ref{eq:glb_flux alt}).
\subsection*{Remark}
All statements of this Section are in the continuous setting. Hence, the
standard discretization, i.e. the Galerkin method, displays the
same correct energy behavior.
\subsection*{Remark}
The various boundary terms may distract the reader and do not
contribute to the goal of this paper. Therefore we only consider
boundary conditions precluding the energy flux $F$ on $\Gamma$.
The homogeneous Dirichlet and periodic boundary conditions
satisfy this purpose. Applying non-homogeneous boundaries
is straightforward.\\
We continue this paper by discretizing the system according to
the dynamic variational multiscale method with the target to
closely resemble energy evolution
(\ref{sec:GE, subsec:EE, energy1})
and (\ref{gal: localized energy evolution}).
\section{Energy evolution of the variational multiscale method
with dynamic small-scales}\label{sec:EESSM}
The convective--diffusive context \cite{EAk17} learns us that
the dynamical structure of the small-scales is a requirement for
the stabilized formulation to display the correct energy behavior.
This allows to skip the static small-scales and to directly apply the
dynamic modeling approach. We follow this road.
\subsection{The multiscale split}\label{sec:The multiscale split}
The variational multiscale split is nowadays a standard execution
\cite{Hug95, Hug98} which we include here for the sake of
completeness and notation. Employing the variational multiscale
methodology the trial and weighting function spaces split into
large- and small-scales as:
\begin{align}
\WW = \WW^h \oplus \WW',
\end{align}
with $\WW^h$ and $\WW'$ containing the large-scales and
small-scales, respectively. The large-scale space is spanned
by the finite dimensional numerical discretization while the
fine-scales are its infinite dimensional complement. The fine-scale
space $\WW'$ is also referred to as subgrid-scales since these
scales are not reproduced by the grid. This decomposition
implies the split of the solution and weighting functions as follows:
\begin{subequations}
\begin{alignat}{2}
\mathbf{U} =&~ \mathbf{U}^h + \mathbf{U}', \label{solution split}\\
\mathbf{W} =&~ \mathbf{W}^h + \mathbf{W}',
\end{alignat}
\end{subequations}
where $\mathbf{U}^h, \mathbf{W}^h \in \WW^h$
and $\mathbf{U}', \mathbf{W}' \in \WW'$
with $\mathbf{U}:=\left\{\mathbf{u},p\right\}, \mathbf{W}:=\left\{\mathbf{w},q\right\}$.
Uniqueness follows when a projector
$\mathscr{P}^h: \WW \rightarrow \WW^h$ is used for the splitting operation:
\begin{subequations}
\label{eq:opt_proj}
\begin{alignat}{1}
\mathbf{U}^h &= \mathscr{P}^h \mathbf{U}, \label{scale sep proj}\\
\mathbf{U}' &= \left(\mathscr{I}-\mathscr{P}^h\right) \mathbf{U},
\end{alignat}
\end{subequations}
where $\mathscr{I}: \WW \rightarrow \WW$ is the identity operator.
Employing both $\mathbf{W} = \mathbf{W}^h$ and
$\mathbf{W} = \mathbf{W}'$, and the solution split
(\ref{solution split}) in (\ref{sec:GE, subsec:SWF, standard weak form1})
leads to the weak formulation:\\
\textit{Find $\mathbf{U}^h \in \WW^h,~\mathbf{U}' \in \WW'$ $\text{for all}~~\mathbf{W}^h \in \WW^h,~\mathbf{W}' \in \WW'$,}
\begin{subequations}
\begin{alignat}{2}
B_{\Omega}\left(\mathbf{U}^h+\mathbf{U}',\mathbf{W}^h\right)
=& \added[id=Rev.1]{L_{\Omega}}(\mathbf{W}^h)_{\Omega}, \quad \text{for all }\mathbf{W}^h\in \WW^h, \label{VMS large-scale eq} \\
B_{\Omega}\left(\mathbf{U}^h+\mathbf{U}',\mathbf{W}'\right)
=& \added[id=Rev.1]{L_{\Omega}}(\mathbf{W}')_{\Omega}, \quad \text{for all }\mathbf{W}'\in \WW'. \label{VMS small-scale eq}
\end{alignat}
\end{subequations}
Note that this is an infinite-dimensional system with unknowns
$\mathbf{U}^h$ and $\mathbf{U}'$. Appropriately parameterizing
the small-scales $\mathbf{U}'$ in terms of $\mathbf{U}^h$
converts (\ref{VMS large-scale eq}) into a solvable finite element problem.
This conversion can be done with inspiration from (\ref{VMS small-scale eq}).
For the technical details of the parameterization consult \cite{HugSan06}.
\subsection{Dynamic small-scales}\label{sec:Dynamic small-scales}
Here we employ the dynamic small-scales, see \cite{Cod02},
demanded by the convective--diffusive context for correct energy
behavior \cite{EAk17}. The fine-scale model
\begin{align}\label{eq:Dynamic small-scales}
\pd_t\left\{\hat{\mathbf{u}}', 0\right\}+\boldsymbol{\tau}^{-1}\left\{\hat{\mathbf{u}}', \hat{p}'\right\}
+ \mathscr{R}\left(\left\{\mathbf{u}^h, p^h\right\}, \hat{\mathbf{u}}' \right)=0,
\end{align}
is an ordinary differential equation. The hat-sign is used to indicate
a small-scale model instead of the actual small-scales. The intrinsic
time scale $\boldsymbol{\tau}$ is a matrix of stabilization parameters,
here $\boldsymbol{\tau} \in \mathbb{R}^{4\times 4}$, with
contributions for the two equations:
\begin{align}
\boldsymbol{\tau}=\begin{pmatrix} \tau_M \B{I}_{3 \times 3} & \B{0}_3 \\ \B{0}_3^T & \tau_C \end{pmatrix}.
\end{align}
The local large-scale residual contains a momentum part
$\B{r}_M$ and continuity part $r_C$ linked to the
incompressibility constraint, respectively, given by
\begin{subequations}
\label{vms: residuals}
\begin{alignat}{1}
\mathscr{R}\left(\left\{\mathbf{u}^h, p^h\right\}, \hat{\mathbf{u}}' \right)&=\left\{\B{r}_M(\left\{ \mathbf{u}^h, p^h\right\},\hat{\mathbf{u}}'), r_C(\mathbf{u}^h) \right\}^T,\\
\B{r}_M &= \pd_t \mathbf{u}^h + \left(\left(\mathbf{u}^h+\hat{\mathbf{u}}'\right)\cdot\nabla\right) \mathbf{u}^h
+ \nabla p^h - \nu \Delta \mathbf{u}^h - \mathbf{f}, \label{vms: residuals mom}\\
r_C &= \nabla \cdot \mathbf{u}^h. \label{vms: residuals cont}
\end{alignat}
\end{subequations}
In the following we ignore the hat-sign. We employ a dynamic version
of the stabilization parameters $\tau_M, \tau_C$ defined in \cite{BaCaCoHu07}.
The details are provided in \ref{Appendix: Definition stabilization parameters}.
The subscripts \textit{M} and \textit{C} refer to \textit{momentum} and \textit{continuity}, respectively.
Mirroring \cite{EAk17}, the momentum residual (\ref{vms: residuals mom}) uses
the full velocity $\mathbf{u}^h+\mathbf{u}'$. This creates a nonlinearity in the system.
Therefore we apply a standard iterative procedure to determine the small-scales.
Assume now that the domain $\Omega$ is partitioned into a set of elements $\Omega_e$.
The domain of element interiors does not include the interior boundaries and denotes
\begin{equation}
\tilde{\Omega} = \displaystyle \bigcup_e \Omega_e.
\end{equation}
The resulting residual-based dynamic VMS weak formulation is\\
\textit{Find $\mathbf{U}^h \in \WW^h$ $\text{for all} ~~\mathbf{W}^h \in \WW^h$}
\begin{subequations}
\label{standard dynamic VMS}
\begin{alignat}{2}
B_{\Omega}^{\text{VMSD}}\left(\mathbf{U}^h,\mathbf{W}^h\right)=& \added[id=Rev.1]{L_{\Omega}}(\mathbf{W}^h),
\end{alignat}
\indent \textit{where}
\begin{alignat}{2}
B_{\Omega}^{\text{VMSD}}\left(\mathbf{U}^h,\mathbf{W}^h\right) =& B_{\Omega}\left(\mathbf{U}^h,\mathbf{W}^h\right)
+ \left(\mathbf{w}^h, \pd_t \mathbf{u}'\right)_{\tilde{\Omega}}
- \left( \nu \Delta \mathbf{w}^h, \mathbf{u}'\right)_{\tilde{\Omega}} \nonumber \\
&-\left(\nabla q^h, \mathbf{u}'\right)_{\tilde{\Omega}}-\left(\nabla \cdot \mathbf{w}^h, p'\right)_{\tilde{\Omega}}\nonumber \\
& -\left(\nabla \mathbf{w}^h, \mathbf{u}^h\otimes \mathbf{u}'\right)_{\tilde{\Omega}}
-\left(\nabla \mathbf{w}^h, \mathbf{u}'\otimes \mathbf{u}^h\right)_{\tilde{\Omega}}
-\left(\nabla \mathbf{w}^h, \mathbf{u}'\otimes \mathbf{u}'\right)_{\tilde{\Omega}},\label{VMS bilinear}\\
& \pd_t\left\{\mathbf{u}', 0\right\}+\boldsymbol{\tau}^{-1}\left\{\mathbf{u}', p'\right\}
+ \mathscr{R}\left(\left\{\mathbf{u}^h, p^h\right\}, \mathbf{u}' \right)=0,\label{VMSD small scales}
\end{alignat}
\end{subequations}
and where the additional D stands for \textit{dynamic}. When examining the last line of
(\ref{VMS bilinear}), we recognize the following contributions. The first term is the
SUPG contribution. The first two terms model the \textit{cross stress}, while the
last term models the \textit{Reynolds stress}. Note that no spatial derivatives act
on the small-scales. Furthermore, in contrast to static small-scales, the dynamic
small-scale model (\ref{VMSD small scales}) is a separate equation and cannot
directly be substituted into the large-scale equation (\ref{VMS bilinear}).
\subsection{Local energy evolution of the VMSD form}
\label{sec:Local energy evolution dynamic VMS}
To arrive at the local energy evolution of (\ref{standard dynamic VMS}),
we extend the weak formulation to a Lagrange multiplier setting to allow
discontinuous functions across subdomains, similar as
(\ref{sec:GE, subsec:LEE, standard weak form1}). The weak statement,
here stated for domain $\omega \subset \Omega$, reads \\
\textit{Find $\left(\mathbf{U}^h, \boldsymbol{\lambda}_{\omega}^h\right) \in \WW \times \VV$
such that for all $\left(\mathbf{W}^h, \boldsymbol{\vartheta}^h\right) \in \WW \times \VV$,}
\begin{subequations}
\label{sec:VMS dyn LM form}
\begin{alignat}{2}
\left(\mathbf{w}^h,\boldsymbol{\lambda}^h_\omega\right)_{\chi_\omega}
&= B_{\omega}^{\text{VMSD}}\left(\mathbf{U}^h,\mathbf{W}^h\right)- \added[id=Rev.1]{L_{\omega}}(\mathbf{W}^h), \\
\left(\boldsymbol{\vartheta}^h,[\![\mathbf{u}^h]\!]\right)_{\chi_\omega}&=0,\\
\pd_t\left\{\mathbf{u}', 0\right\}+\boldsymbol{\tau}^{-1}\left\{\mathbf{u}', p'\right\}
+ \mathscr{R}\left(\left\{\mathbf{u}^h, p^h\right\}, \mathbf{u}' \right)&=0.\label{small scale dynamic VMS}
\end{alignat}
\end{subequations}
To obtain the evolution of the \textit{local total energy}
$E_\omega = \tfrac{1}{2}\left(\mathbf{u}^h + \mathbf{u}',\mathbf{u}^h + \mathbf{u}'\right)_{\tilde{\omega}}$
linked to the variational formulation (\ref{standard dynamic VMS}), we employ
$\mathbf{w}^h = \mathbf{u}^h, q^h = p^h$
and $\boldsymbol{\vartheta}^h=\boldsymbol{\lambda}^h_{\omega} $ in (\ref{sec:VMS dyn LM form}).
Adding $\mathbf{u}'$ times the momentum component of (\ref{small scale dynamic VMS})
integrated over $\tilde{\omega}$ eventually leads to
\begin{align}\label{energy evolution dyn VMS}
\dfrac{{\rm d}}{{\rm d}t} E_\omega=&
- \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\omega}^2
+ (\mathbf{u}^h,\mathbf{f})_{\omega}
- (1,F_\omega^h)_{\chi_\omega}\nonumber \\
& -\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\omega}}^2
+ (\mathbf{u}',\mathbf{f})_{\tilde{\omega}}
+ 2(\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\omega}}\nonumber \\
&+(\nabla \cdot \mathbf{u}^h,p')_{\tilde{\omega}}
+\left(\nabla \mathbf{u}^h , (\mathbf{u}^h
+ \mathbf{u}') \otimes (\mathbf{u}^h
+ \mathbf{u}') \right)_{\tilde{\omega}}
- \left(\mathbf{u}',(\mathbf{u}^h
+\mathbf{u}')\cdot \nabla \mathbf{u}^h \right)_{\tilde{\omega}},
\end{align}
where
\begin{equation}\label{energy flux VMS}
F_\omega^h= - \boldsymbol{\lambda}^h_\omega \cdot \mathbf{u}^h.
\end{equation}
The first line closely resembles the continuous energy evolution relation.
Each one of the other terms appears as a result of the VMS stabilization.
The first term of the second line represents the numerical dissipation due
to the missing small-scales. This contributes to a decay of the energy,
which is favorable from a stability argument. The second term is the
power exerted by the body force on the small-scales, this term closely
resembles its large-scale counterpart. The remaining terms have no
continuous counterpart. With the current small-scale model, the
small-scale pressure term dissipates energy\footnote{The small-scale pressure expression can be substituted into this
term to arrive at $(\nabla \cdot \mathbf{u}^h,p')_{\tilde{\omega}} = -|| \tau_C^{-1/2} p'||^2_{\tilde{\omega}}$.
Note that it vanishes when employing a divergence-conforming discrete velocity space.}.
The signs of the other terms are undetermined and therefore these can create energy artificially.
The term $2(\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\omega}}$ can be bounded by both the
physical dissipation $ \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\omega}^2$ and numerical dissipation
$\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\omega}}^2$ using a standard argument. However, this
results in an overall dissipation that can be smaller than the physical one. This is deemed
undesirable. Note that it is comparable with that of the dynamic VMS stabilized form in the
convective--diffusive context. The contrast occurs in the last line which is linked to the
incompressibility constraint (\ref{sec:GE, subsec:SF, NS Strong, div free eq}) and the
small-scale pressure. Inspired by the convective--diffusive context, the next Section
rectifies the method to closely resemble the energy behavior of the continuous setting.
\subsection*{Remark}
Employing $\omega=\Omega$, and hence $\tilde{\omega}=\tilde{\Omega}$,
provides the global energy evolution of (\ref{standard dynamic VMS}):
\begin{align}\label{energy evolution dyn VMS global}
\dfrac{{\rm d}}{{\rm d}t} E_\Omega=
& - \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\Omega}^2+ (\mathbf{u}^h,\mathbf{f})_{\Omega}\nonumber \\
& -\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\Omega}}^2 + (\mathbf{u}',\mathbf{f})_{\tilde{\Omega}}
+ 2(\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\Omega}}\nonumber \\
&+(\nabla \cdot \mathbf{u}^h,p')_{\tilde{\Omega}}
+\left(\nabla \mathbf{u}^h , (\mathbf{u}^h + \mathbf{u}') \otimes (\mathbf{u}^h + \mathbf{u}') \right)_{\tilde{\Omega}}
- \left(\mathbf{u}',(\mathbf{u}^h+\mathbf{u}')\cdot \nabla \mathbf{u}^h \right)_{\tilde{\Omega}}.
\end{align}
\section{Toward a stabilized formulation with correct energy behavior}
\label{sec:Towards correct energy behavior}
This Section presents the procedure to remedy the incorrect energy behavior
(\ref{energy evolution dyn VMS}) of the dynamic VMS formulation (\ref{standard dynamic VMS}).
The first ingredient is the switch from the conservative form to a skew-symmetric form
with the help of the divergence-free velocity field constraint. Next, we employ the natural
choice of a Stokes-projector and demand divergence-free small-scales. In view of the
convective--diffusive context, we use $H_0^1$ small-scales to treat the small-scale viscous term.
\subsection{Design condition}\label{subsec: Design condition}
We present a design condition which clarifies the desirable energy behavior of the formulation.
The variational weak formulation corresponding to (\ref{sec:GE, subsec:SF, NS Strong})
is demanded to satisfy the local energy behavior:
\begin{align}\label{sec:EESSM, subsec: EE, energy evo design cond}
\dfrac{{\rm d}}{{\rm d}t} E_{\omega}= & - \|\nu^{1/2} \nabla \mathbf{u}^h \|_{\omega}^2
+ (\mathbf{u}^h ,\mathbf{f})_{\omega}-(1,F_\omega^h)_{\chi_\omega} \nonumber\\
& - \|\tau_M^{-1/2} \mathbf{u}'\|_{\tilde{\omega}}^2 + (\mathbf{u}' ,\mathbf{f})_{\tilde{\omega}},
\end{align}
with \textit{exact} divergence-free velocity fields. Note that this requirement is very
similar to that of the convective--diffusive context \cite{EAk17} where the convective velocity is assumed solenoidal.
\added[id=Rev.2]{
\subsection*{Remark}
In the following we use the ingredients mentioned above to convert the VMS formulation (\ref{sec:VMS dyn LM form}) into a method that satisfies the design condition.
It is important to realize that the small-scales employed in the formulation are determined by a model equation.
This implies that these properties are not necessarily valid for the \textit{model} small-scales.
In contrast, the \textit{exact} small-scales do satisfy these properties.
The model small-scales approximate its exact counterpart which justifies the judicious use of these properties to construct a method that satisfies the design condition.
}
\subsection{Skew-symmetric form}
\label{subsec:CF}
We employ a multiscale form of the skew-symmetric formulation
(see (\ref{sec:GE, subsec:SWF, standard weak form1: convective}))
to eliminate the convective contributions in (\ref{energy evolution dyn VMS}).
Considering the convective terms in isolation, we cast them into the following form:
\begin{align}\label{convective derivation}
-(\nabla \mathbf{w}^h, (\mathbf{u}^h+\mathbf{u}')\otimes (\mathbf{u}^h+\mathbf{u}'))_{\tilde{\Omega}}
=&- \left( \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h , \mathbf{u}^h \right)_{\tilde{\Omega}}
- \left( \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h , \mathbf{u}' \right)_{\tilde{\Omega}} \nonumber\\
=& \tfrac{1}{2} \left(\mathbf{w}^h, \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{u}^h\right)_{\tilde{\Omega}}
- \tfrac{1}{2} \left(\left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h, \mathbf{u}^h \right)_{\tilde{\Omega}} \nonumber \\
&+\tfrac{1}{2}\left(\mathbf{u}^h,\mathbf{w}^h\nabla \cdot \left(\mathbf{u}^h+\mathbf{u}'\right)\right)_{\tilde{\Omega}}
- \left( \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h , \mathbf{u}' \right)_{\tilde{\Omega}} \nonumber\\
=& \tfrac{1}{2} \left(\mathbf{w}^h, \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{u}^h\right)_{\tilde{\Omega}}
- \tfrac{1}{2} \left(\left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h, \mathbf{u}^h \right)_{\tilde{\Omega}} \nonumber \\
&- \left( \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h , \mathbf{u}' \right)_{\tilde{\Omega}},
\end{align}
where we have employed the multiscale incompressibility constraint
$\nabla \cdot \mathbf{u}=\nabla \cdot (\mathbf{u}^h+\mathbf{u}')=0$
in the last equality. The last expression is incorporated into the formulation.
The resulting residual-based skew-symmetric VMS weak formulation is\\
\textit{Find $\mathbf{U}^h \in \WW^h$ $\text{such that for all} ~~\mathbf{W}^h \in \WW^h$,}
\begin{subequations}
\label{sec:Towards correct energy, VMSC}
\begin{alignat}{2}
C_{\Omega}^{\text{VMSD}}\left(\mathbf{U}^h,\mathbf{W}^h\right)=& \added[id=Rev.1]{L_{\Omega}}(\mathbf{W}^h),
\end{alignat}
\indent \textit{where}
\begin{alignat}{2}
C_{\Omega}^{\text{VMSD}}\left(\mathbf{U}^h,\mathbf{W}^h\right) =&
C_{\Omega}\left(\mathbf{U}^h,\mathbf{W}^h\right)+ \left(\mathbf{w}^h, \pd_t \mathbf{u}'\right)_{\tilde{\Omega}}
- \left( \nu \Delta \mathbf{w}^h, \mathbf{u}'\right)_{\tilde{\Omega}}\nonumber\\
&-\left(\nabla q^h, \mathbf{u}'\right)_{\tilde{\Omega}}-\left(\nabla \cdot \mathbf{w}^h, p'\right)_{\tilde{\Omega}}\nonumber \\
& +\tfrac{1}{2} \left(\mathbf{w}^h, \mathbf{u}' \cdot \nabla \mathbf{u}^h\right)_{\tilde{\Omega}}
- \tfrac{1}{2} \left( \mathbf{u}'\cdot \nabla \mathbf{w}^h, \mathbf{u}^h \right)_{\tilde{\Omega}} \nonumber \\
&- \left( \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h , \mathbf{u}' \right)_{\tilde{\Omega}},
\label{sec:Towards correct energy, VMSC, bilinear form}\\
\pd_t\left\{\mathbf{u}', 0\right\}+\boldsymbol{\tau}^{-1}\left\{\mathbf{u}', p'\right\} + \mathscr{R}\left(\left\{\mathbf{u}^h, p^h\right\}, \mathbf{u}' \right)&=0.
\end{alignat}
\end{subequations}
This eliminates the convective contributions from the local energy evolution equation:
\begin{align}\label{sec:Towards correct energy, VMSC: energy}
\dfrac{{\rm d}}{{\rm d}t} E_\omega=&
- \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\omega}^2
+ (\mathbf{u}^h,\mathbf{f})_{\omega}- (1,F_\omega^h)_{\chi_\omega}\nonumber \\
& -\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\omega}}^2
+ (\mathbf{u}',\mathbf{f})_{\tilde{\omega}} + 2(\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\omega}}
+(\nabla \cdot \mathbf{u}^h,p')_{\tilde{\omega}}.
\end{align}
\subsection{Stokes projector}
In the convective--diffusive context a $H_0^1$-orthogonality of the small-scale viscous term
is required for correct energy behavior. This is the distinguished limit of $Pe \rightarrow 0$ of
the steady convection--diffusion equations, where $Pe$ is the P\'{e}clet number.
Its Navier--Stokes counterpart is to apply a Stokes-projector which is based on the distinguished
limit $Re \rightarrow 0$ of the steady incompressible Navier--Stokes equations.
Here $Re$ is the Reynolds number. Thus, applying a Stokes projection on the large-scale
equation seems a natural choice. Moreover, it reduces the variational form in the limit
$Re \rightarrow 0$ to the standard Galerkin method. This is a valid and established method
in that regime, provided compatible discretizations for the velocity and pressure spaces are used.
For the scale separation (\ref{eq:opt_proj}) we select the Stokes projector given by\\
$\mathscr{P}_{\text{Stokes}}^h: \mathbf{U} \in \WW \rightarrow \mathbf{U}^h \in \WW^h$: \textit{Find $\mathbf{U}^h \in \mathcal{W}^h$
such that for all $\mathbf{W}^h \in \WW^h$,}
\begin{subequations}\label{Stokes projector}
\begin{alignat}{2}
\left(\nu \Delta \mathbf{w}^h, \mathbf{u}^h\right)_\Omega + \left(\nabla \cdot \mathbf{w}^h,p^h\right)_\Omega
&= \left(\nu \Delta \mathbf{w}^h, \mathbf{u}\right)_\Omega + \left(\nabla \cdot \mathbf{w}^h,p\right)_\Omega,\\
\left(\nabla q^h, \mathbf{u}^h \right)_\Omega &= \left(\nabla q^h, \mathbf{u} \right)_\Omega,
\end{alignat}
\end{subequations}
in the bilinear form (\ref{sec:Towards correct energy, VMSC, bilinear form}). \added[id=Rev.2]{Note that this projector only makes sense if the elements of $\mathcal{W}^h$ are inf--sup stable and the velocities are at least $C^1$-continuous. The numerical results presented in Section \ref{sec:ns_case} fulfill this requirement: quadratic NURBS basis functions are employed. However, note that the final form, given in \ref{Appendix: Galerkin/least-squares formulation with dynamic divergence-free small-scales}, does not have the smoothness restriction.}
As a consequence we assume the modeled small-scales to satisfy the
orthogonality induced by the Stokes operator:
\begin{subequations}\label{Stokes projector orthog}
\begin{alignat}{2}
\left(\nu \Delta \mathbf{w}^h, \mathbf{u}'\right)_{\tilde{\Omega}} + \left(\nabla \cdot \mathbf{w}^h,p'\right)_{\tilde{\Omega}}&= 0,\\
\left(\nabla q^h, \mathbf{u}' \right)_{\tilde{\Omega}} &= 0,
\end{alignat}
\end{subequations}
for all $\mathbf{W}^h \in \WW^h$ . This converts (\ref{sec:Towards correct energy, VMSC})
into the simplified formulation:\\
\textit{Find $\mathbf{U}^h \in \WW^h$ $\text{such that for all} ~~\mathbf{W}^h \in \WW^h$}
\begin{subequations}
\label{VMSDS form}
\begin{alignat}{2}
S_{\Omega}\left(\mathbf{U}^h,\mathbf{W}^h\right)=& \added[id=Rev.1]{L_{\Omega}}(\mathbf{W}^h),\label{large scale eq VMSDS}
\end{alignat}
\indent \textit{where}
\begin{alignat}{2}
S_{\Omega}\left(\mathbf{U}^h,\mathbf{W}^h\right) =&
C_{\Omega}\left(\mathbf{U}^h,\mathbf{W}^h\right)
+ \left(\mathbf{w}^h, \pd_t \mathbf{u}'\right)_{\tilde{\Omega}}\nonumber \\
& +\tfrac{1}{2} \left(\mathbf{w}^h, \mathbf{u}' \cdot \nabla \mathbf{u}^h\right)_{\tilde{\Omega}}
- \tfrac{1}{2} \left( \mathbf{u}'\cdot \nabla \mathbf{w}^h, \mathbf{u}^h \right)_{\tilde{\Omega}} \nonumber \\
&- \left( \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h , \mathbf{u}' \right)_{\tilde{\Omega}},\\
\pd_t \mathbf{u}'+\tau_M^{-1} \mathbf{u}' + \B{r}_M =& 0, \label{small scale eq VMSDS}
\end{alignat}
\end{subequations}
where the $S$ abbreviates \textit{Stokes}. Note that the small-scale pressure terms
have vanished from the formulation. The energy linked to this formulation is
\begin{align}\label{energy evolution VMSDS form}
\dfrac{{\rm d}}{{\rm d}t} E_\omega=&
- \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\omega}^2
+ (\mathbf{u}^h,\mathbf{f})_{\omega}- (1,F_\omega^h)_{\chi_\omega}\nonumber \\
& -\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\omega}}^2 + (\mathbf{u}',\mathbf{f})_{\tilde{\omega}}
+ (\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\omega}} -(\nabla p^h,\mathbf{u}')_{\tilde{\omega}}.
\end{align}
To fulfill the design condition (\ref{sec:EESSM, subsec: EE, energy evo design cond}),
the last two terms of (\ref{energy evolution VMSDS form}) need to be eliminated, i.e.
\begin{align}\label{eq: necessary optimality projector}
(\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\Omega}} -(\nabla p^h,\mathbf{u}')_{\tilde{\Omega}} = 0.
\end{align}
There are various options available to accomplish this.
Before sketching some of these options we first like to note the following.
Augmenting the undesirable terms of (\ref{energy evolution VMSDS form})
with $(\nabla \cdot \mathbf{u}^h,p')$ results in the requirement
\begin{align}
(\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\omega}} - (\nabla p^h,\mathbf{u}')_{\tilde{\omega}}+(\nabla \cdot \mathbf{u}^h,p')_{\tilde{\omega}} =0.
\end{align}
This is a well-defined orthogonality induced by the Stokes operator, given in (\ref{Stokes projector orthog}).
The augmented term would appear if $\nabla p'$ in the small-scale momentum
equation is not neglected\footnote{
Including the small-scale pressure in the residual augments the right-hand
side of (\ref{energy evolution VMSDS form}) with the term $\left(\nabla p',\mathbf{u}'\right)$.
Next, by using the strong form continuity equation weighted with the
small-scale pressure, i.e. $\left(p',\nabla \cdot (\mathbf{u}^h+\mathbf{u}')\right)=0$,
this term converts into $(\nabla \cdot \mathbf{u}^h,p')$.
}.
Note that this term is not (easily) computable and therefore usually omitted in the formulation.
The required orthogonality (\ref{eq: necessary optimality projector}) can be either
\textit{assumed} or \textit{enforced} \cite{EAk17}. We discuss four options here.
\begin{itemize}
\item First we could assume the orthogonality in the small-scale equation (\ref{small scale eq VMSDS}).
This orthogonality has previously been assumed to modify the large-scale equation (\ref{large scale eq VMSDS}).
Assuming it in the small-scale equation results in a stable method with the desired energy property.
However the small-scale model is not residual-based anymore. This results in an inconsistent method.
We do not further consider this option.
\item Alternatively, we could assume the orthogonality in the large-scale equation
(\ref{large scale eq VMSDS}) again. This converts the formulation into a GLS method.
This method includes a PSPG term, $-(\nabla q^h,\mathbf{u}')_{\tilde{\Omega}}$, and therefore
pointwise divergence-free solutions cannot be guaranteed. The formulation harms the design
condition of Section \ref{subsec: Design condition} and is therefore omitted.
\item Another option is to enforce the required orthogonality using Lagrange-multipliers.
This is not straightforward and is deemed unnecessarily expensive.
\item The path we propose is to cure the unwanted terms separately by combining
the second and third options. The approach is to (i) enforce divergence-free small-scales
to eliminate the second term of \added[id=Rev.2]{(\ref{eq: necessary optimality projector})} and (ii) assume an
$H_0^1$-orthogonality to erase the first term of \added[id=Rev.2]{(\ref{eq: necessary optimality projector})}.
Sections \ref{subsec:Divergence-free small-scales} and \ref{subsec: H_0^1-orthogonal small-scales}
respectively describe these steps.
\end{itemize}
\subsection{Divergence-free small-scales}\label{subsec:Divergence-free small-scales}
The last term of (\ref{energy evolution VMSDS form}) disappears when enforcing
divergence-free small-scales. We handle this with a projection operator on the small-scales:\\
$\mathscr{P}^h_{\text{div}}: \mathbf{U} \in \WW \rightarrow \mathbf{U}^h \in \WW^h$:
\textit{Find $\mathbf{U}^h \in \mathcal{W}^h$ such that for all $\mathbf{W}^h \in \WW^h$,}
\begin{align}\label{div projection}
\left(\nabla q^h,\mathbf{u}^h \right)_{\Omega}&= \left(\nabla q^h,\mathbf{u} \right)_{\Omega},
\end{align}
with corresponding orthogonality:
\begin{align}\label{div projection}
\left(\nabla q^h,\mathbf{u}' \right)_{\tilde{\Omega}}&= 0, \quad \text{for all}\quad \mathbf{W}^h \in \WW^h.
\end{align}
This orthogonality defines the fine-scale space $\WW'$ which represents the
orthogonal component of $\WW^h$ in terms of the projection (\ref{div projection}) as
\begin{equation}\label{sec:EEDF, subsec:OSS, constricted fine-scale space}
\begin{array}{l r l}
\WW'=\WW'_{\text{div}}:=\Big\{\left\{\mathbf{u},p\right\} \in \WW;
& \left. \left(\nabla \theta^h,\mathbf{u} \right)_{\Omega}=0, \right.
& \text{for all } \theta^h \in \mathcal{P}^h \Big\},
\end{array}
\end{equation}
where the space $\mathcal{P}^h$ is \added[id=Rev.1]{the} pressure part of $\WW^h=\mathcal{U}^h\times \mathcal{P}^h$.
Directly employing this divergence-free space indeed eliminates the last term of (\ref{energy evolution VMSDS form}).
However the small-scale solution space is infinite dimensional, and therefore not applicable in the numerical method.
As before, we avoid dealing with this space by using a Lagrange-multiplier construction yielding a mixed formulation.
Opening the solution space leads to the formulation:\\
\textit{Find $\left(\mathbf{U}^h, \zeta^h\right) \in \WW^h\times \mathcal{P}^h$ such that for all $\left(\mathbf{W}^h,\theta^h \right) \in \WW^h\times\mathcal{P}^h$,}
\begin{subequations}
\label{sec:Towards correct energy, GLS}
\begin{alignat}{2}
S_{\Omega}^{\text{div}}\left(\left(\mathbf{U}^h, \zeta^h\right),\left(\mathbf{W}^h, \theta^h\right) \right) =&
\added[id=Rev.1]{L_{\Omega}}(\mathbf{W}^h)_{\Omega},
\label{sec:Towards correct energy, GLS, large scale}
\end{alignat}
\indent \textit{where}
\begin{alignat}{2}
S_{\Omega}^{\text{div}}\left(\left(\mathbf{U}^h, \zeta^h\right),\left(\mathbf{W}^h, \theta^h\right) \right)=&
S_{\Omega}\left(\mathbf{U}^h,\mathbf{W}^h\right)+\left( \nabla \theta^h, \mathbf{u}'\right)_{\tilde{\Omega}},\\
\pd_t \mathbf{u}'+\tau_M^{-1} \mathbf{u}' + \nabla \zeta^h + \B{r}_M =& 0.
\end{alignat}
\end{subequations}
Obviously, this form follows the energy evolution
\begin{align}\label{energy evolution VMSDSO form}
\dfrac{{\rm d}}{{\rm d}t} E_\omega=& - \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\omega}^2
+ (\mathbf{u}^h,\mathbf{f})_{\omega}- (1,F_\omega^h)_{\chi_\omega}\nonumber \\
& -\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\omega}}^2
+ (\mathbf{u}',\mathbf{f})_{\tilde{\omega}}
+ (\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\omega}}.
\end{align}
\added[id=Both Rev]{
\subsection*{Remark}
Note that enforcing divergence-free small-scales has introduced an additional equation in the system. The new method has $5$ global variables instead of $4$ leading to a commensurate increase in computational time. The added block diagonal term is a diffusion matrix which does not further complicate the saddle point structure of the problem.}
\subsection{$H_0^1$-orthogonal small-scales}
\label{subsec: H_0^1-orthogonal small-scales}
In the energy evolution (\ref{energy evolution VMSDSO form}) unwanted artificial energy
can only be created by the term $\left(\nu \Delta \mathbf{u}^h,\mathbf{u}'\right)_{\tilde{\omega}}$.
Employing the orthogonality induced by the $H_0^1$-seminorm,
\begin{align}
(\nu \Delta \mathbf{w}^h,\mathbf{u}')_{\tilde{\Omega}} = 0 \quad \text{for~all}~\mathbf{W}^h \in \WW^h,
\end{align}
obviously cancels this term. To avoid dealing with a larger system of equations, we do not enforce the
orthogonality but we assume it in the large-scale equation (\ref{sec:Towards correct energy, GLS, large scale}).
This leads to a consistent GLS method. The resulting GLSDD-formulation reads:\\
\textit{Find $\left(\mathbf{U}^h, \zeta^h\right) \in \WW^h\times \mathcal{P}^h$ such that for all $\left(\mathbf{W}^h,\theta^h \right) \in \WW^h\times\mathcal{P}^h$,}
\begin{subequations}
\label{sec:Towards correct energy, GLSDSD}
\begin{alignat}{2}
S_{\Omega}^{\text{GLSDD}}\left(\left(\mathbf{U}^h, \zeta^h\right),\left(\mathbf{W}^h, \theta^h\right) \right)=& \added[id=Rev.1]{L_{\Omega}}(\mathbf{W}^h),
\end{alignat}
\indent \textit{where}
\begin{alignat}{2}
S_{\Omega}^{\text{GLSDD}}\left(\left(\mathbf{U}^h, \zeta^h\right),\left(\mathbf{W}^h, \theta^h\right) \right) =&
S_{\Omega}^{\text{div}}\left(\left(\mathbf{U}^h, \zeta^h\right),\left(\mathbf{W}^h, \theta^h\right)\right)
+ \left( \nu \Delta \mathbf{w}^h, \mathbf{u}'\right)_{\tilde{\Omega}},\nonumber \\
\pd_t \mathbf{u}'+\tau_M^{-1} \mathbf{u}' + \nabla \zeta^h + \B{r}_M =& 0.
\end{alignat}
\end{subequations}
In the abbreviation GLSDD we follow the same structure as before where the last two D's
stand for \textit{dynamic, divergence-free small-scale velocities}\footnote{\added[id=Rev.2]{The name GLS refers to the convection--diffusion part of the problem.}}.
This method displays the correct-energy behavior:
\begin{equation}\label{energy evol GLSDD}
\begin{array}{l l}
\dfrac{{\rm d}}{{\rm d}t} E_{\omega}= & - \|\nu^{1/2} \nabla \mathbf{u}^h \|_{\omega}^2 + (\mathbf{u}^h ,\mathbf{f})_{\omega}-(1,F_\omega^h)_{\chi_\omega}\\
& - \|\tau_M^{-1/2} \mathbf{u}'\|_{\tilde{\omega}}^2 + (\mathbf{u}' ,\mathbf{f})_{\tilde{\omega}}.
\end{array}
\end{equation}
The full expansion of this novel formulation is included in
\ref{Appendix: Galerkin/least-squares formulation with dynamic divergence-free small-scales} for clarity.
\subsection{Local energy backscatter}
The separate energy evolution of the large- and small-scales deduces in a similar fashion as above.
The large-scale energy $E^h_\omega=\tfrac{1}{2}(\mathbf{u}^h,\mathbf{u}^h)_{\omega}$ and the
small-scale energy $E'_\omega=\tfrac{1}{2}(\mathbf{u}',\mathbf{u}')_{\tilde{\omega}}$ do not add
up to the total energy $E_{\omega}$ because of the missing cross terms. This energy is stored
in \added[id=Rev.1]{an} intermediate (buffer) regime which we denote with
$E^{h'}_\omega=(\mathbf{u}^h,\mathbf{u}')_{\tilde{\omega}}$. The energy evolution takes the form:
\begin{subequations}\label{eq: local energy backscatter}
\begin{alignat}{2}
\dfrac{{\rm d}}{{\rm d}t} E^h_\omega & =
- \| \nu^{1/2} \nabla \mathbf{u}^h \|^2_{\omega}
+ \left(\mathbf{u}^h,\mathbf{f}\right)_{\omega} -(1,F_\omega)_{\chi_\omega}
+ \left(\left(\mathbf{u}^h+\mathbf{u}'\right)\cdot \nabla \mathbf{u}^h,\mathbf{u}'\right)_{\tilde{\omega}} -( \mathbf{u}^h, \partial_t \mathbf{u}' )_{\tilde{\omega}}, \\
\dfrac{{\rm d}}{{\rm d}t} E^{h'}_\omega & = \left(\mathbf{u}^h,\partial_t \mathbf{u}'\right)_{\tilde{\omega}}
+ \left(\mathbf{u}',\partial_t \mathbf{u}^h\right)_{\tilde{\omega}}, \\
\dfrac{{\rm d}}{{\rm d}t} E'_\omega & = - \| \tau_M^{-1/2} \mathbf{u}'\|^2_{\tilde{\omega}}
+ \left(\mathbf{u}' ,\mathbf{f}\right)_{\tilde{\omega}}
- \left(\left(\mathbf{u}^h+\mathbf{u}'\right)\cdot \nabla \mathbf{u}^h,\mathbf{u}'\right)_{\tilde{\omega}}
-( \mathbf{u}', \partial_t \mathbf{u}^h )_{\tilde{\omega}}.
\end{alignat}
\end{subequations}
The result mirrors to the convective--diffusive context with as convective velocity now the
total velocity $\mathbf{u}^h+\mathbf{u}'$. There is a direct exchange of convective
energy between the large-scale and small-scales. Clearly the superposition of
(\ref{eq: local energy backscatter}) yields (\ref{energy evol GLSDD}).
\subsection{Time-discrete energy behavior}\label{subsec:Time-discrete energy behavior}
The generalized-$\alpha$ method serves as time-integrator. Mirroring the convective--diffusive
context \cite{EAk17}, and using the same notation, we eventually obtain for $\alpha_m = \gamma$:
\begin{align}\label{discretized energy}
E_{n+1}=E_n-\Delta t^2 (\alpha_f-\onehalf) \| \dot{\mathbf{u}}_{n+\alpha_m} \|^2_{\Omega}&
- \Delta t\| \nu^{1/2}\nabla \mathbf{u}^h_{n+\alpha_f} \|^2_{\Omega}
-\Delta t\| \tau_{\text{dyn}}^{-1/2} \mathbf{u}_{n+\alpha_f}'\|^2_{\tilde{\Omega}} \nonumber \\
&+ \Delta t(\mathbf{u}^h_{n+\alpha_f} ,f)_{\Omega}+ \Delta t(\mathbf{u}_{n+\alpha_f}' ,f)_{\tilde{\Omega}}.
\end{align}
Hence, we have a decay of the discretized energy when, in absence of forcing,
$\alpha_f \geq \tfrac{1}{2}$. In the numerical implementation we use
$\alpha_f = \alpha_m = \gamma = \tfrac{1}{2}$ for the stability and
second-order accuracy properties \cite{Hul93}.
\section{Conservation properties}
\label{sec:CP}
Conservation of physical quantities in the numerical formulation is an often
sought-after property. In this Section we derive the various conservation
properties (continuity, linear momentum, angular momentum) of the proposed
formulation (\ref{sec:Towards correct energy, GLSDSD}). We \added[id=Rev.1]{prove} these by
selecting the appropriate weighting functions. The conservation properties
hold on both a global and a local scale. Therefore we omit the domain
subscript in the following.
\subsection{Continuity}
\label{sec:CP, subsec:C}
Employing the weighting function $\mathbf{w}^h=\mathbf{0},~\theta^h=0$
in (\ref{sec:Towards correct energy, GLSDSD}) yields
\begin{align}
(q^h,\nabla \cdot \mathbf{u}^h) = 0.
\end{align}
The choice $q^h= \nabla\cdot \mathbf{u}^h$ \added[id=Rev.1]{proves} the \textit{pointwise} satisfaction of incompressibility constraint\footnote{\added[id=Rev.1]{Note that in general this weighting function choice is not allowed. We employ the IGA spaces with stable velocity and pressure pairs that do allow this choice.}}
\begin{align}\label{div free}
||\nabla \cdot \mathbf{u}^h||^2=0 \quad \Rightarrow \quad \nabla \cdot \mathbf{u}^h=0 \quad \text{for all} \quad \mathbf{x} \in \Omega.
\end{align}
\added[id=Authors]{Furthermore,} the choice of weighting functions $\mathbf{w}^h=\mathbf{0},~q^h=0$ leads to divergence-free small-scale velocities in the following sense:
\begin{align}
(\nabla \theta^h, \mathbf{u}') = 0.
\end{align}
\subsection{Linear momentum}
\label{sec:CP, subsec:GLM}
We substitute the weighting functions
$\left(\mathbf{w}^h,q^h, \theta^h\right)=\left(\mathbf{e}_i,0, -\tfrac{1}{2}\mathbf{e}_i\cdot\mathbf{u}^h \right)$
in (\ref{sec:Towards correct energy, GLSDSD}), where $\mathbf{e}_i$ is the $i$th Cartesian basis vector.
Using $\nabla \mathbf{e}_i=\B{0}$ and the pointwise divergence-free velocity (\ref{div free}), all diffusive
and pressure terms drop out and we are left with:
\begin{align}\label{global line ar mom conv term initial}
\left(\mathbf{e}_i, \pd_t \mathbf{u}^h + \pd_t \mathbf{u}' \right)
+\tfrac{1}{2} \left(\mathbf{e}_i, \left(\left(\mathbf{u}^h
+ \mathbf{u}'\right)\cdot \nabla\right) \mathbf{u}^h\right)
+ \left( \nabla \left(- \tfrac{1}{2}\mathbf{e}_i\cdot \mathbf{u}^h\right) , \mathbf{u}' \right)= (\mathbf{e}_i,f).
\end{align}
Consider the convective term in isolation and write
\begin{align}\label{global line ar mom conv term}
\tfrac{1}{2}\left(\mathbf{e}_i, \left(\left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla\right) \mathbf{u}^h\right)
&= \tfrac{1}{2}\left(\mathbf{e}_i , \nabla \cdot \left( \left(\mathbf{u}^h + \mathbf{u}'\right) \otimes \mathbf{u}^h\right) \right)
- \tfrac{1}{2}\left(\mathbf{e}_i , \left(\nabla \cdot \left(\mathbf{u}^h + \mathbf{u}'\right) \right) \mathbf{u}^h\right) \nonumber \\
&= - \tfrac{1}{2}\left(\nabla \mathbf{e}_i , \left(\mathbf{u}^h + \mathbf{u}'\right) \otimes \mathbf{u}^h\right)
-\tfrac{1}{2} \left(\mathbf{e}_i , \left(\nabla \cdot \left(\mathbf{u}^h + \mathbf{u}'\right) \right) \mathbf{u}^h\right)\nonumber \\
&= - \tfrac{1}{2}\left(\mathbf{e}_i\cdot \mathbf{u}^h, \nabla \cdot \mathbf{u}' \right) \nonumber \\
&= \left( \nabla \left(\tfrac{1}{2}\mathbf{e}_i\cdot \mathbf{u}^h\right) , \mathbf{u}' \right).
\end{align}
Combining (\ref{global line ar mom conv term initial}) and (\ref{global line ar mom conv term}) leads to the balance
\begin{align}
(\mathbf{e}_i, \pd_t \mathbf{u}^h + \pd_t \mathbf{u}') = (\mathbf{e}_i,\mathbf{f}).
\end{align}
Linear momentum is thus conserved in terms of the total solution.
\subsection{Angular momentum}\label{subsec:Angular momentum}
Conservation of global angular momentum is a desirable property, certainly in rotating flows.
It has been analyzed by Bazilevs et al. \cite{BaAk10} and Evans et al. \cite{Evans13unsteadyNS}.
When using the appropriate weighting function spaces the formulation conserves angular momentum.
The numerical results of Section \ref{sec:ns_case} are however not computed with these weighting
function spaces. The demonstration of conservation of angular momentum follows the same ideas
as \cite{BaAk10}. We set the weighting functions
$\left(\mathbf{w}^h,q^h, \theta^h \right)=\left(\mathbf{x} \times \mathbf{e}_j,0,-\tfrac{1}{2}\left(\mathbf{x} \times \mathbf{e}_j\right)\cdot\mathbf{u}^h\right)$.
By construction the gradient of the weighting function leads to a skew-symmetric tensor \cite{BaAk10}.
As a result the gradient tensor is orthogonal to any symmetric tensor. Consequently the
divergence, which is the trace of the gradient, is zero.
Employing these weighting functions in the weak form we arrive at
\begin{align}
&( \mathbf{x} \times \mathbf{e}_j, \pd_t \mathbf{u}^h + \pd_t \mathbf{u}')
+\tfrac{1}{2} (\mathbf{x} \times \mathbf{e}_j, ((\mathbf{u}^h + \mathbf{u}')\cdot\nabla) \mathbf{u}^h)_{\Omega}
- \tfrac{1}{2} (\left((\mathbf{u}^h + \mathbf{u}')\cdot \nabla\right) \left( \mathbf{x} \times \mathbf{e}_j\right), \mathbf{u}^h )\nonumber \\
&- \left(\left((\mathbf{u}^h+\mathbf{u}')\cdot \nabla\right) \left(\mathbf{x} \times \mathbf{e}_j\right),\mathbf{u}'\right)
-\tfrac{1}{2}\left(\nabla \left(\left( \mathbf{x} \times \mathbf{e}_j\right)\cdot \mathbf{u}^h\right),\mathbf{u}'\right)
= (\left(\mathbf{x} \times \mathbf{e}_j\right),\mathbf{f}).
\end{align}
Consider again the convective terms in isolation. \added[id=Rev.1]{Switching back to a conservative form, see (\ref{convective derivation}), yields an incompressibility term:}
\begin{align}
&\tfrac{1}{2} (\mathbf{x} \times \mathbf{e}_j, ((\mathbf{u}^h + \mathbf{u}')\cdot\nabla) \mathbf{u}^h)
- \tfrac{1}{2} (\left((\mathbf{u}^h + \mathbf{u}')\cdot \nabla\right) \left( \mathbf{x} \times \mathbf{e}_j\right), \mathbf{u}^h )\nonumber \\
&- \left(\left((\mathbf{u}^h+\mathbf{u}')\cdot \nabla\right) \left(\mathbf{x} \times \mathbf{e}_j\right),\mathbf{u}'\right)\nonumber\\
=&-(\nabla \left(\mathbf{x} \times \mathbf{e}_j\right), (\mathbf{u}^h+\mathbf{u}')\otimes (\mathbf{u}^h+\mathbf{u}'))
-\tfrac{1}{2}\left(\mathbf{u}^h,\left(\mathbf{x} \times \mathbf{e}_j\right)\nabla \cdot \left(\mathbf{u}^h+\mathbf{u}'\right)\right)\nonumber \\
=&-(\nabla \left(\mathbf{x} \times \mathbf{e}_j\right), (\mathbf{u}^h+\mathbf{u}')\otimes (\mathbf{u}^h+\mathbf{u}'))
+ \tfrac{1}{2}\left(\nabla \left( \left(\mathbf{x} \times \mathbf{e}_j\right)\cdot\mathbf{u}^h\right),\mathbf{u}'\right).
\end{align}
The antisymmetric tensor and the symmetric tensor in the first and second argument, \added[id=Authors]{respectively,
cause} the first term to vanish. The \added[id=Rev.1]{incompressibility term} cancels with the choice of $\theta^h$
and the conservation of angular momentum is what remains:
\begin{align}
( \mathbf{x} \times \mathbf{e}_j, \pd_t \mathbf{u}^h + \pd_t \mathbf{u}') =(\mathbf{x} \times \mathbf{e}_j,\mathbf{f}).
\end{align}
\begin{comment}
\subsection{Angular momentum}
Conservation of global angular momentum is a desirable property, certainly in rotating flows.
It has been analyzed by Bazilevs et al. \cite{BaAk10} and Evans et al. \cite{Evans13unsteadyNS}.
When using the appropriate weighting function spaces the formulation conserves angular momentum.
The numerical results of Section \ref{sec:ns_case} are however not computed with these weighting
function spaces. The demonstration of conservation of angular momentum follows the same
ideas as \cite{BaAk10}. We set the weighting functions
$\left(\mathbf{w}^h,q^h, \theta^h \right)_i=\left(\epsilon_{ijk}x_j\mathbf{e}_k,0,-\tfrac{1}{2}\left(\epsilon_{ijk} x_j \mathbf{e}_k\right)\cdot\mathbf{u}^h\right)$,
where $\epsilon_{ijk}$ represents the alternator tensor and $x_j$ are the components of the position vector $\B{x}$.
This choice of the weighting function $\mathbf{w}^h_i$ leads to a skew-symmetric gradient tensor
$\nabla \mathbf{w}^h_i$ which can be written as $\nabla \mathbf{w}^h=\epsilon_{ijk} \mathbf{e}_j \otimes \mathbf{e}_k$ \cite{BaAk10}.
As a result the skew-symmetric tensor is orthogonal to any symmetric tensor.
The divergence $\nabla \cdot \mathbf{w}^h_i=T\left(\nabla \mathbf{w}^h_i\right)=\epsilon_{ill}$,
where $T$ is the trace operator, vanishes due to the symmetric properties of the alternator tensor.
Employing these weighting functions in the weak form we arrive at
\begin{align}
&( \epsilon_{ijk} x_j \mathbf{e}_k, \pd_t \mathbf{u}^h + \pd_t \mathbf{u}')
+\tfrac{1}{2} (\epsilon_{ijk} x_j \mathbf{e}_k, ((\mathbf{u}^h + \mathbf{u}')\cdot\nabla) \mathbf{u}^h)_{\Omega}
- \tfrac{1}{2} (\left((\mathbf{u}^h + \mathbf{u}')\cdot \nabla\right) \left( \epsilon_{ijk} x_j \mathbf{e}_k\right), \mathbf{u}^h )\nonumber \\
&- \left(\left((\mathbf{u}^h+\mathbf{u}')\cdot \nabla\right) \left(\epsilon_{ijk} x_j \mathbf{e}_k\right),\mathbf{u}'\right)
-\tfrac{1}{2}\left(\nabla \left(\left( \epsilon_{ijk} x_j \mathbf{e}_k\right)\cdot \mathbf{u}^h\right),\mathbf{u}'\right)
= (\left(\epsilon_{ijk} x_j \mathbf{e}_k\right),\mathbf{f})
\end{align}
Consider again the convective terms in isolation. We use (\ref{convective derivation}) to switch back to a conservative form
\begin{align}
&\tfrac{1}{2} (\epsilon_{ijk} x_j \mathbf{e}_k, ((\mathbf{u}^h + \mathbf{u}')\cdot\nabla) \mathbf{u}^h)
- \tfrac{1}{2} (\left((\mathbf{u}^h + \mathbf{u}')\cdot \nabla\right) \left( \epsilon_{ijk} x_j \mathbf{e}_k\right), \mathbf{u}^h )\nonumber \\
&- \left(\left((\mathbf{u}^h+\mathbf{u}')\cdot \nabla\right) \left(\epsilon_{ijk} x_j \mathbf{e}_k\right),\mathbf{u}'\right)\nonumber\\
=&-(\nabla \left(\epsilon_{ijk} x_j \mathbf{e}_k\right), (\mathbf{u}^h+\mathbf{u}')\otimes (\mathbf{u}^h+\mathbf{u}'))
-\tfrac{1}{2}\left(\mathbf{u}^h,\left(\epsilon_{ijk} x_j \mathbf{e}_k\right)\nabla \cdot \left(\mathbf{u}^h+\mathbf{u}'\right)\right)\nonumber \\
=&-(\nabla \left(\epsilon_{ijk} x_j \mathbf{e}_k\right), (\mathbf{u}^h+\mathbf{u}')\otimes (\mathbf{u}^h+\mathbf{u}'))
+ \tfrac{1}{2}\left(\nabla \left( \left(\epsilon_{ijk} x_j \mathbf{e}_k\right)\cdot\mathbf{u}^h\right),\mathbf{u}'\right)
\end{align}
The antisymmetric tensor and the symmetric tensor in the first and second
argument causes the first term to vanish. The second term cancels with the
choice of $\theta^h$ and the conservation of angular momentum is what remains:
\begin{align}
( \epsilon_{ijk} x_j \mathbf{e}_k, \pd_t \mathbf{u}^h + \pd_t \mathbf{u}') =(\epsilon_{ijk} x_j \mathbf{e}_k,\mathbf{f}).
\end{align}
\end{comment}
\section{Numerical test case}
\label{sec:ns_case}
In this Section we test the GLSDD method (\ref{sec:Towards correct energy, GLSDSD}) on
a three-dimensional Taylor--Green vortex flow at Reynolds number $Re=1600$. This test
case is challenging and it is often employed to examine the performance of numerical
algorithms for turbulence computations. It serves our purpose because (i) the energy
behavior of a fully turbulent flow can be studied, (ii) reference data is available and
(iii) the domain is periodic. Other boundary conditions than periodic ones are
beyond the scope of this work.
The flow is initially of laminar type. As the time evolves, the vortices begin to
evolve and roll-up. The vortical structures undergo changes and subsequently
their structures breakdown and form distorted vorticity patches. The flow
transitions to one with a turbulence character; the vortex stretching causes
the creation of small-scales.
The Taylor--Green vortex initial conditions are specified as follows:
\begin{subequations}
\label{sec:Towards correct energy, GLS}
\begin{alignat}{2}
u(\mathbf{x},0)&=\sin(x)\cos(y)\cos(z),\\
v(\mathbf{x},0)&= -\cos(x)\sin(y)\cos(z),\\
w(\mathbf{x},0)&=0,\\
p(\mathbf{x},0)&= \tfrac{1}{16}\left(\cos(2 x)+\cos(2 y)\right)\left(\cos(2z)+2\right).
\end{alignat}
\end{subequations}
The physical domain is the cube $\Omega=\left[0,2\pi\right]^3$ with periodic
boundary conditions. For this test case the viscosity is given by $\nu=\frac{1}{Re}$.
Here we consider the transition phase for times $t \leq 10$~s. Figure \ref{fig:TG vis}
shows the iso-surfaces of the z-vorticity of the initial condition (laminar flow)
and the final configuration (fully turbulent flow).\\
\begin{figure}[h!]
\begin{center}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[scale=0.25]{figures/TG_iso_vort_z_0.png}
\caption{Laminar flow at $t=0$~s.}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[scale=0.25]{figures/TG_iso_vort_z_800.png}
\caption{Fully turbulent flow at $t=10$~s.}
\end{subfigure}
\caption{Taylor--Green vortex flow at $Re=1600$. Iso-surfaces of z-vorticity.}\label{fig:TG vis}
\end{center}
\end{figure}
\indent Due to the symmetric behavior of the flow, we are allowed to simulate only
an eighth part of the domain. Hence, we take as computational domain
$\Omega^h=\left[0,\pi\right]^3$ and apply no-penetration boundary conditions.
All the implementations employ NURBS basis functions that are mostly $C^1$-quadratic,
however every velocity space is enriched to be cubic $C^2$ in the \added[id=Rev.2]{associated
direction} \cite{Evans13steadyNS, Evans13unsteadyNS, buffa2011isogeometric, buffa2011isogeometric3D}.
Note that conservation of angular momentum cannot be guaranteed, since the
\added[id=Rev.1]{choice of the weighting function $\theta^h$ in section \ref{subsec:Angular momentum}} is not valid. We apply a standard
$L_2$-projection to set the initial condition on the mesh. For the time-integration
we employ the generalized-$\alpha$ method with the parameter choices of \cite{EAk17}
which yield correct energy evolution. This method is stable and shows second-order temporal accuracy.
\added[id=Rev.1]{The resulting system of equations is solved with the standard flexible GMRES method with additive Schwartz preconditioning provided by Petsc \cite{PETSC,petsc-efficient}.}
We perform simulations with three different methods: (i) the classical Galerkin method,
(ii) the VMS method with static small-scales (VMSS), comparable with \cite{BaCaCoHu07} and
(iii) the novel Galerkin/least-squares formulation with dynamic and divergence-free
small-scales (GLSDD), i.e. form (\ref{sec:Towards correct energy, GLSDSD}).
The DNS results of Brachet et al. \cite{brachet1983small} obtained with a spectral
method on a fine $512^3$-mesh serve as reference data (ref).
First, we perform a brief mesh refinement study for the novel method.
Figure \ref{fig: TG mesh ref} shows mesh refined results for the novel
GLSDD method (\ref{sec:Towards correct energy, GLSDSD}).
For this purpose meshes with $16^3$, $24^3$, $32^3$, $48^3$ \added[id=Authors]{elements} have been employed. Clearly, the energy behavior on the coarsest
two meshes is quite off. The finer meshes are able to closely capture the
turbulence character of the flow. In the following we therefore use
meshes of $32^3$ or $48^3$ elements.
\begin{figure}[h!]
\begin{center}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/CE_o2_dissipation_reynolds.pdf}
\caption{Dissipation rate}
\label{fig: TG dissipation mesh ref}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/CE_o2_energy_reynolds.pdf}
\caption{Energy evolution}
\label{fig: TG energy mesh ref}
\end{subfigure}
\caption{Taylor--Green vortex flow at $Re=1600$ mesh convergence. The
GLS method with dynamic divergence-free small-scales.}
\label{fig: TG mesh ref}
\end{center}
\end{figure}
We compare the results of the novel GLSDD method with the VMSS and the
Galerkin approach. The simulations are carried out on a mesh of $32^3$
elements, i.e. the mesh size is $h=\frac{\pi}{32}$, and on a slightly finer mesh
of $48^3$ elements. The time-step is taken as $\Delta t = \frac{ 4 h}{5\pi}$,
i.e. the initial CFL-number is roughly $0.25$. In the Figures \ref{fig: TG 32}-\ref{fig: TG 48}
we visualize the time history of the kinetic energy and kinetic energy dissipation
rate for each of the three methods and the reference data.
\begin{figure}[h!]
\begin{center}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/o2_dissipation_reynolds.pdf}
\caption{Dissipation rate}
\label{fig: TG dissipation 32}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/o2_energy_reynolds.pdf}
\caption{Energy evolution}
\label{fig: TG energy 32}
\end{subfigure}
\caption{Taylor--Green vortex flow at $Re=1600$ on $32^3$-mesh for
various methods: the Galerkin method, the VMS method with static
small-scales and the GLS method with dynamic divergence-free small-scales.}\label{fig: TG 32}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/48o2_dissipation_reynolds_new.pdf}
\caption{Dissipation rate}
\label{fig: TG dissipation 48}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/48o2_energy_reynolds_new.pdf}
\caption{Energy evolution}
\label{fig: TG energy 48}
\end{subfigure}
\caption{Taylor--Green vortex flow at $Re=1600$ on $48^3$-mesh for various
methods: the Galerkin method, the VMS method with static small-scales and
the GLS method with dynamic divergence-free small-scales.}\label{fig: TG 48}
\end{center}
\end{figure}
The Figure \ref{fig: TG 32} shows that each of the methods is able to roughly
capture the energy behavior on the coarse mesh. The dissipation peek
appears too early in time for each of the simulations. The Galerkin method
displays the least accurate results, it overpredicts the dissipation rate.
The VMSS method performs a bit better at all times. The novel GLSDD
approach demonstrates an even closer agreement with the reference results.
The results on the finer mesh, in Figure \ref{fig: TG 48}, reveal
almost no difference with the reference data.
In the following we further analyze the contributions of the dissipation
rate (\added[id=Rev.1]{on the course mesh}). The dissipation rate of the Galerkin
method only consists of the large-scale/physical dissipation
$\|\nu^{1/2} \nabla \mathbf{u}^h\|_{\Omega}^2$. In contrast,
the dissipation of the GLSDD method is composed of a large-scale and a small-scale contribution:
\begin{align}
\dfrac{{\rm d}}{{\rm d}t} E_\Omega^{\text{GLSDD}}= & - \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\Omega}^2-\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\Omega}}^2.
\end{align}
In Figure \ref{fig: TG 32 split fraction} we display the temporal evolution of
both parts and the small-scale dissipation fraction \newline
$(\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\Omega}}^2)/(\|\nu^{1/2}\nabla \mathbf{u}^h\|_{\Omega}^2+\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\Omega}}^2)$.
In the laminar regime ($t<3$) the small-scale contribution is negligible.
When the flow has a more turbulent character the contribution of the
small-scales is substantial: the maximum of the dissipation fraction exceeds $0.35$.
\begin{figure}[h!]
\begin{center}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/CE_32o2_dissipation_rate_split.pdf}
\caption{Large- and small-scale contributions}
\label{fig: TG dissipation 32 split}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/CE_32o2_dissipation_rate_fraction.pdf}
\caption{The small-scale dissipation fraction}
\label{fig: TG energy 32 rate fraction}
\end{subfigure}
\caption{Taylor--Green vortex flow at $Re=1600$ on $32^3$-mesh with the
GLSDD method: (a) large-scale and small-scale parts of the dissipation
rate and (b) their fraction.}\label{fig: TG 32 split fraction}
\end{center}
\end{figure}
Lastly, we focus on the energy dissipation of the VMSS formulation.
The derivation follows the same steps used throughout this paper.
\added[id=Rev.1]{One might argue that the energy could also be solely based on the large-scales. This is what we do here.} Its evolution reads:
\begin{align}\label{energy evolution stat VMS}
\dfrac{{\rm d}}{{\rm d}t} E_\Omega^{h,\text{VMSS}}=&
- \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\Omega}^2
-\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\Omega}}^2
+ (\nu \Delta \mathbf{u}^h,\mathbf{u}')_{\tilde{\Omega}}
-\left( \mathbf{u}',\partial_t \mathbf{u}^h \right)_{\tilde{\Omega}}\nonumber \\
&+(\nabla \cdot \mathbf{u}^h,p')_{\tilde{\Omega}}
+\left(\nabla \mathbf{u}^h , (\mathbf{u}^h + \mathbf{u}') \otimes (\mathbf{u}^h + \mathbf{u}') \right)_{\tilde{\Omega}}
- \left(\mathbf{u}',(\mathbf{u}^h+\mathbf{u}')\cdot \nabla \mathbf{u}^h \right)_{\tilde{\Omega}}.
\end{align}
Figure \ref{fig: TG VMSS32 smallscaledissipation} shows the contribution of the separate terms.
The two desired dissipation terms are clearly dominant. The small-scale
dissipation is smaller than the large-scale dissipation, however it has a
significant contribution. Although the contributions are small, the unwanted
terms can create artificial energy.
\begin{figure}[h!]
\begin{center}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/VMSS_large0.pdf}
\caption{All terms}
\label{fig: TG dissipation 32 split}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[scale=0.625]{figures/VMSS_large2.pdf}
\caption{Zoom unwanted terms}
\label{fig: TG energy 32 rate fraction}
\end{subfigure}
\caption{Taylor--Green vortex flow at $Re=1600$ on $32^3$-mesh with the
VMSS method: energy dissipation of separate terms.}\label{fig: TG VMSS32 smallscaledissipation}
\end{center}
\end{figure}
\section{Conclusions}
\label{sec:CONC}
We continued the study initiated in \cite{EAk17} \added[id=Rev.1]{concerning} the construction of
methods displaying correct-energy behavior. In this paper we have
applied the developed methodology to the incompressible Navier--Stokes
equations. It clearly shows \added[id=Rev.1]{that the link between the methods VMS, SUPG and GLS, established in \cite{EAk17}, is also valid for the incompressible Navier--Stokes equations.}
The novel GLSDD methodology employs divergence-conforming NURBS
basis functions and uses a Lagrange multiplier setting to ensure divergence-free
small-scales. Furthermore, it enjoys the favorable behavior of the dynamic
small-scales and reduces to the Galerkin method in the Stokes regime.
These properties all emerge from the correct-energy design condition.
A pleasant byproduct of the method is the conservation of linear momentum.
The conservation of angular momentum can be achieved when employing
the appropriate weighting function spaces. The numerical results support the
theoretical framework in that the energy behavior improves upon the VMS
method with static small-scales. The variational multiscale method with
static small-scales has unwanted small-scale contributions
which create artificial energy.
The novel formulation requires a bit more effort to \added[id=Rev.1]{implement} compared
\added[id=Rev.1]{to} the variational multiscale method with static small-scales. One has to
include an additional variable to ensure the divergence-free behavior of the
small-scales. In addition the formulation \added[id=Rev.1]{needs} to be equipped with the
dynamic small-scale model. However, the resulting system of equations
does not demand a sophisticated preconditioner; we have employed the
standard ASM (Additive Schwarz Method) technique. In our opinion, the
accuracy gain outweighs the little \added[id=Rev.1]{extra implementation
effort} and calculation cost.
We see several directions for future work. The first concerns the
development of a method displaying correct energy behavior at the
boundary, in particular \added[id=Rev.1]{when} using the weak imposition of Dirichlet
boundary conditions. This allows to test the effect of correct energy
behavior on wall-bounded turbulent flow problems. Another extension
is correct energy behavior for free-surface flow computations. This is an
important step, since artificial energy creation can yield highly instable
behavior, as demonstrated in \cite{AkBaBeFaKe12}. We have work on both extensions in
progress and aim to report on it in the near future.
\section*{Acknowledgment}
\label{sec:ack}
The authors are grateful to the Delft University of Technology
for its support.
\appendix
\section{Galerkin/least-squares formulation with dynamic divergence-free small-scales}
\label{Appendix: Galerkin/least-squares formulation with dynamic divergence-free small-scales}
We repeat the Galerkin/least-squares formulation with dynamic divergence-free
small-scales (GLSDD), i.e. form (\ref{sec:Towards correct energy, GLSDSD}), to
provide an overview of the separate terms. The formulation is of skew-symmetric type,
applies GLS stabilization and uses divergence-free dynamic small-scales. The method requires a stable velocity--pressure pair and reads:\\
\textit{Find $\left(\mathbf{u}^h,p^h, \zeta^h\right) \in \WW^h\times \mathcal{P}^h$
such that for all $\left(\mathbf{w}^h,q^h,\theta^h \right) \in \WW^h\times\mathcal{P}^h$,}
\begin{subequations}
\label{eq:omegai}
\begin{align}
\label{sec:Towards correct energy, GLSDSD2}
\begin{split}
\left( \mathbf{w}^h, \pd_t \mathbf{u}^h \right)_{\Omega} +{}& \left(\mathbf{w}^h, \pd_t \mathbf{u}'\right)_{\tilde{\Omega}}\\[6pt]
+& \tfrac{1}{2}(\mathbf{w}^h, (\mathbf{u}^h+\mathbf{u}') \cdot \nabla \mathbf{u}^h)_{\Omega}
-\tfrac{1}{2}((\mathbf{u}^h+\mathbf{u}')\cdot\nabla\mathbf{w}^h, \mathbf{u}^h)_{\Omega}
- \left( \left(\mathbf{u}^h + \mathbf{u}'\right)\cdot \nabla \mathbf{w}^h , \mathbf{u}' \right)_{\tilde{\Omega}}\\[6pt]
+&(\nabla \mathbf{w}^h, 2 \nu \nabla^s \mathbf{u}^h )_{\Omega}
+ \left( \nu \Delta \mathbf{w}^h, \mathbf{u}'\right)_{\tilde{\Omega}} \\[6pt]
+&(q^h,\nabla \cdot \mathbf{u}^h)_{\Omega}- (\nabla \cdot \mathbf{w}^h, p^h )_{\Omega}
+\left( \nabla \theta^h, \mathbf{u}'\right)_{\tilde{\Omega}}=(\mathbf{w},\mathbf{f})_{\Omega},
\end{split}
\\[6pt]
\label{eq:omega2}
\begin{split}
\pd_t \mathbf{u}'+\tau_M^{-1} \mathbf{u}' + \nabla \zeta^h {}&+ \B{r}_M =0,
\end{split}
\end{align}
\end{subequations}
where momentum residual is
\begin{equation}
\B{r}_M= \pd_t \mathbf{u}^h + \left(\left(\mathbf{u}^h+\mathbf{u}'\right)\cdot\nabla\right) \mathbf{u}^h
+ \nabla p^h - \nu \Delta \mathbf{u}^h - \mathbf{f}.
\end{equation}
The separate terms of (\ref{sec:Towards correct energy, GLSDSD2}) are from
left to right: the temporal terms, the convective contributions, the viscous contributions,
the incompressibility constraint, the pressure term, the divergence-free small-scale
velocity constraint and the forcing term. This form follows the correct-energy evolution (on a local scale):
\begin{align}\label{energy evolution GLSDD Appendix}
\dfrac{{\rm d}}{{\rm d}t} E_\omega=& - \|\nu^{1/2}\nabla \mathbf{u}^h\|_{\omega}^2+
(\mathbf{u}^h,\mathbf{f})_{\omega}- (1,F_\omega^h)_{\chi_\omega}\nonumber \\
& -\|\tau_M^{-1/2}\mathbf{u}'\|_{\tilde{\omega}}^2 + (\mathbf{u}',\mathbf{f})_{\tilde{\omega}},
\end{align}
and possesses the conservation properties of Section \ref{sec:CP}.
\section{Definition dynamic stabilization parameter}
\label{Appendix: Definition stabilization parameters}
The dynamic stabilization parameter $\tau_{M}$ is the discrete
approximation of the inverse of the convective and viscous parts
of momentum Navier--Stokes operator. It mirrors the dynamic
stabilization parameter of convection--diffusion equation (see \cite{EAk17}).
The continuity stabilization parameter $\tau_C$ is on its turn the discrete
approximation of the inverse of the divergence operator, here we use the
objective definition introduced in \cite{BaAk10}. The parameters take the form:
\begin{subequations}
\begin{alignat}{1} \label{eq:tau_static 2}
\tau_{M}=&\left(\tau^{-2}_{\text{conv}}+\tau^{-2}_{\text{visc}}\right)^{-1/2},\\
\tau_{C}=&\left(\tau_M \sqrt{ \mathbf{G}:\mathbf{G}}\right)^{-1},
\end{alignat}
\end{subequations}
where the convective and viscous contributions of $\tau_{M}$ are
\begin{subequations}
\begin{alignat}{1}
\tau_{\text{conv}}^{-2}=& 4 \mathbf{u}\cdot \mathbf{G} \mathbf{u},\\
\tau_{\text{visc}}^{-2}=& C_I\nu^2 \mathbf{G}:\mathbf{G}.
\end{alignat}
\end{subequations}
Here the following definition is employed:
\begin{subequations}
\begin{alignat}{1}
\mathbf{G}&=\frac{\pd \boldsymbol{\xi}}{\pd \bx}^T\frac{\pd \boldsymbol{\xi}}{\pd \bx},\\
\mathbf{G}:\mathbf{G} &= \displaystyle\sum_{i,j=1}^3 G_{ij} G_{ij},
\end{alignat}
\end{subequations}
where $\pd \boldsymbol{\xi}/\pd \bx$ is the inverse Jacobian of the map
between the elements in the reference and physical domain. The positive
constant $C_I$ is determined by an inverse estimate.
\bibliographystyle{unsrt}
\bibliography{references}
\end{document}
| 11,484
|
\begin{document}
\title[On the unitary structures of VOSAs]{On the unitary structures of vertex operator superalgebras}
\author{Chunrui Ai}
\address{Chunrui Ai, School of Mathematics and Statistics, Zhengzhou University, Henan 450001, China
}
\email{aicr@zzu.edu.cn}
\author{Xingjun Lin}
\address{Xingjun Lin, Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan}
\email{linxingjun@math.sinica.edu.tw}
\begin{abstract}
In this paper, the notion of unitary vertex operator superalgebra is introduced. It is proved that the vertex operator superalgebras associated to the unitary highest weight representations for the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary vertex operator superalgebras. The unitary structures are then used to study the structures of vertex operator superalgebras, it is proved that any unitary vertex operator superalgebra is a direct sum of strong CFT type unitary simple vertex operator superalgebras. The classification of unitary vertex operator superalgebras generated by the subspaces with conformal weights less than or equal to $1$ is also considered.
\end{abstract}
\maketitle
\section{Introduction \label{intro}}
\def\theequation{1.\arabic{equation}}
\setcounter{equation}{0}
Unitary structures of vertex operator algebras were introduced in the early days of vertex operator algebras, the unitary structures of the lattice vertex operator algebras and Moonshine vertex operator algebra were used to study the Monster group \cite{B}, \cite{FLM}. Later, based on a symmetric contravariant bilinear form with respect to a Cartan involution in \cite{B} for a vertex algebra constructed from an even lattice, the notion of invariant bilinear form was introduced and studied in \cite{FHL}, \cite{Li1}.
Recently, it was found in \cite{DLin} that the positive definite Hermitian forms of vertex operator algebras which are invariant with respect to anti-linear involutions of vertex operator algebras can be used to define unitary vertex operator algebras. And it was proved in \cite{DLin} that the vertex operator algebras associated
to the unitary highest weight representations for the Heisenberg algebras, Virasoro algebra and affine Kac-Moody algebras are unitary vertex operator algebras. Moreover, the unitary structures of these vertex operator algebras are induced from the unitary structures of the highest weight modules for the corresponding Lie algebras. The unitary structures of vertex operator algebras were later used to construct conformal nets from vertex operator algebras \cite{C}.
In the first part of this paper, the notion of unitary vertex operator superalgebra is introduced, this is a generalization of the notion of unitary vertex operator algebra.
It is then proved that the vertex operator superalgebras associated to the unitary highest weight representations for the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary vertex operator superalgebras. Also, the unitary structures of the vertex operator superalgebras associated to the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras are induced from the unitary structures of the highest weight modules for the corresponding Lie superalgebras. It is also expected that the unitary structures of vertex operator superalgebras can be used to construct superconformal nets (cf. \cite{CKL}).
In the second part of this paper, the structures of unitary vertex operator superalgebras are studied. It is well-known that a finite dimensional associative algebra over the field of complex numbers is semisimple if and only if the trace form of the algebra is nondegenerate and that any semisimple associative algebra over the field of complex numbers is a direct sum of simple algebras. Since vertex operator algebras are analogues of associative algebras and unitary vertex operator superalgebras have positive definite invariant Hermitian forms, it is natural to expect that any unitary vertex operator superalgebra is a direct sum of unitary simple vertex operator superalgebras. In this paper, it is shown that any unitary vertex operator superalgebra is a direct sum of strong CFT type unitary simple vertex operator superalgebras. And it is found that the strong CFT type properties of vertex operator superalgebras, which have played important roles in the study of vertex operator superalgebras (see \cite{Bu}, \cite{DM1}, \cite{DM2}, \cite{Hu}), are closely related to the unitary structures of vertex operator superalgebras.
As an application of the structure theorem of unitary vertex operator superalgebras, the classification of unitary vertex operator superalgebras generated by the subspaces with conformal weights less than or equal to $1$ is considered. The key point is to classify the simple vertex operator superalgebras in these classes. The simple vertex operator superalgebras generated by the subspaces with conformal weight $\frac{1}{2}$ have been classified in \cite{X}, it was proved that any simple vertex operator superalgebra generated by the subspace with conformal weight $\frac{1}{2}$ is isomorphic to the vertex operator superalgebra associated to the highest weight representation for some Heisenberg superalgebra. So the key point is to classify unitary simple vertex operator algebras generated by the subspaces with conformal weight $1$. Simple vertex operator algebras generated by the subspaces with conformal weight $1$ have been studied in \cite{DM1}, \cite{DM2}, \cite{M} under the condition that the vertex operator algebras are $C_2$-cofinite or regular, and their results have played important roles in the classification of rational and $C_2$-cofinite vertex operator algebras. But their results depend on the condition that the vertex operator algebras are $C_2$-cofinite or regular, and there are unitary vertex operator algebras which are neither $C_2$-cofinite nor regular, so we still need to study the structures of unitary simple vertex operator algebras generated by the subspaces with conformal weight $1$. In this paper, it is shown that each of these unitary vertex operator superalgebras is isomorphic to the tensor product of some vertex operator superalgebras associated to unitary highest weight representations for the affine Kac-Moody algebras and the Heisenberg algebras.
The rest of this paper is organized as follows: In Section 2, we prove some basic facts about unitary vertex operator superalgebras and then show that the vertex operator superalgebras associated to the unitary highest weight representations for the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary vertex operator superalgebras. In Section 3, the structures of unitary vertex operator superalgebras are studied. In Section 4, the classification of unitary vertex operator superalgebras generated by the subspaces with conformal weights less than or equal to $1$ is considered.
\section{Unitary vertex operator superalgebras}
\def\theequation{2.\arabic{equation}}
\setcounter{equation}{0}
\subsection{Preliminaries} In this subsection, we shall recall from \cite{DL}, \cite{FFR}, \cite{KW}, \cite{Li} and \cite{X} some facts about vertex operator superalgebras. First, we recall some notations, let $V=V_{\bar 0}\oplus V_{\bar 1}$ be any ${\Bbb Z}_2$-graded vector space, the element in $V_{\bar 0} $ (resp. $V_{\bar 1}$) is called {\em even} (resp. {\em odd}).
We then define $[v]=i$ for any $v\in V_{\bar i}$ with $i=0,1$. A {\em vertex operator superalgebra} is a quadruple $(V,Y,\1,\w),$ where $V=V_{\bar 0}\oplus V_{\bar 1}$ is a ${\Bbb Z}_2$-graded vector space, ${\bf 1}$ is the {\em vacuum vector}
of $V$, $\w$ is the {\em conformal vector} of $V,$ and $Y$ is a linear map
\begin{align*}
V &\to (\End\,V)[[z,z^{-1}]] ,\\
v&\mapsto Y(v,z)=\sum_{n\in{\Z}}v_nz^{-n-1}\ \ \ \ (v_n\in
\End\,V)
\end{align*}
satisfying the following axioms: \\
(i) For any $u,v\in V,$ $u_nv=0$ for sufficiently large $n$;\\
(ii) $Y({\bf 1},z)=\id_{V}$;\\
(iii) $Y(v,z){\bf 1}=v+\sum_{n\geq 2}v_{-n}{\bf 1}z^{n-1},$ for any $v\in V$;\\
(iv) The component operators of $Y(\w,z)=\sum_{n\in\Z}L(n)z^{-n-2}$ satisfy the Virasoro algebra
relation with {\em central charge} $c\in \C:$
\begin{align*}
[L(m),L(n)]=(m-n)L(m+n)+\frac{1}{12}(m^3-m)\delta_{m+n,0}c,
\end{align*}
and
\begin{align*}
\frac{d}{dz}Y(v,z)=Y(L(-1)v,z),\ \ \text{ for any } v\in V;
\end{align*}
(v) $V$ is $\frac{1}{2}\Z$-graded such that $V=\oplus_{n\in \frac{1}{2}\Z}V_n$, $V_{\bar 0}=\oplus_{n\in \Z}V_n$, $V_{\bar 1}=\oplus_{n\in \frac{1}{2}+\Z}V_n$, $L(0)|_{V_n}=n$, $\dim(V_n)<\infty$ and $V_n=0$ for sufficiently small $n$;\\
(vi) The {\em Jacobi identity} for ${\Bbb Z}_2$-homogeneous $u,v\in V$ holds,
\begin{align*}
\begin{array}{c}
\displaystyle{z^{-1}_0\delta\left(\frac{z_1-z_2}{z_0}\right)
Y(u,z_1)Y(v,z_2)-(-1)^{[u][v]} z^{-1}_0\delta\left(\frac{z_2-z_1}{-z_0}\right)
Y(v,z_2)Y(u,z_1)}\\
\displaystyle{=z_2^{-1}\delta
\left(\frac{z_1-z_0}{z_2}\right)
Y(Y(u,z_0)v,z_2)}.
\end{array}
\end{align*}
This completes the definition of a vertex operator superalgebra and we will denote the vertex operator superalgebra briefly by $V$. An {\em anti-linear automorphism} $\phi$ of $V$ is
an anti-linear isomorphism (as anti-linear morphism) $\phi:V\to V$ such
that $\phi(\bf1)=\bf1, \phi(\omega)=\omega$
and
$\phi(u_nv)=\phi(u)_n\phi(v)$ for any $ u, v\in V$ and $n\in
\mathbb{Z}$. An anti-linear automorphism $\phi$ of $V$ is called an {\em anti-linear involution} if the order of $\phi$ is $2$.
We then define a {\em unitary} vertex operator superalgebra to be a pair $(V,\phi)$, where $V$ is a vertex operator superalgebra and $\phi: V\to V$ is an anti-linear involution of $V$, such that there exists a positive definite Hermitian form $(,): V\times V\to \mathbb{C}$ on $V$ and the following invariant property
$$(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}a, z^{-1})u, v)=(u,
Y(\phi(a), z)v),$$holds for any $a, u, v\in V$.
\begin{remark}
Note that if $V$ is a vertex operator algebra, the definition of unitary vertex operator superalgebra coincides with the definition of unitary vertex operator algebra in \cite{DLin}.
\end{remark}
Similarly, we can define the unitary module for a unitary vertex operator superalgebra (cf. \cite{DLin}). The following results are immediate consequences of the definitions of unitary modules and unitary vertex operator superalgebras.
\begin{proposition}
Let $V$ be a unitary vertex operator superalgebra $V$. Then \\
(1) For any homogenous vectors $u,v \in V$ such that $\wt u\neq \wt v$, we have $(u, v)=0$.\\
(2) Any unitary $V$-module $M$ is completely reducible, that is, $M \cong \oplus_{i\in I}W^i$, where $W^i$, $i\in I$, are irreducible submodules of $M$. Moreover, $(W^i, W^j)_M=0$ for $i\neq j$, where $(,)_M$ is the positive definite Hermitian form on $M$.
\end{proposition}
\pf We only prove the second result. Let $M$ be a unitary $V$-module and $(,)_M$ be the positive definite Hermitian form on $M$. Consider the set $S$ consisting of submodules of $M$ such that each element $W\in S$ satisfies the property $W \cong \oplus_{i\in I}W^i$, and $(W^i, W^j)_M=0$ for $ i\neq j$, where $W^i, i\in I$, are irreducible submodules of $M$.
Note that there exists a maximal object $W^0$ in $S$. We now prove that $W^0=M$. Otherwise, consider the orthogonal complement $(W^0)^\perp$ of $W^0$ with respect to $(,)_M$. It follows from the definition of unitary modules that $(W^0)^\perp$ is also a submodule of $M$. Let $n_0$ be a number such that $(W^0)^\perp_{n_0}\neq 0$ but $(W^0)^\perp_{n}= 0$ if $Re~ n< Re~ n_0$, where $(W^0)^\perp_{n}=\{w\in (W^0)^\perp|L(0)w=nw\}$. Then we know that $(W^0)^\perp_{n_0}$ is a module for the Zhu's algebra $A(V)$ of $V$ (cf. \cite{KWa}). Since $\dim (W^0)^\perp_{n_0}<\infty$, there exists an irreducible $A(V)$-submodule $Z$ of $(W^0)^\perp_{n_0}$, we then prove that the submodule $\langle Z\rangle$ of $(W^0)^\perp$ generated by $Z$ is an irreducible module for $V$. Note that $\langle Z\rangle\cap (W^0)^\perp_{n_0}=Z$ and that any proper submodule of $\langle Z\rangle$ has zero intersection with $Z$, it follows immediately from the invariant property of $(,)_M$ that $\langle Z\rangle$ is irreducible.
As a result, $W^0\oplus \langle Z\rangle$ is an object in $S$, this is a contradiction. The proof is complete. \qed
\vskip.25cm
Recall that for a vertex operator superalgebra $V$ and a $V$-module $M=\oplus_{\lambda\in \C}M_{\lambda}$, the {\em contragredient module} $M'$ of $M$ is defined to be $(M', Y')$, where $M'=\oplus_{\lambda\in \C} (M_\lambda)'$ is the restricted dual of $M$ and $Y'$ is the linear map defined by
\begin{align*}
\langle Y'(a, z)w', w\rangle = \langle w', Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}a, z^{-1})w\rangle,
\end{align*}for any $a\in V, w\in M, w'\in M'$ (cf. \cite{FHL}, \cite{Y}). Recall also that a $V$-module $M$ is called {\em self-dual} if $M$ is isomorphic to $M'$, then we have
\begin{proposition}\label{self-dual}
Let $V$ be a unitary vertex operator superalgebra. Then $V$ is self-dual.
\end{proposition}
\pf It is good enough to prove that $V$ has a nondegenerate invariant bilinear form, that is, a nondegenerate bilinear form $\langle ,\rangle$ such that the following invariant property
\begin{align*}
\langle Y(u, z)v, w\rangle=\langle v, Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}u, z^{-1})w\rangle,
\end{align*}
holds for any $u,v, w\in V$. For any $u,v\in V$, we define $\langle u, v\rangle =(u, \phi(v))$. It is clear that $\langle, \rangle$ is a nondegenerate bilinear form. We then prove that $\langle, \rangle$ is invariant. By definition, we have
\begin{align*}
\langle Y(u, z)v, w\rangle
& =(Y(u, z)v, \phi (w))\\
& =(v, Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}\phi(u), z^{-1})\phi (w))\\
& =(v, \phi(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}u, z^{-1})w))\\
& =\langle v, Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}u, z^{-1})w\rangle,
\end{align*}
as desired.\qed
The following results concern constructing unitary vertex operator superalgebras from known unitary vertex operator superalgebras, the proofs of these results are similar to those of Propositions 2.6, 2.9 in \cite{DLin}.
\begin{proposition}\label{sub}
(1) Let $(V, \phi)$ be a unitary vertex operator superalgebra and $U$ be a subalgebra of $V$ such that the Virasoro element of $U$ is the same as that of $V$ and $\phi(U)=U$. Then $(U, \phi|_U)$ is a unitary vertex operator superalgebra.\\
(2) Let $(V^1, \phi_1),..., (V^p, \phi_p)$ be unitary vertex operator superalgebras. Then there exists an anti-linear involution $\phi$ of $V^1\otimes\dots\otimes V^p $ such that $(V^1\otimes\dots\otimes V^p, \phi)$ is a unitary vertex operator superalgebra.
\end{proposition}
We also need the following result which is useful to prove the unitarity of vertex operator superalgebra.
\begin{proposition}\label{basic}
Let $V$ be a vertex operator superalgebra equipped with a positive
definite Hermitian form $(, ): V\times V\to \mathbb{C}$ and $\phi$ be an anti-linear involution of $V$. Assume that $V$ is
generated by the subset $S\subset V$, i.e.,
$$V=span\{u^1_{n_1}\cdots u^k_{n_k}{\bf 1}| k\in \N, u^1,\cdots,
u^k\in S\}$$ and that the invariant property
\begin{equation*} (Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}a, z^{-1})u,
v)=(u, Y(\phi(a), z)v)
\end{equation*}
holds for any $a\in S, u, v\in V.$ Then $(V, \phi)$ is a unitary vertex operator superalgebra.
\end{proposition}
\pf Note that for a vertex operator superalgebra $V$, we have the following identity which was essentially proved in Theorem 5.2.1 of \cite{FHL},
\begin{align*}
&-(-1)^{[a][b]}z _0^{-1}\delta\left(\frac{z_2-z_1}{-z_0}\right)Y(e^{z_1L(1)}(-1)^{L(0)+2L(0)^2}z_1^{-2L(0)}a, z_1^{-1})\\
&~~\cdot Y(e^{z_2L(1)}(-1)^{L(0)+2L(0)^2}z_2^{-2L(0)}b, z_2^{-1})\\
&+z_0^{-1}\delta\left(\frac{z_1-z_2}{z_0}\right)Y(e^{z_2L(1)}(-1)^{L(0)+2L(0)^2}z_2^{-2L(0)}b, z_2^{-1})\\
&~~\cdot Y(e^{z_1L(1)}(-1)^{L(0)+2L(0)^2}z_1^{-2L(0)}a, z_1^{-1})\\
&\ \ =z_1^{-1}\delta\left(\frac{z_2+z_0}{z_1}\right)Y(e^{z_2L(1)}(-1)^{L(0)+2L(0)^2}z_2^{-2L(0)}Y(a, z_0)b, z_2^{-1}),
\end{align*}
for any $a, b\in V$. Then the proposition follows from the similar argument in the proof of Proposition 2.11 in \cite{DLin}. \qed
\subsection{Unitary Neveu-Schwarz vertex operator superalgebras}
In this subsection, we shall show that the Neveu-Schwarz vertex operator superalgebras associated to unitary highest weight modules for the Neveu-Schwarz Lie superalgebra are unitary vertex operator superalgebras. We first recall from \cite{KWa}, \cite{Li} some facts about the Neveu-Schwarz vertex operator superalgebras. Let $NS$ be the Neveu-Schwarz Lie superalgebra
\begin{align*}
NS=\oplus_{m\in \Z}\C L(m)\oplus\oplus_{n\in \Z}\C G(n+\frac{1}{2})\oplus \C C,
\end{align*}
with the following communication relations:
\begin{align*}
&[L(m), L(n)]=(m-n)L(m+n)+\frac{m^3-m}{12}\delta_{m+n, 0}C,\\
&[L(m), G(n+\frac{1}{2})]=(\frac{m}{2}-n-\frac{1}{2})G(m+n+\frac{1}{2}),\\
&[G(m+\frac{1}{2}), G(n+\frac{1}{2})]_+=2L(m+n)+\frac{1}{3}m(m+1)\delta_{m+n, 0}C,\\
&[L(m), C]=0,~[G(n+\frac{1}{2}), C]=0.
\end{align*}
Set
\begin{align*}
&NS_{\pm}=\oplus_{n\in \Z_+}\C L(\pm n)\oplus\oplus_{m\in \Z_+}\C G(\pm m\mp\frac{1}{2}),\\
&NS_0=\C L(0)\oplus \C C.
\end{align*}
It is clear that $NS_+\oplus NS_0$ is a subalgebra of $NS$. Given any complex numbers $c$ and $h$, we then consider the Verma module
\begin{align*}
M(c, h)=U(NS)/J,
\end{align*}
where $U(\g)$ denotes the universal enveloping algebra of a Lie superalgebra $\g$ and $J$ is the left ideal of $U(NS)$ generated by $NS_+$, $L(0)-h$, $C-c$.
It is well-known that the Verma module has a unique maximal proper submodule $J(c, h)$, we then let $L(c, h)$ be the irreducible $NS$-module $M(c, h)/J(c, h)$. It was proved in \cite{KWa} that $L(c, 0)$ has a vertex operator superalgebra structure such that $\1=1$ and $\w=L(-2)1$.
We next recall from \cite{KW} some facts about the unitary modules for the Neveu-Schwarz Lie superalgebra $NS$. Let $M$ be a highest weight module for $NS$, $M$ is called a {\em unitary} module for $NS$ if there exists a positive definite Hermitian form $(,)$ on $M$ such that
\begin{align*}
(L(n)u,v)=(u, L(-n)v),~(G(n+\frac{1}{2})u, v)=(u,G(-n-\frac{1}{2})v ),
\end{align*}
for any $n\in \Z$ and $u,v \in M$ (cf. \cite{KT}, \cite{KW}). It was proved in \cite{GKO}, \cite{KW} that $L(c, h)$ is a unitary module for $NS$ if $c\geq \frac{3}{2}$ and $h\geq 0$ or
\begin{align*}
&c=c_m=\frac{3}{2}(1-\frac{8}{(m+2)(m+4)}),\\
&h=h^m_{r, s}=\frac{((m+4)r-(m+2)s)^2-4}{8(m+2)(m+4)},
\end{align*}
where $m, r,s\in \Z_+,~ 1\leq s\leq r\leq m+1$ and $r-s\in 2\Z$ if $r\neq 0$.
We now begin to construct unitary structures on the vertex operator superalgebras associated to unitary modules for $NS$. We begin with the definition of the anti-linear involution. Let $T(NS)$ be the tensor algebra of $NS$ and $\Phi: T(NS)\to T(NS)$ be the anti-linear map of $T(NS)$ such that $\Phi(x^1\otimes\cdots \otimes x^n)=x^1\otimes\cdots \otimes x^n$ for any $x^1,\cdots , x^n\in \{L(n), G(m+\frac{1}{2})|n,m\in \Z\}$.
It is clear that $\Phi$ is an anti-linear involution of $T(NS)$. Note that by the definition of $NS$,
we have\begin{align*}
&\Phi(x^1\otimes x^2-(-1)^{[x^1][x^2]}x^2\otimes x^1-[x^1, x^2])\\
&\ \ =\Phi(x^1)\otimes \Phi(x^2)-(-1)^{[x^1][x^2]}\Phi(x^2)\otimes\Phi( x^1)-\Phi([x^1, x^2])\\
&\ \ =\Phi(x^1)\otimes \Phi(x^2)-(-1)^{[x^1][x^2]}\Phi(x^2)\otimes\Phi( x^1)-[\Phi(x^1), \Phi(x^2)],
\end{align*}
for any $x^1, x^2\in NS$. Thus, $\Phi$ induces an anti-linear involution of $U(NS)$, which is still denoted by $\Phi$. Furthermore, if $c\in \R$, we have $\Phi(J)\subset J$. Hence, $\Phi$ induces an anti-linear involution of $M(c, 0)$, which is also denoted by $\Phi$. Finally, note that $\Phi(J(c, 0))\subset J(c, 0)$, we then have an anti-linear involution $\phi$ of $L(c, 0)$ induced from $\Phi$. We are now in a position to prove that $L(c,0)$ has a unitary vertex operator superalgebra structure.
\begin{theorem}
Let $c\in \R$ be a positive real number such that $c\geq \frac{3}{2}$ or $c=c_m$ for some positive integer $m$. Then $(L(c, 0), \phi)$ is a unitary vertex operator superalgebra.
\end{theorem}
\pf Note that if $c\in \R$ is a positive real number such that $c\geq \frac{3}{2}$ or $c=c_m$ for some positive integer $m$, $L(c, 0)$ is a unitary module for $NS$. Let $(,)$ be the positive definite Hermitian form on $L(c, 0)$. We next prove that $(L(c, 0), \phi)$ equipped with the positive definite Hermitian form $(,)$ is a unitary vertex operator superalgebra. Since $L(c, 0)$ is generated by $L(-2)1$ and $G(-\frac{3}{2})1$, by Proposition \ref{basic} it is good enough to prove that the invariant property
\begin{equation*} (Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}a, z^{-1})u,
v)=(u, Y(\phi(a), z)v)
\end{equation*}
holds for $a=L(-2)1\text{ or }G(-\frac{3}{2})1$ and any $ u, v\in L(c, 0).$ When $a=L(-2)1$, the invariant property has been proved in \cite{DLin}. We then assume that $a=G(-\frac{3}{2})1$, by a direct computation,
\begin{align*}
(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}G(-\frac{3}{2})1, z^{-1})u, v)&=z^{-3}(Y(G(-\frac{3}{2})1, z^{-1})u, v)\\
&=z^{-3}\sum_{n\in \Z}z^{n+2}(G(n+\frac{1}{2})u, v)\\
&=\sum_{n\in \Z}z^{n-1}(u,G(-n-\frac{1}{2})v)\\
&=(u, Y(G(-\frac{3}{2})1, z)v)\\
&=(u, Y(\phi(G(-\frac{3}{2})1), z)v),
\end{align*}
as desired.\qed
\subsection{Unitary vertex operator superalgebras associated to the Heisenberg superalgebras}\label{superhe}
In this subsection, we shall show that vertex operator superalgebras associated to the unitary highest weight representations for the Heisenberg superalgebras are unitary vertex operator superalgebras. First, we recall from \cite{KWa}, \cite{Li}, \cite{T} and \cite{X} some facts about highest weight modules for the Heisenberg superalgebras. Let $\h$ be a $n$-dimensional vector space with a nondegenerate symmetric bilinear form $\langle,\rangle$. Consider the infinite dimensional Lie superalgebra
\begin{align*}
\hat \h=\h\otimes \C[t, t^{-1}]t^{\frac{1}{2}}\oplus \C C,
\end{align*}
with $\Z_2$-gradation $\hat \h_{\bar 0}=\C C$, $\hat \h_{\bar 1}=\h\otimes \C[t, t^{-1}]t^{\frac{1}{2}}$ and the communication relations:
\begin{align*}
[u(m), v(n)]_+=\delta_{m+n, 0}\langle u, v\rangle C,~~[u(m), C]=0,
\end{align*}
for any $u, v\in \h$, $m, n\in \frac{1}{2}+\Z$, where $u(m)=u\otimes t^m$. Set $\hat \h_+=t^{\frac{1}{2}}\C[t]\otimes \h$ and $\hat \h_-=t^{-\frac{1}{2}}\C[t^{-1}]\otimes \h$, it is clear that $\hat \h_+$ and $\hat \h_-$ are subalgebras of $\hat \h$. For any complex number $c$, we then consider the Verma module
\begin{align*}
M_{\hat\h}(c, 0)=U(\hat \h)/J,
\end{align*}
where $J$ is the left ideal of $U(\hat \h)$ generated by $\hat \h_+$ and $C-c$. It is well-known that $M_{\hat\h}(c, 0)$ is a highest weight module for $\hat \h$ and that $M_{\hat\h}(c, 0)$ is an irreducible module for $\hat \h$ (cf. \cite{Li}). We then let $u^1, \cdots, u^n$ be an orthonormal basis of $\h$ with respect to $\langle,\rangle$ and set $\1=1$, $\w=\frac{1}{2}\sum_{i=1}^n u^i({-\frac{3}{2}})u^i({-\frac{1}{2}})\1$, it is known that $M_\h(c, 0)$ has a vertex operator superalgebra structure such that $\1$ and $\w$ are the vacuum vector and conformal vector, respectively (cf. \cite{KWa}, \cite{Li}, \cite{X}). Moreover, it was proved in \cite{Li} that the vertex operator superalgebra $M_{\hat\h}(c, 0)$ is isomorphic to the vertex operator superalgebra $M_{\hat\h}(1, 0)$ for any nonzero complex number $c$.
We now begin to construct a unitary structure on $M_{\hat\h}(1, 0)$. We first assume that $\h$ is one-dimensional. Let $\alpha$ be a vector in $\h$ such that $\langle \alpha, \alpha\rangle =1$, then $\h=\C \alpha$. By the PBW Theorem, we know that $M_{\hat\h}(1, 0)$ has a basis consisting of $\alpha(-n_1)\cdots \alpha(-n_k)\1$, $n_1>\cdots>n_k\geq \frac{1}{2}$. It is known that there exists a positive definite Hermitian form $(, )$ on $M_{\hat\h}(1, 0)$ such that $\alpha(-n_1)\cdots \alpha(-n_k)\1$, $n_1>\cdots>n_k\geq \frac{1}{2}$ is an orthonormal basis and that
\begin{align*}
( \alpha(m)u, v )=( u, \alpha(-m)v)
\end{align*}
for any $u, v\in M_{\hat\h}(1, 0)$ (cf. \cite{KR}).
We next construct an anti-linear involution of $M_{\hat\h}(1, 0)$. Let $T(\hat \h)$ be the tensor algebra of $\hat \h$. Define an anti-linear map $\Phi: T(\hat\h)\to T(\hat\h)$ by $\Phi(\alpha(n_1)\otimes \cdots\otimes \alpha(n_k))=(-1)^k\alpha(n_1)\otimes \cdots\otimes \alpha(n_k)$ and $\Phi(C)=C$. It is clear that $\Phi$ is an anti-linear involution of $T(\hat \h)$. Note that
\begin{align*}
&\Phi(x^1\otimes x^2-(-1)^{[x^1][x^2]}x^2\otimes x^1-[x^1, x^2])\\
&\ \ =\Phi(x^1)\otimes \Phi(x^2)-(-1)^{[x^1][x^2]}\Phi(x^2)\otimes\Phi( x^1)-\Phi([x^1, x^2])\\
&\ \ =\Phi(x^1)\otimes \Phi(x^2)-(-1)^{[x^1][x^2]}\Phi(x^2)\otimes\Phi( x^1)-[\Phi(x^1), \Phi(x^2)],
\end{align*}
for any $x^1, x^2\in \hat\h$. It follows that $\Phi$ induces an anti-linear involution $\phi$ of $M_{\hat\h}(1, 0)$.
We now in a position to provide a unitary vertex operator superalgebra structure on $M_{\hat\h}(1, 0)$.
\begin{theorem}
Let $\h$ be a $n$-dimensional vector space with a nondegenerate symmetric bilinear form $\langle, \rangle$. Then $M_{\hat\h}(1, 0)$ has a unitary vertex operator superalgebra structure.
\end{theorem}
\pf Let $u^1, \cdots, u^n$ be an orthonormal basis of $\h$ with respect to $\langle,\rangle$. Then we know that the vertex operator superalgebra $M_{\hat\h}(1, 0)$ is isomorphic to the vertex operator superalgebra $M_{\hat{\C u^1}}(1, 0)\otimes \cdots\otimes M_{\hat{\C u^n}}(1, 0)$.
By Proposition \ref{sub}, it is good enough to prove that $M_{\hat{\C u^i}}(1, 0)$ has a unitary vertex operator superalgebra structure for each $i$. We now prove that $(M_{\hat{\C \alpha}}(1, 0), \phi)$ is a unitary vertex operator superalgebra. Since $M_{\hat{\C \alpha}}(1, 0)$ is generated by $\alpha(-\frac{1}{2})\1$, by Proposition \ref{basic} it is good enough to prove that the invariant property
\begin{equation*} (Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}a, z^{-1})u,
v)=(u, Y(\phi(a), z)v)
\end{equation*}
holds for $a=\alpha(-\frac{1}{2})\1$ and any $ u, v\in M_{\hat{\C \alpha}}(1, 0).$ By a direct computation, \begin{align*}
&(Y(e^{zL(1)}z^{-2L(0)}(-1)^{L(0)+2L(0)^2}\alpha(-\frac{1}{2})\1, z^{-1})u, v)\\
&\ \ =-z^{-1}(Y(\alpha(-\frac{1}{2})\1, z^{-1})u, v)\\
&\ \ =-z^{-1}\sum_{m\in \Z}(\alpha(m+\frac{1}{2})u, v)z^{m+1}\\
&\ \ =-z^{-1}\sum_{m\in \Z}(u, \alpha(-m-\frac{1}{2})v)z^{m+1}\\
&\ \ =-\sum_{m\in \Z}(u, \alpha(-(m+1)+\frac{1}{2})v)z^{m}\\
&\ \ =(u, Y(-\alpha(-\frac{1}{2})\1, z)v)\\
&\ \ =(u, Y(\phi(\alpha(-\frac{1}{2})\1), z)v),
\end{align*}
as desired. \qed
\subsection{Unitary lattice vertex operator superalgebras}In this subsection we shall prove that the lattice vertex operator superalgebras associated to positive definite integral lattices are unitary vertex operator superalgebras.
First, we recall from \cite{FLM}, \cite{K1} some facts about the lattice vertex operator
superalgebras. Let $L$ be a positive definite integral lattice and $(,)$ be the associated positive definite bilinear form.
We then
consider the central extension $\hat{L}$ of $L$ by the cyclic
group $\langle \kappa\rangle$ of order $2$:
$$1\to \langle \kappa\rangle\to \hat{L}\to L\to 1,$$ such that the commutator map
$c(\alpha, \beta)=\kappa^{(\alpha, \beta)+(\alpha, \alpha)(\beta, \beta)}$ for any $\alpha, \beta\in L$. Let $e: L\to \hat{L}$
be a section such that $e_0 = 1$ and $\epsilon_0: L\times L\to \langle \kappa\rangle$ be the corresponding 2-cocycle. Then $\epsilon_0(\alpha,
\beta)\epsilon_0(\beta, \alpha)=\kappa^{(\alpha, \beta)+(\alpha, \alpha)(\beta, \beta)}$ and
$e_{\alpha}e_{\beta}=\epsilon_0(\alpha, \beta)e_{\alpha+\beta}$ for
$ \alpha, \beta \in L$.
Next, we consider the induced
$\hat{L}$-module:$$\C\{L\}=\C[\hat{L}]\otimes_{\langle \kappa\rangle}\C\cong
\C[L]\ \ (\text{linearly}),$$ where $\C[L]$ denotes the group algebra of $L$ and
$\kappa$ acts on $\C$ as multiplication by $-1$. Then $\C[L]$ becomes an
$\hat{L}$-module such that $e_{\alpha}\cdot
e^{\beta}=\epsilon(\alpha, \beta)e^{\alpha+\beta}$ and $\kappa\cdot
e^{\beta}=-e^{\beta}$, where $\epsilon(\alpha, \beta)$ is defined as $\nu\circ\epsilon_0$ and $\nu$ is the an isomorphism from $\langle \kappa\rangle$ to $\langle \pm 1\rangle$ such that $\nu(\kappa)=-1$. We also define an action $h(0)$ on
$\C[L]$ by $h(0)\cdot e^{\alpha}=(h, \alpha)e^{\alpha}$ for $h
\in \h, \alpha\in {L}$ and an action $z^h$ on $\C[L]$ by $z^{h}\cdot e^{\alpha}=z^{(h,
\alpha)}e^{\alpha}$.
We proceed to construct the vector space of the lattice vertex operator superalgebra. Set $\h=\C\otimes_{\Z}L$, and consider the Heisenberg algebra $\hat{\h}=\h\otimes \C[t, t^{-1}]\oplus \C K$ with the communication relations: For any $u, v\in \h$,
\begin{align*}
&[u(m), v(n)]=m\langle u, v\rangle\delta_{m+n, 0}K,\\
&[K, x]=0, \text{ for any } x\in \hat{\h},
\end{align*}
where $u(n)=u\otimes t^n$, $u\in \h$.
Let $M(1)$ be the
Heisenberg vertex operator algebra associated to $\hat\h$. The vector space of the lattice vertex operator superalgebra is defined to be
$$V_L=M(1)\otimes_{\C}\C\{L\}\cong M(1)\otimes_{\C}\C[L]\ \ (\text{linearly}).$$Then $\hat{L}$, $h(n)(n\neq 0)$, $h(0)$ and $z^{h(0)}$ act naturally on $V_L$ by acting on either $M(1)$ or $\C[L]$ as indicated above. It was proved in \cite{FLM}, \cite{K1} that
$V_L$ has a vertex operator superalgebra structure such that
\begin{align*}
&Y(h(-1)1, z)=h(z)=\sum_{n\in \Z}h(n)z^{-n-1}\ \ (h\in \h),\\
&Y(e^{\alpha}, z)=E^-(-\alpha, z)E^+(-\alpha, z)e_{\alpha}z^{\alpha},
\end{align*}
where
\begin{align*}
E^-(\alpha, z)=\exp\left(\sum_{n<0}\frac{\alpha(n)}{n}z^{-n}\right),\ \ \
E^+(\alpha, z)=\exp\left(\sum_{n>0}\frac{\alpha(n)}{n}z^{-n}\right).
\end{align*}
We now begin to construct a unitary vertex operator superalgebra structure on $V_L$. First, we define a positive definite Hermitian form on $V_{L}$. Recall from \cite{DLin} that there is
a unique positive definite Hermitian form $(,)$ on $M(1)$ such
that
$$({\bf 1}, {\bf 1})=1,\ \ \ \ (h(n)u, v)=(u,
h(-n)v)$$
for any $h\in \h, u, v\in M(1).$ Also, there is a positive definite Hermitian form
$(,):\C[L]\times\C[L]\to \C$
on
$\C[L]$ determined by $(e^{\alpha},e^{\beta})=0$ if
$\alpha\neq\beta$ and $(e^{\alpha},e^{\beta})=1$ if
$\alpha=\beta$ (see \cite{FLM}).
We then define a positive definite Hermitian form $(,)$ on $V_{L}$
as follows: For any $u, v\in M(1)$ and $e^{\alpha}, e^{\beta}\in \C[L]$, $$(u\otimes e^{\alpha}, v\otimes e^{\beta})=(u,
v)(e^{\alpha}, e^{\beta}).$$
\begin{lemma}\label{lattv}
Let $(,)$ be the positive definite Hermitian form on $V_{L}$ defined above.
Then we have: For any $\alpha \in L$ and $ w_1, w_2\in
V_{L}$,
\begin{align*}
(e_{\alpha}w_1, w_2)=(w_1,(-1)^{\frac{(\alpha, \alpha)+(\alpha,\alpha)^2}{2}} e_{-\alpha}w_2),\ \ \ \ (z^{\alpha}w_1, w_2)=(w_1, z^{\alpha}w_2).
\end{align*}
\end{lemma}
\pf The second identity is obvious. The first identity follows immediately from the facts that $(e_{\alpha} w_1, e_{\alpha}w_2)=(w_1, w_2)$ for any $w_1, w_2\in V_{L}$ and that $e_\alpha e_{-\alpha}=(-1)^{\frac{(\alpha, \alpha)+(\alpha, \alpha)^2}{2}}$ for any $\alpha\in L$ (see \cite{K1}).\qed
We next construct an anti-linear involution of $V_L$. Let $\phi: V_L\to V_L$ be an anti-linear map determined by:
\begin{align*}
\phi: V_L &\to V_L\\
\alpha_1(-n_1)\cdots\alpha_k(-n_k)\otimes e^{\alpha}&\mapsto (-1)^k\alpha_1(-n_1)\cdots\alpha_k(-n_k)\otimes
e^{-\alpha}.
\end{align*}
Note that we have $$\phi\alpha(n)\phi^{-1}=-\alpha(n),\ \ \ \ \phi Y(e^{\alpha}, z)\phi^{-1}=Y(e^{-\alpha}, z),$$
for any $\alpha\in L$, $n\in \Z.$
By a similar argument as that of Lemma 4.1 in \cite{DLin} we can show that $\phi$ is an anti-linear involution of
$V_L.$
\begin{theorem}\label{latt}
Let $L$ be a positive definite integral lattice and $\phi$ be the anti-linear involution of $V_L$ defined above. Then $(V_L, \phi)$ is a unitary vertex operator superalgebra.
\end{theorem}
\pf Since the lattice vertex operator superalgebra $V_L$ is generated by
$$\{\alpha(-1) | \alpha\in L \}\cup \{e^\alpha |\alpha\in
L\},$$
by Proposition \ref{basic} it is sufficient to prove
the following identities
\begin{equation}\label{1.1}
(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}\alpha(-1)\cdot1, z^{-1})w_1,
w_2)=(w_1, Y(\phi(\alpha(-1)\cdot 1), z)w_2),\end{equation}
\begin{equation}\label{1.2}
(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}e^{\alpha},
z^{-1})w_1, w_2)=(w_1, Y(\phi(e^{\alpha}), z)w_2),\end{equation}
hold for any $w_1,w_2 \in V_L$.
Assume that
$w_1= u\otimes
e^{\gamma_1},$ $w_2= v\otimes e^{\gamma_2}$ for some $u, v\in M(1)$ and $\gamma_1, \gamma_2\in L $. By the
definition of the Hermitian form, we have
\begin{align*}
&(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}\alpha(-1), z^{-1})w_1, w_2)\\
&=-z^{-2}\sum_{n\in \Z}(\alpha(n)w_1, w_2)z^{n+1}\\
&=\sum_{n\in \Z}-(w_1, \alpha(-n)w_2)z^{n-1}\\
&=(w_1, Y(\phi(\alpha(-1)), z)w_2),
\end{align*}
as desired.
To prove the identity (\ref{1.2}), note that we only need to consider the case that $\alpha+\gamma_1= \gamma_2$, by Lemma \ref{lattv},
\begin{align*}
&(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}e^{\alpha}, z^{-1})w_1, w_2)\\
&=(-1)^{\frac{(\alpha, \alpha)+(\alpha, \alpha)^2}{2}}z^{-(\alpha, \alpha)}(Y(e^{\alpha}, z^{-1}) u\otimes
e^{\gamma_1}, v\otimes e^{\gamma_2})\\
&=(-1)^{\frac{(\alpha, \alpha)+(\alpha, \alpha)^2}{2}}z^{-(\alpha, \alpha)}(E^-(-\alpha,
z^{-1})E^+(-\alpha, z^{-1})e_{\alpha}(z^{-1})^{\alpha}u\otimes
e^{\gamma_1}, v\otimes
e^{\gamma_2})\\
&=z^{-(\alpha, \alpha)}(u\otimes
e^{\gamma_1},E^-(\alpha, z)E^+(\alpha,
z)(z^{-1})^{\alpha}e_{-\alpha}v\otimes e^{\gamma_2})\\
&=z^{-(\alpha, \alpha)}(u\otimes
e^{\gamma_1},E^-(\alpha, z)E^+(\alpha,
z)e_{-\alpha}(z^{-1})^{\alpha}(z^{-1})^{(\alpha,-\alpha)}v\otimes e^{\gamma_2})\\
&=(u\otimes e^{\gamma_1},E^-(\alpha,
z)E^+(\alpha,
z)e_{-\alpha}(z^{-1})^{\alpha}v\otimes e^{\gamma_2})\\
&=(w_1, Y(e^{-\alpha}, z)w_2)\\
&=(w_1, Y(\phi(e^{\alpha}), z)w_2).
\end{align*}
Hence, $(V_L, \phi)$ is a unitary vertex operator superalgebra.\qed
\section{Structures of unitary vertex operator superalgebras}
\def\theequation{3.\arabic{equation}}
\setcounter{equation}{0}
\subsection{Unitary structures of the direct sums of unitary vertex operator superalgebras}
In this subsection, we shall show that there exists a unitary vertex operator superalgebra structure on the direct sum of unitary vertex operator superalgebras. We first recall from \cite{FHL}, \cite{LL} some facts about the direct sum of vertex operator superalgebras. Let $(V_1,\1_1, \w_1, Y_1), \cdots, (V_r,\1_r, \w_r, Y_r)$ be vertex operator superalgebras of the same central charge. Consider the vector space $V=V_1\oplus\cdots\oplus V_r$ and define the linear map $Y(\cdot, z)$ from $V$ to $(\End~ V)[[z, z^{-1}]]$ by
\begin{align*}
Y((v^{(1)},\cdots, v^{(r)}), z)=(Y_1(v^{(1)}, z), \cdots, Y_r(v^{(r)}, z))
\end{align*}
for $v^{(i)}\in V_i$, $1\leq i\leq r$. Set $\1=(\1_1, \cdots, \1_r)$ and $\w=(\w_1, \cdots, \w_r)$. It was proved in \cite{FHL}, \cite{LL} that $(V, \1, \w, Y)$ is a vertex operator superalgebra, the so-called direct sum vertex operator superalgebra of $(V_1,\1_1, \w_1, Y_1), \cdots, (V_r,\1_r, \w_r, Y_r)$.
\begin{theorem}\label{direct}
Let $(V_1,\1_1, \w_1, Y_1), \cdots, (V_r,\1_r, \w_r, Y_r)$ be vertex operator superalgebras of the same central charge and $V$ be the direct sum of $V_i$, $1\leq i\leq r$. Then $V$ has a unitary vertex operator superalgebra if each $V_i$, $1\leq i\leq r$, has a unitary vertex operator superalgebra structure.
\end{theorem}
\pf Assume that $(V_1,\1_1, \w_1, Y_1), \cdots, (V_r,\1_r, \w_r, Y_r)$ are unitary vertex operator superalgebras of the same central charge. Let $(,)_1, \cdots, (,)_r$ be the Hermitian forms on $V_1, \cdots, V_r$, respectively, and $\phi_1, \cdots, \phi_r$ be the anti-linear involutions of $V_1, \cdots, V_r$, respectively. We define a Hermitian form $(,)_V$ and an anti-linear involution $\phi$ of $V$ as follows:
\begin{align*}
&((u^{(1)},\cdots, u^{(r)}),(v^{(1)},\cdots, v^{(r)}))_V=\sum_{i=1}^r(u^{(i)}, v^{(i)})_{i},\\
&\phi((u^{(1)},\cdots, u^{(r)}))=(\phi_1(u^{(1)}),\cdots, \phi_r(u^{(r)})),
\end{align*}
for any $(u^{(1)},\cdots, u^{(r)}),(v^{(1)},\cdots, v^{(r)})\in V$. It is clear that $(,)_V$ is a positive definite Hermitian form on $V$ and that $\phi$ is an anti-linear involution of $V$.
Let $L(n), L_i(n), ~1\leq i\leq r,$ be the operators defined by \begin{align*}
&Y_V(\w,z)=\sum_{n\in\Z}L(n)z^{-n-2},\\
&Y_i(\w_i,z)=\sum_{n\in\Z}L_i(n)z^{-n-2},~1\leq i\leq r,
\end{align*}respectively. We then need to verify that the invariant property:
\begin{align*}
&(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}(u^{(1)},\cdots, u^{(r)}), z^{-1})(v^{(1)},\cdots, v^{(r)}), (w^{(1)},\cdots, w^{(r)}))_V\\
&\ \ =((v^{(1)},\cdots, v^{(r)}), Y(\phi((u^{(1)},\cdots, u^{(r)})), z)(w^{(1)},\cdots, w^{(r)}))_V,
\end{align*}
holds for any $(u^{(1)},\cdots, u^{(r)}),(v^{(1)},\cdots, v^{(r)}), (w^{(1)},\cdots, w^{(r)})\in V$. By the definitions,
\begin{align}\label{udirect}
&(Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}(u^{(1)},\cdots, u^{(r)}), z^{-1})(v^{(1)},\cdots, v^{(r)}), (w^{(1)},\cdots, w^{(r)}))_V \nonumber\\
&\ \ =(Y((e^{zL_1(1)}(-1)^{L_1(0)+2L_1(0)^2}z^{-2L_1(0)}u^{(1)},\cdots, e^{zL_r(1)}(-1)^{L_r(0)+2L_r(0)^2}z^{-2L_r(0)}u^{(r)}), z^{-1}) \nonumber \\
&\ \ \ \ \ \ \cdot(v^{(1)},\cdots, v^{(r)}), (w^{(1)},\cdots, w^{(r)}))_V \nonumber \\
&\ \ =\sum_{i=1}^r(Y_i(e^{zL_i(1)}(-1)^{L_i(0)+2L_i(0)^2}z^{-2L_i(0)}u^{(i)}, z^{-1})v^{(i)}, w^{(i)})_i\nonumber \\
&\ \ =\sum_{i=1}^r(v^{(i)}, Y_i(\phi_i(u^{(i)}), z)w^{(i)})_i \nonumber \\
&\ \ =((v^{(1)},\cdots, v^{(r)}), Y((\phi_1(u^{(1)}),\cdots, \phi_r(u^{(r)})), z)(w^{(1)},\cdots, w^{(r)}))_V \nonumber \\
&\ \ =((v^{(1)},\cdots, v^{(r)}), Y(\phi((u^{(1)},\cdots, u^{(r)})), z)(w^{(1)},\cdots, w^{(r)}))_V,
\end{align}
as desired.
\qed
\subsection{Structures of unitary vertex operator superalgebras}
Recall from \cite{DH} that a vertex operator superalgebra $V$ is called a {\em strong CFT type} vertex operator superalgebra if the following conditions hold:\\
(1) $V_0=\C\1$ and $V_n=0$ if $n< 0$;\\
(2) $L(1)V_1=0$.
In this subsection, we shall show that any unitary vertex operator superalgebra is a direct sum of some strong CFT type unitary simple vertex operator superalgebras with the same central charge.
\begin{theorem}\label{CFT}
Let $V$ be a unitary vertex operator superalgebra. Then there exist strong CFT type unitary simple vertex operator superalgebras $V^1, \cdots, V^k$ such that $V$ is isomorphic to the direct sum of the vertex operator superalgebras $V^1, \cdots, V^k$.
\end{theorem}
\pf First, note that $V$ is a unitary module for the Virasoro algebra, it follows immediately that $V_n=0$ for $n< 0$.
We next prove that if $V$ is a unitary vertex operator superalgebra, then $V$ is a direct sum of some vertex operator superalgebras with the same central charge. Since $V$ is a unitary vertex operator superalgebra, we know that $V$ viewed as a $V$-module is completely reducible, then there exist irreducible $V$-submodules $V^1,\cdots, V^k$ of $V$ such that $V\cong V^1\oplus \cdots\oplus V^k$.
Moreover, we have $(V^i, V^j)=0$ if $i\neq j$.
Consider the weight one subspaces $V^i_0$, $1\leq i\leq k$, of $V^i$, $1\leq i\leq k$, respectively. Since $V$ viewed as a $V$-module is generated by the vacuum vector, we know that each $V^i_0$, $1\leq i\leq k$, is not zero. We now prove that each $V^i_0$, $1\leq i\leq k$, is one-dimensional. Note that for any $u\in V^i_0$, we have
\begin{align*}
(L(-1)u, L(-1)u)=(u, L(1)L(-1)u)=(u, 2L(0)u)=0,
\end{align*}
implies $L(-1)u=0$. Hence, $0=Y(L(-1)u, z)=\frac{d}{d z}Y(u, z)$, forces that $Y(u, z)=u_{-1}$. This further implies that $Y(v, z)u=e^{L(-1)z}Y(u, -z)v=e^{L(-1)z}u_{-1}v,$ for any $v\in V$. As a result, $v_ju=0$ for any $v\in V$ and $j\in \Z_{\geq 0}$. Thus, $[v_n, u_{-1}]=\sum_{j\geq 0} (v_ju)_{n-1-j}=0$. Since $V$ has countable dimension and $V^i$ is an irreducible $V$-module, we have $u_{-1}$ acting on $V^i$ is equal to $\lambda_u \id_{V^i}$ for some complex number $\lambda_u$ by the Schur's Lemma (cf. \cite{DY}).
As a result, for any $u, v\in V^i_0$, we have $\lambda_v u=v_{-1}u=u_{-1}v=\lambda_u v$, implies that $V^i_0$ is one-dimensional. Let $e^i\in V^i$, $1\leq i\leq k$, be the vectors such that $e^i$ acting on $V^i$ is equal to $\id_{V^i}$. Then we know that $(V^i, e^i, Y)$, $1\leq i\leq k$, are vertex superalgebras.
For any $n\in \frac{1}{2}\Z$, let $V^i_n=V^i_n\cap V$, $1\leq i\leq k$. Then we know that $V^i=\oplus_{n\in \frac{1}{2}\Z} V^i_n$, $1\leq i\leq k$. We further let $\w^i=L(-2)e^i$, $1\leq i\leq k$. Consider the operator $L^i(n)=\w^i_{n+1}$, we have $L^i(n)=\w^i_{n+1}=(L(-2)e^i)_{n+1}=(\w_{-1}e^i)_{n+1}=\w_{n+1}e^i_{-1}=L(n)e^i_{-1}$. As a result, the operators $L^i(n)$, $n\in \Z$, satisfy the Virasoro algebra relation with central charge $c=c_V$, where $c_V$ denotes the central charge of $V$. Moreover, $L^i(0)|_{V^i_n}=L(0)e^i_{-1}|_{V^i_n}=L(0)|_{V^i_n}=n\id_{V^i}$. Note also that for any $u, v\in V^i$,
\begin{align*}
Y(L^i(-1)u, z)v=Y(L(-1)e^i_{-1}u, z)v=Y(L(-1)u, z)v=\frac{d}{dz}Y(u, z)v.
\end{align*}Therefore, $(V^i, e^i, \w^i, Y)$, $1\leq i\leq k$, are vertex operator superalgebras. Since for any $v\in V_1$ and $u\in V_0$, we have $(L(1)v, u)=(v, L(-1)u)=0$, implies $L(1)v=0$. In particular, for any $v\in V^i_1$, we have $L^i(1)v=L(1)v=0$. Hence, $(V^i, e^i, \w^i, Y)$, $1\leq i\leq k$, are strong CFT type vertex operator superalgebras.
Note that for any $1\leq i, j\leq k$ such that $i\neq j$, and any $v\in V^i, u\in V^j$, we have $e^{L(-1)z}Y(u, -z)v=Y(v, z)u\in V^i\cap V^j$. Hence, $Y(v, z)u=0$ for any $v\in V^i, u\in V^j$. This immediately implies that the vacuum vector $\1$ is equal to $e^1+\cdots+e^k$. Therefore, $\w=L(-2)\1=L(-2)(e^1+\cdots+e^k)=\w^1+\cdots+\w^k$. As a result, $V$ viewed as a vertex operator superalgebra is isomorphic to the direct sum of $V^i$, $1\leq i\leq k$.
We proceed to prove that $(V^i, e^i, \w^i, Y)$, $1\leq i\leq k$, are unitary vertex operator superalgebras. Let $\phi$ be the anti-linear involution associated to the unitary structure. We first prove that $\phi(V^i)=V^i$, $1\leq i\leq k$. Note that for any $u, v\in V$,
\begin{align*}
(\phi(u), v)&=(\phi(u)_{-1}\1, v)\\
&=\Res_z z^{-1}(Y(\phi(u), z)\1, v)\\
&=\Res_zz^{-1}(\1, Y(e^{zL(1)}(-1)^{L(0)+2L(0)^2}z^{-2L(0)}u, z^{-1})v).
\end{align*}
Thus, $(\phi(V^i), V^j)=0$ if $i\neq j$. Note also that $(V^i, V^j)=0$ if $i\neq j$, which forces that $\phi(V^i)=V^i$, $1\leq i\leq k$. As a result, we get anti-linear involutions $\phi_i$, $1\leq i\leq k$, of $V^i$, $1\leq i\leq k$, respectively. Now we can show that $(V^i, e^i, \w^i, Y)$, $1\leq i\leq k$, are unitary vertex operator superalgebras by the formula (\ref{udirect}).
Finally, we prove that each $V^i$, $1\leq i\leq k$, is a simple vertex operator superalgebra. Otherwise, assume that $V^i$ has a nontrivial ideal $I$. Note that for any $u\in I, v\in V^i$,
\begin{align*}
(u, v)=\Res_z z^{-1}(u,Y(v, z)e^i )=\Res_z z^{-1} (Y(e^{zL^i(1)}(-1)^{L^i(0)+2L^i(0)^2}z^{-2L^i(0)}\phi (v), z^{-1})u, e^i).
\end{align*}Since $(,)$ is positive definite, for any $u\in I$, there exists some $v\in V^i$ such that $(u, v)\neq 0$. This implies that $\Res_z z^{-1} (Y(e^{zL^i(1)}(-1)^{L^i(0)+2L^i(0)^2}z^{-2L^i(0)}\phi (v), z^{-1})u, e^i)\neq 0$. Set $w=\Res_z z^{-1} Y(e^{zL^i(1)}(-1)^{L^i(0)+2L^i(0)^2}z^{-2L^i(0)}\phi (v), z^{-1})u$, it follows from the discussion above that there exist some homogenous vectors $w_1, \cdots, w_n$ with different nonzero conformal weights such that $w=\lambda e^i+w_1+\cdots +w_n$ for some nonzero number $\lambda$. We now prove that $e^i\in I$, which is a contradiction. Note that we have
\begin{align*}
& w=\lambda e^i+w_1+\cdots +w_n,\\
&L^i(0)w=(\wt w_1) w_1+\cdots +(\wt w_n)w_n,\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\\
&L^i(0)^nw=(\wt w_1)^n w_1+\cdots +(\wt w_n)^nw_n,
\end{align*}
and $w, L^i(0)w, \cdots, L^i(0)^nw\in I$, it follows immediately from the Vandermende determinant formula that $e^i\in I$.
Hence, each $V^i$, $1\leq i\leq k$, is a simple vertex operator superalgebra. The proof is complete.
\qed
\begin{corollary}
Let $(V, \phi)$ be a unitary simple vertex operator superalgebra. Then the Hermitian form $(,)$ is uniquely determined by the value $(\1, \1)$.
\end{corollary}
\pf Since the unitary vertex operator superalgebra $V$ is simple, it follows from the argument in the proof of Theorem \ref{CFT} that $V$ is of CFT type.
As a result, the Hermitian form is uniquely determined by the value $(\1, \1)$.
\qed
In the following, for any unitary simple vertex operator superalgebra $V$, we will
normalize the Hermitian form $(,)$ on $V$ such that $(\1, \1)=1$.
\section{Classification of unitary vertex operator superalgebras generated by the subspaces with small conformal weights}
\def\theequation{4.\arabic{equation}}
\setcounter{equation}{0}
\subsection{Classification of unitary vertex operator superalgebras generated by the subspaces with conformal weight $\frac{1}{2}$} In the last section, we have shown that the conformal weights of any unitary vertex operator superalgebra are nonnegative. Then it is natural to consider first the classification of unitary vertex operator superalgebras generated by the subspaces with conformal weight $\frac{1}{2}$. In subsection \ref{superhe}, we have proved that the vertex operator superalgebras associated to the Heisenberg superalgebras have unitary vertex operator superalgebra structures. Moreover, these vertex operator superalgebras are generated by the subspaces with conformal weight $\frac{1}{2}$. In this subsection, we shall show that any unitary vertex operator superalgebras generated by the subspaces with conformal weight $\frac{1}{2}$ is a direct sum of vertex operator superalgebras associated to some Heisenberg superalgebras. First, we recall from \cite{X} a characterization of the vertex operator superalgebras associated to the Heisenberg superalgebras.
\begin{theorem}\label{char}
Let $V$ be a simple vertex operator superalgebra generated by the subspace $V_{\frac{1}{2}}$.
Then $V_{\frac{1}{2}}$ has a nondegenerate symmetric bilinear form $\langle, \rangle$ such that $V$ is isomorphic to the vertex operator superalgebra associated to the Heisenberg superalgebra $\hat V_{\frac{1}{2}}$.
\end{theorem}
As a consequence, we have the following classification of unitary vertex operator superalgebras generated by the subspaces with conformal weight $\frac{1}{2}$.
\begin{theorem}
Let $V$ be a unitary vertex operator superalgebra generated by the subspace $V_{\frac{1}{2}}$.
Then $V$ is isomorphic to a direct sum of vertex operator superalgebras associated to some Heisenberg superalgebras.
\end{theorem}
\pf Since $V$ is a unitary vertex operator superalgebra, by Theorem \ref{CFT}, we know that there exists unitary simple vertex operator superalgebras $V^1, \cdots, V^k$ such that $V$ is the direct sum of $V^i$, $1\leq i\leq k$. By assumption, $V$ is generated by the subspace $V_{\frac{1}{2}}$ with conformal weight $\frac{1}{2}$, it follows that each $V^i$, $1\leq i\leq k$, is generated by the subspace with conformal weight $\frac{1}{2}$. Therefore, by Theorem \ref{char}, we know that each $V^i$, $1\leq i\leq k$, is isomorphic to vertex operator superalgebra associated to some Heisenberg superalgebra. The proof is complete.\qed
\subsection{The structures of the weight one subspaces of unitary vertex operator algebras} In the last subsection, we have classified the unitary vertex operator superalgebras generated by the subspaces with conformal weight $\frac{1}{2}$. Our next goal is to classify the unitary vertex operator algebras generated by the subspaces with conformal weight $1$. In this subsection, we shall study first the structures of the weight one subspaces of unitary vertex operator algebras. Let $V$ be a unitary vertex operator algebra such that $V_1\neq 0$. Note that for $u, v\in V_1$, we have $u_1v\in V_0$, it follows from Theorem \ref{CFT} that $L(-1)u_1v=0$. As a result, we know that $V_1$ has a Lie algebra structure with the Lie bracket defined by $[u, v]=u_0v$ for any $u, v\in V_1$ (cf. \cite{B}). We next prove that $V_1$ is a reductive Lie algebra, that is, $V_1$ viewed as a $V_1$-module is completely reducible (cf. \cite{H}).
\begin{theorem}\label{reductive}
Let $V$ be a unitary vertex operator algebra. Then $(V_1, [,])$ is a reductive Lie algebra.
\end{theorem}
\pf Let $(,)$ be the positive definite Hermitian form on $V$, then it follows from Theorem \ref{CFT} that $(L(1)v, u)=(v, L(-1)u)=0$ for any $v\in V_1$ and $u\in V_0$, which implies $L(1)v=0$. Thus, for any $u,v, w\in V_1$,
\begin{align}\label{invariant}
\nonumber([u, v], w)&=(u_0v, w)\\ \nonumber
&=\Res_z(Y(u, z)v, w)\\ \nonumber
&=\Res_z(v, Y(e^{zL(1)}(-z^{-2})^{L(0)}\phi(u), z^{-1})w)\\ \nonumber
&=\Res_z(v, Y((-z^{-2})\phi(u), z^{-1})w)\\
&=-(v, [\phi(u), w]).
\end{align}
Therefore, for any submodule $I$ of $V_1$, the orthogonal complement $I^{\perp}$ of $I$, which is defined by $I^{\perp}=\{u\in V_1|(u, v)=0\text{ for any }v\in I\}$, is also a submodule of $V_1$. This implies that $V_1$ viewed as a $V_1$-module is completely reducible. This completes the proof.\qed
\begin{remark}
The similar result in Theorem \ref{reductive} has been established in \cite{DM1} for the vertex operator algebra $V$ satisfying the following conditions:\\
(1) $V$ is simple, rational and $C_2$-cofinite;\\
(2) $V$ is of strong CFT type.
\end{remark}
\subsection{A characterization of the unitary vertex operator algebra associated to the affine Lie algebra}
We now assume that $V$ is a unitary simple vertex operator algebra such that $V_1$ contains a simple Lie algebra $\g$. In the following, we shall study the structure of the vertex subalgebra $U$ generated by $\g$.
By assumption, we know that $V$ is a self-dual, strong CFT, simple vertex operator algebra. Then there is a unique nondegenerate invariant bilinear form $\langle ,\rangle$ on $V$ such that $\langle\1,\1\rangle=-1$ (see \cite{Li1}). Moreover, the nondegenerate invariant bilinear form $\langle ,\rangle$ is symmetric (see \cite{FHL}). Note that we have defined in Proposition \ref{self-dual} a nondegenerate invariant bilinear form $\langle ,\rangle$ on $V$ such that $\langle u,v\rangle=-(u, \phi(v))$ for any $u,v\in V$ and that $\langle \1,\1\rangle=-(\1, \1)=-1$. Therefore, the nondegenerate invariant bilinear form $\langle ,\rangle$ defined by $\langle u,v\rangle=-(u, \phi(v))$ is symmetric.
By restricting the bilinear form $\langle ,\rangle$ to $V_1$, we then get a nondegenerate symmetric bilinear on $V_1$. We proceed to prove $\langle,\rangle$ is an invariant form, that is, $\langle [u, v], w\rangle=\langle u, [v, w]\rangle$ for any $u, v, w\in V_1$. By the formula (\ref{invariant}), we have $\langle [u, v], w\rangle=-([u, v], \phi(w))=-(u, [\phi(v), \phi(w)])=-(u, \phi([v, w]))=\langle u, [v, w]\rangle$. Hence, $\langle,\rangle$ is a nondegenerate invariant symmetric bilinear form on $V_1$. Since $V_1$ is reductive, the restriction of $\langle,\rangle$ on $\g$ is still nondegenerate.
This implies that there exists a nonzero complex number $c$ such that $\langle,\rangle|_{\g\times \g}=c\kappa (,)$, where $\kappa(,)$ denotes the normalized Killing form of $\g$ such that $\kappa(\theta, \theta)=2$ for the highest root $\theta$ of $\g$.
Note also that for any $u,v\in V_1$, \begin{align*}
(u, v)&=(u, v_{-1}\1)=\Res_zz^{-1}(u, Y(v, z)\1)\\
&=\Res_zz^{-1}(Y(e^{zL(1)}(-z^{-2})^{L(0)}\phi(v), z^{-1})u, \1)\\
&=\Res_z-z^{-3}(Y(\phi(v), z^{-1})u, \1)\\
&=-(\phi(v)_1u, \1).
\end{align*} It follows that $u_1v=-(v, \phi(u))\1=\langle v, u\rangle\1=\langle u, v\rangle\1$. Thus, we have
\begin{align*}
[u_m, v_n]&=\sum_{i\geq 0}{m\choose i}(u_iv)_{m+n-i}\\
&=(u_0v)_{m+n}+m(u_1v)_{m+n-1}\\
&=[u, v]_{m+n}+m\langle u, v\rangle\delta_{m+n, 0}\id.
\end{align*}
Since $U$ is generated by $\g$, we have in fact proved that $U$ is a highest weight module for the affine Lie algebra $\hat{\g}$ with highest weight vector $\1$ and level $c$. In particular, $U$ viewed as a $\hat{\g}$-module is isomorphic to some quotient module of the Verma module $M_{\hat{\g}}(c, 0)$. Furthermore, we have the following
\begin{theorem}\label{affine}
Let $U$ be the vertex subalgebra defined above. Then $U$ viewed as a $\hat{\g}$-module is irreducible, i.e., $U$ isomorphic to $L_{\hat{\g}}(c, 0)$ for some complex number $c$.
\end{theorem}
\pf By assumption, $V$ has a positive definite Hermitian form such that
$$(Y(e^{zL(1)}(-z^{-2})^{L(0)}a, z^{-1})u, v)=(u,
Y(\phi(a), z)v),$$holds for any $a, u, v\in V$. In particular, we have for any $a\in \g$, $u, v\in U$ and $n\in \Z$,
$$(a_nu, v)=-(u,
\phi(a)_{-n}v).$$ Thus we can prove that the radical of $(,)$ is a $\hat{\g}$-module and that the radical of $(,)$ contains the maximal proper submodule which has zero intersection with highest weight subspace. Since the radical of $(,)$ is equal to zero, we get that the maximal proper submodule which has zero intersection with highest weight subspace is zero. Therefore, $U$ is irreducible.\qed
\vskip0.25cm
We proceed to prove that $U$ viewed as a $\hat{\g}$-module is an integrable module, that is, $c$ is a positive integer (cf. \cite{K}).
\begin{theorem}\label{simplelie}
Let $U$ be the vertex subalgebra defined above. Then $U$ viewed as a $\hat{\g}$-module is integrable.
\end{theorem}
\pf Let $\phi$ be the anti-linear involution of $V$ associated to the unitary structure. We first prove that $\phi(\g)=\g$. Note that we have proved in Theorem \ref{reductive} that $V_1$ is reductive and then $V_1\cong \g_1\oplus \cdots\oplus \g_k\oplus \h$, where $\g_i$, $1\leq i\leq k$, are simple Lie subalgebras of $V_1$ and $\h$ is an abelian Lie subalgebra of $V_1$. Moreover, we have $\g_i$, $1\leq i\leq k$, and $\h$ are mutually orthogonal with respect to $(,)$. On the other hand, we have proved that the bilinear form $\langle ,\rangle$ defined by $\langle u,v\rangle=-(u, \phi(v))$ is a nondegenerate invariant symmetric bilinear form. Since $\g_i$, $1\leq i\leq k$, are simple Lie algebras, it follows that $\g_i$, $1\leq i\leq k$, and $\h$ are mutually orthogonal with respect to $\langle ,\rangle$. As a result, $\phi(\g_i)=\g_i$, $1\leq i\leq k$, and $\phi(\h)=\h$.
Note also that $\phi([u, v])=\phi(u_0v)=\phi(u)_0\phi(v)=[\phi(u), \phi(v)]$ for any $u, v\in V_1$. Thus, $\phi$ induces an anti-linear involution of $\g$, which is still denoted by $\phi$. Hence, $\phi$ is a real structure of $\g$ (cf. \cite{O}). Consider the real form $\g^\phi=\{v\in \g|\phi(v)=v\}$ of $\g$ with respect to $\phi$. We then prove that $\g^\phi$ is a compact Lie algebra, that is, there exists a positive definite symmetric bilinear form on $\g^\phi$ (cf. \cite{O}). Note that we have proved that the bilinear form $\langle, \rangle$ defined by $\langle u, v\rangle=(u, \phi(v))$, for any $u, v\in \g$, is a symmetric invariant bilinear form,
we then get a positive definite symmetric bilinear form on $\g^\phi$. Therefore, $\g^\phi$ is a compact Lie algebra and $\phi$ is a compact real structure of $\g$. Since the compact real structure of a simple Lie algebra is unique up to conjugacy by inner automorphism, we can choose a Cartan subalgebra $\h_1$ of $\g$ such that $\phi$ is determined by $\phi(e_i)=-f_i, \phi(f_i)=-e_i, \phi(h)=-h, h\in \h_1$, where $e_i, f_i$ denote the Chevalley generators of $\g$ with respect to $\h_1$ (cf. \cite{O}).
We then consider the compact involution $\hat \phi$ of $\hat{\g}$, which is the anti-linear map $\hat \phi: \hat{\g}\to \hat{\g}$ determined by $\hat\phi(u\otimes t^n)=\phi(u)\otimes t^{-n}, u\in \g,$ and $\hat \phi(C)=-C$ (cf. \cite{K}).
Note that with respect to $\hat \phi$ and the positive definite Hermitian form $(,)$, the $\hat{\g}$-module $U$ is a unitarizable module, i.e.,
$(x\cdot u, v)=-(u, \hat{\phi}(x)\cdot v)$ for any $x\in \hat{\g}$, $u, v\in U$ (cf. \cite{K}). As a result, $U$ viewed as $\hat{\g}$-module is integrable. The proof is complete. \qed
We now in a position to give a characterization of the unitary vertex operator algebra associated to the affine Lie algebra.
\begin{theorem}\label{intvoa}
Let $V$ be a unitary vertex operator algebra such that $V_1$ is a simple Lie algebra and that $V$ is generated by $V_1$. Then $V$ is isomorphic to the affine vertex operator algebra $L_{\hat{V_1}}(c, 0)$.
\end{theorem}
\pf We first prove that $V$ is a simple vertex operator algebra. Since $V$ is a unitary vertex operator algebra, by Theorem \ref{CFT}, we know that there exist unitary simple vertex operator algebras $V^1, \cdots, V^k$ such that $V$ is the direct sum of $V^i$, $1\leq i\leq k$. However, by assumptions, $V$ is generated by the subspace $V_{1}$ and $V_1$ is a simple Lie algebra, it follows that $V$ is isomorphic to some $V^j$, $1\leq j\leq k$. Therefore, by Theorem \ref{CFT}, we know that $V$ is a simple vertex operator algebra.
We now only need to prove that the conformal vector $\w$ of $V$ is equal to that of $L_{\hat{V_1}}(c, 0)$. Consider the component operators $L(n)=\w_{n+1}$, $n\in \Z$, we have: For any $x\in V_1$, $n, m\in \Z$,
\begin{align*}
[L(n), x_m]&=[\w_{n+1}, x_m]\\
&=\sum_{i\geq 0} {n+1\choose i}(\w_ix)_{m+n+1-i}\\
&=(\w_0x)_{m+n+1}+(n+1)(\w_1x)_{m+n}\\
&=-mx_{m+n}.
\end{align*}
On the other hand, let $x_1,\cdots, x_d$ be an orthonormal basis of $V_1$ with respect to $\kappa(, )$, since $c$ is a positive integer, we can define $\w'=\frac{1}{2(c+h)}\sum_{i=1}^dx_i(-1)^2\1$, where $h$ denotes the dual Coxeter number of $V_1$. It is known that the component operators $L'(n)=\w'_{n+1}$, $n\in \Z$, satisfy $[L'(n), x_m]=-mx_{m+n}$ for any $x\in V_1$, $n, m\in \Z$ (see \cite{LL}). As a consequence, we have $\w=\w'$ by Theorem 6.3 in \cite{Lian}. The proof is complete.\qed
\begin{remark}
The structure of the vertex operator algebra $V$ satisfying the following conditions:\\
(1) $V$ is simple and $C_2$-cofinite,\\
(2) $V$ is of strong CFT type,\\
(3) $V_1$ contains a simple Lie subalgebra,\\
has been studied in \cite{DM2} and
the similar results in Theorems \ref{affine}, \ref{simplelie}, \ref{intvoa} have been established for the vertex operator algebra $V$ satisfying these conditions.
\end{remark}
\subsection{Classification of unitary vertex operator algebras generated by the weight one subspaces} In the last subsection, we have studied the structure of the unitary simple vertex operator algebra $V$ under the condition that $V_1$ contains a simple Lie subalgebra. To classify unitary vertex operator algebras generated by the weight one subspaces, it remains to study the structure of the unitary simple vertex operator algebra $V$ under the condition that $V_1$ contains an abelian Lie algebra. In this subsection, we shall study this case and then classify unitary vertex operator algebras generated by the weight one subspaces.
Let $V$ be a unitary simple vertex operator algebra such that $V_1\cong \g_1\oplus \cdots\oplus \g_k\oplus \h$, where $\g_i$, $1\leq i\leq k$, are simple Lie subalgebras of $V_1$ and $\h$ is an abelian Lie subalgebra of $V_1$. Our goal is to study the structure of the vertex subalgebra $U$ generated by $\h$. Note that we have proved in Theorem \ref{simplelie} that there is a nondegenerate symmetric bilinear form $\langle , \rangle$ on $\h$ defined by $\langle u, v\rangle=-(u, \phi (v))$ for any $u, v\in \h$. Then we can consider the Heisenberg algebra $\hat{\h}=\h\otimes \C[t, t^{-1}]\oplus \C K$ with the communication relations: For any $u, v\in \h$,
\begin{align*}
&[u(m), v(n)]=m\langle u, v\rangle\delta_{m+n, 0}K,\\
&[K, x]=0, \text{ for any } x\in \hat{\h}.
\end{align*}
Since $U$ is generated by $\h$, we can prove that $U$ is a highest weight module for the Heisenberg Lie algebra $\hat{\h}$ with highest weight vector $\1$ and level $1$. In particular, $U$ viewed as an $\hat{\h}$ module is isomorphic to the irreducible highest weight $\hat{\h}$-module $V_{\hat{V_1}}(1, 0)$ (cf. \cite{FLM}, \cite{LL}). Moreover, we have
\begin{theorem}\label{abelian}
Let $V$ be a unitary simple vertex operator algebra such that $V$ is generated by $V_1$ and that $V_1$ is an abelian algebra. Then the vertex operator algebra $V$ is isomorphic to $V_{\hat{V_1}}(1, 0)$, where we use $V_{\hat{\h}}(1, 0)$ to denotes the Heisenberg vertex operator algebra associated to a Heisenberg algebra $\hat{\h}$.
\end{theorem}
\pf It is good enough to prove that the conformal vector $\w$ of $V$ coincides with that of $V_{\hat\h}(1,0)$. First, note that we have proved in Theorem \ref{intvoa} that $[L(n), x_m]=-mx_{m+n}$ holds for any $x\in V_1$, $n, m\in \Z$, where $L(n)=\w_{n+1}$. On the other hand, let $\alpha_1, \cdots, \alpha_d$ be an orthonormal basis of $V_1$ with respect to $\langle, \rangle$ and consider $\w'=\frac{1}{2}\sum_{i=1}^d\alpha_i^2(-1)\1$. It is known that $[L'(n), x_m]=-mx_{m+n}$ holds for any $x\in V_1$, $n, m\in \Z$, where $L'(n)=\w'_{n+1}$ (cf. \cite{LL}). As a consequence, we have $\w=\w'$ by Theorem 6.3 in \cite{Lian}. The proof is complete.\qed
\vskip.25cm
We now in a position to classify the unitary vertex operator algebras generated by the weight one subspaces. Let $V$ be a unitary vertex operator algebra such that $V$ is generated by $V_1$. By Theorem \ref{CFT}, we know that $V$ is a direct sum of unitary simple vertex operator algebras $V^1, \cdots, V^k$. Since $V$ is generated by $V_1$, we know that the weight one subspaces $V^i_1$, $1\leq i\leq k$, of $V^i$, $1\leq i\leq k$, respectively, are nonzero and that each $V^i$ is generated by $V^i_1$. As a consequence, we know that $V^i_1$, $1\leq i\leq k$, are reductive Lie algebras. Assume that $V^i_1$, $1\leq i\leq k$, are isomorphic to $\g^i_1\oplus\cdots\oplus \g^i_{l_i}\oplus \h^i$, $1\leq i\leq k$, respectively, where $\g^i_j$ , $1\leq i\leq k$, $1\leq j\leq l_i$, are simple Lie algebras and $\h^i$, $1\leq i\leq k$, are abelian Lie algebras. Then we have
\begin{theorem}\label{all}
The unitary vertex operator algebra $V$ is isomorphic to
\begin{align*}
L_{\hat{\g^1_1}}(c^1_1, 0)\otimes\cdots\otimes L_{\hat{\g^1_{l_1}}}(c^1_{l_1}, 0)\otimes V_{\hat\h^1}(1, 0)\oplus \cdots\oplus L_{\hat{\g^k_1}}(c^k_1, 0)\otimes\cdots\otimes L_{\hat{\g^k_{l_k}}}(c^k_{l_k}, 0)\otimes V_{\hat\h^k}(1, 0)
\end{align*} for some positive integers $c^i_j$, $1\leq i\leq k$, $1\leq j\leq l_i$.
\end{theorem}
\pf We now only need to show that the conformal vectors of $V^i$, $1\leq i\leq k$, coincide with those of $L_{\hat{\g^i_1}}(c^i_1, 0)\otimes\cdots\otimes L_{\hat{\g^i_{l_1}}}(c^i_{l_i}, 0)\otimes V_{\hat\h^i}(1, 0)$, $1\leq i\leq k$, respectively. This follows immediately by the same arguments in the proofs of Theorems \ref{intvoa}, \ref{abelian}. The proof is complete.\qed
\begin{remark}
The similar results in Theorems \ref{abelian}, \ref{all} have been established in \cite{DM2} for the vertex operator algebra $V$ satisfying the following conditions:\\
(1) $V$ is simple and $C_2$-cofinite;\\
(2) $V$ is of strong CFT type.
\end{remark}
\subsection{A characterization of the lattice vertex operator algebra}In this subsection, we use the results obtained in the last subsection to give a characterization of the lattice vertex operator algebra. We will use the notations in the last subsection.
\begin{theorem}
Let $V$ be a unitary simple vertex operator algebra such that $V_1$ is an abelian Lie algebra and $\dim V_1=c$, where $c$ denotes the central charge of $V$. Then there exist an abelian subalgebra $\h_1$ of $V_1$ and a positive definite even lattice $L$ in $V_1$ such that $V$ is isomorphic to $V_{\hat\h_1}(1,0)\otimes V_{L}$.
\end{theorem}
\pf By the discussion above, we know that $V$ is a module for the Heisenberg algebra $\hat V_1$. Thus, by the Stone-Von Neumann Theorem we know that $V$ viewed as a $\hat V_1$-module has the decomposition
$V=V_{\hat V_1}(1, 0)\otimes \Omega_V,$
where $V_{\hat V_1}(1, 0)$ is the highest weight $\hat V_1$-module with highest weight $(1, 0)$ and $\Omega_V=\{v\in V|x(m)v=0, \text{ for any } x\in V_1,\ m>0\}$.
We next show that $x(0)$ acts semisimplely on $V$ for any $x\in V_1$. Note that the action on $V_n$ is closed for any $n\in \Z$, so it is good enough to prove that $x(0)$ acts semisimplely on $V_n$ for any $x\in V_1$. Let $(,)$ be the positive definite Hermitian form on $V$ and $\phi$ be the anti-linear involution of $V$ associated to the unitary structure. Then we have $(x(0)u, v)=(u, -\phi(x)(0)v)$ for any $u, v\in V_n$. Since $\phi$ is an anti-linear involution, by viewing $V_1$ as a vector space $V_{1,\R}$ over the field $\R$ of real numbers, we can view $\phi$ as a linear involution of $V_{1,\R}$. Thus, $V_{1,\R}$ has a decomposition $V_{1,\R}=V_{1,\R}^+\oplus V_{1,\R}^-$, where $V_{1,\R}^+=\{v\in V_{1,\R}| \phi(v)=v\}$ and $V_{1,\R}^-=\{v\in V_{1,\R}| \phi(v)=-v\}$.
We first assume that $x$ is an element in $V_{1,\R}^-$, then we have $(x(0)u, v)=(u, x(0)v)$ for any $u, v\in V_n$. It follows that $x(0)$ acts semisimplely on $V_n$. We next assume that $x\in V_{1,\R}^+$, then we have $\phi(ix)=-i\phi(x)=-ix$, that is, $ix\in V_{1,\R}^-$. Therefore, $ix(0)$ acts semisimplely on $V_n$. Thus, for any $x\in V_1$, we have $x(0)$ acts semisimplely on $V_n$.
As a result, for any $x\in V_1$, $x(0)$ acts semisimplely on $\Omega_V$. For any $\lambda\in V_1$, let $\Omega_V(\lambda)=\{v\in \Omega_V|x(0)v=\langle x, \lambda\rangle v\text{ for any } x\in V_1\}$, then we have
$\Omega_V=\oplus_{\lambda\in V_1}\Omega_V(\lambda).$
Let $L=\{\lambda\in V_1|\Omega_V(\lambda)\neq 0\}$. Since $V$ is a unitary simple vertex operator algebra, by a similar argument in the proof of Theorem 2 in \cite{DM1} we can prove that $L$ is an additive group. We proceed to prove that $L$ is a positive definite even lattice. Let $\alpha_1, \cdots, \alpha_c$ be an orthonormal basis of $V_1$ with respect to $\langle, \rangle$. Then we know that $\w'=\frac{1}{2}\sum_{i=1}^c\alpha_i^2(-1)\1$ is the conformal vector of the Heisenberg vertex operator algebra $V_{\hat V_1}(1, 0)$. Set $\w''=\w-\w'$, where $\w$ is the conformal vector of $V$. We now prove that $\w''$ is a Virasoro element. Set $L(n)=\w_{n+1}$ and $L'(n)=\w'_{n+1}$, $n\in \Z$. Then we have $[L(1), x_{-1}]=x_0$ for any $x\in V_1$. Therefore, $L(1)\w'=0$. It follows that $\w''$ is a Virasoro element with central charge $0$ (see \cite{FZ}). Set
$L''(n)=\w''_{n+1}$, then we have
\begin{align*}
(\w'', \w'')&=(\w-\w',\w-\w')\\
& =((L(-2)-L'(-2))\1, (L(-2)-L'(-2))\1)\\
& =(\1, (L(2)-L'(2))(L(-2)-L'(-2))\1)\\
& =(\1, L''(2)L''(-2)\1)=0.
\end{align*}
Since the Hermitian form is positive definite, $\w''=0$. As a consequence, $\w'$ is equal to the conformal vector of $V$. This forces that $\frac{\langle \lambda, \lambda\rangle}{2}\in \Z^+$ for any $\lambda\in L$ (see \cite{DM1}). Hence, $L$ is a positive definite even lattice.
We now set $\h_2=\C\otimes_{\Z}L$ and $\h_1=\{v\in V_1|\langle x, v\rangle=0 \text{ for any } x\in \h_2\}$. Then $V_1=\h_1\oplus \h_2$ and $\langle, \rangle|_{\h_1\times\h_1}$, $\langle, \rangle|_{\h_2\times\h_2}$ are nondegenerate. Thus we can consider the Heisenberg algebras $\hat \h_1$, $\hat \h_2$. Moreover, we have $V_{\hat V_1}(1, 0)=V_{\hat \h_1}(1, 0)\otimes V_{\hat \h_2}(1, 0)$. Thus, $V$ viewed as a module for $V_{\hat \h_1}(1, 0)\otimes V_{\hat \h_2}(1, 0)$ has the following decomposition:
\begin{align*}
V=\left(V_{\hat \h_1}(1, 0)\otimes V_{\hat \h_2}(1, 0)\right)\otimes \Omega_V=V_{\hat \h_1}(1, 0)\otimes\left(\oplus_{\lambda\in \h_2} V_{\hat \h_2}(1, 0)\otimes\Omega_V(\lambda)\right).
\end{align*}
It follows immediately from Proposition 5.3 of \cite{DM1} that the vertex operator algebra $V$ is isomorphic to $V_{\hat \h_1}(1, 0)\otimes V_L$. This completes the proof.\qed
As an immediate consequence, we have the following
\begin{corollary}\label{clattice}
Let $V$ be a unitary simple vertex operator algebra such that $V_1$ is an abelian Lie algebra and $\dim V_1=c$, where $c$ denotes the central charge of $V$. If we assume in addition that $V$ is rational or $C_2$-cofinite, then there exists a positive definite even lattice $L$ in $V_1$ such that $V$ is isomorphic to $V_L$.
\end{corollary}
\begin{remark}
The similar result in Corollary \ref{clattice} has been established in \cite{DM1}, \cite{M} for the vertex operator algebra $V$ satisfying the following conditions:\\
(1) $V$ is simple, rational and $C_2$-cofinite;\\
(2) $V$ is of strong CFT type.\\
There is also a characterization of vertex algebras associated to even lattices in \cite{LX}.
\end{remark}
{\bf Acknowledgement}. The authors wish to thank Professor Ching Hung Lam for useful comments and
valuable suggestions.
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TITLE: More than 5 but less than 50 rational solutions
QUESTION [0 upvotes]: Can there be system of irreducible polynomial equations over $\mathbb{Q}$
which has more than 5 but less than 50 $\mathbb{Q}$-solutions?
I somehow feel that I have to use the fact that polynomials in more than one variables over an infinite field have infinity many solutions. I'm thinking in terms of algebraic sets but I really can't come up with examples (or counterexamples).
REPLY [3 votes]: For any $a \in \mathbb{Q}^n$, define $P_a(x_1,\ldots,x_n)=(x_1-a_1)^2+\ldots+(x_n-a_n)^2$. Given a finite subset $S \subset \mathbb{Q}^n$, let $P_S=\prod_{a \in S}{P_a}$. Then the zeroes of $P_S$ are exactly the roots of $S$.
Edit: I was recalled that the question was about irreducible polynomials. Then, natural objects to consider are the equations of “elliptic curves” ($y^2=x^3+Ax+B$ with $x^3+Ax+B$ separable as well) and “hyperelliptic curves” ($y^2=P(x)$, $P$ separable of degree at most five).
For elliptic curves, it is a well-known result of Mazur that, when this equation has finitely many solutions, its number of solutions is $15$, or $11$, or at most $9$ (and all cases occur). See for instance https://en.m.wikipedia.org/wiki/Torsion_conjecture .
For hyperelliptic curves (or any curve really but the formulation is a bit trickier), it is a theorem of Faltings (again, an important and difficult one) that the number of rational solutions is finite – and a difficult problem to provide in general a somewhat tight upper bound of the number of solutions (see however the very striking Theorem 1.1 of https://arxiv.org/pdf/2001.10276.pdf ).
Here is an explicit equation (a hyperelliptic curve): $y^2=x(x-1)(x-2)(x-5)(x-6)$ (from http://www-math.mit.edu/~poonen/papers/chabauty.pdf ). It has exactly nine solutions, that are: the five where $y=0$, $(3,\pm 6)$ and $(10,\pm 120)$. The proof in the reference that these are the only solutions is somewhat involved though.
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\begin{document}
\title{Chaotic dynamical systems associated
with tilings of $\R^N$}
\author{
Lionel Rosier \thanks{Institut \'{E}lie Cartan,
UMR 7502 UHP/CNRS/INRIA,
B.P. 239, 54506 Vand\oe uvre-l\`{e}s-Nancy Cedex, France.
({\tt rosier@iecn.u-nancy.fr}). LR was partially
supported by the ``Agence Nationale de la Recherche'' (ANR), Project CISIFS,
grant ANR-09-BLAN-0213-02.}}
\maketitle
\begin{abstract}
In this chapter,
we consider a class of discrete dynamical systems defined
on the homogeneous space associated with a regular tiling of $\R^N$, whose
most familiar example is provided by the $N-$dimensional torus $\T ^N$.
It is proved that any dynamical system in this class
is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent.
Next, a chaos-synchronization mechanism is introduced and used
for masking information in a communication setup.
\end{abstract}
\va\va
{\bf Key words:} Chaotic dynamical system, regular tiling of $\R^N$, ergodicity,
Lyapunov exponent, equidistributed sequence, chaos synchronization, cryptography.
\va
{\bf AMS subject classifications: 34C28, 37A25, 93B55, 94A60}
\va
\section{Introduction}
Chaos synchronization has exhibited an increasing interest in
the last decade since the pioneering works
reported in \cite{PecCar90,PecCar91}, and it has been advocated
as a powerful tool in secure communication
\cite{Spec97a,Spec97b,KolKenChu98b,Spec00a,Spec02a}.
Chaotic systems are indeed characterized by a great sensitivity to the
initial conditions and a spreading out of the trajectories, two properties
which are very close to the Shannon requirements of confusion and diffusion
\cite{Massey92}.
There are basically two approaches when using chaotic dynamical systems for
secure communications purposes. The first one amounts to numerically
computing a great number of iterations of a discrete chaotic system,
in using e.g. the message as initial data
(see \cite{Schmitz01} and the references therein).
The second one amounts
to hiding a message in a chaotic dynamics. Only a part of the state vector
(the ``output'') is conveyed through the public channel. Next, a
synchronization mechanism is designed to retrieve the message at the
receiver part (see \cite{RMB06} and the references therein).
In both approaches, the first difficulty is to ``build'' a
chaotic system appropriate for encryption purposes.
In this context, the corresponding chaotic signals must have
no patterning, a broad-band power spectrum and an auto-correlation
function that quickly drops to zero.
In \cite{PecCar97}, a mean for synthesizing volume-preserving
or volume expanding maps
is provided. For such systems, there are several directions of expansion
(stretching), while the discrete trajectories are folded back into a confined
region of the phase space. Expansion can be carried out by
unstable linear mappings with at least one positive Lyapunov exponent.
Folding can be carried out with modulo functions through shift operations,
or with triangular, trigonometric functions through reflexion operations.
Fully stretching piecewise affine Markov maps have also attracted
interest because such maps are expanding in all directions and
they have uniform invariant probability densities
(see \cite{rovset98,Hasdelro96}).
Besides, we observe that the word ``chaotic'' has not the
same meaning everywhere, and that the chaotic behavior of a
system is often demonstrated only by numerical evidences.
The first aim of this chapter is to provide a rigorous
analysis, based on the definition given by Devaney \cite{Devaney},
of the chaotic behavior of a large class of affine dynamical systems defined
on the homogeneous space associated with a regular tiling of $\R ^N$.
Classical piecewise affine chaotic transformations, as the {\em tent map},
belong to that class. The
dimension $N$ may be arbitrarily large in the theory developed below,
but, for obvious reasons, most of the examples given here will be related
to regular tilings of the plane ($N=2$).
The study of the subclass of (time-invariant or switched) affine systems on $\T ^N$, the $N-$dimensional torus, is done in \cite{RMB04,RMB06}.
The folding for this subclass is carried out with modulo maps, which,
from a geometric point of view, amounts to ``fold back''
$\R ^N$ to $[0,1)^N$ by
means of translations by vectors in $\Z ^N$. Those translations are replaced
here by all the isometries of some crystallographic group for an arbitrary regular
tiling of $\R ^N$. Notice also that the fundamental domain used in
the numerical
implementation may be chosen with some degree of freedom. It may be a
hypercube (as $[0,1)^N$ for $\T ^N$), or a polyhedron, or a more complicated bounded, connected set in $\R ^N$.
For ease of implementation and duplication, a
cryptographic scheme must involve a map for which the parameters
identification is expected to be a difficult task, while computational requirements
for masking and unmasking information are not too heavy. The second aim
of this chapter is to show that all these requirements are fulfilled for
the class of dynamical systems considered here. The way of extracting the masked information is provided through an observer-based
synchronization mechanism with a finite-time
stabilization property.
Let us now describe the content of the chapter. Section 2 is devoted to the
mathematical analysis of the chaotic properties of the following
discrete dynamical system
\begin{equation}
\label{affine}
x_{k+1}=Ax_k+B \quad (\hbox{\rm mod } G)
\end{equation}
where $A\in \Z ^{N\times N}$, $B\in \R^N$, and (mod $G$) means roughly
that $x_{k+1}$ is the point in the fundamental domain $\cal T$ derived from
$A x_k+B$ by some transformation $g$ in the group $G$.
(\ref{affine}) may be viewed as a ``realization'' in ${\cal T}\subset \R ^N$
of an abstract dynamical system on the homogeneous space $\R ^N/G$ of classes modulo $G$.
The torus $\T ^N$ corresponds to the simplest case when
$G$ is the group of all the translations of vectors $u\in \Z ^N$ and
the fundamental domain is ${\cal T}=[0,1)^N$.
Note that most of the examples encountered in the literature are given only
for the torus
$\T ^N$ with $N=1$ and $|A|\ge 2$, or for $N=2$ and $\text{det } A=1$
(see e.g. \cite{KH}).
We give here a sufficient condition for (\ref{affine}) to be
chaotic in the sense of Devaney for any given regular tiling of $\R ^N$ ($N\ge 1$),
and we investigate the Lyapunov exponents of (\ref{affine}) and the
equirepartition of the trajectories of (\ref{affine}).
Finally, a masking/unmasking technique based on a dynamical
embedding is proposed in Section 3.
\section{Chaotic dynamical systems and regular tilings of $\R ^N$}
\subsection{Chaotic dynamical system}
Let $(M,d)$ denote a compact metric space, and let $f:M\to M$ be a
continuous map.
The following definition of a chaotic system
is due to Devaney \cite{Devaney}.
\begin{defi}
\label{def1}
The discrete dynamical system
$$
(\Sigma )\qquad x_{k+1}=f(x_k)
$$
is said to be {\em chaotic}
if the following conditions are fulfilled:\\
{\em (C1) (Sensitive dependence on initial conditions)} There exists a number
$\varepsilon >0$ such that for any $x_0\in M$ and any $\delta >0$, there
exists a point $y_0\in M$ with $d(x_0,y_0)<\delta $ and an integer $k\ge 0$
such that $d(x_k,y_k)\ge \ep$;\\
{\em (C2) (One-sided topological transitivity)} There exists some
$x_0\in M$ with $(x_k)_{k\ge 0}$ dense in $M$;\\
{\em (C3) (Density of periodic points)} The set
$D=\{ x_0\in M; \ \exists k>0,\ x_k=x_0\}$ is dense in $M$.
\end{defi}
Recall \cite[Thm 5.9]{Walters}, \cite[Thm 1.2.2]{Vesentini}
that when $f$ is {\em onto} (i.e., $f(M)=M$), the one-sided
topological transitivity is equivalent to the condition: \\
(C$2'$) For any pair of nonempty open sets $U,V$ in $M$,
there exists an integer $k\ge 0$ such that $f^{-k}(U)\cap V\ne \emptyset$
($\iff U\cap f^k(V)\ne \emptyset$).
\subsection{Regular tiling of $\R^N$}
An {\em isometry} $g$ of $\R^N$ is a map from $\R ^N$ into $\R ^N$ such that
$||g(X)-g(Y)||=||X-Y||$ for all $X,Y\in \R ^N$.
Let $G$ be a group of isometries of $\R ^N$ such that
for any point $X\in \R ^N$
the orbit of $X$ under the action of $G$, namely the set
$$
G\cdot X =\{ g(X);\ g\in G \},
$$
is closed and discrete.
Let $P\subset \R ^N$ be a compact, connected set with a nonempty interior.
Following \cite{Berger},
we shall say that the pair $(G,P)$ constitutes a {\em regular tiling} of $\R^N$ if
the two following conditions are fulfilled:
\begin{eqnarray}
&& \displaystyle\bigcup_{g\in G} g(P) = \R ^N \label{T1}\\
&& \forall g,h\in G\quad
\left( g(\intP ) \cap h(\intP )\ne \emptyset \quad \Rightarrow \quad g=h\right).\label{T2}
\end{eqnarray}
Recall that $\intP$ stands for the {\em interior} of $P$, that is
$$
\intP = \{ x\in P;\ \exists \varepsilon >0,\ B(x,\varepsilon )\subset P\}.
$$
The set $P\subset \R ^N$ is termed a {\em fundamental tile}, and the group $G$
a {\em crystallographic group}.
An example of a regular tiling of $\R^ 2$ with
a triangular fundamental tile is represented in Fig. \ref{triangle}.
\begin{figure}[hbt]
\begin{center}
\epsfig{figure=triangle,width=10cm}
\caption{A regular tiling of $\R^2$ with a triangular fundamental tile.}
\label{triangle}
\end{center}
\end{figure}
Note that a point
$X\in \R ^N$ may in general be obtained in several ways as the transformation
of a point in $P$ by an isometry in $G$. We introduce a set
$\cal T$, called a {\em fundamental domain}, with $\intP \subset {\cal T}\subset P$ and such that
\begin{eqnarray}
&& \displaystyle\bigcup_{g\in G} g({\cal T}) = \R ^N \label{T10}\\
&& \forall X,X'\in {\cal T},\ \forall g\in G\quad
\left( X'=g(X)\quad \Rightarrow \quad X'=X\right).\label{T11}
\end{eqnarray}
Introducing the equivalence relation in $\R ^N$
$$
X\sim Y \quad \iff \quad\exists g\in G,\quad Y=g(X),
$$
we denote by $x=\overline{X}$ the class of $X$ for $\sim$,
i.e. $x=\{ g(X);\ g\in G\} = G\cdot X$.
When several groups are considered at some time, we denote by $\overline{X}^G$ the class of
$X$ modulo $G$.
Finally, we introduce the homogeneous space of cosets $\H =(\R ^N/G) =
\{ x=\overline{X}; X\in \R ^N\}$, and define on it the following metric
$$
d(\overline{X}, \overline{Y} )=\inf_{g\in G} ||Y-g(X)||.
$$
The natural covering mapping
$\pi : \R^N\to \H$, defined by $\pi (X)=\overline{X}$, satisfies
$$
d(\pi (X), \pi (Y)) \le ||X-Y||,
$$
hence it is continuous. It follows that $\H=\pi( P)$ is a compact metric space.
On the other hand, the restriction of $\pi$ to $\cal T$ is a bijection from $\cal
T$ onto $\H$. We may therefore define the projection $\varpi:\R ^N\to {\cal T}$
by $\varpi (X)= (\pi_{\vert {\cal T}})^{-1} \pi (X)$. Note that $\varpi$ is
in general not continuous when $\cal T$ is equipped with the topology induced from $\R ^N$, while it is continuous when $\cal T$ is endowed with the topology inherited from $\H$.
The simplest example of a regular tiling of $\R ^N$ is provided by the
group of translations by vectors with integral coordinates
(which is isomorphic to the lattice subgroup)
\begin{equation}
G=\{ t_u;\ u\in \Z ^N\} \sim \Z^N,
\label{Gtore}
\end{equation}
where $t_u(X)=X+u$.
In such a situation, a fundamental tile (resp. domain) is given by
$P=[0,1]^N$ (resp. ${\cal T}=[0,1)^N$), and the homogeneous space $\H$ is the
standard $N-$dimensional torus $\T ^N$.
A classification (up to isomorphism) of the crystallographic groups
of $\R ^N$ has been
done for a long time for $N\le 3$. There are 17 such groups in $\R ^2$, and
230 groups in $\R ^3$, see \cite{Berger,Bur}.
\subsection{Affine transformation}
We aim to define ``simple'' chaotic dynamical systems on $M=\H$ by
using affine transformations. Assume given a matrix $A\in \Z ^{N\times N}$
and a point $B\in \R^N$. The following hypotheses will be used at
several places in the chapter.\\
(H1)
$$\forall X,X'\in \R^n \quad (X\sim X' \Rightarrow
AX+B \sim AX' +B)
$$
i.e. $X'=g(X)$ for some $g\in G$ implies $AX'+B=g'(AX+B)$ for some
$g'\in G$;\\
(H2) There exist a subgroup $G'\subset G$ of translations and a finite
collection of isometries $(g_i)_{i=1}^k$ in $G$ such that
\begin{enumerate}
\item $G$ is spanned as a group by the isometries in $G'\cup (g_i)_{i=1}^k$;
\item $G'=\{ t_u; u=\sum_{i=1}^N y_iu_i,\ y=(y_i)_{i=1}^N\in \Z^N \}$
for some basis $(u_i)_{i=1}^N$ of $\R ^N$;
\item Setting
$ P':= \cup_{1\le i\le k} \ g_i(P)$
we have that $(G',P')$ is a regular
tiling of $\R ^N$. We denote by ${\cal T}'$ a fundamental domain for $(G',P')$.
\end{enumerate}
(H1) is a compatibility condition needed to define a dynamical system
on $\H$. If $G$ is given by \eqref{Gtore}, then (H1)
holds for any $A\in \Z^{N\times N}$ and any $B\in \R^N$. However, if
\begin{equation}
\label{Greseau}
G=\{ t_u; u=\sum_{i=1}^N y_iu_i,\ y=(y_i)_{i=1}^N\in \Z^N \}
\end{equation}
for some basis $(u_i)_{i=1}^N$ of $\R ^N$, then
(H1) holds if and only if
\begin{equation}
\label{condAreseau}
U^{-1}AU\in \Z ^{N\times N}
\end{equation}
where $U$ is the $N\times N$ matrix with $u_i$ as $i$th column
for $1\le i\le N$.
(H2) allows to decompose the projection $\varpi$ onto $\cal T$ into
a projection onto ${\cal T}'$, a fundamental domain for the
regular tiling $(G',P')$ of $\R ^N$ involving only
translations, followed by a projection from ${\cal T}'$ onto $\cal T$.
\begin{exa}
Let $G=<t_1,t_2,r>$ and $G'=<t_1,t_2>$, where $t_1(X)=X+(1,-1)$,
$t_2(X)=X+(1,1)$, and $r(X_1,X_2)=(-X_2,X_1)$.
Pick $k=4$ and $(g_1,g_2,g_3,g_4)=(r,r^2,r^3,id)$.
Take as fundamental tiles $P=\{ X=(X_1,X_2); 1\le X_1\le 2,\
0\le X_2\le 2-X_1\}$ (solid line) and $P'=P\cup r(P)\cup r^2(P)\cup r^3(P)$
(broken line) (see Fig. \ref{GGprime}).
\begin{figure}[hbt]
\begin{center}
\epsfig{figure=GGprime,width=10cm}
\caption{A regular tiling of $\R^2$ with a triangular fundamental tile.}
\label{GGprime}
\end{center}
\end{figure}
\end{exa}
Assume that (H1) holds. Then we may define
$$A\overline{X} +B :=\overline{AX+B}$$
for any $X\in \R ^N$. Thus we may consider the dynamical system
$(\Sigma _{A,B})$ on $\H$ defined by
\begin{equation}
\label{dyn}
(\Sigma _{A,B})\
\left\{
\begin{array}{rl}
x_{k+1}&=f(x_k):=A x_k+B,\\
x_0&\in \H.
\end{array}
\right.
\end{equation}
The map $f$ is called an {\em affine transformation} of $\H$.
\begin{exa}
Let $N=1$, and let $G=<t,s>$ be the group spanned by the translation
$t(X)=X+2$ and the symmetry $s(X)=2-X$. Set $P=[0,1]$. Then $(G,P)$ constitutes a regular tiling of $\R$. Note that $P$ is also a fundamental domain. Pick
$(A,B)=(2,0)\in \R ^2$. (H1) and (H2) are satisfied with $G'=\{ t_u;\ u\in 2\Z \}$, $k=2$, $g_1=s$ and $g_2=s^2=id$.
Let us write the realization of \eqref{dyn} in $P$.
Obviously, $AX\in P$ for $0\le X<1/2$, while $s(AX)=2(1-X)\in P$ for
$1/2\le X\le 1$. Viewed in $P=[0,1]$, the dynamics reads then
\begin{equation}
\label{tent}
x_{k+1}=h(x_k)
\end{equation}
where $h$ is the familiar tent map (see Fig. \ref{tentfig})
\begin{figure}[hbt]
\begin{center}
\epsfig{figure=tent,width=10cm}
\caption{A~: Action of $s$ and $t$; B~:
the tent map}
\label{tentfig}
\end{center}
\end{figure}
$$
h(x)=\left\{
\begin{array}{ll}
2x \quad &\text{\rm if}\ 0\le x<\frac{1}{2}, \\
2(1-x) &\text{\rm if}\ \frac{1}{2} \le x \le 1.
\end{array}
\right.
$$
It follows from Theorem \ref{thm2} (see below) that \eqref{tent} is chaotic on $[0,1]$.
\end{exa}
When $\H=\T^N$ and $B=0$, $f$ is nothing else than an endomorphism
of the topological group $(\T ^N,+)$, and $f$ is onto (resp., an isomorphism)
if and only if $\hbox{\rm det } A\ne 0$
(resp., $\hbox{\rm det } A=\pm 1$) (see \cite[Thm 0.15]{Walters}).
Let $\text{\rm sp}(A)$ denote the spectrum of the matrix $A$,
that is the set of the eigenvalues of $A$. A {\em root of unity}
is any complex number of the form $\lambda =\exp (2\pi it)$, with $t\in \Q$.
To see whether a dynamical system $(\Sigma _{A,B})$ is chaotic, we
need the following key result \cite[Thm 1.11]{Walters}.
\begin{prop}
\label{ergodic}
Let $f(x)=Ax+b$ ($b\in \T^N$, $A\in \Z^{N\times N}$ with
$\hbox{\rm det }A\ne 0$) be an affine transformation of $\T^N$. Then the
following conditions are equivalent:\\
\begin{tabular}{ll}
(i) &$(\Sigma _{A,b})$ is one-sided topologically transitive;\\
(ii) &(a) $A$ has no proper roots of unity (i.e., other than 1)
as eigenvalues,
and \\
&(b) $(A-I)\T ^N +\Z b$ is dense in $\T ^N$;\\
(iii) &$f$ is {\em ergodic}; that is,
$f$ is {\em measure-preserving} (i.e. for any Borel set
$E\subset \T ^N$, $m(f^{-1}(E))=$ \\
& $m(E)$, where $m$ denotes the Lebesgue measure
on $\T^N$),
and the only Borel sets $E\subset \T ^N$ for \\
&which $f^{-1}(E)=E$ satisfy $m(E)=0$ or $m(E)=1$.
\end{tabular}
\end{prop}
Notice that (ii) reduces to ``$A$ has no roots of unity as eigenvalues''
when $b=0$. Indeed, it may be seen that
$(A-I)\T ^N$ is dense in $\T ^N$ if and only if
$(A-I)$ is invertible.
\subsection{Endomorphism of $\T ^N$}
The first result in this chapter, which comes from \cite{RMB06}, provides a necessary and sufficient condition for $\Sigma _{A,0}$ to be chaotic
in $\T ^N$.
\begin{thm}
\label{thm1}
Let $A\in \Z ^{N\times N}$. Then $(\Sigma _{A,0})$ is chaotic in
$\T ^N$
if, and only
if, $\hbox{\rm det A}\ne 0$ and $A$ has no roots of unity as eigenvalues.
\end{thm}
{\em Proof.\ } Assume first that $(\Sigma _{A,0})$ is chaotic.
We first claim that $A$ is nonsingular.
Indeed, if $\hbox{\rm det }A=0$, then the map $f$ defined in \eqref{dyn}
is not onto \cite[Thm 0.15]{Walters}, i.e. $A\T ^N\ne \T^N$.
As $A\T ^N$ is compact (hence equal to its closure), it is not dense in
$\T ^N$, hence we cannot find some state $x_0\in \T^N$ such that
the sequence $(x_k)=(A^kx_0)$ is dense in $\T ^N$, which contradicts (C2).
Thus $\hbox{\rm det }A\ne 0$.
On the other hand, since $(\Sigma _{A,0})$ is
one-sided topologically transitive, the matrix $A$ has no roots of
unity as eigenvalues by virtue of Proposition \ref{ergodic}.
Conversely, assume that $\hbox{\rm det A}\ne 0$ and that $A$ has
no roots of unity as eigenvalues.
As (C1) is a consequence of (C2) and (C3)
(see \cite{BCDS},\cite[Thm 1.3.1]{Vesentini}), we
only have to establish the later properties. (C2) follows from Proposition
\ref{ergodic}. To prove (C3) we need to prove two lemmas.
\begin{lem}
\label{invertible}
Let $A\in \Z^{N\times N}$ be such that $\hbox{\rm det }A\ne 0$, and pick
any $p\in \N^*$ with $(p,\hbox{\rm det} A)=1$ (i.e. $p$ and
$\hbox{\rm det }A$ are relatively prime).
Then the map
$T:x\in (\Z / p\Z )^N\mapsto Ax\in (\Z/ p\Z )^N$ is invertible.
\end{lem}
{\em Proof of Lemma \ref{invertible}.\ } First, observe that the map
$T$ is well-defined. Indeed, if $X,Y\in \Z ^N$ fulfill $X-Y\in
(p\Z )^N$, then $AX-AY\in (p\Z )^N$ so that $AX$ and $AY$ belong to the same
coset in $(\Z /p\Z )^N=\Z ^N/(p\Z )^N$. As $(\Z /p\Z )^N$ is a finite set,
we only have to prove that $T$ is one-to-one. Let $X,Y\in \Z^N$ be such that
$AX=AY$ in $(\Z /p\Z )^N$ (i.e., $A(X-Y)\in (p\Z )^N$). We aim to show that
$X=Y$ in $(\Z /p\Z )^N$ (i.e., $X-Y\in (p\Z )^N$). Set $U=X-Y$, and
pick a vector $Z\in \Z ^N$ such that $AU=pZ$. It follows that
$U=\frac{p}{\hbox{\rm det }A}{\tilde A}Z$, where
$\tilde A\in \Z^{N\times N}$ denotes the adjoint matrix of $A$ (i.e. the
transpose of the matrix formed by the cofactors).
Since $U\in \Z^N$, each component of
the vector $p{\tilde A}Z$ is divisible by
$\hbox{\rm det } A$.
Since $(p,\hbox{\rm det }A)=1$, we infer the existence of
a vector $V\in \Z ^N$ such that ${\tilde A}Z=(\hbox{\rm det }A) V$. Then
$X-Y=U=pV\in (p\Z )^N$, as desired. \qed
\begin{lem}
\label{dense}
Let $A$ and $p$ be as in Lemma \ref{invertible}, and let
$E_p:=\{\overline{0},\overline{(\frac{1}{p})}, ...,
\overline{(\frac{p-1}{p})}\}\subset \T$. Then each point
$x\in E_p^N$ is periodic for $(\Sigma _{A,0})$. As a consequence, the set
of periodic points of $(\Sigma _{A,0})$ is dense in $\T ^N$ (i.e., (C3)
is satisfied).
\end{lem}
{\em Proof of Lemma \ref{dense}.\ }
First, observe that for any $i,j\in \{0 , ... , p-1\}$,
$i/p \equiv j/p \ (\hbox{\rm mod }1)$ if and only if $i\equiv j\
(\hbox{\rm mod }p)$.
We infer from Lemma \ref{invertible}
that the map $\tilde T: x\in E_p^N\mapsto Ax \in E_p^N$ is well defined
and invertible. Pick any $x\in E_p^N$. As the sequence
$({\tilde T}^kx)_{k\ge 1}$
takes its values in the (finite) set $E_p^N$, there exist two numbers
$k_2>k_1\ge 1$ such that ${\tilde T}^{k_1}x={\tilde T}^{k_2}x$. $\tilde T$
being invertible, we conclude that $A^{k_2-k_1}x=x$ (i.e., $x$ is a
periodic point). Finally, the set $E=\cup \{ E_p^N;\ p\ge 1,\
(p,\hbox{\rm det }A)=1\}$ is clearly dense in $\T^N$
(take for $p$ any large prime number), and all its points
are periodic. This completes the proof of Lemma \ref{dense} and of
Theorem \ref{thm1}.\qed
For an affine transformation, we obtain a result similar to Theorem
\ref{thm1} when $1\not\in \text{\rm sp}(A)$.
\begin{cor}
\label{cor1}
Let $A\in \Z^{N\times N}$ and $b\in \T^N$. Assume that $1$ is not
an eigenvalue of $A$. Then $(\Sigma _{A,b})$ is chaotic in $\T^N$
if, and only if,
$\hbox{\rm det }A\ne 0$ and $A$ has no roots of unity as eigenvalues.
\end{cor}
{\em Proof.\ }
Pick any $B\in \R ^N$ with $\overline{B}=b$. As $1\not\in
\text{\rm sp} (A)$, we
may perform the change of variables
\begin{equation}
\label{cov}
x=r-\overline{(A-I)^{-1}B},
\end{equation}
which transforms (\ref{dyn}) into
\begin{equation}
\label{dynbis}
\left\{
\begin{array}{rl}
r_{k+1}&=A r_k,\\
r_0&=x_0+\overline{(A-I)^{-1}B}.
\end{array}
\right.
\end{equation}
Clearly, the conditions (C2) and (C3) are fulfilled for
$(\Sigma _{A,b})$ if, and only if, they are fulfilled for (\ref{dynbis}).
Therefore, the result is a direct consequence of Theorem \ref{thm1}. \qed
\begin{cor}
\label{cor2}
Let $G$ be defined by \eqref{Greseau} for some basis
$(u_i)_{i=1}^N$ of $\R ^N$.
Let $A\in \Z^{N\times N}$ and $B\in \R ^N$.
Assume that \eqref{condAreseau} holds
and that $1$ is not an eigenvalue of $A$.
Then $(\Sigma _{A,B})$ is chaotic in $\H =\R ^N/G$ if,
and only if, $\hbox{\rm det}\ A\ne 0$ and $A$ has no roots of unity as eigenvalues.
\end{cor}
{\em Proof.\ } From Corollary \ref{cor1}, we know that the dynamical system
on $\T ^N$
\begin{equation}
z_{k+1}={\tilde f} (z_k):=U^{-1}A U z_k + U^{-1}B
\end{equation}
is chaotic if, and only if, ${\rm det}\ A \ne 0$ and $A$ has no roots of
unity as eigenvalues. To prove that the dynamical system on $\H =\R ^N/G$
\begin{equation}
x_{k+1} = f(x_k) := A x_k +B
\end{equation}
is chaotic under the same conditions, it is sufficient to prove that
the maps $f:\H \to \H$ and $\tilde f:\T ^N\to \T ^N$ are topologically
conjugate;
i.e., there exists a homeomorphism $h:\H \to \T ^N$ such that
$h\circ f= \tilde f\circ h$. Define $h$ by $h(\overline{X})=\overline{Z}$
where $Z=U^{-1}X$, $\overline{X}=G\cdot X$ is the class of $X$ in $\H$
and $\overline{Z}$ is the class of $Z$ in $\T ^N$. Note first that $h$ is well
defined and continuous. Indeed, if $X'=X+UK$ with $K\in \Z ^N$, then
$Z'=U^{-1}X'=U^{-1}X+K=Z+K$, so that $h$ is well defined. On the other hand, the map $X\in \R ^N\mapsto \overline{U^{-1}X}\in \T ^N$ is clearly continuous.
Obviously, $h$ is invertible with $h^{-1}(\overline{Z})=\overline{X}$ for
$X=UZ$. $h$ is therefore a homeomorphism from $\H$ onto $\T ^N$. Let us check now
that $h\circ f=\tilde f\circ h$. Pick any $X\in \R ^N$. Then
$$
h\circ f(\overline{X}^G) =h(\overline{AX+B}^{G})
=\overline{U^{-1}(AX+B)}^{\T ^N} = \tilde f (\overline{U^{-1}X}^{\T ^N})
=\tilde f\circ h (\overline{X}^G)
$$
and the result follows.\qed
We are in a position to state and prove the main result of this chapter.
\begin{thm}
\label{thm2}
Let $(G,P)$ be a regular tiling of $\R ^n$, and
let $(A,B)\in \Z^{N\times N}\times \R ^N$ be such that both the assumptions
(H1) and (H2) are fulfilled. Assume in addition that
$\text{\rm det}\ A\ne 0$ and that $A$ has no roots of unity as eigenvalues.
Then the discrete dynamical system in $\R ^N/G$
\begin{equation}
\label{A100}
x_{k+1}=Ax_k+B
\end{equation}
is chaotic.
\end{thm}
{\em Proof.}
Pick any fundamental domain $\cal T$ for $(G,P)$, and let $G'$ and ${\cal T}'$
be as in (H2). In addition to \eqref{A100}, we shall
consider the discrete dynamical system
in $\R ^N/G'$
\begin{equation}
\label{A101}
z_{k+1}=Az_k+B.
\end{equation}
For any given $X_0\in \R ^N$, let $x_0=\overline{X_0}^G$ and
$z_0=\overline{X_0}^{G'}$. Clearly, if $X\sim X'$ (mod $G'$), then
$X\sim X'$ (mod $G$). Therefore, one can define a map $p:\R ^N/G'
\to \R ^N/G$ by $p(\overline{X}^{G'})=\overline{X}^G$. $p$ is continuous
and onto. We need two claims. \\
{\sc Claim 1.} $x_k=p(z_k)$ for all $k$. \\
Indeed, this is true for $k=0$, and if for some $k\ge 0$, $x_k=p(z_k)$
(i.e. for some $X_k\in\R ^N$, $x_k=\overline{X_k}^G$ and $z_k=\overline{X_k}^{G'}$),
then we have that
$$
x_{k+1} = \overline{AX_k+B}^G = p (\overline{AX_k+B}^{G'}) = p(z_{k+1})
$$
which completes the proof of Claim 1.\\
{\sc Claim 2.} The image by $p$ of any dense set in $\R ^N /G'$ is a dense set
in $\R ^N/G$.\\
Let $A\subset \R ^N/{G'}$ be a given dense set. Pick any $X\in \R ^N$ and any $\varepsilon >0$.
Since $A$ is dense in $\R ^N/{G'}$, there exists $Y\in \R ^N$ such that $\overline{Y}^{G'}\in A$ and
$$
d(\overline{X}^{G'},\overline{Y}^{G'})=\inf_{g\in G'}||Y-g(X)||<\varepsilon .
$$
It follows that
$$
d(\overline{X}^G,\overline{Y}^G) =\inf_{g\in G}||Y-g(X)|| <\varepsilon
$$
for $G'\subset G$. Since $\overline{Y}^G = p(\overline{Y}^{G'})\in p(A)$ and the
pair $(X,\varepsilon)$ was arbitrary,
this demonstrates that $p(A)$ is dense in $\R ^N/G$. Claim 2 is proved.
Let us complete the proof of Theorem \ref{thm2}. To prove that \eqref{A100} is chaotic, it is sufficient (see \cite{BCDS})
to check that the conditions (C2) and (C3) are fulfilled. We know from Corollary
\ref{cor2} that \eqref{A101} is chaotic. We may therefore pick $X_0\in \R ^N$ so that, setting $z_0=\overline{X_0}^{G'}$, the sequence $\{z_k\} _{k\ge 0}$
defined by \eqref{A101} is dense in $\R ^N/G'$. By Claim 1 and Claim 2, the
sequence $\{x_k\}$ defined by \eqref{A100} and $x_0=\overline{X_0}^G$ is dense
in $\R ^N/G$; that is, (C2) is fulfilled for \eqref{A100}. On the other hand,
the set of periodic points for \eqref{A101} is dense in $\R ^N/G'$, since
(C3) is fulfilled for \eqref{A101}. By Claim 1, any periodic point $z_0$ for
\eqref{A101} gives rise to a periodic point $x_0=p (z_0)$ for \eqref{A100}.
By Claim 2, the set of periodic points for \eqref{A100} is dense in
$\R ^N/G$; i.e., (C3) is fulfilled for \eqref{A100}. The proof of Theorem \ref{thm2}
is complete. \qed
\begin{exa}
\begin{enumerate}
\item Let $G=<t_{e_1},t_{2e_2},s>$ where $t_{e_1}(X)=X+(1,0)$,
$t_{e_2}(X)=X+(0,2)$, $s(X_1,X_2)=(X_1,-X_2)$, and $P=[0,1]\times [0,1]$.
Pick $G'=<t_{e_1},t_{e_2}>$, $k=2$, $(g_1,g_2)=(s,id)$ (see Fig. \ref{sym1}).
Finally, pick
$A=\left(\begin{array}{cc}-2&0\\0&3\end{array}\right)$ and $B=(0.5, -3.2)$.
Note that $[A,S]:=AS-SA=0$, where
$S=\left( \begin{array}{cc}1&0\\0&-1\end{array}\right)$ is the matrix
corresponding to the symmetry $s$.
Then (H1) and (H2) are satisfied, sp$\, (A)=\{-2,3\}$, and by Theorem \ref{thm2}
the dynamical system \eqref{dyn} is chaotic in $\H =\R ^2/G$.
\begin{figure}[hbt]
\begin{center}
\psfig{figure=tiling1,width=10cm}
\caption{$G=<t_{e_1},t_{2e_2},s>$.}
\label{sym1}
\end{center}
\end{figure}
\item Let $G=<t_{2e_1},t_{2e_2},s_1,s_2>$ where $t_{2e_1}(X)=X+(2,0)$,
$t_{2e_2}(X)=X+(0,2)$, $s_1(X_1,X_2)=(-X_1,X_2)$, $s_2(X_1,X_2)=(X_1,-X_2)
=-s_1(X_1,X_2)$,
and $P=[0,1]\times [0,1]$.
Pick $G'=<t_{2e_1},t_{2e_2}>$, $k=4$, $(g_1,g_2,g_3,g_4)=(s_1,s_2,s_2\circ s_1,id)$.
(see Fig. \ref{sym2}).
Finally, pick
$A=\left(\begin{array}{cc}0&-3\\4&0\end{array}\right)$ and $B=(-0.2,1.7)$.
Note that $AS=-SA$, where $S$ is as above.
Then (H1) and (H2) are satisfied, sp$\, (A)=\{ \pm 2i\
\sqrt{3} \}$, and by Theorem \ref{thm2} the dynamical
system \eqref{dyn} is chaotic in $\H =\R ^2/G$.
\begin{figure}[hbt]
\begin{center}
\epsfig{figure=tiling2,width=10cm}
\caption{$G=<t_{2e_1},t_{2e_2},s_1,s_2>$.}
\label{sym2}
\end{center}
\end{figure}
\end{enumerate}
\end{exa}
\subsection{Lyapunov exponents}
Let $M$ denote a compact differentiable manifold endowed with a Riemann
metric $<u,v>_m$, and let $f:M\to M$ be a
map of class $C^1$. The following definition is
borrowed from \cite{Mane}.
\begin{defi}
\label{def2}
A point $x\in M$ is said to be a {\em regular point} of $f$ if there
exist numbers $\lambda _1(x)>\lambda _2(x) >\cdots > \lambda _m(x)$ and
a decomposition
$$
T_xM=E_1(x)\oplus \cdots \oplus E_m(x)
$$
of the tangent space $T_xM$ of $M$ at $x$
such that
$$\lim_{k\to +\infty} \frac{1}{k}\ln ||(D_x f^k)u||=\lambda _ j(x)$$
for all $0\ne u\in E_j(x)$ and every $1\le j\le m$.
($||v||^2:=<v,v>_x\ \forall v\in T_xM$.) The numbers
$\lambda_j(x)$ and the spaces $E_j(x)$ are termed the
{\em Lyapunov exponents} and the {\em eigenspaces}
of $f$ at the regular point $x$.
\end{defi}
Assume now that the group $G$ is such that each isometry $g\in G$
has no fixed point, i.e. $g(X)\ne X$ for all $X\in \R ^N$. Then $\H=\R ^N/G$ is a smooth
flat Riemannian manifold. Before investigating the Lyapunov exponents of an affine transformation on
$\H$, let us give a few examples.
\begin{exa}
\begin{enumerate}
\item $\H =\T ^N$, and more generally, $\H =\R ^N / G$ where $G$ is as in \eqref{Greseau};
\item $\H=\R ^2 /G$ for $G=<t_{2e_1},t_{e_2},t_{e_1}\circ s>$ where $(e_1,e_2)$ is the canonical basis
of $\R ^2$ and $s(X_1,X_2)=(X_1,-X_2)$
(see Fig. \ref{klein}).
\begin{figure}[hbt]
\begin{center}
\psfig{figure=klein,width=10cm}
\caption{The regular tiling of $\R^2$ associated with the Klein bottle.}
\label{klein}
\end{center}
\end{figure}
$\H$ is then the {\em Klein bottle}. The torus $\T ^2$ and the
Klein bottle $\H$ are the only smooth manifolds obtained in dimension 2. In dimension 3, there are 6 smooth
manifolds (see \cite[Section 3.5.5 p. 117]{Wolf}).
\end{enumerate}
\end{exa}
Consider now an affine transformation $f(\overline{X})=\overline{AX+B}$ of $\H$,
the pair $(A,B)$ fulfilling (H1). Assume also that det $A\ne 0$. Then for any
$k\ge 1$,
$$
f^k (\overline{X}) = \overline{A^k X + A^{k-1} B + \cdots + A B + B }.
$$
Pick a point $X\in \intP$ such that
$$
A^k X + A^{k-1}B + \cdots + AB + B\in \cup_{g\in G}g(\intP )
$$
(note that such a property holds for almost every $X\in \R ^N$), and an isometry $g\in G$ such that
$$
g(A^k X + A^{k-1}B + \cdots + AB + B )\in \intP.
$$
For $||U||$ sufficiently small, we also have that
$$
g(A^k (X+U) + A^{k-1}B + \cdots + AB + B )\in \intP.
$$
Therefore $(D_{\overline{X}} f^k) \overline{U}=\overline{GA^kU}$,
where $G=Dg\in \R ^{N\times N}$. Since $G$ is an orthogonal matrix,
we have that $||\overline{GA^kU}||=||A^kU||$.
Let $\mu _1>\mu _2>\cdots >\mu _m >0$
denote the absolute values of the eigenvalues of $A$, and let
$E_i(x)$ be the direct sum of the generalized eigenspaces
(see \cite{greub}) associated with the
eigenvalues whose absolute value is $\mu _i$, for each
$i\le m$. Then, using the Jordan decomposition of $A$,
we easily see that for any $U\in E_j\setminus \{0\}$
$$\lim_{k\to +\infty}\frac{1}{k}\ln ||A^k U|| =\ln \mu _j .$$
Observe now that if $\sigma (A)$ does not intersect the circle
$\{ z\in \C ;\ |z|=1 \}$, then $A$ has at least one eigenvalue $\lambda$ with
$|\lambda |>1$ (since the product of all the eigenvalues of $A$ is
$\hbox{\rm det } A\in \Z\setminus\{ 0\}$), hence $f$ admits at least
one {\em positive} Lyapunov exponent.
Therefore, we have proved the following
\begin{prop}
\label{prop3}
Let $(G,P)$ be a regular tiling of $\R ^N$ such that any isometry
$g\in G$ has no fixed point. Let $(A,B)\in \R ^{N\times N} \times \R^N$ be
such that (H1) is satisfied, $\text{\rm det}\ A\ne 0$ and each
eigenvalue $\lambda$ of $A$ satisfies $|\lambda | \ne 1$,
and let $f:\H=\R ^N/G \to \H$ be defined
by $f(x)=Ax+B$. Then almost every point $x\in\H$
is regular for $f$, with Lyapunov exponents
$\ln \mu _1 >\cdots > \ln \mu _m$, where
$\mu _1>\cdots >\mu _m$ are the absolute values of the
eigenvalues of $A$. Furthermore, $\ln \mu _1 >0$.
\end{prop}
Notice that the existence of (at least) one positive Lyapunov exponent is often
considered as a characteristic property of a chaotic motion
\cite{WykSteeb}.
That property quantifies the sensitive dependence on initial conditions.
\subsection{Equidistribution}
In this section, $\H=\T ^N$. Let us consider a discrete dynamical system with an output
\begin{equation}
\left\{
\begin{array}{rl}
x_{k+1}&=Ax_k +B\\
y_k&=Cx_k
\end{array}
\right.
\label{output}
\end{equation}
where $x_0\in \T ^N$, $A\in \Z ^{N\times N}$, $b\in \T ^N$
and $C\in \Z ^{1\times N}$.
It should be expected that the output $y_k$
inherits the chaotic behavior of the state $x_k$.
However, Devaney's definition of a chaotic system cannot be tested
on the sequence $(y_k)$, since this sequence is not defined as
a trajectory of a dynamical system.
Rather, we may give a condition ensuring that
the sequence $(y_k)$ is equidistributed (hence dense) in $\T$ for a.e. $x_0$,
a property which may be seen as an {\em ersatz} of (C2).
If $X=(X_1, ... ,X_N),Y=(Y_1, ... ,Y_N)$ are any given points in $[0,1)^N$
and $x=\overline{X}$, $y=\overline{Y}$, then we say that $x<y$
(resp., $x\le y$) if $X_i<Y_i$ (resp., $X_i\le Y_i$)
for $i=1,...,N$. The set of points
$z\in \T^N$ such that $x\le z<y$ will be denoted by $[x,y)$.
Let $(x_k)_{k\ge 0}$ be any sequence in $\T^N$.
For any subset $E$ of $\T ^N$, let
$S_K(E)$ denote the number of points $x_k$, $0\le k\le K -1$, which
lie in $E$.
\begin{defi} \cite{KN}
We say that $(x_k)$ is {\em uniformly distributed modulo 1} (or
{\em equidistributed in $\T^N$}) if
$$
\lim_{K\to \infty} \frac{S_K([x,y))}{K} =m([x,y))=\prod_{i=1}^N(Y_i-X_i)
$$
for all intervals $[x,y)\subset \T^N$.
\end{defi}
The following result is very useful to decide whether a sequence is
equidistributed or not.
\begin{prop} ({\bf Weyl criterion} \cite{KN}, \cite{Rauzy})
The sequence $(x_k)_{k\ge 0}$ is equidistributed in $\T ^N$ if, and only if,
for every lattice point $p\in \Z ^N$, $p\ne 0$
$$
\frac{1}{K}\sum_{0\le k<K} e^{2i\pi p\cdot x_k}\to 0
\qquad \hbox{ as }K\to +\infty .
$$
\end{prop}
The next result shows that under the same assumptions as in Corollary
\ref{cor1} the sequences
$(x_k)$ and $(y_k)$ are respectively equidistributed in $\T ^N$ and $\T$
for a.e. initial state $x_0\in \T ^N$.
\begin{thm}
\label{thm3}
Let $A\in \Z ^{N\times N}$, $b\in \T ^N$ and $C\in \Z ^{1\times N}
\setminus \{ 0\}$.
Assume that $\hbox{\rm det }A\ne 0$ and that $A$ has no roots of unity as
eigenvalues (hence $\Sigma _{A,b}$ is chaotic).
Then for a.e. $x_0\in \T ^N$ the sequence $(x_k)$ (defined in (\ref{output}))
is equidistributed in $\T ^N$, and the sequence $(y_k)=(Cx_k)$ is
equidistributed in $\T$.
\end{thm}
{\em Proof:\ }
By virtue of Theorem \ref{ergodic}, the map $f(x)=Ax+b$ is ergodic
on $\T ^N$. It follows then from Birkhoff Ergodic Theorem
(see e.g. \cite[Thm 1.14]{Walters}) that for any $h\in L^1(\T ^N, dm)$ and
for a.e. $x_0\in \T^N$
$$
\frac{1}{K}\sum_{0\le k < K } h(f^k(x_0)) \to \int_{\T ^N}h(y)\, dm(y)\qquad
\hbox{\rm as } K\to +\infty .
$$
Therefore, for every lattice point $p\in \Z ^N$, $p\ne 0$, and for a.e.
$x_0\in \T ^N$
$$
\frac{1}{K}\sum_{0\le k<K}e^{2\pi i p\cdot f^k(x_0)}
\to \int_{\T ^N} e^{2\pi ip\cdot y}\, dm(y)=0\qquad
\hbox{ as } K\to +\infty .
$$
As $\Z ^N\setminus \{ 0\}$ is countable, the same property
holds for a.e. $x_0\in \T ^N$ and all $p\in \Z ^N\setminus \{ 0\}$.
Therefore, we
infer from Weyl criterion
that the sequence $(x_k)=(f^k(x_0))$ is
equidistributed for a.e. $x_0\in \T ^N$.
Pick any $x_0\in \T ^N$ such that $(x_k)$ is equidistributed, and let us
show that the output sequence $(y_k)=(Cx_k)$ is also
equidistributed provided that $C=(C_1,...,C_N)\ne (0,...,0)$.
Indeed, for any $p\in \Z \setminus \{ 0\}$
$$
\frac{1}{K}\sum_{0\le k<K} e^{2\pi i py_k}
=\frac{1}{K} \sum_{0\le k<K} e^{2\pi i (pC)\, x_k}\to 0
\qquad \hbox{ as }K\to +\infty ,
$$
hence the equidistribution of $(y_k)$ follows again by Weyl criterion.\qed
\begin{rem}
For a regular tiling $(G,P)$ of $\R ^N$, even if the sequence
$(x_k)$ is equidistributed in $\H$, the output
$(y_k)$ fails in general to be equidistributed in $\T$. This is clear when
one considers a regular tiling of $\R ^2$ with the triangle
$P=\{ X=(X_1,X_2);\quad X_1\ge 0,\ X_2\ge 0, \ X_1+X_2\le 1\}$ as fundamental tile,
and $C=(1\quad 0)$.
\end{rem}
\section{Synchronization and information recovering}
The aim of this section is to suggest a chaos-based encryption
scheme involving affine transformations on the homogeneous space $\H$ associated
with some regular tiling of $\R ^ N$.
We shall provide conditions which guarantee a synchronization
with a finite-time stability of the error despite the inherent
nonlinearity of the chaotic systems under study.
\subsection{Encryption setup}
Assume given a regular tiling $(G,P)$ of $\R ^N$ and a pair
$(A,B)\in \R ^{N\times N}\times \R ^N$ fulfilling the assumptions of Theorem
\ref{thm2}. For the sake of simplicity, assume further that $\R ^N/G'=\T ^N$,
so that ${\cal T}'=[0,1) ^N$. Let $\varpi : \R ^N\to {\cal T}$ and
$\varpi ' : \R ^N\to {\cal T}'$ denote the projections on the fundamental
domains of $(G,P)$ and $(G',P')$, respectively. Set for $k\in \N$ and $X\in \R ^N$
\begin{equation}
\label{switch}
\varpi _k(X) =
\left\{
\begin{array}{ll}
\varpi '(X) & \text{ if }\ k\not\in (N+1)\N ; \\
\varpi (X) & \text{ if }\ k\in (N+1)\N.
\end{array}
\right.
\end{equation}
At each discrete time $k$, a symbol $m_k \in \R$ (the {\em plaintext})
of a sequence $(m_k)_{k\ge 0}$
is encrypted by a (nonlinear) encrypting
function $e$ which ``mixes'' $m_k$ and $X_k$ and produces a
{\em ciphertext} $u_k=e(X_k,m_k)$. We also assume given a decrypting
function $d$ such that $m_k=d(X_k,u_k)$ for each $k$.
Next, the ciphertext $u_k$ is embedded in the dynamics \eqref{dyn}. We shall consider the following encryption
\begin{equation}
\label{drive1}
(\Sigma _{A,B,M,C}) \qquad \left\{
\begin{array}{rl}
X_{k+1}&= \varpi _k \{ A (X_k+M u_k) + B\} \\
Y_k&=C (X_k+M u_k)
\end{array}
\right.
\end{equation}
which corresponds to an embedding of the ciphertext
in both the dynamics and the output.
In (\ref{drive1}),
$A\in \Z ^{N\times N}$, $M\in \Z ^{N\times 1}$, and
$C\in \Z ^{1\times N}$ are given matrices, and $B\in \R ^N$.
$Y_k\in \R$ is the output conveyed to the
receiver through the channel.
$\,$From the definition of the decrypting function $d$, it is clear that
to retrieve $m_k$ at the decryption side we need to recover the pair
$(X_k,u_k)$, which in turn calls for reproducing
a chaotic sequence $(\hat{X}_k)$ synchronized with $({X}_k)$ (i.e.,
such that $\hat{X}_k-X_k\to 0$). To this end, we propose a mechanism
based on some suitable unknown input observers, inspired from the
ones given in \cite{MilDaf03c,MilDaf04d,RMB04,RMB06}. We stress
that the gain matrices have to be $\Z$-valued here.
For the encryption considered here, the decryption involves
the following observer-like structure
\begin{equation}
\label{response1}
(\hat{\Sigma} _{A,B,M,C}) \qquad \left\{
\begin{array}{rl}
{\hat X}_{k+1} &= \varpi_k \{ A {\hat X}_k+L (Y_k-{\hat Y}_k) + B \} \\
{\hat Y}_k&=C{\hat X}_k
\end{array}
\right.
\end{equation}
where $L\in \Z ^{N\times 1}$, ${\hat X}_k\in \R ^N$ and ${\hat Y}_k\in \R$
(${\hat X}_0$ being an arbitrary point in $\R ^N$).
Let $\overline{X}$ denote the class of $X$ modulo $G'$, i.e. in $\T ^N$.
Set $e_k=\overline{X_k}-\overline{{\hat X}_k}$ for all $k\ge 0$.
Noticing that for all $X\in \R ^N$
$$
\overline{\varpi _k (X)} = \overline{\varpi '(X)} = \overline{X}\qquad
\text{ for } 1\le k\le N,
$$
we obtain by subtracting (\ref{response1})
from (\ref{drive1}) that the error dynamics reads
\begin{equation}
\label{errordyn}
e_{k+1}=(A -L C )e_k+\overline{(A-L C ) M u_k}, \qquad
1\le k\le N.
\end{equation}
Before proceeding to the design of the observers, we give a few definitions
and a preliminary result.
\subsection{Definitions and preliminary results}
\begin{defi}
A pair $(A ^\flat, C^\flat )$ is said to be in a
{\em companion canonical form} if it takes the form
\begin{equation}
\label{companion}
A^\flat =
\left(
\begin{array}{lcccc}
- \alpha ^{N-1} & 1 & 0 & \cdots & 0 \\
- \alpha ^{N-2} & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
- \alpha ^1 & 0 & 0 & \cdots & 1 \\
-\alpha ^0 & 0 & 0 & \cdots & 0
\end{array}
\right), \qquad
C^\flat =
\left(
\begin{array}{ccccc}
1 &
0 &
\cdots &
0 &
0
\end{array}
\right)\cdot
\end{equation}
\end{defi}
It is well known that the characteristic polynomial of
$A^\flat$ reads
$\chi_{A^\flat}(\lambda )
=\lambda ^N+\alpha ^{N-1}\lambda ^{N-1}+\cdots +\alpha ^1\lambda +\alpha ^0$.
\begin{defi}
Two pairs $(A,C)$ and $(A^\flat, C^\flat )$ in
$\Z ^{N\times N}\times \Z ^{1\times N}$ are said to be {\em similar over
$\Z$} if there exists a matrix $T\in \Z ^{N\times N}$ with
$\hbox{\rm det }T=\pm 1$ (hence $T^{-1}\in \Z ^{N\times N}$ too) such that
\begin{equation*}
A=T^{-1}A^\flat T,\quad C=C^\flat T.
\end{equation*}
\end{defi}
The following result provides a sufficient condition for an observable pair
$(A,C)$ to admit a $\Z$-valued gain matrix $L$ such that $A-LC$ is Hurwitz.
\begin{prop}
\label{prop1}
Let $A\in \Z^{N\times N}$ and $C\in \Z ^{1\times N}$ be two
matrices such that $(A,C)$ is
similar over $\Z$ to a pair
$(A^\flat ,C^\flat )\in \Z^{N\times N}\times \Z ^{1\times N}$ in a companion canonical form. Let us denote by $(-\alpha^{N-1}~\cdots~-\alpha^0)'$ the first
column of $A^\flat$.
Then there exists a unique
matrix $L\in \Z ^{N\times 1}$ such that the matrix $A-LC$ is
Hurwitz (i.e., $\text{sp}(A-LC) \subset \{ z\in \C ;\ |z|<1\}$), namely
$L=T^{-1}L^\flat$ with $L^\flat=(-\alpha^{N-1}~\cdots~-\alpha^0)'$.
Furthermore, $(A-LC)^N=0$.
\end{prop}
{\em Proof.\ } Write $A=T^{-1}A^\flat T$, $C=C^\flat T$, with
$(A^\flat ,C^\flat )$ as in (\ref{companion}) and $T\in \Z^{N\times N}$
with $\hbox{\rm det }T=\pm 1$. For any given matrix
$L\in \Z^{N \times 1}$, we define the matrix
$L^\flat =(l^{N-1} \cdots \ l^0 )'$ by $L^\flat =TL$. Then,
$A-LC=T^{-1}(A^\flat -L^\flat C^\flat)T$ with
$$
A^\flat -L^\flat C^\flat =\left(
\begin{array}{lcccc}
- \alpha ^{N-1}-l^{N-1} & 1 & 0 & \cdots & 0 \\
- \alpha ^{N-2}-l^{N-2} & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-\alpha ^1-l^1 & 0 & 0 & \cdots & 1 \\
- \alpha ^0-l^0 & 0 & 0 & \cdots & 0
\end{array}
\right) .
$$
Its characteristic polynomial reads
$$\chi _{A^\flat -L^\flat C^\flat}(\lambda )=
\lambda ^{N}+(\alpha ^{N-1}+l^{N-1})\lambda ^{N-1}+\cdots
+(\alpha ^1+l^1)\lambda
+(\alpha ^0+l^0).$$
If $L$ is such that $A-LC$ is Hurwitz, then $A^\flat -L^\flat C^\flat
=T(A-LC)T^{-1}$ is Hurwitz too, hence
we may write
$\chi _{A-LC}(\lambda )=\chi _{A^\flat -L^\flat C^\flat}(\lambda )=
\lambda ^p \chi(\lambda )$,
where $p\in \{ 0,...,N \}$
and $\chi \in \Z [\lambda ]$ has its roots $\lambda _1,...,\lambda _{N-p}$ in
the set $\{ z\in \C;\ 0<|z|<1\}$. Assume that $p<N$, and denote by $q$
the constant coefficient of $\chi$. Then $q\ne 0$ (since $\chi (0)\ne 0$),
and $|q|=\prod _{i=1}^{N-p}|\lambda _i| <1$, which is impossible, since
$q\in \Z$. Therefore $p=N$ and $l^j=-\alpha ^j$ for any $j\in \{0,...,N-1\}$
(hence $L^\flat$ and $L$ are unique). On the other hand
\begin{equation}
\label{eq100t2}
A^\flat -L^\flat C^\flat =\left(
\begin{array}{lcccc}
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1 \\
0 & 0 & 0 & \cdots & 0
\end{array}
\right) \cdot
\end{equation}
For this choice of $L$, $\chi _{A-LC}(\lambda )
=\lambda ^{N}$ and $(A-LC)^{N}=0$. \qed
It should be emphasized that the above argument shows that
a $\Z$-valued matrix ${\cal N}$
is Hurwitz if and only if it is nilpotent. In other words, the
system $\nu _{k+1}={\cal N} \nu _k$ is {\em asymptotically stable}
if and only if it is {\em finite-time stable}.
We are now in a position to state the second main result of this chapter.
\begin{thm}
\label{thm4}
Let $(G,P)$ be a regular tiling of $\R ^N$, and let
$(A,B)\in \Z ^{N\times N}\times \R ^N$ be such that (H1) and (H2)
are fulfilled with $\R ^N/G'=\T ^N$. Assume given $C\in \Z ^{1\times N}$ such that
$(A,C)$ is similar over $\Z$ to a pair $(A^\flat, C^\flat )$ in a companion canonical form. Then one can pick two matrices $L\in \Z ^{N\times 1}$ and
$M\in \Z ^{N\times 1}$ so that
$(A-LC)M=0$ and $CM=1$. Furthermore
$$
X_k=\hat X_k \text { and } u_k=Y_k-\hat Y_k \qquad \forall k\ge N+1.
$$
\end{thm}
{\em Proof.} Let $T$, $A^\flat$, $C^\flat$, $L$ and
$L^\flat$ be as in the proof of Proposition \ref{prop1}. Set
$M^\flat =(1\ 0 \ \cdots 0 )'$ and $M=T^{-1}M^\flat $. Then
$(A-LC)M=T^{-1}(A^\flat -L^\flat C^\flat )T\cdot T^{-1}M^\flat =0$
by \eqref{eq100t2}, and $CM=C^\flat T\cdot T^{-1}M^\flat =1$. On the other hand, it follows from \eqref{errordyn} and the choice of $M$ that
$$
e_{k+1}=(A-LC)e_k \qquad \forall k\in \{ 1, ..., N \}
$$
hence $e_{N+1}=(A-LC)^Ne_1=0$. Since $X_{N+1}$ and $\hat X_{N+1}$ belong
to ${\cal T}'$ by construction, we have that $\hat X_{N+1}=X_{N+1}$.
To complete the proof, it is sufficient to prove the following\\
{\sc Claim.} For any $k\ge 0$, $\hat X_k=X_k$ implies $\hat X_{k+1}=X_{k+1}$.\\
Indeed, using the fact that $(A-LC)M=0$ and $\hat X_k=X_k$ we obtain that
\begin{eqnarray*}
\hat X_{k+1}
&=& \varpi _k (A\hat X_k + L C (X_k + M u_k -\hat X_k ) +B) \\
&=& \varpi _k ( A X_k + A M u_k +B) \\
&=& X_{k+1}.
\end{eqnarray*}
This completes the proof of Theorem \ref{thm4}. \qed
\begin{rem}
\begin{enumerate}
\item The projection $\varpi _k(x)$ allows to switch between
the dynamics \eqref{A100} and \eqref{A101} in $\R /G$ and $\R /G'$,
respectively. For a dynamics in $\T ^N$ only ($G'=G$), one can replace
$\varpi _k(x)$ by $\varpi '(x)$ (the projection onto $[0,1)^N$).
\item The result in Theorem \ref{thm4} remains true if we take
$\varpi_k(x)=\varpi '(x)$ for $k\le N$ and $\varpi_k(x)=\varpi (x)$ for $k\ge N+1$.
However, the definition of $\varpi_k(x)$ in \eqref{switch} guarantees that
a finite time synchronization occurs even if the output $Y_k$ is
not transmitted at some times. Such a property may be useful for the secured transmission of video sequences.
\item The output $Y_k=C(X_k+Mu_k)$ may be replaced by $\tilde Y_k=h(Y_k)$,
where $h:\R \to \R$ is a nonlinear invertible map. This renders the analysis
of the dynamics of $Y_k$ much more complicated.
\item In practice, when $\H =\T ^N$, the matrices $A,C,L$ and $M$ may be constructed in the following way. Pick any matrix $\hat T =[\hat T_{i,j}]\in \Z ^{N\times N}$ with $\hat T_{i,j}=0$ for $i>j$ and $\hat T_{i,i}=1$ for all $i$.
We set $T=\hat T'\ \hat T$. Note that $\text{det} \ \hat T=
\text{det} \ T=1$. Next, we pick a pair $(A^\flat, C^\flat )$ in a companion canonical form so that the roots of $\chi _{A^\flat}$ do not belong to
the set $\{ 0\}\cup \{ z\in \C;\ |z|=1\}$. Then $A,C,L$ and $M$ are defined by
$$
A=T^{-1}A^\flat T,\quad C=C^\flat T, \quad L=T^{-1}A^\flat (C^\flat )',\quad
\text{ and } \quad M=T^{-1}(C^\flat )'.
$$
\end{enumerate}
\end{rem}
\subsection{Numerical simulations}
This section is borrowed from \cite{RMBpreprint}.
Assume $\H=\T ^3$ and consider the dynamical system $(\Sigma _{A,b,M,C})$ with
$$
A=
\left(
\begin{array}{ccc}
-19 & 26 & 7\\
-51 & 65 & 17\\
152 & -184 & -47
\end{array}
\right),~~C=(6~-5~-1),~~b=0.
$$
$(\Sigma _{A,b})$ is chaotic by virtue of Theorem~\ref{thm1},
since $\hbox{\rm det } A=3$ (hence det $A\ne 0$) and the eigenvalues of $A$ are
-3, -0.4142, 2.4142 ($A$ has no roots
of unity as eigenvalues). The pair $(A,C)$ is similar over
$\Z$ to the pair $(A^\flat,C^\flat )$ in companion
canonical form, where
$$
A^{\flat}=
\left(
\begin{array}{ccc}
-1 & 1 & 0\\
7 & 0 & 1\\
3 & 0 & 0
\end{array}
\right),\
C^{\flat}=(1\ 0\ 0)
\ \hbox{\rm and }
~~T=
\left(
\begin{array}{ccc}
6 & -5 & -1\\
-5 & 10 & 3\\
-1 & 3 & 1
\end{array}
\right) .
$$
According to Proposition~\ref{prop1}, the unique matrix
$L\in \Z ^{N\times 1}$ such that $A-LC$ is Hurwitz is
$L=T^{-1}L^{\flat}$, with $L^{\flat}=(-1\ 7\ 3)^T$. We obtain
$L=(-2\ -6\ 19)^T$. The corresponding matrix
$M\in \Z ^{3\times 1}$ such that $(A-LC)M=0$ and $CM=1$ is
$M=(1\ 2\ -5)^T$.\\
The information to be masked is a flow corresponding to integers
ranging from 0 to 255. The data are scaled to give an
input $u_k$ ranging from 0 to $1$, and are embedded
into the chaotic dynamics of $(\Sigma_{A,b,M,C})$. From a practical point of
view, the transmitted signal $y_k$ cannot be coded with an infinite
accuracy and so it has to be truncated for throughput purpose. The observer
($\hat{\Sigma}_{A,b,M,C}$) is used in order to recover the information.
Numerical experiments bring out that the number of digits of the conveyed
output can actually be limited without giving rise to recovering errors.
The results
reported in Fig.~\ref{fig_1} show a perfect recovering for a number
of digits of $y_k$ equal to 4 (this is the minimum number required for
perfect retrieving).
\begin{figure}[hbt]
\begin{center}
\epsfig{figure=erreur,width=10cm}
\caption{A~: error on the recovered information $u_k-\hat{u}_k$; B~:
state reconstruction error $X_k-\hat{X}_k$}
\label{fig_1}
\end{center}
\end{figure}
The recovering error reaches zero after 3 steps, a fact which is consistent
with above theoretical results on finite time synchronization ($N=3$).
The figure highlights the fact that even though the state reconstruction may not be
perfect (residual errors due to truncations), a perfect information reconstruction
is nevertheless achieved.
\begin{rem}
Actually, for {\em any} system $\Sigma _{A,B,M,C}$, the numerical computations
can be performed {\em in an exact way}, i.e. without rounding errors, provided that
the number of digits is sufficiently large.
\end{rem}
\subsection{Concluding remarks}
The {\em message-embedding} masking technique studied here
does not originate from the conventional cryptography
(see \cite{Menal96} for a good survey). Nevertheless, it
seems to be highly related to some popular encryption schemes, the so-called
{\em stream ciphers} \cite{Millamigo05a}.
Therefore, it is desirable that the proposed scheme
be robust against both statistical and algebraic attacks. On one hand, the
robustness against statistical attacks follows from the chaotic behavior of the
output. On the other hand, the security against algebraic attacks rests on
the difficulty to identify the parameters of the
system. The identification of the parameters
is here a hard task for two reasons:
\begin{enumerate}
\item
The particular {\em structure} of the encryption system
$(\Sigma _{A,B,M,C})$,
that is the {\em dimension} of the matrix $A$ and the tiling of the space
used, is assumed to be unknown;
\item The ciphertext $u_k$ actually results from a mixing between the
plaintext $m_k$ and the state $X_k$ ($u_k=e(X_k,m_k)$).
This generally results in a {\em nonlinear} dynamics
$(\Sigma _{A,B,M,C})$, rendering the parameters hardly identifiable
\cite{LjungGlad94}.
\end{enumerate}
A real-time
implementation has already been carried out on an experimental platform
involving a secured multimedia communication. (For details about the
platform, see e.g. \cite{Milal03}).
| 80,690
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\begin{document}
\begin{abstract} Let $X$ be a topological space. Let $X_0 \subseteq X$ be a second countable subspace. Also, assume that $X$ is first countable at any point of $X_0$. Then we provide
some conditions under which we ensure that $X_0$ is not Baire.
\end{abstract}
\maketitle Subject Classification: 37B55, 54E52. \\
\maketitle Keywords: Nonautonomous systems, topological transitivity, Baire space, Birkhoff theorem.
\maketitle
\maketitle
\section{Introduction }
A space $X$ is called Baire if the intersection of any sequence of dense open subsets of $X$ is dense in $X$.
Alternatively, this notation can be formulated in terms of second category sets.
The Baire category theory has numerous applications in Analysis and Topology.
Among these applications are, for instance, the open mapping, closed graph theorem and
the Banach-Steinhaus theorem in Functional Analysis \cite {{Aarts}, {Haworth}}.
The aim of this paper is to introduce a trick that concludes the absence of Baire property for some topological spaces using dynamical techniques and tools. Before
stating the main result, we establish some notations.
Let $(X,~\tau)$ be a topological space, $X_n$'s its subspaces and
$$ x_{n+1}=f_n(x_n), ~ n \in \mathbb{N} \cup \{0\}, $$ where
$ f_n: {X_n} \to { X_{n+1}}$ are continuous maps.
The family $\{f_n\}_{n=0}^{\infty}$ is called a nonautonomous discrete system \cite{{Shi}, {Shi-Chen}}.
For given $x_{0}\in X_{0}$, the orbit of $x_0$ is defined as
$$ orb(x_{0}):=\big\{ x_{0}, ~ f_{0}(x_{0}), ~ f_{1}\circ f_{0}(x_{0}), ~\cdots , ~f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}(x_{0}),~\cdots \big\},$$
and we say that this orbit starts from the point $x_0$. The topological structure of the orbit that starts from the point $x_0$ may be complex.
Here, we study the points of $ X_{0}$ whose orbits always intersect around $ X_{0}$. They are formulated as follows:
$$O:= \big\{x \in X_0 | ~~ {\overline{orb(x)}}^{{X}}\cap X_0=X_0 \big\}.$$
The system $\{f_n\}_{n=0}^{\infty}$ is called topologically transitive on $ X_{0}$ if for any two non-empty open sets $U_{0}$ and $ V_{0}$ in $ X_{0}, $ there
exists $ n \in \mathbb{N}$ such that $U_{n}\cap V_{0}\neq \phi $, where $ U_{i+1}=f_{i}(U_i)$ for $ 0 \leq i \leq n-1$, in other word
$(f_{n-1} \circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0})(U_{0}) \cap V_{0}\neq \phi$ \cite {Shi-Chen}.
Our main theorem is as follows:
\begin{theorem} \label{Theorem }
Let $X$ be a topological space. Let $X_0$ be a second countable subspace of $ X$ and let $X$ be first countable at any point of $X_0$. Also, suppose that
the system $\{f_n\}_{n=0}^{\infty}$ is topologically transitive on $X_{0}$ and $ \overline{O}\neq X_0$. Then $X_0$ can not be a Baire subspace.
\end{theorem}
Note that, if $X$ is a metric space, $X_n=X$, and $f_n=f $ for each $n$, then Theorem \ref{Theorem } will be obtained as a direct result of Birkhoff transitivity theorem. This fact was our motivation in writing the paper.
\section{Proof }
So as $X_{0}$ is a second countable subspace and $X$ first countable at any point of $X_0$, it is easy to show that
there exists a collection $\{U_m\}_{m \in N}$ of open sets in $X$ such that
\begin{itemize}
\item [$i$)] $ U_m \cap D_0 \neq\phi$,
\item [$ii$)] the family $\{U_m \cap D_0\}_{m \in \mathbb{N}}$ is a basis for $D_0$,
\item [$iii$)]for each $x_0\in D_0$, the family $\{U_m\}_{m \in {\mathbb{N}}}$ is a local basis for $x_0$ in $X$.
\end{itemize}
We claim that
$$O=\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty}{f_{n-1}}
\circ {f_{n-2}} \circ \cdots \circ {f_{1}} \circ {f_{0}}^{-1}(U_{m}). \eqno{(2.1)}$$
To prove the claim, put $O^*:=\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty}{f_{n-1}\circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0}}^{-1}(U_{m}).$
Firstly, we show that $O \subseteq O^*$. Suppose otherwise, there is $ x \in O $ such that $x \notin O^*$. So as $x \notin O^*$,
there exists $m \in \mathbb{N}$ such that for each $n \in \mathbb{N}$ we have
$$ {(f_{n-1}\circ f_{n-2}\circ \cdots \circ f_{1} \circ f_{0})}(x) \notin {U_{m}}.$$
Hence, $orb(x) \cap U_m=\phi$. Since $U_m\cap D_0 \neq \phi$, there exists an element $z\in U_m\cap X_0$, such that $z\notin {\overline{orb(x)}}^X.$
But $ z \in X_0$ and so $ {\overline{orb(x)}}^X\cap X_0 \neq D_0$. It is concluded that $x \notin O$ which contradicts the choice of $x$.
Now, it is shown that $O^* \subseteq O$. Let $x\in O^*$ but $x \notin O$. So as $x\in O^*$, concluded
for each $m \in \mathbb{N}$, there exists $n \in \mathbb{N}$ such that
$ {(f_{n-1}\circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0})}(x) \in U_m.$
Thus, $orb(x) \cap U_m \neq \phi$. Moreover, the relation $x \notin O$ indicates that there exists $ z\in X_0$ such that $z \notin \overline{orb(x)}^X.$ Consequently,
there exists $U_k$ containing $z$, such that $U_k \cap orb(x)= \phi$ that this contradicts with $ orb(x) \cap U_m \neq \phi$, for each $m\in\mathbb{N}$.
\\ By continuity of $ f_n: {X_n} \rightarrow { X_{n+1}},$
each set $\bigcup_{n=1}^{\infty}{(f_{n-1} \circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0})}^{-1}(U_{m})$ is open and because of transitivity, these open sets are dense in $X_{0}$. If $X_{0}$ be a Baire space, then (2.1) implies that $O$ is a dense $ G_{\delta}$-set. This is a contradict with $ \overline{O}\neq X_0$. Thus $X_0$ is not a Baire subspace, and the proof of the Theorem \ref{Theorem } is complete.
\section{Example }
\begin{example}
Consider $X=\mathbb H(\mathbb C)
=\big\{f:\mathbb C \rightarrow \mathbb C|~ f ~ is ~holomorphic \big\}$ endowed with the metric $d(f,g)=\displaystyle\sum_ {n=1}^{\infty}\frac {1}{2^n} min \big(~1,~p_n(f-g)\big),$
with $p_n(h) ={sup}_{|z| \leq n } |h(z)|$.
Then $X$ is a separable Banach space and besides that the differentiation operator $D:\mathbb H(\mathbb C) \rightarrow \mathbb H(\mathbb C)$ with $ D(f)=f^\prime$ is continuous \cite {Grosse-Erdmann}. Moreover, the space $\mathbb H(\mathbb C)$ is Baire and if we consider the dynamical system $ D:\mathbb H(\mathbb C ) \rightarrow \mathbb H(\mathbb C),$ then Birkhoff theorem guarantees the existence of functions that their orbit is dense in $\mathbb{ H} (\mathbb{ C})$.
Now, assume that
$$ X_0=\big\{\sum_{i=0}^{N} a_iz^i+\alpha g(z)\big |~ a_i , \alpha \in \mathbb{C} \big\}.$$
Then the subspace $ X_0$ is not Baire. To see this, take $\{\alpha_n\}_{n=0}^{\infty}$ be a subsequence
with $\alpha_0=0$ in this way that $D^{\alpha_n}(g)$ is convergent. We consider nonautonomous discrete system $\{f_{n}\}_{n=0}^{\infty}$ with $f_n=D^{\alpha_{n+1}-\alpha_n}$ where
$ X_{n}=\big\{\sum_{i=0}^{N} a_iz^i+\alpha g^{(\alpha_n)}(z)\big | a_i , \alpha \in \mathbb{C} \big\}$. By planning the arguments similar to what employed in the proof of Example 2.21 in \cite{Grosse-Erdmann}, we observe that the
system $\{f_n\}_{n=0}^{\infty}$ is topologically transitive. Now the assertion obtains by using Theorem \ref{Theorem } since the set $ O$ is empty.
\end{example}
| 174,125
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Mescalero Apache Man Pleads Guilty to Federal Methamphetamine Distribution Charge
Defendant is One of 34 Individuals Charged as Part of Investigation into Methamphetamine Trafficking on the Mescalero Apache Reservation
ALBUQUERQUE –.
Rice was one of 34 individuals charged with federal and tribal drug offenses as the result of an 18-month multi-agency investigation led by the DEA and BIA.
Rice was arrested on Dec. 29, 2015, on a criminal complaint charging him with distribution of methamphetamine on March 25, 2015, in Otero County, N.M. According to the complaint, on March 25, 2015, Rice sold approximately one gram of methamphetamine to undercover law enforcement agents in Mescalero.
During today’s proceedings, Rice pled guilty to a felony information charging him with possession of methamphetamine with intent to distribute and admitted that on March 25, 2015, he sold .75 grams of pure methamphetamine to an individual who, unbeknownst to him, was an undercover law enforcement agent for $100.00.
At sentencing, Rice faces a maximum penalty of 20 years in federal prison followed by not less than three years of supervised release. Rice remains in custody pending a sentencing hearing which has yet to be scheduled..
| 262,325
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It has been almost 5 years since I photographed my loft for this Design*Sponge peek. I've never really taken full angle shots of my studio space, located in the loft. Since my studio is currently the cleanest it has been in a few years, I thought it good opportunity to document it. I'm pretty proud of myself that not much as changed in my studio decor-wise in the past 5 years, and I still aesthetically love it.
Wednesday, February 13, 2013
4 comments:
Christie Chase! I was just digging thru your blog and you are inspirational, girlie! Love the ladder on the bookshelves and want to do something similar outside of outright buying one on Restoration hardware. Did you find ladder and attach? I'll keep digging ;) Celeste (St. Amant)
Hi Celeste!
Thanks for the kind words about my blogging : )
The ladder in my studio is actually repurposed from inside my bungalow. It was the attic ladder, but it did not meet current code when we were remodeling, so we turned it into the bookcase ladder for my studio. Besides the ladders at Restoration Hardware, World Market sells a bookcase with ladder.
Here's a link
It also might be possible to have a local carpenter build you a ladder to your specs.
Hope to see you soon!
Christie
Hi! I google searched "art studio table" and found your table from apartment therapy which I read all the time anyway :) anyway I love it!! My husband is going to make me something very similar for my new studio! Also I'm a blogger in Houston too! Love your blog
Also how tall is your work table? Thanks!
Taylor
Hi Taylor,
My table is about 38-40 inches high, which suits my 5'6" height!
Christie
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TITLE: How do we prove that $\sup_{n\geq 1}f_{n}$ is a measurable function when each term $f_{n}$ is measurable?
QUESTION [2 upvotes]: Proposition
For each $n\in\mathbb{N}$, let $f_{n}:(\Omega,\mathcal{F})\to(\overline{\mathbb{R}},\mathcal{B}(\overline{\mathbb{R}}))$ be a $\langle\mathcal{F},\mathcal{B}(\overline{\mathbb{R}})\rangle$-measurable function. Then $\sup_{n\in\mathbb{N}}f_{n}$ is $\langle\mathcal{F},\mathcal{B}(\overline{\mathbb{R}}\rangle$-mesurable.
Proof
Let $g = \sup_{n\geq 1}f_{n}$. To show that $g$ is $\langle\mathcal{F},\mathcal{B}(\overline{\mathbb{R}}\rangle$-measurable, it is enough to show that $\{\omega\in\Omega : g(\omega)\leq r\}\in\mathcal{F}$ for all $r\in\mathbb{R}$.
Now, for any $r\in\mathbb{R}$,
\begin{align*}
\{\omega:g(\omega)\leq r\} & = \bigcap_{n=1}^{\infty}\{\omega:f_{n}(\omega)\leq r\}\\\\
& = \bigcap_{n=1}^{\infty}f^{-1}_{n}((-\infty,r)])\in\mathcal{F}
\end{align*}
since $f^{-1}_{n}((-\infty,r])\in\mathcal{F}$ for all $n\geq 1$, by the measurability of $f_{n}$.
My concerns
I do not know how to interpret the symbol $\sup_{n\in\mathbb{N}}f_{n}$.
As far as I have understood, for each $\omega\in\Omega$, $g(\omega) = \sup_{n\geq 1}f_{n}(\omega)$.
That is to say, for each $\omega\in\Omega$, $\sup_{n\geq 1}f_{n}(\omega)$ is the least upper bound of the sequence $f_{n}(\omega)$.
Is it correct to think so?
If it is not the case, please let me know.
Moreover, is there a more detailed way to write the proof? I've tried the following one.
Since $g(\omega)\geq f_{n}(\omega)$ for every natural $n$, one has that
\begin{align*}
x\in\{\omega:g(\omega)\leq r\} \Rightarrow g(x)\leq r & \Rightarrow (\forall n\in\mathbb{N})(f_{n}(x)\leq g(x) \leq r)\\\\
& \Rightarrow (\forall n\in\mathbb{N})(f_{n}(x)\leq r)\\\\
& \Rightarrow x\in\bigcap_{n=1}^{\infty}\{\omega:f_{n}(\omega)\leq r\}
\end{align*}
Conversely, if $f_{n}(\omega)\leq r$ for every $n\in\mathbb{N}$, taking the sup one obtains that $g(\omega) = \sup f_{n}(\omega)\leq \sup r = r$.
This means that
\begin{align*}
x\in\bigcap_{n=1}^{\infty}\{\omega:f_{n}(\omega)\leq r\} \Rightarrow x\in\{\omega: g(\omega)\leq r\}
\end{align*}
Hence we conclude that both sets are equal.
REPLY [1 votes]: You are on the right track. Just need to use the properties of the supremum to get
$$\{\sup_nf_n>a\}=\bigcup_n\{f_n>a\}$$
where $\{h>a\}:=\{\omega\in \Omega: h(\omega)>a\}$. If each $f_n$ is measurable, then each set $\{f_n>a\}$ is measurable and so the union of all of them.
Recall that a real valued function $g$ is (Borel) measurable iff $\{g>a\}$ is measurable.
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BGMEA
25 January 2020
GREEN GROWTH
For Sustainable Development
BGMEA held its second monthly session on Green Growth. While paradoxically prices have fallen, the industry has continued going green. BGMEA President Dr. Rubana Huq moderated the session. The session started with the presentation on Green Growth of Bangladesh RMG Industry. The presentation focused on global trend of sustainability and fashion, impact of the fashion industry on environment, Bangladesh RMG industry’s strides toward sustainability, BGMEA’s activities in the area of environmental sustainability, factory best green practices, green finance and strategic planning to move forward.
The presentation was followed by group discussions. Discussants divided into four groups to ideate on how to take the following conversations to the next level. The groups were: 1) Green Community, 2) Capacity Development and/Home Grown Solution, 3) Green Project, and 4) Green Financing. The discussion was a vibrant discourse between government, industry, brand, development partners, think tanks and diplomats. Key points from the discussions are as follows:
GREEN COMMUNITY
Identifying cluster would be the first initiative for green community development. For example, 15-20 clusters can be identified only in Gazipur area. Clusters might be identified in the factory neighborhoods. The procedure of cluster mapping was raised as the next question. WARPO and UN agencies can help BGMEA with their knowledge and tools in this activity. BGMEA would circulate a general email which would act as a quick survey for cluster mapping and identifying areas where support is needed. For example, education, housing, etc. After identifying the cluster, Green Ambassadors and champions will be declared; then BGMEA can urge the government to help the community adjacent to factories.
In Bangladesh Water Rules 2018, a roadmap for water stewardship is proposed. This roadmap is designed as Integrated Water Management system to reach the grassroot level. The tool developed for this mapping has been used by business community in developed countries for sustainable business.
Green factory owners can be declared as Green Ambassadors and they will be recognized. The workers from green factories do not maintain a green livelihood. If green factory owners can motivate their workers to adapt green practices and declare them Green Champions. Green ambassadors can work for low costhousing, better education etc. for the worker community. BGMEA can urge to government to help the community to adjacent to factories for their better livelihood.
ACTION ITEMS
Identification of cluster
Quick survey of factory neighborhoods
Mapping of workers livelihood
Form a league of Green Ambassadors (Green Factory Owners) and Green Champion (Green Factory Workers)
Ask for government’s support in the next fiscal budget
CAPACITY DEVELOPMENT/HOME GROWN SOLUTION
A full functional environment cell with adequate resources and knowledge needs to be set up in BGMEA. A platform or center could be established where environmental professional can communicate for troubleshooting regarding ETP, HVAC systems, environment management systems, energy management systems etc. BGMEA could have energy auditors in their expert panel who can contribute in upcoming energy audits, energy rating of equipment net metering requirements etc. A ‘Community of Excellence’ can also be set up instead of a center. One center could not have expertise on all the topics. If a community was developed, they can help the sector on required solutions. It could mobilize the sector with the association of international expertise.
Awareness building activities can be an effective tool to encourage green solutions in the sector. Center like ‘Women Café’ could help raising among workers on environmental topics and ecological challenges as these are cross cutting issue. Survey and mapping green activities had already been initiated by BGMEA. A questionnaire had been prepared by BGMEA and distribute among the members. 35 factories have been participated in this survey so far. BGMEA has been getting support from UNDP and GRI in these activities.
Success stories of green factories, like Cute Dress Industry Ltd. could be showcased. Cute Dresses Limited is a factory of 600 workers and they have achieved LEED Platinum certificate they have showed innovative activities in water recycling, energy efficient cooler, indoor water quality monitoring devices and many more. It’s an empirical evidence of how a garment factory can be small yet sustainable. If such initiatives should be encouraged and incentivized, more factories would come toward to have own green solutions.
RECOMMENDED ACTION ITEMS
Establish and strengthen BGMEA environment cell
Organize awareness raising programs
Encourage and survey factories on best available practices
Incentivizing factories for adopting best available green practices
GREEN PROJECTS
Concern over the upcoming hundred economic zones, how these zones will be going to affect natural resources was expressed by the development partners. Formulation of a regulatory framework to protect natural resources like ground water was proposed by them. Interest expressed by development partners and government representatives to support the readymade garments and textile sector regarding home grown solutions health, people, environment.
Innovative business models, for example, third party service providers for private to private power selling; projects on circular economy; effluent treatment plant (ETP) management; waste management; sludge management; zero liquid discharge (ZLD) system etc., which may not be suitable for many existing factories. But such business models can be introduced to factories which could be set up in special economic zones like Mirsarai. Since it is very difficult to achieve zero discharge goals by 2030; vast, inclusive targets and home-grown solutions are needed to meet this international demand.
There is no international convention regarding environment unlike labor standards. There is a competition between standards and initiatives in this sector. If Bangladesh could have its own standard, apparel industry does not need to follow standards or initiatives imposed to them. Readymade garments and textile mills community can upgrade by redefine and formulate their own standards.
ACTION ITEMS
Showcasing business case of a factory on environmental activities
Revisiting the existing standards by developing Bangladeshi Standards
Identify the need of home-grown solutions
Strengthening Environment component of RSC (RMG sustainability Council)
Innovative business models, for example, business to business power sell; third party of business solution providers for circular economy projects, effluent treatment plant (ETP) management, waste management, sludge management, zero liquid discharge (ZLD) system etc. for the upcoming special economic zones in Mirsarai.
GREEN FINANCING
The discussion on green financing started with the question, whether there is lack of finance or lack of capacity to access the finance? All the participants agreed to the statement that there is no lack of finance but there is definitely lack of capacity to access the available finance. Funds like Global Climate Fund (GCF) comes with a very strict due diligence which is very difficult for SMEs to access due to lack of capacity and knowledge. Green Transformation Fund (GTF) from Bangladesh Bank was poorly utilized due to same reason. Accesses to the available green finance are easier for big enterprises but then again the interest rate (6%-9%) is not helping them to offset the initial risk to be feasible, the interest rates are not set based on the ROI of different environment friendly technologies. The due diligence framework could be revisited so that the SMEs can get access to the green finances. From the discussion the challenges identified were lack of concessional fund, stricter due diligence, reluctances of the financial institutes to understand the worthiness of such finance. The experience of borrower of available green funds and mindset of financiers should be collated and studied. All the available green finance schemes should be mapped and the criteria to access need to be identified as well. Lead of this mapping activities can be taken by BGMEA as the leading trade association of Bangladesh in collaboration with other stakeholders. A pressure group or working group can also be formed by BGMEA for flexible requirements of the green funds.
Another important topic arisen from this discussion that self-assessment of resource consumption could be done by factories. So that they can identify the areas they need support. An insurance mechanism should be established so that resource efficient technology provider would guarantee the factory owners on resource saving made from adopting that technology.
ACTION ITEMS
Industry mapping to identify problems of factories in availing green finance
Revisit fund requirements and make it more inclusive for SMEs
Search for Government support as guarantor for SMEs to avail Green Finance
Guarantee or Insurance of resource efficiency technology from the provider
Formation of working group regarding green financing
WAY FORWARD
BGMEA President Dr. Rubana Huq recommends forming four working groups based on the discussion of the session. All the stakeholders would be virtually connected via Google Drive to share their views and opinions. Dr. Rubana Huq ended the session saying there would be upcoming sessions on innovations and green finance.
PARTICIPANTS
1. Winnie Estrup Petersen, The Ambassador of Denmark to Bangladesh
2. Tuomo Poutiainen, Country Director, ILO
3. Jeremy Opritesco, Deputy Head of Mission, Delegation of EU to Bangladesh
4. Chiara Vidussi, Attaché, Delegation of EU to Bangladesh
5. Mercy MiyangTembon ,Country Director for Bangladesh and Bhutan, World Bank
6. Sanjeev Dua, Vice President, Global Manufacturing and Sourcing, Ralph Lauren
7. Taufiqul Islam, Director, Water Resources Planning Organization,
8. Syed Nazmul Ahsan, Director, DoE
9. Arch. Nafees Rahman, HBRI &Former Deputy Director, SREDA
10. Md. Shahan Reza, Country Relation Manager, Bangladesh, BluesignTechnologies
11. Nishat Chowdhury, Program Manager, PaCT, IFC,
12. Linda Germanis, UNDP
13. Rebecca Peters, Oxford University
14. Dr. Khondaker Golam Moazzem, Research Director, CPD
15. Dr. Nurun Nahar, Deputy Chief, Programming Division, Bangladesh Planning Commission
16. Werner Lange, Coordinator of Textile Cluster, GIZ
17. Dr. Mohammed Abed Hossain, Professor, IWFM, BUET
18. Dr. Sebastian Groh, Managing Director, ME Solshare Ltd.
19. Farhtheeba Rahat Khan, Team Leader, RMG Program, SNV Netherland Development Organization
20. Jamal Uddin, Inclusive Business Advisor, SNV Netherland Development Organization
21. Mosleh Uddin, Unit Head, GCF, IDCOL
22. Faisal Rabbi, GIZ
23. Md. Mahady Hassan, 2030 WRG, World Bank
24. Judith Herbertson, Head, DFID Bangladesh
25. Ms. Tanuja Bhattacharjee, Energy Specialist, World Bank
From BGMEA:
1. Dr. Rubana Huq, President, BGMEA
2. Mr. Arshad Jamal (Dipu), Vice President, BGMEA
3. Mr. Miran Ali, Director, BGMEA
4. Mr. Md. Kamruzzaman Jewel, Chairman, Standing Committee, Subcontracting
5. Md. Abdul Jalil, Member, Chairman, Standing Committee, Floor Price and cost Review
6. Mr. Asif Ashraf, Managing Director, Urmi Group
7. Mr. Mijanur Rahman, Chairman, Standing Committee, Branding BGMEA
8. Mr. Khan Monirul Alam (Shuvo), Chairman, Standing Committee, PR
9. Mr. Wasim Zakaria, Chairman, Standing committee, RDTI and SDGs Affairs
.
© 2021, Bangladesh Garment Manufacturers and Exporters Association
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\begin{document}
\title{Asymptotic stability of ground states in 3D nonlinear Schr\" odinger equation including subcritical cases}
\author{E. Kirr\thanks{Department of Mathematics, University of Illinois at Urbana-Champaign}\ \ and \" O. M\i zrak\footnotemark[1]}
\maketitle
\begin{abstract}
\noindent We consider a class of nonlinear Schr\"{o}dinger equation in three space
dimensions with an attractive potential. The nonlinearity is local but rather general
encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic
stability of the nonlinear bound states, i.e. periodic in time
localized in space solutions. Our result shows that all solutions
with small initial data, converge to a nonlinear bound state.
Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we
obtain for the time dependent, Hamiltonian, linearized dynamics
around a careful chosen one parameter family of bound states that ``shadows" the nonlinear evolution of the
system. Due to the generality of the methods we develop we expect
them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.
\end{abstract}
\section{Introduction}
In this paper we study the long time behavior of solutions of the
nonlinear Schr\" odinger equation (NLS) with potential in three
space dimensions (3-d):
\begin{align}
i\partial_t u(t,x)&=[-\Delta_x+V(x)]u+g(u), \quad t\in\R,\quad x\in\R^3 \label{u} \\
u(0,x)&=u_0(x) \label{ic}
\end{align}
where the local nonlinearity is constructed from the real valued,
odd, $C^2$ function $g:\mathbb{R}\mapsto\mathbb{R}$ satisfying
\begin{equation}\label{gest}
|g''(s)|\leq C(|s|^{\alpha_1}+|s|^{\alpha_2}),\quad s\in\mathbb{R}\
0<\alpha_1\le\alpha_2<3\end{equation} which is then extended to a
complex function via the gauge symmetry:
\begin{equation}\label{gsym}
g(e^{i\theta}s)=e^{i\theta}g(s)
\end{equation}
The equation has important applications in statistical physics
describing certain limiting behavior of Bose-Einstein condensates
\cite{dgps:bec,lsy:3d,htyau:gp}.
It is well known that this nonlinear equation admits periodic in
time, localized in space solutions (bound states or solitary waves).
They can be obtained via both variational techniques
\cite{bl:i,str:sw,rw:bs} and bifurcation methods
\cite{pw:cm,rw:bs,kz:as2d}, see also next section. Moreover the set
of periodic solutions can be organized as a manifold (center
manifold). Orbital stability of solitary waves, i.e. stability
modulo the group of symmetries $u\mapsto e^{-i\theta}u,$ was first
proved in \cite{rw:bs,mw:ls}, see also \cite{gss:i,gss:ii,ss:ins}.
In this paper we show that solutions of \eqref{u}-\eqref{ic} with
small initial data asymptotically converge to the orbit of a certain
bound state, see Theorem \ref{mt}. Asymptotic stability studies of
solitary waves were initiated in the work of A. Soffer and M. I.
Weinstein \cite{sw:mc1,sw:mc2}, see also
\cite{bp:asi,bp:asii,bs:as,sc:as,gnt:as}. Center manifold analysis
was introduced in \cite{pw:cm}, see also \cite{kn:Wed}.
The main contribution of our result is to allow for subcritical and
critical ($L^2$) nonlinearities, $0<\alpha_1\le 1/3$ in
\eqref{gest}. To accomplish this we develop an innovative technique
in which linearization around a one parameter family of bound states
is used to track the solution. Previously a fixed bound state has
been used, see the papers cited in the previous paragraph. By
continuously adapting the linearization to the actual evolution of
the solution we are able to capture the correct effective potential
induced by the nonlinearity $g$ into a time dependent linear
operator. Once we have a good understanding of the semigroup of
operators generated by the time dependent linearization, see Section
\ref{se:lin}, we obtain sharper estimates for the nonlinear dynamics
via Duhamel formula and contraction principles for integral
equations, see Section \ref{se:main}. They allow us to treat a large
spectrum of nonlinearities including, for the first time, the
subcritical ones.
The main challenge is to obtain good estimates for the semigroup of
operators generated by the time dependent linearization that we use.
This is accomplished in Section \ref{se:lin}. The technique is
perturbative, and similar to the one developed by the first author
and A. Zarnescu for 2-D Schr\" odinger type operators in
\cite{kz:as2d}, see also \cite{kz:as2d2}. The main difference is
that in 3-D one needs to remove the non-integrable singularity in
time at zero of the free Schr\" odinger propagator:
$$\|e^{i\Delta t}\|_{L^{1}\mapsto L^\infty}\sim |t|^{-3/2}.$$
We do this by generalizing a Fourier multiplier type estimate first
introduced by Journ\' e, Soffer, and Sogge in \cite{kn:JSS} and by
proving certain smoothness properties of the effective potential
induced by the nonlinearity, see the Appendix.
Since our methods rely on linearization around nonlinear bound states and estimates for integral
operators we expect them to generalize to the case of large
nonlinear ground states, see for example \cite{sc:as}, or the
presence of multiple families of bound states, see for example
\cite{sw:sgs}, where it should greatly reduce the restrictions on
the nonlinearity. We are currently working on adapting the method to
other spatial dimensions. The work in 2-D is almost complete, see
\cite{kz:as2d,kz:as2d2}.
\bigskip
\noindent{\bf Notations:} $H=-\Delta+V;$
$L^p=\{f:\mathbb{R}^2\mapsto \mathbb{C}\ |\ f\ {\rm measurable\
and}\ \int_{\mathbb{R}^2}|f(x)|^pdx<\infty\},$
$\|f\|_p=\left(\int_{\mathbb{R}^2}|f(x)|^pdx\right)^{1/p}$ denotes
the standard norm in these spaces;
$<x>=(1+|x|^2)^{1/2},$ and for $\sigma\in\mathbb{R},$ $L^2_\sigma$
denotes the $L^2$ space with weight $<x>^{2\sigma},$ i.e. the space
of functions $f(x)$ such that $<x>^{\sigma}f(x)$ are square
integrable endowed with the norm
$\|f(x)\|_{L^2_\sigma}=\|<x>^{\sigma}f(x)\|_2;$
$\langle f,g\rangle =\int_{\mathbb{R}^2}\overline f(x)g(x)dx$ is the
scalar product in $L^2$ where $\overline z=$ the complex conjugate
of the complex number $z;$
$P_c$ is the projection on the continuous spectrum of $H$ in $L^2;$
$H^n$ denote the Sobolev spaces of measurable functions having all
distributional partial derivatives up to order $n$ in $L^2,
\|\cdot\|_{H^n}$ denotes the standard norm in this spaces.
\bigskip
\noindent{\bf Acknowledgements:} The authors would like to thank
Wilhelm Schlag and Dirk Hundertmark for useful discussions on this
paper. Both authors acknowledge the partial support from the NSF
grants DMS-0603722 and DMS-0707800.
\section{Preliminaries. The center manifold.}\label{se:prelim}
The center manifold is formed by the collection of periodic solutions for (\ref{u}):
\begin{equation}\label{eq:per}
u_E(t,x)=e^{-iEt}\psi_E(x)
\end{equation}
where $E\in\mathbb{R}$ and $0\not\equiv\psi_E\in H^2(\mathbb{R}^3)$
satisfy the time independent equation:
\begin{equation}\label{eq:ev}
[-\Delta+V]\psi_E+g(\psi_E)=E\psi_E
\end{equation}
Clearly the function constantly equal to zero is a solution of (\ref{eq:ev})
but
(iii) in the following hypotheses on the potential $V$ allows for a
bifurcation
with a nontrivial, one parameter family of solutions:
\bigskip
\noindent{\bf (H1)} Assume that
\begin{itemize}
\item[(i)] There exists $C>0$ and $\rho >3$ such that:
\begin{enumerate}
\item $|V(x)|\le C<x>^{-\rho},\quad {\rm for\ all}\ x\in\mathbb{R}^3;$
\item $\nabla V\in L^p(\mathbb{R}^3)$ for some $2\le p\le\infty$
and $|\nabla V(x)|\rightarrow 0$ as $|x|\rightarrow\infty ;$
\item the Fourier transform of $V$ is in $L^1.$
\end{enumerate}
\item[(ii)] $0$ is a regular point\footnote{see
\cite[Definition 6]{gs:dis} or $M_\mu=\{0\}$ in relation (3.1) in \cite{mm:ae}}
of the
spectrum of
the linear operator $H=-\Delta+V$ acting on $L^2.$
\item [(iii)]$H$ acting on $L^2$ has exactly one
negative eigenvalue $E_0<0$ with corresponding normalized
eigenvector $\psi_0.$ It is well known that $\psi_0(x)$ is exponentially decaying as
$|x|\rightarrow\infty,$ and can be
chosen strictly positive.
\end{itemize}
\par\noindent Conditions (i)1. and (ii) guarantee the applicability of dispersive estimates of Murata \cite{mm:ae} and Goldberg-Schlag \cite{gs:dis} to the
Schr\" odinger group $e^{-iHt}.$ Condition (i)2. implies certain
regularity of the nonlinear bound states while (i)3. allow us to use
commutator type estimates, see Theorem \ref{th:gjss}. All these
are needed to obtain estimates for the semigroup of operators generated by our time dependent linearization,
see Theorems \ref{th:lw} and \ref{th:lp} in
section \ref{se:lin}. In particular (i)1. implies the local well
posedness in $H^1$ of the initial value problem
(\ref{u})-(\ref{ic}), see section \ref{se:main}.
By the standard bifurcation argument in Banach spaces \cite{ln:fa}
for (\ref{eq:ev}) at $E=E_0,$ condition (iii) guarantees existence
of nontrivial solutions. Moreover, these solutions can be organized
as a $C^1$ manifold (center manifold), see \cite[section
2]{kz:as2d}. Since our main result requires, we are going to show in
what follows that the center manifold is $C^2.$ We note that for
three and higher dimensions this has been sketched in \cite{gnt:as},
however they show smoothness by formal differentiation of certain
equations without proof that at least one side has indeed
derivatives.
As in \cite{kz:as2d} we decompose the solution of (\ref{eq:ev}) in
its projection onto the discrete and continuous part of the spectrum
of $H:$
$$\psi_E=a\psi_0+h,\quad a=\langle \psi_0,\psi_E\rangle,\
h=P_c\psi_E.$$ Projecting now (\ref{eq:ev}) onto $\psi_0$ and its
orthogonal complement $={\rm Range}\ P_c$ we get:
{\setlength\arraycolsep{2pt}
\begin{eqnarray}
0&=&h+(H-E)^{-1}P_cg(a\psi_0+h)\label{eq:evc}\\
0&=&E-E_0- a^{-1}\langle\psi_0,g(a\psi_0+h)\rangle\label{eq:evp}
\end{eqnarray}}
Although we are using milder hypothesis on $V$ the argument in the Appendix
of
\cite{pw:cm} can be easily adapted to show that:
$$\mathcal{F}(E,a,h)=h+(H-E)^{-1}P_cg(a\psi_0+h)$$
is a $C^2$ function from $(-\infty,0)\times\mathbb{C}\times
\Ls\cap H^2$ to $L^2_\sigma\cap H^2$ and $\mathcal{F}(E_0,0,0)=0,$ $D_h\mathcal{F}(E_0,0,0)=I.$ Therefore the implicit
function theorem applies to equation (\ref{eq:evc}) and leads to the
existence of $\delta_1>0$ and the $C^2$ function $\tilde h(E,a)$
from $(E_0-\delta_1,E_0+\delta_1)\times \{a\in\mathbb{C}\ :\
|a|<\delta_1\} $ to $L^2_\sigma\cap H^2$ such that (\ref{eq:evc})
has a unique solution $h=\tilde h(E,a)$ for all $E\in
(E_0-\delta_1,E_0+\delta_1)$ and $|a|<\delta_1.$ Note that, by gauge
invariance, if $(a,h)$ solves (\ref{eq:evc}) then
$(e^{i\theta}a,e^{i\theta}h),\ \theta\in[0,2\pi )$ is also a
solution, hence by uniqueness we have:
\begin{equation}\label{eq:hsym}
\tilde h(E,a)=\frac{a}{|a|}\tilde h(E,|a|).
\end{equation}
Because $\psi_0$ is real valued, we could apply the implicit function theorem
to
(\ref{eq:evc}) under the restriction $a\in\mathbb{R}$ and $h$ in the subspace
of
real valued functions as it is actually done in \cite{pw:cm}. By uniqueness of
the
solution we deduce that $\tilde h(E,|a|)$ is a real valued function.
Consider now the restriction of $\tilde h(E,a)$ to $a\in\mathbb{R},\
|a|<\delta_1.$ This is now a real valued $C^2$ function on
$(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1)$ which, by
\eqref{eq:hsym}, is odd in the second variable. We now differentiate
\eqref{eq:evc} with $h=\tilde h(E,a),$ to obtain the following
estimates for the first and second derivatives of $\tilde h$ on
$(E,a)\in(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1):$
\begin{eqnarray}
\frac{\partial\tilde h}{\partial a}(E,a)&=&-(D_h\mathcal{F})^{-1}(E,a,\tilde
h(E,a))[(H-E)^{-1}P_c g'(a\psi_0+\tilde h(E,a))\psi_0]=\mathcal{O}(|a|^{1+\alpha_1})\nonumber\\
\frac{\partial\tilde h}{\partial E}(E,a)&=&(D_h\mathcal{F})^{-1}(E,a,\tilde
h(E,a))[(H-E)^{-2}P_c g(a\psi_0+\tilde h(E,a))]=\mathcal{O}(|a|^{2+\alpha_1})\nonumber\\
\frac{\partial^2\tilde h}{\partial a^2}(E,a)&=&
-(D_h\mathcal{F})^{-1}\left[(H-E)^{-1}P_c g''(a\psi_0+\tilde h(E,a))\left(\psi_0+\frac{\partial\tilde h}{\partial a}\right)^2\right]=\mathcal{O}(|a|^{\alpha_1}) \nonumber \end{eqnarray}
\begin{eqnarray}
\frac{\partial^2\tilde h}{\partial E\partial a}(E,a)=\mathcal{O}(|a|^{1+\alpha_1})&=&
(D_h\mathcal{F})^{-1}\left[(H-E)^{-2}P_c g'(a\psi_0+\tilde h)\left(\psi_0+\frac{\partial\tilde h}{\partial a}\right)\right]\nonumber\\
&-&(D_h\mathcal{F})^{-1}\left[(H-E)^{-1}P_c g''(a\psi_0+\tilde h)\left(\psi_0+\frac{\partial\tilde h}{\partial a}\right)\frac{\partial\tilde h}{\partial E}\right]\nonumber\\
\frac{\partial^2\tilde h}{\partial E^2}(E,a)=\mathcal{O}(|a|^{2+\alpha_1})&=&
+(D_h\mathcal{F})^{-1}\left[(H-E)^{-1}P_c g''(a\psi_0+\tilde h)\left(\frac{\partial\tilde h}{\partial E}\right)^2\right]\nonumber\\
&-&(D_h\mathcal{F})^{-1}\left[2(H-E)^{-3}P_c g(a\psi_0+\tilde h)-2(H-E)^{-2}P_c g'(a\psi_0+\tilde h)\frac{\partial\tilde h}{\partial E}\right]\nonumber
\end{eqnarray}
where we used $D_h\mathcal{F}(E,a,\tilde
h(E,a))$ is invertible with bounded inverse and $D_h\mathcal{F}(E,0,0)=I,$ $(H-E)^{-1}$ is bounded
and analytic operator in $E\in(E_0-\delta_1,E_0+\delta_1),$ and
$g'(s)=\mathcal{O}(s^{1+\alpha_1}),$ $g''(s)=\mathcal{O}(s^{\alpha_1})$ as
$s\rightarrow 0.$
Replacing now $h=\tilde h(E,a),\
(E,a)\in(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1) $ in
(\ref{eq:evp}) we get:
\begin{equation}\label{eq:evp1}
E-E_0=a^{-1}\langle\psi_0,g(a\psi_0+\tilde h(E,a))\rangle.
\end{equation}
To this we can apply again the implicit function theorem by
observing that $ G(E,a)=E-E_0-a^{-1}\langle\psi_0,g(a\psi_0+\tilde
h(E,a))\rangle$ is a $C^1$ function from
$(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1)$ to
$\mathbb{R}$ with the properties $G(E_0,0)=0,$ $\partial_E
G(E_0,0)=1.$ We obtain the existence of $0<\delta\le\delta_1,$ and
the $C^1$ even function $\tilde E:(-\delta , \delta)\mapsto
(E_0-\delta,E_0+\delta)$ such that, for $|E-E_0|,|a|<\delta,$ the
unique solution of (\ref{eq:evp}) with $h=\tilde h(E,a),$ is given
by the $E=\tilde E(a).$ Note that $\tilde E$ is $C^2$ except at
$a=0$ because $G$ is $C^2$ except at $a=0,$ and:
\begin{eqnarray}
\frac{d\tilde E}{da}(a)&=&-\frac{\partial_a G(E(a),a)}{\partial_E
G(E(a),a)}=\mathcal{O}(|a|^{\alpha_1})\nonumber\\
\frac{d^2\tilde E}{da^2}(a)&=&\mathcal{O}(|a|^{\alpha_1-1})\qquad {\rm
for}\ a\not=0,\ {\rm recall\ that\ }0<\alpha_1\le 1.\nonumber
\end{eqnarray}
If we now define the odd function:
$$h(a)\equiv\tilde h(E(a),a),\qquad -\delta< a <\delta$$
we get a $C^2$ function because, for $a\not=0,$ based on the
previous estimates on the derivatives of $\tilde h$ and $\tilde E,$
we have
$$
\frac{d^2h}{da^2}(a)=\frac{\partial\tilde h}{\partial
E}\frac{d^2\tilde E}{da^2}+\frac{\partial^2\tilde h}{\partial
E^2}\left(\frac{d\tilde E}{da}\right)^2+2\frac{\partial^2\tilde
h}{\partial E\partial a}\frac{d\tilde E}{da}+\frac{\partial^2\tilde
h}{\partial a^2} =\mathcal{O}(|a|^{\alpha_1}),
$$
hence, by L'Hospital
$$\frac{d^2h}{da}(0)\stackrel{def}{=}\lim_{a\rightarrow 0}\frac{\frac{dh}{da}(a)-0}{a}=\lim_{a\rightarrow
0}\frac{d^2h}{da^2}(a)=0.$$
We now extend $h$ to complex values via the rotational symmetry
\eqref{eq:hsym}:
$$
h(a)=\frac{a}{|a|}\tilde h(E(|a|),|a|).
$$ We have just proved:
\begin{proposition}\label{pr:cm} There exist $\delta>0,$
the $C^2$ function
$$h:\{a\in\mathbb{R}\times\mathbb{R}\ :\ |a|<\delta\}\mapsto L^2_\sigma\cap H^2
,$$ and the $C^1$ function $E:(-\delta,\delta)\mapsto\mathbb{R}$
such that for $|E-E_0|<\delta$ and $|\langle\psi_0,\psi_E\rangle
|<\delta$ the eigenvalue problem (\ref{eq:ev}) has a unique solution
up to multiplication with $e^{i\theta},\ \theta\in [0,2\pi),$ which
can be represented as a center manifold:
\begin{equation}\label{eq:cm}
\psi_E=a\psi_0+h(a),\ E=E(|a|), \quad \langle\psi_0,h(a)\rangle =0,\quad h(e^{i\theta}a)=e^{i\theta}h(a),
|a|<\delta .\end{equation} Moreover
$E(|a|)=\mathcal{O}(|a|^{1+\alpha_1})$,
$h(a)=\mathcal{O}(|a|^{2+\alpha_1}),$ and for $a\in\mathbb{R},\
|a|<\delta,$ $h(a)$ is a real valued function with
$\frac{d^2h}{da^2}(a)=\mathcal{O}(|a|^{\alpha_1})$
\end{proposition}
Since $\psi_0(x)$ is exponentially decaying as $|x|\rightarrow\infty$ the
proposition implies that $\psi_E\in L^2_\sigma .$ A regularity argument, see
\cite{sw:mc1}, gives a stronger result:
\begin{corollary}\label{co:decay} For any $\sigma\in\mathbb{R},$
there exists a finite constant $C_\sigma$ such that:
$$\|<x>^\sigma\psi_E\|_{H^2}\le C_\sigma\|\psi_E\|_{H^2}.$$
\end{corollary}
\begin{remark}\label{rmk:reg} By standard regularity methods, see for example \cite[Theorem 8.1.1]{caz:bk}, one can show $\psi_E\in H^3.$
Hence by Sobolev imbeddings both $\psi_E$ and $\nabla\psi_E$ are continuous and converge to zero as $|x|\rightarrow\infty.$
\end{remark}
\begin{remark}\label{rmk:pos} By standard variational methods, see for example \cite{rw:bs}, one can show that the real valued solutions of
\eqref{eq:ev} do not change sign. Then Harnack inequality for
$H^2\bigcap C(\mathbb{R}^3)$ solutions of \eqref{eq:ev} implies that
these real solution cannot take the zero value. Hence $\psi_E$ given
by \eqref{eq:cm} for $a\in\mathbb{R}$ is either strictly positive or
strictly negative.
\end{remark}
In section \ref{se:lin} we also need some smoothness for the
effective (linear) potential induced by the nonlinearity which
modulo rotations of the complex plane is given by:
$$Dg|_{\pe}[u+iv]=g'(\pe)u+i\frac{g(\pe)}{\pe}v,\qquad \psi_E\ge 0$$
namely:
\bigskip
\noindent{\bf (H2)} Assume that for the positive solution of
\eqref{eq:ev} we have $\widehat{g'(\pe)},\
\widehat{\frac{g(\pe)}{\pe}}\in L^1(\mathbb{R}^3)$ where $\hat{f}$
stands for the Fourier transform of the function $f.$
\bigskip
\noindent In concrete cases the hypothesis may be checked directly
using the regularity of $\psi_E,$ the solution of an uniform
elliptic e-value problem. In general we can prove the following
result:
\begin{proposition}\label{pr:dg} If the following holds
\bigskip
\noindent{\bf (H2')} $g$ restricted to reals
has third derivative except at zero and $|g'''(s)|<\frac{C}{s^{1-\alpha_1}}+Cs^{\alpha_2 -1},$ $s>0,$ $0<\alpha_1\le\alpha_2;$
\bigskip
\noindent then for the nonnegative solution of \eqref{eq:ev}, $\pe,$ we have
$\widehat{g'(\pe)}\in L^1$ and $\widehat{\frac{g(\pe)}{\pe}}\in
L^1$.
\end{proposition}
We will give the proof in the Appendix.
We are going to decompose the solution of \eqref{u}-\eqref{ic} into
a projection onto the center manifold and a correction. For orbital
stability the projection which minimizes the $H^1$ norm of the
correction is used, see for example \cite{mw:ls}, while for
asymptotic stability one wants to remove periodic in time components
of the correction. Currently there are two different ways to
accomplish this. First and most used one is to keep the correction
orthogonal to the discrete spectrum of a fixed linear Schr\" odinger
operator ``close" to the dynamics, see \cite{kz:as2d,pw:cm}. For
example in \cite{kz:as2d} the linear Schr\" odinger operator is
$-\Delta + V$ and the correction is always orthogonal on its sole
eigenvector $\psi_0,$ hence the decomposition becomes
$$u=a\psi_0+h(a)+correction,\qquad {\rm where}\ a=\langle
\psi_0,u\rangle.$$ Second technique is to use the invariant
subspaces of the actual linearized dynamics at the projection, see
for example \cite{gnt:as}. While more complicated the latter is the
only one capable to render our main result. Since there are slight
mistakes in the previous presentations of this decomposition we are
going to describe it in what follows.
Consider the linearization of \eqref{u} at function on the
center manifold $\psi_E=a\psi_0+h(a),\ a=a_1+ia_2\in\mathbb{C},\
|a|<\delta:$
\begin{equation}\label{eq:ldE}
\frac{\partial w}{\partial t}=-iL_{\psi_E}[w]-iEw
\end{equation}
where
\begin{equation}\label{def:linop}
L_{\psi_E}[w]=(-\Delta+V-E)w+Dg_{\psi_E}[w]=(-\Delta+V-E)w+\lim_{\varepsilon\in\mathbb{R},\ \varepsilon\rightarrow
0}\frac{g(\psi_E+\varepsilon w)-g(\psi_E)}{\varepsilon}
\end{equation}
{\bf Properties of the linearized operator}:
\begin{enumerate}
\item $L_{\psi_E}$ is real linear and symmetric with respect to the
real scalar product $\Re\langle\cdot,\cdot\rangle,$ on
$L^2(\mathbb{R}^3),$ with domain $H^2(\mathbb{R}^3).$
\item Zero is an e-value for $-iL_{\psi_E}$ and its generalized
eigenspace includes $\left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial
a_2}\right\}$
\end{enumerate}
The real linearity of $L_{\psi_E}$ follows from \eqref{def:linop}.
For symmetry consider first the case of a real valued
$\psi_E=a\psi_0+h(a),\ a\in (-\delta,\delta)\subset\mathbb{R}.$ Then
for $w=u+iv\in H^2(\mathbb{R}^3),\ u,v$ real valued we have
$$L_{\psi_E}[u+iv]=L_+[u]+iL_-[v]$$ with $L_+[u],$ $L_-[v]$ being
real valued and symmetric:
\begin{eqnarray}
L_+[u]&=&(-\Delta+V-E)u+g'(\psi_E)u\nonumber\\
L_-[v]&=&(-\Delta+V-E)v+\frac{g(\psi_E)}{\psi_E}v.\nonumber
\end{eqnarray}
To determine the expression for $L_-$ we used the rotational
symmetry \eqref{gsym}:
$$
g(e^{i\theta}\psi_E)=e^{i\theta}g(\psi_E)
$$
and we differentiate it with respect to $\theta$ at $\theta=0$ to
get
\begin{equation}\label{symgen}
Dg_{\psi_E}[i\psi_E]=ig(\psi_E).
\end{equation}
Now,
$$\Re\langle L_{\psi_E}[u+iv],u_1+iv_1\rangle=\Re\langle L_+[u],u_1\rangle
+\Re\langle L_-[v],v_1\rangle=\Re\langle u,L_+[u_1]\rangle
+\Re\langle v,L_-[v_1]\rangle=\Re\langle
u+iv,L_{\psi_E}[u_1+iv_1]\rangle$$ hence $L_{\psi_E}$ is symmetric
for real valued $\psi_E.$
For a complex valued function on the center manifold
$\psi_E=a\psi_0+h(a),\ a\in\mathbb{C},\ |a|<\delta$ there exists
$\theta\in [0,2\pi)$ such that $a=|a|e^{i\theta}$ and
$$\psi_E=e^{i\theta}(|a|\psi_0+h(|a|))=e^{i\theta}\psi_E^{real}$$
where $\psi_E^{real}$ is real valued and on the center manifold.
Using again the rotational symmetry of $g$ \eqref{gsym} we get:
\begin{equation}\label{simlinop}
L_{\psi_E}[w]=e^{i\theta}L_{\psi_E^{real}}[e^{-i\theta}w].
\end{equation}
Since $e^{i\theta}$ is a unitary linear operator on the real Hilbert
space $L^2(\mathbb{R}^3)$ and, due to the argument above,
$L_{\psi_E^{real}}$ is symmetric we get that $L_{\psi_E}$ is
symmetric.
For the second property, we observe that substituting $w=i\psi_E$ in
\eqref{def:linop} and using \eqref{symgen}, \eqref{eq:ev} we get
$$L_{\psi_E}[i\psi_E]=i[(-\Delta+V-E)\psi_E+g(\psi_E)]=0.$$
Hence zero is an e-value for $-iL_{\psi_E}$ and
$i\frac{\psi_E}{|a|}$ for $a\not=0$ and $i\psi_0=\lim_{a\rightarrow
0}i\frac{\psi_E}{a}$ for $a=0$ are the corresponding eigenvectors.
Moreover by differentiating \eqref{eq:ev} with respect to $a_1=\Re
a\in\mathbb{R}$ or $a_2=\Im a\in\mathbb{R}$ we get
$$-iL_{\psi_E}\left[\frac{\partial\psi_E}{\partial a_j}\right]=-\frac{\partial E}{\partial a_j}i\psi_E,\quad j=1,2.$$
Since $\frac{\partial E}{\partial a_j}=E'(|a|)\frac{\partial
|a|}{\partial a_j}\in\mathbb{R}$ we deduce that
$\frac{\partial\psi_E}{\partial a_j},\ j=1,2$ are in the generalized
eigenspace of zero. Note that, by differentiating
$h(e^{i\theta}a)=e^{i\theta}h(a)$ with respect to $\theta$ at
$\theta=0$ we get $Dh|_a[ia]=ih(a)$ and, via \eqref{eq:cm},
$D\psi_E|_a[ia]=i\psi_E.$ Since the differential can be written with
the help of the gradient:
$$i\psi_E=D\psi_E|_a[ia]=\frac{\partial\psi_E}{\partial
a_1}\Re [ia]+\frac{\partial\psi_E}{\partial
a_2}\Im [ia],$$ we infer that
$$i\psi_E\in {\rm span} \left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial
a_2}\right\}\quad {\rm or\ equivalently}\ \psi_E\in {\rm span} \left\{i\frac{\partial\psi_E}{\partial a_1},i\frac{\partial\psi_E}{\partial
a_2}\right\}$$ where the span is taking over the reals\footnote{One can actually show that, for small $|a|,$ zero
is the only e-value of $-iL_{\psi_E}$ and the corresponding
eigenspace is spanned by $\frac{\partial\psi_E}{\partial a_j},\
j=1,2.$ However this is not needed in our argument.}.
One can now decompose $L^2(\mathbb{R}^3)$ into invariant subspaces
with respect to $-iL_{\psi_E}$:
$$L^2(\mathbb{R}^3)={\rm span} \left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial
a_2}\right\}\oplus
{\cal H}_a.$$
The standard choice is to use the projection along the dual basis:
$${\cal H}_a=\{\phi_1,\phi_2\}^\perp$$
where the orthogonality is with respect to the real scalar product,
and $\phi_1,\ \phi_2$ are in the generalized eigenspace of the
adjoint of $-iL_{\psi_E}$ corresponding to the eigenvalue zero, and
$\phi_1$ is orthogonal to $\frac{\partial\psi_E}{\partial a_2}$ but
not to $\frac{\partial\psi_E}{\partial a_1}$ while $\phi_2$ is
orthogonal to $\frac{\partial\psi_E}{\partial a_1}$ but not to
$\frac{\partial\psi_E}{\partial a_2}.$ Since $L_{\psi_E}$ is
symmetric we have $(-iL_{\psi_E})^*=L_{\psi_E}i$ and a direct
calculations shows that one can choose
$$\phi_1=-i\frac{\partial\psi_E}{\partial a_2},\qquad
\phi_2=i\frac{\partial\psi_E}{\partial a_1}$$
as long as $\Re\langle i\frac{\partial \psi_E}{\partial
a_1},\frac{\partial \psi_E}{\partial a_2}\rangle\not=0.$ But
$$\Re\langle i\frac{\partial \psi_E}{\partial
a_1},\frac{\partial \psi_E}{\partial
a_2}\rangle=\Re\langle i\psi_0,i\psi_0\rangle=1,\qquad {\rm at}\ a=0
$$
and since $\psi_E$ is $C^2$ in $a_1,\ a_2$ we have:
\begin{remark} By possible choosing $\delta>0$ smaller than the one
in Proposition \ref{pr:cm} we get:
\begin{equation}\label{jacobi:a0}\Re\langle i\frac{\partial \psi_E}{\partial
a_1},\frac{\partial \psi_E}{\partial
a_2}\rangle=\Re\langle i\psi_0,i\psi_0\rangle\ge\frac{1}{2}.
\end{equation}
\end{remark}
Consequently, for $|a|<\delta,$
\begin{equation}\label{def:ha}
{\cal H}_a=\left\{-i\frac{\partial\psi_E}{\partial a_2},i\frac{\partial\psi_E}{\partial
a_1}\right\}^\perp,\qquad {\rm and}\ L^2(\mathbb{R}^3)={\rm span} \left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial
a_2}\right\}\oplus
{\cal H}_a.
\end{equation}
Our goal is to decompose the solution of \eqref{u} at each
time into:
$$u=\psi_E+\eta=a\psi_0+h(a)+\eta,\qquad \eta\in{\cal H}_a$$
which insures that $\eta$ is not in the non-decaying directions
(tangent space of the central manifold) ${\rm span}
\left\{\frac{\partial\psi_E}{\partial
a_1},\frac{\partial\psi_E}{\partial
a_2}\right\}$ of the linearized equation
\eqref{eq:ldE} around $\psi_E$. The fact that this can be done in an
unique manner is a consequence of the following lemma\footnote{This
is an immediate consequence of the implicit function theorem but we
find the proof in \cite{gnt:as} to be incomplete.}:
\begin{lemma}\label{lem:decomp} There exists $\delta_1>0$ such that
any $\phi\in L^2(\mathbb{R}^3)$ satisfying $\|\phi\|_{L^2}\le\delta_1$
can be uniquely decomposed:
$$\phi =\psi_E+\eta=a\psi_0+h(a)+\eta$$
where $a=a_1+ia_2\in\mathbb{C},\ |a|<\delta,\ \eta\in {\cal H}_a.$
Moreover the maps $\phi\mapsto a$ and $\phi\mapsto \eta$ are $C^1$
and there exist constant $C$ independent on $\phi$ such that
$$|a|\le 2\|\phi\|_{L^2},\qquad \|\eta\|_{L^2}\le C\|\phi\|_{L^2}.$$
\end{lemma}
\smallskip\par{\bf Proof:} Consider the map
$F:\{a=(a_1,a_2)\in\mathbb{R}^2\ :\ |a|<\delta\}\times
L^2(\mathbb{R}^3)\mapsto\mathbb{R}\times\mathbb{R}:$
\begin{equation}\label{eq:jacobi} F(a_1,a_2,\phi)=\left(\Re\langle -i\frac{\partial
\psi_E}{\partial
a_2},\phi-\psi_E\rangle,\Re\langle i\frac{\partial \psi_E}{\partial
a_1},\phi-\psi_E\rangle\right)\end{equation}
where $\psi_E=a\psi_0+h(a),\ a=a_1+ia_2.$ Since $h(a)$ is $C^2,$
$F$ is a $C^1$ map and:
\begin{eqnarray}
F(0,0,0)&=&0\nonumber\\
\frac{\partial F}{\partial
(a_1,a_2)}(0,0,0)&=&\mathbb{I}_{\mathbb{R}^2}\nonumber
\end{eqnarray}
where for the calculation of the Jacobi matrix we used
\eqref{jacobi:a0}.
The implicit function theorem implies that there exist $\delta_2\le
\delta$ and a $C^1$ map:
$$\tilde F=(\tilde F_1,\tilde F_2):B(0,\delta_2)\subset
L^2(\mathbb{R}^3)\mapsto\mathbb{R}\times\mathbb{R}$$ such that the
only solutions of
$$F(a_1,a_2,\phi)=0$$
in $|a|=|a_1+ia_2|<\delta_2,\ \|\phi\|_{L^2}<\delta_2$ are given by
$$(a_1=\tilde F_1(\phi),a_2=\tilde F_2(\phi),\phi).$$
Now, for an arbitrary $\phi\in B(0,\delta_2)\subset
L^2(\mathbb{R}^3),$ since
$$\phi =\psi_E+\eta=a\psi_0+h(a)+\eta$$
with $a=a_1+ia_2\in\mathbb{C},\ |a|<\delta_2\le\delta,\ \eta\in
{\cal H}_a$ is equivalent to $F(a_1,a_2,\phi)=0$ we get that there
is a unique choice:
$$a_1=\tilde F_1(\phi),\quad a_2=\tilde F_2(\phi),\quad
\eta=\phi-a\psi_0-h(a).$$ Moreover, by choosing
$\delta_1\le\delta_2$ such that
$$\|D\tilde F_\phi\|\le 2\qquad \forall\phi\in
L^2(\mathbb{R}^3),\ \|\phi\|_{L^2}\le\delta_1$$ where the norm is
the operator norm from $L^2(\mathbb{R}^3)$ into
$\mathbb{R}\times\mathbb{R},$ we get, for all $\phi\in
L^2(\mathbb{R}^3),\ \|\phi\|_{L^2}\le\delta_1:$
$$|a|=\sqrt{a_1^2+a_2^2}\le 2\|\phi\|_{L^2}$$
and
$$\|\eta\|_{L^2}\le \|\phi\|_{L^2}+\|\psi_E\|_{L^2}\le
\|\phi\|_{L^2}+|a|+\|h(a)\|_{L^2}\le C\|\phi\|_{L^2}$$ where $C\ge
3+2\sup_{a\in\mathbb{C},|a|\le\delta_2}\|Dh_a\|.$ Note that the
existence of $\delta_1$ is insured by the continuity of $D\tilde F$
and, from the implicit function theorem:
$$D\tilde F_0=D_\phi F|_{\phi=0}$$
and the latter has norm one being the projection operator onto
$\psi_0.$
This finishes the proof of Lemma \ref{lem:decomp}. $\Box$
\begin{remark}\label{rmk:h-1}
Both the decomposition \eqref{def:ha} and Lemma \ref{lem:decomp} can
be extended without modifications to $H^{-1}(\mathbb{R}^3)$ the dual
of $H^1$ because $\frac{\partial \psi_E}{\partial a_j}\in H^1,\
j=1,2.$ In this case $\langle u, \phi\rangle $ denotes the
evaluation of the functional $\phi\in H^{-1} $ at $u\in H^1.$
\end{remark}
We need one more technical result relating the spaces ${\cal H}_a$
and the space corresponding to the continuous spectrum of
$-\Delta+V:$
\begin{lemma}\label{le:pcinv}
There exists $\delta>\delta_2>0$ such that for any $a\in\mathbb{C},\ |a|\le\delta_2$ the linear map $P_c|_{{\cal H}_a}:{\cal H}_a\mapsto {\cal
H}_0$ is invertible, and its inverse $R_a :{\cal H}_0\mapsto {\cal
H}_a$ satisfies:
\begin{eqnarray}
\|R_a\zeta\|_{L^2_{-\sigma}}&\le
&C_{-\sigma}\|\zeta\|_{L^2_{-\sigma}},\qquad \sigma\in\mathbb{R}\ {\rm and\ for\ all}\ \zeta\in {\cal H}_0\cap L^2_{-\sigma}\label{raest1}\\
\|R_a\zeta\|_{L^p}&\le
&C_p\|\zeta\|_{L^p},\qquad 1\le p<\infty\ {\rm and\ for\ all}\ \zeta\in {\cal H}_0\cap
L^p\label{raest2}\\
\overline{R_a\zeta}&=&R_a\overline\zeta\label{racc}
\end{eqnarray}
where the constants $C_{-\sigma},\ C_p>0$ are independent of
$a\in\mathbb{C},\ |a|\le\delta_2.$
\end{lemma}
\smallskip\par{\bf Proof:} Since $\psi_0$ is orthogonal to ${\cal H}_0,$ by continuity we can choose
$\delta>\tilde\delta_2>0$ such that $\psi_0\notin {\cal H}_a$ for
$|a|<\tilde\delta_2.$ Consequently $P_c|_{{\cal H}_a}$ is one to
one, otherwise from $\phi\in {\cal H}_a,\ \phi\neq 0,\ P_c\phi=0$ we
get $\phi=z\psi_0$ for some $z\in\mathbb{C},\ z\neq 0$ which
contradicts $\psi_0\notin {\cal H}_a.$
Next, for $|a|<\tilde\delta_2$ we construct $R_a :{\cal H}_0\mapsto
{\cal H}_a$ such that:
\begin{equation}\label{pcinv}
P_cR_a\zeta=\zeta,\qquad \forall\zeta\in {\cal H}_0.
\end{equation}
Since $P_c$ is the projection onto $\{\psi_0\}^\perp,$ condition
\eqref{pcinv} is equivalent to
\begin{equation}\label{radef} R_a\zeta=\zeta+z\psi_0\end{equation}
for some $z\in\mathbb{C}.$ To insure that the range of $R_a$ is in
${\cal H}_a$ we impose
\begin{equation}\label{zsys}
\Re z\langle -i\frac{\partial\psi_E}{\partial a_2},\psi_0\rangle=-\Re\langle -i\frac{\partial\psi_E}{\partial
a_2},\zeta\rangle,\qquad \Re z\langle i\frac{\partial\psi_E}{\partial a_1},\psi_0\rangle=-\Re\langle i\frac{\partial\psi_E}{\partial
a_1},\zeta\rangle.
\end{equation}
This linear system of two equations with two unknowns, $\Re z$ and
$\Im z,$ is uniquely solvable whenever $\psi_0\notin {\cal H}_a.$
Note that for $a=0$ the system becomes:
$z=\langle\psi_0,\zeta\rangle.$
In \eqref{radef} we now choose $z$ to be the unique solution of
\eqref{zsys} and obtain a well defined linear map $R_a :{\cal
H}_0\mapsto {\cal H}_a$ satisfying \eqref{pcinv}.
Consequently, $P_c|_{{\cal H}_a}$ is also onto, hence invertible and
its inverse is $R_a.$ Moreover, by the continuity of the
coefficients of \eqref{zsys} with respect to $a$ we can choose
$\delta_2\le\tilde\delta_2$ such that, for all $|a|\le\delta_2:$
\begin{equation}\label{z:est}|z|\le 2\ \sqrt{(\Re\langle -i\frac{\partial\psi_E}{\partial
a_2},\zeta\rangle)^2+(\Re\langle i\frac{\partial\psi_E}{\partial
a_1},\zeta\rangle)^2}.\end{equation} Hence, via \eqref{radef} and H\" older inequality we get:
$$\|R_a\zeta\|_Y\le\|\zeta\|_Y+2\|\psi_0\|_Y\|\zeta\|_Y\sqrt{\left\|\frac{\partial\psi_E}{\partial
a_2}\right\|_{Y^*}^2+\left\|\frac{\partial\psi_E}{\partial
a_1}\right\|_{Y^*}^2},$$
which, for the choice $Y=L^2_{-\sigma}(\mathbb{R}^3),\
Y^*=L^2_{-\sigma}(\mathbb{R}^3)$ respectively $Y=L^p(\mathbb{R}^3),\
Y^*=L^{p'}(\mathbb{R}^3),\ \frac{1}{p}+\frac{1}{p'}=1$ give
\eqref{raest1}, respectively \eqref{raest2}. The constants are
independent of $a$ due to the continuous dependence of
$\frac{\partial \psi_E}{\partial a_j},\ j=1,2$ on $a\in\mathbb{C}$
in the compact $|a|\le\delta_2,$ and their exponential decay in
time, see proposition \ref{pr:cm} and corollary \ref{co:decay}.
Now, $P_c$ commutes with complex conjugation because it is the
orthogonal projection onto ${\psi_0}^\perp$ and $\psi_0$ is real
valued. Then \eqref{racc} follows from $R_a$ being the inverse of
$P_c.$
The proof of Lemma \ref{le:pcinv} is now complete. $\Box$
We are now ready to prove our main result.
\section{Main Result}\label{se:main}
\begin{theorem}\label{mt}
Assume that the nonlinear term in \eqref{u} satisfies \eqref{gest}
and \eqref{gsym}. In addition assume that hypothesis (H1) and either
(H2) or (H2') hold. Let $p_1=3+\alpha_1,\ p_2=3+\alpha_2$. Then
there exists an $\varepsilon_0$ such that for all initial conditions
$u_0(x)$ satisfying
$$\max\{\|u_0\|_{L^{p_2'}},\|u_0\|_{H^1}\}\leq\varepsilon_0,\qquad \frac{1}{p_2'}+\frac{1}{p_2}=1$$
the initial value problem (\ref{u})-(\ref{ic}) is globally
well-posed in $H^1$ and the solution decomposes into a radiative
part and a part that asymptotically converges to a ground state.
More precisely, there exist a $C^1$ function
$a:\mathbb{R}\mapsto\mathbb{C}$ such that, for all $t\in\R$ we have:
\begin{equation}
u(t,x)=\underbrace{a(t)\psi_0(x)+h(a(t))}_{\psi_E(t)}+\eta(t,x)
\label{dc} \end{equation} where $\psi_E(t)$ is on the central
manifold (i.e it is a ground state) and $\eta(t,x)\in {\cal
H}_{a(t)},$ see Proposition \ref{pr:cm} and Lemma \ref{lem:decomp}.
Moreover there exists the ground states states
$\psi_{E_{\pm\infty}}$ and the $C^1$ function $\theta:\R\mapsto \R$
such that $\lim_{|t|\rightarrow\infty}\theta(t)=0$ and:
$$\lim_{t\rightarrow\pm\infty}\|\psi_E(t)-e^{-it(E_\pm-\theta(t))}\psi_{E_{\pm\infty}}\|_{H^2\bigcap
L^2_\sigma}=0,$$ while $\eta$ satisfies the following decay
estimates:
\begin{eqnarray}
\|\eta(t)\|_{L^2}&\leq &C_0(\alpha_1,\alpha_2)\varepsilon_0\nonumber\\
\|\eta(t)\|_{L^{p_1}}&\leq &
C_1(\alpha_1,\alpha_2)\frac{\varepsilon_0}{(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_1})}},\quad p_1=3+\alpha_1\nonumber
\end{eqnarray}
and, for $p_2=3+\alpha_2:$
\begin{enumerate}
\item [(i)] if $\alpha_1\geq \frac{1}{3}$ or $\frac{1}{3} >\alpha_1 >
\frac{2\alpha_2}{3(3+\alpha_2)}$ then
$$\|\eta(t)\|_{L^{p_2}}\leq
C_2(\alpha_1,\alpha_2)\frac{\varepsilon_0}{(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_2})}}
$$
\item [(ii)] if $\alpha_1 =
\frac{2\alpha_2}{3(3+\alpha_2)}$ then
$$\|\eta(t)\|_{L^{p_2}}\leq
C_2(\alpha_1,\alpha_2)\varepsilon_0\frac{\log(2+|t|)}{(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_2})}}
$$
\item [(iii)] if $\alpha_1 <
\frac{2\alpha_2}{3(3+\alpha_2)}$ then
$$\|\eta(t)\|_{L^{p_2}}\leq
C_2(\alpha_1,\alpha_2)\frac{\varepsilon_0}{(1+|t|)^{\frac{1+3\alpha_1}{2}}}
$$
\end{enumerate}
where the constants $C_0,\ C_1$ and $C_2$ are independent of
$\varepsilon_0$.
\end{theorem}
\begin{remark}
Note that the critical and supercritical cases $\frac{1}{3}\leq\p_1< 3$ are contained in
$(i)$. Our results for these cases are stronger than the ones in
\cite{pw:cm,sw:mc1,sw:mc2} because we do not require the initial
condition to be in $L^2_\sigma,\ \sigma>1.$ Compared to
\cite{gnt:as} we have sharper estimates for the asymptotic decay to the ground state but we
require the initial data to be in $L^{p_2'}.$ To the best of our knowledge the
subcritical case
$\alpha_1<1/3$ has not been treated previously.
\end{remark}
\begin{remark}
One can obtain estimates for the radiative part $\eta$ in $L^p$, $2\leq p\leq p_1=3+\p_1$, or $p_1\leq p\leq p_2=3+\p_2$ by Riesz-Thorin interpolation between
$L^2$ and $L^{p_1}$ respectively between $L^{p_1}$ and $L^{p_2}.$
\end{remark}
\noindent\textbf{Proof of Theorem \ref{mt}} It is well known that
under hypothesis (H1)(i) the initial value problem (1)-(2) is
locally well posed in the energy space $H^1$ and its $L^2$ norm is
conserved, see for example \cite[Cor. 4.3.3 at p. 92]{caz:bk}.
Global well posedness follows via energy estimates from
$\|u_0\|_{H^1}$ small, see \cite[Remark 6.1.3 at p. 165]{caz:bk}.
We choose $\varepsilon_0\le \delta_1$ given by Lemma
\ref{lem:decomp}. Then, for all times, $\|u(t)\|_{L^2}\le\delta_1$
and we can decompose the solution into a solitary wave and a
dispersive component as in \eqref{dc}:
$$u(t)=a(t)\psi_0+h(a(t))+\eta(t)=\psi_E(t)+\eta(t)$$ Moreover, by possible making $\varepsilon_0$ smaller we can insure that
that $\|u(t)\|_{L^2}\le\varepsilon_0$ implies $|a(t)|\le\delta_2,\
t\in\mathbb{R}$ where $\delta_2$ is given by Lemma \ref{le:pcinv}.
In addition,
since
$$u\in C(\mathbb{R},H^{1}(\mathbb{R}^3))\cap
C^1(\mathbb{R},H^{-1}(\mathbb{R}^3)),$$
and $u\mapsto a$ respectively $u\mapsto \eta$ are $C^1,$ see Remark
\ref{rmk:h-1}, we get that $a(t)$ is $C^1$ and $\eta\in
C(\mathbb{R},H^{1})\cap
C^1(\mathbb{R},H^{-1}).$
The solution is now described by the $C^1$ function
$a:\mathbb{R}\in\C$ and $\eta(t)\in C(\R,H^1)\cap C^1(\R,H^{-1}).$
To obtain their equations we plug in (\ref{dc}) into (\ref{u}). Then
we get
\begin{align}
\frac{\partial\eta}{\partial t}+D\pe|_a a'&=-i(L_{\pe}+E)\eta-Ei\pe-iF_2(\pe,\eta) \label{eq:udecomp}
\end{align}
where $L_{\pe}$ is defined by \eqref{def:linop}
$$L_{\pe}\eta=(-\Delta+V-E)\eta-i\frac{d}{d\varepsilon}g(\pe+\varepsilon\eta)|_{\varepsilon=0}$$
and $F_2(\pe,\eta)$ denotes the nonlinear terms in $\eta$
\begin{equation}
F_2(\psi_E,\eta)=g(\psi_E+\eta)-g(\psi_E)-\underbrace{\frac{d}{d\varepsilon}g(\pe+\varepsilon\eta)|_{\varepsilon=0}}_{F_1(\psi_E,\eta)}
\label{g} \end{equation} Then projecting \eqref{eq:udecomp} onto the
invariant subspaces of $-iL_{\psi_E},$ ${\cal H}_{a},$ see
\eqref{def:ha} and the ${\rm span} \{\frac{\partial\psi_E}{\partial
a_1},\frac{\partial\psi_E}{\partial
a_2}\}$, we obtain the equations for $\eta(t)$ and
$a(t):$
\begin{align}
\frac{\partial\eta}{\partial t}&=-i(L_{\pe}+E)\eta-iF_2(\pe,\eta)-\tilde{F}_2(\pe,\eta) \label{eq:eta} \\
D\pe|_a a'&=-Ei\pe+\tilde{F}_2(\pe,\eta) \label{eq:a}
\end{align}
where \begin{equation}
\tilde{F}_2(\pe,\eta)=\underbrace{\frac{\Re\<-i\frac{\partial\pe}{\partial a_2},-iF_2(\pe,\eta)\>}{\Re\<-i\frac{\partial\pe}{\partial a_2},\frac{\partial\pe}{\partial a_1}\>}}_{\beta_1(\pe,\eta)}\cdot\frac{\partial\pe}{\partial a_1}+\underbrace{\frac{\Re\<i\frac{\partial\pe}{\partial a_1},-iF_2(\pe,\eta)\>}{\Re\<i\frac{\partial\pe}{\partial a_1},\frac{\partial\pe}{\partial a_2}\>}}_{\beta_2(\pe,\eta)}\cdot\frac{\partial\pe}{\partial a_2} \label{eq:f2t}
\end{equation}
\noindent In order to obtain the estimates for $\eta(t)$, we analyze \eqref{eq:eta}. The linear part of \eqref{eq:eta} is:
\begin{align}
\frac{\partial\zeta}{\partial t}&=-i(L_{\pe}+E)\zeta=(-\Delta+V)\zeta-i\frac{d}{d\varepsilon}g(\pe(t)+\varepsilon\zeta)|_{\varepsilon=0} \label{zeta} \\
\zeta(s)&=v \nonumber
\end{align}
Define $\Omega(t,s)v=\zeta(t)$. Then using Duhamel's principle \eqref{eq:eta} becomes
\begin{equation}
\eta(t)=\om(t,0)\eta(0)-\int_0^t \om(t,s)[iF_2(\pe,\eta)+\tilde{F}_2(\pe,\eta)]ds \label{eq:duheta}
\end{equation}
It is here where we differ from the approach \cite{sc:as,pw:cm,sw:mc1,sw:mc2}. The right-hand side of our equation contains only nonlinear terms in $\eta$. However the challenge is to obtain good dispersive estimates for the propagator $\om(t,s)$ of the linearization \eqref{zeta}, see Theorems \ref{th:lw} and \ref{th:lp}.
In order to apply a contraction mapping argument for
(\ref{eq:duheta}) we use the following Banach spaces. Let
$p_1=3+\p_1$ and $p_2=3+\p_2$, $$Y_i=\{u\in L^2\cap L^{p_1}\cap
L^{p_2}:\sup_t
(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_1})}\|u\|_{L^{p_1}}<\infty, \sup_t
\frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\|u\|_{L^{p_2}}<\infty,
\sup_t \|u\|_{L^2}<\infty \}$$ endowed with the norm $$
\|u\|_{Y_i}=\max\{\sup_t
(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_1})}\|u\|_{L^{p_1}}, \sup_t
\frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\|u\|_{L^{p_2}}, \sup_t
\|u\|_{L^2} \}$$ for $i=1,2,3$, where
$n_1=n_2=3(\frac{1}{2}-\frac{1}{p_2})$, $n_3=\frac{1+3\p_1}{2}$,
$m_1=m_3=0$ and $m_2=1$.
Consider the nonlinear operator in (\ref{eq:duheta}):
$$N(u)=\int_0^t \om(t,s)[iF_2(\pe,u)+\tilde{F}_2(\pe,u)]ds$$
\begin{lemma}\label{lm}
Consider the cases:
$$ 1.\ \alpha_1\geq \frac{1}{3}\ {\rm or}\ \frac{1}{3} >\alpha_1 >
\frac{2\alpha_2}{3(3+\alpha_2)}; \quad 2.\ \alpha_1 =
\frac{2\alpha_2}{3(3+\alpha_2)}; \quad 3.\ \alpha_1 <
\frac{2\alpha_2}{3(3+\alpha_2)}.$$ Then, for each case number i:
$N : Y_i\rightarrow Y_i$ is well defined, and locally Lipschitz,
i.e. there exists $\tilde{C_i}>0$, such that $$\|Nu_1
-Nu_2\|_{Y_i}\leq\tilde{C_i}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i}+\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1}+\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1
-u_2\|_{Y_i}. $$
\end{lemma}
Note that the Lemma gives the estimates for $\eta $ in the Theorem
\ref{mt}. Indeed, if we denote:$$v=\Omega(t,0)\eta(0),$$ then
$$\|v\|_{Y_i}\leq C_0\|\eta(0)\|_{L^{p_2'}\cap H^1},$$ where
$C_0=\max\{C,C_p\}$, see theorem \ref{th:lw}. We choose $\epsilon_0$
in the hypotheses of theorem \ref{mt}, such that
$$C_0\epsilon_0\le\frac{1}{2}\Big(\sqrt{1+2/\tilde{C_i}}-1\Big)$$
Then by continuity there exists $0\le Lip\le 1$ such that:
$$\|v\|_{Y_i}\leq\frac{2-Lip}{4}\Big(\sqrt{1+2Lip/\tilde{C_i}}-1\Big).$$
Let $R=L\|v\|_{Y_i}/(2-Lip)$ and $B(v,R)$ be the closed ball in
$Y_i$ with center $v$ and radius $R$. A direct calculation shows
that the right-hand side of (\ref{eq:duheta}): $$Ku=v+Nu$$ leaves
$B(v,R)$ invariant, i.e. $K:B(v,R)\mapsto B(v,R)$, and it is a
contraction with Lipschitz constant $Lip$ on $B(v,R)$.
By the contraction mapping argument, (\ref{eq:duheta}) has a unique
solution in $Y_i$. We now have two solutions of (\ref{eq:eta}), one
in $C(\R,H^1)$ from classical well posedness theory and one in
$C(\R,L^2\cap L^{p_1}\cap L^{p_2})$, $p_1=3+\p_1$, $p_2=3+\p_2$ from
the above argument. Using uniqueness and the continuous embedding of
$H^1$ in $L^2\cap L^{p_1}\cap L^{p_2}$, we infer that the solutions
must coincide. Therefore, the time decaying estimates in the spaces
$Y_{1-3}$ hold also for the $H^1$ solution.
\noindent\textbf{Proof of Lemma \ref{lm}} Let $u_1,u_2$ be in one of
the spaces $Y_i,\ i=1,2,3.$ Then at each $s\in\R$ we have:
\begin{align}
F_2(\pe(s),u_1(s))-&F_2(\pe(s),u_2(s))=g(\pe+u_1)-g(\pe+u_2)-F_1(\pe,u_1)+F_1(\pe,u_2) \nonumber \\
&=\int_0^1\Big[\frac{d}{d\tau}g(\pe+u_2+\tau(u_1-u_2))-\frac{d}{d\tau}g(\pe+\tau(u_1-u_2))|_{\tau=0}\Big]d\tau \nonumber \\
&=\int_0^1\int_0^1\frac{d}{ds}\cdot\frac{d}{d\tau}g\big(\pe+s(u_2+\tau(u_1-u_2))\big)ds\; d\tau \nonumber \end{align}
Using the hypothesis \eqref{gest} we have $|g(u)|\leq C(|u|^{2+\p_1}+|u|^{2+\p_2})$, then taking the derivatives with respect to $\tau$ and $s$ and estimating the integral we get:
\begin{align}
|F_2(\pe,u_1)-F_2(\pe,u_2)|&\leq C\big[\underbrace{(|\pe|^{\p_1}+|\pe|^{\p_2})(|u_1|+|u_2|)|u_1-u_2|}_{A_1}\label{def:a13} \\
&+\underbrace{(|u_1|^{1+\p_1}+|u_2|^{1+\p_1})|u_1-u_2|}_{A_2}+\underbrace{(|u_1|^{1+\p_2}+|u_2|^{1+\p_2})|u_1-u_2|}_{A_3}\big].\nonumber
\end{align} By \eqref{eq:f2t} and H\" older inequality, for any $1\le q\le \infty$ we have:
\begin{eqnarray} \|\tilde{F}_2(\pe,u_1)-\tilde{F}_2(\pe,u_2)\|_{L^q}&\leq &\tilde
C \Big(\Big\|\frac{\partial\pe}{\partial
a_2}\Big\|_{L^{p_2}}\Big\|\frac{\partial\pe}{\partial
a_1}\Big\|_{L^q}+\Big\|\frac{\partial\pe}{\partial
a_1}\Big\|_{L^{p_2}}\Big\|\frac{\partial\pe}{\partial
a_2}\Big\|_{L^q}\Big)(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})\nonumber \\
&+&\tilde C \Big(\Big\|\frac{\partial\pe}{\partial
a_2}\Big\|_{L^{p_1}}\Big\|\frac{\partial\pe}{\partial
a_1}\Big\|_{L^q}+\Big\|\frac{\partial\pe}{\partial
a_1}\Big\|_{L^{p_1}}\Big\|\frac{\partial\pe}{\partial
a_2}\Big\|_{L^q}\Big)\|A_2\|_{L^{p'_1}}\label{est:tildef}\\
&\leq
&C(\|A_1\|_{L^{p'_2}}+\|A_2\|_{L^{p'_1}}+\|A_3\|_{L^{p'_2}}),\nonumber
\end{eqnarray} where the
uniform bounds on $\frac{\partial\pe}{\partial a_j}\in H^2(\R^3),\
j=1,2,$ follow from their continuous dependence on scalar $a,$ and
$|a(t)|\le\delta_2,\ t\in\R.$
Now let us consider the difference $Nu_1-Nu_2$
\begin{equation}\label{eq:nu12}
(Nu_1-Nu_2)(t)=\int_0^t \Omega(t,s) \big[iF_2(\pe
(s),u_1(s))-iF_2(\pe (s),u_2(s))+\tilde{F}_2(\pe
(s),u_1(s))-\tilde{F}_2(\pe (s),u_2(s))]ds
\end{equation}
\begin{itemize}
\item \textbf{$L^{p_2}$ Estimate :}
\begin{align}
\|Nu_1-Nu_2\|_{L^{p_2}}&\leq\int_0^t \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}C\Big(2\|A_1\|_{L^{p'_2}}+\|A_2\|_{L^{p'_2}}+\|A_2\|_{L^{p'_1}}+2\|A_3\|_{L^{p'_2}}\Big)ds \nonumber \end{align}
To estimate the term containing $A_1$, observe that
$$\|(|\pe|^{\p_1}+|\pe|^{\p_2})(|u_1|+|u_2|)|u_1-u_2|\|_{L^{p'_2}}\leq\||\pe|^{\p_1}+|\pe|^{\p_2}\|_{L^{\beta}}(\|u_1\|_{L^{p_2}}+\|u_2\|_{L^{p_2}})\|u_1-u_2\|_{L^{p_2}}$$
with $\frac{1}{\beta}+\frac{2}{p_2}=\frac{1}{p'_2}$. Using Theorem
\ref{th:lp} (see also Remark \ref{rmk:lp}), we have for each case
number i: and $u_1,\ u_2\in Y_i:$
\begin{align}
\int_0^t & \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}\|A_1\|_{L^{p'_2}}ds \nonumber \\
&\leq\int_0^t \frac{C(p_2)}{|t-s|^{3\ipt}}\||\pe|^{\p_1}+|\pe|^{\p_2}\|_{L^{\beta}}\frac{[\log(2+|s|)]^{2m_i}}{(1+|s|)^{2n_i}}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i} ds \nonumber \\
&\leq\frac{C(p_2)C_1 C_2}{(1+|t|)^{3\ipt}}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i} \nonumber \end{align}
where $C_2=\sup_t \frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\int_0^t
\frac{[\log(2+|s|)]^{2m_i}ds}{|t-s|^{3\ipt}(1+|s|)^{2n_i}}<\infty$
since $2n_i >1$ and $C_1=\sup_t
\||\pe|^{\p_1}+|\pe|^{\p_2}\|_{L^{\beta}}.$ The uniform bounds in
$t\in\R$ for $\|\pe\|_{L^{\alpha_j\beta}}^{\alpha_j},\ j=1,2$
follow from the continuous dependence of $\pe=a(t)\po+h(a(t))\in
H^2(\R^3)$ on $a(t)$ and $|a(t)|\le \delta_2,\ t\in\R.$
To estimate the terms containing $A_2$, observe that
$$\|(|u_1|^{1+\p_1}+|u_2|^{1+\p_1})|u_1-u_2|\|_{L^{p'_1}}\leq\big(\|u_1\|_{L^{p_1}}^{1+\p_1}+\|u_2\|_{L^{p_1}}^{1+\p_1}\big)\|u_1-u_2\|_{L^{p_1}}$$
since $\frac{1}{p'_1}=\frac{2+\p_1}{p_1}$ and
\begin{eqnarray} \|(|u_1|^{1+\p_1}+|u_2|^{1+\p_1})|u_1-u_2|\|_{L^{p'_2}}&\leq &\left(\|u_1\|_{L^{p_1}}^{\theta(1+\p_1)}\|u_1\|_{L^{2}}^{(1-\theta)(1+\p_1)}+\|u_2\|_{L^{p_1}}^{\theta(1+\p_1)}\|u_2\|_{L^{2}}^{(1-\theta)(1+\p_1)}\right) \nonumber \\
&&\times\|u_1-u_2\|^\theta_{L^{p_1}}\|u_1-u_2\|^{1-\theta}_{L^{2}}\nonumber
\end{eqnarray}
where $\frac{1}{p'_2}=(2+\p_1)(\frac{1-\theta}{2}+\frac{\theta}{p_1}),\ 0\le\theta\le 1. $ Again using Theorem \ref{th:lp} (see also Remark \ref{rmk:lp}), we have
\begin{align}
\int_0^t & \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}\|A_2\|_{L^{p'_2}}ds \nonumber \\
&\leq\int_0^t \frac{C(p_2)}{|t-s|^{3\ipt}}\cdot\frac{(\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i}}{(1+|s|)^{3(\frac{\p_1}{2}+\frac{1}{p_2})}} ds \nonumber \\
&\leq\frac{C(p_2)C_3[\log(2+|t|)]^{m_i}}{(1+|t|)^{n_i}}(\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i}
\nonumber \end{align} where the different decay rates $n_i$ depend
on the case number in the hypotheses of this Lemma:
\begin{enumerate}
\item corresponds to
$3(\frac{\p_1}{2}+\frac{1}{p_2})>1$, and $C_3=\sup_t
(1+|t|)^{3\ipt}\int_0^t
\frac{ds}{|t-s|^{3\ipt}(1+|s|)^{3(\frac{\p_1}{2}+\frac{1}{p_2})}}<\infty;$
\item corresponds to $3(\frac{\p_1}{2}+\frac{1}{p_2})=1$, and $C_3=\sup_t
\frac{(1+|t|)^{3\ipt}}{\log(2+|t|)}\int_0^t
\frac{ds}{|t-s|^{3\ipt}(1+|s|)}<\infty;$
\item corresponds to $3(\frac{\p_1}{2}+\frac{1}{p_2})<1$, and $C_3=\sup_t
(1+|t|)^{\frac{1+3\p_1}{2}}\int_0^t
\frac{ds}{|t-s|^{3\ipt}(1+|s|)^{3(\frac{\p_1}{2}+\frac{1}{p_2})}}<\infty.$
\end{enumerate}
To estimate the term containing $A_3$, observe that
$$\|(|u_1|^{1+\p_2}+|u_2|^{1+\p_2})|u_1-u_2|\|_{L^{p'_2}}\leq\big(\|u_1\|_{L^{p_2}}^{1+\p_2}+\|u_2\|_{L^{p_2}}^{1+\p_2}\big)\|u_1-u_2\|_{L^{p_2}}$$ since $\frac{1}{p'_2}=\frac{2+\p_2}{p_2}$. Again using Theorem \ref{th:lp} (see also Remark \ref{rmk:lp}), we have
\begin{align}
\int_0^t & \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}\|A_3\|_{L^{p'_2}}ds \nonumber \\
&\leq\int_0^t \frac{C(p_2)}{|t-s|^{3\ipt}}\cdot\frac{[\log(2+|s|)]^{(2+\p_2)m_i}}{(1+|s|)^{(2+\p_2)n_i}}(\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i} ds \nonumber \\
&\leq\frac{C(p_2)C_4C_5[\log(2+|t|)]^{m_i}}{(1+|t|)^{n_i}}(\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i} \nonumber \end{align}
where $C_5=\sup_t \frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\int_0^t \frac{[\log(2+|s|)]^{(2+\p_2)m_i}ds}{|t-s|^{3\ipt}(1+|s|)^{(2+\p_2)n_i}}<\infty$ since $(2+\p_2)n_i >1$.
\item \textbf{$L^{p_1}$ Estimate :}
From \eqref{eq:nu12} we have
\begin{eqnarray}
\|Nu_1-Nu_2\|_{L^{p_1}}(t)&\le &\|\int_0^t \Omega(t,s)
[iF_2(\pe(s),u_1(s))-iF_2(\pe(s),u_2(s))]ds\|_{L^{p_1}}\nonumber\\
&+&C\int_0^t\|\Omega(t,s)\|_{L^{p'_1}\mapsto
L^{p_1}}\|\tilde{F}_2(\pe(s),u_1(s))-\tilde{F}_2(\pe(s),u_2(s))\|_{L^{p'_1}}ds\nonumber
\end{eqnarray}
For the second integral we use \eqref{est:tildef} with $q=p'_1$ and
the previous estimates on $A_1,\ A_2,\ A_3$ to obtain the required
bound. For the first integral moving the norm inside the integration
and applying $L^{p'_1}\mapsto L^{p_1}$ estimates for $\Omega(t,s)$
and \eqref{def:a13} for the nonlinear term would require the control
of $A_3$ in $L^{p'_1}.$ The latter, unfortunately, can no longer be
interpolated between $L^2$ and $L^{p_2}.$ To avoid this difficulty
we separate and treat differently the part of the nonlinearity
having an $A_3$ like behavior by decomposing $\R^3$ in two disjoints
measurable sets related to the inequality \eqref{def:a13}:
$$
V_1(s)=\{x\in\R^3\ |\
|F_2(\pe(s,x),u_2(s,x))-F_2(\pe(s,x),u_1(s,x))|\le C
A_3(s,x)\},\qquad
V_2(s)=\R^3\setminus V_1(s)$$
On $V_2(s),$ using polar representation of complex numbers, we
further split the nonlinear term into:
\begin{eqnarray}
\lefteqn{iF_2(\pe(s,x),u_1(s,x))-iF_2(\pe(s,x),u_2(s,x))=e^{i\theta(s,x)}CA_3(s,x)}\nonumber\\
&&+\underbrace{e^{i\theta(s,x)}[|iF_2(\pe(s,x),u_1(s,x))-iF_2(\pe(s,x),u_2(s,x))|-CA_3(s,x)]}_{G(s,x)}\nonumber\end{eqnarray}
where, due to inequality \eqref{def:a13}, $|G(s,x)|\le
C(A_1(s,x)+A_2(s,x))$ on $V_2(s).$ Then we have:
\begin{eqnarray}\lefteqn{\int_0^t \Omega(t,s)
[iF_2(\pe(s),u_1(s))-iF_2(\pe(s),u_2(s))]ds=\int_0^t \Omega(t,s)
(1-\chi (s))G(s)ds}\nonumber\\
&&+\underbrace{\int_0^t \Omega(t,s)
[\chi(s)(iF_2(\pe(s),u_1(s))-iF_2(\pe(s),u_2(s)))+(1-\chi(s))e^{i\theta(s)}CA_3(s)]ds}_{I(t)},\nonumber\end{eqnarray}
where $\chi(s)$ is the characteristic function of $V_1(s).$ Now
$$\|\int_0^t \Omega(t,s)
(1-\chi (s))G(s)ds\|_{L^{p_1}}\le \int_0^t\| \Omega(t,s)\|_{L^{p'_1}\mapsto
L^{p_1}}C(\|A_1(s)\|_{L^{p'_1}}+\|A_2(s)\|_{L^{p'_1}})ds$$
and estimates as in the previous step for $A_1$ and $A_2$ give the required
decay. For $I(t)$ we use interpolation:
$$\|I(t)\|_{L^{p_1}}\leq\|I(t)\|_{L^2}^{1-\theta}\|I(t)\|_{L^{p_2}}^{\theta}\leq\|I(t)\|_{L^2}^{1-\theta}\left(\int_0^t \|\om(t,s)\|_{L^{p'_2}\mapsto L^{p_2}}\|A_3\|_{L^{p'_2}} ds\right)^{\theta}$$
where $\frac{1}{p_1}=\frac{1-\theta}{2}+\frac{\theta}{p_2}$. We know
from previous step that the above integral decays as
$(1+|t|)^{-3\ipt}$ and below we will show its $L^2$ norm will be
bounded. Therefore $$\sup_t
(1+|t|)^{3\ipo}\|I(t)\|_{L^{p_1}}<\infty$$ and the $L^{p_1}$
estimates are complete.
\item \textbf{$L^2$ Estimate :}
To estimate $L^2$ norm we cannot use $L^2\rightarrow L^2$ estimate
for $\om(t,s)$ because that would force us to control
$L^{2(\alpha_2+2)}$ which cannot pe interpolated between $L^2$ and
$L^{p_2},\ p_2=\alpha_2+3.$ We avoid this by using the
decomposition:
$$T(t,s)v=[P_c\om(t,s)-\ehs P_c]v \quad \text{i.e.}\quad \om(t,s)=R_{a(t)} T(t,s)+R_{a(t)} \ehs P_c$$
For $T(t,s)$ we will use $L^{p'}\rightarrow L^2$ estimates, see
Theorem \ref{th:lw}, while for $\ehs P_c$ we will use Stricharz
estimates $L^\infty_tL_x^2$. We will also use a decomposition of the
nonlinear term similar to the one for $L^{p_1}$ estimates that will
allow us to estimate in a different manner this time the terms
behaving like $A_2,$ see \eqref{def:a13}. All in all we have:
\begin{align}
\|Nu_1-Nu_2\|_{L^2}&\leq \int_0^t \|\om(t,s)\|_{L^2\rightarrow L^2}\|\tilde{F}_2(\pe,u_1)-\tilde{F}_2(\pe,u_2)\|_{L^2}ds
\nonumber\\
&+\|R_{a(t)}\|_{L^2\mapsto L^2}\int_0^t \|T(t,s)\|_{L^{p'_2}\rightarrow L^2}C(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})ds \nonumber \\
&+\|R_{a(t)}\|_{L^2\mapsto L^2}\int_0^t \|T(t,s)\|_{L^{p'_1}\rightarrow L^2}C\|A_2\|_{L^{p'_1}}ds \nonumber \\
&+\|R_{a(t)}\|_{L^2\mapsto L^2}\|\int_0^t \ehs P_c (A_1(s)+A_3(s) ds\|_{L^2}+\|R_{a(t)}\|_{L^2\mapsto L^2}\|\int_0^t \ehs P_c A_2 ds\|_{L^2}\nonumber
\end{align}
For the first integral we use Theorem \ref{th:lp} part (i),\
\eqref{est:tildef} with $q=2$ and the estimates we have already
obtained for $A_1,\ A_2$ and $A_3.$ We deduce that this integral is
uniformly bounded in $t\in\R.$ Similarly we get uniform boundedness
of the second and third integral by using Theorem \ref{th:lw} part
(iv).
For the fourth integral we use Stricharz estimate:
$$\sup_{t\in\R}\|\int_0^t \ehs P_c A_1ds\|_{L^2}\leq C_s\left[\left(\int_\R \|A_1(s)\|_{L^{p'_2}}^{\gamma_2'}ds\right)^{\frac{1}{\gamma_2'}}+\left(\int_\R \|A_3(s)\|_{L^{p'_2}}^{\gamma_2'}ds\right)^{\frac{1}{\gamma_2'}}\right]$$
where $\frac{1}{\gamma_2'}+\frac{1}{\gamma_2}=1,$ and
$\frac{2}{\gamma_2}=3\left(\frac{1}{2}-\frac{1}{p_2}\right).$ Using
again the estimates we obtained before for $A_1$ and $A_3.$ we get:
\begin{align}
\| A_1\|_{L^{\gamma_2'}_sL^{{p'_2}}}&\leq C_{11}\Big[\int_\R \frac{(\log(2+|s|))^{2m_i\gamma_2'}}{(1+|s|)^{2n_i\gamma_2'}}ds\Big]^{\frac{1}{\gamma_2'}}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i} \nonumber \\
&\leq C_{11} C_8 (\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i}
\nonumber \end{align} where $C_8= \int_\R
\frac{(\log(2+|s|))^{2m_i\gamma_1'}}{(1+|s|)^{2n_i\gamma_1'}}ds<\infty$
since $2n_i\gamma'>1$ and:
\begin{align}
\| A_3\|_{L^{\gamma_2'}_sL^{{p'_2}}}&\leq C_{12}\Big[\int_\R \frac{(\log(2+|s|))^{(2+\p_2)m_i\gamma_3'}}{(1+|s|)^{(2+\p_2)n_i\gamma_3'}}ds\Big]^{\frac{1}{\gamma_3'}}(\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i} \nonumber \\
&\leq C_9
(\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i}
\nonumber \end{align} where $C_9=\int_\R
\frac{(\log(2+|s|))^{(2+\p_2)m_i\gamma_3'}}{(1+|s|)^{(2+\p_2)n_i\gamma_2'}}ds<\infty$
since $(2+\p_2)n_1\gamma_3'>1$.
Similarly, for the fifth integral:
$$\sup_{t\in\R}\|\int_0^t \ehs P_c A_2 ds\|_{L^2}\leq C_s\Big(\int_\R \|A_2\|_{L^{p_1'}}^{\gamma_1'}ds\Big)^{\frac{1}{\gamma_1'}}$$
where $\frac{1}{\gamma_1'}+\frac{1}{\gamma_1}=1,$ and
$\frac{2}{\gamma_1}=3\left(\frac{1}{2}-\frac{1}{p_1}\right).$
Furthermore we have
\begin{align}
\| A_3\|_{L^{\gamma_1'}_sL^{{p'_1}}}&\leq C_{13}\Big[\int_\R \frac{ds}{(1+|s|)^{3(2+\p_1)\gamma_2'\ipo}}\Big]^{\frac{1}{\gamma_2'}}(\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i} \nonumber \\
&\leq C_{13}C_{10}
(\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i}
\nonumber \end{align} where $C_{10}=\int_\R
\frac{ds}{(1+|s|)^{3(2+\p_1)\gamma_2'\ipo}}ds<\infty$ since
$3(2+\p_1)\gamma_2'\ipo>1$ .
\end{itemize}
The $L^2$ estimates are now complete and the proof of Lemma \ref{lm}
is finished. $\Box$
We now finish the proof of Theorem \ref{mt} by analyzing the
dynamics on the center manifold and showing it converges to a ground
state. Using the fact that
$$i\psi_E=D\psi_E|_a[ia]=\frac{\partial\psi_E}{\partial
a_1}\Re [ia]+\frac{\partial\psi_E}{\partial
a_2}\Im [ia]$$ equation \eqref{eq:a} becomes $$D\pe|_a(a'+iEa)=\frac{\partial\psi_E}{\partial
a_1}\Re [a'+iEa]+\frac{\partial\psi_E}{\partial
a_2}\Im [a'+iEa]=\tilde{F}_2(\pe,\eta)=\beta_1(\pe,\eta)\frac{\partial\psi_E}{\partial
a_1}+\beta_2(\pe,\eta)\frac{\partial\psi_E}{\partial
a_2}$$ Hence $$|a'+iEa|=\sqrt{\beta_1^2+\beta_2^2}=b(t)$$
and
$$
\left|[a(t)e^{i\int_0^t E(s)ds}]'\right|=b(t)
$$
Since $b(t)=\sqrt{\beta_1^2+\beta_2^2}$, and
$$\beta_1\leq\left\|\frac{\partial\pe}{\partial a_2}\right\|_{L^{p_2}}(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})+\left\|\frac{\partial\pe}{\partial a_2}\right\|_{L^{p_1}}\|A_2\|_{L^{p'_1}}\leq C(\|\eta\|^{2}_{L^{p_2}}+\|\eta\|^{2+\p_2}_{L^{p_2}}+\|\eta\|^{2+\p_1}_{L^{p_1}})$$
$$\beta_2\leq\left\|\frac{\partial\pe}{\partial a_1}\right\|_{L^{p_2}}(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})+\left\|\frac{\partial\pe}{\partial a_1}\right\|_{L^{p_1}}\|A_2\|_{L^{p'_1}}\leq C(\|\eta\|^{2}_{L^{p_2}}+\|\eta\|^{2+\p_2}_{L^{p_2}}+\|\eta\|^{2+\p_1}_{L^{p_1}})$$
we get $0\le b(t)\le C(1+|t|)^{1+\delta}$ for some $\delta>0,$ in
each of the cases $(i)$, $(ii)$ and $(iii)$ in the Theorem \ref{mt}.
Then, for any $\varepsilon>0$ we have
\begin{equation}
\left|a(t)e^{i\int_0^{t} E(s)ds}-a(t')e^{i\int_0^{t'}
E(s)ds}\right|\le \int_{t'}^{t} b(s)ds<\varepsilon
\end{equation}
for $t,\ t'$ sufficiently large respectively sufficiently small.
Therefore $a(t)e^{i\int_0^t E(s)ds}$ has a limit when
$t\rightarrow\pm\infty$. This means
$$e^{i\int_0^t E(s)ds}\pe=a(t)e^{i\int_0^t E(s)ds}\po+e^{i\int_0^t E(s)ds}h(a(t))=a(t)e^{i\int_0^t E(s)ds}\po+h(a(t)e^{i\int_0^t E(s)ds})\rightarrow\psi_{E_{\pm\infty}}$$
Above we used $h(e^{i\theta}a)=e^{i\theta}h(a)$, see Proposition
\ref{pr:cm}. In addition $|a(t)|\rightarrow a_{\pm}$ as
$t\rightarrow\pm$ at a rate $|t|^{-\delta}.$ Since $E(s)=E(|a(s)|$
is $C^1$ in $|a|$ on $|a|\le\delta_2,$ we deduce $|E(\pm
s)-E_\pm|\le C(1+s)^{-\delta}$ for $s\ge 0$ and some constant $C>0.$
If we denote
$$\theta(\pm t)=\frac{1}{\pm t}\int_0^{\pm t}E(s)-E_\pm ds,\qquad t\ge 0$$
then $\lim_{|t|\rightarrow\infty}\theta(t)=0$ and
$$\lim_{t\rightarrow\pm}e^{it(E_\pm-\theta(t))}\psi_E(t)=\psi_{E_\pm}.$$
This finishes the proof of Theorem \ref{mt}. $\Box$
\section{Linear Estimates}\label{se:lin}
Consider the linear Schr\" odinger equation with a potential in
three space dimensions:
\begin{align}
i\frac{\partial u}{\partial t}&=(-\Delta+V(x))u \nonumber \\
u(0)&=u_0 \nonumber \end{align}
It is known that if V satisfies hypothesis (H1) (i) and (ii) then the radiative part of the solution, i.e. its projection onto the continuous spectrum of $H=-\Delta+V$, satisfies the estimates:
\begin{equation}
\|e^{-iHt}P_c u_0\|_{\Lsn}\leq C_M\frac{1}{|t|^{\frac{3}{2}}}\|u_0\|_{\Ls}
\label{eq:ls}
\end{equation}
for $\sigma>1$ and some constant $C_M>0$ independent of $u_0$ and $t\in\R$, and
\begin{equation}
\|e^{-iHt}P_c u_0\|_{L^p}\leq C_p\frac{1}{|t|^{3(\frac{1}{2}-\frac{1}{p})}}\|u_0\|_{L^{p'}}
\label{eq:lp}
\end{equation}
for some constant $C_p>0$ depending only on $2\leq p$. The case
$p=\infty$ in (\ref{eq:lp}) is proved by Goldberg and Schlag in \cite{gs:dis}.
The conservation of the $L^2$ norm gives the $p=2$ case:
$$\|e^{-iHt}P_c u_0\|_{L^2}=\|u_0\|_{L^2}$$ The general result
(\ref{eq:lp}) follows from Riesz-Thorin interpolation.
We would like to extend these estimates to the linearized dynamics around the center manifold. We consider the linear equation, with initial data at time $s$,
\begin{align}
i\frac{d\zeta}{dt}&=H\zeta+F_1(\psi_E,\zeta) \nonumber \\
\zeta(s)&=v \nonumber \end{align}
where $F_1(\pe,\zeta)=\frac{d}{d\varepsilon}g(\pe+\varepsilon\zeta)|_{\varepsilon=0}=\frac{\partial}{\partial u}g(u)|_{u=\pe}\zeta+\frac{\partial}{\partial \bar{u}}g(u)|_{u=\pe}\overline{\zeta}$. For the sake of simpler notation, we will use $F_1(\zeta)$.
By Duhamel's principle we have: \begin{equation} \zeta(t)=\ehs v(s)-i\int_s^t \eht F_1(\zeta)d\tau \label{soz} \end{equation}
In the next theorems we will extend estimates of type (\ref{eq:ls})-(\ref{eq:lp}) to the operators $\om(t,s)$ and
$T(t,s)$ considering the fact that $\pe(t)$ is small. Recall that
$$T(t,s)=P_c\om(t,s)-\ehs P_c\quad\textrm{i.e.}\quad \om(t,s)=R_{a(t)} T(t,s)+R_{a(t)}\ehs P_c$$
\begin{theorem}\label{th:lw}
There exists $\varepsilon_1>0$ such that for
$\|\xs\pe\|_{H^2}<\varepsilon_1$ there exist constants $C$, $C_p >0$
with the property that for any t, s $\in\R$ the followings hold:
\begin{align}
(i)\quad &\|\om(t,s)\|_{\Ls\rightarrow\Lsn}\leq\frac{C}{(1+|t-s|)^{\frac{3}{2}}} \nonumber \\
(ii) \quad &\|T(t,s)\|_{L^1\rightarrow\Lsn}\leq\left\{ \begin{array}{ll}
\frac{C}{|t-s|^{\frac{1}{2}}} & \textrm{for $s\leq t\leq s+1$} \\
\frac{C}{(1+|t-s|)^{\frac{3}{2}}} & \textrm{for $t>s+1$} \end{array} \right.
\nonumber \\
(iii)\quad &T(t,s)\in L_t^2(\R,L^2\rightarrow\Lsn)\cap L_t^\infty(\R,L^2\rightarrow\Lsn) \nonumber \\
(iv)\quad&\|\om(t,s)\|_{L^{p'}\rightarrow\Lsn}\leq\frac{C}{|t-s|^{3(\frac{1}{2}-\frac{1}{p})}}\quad\textrm{for all $2\leq p\leq L^\infty$} \nonumber \\
&\|T(t,s)\|_{L^{p'}\rightarrow\Lsn}\leq\left\{ \begin{array}{ll}
\frac{C}{|t-s|^{(\frac{1}{2}-\frac{1}{p})}} & \textrm{for $s\leq t\leq s+1$} \\
\frac{C}{(1+|t-s|)^{3(\frac{1}{2}-\frac{1}{p})}} & \textrm{for $t>s+1$} \end{array} \right.
\nonumber \end{align}
\end{theorem}
\textbf{Proof of Theorem \ref{th:lw}} Fix $s\in\R$.
$(i)$ By definition, we have $\om(t,s)v=\zeta(t)$ where $\zeta(t)$
satisfies equation \eqref{soz}. We project \eqref{soz} onto
continuous spectrum of $H=-\Delta+V:$ \begin{equation} \xi(t)=\ehs
P_c v-i\int_s^t \eht P_c F_1(R_a\xi)d\tau \label{sox} \end{equation}
where $\xi=P_c\zeta$. We are going to prove the estimate for
$P_c\Omega(t,s)$ by showing that the nonlinear equation \eqref{sox}
can be solved via contraction principle argument in an appropriate
functional space. To this extent let us consider the functional
space
$$X_1:=\{u\in C(\R,L_{-\sigma}^2(\R^3))|\sup_{t>s}(1+(t-s))^{\frac{3}{2}}\|u(t)\|_{L_{-\sigma}^2}<\infty\}$$ endowed with the norm $$\|u\|_{X_1}:=\sup_{t>s}\{(1+(t-s))^{\frac{3}{2}}\|u(t)\|_{L_{-\sigma}^2}\}<\infty$$ Note that the inhomogeneous term in (\ref{sox}) $\xi_0=\ehs P_c v$ satisfies $\xi_0\in X_1$ and
\begin{equation}
\|\xi_0\|_{X_1}\leq C_M\|v\|_{\Ls}
\label{eq:z0}
\end{equation}
because of (\ref{eq:ls}). We collect the $\xi$ dependent part of the right hand side of (\ref{sox}) in a linear operator $L(s):X_1\rightarrow X_1$
\begin{equation}
[L(s)\xi](t)=-i\int_s^t \eht P_c[F_1(R_a\xi)]d\tau \label{L} \end{equation}
We will show that $L$ is a well defined bounded operator from $X_1$ to $X_1$ whose operator norm can be made less or equal to 1/2 by choosing $\varepsilon_1$ sufficiently small. Consequently $Id-L$ is invertible and the solution of the equation (\ref{sox}) can be written as $\xi=(Id-L)^{-1}\xi_0$. In particular $$\|\xi\|_{X_1}\leq(1-\|L\|)^{-1}\|\xi_0\|_{X_1}\leq 2\|\xi_0\|_{X_1}$$
which in combination with the definition of $\om$, the definition of the norm $X_1$ and the estimate (\ref{eq:z0}), finishes the proof of $(i)$.
It remains to prove that $L$ is a well defined bounded operator from $X_1$ to $X_1$ whose operator norm can be made less than 1/2 by choosing $\varepsilon_1$ sufficiently small.
\begin{align}
\|L(s)\xi(t)\|_{L_{-\sigma}^2}\leq & \int_s^t \|\eht P_c\|_{L_{\sigma}^2\rightarrow L_{-\sigma}^2}\|F_1(R_a\xi)\|_{\Ls}d\tau \nonumber \\
\end{align}
On the other hand
$$\|F_1(R_a\xi)\|_{\Ls}\leq\|\xst(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})\|_{L^{\infty}}\|R_a \xi\|_{\Lsn}\leq(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2})\|\xi\|_{\Lsn}$$ and using the last three relations, as well as the estimate (\ref{eq:ls}) and the fact that $\xi\in X_1$ we obtain that
\begin{align}
\|L(s)\|_{X_1\rightarrow X_1}&\leq(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2}) \sup_{t>0}(1+|t-s|)^{\frac{3}{2}}\int_s^t \frac{1}{(1+|t-\tau|)^{\frac{3}{2}}} \cdot \frac{1}{(1+|\tau-s|)^{\frac{3}{2}}}{\ } d\tau \nonumber \\
&\leq(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2}) \sup_{t>0}(1+|t-s|)^{\frac{3}{2}}\frac{1}{(1+|\frac{t-s}{2}|)^{\frac{3}{2}}}\leq C(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2}) \nonumber \end{align}
Now choosing $\varepsilon_1$ small enough we get
$$\|L\|_{X_1\rightarrow X_1}<\frac{1}{2}$$ Therefore
$$\|P_c\Omega(t,s)\|_{\Ls\rightarrow\Lsn}\leq\frac{\tilde C}{(1+|t-s|)^{\frac{3}{2}}}$$
and
$$\|\Omega(t,s)\|_{\Ls\rightarrow\Lsn}\leq \|R_{a(t)}\|_{\Lsn\rightarrow\Lsn}\|P_c\Omega(t,s)\|_{\Ls\rightarrow\Lsn}\le
\frac{C}{(1+|t-s|)^{\frac{3}{2}}}$$ by Lemma \ref{le:pcinv}.
$(ii)$ Recall that \begin{equation}
P_c\om(t,s)v=T(t,s)v+\ehs P_c v
\label{omt}
\end{equation}
Denote: \begin{equation}
T(t,s)v=W(t)
\label{w}
\end{equation}
then, by plugging in (\ref{sox}), $W(t)$ satisfies the following equation:
\begin{equation}
W(t)=\underbrace{-i\int_s^t \eht P_c [F_1(R_a\ehts P_c v)]d\tau}_{f(t)}+[L(s)W](t) \label{sow} \end{equation} By definition of $T(t,s)$ (\ref{w}) it is sufficient to prove that the solution of (\ref{sow}) satisfies $$\|W(t)\|_{\Lsn}\leq\left\{ \begin{array}{ll}
\frac{C\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}} & \textrm{for $s\leq t\leq s+1$} \\
\frac{C\|v\|_{L^1}}{(1+|t-s|)^{\frac{3}{2}}} & \textrm{for $t>s+1$} \end{array} \right.$$
Let us also observe that it suffices to prove this estimate only for the forcing terms $f(t)$ because then we will be able to do the contraction principle in the functional space in which $f(t)$ will be, and thus obtain the same decay for $W$ as for $f(t)$.
This time we will consider the functional space $$X_2=\{u\in C(\R,\Lsn{\R^3})| \sup_{|t-s|>1} (1+|t-s|)^\frac{3}{2}\|u\|_{\Lsn}<\infty, \sup_{|t-s|\leq1} |t-s|^\frac{1}{2}\|u\|_{\Lsn}<\infty \}$$
endowed with the norm $$\|u\|_{X_2}=\left\{ \begin{array}{ll}
\sup_t |t-s|^\frac{1}{2}\|u\|_{\Lsn} & \textrm{for $|t-s|\leq 1$} \\
\sup_t (1+|t-s|)^\frac{3}{2}\|u\|_{\Lsn} & \textrm{for $|t-s|>1$} \end{array} \right.$$
Now we will estimate $f(t)$. First we will investigate the short time behavior of this term. If $s\leq t\leq s+1$. Recall that $F_1(u)=\frac{d}{d\tau}g(\pe+\tau u)=\frac{\partial}{\partial u}g(u)|_{u=\pe}u+\frac{\partial}{\partial \bar{u}}g(u)|_{u=\pe}\overline{u}=g_{u}u+g_{\bar{u}}\overline{u}.$
$$\|f(t)\|_{\Lsn}\leq\|\xsn\int_s^t \eht P_c F_1(R_a\ehts P_c v) d\tau\|_{L^2}$$
For the term $g_u R_a\ehts P_c v$ we have
\begin{align}
&\|\xsn\int_s^t \|\eht P_c g_u R_a\ehts P_c v d\tau\|_{L^2} \nonumber\\
\leq&\|\xsn\|_{L^2}\int_s^t \|\ehs e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^\infty} d\tau \nonumber\\
&\leq \int_s^t\frac{C}{|t-s|^{\frac{3}{2}}}\sup \|\widehat{g_u}\|_{L^1}\|v\|_{L^1}d\tau\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}}\sup\|\widehat{g_u}\|_{L^1} <\infty \nonumber
\end{align} and for the term $g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v$ we have
\begin{align}
&\|\xsn\int_s^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v d\tau\|_{L^2} \nonumber\\
&\leq\|\xsn\|_{L^2}\int_s^{s+\frac{t-s}{4}} \|e^{-iH(t+s-2\tau)} \ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v\|_{L^\infty} d\tau \nonumber\\
&\quad+\int_{s+\frac{t-s}{4}}^t \|\eht P_c\|_{\Ls\rightarrow\Lsn}\|\xs g_{\bar{u}}\overline{R}_a \ehts P_c v\|_{L^2} d\tau \nonumber \\
&\leq\int_s^{s+\frac{t-s}{4}}\frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\sup \|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau+\int_{s+\frac{t-s}{4}}^t \frac{C}{(1+|t-\tau)^{3/2}}\|\xs g_{\bar{u}}\|_{L^2} \|\ehts P_c v\|_{L^\infty} d\tau \nonumber \\
&\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}}\sup (\|\widehat{g_{\bar{u}}}\|_{L^1}+\|\xs g_{\bar{u}}\|_{L^2})<\infty \nonumber \end{align} There we used J-S-S type estimate; see Appendix for $t=\tau-s$; $|\tau-s|\leq 1$. For the long time behavior of $f(t)$, we will split this integral into three parts to be estimated differently. For $t>s+1$, $$f(t)=\underbrace{\int_s^{s+\frac{1}{2}}\cdots}_{I_{1}}+\underbrace{\int_{s+\frac{1}{2}}^{\frac{t+s}{2}}\cdots}_{I_{2}}+\underbrace{\int_{\frac{t+s}{2}}^t \cdots}_{I_{3}}$$
Then for $t>s+1$,
\begin{align} \|I_1\|_{\Lsn}&\leq\|\xsn\int_s^{s+\frac{1}{2}} \ehs P_c F_1(R_a \ehts P_c v) d\tau\|_{L^2} \nonumber \\
&\leq\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \|\ehs e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^\infty} d\tau \nonumber \\
&\quad+\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \|e^{-iH(t+s-2\tau)} \ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v\|_{L^\infty} d\tau \nonumber \\
&\leq\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \frac{C}{|t-s|^{\frac{3}{2}}}\|e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^1} d\tau \nonumber \\
&\quad+\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v\|_{L^1} d\tau \nonumber \\
&\leq C\|\xsn\|_{L^2}\Big(\frac{1}{|t-s|^{\frac{3}{2}}}+\frac{1}{|t-s-1|^{\frac{3}{2}}}\Big)\int_s^{s+\frac{1}{2}} \sup (\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})\|v\|_{L^1}d\tau \nonumber \\
&\leq C\|\xsn\|_{L^2}\sup(\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})\frac{1}{(1+|t-s|)^{\frac{3}{2}}}\|v\|_{L^1} \nonumber \end{align}
For the second integral we have \begin{align}
\|I_2\|_{\Lsn}&\leq\int_{s+\frac{1}{2}}^{\frac{t+s}{2}} \|\eht P_c\|_{\Ls\rightarrow\Lsn}\|F_1(R_a\ehts P_c v)\|_{\Ls} d\tau \nonumber \\
&\leq\int_{s+\frac{1}{2}}^{\frac{t+s}{2}} \frac{C}{(1+|t-\tau|)^{\frac{3}{2}}}\|\xs|\pe|^{1+\p}\|_{L^2}\|\ehts P_c v\|_{L^\infty}d\tau \nonumber \\
&\leq\frac{C\|v\|_{L^1}}{(1+|\frac{t-s}{2}|)^{\frac{3}{2}}}\int_{s+\frac{1}{2}}^{\frac{t+s}{2}} \frac{d\tau}{|\tau-s|^{\frac{3}{2}}}\leq\frac{C\|v\|_{L^1}}{(1+|t-s|)^{\frac{3}{2}}} \nonumber \end{align}
$I_3$ is estimated similiar to $I_2$.
$(iii)$ From \eqref{sow} we have \begin{align} \xsn W(t)=&\int_s^t \xsn\eht P_c[F_1(R_a\ehts P_c v)]d\tau+\int_s^t \xsn\eht P_c[F_1(R_a W(\tau))]d\tau \nonumber \end{align}
Then
\begin{align} \|\xsn W(t)\|_{L_t^2 L_x^2}&\leq\Big\|\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}} \|\xs F_1(R_a\ehts P_c v)\|_{L^2}d\tau\Big\|_{L^2_t} \nonumber \\
&+\Big\|\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}}(\|\xst g_u\|_{L^\infty}+\|\xst g_{\bar{u}}\|_{L^\infty})\|\xsn W(\tau)\|_{L^2_x}d\tau \Big\|_{L^2_t} \nonumber \\
&\leq C\|K\|_{L^1}\|v\|_{L^2}+\varepsilon_1 C\|K\|_{L^1}\|\xsn W\|_{L^2_x L^2_t} \nonumber\end{align}
Where $K(t)=(1+|t|)^{-3/2}$. For the term $\xs F_1(R_a e^{-iHt} P_c v)=\xs (g_u R_a e^{-iHt} P_c v+g_{\bar{u}}\overline{R}_a e^{iHt} P_c v)$ we used $\|\xst g_u\|_{L^\infty}$ and $\|\xst g_{\bar{u}}\|_{L^\infty}$ is uniformly bounded in $t$ since $|g_u|=|g_{\bar{u}}|\leq C(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})$ and the Kato smoothing estimate $\|\xsn e^{-iHt} P_c v\|_{L^2_t(\R,L^2_x)}\leq C\|v\|_{L^2_x}$. Choosing $\varepsilon_1$ small enough we get $\|\xsn W\|_{L^2_x L^2_t}<\infty$. In other words $T(t,s)\in L_t^2(\R, L^2\rightarrow\Lsn)$. And similarly \begin{align} \|\xsn W(t)\|_{L_x^2}&\leq\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}} \|\xs F_1(R_a\ehts P_c v)\|_{L^2}d\tau \nonumber \\
&+\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}}(\|\xst g_u\|_{L^\infty}+\|\xst g_{\bar{u}}\|_{L^\infty})\|\xsn W(\tau)\|_{L^2_x}d\tau \nonumber \\
&\leq C\|v\|_{L^2}+\varepsilon_1 C\|\xsn W\|_{L^2_x} \nonumber\end{align} This finishes the proof of $(iii)$, $T(t,s)\in L_t^2(\R, L^2\rightarrow\Lsn)\cap L_t^\infty(\R, L^2\rightarrow\Lsn).$
$(iv)$ By Riesz-Thorin interpolation between $(ii)$ and $(iii)$ (the $L_t^\infty$ part) we get the desired estimates. $\Box$
The next step is to obtain estimates for $\om(t,s)$ and $T(t,s)$ in unweighted $L^p$ spaces.
\begin{theorem}{\label{th:lp}}
Assume that $\|\xs\pe\|_{H^2} < \varepsilon_1$ (where $\varepsilon_1$ is the one used in Theorem \ref{th:lw}).
Then there exist constants $C_2$, $C_2'$ and $C_\infty$ for all $t$, $s$ $\in$ $\R$ the following estimates hold:
\begin{align}
(i)\quad &\|\om(t,s)\|_{L^2\rightarrow L^2}\leq C_2,\quad \|T(t,s)\|_{L^2\rightarrow L^2}\leq C_2 \nonumber \\
(ii)\quad &\|\om(t,s)\|_{L^1\rightarrow L^\infty}\leq\frac{C_\infty}{|t-s|^{\frac{3}{2}}}, \quad \|T(t,s)\|_{L^1\rightarrow L^\infty}\leq\left\{ \begin{array}{ll}
C_\infty|t-s|^{\frac{1}{2}} & \textrm{for $|t-s|\leq 1$} \\
\frac{C_\infty}{|t-s|^{\frac{3}{2}}} & \textrm{for $|t-s|>1$} \end{array} \right. \nonumber \\
(iii)\quad &\|T(t,s)\|_{L^{p'}\rightarrow L^2}\leq C_2',
\quad\textrm{for }\quad p=6 \nonumber\end{align}
\end{theorem}
\begin{remark}\label{rmk:lp}
By Riesz-Thorin interpolation from $(i)$ and $(ii)$, and from $(i)$ and $(iii)$ we get
\begin{eqnarray}
\|\om(t,s)\|_{L^{p'}\rightarrow L^p}&\leq&\frac{C_p}{|t-s|^{3\ip}},\quad\textrm{for all}\quad 2\leq p\leq\infty \nonumber \\
\|T(t,s)\|_{L^{p'}\rightarrow L^p}&\leq&\left\{ \begin{array}{ll}
C_p|t-s|^{\ip} & \textrm{for $|t-s|\leq 1$} \\
\frac{C}{|t-s|^{3\ip}} & \textrm{for $|t-s|>1$} \end{array} \right.,\quad\textrm{for all}\quad 2\leq p\leq\infty \nonumber \\
\|T(t,s)\|_{L^{p'}\rightarrow L^2}&\leq &C_p,
\quad\textrm{for all}\quad 2\leq p\leq6 \nonumber
\end{eqnarray}
\end{remark}
\noindent\textbf{Proof of Theorem \ref{th:lp}} Because of the estimate (\ref{eq:lp}) and relation $P_c\om=T+\ehs P_c$, It suffices to prove the theorem for $T(t,s)$.
$(i)$ To estimate the $L^2$ norm we will use duality argument to make use of cancelations. \begin{align}
&\|f(t)\|_{L^2}^2=\< f(t),f(t)\> \nonumber \\
&=\int_s^t\int_s^t \< \eht P_c F_1(R_a\ehts P_c v),e^{-iH(t-\tau')} P_c F_1(R_a e^{-iH(\tau' -s)} P_c v)\> d\tau' d\tau \nonumber \\
&=\int_s^t \int_s^t \< F_1(R_a\ehts P_c v),e^{-iH(\tau-\tau')} P_c F_1(R_a e^{-iH(\tau'-s)} P_c v)\> d\tau' d\tau \nonumber \\
&=\int_s^t \int_s^t \< \xs F_1(R_a\ehts P_c v), \xsn e^{-iH(\tau-\tau')} P_c F_1(R_ae^{-iH(\tau'-s)} P_c v) \> d\tau' d\tau \nonumber \\
&\leq\int_s^t \int_s^t \|F_1(R_a\ehts P_c v)\|_{\Ls}\|e^{-iH(\tau-\tau')} P_c F_1(R_a e^{-iH(\tau'-s)} P_c v)\|_{\Lsn} d\tau' d\tau \nonumber \\
&\leq\int_s^t \|\xs F_1(R_a\ehts P_c v)\|_{L^2}\int_s^t\frac{C}{(1+|\tau-\tau'|)^{3/2}}\|\xs F_1(R_a e^{-iH(\tau'-s)} P_c v)\|_{L^2} d\tau' d\tau\nonumber \\
&\leq C\|\xs F_1(R_a\ehts P_c v)\|_{L_{\tau}^2 L_x^2}
\Big\|\int_s^t\frac{C}{(1+|\tau-\tau'|)^{3/2}} \|\xs F_1(R_a e^{-iH(\tau'-s)} P_c v)\|_{L_x^2} d\tau\Big\|_{L^2_{\tau}}\nonumber \\
&\leq C\|K\|_{L^1}\|\xs F_1(R_a e^{-iHt} P_c v)\|_{L_t^2 L^2_x}^2\leq C\|v\|^2_{L^2}<\infty \nonumber \end{align}
At the last line, $K(t)=(1+|t|)^{-3/2}$ and we used convolution estimate. For the term $\xs F_1(R_a e^{-iHt} P_c v)=\xs (g_u R_a e^{-iHt} P_c v+g_{\bar{u}}\overline{R}_a e^{iHt} P_c v)$ we used the Kato smoothing estimate $\|\xsn e^{-iHt} P_c v\|_{L^2_t(\R,L^2_x)}\leq C\|v\|_{L^2_x}$.
We will estimate $L^2$ norm of $L$ similiar to $f$.
\begin{align}
&\|L(s)W\|_{L^2}^2=\<L(s)W,L(s)W\> \nonumber \\
&=\<\int_s^t \eht P_c F_1(W(\tau))d\tau,\int_s^t e^{-iH(t-\tau')} P_c F_1(W(\tau'))d\tau'\> \nonumber \\
&=\int_s^t \int_s^t \< F_1(W(\tau)),e^{-iH(\tau-\tau')} P_c F_1(W(\tau'))\>] d\tau' d\tau \nonumber \\
&\leq\int_s^t (\|\xs g_u\|_{L^\infty}+\|\xs g_{\bar{u}}\|_{L^\infty})\|\xsn W\|_{L^2} \nonumber \\
&\quad\quad\quad\quad\quad\quad\times\int_s^t C K(\tau-\tau')(\|\xs g_u\|_{L^\infty}+\|\xs g_{\bar{u}}\|_{L^\infty})\|\xsn W\|_{L^2} d\tau' d\tau\nonumber \\
&\leq C\|\xsn W\|_{L_x^2 L_\tau^2}
\Big\|\int_s^t CK(\tau-\tau') \|\xsn W\|_{L_x^2} d\tau'\Big\|_{L^2_{\tau}}\nonumber \\
&\leq C\|K\|_{L^1}\|\xsn W\|_{L_\tau^2 L^2_x}<\infty \nonumber \end{align} By Theorem \ref{th:lw} $(iii)$, $\|\xsn W\|_{L_\tau^2 L^2_x}<\infty$.
\noindent Therefore we conclude $\|T(s,t)\|_{L^2\rightarrow L^2}\leq C$ and $\|\om(s,t)\|_{L^2\rightarrow L^2}\leq C$
$(ii)$ Let us first investigate the short time behavior of the forcing term $f(t)$. We will assume $s\leq t\leq s+1$,
\begin{align}
\|f(t)\|_{L^\infty}&=\|\int_s^t \eht P_c F_1(R_a \ehts P_c v)d\tau\|_{L^\infty} \nonumber \\
&\leq\int_s^t \|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty}\|e^{iH(\tau-s)}P_c g_u R_a\ehts P_c v\|_{L^1}d\tau \nonumber \\
&+\int_s^{s+\frac{t-s}{4}} \|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\
&+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}} \|e^{-iH(t-\tau)}P_c\|_{L^1\rightarrow L^infty}\|g_{\bar{u}} \overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\
&+\int_{t-\frac{t-s}{4}}^t \|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\
&\leq\int_s^t \frac{C}{|t-s|^{\frac{3}{2}}}\sup\|\widehat{g_u}\|_{L^1}\|v\|_{L^1}d\tau+\int_s^{s+\frac{t-s}{4}} \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\sup \|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau \nonumber \\
&+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}} \frac{C}{|t-\tau|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1}\|v\|_{L^1}d\tau+\int_{t-\frac{t-s}{4}}^t \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\sup \|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau \nonumber \\ &\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}}(\|g_{\bar{u}}\|_{L^1}+\sup(\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})) \nonumber \end{align}
Now let us investigate the long time bevaviour of the forcing term $f(t)$. We will assume $t>s+1$ and seperate $f(t)$ into four parts as follows,
$$f(t)=\underbrace{\int_s^{s+\frac{1}{4}}\cdots}_{I_1}+\underbrace{\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\cdots}_{I_2}+\underbrace{\int_{t-\frac{1}{4}}^t\cdots}_{I_3}$$ We will start with $I_2$ for which we are away from the singularities around $\tau=s$ and $\tau=t$. Then for $I_1$ and $I_3$ we will use J-S-S type estimate to remove the singularities.
\begin{align}
\|I_2\|_{L^\infty}&\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}} \|\eht P_c (g_u R_a\ehts P_c v+g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v})\|_{L^\infty}d\tau \nonumber \\
&\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}} \frac{C}{|t-\tau|^{\frac{3}{2}}} (\|g_u R_a\ehts P_c v\|_{L^1}+\|g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1})d\tau \nonumber \\
&\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}}( \|g_u\|_{L^1}+\|g_{\bar{u}}\|_{L^1})\|\ehts P_c v\|_{L^\infty}d\tau \nonumber \\
&\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}}( \|g_u\|_{L^1}+\|g_{\bar{u}}\|_{L^1}) \frac{\|v\|_{L^1}}{|\tau-s|^{\frac{3}{2}}}d\tau \nonumber \\
&\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}}(\|g_u\|_{L^1}+\|g_{\bar{u}}\|_{L^1}) \nonumber \end{align}
\begin{align}
\|I_1\|_{L^\infty}&\leq\int_s^{s+\frac{1}{4}} \|\ehs P_c e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^\infty}d\tau \nonumber \\
&\quad+\int_s^{s+\frac{1}{4}} \|e^{-iH(t+s-2\tau)} \ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^\infty}d\tau \nonumber \\
&\leq\int_s^{s+\frac{1}{4}}\frac{C}{|t-s|^{\frac{3}{2}}}\|e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^1}d\tau \nonumber \\
&\quad+\int_s^{s+\frac{1}{4}}\frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\
&\leq\int_s^{s+\frac{1}{4}} C \Big(\frac{1}{|t-s|^{\frac{3}{2}}}+\frac{1}{|t-s-1|^{\frac{3}{2}}}\Big)\sup(\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})\|v\|_{L^1}d\tau \nonumber \\
&\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align}
\begin{align}
\|I_3\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)} P_c e^{-iH(t-s)} P_cv\|_{L^\infty}d\tau \nonumber \\
&\quad+\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)} P_c e^{iH(t+s-2\tau)} P_c \bar{v}\|_{L^\infty}d\tau \nonumber \\
&\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align}
Now it remains to show that $L(s)W$ is bounded in $L^\infty$. Again to remove the singularities we will split the integral in different parts. Let us consider $s\leq t\leq s+1$,
\begin{align}
&L(s)W=\int_s^t \eht P_c F_1(R_a W(\tau)) d\tau\nonumber \\
&=\int_s^t \eht P_c g_u R_a\Big[\int_s^\tau e^{-iH(\tau-\tau')} P_c [F_1(R_a e^{-iH(\tau'-s)}P_c v)+F_1(R_a W(\tau'))]d\tau'\Big]d\tau \nonumber \\
&\quad+\int_s^t \eht P_c g_{\bar{u}}\overline{R}_a\Big[\int_s^\tau e^{iH(\tau-\tau')}P_c[\overline{ F_1(R_a e^{-iH(\tau'-s)}P_c v)}+\overline{ F_1(R_a W(\tau'))}]d\tau'\Big]d\tau \nonumber \end{align}
All the terms will be either of the following forms
$$L_1=\int_s^t \eht P_c g_u R_a\int_{s}^\tau e^{-iH(\tau-\tau')} P_c X(\tau')d\tau'd\tau$$
$$L_2=\int_s^t \eht P_c g_{\bar{u}}\overline{R}_a\int_{s}^\tau e^{iH(\tau-\tau')} P_c \overline{X(\tau')}d\tau'd\tau$$
where $X(\tau')=g_u R_a e^{-iH(\tau'-s)}P_c v, g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}, g_u R_a W(\tau'), g_{\bar{u}}\overline{R_a W(\tau')}$
In what follows we will add $e^{iH(t-\tau)}$ and $\eht$ terms after $g_u R_a$ and $g_{\bar{u}}\overline{R}_a$ then we will estimate the terms in a similiar way as we estimated $I_1$ and $I_3$.
\begin{equation}
L_1=\int_s^t \eht P_c g_u R_a e^{iH(t-\tau)}\int_{s}^\tau e^{-iH(t-\tau')} P_c X(\tau')d\tau'd\tau \label{l1} \end{equation}
\begin{equation}L_2=\int_s^t \eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\int_{s}^\tau e^{-iH(t-2\tau+\tau')} P_c \overline{X(\tau')}d\tau'd\tau \label{l2} \end{equation}
\begin{itemize}
\item For $X(\tau')=g_u R_a e^{-iH(\tau'-s)}P_c v$ we have
\begin{align}
\|L_1\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\int_s^{\tau}\|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty} \|e^{iH(\tau'-s)}P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau'd\tau \nonumber \\
&\leq\int_s^t \|\widehat{g_u}\|_{L^1}\int_s^\tau \frac{C}{|t-s|^{\frac{3}{2}}}\|\widehat{g_u}\|_{L^1}\|v\|_{L^1}d\tau'd\tau\leq C\sqrt{t-s}\|v\|_{L^1}\leq C\|v\|_{L^1}\quad\textrm{for }s\leq t\leq s+1 \nonumber \end{align}
\begin{align}
\|L_2\|_{L^\infty}&\leq\int_s^{t-\frac{t-s}{4}} \|\eht P_c\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}}\overline{R}_a\int_s^{\tau}e^{-iH(\tau-s)}P_c e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'd\tau \nonumber \\
&+\int_{t-\frac{t-s}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty} \nonumber \\
&\quad\quad\quad\quad\times \int_s^{\tau}\|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'd\tau \nonumber \\
&\leq\int_s^{t-\frac{t-s}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}} \|g_{\bar{u}}\|_{L^1}\int_s^\tau \frac{C}{|\tau-s|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau'd\tau \nonumber \\
&+\int_{t-\frac{t-s}{4}}^t \|\widehat{g_{\bar{u}}}\|_{L^1}\int_s^\tau \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau'd\tau\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber \end{align}
\item For $X(\tau')=g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}$ we have
\begin{align}
\|L_1\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\nonumber \\ &\quad\quad\quad\times\Big[\int_{s}^{s+\frac{t-s}{4}} \|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\nonumber \\
&\quad\quad\quad+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}}\|e^{-iH(t-\tau')}\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\
&\quad\quad\quad+\int_{t-\frac{t-s}{4}}^{\tau}\|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\Big]d\tau \nonumber \\
&\leq\int_s^t \|\widehat{g_u}\|_{L^1}\Big[\int_s^{s+\frac{t-s}{4}}\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1}\frac{C}{|\tau-s|^{\frac{3}{2}}}d\tau'\nonumber \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{t-\frac{t-s}{4}}^\tau\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'\Big]\|v\|_{L^1}d\tau\nonumber \\
&\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber\end{align}
\begin{align}
\|L_2\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\Big[\int_{s}^{s+\frac{t-s}{4}}\| e^{-iH(t-2\tau+2\tau'-s)}e^{iH(\tau'-s)} P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^\infty}d\tau' \nonumber \\
&\quad\quad\quad+\int_{s+\frac{t-s}{4}}^\tau \|e^{-iH(t-2\tau+\tau')} P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau' \Big]d\tau \nonumber \\
&\leq\int_s^t \|\widehat{g_{\bar{u}}}\|_{L^1}\Big[\int_s^{s+\frac{t-s}{4}} \frac{C}{|t-2\tau+2\tau'-s|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1} \|v\|_{L^1}d\tau'+\int_{s+\frac{t-s}{4}}^\tau \frac{C}{|t-2\tau+\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1} \frac{\|v\|_{L^1}}{|\tau'-s|^{\frac{3}{2}}}d\tau'\Big] d\tau \nonumber \\
&\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber \end{align}
\item For $X(\tau')=g_u R_a W(\tau')$ and $g_{\bar{u}} \overline{R_a W(\tau')}$ we will change the order of integration,
\begin{align}
\|L_1\|_{L^\infty}&\leq\int_s^t\int_{\tau'}^t\|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\|e^{-iH(t-\tau')}P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a W(\tau')\|_{L^1}d\tau d\tau' \nonumber \\
&\leq\int_s^t\int_{\tau'}^t \|\widehat{g_u}\|_{L^1}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|\xs g_u\|_{L^2}\|W\|_{\Lsn}d\tau d\tau' \nonumber \\
&\leq\int_s^t\|\widehat{g_u}\|_{L^1}\frac{C}{|t-\tau'|^{\frac{1}{2}}}\|\xs g_u\|_{L^2}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{1}{2}}} d\tau' \nonumber \\
&\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber \end{align}
\begin{align}
\|L_2\|_{L^\infty}&\leq\int_s^t\int_{\tau'}^{t-\frac{t-\tau'}{4}} \|\eht P_c\|_{L^1\rightarrow L^\infty}\|g_u R_a e^{-iH(\tau-\tau')} P_c g_u R_a W(\tau')\|_{L^1}d\tau d\tau' \nonumber \\
&+\int_s^t\int_{t-\frac{t-\tau'}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\| e^{-iH(t+\tau'-2\tau)} P_c g_u R_a W(\tau')\|_{L^\infty}d\tau d\tau' \nonumber \\
&\leq\int_s^t\int_{\tau'}^{t-\frac{t-\tau'}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^2}\|e^{-iH(\tau-\tau')}P_c\|_{L^2\rightarrow L^2}\|g_u R_a W(\tau')\|_{L^2}d\tau d\tau' \nonumber \\
&+\int_s^t\int_{\tau'}^{t-\frac{t-\tau'}{4}} \|\widehat{g_u}\|_{L^1}\frac{C}{|t+\tau'-2\tau|^{\frac{3}{2}}}\|g_u R_a W(\tau')\|_{L^1}d\tau d\tau' \nonumber \\
&\leq\int_s^t\frac{C}{|t-\tau'|^{\frac{1}{2}}}\|g_{\bar{u}}\|_{L^2}\|\xs g_u\|_{L^\infty}\|W(\tau')\|_{\Lsn}d\tau' \nonumber \\
&+\int_s^t \|\widehat{g_u}\|_{L^1}\frac{C}{|t-\tau'|^{\frac{1}{2}}}\|\xs g_u\|_{L^2}\|W(\tau')\|_{\Lsn}d\tau' \nonumber \\
&\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber
\end{align}
\end{itemize}
Similarly we will investigate the long time behavior of the operator $L(s)$ for $t>s+1$.
$$L(s)W(t)=\underbrace{\int_s^{t-\frac{1}{4}}\cdots}_{L_3}+\underbrace{\int_{t-\frac{1}{4}}^t\cdots}_{L_4}$$
\begin{align}
\|L_3\|_{L^\infty}&\leq\int_s^{t-\frac{1}{4}} \|\eht P_c F_1(R_a W(\tau))\|_{L^\infty} d\tau \nonumber \\
&\leq\int_s^{t-\frac{1}{4}} \|\eht P_c\|_{L^1\rightarrow L^\infty}\| F_1(R_a W(\tau))\|_{L^1} d\tau \nonumber \\
&\leq\int_s^{t-\frac{1}{4}} \frac{C}{|t-s|^{\frac{3}{2}}}(\|\xs g_u\|_{L^2}+\|\xs g_{\bar{u}}\|_{L^2})\|W\|_{\Lsn} d\tau \nonumber \\
&\leq\frac{C}{|t-s|^{\frac{3}{2}}}\int_s^{t-\frac{1}{4}} \frac{\|v\|_{L^1}}{(1+|\tau-s|)^{\frac{3}{2}}} d\tau\leq\frac{C}{|t-s|^{\frac{3}{2}}}\|v\|_{L^1} \nonumber \end{align}
In $L_4$ we will plug in \eqref{sow} once more:
\begin{align}
L_4&=\int_{t-\frac{1}{4}}^t \eht P_c F_1(R_a W(\tau)) d\tau\nonumber \\
&=\int_{t-\frac{1}{4}}^t \eht P_c g_u R_a\Big[\int_s^\tau e^{-iH(\tau-\tau')} P_c [F_1(R_a e^{-iH(\tau'-s)}P_c v)+ F_1(R_a W(\tau'))]d\tau'\Big]d\tau \nonumber \\
&\quad+\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a\Big[\int_s^{\tau} e^{iH(\tau-\tau')}P_c[\overline{F_1(R_a e^{-iH(\tau'-s)}P_c v)}+\overline{F_1(R_a W(\tau'))}]d\tau'\Big]d\tau \nonumber \end{align}
Again we will add $e^{iH(t-\tau)}$ and $\eht$ terms after $g_u R_a$ and $g_{\bar{u}}\overline{R}_a$. Then all the terms will be similar to $L_1$, $L_2$, $(\ref{l1})-(\ref{l2})$ respectively. After seperating the the inside integrals into pieces, we will estimate short time step integrals exactly the same way we did short time behavior by using JSS estimate, and the other integrals will be estimated using the usual norms.
\begin{itemize}
\item For $X(\tau')=g_u R_a e^{-iH(\tau'-s)}P_c v$ we have
\begin{align}
\|L_1\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\nonumber \\
&\quad\quad\Big[\int_s^{s+\frac{1}{4}}\|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty} \|e^{iH(\tau'-s)}P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau' \nonumber \\
&\quad\quad+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}} \|e^{-iH(t-\tau')} P_c\|_{L^1\rightarrow L^\infty}\|g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^\infty}d\tau' \nonumber \\
&\quad\quad+\int_{t-\frac{1}{4}}^{\tau}\|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty} \|e^{iH(\tau'-s)}P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau'\Big]d\tau \nonumber \\
&\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align}
\begin{align}
\|L_2\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty} \nonumber \\
&\quad\quad\quad\quad\times\Big[ \int_s^{s+\frac{1}{4}}\|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\
&\quad\quad\quad+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\|e^{-iH(t-2\tau+\tau')}P_c\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}} \overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\
&\quad\quad\quad+\int_{t-\frac{1}{4}}^{\tau}\|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\Big]d\tau \nonumber \\
&\leq\frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align}
\item For $X(\tau')=g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}$ we have
\begin{align}
\|L_1\|_{L^\infty}\leq&\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\nonumber \\ &\times\Big[\int_{s}^{s+\frac{1}{4}} \|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\nonumber \\
&\quad\quad\quad+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\|e^{-iH(t-\tau')}\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\
&\quad\quad\quad+\int_{t-\frac{1}{4}}^{\tau}\|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\Big]d\tau \nonumber \\
\leq&\int_{t-\frac{1}{4}}^t \|\widehat{g_u}\|_{L^1}\Big[\int_s^{s+\frac{1}{4}}\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1}\frac{C}{|\tau-s|^{\frac{3}{2}}}d\tau'\nonumber \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{t-\frac{1}{4}}^\tau\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'\Big]\|v\|_{L^1}d\tau\nonumber \\
\leq& C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber\end{align}
\begin{align}
\|L_2\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\Big[\int_{s}^{s+\frac{1}{4}}\| e^{-iH(t-2\tau+2\tau'-s)}e^{iH(\tau'-s)} P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^\infty}d\tau' \nonumber \\
&\quad\quad\quad+\int_{s+\frac{1}{4}}^\tau \|e^{-iH(t-2\tau+\tau')} P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau' \Big]d\tau \nonumber \\
&\leq\int_s^t \|\widehat{g_{\bar{u}}}\|_{L^1}\Big[\int_s^{s+\frac{1}{4}} \frac{C}{|t-2\tau+2\tau'-s|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1} \|v\|_{L^1}d\tau'+\int_{s+\frac{1}{4}}^\tau \frac{C}{|t-2\tau+\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1} \frac{\|v\|_{L^1}}{|\tau'-s|^{\frac{3}{2}}}d\tau'\Big] d\tau \nonumber \\
&\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align}
\item $L_1$, $L_2$ terms corresponding to $X(\tau')=g_u R_a W(\tau')$ and $g_{\bar{u}} \overline{R_a W(\tau')}$
\begin{align}
\|L_1\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\int_s^{\tau}\|e^{-iH(t-\tau')}P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a W(\tau')\|_{L^1}d\tau'd\tau \nonumber \\
&\leq\int_{t-\frac{1}{4}}^t \|\widehat{g_u}\|_{L^1}\int_s^{\tau}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|\xs g_u\|_{L^2}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau' d\tau \nonumber \\
&\leq\int_{t-\frac{1}{4}}^t\Bigg[\int_s^{\frac{t+s}{2}} \frac{C}{|t-\tau'|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau +\int_{\frac{t+s}{2}}^\tau \frac{C}{|t-\tau'|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau\Bigg] d\tau' \nonumber \\
&\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber
\end{align}
\begin{align}
\|L_2\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R_a} e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\int_s^{\tau}\|e^{-iH(t+\tau'-2\tau)}P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a W(\tau')\|_{L^1}d\tau'd\tau \nonumber \\
&\leq\int_{t-\frac{1}{4}}^t \|\widehat{g_{\bar{u}}}\|_{L^1}\int_s^{\tau}\frac{C}{|t+\tau'-2\tau|^{\frac{3}{2}}}\|\xs g_u\|_{L^2}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau' d\tau \nonumber \\
&\leq\int_{t-\frac{1}{4}}^t\Bigg[\int_s^{\frac{\tau+s}{2}} \frac{C}{|t+\tau'-2\tau|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau +\int_{\frac{\tau+s}{2}}^\tau \frac{C}{|t-\tau'|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau\Bigg] d\tau' \nonumber \\
&\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber
\end{align}
\end{itemize}
\noindent Now combining all the above estimates we get $$\|W(t)\|_{L^\infty}\leq\left\{ \begin{array}{ll}
C |t-s|^{\frac{1}{2}} & \textrm{for $|t-s|\leq 1$} \\
\frac{C}{|t-s|^{\frac{3}{2}}} & \textrm{for $|t-s|>1$} \end{array} \right.$$ This finishes the proof of $(ii)$.
$(iii)$ We split $f$ given by \eqref{sow}:
$$f=\underbrace{\int_s^{s+1}\cdots}_{I_1}+\underbrace{\int_{s+1}^t\cdots}_{I_2}$$
For $I_1$ integral, it suffices to show that $\|g_u(\tau)R_a\ehts P_c v\|_{L^2}\in L^1_\tau[s,s+1]$. Since $g_u(\tau)$ has bounded derivatives we have $\|g_u(\tau)-g_u(s)\|_{L^3}\leq C|\tau-s|$, then by H\"{o}lder inequality in space, $$\|(g_u(\tau)-g_u(s))R_a\ehts P_c v\|_{L^2}\leq\|g_u(\tau)-g_u(s)\|_{L^3}\|\ehts P_c v\|_{L^6} \in L^1_\tau$$ Now it suffices to show $\|g_u(s)R_a\ehts P_c v\|_{L^2}\in L^1_\tau$. For any $\tilde{v}\in L^2$ we have $$\|g_u(s)R_a\ehts P_c v\|_{L^2}=\<\tilde{v},g_u(s)R_a\ehts P_c v\>=\<e^{iH(\tau-s)}P_c g_u(s)R_a\tilde{v},v\>\leq\underbrace{\|e^{iH(\tau-s)}P_c R_a g_u(s)\tilde{v}\|_{L^6}}_{\in L^2_\tau}\|v\|_{L^{6/5}}$$ Since $L^2_\tau[s,s+1]\hookrightarrow L^1_\tau[s,s+1]$, $\|e^{iH(\tau-s)}P_c R_a g_u(s)\tilde{v}\|_{L^6}\in L^1_\tau$.
\begin{align}
\|I_2\|_{L^2}&\leq C\Big(\int_{s+1}^t \|g_u R_a\ehts P_c v\|_{L^{\rho'}}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}}+C\Big(\int_{s+1}^t \|g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^{\rho'}}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}} \nonumber \\
&\leq C\Big(\int_{s+1}^t \|\xs g_u\|_{L^{\frac{3\gamma}{2}}}^{\gamma'}\|\ehts P_c v\|_{L_{-\sigma}^2}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}}+C\Big(\int_{s+1}^t \|\xs g_{\bar{u}}\|_{L^{\frac{3\gamma}{2}}}^{\gamma'}\|e^{iH(\tau-s)} P_c \bar{v}\|_{L_{-\sigma}^2}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}} \nonumber \\
&\leq C\Big(\int_{s+1}^t \frac{d\tau}{|\tau-s|^{3(\frac{1}{2}-\frac{1}{p})\gamma'}}\Big)^{\frac{1}{\gamma'}}<\infty \nonumber
\end{align}
At the first inequality we used Strichartz estimate with $(\gamma,\rho)$ with $\gamma>2$ and the last inequality holds since $3(\frac{1}{2}-\frac{1}{p})\gamma'>1$ for $p=6$ and $\gamma>2$.
Similarly we will estimate $L(s)W$.
\begin{align}
\|L(s)W(t)\|_{L^2}&\leq C\Big(\int_s^t \|g_u R_a W+g_{\bar{u}}\overline{R_aW}\|_{L^{\rho'}}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}} \nonumber \\
&\leq C\Big(\int_s^t \|\xs (g_u+g_{\bar{u}})\|_{L^{\frac{3\gamma}{2}}}^{\gamma'}\|W\|_{\Lsn}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}}\nonumber \\
&\leq C\Big(\int_s^t \frac{1}{(1+|\tau-s|)^{3(\frac{1}{2}-\frac{1}{p})\gamma'}}\Big)^{\frac{1}{\gamma'}}<\infty \nonumber \end{align}
Hence $T(t,s):L^{p'}\rightarrow L^2$ is bounded for $p=6$.
This finishes the proof of part $(iii)$ and the theorem. $\Box$
\section{Appendix}\label{se:ap}
\subsection{J-S-S type estimates}
In \cite{kn:JSS} the authors obtain the following
estimate\footnote{Their theorem is stated differently but the proof
can be easily adapted to obtain the advertised estimate}:
\begin{theorem}\label{th:jss} If $W:\mathbb{R}^n\mapsto\mathbb{C}$ has
Fourier transform $\widehat{W}\in L^1(\mathbb{R}^n)$ then for any
$t\in\mathbb{R}$ and any $1\le p\le\infty$ we have:
$$\|e^{-i\Delta t}We^{i\Delta t}\|_{L^p\mapsto L^p}\le \|\widehat{W}\|_{L^1}$$
\end{theorem}
In what follows we are going to generalize the estimate to the
semigroup of operators generated by $-\Delta+V:$
\begin{theorem}\label{th:gjss} Assume $V:\mathbb{R}^n\mapsto\mathbb{R}$
and $W:\mathbb{R}^n\mapsto\mathbb{C}$ have Fourier transforms in
$L^1(\mathbb{R}^n).$ Then for any $T>0$ there exist a constant $C_T$
independent of $W$ such that for any $-T\le t\le T$ and any $1\le
p\le\infty$ we have:
$$\|e^{-i(-\Delta +V) t}We^{i(-\Delta +V)t}\|_{L^p\mapsto L^p}\le C_T\|\widehat{W}\|_{L^1}.$$
One can choose $C_T=\exp(2\|\widehat{V}\|_{L^1}T).$
\end{theorem}
The proof relies on existence of finite time wave operators:
\begin{lemma}\label{lem:waveop}
If $V:\mathbb{R}^n\mapsto\mathbb{R}$ has Fourier transform in
$L^1(\mathbb{R}^n)$ then for any $T>0$ there exist a constant $C_T$
such that for any $-T\le t\le T$ and any $1\le p\le\infty$ we have:
$$\|e^{-i(-\Delta +V) t}e^{-i\Delta t}\|_{L^p\mapsto L^p}\le C_T\|\widehat{W}\|_{L^1},\quad
\|e^{i\Delta t}e^{i(-\Delta +V) t}\|_{L^p\mapsto L^p}\le
C_T\|\widehat{W}\|_{L^1}.$$ One can choose
$C_T=\exp(\|\widehat{V}\|_{L^1}T).$
\end{lemma}
{\bf Proof of Lemma:} Let
$$H=-\Delta+V$$ then $H$ is a self adjoint operator on $L^2$ with
domain $H^2,$ (note that $V\in L^\infty,$) hence it generates a
group of isometric operators:
$$e^{-iHt}:L^2\mapsto L^2,\qquad t\in\mathbb{R}.$$
Consequently:
\begin{equation}\label{def:qt}Q(t)=e^{-iHt}e^{-i\Delta
t}:L^2\mapsto L^2,\qquad t\in\mathbb{R},
\end{equation} is also a
family of isometric operators. Their infinitesimal generators are:
$$\frac{dQ}{dt}=-ie^{-iHt}Ve^{-i\Delta
t}=-i\underbrace{e^{-iHt}e^{-i\Delta t}}_{Q(t)}\
\underbrace{e^{i\Delta t}Ve^{-i\Delta t}}_{Q_0(t)}$$ Hence
\begin{equation}\label{eq:qt}
Q(t)=\mathbb{I}_d-i\int_0^tQ(s)Q_0(s)ds
\end{equation}
where $$Q_0(t)=e^{i\Delta t}Ve^{-i\Delta t}:L^p\mapsto L^p,\qquad
1\le p\le \infty$$ is bounded uniformly by $\|\widehat V\|_{L^1},$
see Theorem \ref{th:jss}.
The contraction principle shows that for any $T>0$ and any $1\le
p\le\infty$ the linear equation \eqref{eq:qt} has a unique solution
in the Banach space $C([-T,T],B(L^p,L^p)).$ Since on $L^2\bigcap
L^p$ the solution is given by \eqref{def:qt} and $L^2\bigcap L^p$ is
dense in $L^p$ we obtain that for any $-T\le t\le T$ and any $1\le
p\le\infty,$ $e^{-iHt}e^{-i\Delta t}$ has a unique extension to a
bounded operator on $L^p.$ Applying the $L^p$ norm in \eqref{eq:qt}
we get:
$$\|Q(t)\|_{L^p}\le 1+\int_0^t\|Q(s)\|_{L^p} \|\widehat V\|_{L^1}ds$$
and by Gronwall inequality:
$$\|Q(t)\|_{L^p}\le e^{\|\widehat V\|_{L^1} |t|}\le e^{\|\widehat
V\|_{L^1} T},\quad {\rm for} -T\le t\le T.$$
A similar argument can be made for $Q^*(t)=e^{i\Delta t}e^{iHt}.$
The Lemma is now completely proven.
\bigskip
\noindent {\bf Proof of Theorem \ref{th:gjss}:} For $H,$ $Q$ and
$Q^*$ as in the proof of the previous Lemma we have:
$$
e^{-iHt}WE^{iHt}=\underbrace{e^{-iHt}e^{-i\Delta
t}}_{Q(t)}\underbrace{e^{i\Delta t}We^{-i\Delta t}}_{L^p\mapsto L^p\
{\rm bounded}}\underbrace{e^{i\Delta t}e^{iHt}}_{Q^*(t)}.$$ Hence
using Theorem \ref{th:jss} and Lemma \ref{lem:waveop} we get for any
$1\le p\le\infty:$
$$\|e^{-iHt}WE^{iHt}\|_{L^p\mapsto L^p}\le e^{2\|\widehat
V\|_{L^1} T}\|\widehat W\|_{L^1},\qquad {\rm for}\ -T\le t\le T. $$
The theorem is now completely proven.
\begin{remark}\label{jss} To obtain the linear estimates in Section
\ref{se:lin} we used Theorem \ref{th:gjss} in the form:
$$\|e^{iHt}WR_ae^{-iHt}\|_{L^p\mapsto L^p}\le C\|\widehat
W\|_{L^1},\qquad {\rm for}\ 0\le t\le 1$$ where $W$ is the effective
potential induced by the nonlinearity, see next subsection, while
$R_a$ is the linear operator defined in Lemma \ref{le:pcinv}.
\end{remark}
To see why the above estimate holds consider $f\in L^p\bigcap
L^2\bigcap {\cal H}_0.$ Then by \eqref{radef} we have for a certain
$z=z(f)\in\mathbb{C}:$
$$e^{iHt}WR_ae^{-iHt}f=e^{iHt}We^{-iHt}f+ze^{iHt}W\psi_0.$$
Theorem \ref{th:gjss} applies directly to the first term on the
right hand side, while for the second term we use, see
\eqref{z:est}:
$$|z|\le 2\|f\|_{L^p}\sqrt{\left\|\frac{\partial \psi_E}{\partial
a_2}\right\|_{L^{p'}}+\left\| \frac{\partial \psi_E}{\partial
a_1}\right\|_{L^{p'}}},\qquad \frac{1}{p}+\frac{1}{p'}=1$$ and the
fact that $\psi_0$ is an e-vector of $H$ with e-value $E_0<0$ hence
$$\|e^{iHt}W\psi_0\|_{L^p}=\|e^{iHt}We^{-iHt}e^{iE_0t}\psi_0\|_{L^p}\le
C\|\widehat W\|_{L^1}\|\psi_0\|_{L^p},$$ where again we used Theorem \ref{th:gjss}.
\subsection{Smoothness of the effective potential}
In this section we will prove Proposition \ref{pr:dg} i.e. $\widehat{g'(\pe)}$ and $\widehat{(\frac{g(\pe)}{\pe})}$
\bigskip
\noindent From by Corollary \ref{co:decay}, we have $\pe\in H^2$ which implies $\pe\in L^p$ for $2\leq p\leq\infty$. Also from \eqref{gest}, by integrating, we get $|g'(s)|\leq C(|s|^{1+\p_1}+|s|^{1+\p_2})$. Hence $|g'(\pe)|\leq C(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})\in L^2$ and $|g''(\pe)|\leq C(|\pe|^{\p_1}+|\pe|^{\p_2})\in L^\infty$. Now we have
\begin{align}
\|\widehat{g'(\pe)}\|_{L^1}&=\|\frac{1}{1+|\xi|^2}(1+|\xi|^2)\widehat{g'(\pe)}\|_{L^1}\nonumber \\
&\leq\|\frac{1}{1+|\xi|^2}\|_{L^2}\|(1+|\xi|^2)\widehat{g'(\pe)}\|_{L^2}\nonumber \\
&\leq C(\|\widehat{g'(\pe)}\|_{L^2}+\|\widehat{\Delta g'(\pe)}\|_{L^2}) \nonumber \\
&\leq C(\|\underbrace{g'(\pe)}_{\in L^2}\|_{L^2}+\|\Delta
g'(\pe)\|_{L^2}) \nonumber\end{align} So it suffices to show that
$\Delta g'(\pe)\in L^2$. Similarly it is enough to show that $\Delta
(\frac{g(\pe)}{\pe})\in L^2$.
\begin{equation}
\Delta g'(\pe)=g'''(\pe)|\nabla \pe|^2+\underbrace{g''(\pe)}_{\in
L^\infty}\underbrace{\Delta\pe}_{\in L^2}\label{eq:dgpe}
\end{equation} and
\begin{equation}
\Delta
(\frac{g(\pe)}{\pe})=(\frac{g''(\pe)}{\pe}-2\frac{g'(\pe)}{\pe^2}+2\frac{g(\pe)}{\pe^3})|\nabla
\pe|^2+(\underbrace{\frac{g'(\pe)}{\pe}-\frac{g(\pe)}{\pe^2}}_{\in
L^\infty})\underbrace{\Delta\pe}_{\in L^2}\label{eq:gpe}
\end{equation} We will use the following comparison theorem proved in
\cite[Theorem 2.1]{deiftsimon} to get the upper bound for the
$\nabla\pe$ and lower bound for $\pe$:
\begin{theorem}\label{th:comp} Let $\varphi\geq0$ be continuous on $\overline{\R^3\setminus K}$ and $A\geq B\geq0$ for some closed set $K$. Suppose that on $\overline{\R^3\setminus K}$, in the distributional sense, $$\Delta |\psi|\geq A|\psi|;\quad\quad \Delta \varphi\leq B\varphi$$ and that $|\psi|\leq\varphi$ on $\partial K$ and $\psi$, $\varphi\rightarrow0$ as $x\rightarrow\infty$. Then $|\psi|\leq\varphi$ on all of $\R^3\setminus K$.
\end{theorem}
\noindent Note that $\frac{\partial \pe}{\partial x_1}$ and $\pe$ are continuous and $\frac{\partial \pe}{\partial x_1},\ \pe\rightarrow0$ as $|x|\rightarrow\infty$. Hence $\left|\frac{g(\pe)}{\pe}\right|\leq C(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})\rightarrow0$ as $x\rightarrow\infty$.
\bigskip
\noindent First we need the standard upper bound for $\pe\geq0$. For any $A<-E$, there exists $C_A$ depending on $A$ such that $\pe\leq C_A e^{-\sqrt{A}|x|}$. Indeed if $R$ is sufficiently large, on $\overline{\R^3\setminus B(0,R)}$ we have $$\Delta\pe=[-E+V(x)+\frac{g(\pe)}{\pe}]\pe\geq A\pe,\quad\text{and}\quad \Delta\varphi=A\varphi-\frac{2\sqrt{A}}{|x|}\varphi\leq A\varphi$$ and on $\partial B(0,R)$ we have $\pe\leq C_A e^{-\sqrt{A}|x|}$ for $C_A$ big enough. Then by Theorem \ref{th:comp} we have $\pe\leq C_A e^{-\sqrt{A}|x|}$ on $\R^3\setminus B(0,R)$.
\noindent To get the lower bound for $\pe$ we will choose $\varphi=\pe$ and $\psi=C e^{-\sqrt{A_2}|x|}$ in Theorem \ref{th:comp}. On $\overline{\R^3\setminus B(0,R)}$, fix $\varepsilon>0$, $A_2\geq-E+2\varepsilon$ and choose $R$ large enough such that $\frac{2\sqrt{A_2}}{|x|}\leq\varepsilon$ for $|x|\geq R$. Then from \eqref{eq:ev} we have $$\Delta \pe=[-E+V(x)]\pe+g(\pe)
\leq[-E+V+\frac{g(\pe)}{\pe}]\pe\leq (-E+\varepsilon)\pe$$ and for $A_2\geq -E+2\varepsilon$ we have $$\Delta
\psi=A_2\psi-\frac{2\sqrt{A_2}}{|x|}\psi\geq(-E+\varepsilon)\psi$$
Choose $C$ such that $C
e^{-\sqrt{A_2}|x|}\leq\pe$ on $\partial B(0,R)$. Then by theorem
\ref{th:comp}, we have $C e^{-\sqrt{A_2}|x|}\leq\pe$ for $|x|>R$.
\noindent We will show that for $\psi=\frac{\partial \pe}{\partial x_1}$ and
$\varphi=C e^{-\sqrt{A_1}|x|}$ where $A_1<-E$ hypothesis of the
theorem \ref{th:comp} is satisfied.
Differentiating the eigenvalue equation \eqref{eq:ev} with respect to $x_1$ we
get $$\Delta \frac{\partial \pe}{\partial
x_1}=[-E+V(x)]\frac{\partial \pe}{\partial
x_1}+g'(\pe)\frac{\partial \pe}{\partial x_1}+\frac{\partial
V}{\partial x_1}\pe$$ Let $$f^{\pm}=\max\{ 0,\pm f\}\quad\text{and}\quad
S_{\leq}=\{x\in\R^3|\Big|\frac{\partial \pe}{\partial x_1}\Big|\leq\pe\}\quad\text{and}\quad
S_{\geq}=\{x\in\R^3|\Big|\frac{\partial \pe}{\partial x_1}\Big|\geq\pe\}$$ Fix $A_1<-E$, choose $R$ large enough such that $-E+V(x)+g'(\pe)-\Big|\frac{\partial V}{\partial x_1}\Big|\geq A_1$ on $|x|\geq R$. Let $S=S_{\leq}\cup B(0,R)$, then on $\overline{\R\setminus S}$ we have
$$\Delta \Big|\frac{\partial \pe}{\partial x_1}\Big|\geq A_1\Big|\frac{\partial \pe}{\partial x_1}\Big|$$ Now, by continuity of $\frac{\partial \pe}{\partial x_1}$ there exists $C_1$ such that $\Big|\frac{\partial \pe}{\partial x_1}\Big|e^{\sqrt{A_1}|x|}\leq C_1$ on $|x|=R$. Since both on $\frac{\partial \pe}{\partial x_1}$ and $\pe$ are continuous we have $\Big|\frac{\partial \pe}{\partial x_1}\Big|=\pe\leq C_2 e^{\sqrt{A_1}|x|}$ on $\partial S_\leq$. So on $\partial S$, we have $\Big|\frac{\partial \pe}{\partial x_1}\Big|\leq\max\{C_1,C_2\} e^{\sqrt{A_1}|x|}$
\noindent Therefore by
theorem \ref{th:comp}, we have $|\nabla \pe|\leq C
e^{-\sqrt{A_1}|x|}$
Now we can prove Proposition \ref{pr:dg}
\textbf{Proof of Proposition \ref{pr:dg}} By {\bf (H2')} we have $|g'''(s)|<\frac{C}{s^{1-\alpha_1}}+Cs^{\alpha_2 -1},$ $s>0,$ $0<\alpha_1\le\alpha_2;$ then $$|g'''(\pe)|\nabla
\pe|^2|\leq\frac{C}{\pe^{1-\p_1}}|\nabla \pe|^2$$ and
$$|(\frac{g''(\pe)}{\pe}-2\frac{g'(\pe)}{\pe^2}+2\frac{g(\pe)}{\pe^3})|\nabla
\pe|^2|\leq\frac{C}{\pe^{1-\p_1}}|\nabla \pe|^2$$
Using the estimates for $|\nabla \pe|$ and $\pe$ and choosing
$2\sqrt{A_1}>\sqrt{A_2}$, we get that $\Delta g'(\pe),\Delta
(\frac{g(\pe)}{\pe})\in L^2$. Hence we get the desired estimates for
$\widehat{g'(\pe)}$ and $\widehat{\frac{g(\pe)}{\pe}}$.
\bibliographystyle{plain}
\bibliography{ref}
\end{document}
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Summary ICRWorlds Enteric Empty Capsules Enteric Empty Capsules Industry 1.1 Industry Definition and Types 1.1.1 Gelatin Type 1.1.2 HPMC Type 1.2 Main Market Activities 1.3 Similar Industries 1.4 Industry at a Glance Chapter 2 World Market Competition Landscape 2.1 Enteric Empty Capsules Markets by Regions 2.1.1 USA Market Revenue (M USD) and Growth Rate 2012-2022 Sales and Growth Rate 2012-2022 Ma
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Submitted by phyllidalewis • February 25, 2014 allmakedigital.com.au
Xerox colorqube 8900 from All Make Digital improves your business appeal and reduces costs and most importantly it is environmentally very friendly. A special feature of colour multifunction with Xerox Colour Qube completely engineered to deliver outstanding performance at your needs.
- Category: World News
- Tags: colorqube 8900, xerox colorqube 8900, xerox colour qube
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Donna and crew return — and catch a blue-sky day!
Joe and Grace visit for a third time (and are planning a wedding here, too!) Yay!
Devoted fisherman Keichi caught mostly rain on his stay.
Jason, Rachel, Bob, and Teresa were supremely lucky with the weather. Smiles and sun go together, eh?
Nader and Shyva shared news of Texas, smiles, and a basket of oranges.
Curtis and Dottie brought their laid back southern style to the bnb — ah, mellowness.
Brothers and fishermen Carlo and Michael pose during a rare, rain-free moment.
Almost no room for guests LaDonna and Joe! Two of my faves came for some R&R. So glad they did.
Bruce and Patitta mix it up at the bnb (well-chilled Cosmopolitans in case you were wondering).
Patitta cooking up a mean pad thai — on her honeymoon!
Looking for a tasty dungeness crab in Crescent City? Head to the Crab Shack at the harbor.
Co-owner and long-time crab fisherman Windy knows how to cook ’em, chill ’em, and serve ’em.
River guide Kevin Brock backs to shore with a just-caught steelie in the client’s net. Nice work!
Hero shot…a wild buck joins the pose. Reeled in like a pro by the young lady on the right..
Nice addition to the pasta sauce — wild-picked yellow foot.
Two of the biggest hedgehogs ever. Funny, I can’t quite remember the location…
Looks like, smells like, feels like a Matsutake. A rare find if it is.
Beautiful, clean, fresh hedgehog mushrooms — note the unmistakable spikes. That’s the giveaway.
Frittata anyone? Wild mushrooms, garlic, and purple onion. Buttery. Crisp from the broiler. Gotta say I nailed it this time.
January 2016 will go down as one for the record books.
It was a month of record precipitation with at least half the days sporting wind, thunder and roof-pounding rain. Often, driving was turned into a death sport (grim reminder is the swamped car that remains submerged in the middle fork gorge with its driver and passenger unaccounted for and presumed dead). Often, during the heaviest nighttime rains, sleep was impossible.
Another one for the record book is fishing. It’s been off the charts with guides crowding the nearby Hiouchi Hamlet in the early hours as they meet up with clients. Like old times. True, the catch has been predominately wild steelhead (as opposed to hatchery raised) and therefore not ones that can be kept, but it’s been amazing to watch sportfishermen pull in steelhead after steelhead.
And mushroom foraging — basically a non-event last winter with hardly a single fungi found — has been solid. Need mushrooms for a pizza? Take a walk in the redwood forest. Thinking of adding mushrooms to a salad, frittata, or red sauce? Grab Molly and head for the hills. Forty minutes later and you’ve got yourself a pound of hedgehogs and yellow foot to saute up. Very nice.
The bnb had a record January, too, as did the local motel (based on informal sightings of a parking lot that has been uncharacteristically full).
Setting our sights for the future, we are aiming for a -0.9 low tide in about a week when razor clam will just barely be present, so with luck we’ll extend this batting streak of riches to that endeavor as well.
The one blemish so far this year is the absence of crabbing and therefore crabs. Warm waters last summer resulted in an algae bloom that has adversely affected crabs. For now, crabs remain unfishable. (If you have a yen for eating crabs, however, I suggest you head for the Crab Shack in the Crescent City harbor. A first time visit for Meg and I netted the most delicious two-pound dungeness; it blew us away. Claws packed with meat and a flavor to die for. Of course the crabs are sourced from Oregon waters so they are safer to eat than ours. Bravo to Crab Shack partners Windy and Debbie who ply their trade without a lot of fanfare. It’s a destination worth noting and noting still that the crabs are served quite naked — no lemon or sauce; only napkins — but really, we didn’t miss any of that the taste was so outstanding.)
On to a recap of January’s guests…it was a stellar bunch indeed. Returnees Carlo and Donna made haste and completed their second stay at the bnb each. Carlo returned with his brother Michael for three days of bank fishing; Donna returned with husband Joe and son David and David’s gf Marley. Both are angling to return for third stays as well. Joe and Grace returned for their third stay with us. It’s getting to be like family around here!
To recap the recap…January was the month us river folk were supposed to have been recharging the batteries and generally catching up on sleep. Netflix, slow cooker brisket, mornings with sunshine not the sound of an alarm to wake us.
But no. This was really one for the records.
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Numbers 18-19; Mark 7:1-23
Key Verses
Numbers 19.”
These two passages seem to be in stark contrast to one another. Numbers restates and adds to the cleanliness laws – while in Mark, Jesus chides the Pharisees for their hypocritical observance of the cleanliness laws. Who was right? Well, both of them were…
Remember… God was establishing a people with a law that was a physical representation of His future spiritual kingdom. Cleanliness was a HUGE deal to God. The people were taught through the word pictures presented in the law that uncleanliness was connected with death – whereas cleanliness was associated with life. This is why a person was deemed “unclean” when touching any animal or person that was dead.
Jesus teaches that it is not the physical things that make a person unclean – but rather it is the sinful actions of the heart that defile a person. Our hearts are unclean… which means they are associated with death – spiritual death.
How do we reverse the spiritual death in our hearts? Our hearts must be made clean! …But how? We know that the blood of animals is insufficient to cleanse the heart! But listen to the writer of Hebrews…)
Jesus’ once-and-for-all-sacrifice makes life possible – not just the physical representation of life – No! His sacrifice opens the way for us to have true, spiritual life. The real life – the forever kind of life! I don’t know about you, but I want some of that life!! I find it in Jesus…
“I am the way, and the truth, and the life. No one comes to the Father except through me” (John 14:6).
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Published in Hepatitis Weekly, February 1st, 2010
"Program directors (N = 223) reported that, on average, 35% of their patients were tested for HIV (median = 10%) and 57% were tested for HCV (median = 80%). Of the quality improvement initiatives examined, computerized reminders to clinicians (p = .02) and a designated....
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TITLE: Combinatorics and Probability- where am I wrong?
QUESTION [4 upvotes]: Let there be a cube with $n$ sides denoted $1,...,n$ each. The cube is tossed $n+1$ times. For $1\le k\le n$, what is the probability that exactly $k$ first tosses give different number (i.e, the $(k+1)$-st toss give a number that was already gotten.) I really need to know why I got a slightly different answer from the official one.
My attempt: Let us build a uniform sample space. $\Omega=\{a_i=(i_1,...,i_k)|1\le i_j\le n\}$. $|\Omega|=(n+1)^n$, $\forall \omega\in \Omega, P(\omega)={1\over |\Omega|}$.
We seek for the event $A=\{(i_1,...,i_k,i_{k+1},...,i_{n+1})|i_t\ne i_s, \forall 1\le t\ne s\le k, k\in \{i_1,...,i_k\}\}$.
This is the problematic part: $|A|={n\choose k}\cdot k!\cdot k \cdot n^{n-k-1} $. (Then I and the answer use the formula for probability of an even it a uniform sample space.)
The point is, the answer says: $|A|={n\choose k}\cdot k!\cdot k \cdot n^{n-k} $. I don't understand why; First I pick $k$ numbers, count all their permutations, then pick one of them for the $(k+1)$-th toss, and then I have $n-k-1$ tosses left, each of which has $n$ possibilities. I would appreciate your help.
REPLY [1 votes]: Community wiki answer so the question can be marked as answered:
As pointed out by grand_chat in a comment, the discrepancy is resolved by noting that the cube is cossed $n+1$ times, not $n$ times.
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Contents
Obtaining Limited Dragons
Purchasing Galaxy Habitats
Unable to Buy Habitats
I’m unable to buy any new DragonVale Habitats. They are greyed out in the DragonVale Market.
It sounds like you have reached the maximum number of allowed habitats for the current level of your park. If you look at any habitat in the Market, you’ll see a number in the lower left hand corner that shows your habitat quota. For example, if you see “13/14”, it means that you have built 13 of the 14 allowable habitats at your current level. “14/14” means that you have maxed out all allowable habitats for your current level. Try leveling up your park to unlock additional gift each day?
By default, players can give 3 Gems from each park per day. If you have the Dragonsai Tree or Dragonsai Bush in your park you can send Gems to additional friends. The Dragonsai Tree allows you to send Gems to 3 additional friends per day, and the Dragonsai Bush allows you to send a Gem to 1 additional friend per day. Please note that you cannot send more than 1 Gem to the same friend from the same park.
If you have multiple parks, you can send a maximum of 21 Gems per day from your device. Once you have sent 21 Gems you will not be able to send any more Gems from that device until the following day.
Can I give more Gems per day if I have the Dragonsai Tree or Dragonsai Bush?
Purchasing the Dragonsai Tree or Dragonsai Bush will not allow you to bypass the Gem gifting cap. Once you have sent 21 Gems you will not be able to send any more Gems from that device until the following day. This includes any Gems you may have from the Dragonsai Tree or Dragonsai Bush.
What time do my Gems reset each day?
The Gems you can give to your friends now reset daily at 11am MST. Once you've given out all your Gems for the day you will not be able to give any more until 11am Mountain Time on the following day.
Can I give more than one Gem to a single friend?
If you only have one park you can only give 1 Gem to each friend per day. If you have multiple parks, you can gift a single friend one Gem from each of those parks, up to the daily device limit.!
Missing Twins
Why didn't I receive a twin dragon that my friend bred with my park?
Please note that you can only receive up to 10 twins while you are away from the game. If your friends breed 10 twins with you while you aren't playing, you will need to log in and acknowledge the existing twins before you will start to receive new twins.
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Happy 2011
Andy Cohen recounts his holiday and New Year's shenanigans.
Happy New Year! I am about to take off for Los Angeles, and the Beverly Hills Housewives Reunion.... It's gonna be a good one.
There's so much to discuss and so little time. The holidays were fantastic and fast and cozy and snowy. I went to Sag Harbor on Christmas night and hunkered down for the massive Nor'easter. I essentially lit fires, cooked, watched movies, and played with my dj program on my Mac. I loved The Kings Speech SO MUCH. I think my fave of the year. I liked Black Swan and The Kids are Alright and True Grit. nd I thought Somewhere was a painfully boring tone poem.
Thursday night I came back to the city and dj'd at Bedlam in the East Village. I had a BLAST and I want to thank everybody who came out to dance and hear music and play. What a great time; I can't wait to do it again. I always love celebrating NYE the night BEFORE New Years because it takes the pressure off the actual night.
The pressure, of course, was ON on New Year's Eve because I was hosting a live two-=hour house party on Bravo featuring a fantastic crew of our Bravo family plus Sandra Bernhard and Megan McCain. I think the best way to describe the show was a mix of fellini and The Robin Byrd Show. If you live in NYC, you will understand the reference to the latter show, a public access full frontal affair.
Some of my highlights were NeNe and Lisa getting into it, Jeff Lewis having fun with Ellen Barkin's drunk-dialed "Where's Lexi?" query, kinda making out with Tamra, Jimmy Fallon (as Neil Young) hilariously lampooning "Tardy for the Party" before Kim sang the real thing at midnight, the wedding of Giggy and Grandma Wrinkles, Jenni's rap, all the Mazel winners, and Sandra Bernhard. It was live, unscripted, weird, and fun. After the show, the party went on in another room and we all kicked it hard.
I spent the rest of 2011 watching OWN. Did you? I actually only watched (and loved) the Behind the Scenes of Oprah's last season. I watched a little of the Master Class show but Diane Sawyer lost me when she called herself a geek as a kid. Its kinda ridiculous. She was a beauty queen -- what's geeky about that???
Anyway, here are some clips from our geeky New Year's Show:
Want to reach Andy? E-mail him
Who's Andy? Read his bio
On Facebook? Join Andy
On Twitter? Follow Andy
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TITLE: Finding $\lim\limits_{n→∞}n\cos x\cos(\cos x)\cdots\underbrace{\cos(\cos(\cdots(\cos x)))}_{n\text{ times of }\cos}$
QUESTION [4 upvotes]: Find$$\lim_{n→∞}n\cos x\cos(\cos x)\cdots\underbrace{\cos(\cos(\cdots(\cos x)))}_{n \text{ times of } \cos}.$$
I approximated cos(cosx) to cos x, but i don't think it is the proper approach.
I got answer as 0 on the approximation.
It is clear that it is a 0/0 form, but how can the l's Hopital rule be applied? I tried using the sandwich theorem but I am unable to reach the answer.
I plotted the graph on desmos. But I got the resultant graph covering the entire area.
please help me reach the proper answer. Thanks in advanced to all.
REPLY [1 votes]: We can say few more things. With the same notations as before: consider the sequence $x_{n+1} = \cos(x_n)$ with $0 < x_0 \leq 1$, and $x_*= \cos(x_*)$.
Heuristically, $x_n \to x^*$ very quickly. Thus, $x_0 x_1 ... x_n$ behaves like a geometric sequence :
$$x_0 x_1 ... x_n \propto x_*^n \simeq 0.73^n \to 0$$
(consequently $n^k x_0 x_1 ... x_n \to 0$ for all $k > 0$)
To be more precise, consider:
$$y_n := \frac1{x_*^{n+1}}x_0 x_1 ... x_n = \frac{x_0}{x_*} \frac{x_1}{x_*} ... \frac{x_n}{x_*}> 0$$
It's better to consider $\log y_n$ :
$$\log y_n = \sum_{i = 0}^n \log(\frac{x_i}{x_*})$$
This série is really cool. First, the sign of $(-1)^i \log(\frac{x_i}{x_*})$ is constant. Indeed, if $x_n < x_*$ then $x_{n+1} > x_*$ :
$$x_{n+1} - x_* = \cos(x_n) - \cos(x_*)$$
In fact, this série is geometric :
$$|\log(\frac{x_i}{x_*})| = |\log(x_i) - \log(x_*)| = |\log(\cos(x_{i-1})) - \log(\cos(x_*))| \leq k |x_{i-1} - x_0|$$
with $k = \displaystyle\sup_{[0, 1]} |\tan| < \infty$. And, with a Taylor expansion :
$$|\log(\frac{x_i}{x_*})| \leq k (\cos(1))^{i-1} |x_1 - x_0|$$
Thus, $y_n$ converges.
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Real People Doing Great Things – Eileen H
Real People. Real Progress. Real Fat Loss.
Since opening Catalyst 4 Fitness, I’ve had the opportunity to meet some incredible people. Each has her/his own story, obstacles, and successes. One of these women is Eileen Hopkins.
Tell us about yourself. I am 52 years old and have been married for 19 years to my husband Matt. I have 2 stepsons Jason (33) and Michael (31). I am Gigi to Michael’s 2 1/2 year old daughter Zoe and his stepdaughter Alyssa (10). My husband and I share our home with four dogs: Lilly, Holly, Thor and Penny. I have been employed at the YWCA of Greater Cincinnati for 10 years. I enjoying running, working in the yard and vacationing in the Great Smoky Mountains and Indian Rocks Beach, Florida.
How long have we worked together? I started taking Boot Camp in March of 2014. I did that for one session then switched to Metabolic Explosion Training 2x/week.
What did you expect when you started on each? I started Boot Camp because I was feeling stuck with my normal exercise program (maybe because of hitting the big 50!). Running is my main thing but I also had been doing weights and used a personal trainer from time to time but was not happy with the results. I was looking for a workout that would challenge me and help me to look and feel better. Also I had recently moved to the east side of town and was hoping to make some new friends. Score!!!
How do you feel now? I feel great!! I love MET. It is the only group fitness program that I have ever stuck with!! It provides me with a challenging and fun workout with great people who I now call friends!!
How has your body changed? I look and feel more fit. I look and feel strong.
What non-scale victories have you experienced? I just feel good I my clothes and I like my body now better than ever! I cannot even believe I am saying this as body image has been a longtime struggle for me.
Are you stronger? I am definitely stronger. When I first started Boot Camp I thought I was going to puke or die!! Now I can complete MET and still have energy to spare!!
How has your health improved? I have always had good cholesterol and blood pressure but since finding Sharon my numbers have gotten even better!!
What are your goals? I just want to feel and look strong, healthy and fit!
If you have tried fitness and nutrition plans in the past, what did you try? Why didn’t they work? How is this different? I have done lots of exercise classes and programs over the years. MET is by far the best fit for me!!!
What is the best part about working with Sharon? Sharon challenges you and makes class fun! I feel like she really cares about me and my fitness goals!! She’s just not a trainer but also a friend.
What is the worst part about working with Sharon? She pushes you even when you don’t feel like being pushed!! This is not bad but it does sometimes result in the use of some bad words!!!
What would you tell others who are hesitant to start? Don’t hesitate!!! Just do it!! It will change your life for the better!!
Have your say
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Hendy Zelishovsky English II Professor Spollen February 10 2013 Those Winter Sundays By Robert Haydon Sundays too my father got up early Even on weekends, when it was not a work day, his father woke up early in the morning And put his clothes on the blueback cold He would have to dress in the cold. Then with cracked hands that ached Cracked from cold and labor from labor in the weekday weather made banked fires blaze. No one ever thanked him. He’d light a fire for everyone else’s warmth. I’d wake and hear the cold splintering, breaking. Probably what made his father’s hands cracked. When the rooms were warm, he’d call, And slowly I would rise and dress, Fearing the chronic angers of that house, Speaking indifferently to him, The son did not acknowledge what his father did for him Who had driven out the cold By waking early and lighting a fire And polished my good shoes as well. What did I know, what did I know He is eating himself up about treating his father the way he did- that is why it is repeated twice.
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By Zoe
Here is a drawing I did of the character Eugene Meltsner from the audio show Adventures in Odyssey. I have tried to draw him in the past a couple of times, but it never really worked out until now, haha. I tried to go with some simple shading/lighting. I really like how the warm tones look. I’m pretty satisfied with how it came out. My art block that I talked about last month still continues, sadly, so this isn’t too recent of a drawing. But it’s recent enough. I don’t have much else to say about this drawing but I hope you enjoy it!
Commission me at @Buffaloh.daisy on instagram or zachzoe409@gmail.com
Zoe Olson, 14, has been homeschooled since age five.
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SetPoint Consultants is an international, full-service, professional recruiting firm. We provide contract and direct-hire employment solutions.
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Title Curative Associate and Loan Closing Coordinator- Irvine, CA Setpoint Consultants
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Espai Triadú
Espai Triadú is located inside the Terra Baixa library, in Ribes de Freser. Different personal objects and documents, and a 30-minutes audio-visual about the life of Joan Triadú round off the exhibition “Llegir com viure. Homenatge a Joan Triadú 1921-2010 [Read how to live. A tribute to Joan Triadú 1921-2010]. The visitors can also access to the book depot written by Joan Triadú, created to make available to everyone the necessary tools to understand and deepen in the life, work and thoughts of this eminent educator, cultural firebrand and literary critic born in Ribes de Freser. Opening hours are Monday to Friday from 5.00 p.m. to 8.00 p.m. Wednesday from 10.00 a.m.to 1.00 p.m. Prior booking on the weekends.
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Link to the original thread can be found here.
The Anthony Nolan website is here.
Thank you so much for all of your contributions both to the thread and to the JustGiving page we managed to raise a huge £1373.43 for Anthony Nolan, so many many thanks.
Northernmonkey’s name was picked our of the hat to win the fabulous MN blanket, congratulations!
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TITLE: Absolutely irreducible/simple $A$-module iff Endomorphism ring consists of scalar matrices
QUESTION [1 upvotes]: Let $A$ be a non-commutative $K$-algebra (where $K$ a field),
whose underlying $K$-vector space is finite dimensional.
Definition An $A$-module $M$ is said to be
absolutely irreducible or abs. simple if for every extension field $E$ of $K$,
the $M \otimes_K E$ is a irreducible $A \otimes_K E$-module.
Remark. The terminology simple and irreducible module means the same.
Theorem 2.3. (from these notes:
https://www.imsc.res.in/~amri/topics/modular.pdf )
An irreducible $A$-module $M$ is absolutely irreducible if
and only if every $A$-module endomorphism of $M$ is multiplication by a
scalar in the ground field, ie $\operatorname{End}_A(M) =K$.
I not completely understand the proof of the direction
$\operatorname{End}_A(M) =K$ implies
$M$ absolutely irreducible:
Proof. We know from Schur’s lemma that
$D := \operatorname{End}_A(M)$ is a division
ring. This division ring is clearly a finite dimensional vector space over
$K$ (in fact a subspace of $\operatorname{End}_K(M)=K^{d^2}$).
The image $B:=s(A)$ of $A$ in $\operatorname{End}_K(M)$ under canonical
map $s:A \to \operatorname{End}_K(M)$ induced by $A$-module structure is
a matrix algebra $M_n(D)$ over $D$. Why?
$M$ can be realised as a minimal left
ideal in $M_n(D)$. M is an absolutely irreducible $A$-module if and only
if it is an absolutely irreducible $B$-module.
If $\operatorname{End}_A(M) = K$, then $B = M_n(K)$, and $M \cong K^n$.
$B
\otimes_K E = M_n(E)$, and $M \otimes_K E = E^n$.
Thus $ M \otimes_K E$ is clearly an irreducible
$B \otimes_K E $ module. Therefore, $M$ is absolutely irreducible.
Several steps appear nebulous to me.
Why is the image $B:=s(A)$ of $A$ in $\operatorname{End}_K(M)$ is
a matrix algebra $M_n(D)$ over $D$? Wedderburn–Artin theorem implies that $B$ is a direct product of matrix algebras $\operatorname{Mat}_{n_i}(D_i)$, $D_i$ divisor rings. Why $B$ consists of only one such factor?
How $M$ should be realised as a minimal left ideal in $M_n(D)$. Is there used tacitly certain structure result?
REPLY [1 votes]: For your first question, to go from a product of $M_n(D_i)$ to a single one, observe that the representation of $B$ is both faithful and simple. If we had a product of matrices over division rings, then we would have nontrivial central idempotents which act non trivially by faithfulness, contradicting simplicity.
For your second question, I think we are tacitly using the structure theorem of modules over $M_n(D)$, that they are all direct sums of the standard module $D^n$. It doesn’t seem relevant that we can realise $M$ as a minimal left ideal, aside from potentially making things more concrete (thinking of $M$ as matrices supported on a single column).
Overall one can think of this proof as being entirely a fact about the representation theory of matrices over division algebras, and the first part is the reduction step to this case.
It’s also probably worth keeping in mind that as categories, modules over $D$ and modules over $M_n(D)$ are equivalent, given by tensoring with the standard binodule. So after this first reduction step, one can pretend $n=1$ to make things a bit more concrete.
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>>.
3 comments:
Hey good evening Hayami,.
Please keep producing these lovely pieces, and bravo to everyone at the Hobby Shop who helped you out. Hope you get some commissions out of this exhibit.
Take care man and thanks for the unexpected philosophy and high art at MIT...
CM
Hayami:
Just to say that I ran into your both-ways benches in the Stata Center for the first time recently. Really a beautiful and thought provoking setup. Congratulations!
(This comes from Utreht, Neth.) -- Ralph
I was fascinated by Hayami Arakawa's "Intellectuals Circle" variation
on the classical loveseat; it's a beautiful piece of furniture, that
will look good in any public space. Have him copyright and license
the design before some manufacturer rips him off.
However as the picture illustrates, adults will feel comfortable only
sitting on the inner row, as it has the wider back support; children
will gravitate to the outer row and pull their legs up onto the seat.
Barry S. Gloger, MD, FAAOS
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Items for Nonprofit Notebook need to be received by the 15th of the month prior to publication.
Garrison Keillor event to support Pickford Film Center
A “Prairie Home Companion” star and well-known author Garrison Keillor will join the cast of The Chuckanut Radio Hour for a special performance to benefit Pickford Film Center.
The show will take place at Western Washington University’s Performing Arts Center Mainstage. It will feature musical guests The Senate and Bellingham’s own Courtney Fortune. A brand new episode of The Bellingham Bean and an appearance by essayist Alan Rhodes will conclude the evening.
This event, hosted by Village Books, will raise funds to build the new Pickford Film Center.
Founded in 1998 in an effort to keep independent cinema alive in Bellingham, The Pickford Cinema is the region’s only independent year-round movie theater, playing independent, foreign, and documentary features.
Mother Baby Center reaches fundraising goal
The Mother Baby Center was thankful to receive a $10,000 anonymous donation.
With this significant donation, the center was able to reach its goal of raising $60,000.
When it receives several matching funds donations, which have been submitted to corporations by employees donating monies to Mother Baby Center, the center will have exceeded its goal. years.
BP Cherry Point Refinery has allotted $15,000 to the Real Heroes Celebration dinner in November. This awards banquet dinner is the primary annual fundraiser of the local Red Cross Chapter with a 2008 goal of raising $120,000.
All proceeds support local disaster relief services.
The American Red Cross is not a government agency and receives no federal funding. All programs offered locally are funded locally. Corporate partners such as BP Cherry Point Refinery are key to providing these Red Cross services to the community.
DeWaard & Bode donates beds to local fire victims
DeWaard and Bode, a locally owned business with two stores in Bellingham, recently donated mattresses and box springs with a retail value of $24,000 to local fire victims.
The 15 mattresses and six box springs were distributed through the Mount Baker Chapter of the American Red Cross, which serves Whatcom County and Skagit Valley.
Founded in 1946, DeWaard and Bode specializes in major appliances and mattresses.
Lighthouse Mission Agape walk raises $9,900
More than 100 men, women, and children raised $9,900 to help finance the new Agape Home, which will be built at the corner of Holly and F streets. Sponsors included Peoples Bank, Clinic Health Apparel, Fast Cap, Margery Jones CPA, MJB Jewelers and Runningshoes.com.
The construction of the new 100-bed women’s and children’s home is scheduled to being this month. The mission still needs to raise $500,000.
Open 24 hours a day since 1923, Lighthouse Mission has existed in Bellingham to help the homeless and hurting. The mission does everything in its power to assess and address the underlying causes that lead to homelessness.
Mount Baker Theatre renames Studio Theatre
after Waltons
The Mount Baker Theatre has formally changed the name of the Studio Theatre to the Harold and Irene Walton Theatre.
The Studio Theatre name change was in recognition of a $1 million gift from the Walton family that enabled the establishment of the Mount Baker Theatre endowment fund.
Local businesses support Blue Skies for Children
LouArestad.com and Mumm’s Heating will host the Annual Little Wishes Golf Tournament Sept. 7 at Shuksan Golf Club to benefit Blue Skies for Children.
Cost to participate is $85 per individual or $340 for a four-person team. Registration will include 18 holes of golf, cart for two, dinner at Shuksan, T-shirt, awards and prizes. Hole sponsorships are available for $100.00.
For more information call Lou Arestad at (360) 312-9590.
WECU donates more than $10,000 to local nonprofits
Whatcom Educational Credit Union (WECU) donated a total of $10,800, dividing the funds among 13 local non-profits including American Museum of Radio and Electricity, Animals as Natural Therapy, Western Washington University’s anthropology department, and Womencare Shelter.
WECU’s social responsibility committee oversees monetary donations and participation in volunteer projects. WECU’s Social Responsibility Committee supports education, health and community-related projects, actively seeking those that address acute community concerns such as violence, homelessness and drug abuse.
With more than 54,600 members, WECU is Whatcom County’s largest member-owned, not-for-profit financial cooperative..
State educational agencies, the governor and school districts have all designated the recruitment of minority teachers as a top priority for teacher education programs, in keeping with the greater need created by increasing numbers of minority students.
Woodring’s Promise Scholarship program supports students just prior to entering Woodring and may be renewed for their first year in the teacher preparation program. To date, $70,500 from various private funding sources, in collaboration with the Western Washington University Foundation, has been awarded to 28 students.
Larson Gross accounting scholarship awarded to
WWU student
Larson Gross donated $1,650 to the Western Washington University Foundation for the 2008-09 academic year. This donation was awarded to Western student Jessica Poling as a scholarship.
Scholarships are awarded to juniors and seniors with a declared accounting major or to a master’s student with a bachelor’s degree in accounting. Preference is given to students from Whatcom or Skagit County who have expressed interest in establishing a career in Whatcom or Skagit County.
Based in Bellingham, Larson Gross is a full-service accounting firm providing comprehensive accounting, tax and consulting services to individual and business clients throughout Northwest Washington and British Columbia.
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TITLE: Spaces of closed subgroups of a profinite group up to conjugacy
QUESTION [3 upvotes]: $\DeclareMathOperator{\Sub}{\operatorname{Sub}}$ Let $G$ be a profinite group and consider the space $\Sub(G)$ of closed subgroups of $G$ equipped with the profinite topology. That is, we have $G = \underleftarrow\lim(G_i)$ for finite groups $G_i$, and we construct $\Sub(G)$ as $\lim(\Sub(G_i))$. The space $\Sub(G)$ admits an action of $G$ by conjugation, therefore we may also consider the quotient space $\Sub(G)/G$.
In a pair of papers [1,2], Gartside—Smith consider various topological properties of the space $\Sub(G)$. I am interested in some of the corresponding statements for $\Sub(G)/G$.
Explicitly, assume that $G$ is second countable (which is equivalent to $\Sub(G)$ or $\Sub(G)/G$ being second countable). Is is true that if $\Sub(G)$ is uncountable then $\Sub(G)/G$ is also uncountable? Note that this is clearly true for abelian groups.
This question is aiming towards having a classification of those profinite $G$ such that $\Sub(G)/G$ is scattered, in analogy to the Gartside—Smith results for $\Sub(G)$.
[1] Counting the closed subgroups of profinite groups J. Group Theory (2010)
[2] Classifying the closed subgroups of profinite groups J. Group Theory (2010)
REPLY [4 votes]: Just to restate the question concisely:
Let $G$ be a second-countable profinite group. If $G$ has uncountably many subgroups, does it have uncountably many closed subgroups modulo conjugacy?
The answer is no. A counterexample is the $p$-adic group $\mathrm{SL}_2(\mathbf{Z}_p)$ for $p$ prime.
It clearly has uncountably many closed subgroups (mapping a line of $\mathbf{Q}_p^2$ to its stabilizer in $G$ is injective). So the main claim is
$\mathrm{SL}_2(\mathbf{Z}_p)$ has countably many conjugacy classes of closed subgroups.
Since $\mathrm{SL}_2(\mathbf{Z}_p)$ has countable index in $H=\mathrm{SL}_2(\mathbf{Q}_p)$, it is enough to check that the latter has only countably many compact subgroups up to conjugacy.
Note that every closed subgroup of $H$ is $p$-adic analytic, so is locally determined by its Lie algebra. The action of $\mathbf{SL}_2(\mathbf{Q}_p)$ on its Lie algebra has countably many orbit on the Grassmanian, so it is enough to check that for a given Lie subalgebra $\mathfrak{k}$, there are only countably many possible subgroups $K$.
Write $d=\dim(\mathfrak{k})=\dim(K)$.
If $d=3$ $K$ is a compact open subgroup and there are only countably many.
If $d=0$, $K$ is finite and it is known that there are only finitely many conjugacy classes of finite subgroups (e.g., as every finite group has only finitely many representations on $\mathbf{Q}_p^2$ modulo $\mathrm{SL}_2(\mathbf{Q}_p)$-conjugation).
If $d=2$, the Lie algebra is a line stabilizer, isomorphic to $\mathbf{Q}_p^*\ltimes\mathbf{Q}_p$ (with action $t\cdot x=t^2x$); a compact open subgroup has to lie in $\mathbf{Z}_p^*\ltimes\mathbf{Q}_p$. It intersects $\mathbf{Q}_p$ in $p^n\mathbf{Z}_p$ for some $n\in\mathbf{Z}$. The intersection being given, the subgroup is determined by some compact open subgroup of the quotient $\mathbf{Z}_p^*\ltimes \mathbf{Q}_p/p^n\mathbf{Z}_p$, necessarily contained in $\mathbf{Z}_p^*\ltimes p^m\mathbf{Z}_p/p^n\mathbf{Z}_p$ for some $m\le n$. This is a semidirect product $\mathbf{Z}_p^*\ltimes\mathbf{Z}/p^k\mathbf{Z}$ for some $k$. In turn, this has to contain some congruence subgroup in $\mathbf{Z}_p$, and then there are only finitely many possibilities.
If $d=1$, and the Lie algebra is diagonalizable over an extension, then the normalizer of the Lie algebra is a one-dimensional group $M$, with a compact open normal subgroup isomorphic to $\mathbf{Z}_p$. The intersection with $\mathbf{Z}_p$ is some $p^n\mathbf{Z}_p$, the quotient being finite or virtually isomorphic to $\mathbf{Z}$, hence has countably many subgroups.
Finally if $d=1$ and the Lie algebra is nilpotent, it is conjugate to the Lie algebra of strictly upper triangular matrices. So it's a compact subgroup of its normalizer, which is the group of upper triangular matrices of determinant 1 (as when $d=2$). So as when $d=2$, $K\subset\mathbf{Z}_p^*\rtimes\mathbf{Q}_p$. Then $K$ contains $e_{12}(p^n\mathbf{Z}_p)=I_2+p^n\mathbf{Z}_pE_{12}$ for some $n$, so is determined by its image in the quotient, a finite subgroup of $\mathbf{Z}_p^*\ltimes p^m\mathbf{Z}_p/p^n\mathbf{Z}_p$ for some $m\le n$. Since $\mathbf{Z}_p^*\simeq \mathbf{Z}_p\times F$ with $F$ finite and $\mathbf{Z}_p$ is torsion-free, the subgroup has to lie in the finite group $F\ltimes p^m\mathbf{Z}_p/p^n\mathbf{Z}_p$. So for given $m,n$, this leaves only finitely many possibilities.
Remark 1: it can be shown along the same lines that $\mathrm{SL}_2(\mathbf{Q}_p)$ has countably many conjugacy classes of closed subgroups that are not (infinite discrete). While it has uncountably many conjugacy classes of discrete infinite cyclic subgroups (e.g., those generated by the diagonal matrices $\mathrm{diag}(t,t^{-1})$ when $|t|>1$ are pairwise non-conjugate).
Remark 2: in $\mathrm{SL}_3(\mathbf{Z}_p)$ there are uncountably many conjugacy classes of closed subgroups, just because there is a closed subgroup isomorphic to $\mathbf{Z}_p^2$ (in which the subgroups $L\cap\mathbf{Z}_p^2$, when $L$ ranges among lines in $\mathbf{Q}_p^2$, are pairwise distinct).
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MVP Health Care Selects Pitney Bowes Group 1 Software for Enhanced Customer Communication Management
EngageOne™ Interactive Communications to Help Top Health Care Provider Raise the Bar for Quality and Customer Service
LANHAM, Md.--(BUSINESS WIRE)--Pitney Bowes Group 1 Software today announced that MVP Health Care, one of the top-rated health plans in the nation in the current U.S. News and World Report/NCQA listing of “America’s Best Health Plans,” has selected its EngageOne™ Interactive Communications solution of the Customer Communication Management (CCM) suite to create, deliver and manage real-time interactive communications with its members, resulting in enhanced customer service and reduced costs.
Following a merger in 2006 with Preferred Care, MVP Health Care serves more than 700,000 members throughout New York state, Vermont and New Hampshire. As a result of this organizational growth coupled with its 25-year history, MVP Health Care had amassed more than 15,000 different document templates for business letters and customer correspondence. For customer service representatives, determining what letter should be sent was a manually-intensive, time-consuming process. In addition, customer service representatives did not have the ability to view documents that were mailed to members, resulting in lengthy phone conversations.
With EngageOne Interactive, MVP Health Care employees now have the ability to easily create, deliver and manage real-time personalized, interactive customer communications, such as correspondence, new business applications and negotiated documents. Customer service representatives can now quickly and easily find the correct template and tailor the communication to the specific needs of the customer interaction. The result is faster, more efficient and cost-effective communications across the enterprise.
“At MVP Health Care, we are always looking for ways to make health care easier for our members—whether it is searching online for available physicians or speaking with a customer service representative,” said Jack Van Graafeiland, chief information officer of MVP Health Care. “EngageOne Interactive provides us with an efficient and effective solution to control the content and design, as well as the management of our important real-time, highly personalized customer correspondence. With EngageOne Interactive, we are able to make sure that our members’ questions are not only answered, but answered in a timely and efficient manner.”
In addition to EngageOne Interactive, MVP Health Care will use Pitney Bowes Group 1’s OpenEDMS, an enterprise content management solution that provides a Web-optimized tool for managing documents and improving automated workflow. For document storage and retrieval, MVP will use Pitney Bowes Group 1’s e2 Vault and Service, part of the e2™ Suite, a platform that facilitates internal customer support and external customer self-service by integrating e-presentment, archiving, online account management and e-service technologies.
“The health insurance industry is marked by mergers and acquisitions, intense competition and rising customer expectations, and insurers need the ability to communicate more effectively with each and every member,” said Bill Sinn, strategic industry manager, insurance, for Pitney Bowes Group 1 Software. “As one of the leading health plans in the country, MVP understands that effective customer communication is key to member satisfaction and financial success. EngageOne Interactive allows MVP to efficiently streamline the process of health care communication by enabling its employees to more quickly and intelligently respond to policy holders with personalized, timely and accurate information.”.
About MVP Health Care.
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The Akumal Yoga Studio offers daily classes throughout the year with meditation, pranayama, hatha, and hatha flow. Please contact them directly through their website, or drop by the studio for a weekly timetable. Akumal Yoga is located upstairs above the Las Casitas booking office at the arch entranceway to the village.
Akumal Yoga also offers private sessions in your villa or condo.
WEBSITE:
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A selection of popular Ramadan delicacies in Malaysia.
Malaysians love to break fast with an assortment of their favourite food.
Fact: Malaysians love their food. Wherever you go, you’ll find 24-hour restaurants, fast food outlets and roadside stalls. Malls with cafés and eateries serve everything from local cuisine to Thai fare to Greek food.
Even during Ramadan, when Muslims fast from dawn to dusk for a whole month, the subject of enjoying a good meal is taken very seriously.
What’s special about the fasting month is that it is as much about tradition as it is about food. During Ramadan, people sit together with family and friends during the breaking of fast, facing a table groaning with various delicacies. Mosques and business premises hold mass buka puasa events to share food and bring people closer together.
There are hundreds of dishes, varied in their texture, shape and taste, making each offering unique. And they are sold at Ramadan bazaars all over Malaysia.
Get the details on some great favourites below:
Bubur lambuk
The bubur lambuk is a bowl of comfort and goodness.
This dish has become a real classic and it is easily the king of the buka puasa dishes. There are many variations depending on personal tastes, regional preferences and family traditions. The basic ingredients are rice, coconut milk, spices and some meat or seafood.
The bubur lambuk of Kampung Baru is the most renowned version due to its decades-old history and simple but tasty ingredients.
Murtabak
The delectable murtabak is convenient to eat.
The murtabak or martabak is a ubiquitous sight at any evening and weekend markets throughout the year. It’s a savoury dish that’s also very filling, making it ideal for the fasting month when there isn’t much appetite for the usual daily grub.
How is it made? A runny mixture of meat, eggs, chopped onions and spices is poured onto paper-thin wheat flour dough and folded into a neat square. It is then fried on a flat griddle with ghee (clarified butter) or oil. Frequent accompaniments are pickled red onions and dhal gravy.
Popia
Tasty and crunchy, the popia is also a winner.
Another popular snack which can be just as filling, the popia is a type of spring roll which consists of vegetables and chicken or prawns rolled in a very thin wheat flour crepe square and fried. The crispy concoction is then dipped into a sweet and spicy chilli sauce.
A popular variation is popia basah (literally wet popia) where a softer crepe is used and eaten fresh instead of fried. You can be sure of long snaking queues during Ramadan wherever these nibbles are sold.
Ayam percik
The original ayam percik is a Kelantanese invention.
Ayam percik is a dish of chicken grilled with thickened and delicately spiced coconut milk. This dish comes from the state of Kelantan and is a typical side dish for nasi kerabu (a blue rice dish with fresh herbs, vegetables and fish crackers).
Some vendors, especially in cities and towns outside Kelantan, sell ayam percik which is spiced with chilli, giving it a reddish colour. But purists swear by the original version, which is more mellow in flavour.
Last but not least – Drinks!
At the end of the fasting day, nothing beats a cold drink.
Yes, it really does deserve a category of its own. And why not? After fasting from sunrise to sunset, this is the first item that people reach out for in order to slake their thirst.
Just like the eats, the local drinks are a real delight and come in a kaleidoscope of colours. Discover some of the popular ones below:
Sirap bandung — Very popular among children and young people due to its attractive pink colour and amazing burst of sweetness. Its basic ingredients are rose syrup and evaporated milk, though some variations include grass jelly and basil seeds.
Katira — Originating from Johor, katira (local name for almond gum) is one of the ingredients used in the drink, hence the name. Other components include Chinese red dates, barley, candied winter melon, grass jelly, cendol (rice flour vermicelli) and pandan-flavoured sugar syrup.
A glass of Limau asam boi.
Limau asam boi — Comprising blended calamansi lime and sour plum, this thirst quencher is also the best drink to consume with rich greasy food.
Air kelapa — With a little bit of ice and flesh of the young coconut, having this drink after a day of fasting is akin to discovering an oasis in the desert. It is refreshing, and has marvellous cooling properties and a clean taste.
Check out the list of restaurants and cafés on Yellow Pages Malaysia if you want to plan a quick buka puasa at great eateries across the country.
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\begin{document}
\title{Heyting Algebra and G\"{o}del Algebra vs. various Topological Systems and Esakia Space: a Category Theoretic Study}
\author[ A. Di Nola, R. Grigolia, P. Jana]{ A. Di Nola \corrauth, R. Grigolia, P. Jana}
\begin{abstract}
This paper introduces a notion of intuitionistic topological system. Properties of the proposed system is studied in details. Categorical interrelationships among Heyting algebra, G\"{o}del algebra, Esakia space and proposed intuitionistic topological systems have also been studied. A flavour of Kripke model is given.
\end{abstract}
\keywords{Heyting algebra, G\"{o}del algebra, Esakia space, Intuitionistic logic, Topological system}
\ams{06D20, 97H50, 03G25, 03B20, 54H10}
\maketitle
\section{Introduction}
Topological system was introduced by S. Vickers in his book ``Topology via Logic" \cite{SV} in 1989. A topological system is a triple $(X,\models, A)$, consisting of a non empty set $X$, a frame $A$ and a binary relation between the set and the frame, which matches the logic of finite observations or geometric logic. Topological system is a mathematical object which unifies the concepts of topological space and frame in one framework. Hence such a structure allows us to switch among the concepts of frame, topological space and corresponding logic freely.
Concepts of a topological system and geometric logic or logic of finite observations have a deep connection. It is well known that the Lindenbaum algebra of geometric logic is a frame, likewise the Lindenbaum algebra of classical logic is Boolean algebra and that of intuitionistic logic is Heyting algebra etc. One may notice that any topological system is a model of geometric logic.
In \cite{SV1}, it may be noticed that ``Logically, spatiality is the same as completeness, but there is a difference of emphasis. Completeness refers to the ability of the logical reasoning (from rules and axioms) to generate all the equivalences that are valid for the models: if not, then it is the logic that is considered incomplete. Spatiality refers to the existence of enough models to discriminate between logically inequivalent formulae: if not, then the class of models is incomplete." In this respect we may recall that there exist adjunction between category of topological systems and the category of topological spaces, which leads to the concept that not every topological system comes from a topological space. To elaborate the fact one may notice that every topological space can be considered as a topological system because of the following fact: if $(X,\tau)$ is a topological space then $(X,\vdash,\tau)$ is the corresponding topological system, where $x\vdash T$ represents that $x$ is an element of $T(\in \tau$). Hence not every topological system is spatial and correspondingly we arrive at the conclusion (logical fact) that the corresponding logic (i.e., geometric logic) is not complete. On the contrary whenever we deal with a logic which is complete then we can expect categorical equivalence or duality between categories of mathematical structures which are the models of the logic.
Topological system is an important mathematical structure in its own right. It is already mentioned earlier that this kind of structure reflects the corresponding topological and algebraic structures simultaneously. In fact it is closely connected to the corresponding logic. On the other hand topological system plays important roles in computer science and (quantum) physics \cite{SV, DI}.
It is well known that the category of Heyting algebras are dually equivalent to the category of Esakia spaces. Consequently both Heyting algebra and Esakia space are models of intuitionistic logic. Our main goal of this paper is to introduce a notion of I-topological system such that it will able to unify the notions of Heyting algebra, Esakia space and I-topological system in it. The similar study for G\"{o}del algebra and related structures is also a focus point for the present paper. It is quite expected that the proposed notions will have its impact in the areas of computer science and physics.
This paper is organised as follows. Section 2 contains the required preliminary notions to make the paper self contained. Notion of I-topological system is introduced and studied in details in Section 3. This section gives a cue to connect the proposed system with Kripke model. A detailed categorical study of the proposed systems with corresponding topological and algebraic structure has also done in this section. Section 4 contributes some concluding remarks.
\section{Preliminaries}
In this section we include a brief outline of relevant notions to develop our proposed mathematical structures and results. In \cite{AJ, LE, PJ, SV}, one may found the details of the notions stated here.
\begin{definition}[$G$-structured arrow and $G$-costructured arrow]\label{2.1} Let $G:\mathbb{A}\to\mathbb{B}$ be a functor, where $\mathbb{A},\ \mathbb{B}$ are two categories and let $B$ be a $\mathbb{B}$-object. Then the concepts of $G$-structured arrow and $G$-costructured arrow are defined as follows.
\begin{enumerate}
\item A $G$-$\mathbf{structured\ arrow\ with\ domain}$ \index{$G$-structured arrow} $B$ is a pair $(f,A)$ consisting of an $\mathbb{A}$-object $A$ and a $\mathbb{B}$-morphism $ f:B\longrightarrow GA$.
\item A $G$-$\mathbf{costructured\ arrow\ with\ codomain}$ \index{$G$-costructured arrow} $B$ is a pair $(A,f)$ consisting of an $\mathbb{A}$-object $ A$ and a $\mathbb{B}$-morphism $ f:GA\longrightarrow B$.
\end{enumerate}
\end{definition}
\begin{definition}[$G$-universal arrow and $G$-couniversal arrow]\label{2.2}
$G$-universal arrow and $G$-couniversal arrow are defined as follows:
\begin{enumerate}
\item A $G$-structured arrow $(g,A)$ with domain $B$ is called $G$-$\mathbf{universal}$ \index{$G$-universal arrow} for $B$ provided that for each $G$-structured arrow $(g',A')$ with domain $B$, there exists a unique $\mathbb{A}$-morphism $\hat f:A \longrightarrow A'$ with $g'=G(\hat f)\circ g$. i.e., s.t. the triangle
\begin{center}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{B & &GA \\
& & GA' \\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$g$} (m-1-3)
(m-1-1) edge node[auto,swap] {$g'$} (m-2-3)
(m-1-3) edge node[auto] {$G\hat f$} (m-2-3);
\end{tikzpicture}
\end{center}
commutes.
\\ We can also represent the above statement by the following diagram
\begin{center}
\begin{tabular}{ l | r }
$\mathbb{B}$ & $\mathbb{A}$\\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{B & &GA \\
& & GA' \\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$g$} (m-1-3)
(m-1-1) edge node[auto,swap] {$g'$} (m-2-3)
(m-1-3) edge node[auto] {$G\hat f$} (m-2-3);
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &A \\
& &A' \\ };
\path[->,font=\scriptsize]
(m-1-3) edge node[auto] {$\hat f$} (m-2-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
The diagram above indicates the fact that $g:B\longrightarrow GA$ is the $G$-universal arrow provided that for given $g':B\longrightarrow GA'$ there exist a unique $\mathbb{A}-morphisim$ $\hat f:A\longrightarrow A'$ s.t. the triangle commutes.
\item A $G$-costructured arrow $(A,g)$ with codomain $B$ is called $G$-$\mathbf{couniversal}$ \index{$G$-couniversal arrow} for $B$ provided that for each $G$-costructured arrow $(A',g')$ with codomain $B$, there exists a unique $\mathbb{A}$-morphism $\hat f:A' \longrightarrow A$ with $g'=g\circ G(\hat f)$. i.e., s.t. the triangle
\begin{center}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{ GA&&B \\
GA'\\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$g$} (m-1-3)
(m-2-1) edge node[auto] {$G\hat f$} (m-1-1)
(m-2-1) edge node[auto,swap] {$g'$} (m-1-3)
;
\end{tikzpicture}
\end{center}
commutes.
\\ We can also represent the above statement by the following diagram
\begin{center}
\begin{tabular}{ l | r }
$\mathbb{B}$ & $\mathbb{A}$\\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{ GA&&B \\
GA'\\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$g$} (m-1-3)
(m-2-1) edge node[auto] {$G\hat f$} (m-1-1)
(m-2-1) edge node[auto,swap] {$g'$} (m-1-3)
;
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &A \\
& &A' \\ };
\path[->,font=\scriptsize]
(m-2-3) edge node[auto] {$\hat f$} (m-1-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
The diagram above indicates the fact that $g:GA\longrightarrow B$ is the $G$-couniversal arrow provided that for given $g':GA'\longrightarrow B'$ there exist a unique $\mathbb{A}-morphism$ $\hat f:A'\longrightarrow A$ s.t. the triangle commutes.
\end{enumerate}
\end{definition}
\begin{definition}[Left Adjoint and Right Adjoint]\label{2.3}
Left Adjoint and Right Adjoint are defined as follows.
\begin{enumerate}
\item A functor $G:\mathbb{A}\longrightarrow \mathbb{B}$ is said to be $\mathbf{left\ adjoint}$ \index{left adjoint} provided that for every $\mathbb{B}$-object $B$, there exists a $G$-couniversal arrow with codomain $B$.
\\As a consequence, there exists a natural transformation $\eta:id_A\longrightarrow FG$ ($id_A$ is the identity morphism from $A$ to $A$), where $F:\mathbb{B}\longrightarrow \mathbb{A}$ is a functor s.t. for given $f:A\longrightarrow FB$ there exists a unique $\mathbb{B}$-morphism $\hat f:GA\longrightarrow B$ s.t. the triangle
\begin{center}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{ A&&FGA \\
&&FB\\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\eta_A$} (m-1-3)
(m-1-1) edge node[auto,swap] {$f$} (m-2-3)
(m-1-3) edge node[auto] {$F\hat f$} (m-2-3)
;
\end{tikzpicture}
\end{center}
commutes.
\\ This $\eta$ is called the unit \index{unit} of the adjunction.
\\ Hence, we have the diagram of unit \index{diagram of!unit} as follows:
\begin{center}
\begin{tabular}{ l | r }
$\mathbb{A}$ & $\mathbb{B}$\\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{A & &FGA \\
& &FB \\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\eta$} (m-1-3)
(m-1-1) edge node[auto,swap] {$f$} (m-2-3)
(m-1-3) edge node[auto] {$F\hat f$} (m-2-3);
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &GA \\
& &B \\ };
\path[->,font=\scriptsize]
(m-1-3) edge node[auto] {$\hat f$} (m-2-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
\item A functor $G:\mathbb{A}\longrightarrow \mathbb{B}$ is said to be $\mathbf{right\ adjoint}$ \index{right adjoint} provided that for every $\mathbb{B}$-object $B$, there exists a $G$-universal arrow with domain $B$.
\\ From the definition above, it follows that there exists a natural transformation $\xi:FG\longrightarrow id_A$ ($id_A$ is the identity morphism from $A$ to $A$), where $F:\mathbb{B}\longrightarrow\mathbb{A}$ is a functor s.t. for given $f':FB\longrightarrow A$, there exists a unique $\mathbb{B}$-morphism $\hat f:B\longrightarrow GA$ s.t the triangle
\begin{center}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{ FGA&&A \\
FB\\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\xi_A$} (m-1-3)
(m-2-1) edge node[auto] {$F\hat f$} (m-1-1)
(m-2-1) edge node[auto,swap] {$f'$} (m-1-3)
;
\end{tikzpicture}
\end{center}
commutes.
\\ This $\xi$ is called the co-unit \index{co-unit} of the adjunction.
\\ Hence, we have the diagram of co-unit \index{diagram of!co-unit} as follows:
\begin{center}
\begin{tabular}{ l | r }
$\mathbb{A}$ & $\mathbb{B}$\\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{ FGA&&A \\
FB\\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\xi$} (m-1-3)
(m-2-1) edge node[auto] {$F\hat f$} (m-1-1)
(m-2-1) edge node[auto,swap] {$f'$} (m-1-3)
;
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &GA \\
& &B \\ };
\path[->,font=\scriptsize]
(m-2-3) edge node[auto] {$\hat f$} (m-1-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
\end{enumerate}
\end{definition}
\begin{definition}[Heyting algebra]\label{gf}
An algebra $(A, \vee, \wedge, \rightarrow, \mathbf{1}, \mathbf{0})$ with three binary and two nullary operations is said to be $\textbf{Heyting algebra}$ if $(A, \vee, \wedge, \mathbf{1}, \mathbf{0})$ is a bounded distributive lattice and $\rightarrow$ is a binary operation which is adjoint to
$\wedge$.
\end{definition}
\begin{definition}[G\"{o}del algebra]\label{ga}
A Heyting algebra $A$ satisfying the prelinearity property viz. $(a\rightarrow b)\vee (b\rightarrow a)=\mathbf{1}$, for any $a,b\in A$ is said to be a \textbf{G\"{o}del algebra}.
\end{definition}
\begin{definition}[Heyting homomorphism]\label{hhm}
Let $A$, $B$ be two Heyting algebras. A map $f:A\longrightarrow B$ is said to be $\textbf{Heyting homomorphism}$ if the following conditions hold:\\
(i) $f(a_1\wedge a_2)=f(a_1)\wedge f(a_2)$;\\
(ii) $f(a_1\vee a_2)=f(a_1)\vee f(a_2)$;\\
(iii) $f(a_1\rightarrow a_2)=f(a_1)\rightarrow f(a_2)$;\\
(iv) $f(\mathbf{0})=\mathbf{0}$.
\end{definition}
\textbf{\underline{Note:}} The set of bounded distributive lattice homomorphisms from a Heyting algebra $A$ to the Heyting algebra $(\{0,1\},\vee,\wedge,\rightarrow,1,0)$ will be denoted by $Hom(A,\{0,1\})$ in this paper.
Let us consider the example:
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=.7]
\node (one) at (0,2) {$1$};
\node (a) at (0,0) {$a$};
\node (zero) at (0,-2) {$0$};
\node (1) at (4,1) {$1$};
\node (0) at (4,-1) {$0$};
\draw (zero) -- (a) -- (one);
\draw (0) -- (1);
\draw [-latex,blue] (one) -- (1);
\draw [-latex,blue] (a) -- (1);
\draw [-latex,blue] (zero) -- (0);
\node (onen) at (8,2) {$1$};
\node (an) at (8,0) {$a$};
\node (zeron) at (8,-2) {$0$};
\node (1n) at (12,1) {$1$};
\node (0n) at (12,-1) {$0$};
\draw (zeron) -- (an) -- (onen);
\draw (0n) -- (1n);
\draw [-latex,blue] (onen) -- (1n);
\draw [-latex,blue] (an) -- (0n);
\draw [-latex,blue] (zeron) -- (0n);
\end{tikzpicture}
\caption{}
\end{center}
\end{figure}
We have two lattice homomorphisms
$$h_1(1)=h_1(a)=1\ , h_1(0)=0 \ \text{and}\ h_2(1)=1,\ h_2(a)=h_2(0)=0,\ h_2\leq h_1.$$
Here $h_2\leq h_1$ iff $h_2(a)\leq h_1(a)$ for any $a\in A$. $h_2$ is not Heyting homomorphism. $h_1$ is the only Heyting homomorphism, which is maximal. But there exists two prime filters in the Heyting algebra (and in the lattice as well):
$$F_1=\{1,a\}\ \text{and}\ F_2=\{1\},\ F_2\subseteq F_1.$$
It is well known Priestley duality between bounded distributive lattices and Priestley spaces $(X,R)$ \cite{BD,HA}. Priestley space is Heyting space (or Esakia space) \cite{MA} if and only if
$$(*) R^{-1}(U)\ \text{is open for every open set}\ U.$$ So in the construction of Heyting space (or Esakia space) we use Priestley space with the condition $(*)$.
Notice, the restricted Priestley duality for Heyting algebras states that a bounded distributive lattice $A$ is a Heuting algebra if and only if the Priestley dual of $A$ is a Heyting space and a $\{0,1\}$-lattice homomorphism $h$ between Heyting algebras preserves the implication $\rightarrow$ if and only if the Priestley dual of $h$ is a Heyting morphism.
\begin{definition}[{$\mathbf{HA}$}]\label{Grfrm}
Heyting algebras together with Heyting homomorphisms form a category, which is well known as a category of Heyting algebras and denoted by $\mathbf{HA}$.
\end{definition}
\begin{definition}[{$\mathbf{GA}$}]\label{GA}
G\"{o}del algebras together with corresponding Heyting homomorphisms form a category, which is well known as a category of G\"{o}del algebras and denoted by $\mathbf{GA}$.
\end{definition}
\begin{definition}[Esakia Space]\label{ES}
An ordered topological space $(X,\leq,\tau)$ is called an $\textbf{Esakia space}$ if
\begin{itemize}
\item $(X,\tau)$ is compact;
\item for any $x, y\in X$ with $x\nleq y$ there exists a clopen up-set $U\subseteq X$ with $x\in U$, $y\notin U$;
\item for any clopen set $U$, the down-set $\downarrow U$ is also clopen.
\end{itemize}
\end{definition}
Note that an ordered topological space $(X,\leq,\tau)$ together with the first two conditions of Definition \ref{ES} is known as Priestley space.
\begin{definition}[Esakia morphism]
Let $(X,\leq,\tau)$ and $(Y,\leq,\tau')$ be Esakia spaces. Then a map $f:X\longrightarrow Y$ is called an $\textbf{Esakia morphism}$ if $f$ is a continuous bounded morphism (p-morphism), i.e., if for each $x\in X$ and $y\in Y$, if $f(x)\leq y$, then there exists $z\in X$ such that $x\leq z$ and $f(z)=y$.
\end{definition}
\begin{definition}
Esakia spaces together with Esakia morphisms forms a category of Esakia spaces and denoted by $\mathbf{ESA}$.
\end{definition}
\begin{theorem}\label{ESD} \cite{LE}
$\mathbf{HA}$ is dually equivalent with $\mathbf{ESA}$.
\end{theorem}
\section{Categories: I Top, I TopSys, HA and their interrelationships}
Suppose that we have the algebras $A$ and $B$, and two homomorphisms $h_1$, $h_2$ from $A$ to $B$. Then we can define the ordering $R$ on the set of all homomorphisms from $A$ to $B$:
$$h_1Rh_2\ \text{iff}\ h_1(a)\leq h_2(a)\ \text{for all}\ a\in A.$$
So, (Hom(A,\{0,1\}),R) is a poset, where $A$ is a Heyting algebra and $Hom(A,\{0,1\})$ is the set of all \textbf{bounded distributive lattice} homomorphisms from $A$ to $(\{0,1\},\vee,\wedge,\rightarrow,1,0)$.
\begin{definition}[I-topological system]\label{gftsy}
An \textbf{I-topological system} is a triple $(X,\models,A)$ consisting of a nonempty set $X$, a Heyting algebra $A$ and a relation $\models$ from $X$ to $A$ such that
\begin{enumerate}
\item $x\models \mathbf{0}$ for no $x\in X$;
\item $x\models a\wedge b$ iff $x\models a$ and $x\models b$;
\item $x\models a\vee b$ iff $x\models a$ or $x\models b$;
\item $x\models a\rightarrow b$ iff for all $y\in X$ such that $p^*(x)Rp^*(y)$, $y\not\models a$ or $y\models b$, where $p^*:X\to Hom(A,\{0,1\})$ such that $p^*(x)(a)=1\ \text{iff}\ x\models a$.
\end{enumerate}
\end{definition}
From Definition \ref{gftsy} it is easy to deduce that:
$$x\models\neg a\ \text{iff}\ \text{for all}\ y\in X\ \text{such that}\ p^*(x)Rp^*(y), y\not\models a.$$
Now, let us show that $x\not\models a\vee\neg a$, for some $x\in X$. Let $X=\{x,y\}$ and $A=(\{0,a,1\},\vee,\wedge,\rightarrow,1,0)$, where $0\leq a\leq 1$. Then we have two bounded distributive lattice homomorphisms $p'(x)(=h_2)$ and $p'(y)(=h_1)$ ($h_1$ and $h_2$ are represented in Figure 1) and $p'(x)\leq p'(y)$. Let us consider $$x\models a\ \text{iff}\ p'(x)(a)=1.$$ Then clearly $y\models a$ and $x\not\models a$. So it can be derived that $x\not\models \neg a$. Hence $y\models a\vee\neg a$ but $x\not\models a\vee\neg a$. Consequently we may conclude that for this choice of $x\in X$, $x\not\models a\vee\neg a$.
\begin{proposition}\label{prop1}
$x\models \mathbf{1}$ for any $x\in X$.
\end{proposition}
\begin{proof}
$x\models \mathbf{1}$ iff $x\models a\rightarrow a$ iff for all $y\in X$ such that $p^*(x)Rp^*(y)$, $y\not\models a$ or $y\models a$. As for any $x\in X$ and $a\in A$ either $x\models a$ or $x\not\models a$ holds, $x\models \mathbf{1}$ for any $x\in X$.
\end{proof}
\begin{definition}[Heyting algebraic I-topological system]\label{HITopSys}
An I-topological system $(X,\models,A)$ is said to be \textbf{Heyting algebraic} if the map $p^*:X\longrightarrow Hom(A,\{0,1\})$ defined by, $p^*(x)(a)=1$ iff $x\models a$ for $x\in X$ and $a\in A$, is a bijective mapping.
\end{definition}
\begin{definition}
An I-topological system $(X,\models, A)$ is said to be $\mathbf{T_0}$ iff (if $x_1\neq x_2$ then there exist some $a\in A$ such that $x_1\models a$ but $x_2\not\models a$).
\end{definition}
\begin{proposition}
Any Heyting algebraic I-topological system is $T_0$.
\end{proposition}
\begin{proof}
For Heyting algebraic I-topological system $(X,\models,A)$, the map $p^*:X\longrightarrow Hom(A,\{0,1\})$ is bijective and consequently injective. Hence if $x_1\neq x_2$ then $p^*(x_1)\neq p^*(x_2)$ and hence there exist $a\in A$ such that $p^*(x_1)(a)\neq p^*(x_2)(a)$. So as per the definition of $p^*$ it is clear that the system is $T_0$.
\end{proof}
\begin{definition}[G\"{o}del algebraic I-topological system]\label{GITopSys}
A \textbf{G\"{o}del algebraic I-topological system} is a triple $(X,\models,A)$ consisting of a non empty set $X$, a G\"{o}del algebra $A$ and a binary relation $\models$ from $X$ to $A$ such that
\begin{enumerate}
\item $x\models \mathbf{0}$ for no $x\in X$;
\item $x\models a\wedge b$ iff $x\models a$ and $x\models b$;
\item $x\models a\vee b$ iff $x\models a$ or $x\models b$;
\item $x\models a\rightarrow b$ iff for all $y\in X$ such that $p^*(x)Rp^*(y)$, $y\not\models a$ or $y\models b$, where $p^*:X\to Hom(A,\{0,1\})$ such that $p^*(x)(a)=1\ \text{iff}\ x\models a$;
\item the map $p^*:X\longrightarrow Hom(A,\{0,1\})$ defined by, $p^*(x)(a)=1$ iff $x\models a$ for $x\in X$ and $a\in A$, is a bijective mapping.
\end{enumerate}
\end{definition}
\subsection{Kripke model for intuitionistic logic and I-topological system}
In this subsection we will indicate the connection of the notion of I-topological system with Kripke model for intuitionistic logic \cite{AM}.
\begin{definition}
A Kripke frame $\mathscr{F}$ is a pair $(W,\mathscr{R})$ consisting of a nonempty set of worlds (or points), $W$, and a partial order relation $\mathscr{R}$ on $W$ ($\mathscr{R}\subseteq W\times W$).
\end{definition}
\begin{definition}
A Kripke model $\mathscr{M}$ is a pair $(\mathscr{F},v)$ consisting of a Kripke frame $\mathscr{F}$ and a valuation map $v:W\times \mathbf{V}\to \{0,1\}$, where $\mathbf{V}$ is the set of propositional variables such that:
\begin{enumerate}
\item for all $w\in W$ and for all propositional variables $p\in \mathbf{V}$, if $v(w,p)=1$ and $w\mathscr{R} u$ then $v(u,p)=1$;
\item $v(w,\bot)=0$ for all $w\in W$.
\end{enumerate}
\end{definition}
\begin{definition}
Let $\mathscr{M}$ be a Kripke model for intuitionistic logic and $w$ be a world in the frame $\mathscr{F}$. By induction on the construction of a formula $a$ we define a relation $(\mathscr{M},w)\Vdash a$, which is read as ``$a$ is true at $w$ in $\mathscr{M}$":
\begin{itemize}
\item $\mathscr{M},w\Vdash p\ \text{iff}\ v(w,p)=1$;
\item $\mathscr{M},w\Vdash a\wedge b\ \text{iff}\ \mathscr{M},w\Vdash a\ \text{and}\ \mathscr{M},w\Vdash b$;
\item $\mathscr{M},w\Vdash a\vee b\ \text{iff}\ \mathscr{M},w\Vdash a\ \text{or}\ \mathscr{M},w\Vdash b$;
\item $\mathscr{M},w\Vdash \neg a\ \text{iff}\ \forall u\geq w,\ \mathscr{M},u\not\Vdash a$;
\item $\mathscr{M},w\Vdash a\rightarrow b\ \text{iff}\ \forall w\mathscr{R} u,\ \text{if}\ \mathscr{M},u\Vdash a \ \text{then} \mathscr{M},u\Vdash b$;
\item $\mathscr{M},w\not\Vdash\bot$.
\end{itemize}
\end{definition}
Let $(X,\models , A)$ be an I-topological system. Then consider the relation $\mathscr{R}$ on $X$ such that
$$x\mathscr{R}y\ \text{iff}\ p^*(x)Rp^*(y),\ \text{where}\ p^*(x)(a)=1\ \text{iff}\ x\models a.$$
It may be noticed that $(X,\mathscr{R})$ is a partially ordered set. Hence $(X,\mathscr{R})$ is a Kripke frame.
Moreover if we consider $v:X\times A\to \{0,1\}$ such that $v(x,a)=1\ \text{iff}\ x\models a$ then the following holds.
\begin{enumerate}
\item For all $x\in X$ and for all $a\in A$ let us assume that $v(x,a)=1$ and $x\mathscr{R}y$. Then we have $x\models a$ and $p^*(x)Rp^*(y)$ i.e. $p^*(x)(a)\leq p^*(y)(a)$. As $x\models a$, $p^*(x)(a)=1=p^*(y)(a)$. Hence $y\models a$. Therefore for all $x\in X$ and for all $a\in A$, $v(x,a)=1$ and $x\mathscr{R}y$ implies $v(y,a)=1$.
\item We know $v(x,\mathbf{0})=1\ \text{iff}\ x\models \mathbf{0}$. But $x\models \mathbf{0}$ for no $x\in X$. Hence for all $x\in X$, $v(x,\mathbf{0})=0$.
\end{enumerate}
Consequently $(X,\mathscr{R},v)$ is a Kripke model.
Now let us define $x\Vdash a$ iff $x\models a$. Then,
\begin{itemize}
\item $x\Vdash a$ iff $x\models a$ iff $v(x,a)=1$;
\item $x\Vdash a\wedge b$ iff $x\models a\wedge b$ iff $x\models a$ and $x\models b$ iff $x\Vdash a$ and $x\Vdash b$;
\item $x\Vdash a\vee b$ iff $x\models a\vee b$ iff $x\models a$ or $x\models b$ iff $x\Vdash a$ or $x\Vdash b$;
\item Let $x\Vdash a\rightarrow b$. Then,
\begin{align*}
x\Vdash a\rightarrow b\ & \text{iff}\ x\models a\rightarrow b \\
& \text{iff for all}\ y\in X\ \text{such that}\ p^*(x)Rp^*(y), y\not\models a\ \text{or}\ y\models b \\
& \text{iff} \text{ for all}\ y\in X \ \text{and}\ p^*(x)Rp^*(y),\ y\not\Vdash a\ \text{or}\ y\Vdash b \\
& \text{iff for all}\ y\in X\ \text{and}\ x\mathscr{R}y,\ \text{if}\ y\Vdash a\ \text{then}\ y\Vdash b;
\end{align*}
\item As $x\models \mathbf{0}$ for no $x\in X$, $x\not\Vdash \mathbf{0}$.
\end{itemize}
Summarizing all sayings above we can deduce the following
Theorem.
\begin{theorem}
Let $(X,\models,A)$ be an I-topological system. Then $(X,\mathscr{R},v)$, defined as above, is an intuitionistic Kripke model.
\end{theorem}
\subsection{Categories}
\begin{definition}[{$\mathbf{I-TopSys}$}]\label{Grftsy}
The category $\mathbf{I-TopSys}$\index{category!Graded Fuzzy TopSys} is defined thus.
\begin{itemize}
\item The objects are I-topological systems $(X,\models,A)$, $(Y,\models,B)$ etc. (c.f. Definition \ref{gftsy}).
\item The morphisms are pair of maps satisfying the following continuity properties: If $(f_1,f_2):(X,\models ,A)\longrightarrow (Y,\models',B)$ then\\
(i) $f_1:X\longrightarrow Y$ is a set map;\\
(ii) $f_2:B\longrightarrow A$ is a Heyting homomorphism;\\
(iii) $x\models f_2(b)$ iff $f_1(x)\models' b$.
\item The identity on $(X,\models,A)$ is the pair $(id_X,id_A)$, where $id_X$ is the identity map on $X$ and $id_A$ is the identity Heyting homomorphism. That this is an $\mathbf{I-TopSys}$ morphism\index{Graded Fuzzy TopSys!morphism} can be proved.
\item If $(f_1,f_2):(X,\models,A)\longrightarrow (Y,\models',B)$ and
$(g_1,g_2):(Y,\models',B)\longrightarrow (Z,\models'',C)$ are morphisms in $\mathbf{I-TopSys}$, their
composition $(g_1,g_2)\circ (f_1,f_2)=(g_1\circ f_1,f_2\circ g_2)$ is the pair of composition of functions between two sets and composition of Heyting homomorphisms between two Heyting algebras. It can be verified that
$(g_1,g_2)\circ (f_1,f_2)$ is a morphism in $\mathbf{I-TopSys}$.
\end{itemize}
\end{definition}
\begin{definition}[{$\mathbf{HI-TopSys}$}]\label{hits}
Heyting algebraic I-topological systems (c.f. Definition \ref{HITopSys}) together with corresponding $\mathbf{I-TopSys}$ morphisms form a category and called $\mathbf{HI-TopSys}$.
\end{definition}
\begin{definition}[{$\mathbf{GI-TopSys}$}]\label{gits}
G\"{o}del algebraic I-topological systems (c.f. Definition \ref{GITopSys}) together with corresponding $\mathbf{I-TopSys}$ morphisms form a category and called $\mathbf{GI-TopSys}$.
\end{definition}
\subsection{Functors }
Let us construct suitable functors among the above mentioned categories as follows to establish their interrelations.
\begin{definition}
$H$ is a functor from $\mathbf{HI-TopSys}$ to $\mathbf{HA^{op}}$ defined as follows:\\
$H$ acts on an object $(X,\models, A)$ as $H((X,\models , A))=A$ and on a morphism $(f_1,f_2)$ as $H((f_1,f_2))=f_2$.
\end{definition}
It is easy to verify that $H$ is indeed a functor.
\begin{definition}
$\mathscr{G}$ is a functor from $\mathbf{GI-TopSys}$ to $\mathbf{GA^{op}}$ defined as follows:\\
$\mathscr{G}$ acts on an object $(X,\models, A)$ as $\mathscr{G}((X,\models , A))=A$ and on a morphism $(f_1,f_2)$ as $\mathscr{G}((f_1,f_2))=f_2$.
\end{definition}
It is easy to verify that $\mathscr{G}$ is indeed a functor.
\begin{lemma}\label{lemma1}
$(Hom(A,\{0,1\}),\models^*,A)$, where $A$ is a Heyting algebra and $v\models^* a$ iff $v(a)=1$, is an $I$-topological system.
\end{lemma}
\begin{proof}
Let us proceed in the following way.
(i) $v\models^*\mathbf{0}$ iff $v(\mathbf{0})=1$, but as $v$ is a bounded distributive lattice homomorphism so, $v(\mathbf{0})=0$. Hence $v\models^*\mathbf{0}$ for no $v\in Hom(A,\{0,1\})$.
(ii) $v\models^* a\wedge b$ iff $v(a\wedge b)=1$ iff $v(a)\wedge v(b)=1$ iff $v(a)=1$ and $v(b)=1$ iff $v\models^* a$ and $v\models^* b$.
(iii) $v\models^* a\vee b$ iff $v(a\vee b)=1$ iff $v(a)\vee v(b)=1$ iff $v(a)=1$ or $v(b)=1$ iff $v\models^* a$ or $v\models^* b$.
(iv) Let us assume that $v\models^* a\rightarrow b$. We have $v\models^* a\rightarrow b$ iff $v(a\rightarrow b)=1$. Now for any $v'\in Hom(A,\{0,1\})$ such that $v\leq v'$, we have $v'(a\rightarrow b)=1$. So $v'(a)\rightarrow v'(b)=1$. Hence $v'(a)=0$ or $v'(b)=1$. Consequently $v'\not\models^* a$ or $v'\models^* b$ for any $v'\in Hom(A,\{0,1\})$ such that $vRv'$.
Let for all $v'\in Hom(A,\{0,1\})$ such that $vRv'$, $v'\not\models^*a$ or $v'\models^* b$, i.e., $v'(a)=0$ or $v'(b)=1$. In particular we have $v(a)=0$ or $v(b)=1$. We need to show that $v(a\rightarrow b)=1$ i.e., $v\models^* a\rightarrow b$. For any Heyting algebra $A$ and $a,b\in A$ it is known that $b\leq a\rightarrow b$ and so $v(b)\leq v(a\rightarrow b)$. Hence for $v(b)=1$, $v(a\rightarrow b)=1$. Now when $v(b)=0$, if possible let us assume that $v(a\rightarrow b)=0$. Now $v^{-1}(0)$ is an ideal so $v(a\rightarrow b)=0$ and $v(b)=0$ implies $v(a)=0$. In this case $v(a)\rightarrow v(b)=1$ but it is possible to choose $w\in Hom(A,\{0,1\})$ such that $w(a)=1$ and $w(b)=0$. For this choice of $w$ it is clear that $vRw$ but $w\models^* a$ and $w\not\models^* b$, which contradicts our assumption. Hence $v(a\rightarrow b)=1$ for this case.
Hence we can conclude that $v\models^* a\rightarrow b$ iff for all $v'\in Hom(A,\{0,1\})$ such that $vRv'$, $v'\not\models^*a$ or $v'\models^* b$.
\end{proof}
\begin{corollary}\label{cor1}
$(Hom(A,\{0,1\}),\models^*,A)$, where $A$ is a G$\ddot{o}$del algebra and $v\models^* a$ iff $v(a)=1$, is an $I$-topological system.
\end{corollary}
\begin{lemma}\label{lemma2}
For any Heyting algebra $A$, $(Hom(A,\{0,1\}),\models^*,A)$ have the following properties.\\
(i) if for any $a,b\in A$, $(v\models^* a$ iff $v\models^*b$ for any $v\in Hom(A,\{0,1\}))$ then $a=b$.\\
(ii) if $v_1\neq v_2$ then there exist $a\in A$ such that $v_1\models^* a$ but $v_2\not\models^*a$.\\
(iii) $p^*:Hom(A,\{0,1\})\longrightarrow Hom(A,\{0,1\})$ defined by, $p^*(v)(a)=1$ iff $v\models^*a$ is a bijection.
\end{lemma}
\begin{proof}
(i) Let for any $a,b\in A$ and $v\in Hom(A,\{0,1\})$, $v\models^* a$ iff $v\models^*b$. So, $v(a)=1$ iff $v(b)=1$ for any $v\in Hom(A,\{0,1\})$. Hence $a=b$ can be concluded.
\\
Properties (ii) and (iii) can be verified by routine check.
\end{proof}
\begin{corollary}\label{cor2}
For any G\"{o}del algebra $A$, $(Hom(A,\{0,1\}),\models^*,A)$ have the following properties.\\
(i) if for any $a,b\in A$, $(v\models^* a$ iff $v\models^*b$ for any $v\in Hom(A,\{0,1\}))$ then $a=b$.\\
(ii) if $v_1\neq v_2$ then there exist $a\in A$ such that $v_1\models^* a$ but $v_2\not\models^*a$.
\end{corollary}
\begin{lemma}\label{lemma3}
If $f:B\longrightarrow A$ is a Heyting homomorphism then $(\_\circ f,f):(Hom(A,\{0,1\}),\models^*,A)\longrightarrow (Hom(B,\{0,1\}),\models^*,B)$ is continuous.
\end{lemma}
\begin{proof}
We have $v\models^* f(b)$ iff $v(f(b))=1$ iff $v\circ f(b)=1$ iff $(\_\circ f(v))(b)=1$ iff $\_\circ f(v)\models^* b$.
\end{proof}
\begin{definition}
$S$ is a functor from $\mathbf{HA^{op}}$ to $\mathbf{I-TopSys}$ defined as follows. $S$ acts on an object $A$ as $S(A)=(Hom(A,\{0,1\}),\models^*,A)$ and on a morphism $f$ as $S(f)=(\_\circ f,f)$ (it is indeed a functor follows from Lemma \ref{lemma1} and Lemma \ref{lemma3}).
\end{definition}
\begin{proposition}\label{proposition1}
$S$ is a functor from $\mathbf{HA^{op}}$ to $\mathbf{HI-TopSys}$.
\end{proposition}
\begin{proof}
Follows from Lemma \ref{lemma1}, Lemma \ref{lemma2} and Lemma \ref{lemma3}.
\end{proof}
\begin{definition}
$\mathscr{S}$ is a functor from $\mathbf{GA^{op}}$ to $\mathbf{GI-TopSys}$ defined as follows. $\mathscr{S}$ acts on an object $A$ as $\mathscr{S}(A)=(Hom(A,\{0,1\}),\models^*,A)$ and on a morphism $f$ as $\mathscr{S}(f)=(\_\circ f,f)$ (it is indeed a functor follows from Corollary \ref{cor1}, Corollary \ref{cor2} and Lemma \ref{lemma3}).
\end{definition}
\begin{theorem}\label{theorem1}
$\mathbf{HI-TopSys}$ is dually equivalent to $\mathbf{HA}$.
\end{theorem}
\begin{proof}
First we will prove that $H$ is the left adjoint to the functor $S$ by presenting the unit of the adjunction.
Recall that $S(A)=(Hom(A,\{0,1\}),\models^*,A)$ where $v\models^* a$ iff $v(a)=1$ and $H((X,\models,A))=A$.
Hence $S(H((X,\models, A)))=(Hom(A,\{0,1\}),\models^*, A)$.
\begin{center}
\begin{tabular}{ l | r }
$\mathbf{HI-TopSys}$ & $\mathbf{HA^{op}}$ \\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{(X,\models ,A) & &S(H((X, \models ,A))) \\
& & S(B) \\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\eta$} (m-1-3)
(m-1-1) edge node[auto,swap] {$f(\equiv(f_1,f_2))$} (m-2-3)
(m-1-3) edge node[auto] {$S\hat{f}$} (m-2-3);
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &H((X,\models ,A)) \\
& &B \\ };
\path[->,font=\scriptsize]
(m-1-3) edge node[auto] {$\hat{f}(\equiv f_2)$} (m-2-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
Then unit is defined by $\eta =(p^*,id_A)$.\\
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{i.e.\ (X,\models ,A) & &S(H((X, \models ,A))) \\
};
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\eta$} (m-1-3)
(m-1-1) edge node[auto,swap] {$(p^*,id_A)$} (m-1-3)
;
\end{tikzpicture}
where $\\p^*:X\longrightarrow Hom(A,\{0,1\})$,
$x\longmapsto p_x:A\longrightarrow \{0,1\}$
such that $p_x(a)=1$ iff $x\models a$.
If possible let $p_x(\mathbf{0})=1$. Then we have $x\models 0$, which is a contradiction as $x\models 0$ for no $x\in X$. Hence $p_x(\mathbf{0})=0$. Also we have $p_x(a_1\wedge a_2)=1$ iff $x\models a_1\wedge a_2$ iff $x\models a_1$ and $x\models a_2$ iff $p_x(a_1)=1$ and $p_x(a_2)=1$ iff $p_x(a_1)\wedge p_x(a_2)=1$. Similarly it can be shown that $p_x(a_1\vee a_2)=p_x(a_1)\vee p_x(a_2)$ and $p_x(a_1\rightarrow a_2)=p_x(a_1)\rightarrow p_x(a_2)$. Hence for each $x\in X,$ $p_x:A\longrightarrow \{0,1\}$ is a Heyting homomorphism.
It may be observed that $x\models id_A(a)$ iff $x\models a$ iff $p_x(a)=1$ iff $(p^*(x))(a)=1$ iff $p^*(x)\models^* a$. Consequently we can conclude that $(p^*,id_A):(X,\models,A)\longrightarrow S(H((X,\models,A))) $ is a continuous map of Heyting algebraic I-topological system.
Let us define $\hat{f}$ as follows:
$(f_1,f_2):(X,\models ,A)\longrightarrow (Hom(B,\{0,1\}),\models_*,B)$
\\then $\hat{f}=f_2$.
Recall that $S(\hat{f})=(\_\circ f_2,f_2)$.
It suffices to show that the triangle on the left commute, i.e., $(f_1,f_2)=S(\hat{f})\circ \eta$. Now, $S(\hat{f})\circ \eta=(\_\circ f_2,f_2)\circ (p^*,id_A)=((\_\circ f_2)\circ p^*,id_A\circ f_2)=((\_\circ f_2)\circ p^*, f_2)$. For any $x\in X,$ $f_1(x)=(\_\circ f_2)\circ p^*(x)=(\_\circ f_2)\circ p_x=p_x\circ f_2$. Consequently, for all $b\in B$, $f_1(x)(b)=1$ iff $f_1(x)\models^* b$ iff $x\models f_2(b)$ iff $p_x(f_2(b))=1$ iff $(p_x\circ f_2)(b)=1$ iff $((\_\circ f_2)\circ p_x)(b)=1$ iff $((\_\circ f_2)\circ p^*)(x)(b)=1$. Therefore $f_1=(\_\circ f_2)\circ p^*$. Hence $\eta(\equiv(p^*,id_A)):(X,\models ,A)\longrightarrow S(H((X,\models ,A)))$ is the unit, consequently $H$ is the left adjoint to the functor $S$.
Diagram of the co-unit of the above adjunction is as follows.
\begin{center}
\begin{tabular}{ l | r }
$\mathbf{HA^{op}}$ & $\mathbf{HI-TopSys}$ \\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{H(S(A)) & & A \\
H((Y,\models ,B)) \\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\xi(\equiv id_A)$} (m-1-3)
(m-2-1) edge node[auto] {$f$} (m-1-1)
(m-2-1) edge node[auto,swap] {$H\hat{f}$} (m-1-3);
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &S(A) \\
& &(Y,\models,B) \\ };
\path[->,font=\scriptsize]
(m-2-3) edge node[auto,swap] {$\hat{f}(\equiv\_\circ f)$} (m-1-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
From the construction it can be easily shown that $\xi$ and $\eta$ are natural isomorphism and hence the theorem holds.
\end{proof}
\begin{corollary}\label{CA}
There exist adjoint functors between $\mathbf{HA^{op}}$ and $\mathbf{I-TopSys}$.
\end{corollary}
\begin{theorem}\label{ED}
There exist adjoint functors between $\mathbf{ESA}$ and $\mathbf{HA^{op}}$.
\end{theorem}
\begin{proof}
Follows from Theorem \ref{ESD}.
\end{proof}
\begin{theorem}
There exist adjoint functors between $\mathbf{ESA}$ and $\mathbf{I-Top Sys}$.
\end{theorem}
\begin{proof}
Follows from Corollary \ref{CA} and Theorem \ref{ED}.
\end{proof}
\begin{theorem}
Category $\mathbf{HI-TopSys}$ is equivalent to $\mathbf{ESA}$.
\end{theorem}
\begin{proof}
Follows from Theorem \ref{theorem1} and Theorem \ref{ESD}.
\end{proof}
\begin{theorem}\label{GIGA}
$\mathbf{GI-TopSys}$ is dually equivalent to $\mathbf{GA}$.
\end{theorem}
\begin{proof}
First we will prove that $\mathscr{G}$ is the left adjoint to the functor $\mathscr{S}$ by presenting the unit of the adjunction.
Recall that $\mathscr{S}(A)=(Hom(A,\{0,1\}),\models^*,A)$ where $v\models^* a$ iff $v(a)=1$ and $\mathscr{G}((X,\models,A))=A$.
Hence $\mathscr{S}(\mathscr{G}((X,\models, A)))=(Hom(A,\{0,1\}),\models^*, A)$.
\begin{center}
\begin{tabular}{ l | r }
$\mathbf{GI-TopSys}$ & $\mathbf{GA^{op}}$ \\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{(X,\models ,A) & &\mathscr{S}(\mathscr{G}((X, \models ,A))) \\
& & \mathscr{S}(B) \\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\eta$} (m-1-3)
(m-1-1) edge node[auto,swap] {$f(\equiv(f_1,f_2))$} (m-2-3)
(m-1-3) edge node[auto] {$\mathscr{S}\hat{f}$} (m-2-3);
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &\mathscr{G}((X,\models ,A)) \\
& &B \\ };
\path[->,font=\scriptsize]
(m-1-3) edge node[auto] {$\hat{f}(\equiv f_2)$} (m-2-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
Then unit is defined by $\eta =(p^*,id_A)$.\\
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{i.e.\ (X,\models ,A) & &\mathscr{S}(\mathscr{G}((X, \models ,A))) \\
};
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\eta$} (m-1-3)
(m-1-1) edge node[auto,swap] {$(p^*,id_A)$} (m-1-3)
;
\end{tikzpicture}
where $\\p^*:X\longrightarrow Hom(A,\{0,1\})$,
$x\longmapsto p_x:A\longrightarrow \{0,1\}$
such that $p_x(a)=1$ iff $x\models a$.
If possible let $p_x(\mathbf{0})=1$. Then we have $x\models 0$, which is a contradiction as $x\models 0$ for no $x\in X$. Hence $p_x(\mathbf{0})=0$. Also we have $p_x(a_1\wedge a_2)=1$ iff $x\models a_1\wedge a_2$ iff $x\models a_1$ and $x\models a_2$ iff $p_x(a_1)=1$ and $p_x(a_2)=1$ iff $p_x(a_1)\wedge p_x(a_2)=1$. Similarly it can be shown that $p_x(a_1\vee a_2)=p_x(a_1)\vee p_x(a_2)$ and $p_x(a_1\rightarrow a_2)=p_x(a_1)\rightarrow p_x(a_2)$. Hence for each $x\in X,$ $p_x:A\longrightarrow \{0,1\}$ is a Heyting homomorphism.
It may be observed that $x\models id_A(a)$ iff $x\models a$ iff $p_x(a)=1$ iff $(p^*(x))(a)=1$ iff $p^*(x)\models^* a$. Consequently we can conclude that $(p^*,id_A):(X,\models,A)\longrightarrow \mathscr{S}(\mathscr{G}((X,\models,A))) $ is a continuous map of G$\ddot{o}$del algebraic I-topological system.
Let us define $\hat{f}$ as follows:
$(f_1,f_2):(X,\models ,A)\longrightarrow (Hom(B,\{0,1\}),\models_*,B)$
\\then $\hat{f}=f_2$.
Recall that $\mathscr{S}(\hat{f})=(\_\circ f_2,f_2)$.
It suffices to show that the triangle on the left commute, i.e., $(f_1,f_2)=\mathscr{S}(\hat{f})\circ \eta$. Now, $\mathscr{S}(\hat{f})\circ \eta=(\_\circ f_2,f_2)\circ (p^*,id_A)=((\_\circ f_2)\circ p^*,id_A\circ f_2)=((\_\circ f_2)\circ p^*, f_2)$. For any $x\in X,$ $f_1(x)=(\_\circ f_2)\circ p^*(x)=(\_\circ f_2)\circ p_x=p_x\circ f_2$. Consequently, for all $b\in B$, $f_1(x)(b)=1$ iff $f_1(x)\models^* b$ iff $x\models f_2(b)$ iff $p_x(f_2(b))=1$ iff $(p_x\circ f_2)(b)=1$ iff $((\_\circ f_2)\circ p_x)(b)=1$ iff $((\_\circ f_2)\circ p^*)(x)(b)=1$. Therefore $f_1=(\_\circ f_2)\circ p^*$. Hence $\eta(\equiv(p^*,id_A)):(X,\models ,A)\longrightarrow \mathscr{S}(\mathscr{G}((X,\models ,A)))$ is the unit, consequently $\mathscr{G}$ is the left adjoint to the functor $\mathscr{S}$.
Diagram of the co-unit of the above adjunction is as follows.
\begin{center}
\begin{tabular}{ l | r }
$\mathbf{GA^{op}}$ & $\mathbf{GI-TopSys}$ \\
\hline
{\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{\mathscr{G}(\mathscr{S}(A)) & & A \\
\mathscr{G}((Y,\models ,B)) \\ };
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\xi(\equiv id_A)$} (m-1-3)
(m-2-1) edge node[auto] {$f$} (m-1-1)
(m-2-1) edge node[auto,swap] {$\mathscr{G}\hat{f}$} (m-1-3);
\end{tikzpicture}} & {\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em]
{& &\mathscr{S}(A) \\
& &(Y,\models,B) \\ };
\path[->,font=\scriptsize]
(m-2-3) edge node[auto,swap] {$\hat{f}(\equiv\_\circ f)$} (m-1-3);
\end{tikzpicture}}\\
\end{tabular}
\end{center}
From the construction it can be easily shown that $\xi$ and $\eta$ are natural isomorphism and hence the theorem holds.
\end{proof}
\begin{theorem}\cite{CH}\label{GEF}
$\mathbf{GA}$ is dually equivalent with category of Esakia spaces whose order structure is a forest and Esakia morphisms ($\mathbf{FESA}$).
\end{theorem}
From Theorem \ref{GIGA} and Theorem \ref{GEF} the following theorem holds.
\begin{theorem}
$\mathbf{GI-TopSys}$ is equivalent to $\mathbf{FESA}$.
\end{theorem}
We can summarize our results by the following diagram.
\def\firstcircle{(3,8cm) circle (1.2cm)}
\def\secondcircle{(4,8.65) circle (1.5cm)}
\def\firscircle{(0,4cm) circle (1.2cm)}
\def\seconcircle{(0,4.65) circle (1.5cm)}
\def\fircircle{(6,4cm) circle (1.2cm)}
\def\secocircle{(8,4.65) circle (1.5cm)}
\begin{center}
\begin{tikzpicture}
\centering
\draw \firstcircle node[name=A] {$\mathbf{GI-TopSys}$};
\draw \firscircle node[name=C] {$\mathbf{FESA}$};
\draw \fircircle node[name=E] {$\mathbf{GA^{op}}$};
\node[draw, thick, rounded corners, inner xsep=0.3em, inner ysep=2.9em, fit=(A)] (B) {};
\node[draw, thick, rounded corners, inner xsep=1.8em, inner ysep=3.0em, fit=(C)] (D) {};
\node[draw, thick, rounded corners, inner xsep=1.8em, inner ysep=3.0em, fit=(E)] (F) {};
\node (GFTS) at (3,9.6) {$\mathbf{HI-TopSys}$};
\node (GFT) at (0,2.4) {$\mathbf{ESA}$};
\node (GFr) at (6,2.4) {$\mathbf{HA^{op}}$};
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\end{center}
\section{Conclusion}
This paper suggest a new approach (new view) of representation of Heyting algebra as I-topological system. Moreover, relationship between the I-topological system and Esakia space and its particular case G\"{o}del space are shown. Connection of Kripke model with proposed system is indicated. It is expected that the proposed notion will play vital roles in the field of computer science and physics.
\subsection*{Acknowledgement}
The research work of the third author was supported by Dipartimento di Matematica, Universit\'a degli Studi di Salerno, the \emph{Indo-European Research Training Network in Logic} (IERTNiL) funded by the \emph{Institute of Mathematical Sciences, Chennai}, the \emph{Institute for Logic, Language and Computation} of the \emph{Universiteit van Amsterdam} and the \emph{Fakult\"at f\"ur Mathematik, Informatik und Naturwissenschaften} of the \emph{Universit\"at Hamburg} and Department of Science $\&$ Technology, Government of India under Women Scientist Scheme (reference no. SR/WOS-A/PM-52/2018). Part of this work was done when the second and the third author was visiting Dipartimento di Matematica, Universit\`a degli Studi di Salerno and they are thankful to the department for the warm hospitality.
| 169,577
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TITLE: Question regarding the dimension of a vector space of all functions from a finite set .
QUESTION [0 upvotes]: $\mathbf {The \ Problem \ is}:$ Let, $E=\{1,2,3,....,n\}$ where $n$ is odd positive integer. Let, $V$ be the vector space (over $\mathbb R$) of the set of all possible functions from $E$ to $\mathbb R^3$ under the usual addition and multiplication of functions. Find $\operatorname{dim V} .$
$\mathbf {My \ approach} :$ Actually, I couldn't try much of the problem but if $n=1$, then obviously, dimension is $3.$
Now, I am thinking about a spanning set to span $V$ and I think $\operatorname{dim V} \leq 3n$ but I am unable to show it .
REPLY [0 votes]: So, you’ve figured out that when $n=1$, your vector space has as basis the functions
$$\begin{align*}
f_1\colon\{1\}\to\mathbb{R}^3,&\qquad 1\longmapsto (1,0,0),\\
f_2\colon\{1\}\to\mathbb{R}^3,&\qquad 1\longmapsto (0,1,0),\\
f_3\colon\{1\}\to\mathbb{R}^3,&\qquad 1\longmapsto (0,0,1).
\end{align*}$$
Okay, what happens if $n=3$? Well, we could extend these three functions by making them send $2$ and $3$ to $(0,0,0)$. Will they stay linearly independent if we do that? If so, can we do something similar with functions that send $1$ and $3$ to $(0,0,0)$, and do something else to $2$? Or send $1$ and $2$ to $(0,0,0)$ and map $3$ to something else? Will the result be linearly independent? Will it span?
| 189,187
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\begin{document}
\begin{abstract}
We show that every non-trivial compact connected group and every non-trivial general or special linear group over an infinite field admits a generating set such that the associated Cayley graph has infinite diameter.
\end{abstract}
\maketitle
A group $G$ is said to have the \emph{Bergman property} if for any symmetric generating subset $S\subseteq G$ containing the identity of $G$ there exists $k\in\nats$ such that $S^{\ast k}=G$, where $S^{\ast k}\defeq\set{s_1\cdots s_k}[s_1,\ldots,s_k\in S]$. The smallest such $k$ is called the \emph{width} of $S$ or the \emph{diameter} of the associated Cayley graph $\Cay(G,S)$. If there is no such $k$, $S$ is said to have infinite width.
Examples of such groups include finite groups, where the study of worst case diameters has a long history, but also infinite groups such as $\Sym(\nats)$ (a result due to Bergman \cite{bergman2006generating}). The first example of an infinite group with uniformly bounded width with respect to any generating set has been constructed by Shelah \cite[Theorem~2.1]{shelah1980problem}.
In \cite{dowerk2018bergman} Dowerk shows that unitary groups of \emph{infinite-dimensional} von Neumann algebras admit \emph{strong uncountable cofinality}.
A group $G$ admits this property if for any exhausting chain $W_0\subseteq W_1\subseteq\cdots\subseteq G=\bigcup_{n=0}^\infty{W_n}$ of subsets there exist $n,k\in\nats$ such that $W_n^{\ast k}=G$ (see \cite[Definition~1.1]{rosendal2009topological}). It is apparent that, setting $W_n\defeq S^{\ast n}$, where $S$ is any symmetric generating set containing the identity, strong uncountable cofinality implies the Bergman property. Indeed, for uncountable groups it is actually equivalent to admitting both the Bergman property and \emph{uncountable cofinality} (see \cite[Proposition 2.2]{drostegoebel2005uncountable} or \cite{drosteholland2005generating}).
Regarding his result, Dowerk asked whether unitary groups of \emph{finite-dimensional} von Neumann algebras have the Bergman property.
In this note we answer this question in the negative by showing that any non-trivial compact connected group fails to have this property. We also show this for groups of type $\GL_n(K)$ or $\SL_n(K)\neq\trivgrp$ ($n\geq 1$), where $K$ is an infinite field.
Note that all of the above groups also fail to have uncountable cofinality. This is shown for $\SL_n(K)$ over an uncountable field in \cite[Proposition]{thomaszapletal2012steinhaus} and can easily be extended to all groups admitting a finite-dimensional representation with uncountable image. Such groups encompass compact groups by the Peter--Weyl theorem. This was pointed out by Cornulier in private communication.
Our main result is the following.
\begin{theorem}\label{thm:cpt_cn_grps} The following are true
\begin{enumerate}[(i)]
\item Any non-trivial compact connected group does not admit the Bergman property.
\item Any group of type $\GL_n(K)$ ($n\geq 1$) or $\SL_n(K)$ ($n\geq 2$), for an infinite field $K$, does not admit the Bergman property.
\end{enumerate}
\end{theorem}
The proofs use a few non-trivial facts from the theory of fields, for which we refer the reader to \cite{lang2002algebra}. For facts like the existence of a transcendence basis of $\reals$ over $\rats$ we need to assume the Axiom of Choice.
\section{Proof of Theorem~\ref{thm:cpt_cn_grps}}
This section is devoted to the proof of Theorem~\ref{thm:cpt_cn_grps}.
\begin{proof}[Proof of Theorem~\ref{thm:cpt_cn_grps}(\textnormal{i}):]
Let $\abs{\bullet}\colon\complex\to\reals_{\geq 0}$ be a \emph{norm} such that the following hold
\begin{enumerate}[(i)]
\item $\abs{x}=0$ if and only if $x=0$ (identity of indiscernibles);
\item $\abs{a+b}\leq\abs{a}+\abs{b}$ (subadditive);
\item $\abs{ab}\leq\abs{a}\abs{b}$ (submultiplicative);
\item $\abs{x^2}=\abs{x}^2$ (compatible with squaring);
\item there exists $x\in[0,1]$ such that $\abs{x}>1$ (small elements with large norm).
\end{enumerate}
Examples of such norms are presented in Lemma~\ref{lem:cnst_nrms} below.
We now consider the matrix group $\SO(2,\reals)$. Set $C\defeq 4\abs{1/2}$. Then $C\geq 2\abs{1}=2$ by (ii) (as $\abs{1}=1$; see below). Define $S$ to be the set of elements of $\SO(2,\reals)$ with coefficients in $B\defeq\set{x\in\reals}[\abs{x}\leq C]$. Observe that $1\in S$, as (i) and (iv) imply that $\abs{1}=1$, and $S=S^{-1}$, as the inverse of $g\in \SO(2,\reals)$ is just $g^\top$. We claim that $S$ is a generating set and that $\SO(2,\reals)$ has infinite diameter with respect to $S$.
Assume that $g=(\begin{smallmatrix} a&b\\ -b&a \end{smallmatrix})\in \SO(2,\reals)$ corresponds to an element $z\in \U(1)\subseteq\complex$ via the isomorphism $(\begin{smallmatrix} a&b\\ -b&a \end{smallmatrix}) \mapsto a+bi$.
Take $k\in\nats$ large enough such that
$$
\abs{z^{1/2^k}},\abs{z^{-1/2^k}}\leq 2.$$
This is possible by (iv).
Set $a'\defeq\Re(z^{1/2^k})$ and $b'\defeq\Im(z^{1/2^k})$. By property (iv) we have $\abs{i}=1$ as $\abs{i^4}=\abs{i}^4=\abs{1}=1$. From (ii), (iii) we derive that
$$
\abs{a'}=\lrabs{\frac{1}{2}(z^{1/2^k}+z^{-1/2^k})}\leq\lrabs{\frac{1}{2}}(\abs{z^{1/2^k}}+\abs{z^{-1/2^k}})\leq C.
$$
Similarly, using that $\abs{i}=1$, we obtain
$$
\abs{b'}=\lrabs{\frac{1}{2i}(z^{1/2^k}-z^{-1/2^k})}\leq\lrabs{\frac{1}{2}}(\abs{z^{1/2^k}}+\abs{z^{-1/2^k}})\leq C.
$$
Since $(a'+b'i)^{2^k}=z$, we conclude that $g\in S^{\ast2^k}$. Since $g$ was arbitrary, $S$ follows to be a generating set of $\SO(2,\reals)$.
It remains to show that $\SO(2,\reals)$ does not have finite diameter with respect to $S$. Indeed, for any $R\in\reals_{\geq 0}$ on finds $k$ large enough such that, if $x$ is an element as in (v), $\abs{x^{2^k}}=\abs{x}^{2^k}\geq R$. Now putting $a\defeq x^{2^n}$ and $b\defeq\sqrt{1-a^2}$, we obtain an element $g=a+bi\in\U(1)\cong\SO(2,\reals)$ which needs arbitrarily many factors in $S$ to be represented (taking $R$ large enough). Indeed, by induction, all coordinate entries of an element in $S^{\ast k}$ have norm at most $2^{k-1}C^k$ ($k\in\ints_+$).
Now let $H$ be a non-trivial compact connected Lie group represented as a matrix subgroup of $\O(n)$ for some $n\in\nats$. We observe that finitely many copies $H_i$ ($i=1,\ldots,m$) of $\SO(2,\reals)$ in $H$ generate $H$ as a group, e.g., see \cite[Theorem~2]{dalessandro2002uniform}.
This allows us to extend our argument for $\SO(2,\reals)$ above to the group $H$ as follows.
Assume that $H$ is generated by
$$
H_i= \lrset{g_i^{-1}\left(\begin{pmatrix} a & b\\-b&a \end{pmatrix}\oplus\id_{n-2}\right) g_i}[a,b\in\reals,\ a^2+b^2=1]
$$
for some $g_i\in\O(n)$ ($i=1,\ldots,m$). After conjugating by $g_1^{-1}$ we may assume that $g_1=1$.
As a generating set $S$ for $H$ it now suffices to take the set of elements in $H_i$ ($i=1,\ldots,m$) with coefficients $a,b\in B$, where $B$ is defined as above. Apparently, $S$ generates $H_i$ ($i=1,\ldots,m$) and hence $H$.
Let $D$ be the maximum of the norms of the matrix entries of the $g_i$ ($i=1,\ldots,m$; and so it is the maximum of the norms of the entries of the $g_i^{-1}=g_i^\top$). Then any coordinate entry of an element in $S\cap H_i$ has norm bounded by $c\defeq n^2D^2C$. Thus, by induction, for $g\in S^{\ast k}$, all coordinate entries of $g$ have norm at most $n^{k-1}c^k$ ($k\in\ints_+$). Therefore, $S$ cannot generate $H$ in finitely many steps, since, as above, the coordinate entries of $H_1\cong\SO(2,\reals)$ are unbounded with respect to the norm.
If $H$ is now any non-trivial compact connected group, then by the Peter--Weyl theorem $H$ has a non-trivial finite-dimensional unitary representation, say $\pi \colon H\rightarrow \U(n) \subseteq\O(2n)$. Let $S$ be a generating set in $\O(2n)$ witnessing that $\pi(H)$ does not have the Bergman property. Then $\pi^{-1}(S)$ is a generating set witnessing that $H$ does not have the Bergman property.
\end{proof}
It remains to construct norms satisfying (i)--(v). This is done in the subsequent lemma.
\begin{lemma}\label{lem:cnst_nrms}
Let $T$ be a transcendence basis of $\reals$ over $\rats$. Consider $K\defeq\rats(T)$ and let $G$ denote the Galois group of the field extension $\complex/K$. The following norms $\abs{\bullet}\colon\complex\to\reals_{\geq 0}$ satisfy properties (i)--(v) from the proof Theorem~\ref{thm:cpt_cn_grps} (ii).
\begin{enumerate}[(i)]
\item Since, $K\subseteq\complex$ is algebraic, $G$ is a profinite group and each element in $\complex$ has a finite orbit under the action of $G$.
For $x\in \complex$ we define the \emph{Galois radius} of $x$ to be
$\rho(x)\defeq\max_{\sigma\in G}\abs{x^\sigma}$. Then $\abs{\bullet}\defeq \rho$ defines a norm with the desired properties.
\item Choose $t\in T$. Set $L\defeq \rats(T\setminus\set{t})$. Let $\nu\colon K\to\ints$ be the degree valuation corresponding to $t$ on $K$. Extend $\nu$ to a valuation $\omega\colon\complex\to\rats$ (by using Zorn's lemma). Setting $\abs{x}\defeq\exp(-\omega(x))$ gives a norm on $\complex$ satisfying the above properties.
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:cnst_nrms}]
In both cases, we verify properties (i)--(v).
\emph{(i):} Properties (i)--(iv) are immediate from the definition.
We now show that also property (v) is fulfilled. Choose arbitrary real numbers $a\in(0,1)$ and $b>1$. Set $p(X)\defeq (X-a)(X-b)\in\reals[X]$. We claim that by density of $K\supseteq\rats$ in $\reals$ we can find an irreducible polynomial over $K$ with coefficients arbitrarily close to those of $p(X)$. Indeed, by Gauss' lemma, an irreducible polynomial over $\rats$ remains irreducible over $K$ and hence, it suffices to approximate by irreducible rational polynomials. Using Eisenstein's criterion, we can find an irreducible monic rational polynomial $q(X)$ which has coefficients arbitrarily close to the coefficients of $p(X)$. Indeed, Eisenstein's criterion implies that for $\alpha,\beta,\gamma\in\mathbb Z$, $\gamma>0$, the polynomial $q(X)=X^2+(\alpha/\gamma)X+(\beta/\gamma)$ is irreducible if $p^2$ does not divide $\beta$, $p$ divides $\alpha$ and $\beta$ and does not divide $\gamma$. Choosing $\gamma$ large enough and coprime to $p$, we can easily find $\alpha$ and $\beta$ with the desired properties such that $\alpha/\gamma$ is close to $-(a+b)$ and $\beta/\gamma$ is close to $ab$.
By the implicit function theorem, the zeroes of $q$, say $x$ and $y$, are arbitrarily close to $a$ and $b$, respectively. Hence $\rho(x)$ is close to $b>1$, as desired.
\emph{(ii):} Properties (i)--(iv) follow from the definition of a valuation. For property (v) observe that $\abs{t^{-1}}=\exp(\nu(t^{-1}))=e>1$. Also, for $y\in\rats$ we have $\abs{y}=\exp(\nu(y))=1$. Taking $y$ so that $x\defeq t^{-1}y\in[0,1]$, we obtain that for $\abs{x}=\abs{y}\abs{t^{-1}}=\abs{t^{-1}}=e>1$, as wished.
\end{proof}
\begin{remark}
In the proof of Theorem~1(i) and Lemma~\ref{lem:cnst_nrms} the field $\reals$ can be replaced by a Euclidean field $R$ and $\complex$ by $R[i]$. Then, if $T=\emptyset$ in Lemma~\ref{lem:cnst_nrms}(ii) we need to take a $p$-adic valuation, instead of the degree valuation, on $K=\rats$ and extend it. Then, for $y$, in the above argument, we have to take a suitable element $r/s\in\rats$ such that $p\nmid r,s$.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:cpt_cn_grps}\textnormal{(ii)}]
At first we consider the case $G=\SL_n(K)$. We distinguish two cases.
\emph{Case~1:} Assume first that a transcendence basis $T$ of $K$ over its prime field $k$ is not empty. Define for $\lambda\in K^\times$, $\mu\in K$ the matrices
$$
D(\lambda)\defeq\begin{pmatrix}
\lambda^{-1} & 0\\
0 & \lambda
\end{pmatrix}
\text{ and }
E_{12}(\mu)\defeq\begin{pmatrix}
1 & \mu\\
0 & 1
\end{pmatrix}.
$$
Note that
\begin{equation}\label{eq:el_mats_id}
E_{12}(\mu)^{D(\lambda_1)\cdots D(\lambda_n)}=E_{12}(\mu)^{D(\lambda_1\cdots\lambda_n)}=E_{12}((\lambda_1\cdots\lambda_n)^2\mu)
\end{equation}
for $\lambda_1,\ldots,\lambda_n\in K^\times$, $\mu\in K$. Choose $t\in T$. For $L\defeq k(T\setminus\set{t})$ we have that $K$ is algebraic over $L(t)$.
Take the normed degree valuation on $\nu\colon L(t)\to\ints$ and extend it to a valuation $\omega\colon K\to\rats$. Define $B\defeq\set{b\in K}[\abs{\omega(b)}\leq 1]$.
Set
$$
S\defeq\set{g\in\SL_n(K)}[\text{all entries of }g\text{ are in }B].
$$
Then $S^{\ast k}\neq\SL_n(K)$, since one sees easily by induction that for $g\in S^{\ast k}$ $\abs{\omega(g_{ij})}\leq k$ (from the strong triangle inequality). We show that $S$ generates all elementary matrices, and hence $\SL_n(K)$.
Indeed, we may assume that $n=2$, via the embeddings $\SL_2(K)\hookrightarrow\SL_n(K)$. We show that $S$ generates any matrix $M\defeq E_{12}(\alpha)$ for $\alpha\in K$. Indeed, if $\alpha=0$, then $M\in S$. In the opposite case $v\defeq\omega(\alpha)\in\rats$ and we find an integer $n\in\ints$ such that $\abs{v-2n}\leq 1$.
Then choose $\lambda\in K$ such that $\omega(\lambda)=1$ (which exists even in $L(t)\subseteq K$, since $\nu$ was normed) and set $\mu\defeq\alpha\lambda^{-2n}$. By construction, we have $\abs{\omega(\mu)}\leq 1$, so that $D(\lambda),E_{12}(\mu)\in S$ and $M=E_{12}(\alpha)=E_{12}(\mu)^{D(\lambda)^{2n}}\in\gensubgrp{S}$ by Equation~\eqref{eq:el_mats_id}. This completes the proof of Case~1.
\emph{Case~2:} In this case $K$ is an algebraic extension of its prime field $k=\rats$ or $k=\finfield_p$. In the first case, $K$ embeds into $\complex$ and we can define
$$
S\defeq\set{g\in\SL_n(K)}[\text{all entries of }g\text{ are in }B],
$$
where $B$ is the unit ball of $\complex$ intersected with $K\subseteq\complex$.
Hence we assume that $k=\finfield_p$ and $K$ is an infinite algebraic extension of $k$.
In this case, we construct a set $B\subseteq K$ with the following properties
\begin{enumerate}[(i)]
\item $\gensubgrp{B}_{+}=K$, i.e., $B$ generates $K$ as an abelian group;
\item $P(B)\defeq\set{p(b_1,\ldots,b_m)}[b_1,\ldots,b_m\in B, p\in P]\neq K$ for each finite set $P\subseteq\ints[X_1,\ldots,X_m]$ of polynomials over $\ints$ and $m\in\nats$.
\end{enumerate}
Indeed, if we have such a set $B$, we can use the same definition for $S$ as above. This is due to the fact that $E_{12}(\mu)E_{12}(\lambda)=E_{12}(\mu+\lambda)$ and the elementary matrices generate $\SL_n(K)$. On the other hand $S^{\ast k}\neq\SL_n(K)$ by condition (ii), as the matrix entries are bounded degree polynomials over $\ints$ in the entries of the matrices.
The set $B$ is now inductively constructed in Lemma~\ref{lem:comb_finflds} and Corollary~\ref{cor:cnst_B} below.
In the case $G=\GL_n(K)$ we add matrices $\diag(\lambda,1,\ldots,1)$ to the generating set $S$ with $\abs{\omega(\lambda)}\leq 1$.
\end{proof}
\begin{lemma}\label{lem:comb_finflds}
Fix a set $P\subseteq\finfield_p[X_1,\ldots,X_m]$ of non-constant polynomials of total degree at most $n$, i.e., especially $P$ is finite.
Consider the inclusion of finite fields $\finfield_{p^e}\subseteq\finfield_{p^{ef}}$ for $e,f\in\ints_+$ and let $E\subseteq\finfield_{p^e}$ be a subset of cardinality $e$. Define $P_E\subseteq\finfield_{p^e}[X_1,\ldots,X_m]$ as the set of non-constant polynomials which arise from the polynomials from $P$ by substituting a subset of the variables $X_1,\ldots,X_m$ by elements from $E$. Then for $e$ sufficiently large there exists a set $F\subseteq\finfield_{p^{ef}}$ such that $\gensubsp{F}_{\finfield_p}$ is a complement to $\finfield_{p^e}$ in $\finfield_{p^{ef}}$ as $\finfield_p$-vector spaces and $r(f_1,\ldots,f_m)\not\in\finfield_{p^e}$ for all $f_1,\ldots,f_m\in F$ and $r\in P_E$.
\end{lemma}
\begin{proof}
At first note that the set $C$ of $e(f-1)$-tuples with entries in $\finfield_{p^{ef}}$ which span an $\finfield_p$-complement of $\finfield_{p^e}$ in $\finfield_{p^{ef}}$ is of cardinality
$(p^{ef}-p^e)\cdots(p^{ef}-p^{ef-1})$, so its portion in the set $T$ of all $e(f-1)$-tuples with entries in $\finfield_{p^{ef}}$ is equal to
$$
\card{C}/p^{e^2f(f-1)}=(1-p^{-e(f-1)})\cdots(1-p^{-1})
$$
but for $c\defeq 2\log(2)$ we have $e^{-cx}\leq 1-x$ for $0\leq x\leq 1/2$, so that the above expression is bounded from below by
$$
e^{-c\sum_{i=1}^{e(f-1)}{p^{-i}}}\geq d\defeq e^{-\frac{c}{p-1}}\in (0,1).
$$
Now let us estimate how many tuples $t=(t_1,\ldots,t_{e(f-1)})\in T$ have the property that $r(s_1,\ldots,s_m)\not\in\finfield_{p^e}$ for all $s_1,\ldots,s_m\in\overline{t}\defeq\set{t_1,\ldots,t_{e(f-1)}}$ and $r\in P_E$. By the Schwartz-Zippel lemma, for all $x\in\finfield_{p^e}$ we have
$$
\Prob_{s_1,\ldots,s_n\in\finfield_{p^{ef}}}[r(s_1,\ldots,s_m)=x]\leq n/p^{ef},
$$
so that
$$
\Prob_{s_1,\ldots,s_n\in\finfield_{p^{ef}}}[r(s_1,\ldots,s_m)\in\finfield_{p^e}]\leq n/p^{e(f-1)},
$$
and hence
$$
\Prob_{t\in T}[\exists r\in P_E,s_1,\ldots,s_m\in\overline{t}:r(s_1,\ldots,s_m)\in\finfield_{p^e}]\leq \card{P_E}(e(f-1))^m n/p^{e(f-1)}.
$$
Putting both estimates together, we obtain that the set of $t\in T$ such that $F\defeq\overline{t}$ satisfies the hypothesis of the lemma has portion bounded from below by $d-\card{P_E}(e(f-1))^m n/p^{e(f-1)}\geq d-(e+1)^m\card{P}(e(f-1))^m n/p^{e(f-1)}$ in $T$. But this term is clearly positive for all $f>1$ when $e$ is large enough.
\end{proof}
\begin{corollary}\label{cor:cnst_B}
Let $K\subseteq\overline{\finfield}_p$ be infinite. There exists a set $B\subseteq K$ satisfying (i) and (ii) in the proof of Theorem~\ref{thm:cpt_cn_grps}(ii).
\end{corollary}
\begin{proof}
Set $P_i\defeq\finfield_p[X_1,\ldots,X_i]_{\deg\leq i}$ for $i\in\nats$, so that the $P_i$ exhaust the polynomial ring over $\finfield_p$ in the countably many variables $X_i$ ($i\in\ints_+$).
Apply Lemma~\ref{lem:comb_finflds} to $P=P_0$, and choose $b_0\defeq e$ large enough and an $\finfield_p$-basis $B_0\defeq E$ of $\finfield_{p^{b_0}}$ such that for all $f>1$ and an appropriate extension $F\subseteq\finfield_{p^{b_0f}}$ to a basis $B_1\defeq B_0\cup F$ of this field, we have $P_{0,B_0}(F)\not\in\finfield_{p^{b_0}}$. Then, again using Lemma~\ref{lem:comb_finflds}, choose $f>1$ large enough, set $b_1\defeq b_0f$ and $B_1= B_0\cup F$ such that the same as above holds, replacing $P_0$ by $P_1$ and $b_0$ by $b_1$.
Proceed by induction to get $B\defeq\bigcup_{i=0}^\infty{B_i}$.
Apparently, $\gensubsp{B}_+=K$. Assume now that for $P\subseteq\finfield_p[X_1,X_2,\ldots]$ finite, we have $P(B)=K$.
Then choose $m$ large enough such that $P\subseteq P_m$.
Now note that
$$
P_m(B)=P_m(B_m)\cup\bigcup_{i=m}^\infty{P_{m,B_i}(B_{i+1}\setminus B_i)}.
$$
But $P_m(B_m)\subseteq\finfield_{p^{b_m}}$ and $P_{m,B_i}(B_{i+1}\setminus B_i)\subseteq P_{i,B_i}(B_{i+1}\setminus B_i)\subseteq\finfield_{p^{b_{i+1}}}\setminus\finfield_{p^{b_i}}$ for $i\geq m$ by construction.
Hence for $i\geq m$ we have $P_m(B)\cap\finfield_{p^{b_i}}=P_m(B_i)$. But the size of this set is bounded by $\card{P_m}\card{B_i}^m=\card{P_m}b_i^m$, whereas $\card{\finfield_{p^{b_i}}}=p^{b_i}$, which is eventually larger than the first.
This shows that $P(B)\subseteq P_m(B)\neq K$, as desired.
\end{proof}
We end up with a question: Does there exist a countably infinite group admitting the Bergman property?
\section*{Acknowledgments}
The author wants to thank Philip Dowerk and Andreas Thom for interesting discussions and Yves Cornulier for a comment concerning a first version of this article.
This research was supported by ERC Consolidator Grant No.\ 681207.
\end{document}
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TITLE: Rational points of non-rational curves
QUESTION [1 upvotes]: An algebraic curve (in this question) is the zero set $C = f^{-1}(X\ Y)$ of any polynomial $f\in\mathbb R[X\ Y]$; we say then that $f$ represents $C$. An algebraic curve $C$ is non-rational $\ \Leftrightarrow\ $ there does not exist any polynomial $f\in \mathbb Q[X\ Y]$ which represents $C$. An algebraic curve $C$ is irreducible $\ \Leftrightarrow\ $ it is not a union of any two curves, different from $C$. The following problem is open to me:
Question: does there exist a non-rational irreducible curve which contains infinitely many rational points (i.e. when $C\cap\mathbb Q^2$ is infinite)?
COMMENT: Curve $(X-\frac 1{\sqrt 2})^2 + (Y-\frac 1{\sqrt 2})^2 = 1$ has exactly one rational point. This promises a taste for trying the rational points of non-rational curves, and of the geometric-combinatorial considerations related to them (other fields and dimensions are possible too).
REPLY [5 votes]: Using a basis for $\mathbb R$ over $\mathbb Q$ (or just the sub-vector space generated by the coefficients of $f$), we can write the equation for a point being on $C$, $f(x,y)=0$, as a set of equations with rational coefficients - we view $f(x,y)$ as a vector, and set the entries to $0$ indepedently.
This gives a list of rational polynomial equations that must be satisfied by all the rational points. If they have no common factor, they have finitely many common roots, by Bezout's theorem. If they do have a common factor, it is also a factor of $f$, so $C$ is either reducible or rational.
| 66,304
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