text stringlengths 4 2.78M |
|---|
---
abstract: 'We present a collection of results concerning the structure of reversible gate classes over non-binary alphabets, including (1) a reversible gate class over non-binary alphabets that is not finitely generated (2) an explicit set of generators for the class of all gates, the class of all conservative gates, and a class of generalizations of the two (3) an embedding of the poset of reversible gate classes over an alphabet of size $k$ into that of an alphabet of size $k+1$ (4) a classification of gate classes containing the class of $(k-1,1)$-conservative gates, meaning gates that preserve the number of occurrences of a certain element in the alphabet.'
author:
- 'Yuzhou Gu[^1]'
bibliography:
- 'ref.bib'
title: 'Some Results on Reversible Gate Classes Over Non-Binary Alphabets'
---
Introduction
============
The “pervasiveness of universality”, described in Aaronson et al. [@AGS15], is the phenomenon that in a class of operations, a small number of simple operations is likely to generate all operations. This phenomenon is central in the theory of computation. Yet in some cases, systems fail to be universal, and different ways of failure reveal a rich structure. An example is Post’s lattice [@Pos41], one of the most remarkable results in the early age of computer science, which is a complete description of all ways in which a set of classical gates over the binary alphabet can fail to be universal.
Inspired by Post’s lattice, Aaronson et al. [@AGS15] proposed the problem of classifying all quantum gate classes. However, quantum gate classes turn out to be much more complicated than classical gates classes. Therefore, instead of studying quantum gates directly, they studied reversible gates, which are in some sense the correct classical analogue of quantum gates. In op. cit., Aaronson et al. gave a complete classification of reversible gate classes over the binary alphabet. They also pointed out several directions for further research. One direction is to classify stabilizer operations over qubits, which has been completed recently by Grier and Schaeffer [@GS16]. The current paper, taking another direction, studies the behavior of reversible gate classes when the alphabet is not binary.
Our results
-----------
Our results include the following:
1. A reversible gate class over non-binary alphabets that is not generated by a finite subset of reversible gates (Theorem \[NonFinGenClass\]). This answers a question of Aaronson et al. [@AGS15].
2. An explicit set of generators for the class of all gates, the class of all conservative gates (Theorem \[AllGenerate\] and Theorem \[ConsGen\]). We define generalizations of these two classes (Definition \[LmCons\]), and give an explicit set of generators for them (Theorem \[LmConsGeneration\]).
3. An embedding of the poset of reversible gate classes over an alphabet of size $k$ into the poset of reversible gate classes over an alphabet of size $k+1$ (Theorem \[PosetEmbedding\]).
4. A classification of reversible gate classes containing the class of reversible $(k-1,1)$-conservative gates (Theorem \[CONSInteger\]).
These results show that there are differences and similarities between reversible gate classes over the binary alphabet and those over a non-binary alphabet. It is known that 1) and 4) do not hold in the binary case. Yet, 2) and 4) can be seen as a generalization of the corresponding results in the binary case.
Related work
------------
Aaronson et al. [@AGS15] completely classified reversible gate classes over the binary alphabet. Many of their methods for proving generation results apply to non-binary case, e.g. [@AGS15] Theorem 23, Theorem 25, Theorem 41. Je[ř]{}[á]{}bek [@Jer14] gave a reversible clone-coclone duality (called “master clone-coclone duality” in that work), generalizing the classical clone-coclone duality. Je[ř]{}[á]{}bek’s result works for non-binary alphabets.
In this paper reversible gate classes are closed under the ancilla rule by definition (Definition \[GenRules\]). In some previous work the ancilla rule is removed. Boykett [@Boy15] gave a finite set of generators with no ancillas for the class of all reversible gates when the alphabet size is odd. Boykett’s result implies Theorem \[AllGenerate\] when $|A|$ is odd. Boykett et al. [@BKS16] gave several non-finitely generated gate classes when borrowed symbols (called “borrowed bits” in the paper) are allowed but ancilla symbols are not allowed. Because borrowed symbols are weaker than ancilla symbols, Boykett et al.’s result does not give a non-finitely generated gate class when ancilla symbols are allowed.
Aaronson et al. [@AGS15] and Xu [@Xu15] studied the number of ancilla bits (or borrowed bits) used in generation over the binary alphabet. In this paper we do not try to minimize the number of ancilla symbols used.
Acknowledgments
===============
This work is supported in part by MIT SuperUROP Undergraduate Research Program and MIT Lincoln Laboratory. The author wishes to express his thanks to Prof. Scott Aaronson for proposing this interesting topic and giving useful advice, and to Luke Schaeffer for helpful mentoring. The author also thanks Cameron Musco, Dr. Kevin Obenland and Prof. Yury Polyanskiy for helpful discussions and Thalia Rubio for writing advice.
Preliminaries
=============
In this section we give basic definitions of the object of study.
Fix a finite set $A$ with $|A|\ge 2$. $A$ is called the **alphabet**. A theory of reversible gates can be defined for arbitrary $A$, including infinite $A$. However, we restrict our attention to the case $A$ is finite in this paper.
By a **classical gate** we mean a function of sets $A^k\to A$ for some $k\in \bN$. (We take $\bN=\bZ_{\ge 0}$.) In contrast, a reversible gate is a reversible function, which then requires the domain and the codomain to have the same cardinality.
A **reversible gate** is a bijective function $A^k\to A^k$ for some $k\in \bN$.
\[GateDefn\] We give some examples of gates and build notations that will be used in later sections.
Let $u,v\in A^k$ be two strings. Define reversible gate $\tau_{u,v}:A^k\to A^k$ to be the function that maps $u$ to $v$, $v$ to $u$, and all other inputs to themselves.
Let $u\in A^k$, $v\in A^l$ be two strings. Define reversible gate $\SWAP_{u,v}:A^{k+l}\to A^{k+l}$ to be the function that maps $uv$ to $vu$, $vu$ to $uv$, and all other inputs to themselves. In other words, $\SWAP_{u,v}=\tau_{uv,vu}$.
Let $F:A^k\to A^k$ be a gate and $w\in A^l$ be a string. Define reversible gate $w\hyp F:A^{k+l}\to A^{k+l}$ that maps $wu$ to $wF(u)$ where $u\in A^k$, and maps all other inputs to themselves.
Let $A=\{0,1\}$. Then $11\hyp \tau_{0,1}$ is the Toffoli gate. $1\hyp \SWAP_{0,1}$ is the Fredkin gate.
In the theory of classical gates, there is a notion of generation of gates, which refers to the process of creating new gates from existing gates. Similarly, in the reversible case, we also have generation of gates.
\[GenRules\] There are several ways in which reversible gates can be generated.
1. **Permutation rule.** Let $k$ be a non-negative integer and $\sm$ be a permutation of $\{1,\ldots,k\}$. We define the permutation gate $P_\sm:A^k\to A^k$ that maps $(a_1,\ldots,a_k)\in A^k$ to $(a_{\sm(1)},\ldots,a_{\sm(k)})\in A^k$. Permutations of symbols come for free: we can generate $P_\sm$ from nothing.
2. **Tensor product rule.** Assume we have two reversible gates $F:A^k\to A^k$, $G:A^l\to A^l$. Then we can generate their tensor product $F\ot G:A^{k+l}\to A^{k+l}$ which sends $(a_1,\ldots,a_k,b_1,\ldots,b_l)\in A^{k+l}$ to $(F(a_1,\ldots,a_k),G(b_1,\ldots,b_l))$.
3. **Composition rule.** Assume that we have two reversible gates $F:A^k\to A^k$, $G:A^k\to A^k$. Then we can generate composition $F\circ G:A^k\to A^k$ which sends $(a_1,\ldots,a_k)\in A^k$ to $F(G(a_1,\ldots,a_k))$.
4. **Ancilla rule.** Assume that we have a reversible gate $F:A^k\to A^k$. Assume there exists $0\le l\le k$, $(a_1,\ldots,a_l)\in A^l$ and a reversible gate $F^\p:A^{k-l}\to A^{k-l}$ such that for any $(b_1,\ldots,b_{k-l})\in A^l$, $F$ maps $(a_1,\ldots,a_l,b_1,\ldots,b_{k-l})$ to $(a_1,\ldots,a_l,F^\p(b_1,\ldots,b_{k-l}))$. Then $F^\p$ can be generated from $F$.
Generation of reversible gates can be understood as the following process.
1. Start with $n$ symbols. These are the inputs and we do not have control over their initial values.
2. Introduce $m$ ancilla symbols. The initial values of these symbols are decided by us.
3. Apply some sequence of reversible gates. An application of a reversible gate $G$ with $k\le n+m$ inputs is first choosing $k$ different symbols among the $n+m$ ones, and then applying $G$ on the $k$ chosen symbols.
4. After step (3) finishes, make sure that the values of the $m$ ancilla symbols are the same as their initial values. (However, their values can change during step (3).) Remove the $m$ ancilla symbols.
5. The set of all configurations of the $n$ input symbols is $A^n$. The above process gives a function from the set of initial configurations to the set of final configurations, written as $F:A^n\to A^n$. $F$ is actually a reversible gate, and is the reversible gate generated by this process.
This way of understanding generation of reversible gates is very helpful and is used throughout this paper.
The first three generation rules are natural and easy to understand. For a detailed discussion of the ancilla rule, see [@AGS15], Section 1.2.
The set of generation rules given in Definition \[GenRules\] is equivalent to the one in [@AGS15], Section 2.2.
With the generation rules, we can define gate classes.
Let $S$ be a set of reversible gates. We say $S$ is a **reversible gate class** if $S$ is closed under the generation rules in Definition \[GenRules\].
Let $S$ be a set of reversible gates. Define $\la S\ra$ to be the smallest reversible gate class containing $S$. We say $\la S\ra$ is the **reversible gate class generated by** $S$. Let $F$ be a reversible gate. We say **$S$ generates $F$** if $F\in \la S\ra$.
$\la S\ra$ is equal to the intersection of all reversible gate classes containing $S$.
We give some examples of reversible gate classes.
Let $\ALL$ be the set of all reversible gates. Then $\ALL$ is a reversible gate class. When $|A|=2$, $\ALL$ is the reversible gate class generated by the Toffoli gate.
Let $\CONS$ be the set of all conservative gates, i.e. reversible gates $F$ such that for every $a\in A$ and every input to $F$, the number of $a$’s in the input of $F$ is the same as the number of $a$’s in the output. Then $\CONS$ is a reversible gate class. When $|A|=2$, $\CONS$ is the reversible gate class generated by the Fredkin gate.
The set of reversible gate classes has a natural structure of a poset.
We give the set of all reversible gate classes a partial order $\le$. For two reversible gate classes $S$, $T$, we have $S\le T$ if and only if $S\sse T$ as sets.
The structure is actually more than a poset, as shown in the following proposition.
Let $L$ be the poset of reversible gate classes. Then $L$ is a complete lattice.
Let $\{S_i:i\in I\}$ be a set of reversible gate classes where $I$ can be infinite. Then the join of $\{S_i\}$ is $\la \bup_{i\in I} S_i\ra$ and the meet of $\{S_i\}$ is $\bap_{i\in I} S_i$.
In the following sections, when there is no ambiguity, we sometimes say “gate” instead of “reversible gate”, and say “gate class” instead of reversible gate class.
Non-finite generation
=====================
When $|A|=2$, it follows from Aaronson et al.’s classification that every gate class is generated by a single gate ([@AGS15], Corollary 5). In this section we show that this fails when $|A|>2$. Actually, we prove that there are gate classes that are not generated by a finite set of gates.
\[NonFinGenClass\] If $|A|\ge 3$, there exists a gate class that is not finitely generated.
Assume without loss of generality that $A=\{1,\ldots,|A|\}$. For $k\ge 1$, define $T_k=\tau_{1^k2,1^k3}$. Clearly $T_{k+1}$ generates $T_k$ by the ancilla rule. Let $T=\la T_k:k\ge 0\ra$. We claim that $T$ is not finitely generated.
Assume for the sake of contradiction that $T$ is finitely generated. Then $T$ is generated by some finite set $I\sse T$. Let $F\in I$ be a gate. Then $F\in I\sse T=\la T_k:k\ge 0\ra$. So $F$ is generated by a finite subset of gates in the form $T_k$. $T_{k+1}$ generates $T_k$, so $F$ is generated by a single gate $T_n$ for some $n$. Because $I$ is finite, we can take $n$ to be large enough, so that all gates in $I$ are generated by $T_n$. We know that $T$ is generated by $I$. So $T$ is generated by $T_n$.
In particular, this means that $T_n$ generates $T_{n+1}$. We prove that this cannot be the case. We consider how ancilla symbols can be used when using $T_n$ to generate $T_{n+1}$.
1. If an ancilla symbol is initially $1$, then it is $1$ all the time. Any $T_n$ gate whose last input is this ancilla symbol has not effect. So we can assume this ancilla symbol only acts as one of the first $n$ inputs of $T_n$.
2. If an ancilla symbol is initially $2$ or $3$, then it is $2$ or $3$ all the time. If it acts as one of the first $n$ inputs of some $T_n$, then that $T_n$ gate has no effect. So we can assume this ancilla symbol only acts as one of the last input of $T_n$. On the other hand, if a $T_n$ gate has last symbol the ancilla symbol, then only that ancilla symbol is affected. This means we do not need this ancilla symbol at all.
3. If an ancilla symbol is initially larger than $3$, then any $T_n$ gate acting on this symbol does not have any effect. So we do not need this ancilla symbol.
By the above discussion, we can assume that all ancilla symbol are initially $1$’s, and the only use of them is to act as one of the first $n$ inputs of $T_n$. Therefore we only need to show that, using $T_0,\ldots,T_n$ and no ancilla symbols, it is impossible to generate $T_{n+1}$.
Let $a_1,\ldots,a_n,b$ be the inputs of $T_{n+1}$. Consider all $T_i$ ($0\le i\le n$) gates whose last input is not $b$. We can remove these $T_i$ actions, because they do not change the effect of all other $T_i$ actions. So we can assume that the last input of each $T_i$ is $b$. In other words, every $T_i$ acts by choosing a subset of size $i$ among $a_1,\ldots,a_n$ as the first $i$-inputs, and $b$ as the last input.
Consider $2^n$ different inputs $(a_1,\ldots,a_n,b)$, where the $k$-th input $0\le k<2^n$ has
1. $a_i=1$ if the $i$-th lowest bit in the binary representation of $k$ is $1$;
2. $a_i=2$ otherwise;
3. $b=2$.
$T_{n+1}$ changes the value of $b$ for exactly one of the inputs, namely the input $2^n-1$. So we only need to prove that for any gate $A^{n+1}\to A^{n+1}$ built using $T_0,\ldots,T_n$ and no ancilla symbols, there are always an even number of inputs among the $2^k$ ones defined above, on which the value of $b$ is changed.
Each gate $T_i$ ($0\le i\le n$) changes the value of $b$ for an even number of inputs. The symmetric difference of several sets of even size must have even size. So the overall effect is that for an even number of inputs, the value $b$ is changed.
\[UncountablyMany\] A natural question to ask is whether there are uncountably many gate classes over a non-binary alphabet. We expect this to be true because this is true for classical gate classes over non-binary alphabets [@IM59].
A possible approach to proving this is to find a countable collection of gates $F_1,F_2,\ldots$ such that $F_i\not \in \la F_j:j\ne i\ra$ for all $i\ge 1$. If we have such a collection, then for different subsets $I\sse \{1,2,\cdots\}$, the gate classes $F_I=\la F_i:i\in I\ra$ are different. However, we have not been able to find such a collection.
Some generation results
=======================
In this section, let $A=\{1,\ldots,k\}$ with $k\ge 3$. We prove several generation results which can be seen as generalizations of generation results over the binary alphabet. Nevertheless there are some subtle differences between the binary alphabet and non-binary alphabets.
Recall Notation \[GateDefn\] where we defined the reversible gates $\tau_{u,v}$, $\SWAP_{u,v}$, and $w\hyp F$.
Generating $\ALL$
-----------------
Recall that $\ALL$ is the class of all gates.
\[AllGenerate\] Let $S$ be the following set of gates:
1. $\tau_{a,b}$ for all $a\ne b\in A$;
2. $\tau_{11,12}$.
Then $\la S\ra=\ALL$.
Over the binary alphabet $\{0,1\}$, $\ALL$ is generated by the Toffoli gate $11\hyp\tau_{0,1}$, but not by the CNOT gate $\tau_{10,11}$. Therefore Theorem \[AllGenerate\] does not hold over the binary alphabet.
We prove the theorem in several steps.
\[TauABCABD\] $S$ generates $\tau_{abc,abd}$ for all $a,b,c,d\in A$.
**Step 1.** $S$ generates $\tau_{11,1c}$ for all $c\in A$. Assume we have two input symbols $x$, $y$. The following sequence of operations implements $\tau_{11,1c}$.
1. Apply $\tau_{2,c}$ on $y$.
2. Apply $\tau_{11,12}$ on $xy$.
3. Apply $\tau_{2,c}$ on $y$.
**Step 2.** $S$ generates $\tau_{1b,1c}$ for all $b,c\in A$. This is similar to step 1 and omitted.
**Step 3.** $S$ generates $\tau_{ab,ac}$ for all $a,b,c\in A$. Assume we have two input symbols $x$, $y$. The following sequence of operations implements $\tau_{ab,ac}$.
1. Apply $\tau_{1,a}$ on $x$.
2. Apply $\tau_{1b,1c}$ on $xy$.
3. Apply $\tau_{1,a}$ on $x$.
**Step 4.** $S$ generates $\tau_{abc,abd}$. Assume we have three input symbols $x$, $y$, $z$. We introduce an ancilla symbol $w$ which is initially $1$. The following sequence of operations implements $\tau_{abc,abd}$ and fixes all ancilla symbols.
1. Apply $\tau_{a1,a2}$ on $xw$.
2. Apply $\tau_{b2,b3}$ on $yw$.
3. Apply $\tau_{3c,3d}$ on $wz$.
4. Apply $\tau_{b2,b3}$ on $yw$.
5. Apply $\tau_{a1,a2}$ on $xw$.
\[WControlTrans\] $S$ generates $w\hyp \tau_{a,b}$ for all strings $w$, and $a,b\in A$.
If $|w|\le 2$, then $S$ generates $w$ by Lemma \[TauABCABD\]. In the following, assume $|w|\ge 3$.
Let $|w|=n$. Assume we have $n+1$ inputs $x_1,\ldots,x_n, y$. Introduce $n-1$ ancilla symbols $z_1,\ldots, z_{n-1}$. Initially all $z_i=1$. The following sequence of operations implements $w\hyp\tau_{a,b}$ and fixes all ancilla symbols.
1. Apply $\tau_{w_1w_21,w_1w_22}$ on $x_1x_2z_1$.
2. For $i=3,\ldots,n$, apply $\tau_{w_i21,w_i22}$ on $x_iz_{i-2}z_{i-1}$.
3. Apply $\tau_{2a,2b}$ on $z_{n-1}y$.
4. For $i=n,\ldots,3$, apply $\tau_{w_i21,w_i22}$ on $x_iz_{i-2}z_{i-1}$.
5. Apply $\tau_{w_1w_21,w_1w_22}$ on $x_1x_2z_1$.
We would like to show that for all $l$, all bijections $A^l\to A^l$ are in $\la S\ra$. The group of bijections $A^l\to A^l$ is the symmetric group $S_{|A|^l}$, so it is generated by transpositions. The composition in the symmetric group is the same as the one in composition rule of reversible gates. So we only need to prove that every transposition is in $\la S\ra$. Moreover, we do not need all transpositions. We only need a set of transpositions $T$ so that the graph whose vertices are $A^k$ and edges are $\{(u,v):\tau_{u,v}\in T\}$ is connected. We can take $T$ to be the set of $\tau_{u,v}$ where $u$ and $v$ differ by one position.
We can assume without loss of generality that $u$ and $v$ differ in the last position. Then $\tau_{u,v}$ is in the form $w\hyp\tau_{a,b}$. The theorem follows from Lemma \[WControlTrans\].
Generating $\CONS$
------------------
Recall that $\CONS$ is the set of conservative gates.
\[ConsGen\] Let $S$ be the set of $1\hyp \SWAP_{a,b}$ for all $a\ne b\in A$. Then $\la S\ra=\CONS$.
Over the binary alphabet $\{0,1\}$, $\CONS$ is generated by the Fredkin gate $1\hyp\SWAP_{0,1}$. So the statement of Theorem \[ConsGen\] is true over the binary alphabet. However, the proof presented here is for non-binary alphabets.
We prove the theorem in several steps.
$S$ generates $c\hyp \SWAP_{a,b}$ for all $a,b,c\in A$.
If $c=1$, then the lemma follows from definition of $S$. In the following, assume $c\ne 1$.
Assume we have three input symbols $x$, $y$, $z$. Introduce an ancilla symbol $w$ which is initially $1$. Introduce an ancilla symbol $u$ which is initially $d$ where $d\ne 1,c$. The following sequence of operations implements $c\hyp \SWAP_{a,b}$ and fixes all ancilla symbols.
1. Apply $\SWAP_{1,d}$ on $ux$.
2. Apply $\SWAP_{1,c}$ on $wx$.
3. Apply $1\hyp \SWAP_{a,b}$ on $xyz$.
4. Apply $\SWAP_{1,c}$ on $wx$.
5. Apply $\SWAP_{1,d}$ on $ux$.
\[WControlSWAP\] $S$ generates $w\hyp \SWAP_{a,b}$ for all strings $w$ and $a,b\in A$.
Let $|w|=n$. Assume that we have $n+2$ inputs $x_1,\ldots,x_n,y_1,y_2$. Introduce $n+1$ ancilla symbols $z_1,\ldots,z_{n+1}$. Initially $z_1=1$ and $z_i=2$ for $i\ge 2$. The following sequence of operations implements $w\hyp \SWAP_{a,b}$ and fixes all ancilla symbols.
1. For $i=1,\ldots,n$, apply $w_i\hyp\SWAP_{1,2}$ on $x_iz_iz_{i+1}$.
2. Apply $1\hyp\SWAP_{a,b}$ on $z_{n+1}y_1y_2$.
3. For $i=n,\ldots,1$, apply $w_i\hyp\SWAP_{1,2}$ on $x_iz_iz_{i+1}$.
Clearly $S\sse \CONS$. So we only need to prove that $S$ generates $\CONS$.
Similar to the proof of Theorem \[AllGenerate\], we only need to generate all transpositions $\tau_{u,v}$ where $u$ is a permutation of $v$. If we have such $u,v$, then we can find a sequence $u=w_0,w_1,\ldots,w_m=v$ where $w_i$ and $w_{i+1}$ differ by exchanging two positions. So we can assume that $u,v$ differ by exchanging two positions.
We can assume without loss of generality that the positions at which $u$ and $v$ differ are the last two symbols. Then $\tau_{u,v}$ is in the form $w\hyp\SWAP_{a,b}$. The theorem follows from Lemma \[WControlSWAP\].
Generating $\CONS_\lm$
----------------------
$\ALL$ and $\CONS$ fall into a more general class of gate classes.
\[LmCons\] Let $\lm$ be a partition of $A$, which means $\lm=\{\lm_1,\ldots,\lm_l\}$ where $\lm_1,\ldots,\lm_l$ are disjoint nonempty subsets of $A$ whose union is $A$. Define $\CONS_\lm$ to be the class of **$\lm$-conservative gates**, which are gates that preserve, for each $\lm_i\in \lm$, the total number of occurrences of elements in $\lm_i$.
When $\lm=\{A\}$, $\CONS_\lm=\ALL$. When $\lm=\{\{a\}:a\in A\}$, $\CONS_\lm=\CONS$.
The following theorem is a simultaneous generalization of Theorem \[AllGenerate\] and Theorem \[ConsGen\].
\[LmConsGeneration\] Let $S$ be the following set gates:
1. $\tau_{a,b}$ for all $1\le i\le l$ and all $a\ne b\in \lm_i$;
2. $\tau_{aa,ab}$ for all $1\le i\le l$ with $|\lm_i|>1$ and one pair of $a\ne b\in \lm_i$;
3. $1\hyp\SWAP_{a,b}$ for all $1\le i<j\le l$ and one pair of $a\in \lm_i$, $b\in \lm_j$.
Then $\la S\ra=\CONS_\lm$.
We prove the theorem in several steps.
\[LmConsGenAll\] For all $1\le i\le l$, $S$ generates all gates that fix the input if at least one symbol of the input is not in $\lm_i$.
This is almost identical to the proof of Theorem \[AllGenerate\]. Omitted.
\[LmConsGenCons\] $S$ generates $\CONS$.
**Step 1.** Assume in (3) of the definition of $S$ in Theorem \[LmConsGeneration\], we choose $a=a_{ij}$, $b=b_{ij}$. Then for all $1\le i<j\le l$, $S$ generates $1\hyp \SWAP_{a_{ij},b_{ij}}$.
**Step 2.** $S$ generates $1\hyp \SWAP_{a_{ij},c}$ for all $1\le i<j\le l$ and $c\in \lm_j$. Assume we have input symbols $x$, $y$, $z$. By Lemma \[LmConsGenAll\], $S$ generates $\tau_{b_{ij},c}$. The following sequence of operations implements $1\hyp \SWAP_{a_{ij},c}$.
1. Apply $\tau_{b_{ij},c}$ on $z$.
2. Apply $\tau_{b_{ij},c}$ on $y$.
3. Apply $1\hyp \SWAP_{a_{ij},b_{ij}}$ on $xyz$.
4. Apply $\tau_{b_{ij},c}$ on $z$.
5. Apply $\tau_{b_{ij},c}$ on $y$.
**Step 3.** $S$ generates $1\hyp \SWAP_{a,b}$ for all $1\le i<j\le l$ and $a\in \lm_i$, $b\in \lm_j$. This is similar to step 2 and omitted.
**Step 4.** $S$ generates $1\hyp\SWAP_{a,b}$ for all $1\le i\le l$ and $a,b\in \lm_i$. If $1\in \lm_i$ then this follows from Lemma \[LmConsGenAll\]. Assume $1\not \in \lm_i$. By step 3, $S$ generates $\SWAP_{1,a}$. By Lemma \[LmConsGenAll\], $S$ generates $a\hyp\SWAP_{a,b}$.
Assume we have input symbols $x$, $y$, $z$. Introduce an ancilla symbol $w$ which is initially $a$. The follows sequence of operations implements $1\hyp\SWAP_{a,b}$ and fixes all ancilla symbols.
1. Apply $\SWAP_{1,a}$ to $wx$.
2. Apply $a\hyp\SWAP_{a,b}$ to $xyz$.
3. Apply $\SWAP_{1,a}$ to $wx$.
**Step 5.** Apply Theorem \[ConsGen\].
\[WControlTransLm\] $S$ generates $\tau_{wa,wb}$ where $w$ is a string and $a,b\in \lm_i$ for some $1\le i\le l$.
Assume $|w|=n$. Assume we have $n+1$ inputs $x_1,\ldots,x_n,y$. Introduce two ancilla symbols $z_1,z_2$. Initially, $z_1=a$, $z_2=b$. $S$ generates $w\hyp\SWAP_{a,b}$ by Lemma \[LmConsGenCons\]. $S$ generates $\tau_{aa,ab}$ by Lemma \[LmConsGenAll\].
The following sequence of operations implements $\tau_{wa,wb}$ and fixes all ancilla symbols.
1. Apply $w\hyp\SWAP_{a,b}$ on $x_1\ldots x_nz_1z_2$.
2. Apply $\tau_{aa,ab}$ on $z_2y$.
3. Apply $w\hyp\SWAP_{a,b}$ on $x_1\ldots x_nz_1z_2$.
Clearly $S\sse \CONS_\lm$. We only need to prove that $S$ generates $\CONS_\lm$.
Similar to the proof of Theorem \[AllGenerate\], we only need to prove that $S$ generates transpositions $\tau_{u,v}$ where $u,v\in A^m$ for some $m$, and for all $1\le i\le l$, the number of occurrences in $u$ of elements in $\lm_i$ is the same as that of $v$.
Assume $u,v\in A^m$ has the above property. We can find $w\in A^m$ such that $w$ is a permutation of $v$, and for all $1\le j\le m$, $w_j$ and $u_j$ are both in $\lm_i$ for some $1\le i\le l$. $S$ generates $\tau_{w,v}$ by Lemma \[LmConsGenCons\]. So we can replace $v$ with $w$ and assume that for all $1\le j\le m$, $u_j$ and $v_j$ are both in $\lm_i$ for some $1\le i\le l$.
Assume we have such $u,v$. Then we can find a sequence $u=w_0,w_1,\ldots,w_p=v$ such that $w_i$ and $w_{i+1}$ differ by one position $j$, and $w_{i,j}$ and $w_{i+1,j}$ are both in $\lm_h$ for some $1\le h\le l$. So we can assume that $u$ and $v$ satisfies this.
So we only need to generate $\tau_{wa,wb}$ where $w$ is some string, and $a,b\in \lm_i$ for some $1\le i\le l$. This follows from Lemma \[WControlTransLm\].
Single gate generation
----------------------
The following proposition points out the reason why finitely-generated gate classes are usually generated by a single gate.
\[SingleGeneration\] Let $S$ be a finite set of reversible gates such that each gate in $S$ has a fixed point. Then $\la S\ra$ is generated by a single gate.
Let $S=\{F_1,\ldots,F_n\}$. Consider the gate $F=F_1\ot \cdots \ot F_n$. By the tensor product rule of generation, $F\in \la S\ra$. So we only need to prove that $F$ generates $F_i$ for all $i$.
Fix an $i$. We use the ancilla rule of generation. Because every $F_j$ has a fixed point, for $j\ne i$, we can take the input gates for $F$ that correspond to $F_j$ to be a fixed point of $F_j$. Then $F$ fixes all inputs symbols except for the inputs corresponding to $F_i$.
$\CONS_\lm$ is generated by a single gate for all $\lm$.
Each gate in the set $S$ in Theorem \[LmConsGeneration\] has fixed points.
Base change
===========
\[SkAction\] Let $L_k$ be the lattice of gate classes when $|A|=k$. Then $L_k$ is naturally equipped with a nontrivial $S_k$-action.
Let $\sm\in S_k$ be a permutation of $A$. For a gate $F:A^l\to A^l$, define reversible gate $F^\sm:A^l\to A^l$ that maps $(a_1,\ldots,a_l)\in A^l$ to $\sm(F(\sm^{-1}(a_1),\ldots,\sm^{-1}(a_l)))$, where the outer $\sm$ means applying $\sm$ on every input symbol. For every gate class $S$, define $S^\sm=\{F^\sm:F\in S\}$. It is easy to verify that $S^\sm$ is also a gate class. Then $\sm:L_k\to L_k$ gives the desired $S_k$-action.
This generalizes the notion of dual gate and dual gate class when $|A|=2$ defined in [@AGS15].
\[PosetEmbedding\] Let $L_k$ be the lattice of gate classes when $|A|=k$. Then there is a poset embedding $P:L_k\to L_{k+1}$ for $k\ge 2$.
Let $F$ be a reversible gate over alphabet $A=\{1,\ldots,k\}$. Define $PF$ to be the reversible gate over alphabet $A^\p=\{1,\ldots,k+1\}$ that fixes the input when at least one of the input symbols is $k+1$, and acts as $F$ otherwise.
Let $S\in L_k$ be a gate class. Define $PS=\la PF:F\in S\ra$ to be a gate class over $A^\p$. This defines a map between sets $P:L_k\to L_{k+1}$. $P$ is clearly order-preserving, so $P$ is a map between posets.
So we only need to prove that $P$ is an embedding, i.e. $P$ is an injection of sets. This means for every two different gate classes $S,T\in L_k$, we have $PS\ne PT$. $PT=\la PF:F\in T\ra$. So we only need to prove that there exists some $F\in T$ such that $PF\not \in PS$. Actually, we prove that for any reversible gate $F$ over $A$, if $PF\in PS$, then $F\in S$.
$PF\in PS$, so $PF$ can be generated by $\{PG:G\in S\}$. Consider how ancilla symbols are used to perform this generation.
If an ancilla symbol is initially $k+1$, then it remains $k+1$ all the time. If this ancilla symbol is used as an input of some gate $PG$, then this $PG$ action has no effect. So we do not need this ancilla symbol .
We can perform the generation with all ancilla symbols initially in $A$. This gives a way to generate $F$ using $S$.
\[FindLatticeEmbedding\] We do not see the reason for $P$ to be either a join-lattice embedding or a meet-lattice embedding. It is an interesting question whether we can find some map $Q:L_{k}\to L_{k+1}$ that is a lattice embedding (or even a complete lattice embedding).
Gate classes containing $\CONS_{k-1,1}$
=======================================
Let $A=\{1,\ldots,k\}$ where $k\ge 3$. By Proposition \[SkAction\], there is a natural action of $S_k$ on $L_k$. Therefore if we have two partitions $\lm=\{\lm_1,\ldots,\lm_l\}$ and $\mu=\{\mu_1,\ldots,\mu_l\}$ of $A$ such that $|\lm_i|=|\mu_i|$ for all $i$, then the gate classes $\CONS_\lm$ and $\CONS_\mu$ are “isomorphic” in a certain sense.
Let $\tau$ be a partition of $k$, i.e. $\tau=\{\tau_1,\ldots,\tau_l\}$ where $\tau_i\ge 1$ and $\sum \tau_i=k$. Define $\CONS_\tau$ to be any $\CONS_\lm$ where $\lm=\{\lm_1,\ldots,\lm_l\}$ is a partition of $A$ and $|\lm_i|=\tau_i$ for all $i$. $\CONS_\lm$ is well-defined up to $S_k$-action.
In this section we study the properties of $\CONS_{k-1,1}$. We only need to study $\CONS_\lm$ where $\lm=\{\{1,\ldots,k-1\},\{k\}\}$.
We adapt the methods in [@AGS15] and prove the following theorem.
\[CONSInteger\] The lattice of gate classes containing $\CONS_{k-1,1}$ is anti-isomorphic to the lattice $(\bN,|)$ of non-negative integers, where $n\le m$ in the lattice order relation if and only if $n|m$. The gate class corresponding to $0$ is $\CONS_{k-1,1}$, and the gate class corresponding to $m\ge 1$ is $\la \CONS_{k-1,1},\CC_m\ra$, where $\CC_m=1\hyp \tau_{1^m,k^m}$.
The Hasse diagram of the lattice of gate classes containing $\CONS_{k-1,1}$ is shown in Figure \[FigCONSInteger\].
Theorem \[CONSInteger\] fails when $k=2$ because there are parity-flipping gates in that case. In particular, Lemma \[NoModShifter\] fails when $k=2$.
; (ALL) edge (MOD2) (ALL) edge (MOD3) (ALL) edge (MOD5) (MOD2) edge (MOD4) (MOD2) edge (MOD6) (MOD3) edge (MOD6) (MOD5) edge (MOD10) (MOD4) edge (MOD8) (MOD6) edge (MOD18) (MOD10) edge (MOD20) (MOD8) edge (CONS) (MOD18) edge (CONS) (MOD20) edge (CONS) ;
The theorem is proved in several steps.
For any finite string $w$ of elements in $A$, define $c_k(w)$ to be the number of $k$’s in the string.
A gate $F:A^n\to A^n$ is called **mod-$m$-respecting** if for every $w\in A^n$, $c_k(F(w))=c_k(w)+j \pmod m$ for some $0\le j\le m-1$. When $j=0$, $F$ is called **mod-$m$-preserving**.
\[NoModShifter\] If $F$ is mod-$m$-respecting, then it is mod-$m$-preserving.
Let $z$ be a variable. $c_k(F(w))=c_k(w)+j\pmod k$ means $z^{c_k(F(w))}=z^{c_k(w)+j} \pmod{(z^m-1)}$. We sum over all $w\in A^n$ and get $$\begin{aligned}
\sum_{w\in A^n} z^{c_k(F(w))}=z^j \sum_{w\in A^n} z^{c_k(w)} \pmod {(z^m-1)}.
\end{aligned}$$ On the other hand, we have $$\begin{aligned}
\sum_{w\in A^n} z^{c_k(F(w))}=\sum_{w\in A^n} z^{c_k(w)}=\sum_{0\le i\le n} \binom ni(k-1)^{n-i}z^i=(z+k-1)^n
\end{aligned}$$ From the above two equalities, we see $(z+k-1)^n(z^j-1)=0\pmod {(z^m-1)}$. $k\ge 3$ so $z+k-1$ is coprime with $z^m-1$. Therefore $z^j-1=0\pmod{(z^m-1)}$. This is true only when $j=0$.
\[ModPreserverM\] For a gate $F$, define $m(F)$ to be the largest $m$ such that $F$ is a mod-$m$-preserving. If there is no such largest $m$, define $m(F)=0$.
\[TensorModPreserver\] For two gates $F$, $G$, we have $m(F\ot G)=\gcd(m(F),m(G))$.
Clearly $\gcd(m(F),m(G))|m(F\ot G)$. Assume $m(F\ot G)=m^\p\ne \gcd(m(F),m(G))$. Then either $m^\p\nmid m(F)$ or $m^\p\nmid m(G)$. Assume without loss of generality $m^\p\nmid m(F)$. If $F$ is mod-$m^\p$-preserving, then it is mod-$\lcm(m^\p,m(F))$-preserving, and there is contradiction. So $F$ is not mod-$m^\p$-preserving. By Lemma \[NoModShifter\], $F$ is not mod-$m^\p$-respecting. So we can find two inputs $u$, $v$ of $F$ such that $c_k(Fu)-c_k(u)\ne c_k(Fv)-c_k(v)\pmod {m^\p}$. Then for any input $w$ of $G$, $c_k((F\ot G)uw)-c_k(uw)\ne c_k((F\ot G)vw)-c_k(vw)\pmod {m^\p}$, which means that $F\ot G$ is not mod-$m^\p$-preserving. Contradiction. So $m(F\ot G)=\gcd(m(F),m(G))$.
\[ModPreserverFromCCm\] For any $m\ge 1$, $\CONS_{k-1,1}$ together with $\CC_m$ generates the class of all mod-$m$-preserving gates.
Similar to the proof of Theorem \[AllGenerate\], we only need to show that $\CONS_{k-1,1}+\CC_m$ generates transpositions $\tau_{u,v}$ where $u,v\in A^n$ for some $n$ and $c_k(u)=c_k(v)\pmod m$.
If $c_k(u)=c_k(v)$, then $\tau_{u,v}\in \CONS_{k-1,1}$. So we can assume $c_k(u)\ne c_k(v)$. Assume without loss of generality $c_k(u)<c_k(v)$. Then we can find a sequence $u=w_0,w_1,\ldots,w_p=v$ such that $c_k(w_i)=c_k(w_{i+1})-m$. So we can assume $c_k(v)-c_k(u)=m$.
Choose a string $w\in A^n$ such that $w$ is a permutation of $v$, and the set of positions $j$ where $w_j=k$ contains the set of positions $j$ where $u_j=k$. $\tau_{v,w}\in \CONS_{k-1,1}$, so we can replace $v$ with $w$.
Therefore we can assume the set of positions $j$ where $v_j=k$ contains the set of positions $j$ where $u_j=k$. Assume without loss of generality that for $1\le j\le m$, $v_j=k$ but $u_j\ne k$. Let $w$ be the string whose first $m$ symbols are $k$, and the last $n-m$ symbols are the same as $u$. Then $\tau_{w,v}\in \CONS_{k-1,1}$. So we can replace $v$ with $w$ and assume that the last $n-m$ symbols of $u$ are the same as that of $v$.
Let $w$ be the string whose whose first $m$ symbols are $1$, and the last $n-m$ symbols are the same as $u$. Then $\tau_{w,u}\in \CONS_{k-1,1}$. So we can replace $u$ with $w$ and assume that the first $m$ symbols of $u$ are all $1$.
Now we generate $\tau_{u,v}$. Let $x_1,\ldots,x_n$ be inputs. Introduce an ancilla symbol $z$, initialized to be $2$. The following sequence of operations implements $\tau_{u,v}$ and fixes all ancilla symbols.
1. Apply $u_{m+1}\ldots u_n\hyp \tau_{1,2}$ on $x_{m+1}\ldots x_nz$.
2. Apply $\CC_m$ on $zx_1\ldots x_m$.
3. Apply $u_{m+1}\ldots u_n\hyp \tau_{1,2}$ on $x_{m+1}\ldots x_nz$.
\[TensorW\] Let $F:A^n\to A^n$ be a gate with $m(F)\ge 1$. Then there exists $t\ge 1$ and string $w\in A^{nt}$ such that $c_k(F^{\ot t}(w))-c_k(w)=m$.
Let $p=\gcd_{w\in A^n} (c_k(F(w))-c_k(w))$. By definition $F$ is mod-$p$-respecting. So $F$ is mod-$p$-preserving, i.e. $p|m(F)$, by Lemma \[NoModShifter\]. There are both positive and negative elements in $\{c_k(F(w))-c_k(w):w\in A^n\}$. So we can find $w_1,\ldots,w_t\in A^n$ such that $\sum_{1\le i\le t} (c_k(F(w_i))-c_k(w_i))=m$. Then $w_1\ldots w_t\in A^{nt}$ is the desired string.
\[CCmFromModPreserver\] For any $m\ge 1$, $\CONS_{k-1,1}$ together with any gate $F$ with $m(F)=m$ generates $\CC_m$.
Let $G$ be $F\ot F^{-1}$ followed by swapping the inputs of two $F$’s. Then $G^2=\id$. By Lemma \[TensorModPreserver\], $m(G)=m$. By Lemma \[TensorW\], there exists $t\ge 1$ and two strings $u$ and $v$ (whose lengths are the same as the number of inputs of $G^{\ot t}$) such that $v=G^{\ot t}(u)$ and $c_k(v)=c_k(u)+m$. We also have $G^{\ot t}(v)=u$ because $G$ is an involution. Let $|u|=n$.
Assume without loss of generality that the first $c_k(v)$ symbols of $v$ are $k$. Because we have $\CONS_{k-1,1}$, we have a gate $H:A^n\to A^n$ satisfying the following:
1. $H(u)=H^{-1}(u)=k^{c_k(u)}1^{n-c_k(u)}$;
2. $H(v)=H^{-1}(v)=k^{c_k(v)}1^{n-c_k(v)}$.
Let $w$ be any string of length $n$ of form $c_1,F(c_1),c_2,F(c_2),\ldots,c_t,F(c_t)$ where $c_i$ are strings of length equal to the number of inputs of $F$. Then $G^{\ot t}(w)=w$.
Let $R=H\circ G^{\ot t}\circ H^{-1}$. Let $w^\p=H(w)$, $u^\p=H(u)$ and $v^\p=H(v)$. Direct calculation shows that $R(w^\p)=w^\p$, $R(u^\p)=v^\p$ and $R(v^\p)=u^\p$.
Now we use $R$ to generate $\CC_m$. Let $c,x_{c_k(u)+1},\ldots,x_{c_k(v)}$ be inputs. Introduce ancilla symbols:
1. $x_1,\ldots,x_{c_k(u)}$, initialized to $1$;
2. $x_{c_k(v)+1},\ldots,x_n$, initialized to $k$;
3. $y_1,\ldots,y_n$, initialized to $w^\p$;
4. $z_1,z_2$ where $z_1$ initialized to $1$, $z_2$ initialized to $2$.
The following sequence of operations implements $\CC_m$ and fixes all ancilla symbols.
1. Swap $z_1$ and $z_2$ if $c=1$ and $x_{c_k(u)+1}\ldots x_{c_k(v)}=1^m$ or $k^m$.
2. Swap $x_1\ldots x_n$ with $y_1\ldots y_n$ if $z_1=1$.
3. Apply $R$ on $x_1\ldots x_n$.
4. Swap $x_1\ldots x_n$ with $y_1\ldots y_n$ if $z_1=1$.
5. Swap $z_1$ and $z_2$ if $c=1$ and $x_{c_k(u)+1}\ldots x_{c_k(v)}=1^m$ or $k^m$.
Let $S$ be a gate class containing $\CONS_{k-1,1}$. Assume $S\ne \CONS_{k-1,1}$. Define $m=\gcd_{F\in S} m(F)$. By Proposition \[ModPreserverFromCCm\], $S$ is contained in $\la \CONS_{k-1,1}, \CC_m\ra$.
We can find a finite number of gates $F_1,\ldots,F_n\in S$ such that $m=\gcd_{1\le i\le n} m(F_i)$. By Lemma \[TensorModPreserver\], $m(F_1\ot \cdots \ot F_n)=m$. By Proposition \[CCmFromModPreserver\], $\CONS_{k-1,1}$ with $F_1\ot \cdots \ot F_n$ generates $\la \CONS_{k-1,1},\CC_m\ra$. So $S=\la \CONS_{k-1,1},\CC_m\ra$. .
Further directions
==================
There are many possible directions for further research on reversible gate classes over non-binary alphabets. As discussed in Remark \[UncountablyMany\], it remains open whether there are uncountably many reversible gate classes over a non-binary alphabet. As discussed in Remark \[FindLatticeEmbedding\], we do not know whether there is a lattice embedding from $L_k$ to $L_{k+1}$.
We have seen from Theorem \[NonFinGenClass\] that reversible gate classes over non-binary alphabets can be much more complicated than those over the binary alphabet. Therefore, it seems very hard to give a complete classification of reversible gate classes over non-binary alphabets. However, when restricted to certain kinds of gate classes, classification can be done. Theorem \[CONSInteger\] is one result in this spirit. We expect there to be more such classification to reveal more structure of the huge lattice $L_k$. For example, it would be interesting to have a classification of gate classes containing $\CONS_{k-2,1,1}$, or of gate classes containing all one-input gates.
[^1]: MIT. Email: yuzhougu@mit.edu.
|
---
abstract: |
Let $p_n$ be a polynomial of degree $n$ having $n$ distinct, real roots distributed according to a nice probability distribution $u(0,x)dx$ on $\mathbb{R}$. One natural problem is to understand the density $u(t,x)$ of the roots of the $(t\cdot n)-$th derivative of $p_n$ where $0 < t < 1$ as $n \rightarrow \infty$. The author suggested that these densities might satisfy the partial differential equation of transport type $$\frac{\partial u}{\partial t} + \frac{1}{\pi} \frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} = 0 \qquad \mbox{on}~\left\{x: u(x) > 0\right\},$$ where $H$ is the Hilbert transform. Conditional on this being the case, we derive an *infinite* number of conversation laws of which the first three are $$\begin{aligned}
\int_{\mathbb{R}}{ u(t,x) ~ dx} = 1-t, \qquad \qquad \int_{\mathbb{R}}{ u(t,x) x ~ dx} = \left(1-t\right)\int_{\mathbb{R}}{ u(0,x) x~ dx}, \qquad\\
\int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy = (1-t)^3 \int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy.\end{aligned}$$ In particular, we present two closed-form solutions for which all infinitely many conservation laws are valid. This raises a number of interesting questions.
address: 'Department of Mathematics, Yale University, New Haven, CT 06511, USA'
author:
- Stefan Steinerberger
title: Conservation Laws for an equation modeling roots of polynomials under differentiation
---
[^1]
Introduction
============
Introduction.
-------------
Let $p_n:\mathbb{R} \rightarrow \mathbb{R}$ be a polynomial of degree $n$ having $n$ distinct real roots. Rolle’s theorem implies that the derivative $p_n'$ has exactly $n-1$ real roots. Moreover, between any two roots of $p_n$ there is exactly one root of $p_n'$.
(-1.6,0) – (1.6,0); plot (,[ 0.05\*(-12\*+8\*]{}); at (-1.7, -0.4) [$H_3$]{}; plot (,[ 0.15\*(-2+4\*]{}); plot (,[ 0.3\*(2\*]{}); at (-1.2, -1) [$H_1$]{}; at (-1.7, 1) [$H_2$]{}; (-1.22, 0) circle (0.04cm); (1.22, 0) circle (0.04cm); (0, 0) circle (0.04cm); (-0.7, 0) circle (0.04cm); (0.7, 0) circle (0.04cm);
These objects are classical and much is known about them. Indeed, the study of the distribution of roots of $p_n'$ depending on $p_n$ is an active field [@bruj; @bruj2; @branko; @dimitrov; @gauss; @han; @kab; @lucas; @malamud; @or; @pem; @ravi; @totik; @ull; @van]. A basic result is commonly attributed to Riesz [@farmer; @riesz]: denoting the smallest gap of a polynomial $p_n$ having $n$ real roots $\left\{x_1, \dots, x_n\right\}$ by $$G(p_n) = \min_{i \neq j}{|x_i - x_j|},$$ we have $G(p_n') \geq G(p_n)$: the minimum gap grows under differentiation. A simple proof is given by Farmer & Rhoades [@farmer]. Moreover, the Gauss-Lucas theorem [@gauss; @lucas] states that the convex hull of the roots of $p_n'$ is contained in the convex hull of the roots of $p_n$: the roots of $p_n'$ are trapped by the roots of $p_n$. Put differently, the roots are localized in space and the minimal gaps between roots do not shrink under differentiation, they spread out. This motivates a simple question: do the roots of such a polynomial become ‘more regular’ under iterated differentiation? This could be made precise in many different ways and many of these different formulations would be interesting. Polya [@pol] has asked similar problems for transcendental functions, we refer to Farmer & Rhoades [@farmer]. We note this informally as
> **Conjecture A** (Polya [@pol], Farmer & Rhoades [@farmer])**.** The roots of a polynomial $p_n$ having $n$ distinct roots become more regular under iterated differentiation.
The formulation of Conjecture A is intentionally vague, to the best of our knowledge no precise formulation exists. One possible weak formulation is, for example, as follows (with scales chosen more or less at random): assuming some mild conditions on the roots of $p_n$ (say, their minimal distance is $\sim n^{-1}$), for any $1/6 < t < 1/3$, the $n/3-$th largest root of the $t \cdot n-$th derivative of $p_n$ and its $\sim \log{n}$ nearest neighbors form an arithmetic progression (up to a small error of size $o(n^{-1})$). To the best of our knowledge, there are no rigorous results in this direction. Farmer & Rhoades [@farmer] discuss the case of entire functions and obtain results in this setting. Having already mentioned the result of Riesz [@riesz], we also mention a little known result of Sz-Nagy [@sz] (see also Rahman & Schmeisser [@rahman]): Sz-Nagy [@sz] proves that the average distance between consecutive pairs of roots increases under differentiation whereas the square mean and the variance decrease. One of these observations is equivalent to the third conservation law in the infinite family that we describe.
at (0,0) ;
at (0,0) ;
A Partial Differential Equation
-------------------------------
Recently, the author [@steini] studied the following question: if the roots of $p_n(x)$ are distributed according to some nice smooth $C^{\infty}_c(\mathbb{R})$ function $u(0,x)$, what can be said about the distribution of the roots of the $(t \cdot n)-$th derivative of $p_n^{}$, where $0 < t < 1$? If we denote their distribution by $u(t,x)$, how is $u(t,x)$ connected to the original distribution $u(0,x)$? The author [@steini] suggested that if $\left\{x \in \mathbb{R}: u(0,x) > 0 \right\}$ is a finite interval, then this process may be governed by the nonlinear partial differential equation $$\frac{\partial u}{\partial t} + \frac{1}{\pi} \frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} = 0 \qquad \mbox{on}~\left\{x: u(x) > 0\right\},$$ where $$Hf(x) = \mbox{p.v.}\frac{1}{\pi} \int_{\mathbb{R}}{\frac{f(y)}{x-y} dy} \qquad \mbox{is the Hilbert transform.}$$ The partial differential equation gives the correct prediction for Hermite polynomials, Laguerre polynomials and a class of orthogonal polynomials, however, a rigorous derivation is still outstanding. The derivation of the partial differential equation implicitly assumes a smoothing phenomenon at the level of the roots, one would assume that this is reflected in the behavior of the partial differential equation itself: that it has smoothing properties. This we state as
> **Conjecture B** ([@steini])**.** The partial differential equation $$\frac{\partial u}{\partial t} + \frac{1}{\pi} \frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} = 0 \qquad \mbox{on}~\left\{x: u(x) > 0\right\},$$ has smoothing properties.
This is intentionally stated in a vague sense and could again be made precise in various ways. Granero-Belinchon [@granero] has studied a similar partial differential equation on $\mathbb{S}^1$. Conjecture B should be more accessible than Conjecture A. Conjecture B, or rather the underlying smoothing mechanisms, could also point towards the kind of quantitative results one would hope to obtain for Conjecture A.
The Complex Case.
-----------------
The same question is meaningful for general polynomials $p_n:\mathbb{C} \rightarrow \mathbb{C}$ having a distribution of roots being given by a smooth probability distribution $u(0,z):\mathbb{C} \rightarrow \mathbb{R}_{\geq 0}$ in the complex plane. In case the limiting measure $u(0,z)$ is radial, the problem was studied by O’Rourke and the author [@or2] who (non-rigorously) derived a nonlocal transport equation $$\frac{\partial \psi}{\partial t} = \frac{\partial}{\partial x} \left( \left( \frac{1}{x} \int_{0}^{x} \psi(s) ds \right)^{-1} \psi(x) \right)$$ for the evolution of the radial profile. We emphasize that the conservation laws we derive in this paper should also apply to any evolution equation derived for the general complex case $p_n:\mathbb{C} \rightarrow \mathbb{C}$ (since our derivation is algebraic in nature and does not distinguish between the real and complex numbers); one would expect them to be less interesting for radial data which is why we are not pursuing this question further at this point. We emphasize that a rigorous understanding of the complex case (both a rigorous derivation of a partial differential equations as well as a study of its properties) remains open and seems to be an interesting problem.
Main Results
============
A Word of Warning
-----------------
The original derivation of the partial differential equation is only valid for data for which $\left\{x \in \mathbb{R}: u(0,x) > 0\right\}$ is a single compact interval. More precisely, if $\left\{x \in \mathbb{R}: u(0,x) > 0\right\}$ is, say, the union of two disjoint intervals, then the currently existing analysis would yield a prediction for the flow of roots within the two intervals but it is not at all clear what would happen in the (initially empty) interval between the two intervals on which the distribution is supported. Roots of the iterative derivatives are going to move there but it is not currently clear how this happens. In particular, since our approach predicts conservation laws as well as Hilbert transform identities, these statements are a priori only to be expected for functions for which $\left\{x \in \mathbb{R}: u(0,x) > 0\right\}$ is an interval. If the more general case was understood, it would imply conservation laws and Hilbert transform identities by the same mechanism that we describe here.
(0,0) – (6,0); (1,0) to\[out=60, in=110\] (3,0); (4,0) to\[out=80, in=110\] (5,0); at (3.5, 0.3) [$?$]{};
Infinitely Many Conservation Laws.
----------------------------------
This section presents the main contribution of the paper: if the equation does indeed describe the roots of polynomials under differentiation, then it has infinitely many conservation laws.
Conditional on the equation $$\frac{\partial u}{\partial t} + \frac{1}{\pi} \frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} = 0 \qquad \mbox{on}~\left\{x: u(x) > 0\right\}$$ describing the roots of polynomials under differentiation, there are infinitely many conservation laws for which we can give a closed form expression.
Any smooth solution has to vanish at time $t=1$, so we understand ‘conservation law’ in a flexible sense, i.e. an algebraic identity involving the solution at time $0<t<1$ and the solution at time 0. The first three laws, which are somewhat easier to write down than the subsequent ones, are $$\begin{aligned}
\int_{\mathbb{R}}{ u(t,x) ~ dx} &= 1-t\\
\int_{\mathbb{R}}{ u(t,x) x ~ dx} &= \left(1-t\right)\int_{\mathbb{R}}{ u(0,x) x~ dx}\\
\int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy &= (1-t)^3 \int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy\end{aligned}$$
We will derive the conservation laws in §3. We also emphasize that Theorem 1 is not purely conditional: as derived in [@steini], there are two explicit closed-form solutions that indeed accurately describe the asymptotic behavior of roots: the asymptotic distributions of Hermite polynomials and associated Laguerre polynomials satisfy the partial differential equation. Theorem 1 therefore applies to these two closed-form solutions for which we have thus found an infinite number of conservation laws. Naturally, this raises the question of whether (1) the partial differential equation only correctly predicts the roots of polynomials under differentiation for some very particular classes of polynomials or (2) whether the partial differential equation might have other properties reminiscent of completely integrable systems as well.
Hilbert Transform Identities.
-----------------------------
The classical way to prove a conservation law is to differentiate in time, use the partial differential equation to replace the $\partial_t u$ term and simplify the arising terms (often by integration by parts). This is how we showed $$\int_{\mathbb{R}}{ u(t,x) ~ dx} = 1-t$$ in [@steini]. However, once we proceed to the next conservation law $$\int_{\mathbb{R}}{ u(t,x) x ~ dx} = \left(1-t\right)\int_{\mathbb{R}}{ u(0,x) x~ dx}$$ this becomes slightly more nontrivial. Differentiating in time for $t=0$, we obtain $$\begin{aligned}
\int_{\left\{u > 0 \right\}}{ u(0,x) x~ dx} &= - \frac{\partial}{\partial t}\big|_{t=0} \int_{\left\{u > 0 \right\}}{ u(t,x) x ~ dx}\\
&= \frac{1}{\pi} \int_{\left\{u > 0 \right\}}{ x\frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} ~ dx}\end{aligned}$$ resulting in the identity, for probability distributions $u(0,x)$ for which the support $\left\{x \in \mathbb{R}: u(0,x) > 0\right\}$ is a finite interval, $$\frac{1}{\pi} \int_{\left\{u > 0 \right\}}{ x\frac{\partial}{\partial x}\arctan{ \left( \frac{Hu(0,\cdot)}{ u(0,x)}\right)} ~ dx} = \int_{\left\{u > 0 \right\}}{ u(0,x) x~ dx}.$$ We can remove the scaling condition and obtain a general identity for functions $u:\mathbb{R} \rightarrow \mathbb{R}$ for which $\left\{x: u(x) > 0 \right\}$ is an interval $$\left( \int_{\left\{u > 0 \right\}}{ x\frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} ~ dx} \right) \left( \int_{\mathbb{R}}{ u(x) ~dx}\right) = \pi\int_{\left\{u > 0 \right\}}{ x\cdot u(x)~ dx}.$$ We are not aware of any statement of such a flavor being stated anywhere. Similarly, plugging in the third conservation law suggests, after some minor computation, that, again for smooth, compactly supported, probability densities $u(x)$ for which $\left\{x \in \mathbb{R}: u(x) > 0\right\}$ is a finite interval, $$\begin{aligned}
\int_{\left\{u > 0 \right\}}{ x^2\frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} ~ dx} = 2\pi \int_{\left\{u > 0 \right\}}{ x^2 \cdot u(x) dx} - \pi \left( \int_{\left\{u > 0 \right\}}{ x \cdot u(x) dx} \right)^2\end{aligned}$$ We remark that if any of these identities were to fail on an explicit function $u(x)$ (satisfying all the conditions), then this would imply that the partial differential equation does not accurately model the roots of polynomials distributed according to $u(x)$. Conversely, establishing these identities would prove that the conservation laws are valid along solutions moving in the class of functions for which the identities have been established.
A Monotone Quantity for the Size of the Support.
------------------------------------------------
This section quickly discusses a consequence of a little-known result of Sz-Nagy [@sz]. Conditional on our partial differential equation describing the evolution of the distribution accurately, it will turn into a monotone quantity. We first summarize the original argument of Sz-Nagy. Let $p_n:\mathbb{R} \rightarrow \mathbb{R}$ denote a polynomial of degree $n$ with $n$ roots on the real line $x_1 < x_2 < \dots < x_n$. Sz-Nagy introduces the average distance between consecutive roots $$\mbox{av}(p_n) = \frac{x_n - x_1}{n-1}.$$
If $p_n:\mathbb{R} \rightarrow \mathbb{R}$ is a polynomial of degree at least 3 having $n$ distinct real roots, then $$\emph{av}(p_n') \geq \emph{av}(p_n).$$
The proof is a very simple consequence of a result proved by Sz-Nagy [@sz0] three decades earlier in 1918: let us define the span of such a polynomial by $$\mbox{span}(p_n) = x_n - x_1 = (n-1) \mbox{av}(p_n).$$ The earlier result of Sz-Nagy [@sz0] shows that the span cannot shrink too quickly under differentiation and $$\frac{\mbox{span}(p_n^{(k)})}{ \mbox{span}(p_n)} \geq \sqrt{ \frac{(n-k)(n-k-1)}{n(n-1)}}$$ with equality if and only if (up to symmetries) $$p_n(x) = \left( x^2 - c\right) x^{n-2} \qquad \mbox{for suitable}~c.$$ This, in turn, follows rather easily from Gauss’ electrostatic interpretation (a proof can be found in the book of Rahman & Schmeisser [@rahman]) and implies the original claim. From this we deduce the following monotone quantity.
Conditional on the partial differential equation describing the evolution of roots of polynomials under differentiation, the quantity $$\frac{\left|\left\{x \in \mathbb{R}: u(t,x) > 0 \right\}\right|}{1-t} \qquad \mbox{is non-decreasing in time.}$$
This statement implies that the support cannot shrinker faster than linearly. We observe that this extremal example in the Theorem of Sz-Nagy is quite degenerate, its roots have a singular limiting distribution. This suggests that stronger monotone quantities might exist. We recall that $ u(t,x) = \sqrt{1-t-x^2} $ is a particular solution of the partial differential equation (corresponding to Hermite polynomials) and $$\frac{\left|\left\{x \in \mathbb{R}: u(t,x) > 0 \right\}\right|}{\sqrt{1-t}} = 1.$$ Interestingly, a similar relation holds for the other closed-form solution as well. We recall that the Marchenko-Pastur solution has the following form: for $c>0$, we introduce the function $$v(c,x)= \frac{ \sqrt{(x_+ - x)(x - x_{-})}}{2 \pi x} \chi_{(x_{-}, x_+)} \quad \mbox{where} \quad x_{\pm} = (\sqrt{c + 1} \pm 1)^2.$$ The corresponding Marchenko-Pastur solution of the equation is then given by $$u_c(t,x) = v\left(\frac{c+t}{1-t}, \frac{x}{1 - t}\right).$$ We observe that $$\left\{ u_c(t,x) > 0\right\} = \left( \left(\sqrt{\frac{c+t}{1-t}+1}-1\right)^2, \left(\sqrt{\frac{c+t}{1-t}+1}+1\right)^2 \right)$$ and thus $$\frac{ \left| \left\{ u_c(t,x) > 0\right\} \right|}{\sqrt{1-t}} = 4\sqrt{1+c}$$ which is constant along the flow. One would assume, also in light of the conservation laws discussed above, that the actual scaling should be on the order of $\sqrt{1-t}$. While we do not know whether monotonicity at that scale is generally true, we can show that it is the right order of magnitude.
Conditional on the partial differential equation describing the evolution of roots of polynomials under differentiation, for each sufficiently smooth initial distribution for which $\left\{x: u(t,x) > 0\right\}$ is an interval, we have, for $0 \leq t < 1$, $$\frac{ \left| \left\{ u(t,x) > 0\right\} \right|}{\sqrt{1-t}} \geq \left(\int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy\right)^{1/2} > 0.$$
We use the first conservation law $$\int_{\mathbb{R}}{ u(t,x) ~ dx} = \left(1-t\right)\int_{\mathbb{R}}{ u(0,x) ~ dx}$$ to rewrite the third one as $$\int_{\mathbb{R}} \int_{\mathbb{R}} \frac{u(t,x)}{1-t} (x-y)^2 \frac{u(t,y)}{1-t} ~ dx dy = (1-t) c(u(0,x)),$$ where $c(u(0,x))$ is the value of the left-hand side for $t=0$. We note that $u(t,x)/(1-t)$ can be interpreted as a probability distribution. The integral is invariant under translations, so we can assume that the mean value of the probability distribution is 0. (The first conservation law would imply that this is preserved under the flow but this is not important here). Then the integral simplifies to $$\int_{\mathbb{R}} \int_{\mathbb{R}} \frac{u(t,x)}{1-t} (x-y)^2 \frac{u(t,y)}{1-t} ~ dx dy = 2 \int_{\mathbb{R}}{\frac{u(t,x)}{1-t} x^2 dx}.$$ This quantity is twice the variance of the distribution given by $u(t,x)/(1-t) dx$. An elementary inequality of Bhatia & Davis [@bhatia] implies that if $u(t,x)/(1-t)$ is a probability distribution with mean 0 and $\left\{x:u(t,x)>0\right\} = (m,M)$ (where $m < 0 < M$), then $$2 \int_{\mathbb{R}}{u(t,x) x^2 dx} \leq 2 M(-m).$$ The Cauchy-Schwarz inequality implies $$2 \int_{\mathbb{R}}{u(t,x) x^2 dx} \leq 2 M(-m) \leq m^2 + M^2.$$ Altogether, we have $$(1-t) c(u(0,x)) \leq m^2 + M^2.$$ Altogether $$\left| \left\{ u(t,x) > 0\right\} \right| = |m| + M \geq \sqrt{m^2 + M^2} \geq \sqrt{1-t} \sqrt{ c(u(0,x)) }.$$
Proof of Theorem 1
==================
We explain how to derive these infinitely many conservation laws. We first explain the rough idea and then explicitly derive the first three conservation laws. After that, we explain the general principle in greater detail. The main idea, in short, is as follows. Let $$p_n(x) = x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = \prod_{k=1}^{n}{(x-x_k)}.$$ Multiplication shows, for every $1 \leq k \leq n$, the Vieta formula $$\sum_{1 \leq i_1 < i_2 < \dots < i_k \leq n} \left(\prod_{\ell=1}^{k} x_{i_{\ell}} \right) = (-1)^k a_{n-k}.$$ Let us now consider the $\ell-$th derivative of the polynomial $$p_n^{(\ell)}(x) = \frac{n!}{(n-\ell)!} x^{n-\ell} + \dots = \frac{n!}{(n-\ell)!} \prod_{i=1}^{n-\ell}{(x-y_i)}.$$ Let us fix an integer $k$. The Vieta formula relates the size of all $k-$products of the roots of a polynomial of degree $n$ to the $a_{n-k}-$th coefficient. However, this coefficient $a_{n-k}$ is preserved under differentiation (except for multiplication with explicit factors). This allows us to obtain algebraic expression of the roots and the same expression for the roots of an arbitrary derivative. Algebraic manipulation allows to rewrite these algebraic terms into expressions that converge to integral expressions (if the roots of a polynomial indeed approximate a certain density in the limit). These, in turn, have to be laws that are also obeyed by the partial differential equation if the equation does indeed model roots under differentiation. Special cases are rather easily illustrated.
### The case $k=0$.
The case ‘$k=0$’ can be interpreted as saying that a polynomial of degree $n$ has $n$ roots. Under differentiation, we lose a root at each step. Taking an appropriate limit, we see that we would expect $$\int_{\mathbb{R}}{ u(t,x) ~ dx} = 1-t$$ which was already shown in [@steini].
### The case $k=1$.
The case $k=1$ turns into the statement $$\sum_{i=1}^{n}{x_i} = - a_{n-1}.$$ Let us now consider the $\ell-$th derivative of the polynomial $p_n(x)$ $$p_n^{(\ell)}(x) = \frac{n!}{(n-\ell)!} x^{n-\ell} + a_{n-1} \frac{(n-1)!}{(n-1 - \ell)!} x^{n- \ell -1} + \dots =\frac{n!}{(n-\ell)!} \prod_{i=1}^{n-\ell}{ (x - y_i)}.$$ Normalizing the polynomial to be monic (which has no effect on the roots) and applying again Vieta’s formula shows that $$\sum_{i=1}^{n-\ell}{y_i} = - a_{n-1} \frac{n-\ell}{n}.$$ Letting both $n$ and $\ell$ tend to infinity in such a way that $\ell/n \rightarrow t$, we obtain the identity $$\boxed{ \int_{\mathbb{R}}{ u(t,x) x ~ dx} = \left(1-t\right)\int_{\mathbb{R}}{ u(0,x) x~ dx}.}$$ In particular, if $u(0,x)$ has mean value 0, then this is preserved by the evolution.
### The case $k=2$.
The case $k=2$ presents one with greater algebraic flexibility since the previous laws can be combined in different ways; nonetheless, there is only one conservation law. The Vieta identity for $k=2$ can be written as $$\left(\sum_{i=1}^{n}{x_i}\right)^2 - \sum_{i=1}^{n}{x_i^2} = 2 a_{n-2}.$$ Using the conservation law from $k=1$, we can assume w.l.o.g. that the mean value of the roots is 0 and this is then also preserved under the evolution. Differentiating and arguing as above, we obtain $$\left(\sum_{i=1}^{n-\ell}{y_i}\right)^2 - \sum_{i=1}^{n-\ell}{y_i^2} = \frac{(n-\ell)(n-\ell -1)}{n(n-1)} \left[ \left(\sum_{i=1}^{n}{x_i}\right)^2 - \sum_{i=1}^{n}{x_i^2} \right]$$ Taking again appropriate limits and using the predicted conservation law for $k=1$, we can simplify the statement to $$\int_{\mathbb{R}}{u(0,x)x dx} = 0 \implies\int_{\mathbb{R}}{ u(t,x) x^2 ~ dx} = (1-t)^2 \int_{\mathbb{R}}{ u(0,x) x^2 ~ dx}.$$ A different formulation is in terms of variance and can be found in the book of Rahman & Schmeisser [@rahman Lemma 6.1.5]. Using $y_i$ to denote the roots of $\ell-$th derivative, we have $$\frac{1}{n^2(n-1)} \sum_{1 \leq i < j \leq n}{ (x_i - x_j)^2} = \frac{1}{(n-\ell)^2(n-\ell-1)} \sum_{1 \leq i < j \leq n-\ell}{ (y_i - y_j)^2}.$$ Taking appropriate limits, this predicts the conservation law for $k=2$ in the form $$\boxed{ \int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy = (1-t)^3 \int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy}$$ We emphasize that this second conservation law, coupled with the first two, already gives some insight into what the dynamic looks like. The density increases linearly according to $1-t$ (the $k=0$ conservation law). This, coupled with the conservation law for $k=1$, implies that the mean value is preserved. The conservation law for $k=2$ coupled with the conservation law for $k=0$ implies $$\frac{ \int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy}{ \int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) u(t,y) ~ dx dy} = \mbox{const}(u_0) \cdot (1-t),$$ where the constant $\mbox{const}(u_0)$ depends on the explicit solution. This means that the typical scale of mass distribution as $t \rightarrow 1$ is $(x-y)^2 \sim 1-t$ which shows shrinking at scale $\sqrt{1-t}$ (much like the semicircle solution).
### The case $k=3$.
This is the last case that we derive ‘by hand’. Let us assume the distinct roots are given by $x_1, \dots, x_n$. An algebraic expansion shows that $$\begin{aligned}
\left( \sum_{i=1}^{n}{x_i} \right)^3 &= \sum_{i=1}^{n}{x_i^3} + 3 \sum_{i \neq j}{x_i^2 x_j} + \sum_{i \neq j \neq k}{x_i x_j x_k}\\
&= -2 \sum_{i=1}^{n}{x_i^3} + 3 \sum_{i, j}{x_i^2 x_j} + \sum_{i \neq j \neq k}{x_i x_j x_k} \\
&= -2 \sum_{i=1}^{n}{x_i^3} + 3 \left( \sum_{i=1}^{n}{x_i} \right) \left( \sum_{i=1}^{n}{x_i^2} \right) + \sum_{i \neq j \neq k}{x_i x_j x_k}.\end{aligned}$$ We recall, from the Vieta identities, that $$\sum_{i=1}^{n}{x_i} = - a_{n-1} \qquad \mbox{and} \qquad \sum_{i=1}^{n}{x_i^2} = a_{n-1}^2 - 2a_{n-2}.$$ Moreover, as the new ingredient, $$\sum_{i \neq j \neq k}{x_i x_j x_k} = -6 a_{n-3}.$$ Collecting all these estimates results in $$\sum_{i=1}^{n}{x_i^3} = - a_{n-1}^3 + 3 a_{n-1} a_{n-2} - 3 a_{n-3}.$$ Rewriting everything in terms of power sums, we obtain $$\begin{aligned}
3 a_{n-3} = - \left( \sum_{i=1}^{n}{x_i} \right)^{3} + 3 \left(- \sum_{i=1}^{n}{x_i} \right) \frac{1}{2}\left( \left( \sum_{i=1}^{n}{x_i} \right)^2 - \sum_{i=1}^{n}{x_i^2} \right) - \sum_{i=1}^{n}{x_i^3}.\end{aligned}$$ We know that $a_{n-3}$ behaves under iterated differentiation and normalization to a monic polynomial like $(1-t)^3$. This then results in the same decay behavior for the functional $$\begin{aligned}
J(f(x)) &= \frac{3}{2} \left( \int_{\mathbb{R}}{ f(x) x dx} \right) \left( \left( \int_{\mathbb{R}}{ f(x) x dx} \right)^2 - \int_{\mathbb{R}}{ f(x) x^2 dx} \right) \\
&+ \left( \int_{\mathbb{R}}{ f(x) x dx} \right)^3 + \left( \int_{\mathbb{R}}{ f(x) x^3 dx} \right) \end{aligned}$$ which then satisfies $$J(u(t, x)) = (1-t)^3 \cdot J(u(0, x)).$$ We note that, as is frequently the case, the higher-order conservation laws do not seem to have as straight-forward an interpretation as the first few.
### The general case.
We are now ready to discuss the general case. We use the notation $e_k(x_1, \dots, x_n)$ to denote the $k-$th elementary symmetric polynomial on $n$ variables, i.e. $$\begin{aligned}
e_0(x_1, \dots, x_n) &= 1\\
e_1(x_1, \dots, x_n) &= x_1 + \dots + x_{n}\\
e_2(x_1, \dots, x_n) &= \sum_{i < j}{x_i x_j}\end{aligned}$$ and so on for $k \leq n$. For $k>n$, we set $e_k(x_1, \dots, x_n) = 0$. We define the $k-$th power sum as $$p_k(x_1, \dots, x_n) = x_1^k + x_2^k + \dots + x_n^k.$$ The elementary symmetric polynomials arise naturally from $$\prod_{i=1}^{n}{(x-x_i)} = \sum_{k=0}^{n} (-1)^k e_k(x_1, \dots, x_n) x^{n-k}.$$ If we differentiate such a polynomial $\ell-$times and obtain the roots $y_1, \dots, y_{n- \ell}$, then $$\frac{d}{dx^{\ell}} \sum_{k=0}^{n} (-1)^k e_k(x_1, \dots, x_n) x^{n-k} = \sum_{k=0}^{n-\ell} (-1)^k e_k(x_1, \dots, x_n) \frac{(n-k)!}{(n-k-\ell)!} x^{n-k-\ell}$$ which, normalized to be a monic polynomial, then has the form $$\prod_{i=1}^{n-\ell}{(x-y_i)} = \frac{(n-\ell)!}{n!}\sum_{k=0}^{n-\ell} (-1)^k e_k(x_1, \dots, x_{n}) \frac{(n-k)!}{(n-k-\ell)!} x^{n-k-\ell}.$$ If $k \in \mathbb{N}$ is fixed and $n, \ell$ tend to infinity in such a way that $\ell/n \rightarrow t$ for some $0 < t < 1$, then $$\frac{(n-\ell)!}{n!} \frac{(n-k)!}{(n-k-\ell)!} \sim \frac{(n-\ell)^k}{n^k} \rightarrow (1-t)^k.$$ At the same time, by definition, we have $$\prod_{i=1}^{n-\ell}{(x-y_i)} = \sum_{k=0}^{n-\ell} (-1)^k e_k(y_1, \dots, y_{n-\ell}) x^{n-\ell-k}.$$ This shows that, as $n$ and $\ell$ get large in such a way that $\ell/n\rightarrow t$ converges, $$\frac{e_k(y_1, \dots, y_{n-\ell})}{e_k(x_1, \dots, x_n)} \rightarrow (1-t)^k.$$ Our final ingredient are the Newton’s identities: we can expand the elementary symmetric polynomials in terms of power sums via $$k e_k(x_1, \dots, x_n) = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i}(x_1, \dots, x_n) p_i(x_1, \dots, x_n).$$ We observe that this representation contians also other elementary symmetric polynomials but of lower degree; in particular, an iterative application of the formula yields representation formulas of $e_k$ purely in terms of $p_i$ for $i \leq k$. For example, we have $$\begin{aligned}
e_1 &= p_1\\
2e_2 &= p_1^2 - p_2\\
3e_3 &= \frac{p_1^3}{2} - \frac{3}{2} p_1 p_2 + p_3 \\
4e_4 &= \frac{p_1^4}{6} - p_1^2p_2 + \frac{4 p_1 p_3}{3} + \frac{p_2^2}{2} - p_4.\end{aligned}$$ However, the power sums have a clear limiting behavior since $$p_k(x_1, \dots, x_n) \rightarrow n\int_{\mathbb{R}}{ u(0,x) x^k dx}$$ as $n \rightarrow \infty$ whenever the roots are indeed distributed according to a nice distribution $u(0,x)$. The conservation law for $e_k$, expressible in terms of quantities that are meaningful, decays like $(1-t)^k$ along solutions of the flow.
More on Hilbert Transform Identities
====================================
Some further evidence.
----------------------
We study the conjectured identity $$\left( \int_{\left\{u > 0 \right\}}{ x\frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} ~ dx} \right) \left( \int_{\mathbb{R}}{ u(x) ~dx}\right) = \pi\int_{\left\{u > 0 \right\}}{ x\cdot u(x)~ dx}$$ and linearize it around the explicit function $u(x) =(2/\pi)\sqrt{1-x^2}$. More precisely, we assume the identities are valid and consider the function $$u(x) = \frac{2}{\pi}\sqrt{1-x^2} + \varepsilon f(x)$$ where $f \in C^{\infty}_c(-1,1)$ and we consider the limit $\varepsilon \rightarrow 0$. We use that $$H \frac{2}{\pi}\sqrt{1-x^2} =\frac{2 x}{\pi} \qquad \mbox{for}~-1 < x < 1$$ and Taylor series expansion to compute that, in the limit $\varepsilon \rightarrow 0$, the arising expression is $$\int_{-1}^{1}{ x \frac{\partial}{\partial x} \left( \sqrt{1-x^2} (Hf)(x) \right) dx} = \int_{-1}^{1}{ x f(x) dx}.$$ As it turns out, the arising linearized relation can be explicitly proven.
Let $f \in C^{\infty}_c(-1,1)$. Then $$\int_{-1}^{1}{ x \frac{\partial}{\partial x} \left( \sqrt{1-x^2} (Hf)(x) \right) dx} = \int_{-1}^{1}{ x f(x) dx}.$$
We introduce the Chebyshev polynomials $T_k$ $$T_{0}(x) =1, T_{1}(x) = x \quad \mbox{and} \quad T_{k+1}(x) = 2x T_{k}(x) - T_{k-1}(x).$$ as well as Chebyshev polynomials of the second kind $U_k$ given by $$U_{0}(x) =1, U_{1}(x) = 2x \quad \mbox{and} \quad U_{k+1}(x) = 2x U_{k}(x) - U_{k-1}(x).$$ These sequences of polynomials satisfy for $n,m \geq 1$, $$\frac{2}{\pi}\int_{-1}^{1}{ T_n(x) T_m(x) \frac{dx}{\sqrt{1-x^2}}} = \delta_{nm}$$ and $$\frac{2}{\pi}\int_{-1}^{1}{ U_n(x) U_m(x) \sqrt{1-x^2}dx} = \delta_{nm}.$$ A crucial identity is $$\frac{1}{\pi}\int_{-1}^{1}{\frac{a_k T_k(y)}{(x-y) \sqrt{1-y^2}} dy} = a_k U_{k-1}(x).$$ We now introduce $$g(x) = f(x) \sqrt{1-x^2}$$ and exand $g$ into Chebychev polynomials via $$g(x) = \sum_{k=0}^{\infty}{a_k T_k(x)}.$$ Smoothness of $f$ implies decay of the coefficients allowing us to write $$\begin{aligned}
Hf = H \frac{g}{\sqrt{1-x^2}} = \sum_{k=0}^{\infty} \frac{1}{\pi}{ \int_{-1}^{1}{ \frac{a_k T_k(y)}{(x-y) \sqrt{1-y^2}} dy}} =\sum_{k=1}^{\infty} a_k U_{k-1}(x).
\end{aligned}$$ Thus $$\begin{aligned}
\frac{\partial}{\partial x} \sqrt{1-x^2} (Hf)(x) &= \frac{\partial}{\partial x} \sqrt{1-x^2} \sum_{k=1}^{\infty} a_k U_{k-1}(x) \\
&= -\frac{x}{\sqrt{1-x^2}} \sum_{k=1}^{\infty} a_k U_{k-1}(x) + \sqrt{1-x^2} \sum_{k=1}^{\infty} a_k \frac{\partial}{\partial x} U_{k-1}(x).\end{aligned}$$ We use the identity $$\frac{\partial}{\partial x} U_{k-1}(x) = \frac{k T_k(x) - x U_{k-1}(x)}{x^2-1}$$ to simplify $$\begin{aligned}
\frac{\partial}{\partial x} \sqrt{1-x^2} (Hf)(x) &= \frac{1}{\sqrt{1-x^2}} \sum_{k=1}^{\infty} a_k k T_k(x).\end{aligned}$$ This allows us to compute the contribution of the first term of the first integral since $$\begin{aligned}
\int_{-1}^{1}{ x \frac{\partial}{\partial x} \sqrt{1-x^2} (Hf)(x) dx} &= \int_{-1}^{1}{ x \left( \sum_{k=1}^{\infty} a_k k T_k(x) \right) \frac{dx}{\sqrt{1-x^2}}} \\
&= \int_{-1}^{1}{ T_1(x) \left( \sum_{k=1}^{\infty} a_k k T_k(x) \right) \frac{dx}{\sqrt{1-x^2}}} \\
&= \sum_{k=1}^{\infty} a_k \int_{-1}^{1}{ T_1(x) \left( k T_k(x) \right) \frac{dx}{\sqrt{1-x^2}}} \\
&= \frac{a_1 \pi}{2}.\end{aligned}$$ It remains to compute the integral on the other side. We obtain $$\begin{aligned}
\int_{-1}^{1}{ x f(x) dx} &= \sum_{k=0}^{\infty}a_k \int_{-1}^{1}{ x T_k(x) \frac{dx}{\sqrt{1-x^2} }} \\
&= \sum_{k=0}^{\infty}a_k \int_{-1}^{1}{ T_1(x) T_k(x) \frac{dx}{\sqrt{1-x^2} }} = \frac{a_1 \pi}{2}.\end{aligned}$$
Unbounded Support.
------------------
An interesting question is how many of these arguments would also hold in the case of unbounded support. The derivation of the partial differential equation is, a priori, also meaningful for functions with unbounded support, however, it becomes more difficult to interpret what this could mean in terms of roots of polynomials. However, when the function $u(0,x)$ has sufficient decay, one would certainly expect things to remain meaningful in a certain sense. We illustrate this with the density $$u(0,x) = \frac{e^{-(x-1)^2}}{\sqrt{\pi}}.$$ The translation by 1 is to avoid trivial symmetries further below. The Hilbert transform of $u(0,x)$ has an explicit representation in terms of the Dawson function $$Hu(0,x) = \frac{2}{\pi} \mbox{DawsonF}(x-1).$$ We can explicitly compute $$\frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} = \frac{2}{\sqrt{\pi} }\frac{e^{(x-1)^2}}{1+\mbox{erfi}^2(1-x)}$$ which is sufficient to show that $$\left( \int_{\left\{u > 0 \right\}}{ x\frac{\partial}{\partial x}\arctan{ \left( \frac{Hu}{ u}\right)} ~ dx} \right) \left( \int_{\mathbb{R}}{ u(x) ~dx}\right) = \pi\int_{\left\{u > 0 \right\}}{ x\cdot u(x)~ dx}.$$ In particular, there exists a nonempty set of functions with unbounded support for which some of our arguments apply.
Another Interpretation
----------------------
The purpose of this section is to connect the partial differential equation to complex analysis methods in signal processing and the notion of *instantaneous frequency*. Let us assume $u:[-1,1] \rightarrow \mathbb{R}$ is a smooth function that is positive in $(-1,1)$ and vanishes at the boundary. One thing that would be nice to have is a local measure of how much the function oscillates locally at a point. For example, if $u(x) = \sin{(ax)}$ for some $a > 0$, then this *instantaneous frequency* should be $a$. There is no canonical way to meaningfully defined this notion because of the uncertainty principle (nicely explained in the book by Gröchenig [@gr]). Nonetheless, given the importance of the question in signal processing (which dates back to the 1920s), various methods have been proposed. One basic notion in that context is the *analytic signal*: the function $$f(x) = u(x) + i (Hu)(x)$$ is a complex-valued function on $\mathbb{R}$ that admits a holomorphic extension to the upper-half plane. We can then write this expression in polar coordinates $$f(x) = r(x) e^{i \phi(x)} \qquad \mbox{and} \qquad \frac{d}{dx} \phi(x)$$ is then a possible definition for the instantaneous frequency. We refer to the book of Cohen [@cohen] for more details. However, this can also be expressed as $$\frac{d}{dx} \phi(x) = \frac{d}{dx} \arctan\left(\frac{Hu}{u} \right)$$ which is exactly the driving term in the partial differential equation. Put differently, the speed of transport is determined by the instantaneous frequency. We believe this to be more than just an algebraic coincidence and fundamental for any deeper understanding of the partial differential equation. It presumably also suggests a way of establishing the identities for the Hilbert transform though it is not clear to us at this point how one would establish all of them in a unified manner.
A Connection to Random Matrices?
================================
Similarities.
-------------
Various objects that appear in this paper also appear in the context of random matrices. In particular, both the semircircle distribution $$\frac{2}{\pi} \sqrt{1-x^2}$$ as well as the Marchenko-Pastur distribution $$\frac{ \sqrt{(x_+ - x)(x - x_{-})}}{2 \pi x} \chi_{(x_{-}, x_+)} \quad \mbox{where} \quad x_{\pm} = (\sqrt{c + 1} \pm 1)^2$$ arise naturally as limiting statistics of random matrices (Gaussian random matrices and Wishart random matrices, respectively). Both in random matrix theory and our approach, the Hilbert transform (or Cauchy transform, respectively) play a crucial role. Another common occurence is the appearance of $$\mbox{the $k-$th moment}~\int_{\mathbb{R}}{ x^k d\mu},$$ which is both a crucial ingredient in random matrix theory but also the elementary quantities in which our conservation laws are phrased. The following fact was pointed out to us by Christopher Xue [@xue]: if $A \in \mathbb{R}^{n \times n}$ is a symmetric matrix and $p_A(x)$ is its characteristic polynomial, then $$p_A'(x) = \sum_{i=1}^{n}{ p_{A_i}(x)},$$ where $p_{A_i}$ is the characteristic polynomial of the $i-$th minor of $A$ (obtained from $A$ by deleting the $i-$th row and column). Moreover, by Cauchy’s interlacing theorem, the eigenvalues of each minor are interlacing. This strongly suggests a connection between our partial differential equation and the eigenvalue distribution of submatrices of random matrices.
A Conjecture.
-------------
One specific interpretation could be the following: if we take a *random* minor $A_i$ (by picking $i \in \left\{1, \dots, n\right\}$ uniformly at random), then a first guess would be that the expected eigenvalue of $A_i$ between two consecutive eigenvalues of $A$ should behave approximately like in the case of differentiation. In particular, if we pick $k$ successive minors at random where $k \sim t n$ is large compared to $n$, then there might be an emerging smoothing effect of the same type. The simple example of a diagonal matrix shows that any type of limiting statement cannot be true for fixed matrices. We arrive at the following question.
> **Question.** Let $\lambda_1, \dots, \lambda_n$ be $n$ distinct numbers drawn from a sufficiently smooth distribution $u(0,x)$. Fix a randomly chosen orthonormal basis $\left\{v_1, \dots, v_n\right\}$ of $\mathbb{R}^n$ and consider the matrix $$A = \sum_{i=1}^{n}{ \lambda_i v_i \otimes v_i}.$$ Let $J \subset \left\{1, 2, \dots, n\right\}$ be a subset of size $(1-t) \cdot n$ where $0 < t < 1$ and let $B$ be matrix induced by restricting $A$ onto rows and columns indexed by $J$. Are the eigenvalues of $B$ predicted by the PDE started at $u(0,x)$ at time $t$?
We carried out the following explicit experiment: we fixed $n=200$ and created a random orthonormal basis $\left\{v_1, \dots, v_{200}\right\}$. We then considered a specific matrix $$A = \sum_{k=1}^{200} \frac{k}{200} v_i \otimes v_i$$ and then considered 1000 random principal submatrices of size 100. We sort their eigenvalues by size and then average the $\ell-$th eigenvalue for $1 \leq \ell \leq 100$.
at (0,0) ; (-2.4,-1.4) – (2.5,-1.4); (-2.05,-1.4) circle (0.04cm); at (-2.05, -1.7) [0]{}; (2.05,-1.4) circle (0.04cm); at (2.05, -1.7) [1]{};
at (0,0) ; (-2.4,-1.4) – (2.5,-1.4); (-2.05,-1.4) circle (0.04cm); at (-2.05, -1.7) [0]{}; (2.05,-1.4) circle (0.04cm); at (2.05, -1.7) [1]{};
We compare this to the roots of the $100-$th derivative of $$q(x) = \prod_{k=1}^{200}{ \left(x-\frac{k}{200}\right)}.$$ The prediction would be that the eigenvalues following from a random submatrix follow exactly the distribution of roots of the derivative of $q$. Both distributions are shown in Figure 4 and seem to confirm this suspicion. The question can be interpreted in a variety of ways. Possibly the most accessible is to interpret it as a statement for the arising distribution *averaged* over all choices of orthonormal basis. However, we point out that the numerical experiments reported above were carried out on a single explicit matrix generated by a single explicit orthonormal basis in $\mathbb{R}^n$. A stronger formulation of the problem would thus be the existence of a concentration phenomenon: is it true that among all bases in $\mathbb{R}^n$ all but an exponentially small (in $n$) set has the property that the eigenvalue distribution of a random principial matrix is approximately described by the solution of the partial differential equation? A simpler version, circumventing the connection between the PDE and the roots of a polynomial, would be to phrase it in terms of roots of the characteristic polynomial: is there a smoothing effect acting on iterated random choice of principial minors that has an effective limit given by differentiation of the characteristic polynomial?
[10]{}
M. Aigner and G. Ziegler, Proofs from The Book. Springer-Verlag, Berlin, 1999.
R. Bhatia and C, Davis, A Better Bound on the Variance. American Mathematical Monthly. Mathematical Association of America. 107 (4): 353–357
S. Bochner: Book review of “Gesammelte Schriften” by Gustav Herglotz, Bulletin Amer. Math. Soc. 1 (1979), 1020-1022
N. G. de Bruijn. On the zeros of a polynomial and of its derivative. Nederl. Akad. Wetensch., Proc., 49:1037–1044 = Indagationes Math. 8, 635–642 (1946), 1946.
N. G. de Bruijn and T. A. Springer. On the zeros of a polynomial and of its derivative. II. Nederl. Akad. Wetensch., Proc., 50:264–270=Indagationes Math. 9, 458–464 (1947), 1947.
B. Curgus and V. Mascioni, A Contraction of the Lucas Polygon, Proc.Amer.Math.Soc. 132, 2004, 2973–2981.
L. Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, 1994.
D. Dimitrov, A Refinement of the Gauss-Lucas Theorem, Proc. Amer. Math. Soc. 126, 1998, 2065–2070.
P. Erdős and P. Turán, On interpolation. III. Interpolatory theory of polynomials. Ann. of Math. (2) 41, (1940). 510–553.
P. Erdős and G. Freud, On orthogonal polynomials with regularly distributed zeros. Proc. London Math. Soc. (3) 29 (1974), 521–537.
D. Farmer and R. Rhoades, Differentiation evens out zero spacings. Trans. Amer. Math. Soc. 357 (2005), no. 9, 3789–3811.
C.F. Gauss: Werke, Band 3, Göttingen 1866, S. 120:112
W.Gawronski, Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials $L_n^{(\alpha n+\alpha)}$ and Hermite polynomials $H_n^{(\alpha n+\alpha)}$ , Analysis 13 (1-2) (1993) 29–67.
R. Granero-Belinchon, On a nonlocal differential equation describing roots of polynomials under differentiation, arXiv:1812.00082
K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser (Basel), 2001.
B. Hanin, Pairing of zeros and critical points for random polynomials, Ann. Inst. H. Poincaré, Probab. Statist. 53(2017), 1498–1511.
Z. Kabluchko, Critical points of random polynomials with independent identically distributed roots. Proc. Amer. Math. Soc. 143 (2015), no. 2, 695–702.
M. Kornyik and G. Michaletzky, On the moments of roots of Laguerre-polynomials and the Marchenko-Pastur law. Ann. Univ. Sci. Budapest. Sect. Comput. 46 (2017), 137–151.
F. Lucas: Sur une application de la Mécanique rationnelle à la théorie des équations. in: Comptes Rendus de l’Académie des Sciences (89), Paris 1979, S. 224–226
S. M. Malamud, Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc., 357(2005), 4043–4064.
A. Martinez-Finkelshtein, P. Martinez-Gonzalez and R. Orive, On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters. Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999). J. Comput. Appl. Math. 133 (2001), no. 1-2, 477–487.
S. O’Rourke and N. Williams, Pairing between zeros and critical points of random polynomials with independent roots, arXiv:1610.06248, to appear in Trans. Amer. Math. Soc.
S. O’Rourke and S. Steinerberger, A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation, arXiv:1910.12161
R. Pemantle, and I. Rivlin. The distribution of the zeroes of the derivative of a random polynomial. Advances in Combinatorics. Springer 2013. pp. 259–273.
R. Pereira, Which quartic polynomials have a hyperbolic antiderivative?, in:Analysis of Operators on Function Spaces: The Serguei Shimorin Memorial Volume, Birkhauser (Basel), 2019
G. Polya, Some Problems Connected with Fourier’s Work on Transcendental Equations, The Quarterly Journal of Mathematics 1, 1930, p. 21–34.
Q. Rahman and G. Schmeisser, Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties, London Mathematical Society Monographs, Clarendon Press, 2002.
M. Ravichandran, Principal Submatrices, Restricted Invertibility, and a Quantitative Gauss–Lucas Theorem, IMRN, to appear
S. Steinerberger, A Nonlocal Transport Equation Describing Roots of Polynomials Under Differentiation, Proc. Amer. Math. Soc, to appear
A. Stoyanoff, Sur un Theorem de M. Marcel Riesz, Nouv. Annal. de Mathematique, 1 (1926), 97–99.
D. M. Souroujon and T. S. Stoyanov. About the primitive polynomials of polynomials with real zeros. Journal of Analysis and Applications, 14(1):21–31, 2016.
G. Sz-Nagy, Uber algebraische Gleichungen mit lauter reellen Nullstellen, Jahresbericht der D. M. V., 27 (1918), S. 37–43
G. Sz-Nagy, Uber Polynome mit lauter reellen Nullstellen, Acta Mathematica Academiae Scientiarum Hungarica 1, p. 225–228, (1950).
V. Totik, The Gauss-Lucas theorem in an asymptotic sense, Bull. London Math. Soc. 48, 2016, p. 848–854.
J. L. Ullman, On the regular behaviour of orthogonal polynomials. Proc. Lond. Math. Soc. 24 (1972), 119–148.
W. Van Assche, Asymptotics for orthogonal polynomials. Lecture Notes in Mathematics, 1265. Springer-Verlag, Berlin, 1987
C. Xue, private communication, 17. Sep 2019
[^1]: S.S. is supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.
|
---
abstract: 'We tackle a novel few-shot learning challenge, which we call few-shot semantic edge detection, aiming to localize crisp boundaries of novel categories using only a few labeled samples. We also present a Class-Agnostic Few-shot Edge detection Network (CAFENet) based on meta-learning strategy. CAFENet employs a semantic segmentation module in small-scale to compensate for lack of semantic information in edge labels. The predicted segmentation mask is used to generate an attention map to highlight the target object region, and make the decoder module concentrate on that region. We also propose a new regularization method based on multi-split matching. In meta-training, the metric-learning problem with high-dimensional vectors are divided into small subproblems with low-dimensional sub-vectors. Since there is no existing dataset for few-shot semantic edge detection, we construct two new datasets, FSE-1000 and SBD-$5^i$, and evaluate the performance of the proposed CAFENet on them. Extensive simulation results confirm the performance merits of the techniques adopted in CAFENet. few-shot edge detection, few-shot learning, semantic edge detection'
author:
- 'Young-Hyun Park'
- Jun Seo
- Jaekyun Moon
bibliography:
- 'egbib.bib'
title: |
CAFENet: Class-Agnostic Few-Shot\
Edge Detection Network
---
Introduction
============
Semantic edge detection aims to identify pixels that belong to boundaries of predefined categories. It is shown that semantic edge detection is useful for a variety of computer vision tasks such as semantic segmentation [@arbelaez2010contour; @bertasius2015high; @bertasius2016semantic; @chen2016semantic; @yu2015generalized], object reconstruction [@ferrari2007groups; @ullman1989recognition; @zhu2018semantic], image generation [@isola2017image; @wang2018high] and medical imaging [@abbass2017edge; @mehena2019medical]. Early edge detection algorithms interpret the problem as a low-level grouping problem exploiting hand-crafted features and local information [@canny1986computational; @hancock1990edge; @sugihara1986machine]. Recently, there have been significant improvements on edge detection thanks to the advances in deep learning [@bertasius2015deepedge; @he2019bi; @hwang2015pixel; @xie2015holistically]. Moreover, beyond previous boundary detection, category-aware semantic edge detection became possible [@acuna2019devil; @hu2019dynamic; @yu2017casenet; @yu2018simultaneous]. However, it is impossible to train deep neural networks without massive amounts of annotated data.
To overcome the data scarcity issue in image classification, few-shot learning has been actively discussed for recent years [@finn2017model; @lifchitz2019dense; @snell2017prototypical; @vinyals2016matching]. Few-shot learning algorithms train machines to learn previously unseen classification tasks using only a few relevant labeled examples. More recently, the idea of few-shot learning is applied to computer vision tasks requiring highly laborious and expensive data labeling such as semantic segmentation [@dong2018few; @shaban2017one; @wang2019panet] and object detection [@fu2019meta; @kang2019few; @karlinsky2019repmet]. Based on meta-learning across varying tasks, the machines can adapt to unencountered environments and demonstrate robust performance in various computer vision problems. In this paper, we consider a novel few-shot learning challenge, few-shot semantic edge detection, to detect the semantic boundaries using only a few labeled samples. To tackle this elusive challenge, we also propose a class-agnostic few-shot edge detector (CAFENet) and present new datasets for evaluating few-shot semantic edge detection.
![Architecture overview of the proposed CAFENet. The feature extractor or encoder extracts feature from the image, the segmentator generates a segmentation mask based on metric learning, and the edge detector detects semantic boundaries using the segmentation mask and query features.[]{data-label="fig:fig_intro_overview"}](Intro_overview.pdf){width="\textwidth"}
Fig. \[fig:fig\_intro\_overview\] shows the architecture of the proposed CAFENet. Since the edge labels do not contain enough semantic information due to the sparsity of labels, performance of the edge detector severely degrades when the training dataset is very small. To overcome this, we adopt the segmentation process in advance of detecting edge with downsized feature and segmentation labels generated from boundaries labels. We utilize a simple metric-based segmentator generating a segmentation mask through pixel-wise non-parametric feature matching with class prototypes, which are computed by masked average pooling of [@zhang2018sg]. The predicted segmentation mask provides the semantic information to the edge detector. The multi-scale attention maps are generated from the segmentation mask, and applied to corresponding multi-scale features. The edge detector predicts the semantic boundaries using the attended features. Using this attention mechanism, the edge detector can focus on relevant regions while alleviating the noise effect of external details.
For meta-training of CAFENet, we introduce a simple yet powerful regularization method, Multi-Split Matching Regularization (MSMR), performing metric learning on multiple low-dimensional embedding sub-spaces. During meta-training, the model is meta-learned to minimize distances between the pixels of query features and the prototype of same class for segmentation. At the same time, the prototypes and pixels of feature vectors are divided into multiple low-dimensional splits and the model also learns to minimize distances between the pixels of query feature splits and their corresponding prototype splits of the same class. The proposed MSMR method can achieve significant performance gain without additional learnable parameters. To sum up, the main contributions of this paper are as follows:
We introduce the few-shot semantic edge detection problem for performing semantic edge detection on previously unseen objects using only a few training examples.
We propose to generate an attention map using a segmentation prediction mask and attend convolutional features so as to localize semantically important regions prior to edge detection.
We introduce a novel MSMR regularization method dividing high-dimensional vectors into low-dimensional sub-vectors and conduct metric-learning sub-problems for few-shot semantic edge detection.
We introduce two datasets for few-shot semantic edge detection and validate the proposed CAFENet techniques.
Related Work
============
Few-shot Learning
-----------------
To tackle the few-shot learning challenge, many methods have been proposed based on meta-learning. Optimization-based methods [@finn2017model; @ravi2016optimization; @santoro2016meta] train the meta-learner which updates the parameters of the actual learner so that the learner can easily adapt to a new task within a few labeled samples. Metric-based methods [@fink2005object; @koch2015siamese; @snell2017prototypical; @sung2018learning; @vinyals2016matching; @yoon2019tapnet] train the feature extractor to assemble features from the same class together on the embedding space while keeping features from different classes far apart. Recent metric-based approaches propose dense classification [@hou2019cross; @kye2020transductive; @lifchitz2019dense; @qi2018low]. Dense classification trains an instance-wise classifier on pixel-wise classification loss which imposes coherent predictions over the spatial dimension and prevents overfitting as a result. Our model adopts the metric-based method for few-shot learning. Inspired by dense classification, we propose multi-split matching regularization which divides the feature vector into sub-vector splits and performs split-wise classification for regularization in meta-learning
Few-shot Semantic Segmentation
------------------------------
The goal of few-shot segmentation is to perform semantic segmentation within a few labeled samples based on meta-learning [@dong2018few; @rakelly2018conditional; @shaban2017one; @wang2019panet; @zhang2019canet]. OSLSM of [@shaban2017one] adopts a two-branch structure: conditioning branch generating element-wise scale and shift factor using the support set and segmentation branch performing segmentation with a fully convolutional network and task-conditioned features. Co-FCN [@rakelly2018conditional] also utilizes a two-branch structure. The globally pooled prediction is generated using support set in conditioning branch, and fused with query features to predict segmentation mask in segmentation branch. SG-One of [@zhang2018sg] proposes a masked average pooling to compute prototypes from pixels of support features. The cosine similarity scores are computed between the prototypes and pixels of query feature, and the similarity map guides the segmentation process. CANet of [@zhang2019canet] also adopts masked average pooling to generate the global feature vector, and concatenate it with every location of the query feature for dense comparison in predicting the segmentation mask. PANet of [@wang2019panet] introduces prototype alignment for regularization, to predict the segmentation mask of support samples using query prediction results as labels of query samples.
Semantic Edge Detection
-----------------------
Semantic edge detection aims to find the boundaries of objects from image and classify the objects at the same time. The history of semantic edge detection [@acuna2019devil; @hariharan2011semantic; @hu2019dynamic; @liu2018semantic; @prasad2006learning; @yu2017casenet; @yu2018simultaneous] dates back to the work of [@prasad2006learning] which adopts the support vector machine as a semantic classifier on top of the traditional canny edge detector. Recently, many semantic edge detection algorithms rely on deep neural network and multi-scale feature fusion. CASENET of [@yu2017casenet] addresses the semantic edge detection as a multi-label problem where each boundary pixel is labeled into categories of adjacent objects. Dynamic Feature Fusion (DFF) of [@hu2019dynamic] proposes a novel way to leverage multi-scale features. The multi-scale features are fused by weighted summation with fusion weights generated dynamically for each images and each pixel. Meanwhile, Simultaneous Edge Alignment and Learning (SEAL) of [@yu2018simultaneous] deals with severe annotation noise of the existing edge dataset [@hariharan2011semantic]. SEAL treats edge labels as latent variables and jointly trains them to align noisy misaligned boundary annotations. Semantically Thinned Edge Alignment Learning (STEAL) of [@acuna2019devil] improves the computation efficiency of edge label alignment through a lightweight level set formulation. In addition, STEAL optimizes the model for non-maximum suppression (NMS) during training while previous works use NMS at the postprocessing step.
Problem Setup
=============
For few-shot semantic edge detection, we use train set $D_{train}$ and test set $D_{test}$ consisting of non-overlapping categories $C_{train}$ and $C_{test}$. The model is trained only using $C_{train}$, and the test categories $C_{test}$ are never seen during the training phase. For meta-training of the model, we adopt episodic training as done in many previous few-shot learning works. Each episode is composed of a support set with a few-labeled samples and a query set. When an episode is given, the model adapts to the given episode using the support set and detect semantic boundaries of the query set. By episodic training, the model is learned so that it adapts to the unseen class using only a few labeled samples and predict semantic edges of query samples.
For $N_c$-way $N_s$-shot setting, each training episode is constructed by $N_c$ classes sampled from $C_{train}$. When $N_c$ categories are given, $N_s$ support samples and $N_q$ query samples are randomly chosen from $D_{train}$ for each class. In evaluation, the performance of the model is evaluated using test episodes. The test episodes are constructed in the same way as the training episodes, except $N_c$ classes and corresponding support and query samples are sampled from $C_{test}$ and $D_{test}$.
In this work, we address $N_c$-way $N_s$-shot semantic edge detection. The goal is training the model to generalize to $N_c$ unseen classes given only $N_s$ images and their edge labels. Based on the few labeled support samples, the model should produce edge predictions of query images which belong to $N_c$ unencountered classes.
![Network architecture overview of proposed CAFENet. ResNet-34 encoder $E^{(1)} \sim E^{(4)}$ extracts multi-level semantic features. The segmentator module generates a segmentation prediction using query feature from $E^{(4)}$ and prototypes $P_{FG}, P_{BG}$ from support set features. Small bottleneck blocks ${S}^{(0)}\sim {S}^{(4)}$ transform the original image and multi-scale features from encoder blocks to be more suitable for edge detection. The attention maps generated from segmentation prediction are applied to multi-scale features to localize the semantically related region. Decoder $D^{(0)} \sim D^{(4)}$ takes attentive multi-scale features to give edge prediction. []{data-label="fig:overview"}](model_overview.pdf){width="90.00000%"}
Method
======
We propose a novel algorithm for few-shot semantic edge detection. Fig. \[fig:overview\] illustrates the network architecture of the proposed method. The proposed CAFENet adopts the semantic segmentation module to compensate for the lack of semantic information in edge labels. The predicted segmentation mask is utilized for attention in skip connection. The final edge detection is done using attentive multi-scale features.
Semantic Segmentator
--------------------
Most previous works on semantic edge detection directly predict edges from the given input image. However, direct edge prediction is a hard task when only a few labeled samples are given. To overcome this difficulty in few-shot edge detection, we adopt a semantic segmentation module in advance of edge prediction. With the assistance of the segmentation module, CAFENet can effectively localize the target object and extract semantic features from query samples. For few-shot segmentation, we employ the metric-learning which utilizes prototypes for foreground and background as done in [@dong2018few; @wang2019panet]. Given the support set $S=\{x^{s}_{i},y^{s}_{i}\}^{N_s}_{i=1}$, the encoder $E$ extracts features $\{E(x^{s}_{i})\}^{N_s}_{i=1}$ from $S$. Also, for support labels $\{y^{s}_{i}\}^{N_s}_{i=1}$, we generate the dense segmentation mask $\{M^{s}_{i}\}^{N_s}_{i=1}$ using a rule-based preprocessor, considering the pixels inside the boundary as foreground pixels in the segmentation label. Using down-sampled segmentation labels $\{m^{s}_{i}\}^{N_s}_{i=1}$, the prototype for foreground pixels $P_{FG}$ is computed as $$\begin{aligned}
P_{FG} = \frac{1}{N_s}\frac{1}{H\times{W}}\sum_{i}\sum_{j}E_j(x^{s}_{i})m^{s}_{i,j}
\end{aligned}$$ where $j$ indexes the pixel location, $E_j(x)$ and $m^{s}_{i,j}$ denote the $j$th pixel of feature $E(x)$ and segmentation mask $m^{s}_{i}$. $H,W$ denote height and width of the images. Likewise, the background prototype $P_{BG}$ is computed as $$\begin{aligned}
P_{BG} = \frac{1}{N_s}\frac{1}{H\times{W}}\sum_{i}\sum_{j}E_j(x^{s}_{i})(1-m^{s}_{i,j}).
\end{aligned}$$
Following the prototypical networks of [@snell2017prototypical], the probability that pixel $j$ belongs to foreground for the query sample $x^{q}_{i}$ is [ $$\begin{aligned}
p(y_{i,j}^q=FG|x^{q}_{i};E) = \frac{exp( -\tau d(E_j(x^{q}_{i}),P_{FG}))}{exp(-\tau d(E_j(x^{q}_{i}),P_{FG}))+exp(-\tau d(E_j(x^{q}_{i}),P_{BG}))}
\label{eq:sm}
\end{aligned}$$]{} where $d(\cdot,\cdot)$ is squared Euclidean distance between two vectors and $\tau$ is a learnable temperature parameter used in [@gidaris2018dynamic; @qi2018low]. With query samples $\{x^{q}_{i}\}^{N_q}_{i=1}$ and the down-sampled segmentation labels for query $\{m^{q}_{i}\}^{N_q}_{i=1}$, the segmentation loss $L_{Seg}$ is calculated as the mean-squared error (MSE) loss between predicted probabilities and the down-sized segmentation mask $$\begin{aligned}
L_{Seg} = \frac{1}{N_q}\frac{1}{H\times{W}}\sum_{i=1}^{N_q}\sum_{j=1}^{H\times{W}}\{(p(y_{i,j}^q=FG|x^{q}_{i};E)-m_{i,j}^q)^{2}\}.
\label{eq:original_seg_loss}
\end{aligned}$$ Note that the segmentation mask is generated in down-sized scale so that any pixel near the boundaries can be classified into the foreground to some extent, as well as the background. Therefore, we approach the problem as a regression using MSE loss rather than cross entropy loss.
[0.47]{} ![Comparison between (a) High-dimensional feature matching used in [@dong2018few; @wang2019panet] and (b) split-wise feature matching in MSMR[]{data-label="fig:image2"}](Pixel-wise_Matching.pdf "fig:"){width="1\linewidth"}
[0.47]{} ![Comparison between (a) High-dimensional feature matching used in [@dong2018few; @wang2019panet] and (b) split-wise feature matching in MSMR[]{data-label="fig:image2"}](Multi-split_Matching.pdf "fig:"){width="1\linewidth"}
Multi-Split Matching Regularization
-----------------------------------
The metric-based few-shot segmentation method utilizes distance metrics between the high-dimensional feature vectors and prototypes, as seen in Fig. \[fig:subim1\]. However, this approach is prone to overfit due to the massive number of parameters in feature vectors. To get around this issue, we propose a novel regularization method, multi-split matching regularization (MSMR). MSMR inherits the spirit of dense feature matching [@kye2020transductive] where pixel-wise feature matching acts as a regularizer for high-dimensional embedding. In MSMR, high-dimensional feature vectors are split into several low-dimensional feature vectors, and the metric learning is conducted on each vector split as Fig. \[fig:subim2\].
With the query feature $E(x_i^q) \in \mathbb{R}^{C\times{W}\times{H}} $, where $C$ is channel dimension and $H,W$ are spatial dimensions, we divide $E(x_i^q)$ into $K$ sub-vectors $\{E^{k}(x_i^q)\}^{K}_{k=1}$ along channel dimension. Each sub-vector $E^{k}(x_i^q)$ is in $\mathbb{R}^{\frac{C}{K}\times{W}\times{H}}$. Likewise, the prototypes $P_{FG}$ and $P_{BG}$ are also disassembled into $K$ sub-vectors $\{P^{k}_{FG}\}^{K}_{k=1}$ and $\{P^{k}_{BG}\}^{K}_{k=1}$ along channel dimension where $P^{k}_{FG}, P^{k}_{BG} \in{\mathbb{R}^{\frac{C}{K}}}$ . For the $k^{th}$ sub-vector of query feature $E^{k}(x_i^q)$, the probability that the $j^{th}$ pixel belongs to the foreground class is computed as follows: [ $$\begin{aligned}
p^{k}(y_{i,j}^q=FG|x_{i}^q;E) = \frac{exp( -\tau d(E_j^k(x^{q}_{i}),P^k_{FG}))}{exp(-\tau d(E_j^k(x^{q}_{i}),P^k_{FG}))+exp(-\tau d(E_j^k(x^{q}_{i}),P^k_{BG}))}.
\label{eq:sm2}
\end{aligned}$$]{} Multi-split matching regularization divides the original metric learning problem into $K$ small sub-problems composed of a fewer parameters and acts as regularizer for high-dimensional embeddings. The prediction results of $K$ sub-problems are reflected on learning by combining the split-wise segmentation losses to original segmentation loss in Eq.. The total segmentation loss is calculated as $$\begin{aligned}
L_{Seg} = \frac{1}{N_q}\frac{1}{H\times{W}}\sum_{i=1}^{N_q}\sum_{j=1}^{H\times{W}}\{(p_{i,j}-m_{i,j}^q)^{2}+\sum_{i=1}^{K} (p^{k}_{i,j}-m_{i,j}^q)^{2}\}.
\end{aligned}$$ where $p_{i,j}=p(y_{i,j}^q=FG|x^{q}_{i};E)$ and $p^k_{i,j}=p^k(y_{i,j}^q=FG|x^{q}_{i};E)$.
Attentive Edge Detector
-----------------------
As shown in Fig. \[fig:overview\], we adopt the nested encoder structure of [@liu2017richer; @xie2015holistically] to extract rich hierarchical features. The multi-scale side outputs from encoder $E^{(1)}\sim E^{(4)}$ are post-processed through bottleneck blocks $S^{(1)}\sim S^{(4)}$. Since ResNet-34 gives side outputs of down-sized scale, we pass the original image through bottleneck block $S^{(0)}$ to extract local details in original scale. In front of $S^{(3)}$, we employ the Atrous Spatial Pyramid Pooling (ASPP) block of [@chen2017deeplab]. We have empirically found that locating ASPP there shows better performance.
In utilizing multi-scale features, we employ the predicted segmentation mask $\hat{M}$ from the segmentator where the $j^{th}$ pixel of $\hat{M}$ is the predicted probability from Eq. . Note that we generate $\hat{M}$ based on the entire feature vectors and the prototypes instead of utilizing sub-vectors, since the split-wise metric learning is used only for regularizing the segmentation module. For each layer $l$, $\hat{M}^{(l)}$ denotes the segmentation mask upscaled to the corresponding feature size by bilinear interpolation. Using segmentation prediction mask $\hat{M}^{(l)}$, we generate attention map $A^{(l)}$, as follows. First, the prediction with a value lower than threshold $\lambda$ is rounded down to zero, to ignore activation in regions with low confidence. Second, we broaden the attention map using morphological dilation of [@feng2019attentive] as a second chance, since the segmentation module may not always guarantee fine results. The final attention map of $l^{th}$ layer $A^{(l)}$ is computed as follows $$\begin{aligned}
A^{(l)} = \mathds{1}(\hat{M}^{(l)}>\lambda){\hat{M}}^{(l)} + Dilation(\mathds{1}(\hat{M}^{(l)}>\lambda){\hat{M}}^{(l)})
\end{aligned}$$ where $\mathds{1}(\hat{M}^{(l)}>\lambda){\hat{M}}^{(l)}$ is the rounded value of prediction mask $\hat{M}^{(l)}$. The attention maps are applied to the multi-scale features of corresponding bottleneck blocks $S^{(0)}\sim S^{(4)}$. We apply the residual attention of [@hou2019cross], where the initial multi-level side outputs from $S^{(l)}$ are pixel-wisely weighted by $1 + A^{(l)}$, to strengthen the activation value of the semantically important region. We visualize the effect of semantic attention in Fig. \[fig:fig\_attention\].
![An example of activation map of [@yosinski2015understanding] before and after pixel-wise semantic attention (warmer color has higher value). As seen, the attention mechanism makes encoder side-outputs attend to the regions of the target object (*horse* in the figure). \[fig:fig\_attention\] ](Semantic_Attention.pdf){width="\textwidth"}
As shown in Fig. \[fig:overview\], the decoder network is composed of five consecutive convolutional blocks. Each decoder block $D^{(l)}$ contains three $3\times{3}$ convolution layers. The outputs of decoder blocks $D^{(1)}\sim D^{(4)}$ are bilinearly upsampled by two and passed to the next block. Similar to [@feng2019attentive], the up-sampled decoder outputs are then concatenated to the skip connection features from bottleneck blocks $S^{(0)}\sim S^{(4)}$ and previous decoder blocks. Multi-scale semantic information and local details are transmitted through skip architectures. The hierarchical decoder network in turn refines the outputs of the previous decoder blocks and finally produces the edge prediction $\hat{y}^{q}_{i}$ of query samples $x^{q}_{i}$.
Following the work of [@deng2018learning], we combine cross-entropy loss and Dice loss to produce crisp boundaries. Given a query set $Q=\{x^{q}_{i},y^{q}_{i}\}^{N_{q}}_{i=1}$ and prediction mask $\hat{y}^{q}_{i}$, the cross-entropy loss is computed as $$\begin{aligned}
L_{CE} = - \sum_{i=1}^{N_{q}}\{\sum_{j\in Y_{+}}log(\hat{y}^{q}_{i}) + \sum_{j\in Y_{-}}log(1-\hat{y}^{q}_{i})
\}
\end{aligned}$$ where $Y_{+}$ and $Y_{-}$ denote the sets of foreground and background pixels. The Dice loss is then computed as $$\begin{aligned}
L_{Dice} = \sum_{i=1}^{N_{q}}\{\frac{\sum_{j}(\hat{y}^{q}_{i,j})^2 + \sum_{j}({y}^{q}_{i,j})^2}{2\sum_{j}\hat{y}^{q}_{i,j}{y}^{q}_{i,j}} \}
\end{aligned}$$ where j denotes the pixels of a label. The final loss for meta-training is given by
$$\begin{aligned}
L_{final} = L_{Seg} + L_{CE} + L_{Dice}.
\end{aligned}$$
Experiments
===========
Datasets
--------
### FSE-1000
The datasets used in previous semantic edge detection research such as SBD of [@hariharan2011semantic] and Cityscapes of [@cordts2016cityscapes] are not suitable for few-shot learning as they have only 20 and 30 classes, respectively. We propose a new dataset for few-shot edge detection, which we call FSE-1000, based on FSS-1000 of [@wei2019fss]. FSS-1000 is a dataset for few-shot segmentation and composed of 1000 classes and 10 images per class with foreground-background segmentation annotation. From the images and segmentation masks of FSS-1000, we build FSE-1000 by extracting boundary labels from segmentation masks. In the light of difficulty associated with few-shot setting, we extract thick edges of which thickness is around 2 $\sim$ 3 pixels on average. For dataset split, we split 1000 classes into 800 training classes and 200 test classes. We will provide the detailed class configuration in the Supplementary Material.
![Qualitative examples of 5-shot edge detection on FSE-1000 dataset.[]{data-label="fig:Qualitative_result_FSE"}](Qualitative_Result_FSE.pdf){width="\textwidth"}
### SBD-$5^i$
Based on the SBD dataset of [@hariharan2011semantic] for semantic edge detection, we propose a new SBD-$5^i$ dataset. With reference to the setting of Pascal-$5^i$, 20 classes of the SBD dataset are divided into 4 splits. In the experiment with split $i$, 5 classes in the $i$th split are used as test classes $C_{test}$. The remaining 15 classes are utilized as training classes $C_{train}$. The training set $D_{train}$ is constructed with all image-annotation pairs whose annotation include at least one pixel from the classes in $C_{train}$. For each class, the boundary pixels which do not belong to that class are considered as background. The test set $D_{test}$ is also constructed in the same way as $D_{train}$, using $C_{test}$ this time. Considering the difficulty of few-shot setting and severe annotation noise of the SBD dataset, we extract thicker edges as done in FSE-1000. We utilize edges extracted from the segmentation mask as ground truth instead of original boundary labels of the SBD dataset, and thickness of extracted edge lies between $3 \sim 4 $ pixels on average. We conduct 4 experiments with each split of $i=0 \sim 3$, and report performance of each split as well as the averaged performance. Note that unlike Pascal-$5^i$, we do not consider division of training and test samples of the original SBD dataset. As a result, the images in $D_{train}$ might appear in $D_{test}$ with different annotation from class in $C_{test}$.
![Qualitative examples of 5-shot edge detection on SBD-$5^{i}$ dataset.[]{data-label="fig:Qualitative_result_SBD"}](Qualitative_Result_SBD.pdf){width="\textwidth"}
Evaluation Settings
-------------------
We use two evaluation metrics to measure the few-shot semantic edge detection performance of our approach: the Average Precision (AP) and the maximum F-measure (MF) at optimal dataset scale(ODS).
In evaluation, we compare the unthinned raw prediction results and the ground truths without Non-Maximum Suppression (NMS) following [@acuna2019devil; @yu2018simultaneous]. For the evaluation of edge detection, an important parameter is matching distance tolerance which is an error threshold between the prediction result and the ground truth. Prior works on edge detection such as [@acuna2019devil; @hariharan2011semantic; @yu2017casenet; @yu2018simultaneous] adopt non-zero distance tolerance to resolve the annotation noise of edge detection datasets. However, the proposed datasets for few-shot edge detection utilize thicker boundaries to overcome the annotation noise issue instead of adopting distance tolerance. Moreover, evaluation with non-zero distance tolerance requires additional heavy computation. This becomes more problematic under few-shot setting where the performance should be measured on the same test image multiple times due to the variation in the support set. For these reasons, we set distance tolerance to be 0 for both FSE-1000 and SBD-$5^i$. In addition, we evaluate the positive predictions from the area inside an object and zero-padded region as false positives, which is stricter than the evaluation protocol in prior works of [@hariharan2011semantic; @yu2017casenet].
Implementation Detail
---------------------
We implement our framework using Pytorch library and adopt Scikit-learn library to construct the precision-recall curve and compute average precision (AP). For the encoder, ResNet-34 pretrained on ImageNet is adopted. All parameters except the encoder parameters are learned from scratch. The entire network is trained using the Adam optimizer of [@kingma2014adam] with weight decay regularization of [@loshchilov2017decoupled]. In both experiments on FSE-1000 and SBD-$5^i$, we use a learning rate of $10^{-4}$ and an $l2$ weight decay rate of $10^{-2}$. For FSE-1000 experiments, the model is trained with 40,000 episodes and the learning rate is decayed by 0.1 after training 38,000 episodes. For SBD-$5^i$ experiments, 30,000 episodes are used for training, and the learning rate is decayed by 0.1 after training 28,000 episodes. Higher shot training of [@liu2018learning] is employed in 1-shot experiments for both datasets. In every experiment of our paper, single NVIDIA GeForce GTX 1080ti GPU is used for computation.
### Data preprocessing
During training, we adopt data augmentation with random rotation by multiples of 90 degrees for both FSE-1000 and SBD-5i. We additionally resize SBD-$5^i$ data to 320$\times$320, while no such resizing is performed on FSE-1000. During evaluation, images of SBD-$5^i$ are zero-padded to 512$\times$512. Again, the original image size is used for FSE-1000.
Experiment Result
-----------------
Table \[table:result\_FSE-1000\] shows the experiment results on the FSE-1000 dataset. To examine the impact of proposed MSMR and attentive decoder, we show the results of ablation experiments together. The baseline method conducts edge prediction in low resolution and utilizes the loss from edge prediction for meta-training. The edge prediction is done using a metric-based method utilizing prototypes which are computed using down-sampled edge labels. The method dubbed as **Seg** utilizes a segmentation module without MSMR or attentive decoding. **Seg** directly matches high-dimensional query feature vectors with prototypes in both training and evaluation. In **Seg**, the segmentation module is utilized only to provide the segmentation loss that assists for model learning to extract semantic features. **Seg + Att** employs the predicted segmentation mask for the additional attention process in skip architecture. **Seg + MSMR + Att** additionally utilizes the MSMR regularization for training. For fair comparison, all methods use the same network architecture and training hyperparameters. For SBD-$5^i$ datasets, the ablation experiments are done with same model variations as FSE-1000. The results on SBD-$5^i$ are shown in Table \[table:result\_SBD-5i\].
-------- ------------------ -------------- --------------
Metric Method 1-way 1-shot 1-way 5-shot
baseline 52.71 53.52
Seg 56.89 59.65
Seg + Att 58.00 60.14
Seg + Att + MSMR **58.47** **60.63**
baseline 53.66 54.59
Seg 58.80 61.87
Seg + Att 59.81 62.37
Seg + Att + MSMR **60.54** **63.92**
-------- ------------------ -------------- --------------
: []{data-label="table:result_FSE-1000"}
i=0 i=1 i=2 i=3
----- ----- ----- -----
: []{data-label="table:result_SBD-5i"}
-------- ------------------ ------------- ------------- ------------- ------------- -----------
Metric Method(5-shot) SBD-$5^{0}$ SBD-$5^{1}$ SBD-$5^{2}$ SBD-$5^{3}$ Mean
baseline 22.27 19.64 20.41 20.41 20.20
Seg 30.61 31.62 28.06 24.97 28.82
Seg + Att 31.75 33.41 28.44 26.03 29.91
Seg + Att + MSMR **34.71** **36.81** **32.02** **28.37** **32.98**
baseline 18.68 15.57 14.97 14.05 15.82
Seg 26.14 26.78 21.92 18.43 23.32
Seg + Att 27.61 28.39 22.66 20.11 24.69
Seg + Att + MSMR **30.47** **32.40** **27.01** **23.06** **28.24**
-------- ------------------ ------------- ------------- ------------- ------------- -----------
: []{data-label="table:result_SBD-5i"}
-------- ------------------ ------------- ------------- ------------- ------------- -----------
Metric Method(1-shot) SBD-$5^{0}$ SBD-$5^{1}$ SBD-$5^{2}$ SBD-$5^{3}$ Mean
baseline 21.81 19.49 20.34 18.06 19.93
Seg 29.89 31.64 27.89 24.41 28.46
Seg + Att 30.72 33.03 28.63 25.04 29.36
Seg + Att + MSMR **31.54** **34.75** **29.47** **26.68** **30.61**
baseline 18.11 15.47 14.73 13.89 15.55
Seg 25.16 26.15 21.52 18.52 22.84
Seg + Att 26.10 27.21 22.47 18.81 23.65
Seg + Att + MSMR **26.81** **29.08** **23.77** **20.44** **25.03**
-------- ------------------ ------------- ------------- ------------- ------------- -----------
: []{data-label="table:result_SBD-5i"}
Tables \[table:result\_FSE-1000\] and \[table:result\_SBD-5i\] demonstrate that the use of the segmentation module in **Seg** gives significant performance advantages over baseline for both FSE-1000 and SBD-$5^i$ datasets. It is also seen that the additional use of attentive decoding, **Seg + Att**, generally improves the performance over **Seg**. Finally, adding the effect of MSMR regularization gives substantial extra gains, as seen by the scores associated with **Seg + MSMR + Att**. Clearly, when compared with baseline, our overall approach **Seg + MSMR + Att** provides large gains.
Experiments on Multi-Split Matching Regularization
---------------------------------------------------
### Feature matching method for segmentation
In Table \[table:Feature Matching Method\], we have compared various feature matching methods between prototypes and query feature vectors for producing segmentation prediction on SBD-$5^i$. The method **baseline** refers to the original method generating segmentation prediction using only the similarity metric between high-dimensional vectors as done in Eq. . For the method **average**, segmentation predictions from low-dimensional feature splits (Eq.) and original high-dimensional feature vectors (Eq.) are averaged to generate the final prediction mask. The **average** method can be understood as a method utilizing MSMR not only for regularization, but also for inference. In the **weighted sum** method, the above five segmentation masks are combined using a weighted sum with learnable weights. As we can see in Table \[table:Feature Matching Method\], the MSMR method shows the best performance when employed for regularization.
------------------------- ----------- --------- --------------
Feature matching method baseline average weighted sum
AP **34.61** 31.05 31.44
MF(ODS) **29.91** 26.20 26.46
------------------------- ----------- --------- --------------
: Comparison of different feature matching method on SBD-$5^{i}$ under 1-way 5-shot setting. MF and AP scores are averaged over 4 splits[]{data-label="table:Feature Matching Method"}
### Number of vector splits
MSMR divides the high-dimensional feature into multiple splits. Table \[table:number of splits\] shows the performance of proposed CAFENet with varying numbers of splits $K$. Comparing the $K=1$ case with other cases, we can see that applying MSMR regularization consistently improves performance. We can see that $K=4$ results in the best AP and MF performance. The performance gain is marginal when we divide the embedding dimension into too small ($K=16$) or too big ($K=2$) a pieces.
------------------ ------- ------- ----------- ------- --------
Number of splits $K=1$ $K=2$ $K=4$ $K=8$ $K=16$
AP 24.69 26.48 **29.91** 27.68 23.83
MF(ODS) 29.91 31.62 **32.30** 31.58 30.77
------------------ ------- ------- ----------- ------- --------
: Comparison of different numbers of vector splits $K$ on SBD-$5^{i}$ under 1-way 5-shot setting. MF and AP scores are averaged over 4 splits[]{data-label="table:number of splits"}
Conclusion
==========
In this paper, we establish the few-shot semantic edge detection problem. We proposed the Class-Agnostic Few-shot Edge detector (CAFENet) based on a skip architecture utilizing multi-scale features. To compensate the shortage of semantic information in edge labels, CAFENet employs a segmentation module in low resolution and utilizes segmentation masks to generate attention maps. The attention maps are applied to multi-scale skip connection to localize the semantically related region. We also present the MSMR regularization method splitting the feature vectors and prototypes into several low-dimension sub-vectors and solving multiple metric-learning sub-problems with the sub-vectors. We built two novel datasets of FSE-1000 and SBD-$5^i$ well-suited to few-shot semantic edge detection. Experimental results demonstrate that the proposed techniques significantly improve the few-shot semantic edge detection performance relative to a baseline approach.
|
---
abstract: 'Hierarchical turbulent structure constituting a jet is considered to reproduce energy-dependent variability in blazars, particularly, the correlation between X- and gamma-ray light curves measured in the TeV blazar Markarian 421. The scale-invariant filaments are featured by the ordered magnetic fields that involve hydromagnetic fluctuations serving as electron scatterers for diffusive shock acceleration, and the spatial size scales are identified with the local maximum electron energies, which are reflected in the synchrotron spectral energy distribution (SED) above the near-infrared/optical break. The structural transition of filaments is found to be responsible for the observed change of spectral hysteresis.'
author:
- Mitsuru Honda
title: |
PHASE-TRANSIENT HIERARCHICAL TURBULENCE AS AN ENERGY CORRELATION\
GENERATOR OF BLAZAR LIGHT CURVES
---
INTRODUCTION
============
A noticeable feature associated with blazars is that the updated shortest variability timescale reaches a few minutes [e.g., Mrk 421: @cui; @blazejowski], not likely to be reconciled with the light-crossing time at the black hole horizon. One possible explanation for this fact is that small-scale structure does exist in the parsec-scale jet anchored in the galactic core [@hh04]. Indeed, in the plausible circumstance that the successive impingement of plasma blobs (ejected from the core) into the jet bulk engenders collisionless shocks, electromagnetic current filamentation (characterized by the skin depth) could be prominent [@medvedev]. It is known that the merging of smaller filaments leads eventually to accumulation of magnetic energy in larger scales [@honda00a; @silva].
Reflecting the self-similar (power law) characteristic in the inertially cascading range, the local magnetic intensity of the self-organized filaments will obey $|{\bf B}|\sim B_{m}(\lambda/d)^{(\beta-1)/2}$, where $\lambda$ and $d$ reflect the transverse size scale of a filament and the maximum, respectively, $B_{m}\equiv |{\bf B}|_{\lambda=d}$, and $\beta$ $(>1)$ corresponds to the filamentary turbulent spectral index. The value of $d$ is limited by the transverse size of jet (or blob size; $D$). Then, it is reasonable to consider that in fluid timescales, the well-developed coherent fields are sure to actually meet hydromagnetic disturbance independent of the filamentation; that is, the turbulent hierarchy is established (see Fig. 1). The spectral index of the superposed fluctuations \[denoted as $\beta^{\prime}$ $(>1)$\] could be different from $\beta$, and the correlation length scale is presumably limited by $\sim\lambda$.
At this site, the electrons bound to the local mean fields suffer scattering by the fluctuations, to be diffusively accelerated by the collisionless shocks [see @hh07]. When the acceleration and cooling efficiency depend on the spatial size scales, the local maximum energies of accelerated electrons will be identified by $\lambda$ (§2), to be reflected in the synchrotron SED extending to the X-ray region. More interestingly, the spatially inhomogeneous property of particle energetics is expected to cause the energy-dependent variability of broadband SEDs. Here the naive question arises whether or not this idea is responsible for the observed elusive patterns of energy correlation of light curves [e.g., @takahashi; @fossatiI; @fossatiII; @blazejowski]: this is the original motivation of the current work.
In the present simplistic model, light travel time effects would still prevent the detection of variability signatures on timescales shorter than $D/(c\delta_z)$, where $\delta_{z}=\delta/(1 + z)$, and $\delta$, $z$, and $c$ are the beaming factor of the jet, redshift, and speed of light, respectively. However, if a filamented piece is isolated, having loose causal relation with the dynamics of a bulk region serving as a dominant emitter, an intrinsic rapid variability involved in the subsystem would be viable. Namely, it is inferred that the shorter timescale is at least potentially realized, and observable, unless energetic emissions from such a compact domain are crucially degraded by synchrotron self-absorption and/or $\gamma\gamma$ absorption [e.g., @aharonian]. As is, the basic notion of the present model seems to provide a vital clue to settle the debate as to the causality problem incidental to observed rapid variabilities.
In this Letter, I demonstrate that the hierarchical system incorporated with the synchrotron self-Compton (SSC) mechanism accurately generates the time lag of gamma-ray flaring activity behind the X-ray, confirmed in the high-frequency-peaked BLLac object Mrk 421 [@blazejowski]. We address that in general, both lag and lead can appear in X-ray interband correlations, accompanying the structural transition. The major transition history is argued in light of the observed spectral hysteresis patterns. We also work out $(B_{m},d)$, to provide the constraint on the field strength and $D$ that should be compared with those of previous models.
AN IMPROVED EMITTER MODEL WITH\
HIERARCHICAL STRUCTURE
===============================
We consider a circumstance in which relativistic shocks propagate through a relativistic jet with the Lorentz factor $\Gamma$, such that the shock viewed upstream (jet frame) is weakly to mildly relativistic. Note the relation of $\delta\sim\Gamma$. The overall geometry and relative size scales of the aforementioned hierarchy are sketched in Figure 1. Provided that the gyrating electrons trapped in the filament (with the size $\lambda$) are resonantly scattered by the magnetic fluctuations, the mean acceleration time upstream is approximately given by $\tau_{\rm acc}\simeq(3\eta r_{\rm g}/c)[r/(r-1)]$, where $\eta=(3/2b)(\lambda/2r_{\rm g})^{\beta^{\prime}-1}$, $b$ is the energy density ratio of fluctuating/local mean magnetic fields (assumed to be $b\ll 1$), $r_{\rm g}(\gamma,|{\bf B}|)$ is the electron gyroradius ($\gamma$ being the Lorentz factor), and $r$ is the shock compression ratio. In the regime in which flares saturate, $\tau_{\rm acc}$ will be comparable to synchrotron cooling time $\tau_{\rm syn}(\gamma,|{\bf B}|^{2})$. Balancing these timescales gives the (local) maximum $\gamma$ of an accelerated electron, described as $\gamma^{\ast}(\lambda)=\{g_{0}^{-(\beta^{\prime}-1)}g_{1}
\left(\lambda/d\right)^{-[(\beta+1)\beta^{\prime}-2]/2}\}^{1/(3-\beta^{\prime})}$, where $g_{0}=eB_{m}d/(2m_{e}c^{2})$, $g_{1}=8\pi^{2}\xi m_{e}^{2}c^{4}/(e^{3}B_{m})$, $\xi=b(r-1)/r$, and the other notations are standard. At $\gamma=\gamma^{\ast}$, the electron energy distribution of the power-law form $n(\gamma)d\gamma=\kappa\gamma^{-p}d\gamma$ is truncated.
For simplicity, $\kappa$ and $\xi$ are assumed to be spatially constant at the moment. Then, for $\beta^{\prime}<3$ (see § 3.1), $\gamma^{\ast}$ decreases as $\lambda$ increases (reflecting likely prolonged $\tau_{\rm acc}$ and shortened $\tau_{\rm syn}$), to take a minimum value at $\lambda=d$, where the synchrotron flux density makes, up to the frequency of $(3/4\pi)(\delta_{z}\gamma^{\ast}|_{\lambda=d}^{2}eB_{m}/m_{e}c)(\equiv\nu_{b})$, a dominant contribution to the $F_{\nu}$ spectrum (owing to the maximum magnetic intensity at the outer scale). As $\lambda$ decreases, the flux density tends to decrease, extending the spectral tail (due to the $\gamma^{\ast}$ increase). Apparently, this property has the $F_{\nu}$ spectrum steepening above $\nu_{b}$, whereas below $\nu_{b}$ the spectrum retains $F_{\nu}\propto\nu^{-(p-1)/2}$. The frequency $\nu_{b}$ characterizing the spectral break can be expressed as $\nu_{b}=7.5\times 10^{14}\delta_{50}
B_{m,10}^{-3/2}\xi_{-4}^{3/2}d_{16}^{-1}~{\rm Hz}$ (for $\beta^{\prime}=5/3$; see § 3.1), where $\delta_{50}=\delta_{z}/50$, $B_{m,10}=B_{m}/10\,{\rm G}$, $d_{16}=d/10^{16}\,{\rm cm}$, $\xi_{-4}=\xi/10^{-4}$, and $\xi=b(r-1)/r$.
The increase of $\gamma^{\ast}$ in smaller $\lambda$ is limited at a critical $\lambda_{c}$, below which escape loss dominates the radiative loss: the equation for the spatial limit, $r_{g}(\gamma^{\ast})\sim\lambda/2$, yields $\lambda_{c}/d\sim(g_{0}^{-2}g_{1})^{2/(3\beta+1)}$. By using this expression, one can evaluate the achievable maximum $\gamma^{\ast}$ value as $\gamma^{\ast}|_{\lambda=\lambda_{c}}=g_{0}(\lambda_{c}/d)^{(\beta+1)/2}$, for which the corresponding synchrotron cutoff frequency is $\nu_{c}=\delta_{z}\nu_{0}g_{0}^{2}(\lambda_{c}/d)^{(3\beta+1)/2}
=\delta_{z}\nu_{0}g_{1}$, where $\nu_{0}=(3/4\pi)(eB_{m}/m_{e}c)$. Also, by combining $\lambda_{c}\propto\nu_{c}^{2/(3\beta+1)}$ with $F_{\nu_{c}}\propto|{\bf B}(\lambda_{c})|^{(p+1)/2}\nu_{c}^{-(p-1)/2}$, we read $\ln(\nu F_{\nu})_{c}/\ln\nu_{c}\propto[(5-p)\beta-(p-1)]/(3\beta+1)$ at $\nu\sim\nu_{c}$. More speculatively, this scaling might be reflected in $\ln(\nu F_{\nu})_{p}/\ln\nu_{p}$ for measured synchrotron flux peaks.
PROPERTIES OF ENERGY-DEPENDENT SPECTRAL\
VARIABILITY AND HYSTERESIS
========================================
*X-Ray Interband Correlation*
-----------------------------
In this context, we derive the $\nu$-dependence of the flaring activity timescale (denoted as $\tau$). In $\nu_{b}<\nu<\nu_{c}$, which typically covers the X-ray band, we have $\gamma^{\ast}=[(4\pi/3)(\nu/\delta_{z})(m_{e}c/e|{\bf B}|)]^{1/2}$, which is written as $\gamma^{\ast}(\lambda,\nu)=(\nu/\delta_{z}\nu_{0})^{1/2}
(\lambda/d)^{-(\beta-1)/4}$. Utilizing this, the expression of $\tau_{\rm syn}(\gamma^{\ast},|{\bf B}|^{2})$ is recast into $\tau_{\rm syn}(\lambda,\nu)(=\tau_{\rm acc}\sim\delta_{z}\tau)
=(\tau_{0}/\delta_{z})
(\delta_{z}\nu_{0}/\nu)^{1/2}(\lambda/d)^{-3(\beta-1)/4}$, where $\tau_{0}=36\pi^{2}m_{e}^{3}c^{5}/(e^{4}B_{m}^{2})$. The relation of $\lambda$ to $\nu$ can be derived from the equality of $\gamma^{\ast}(\lambda,\nu)=\gamma^{\ast}(\lambda)$, such that $\lambda(\nu)/d=\{g_{0}^{-1}g_{1}^{1/(\beta^{\prime}-1)}
(\delta_{z}\nu_{0}/\nu)^{(3-\beta^{\prime})
/[2(\beta^{\prime}-1)]}\}^{4/(3\beta+1)}$. Substituting this into $\tau(\lambda,\nu)$, we arrive at the result $\tau(\nu)\sim(\tau_{0}/\delta_{z})
(\nu/\delta_{z}\nu_{0})^{(\sigma-1)/2}
(g_{0}^{\beta^{\prime}-1}g_{1}^{-1})^{\sigma/(3-\beta^{\prime})}$, where $\sigma(\beta,\beta^{\prime})
=3(\beta-1)(3-\beta^{\prime})/[(3\beta+1)(\beta^{\prime}-1)]$; that is, $$\tau\propto\nu^{-(1/2)(1-\sigma)}.
\label{eq:1}$$ Note that $\sigma=0$ (for $\beta=1$) leads to $\tau\propto\nu^{-1/2}$, formally recovering the scaling for a homogeneous model. Significantly, the states of $\sigma<1$ and $>1$ imply the appearance of the modes for which the X-ray activity in a lower $\nu$ lags that in a higher $\nu$ [“soft lag”; @takahashi; @rebillot] and vice versa [“hard lag”; @fossatiI], respectively, and $\sigma=1$ is the tight-correlation mode [@sembay]. The mode flipping comes about through competing $\lambda$-dependence of cooling and acceleration efficiency. In particular, the soft lag appears if $$\beta<\beta_{c}=(4-\beta^{\prime})/[3(2-\beta^{\prime})].
\label{eq:2}$$
The critical function $\beta_{c}(\beta^{\prime})$ is, for the key range of $(1<)\beta^{\prime}<2$, plotted in Figure 2. Note that for the special $\beta^{\prime}=2$ case, $\sigma[=3(\beta-1)/(3\beta+1)]<1$ is always satisfied, and $\beta^{\prime}>2$ ensures $\beta>\beta_{c}$ (because of $\beta_{c}<1$): $\beta^{\prime}=2$ and $>2$ lead to soft and hard lag, respectively, irrespective of the $\beta$ value. While the index $\beta$ is expected to be variable (reflecting the long-term structural evolution of filaments; see §3.3 for details), $\beta^{\prime}$ would be a constant since a mechanism of superimposed magnetic fluctuations (§1) perhaps has universality. One can exclude $\beta^{\prime}>2$, which yields by no means soft lag, which is at odds with the observational facts, whereupon we can take the modified upper bound (indicated in Fig. 2, [*arrows*]{}) into account. With these ingredients, I conjecture the preferential appearance of $\beta^{\prime}=5/3$ (the Kolmogorov-type turbulence), for which $\beta_{c}=7/3$. As for $\beta$, it appears that for $\beta=2$ [@montgomery] and $5/3$ (and given $\beta^{\prime}=5/3$), the model synchrotron spectra of $\nu F_{\nu}\propto\nu^{0.32}$ and $\nu^{0.40}$ (in $\nu_{b}<\nu<\nu_{p}$; $\nu F_{\nu}\propto\nu^{0.69}$ in $\nu<\nu_{b}$) provide a reasonable fit to the measured ones (at flares) in the mid state (2002–2003) and high state (2004–2005), respectively, of Mrk 421, suggesting $\nu_{b}\sim 2\times 10^{14}\,{\rm Hz}$ (not shown in figure), as compatible with smaller variability in the $\nu$-range below [*R*]{} band [@blazejowski]. Below, we refer to these possible phases $(\beta,\beta^{\prime})=(2,5/3)$ and $(5/3,5/3)$, which satisfy equation (\[eq:2\]), as “$\Phi_{2}$” and “$\Phi_{5/3}$”, respectively. Also, we compare with the detailed data of burst decay time [in 1998; @fossatiI]. The guideline is given in Figure 2: use is made of the translation of the measured timescale $\propto\nu^{s}$ to $\beta=[4-\beta^{\prime}+s(\beta^{\prime}-1)]/
\{3[2-\beta^{\prime}-s(\beta^{\prime}-1)]\}$. This characteristic curve for $s\simeq 0.1$ indicates $\beta\simeq 3$ at $\beta^{\prime}=5/3$, and $\beta>\beta_{c}$, as consistent with the measured hard lag. For these $\beta=5/3$, $2$, and $3$, we anticipate $(\nu F_{\nu})_{p}\propto\nu_{p}^{0.8}$, $\nu_{p}^{0.9}$, and $\nu_{p}$ [for $p\simeq 1.6$; e.g., @macomb], amenable to the full X-ray data analysis of Mrk 421 flares [@tramacere].
Let us now estimate the time lag of a soft energy band $\epsilon_{\rm L}$($>h\nu_{b}$; $h$ is the Planck constant) behind a hard band $\epsilon_{\rm H}(<h\nu_{c})$ by $\Delta\tau=\tau(\epsilon_{\rm L})-\tau(\epsilon_{\rm H})$. Here it is instructive to note the relation of $(\lambda_{c}<)$ $\lambda(\epsilon_{\rm H})<\lambda(\epsilon_{\rm L})$ $(<d)$. Using the expression of $\nu_{b}$ (eliminating $\xi^{3/2}d^{-1}$), we obtain $$\Delta\tau=1.8\,\delta_{50}^{-1/2}B_{m,10}^{-3/2}
\nu_{b,14}^{-3/7}\epsilon_{{\rm L},1}^{-1/14}\eta_{-1}~{\rm hr}
\label{eq:3}$$ for the structural phase $\Phi_{2}$, where $\nu_{b,14}=\nu_{b}/10^{14}\,{\rm Hz}$, $\eta_{-1}=[1-(\epsilon_{\rm L}/\epsilon_{\rm H})^{1/14}]/10^{-1}$, and $\epsilon_{{\rm L},1}=\epsilon_{\rm L}/1\,{\rm keV}$. Concerning the validity, it has been checked that, e.g., for a soft-lag episode [in 1994 May; @takahashi], the measured time lag plotted against $\epsilon_{\rm L}$ could be more naturally fitted by the function (\[eq:3\]) of $\Delta\tau(\epsilon_{\rm L},\epsilon_{\rm H})$ (given $\epsilon_{\rm H}\simeq 4-5\,{\rm keV}$ for [*ASCA*]{}), rather than the function of $\sim\epsilon_{\rm L}^{-1/2}
[1-(\epsilon_{\rm L}/\epsilon_{\rm H})^{1/2}]$ for the homogeneous ($\sigma=0$) model.
*X/Gamma-Ray Cross-Band Correlation*
------------------------------------
The interband correlation property is reflected in the cross-band correlation between X- and gamma-rays, provided the SSC mechanism as a dominant gamma-ray emitter [e.g., @maraschi; @dermer]. Along the heuristic (time independent) manner, we suppose $\gamma\sim\gamma^{\ast}$ for scattering electrons, and examine the correlation between an X-ray band $\epsilon_{\rm x}$ (compared to $\epsilon_{\rm H}$) and gamma-ray band $\epsilon_{\gamma}$ susceptible to the inverse Comptonization of low-energy synchrotron photons (with $\epsilon_{\rm L}$). Here we focus on the feasible, Thomson regime of $(\epsilon_{\rm L}/\delta_{z})\gamma^{\ast}<m_{e}c^{2}$; note that using the expression of $\gamma^{\ast}[\lambda(\epsilon_{\rm L})]$ (§3.1), this range can be written as $\epsilon_{\rm L}<\delta_{50}[B_{m,10}^{2}(\xi_{-4}^{3/2}
d_{16}^{-1})]^{2/23}~{\rm keV}$ (for $\Phi_{2}$). The Lorentz factor of the electrons that execute the boost of $\epsilon_{\gamma}/\epsilon_{\rm L}=(\gamma^{\ast})^{2}$ is denoted as $\gamma_{s}^{\ast}=(\epsilon_{\gamma}/\delta_{z}h\nu_{0})^{1/4}
[\lambda(\epsilon_{\rm L})/d]^{-(\beta-1)/8}$. Then, simply estimating $\Delta\tau_{\gamma{\rm x}}
=\tau[\epsilon_{\gamma}/(\gamma_{s}^{\ast})^{2}]-\tau(\epsilon_{\rm x})$ ($>0$, for $\beta<\beta_{c}$) would be adequate for the present purpose. For convenience, one may eliminate $\epsilon_{\rm L}$ from $\gamma_{s}^{\ast}$ \[transform $\lambda(\epsilon_{\rm L})$ into $\lambda(\epsilon_{\gamma})$\], and adopt the positive soft-lag representation of $\Delta\tau_{{\rm x}\gamma}(=-\Delta\tau_{\gamma{\rm x}})$, so that the negative sign indicates gamma-ray lag. Again using $\nu_{b}$, we find for $\Phi_{2}$ $$\Delta\tau_{{\rm x}\gamma}=-1.7\,\delta_{50}^{-5/32}B_{m,10}^{-7/8}
\nu_{b,14}^{-7/16}\epsilon_{\gamma,1}^{-1/32}
\eta_{\gamma{\rm x},-1}~{\rm days},
\label{eq:4}$$ where $\eta_{\gamma{\rm x},-1}=\{1-0.79\epsilon_{{\rm x},25}^{-1/14}
[(\epsilon_{\gamma,1}\delta_{50}B_{m,10})^{1/2}\times$ $\nu_{b,14}^{1/7}]^{1/16}\}/10^{-1}$, $\epsilon_{\gamma,1}=\epsilon_{\gamma}/1\,{\rm TeV}$, and $\epsilon_{{\rm x},25}=\epsilon_{\rm x}/25\,{\rm keV}$. The simultaneous equations (\[eq:3\]) and (\[eq:4\]) contain the solutions $(\delta,B_{m})$, for given observable quantities $\nu_{b}$ and $(\Delta\tau,\Delta\tau_{{\rm x}\gamma})$, as well as $(\epsilon_{\rm L},\epsilon_{\rm H};\epsilon_{\rm x},\epsilon_{\gamma})$ inherent in detectors.
In Figure 3 ([*top*]{}) for $\nu_{b,14}=2$, $\Delta\tau=1\,{\rm hr}$, and $(\epsilon_{{\rm L},1},\epsilon_{\rm L}
/\epsilon_{\rm H};\epsilon_{{\rm x},25})=(1,0.2;1)$, compared to Mrk 421 ($z=0.031$) measurements [@takahashi; @blazejowski], the self-consistent numerical solution $\delta$ is plotted against $\Delta\tau_{{\rm x}\gamma}$, given $\epsilon_{\gamma}$ that covers a gamma-ray band associated with the Whipple observation [@catanese]. For the allowed domain of $\delta>1$ [@piner], a typical TeV range of $\epsilon_{\gamma,1}\simeq 1-2$ (susceptible to the significant variation in the mid state) is found to a priori restrict the domain of the observable $-\Delta\tau_{{\rm x}\gamma}$ to $1.4-2.2\,{\rm days}$. Surprisingly, this quantitatively agrees with $\Delta\tau_{{\rm x}\gamma}=-1.8\pm 0.4\,{\rm days}$ that has been revealed by multiband monitoring in the 2002/2003 season [@blazejowski]. In order to solidify the argument, the solutions for the high state with $\Phi_{5/3}$ have also been sought. The results show that the upper bound of $-\Delta\tau_{{\rm x}\gamma}$, at which $\delta$ diverges, shifts (from $2.2\,{\rm days}$) to $1.7\,{\rm days}$ and the Whipple coverage $\epsilon_{\gamma,1}<10$ restricts to $-\Delta\tau_{{\rm x}\gamma}>0.7\,{\rm days}$; these combination yields $-\Delta\tau_{{\rm x}\gamma}\simeq 0.7-1.7\,{\rm days}$. This is certainly compatible with the measured $\Delta\tau_{{\rm x}\gamma}=-1.2\pm 0.5\,{\rm days}$ [in the 2003/2004 season; for the significance, see @blazejowski].
*Hysteresis Reversal via Structural Transition*
-----------------------------------------------
From the view point of activity history, it is claimed that, involving the fluctuations with a common $\beta^{\prime}=5/3$, the coherent structure, at least, in the dominant emission region has been in the $\beta=2$ phase [1994 May; @takahashi], $\beta=3$ [1998 April; @fossatiI; @fossatiII], an intermediate phase around $\beta=7/3$ [2000 May and November; @sembay], $\beta=2$ [2002/2003 season; @blazejowski], and $\beta=5/3$ [2003/2004 season; @blazejowski], to give rise to a hard and soft X-ray lag for $\beta\gtrless\beta_{c}=7/3$, respectively, and no lag for $\beta=\beta_{c}$, as consistent with the observed correlation properties in each epoch (Fig. 2). At this juncture, the confirmed reversal between clockwise [@takahashi; @rebillot] and anticlockwise [@fossatiII] hysteresis loops in the flux–spectral index plane is ascribed to the phase transition between $\beta<\beta_{c}$ and $>\beta_{c}$, respectively. Physically, the likely $\beta=2$ is associated with the prominence of filamentation [@montgomery]. The smaller $\beta=5/3$ in a high state arguably reflects strong structural deformation, while the larger $\beta=3$ can be interpreted as the dual-cascade phase of two-dimensional turbulence (e.g., @krommes and references therein) transverse to pronounced filaments [@honda00b].
DISCUSSION AND CONCLUDING REMARKS
=================================
The practical formula that constrains magnetic field strength is readily obtained from equation (\[eq:3\]), and in parallel, one for $\Phi_{5/3}$ can be derived as well. We find the outcome that for $\Phi_{2}$ and $\Phi_{5/3}$, $B_{m}$ must satisfy $$B_{m}\delta_{z}^{1/3}=\left\{
\begin{array}{l}
54\,\nu_{b,14}^{-2/7}
(\Delta\tau^{-1}\epsilon_{{\rm L},1}^{-1/14}
\eta_{-1})^{2/3}~{\rm G},\\
33\,\nu_{b,14}^{-2/9}
(\Delta\tau^{-1}\epsilon_{{\rm L},1}^{-1/6}
\eta_{-1}^{\ast})^{2/3}~{\rm G},\\
\end{array}
\right.
\label{eq:5}$$ respectively, where $\eta_{-1}^{\ast}=[1-(\epsilon_{\rm L}/\epsilon_{\rm H})^{1/6}]/10^{-1}$ and $\Delta\tau$ is in hours. In Figure 3 ([*bottom*]{}), we plot the self-consistent solution $B_{m}$ (against $\Delta\tau_{{\rm x}\gamma}$; corresponding to $\delta$-$\Delta\tau_{{\rm x}\gamma}$ in Fig. 3 \[[*top*]{}\]) that obeys equation (\[eq:5\]) for $\Phi_{2}$ ([*inset*]{}) with the same parameter values as the top panel. We see that the observed $\delta>1$ [@piner] provides the constraint for which local magnetic intensity ($|{\bf B}|$) never exceeds $47\,{\rm G}$ for $\Phi_{2}$ ($51\,{\rm G}$ for $\Phi_{5/3}$). Whereas a mean magnetic intensity $\bar B$ is not well defined within the present framework, the obtained scaling of $B_{m,10}\delta^{1/3}\simeq 5$ seems to be reconciled with the conventional ${\bar B}\delta^{1/3}\simeq 0.1-1\,{\rm G}$ derived from fitting a variety of homogeneous SSC models to the measured broadband SEDs [e.g., @ghisellini; @tavecchio; @krawczynski].
In turn, the quantity of $\xi_{-4}^{-3/2}d_{16}=7.5\nu_{b,14}^{-1}
\delta_{50}B_{m,10}^{-3/2}$ (valid for $\beta^{\prime}=5/3$; §2) is self-consistently determined. Making use of equation (\[eq:5\]) to eliminate $B_{m}$, we have $d=2.6\times 10^{16}(\delta_{50}\xi_{-4})^{3/2}\,{\rm cm}$ for $\Phi_{2}$ \[$2.4\times 10^{16}(\delta_{50}\xi_{-4})^{3/2}\,{\rm cm}$ for $\Phi_{5/3}$\], given the common parameter values (such as $\nu_{b,14}=2$). To estimate $d$, here we call for another expression, $\nu_{c}=1.0\times 10^{22}\delta_{50}\xi_{-4}\,{\rm Hz}$ \[independent of $(\beta,\beta^{\prime})$; §2\]. Using this to eliminate $\delta\xi$ from the $d$-expression, we obtain the simple scaling of $d=8.2\times 10^{14}\nu_{c,21}^{3/2}\,{\rm cm}$ for $\Phi_{2}$ ($7.5\times 10^{14}\nu_{c,21}^{3/2}\,{\rm cm}$ for $\Phi_{5/3}$), where $\nu_{c,21}=\nu_{c}/10^{21}\,{\rm Hz}$. The size $d$ implies the allowable minimum of $D$; e.g., $\nu_{c,21}=0.1-10$ (yet involving the large observational uncertainty) provides $D_{16}\gtrsim 10^{-3}$ to $1$ (where $D_{16}=D/10^{16}\,{\rm cm}$), as reconciled with the previous results [e.g., @fossatiII; @krawczynski; @blazejowski]. It also turns out, from the $\nu_{c}$-scaling, that the range of $\nu_{c,21}<10^{2}$ accommodates $\xi\ll 1$, and thereby the assumption of $b\ll 1$ (§2).
In addition, given an energy input into the jet, particle density $n$ is estimated. Assuming that electron injection operates at $\gamma_{\rm inj}\ll\gamma^{\ast}|_{\lambda=d}$ $(\leq\gamma^{\ast})$, we approximately get $n\simeq(\kappa/0.6)\gamma_{\rm inj}^{-0.6}$ (for $p=1.6$), to find that the steady luminosity of $10^{44}\,{\rm ergs\,s^{-1}}$, which appears to retain a dominant portion around the $\nu_{b}$, requires $n\gtrsim 6\times 10^{4}\gamma_{\rm inj}^{-0.6}
D_{16}^{-3}B_{m,10}^{-1.3}\,{\rm cm^{-3}}$ (when supposing a spherical emitting volume with the diameter of $D$). Recalling $B_{m,10}\lesssim 5$, we thus read $n\gtrsim 10^{3}D_{16}^{-3}\,{\rm cm^{-3}}$ for ordinary $\gamma_{\rm inj}\sim O(1)$; note that an upper bound can be given by imposing the conditions of, e.g., pair-plasma production ($T\gtrsim 1\,{\rm MeV}$) and radial confinement ($nT\lesssim B_{m}^{2}/8\pi$), such that $n\lesssim 10^{8}\,{\rm cm^{-3}}$ (suggesting $D_{16}\gtrsim 10^{-2}$).
In conclusion, the gamma-ray lags of $1-2\,{\rm days}$ measured in Mrk 421 have been nicely reproduced by the hierarchical turbulent model of a jet. The crucial finding is that the structural transition $\Phi_{2}\rightarrow\Phi_{5/3}$ results in downshifting the upper bound of the observable lag \[in a TeV ($\epsilon_{\gamma,1}\simeq 1$) band\] from $2.2$ to $1.7\,{\rm days}$, in accordance with a closer inspection from 2002 to 2004 by @blazejowski. A typical $1.8\,{\rm day}$ lag (in the 2002/2003 season) suggests $\delta=10-92$ and $B_{m}=10-22\,{\rm G}$ (Fig. 3); the latter provides an upper limit of local magnetic intensity. The present model as a possible alternative to the previous leptonic [e.g., @sikora; @bednarek; @konopelko] and hadronic scenarios [e.g., @muecke] will shed light on puzzling aspects of broadband spectral variability.
Aharonian, F. A. 2004, Very High Energy Cosmic Gamma Radiation (River Edge: World Scientific) Bednarek, W., & Protheroe, R. J. 1997, , 292, 646 B[ł]{}ażejowski, M., et al. 2005, , 630, 130 Catanese, M., & Weekes, T. C. 1999, , 111, 1193 Cui, W. 2004, , 605, 662 Dermer, C. D., & Schlickeiser, R. 1993, , 416, 458 Fossati, G., et al. 2000a, , 541, 153 ——–. 2000b, , 541, 166 Ghisellini, G., Celotti, A., Fossati, G., Maraschi, L., & Comastri, A. 1998, , 301, 451 Honda, M., & Honda, Y. S. 2004, , 617, L37 ——–. 2007, , 654, 885 Honda, M., Meyer-ter-Vehn, J., & Pukhov, A. 2000a, Phys. Plasmas, 7, 1302 ——–. 2000b, , 85, 2128 Konopelko, A., Mastichiadis, A., Kirk, J., De Jager, O. C., & Stecker, F. W. 2003, , 597, 851 Krawczynski, H., et al. 2001, , 559, 187 Krommes, J. A. 2002, , 360, 1 Macomb, D. J., et al. 1995, , 449, L99 Maraschi, L. Ghisellini, G., & Celotti, A. 1992, , 397, L5 Medvedev, M. V., & Loeb, A. 1999, , 526, 697 Montgomery, D., & Liu, C. S. 1979, Phys. Fluids, 22, 866 M[" u]{}cke, A., & Protheroe, R. J. 2001, Astropart. Phys., 15, 121 Piner, B. G., et al. 1999, , 525, 176 Rebillot, P. F., et al. 2006, , 641, 740 Sembay, S., et al. 2002, , 574, 634 Sikora, M., Begelman, M. C., & Rees, M. J. 1994, , 421, 153 Silva, L. O., et al. 2003, , 596, L121 Takahashi, T., et al. 1996, , 470, L89 Tavecchio, F., Maraschi, L., & Ghisellini, G. 1998, , 509, 608 Tramacere, A., Massaro, F., & Cavaliere, A. 2007, , 466, 521
|
---
abstract: 'In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalises strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some elementary properties of $q$-quasiadditive and $q$-quasimultiplicative functions, we prove characterisations of $q$-quasiadditivity and $q$-quasimultiplicativity for the special class of $q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.'
address: |
Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Austria, sara.kropf@aau.at\
Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, sarakropf@stat.sinica.edu.tw\
Department of Mathematical Sciences, Stellenbosch University, South Africa, swagner@sun.ac.za
author:
- 'Sara Kropf[^1] '
- 'Stephan Wagner[^2] [^3]'
bibliography:
- 'lit.bib'
title: '$q$-Quasiadditive Functions'
---
Introduction
============
Arithmetic functions based on the digital expansion in some base $q$ have a long history (see, e.g., [@Bellman-Shapiro:1948; @Gelfond:1968:sur; @Delange:1972:q-add-q-mult; @Delange:1975:chiffres; @Cateland:digital-seq; @Bassily-Katai:1995:distr; @Drmota:2000]) The notion of a $q$-*additive* function is due to [@Gelfond:1968:sur]: an arithmetic function (defined on nonnegative integers) is called $q$-additive if $$f(q^k a + b) = f(q^k a) + f(b)$$ whenever $0 \leq b < q^k$. A stronger version of this concept is *strong* (or *complete*) $q$-additivity: a function $f$ is said to be strongly $q$-additive if we even have $$f(q^k a + b) = f(a) + f(b)$$ whenever $0 \leq b < q^k$. The class of (strongly) $q$-*multiplicative* functions is defined in an analogous fashion. Loosely speaking, (strong) $q$-additivity of a function means that it can be evaluated by breaking up the base-$q$ expansion. Typical examples of strongly $q$-additive functions are the $q$-ary sum of digits and the number of occurrences of a specified nonzero digit.
There are, however, many simple and natural functions based on the $q$-ary expansion that are not $q$-additive. A very basic example of this kind are *block counts*: the number of occurrences of a certain block of digits in the $q$-ary expansion. This and other examples provide the motivation for the present paper, in which we define and study a larger class of functions with comparable properties.
An arithmetic function (a function defined on the set of nonnegative integers) is called $q$-*quasiadditive* if there exists some nonnegative integer $r$ such that $$\label{eq:q-add}
f(q^{k+r}a + b) = f(a) + f(b)$$ whenever $0 \leq b < q^k$. Likewise, $f$ is said to be $q$-*quasimultiplicative* if it satisfies the identity $$\label{eq:q-mult}
f(q^{k+r}a + b) = f(a)f(b)$$ for some fixed nonnegative integer $r$ whenever $0 \leq b < q^k$.
We remark that the special case $r = 0$ is exactly strong $q$-additivity, so strictly speaking the term “strongly $q$-quasiadditive function” might be more appropriate. However, since we are not considering a weaker version (for which natural examples seem to be much harder to find), we do not make a distinction. As a further caveat, we remark that the term “quasiadditivity” has also been used in [@allouche:1993] for a related, but slightly weaker condition.
In the following section, we present a variety of examples of $q$-quasiadditive and $q$-quasimultiplicative functions. In Section \[sec:elem-properties\], we give some general properties of such functions. Since most of our examples also belong to the related class of $q$-regular functions, we discuss the connection in Section \[sec:q-regular\]. Finally, we prove a general central limit theorem for $q$-quasiadditive and -multiplicative functions that contains both old and new examples as special cases.
Examples of $q$-quasiadditive and $q$-quasimultiplicative functions {#sec:exampl-q-quasiadd}
===================================================================
Let us now back up the abstract concept of $q$-quasiadditivity by some concrete examples.
Block counts {#block-counts .unnumbered}
------------
As mentioned in the introduction, the number of occurrences of a fixed nonzero digit is a typical example of a $q$-additive function. However, the number of occurrences of a given block $B = \epsilon_1\epsilon_2
\cdots \epsilon_{\ell}$ of digits in the expansion of a nonnegative integer $n$, which we denote by $c_B(n)$, does not represent a $q$-additive function. The reason is simple: the $q$-ary expansion of $q^ka + b$ is obtained by joining the expansions of $a$ and $b$, so occurrences of $B$ in $a$ and occurrences of $B$ in $b$ are counted by $c_B(a) + c_{B}(b)$, but occurrences that involve digits of both $a$ and $b$ are not.
However, if $B$ is a block different from $00\cdots0$, then $c_B$ is $q$-quasiadditive: note that the representation of $q^{k+\ell} a + b$ is of the form $$\underbrace{a_1 a_2 \cdots a_{\mu}}_{\text{expansion of } a} \underbrace{0 0 \cdots 0_{\vphantom{\mu}}}_{\ell \text{ zeros}} \underbrace{b_1 b_2 \cdots {b_{\nu}}_{\vphantom{\mu}}}_{\text{expansion of } b}$$ whenever $0 \leq b < q^k$, so occurrences of the block $B$ have to belong to either $a$ or $b$ only. This implies that $c_B(q^{k+\ell} a + b) = c_B(a) + c_B(b)$, with one small caveat: if the block starts and/or ends with a sequence of zeros, then the count needs to be adjusted by assuming the digital expansion of a nonnegative integer to be padded with zeros on the left and on the right.
For example, let $B$ be the block $0101$ in base $2$. The binary representations of $469$ and $22$ are $111010101$ and $10110$, respectively, so we have $c_B(469) = 2$ and $c_B(22) = 1$ (note the occurrence of $0101$ at the beginning of $10110$ if we assume the expansion to be padded with zeros), as well as $$c_B(240150) = c_B(2^9 \cdot 469 + 22) = c_B(469) + c_B(22) = 3.$$ Indeed, the block $B$ occurs three times in the expansion of $240150$, which is $111010101000010110$.
The number of runs and the Gray code {#the-number-of-runs-and-the-gray-code .unnumbered}
------------------------------------
The number of ones in the Gray code of a nonnegative integer $n$, which we denote by $\hg(n)$, is also equal to the number of runs (maximal sequences of consecutive identical digits) in the binary representations of $n$ (counting the number of runs in the representation of $0$ as $0$); the sequence defined by $\hg(n)$ is [A005811](http://oeis.org/A005811) in Sloane’s On-Line Encyclopedia of Integer Sequences [@OEIS:2016]. An analysis of its expected value is performed in [@Flajolet-Ramshaw:1980:gray]. The function $\hg$ is $2$-quasiadditive up to some minor modification: set $f(n) = \hg(n)$ if $n$ is even and $f(n) = \hg(n)
+ 1$ if $n$ is odd. The new function $f$ can be interpreted as the total number of occurrences of the two blocks $01$ and $10$ in the binary expansion (considering binary expansions to be padded with zeros at both ends), so the argument of the previous example applies again and shows that $f$ is $2$-quasiadditive.
The nonadjacent form and its Hamming weight {#the-nonadjacent-form-and-its-hamming-weight .unnumbered}
-------------------------------------------
The nonadjacent form (NAF) of a nonnegative integer is the unique base-$2$ representation with digits $0,1,-1$ ($-1$ is usually represented as $\overline{1}$ in this context) and the additional requirement that there may not be two adjacent nonzero digits, see [@Reitwiesner:1960]. For example, the NAF of $27$ is $100\overline{1}0\overline{1}$. It is well known that the NAF always has minimum Hamming weight (i.e., the number of nonzero digits) among all possible binary representations with this particular digit set, although it may not be unique with this property (compare, e.g., [@Reitwiesner:1960] with [@Joye-Yen:2000:optim-left]).
The Hamming weight $\hn$ of the nonadjacent form has been analysed in some detail [@Thuswaldner:1999; @Heuberger-Kropf:2013:analy], and it is also an example of a $2$-quasiadditive function. It is not difficult to see that $\hn$ is characterised by the recursions ${h_{\mathsf{NAF}}}(2n) = {h_{\mathsf{NAF}}}(n)$, ${h_{\mathsf{NAF}}}(4n+1) = {h_{\mathsf{NAF}}}(n) + 1$, ${h_{\mathsf{NAF}}}(4n-1) = {h_{\mathsf{NAF}}}(n) + 1$ together with the initial value $\hn(0) = 0$. The identity $${h_{\mathsf{NAF}}}(2^{k+2}a + b) = {h_{\mathsf{NAF}}}(a) + {h_{\mathsf{NAF}}}(b)$$ can be proved by induction. In Section \[sec:q-regular\], this example will be generalised and put into a larger context.
The number of optimal $\{0,1,-1\}$-representations {#the-number-of-optimal-01-1-representations .unnumbered}
--------------------------------------------------
As mentioned above, the NAF may not be the only representation with minimum Hamming weight among all possible binary representations with digits $0,1,-1$. The number of optimal representations of a given nonnegative integer $n$ is therefore a quantity of interest in its own right. Its average over intervals of the form $[0,N)$ was studied by Grabner and Heuberger [@Grabner-Heuberger:2006:Number-Optimal], who also proved that the number $\ro(n)$ of optimal representations of $n$ can be obtained in the following way:
\[lemma:opt-representations-recursion\] Let sequences $u_i$ ($i=1,2,\ldots,5$) be given recursively by $$u_1(0) = u_2(0) = \cdots = u_5(0) = 1, \qquad u_1(1) = u_2(1) = 1,\ u_3(1) = u_4(1) = u_5(1) = 0,$$ and $$\begin{aligned}
u_1(2n) = u_1(n), \qquad & u_1(2n+1) = u_2(n) + u_4(n+1), \\
u_2(2n) = u_1(n), \qquad & u_2(2n+1) = u_3(n), \\
u_3(2n) = u_2(n), \qquad & u_3(2n+1) = 0, \\
u_4(2n) = u_1(n), \qquad & u_4(2n+1) = u_5(n+1), \\
u_5(2n) = u_4(n), \qquad & u_5(2n+1) = 0.\end{aligned}$$ The number $\ro(n)$ of optimal representations of $n$ is equal to $u_1(n)$.
A straightforward calculation shows that $$\label{eq:8n_a}
\begin{aligned}
&u_1(8n) = u_2(8n) = \cdots = u_5(8n) = u_1(8n+1) = u_2(8n+1) = u_1(n),\\
&u_3(8n+1) = u_4(8n+1) = u_5(8n+1) = 0.
\end{aligned}$$ This gives us the following result (see the full version of this extended abstract for a detailed proof):
\[lem:optrep\] The number of optimal $\{0,1,-1\}$-representations of a nonnegative integer is a $2$-quasimultiplicative function. Specifically, for any three nonnegative integers $a,b,k$ with $b < 2^k$, we have $${r_{\mathsf{OPT}}}(2^{k+3}a + b) = {r_{\mathsf{OPT}}}(a){r_{\mathsf{OPT}}}(b).$$
In Section \[sec:q-regular\], we will show that this is also an instance of a more general phenomenon.
The run length transform and cellular automata {#the-run-length-transform-and-cellular-automata .unnumbered}
----------------------------------------------
The *run length transform* of a sequence is defined in a recent paper of Sloane [@Sloane:number-on]: it is based on the binary representation, but could in principle also be generalised to other bases. Given a sequence $s_1,s_2,\ldots$, its run length transform is obtained by the rule $$t(n) = \prod_{i \in \mathcal{L}(n)} s_i,$$ where $\mathcal{L}(n)$ is the multiset of run lengths of $n$ (lengths of blocks of consecutive ones in the binary representation). For example, the binary expansion of $1910$ is $11101110110$, so the multiset $\mathcal{L}(n)$ of run lengths would be $\{3,3,2\}$, giving $t(1910) = s_2 s_3^2$.
A typical example is obtained for the sequence of Jacobsthal numbers given by the formula $s_n = \frac13 (2^{n+2} - (-1)^n)$. The associated run length transform $t_n$ (sequence [A071053](http://oeis.org/A071053) in the OEIS [@OEIS:2016]) counts the number of odd coefficients in the expansion of $(1+x+x^2)^n$, and it can also be interpreted as the number of active cells at the $n$-th generation of a certain cellular automaton. Further examples stemming from cellular automata can be found in Sloane’s paper [@Sloane:number-on].
The argument that proved $q$-quasiadditivity of block counts also applies here, and indeed it is easy to see that the identity $$t(2^{k+1}a + b) = t(a)t(b),$$ where $0 \leq b < 2^k$, holds for the run length transform of any sequence, meaning that any such transform is $2$-quasimultiplicative. In fact, it is not difficult to show that every $2$-quasimultiplicative function with parameter $r=1$ is the run length transform of some sequence.
Elementary properties {#sec:elem-properties}
=====================
Now that we have gathered some motivating examples for the concepts of $q$-quasiadditivity and $q$-quasimultiplicativity, let us present some simple results about functions with these properties. First of all, let us state an obvious relation between $q$-quasiadditive and $q$-quasimultiplicative functions:
\[prop:trivial\] If a function $f$ is $q$-quasiadditive, then the function defined by $g(n) = c^{f(n)}$ for some positive constant $c$ is $q$-quasimultiplicative. Conversely, if $f$ is a $q$-quasimultiplicative function that only takes positive values, then the function defined by $g(n) = \log_c f(n)$ for some positive constant $c \neq 1$ is $q$-quasiadditive.
The next proposition deals with the parameter $r$ in the definition of a $q$-quasiadditive function:
If the arithmetic function $f$ satisfies $f(q^{k+r}a + b) = f(a) + f(b)$ for some fixed nonnegative integer $r$ whenever $0 \leq b < q^k$, then it also satisfies $f(q^{k+s}a + b) = f(a) + f(b)$ for all nonnegative integers $s \geq r$ whenever $0 \leq b < q^k$.
If $a,b$ are nonnegative integers with $0 \leq b < q^k$, then clearly also $0 \leq b < q^{k+s-r}$ if $s \geq r$, and thus $$f(q^{k+s}a + b) = f(q^{(k+s-r)+r}a + b) = f(a) + f(b).$$
\[cor:lin\_comb\] If two arithmetic functions $f$ and $g$ are $q$-quasiadditive functions, then so is any linear combination $\alpha f + \beta g$ of the two.
In view of the previous proposition, we may assume the parameter $r$ in to be the same for both functions. The statement follows immediately.
Finally, we observe that $q$-quasiadditive and $q$-quasimultiplicative functions can be computed by breaking the $q$-ary expansion into pieces. A detailed proof can be found in the full version:
\[lem:simplefacts\] If $f$ is a $q$-quasiadditive ($q$-quasimultiplicative) function, then
- f(0) = 0
($f(0) = 1$, respectively, unless $f$ is identically $0$),
- f(qa) = f(a)
for all nonnegative integers $a$.
\[prop:split\] Suppose that the function $f$ is $q$-quasiadditive with parameter $r$, i.e., $f(q^{k+r}a + b) = f(a) + f(b)$ whenever $0 \leq b < q^k$. Going from left to right, split the $q$-ary expansion of $n$ into blocks by inserting breaks after each run of $r$ or more zeros. If these blocks are the $q$-ary representations of $n_1,n_2,\ldots,n_{\ell}$, then we have $$f(n) = f(n_1) + f(n_2) + \cdots + f(n_{\ell}).$$ Moreover, if $m_i$ is the greatest divisor of $n_i$ which are not divisible by $q$ for $i=1,\ldots,\ell$, then $$f(n) = f(m_1) + f(m_2) + \cdots + f(m_{\ell}).$$ Analogous statements hold for $q$-quasimultiplicative functions, with sums replaced by products.
This is obtained by a straightforward induction on $\ell$ together with the fact that $f(q^{h} a) = f(a)$, which follows from the previous lemma.
Recall that the Hamming weight of the NAF (which is the minimum Hamming weight of a $\{0,1,-1\}$-representation) is $2$-quasiadditive with parameter $r=2$. To determine $\hn(314\,159\,265)$, we split the binary representation, which is $10010101110011011000010100001,$ into blocks by inserting breaks after each run of at least two zeros: $$100|101011100|110110000|1010000|1.$$ The numbers $n_1,n_2,\ldots,n_{\ell}$ in the statement of the proposition are now $4,348,432,80,1$ respectively, and the numbers $m_1,m_2,\ldots,m_{\ell}$ are therefore $1,87,27,5,1$. Now we use the values $\hn(1) = 1$, $\hn(5) = 2$, $\hn(27) = 3$ and $\hn(87) = 4$ to obtain $${h_{\mathsf{NAF}}}(314\,159\,265) = 2{h_{\mathsf{NAF}}}(1) + {h_{\mathsf{NAF}}}(5) + {h_{\mathsf{NAF}}}(27) + {h_{\mathsf{NAF}}}(87) = 11.$$
In the same way, we consider the number of optimal representations $\ro$, which is $2$-quasimultiplicative with parameter $r=3$. Consider for instance the binary representation of $204\,280\,974$, namely $1100001011010001010010001110$. We split into blocks: $$110000|101101000|101001000|1110.$$ The four blocks correspond to the numbers $48 = 16 \cdot 3$, $360 = 8 \cdot 45$, $328 = 8 \cdot 41$ and $14 = 2 \cdot 7$. Since $\ro(3) = 2$, $\ro(45) = 5$, $\ro(41) = 1$ and $\ro(7) = 1$, we obtain $\ro(204\,280\,974) = 10$.
$q$-Regular functions {#sec:q-regular}
=====================
In this section, we introduce $q$-regular functions and examine the connection to our concepts. See [@Allouche-Shallit:2003:autom] for more background on $q$-regular sequences.
A function $f$ is *$q$-regular* if it can be expressed as $f=\bfu^{t}\bff$ for a vector $\bfu$ and a vector-valued function $\bff$, and there are matrices $M_{i}$, $0\leq i<q$, satisfying $$\label{eq:q-regular-recursive}
{\boldsymbol{f}}(qn+i)=M_{i}{\boldsymbol{f}}(n)$$ for $0\leq i<q$, $qn+i>0$. We set $\bfv=\bff(0)$.
Equivalently, a function $f$ is $q$-regular if and only if $f$ can be written as $$\label{eq:q-regular}
f(n)={\boldsymbol{u}}^{t} \prod_{i=0}^{L} M_{n_{i}}{\boldsymbol{v}}$$ where $n_{L}\cdots n_{0}$ is the $q$-ary expansion of $n$.
The notion of $q$-regular functions is a generalisation of $q$-additive and $q$-multiplicative functions. However, we emphasise that $q$-quasiadditive and $q$-quasimultiplicative functions are not necessarily $q$-regular: a $q$-regular sequence can always be bounded by $O(n^{c})$ for a constant $c$, see [@Allouche-Shallit:2003:autom Thm.16.3.1]. In our setting however, the values of $f(n)$ can be chosen arbitrarily for those $n$ whose $q$-ary expansion does not contain $0^{r}$. Therefore a $q$-quasiadditive or -multiplicative function can grow arbitrarily fast.
We call $(\bfu, (M_{i})_{0\leq i<q}, \bfv)$ a *linear representation* of the $q$-regular function $f$. Such a linear representation is called *zero-insensitive* if $M_{0}\bfv=\bfv$, meaning that in , leading zeros in the $q$-ary expansion of $n$ do not change anything. We call a linear representation *minimal* if the dimension of the matrices $M_{i}$ is minimal among all linear representations of $f$.
Following [@Dumas:2014:asymp], every $q$-regular function has a zero-insensitive minimal linear representation.
When is a $q$-regular function $q$-quasimultiplicative?
-------------------------------------------------------
We now give a characterisation of $q$-regular functions that are $q$-quasimultiplicative. Proofs of the results in this and the following subsection can be found in the full version.
\[theorem:reg-mult\] Let $f$ be a $q$-regular sequence with zero-insensitive minimal linear representation . Then the following two assertions are equivalent:
- The sequence $f$ is $q$-quasimultiplicative with parameter $r$.
- M\_[0]{}\^[r]{}=\^[t]{}
.
The number of optimal $\{0,1,-1\}$-representations as described in Section \[sec:exampl-q-quasiadd\] is a $2$-regular sequence by Lemma \[lemma:opt-representations-recursion\]. A minimal zero-insensitive linear representation for the vector $(u_{1}(n),
u_{2}(n), u_{3}(n), u_{1}(n+1), u_{4}(n+1),
u_{5}(n+1))^{t}$ is given by $$M_{0}=
\begin{pmatrix}
1&0&0&0&0&0\\
1&0&0&0&0&0\\
0&1&0&0&0&0\\
0&1&0&0&1&0\\
0&0&0&0&0&1\\
0&0&0&0&0&0
\end{pmatrix},\quad
M_{1}=
\begin{pmatrix}
0&1&0&0&1&0\\
0&0&1&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0
\end{pmatrix},$$ $\bfu^{t}=(1,0,0,0,0,0)$ and $\bfv=(1,1,1,1,0,0)^{t}$.
As $M_{0}^{3}=vu^{t}$, this sequence is $2$-quasimultiplicative with parameter $3$, which is the same result as in Lemma \[lem:optrep\].
The condition on the minimality of the linear representation in Theorem \[theorem:reg-mult\] is necessary as illustrated by the following example:
Consider the sequence $f(n)=2^{s_{2}(n)}$ where $s_{2}(n)$ is the binary sum of digits function. This sequence is $2$-regular and $2$-(quasi-)multiplicative with parameter $r=0$. A minimal linear representation is given by $M_{0}=1$, $M_{1}=2$, $v=1$ and $u=1$. As stated in Theorem \[theorem:reg-mult\], we have $M_{0}^{0}=vu^{t}=1$.
If we use the zero-insensitive non-minimal linear representation defined by $M_{0}=\big(
\begin{smallmatrix}
1&13\\0&2
\end{smallmatrix}\big)$, $M_{1}=\big(
\begin{smallmatrix}
2&27\\0&5
\end{smallmatrix}
\big)$, $v=(1, 0)^{t}$ and $u^{t}=(1, 0)$ instead, we have $\rank M_{0}^{r}=2$ for all $r\geq 0$. Thus $M_{0}^{r}\neq vu^{t}$.
When is a $q$-regular function $q$-quasiadditive?
-------------------------------------------------
The characterisation of $q$-regular functions that are also $q$-quasiadditive is somewhat more complicated. Again, we consider a zero-insensitive (but not necessarily minimal) linear representation. We let $U$ be the smallest vector space such that all vectors of the form $\bfu^{t}\prod_{i\in I}
M_{n_{i}}$ lie in the affine subspace $\bfu^{t} + U^t$ ($U^t$ is used as a shorthand for $\{\bfx^{t} \,:\, \bfx \in U\}$). Such a vector space must exist, since $\bfu^{t}$ is a vector of this form (corresponding to the empty product, where $I = \emptyset$). Likewise, let $V$ be the smallest vector space such that all vectors of the form $\prod_{j\in J}M_{n_{j}}\bfv$ lie in the affine subspace $\bfv +
V$.
\[thm:q-reg-q-quasiadd\] Let $f$ be a $q$-regular sequence with zero-insensitive linear representation . The sequence $f$ is $q$-quasiadditive with parameter $r$ if and only if all of the following statements hold:
- \^t = 0
,
- U\^t
is orthogonal to $(M_0^r - I)\bfv$, i.e., $\bfx^t(M_0^r - I)\bfv = \bfx^tM_0^r\bfv - \bfx^t\bfv = 0$ for all $\bfx \in U$,
- V
is orthogonal to $\bfu^t(M_0^r - I)$, i.e., $\bfu^t(M_0^r - I)\bfy = \bfu^tM_0^r\bfy - \bfu^t\bfy = 0$ for all $\bfy \in V$,
- U\^t M\_0\^r V = 0
, i.e., $\bfx^t M_0^r \bfy = 0$ for all $\bfx \in U$ and $\bfy \in V$.
For the Hamming weight of the nonadjacent form, a zero-insensitive (and also minimal) linear representation for the vector $(\hn(n),\hn(n+1),\hn(2n+1),1)^{t}$ is $$M_{0}=
\begin{pmatrix}
1&0&0&0\\0&0&1&0\\1&0&0&1\\0&0&0&1
\end{pmatrix},\quad
M_{1}=
\begin{pmatrix}
0&0&1&0\\0&1&0&0\\0&1&0&1\\0&0&0&1
\end{pmatrix},$$ $\bfu^{t}=(1,0,0,0)$ and $\bfv=(0,1,1,1)^{t}$.
The three vectors $\mathbf{w}_1 = \bfu^{t}M_{1}-\bfu^{t}$, $\mathbf{w}_2 = \bfu^{t}M_{1}^{2}-\bfu^{t}$ and $\mathbf{w}_3 = \bfu^{t}M_{1}M_{0}M_{1}-\bfu^{t}$ are linearly independent. If we let $W$ be the vector space spanned by those three, it is easily verified that $M_{0}$ and $M_{1}$ map the affine subspace $\bfu^{t}+ W^t$ to itself, so $U=W$ is spanned by these vectors.
Similarly, the three vectors $M_{1}\bfv-\bfv$, $M_{1}^{2}\bfv-\bfv$ and $M_{1}M_{0}M_{1}\bfv-\bfv$ span $V$.
The first condition of Theorem \[thm:q-reg-q-quasiadd\] is obviously true. We only have to verify the other three conditions with $r=2$ for the basis vectors of $U$ and $V$, which is done easily. Thus $\hn$ is a $2$-regular sequence that is also $2$-quasiadditive, as was also proved in Section \[sec:exampl-q-quasiadd\].
Finding the vector spaces $U$ and $V$ is not trivial. But in a certain special case of $q$-regular functions, we can give a sufficient condition for $q$-additivity, which is easier to check. These $q$-regular functions are output sums of transducers as defined in [@Heuberger-Kropf-Prodinger:2015:output]: a transducer transforms the $q$-ary expansion of an integer $n$ (read from the least significant to the most significant digit) deterministically into an output sequence and leads to a state $s$. The output sum is then the sum of this output sequence together with the final output of the state $s$. This defines the value of the $q$-regular function evaluated at $n$. The function $\hn$ discussed in the example above, as well as many other examples, can be represented in this way.
\[proposition:q-add-transducer\] The output sum of a connected transducer is $q$-additive with parameter $r$ if the following conditions are satisfied:
- The transducer has the reset sequence $0^{r}$ going to the initial state, i.e., reading $r$ zeros always leads to the initial state of the transducer.
- For every state, the output sum along the path of the reset sequence $0^{r}$ equals the final output of this state.
- Additional zeros at the end of the input sequence do not change the output sum.
A central limit theorem for $q$-quasiadditive and -multiplicative functions
===========================================================================
In this section, we prove a central limit theorem for $q$-quasimultiplicative functions taking only positive values. By Proposition \[prop:trivial\], this also implies a central limit theorem for $q$-quasiadditive functions.
To this end, we define a generating function: let $f$ be a $q$-quasimultiplicative function with positive values, let $\M_k$ be the set of all nonnegative integers less than $q^k$ (i.e., those positive integers whose $q$-ary expansion needs at most $k$ digits), and set $$F(x,t) = \sum_{k \geq 0} x^k \sum_{n \in {\mathcal{M}}_k} f(n)^t.$$ The decomposition of Proposition \[prop:split\] now translates directly to an alternative representation for $F(x,t)$: let $\B$ be the set of all positive integers not divisible by $q$ whose $q$-ary representation does not contain the block $0^{r}$, let $\ell(n)$ denote the length of the $q$-ary representation of $n$, and define the function $B(x,t)$ by $$B(x,t) = \sum_{n \in {\mathcal{B}}} x^{\ell(n)} f(n)^t.$$ We remark that in the special case where $q=2$ and $r=1$, this simplifies greatly to $$\label{eq:q2_r1}
B(x,t) = \sum_{k \geq 1} x^{k} f(2^k-1)^t.$$
\[prop:gf\] The generating function $F(x,t)$ can be expressed as $$F(x,t) = \frac{1}{1-x} \cdot \frac{1}{1 - \frac{x^r}{1-x} B(x,t)} \Big( 1 + (1+x+\cdots+x^{r-1})B(x,t) \Big) = \frac{1+(1+x+\cdots+x^{r-1})B(x,t)}{1-x-x^rB(x,t)}.$$
The first factor stands for the initial sequence of leading zeros, the second factor for a (possibly empty) sequence of blocks consisting of an element of $\B$ and $r$ or more zeros, and the last factor for the final part, which may be empty or an element of $\B$ with up to $r-1$ zeros (possibly none) added at the end.
Under suitable assumptions on the growth of a $q$-quasiadditive or $q$-quasimultiplicative function, we can exploit the expression of Proposition \[prop:gf\] to prove a central limit theorem in the following steps (full proofs can again be found in the full version).
We say that a function $f$ has *at most polynomial growth* if $f(n)=O(n^{c})$ and $f(n) = \Omega(n^{-c})$ for a fixed $c\geq
0$. We say that $f$ has *at most logarithmic growth* if $f(n)=O(\log n)$.
Note that our definition of at most polynomial growth is slightly different than usual: the extra condition $f(n) =
\Omega(n^{-c})$ ensures that the absolute value of $\log f(n)$ does not grow too fast.
\[lemma:singularity\] Assume that the positive, $q$-quasimultiplicative function $f$ has at most polynomial growth.
There exist positive constants $\delta$ and $\epsilon$ such that
- B(x,t)
has radius of convergence $\rho(t) > \frac1q$ whenever $|t| \leq \delta$.
- For $|t| \leq \delta$, the equation $x + x^r B(x,t) = 1$ has a complex solution $\alpha(t)$ with $|\alpha(t)| < \rho(t)$ and no other solutions with modulus $\leq (1+\epsilon)|\alpha(t)|$.
- Thus the generating function $F(x,t)$ has a simple pole at $\alpha(t)$ and no further singularities of modulus $\leq (1+ \epsilon)|\alpha(t)|$.
- Finally, $\alpha$ is an analytic function of $t$ for $|t| \leq \delta$.
\[lem:sing\_anal\] Assume that the positive, $q$-quasimultiplicative function $f$ has at most polynomial growth.
With $\delta$ and $\epsilon$ as in the previous lemma, we have, uniformly in $t$, $$F(x,t) = \kappa(t) \cdot \alpha(t)^{-k} \big(1 + O((1+\epsilon)^{-k})\big)$$ for some function $\kappa$. Both $\alpha$ and $\kappa$ are analytic functions of $t$ for $|t| \leq \delta$, and $\kappa(t) \neq 0$ in this region.
\[thm:clt-mult\] Assume that the positive, $q$-quasimultiplicative function $f$ has at most polynomial growth.
Let $N_k$ be a randomly chosen integer in $\{0,1,\ldots,q^k-1\}$. The random variable $L_k = \log f(N_k)$ has mean $\mu k + O(1)$ and variance $\sigma^2 k + O(1)$, where the two constants are given by $$\mu = \frac{B_t(1/q,0)}{q^{2r}}$$ and $$\begin{gathered}
\sigma^2= -B_{t}(1/q,0)^{2} {q}^{-4r+1}(q-1)^{-1} + 2B_{t}(1/q,0)^{2} {q}^{-3r+1}(q-1)^{-1} -B_{t}(1/q,0)^{2}{q}^{-4r}(q-1)^{-1} \\-
4rB_{t}(1/q,0)^{2} {q}^{-4r} + B_{tt}(1/q,0){q}^{-2r}
- 2B_{t}(1/q,0)
B_{tx}(1/q,0)
{q}^{-4r-1} .\end{gathered}$$ If $f$ is not the constant function $f \equiv 1$, then $\sigma^2 \neq 0$ and the normalised random variable $(L_k - \mu k)/(\sigma \sqrt{k})$ converges weakly to a standard Gaussian distribution.
\[cor:clt-add\] Assume that the $q$-quasiadditive function $f$ has at most logarithmic growth.
Let $N_k$ be a randomly chosen integer in $\{0,1,\ldots,q^k-1\}$. The random variable $L_k = f(N_k)$ has mean $\hat\mu k + O(1)$ and variance $\hat\sigma^2 k + O(1)$, where the two constants $\mu$ and $\sigma^2$are given by the same formulas as in Theorem \[thm:clt-mult\], with $B(x,t)$ replaced by $$\hat B(x,t) = \sum_{n \in {\mathcal{B}}} x^{\ell(n)} e^{f(n)t}.$$
If $f$ is not the constant function $f \equiv 0$, then the normalised random variable $(L_k - \hat\mu k)/(\hat\sigma \sqrt{k})$ converges weakly to a standard Gaussian distribution.
By means of the Cramér-Wold device (and Corollary \[cor:lin\_comb\]), we also obtain joint normal distribution of tuples of $q$-quasiadditive functions.
We now revisit the examples discussed in Section \[sec:exampl-q-quasiadd\] and state the corresponding central limit theorems. Some of them are well known while others are new. We also provide numerical values for the constants in mean and variance.
The number of blocks $0101$ occurring in the binary expansion of $n$ is a $2$-quasiadditive function of at most logarithmic growth. Thus by Corollary \[cor:clt-add\], the standardised random variable is asymptotically normally distributed, the constants being $\hat\mu = \frac1{16}$ and $\hat\sigma^2 = \frac{17}{256}$.
The Hamming weight of the nonadjacent form is $2$-quasiadditive with at most logarithmic growth (as the length of the NAF of $n$ is logarithmic). Thus by Corollary \[cor:clt-add\], the standardised random variable is asymptotically normally distributed. The associated constants are $\hat\mu = \frac13$ and $\hat\sigma^2 = \frac2{27}$.
The number of optimal $\{0,1,-1\}$-representations is $2$-quasimultiplicative. As it is always greater or equal to $1$ and $2$-regular, it has at most polynomial growth. Thus Theorem \[thm:clt-mult\] implies that the standardised logarithm of this random variable is asymptotically normally distributed with numerical constants given by $\mu\approx 0.060829$, $\sigma^{2}\approx 0.038212$.
Suppose that the sequence $s_1,s_2,\ldots$ satisfies $s_{n}\geq 1$ and $s_{n}=O(c^{n})$ for a constant $c\geq 1$. The run length transform $t(n)$ of $s_{n}$ is $2$-quasimultiplicative. As $s_{n}\geq 1$ for all $n$, we have $t(n)\geq
1$ for all $n$ as well. Furthermore, there exists a constant $A$ such that $s_n \leq A c^n$ for all $n$, and the sum of all run lengths is bounded by the length of the binary expansion, thus $$t(n)=\prod_{i\in\mathcal L(n)}s_{i} \leq \prod_{i \in \mathcal{L}(n)} (A c^i) \leq (Ac)^{1+\log_2 n}.$$ Consequently, $t(n)$ is positive and has at most polynomial growth. By Theorem \[thm:clt-mult\], we obtain an asymptotic normal distribution for the standardised random variable $\log t(N_{k})$. The constants $\mu$ and $\sigma^2$ in mean and variance are given by $$\mu = \sum_{i \geq 1} (\log s_i) 2^{-i-2}$$ and $$\sigma^2 = \sum_{i \geq 1} (\log s_i)^2 \big(2^{-i-2} - (2i-1)2^{-2i-4} \big) - \sum_{j > i \geq 1} (\log s_i)(\log s_j) (i+j-1) 2^{-i-j-3}.$$ These formulas can be derived from those given in Theorem \[thm:clt-mult\] by means of the representation , and the terms can also be interpreted easily: write $\log t(n) = \sum_{i \geq 1} X_i(n) \log s_i$, where $X_i(n)$ is the number of runs of length $i$ in the binary representation of $n$. The coefficients in the two formulas stem from mean, variance and covariances of the $X_i(n)$.
In the special case that $s_{n}$ is the Jacobsthal sequence ($s_n = \frac13(2^{n+2} - (-1)^n)$, see Section \[sec:exampl-q-quasiadd\]), we have the numerical values $\mu \approx 0.429947$, $\sigma^{2} \approx 0.121137$.
[^1]: The first author is supported by the Austrian Science Fund (FWF): P 24644-N26.
[^2]: The second author is supported by the National Research Foundation of South Africa under grant number 96236.
[^3]: The authors were also supported by the Karl Popper Kolleg “Modeling–Simulation–Optimization” funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF). Part of this paper was written while the second author was a Karl Popper Fellow at the Mathematics Institute in Klagenfurt. He would like to thank the institute for the hospitality received.
|
---
abstract: 'A multiple maneuvering target system can be viewed as a Jump Markov System (JMS) in the sense that the target movement can be modeled using different motion models where the transition between the motion models by a particular target follows a Markov chain probability rule. This paper describes a Generalized Labelled Multi-Bernoulli (GLMB) filter for tracking maneuvering targets whose movement can be modeled via such a JMS. The proposed filter is validated with two linear and non-linear maneuvering target tracking examples.'
author:
-
-
-
title: 'A Generalized Labeled Multi-Bernoulli Filter for Maneuvering Targets'
---
Introduction
============
Multiple target tracking is the problem of estimating an unknown and time varying number of trajectories from observed data. There are two main challenges in this problem. The first is the time-varying number of targets due to the appearance of new targets and deaths of existing targets, while the second is the unknown association between measurements and targets, which is further confounded by false measurements and missed detections of actual targets [@YRK01; @YPX11; @Mahler1; @Mahler14; @Blackman; @Barshalom].
The Bayes optimal approach to the multi-target tracking problem is the Bayes multi-target filter that recursively propagates the multi-target posterior density forward in time [@Mahler1] incorporating both the uncertainty in the number of objects as well as their states. Under the standard multi-target system model (which takes into account target births,deaths,survivals and detections,misdetections and clutter), the multi-target posterior densities at each time are Generalized Labeled Multi-Bernoulli (GLMB) densities [@GLMB1]. The $\delta $-GLMB filter [@GLMB2; @HVV15_1; @HVV15_2] is an analytic solution to the multi-target Bayes filter.
While a non-maneuvering target motion can be described by a fixed model, a combination of motion models that characterise different maneuvers may be needed to describe the motion of a maneuvering target.Tracking a maneuvering target in clutter is a challenging problem and is the subject of numerous works [@Kiruba00; @Doucet; @Verca1; @YRK01; @YPX11; @LiVSMM00; @LiJilkovMM05; @IMM; @Pasha; @Dunne13; @IMM_CBMEMBER; @Reuter15]. Tracking multiple maneuvering targets involves jointly estimating the number of targets and their states at each time step in the presence of noise, clutter, uncertainties in target maneuvers, data association and detection. As such, this problem is extremely challenging in both theory and implementation.
The jump Markov system (JMS) or multiple models approach has proven to be an effective tool for single maneuvering target tracking [@Doucet; @Verca1; @Pasha; @Dunne13; @IMM_CBMEMBER; @Reuter15]. In this approach, the target can switch between a set of models in a Markovian fashion. The interacting multiple model(IMM) and variable-structure IMM (VS-IMM) estimators [@YRK01; @YPX11; @LiVSMM00; @LiJilkovMM05; @IMM] are two well known single-target filtering algorithms for maneuvering targets. The number of modes in the IMM is kept fixed, whereas in the VS-IMM the number of modes are adaptively selected from a fixed set of modes for improved estimation accuracy and computational efficiency.
A Probability Hypothesis Density (PHD) filter [@MahlerPHD2] for maneuvering target tracking was derived in [@Pasha] together with a Gaussian mixture implementation and particle implementation. As shown by Mahler in [@MahlerJMS12], this was the only mathematically valid filter amongst severval PHD (and Cardinalized PHD) filters proposed for jump Markov systems (JMSs) [@Punith08],[@Georgescu12]. Recently, multi-Bernoulli and labeled multi-Bernoulli [@VVC09], [@Reuter14] filters were also derived for JMSs in [@Dunne13; @IMM_CBMEMBER; @Reuter15]. These filters, however, are only approximate solutions to the Bayes multi-target filter for maneuvering targets, and at present there are no exact solutions in the literature.
In this paper, we propose an analytic solution to the Bayes multi-target filter for maneuvering target tracking using JMSs. Specifically, we extend the GLMB filter to JMSs that can be implemented via Gaussian mixture or sequential Monte Carlo methods. In addition to being an analytic solution and hence more accurate than approximations, the proposed solution outputs tracks or trajectories of the targets, whereas the PHD and (unlabeled) multi-Bernoulli filters do not. The proposed technique is verified via numerical examples.
Background
==========
We review JMS and the Bayes multi-target tracking filter in this section.
JMS model for maneuvering targets
---------------------------------
A JMS consists of a set of parameterised state space models, whose parameters evolve with time according to a finite state Markov chain. An example of a maneuvering target scenario which can be successfully represented using a JMS model is the dynamics of an aircraft, which can fly with a nearly constant velocity motion, accelerated/decelerated motion, and coordinated turn [@YRK01; @YPX11]. Under a JMS framework for such a system a target that is moving under a certain motion model at any time step are assumed to follow the same motion model with a certain probability or switch to a different motion model (that belongs to a set of pre-selected motion models) with a certain probability in the next time step.
A Markovian transition probability matrix describes the probabilities with which a particular target changes/retains the motion model in the next time step given the motion model at current time step. Let $\vartheta
(r|r^{\prime })$ denote the probability of switching to motion model $r$ from $r^{\prime }$ as given by this markovian transition matrix, in which the sum of the conditional probabilities of all possible motion models in the next time step given the current model adds upto 1, i.e.,$$\sum_{r\in \mathcal{R}}\vartheta (r|r^{\prime })=1$$where $R$ is the (discrete) set of motion models in the system.
Suppose that model $r$ is in effect at time $k$, then the state transition density from $\zeta ^{\prime }$, at time $k-1$, to $\zeta $, at time $k$, is denoted by $\phi _{k|k-1}(\zeta |\zeta ^{\prime },r)$, and the likelihood of $\zeta $ generating the measurement $z$ is denoted by $\gamma _{k}(z|\zeta
,r)$ [@YRK01; @YPX11; @Ristic04]. Moreover, the joint transition of the state and the motion model assumes the form:$$f_{k|k-1}(\zeta ,r|\zeta ^{\prime },r^{\prime })=\phi _{k|k-1}(\zeta |\zeta
^{\prime },r)\vartheta (r|r^{\prime }).$$In general, the measurement can also depend on the model $r$ and hence the likelihood function becomes $g_{k}(z|\zeta ,r)$. Note that by defining the augmented system state as $x=(\zeta ,r\mathbf{)}$ a JMS model can be written as a standard state space model.
JMS models are not only useful for tracking maneuvering targets, but are also useful in the estimation of unknown clutter parameters [@MahlerclutterCPHD11; @RobustMB13].
Bayes multi-target tracking filter
----------------------------------
In the Bayes multi-target tracking filter, the state of a target includes an ordered pair of integers $\ell =(k,i)$, where $k$ is the time of birth, and $%
i$ is a unique index to distinguish targets born at the same time. The label space for targets born at time $k$ is denoted as $\mathbb{L}_{k}$ and the label space for targets at time $k$ (including those born prior to $k$) is denoted as $\mathbb{L}_{0:k}$. Note that $\mathbb{L}_{0:k}=\mathbb{L}%
_{0:k-1}\cup \mathbb{L}_{k}$, and that $\mathbb{L}_{0:k-1}$ and $\mathbb{L}%
_{k}$ are disjoint. An existing target at time $k$ has state $\mathbf{x}%
=(x,\ell )$ consisting of the kinematic/feature $x$ and label $\ell \in
\mathbb{L}_{0:k}$. A multi-target state $\mathbf{X}$ (uppercase notation) is a finite set of single-target states.
All information about the multi-target state at time $k$ is contained in $%
\mathbf{\pi }_{k}$, the posterior density of the multi-target state conditioned on $Z_{1:k}=(Z_{1},...,Z_{k})$, the measurement history upto time $k$, where $Z_{k}$ is the finite set of measurements received at time $%
k $. The Bayes multi-target tracking filter consists of a prediction step and an update step , which propagate the multi-target posterior/filtering density forward in time. Note that the integral in this case is the *set integral* from finite set statistics [@Mahler1]. $$\mathbf{\pi }_{k|k-1}(\mathbf{X}_{k})\!=\!\!\int \!\mathbf{f}_{k|k-1}(%
\mathbf{X}_{k}|\mathbf{X})\mathbf{\pi }_{k-1}(\mathbf{X})\delta \mathbf{X}
\label{eq1}$$$$\mathbf{\pi }_{k}(\mathbf{X}_{k})=\frac{g_{k}(Z_{k}|\mathbf{X}_{k})\mathbf{%
\pi }_{k|k-1}(\mathbf{X}_{k})}{\int g_{k}(Z_{k}|\mathbf{X})\mathbf{\pi }%
_{k|k-1}(\mathbf{X})\delta \mathbf{X}} \label{eq2}$$where $\mathbf{f}_{k|k-1}(\cdot |\cdot )$ denotes the multi-target transition kernel from time $k-1$* *to $k$, and $g_{k}(\cdot |\cdot
) $ denotes the likelihood function at time $k$. Note that for compactness we omitted dependence on the measurement history from $\mathbf{\pi }_{k|k-1}$ and $\mathbf{\pi }_{k}$. Note that the same multi-target recursion - also holds for multi-target states without labels.
A generic particle implementation of the multi-target Bayes recursions - (for both labeled and unlabeled multi-target states) was given in [@VoAES], while analytic approximations for unlabeled multi-target states, such as the PHD, Cardinalized PHD and multi-Bernoulli filters were proposed in [Mahler1,MahlerCPHDAES,VoMaGMPHD05,VoGaussianCPHD07,VVC09,VVPS10]{}. The GLMB filter [@GLMB1], [@GLMB2] is an analytic solution to the multi-target Bayes recursions -.
JMS-GLMB filtering
==================
We start this subsection with some notations. For the labels of a multi-target state $\mathbf{X}$ to be distinct, we require $\mathbf{X}$ and the set of labels of $\mathbf{X}$, denoted as $\mathcal{L}(\mathbf{X})$, to have the same cardinality, .i.e. the same number of elements. Hence, we define the *distinct label indicator* as the function$$\Delta (\mathbf{X})\triangleq \delta _{|\mathbf{X}|}[|\mathcal{L(}\mathbf{X}%
)|],$$where $|Y|$ denotes the cardinality of the set $Y$, and $\delta _{n}[m]$ denotes the Kronecker delta. The *indicator function* is defined as as $$1_{Y}(x)\triangleq \Big\{{}%
\begin{array}{ll}
1, & \text{if }x{\in Y} \\
0, & \text{otherwise }{}%
\end{array}%
.$$For any finite set $Y$, and test function $h\leq 1$, the multi-object exponential is defined by$$h^{Y}\triangleq \prod_{y\in Y}h(y),$$with $h^{\emptyset }$ $=1$ by convention. We also use the standard inner production notation$$\left\langle f,g\right\rangle =\int f(x)g(x)dx,$$for any real functions $f$ and $g$.
An *association map* at time $k$ is a function $\theta :\mathbb{L}%
_{0:k}\rightarrow \{0,1,...,|Z_{k}|\}$ such that $\theta (\ell )=\theta
(\ell ^{\prime })>0~$implies$~\ell =\ell ^{\prime }$. Such a function can be regarded as an assignment of labels to measurements, with undetected labels assigned to $0$. The set of all such association maps is denoted as $\Theta
_{k}$; the subset of association maps with domain $L$ is denoted by $\Theta
_{k}(L)$; and $\Theta _{0:k}\triangleq \Theta _{0}\times ...\times \Theta
_{k}$ denotes the space of association map history.
GLMB filter
-----------
In the GLMB filter, the multi-target filtering density at time $k-1$ is a GLMB of the form: $$\mathbf{\pi }_{k-1}(\mathbf{X})=\Delta (\mathbf{X})\!\!\!\sum_{\xi \in
\Theta _{0:k\!-\!1}}\!\!\!\!w_{k-1}^{(\xi )}(\mathcal{L(}\mathbf{X}%
))[p_{k-1}^{(\xi )}]^{\mathbf{X}}, \label{eq:GLMB_prev}$$where each $\xi =(\theta _{0},...,\theta _{k-1})\in \Theta _{0:k-1}$ represents a history of association maps up to time $k-1$; each weight $%
w_{k-1}^{(\xi )}(L)$ is non-negative with $$\sum_{L\subseteq \mathbb{L}_{0:k-1}}\sum_{\xi \in \Theta
_{0:k-1}}w_{k-1}^{(\xi )}(L)=1,$$and each $p_{k-1}^{(\xi )}(\cdot ,\ell )$ is a probability density.
Given a GLMB filtering density, a tractable suboptimal multi-target estimate is obtained by the following proceedure: determine the maximum a posteriori cardinality estimate $n^{\ast }$ from the cardinality distribution$$\rho _{k-1}(n)=\sum_{L\subseteq \mathbb{L}_{0:k-1}}\sum_{\xi \in \Theta
_{0:k-1}}\delta _{n}[|L|]w_{k-1}^{(\xi )}(L); \label{eq:GLMBCard}$$determine the label set $L^{\ast }$ and $\xi ^{\ast }$ with highest weight $%
w_{k-1}^{(\xi ^{\ast })}(L^{\ast })$ among those with cardinality $n^{\ast }$; determine the expected values of the states from $p_{k-1}^{(\xi ^{\ast
})}(\cdot ,\ell )$, $\ell \in L^{\ast }$ [@GLMB1].
The GLMB density is a conjugate prior with respect to the standard multi-target likelihood function and is also closed under the multi-target prediction [@GLMB1]. Under the standard multi-target transition model, if the multi-target filtering density, at the previous time, $\mathbf{\pi }%
_{k-1}$ is a GLMB of the form (\[eq:GLMB\_prev\]), then the multi-target prediction density $\mathbf{\pi }_{k|k-1}$ is a GLMB of the form ([eq:GLMBpred]{}) given by [@GLMB1].$$\mathbf{\pi }_{k|k-1}(\mathbf{X})=\Delta (\mathbf{X})\!\!\!\sum_{\xi \in
\Theta _{0:k-1}}\!\!w_{k|k-1}^{(\xi )}(\mathcal{L(}\mathbf{X}%
))[p_{k|k-1}^{(\xi )}]^{\mathbf{X}}, \label{eq:GLMBpred}$$where$$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!w_{k|k\!-\!1\!}^{(\xi )}(L)\!\!\!\!
&=&\!\!\!\!w_{S,k|k-1}^{(\xi )}(L\cap \mathbb{L}_{0:k-1})w_{B,k}(L\cap
\mathbb{L}_{k}), \\
\!\!\!\!\!\!\!\!\!\!p_{k|k\!-\!1\!}^{(\xi )}(x,\ell )\!\!\!\! &=&\!\!\!\!1_{%
\mathbb{L}_{0:k\!-\!1}\!}(\ell )p_{S,k|k\!-\!1\!}^{(\xi )\!}(x,\ell )\!+\!1_{%
\mathbb{L}_{k}\!}(\ell )p_{B,k}(x,\ell ), \\
\!\!\!\!\!\!\!\!\!\!w_{S,k|k\!-\!1\!}^{(\xi )}(L)\!\!\!\! &=&\!\!\!\![\bar{P}%
_{S,k|k\!-\!1}^{(\xi )}]^{L}\!\sum_{I\supseteq L}[1\!-\!\bar{P}%
_{S,k|k\!-\!1}^{(\xi )}]^{I\!-\!L}w_{k\!-\!1\!}^{(\xi )}(I), \\
\!\!\!\!\!\!\!\!\!\!\bar{P}_{S,k|k\!-\!1\!}^{(\xi )}(\ell )\!\!\!\!
&=&\!\!\!\!\left\langle P_{S,k|k-1}(\cdot ,\ell ),p_{k-1}^{(\xi )}(\cdot
,\ell )\right\rangle , \\
\!\!\!\!\!\!\!\!\!\!p_{S,k|k\!-\!1\!}^{(\xi )}(x,\ell )\!\!\!\! &=&\!\!\!\!%
\frac{\left\langle P_{S,k|k\!-\!1}(\cdot ,\ell )f_{k|k\!-\!1\!}(x|\cdot
,\ell ),p_{k\!-\!1}^{(\xi )}(\cdot ,\ell )\right\rangle }{\bar{P}%
_{S,k|k-1}^{(\xi )}(\ell )}, \\
\!\!\!\!\!\!\!\!\!\!P_{S,k|k-1}(x,\ell )\!\!\!\! &=&\!\!\!\!\text{%
probability of survival to time }k\text{ of a target } \\
&&\!\!\!\!\text{with previous state }(x,\ell ), \\
f_{k|k\!-\!1\!}(x|x^{\prime },\ell )\!\!\!\! &=&\!\!\!\!\text{transition
density of feature }x^{\prime }\text{ at time } \\
&&\!\!\!\!k-1\text{ to }x\text{ at time }k\text{ for target with label }\ell
\text{, } \\
\!\!\!\!\!\!\!\!\!\!w_{B,k}(L)\!\!\!\! &=&\!\!\!\!\text{probability of
targets with labels }L\text{ being } \\
&&\!\!\!\!\text{born at time }k\text{,} \\
\!\!\!\!\!\!\!\!\!\!p_{B,k}(x,\ell )\!\!\!\! &=&\!\!\!\!\text{probability
density of the feature }x\text{ of a} \\
&&\!\!\!\!\text{new target born at time }k\text{ with label }\ell \text{.}\end{aligned}$$
Moreover, under the standard multi-target measurement model, the multi-target filtering density $\mathbf{\pi }_{k}$ is a GLMB given by$$\mathbf{\pi }_{k}\!(\mathbf{X})=\Delta \!(\mathbf{X})\!\!\!\!\!\!\sum_{\xi
\in \Theta _{0:k\!-\!1}}\sum\limits_{\theta \in \Theta _{k}}\!w_{k}^{\!(\xi
,\theta )\!}(\mathcal{L(}\mathbf{X})|Z_{k})[p^{\!(\xi ,\theta )\!}(\cdot
|Z_{k})]^{\mathbf{X}}\!\!, \label{eq:GLMBupdate}$$where $$\begin{aligned}
w_{k}^{(\xi ,\theta )\!}(L|Z)\!\!\! &\propto &\!\!\!1_{\Theta
_{k}\!(L)}(\theta )[\bar{\Psi}_{Z,k}^{(\xi ,\theta )}]^{L}w_{k|k-1}^{(\xi
)}(L), \\
p_{k}^{\!(\xi ,\theta )\!}(x,\ell |Z)\!\!\! &=&\!\!\!\frac{\Psi
_{Z,k}^{(\theta )}(x,\ell )p_{k|k-1}^{(\xi )}(x,\ell )}{\bar{\Psi}%
_{Z,k}^{(\xi ,\theta )}(\ell )} \\
\bar{\Psi}_{Z,k}^{(\xi ,\theta )}(\ell )\!\!\! &=&\!\!\!\left\langle \Psi
_{Z,k}^{(\theta )}(\cdot ,\ell ),p_{k|k-1}^{(\xi )}(\cdot ,\ell
)\right\rangle , \\
\Psi _{\{z_{1},...,z_{m}\},k}^{(\theta )}(x,\ell )\!\!\! &=&\!\!\!\left\{
\begin{array}{ll}
\frac{P_{D,k}(x,\ell )g_{k}(z_{\theta (\ell )}|x,\ell )}{\kappa
_{k}(z_{\theta (\ell )})}, & \text{if }{\small \theta (\ell )>0} \\
{\small 1-P}_{D,k}{\small (}x{\small ,\ell )}, & \text{if }{\small \theta
(\ell )=0}%
\end{array}%
\right. \\
P_{D,k}(x,\ell )\!\!\! &=&\!\!\!\text{probability of detection at time }k \\
&&\!\!\!\text{of a target with state }(x,\ell ), \\
g_{k}(z|x,\ell )\!\!\! &=&\!\!\!\text{likelihood that at time }k\text{
target with} \\
&&\!\!\!\text{state }(x,\ell )\text{ generate measurement }z, \\
\kappa _{k}\!\!\! &=&\!\!\!\text{intensity function of Poisson clutter} \\
&&\!\!\!\text{at time }k\text{ }\end{aligned}$$
The GLMB recursion above is the first analytic solution to the Bayes multitarget filter. Truncating the GLMB sum is needed to manage the growing the number of components in the GLMB filter [@GLMB2].
GLMB filter for Manuevering Targets
-----------------------------------
We define the (labeled) state of a manuevering target to include the kinematic/feature $\zeta $, the motion model index $r$, and the label $\ell $, i.e., $\mathbf{x}=(\zeta ,r,\ell )$, which can be modeled as a JMS. Note that the label of each target remains constant throughout it’s life even though it is part of the state vector. Hence the JMS state equations for a target with label $\ell $ are indexed by $\ell $, i.e., $\phi
_{k|k-1}^{(\ell )}(\zeta |\zeta ^{\prime },r)$ and $\gamma _{k}^{(\ell
)}(z|\zeta ,r)$. The new state of a surviving target will also be governed by the probability of the target transitioning to that motion model from the previous model in addition to the probability of survival and the relevant state transtition function. Consequently, the joint transition and likelihood function for the state and the model index are given by, $$\begin{aligned}
f_{k|k-1}(\zeta ,r|\zeta ^{\prime },r^{\prime },\ell ) &=&\phi
_{k|k-1}^{(\ell )}(\zeta |\zeta ^{\prime },r)\times \vartheta (r|r^{\prime })
\label{eq:JMS-GLMB} \\
g_{k}(z|\zeta ,r,\ell ) &=&\gamma _{k}^{(\ell )}(z|\zeta ,r)
\label{eq:JMS-GLMB1}\end{aligned}$$
Substituting (\[eq:JMS-GLMB\]) and (\[eq:JMS-GLMB1\]) into the GLMB prediction and update equations yields the GLMB filter for maneuvering targets. Note that since $x=(\zeta ,r)$ $$\int f(x)dx=\sum_{r\in \mathcal{R}}\int f(\zeta ,r)d\zeta .$$
The state extraction is akin to the single model system. To estimate the motion model for each label, we select the motion model that maximizes the marginal probability of that model over the entire density for that label, i.e., for label $\ell $ of component $\xi $, the estimated motion model $%
\hat{r}$ is given by . $$\hat{r}=\underset{r}{\mbox{argmax}}\;\int p^{(\xi )}( \zeta,r,\ell )d\zeta
\label{eq8}$$
Analytic Solution
-----------------
Consider the special case where the target birth model, motion models and observation model are all linear models with Gaussian noise. Given that the posterior density at time $k-1$ is of the form (\[eq:GLMB\_prev\]) with $%
\mathbf{x}=(\zeta ,r,\ell )$, the GLMB filter prediction equation can be explicitly written as$$\mathbf{\pi }_{k|k-1}(\mathbf{X})=\Delta (\mathbf{X})\!\!\!\sum_{\xi \in
\Theta _{0:k-1}}\!\!w_{k|k-1}^{(\xi )}(\mathcal{L(}\mathbf{X}%
))[p_{k|k-1}^{(\xi )}]^{\mathbf{X}}, \label{eq:GLMBpred}$$where$$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!w_{k|k\!-\!1\!}^{(\xi )}(L)\!\!\!\!
&=&\!\!\!\!w_{S,k|k-1}^{(\xi )}(L\cap \mathbb{L}_{0:k-1})w_{B,k}(L\cap
\mathbb{L}_{k}), \\
\!\!\!\!\!\!\!\!\!\!p_{k|k\!-\!1\!}^{(\xi )}(\zeta \!,r\!,\ell \!)\!\!\!\!
&=&\!\!\!\!1_{\mathbb{L}_{0:k\!-\!1}\!}(\ell )p_{S,k|k\!-\!1\!}^{(\xi
)\!}(\zeta \!,r\!,\ell \!)\!\!+\!\!1_{\mathbb{L}_{k}\!}(\ell )p_{B,k}(\zeta
\!,r\!,\ell \!), \\
\!\!\!\!\!\!\!\!\!\!w_{S,k|k\!-\!1\!}^{(\xi )}(L)\!\!\!\! &=&\!\!\!\![\bar{P}%
_{S,k|k\!-\!1}^{(\xi )}]^{L}\!\sum_{I\supseteq L}[1\!-\!\bar{P}%
_{S,k|k\!-\!1}^{(\xi )}]^{I\!-\!L}w_{k\!-\!1\!}^{(\xi )}(I), \\
\!\!\!\!\!\!\!\!\!\!\bar{P}_{S,k|k\!-\!1\!}^{(\xi )}(\ell )\!\!\!\!
&=&\!\!\!\!\sum_{r\in R}\bar{P}_{S,k|k\!-\!1\!}^{(\xi )}(r,\ell ), \\
\!\!\!\!\!\!\!\!\!\!\bar{P}_{S,k|k\!-\!1\!}^{(\xi )}(r,\ell )\!\!\!\!
&=&\!\!\!\!\left\langle P_{S,k|k-1}(\cdot ,r,\ell ),p_{k-1}^{(\xi )}(\cdot
,r,\ell )\right\rangle , \\
\!\!\!\!\!\!\!\!\!\!p_{S,k|k\!-\!1\!}^{(\xi )}(\zeta ,r,\ell )\!\!\!\!
&=&\!\!\!\!\!\!\frac{\underset{r^{\prime }\in R}{\sum }\!\!\!\left\langle
\!\!P_{S,k|k\!-\!1}(\cdot ,\ell )f_{k|k\!-\!1\!}(\!\zeta ,r|\!\cdotp\!,r^{\prime
}\!,\ell ),p_{k\!-\!1}^{(\xi )}(\!\cdot\!,\!\ell )\!\!\right\rangle \!\!}{\bar{P}%
_{S,k|k-1}^{(\xi )}(\ell )}, \\
\!\!\!\!\!\!\!\!\!\!P_{S,k|k-1}(\zeta ,r,\ell )\!\!\!\! &=&\!\!\!\!\text{%
probability of survival to time }k\text{ of a } \\
&&\!\!\!\!\text{target with previous labeled state }(\zeta ,r,\ell ), \\
f_{k|k\!-\!1\!}(\zeta ,r|\zeta ^{\prime },r^{\prime },\ell )\!\!\!\!
&=&\!\!\!\mathcal{N}(\zeta ;F^{(r)}\zeta ^{\prime },Q_{F}^{(r)})\times
\vartheta (r|r^{\prime }) \\
F^{(r)}\!\!\! &=&\!\!\!\text{state transition matrix of motion mode\!l }r, \\
Q_{F}^{(r)}\!\!\! &=&\!\!\!\text{covariance matrix of motion model }r, \\
\!\!\!\!\!\!\!\!\!\!w_{B,k}(L)\!\!\!\! &=&\!\!\!\!\text{probability of
targets with labels }L\text{ being } \\
&&\!\!\!\!\text{born at time }k, \\
\!\!\!\!\!\!\!\!\!\!p_{B,k}(\zeta ,r,\ell )\!\!\!\! &=&\!\!\!\!\mathcal{N}%
(\zeta ;m^{(i)},Q_{B}^{(i)})\times \vartheta ^{(i)}(r) \\
\vartheta ^{(i)}(r) &=&\!\!\text{probability that a target born at birth} \\
&&\!\!\!\!\text{ region }i\text{ possesses motion model }r, \\
m^{(i)}\!\!\! &=&\!\!\!\text{mean of birth region }i, \\
Q_{B}^{(i)}\!\!\! &=&\!\!\!\text{covariance of birth region }i,\end{aligned}$$
Moreover, GLMB update formula can be written explicitly as$$\mathbf{\pi }_{k}\!(\mathbf{X})=\Delta \!(\mathbf{X})\!\!\!\!\!\!\sum_{\xi
\in \Theta _{0:k\!-\!1}}\sum\limits_{\theta \in \Theta _{k}}\!w_{k}^{\!(\xi
,\theta )\!}(\mathcal{L(}\mathbf{X})|Z_{k})[p^{\!(\xi ,\theta )\!}(\cdot
|Z_{k})]^{\mathbf{X}}\!\!, \label{eq:GLMBupdate}$$where $$\begin{aligned}
w_{k}^{(\xi ,\theta )\!}(L|Z)\!\!\! &\propto &\!\!\!1_{\Theta
_{k}\!(L)}(\theta )[\bar{\Psi}_{Z,k}^{(\xi ,\theta )}]^{L}w_{k|k-1}^{(\xi
)}(L), \\
p_{k}^{\!(\xi ,\theta )\!}(\zeta ,r,\ell |Z)\!\!\!\! &=&\!\!\!\frac{\Psi
_{Z,k}^{(\theta )}(\zeta ,r,\ell )p_{k|k-1}^{(\xi )}(\zeta ,r,\ell )}{\bar{%
\Psi}_{Z,k}^{(\xi ,\theta )}(\ell )} \\
\bar{\Psi}_{Z,k}^{(\xi ,\theta )}(\ell )\!\!\!\! &=&\!\!\!\sum_{r\in
R}\left\langle \Psi _{Z,k}^{(\theta )}(\cdotp,r,\ell ),p_{k|k-1}^{(\xi
)}(\cdot ,r,\ell )\right\rangle , \\
\Psi _{\{z_{1},...,z_{m}\},k}^{(\theta )}(\zeta ,r,\ell )\!\!\!\!
&=&\!\!\!\!\left\{
\begin{array}{ll}
\!\!\!\frac{P_{D,k}(\zeta ,r,\ell )g_{k}(z_{\theta (\ell )}|\zeta ,r,\ell )}{%
\kappa _{k}(z_{\theta (\ell )})}, & \!\!\!\!\text{if }{\small \theta (\ell
)>0} \\
\!\!\!{\small 1-P}_{D,k}{\small (}\zeta ,r{\small ,\ell )}, & \!\!\!\!\text{%
if }{\small \theta (\ell )=0}%
\end{array}%
\right. \\
P_{D,k}(\zeta ,r,\ell )\!\!\! &=&\!\!\!\!\text{probability of detection at
time }k \\
&&\!\!\!\text{of a target with state }(\zeta ,r,\ell ), \\
g_{k}(z|\zeta ,r,\ell )\!\!\! &=&\!\!\!\!\mathcal{N}(z;H^{(r)}\zeta
,Q_{H}^{(r)}) \\
\kappa _{k}\!\!\! &=&\!\!\!\!\text{intensity function of Poisson clutter}, \\
H^{(r)}\!\!\! &=&\!\!\!\!\text{likelihood matrix for targets } \\
&&\!\!\!\!\text{moving under motion model r}, \\
Q_{H}^{(r)}\!\!\! &=&\!\!\!\!\text{covariance matrix of likelihood for } \\
&&\!\!\!\!\text{targets \negthinspace moving \negthinspace under
\negthinspace motion \negthinspace model r}.\end{aligned}$$
For mildly non-linear motion models and measurement models, the unscented Kalman Filter (UKF) [@Ristic04; @UKF] can been utilized for predicting and updating each Gaussian component in the mixture forward. Alternatively, instead of a making use of a Gaussian mixture to represent the posterior density of each track in a hypothesis, a particle filter can be employed. Instead of a Gaussian mixture, the density is represented using a set of particles which are propagated forward under the different motion models with adjusted weights for each particle. As in the case of the Gaussian mixture, the number of particles in the density increase by threefold during each prediction forward. Thus resampling needs to be carried out to discard particles with negligible weights and keep the total count of particles in control.
Implementation Issues
---------------------
In the above solution it is evident that the posterior density for each track is a Gaussian mixture, with each mixture component relating to one of the motion models present. For a particular track, at each new time step the posterior is predicted forward for all motion models present in the system, thereby generating a new Gaussian mixture. The weight of each new component will be the weight of the parent component multiplied by the probability of switching to the corresponding motion model. As a result the number of mixture components escalates exponentially. Hence extensive pruning and merging must be carried out for each track in each GLMB hypothesis after the update step to keep the computation managable.
Simulation Results
===================
In this section we demonstrate the use of the proposed JMS-GLMB solution via two multiple manuevering target tracking examples.
*Linear Example*: The kinematic state of each target in this example consists of cartesian x and y coordinates and their respective velocities. $%
T=5s$ is the sampling interval. The observation area is a \[-60, 60\] $\times $ \[-60, 60\] $km^{2}$ area. The JMS used in the simulation consists of three types of motion models viz. constant velocity, right turn (coordinated turn with a $3^{\circ }$ angle), and left turn (coordinated turn with a $%
-3^{\circ }$ angle). The state transition matrices for the three models are obtained via substituting $\omega =0$, $\omega =5\pi /180$ and $\omega
=-5\pi /180$ in equation respectively.The process noise co-variance $Q_{L}$ is given in with $\sigma
_{v1}=5ms^{-1},\sigma _{v2}=\sigma _{v3}=20ms^{-1}$. The markovian motion model switching probability matrix is given in .
$$\label{m1}
F_1 =
\begin{bmatrix}
1 & T & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & T \\
0 & 0 & 0 & 1%
\end{bmatrix}%$$
$$F_{2}(\omega)=%
\begin{bmatrix}
1 & sin(T\omega )/\omega & 0 & (cos(T\omega )-1)/\omega \\
0 & cos(T\omega ) & 0 & -sin(T\omega ) \\
0 & -(cos(T\omega )-1)/\omega & 1 & sin(T\omega )/\omega \\
0 & sin(T\omega ) & 0 & cos(T\omega )%
\end{bmatrix}
\label{m2}$$
$$\label{procnoise}
Q_L = \sigma_{vr}^2
\begin{bmatrix}
T^4/4 & T^3/2 & 0 & 0 \\
T^3/2 & T^2 & 0 & 0 \\
0 & 0 & T^4/4 & T^3/2 \\
0 & 0 & T^3/2 & T^2%
\end{bmatrix}%$$
$$\label{mvt}
\vartheta(r^{\prime }|r) = M( r, r^{\prime }) \text{ where } M =
\begin{bmatrix}
0.8 & 0.1 & 0.1 \\
0.2 & 0.8 & 0 \\
0.2 & 0 & 0.8%
\end{bmatrix}%$$
Targets are spontaneously born at three pre-defined Gaussian birth locations $\mathcal{N}(m_{1},P_{L}),\mathcal{N}(m_{2},P_{L}),\mathcal{N}(m_{3},P_{L})$ where. $$m_{1}=[:40000,0,-50000,0],::m_{2}=[:-50000,0,40000,0]$$$$m_{3}=[-10000,0,0,0],P_{L}\!=\!diag([\!1000,\!300,\!1000,\!300]).$$Targets are born from each location at each time step with a probability of 0.2 and the initial motion model is model 1.
The $x$ and $y$ corrdinates of the targets are observed by a single sensor located at (0, 0) with probability of detection $P_{D}=0.97$ (observation matrix H given in ). The measurements are subjected to zero mean noise with a covariance of $\sigma _{h}^{2}I_{2}$ where $\sigma _{h}=40m $ and $I_{2}$ is the identity matrix of dimestion 2. Clutter is modeled as a uniform Poisson with an average number of 60 measurements per scan.
$$H=%
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0%
\end{bmatrix}
\label{obs}$$
Figure (\[traj1\]) shows the trajectories of three targets born at different time steps in a simlation run. Fig.(\[est\]) illustrates the estimated coordinates colour coded in red (constant velocity), blue (right turn) and green (left turn) to indicate the estimated motion models along with the true path (coninous lines) and measurements (grey crosses).
The Optimal Subpattern Assignment Metric (OSPA)[@OSPApaper] values calculated for 100 monte carlo runs for the linear example are shown in the top graph of fig.(\[ospa\]). The top graph of figure (\[tgt1\]) shows the probabilities of estimating each motion model (colour coded) in each time step for target 1. For example, between time steps 1 to 30, constant velocity model (red) has a higher probability (above 0.9 in most time steps) of being the motion model which guided the target. It can be observed that the the actual motion model under which the target was simulated to move and the estimated model are the same in most time steps.
*Nonlinear Example*: In this case the motion models and the measurement models are non-linear, and the unscented Kalman Filter (UKF) [@Ristic04; @UKF] is used for predicting and updating each Gaussian component in the mixture forward.
The motion models under which the targets are moving are the constant velocity model and the coordinated turn model with unknown turn rate. The birth locations are given by $\mathcal{N}( m_4, P_{NL}), \mathcal{N}( m_5,
P_{NL}), \mathcal{N}( m_6, P_{NL})$ where, $$m_4= [ 40000, 0, -50000, 0, 0], \:\: m_5 = [ -50000, 0, 40000, 0, 0],$$ $$m_6\!=\![\!-10000,\!0,\!0,\!0,\!0],\!P_{NL}\!\!=\!diag([\!1000,\!300,\!1000,%
\!300,\!-1 \times 10^{-4}]).$$ The state vector includes the turn rate in addition to the positions and velocities in $x,y$ directions and $Q_{NL}$ is the process noise co-variance matrix.
The observation region is the same as in the linear example. The measurements are obtained using a bearing and range sensor at (0,0) position Clutter is poisson distributed uniformly with an average value of 60. The measurement noise covariance is $diag([\sigma_\theta^2, \sigma_r^2]$ with $%
\sigma_\theta = \pi/180 rads^{-1}$ and $\sigma_r = 20m$. The markovian transition matrix is given in (\[mvt2\]).
$$\label{mvt2}
\vartheta(r^{\prime }|r) = M( r, r^{\prime }) \text{ where } M =
\begin{bmatrix}
0.8 & 0.2 \\
0.2 & 0.8%
\end{bmatrix}%$$
$$\label{procnoise}
Q_{NL} = \sigma_{vr}^2
\begin{bmatrix}
T^4/4 & T^3/2 & 0 & 0 & 0 \\
T^3/2 & T^2 & 0 & 0 & 0 \\
0 & 0 & T^4/4 & T^3/2 & 0 \\
0 & 0 & T^3/2 & T^2 & 0 \\
0 & 0 & 0 & 0 & T^2%
\end{bmatrix}%$$
The Optimal Subpattern Assignment Metric (OSPA)[@OSPApaper] values calculated for 100 monte carlo runs for the non linear example are shown in the bottom graph of fig.(\[ospa\]). The bottom graph of figure (\[tgt1\]) shows the probabilities of estimating each motion model (colour coded) in each time step for target 1 in the non-linear example. It can be observed that the the actual motion model under which the target was simulated to move has the higher probability.
Conclusion
==========
An algorithm for tracking multiple maneuvering targets is proposed using the GLMB multi-target tracking filtering with JMS motion models. Analytic prediction and update equations are derived along with Linear Gaussian and Unscented implementations. Simulation results verify accurate tracking and motion model estimation.
[1]{} Y. Bar-Shalom, X. Li and T. Kirubarajan, *Estimation with Applications to Tracking and Navigation*, Wiley, New York, 2001. Y. Bar-Shalom, P. Willett, and X. Tian, *Tracking and Data Fusion: A Handbook of Algorithms*, YBS Publishing, 2011. R. Mahler, *Statistical Multisource-Multitarget Information Fusion*, Norwood, MA: Artech House, 2007. R. Mahler, *Advances in Statistical Multisource-Multitarget Information Fusion*, Norwood, MA: Artech House, 2014. S. Blackman, *Multiple Target Tracking With Radar Applications*," Norwood, MA: Artech House; 1986. Y. Bar-Shalom and T. E. Fortmann, *Tracking and Data Association*, San Diego, CA: Academic, 1988.
B.-T. Vo and B.-N. Vo “Labeled Random Finite Sets and Multi-Object Conjugate Priors”, * IEEE Trans. Signal Processing*, vol 61, no 13, 2013. pp. 3460-3475.
B.-N. Vo, B.- T. Vo and D. Phung, “Labeled Random Finite Sets and the Bayes Multi-target Tracking Filter”, * IEEE Trans. Signal Processing*, 2014.
H. G. Hoang, B.-T. Vo, B.-N. Vo, “A Generalized Labeled Multi-Bernoulli Filter Implementation using Gibbs Sampling,” arXiv preprint arXiv:1506.00821
H. Hoang, B.-T. Vo and B.-N. Vo, “A Fast Implementation of the Generalized Labeled multi-Bernoulli Filter with Joint Prediction and Update,” 18th Int. Conf. Inf. Fusion, Washington DC, July 2014.
T. Kirubarajan, Y. Bar-Shalom, K.,R. Pattipati, and I. Kadar, “Ground target tracking with variable structure IMM estimator,” IEEE Trans. Aerospace and Electronic Systems, vol. 36, no. 1 pp. 26-46, 2000.
A. Doucet, N. J. Gordon, V. Krishnamurthy, “Particle filters for state estimation of jump Markov linear systems,” *IEEE Trans. Signal Processing*, vol. 47, Issue 3, pp. 613 - 624, Mar. 2001.
T. Vercauteren , D. Guo and X. Wang, “Joint multiple target tracking and classification in collaborative sensor networks”, IEEE J. Select. Areas in Communications, vol. 23, no. 4, pp.714 -723 2005.
X. R. Li, Engineer’s guide to variable-structure multiple-model estimation for tracking, Chapter 10, in *Multitarget-Multisensor Tracking: Applications and Advances*, Volume III, Ed. Y. Bar-Shalom and W. D. Blair, pp. 449–567, Aetech House, 2000.
X. R. Li and V. P. Jilkov, A survey of maneuvering target tracking, Part V: Multiple-Model methods, *IEEE Trans. Aerospace & Electronic Systems*, vol. 41, no. 4, pp. 1255–1321, 2005.
E. Mazor, A. Averbuch, Y. Bar-Shalom and J. Dayan, “Interacting Multiple Model Methods in Target Tracking:A Survey,” *IEEE Trans. Aerospace and Electronic Systems*, vol. 34, no. 1, Jan. 1998.
R. Mahler, Multitarget [Bayes]{} filtering via first-order multitarget moments, *IEEE Trans. Aerospace & Electronic Systems*, vol. 39, no. 4, pp. 1152–1178, 2003.
A. Pasha, B.-N. Vo, H. D. Tuan and W.K. Ma, “A Gaussian Mixture PHD filter for Jump Markov Systems models,” *IEEE Trans. Aerospace and Electronic Systems*, vol. 45, Issue 3, pp. 919-936, 2009.
R. Mahler, “On multitarget jump-Markov filters.” 15th Int. Conf. Inf. Fusion, Singapore 2012.
K Punithakumar, T Kirubarajan, and A. Sinha, “Multiple-model probability hypothesis density filter for tracking maneuvering targets,” IEEE Trans. Aerospace and Electronic Systems, vol. 44, no. 1, pp. 87-98, 2008.
R. Georgescu and P. Willett, “The multiple model CPHD tracker,” IEEE Trans. Signal Processing, vol. 60, no. 4, pp. 1741-1751, 2012.
B.-T. Vo, B.-N. Vo, and A. Cantoni, The Cardinality Balanced Multitarget Multi-Bernoulli filter and its implementations, *IEEE Trans. Signal Processing*, vol. 57, no. 2, pp. 409–423, Feb. 2009.
S. Reuter, B.-T. Vo, B.-N. Vo, and K. Dietmayer, The labelled multi-Bernoulli filter, *IEEE Trans. Signal Processing*, vol. 62, no. 12, pp. 3246–3260, 2014.
D. Dunne, and T. Kirubarajan, “Multiple model multi-Bernoulli filters for manoeuvering targets,” IEEE Trans. Aerospace and Electronic Systems, vol. 49, no. 4, pp. 2679-2692, 2013.
X. Yuan, F. Lian, and C. Z. Han, “Multiple-Model Cardinality Balanced Multi-target Multi-Bernoulli Filter for Tracking Maneuvering Targets,” *Journal of Applied Mathematics*, vol. 2013, 16 pages, 2013.
S. Reuter, A. Scheel, and K. Dietmayer, The Multiple Model Labeled Multi-Bernoulli Filter, Int. Conf. Inf. Fusion, WA, 2015
R. Mahler, B.-T. Vo, and B.-N. Vo. “CPHD filtering with unknown clutter rate and detection profile.” IEEE Trans. Signal Processing, vol. 59, no. 8, pp. 3497-3513, 2011.
B.-T. Vo, B.-N. Vo, R., Hoseinnezhad, R. Mahler “Robust multi-Bernoulli filtering,” IEEE J. Selected Topics in Signal Processing, vol. 7, no. 3, pp. 399-409, 2013.
B.-N. Vo, S. Singh, and A. Doucet, Sequential [Monte Carlo]{} methods for multitarget filtering with random finite sets, *IEEE Trans. Aerospace & Electronic Systems*, vol. 41, no. 4, pp. 1224–1245, 2005.
R. Mahler, filters of higher order in target number, *IEEE Trans. Aerospace & Electronic Systems*, vol. 43, no. 4, pp. 1523–1543, 2007.
B.-N. Vo and W.-K. Ma, The Gaussian mixture probability hypothesis density filter, *IEEE Trans. Signal Processing*, vol. 54, no. 11, pp. 4091–4104, 2006.
B.-T. Vo, B.-N. Vo, and A. Cantoni, Analytic implementations of the cardinalized probability hypothesis density filter, *IEEE Trans. Signal Processing*, vol. 55, no. 7, pp. 3553–3567, 2007.
B.-N. Vo, B.-T. Vo, N.-T. Pham and D. Suter, Joint detection and estimation of multiple objects from image observations, *IEEE Trans. Signal Procesing*, vol. 58, no. 10, pp. 5129–5241, 2010.
S. J. Julier and J. K. Uhlmann, A new extension of the Kalman filter to nonlinear systems,” 11th Int. Symp. Aerospace/Defense Sensing, Simulation and Controls, 1997, pp. 182-193.
B. Ristic, S. Arulampalam, and N. J. Gordon, *Beyond the Kalman Filter: Particle Filters for Tracking Applications*. Artech House, 2004.
D. Schuhmacher, B.-T. Vo, and B.-N. Vo, “A consistent metric for performance evaluation in multi-object filtering,” * IEEE Trans. Signal Processing*, Vol. 56, No. 8 Part 1, pp. 3447-3457, 2008.
|
---
abstract: 'We consider a class of martingales on Cartan-Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence and almost sure non-convergence of the angular component (in polar coordinates) of a martingale in this class, when the Cartan-Hadamard manifold is radially symmetric. Applied to minimal submanifolds, this gives upper bounds on the curvature of the ambient Cartan-Hadamard manifold under which any minimal submanifold admits a non-constant, bounded, harmonic function. Though our discussion is motivated by minimal submanifolds, this class of martingales includes diffusions naturally associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures, and we briefly discuss these contexts. Our techniques are elementary, consisting mainly of comparison geometry and Ito’s rule.'
address: 'Department of Mathematics, Lehigh University, Bethlehem, PA'
author:
- 'Robert W. Neel'
title: Martingales arising from minimal submanifolds and other geometric contexts
---
Introduction
============
We study a class of degenerate martingales on Cartan-Hadamard manifolds, specifically their transience and angular behavior in large time. The results we obtain (as well as the methods of proof) are similar to those for Brownian motion on Cartan-Hadamard manifolds. The motivation is that such martingales arise in multiple geometric contexts, such as Brownian motion on minimal submanifolds, Brownian motion along a smooth mean curvature flow, and the canonical diffusion associated with certain sub-Riemannian geometries. This allows us to re-prove and extend geometric results, such as results on the transience of minimal submanifolds, and also provides a common perspective on what might otherwise seem like disparate objects in geometry, at least at the level of these relatively coarse properties.
Our discussion is motivated by the case of minimal submanifolds. In particular, let $M$ be a (smooth, complete) Cartan-Hadamard manifold of dimension $m$. Markvorsen and Palmer [@MarkPalmGAFA] prove that if $N$ is a complete, $n$-dimensional, minimally immersed submanifold of M and either $n = 2$ and the sectional curvatures of M are bounded above by $-a^2 < 0$, or $n \geq 3$, then $N$ is transient. They prove this result by first deriving non-trivial capacity estimates. (They note there is a simpler argument using the first Dirichlet eigenvalue for the more restrictive case when $n\geq 2$ and the sectional curvatures of M are bounded above by $-a^2 < 0$.) We also mention that Stroock (see Theorem 5.23 of [@StroockGeo]) proves that minimal submanifolds of dimension 3 or more of Euclidean space are transient (that is, the case $n\geq 3$ and $M={\mathbb{R}}^m$) using a simple stochastic argument which is very much in the spirit of the present work.
We wish to further this line of inquiry. Our first main result (concerning transience), applied to minimal submanifolds, yields the following.
\[Cor:MinimalTrans\] Let $M$ be a Cartan-Hadamard manifold of dimension $m\geq 3$, and let $N$ be a properly immersed minimal submanifold of dimension $n$, for $2\leq n < m$. Then if either of the following two conditions hold:
1. $n=2$ and, in polar coordinates around some point, $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -\frac{1+2{\varepsilon}}{r^2\log r} \quad \text{for $r>R$, and for all $\theta$ and the only $\Sigma$,}$$ for some ${\varepsilon}>0$ and $R>1$, or
2. $n\geq 3$,
we have that $N$ is transient.
Here $K(r,\theta,\Sigma)$ is the sectional curvature of a plane $\Sigma$ in $T_{(r,\theta)}M$ (in polar coordinates). Our choice of $2{\varepsilon}$ rather than ${\varepsilon}$ in the bound in the $n=2$ is simply for convenience.
Obviously, this improves the curvature bound in the case $n=2$. Indeed, the bound given above is sharp (in the sense that the theorem is not true for ${\varepsilon}=0$), as discussed after Theorem \[THM:Trans\]. Further, this result is a special case of a transience result for a broader class of martingales (given in Theorem \[THM:Trans\], of which the above is an immediate corollary) that we call rank-$n$ martingales and that also includes diffusions associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures (more on this in a moment).
Our second main class of results consists of a pair of theorems giving conditions for the almost sure angular convergence of a rank-$n$ martingale on a radially symmetric Cartan-Hadamard manifold (contained in Theorem \[THM:AngleCon\]) and conditions for the almost sure angular non-convergence of a rank-$n$ martingale on a radially symmetric Cartan-Hadamard manifold (contained in Theorem \[THM:AngleNoCon\]). The immediate geometric application is again to minimal submanifolds, where we get the following.
\[Cor:MinHarm\] Suppose that $M$ is Cartan-Hadamard manifold of dimension $m\geq 3$, and that $M$ is radially symmetric around some point $p$. Let $(r,\theta)$ be polar coordinates around $p$. Let $N$ be an $n$-dimensional, properly immersed minimal submanifold (in $M$). Then if either of the following conditions hold:
1. $n=2$ and $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -a^2 \quad \text{for all $\theta$ and (the only) $\Sigma$}$$ for some $a>0$, or
2. $n\geq 3$ and $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -\frac{\frac{1}{2}+{\varepsilon}}{r^2\log r} \quad\text{when $r>R$, and for all $\theta$ and $\Sigma\ni\partial_r$,}$$ for some ${\varepsilon}>0$ and $R>1$,
we have that $N$ admits a non-constant, bounded, harmonic function.
As mentioned, the properties of rank-$n$ martingales which we establish are motivated by minimal submanifolds, one consequence being that the applications in other geometric contexts, namely to mean curvature flow and certain sub-Riemannian structures, appear less immediately compelling. For example, it’s not clear how the transience of an associated diffusion along an ancient solution to mean curvature flow fits into the theory of such flows. In the sub-Riemannian case, the transience of the associated diffusion is a more natural property (akin to the transience of Brownian motion on a Riemannian manifold), but our discussion only applies to a rather restricted class of sub-Riemannian structures (ones where the sub-Laplacian is a sum of squares, as explained in Section \[Sect:SubR\]). Nonetheless, we explain how these results apply in these two cases to the extent feasible, for a couple of reasons. Even though these particular properties of rank-$n$ martingales aren’t chosen for their immediate relevance in these cases, we hope that this demonstrates the natural use of stochastic methods in these areas and indicates that further developments might be possible. It also highlights the common aspects of these geometric situations, in terms of basic properties accessible to stochastic analysis, and clarifies the role of various hypotheses. For instance, the fact that the proof of transience for a minimal submanifold applies unchanged to a class of sub-Riemannian structures shows that, in a sense, only the minimality of the submanifold is relevant, and not the fact that it’s a submanifold. (Said differently, the integrability of the tangent spaces or lack thereof, in the sense of Frobenious, is irrelevant.)
From a technical point of view, not only are we able to treat a variety of geometric objects simultaneously, but our techniques are more or less completely elementary. On the geometric side, we use standard comparison results to pass from curvature estimates to estimates on Jacobi fields and their indices. In terms of stochastic analysis, Ito’s rule (with the right choice of function to compose with our rank-$n$ martingale) and basic stopping time arguments are all that we use.
Definitions and preliminary results
===================================
Rank-$n$ martingales
--------------------
Let $M$ be a Cartan-Hadamard manifold of dimension $m$; we will always assume that $M$ is smooth and complete. The main object of study is what we will call a rank-$n$ martingale, where $n\in{\left\{}
\newcommand{\rc}{\right\}}1, \ldots, m-1 \rc$. This is a (continuous) process $X_t$ on $M$, possibly defined up to some explosion time $\zeta$, which (informally) is infinitesimally a Brownian motion on $\Lambda_t$, where $\Lambda_t$ is a (path) continuous, adapted choice of $n$-dimensional subspace of $T_{X_t}M$. To be be more specific, we suppose that $X_t$ locally (in space and time) satisfies an SDE of the form $$\label{Eqn:RankN}
dX_t = \sum_{i=1}^n v_{i,t} \, dW^i_t$$ where $(v_{1,t}, \ldots, v_{n,t})$ is a continuous, adapted $n$-tuple of orthonormal vectors (in $T_{X_t}M$). (Note that we’re not claiming that a unique solution necessarily exists for any such choice of $v_{i,t}$, just that $X_t$ is a solution to such an equation.) Naturally, we think of the $v_{i,t}$ as a orthonormal basis for the $n$-dimensional subspace $\Lambda_t$ mentioned above. We generally think of $X_t$ as starting from a single point $X_0$, but this is not necessary (and it’s occasionally useful to consider a more general initial distribution).
For clarity, consider our main example of such a process, Brownian motion on a (properly embedded) $n$-dimensional minimal submanifold. In that case, we can, on some local chart, let $v_{i,t} = v_i(X_t)$ for a smooth, orthonormal frame $v_1,\ldots,v_n$. Alternatively, if we want a smooth global (but non-Markov) choice of $v_{i,t}$, we can let them be given by the parallel transport of an orthonormal frame at $X_0$ along $X_t$, in the spirit of the Ells-Elworthy-Malliavin construction of Brownian motion on a manifold.
Returning to the general case of Equation , note that the differentials are Ito differentials, and thus $X_t$ is in fact an $M$-martingale (see [@HsuBook] for basic definitions in stochastic differential geometry). Also note that the law of $X_t$ should generally be thought of as depending on $\Lambda_t$, rather than on the specific choice of frame $v_{i,t}$. In particular, let $\tilde{v}_{i,t}$ be another set of orthonormal frames for $\Lambda_t$. Then write $\tilde{v}_{i,t} = \sum_{j=1}^n c_{i,j,t} v_{j,t}$ for $i,j=1,\ldots, n$, where ${\left[}c_{i,j}{\right]}_t$ is a continuous, adapted $O(n)$-valued process, and suppose that $\tilde{W}^i_t$ for $i=1,\ldots, n$ are independent Brownian motions satisfying the system $$\begin{bmatrix} d\tilde{W}_t^1 \\ \vdots \\ d\tilde{W}^n_t \end{bmatrix} = {\left[}c_{i,j}{\right]}_t
\begin{bmatrix} dW_t^1 \\ \vdots \\ dW^n_t \end{bmatrix} .$$ Then we see that $X_t$ also satisfies the SDE $$dX_t = \sum_{i=1}^n \tilde{v}_{i,t} \, d\tilde{W}^i_t .$$
Finally, we mention that our results extent in a natural way to the situation where our rank-$n$ martingale is (possibly) stopped at some stopping time prior to explosion. In particular, our results will apply on the set of paths that survive until explosion (which may mean they survive for all time, if the explosion time is infinite). We discuss this a bit further after the proof of Lemma \[BasicLemma\] and at the end of Section \[Sect:MinSub\]. Note that this is a fairly natural situation. For example, if $N$ is a minimal submanifold-with-boundary, then Brownian motion on $N$ stopped at the boundary will be a rank-$n$ martingale stopped at the first hitting time of the boundary, as was used in [@MyJGA]. Nonetheless, to make the exposition cleaner, we will simply deal with rank-$n$ martingales as introduced above, aside from the two brief mentions just indicated.
Comparison geometry
-------------------
Again, let $M$ be a (smooth, complete) Cartan-Hadamard manifold of dimension $m$. For some point $p\in M$, let $(r,\theta)\in [0,\infty)\times {\mathbb{S}}^{m-1}$ be polar coordinates; this gives a global coordinate system for $M$. Let $X_t$ be a rank-$n$ martingale on $M$, started at some point $X_0$. We can write $X_t$ in coordinates as $(r_t,\theta_t)= {\left(}r(X_t),\theta(X_t){\right)}$.
We wish to understand the behavior of $r_t$ in terms of the SDE it satisfies. We note that $r$ is smooth everywhere on $M$ except for $p$. We’ll see in a moment (in Lemma \[BasicLemma\]) that if $X_t$ starts at $p$ it immediately leaves, and that, from anywhere else, the process almost surely never hits $p$. Thus we will assume that the process is not at $p$ (equivalently that that $r_t$ is positive), so that the behavior of $r$ at $p$ won’t bother us. (The situation is exactly analogous to what one sees for the the radial component of Brownian motion on Euclidean space.) As usual, let $\partial_r=\nabla r$ be the unit radial vector field. Since we’re free to rotate our orthonormal frame without changing the law of $X_t$, as discussed above, assume for convenience that $v_{2,t}, \ldots, v_{n,t}$ are perpendicular to $\partial_r$. Next, let ${\varphi}_t$ be the angle between $v_{1,t}$ and $\partial_r$, so that ${\left\langlev_{1,t},\partial_r\right\rangle}=\cos{\varphi}_t$. (While ${\varphi}_t$ is not uniquely determined, $\cos{\varphi}_t$ is, and we will see that it is only $\cos{\varphi}_t$ that matters in what follows. Nonetheless, we find it geometrically appealing to make reference to ${\varphi}_t$, and it causes no harm.)
Applying Ito’s formula, we first see that that martingale part of $r_t$ evolves as $\cos{\varphi}_t \,d W_t$ for some Brownian motion $W_t$. (If we always choose our orthonormal frame as just discussed, then $W_t=W^1_t$, but in general we only care about the law of $r_t$, so this isn’t necessary. More generally, $W_t$ is adapted to the filtration generated by $W^1_t,\ldots, W^n_t$, and $\cos{\varphi}_t$ is the angle between $\Lambda_t$ and $\partial_r$.) In order to understand the bounded variation (or drift) term in the SDE, let $\gamma$ be the (unique) geodesic from $p$ to $X_t$. Then for $i=2,\ldots n$, let $J_i(s), s\in [0,r_t]$ be the (unique) Jacobi field along $\gamma$ with $J_i(0)=0$ and $J_i(r_t)=v_{i,t}$. For $i=1$ let $J_1(s), s\in [0,r_t]$ be the Jacobi field which is $0$ at $p$ and equal to the projection of $v_{1,t}$ onto the orthogonal complement of $\partial_r$ (as a subspace of $T_{r_t}M$). In particular, $J_1(r_t)$ has length $\sin{\varphi}_t$ (assuming ${\varphi}_t$ is chosen to make this non-negative). Let $I(J_i)$ be the index of the Jacobi field $J_i$. Since the Hessian of $r$ is given in terms of these indices, we see that $r_t$ satisfies the SDE $$\label{Eqn:BasicDecomp}
dr_t = \cos{\varphi}_t \,d W_t + \frac{1}{2} {\left(}\sum_{i=1}^n I(J_i){\right)}\, dt .$$
We will deal with the indices $I(J_i)$ via standard comparison geometry (see Theorem 1.1 of [@SchoenYau] for the relevant version of the Hessian comparison theorem, and see Section 3.4 of [@HsuBook] for the application to Brownian motion). Let $\hat{K}(r)$ be a continuous function on $[0,\infty)$ such that $$\hat{K}(r) \geq \max_{\theta, \Sigma\ni\partial_r} K(r,\theta,\Sigma) ,$$ where the maximum is taken over all $\theta\in{\mathbb{S}}^{m-1}$ and all two-planes $\Sigma$ in the tangent space at $(r,\theta)$ that contain $\partial_r$ (so that we deal with estimates on what are commonly called the radial curvatures). Further, let $\hat{G}(r)$ be the solution to the (scalar) Jacobi equation $$\hat{G}^{\prime\prime}(r) + \hat{K}(r) \hat{G}(r)=0, \quad \hat{G}(0)=0, \hat{G}^{\prime}(0)=1, \quad\text{on $r\in[0,\infty)$.}$$ Then we have the comparison $$I(J_i) \geq |J_i(r_t)|^2\frac{\hat{G}^{\prime}(r_t)}{\hat{G}(r_t)} \quad \text{for each $i=1,\ldots,n$.}$$ Similarly, let $\check{K}(r)$ be a continuous function on $[0,\infty)$ such that $$\check{K}(r) \leq \min_{\theta, \Sigma\ni\partial_r} K(r,\theta,\Sigma) ,$$ and let $\check{G}(r)$ be the solution to the (scalar) Jacobi equation $$\check{G}^{\prime\prime}(r) + \check{K}(r) \check{G}(r)=0, \quad \check{G}(0)=0, \check{G}^{\prime}(0)=1, \quad\text{on $r\in[0,\infty)$.}$$ Then we have the comparison $$I(J_i) \leq |J_i(r_t)|^2\frac{\check{G}^{\prime}(r_t)}{\check{G}(r_t)} \quad \text{for each $i=1,\ldots,n$.}$$ (We can think of $\hat{G}$ or $\check{G}$ as giving the analogous Jacobi fields of a radially symmetric comparison manifold.) Observe that $|J_i(r_t)|^2 =1$ for $i=2,\ldots,n$ and $|J_1(r_t)|^2 =\sin^2{\varphi}_t$. Thus we have that (refer to Equation ) $$\begin{split}
& \frac{1}{2} {\left(}\sum_{i=1}^n I(J_i){\right)}\geq \frac{n-1+\sin^2{\varphi}_t}{2} \frac{\hat{G}^{\prime}(r_t)}{\hat{G}(r_t)} \\
\text{or} \quad & \frac{1}{2} {\left(}\sum_{i=1}^n I(J_i){\right)}\leq \frac{n-1+\sin^2{\varphi}_t}{2} \frac{\check{G}^{\prime}(r_t)}{\check{G}(r_t)} .
\end{split}$$ Note that $\hat{G}$ and $\check{G}$ depend only on the geometry of $M$ (and $r$). Further, these expressions depend only on $\Lambda_t$ and not on the particular choice of the $v_{i,t}$ (so they are “invariant” in the geometric language sometimes used), which justifies the local (or even pointwise) nature of our choice of frame.
As mentioned, we will frequently denote $ \frac{1}{2} {\left(}\sum_{i=1}^n I(J_i){\right)}$ by $v_t$. We now collect the specific estimates on the lengths of Jacobi fields (given in terms of $\hat{G}$ and $\check{G}$) and on $v_t$ that we will use. To make the notation less cumbersome, throughout the rest of the paper we will write ${\log_{(2)}}r$ for $\log(\log r)$ and ${\log_{(3)}}r$ for $\log( \log(\log r ))$.
First, because we assume that $M$ is a Cartan-Hadamard manifold, we can always take $\hat{K}\equiv 0$, for which we have $$\hat{G}(r)=r \quad\text{and}\quad v_t \geq \frac{n-1+\sin^2{\varphi}_t}{2r_t}$$ (and with equality for $v_t$ if $K\equiv 0$, that is, if $M$ is in fact Euclidean space).
Estimates useful when $n=2$ {#Sect:N2}
---------------------------
The following estimates, though stated in general, will be used in the case $n=2$.
Suppose that $K\leq -a^2 <0$. Then we take $\hat{K}\equiv -a^2$, for which $$\hat{G}(r)=\frac{1}{a}\sinh(ar) \quad\text{and}\quad v_t \geq a\frac{n-1+\sin^2{\varphi}_t}{2}\frac{\cosh(ar_t)}{\sinh(ar_t)} .$$
Next, suppose that for some ${\varepsilon}>0$ and some $R>1$, we have $$K(r,\theta,\Sigma) \leq -\frac{1+2{\varepsilon}}{r^2\log r} \quad\text{when $r>R$, and for all $\theta$ and $\Sigma\ni\partial_r$.}$$ (Here we use “$2{\varepsilon}$” rather than “${\varepsilon}$” simply to have a more convenient constant in the following estimates.) Then we can take $\hat{K}(r)$ to be non-positive and equal to $$-\frac{1+{\varepsilon}}{r^2\log r} {\left(}1+\frac{{\varepsilon}}{\log r}{\right)}\quad \text{for $r\geq A$,}$$ for some $A>R$. We see that $$G_1(r) = r{\left(}\log r{\right)}^{1+{\varepsilon}} \quad\text{and}\quad
G_2(r) = \int^r \frac{1}{s^2{\left(}\log s {\right)}^{2+2{\varepsilon}}} \, ds G_1(r)$$ are a basis for the space of solutions to the Jacobi equation over $r\in[A,\infty)$. Thus $$\hat{G}(r) = c_1 G_1(r) + c_2 G_2(r) \quad\text{for $r\in[A,\infty)$,}$$ for some constants $c_1$ and $c_2$. These constants are determined by the initial conditions at $r=A$. Nonetheless, $ \int^r 1/(s^2{\left(}\log s {\right)}^{2+2{\varepsilon}}) \, ds$ is increasing and bounded, so let $\alpha\in(0,\infty)$ be its limit as $r\rightarrow \infty$. Then $$\hat{G}(r) \sim (c_1 +c_2\alpha) G_1(r) \quad\text{as $r\rightarrow\infty$,}$$ where “$\sim$” means that the ratio of the two sides approaches 1. We know that $\hat{G}(r)$ is positive and increasing for positive $r$, so if we let $c_3=c_1 +c_2\alpha$, then $c_3>0$. Explicit computation gives $$\begin{split}
\hat{G}^{\prime}(r) &= {\left(}c_1+c_2 \int^r \frac{1}{s^2{\left(}\log s {\right)}^{2+2{\varepsilon}}} \, ds {\right)}{\left[}{\left(}\log r{\right)}^{1+{\varepsilon}} + (1+{\varepsilon}){\left(}\log r{\right)}^{{\varepsilon}} {\right]}+\frac{c_2}{r{\left(}\log r{\right)}^{1+{\varepsilon}}} \\
&\sim c_3 {\left[}{\left(}\log r{\right)}^{1+{\varepsilon}} + (1+{\varepsilon}){\left(}\log r{\right)}^{{\varepsilon}} {\right]}\quad\text{as $r\rightarrow \infty$}.
\end{split}$$ Dividing this by $\hat{G}$ (and considering the large $r$ behavior) shows that for some $B>A$ and $c>0$, we have $$v_t \geq c{\left(}n-1+\sin^2{\varphi}{\right)}{\left(}\frac{1}{2r_t}+\frac{1+{\varepsilon}}{2r_t \log r_t} {\right)}\quad\text{for $r_t\geq B$}.$$
Estimates useful when $n\geq 3$ {#Sect:NBig}
-------------------------------
The previous estimates will be useful to us in the $n=2$ case. We now develop similar estimates for use in the $n\geq 3$ case.
Suppose that for some ${\varepsilon}>0$ and some $R>1$, we have $$K(r,\theta,\Sigma) \leq -\frac{\frac{1}{2}+{\varepsilon}}{r^2\log r} \quad\text{when $r>R$, and for all $\theta$ and $\Sigma\ni\partial_r$.}$$ We may as well assume that ${\varepsilon}<1/2$, since if the above holds for some ${\varepsilon}$, it holds for any smaller ${\varepsilon}$. Then we can take $\hat{K}(r)$ to be non-positive and equal to $$-\frac{\frac{1}{2}+{\varepsilon}}{r^2\log r} {\left(}1-\frac{\frac{1}{2}-{\varepsilon}}{\log r}{\right)}\quad \text{for $r> R$.}$$ We see that $$G_1(r) = r{\left(}\log r{\right)}^{\frac{1}{2}+{\varepsilon}} \quad\text{and}\quad
G_2(r) = \int^r \frac{1}{s^2{\left(}\log s {\right)}^{1+2{\varepsilon}}} \, ds G_1(r)$$ are a basis for the space of solutions to the Jacobi equation over $r\in(R,\infty)$. As before, $ \int^r 1/(s^2{\left(}\log s {\right)}^{1+2{\varepsilon}}) \, ds$ is increasing and bounded, so let $\alpha\in(0,\infty)$ be its limit. Then $$\hat{G}(r) \sim (c_1 +c_2\alpha) G_1(r) \quad\text{as $r\rightarrow\infty$.}$$ We know that $\hat{G}(r)$ is positive and increasing for positive $r$, so if we let $c_3=c_1 +c_2\alpha$, then $c_3>0$. Explicit computation analogous to the above gives $$\hat{G}^{\prime}(r) \sim c_3 {\left[}{\left(}\log r{\right)}^{\frac{1}{2}+{\varepsilon}} + {\left(}\frac{1}{2}+{\varepsilon}{\right)}\frac{1}{{\left(}\log r{\right)}^{\frac{1}{2}-{\varepsilon}}} {\right]}\quad\text{as $r\rightarrow \infty$}.$$ Dividing this by $\hat{G}$ (and considering the large $r$ behavior) shows that for some $B>R$, we have $$v_t > \frac{\frac{3}{4}{\left(}n-1+\sin^2{\varphi}{\right)}}{2r_t}{\left(}1+\frac{\frac{1}{2}+{\varepsilon}}{\log r} {\right)}\quad\text{for $r_t\geq B$}.$$ (The $3/4$ could be replaced by any positive real less than 1 by changing $B$, but it’s less hassle for us to just pick an explicit coefficient here.)
Finally, assume that for some $R>1$, we have $$K(r,\theta,\Sigma) \geq -\frac{1/2}{r^2\log r} \quad\text{when $r>R$, and for all $\theta$ and $\Sigma\ni\partial_r$.}$$ Then we can take $\check{K}(r)$ to be equal to $$-\frac{1/2}{r^2\log r} {\left[}1+ \frac{1}{{\log_{(2)}}r}-\frac{1}{2\log r}-\frac{1}{2\log r {\left(}{\log_{(2)}}r {\right)}} {\right]}\quad \text{for $r> A$,}$$ for some $A>R$.
We see that $$G_1(r) = r{\left(}\log r {\right)}^{\frac{1}{2}}{\left(}{\log_{(2)}}r {\right)}^{\frac{1}{2}}
\quad\text{and}\quad
G_2(r) = \int^r \frac{1}{s^2{\left(}\log s{\right)}{\left(}{\log_{(2)}}s {\right)}} \, ds G_1(r)$$ are a basis for the space of solutions to the Jacobi equation over $r\in(A,\infty)$. Just as above, there is $c_3>0$ such that $$\check{G}(r)= c_1G_1(r)+c_2G_2(r) \sim c_3 G_1(r) \quad\text{as $r\rightarrow\infty$.}$$ Another explicit computation gives $$\check{G}^{\prime}(r) \sim c_3 {\left(}\log r {\right)}^{\frac{1}{2}}{\left(}{\log_{(2)}}r {\right)}^{\frac{1}{2}}
{\left[}1+ \frac{1/2}{\log r} +\frac{1/2}{\log r {\left(}{\log_{(2)}}r {\right)}} {\right]}\quad\text{as $r\rightarrow \infty$}.$$ Dividing this by $\check{G}$ (and considering the large $r$ behavior) shows that for some $B>A$, we have $$v_t \leq \frac{5}{4} \frac{n-1+\sin^2{\varphi}_t}{2r_t}{\left[}1+\frac{1/2}{\log r_t}+\frac{1/2}{\log r_t{\left(}{\log_{(2)}}r_t{\right)}} {\right]}\quad\text{for $r_t\geq B$.}$$ (Again, the $5/4$ is chosen for convenience; any positive real greater than 1 could be used by changing $B$.)
Basic properties
----------------
We now give a lemma showing that rank-$n$ martingales share several basic properties of Brownian motion.
\[BasicLemma\] Let $M$ be a Cartan-Hadamard manifold of dimension $m$, and let $X_t$ be a rank $n$ martingale on $M$ (for $2\leq n < m$), started from any initial point $X_0$. For any $p\in M$, let $(r,\theta)$ be polar coordinates around $p$. Then
1. $\limsup_{t\rightarrow \zeta} r_t =\infty$, almost surely;
2. and $r_t>0$ for all $t\in(0,\zeta)$, almost surely.
Note that the sectional curvatures of $M$ are all non-positive, and thus we have that $$dr_t = \cos{\varphi}_t \,d W_t + v_t \, dt$$ where, as just discussed, $v_t=\sum_{i=1}^n I(J_i)$ is time-continuous (and adapted, of course) and satisfies $$\label{Eqn:v_t}
v_t\geq \frac{n-1+\sin^2{\varphi}_t}{2r_t} .$$ Let $\sigma_x$ be the first hitting time of $\{r=x\}$, for $x\geq 0$. Ito’s rule gives that $$d{\left(}r^2 {\right)}_t = 2 r_t \cos{\varphi}_t \,d W_t +{\left(}2 r_t v_t +\cos^2{\varphi}_t{\right)}\, dt .$$
Equation implies $2 r_t v_t +\cos^2{\varphi}_t \geq n$, and thus, for $C>r_0$, $${\mathbb{E}}{\left[}r^2_{\sigma_C\wedge t} {\right]}\geq r^2_0 + n{\mathbb{E}}{\left[}\sigma_C \wedge t {\right]}.$$ Using that ${\mathbb{E}}{\left[}r^2_{\sigma_C\wedge t} {\right]}\leq C^2$, dominated convergence lets us take $t\rightarrow \zeta$, and since $\sigma_C\leq \zeta$ we see that $${\mathbb{E}}{\left[}\sigma_C {\right]}\leq \frac{1}{n}{\left(}C^2-r^2_0 {\right)}.$$ In particular, ${\mathbb{P}}{\left(}\sigma_C <\infty{\right)}=1$. Because $\sigma_C=\zeta$ can only happen if both are infinite by path continuity, we also have ${\mathbb{P}}{\left(}\sigma_C <\zeta{\right)}=1$. Since this holds for all $C>r_0$, and since $r_t$ has continuous paths, we conclude that $\limsup_{t\rightarrow \zeta} r_t =\infty$, almost surely.
The proof of the second part mimics that of Proposition 3.22 of [@KS]. It is immediate from Equation \[Eqn:RankN\] (say, by using normal coordinates around $p$) that if $X_0=p$, the process immediately leaves $p$, almost surely, which is equivalent to $r_t$ immediately becoming positive, almost surely. Thus it is enough to prove the result under the assumption that $r_0>0$, and we now assume this.
From the first part and the definition of explosion, we see that $$\label{Eqn:Lem1First}
{\mathbb{P}}{\left(}\text{$\sigma_k<\zeta$ for all integers $k>r_0$, and $\lim_{k\rightarrow\infty}\sigma_k =\zeta$}{\right)}=1.$$ Ito’s rule shows that $$\label{Eqn:LogDecomp}
d{\left(}\log r{\right)}_{t} = \frac{1}{r_t}\cos{\varphi}_t \, dW_t +{\left(}\frac{1}{r_t}v_t - \frac{1}{2r^2}\cos^2{\varphi}_t{\right)}\, dt ,$$ at least when $r>0$, which is all we will need. Equation implies that the coefficient of $dt$ is greater than or equal to $$\frac{n-1+\sin^2{\varphi}_t -\cos^2{\varphi}_t}{2r^2_t} ,$$ and $n\geq 2$ means that this is always non-negative. Thus, $\log r_t$ is a (local) sub-martingale.
Thus, if $k$ is an integer such that $(1/k)^k<r_0<k $ (which will be true for all sufficiently large $k$), by the first part of the lemma, we know that ${\mathbb{P}}{\left(}\sigma_{(1/k)^k}\wedge\sigma_k<\infty {\right)}=1$. So dominated convergence and the fact that $\log r_t$ is a (local) sub-martingale give $$\begin{split}
\log r_0 &\leq {\mathbb{E}}{\left[}\log r_{\sigma_{(1/k)^k}\wedge\sigma_k}{\right]}\\
& = -k\log k {\mathbb{P}}{\left(}\sigma_{(1/k)^k}
\leq \sigma_k {\right)}+ \log k {\left(}1- {\mathbb{P}}{\left(}\sigma_{(1/k)^k}\leq \sigma_k {\right)}{\right)}.
\end{split}$$ Algebra yields $${\mathbb{P}}{\left(}\sigma_{(1/k)^k}\leq \sigma_k {\right)}\leq \frac{\log k-\log r_0}{(k+1)\log k} ,$$ and letting $k\rightarrow\infty$ shows that $$\label{Eqn:Lem1Second}
\lim _{k\rightarrow\infty} {\mathbb{P}}{\left(}\sigma_{(1/k)^k}\leq \sigma_k {\right)}=0.$$
Now $\sigma_0$ is the first hitting time of $\{r_t=0\}$, and we see that $\sigma_0\leq \sigma_{(1/k)^k}$ for all $k$. Then Equations and imply that $${\mathbb{P}}{\left(}\sigma_0<\zeta {\right)}= \lim_{k\rightarrow\infty} {\mathbb{P}}{\left(}\sigma_0<\sigma_k {\right)}\leq
\lim _{k\rightarrow\infty} {\mathbb{P}}{\left(}\sigma_{(1/k)^k}\leq \sigma_k {\right)}=0.$$ This is equivalent to the desired result, namely that $r_t>0$ for all $t\in(0,\zeta)$, almost surely.
The first part of the lemma says that a rank-$n$ martingale almost surely leaves every compact set. We will routinely use this, in much the way that we did in the second part of the proof where it implied that $\log r_t$ left any interval of the form $((1/k)^k,k)$ (prior to $\zeta$). Since $p$ was arbitrary, the second part means that, like Brownian motion, a rank-$n$ martingale does not charge points. It also justifies our assertion from the introduction that, since are interested in the long-time behavior of our rank-$n$ martingales, we need not worry about the singularity of our polar coordinates at $p$, because the process will avoid $p$ at all positive times almost surely.
We also note that, in light of the above lemma and its proof, our earlier comment about allowing our rank-$n$ martingales to be stopped prior to (possible) explosion becomes clearer. In this case, if $\eta$ is such a stopping time, then (for example) the first part of the lemma becomes the statement that $\limsup_{t\rightarrow \zeta} r_t =\infty$ almost surely on the set of paths with $\eta=\infty$. The proof is a straightforward modification of the above.
Transience of rank-$n$ martingales
==================================
Our goal here is to determine conditions for $X_t$ to be transient, meaning that $\lim_{t\rightarrow \zeta} r(X_t)=\infty$ almost surely.
\[THM:Trans\] Let $M$ be a Cartan-Hadamard manifold of dimension $m$, and let $X_t$ be a rank $n$ martingale on $M$ (for $2\leq n < m$). Then if either of the following two conditions hold:
1. $n=2$ and, in polar coordinates around some point, $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -\frac{1+2{\varepsilon}}{r^2\log r} \quad \text{for $r>R$, and for all $\theta$ and the only $\Sigma$,}$$ for some ${\varepsilon}>0$ and $R>1$, or
2. $n\geq 3$,
we have that $X_t$ is transient.
Note that the sectional curvatures of $M$ are all non-positive, and thus in either case we have that $$dr_t = \cos{\varphi}_t \,d W_t + v_t \, dt$$ where $v_t=\sum_{i=1}^n I(J_i)$ is time-continuous (and adapted, of course) and satisfies $$v_t\geq \frac{n-1+\sin^2{\varphi}_t}{2r_t} .$$
For $n\geq 3$, this is enough. In particular, in this case we have that $$\label{Eqn:TransEstimate}
v_t\geq \frac{2+\sin^2{\varphi}_t}{2r_t} \geq \frac{1}{r_t}.$$ (The intuitive point is that the drift is at least as large as for a 3-dimensional Bessel process, while the quadratic variation of the martingale part grows no faster than for a 3-dimensional Bessel process, and thus one expects $r_t$ to be “at least as transient” as a 3-dimensional Bessel process.) Ito’s rule followed by an application of Inequality and algebra gives $$\begin{split}
& d{\left(}\frac{1}{r}{\right)}_t = -\frac{1}{r^2_t}\cos{\varphi}_t \, dW_t +{\left(}\frac{1}{r^3_t}\cos^2{\varphi}_t -\frac{v_t}{r^2_t}{\right)}\, dt \\
\text{where}\quad& \frac{1}{r^3_t}\cos^2{\varphi}_t -\frac{v_t}{r^2_t} \leq -\frac{\sin^2{\varphi}_t}{r^3_t}
\leq 0 .
\end{split}$$ In particular, $1/r_t$ is a (local) supermartingale, at least for $r>0$.
We know that $\sigma_k$ is finite for all integers $k>r_0$, almost surely. Choose any $a>0$. Then the event $\liminf_{t\rightarrow \zeta} r_t \leq a$ coincides, up to a set of probability zero, with the event that $r_t$ hits the level $a$ after $\sigma_k$ for all $k$ that are also larger than $a$. Choose such a $k$ and a $b>k$. Let $\tilde{\sigma}_a$ be the first hitting time of $a$ (for $r_t$) after $\sigma_k$, and similarly for $\tilde{\sigma}_b$. We know that $\tilde{\sigma}_b$ is almost surely finite. This, along with the fact that $1/r_t$ is a (local) supermartingale and our choice of stopping times implies $$\frac{1}{k} = {\mathbb{E}}{\left[}\frac{1}{r_{\sigma_k}}{\right]}\geq {\mathbb{E}}{\left[}\frac{1}{r_{\tilde{\sigma}_a\wedge\tilde{\sigma}_b}}{\right]}= \frac{1}{a} {\mathbb{P}}{\left(}\tilde{\sigma}_a<\tilde{\sigma}_b {\right)}+ \frac{1}{b}{\left(}1- {\mathbb{P}}{\left(}\tilde{\sigma}_a<\tilde{\sigma}_b {\right)}{\right)}.$$ This, in turn, yields $${\mathbb{P}}{\left(}\tilde{\sigma}_a<\tilde{\sigma}_b {\right)}\leq \frac{\frac{1}{k}-\frac{1}{b}}{\frac{1}{a}-\frac{1}{b}} .$$ If $\tilde{\sigma}_a$ is finite, it must be less than $\tilde{\sigma}_b$ for all sufficiently large $b$ (up to a set of probability zero), so letting $b\rightarrow \infty$ shows that $${\mathbb{P}}{\left(}\tilde{\sigma}_a<\infty {\right)}\leq \frac{a}{k} .$$ Because the right-hand side of the above goes to zero as $k\rightarrow \infty$, we see that the probability of $r_t$ returning to the level $a$ after every $\sigma_k$ (for sufficiently large $k$) is zero. It follows that $\liminf_{t\rightarrow \zeta} r_t > a$ almost surely. Since $a$ was arbitrary, we conclude that $\liminf_{t\rightarrow \zeta} r_t=\infty$ almost surely. Then $\lim_{t\rightarrow \zeta} r_t=\infty$ almost surely, which is equivalent to the transience of $X_t$.
The first part (the $n=2$ case) is similar, the difference being that we must replace $1/r$ with a more suitable function. Using the curvature bound and the results of Section \[Sect:N2\], we see that, for some $B>1$, we have $$v_t \geq {\left(}1+\sin^2{\varphi}_t{\right)}{\left(}\frac{1}{2r_t} + \frac{1+{\varepsilon}}{2r_t \log r_t} {\right)}\quad \text{for $r_t>B$.}$$ Now Ito’s rule gives (for $r_t>1$) $$\begin{split}
d{\left(}\frac{1}{{\left(}\log r{\right)}^{{\varepsilon}}}{\right)}_t &= \frac{-{\varepsilon}\cos{\varphi}_t}{r {\left(}\log r_t{\right)}^{(1+{\varepsilon})}} \, dW_t \\
& + \frac{{\varepsilon}}{r {\left(}\log r_t{\right)}^{(1+{\varepsilon})}} {\left[}\frac{\cos^2{\varphi}_t}{2r_t}{\left(}\frac{1+{\varepsilon}}{\log r_t}+1{\right)}-v_t {\right]}\, dt.
\end{split}$$ Combing this with the upper bound for $v_t$, we see that, for $r_t>B$, the coefficient of $dt$ (in other words, the infinitesimal drift) is less than or equal to $$\frac{{\varepsilon}}{2r^2 {\left(}\log r_t{\right)}^{(1+{\varepsilon})}} {\left(}1+ \frac{1+{\varepsilon}}{\log r_t}{\right)}{\left(}\cos^2{\varphi}_t -\sin^2{\varphi}_t -1{\right)}\leq 0.$$ It follows that $1/(\log r_t)^{{\varepsilon}}$ is a (local) supermartingale for $r_t>A$.
If we now choose $k$, $a$, and $b$ (and the corresponding notation) as above, with the additional stipulation that $a>B$, similar logic gives $$\begin{split}
& \frac{1}{{\left(}\log k{\right)}^{1+{\varepsilon}}} = {\mathbb{E}}{\left[}\frac{1}{{\left(}\log r_{\sigma_k}{\right)}^{1+{\varepsilon}}} {\right]}\geq {\mathbb{E}}{\left[}\frac{1}{{\left(}\log r_{\tilde{\sigma}_a\wedge\tilde{\sigma}_b}{\right)}^{1+{\varepsilon}}} {\right]}\\
& \quad = \frac{1}{{\left(}\log a{\right)}^{1+{\varepsilon}}} {\mathbb{P}}{\left(}\tilde{\sigma}_a<\tilde{\sigma}_b {\right)}+
\frac{1}{{\left(}\log b{\right)}^{1+{\varepsilon}}} {\left(}1- {\mathbb{P}}{\left(}\tilde{\sigma}_a<\tilde{\sigma}_b {\right)}{\right)}.
\end{split}$$ It follows that $${\mathbb{P}}{\left(}\tilde{\sigma}_a<\tilde{\sigma}_b {\right)}\leq \frac{\frac{1}{{\left(}\log k{\right)}^{1+{\varepsilon}}} -\frac{1}{{\left(}\log b{\right)}^{1+{\varepsilon}}} }{\frac{1}{{\left(}\log a{\right)}^{1+{\varepsilon}}} -\frac{1}{{\left(}\log b{\right)}^{1+{\varepsilon}}} } ,$$ and letting $b\rightarrow \infty$ shows that ${\mathbb{P}}{\left(}\tilde{\sigma}_a<\infty {\right)}\leq (\log a/\log k)^{1+{\varepsilon}}$. Since this last quantity goes to zero as $k\rightarrow \infty$, just as before we conclude that $\liminf_{t\rightarrow \zeta} r_t > a$ almost surely. Because this holds for any $a>B$, it follows that $\lim_{t\rightarrow \zeta} r_t=\infty$ almost surely, which is equivalent to the transience of $X_t$.
Transience in geometric contexts
================================
We now establish the connection between rank-$n$ martingales and various geometric objects.
Minimal submanifolds {#Sect:MinSub}
--------------------
First, let $N$ be an $n$-dimensional, properly immersed, minimal submanifold of $M$. Let $X_t$ be Brownian motion on $N$, viewed as a process in $M$ (under the immersion, of course). Then $X_t$ is a rank-$n$ martingale in $M$, as mentioned in the introduction.
To see this, let $\tilde{v}_{i,t}\subset TN$ be such that the solution to $$d\tilde{X}_t = \sum_{i=1}^n \tilde{v}_{i,t} dW^i_t$$ is Brownian motion on $N$. (Locally, we can just let $\tilde{v}_{i,t} = \tilde{v}_i(\tilde{X}_t)$ for a smooth orthonormal frame $\tilde{v}_1,\ldots,\tilde{v}_n$. Alternatively, we can let the $\tilde{v}_{i,t} $ be given by the parallel transport of an orthonormal frame at $\tilde{X}_0$ along $\tilde{X}_t$, in the spirit of the Ells-Elworthy-Malliavin construction of Brownian motion on a manifold.) Now let $X_t$ and $v_{i,t}$ be the images of $\tilde{X}_t$ and $\tilde{v}_{i,t}$ under the immersion. Then in general, Ito’s rule shows that they satisfy the SDE $$dX_t = \sum_{i=1}^n v_{i,t} dW^i_t -\frac{1}{2}H(X_t)\, dt ,$$ where $H$ is the mean curvature vector of $N$ (as a submanifold of $M$). Here, we normalize $H$ so that, if $y_1,\ldots,y_m$ are normal coordinates for $M$ centered at a point $p$ in (the image of) $N$, then $$H(p) = -\sum_{i=1}^m {\left(}{\Delta}_N y_i|_N{\right)}\partial_{y_i} .$$ Since we’re assuming that $N$ is minimal (and, of course, the $v_{i,t}$ are orthonormal since the immersion is isometric), $H\equiv 0$, and we see that $X_t$ is a rank-$n$ martingale.
Properness of $N$ implies that $X_t$ explodes relative to the topology of $N$ if and only if it explodes relative to the topology of $M$. (In particular, $\zeta$ is independent of whether we view $X_t$ as a process on $N$ or on $M$.) Thus Theorem \[THM:Trans\], applied to this case, implies Corollary \[Cor:MinimalTrans\]. Note that the assumption of properness can be dropped, via a natural application of allowing a rank-$n$ martingale to be stopped prior to explosion. If $N$ is not properly immersed, then Brownian motion on $N$ might explode (relative to the topology of $N$) without also exploding in $M$ (relative to the topology of $M$). Let $\eta$ be the explosion time relative to the topology of $N$. Clearly $\eta$ is a stopping time, and we consider $X_t$ run until $\eta$. On the set of paths where $\eta=\zeta$, Theorem \[THM:Trans\] (or more precisely, a simple modification of its proof) shows that $X_t$ is transient (noting that it’s still true that transience on $M$ implies transience on $N$). On the set of paths where $\eta<\zeta$ (which is the only other possibility), we have in particular that $\eta<\infty$. Thus $X_t$ is almost surely transient on this set as well, and the claim that the assumption of properness can be dropped follows.
Mean curvature flow
-------------------
Next, we note that this idea generalizes to mean curvature flow. To describe this, for a manifold $N$, with a smooth structure, of dimension $n$ (for $2\leq n < m$), let $$f_{\tau}(y):(-\infty,0]\times N\rightarrow M$$ be an ancient solution to mean curvature flow. (The term “ancient” refers to the fact that the solution is defined at all past times. Solutions defined for $\tau\in[0,\infty)$ are called immortal, and solutions defined for all $\tau$, eternal.) In particular, let $g_{\tau}$ be the (smoothly-varying) metric induced on $N$ at time $\tau$ by the immersion, let ${\Delta}_{\tau}$ be the associate Laplacian, and let $H_{\tau}$ be the associated mean curvature vector (with the same normalization as above, for any fixed time) for $f_{\tau}(N)$. Then $f_{\tau}$ is a smooth function, proper as a map from $N$ to $M$ at every fixed time $\tau$, satisfying the differential equation $$\partial_{\tau} f_{\tau}(y) = -\frac{1}{2} H_{\tau}(y) \quad\text{for all $\tau\in[0,\infty)$ and $y\in N$.}$$ Note that the factor of $1/2$ in front of $H_{\tau}$ is non-standard. This is the same difficulty as encountered in normalizing the heat equation; the factor of $1/2$ is better suited to stochastic analysis, but analysts and geometers prefer not to include it. Rescaling time allows one to recover the standard normalization. Since our results are all for the asymptotic behavior of our process, though, rescaling time doesn’t change them, and thus the unusual normalization causes no particular trouble.
Note that we don’t require the solution to develop a singularity at some positive time (so that eternal solutions are a special case of ancient solutions). Further, we assume that the solution is smooth even at time 0 (which is the meaning of our assumption $\tau\in[0,\infty)$). In other words, if there is a singularity (such as collapse to a “round point”), we don’t start our rank-$n$ martingale from there. This is so that we can make our choice of $\Lambda_t$ continuous, in keeping with our desire to avoid technical difficulties here.
Now suppose that $\tilde{X}_t$ is a process on $N$ that satisfies $$d\tilde{X}_t = \sum_{i=1}^n \tilde{v}_{i,t} dW^i_t ,$$ where the $\tilde{v}_{i,t}$ are an orthonormal frame at $\tilde{X}_t$ with respect to the metric $g_{-t}$. We can always find such a process; for example, by letting the $\tilde{v}_{i,t}$ come from a local (in both space and time) smooth choice of time-varying orthonormal frame. We think of $\tilde{X}_t$ as being Brownian motion along the mean curvature flow, run backwards in time. Indeed, note that $\tilde{X}_t$ is the (inhomogeneous) diffusion associated to $\frac{1}{2}{\Delta}_{-t}$.
If we again let $X_t$ be the image of $\tilde{X}_t$ under $f_{-t}$, then $X_t$ is a rank-$n$ martingale (on $M$). To see this, let $v_{i,t}$ be the pushforward of $\tilde{v}_{i,t}$, and note that the $v_{i,t}$ are orthonormal since $f_{-t}$ is an isometric immersion for all $t$. Then Ito’s formula gives $$dX_t = \sum_{i=1}^n v_{i,t} dW^i_t -\frac{1}{2}H_{-t}(X_t)\, dt - \left.\frac{\partial f}{\partial \tau}\right|_{\tau=-t}{\left(}\tilde{X}_t{\right)}\, dt .$$ But using that $f_{\tau}$ is a solution to the mean curvature flow, this reduces to $$dX_t = \sum_{i=1}^n v_{i,t} dW^i_t$$ as desired. (The minimal submanifold situation above is just the special case of a constant solution to the mean curvature flow.) This also explains why we consider our “inhomogeneous Brownian motion” to be run backwards in time with respect to the flow. This phenomenon (namely, “process time” running in the opposite direction from “PDE time”) is familiar, arising even in the standard approach to the heat equation on the real line via Brownian motion.
Of course, for a non-ancient solution to the mean curvature flow, the same procedure leads to a rank-$n$ martingale run for a finite time. However, since all of our results in the current paper concern the asymptotic behavior of rank-$n$ martingales, we don’t consider this case.
A geometric interpretation of the transience of this process is less obvious than it was for minimal submanifolds. Certainly, though, it contains some information about the flow, such as the (fairly basic) fact that an ancient solution cannot be contained in a compact subset of $M$ (for all time). For a refinement of this idea, we mention the following. In [@MyJGA], rank-2 martingales in ${\mathbb{R}}^3={\left\{}
\newcommand{\rc}{\right\}}(x_1, x_2, x_3):x_i\in{\mathbb{R}}\rc$ were studied, with an eye toward classical minimal surfaces. Let $r=\sqrt{x_1^2+x_2^2}$. It was proved that for any $c>0$, there is a positive integer $L$ such that any rank-2 martingale (as considered in the present paper) exits the region $$A={\left\{}
\newcommand{\rc}{\right\}}r>e^L\text{ and } |x_3|<\frac{cr}{\sqrt{\log r {\log_{(2)}}r}} \rc$$ in finite time, almost surely (this is a restatement of Theorem 2 of [@MyJGA]). Thus, we see that an ancient solution to mean curvature flow, for surfaces in ${\mathbb{R}}^3$, cannot be contained in $A$ for all $\tau\in(-\infty,0]$.
Finally, we mention that the relationship between rank-$n$ martingales and mean curvature flow can be used to represent mean curvature flow in terms of a type of stochastic target problem. This is done, in the case when the ambient space is ${\mathbb{R}}^n$, in [@SonerTouzi].
Sub-Riemannian geometry {#Sect:SubR}
-----------------------
Our final example of rank-$n$ martingales arising in geometry is as follows. Again, starting with a Cartan-Hadamard manifold $M$ (of dimension $m\geq 3$), choose a smooth rank-$n$ distribution satisfying the bracket-generating property. That is, let ${\mathcal{D}}$ be a smooth map which assigns an $n$-dimensional subspace ${\mathcal{D}}_y$ of $T_yM$ to each $y\in M$. Further, if $${\mathcal{D}}_y^k = \operatorname{span}{\left\{}
\newcommand{\rc}{\right\}}[w_1,[\ldots[w_{k-1},w_k]]]_y : w_i(z)\in {\mathcal{D}}_z \text{ for all $z\in M$ and $w_i$ is smooth}\rc ,$$ we assume that for each $y\in M$, there exists an integer $k(y)\geq 2$ such that $T_yM={\mathcal{D}}_y^{k(y)}$ (this is the bracket-generating property). Each subspace ${\mathcal{D}}_y$ can be given a Riemannian metric by restricting the metric on $M$ to ${\mathcal{D}}_y$. This gives a sub-Riemannian structure on $M$. (See [@Montgomery] for background on sub-Riemannian geometry.)
Further, suppose we choose a smooth volume form on $M$ (in general, there is no canonical volume associated to a sub-Riemannian structure, although intrinsic choices, such as the Popp volume, can be considered). Then let ${\Delta}_{s}$ be the associated sub-Laplacian (defined as the divergence of the horizontal gradient). In general, if $v_1,\ldots v_n$ is a local orthonormal frame for ${\mathcal{D}}$, then the sub-Laplacian will be locally given by $\sum v_i^2$ plus a first-order term.
We are interested in the case when the sub-Laplacian is a sum of squares, that is, when the first-order term vanishes identically. While this will certainly not be true in general, it will be, for example, for unimodular Lie groups as discussed in [@Ugo]. In this case, the diffusion associated to $\frac{1}{2}{\Delta}_s$ is a rank-$n$ martingale (on $M$ with the original Riemannian metric), by assumption. (In our earlier notation, we have $\Lambda_t = {\mathcal{D}}_{X_t}$.) This implies the following.
Let $M$ be a Cartan-Hadamard manifold of dimension $m\geq 3$. With $2\leq n < m$, let the rank-$n$ distribution ${\mathcal{D}}$ (as above) with the restriction metric be a sub-Riemannian structure on $M$, and let this sub-Riemannian structure be given a volume form such that the associated sub-Laplacian ${\Delta}_s$ is a sum of squares. Then if either of the following two conditions hold:
1. $n=2$ and, in polar coordinates around some point, $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -\frac{1+2{\varepsilon}}{r^2\log r} \quad \text{for $r>R$, and for all $\theta$ and the only $\Sigma$,}$$ for some ${\varepsilon}>0$ and $R>1$, or
2. $n\geq 3$,
we have that the diffusion associated to $\frac{1}{2}{\Delta}_s$ (from any initial point) is transient.
Angular behavior
================
In this final section, we consider the asymptotic behavior of $\theta_t=\theta(X_t)$, in the special case when $M$ is radially symmetric (about some pole $p$, which we also use as the origin for our polar coordinates). As mentioned, the motivation, as well as the approach (to an extent), is modeled on the case of Brownian motion on a radially symmetric manifold, for which the angular behavior is closely tied to the solvability of the Dirichlet problem at infinity (see March [@March] or Hsu [@HsuBook] for this relationship and the study of the angular behavior of Brownian motion). While we don’t attempt to formulate an analogue of the Dirichlet problem at infinity for rank-$n$ martingales, we will nonetheless give analogous results for angular convergence and use them to draw some conclusions about bounded harmonic functions on minimal submanifolds, and similarly for sub-Riemannian geometries.
Results for rank-$n$ martingales
--------------------------------
With the same assumptions and notation as before, we now also assume that $M$ is radially symmetric about $p$. The upshot of this assumption is that $\theta_t$ is a martingale (on ${\mathbb{S}}^{m-1}$). Moreover, we can write the metric on $M$ in polar coordinates as $$dr^2 + G^2(r) d\theta^2 \quad\text{where $d\theta^2$ is the standard metric on ${\mathbb{S}}^{m-1}$,}$$ for a (smooth) function $G:[0,\infty)\rightarrow[0,\infty)$ with $G(0)=0$ and $G^{\prime}(0)=1$. Then if we let $w_{i,t}$ be the component of $v_{i,t}$ orthogonal to $\partial_r$ (that is, the component in the tangent space to $\{r=r_t\}$), we can think of the $w_{i,t}$ as vector fields on ${\mathbb{S}}^{m-1}$, and $\theta_t$ satisfies the SDE $$\label{Eqn:ThetaProcess}
d\theta_t = \sum_{i=1}^n \frac{w_{i,t}}{G(r_t)} \, dW^i_t .$$ Without the assumption of radial symmetry, this is still the correct martingale part of $\theta_t$, but there will also be a drift term. This drift is not directly controlled by the sort of curvature assumptions that we’re working with, and so angular convergence becomes a much harder question. Indeed, for the motivating case of Brownian motion, see [@EltonArticle] for an approach to angular convergence without radial symmetry (and note that the techniques are more difficult than for the corresponding results with radial symmetry that we cited earlier). It seems likely that those techniques could be adapted to rank-$n$ martingales, but we don’t pursue that here.
Continuing, suppose, as before, that we rotate our vector fields at some instant so that $v_{2,t}, \ldots, v_{n,t}$ are perpendicular to $\partial_r$, and we let ${\varphi}_t$ be the angle between $v_{1,t}$ and $\partial_r$, so that ${\left\langlev_{1,t},\partial_r\right\rangle}=\cos{\varphi}_t$. Then, at this instant, $w_{2,t}, \ldots, w_{n,t}$ all have length 1, while $w_{1,t}$ has length $\sin{\varphi}_t$. If we consider the special case when $n=m-1$ and ${\varphi}_t\equiv \pi/2$ (which we see for the mean curvature flow associated to a sphere around $p$), then the $w_{i,t}$ are an orthonormal frame on ${\mathbb{S}}^{m-1}$ for all time, and $\theta_t$ is time-changed Brownian motion on ${\mathbb{S}}^{m-1}$, with the time-change given by the integral of $1/G^2(r_t)$ along paths. In this case, it’s clear that asking whether $\theta_t$ converges is equivalent to asking whether the “changed time” converges to a finite limit, or alternatively, whether the integral of $1/G^2(r_t)$ (over all time) is finite.
It turns out that even when $\theta_t$ is not time-changed Brownian motion on the sphere, the convergence or divergence of the integral of $1/G^2(r_t)$ still controls the convergence or divergence of $\theta_t$. The question is only interesting when $X_t$ is transient, so we now assume this as well. Let $y_1,\ldots, y_m$ be the functions on ${\mathbb{S}}^{m-1}$ arising from the standard embedding into ${\mathbb{R}}^m$. Further, let $y_{i,t}=y_i(\theta_t)$. Then it’s clear that $\theta_t$ converges if and only if all of the $y_{i,t}$ converge. For each $i$, the gradient of $y_i$ has length bounded above by 1, and Laplacian bounded in absolute value by $2c$ for some $c>0$ (independent of $i$). For convenience, let $Z$ be the random variable taking values in $(0,\infty]$ given by $$Z= \int_0^{\zeta} \frac{1}{G^2(r_s)} \, ds .$$
Now suppose that $Z$ is almost surely finite. Then Ito’s rule shows that, for each $i$, the martingale part of $y_{i,t}$ has quadratic variation less than or equal to $Z$ (for all $t$), and the bounded variation part of $y_{i,t}$ has total variation (on $[0,\zeta)$) less than or equal to $cZ$. In particular, all of the $y_{i,t}$ converge, and so does $\theta_t$ (almost surely). We denote this limit by $\theta_{\zeta}$. Further, suppose that, by choosing $r_0$ large enough, we can make the expectation of $Z$ as close to zero as we wish. Then, for any $\delta>0$ we can make the probability that $|y_{i,t}-y_{i,0}|$ ever exceeds $\delta$, for any $i$, less than $\delta$. In other words, by making $r_0$ large enough, we can ensure that $\theta_{\zeta}\in B_{\delta}(\theta_0) \subset{\mathbb{S}}^{m-1}$ with probability at least $1-\delta$, where $ B_{\delta}(\theta)$ denotes the ball of radius $\delta$ around $\theta\in{\mathbb{S}}^{m-1}$.
In the other direction, suppose that $Z$ is almost surely infinite. At any instant, we can assume that $w_{2,t}$ has length 1 (as above). Consider the gradients of the $y_i$, it follows that there is some $c>0$ such that, for some $i$, we have ${\left\langlew_{2,t},\nabla_{{\mathbb{S}}^{m-1}} y_i\right\rangle} >c/G(r_t)$. In other words, at every time, there is always at least one $y_{i,t}$ with quadratic variation growing at a rate greater than $c^2/G^2(r_t)$. Since there are only finitely many $y_{i,t}$ (and recall that all of the functions under consideration are smooth), we see that at least one of the $y_{i,t}$ has infinite (unbounded) quadratic variation. Thus this $y_{i,t}$ doesn’t converge, and neither does $\theta_t$, almost surely.
Before proving our next theorem, we make one technical observation. Suppose we show that, for any $\beta>0$, there exists $\tilde{B}> 0$ and $\rho>\tilde{B}$ such that $${\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{G^2(r_s)} \, ds{\right]}<\beta$$ whenever $r_0>\rho$. Then this is enough to prove that $\theta_t$ converges almost surely. Further, for any $\delta>0$, we can find $\rho$ (perhaps larger than before), such that $${\mathbb{P}}{\left(}\theta_{\zeta}\in B_{\delta}(\theta_0) {\right)}>1-\delta ,$$ whenever $r_0>\rho$. To see this, recall that we’re assuming that $X_t$ is transient, and thus for any $\tilde{B}$, we can make ${\mathbb{P}}{\left(}\sigma_{\tilde{B}}<\infty{\right)}$ as close to zero as we wish by making $r_0$ large. Since our previous observations apply (up to a set of probability zero) on the set $\{\sigma_{\tilde{B}}=\infty\}$, for any $\delta>0$, we can choose $\rho$ so that $${\mathbb{P}}{\left(}\sigma_{\tilde{B}}=\infty\text{ and }\theta_{\zeta}\in B_{\delta}(\theta_0) {\right)}>1-\delta$$ whenever $r_0>\rho$. So the only issue that remains is seeing that $\theta_t$ also converges on the set $\{\sigma_{\tilde{B}}<\infty\}$. Again by transience, if $X_t$ hits $\{r=\tilde{B}\}$, then it almost surely hits, say, $\{r=\rho+1\}$ at some later time. At this point, the same result applies, so that $r_t$ stays above $\tilde{B}$ at all future times, and $\theta_t$ converges, with probability at least $1-\delta$. Iterating this argument, we see that $\theta_t$ converges almost surely, as desired.
\[THM:AngleCon\] Suppose that $M$ is Cartan-Hadamard manifold of dimension $m\geq 3$, and that $M$ is radially symmetric around some point $p$. Let $(r,\theta)$ be polar coordinates around $p$, and let $X_t$ be a rank-$n$ martingale, for $2\leq n< m$. Then if either of the following conditions hold:
1. $n=2$ and $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -a^2 \quad \text{for all $r$ and $\theta$ and (the only) $\Sigma$}$$ for some $a>0$, or
2. $n\geq 3$ and $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -\frac{\frac{1}{2}+{\varepsilon}}{r^2\log r} \quad\text{when $r>R$, and for all $\theta$ and $\Sigma\ni\partial_r$,}$$ for some ${\varepsilon}>0$ and $R>1$,
we have that $\theta_t=\theta(X_t)$ converges, almost surely, as $t\rightarrow\zeta$. Further, for any $0<\delta<1$, there exists $\rho$ (depending only on $M$ and $n$) such that, if $r_0>\rho$, then $\theta_{\zeta}\in B_{\delta}(\theta_0) \subset{\mathbb{S}}^{m-1}$ with probability at least $1-\delta$.
First note that in either case, Theorem \[THM:Trans\] implies that $X_t$ is transient.
We begin with the first part. The Rauch comparison theorem and the estimates of Section \[Sect:N2\] (along with basic facts about the hyperbolic trigonometric functions) imply that $$G(r)\geq \frac{1}{a}\sinh(ar) \quad\text{and}\quad v_t> \frac{1+\sin^2{\varphi}_t}{2} a .$$ Thus $${\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}} \frac{1}{G^2(r_s)} \, ds{\right]}\leq a^2 {\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}}\frac{1}{\sinh^2(ar_s)} \, ds{\right]}$$ for any $\tilde{B}>0$. Choose $\beta>0$. In light of the discussion preceding the theorem, it’s enough to show that for some $\tilde{B}> 0$, there exists $\rho>\tilde{B}$ such that $${\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{\sinh^2(ar_s)} \, ds{\right]}<\beta$$ whenever $r_0>\rho$.
Now Ito’s rule gives $$d{\left(}\frac{-1}{r}{\right)}_t = \frac{\cos{\varphi}_t}{r^2_t}\, dW_t + \gamma_t \, dt
\quad\text{where}\quad \gamma_t =
\frac{v_t}{r^2_t} -\frac{\cos^2{\varphi}_t}{r^3_t} .$$ The bound for $v_t$ then implies $$\gamma_t \geq \frac{1+\sin^2{\varphi}_t}{2r^2_t} a -\frac{\cos^2{\varphi}_t}{r^3_t} .$$ For large enough $r$, the right-hand side is at least $a/(3r^2)$, and by the exponential growth of hyperbolic sine, it follows that there is some $\tilde{B}>0$ such that $$\gamma_t > \frac{1}{\sinh^2(ar_t)} \quad \text{for $r_t>\tilde{B}$.}$$
Putting this together, for $r_0>\tilde{B}$, a standard dominated convergence argument shows that $$\begin{split}
{\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{\sinh^2(ar_s)} \, ds{\right]}&<
{\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \gamma_s \, ds{\right]}\\
&= {\mathbb{E}}{\left[}\frac{-1}{r_{\sigma_{\tilde{B}}\wedge\zeta} } {\right]}- \frac{-1}{r_0} \\
& = \frac{1}{r_0} - \frac{1}{\tilde{B}}{\mathbb{P}}{\left(}\sigma_{\tilde{B}} <\infty {\right)},
\end{split}$$ where in the last line we’ve used that $\lim_{t\rightarrow\zeta}r_t=\infty$ to see that $1/r_{\sigma_{\tilde{B}}\wedge\zeta}$ is zero on the set where $\sigma_{\tilde{B}} =\infty$. Also note that it’s the second-to-last line where our central idea of controlling an integral along paths by recognizing it as the drift of a semi-martingale (with well-controlled asymptotic behavior) is used. At any rate, the last line of the above is certainly less than $1/r_0$, and so it’s clear that we can find $\rho>\tilde{B}$ such that $${\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{\sinh^2(ar_s)} \, ds{\right]}<\beta$$ whenever $r_0>\rho$. This completes the proof of the first part.
The second part is analogous, just with a different bound on $v_t$ and a choice of semi-martingale different from $-1/r_t$. Without loss of generality, we can assume that ${\varepsilon}<1/2$. Here, using the estimates in Section \[Sect:NBig\], we have that for some $c>0$ and $B>1$, $$G(r) \geq cr{\left(}\log r{\right)}^{\frac{1}{2}+{\varepsilon}} \quad\text{for $r>B$.}$$ Thus $${\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}} \frac{1}{G^2(r_s)} \, ds{\right]}\leq \frac{1}{c^2} {\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}}\frac{1}{r^2_s{\left(}\log r_s {\right)}^{1+2{\varepsilon}}} \, ds{\right]}$$ for any $\tilde{B}>B$. Choose $\beta>0$. Again, it’s enough to show that for some $\tilde{B}\geq B$, there exists $\rho>\tilde{B}$ such that $${\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{r^2_s{\left(}\log r_s {\right)}^{1+2{\varepsilon}}} \, ds{\right]}<\beta$$ whenever $r_0>\rho$.
Recall that, in this case, $$v_t > \frac{\frac{3}{4}{\left(}n-1+\sin^2{\varphi}{\right)}}{2r_t}{\left(}1+\frac{\frac{1}{2}+{\varepsilon}}{\log r} {\right)}\quad\text{for $r_t\geq B$}.$$ Then for $\alpha>0$, Ito’s rule plus the bound on $v_t$ gives (for $r_t>B$) $$\begin{split}
& \quad\quad d{\left(}\frac{-1}{{\left(}\log r_t {\right)}^{\alpha}} {\right)}_t = \frac{\alpha\cos{\varphi}_t}{r{\left(}\log r{\right)}^{1+\alpha}}\, dW_t
+\gamma_t \, dt \quad \text{where} \\
& \gamma_t \geq \frac{\alpha}{2r_t^2 {\left(}\log r_t{\right)}^{1+\alpha}}
{\left[}\frac{3}{4}{\left(}1+\frac{\frac{1}{2}+{\varepsilon}}{\log r_t} {\right)}{\left(}n-1+\sin^2{\varphi}_t {\right)}-
\cos^2{\varphi}_t{\left(}1+\frac{\alpha+1}{\log r_t} {\right)}{\right]}.
\end{split}$$ We now take $\alpha<2{\varepsilon}$. Then using that $n\geq 3$, it’s easy to see that we can find $\tilde{B}$ (depending only on $\alpha$ and ${\varepsilon}$) such that the quantity in brackets in the above expression is at least $1/4$ whenever $r_t>\tilde{B}$. Since we also have $1+\alpha<1+2{\varepsilon}$, we see that for some $D>0$, $${\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{r^2_s{\left(}\log r_s {\right)}^{1+2{\varepsilon}}} \, ds{\right]}<
D {\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \gamma_s \, ds{\right]}.$$ We further have (again, via a standard dominated convergence argument, and the fact that $r_t\rightarrow\infty$) $$\begin{split}
{\mathbb{E}}{\left[}\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \gamma_s \, ds{\right]}&=
{\mathbb{E}}{\left[}\frac{-1}{{\left(}\log r_{\sigma_{\tilde{B}}\wedge\zeta} {\right)}^{\alpha}} {\right]}- \frac{-1}{{\left(}\log r_0 {\right)}^{\alpha}} \\
& = \frac{1}{{\left(}\log r_0 {\right)}^{\alpha}} - \frac{1}{{\left(}\log \tilde{B} {\right)}^{\alpha}}{\mathbb{P}}{\left(}\sigma_{\tilde{B}}
<\infty {\right)}.
\end{split}$$ Since this last line can be made arbitrarily close to zero by taking $r_0$ large, it’s clear that we can complete the proof of this second part just as we did the first part.
To get the complementary results for non-convergence of $\theta_t$, we again make some preliminary observations. Suppose we show that there exists $\tilde{B}> 0$ such that $$\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{G^2(r_s)} \, ds=\infty \quad\text{almost surely on the set $\{\sigma_{\tilde{B}}=\infty\}$,}$$ whenever $r_0>\tilde{B}$. Then this is enough to prove that $\theta_t$ does not converge, almost surely. To see this, note that our earlier discussion shows that $\theta_t$ doesn’t converge on $\{\sigma_{\tilde{B}}=\infty\}$ (up to a set of probability zero), and thus we need to consider the set $\{\sigma_{\tilde{B}}<\infty\}$. The point is that, analogously to the above, transience implies that almost every path that hits $\{r=\tilde{B}\}$ subsequently hits $\{r=r_0\}$. From there, the same result applies, so that paths which don’t hit $\{r=\tilde{B}\}$ again almost surely have the property that $\theta_t$ doesn’t converge. Iterating this argument (and noting that almost every path eventually has a last exit time from $\{r\leq \tilde{B}\}$), we see that $\theta_t$ fails to converge, almost surely.
\[THM:AngleNoCon\] Suppose that $M$ is Cartan-Hadamard manifold of dimension $m\geq 3$, and that $M$ is radially symmetric around some point $p$. Let $(r,\theta)$ be polar coordinates around $p$, and let $X_t$ be a rank-$n$ martingale, for $2\leq n< m$. Then if either of the following conditions hold:
1. $n=2$ and $K\equiv 0$ (so that $M={\mathbb{R}}^m$), or
2. $n\geq 3$ and, in polar coordinates around some point, $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \geq -\frac{1/2}{r^2\log r} \quad\text{when $r>R$, and for all $\theta$ and $\Sigma\ni\partial_r$,}$$ for some $R>1$,
we have that $\theta_t=\theta(X_t)$ almost surely does not converge, as $t\rightarrow\zeta$.
As before, we begin with the first part. From Section \[Sect:N2\] , we have $$G(r) = r \quad\text{and}\quad v_t = \frac{1+\sin^2{\varphi}_t}{2r_t} .$$ In light of the discussion before the theorem (and the fact that $1/G^2 = 1/r^2$), it’s enough for us to show that for some $\tilde{B}> 0$, $$\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{r_s^2}\, ds = \infty \quad\text{almost surely on the set $\{\sigma_{\tilde{B}}=\infty\}$,}$$ whenever $r_0>\tilde{B}$.
Ito’s rule (see Equation ) plus the above expression for $v_t$ give $$d{\left(}\log r{\right)}_t = \frac{1}{r_t}\cos{\varphi}_t \, dW_t + \gamma_t \, dt \quad\text{where}\quad
0\leq \gamma_t \leq \frac{1}{r^2_t} .$$
By transience of $r_t$, it follows that $\log r_{\sigma_{\tilde{B}}\wedge\zeta}=\infty$ almost surely on the set $\{\sigma_{\tilde{B}}=\infty\}$ (we assume that $r_0>\tilde{B}$). Now a (local) submartingale can diverge to $\infty$ only if its bounded variation part diverges to infinity (up to a set of probability zero). This follows from the fact that the martingale part either converges (if the quadratic variation remains bounded) or hits every real value infinitely often (if the quadratic variation increases without bound), up to a set of probability zero. Thus, for any $\tilde{B}> 0$, $$\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \gamma_s \, ds = \infty \quad\text{almost surely on the set $\{\sigma_{\tilde{B}}=\infty\}$,}$$ whenever $r_0>\tilde{B}$.
Since $\gamma_t \leq 1/r^2_t$, we’ve succeeded in proving the first part.
Again, the second part is similar with more sophisticated estimates. Now, using the estimates of Section \[Sect:NBig\], we have that, for some $c>0$ and $B>R$, $$\begin{split}
& G(r) \leq c r{\left(}\log r {\right)}^{\frac{1}{2}}{\left(}{\log_{(2)}}r {\right)}^{\frac{1}{2}} \quad\text{for $r>B$, and} \\
& \frac{3}{4} \frac{n-1+\sin^2{\varphi}_t}{2r_t}\leq v_t\leq \frac{5}{4} \frac{n-1+\sin^2{\varphi}_t}{2r_t}
\quad\text{for $r_t>B$.}
\end{split}$$ Thus, it’s enough for us to show that for some $\tilde{B}> B$, $$\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{r_s^2 {\left(}\log r_s{\right)}{\left(}{\log_{(2)}}r_s{\right)}}\, ds = \infty \quad\text{almost surely on the set $\{\sigma_{\tilde{B}}=\infty\}$,}$$ whenever $r_0>\tilde{B}$.
Note that $$\begin{split}
{\left(}{\log_{(3)}}{\right)}^{\prime}(r) &= \frac{1}{r {\left(}\log r{\right)}{\left(}{\log_{(2)}}r{\right)}} \quad\text{and}\\
{\left(}{\log_{(3)}}{\right)}^{\prime\prime}(r) &= \frac{-1}{r^2 {\left(}\log r{\right)}{\left(}{\log_{(2)}}r{\right)}}{\left(}1+\frac{1}{\log r}+\frac{1}{{\left(}\log r{\right)}{\left(}{\log_{(2)}}r{\right)}} {\right)}.
\end{split}$$ Then Ito’s rule plus the two-sided bound on $v_t$ lets us compute that, for $r_t>B$, $$\begin{split}
& \quad d{\left(}{\log_{(3)}}{\right)}_t r = \frac{\cos{\varphi}_t}{r_t {\left(}\log r_t{\right)}{\left(}{\log_{(2)}}r_t{\right)}} \, dW_t +\gamma_t \, dt \quad\text{where} \\
& \frac{ \frac{3}{4}{\left(}n-1+\sin^2{\varphi}_t{\right)}-\cos^2{\varphi}_t{\left(}1+\frac{1}{\log r_t}+\frac{1}{{\left(}\log r_t{\right)}{\left(}{\log_{(2)}}r_t{\right)}}{\right)}}{2r^2_t {\left(}\log r_t{\right)}{\left(}{\log_{(2)}}r_t{\right)}}
\leq \gamma_t \\
& \leq
\frac{ \frac{5}{4}{\left(}n-1+\sin^2{\varphi}_t{\right)}-\cos^2{\varphi}_t{\left(}1+\frac{1}{\log r_t}+\frac{1}{{\left(}\log r_t{\right)}{\left(}{\log_{(2)}}r_t{\right)}}{\right)}}{2r^2_t {\left(}\log r_t{\right)}{\left(}{\log_{(2)}}r_t{\right)}}
\end{split}$$ Observe that we can find $\tilde{B}>B$ such that the lower bound on $\gamma_t$ is positive when $r_t>\tilde{B}$. Thus ${\log_{(3)}}r_t$ is a (local) sub-martingale (on the set where $r>\tilde{B}$). Further, the upper bound implies that, perhaps after increasing $\tilde{B}$, we can find $D>0$ such that $$\frac{1}{D} \int_0^{\sigma_{\tilde{B}}\wedge\zeta} \gamma_s \, ds \leq
\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{r_s^2 {\left(}\log r_s{\right)}{\left(}{\log_{(2)}}r_s{\right)}}\, ds$$ whenever $r_0>\tilde{B}$.
As above, by transience of $r_t$, it follows that ${\log_{(3)}}r_{\sigma_{\tilde{B}}\wedge\zeta}=\infty$ almost surely on the set $\{\sigma_{\tilde{B}}=\infty\}$ (we assume that $r_0>\tilde{B}$). Thus the bounded variation part also almost surely diverges to infinity on $\{\sigma_{\tilde{B}}=\infty\}$. Sine $1/D$ is just a positive constant, we’ve succeeded in showing that $$\int_0^{\sigma_{\tilde{B}}\wedge\zeta} \frac{1}{r_s^2 {\left(}\log r_s{\right)}{\left(}{\log_{(2)}}r_s{\right)}}\, ds = \infty \quad\text{almost surely on the set $\{\sigma_{\tilde{B}}=\infty\}$,}$$ whenever $r_0>\tilde{B}$. This completes the proof.
The approach used to prove both of these theorems differs from the method used to study Brownian motion in [@March] and [@HsuBook], which transforms the question of angular convergence to the question of whether or not a certain one-dimensional diffusion, coming from a time-change of the radial process, has finite lifetime (for which an answer is essentially known). In the case of rank-$n$ martingales, the appearance of ${\varphi}_t$ (which functions here mostly as a nuisance parameter) prevents us from being able to derive a simple one-dimensional diffusion, and it’s not immediately clear how to adapt this method. Our approach gets around this, and also has the advantage of (arguably) being somewhat more elementary.
On the other hand, we are only able to get sharp results for the $n\geq 3$ case. Indeed, comparing the second parts of Theorems \[THM:AngleCon\] and \[THM:AngleNoCon\], we see that neither curvature condition can be improved, in terms of bounds of the form $-c/r^2\log r$. Naturally, these are the same bounds which one finds for the analogous results for Brownian motion on Cartan-Hadamard manifolds. In fact, take $n \geq 2$ and let $m=n+1$. Then if $y_1,\ldots,y_{n+1}$ are normal coordinates around $p$, the $n$-dimensional submanifold $N=\{y_{n+1}=0\}$ is totally geodesic, because of radial symmetry. Then $N$ is an $n$-dimensional Cartan-Hadamard manifold satisfying the same curvature bound as $M$, and Brownian motion on $N$ is a rank-$n$ martingale on $M$. Thus we see that Brownian motion on a radially symmetric Cartan-Hadamard manifold is actually a special case of the above, in spite of our taking $n<m$ in our definition of a rank-$n$ martingale.
Given this (and the general pattern of this paper), we expect the $n=2$ case to directly extend what is known for Brownian motion on surfaces. Namely, the first parts of Theorems \[THM:AngleCon\] and \[THM:AngleNoCon\] should hold under the curvature bounds $$\begin{split}
K(r,\theta,\Sigma) & \leq -\frac{1+{\varepsilon}}{r^2\log r} \quad\text{when $r>R$, for some ${\varepsilon}>0$ and $R>1$, or} \\
K(r,\theta,\Sigma) & \geq -\frac{1}{r^2\log r} \quad\text{when $r>R$, for some $R>1$,} \\
\end{split}$$ respectively. However, we were unable to prove that by our current methods, and so we have contented ourselves with constant curvature bounds in the $n=2$ case.
Geometric consequences
----------------------
The geometric implications of the theorems in the previous section are not obvious in every context (such as for ancient solutions of the mean curvature flow). Nonetheless, there are some things we can say.
First, again consider a rank-$n$ sub-Riemannian structure on $M$, with the restriction metric and a volume form such that ${\Delta}_s$ is a sum of squares. Then let $X_t$ be the associated diffusion. In this case, $X_t$ is Markov (indeed, the $v_{i,t}$ in Equation can and should be chosen to be smooth, so that the equation has a unique solution by standard results for SDEs) and has a positive density with respect to the volume form at any positive time (by a famous result of Hörmander and later via the Malliavin calculus). Assume that $M$ (and the sub-Riemannian structure) satisfy the hypotheses of Theorem \[THM:AngleCon\]. We now let $U$ be an open, nonempty subset of ${\mathbb{S}}^{m-1}$ such that the complement $U^c$ has non-empty interior, and let $p(y)$ be the probability that the diffusion, started from any $y=X_0\in M$, has $X_{\zeta}\in U$. Then $p$ is a non-constant, bounded ${\Delta}_s$-harmonic function on $M$. (The ${\Delta}_s$-harmonicity is a consequence of the facts that $X_t$ is Markov and the event $X_{\zeta}\in U$ is tail-measurable.) Indeed, for $\tilde{\theta}\in U$, we see that $p(\tilde{r},\tilde{\theta})\rightarrow 1$ as $\tilde{r}\rightarrow \infty$, and similarly, for $\tilde{\theta}$ in the interior of $U^c$, $p(\tilde{r},\tilde{\theta})\rightarrow 0$ as $\tilde{r}\rightarrow \infty$. These observations form the basis for a probabilistic approach to a version of the Dirichlet problem at infinity, relative to the sphere at infinity determined by the original Riemannian structure on $M$. However, we don’t pursue this (as already mentioned) and instead simply give the following corollary of Theorem \[THM:AngleCon\].
\[Cor:sRHarm\] Suppose that $M$ is Cartan-Hadamard manifold of dimension $m\geq 3$, and that $M$ is radially symmetric around some point $p$. Let $(r,\theta)$ be polar coordinates around $p$. Consider a rank-$n$ sub-Riemannian structure on $M$, with the restriction metric and a volume form such that ${\Delta}_s$ is a sum of squares. Then if either of the following conditions hold:
1. $n=2$ and $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -a^2 \quad \text{for all $\theta$ and (the only) $\Sigma$}$$ for some $a>0$, or
2. $n\geq 3$ and $M$ satisfies the curvature estimate $$K(r,\theta,\Sigma) \leq -\frac{\frac{1}{2}+{\varepsilon}}{r^2\log r} \quad\text{when $r>R$, and for all $\theta$ and $\Sigma\ni\partial_r$,}$$ for some ${\varepsilon}>0$ and $R>1$,
we have that $M$ admits a non-constant, bounded, ${\Delta}_s$-harmonic function.
Similar logic shows that, under the same hypotheses on curvature, an $n$-dimensional minimal surface $N$ will admit a non-constant, bounded, harmonic function (where the harmonicity is with respect to the induced Laplacian on $N$, of course). The only additional complication is that it’s less obvious that we can find some set $U$ as above for which we can always find points with $\theta$-components in both $U$ and the interior of $U^c$ for arbitrarily large $r$-components. (That is, we might worry that $X_{\zeta}$ is constant, independent of starting point.) The only way we could fail to be able to find a set $U$ as desired is if there were some $\tilde{\theta}$ such that for all sequences $x_i \in N$ with $r(x_i)\rightarrow \infty$, we had $\theta(x_i)\rightarrow \tilde{\theta}$. However, one can show fairly easily that this is not possible. One way to do this, in the spirit of this paper, is to use Equation . This, along with the properties of the $w_{i,t}$ and the fact that the integral of $1/G^2(r_t)$ along paths is almost surely finite (because of the curvature assumptions of Theorem \[THM:AngleCon\]) and independent of the $\theta_t$-process, shows that $\theta_t$ cannot converge to a single point. Thus there can be no $\tilde{\theta}$ as just described. With this in mind, we have established Corollary \[Cor:MinHarm\].
Finally, we note that the hypotheses of Theorem \[THM:AngleNoCon\] do not imply, either for sub-Riemannian structures of the type under consideration or for minimal submanifolds, that there are no non-constant harmonic functions.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[99]{}
Andrei Agrachev, Ugo Boscain, Jean-Paul Gauthier, and Francesco Rossi, *The intrinsic hypoelliptic [L]{}aplacian and its heat kernel on unimodular [L]{}ie groups*, J. Funct. Anal. **256** (2009), no. 8, 2621–2655.
Elton P. Hsu, *Stochastic analysis on manifolds*, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002.
, *Brownian motion and [D]{}irichlet problems at infinity*, Ann. Probab. **31** (2003), no. 3, 1305–1319.
Ioannis Karatzas and Steven E. Shreve, *Brownian motion and stochastic calculus*, second ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991.
Peter March, *Brownian motion and harmonic functions on rotationally symmetric manifolds*, Ann. Probab. **14** (1986), no. 3, 793–801.
S. Markvorsen and V. Palmer, *Transience and capacity of minimal submanifolds*, Geom. Funct. Anal. **13** (2003), no. 4, 915–933.
Richard Montgomery, *A tour of subriemannian geometries, their geodesics and applications*, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002.
Robert W. Neel, *On parabolicity and area growth of minimal surfaces*, J. Geom. Anal. (2011), 10.1007/s12220–011–9280–2.
R. Schoen and S.-T. Yau, *Lectures on differential geometry*, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994, Lecture notes prepared by Wei Yue Ding, Kung Ching Chang \[Gong Qing Zhang\], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso.
H. Mete Soner and Nizar Touzi, *A stochastic representation for mean curvature type geometric flows*, Ann. Probab. **31** (2003), no. 3, 1145–1165.
Daniel W. Stroock, *An introduction to the analysis of paths on a [R]{}iemannian manifold*, Mathematical Surveys and Monographs, vol. 74, American Mathematical Society, Providence, RI, 2000.
|
---
abstract: 'We observe a *dc* electric current in response to terahertz radiation in lateral inter-digitated double-comb graphene $p$-$n$ junctions. The junctions were fabricated by selective ultraviolet irradiation inducing $p$-type doping in intrinsic $n$-type epitaxial monolayer graphene. The photocurrent exhibits a strong polarization dependence and is explained by electric rectification in $p$-$n$ junctions.'
author:
- 'Yu. B. Vasilyev,$^1$ G. Yu. Vasileva,$^1$ S. Novikov,$^2$ S. A. Tarasenko,$^1$ S. N. Danilov,$^3$ and S. D. Ganichev,$^3$'
title: 'High-frequency rectification in graphene lateral $p$-$n$ junctions'
---
The development of graphene electronic circuits is one of the important tasks of modern solid-state electronics. Doping or electrical gating of graphene enable the fabrication of $p$-$n$ junctions, which are the subject of intensive study, see e.g. [@1; @2; @5; @3; @4]. It was found experimentally that graphene structures with lateral $p$-$n$ junctions demonstrate almost symmetric current-voltage ($I$-$V$) characteristics, which are very similar for the negative and positive polarities of the source-drain voltage [@1; @2; @5; @3]. The bi-directional charge transport is attributed to the Klein tunneling of Dirac fermions [@6]. The symmetric character of the $I$-$V$ characteristic results in a lack of a pronounced rectification of electric signals in graphene $p$-$n$ devices being an obstacle for their application in electronics.
In optical measurements, however, it was observed that the illumination of graphene $p$-$n$ junctions leads to an electric response, which is a prerequisite for the fabrication of photodiodes [@8; @9]. At high photon energy the photoresponse is caused by the creation of electron-hole pairs, which are separated by a built-in electric field of the $p$-$n$ junction [@10]. This mechanism, however, fails for lower frequencies of terahertz (THz) range, for which the photon energy is typically much lower than the Fermi energy and the generation of electron-hole pairs is blocked. Nevertheless, irradiating sharp lateral graphene $p$-$n$ junctions by THz laser radiation we observe a $dc$ current. The photocurrent is detected in epitaxial graphene on a SiC substrate with inter-digitated dual-comb $p$-$n$ junctions and is characterized by a strong dependence on the radiation polarization. The observations suggest the rectification of THz electric field in graphene $p$-$n$ junctions.
![ (a) Top view of graphene photodiode, where the source and drain electrodes (yellow) are in contact with the $p$- (red) and $n$-type (blue) areas. (b) Experimental setup. []{data-label="fig_1"}](fig1.eps){width="\linewidth"}
Epitaxial graphene samples for the subsequent $p$-$n$ junction fabrication were prepared by high-temperature Si sublimation of SiC [@11]. In order to reduce the density of carriers we exposed the prepared samples in hot air [@25]. This resulted in the initial $n$-type carrier density in the range $(1.6 \div
3.1)\times 10^{11}$ cm$^{-2}$ and the mobility of about $(1400
\div 2700)$ cm$^2$/(V s). Then, the graphene was covered with polymers PMMA and ZEP500. This enable the adjusting of the carrier density by photochemical gating in which ZEP500 provide potent acceptors under deep ultraviolet (UV) light [@26]. By irradiation with UV through the shadow mask graphene was patterned into $p$-$n$ structures consisting of inter-digitated finger stripes of $p$- and $n$-types, see Fig. \[fig\_1\]. The irradiation dose was chosen high enough to inverse the carrier type from $n$- to $p$-type [@12]. Consequently, $p$-$n$ junctions are formed along the boundary of the illuminated and not illuminated areas. Inter-digitated devices with finger length $l=1.8$ mm and width of $d=50$ and 100 $\mu$m for samples $\# A$ (68 junctions), and $\#
B$ (34 junctions), respectively, were fabricated to enlarge the area of the $p$-$n$ junctions. The type and degree of doping in the fingers were extracted from the Hall measurements. The obtained carrier densities are $3\times 10^{11}$ cm$^{−2}$ for the $n$-doped part and $5\times 10^{10}$ cm$^{−2}$ for the $p$-doped part. The corresponding Fermi energies $E_F$ are 70 and 25 meV, respectively.
The experiments on photocurrents are performed applying radiation of NH$_3$ laser [@Ganichev93; @Schneider04] operating at the frequencies $f=2$ and 1.1 THz. We use single pulses with a peak power of $P \approx 10$ kW, a duration of about 100 ns and a repetition rate of 1 Hz. The beam had an almost Gaussian shape, as measured by a pyroelectric camera [@LechnerAPL2009], and was focused at a spot with the diameter $\approx 2$ mm. All experiments are performed at normal incidence of radiation, Fig. \[fig\_1\](b), and temperature $T = 4.2$ K. We use linearly polarized radiation with the direction of the radiation electric field $\bm E$ relative to the $p$-$n$ junction described by the angle $\alpha$, see Fig. \[fig\_1\](a). The latter was varied applying $\lambda/2$-plates [@15a].
![ $I$-$V$ characteristics. Insets illustrate the mechanism of the current flow in direct- and reverse-biased $p$-$n$ junctions.[]{data-label="fig_2"}](fig2.eps){width="\linewidth"}
First, we study $I$-$V$ characteristics applying $dc$ voltage, $V_{SD}$, between the source and drain contacts, see Fig. \[fig\_2\]. The data reveal that there is no asymmetry in the behavior of the current for the forward- and reverse-biased junctions. The symmetry indicates that our graphene structures as a whole do not exhibit rectifying properties, which is in contrast to conventional diodes based on gaped semiconductors. A substantial current flow at the reverse-biased $p$-$n$ junction is attributed to the absence of a band gap in graphene and the related efficient Klein tunneling [@6]. The corresponding band diagrams, sketched in Fig. \[fig\_2\], show that the electric current can flow for both polarities of the applied voltage.
The ability of a graphene $p$-$n$ junction to conduct electric current in both directions, however, implies neither that the $I$-$V$ characteristic itself is symmetric nor that the $p$-$n$ junction cannot rectify $ac$ electric fields. In fact, the real $I$-$V$ characteristic of the $p$-$n$ junction and its possible asymmetry can be hidden in transport experiments and its measurement in the lateral graphene $p$-$n$ junction is a challenging task. This is because the current in such samples is determined by the large resistance of the $p$- and $n$-type areas rather than the small resistance of the thin $p$-$n$ junction. Note, in vertical graphene $p$-$n$ junctions, where the lateral resistance of the $p$-type and $n$-type areas does not play a role, an asymmetry of the $I$-$V$ characteristic was indeed observed [@16; @17].
The conclusion that the $I$-$V$ characteristics in our lateral graphene samples are determined by the $p$- and $n$-type areas is supported by the fact that they are well described by the dependence $I\propto V^\delta$ with $1<\delta <1.5$, see Fig. \[fig\_2\]. Similar superlinear dependence was obtained earlier in experiments on homogeneous graphene samples and explained by the interplay of the intraband and interband contributions to the charge transport [@18]. The dependence $I\propto V^{1.5}$ also follows from theoretical calculations in the ballistic regime, see Refs. [@19; @20].
![ Dependence of the photoresponse on the angle $\alpha$ which determines the in-plane orientation of the THz electric field. The photoresponse is measured at the normal incidence of radiation in two samples for two different wavelengths. []{data-label="fig_3"}](fig3.eps){width="\linewidth"}
To measure the response of the lateral $p$-$n$ junctions to a high-frequency electric field we excite the structure by THz radiation in the absence of an applied bias voltage (photovoltaic mode). Illuminating the structure at normal incidence we detect a *dc* photocurrent between the source and drain contacts which linearly scales with the radiation intensity and has a strong polarization dependence, see Fig. \[fig\_3\]. The fact that the photoresponse emerges at normally incident radiation unambiguously indicates that it originates from the $p$-$n$ junction. Indeed, all known mechanisms of the photocurrent formation in homogeneous graphene structures, such as photon drag or photogalvanic effect, require the oblique incidence of radiation or breaking the symmetry by a magnetic field [@jiangPRB2010; @22]. The photocurrent also cannot be caused by ratchet [@Olbrich16] or plasmonic [@10] effects recently observed in graphene because they require metal gates superlattices.
The role of the $p$-$n$ junction also follows from the dependence of the photoresponse on the angle $\alpha$ between the electric field of the radiation and the normal to the $p$-$n$ junctions, see \[fig\_3\]. For both wavelengths studied the photocurrent reaches a maximum at $\alpha = 0$ and $180^\circ$, when the $ac$ electric field of the incident radiation is perpendicular to the $p$-$n$ junctions, and a minimum at $\alpha = 90^\circ$, when the radiation is polarized along the $p$-$n$ junctions. The overall polarization dependence is well fitted by $$\label{FITalpha}
J(\alpha) = (a + b \cos^2{\alpha}) P \:.$$ Note that the observed variation of the photoresponse is very large so that $(J_{\rm max} - J_{\rm min})/(J_{\rm max} + J_{\rm min}) = 0.8 \div 0.9$ being close to unity. Furthermore, increasing the number of $p$-$n$ junctions by decreasing the width of the fingers in the combs (samples $\# A$ vs. $\# B$) we observed an increase of the signal for low radiation frequencies. Comparing the photocurrent excited by $f=1.1$ and 2 THz radiation we see that in sample $\# A$ it decreases with raising frequency whereas in sample $\# B$ it remains unchanged.
Now we discuss the origin of the photoresponse. Since the energy of photons of THz radiation is much smaller than the Fermi energies of carriers in $p$-type and $n$-type areas, as well as in $p$-$n$ junction areas, the interband absorption of radiation with the creation of electron-hole pairs is blocked. In this case, the Drude absorption dominates and the quasi-classical approach to the description of electron transport through the $p$-$n$ junction becomes valid. The observed $dc$ photocurrent can be interpreted as a result of the rectification: The $ac$ electric field of the THz radiation incident upon the $p$-$n$ junction causes a high-frequency electric current, which is partially rectified due to the asymmetry of the $I$-$V$ characteristic of the $p$-$n$ junction. The rectification occurs in the area of the $p$-$n$ junction while, as discussed above, the homogenous $p$- and $n$-doped parts do not contribute to the formation of the $dc$ signal. To the lowest order in the electric field amplitude, the $dc$ current is described by the second order nonlinear term in the $I$-$V$ characteristic: $$\label{FITalpha}
I_{dc} = \sigma_2 E^2_{\perp} \:,$$ where $\sigma_2$ is the nonlinear conductance and $E_{\perp} $ is the amplitude of the $ac$ electric field across the $p$-$n$ junctions. The photocurrent scales quadratically with the electric field amplitudes as detected in experiment. The rectification mechanism also explains the observed polarization dependence. The $dc$ electric current is generated by the component of the $ac$ electric field normal to the $p$-$n$ junction varying as $E^2_{\perp} \propto \cos^2(\alpha) $ which is in agreement with the experimental data, Fig. \[fig\_3\]. The presence of an additional small polarization-independent signal may be related to the photo-thermoelectric effect [@24]. The observed weak frequency dependence of the photosignal in sample $\# A$ and its absence in the sample $\# B$ are due to low mobility of carriers in our samples. Indeed, the frequency dependence is determined by the parameter $\omega \tau$, where $\omega = 2 \pi f$ and $\tau$ is the momentum relaxation time. For the carrier density $3\times 10^{11}$ cm$^{-2}$, mobility $10^3$ cm$^2$/(Vs), and the radiation frequency $f=1$ THz, the estimation yields $\omega \tau \approx 0.05$. It shows that the $ac$ transport of carries induced by THz radiation in the $p$-$n$ junctions is, in fact, quasi-stationary and depends on the $ac$ electric field amplitude rather than on its frequency.
In conclusion, we have reported photocurrent measurements in graphene lateral $p$-$n$ junctions formed by selective deep UV illumination the graphene monolayer fabricated on a SiC substrate. We have observed a pronounced photoresponse in the THz range, which has a strong dependence on the radiation polarization. The observations are explained by the rectification mechanism of the photocurrent formation in graphene $p$-$n$ junctions.
We thank V. Belkov and V. Kachorovskii for fruitful discussions. The support from the RFBR (16-02-00326), and the DFG (SFB 1277-A04) is acknowledged. S.T. acknowledges the support from the RSF (14-12-01067).
|
---
abstract: 'We present a scheme to conditionally engineer an optical quantum system via continuous-variable measurements. This scheme yields high-fidelity squeezed single photon and superposition of coherent states, from input single and two photon Fock states respectively. The input Fock state is interacted with an ancilla squeezed vacuum state using a beam splitter. We transform the quantum system by post-selecting on the continuous-observable measurement outcome of the ancilla state. We experimentally demonstrate the principles of this scheme using coherent states and measure experimentally fidelities that are only achievable using quantum resources.'
author:
- 'Andrew M. Lance'
- Hyunseok Jeong
- 'Nicolai B. Grosse'
- Thomas Symul
- 'Timothy C. Ralph'
- Ping Koy Lam
title: 'Quantum State Engineering with Continuous-Variable Post-Selection '
---
[*Introduction -* ]{} The transformation or engineering of quantum states via measurement induced conditional evolution is an important technique for discrete variable systems, particularly in the field of quantum information [@NIE00] Typically, the quantum system of interest is interacted with a prepared ancilla state, which is then measured in a particular basis. The system state is retained, or discarded, depending on the measurement outcome, resulting in the controlled conditional evolution of the quantum system. It is a necessary condition for inducing a non-trivial conditional evolution that the interaction of the ancilla and system produces entanglement between them.
In optical systems, highly non-linear evolutions, which are difficult to induce directly, can be induced conditionally on systems by post-selecting on particular photon counting outcomes [@OBR03]. In principle, a near deterministic, universal set of unitary transformations can be induced on optical qubits in this way [@KNI01]. Importantly, it was shown that arbitrary optical states can be engineered conditionally, based on discrete single photon measurements [@DAK99].
Recently, there has been increased interest in conditional evolution based on continuous-variable measurements [@Lau03; @Bab05; @RAL05]. In these schemes a quantum system is interacted with a prepared ancilla, which is measured via a [*continuous*]{} observable, e.g. the amplitude or phase quadratures of the electromagnetic field. This has been experimentally demonstrated for a system using a beamsplitter as the interaction and a vacuum state as the ancilla, with conditioning based on homodyne detection [@Bab05]. A similar system using conditioning of adaptive phase measurements has also been studied [@RAL05].
In this letter, we investigate a continuous-variable conditioning scheme based on a beam splitter interaction, homodyne detection and an ancilla squeezed vacuum state. We theoretically show that for input one and two photon Fock states, this scheme yields high fidelity squeezed single photon Fock states and superposition of coherent states (SCS) respectively, which are highly non-classical and interesting quantum states that have potentially useful applications in quantum information processing [@catapply]. We experimentally demonstrate the principles of this scheme using input coherent states, and measure experimental fidelities that are only achievable using quantum resources.
![(Color online) (a) Schematic of the post-selection protocol. $\hat{X}_{\rm in}^{\pm}$: amplitude (+) and phase (-) quadratures of the input state; (anc) ancilla state; (r) reflected; (t) transmitted; and (out) post-selected output state. $R$: beam splitter reflectivity; GD: gate detector; PS: post-selection protocol. (b) Standard deviation contours of the Wigner functions of an input coherent state (blue) and post-selected output states (green) for $R=0.75$ and varying ancilla state squeezing of (i) $s=0$, (ii) $s=0.35$, (iii) $s=0.69$, (iv) $s=1.03$ and (v) ideal squeezing. []{data-label="ExptSetup"}](Figure1_Lance_PRA.eps){width="\columnwidth"}
[*Theory -* ]{} The squeezed vacuum ancilla state used in our scheme is represented as ${\hat S}(s)|0\rangle$ with the squeezing operator ${\hat S}(s) = {\rm exp}[-(s/2)(\hat{a}^2 - \hat{a}^{\dagger 2})]$, where $s$ is the squeezing parameter and $\hat a$ is the annihilation operator. The Wigner function of the squeezed vacuum is $W_{\rm sqz}(\alpha;s)=2\exp[-2(\alpha^{+})^2e^{-2s}-2(\alpha^{-})^2e^{2s}]/\pi$, where $\alpha=\alpha^{+}+i\alpha^{-}$ with real quadrature variables $\alpha^{+}$ and $\alpha^{-}$. The first step of our transformation protocol is to interfere the input field with the ancilla state on a beam splitter as shown in Fig. \[ExptSetup\] (a). The beam splitter operator ${\hat{B}}$ acting on modes $a$ and $b$ is represented as $\hat{B}(\theta)=\exp \{(\theta /2)
(\hat{a}^{\dagger }\hat{b}
-\hat{b}^{\dagger }\hat{a})\}$, where the reflectivity is defined as $R=\sin^2(\theta /2)$ and $T=1-R$. A homodyne measurement is performed on the amplitude quadrature on the reflected field mode, with the measurement result denoted as $X^+_{\rm r}$. The transmitted state is post-selected for $|X^+_{\rm r}|<x_0$, where the post-selection threshold $x_0$ is determined by the required fidelity between the output state and the ideal target state.
We first consider a single-photon state input, $|1\rangle$, and a squeezed single photon, ${\hat S}(s^\prime)|1\rangle$, as the target state. The Wigner function of the single photon state is $W^{|1\rangle}_{\rm in}(\alpha)=2\exp[-2|\alpha|^2](4|\alpha|^2-1)/\pi$. After interference via the beam splitter, the resulting two-mode state becomes $W(\alpha,\beta)=W^{|1\rangle}_{\rm in}\big( \sqrt{T}\alpha+\sqrt{R}\beta\big)
W_{\rm anc}\big(-\sqrt{R}\alpha+\sqrt{T}\beta\big)$, where $W_{\rm anc}= W_{\rm sqz}(\alpha;s)$ and $\beta=\beta^{+}+i\beta^{-}$. The transmitted state after the homodyne detection of the reflected state is $W_{\rm out}(\alpha;X^+_{\rm r})=P_1(X^+_{\rm r})^{-1}\int_{-\infty}^{\infty}
d\beta^{-}W(\alpha,\beta^{+}=X^+_{\rm r},\beta^{-})$, where the normalization parameter is $P_1(X^+_{\rm r})=\int_{-\infty}^\infty d^2\alpha
d\beta^{-}W(\alpha,\beta^{+}=X^+_{\rm r},\beta^{-})$. If the measurement result is $X^+_{\rm r}=0$, the Wigner function of the output state becomes $$\begin{aligned}
\label{eq:Wout1}
W_{\rm out}(\alpha)
&=&\frac{2}{\pi}e^{-2[e^{-2s^\prime}(\alpha^{+})^2+e^{2s^\prime}(\alpha^{-})^2]}\\ \nonumber
&&\times\big(4e^{-2s^\prime}(\alpha^{+})^2+4e^{2s^\prime}(\alpha^{-})^2-1\big), \end{aligned}$$ where $s^\prime=-\ln [(T + e^{-2s}R)^2]/4$. One can immediately notice that the output state in Eq. (\[eq:Wout1\]) is [*exactly*]{} the Wigner function of a squeezed single photon, $\hat{S}(s^\prime)|1\rangle$. We note that the output squeezing $s^\prime$ can be arbitrarily close to the squeezing of the ancilla state $s$ by making $R$ close to zero. For the nonzero post-selection threshold criteria $|X^+_{\rm r}|<x_0$, the corresponding success probability is given by $P_s(x_0)=\int_{-x_0}^{x_0} d X^+_{\rm r} P_1(X^+_{\rm r})$.
![ (Color online) (a) The average fidelity ${\cal F}_{\rm ave}$ between the post-selected output state of an input single photon state $|1\rangle$, and the squeezed single photon state ${\hat S}(s^\prime)|1\rangle$, for varying threshold $x_0$. The beam splitter reflectivity is $R=0.98$, ancilla state squeezing is $s=0.7$ and target state squeezing is $s^\prime=0.67$. (b) The average Wigner function $W_{\rm ave}$ of the output state for $F_{\rm ave}=0.99$, for $x_0=0.025$ and $P_s=0.003$. []{data-label="1photon_Input"}](Figure2_Lance_PRA.eps){width="\columnwidth"}
We calculate the fidelity between the output state (with the measurement result $X^+_{\rm r}$) and the ideal target state given by ${\cal F}_1(X^+_{\rm r})=\pi\int^\infty_{-\infty}
d^2\alpha W_{\rm out}(\alpha;X^+_{\rm r})W_{\rm out}(\alpha)$. From this we can determine the average fidelity for the threshold $x_0$ defined as ${\cal F}_{\rm ave}(x_0)=\int_{-x_0}^{x_0} dx P_1(X^+_{\rm r}){\cal F}_1(X^+_{\rm r})/
\int_{-x_0}^{x_0}d{\tilde X^+_{\rm r}}
P_1({\tilde X^+_{\rm r}})$. We use this average fidelity measure to characterize the efficacy of our protocol for nonzero thresholds. We can likewise determine the average Wigner function $W_{\rm ave}(\alpha;x_0)$ for the threshold $x_0$.
Fig. \[1photon\_Input\] (a) shows the average fidelity ${\cal F}_{\rm ave}$ for varying post-selection threshold $x_0$. This figure illustrates that high fidelity squeezed single photon states can be produced using experimentally realizable squeezing of the ancilla state and finite thresholds. The average Wigner function corresponding to an average fidelity ${\cal F}_{\rm ave}=0.99$ is shown in Fig. \[1photon\_Input\] (b). We point out that post-selection around $X^{+}_{\rm r}=0$ preserves the non-Gaussian features of the input state. Our scheme enables one to perform the squeezing of a single photon with high fidelities using any [*finite*]{} degree of squeezing of the ancilla state. We point out that this squeezed single photon state is a good approximation to an odd SCS, which has applications in quantum information processing [@catapply]. We emphasize that this interesting result [*cannot*]{} be achieved by continuous electro-optic feed-forward methods [@Fil05] with finite ancilla state squeezing.
![ (Color online) (a) The average fidelity ${\cal F}_{\rm ave}$ between the output state of the input two photon state, $|2\rangle$, and the ideal superposition of coherent states (SCS), for varying threshold $x_0$. The beam splitter reflectivity is $R=1/2$, the ancilla state squeezing is $s=-0.37$, and the amplitude of the SCS is $\gamma=1.1i$. (b) The average Wigner function $W_{\rm ave}$ of the output state for $F_{\rm ave}=0.99$, for $x_0=0.084$ and $P_s=0.052$.[]{data-label="2photon_Input"}](Figure3_Lance_PRA.eps){width="\columnwidth"}
Another example of our post-selection protocol is for the case of input two-photon Fock states, $|2\rangle$. In this case, our target state is an even SCS, $|\gamma\rangle+|-\gamma\rangle$ (unnormalized), where $|\gamma\rangle$ is a coherent state of amplitude $\gamma=\gamma^{+}+i\gamma^{-}$. The Wigner representation of the SCS is $$\begin{aligned}
W_{\rm scs}(\alpha)=N_{1}\Big\{e^{-2|\alpha-\gamma|^2}
+e^{-2|\alpha+\gamma|^2}~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+ e^{-2|\gamma|^2}(e^{-2(\alpha+\gamma)^*(\alpha-\gamma)}
+e^{-2(\alpha+\gamma)(\alpha-\gamma)^*})
\Big\},\end{aligned}$$ where $N_{1}=\{\pi(1+ e^{-2|\gamma|^2})\}^{-1}$. For an input two photon Fock state, the fidelity between the post-selected output state (with the measurement result $X^+_{\rm r}$) and the ideal SCS target state is ${\cal F}_2(X^+_{\rm r})=\pi\int^\infty_{-\infty}
d^2\alpha W_{\rm out}(\alpha;X^+_{\rm r})W_{\rm scs}(\alpha)$. From this expression, the average fidelity $\mathcal{F}_{\rm ave}$ and average Wigner function $W_{\rm ave}$ for a post-selection threshold $x_0$ can be be calculated. Fig. \[2photon\_Input\] shows the average fidelity of the output state for varying threshold, which illustrates that high fidelity SCS can be obtained with experimentally realizable ancilla state squeezing and finite thresholds. The average Wigner function corresponding to an average fidelity of ${\cal F}_{\rm ave}=0.99$ is shown in Fig. \[2photon\_Input\] (b). Once such SCSs are obtained, they can be conditionally amplified for SCSs of larger amplitudes using only linear optics schemes [@Lund04].
We now consider the case of a Gaussian state, i.e., an [*unknown*]{} coherent state, $|\gamma\rangle$, as the input. The post-selection scheme for $X^+_{\rm r}=0$ transforms the coherent state as $$D(\gamma)|0\rangle\longrightarrow
D\big(\sqrt{T}[e^{2s^\prime}
\gamma^{+}+i \gamma^{-} ]\big)S(s^\prime)|0\rangle,
\label{eq:ppt}$$ where $D(\gamma)={\rm exp}[\gamma\hat{a}^\dagger-\gamma^{*}\hat{a}]$ is the displacement operator. Figure \[ExptSetup\] (b) illustrates that the squeezing and the displacement transformation of the post-selected output state in Eq. (\[eq:ppt\]) is dependent on the ancilla state squeezing, and that the output state is a minimum uncertainty state independent of the ancilla state squeezing. In the limit of ideal ancilla state squeezing, the post-selection scheme works as an ideal single-mode squeezer for arbitrary input coherent states $D(\gamma)|0\rangle\rightarrow S(s^\prime)D(\gamma)|0\rangle $. In this case, the output squeezing is $s^\prime\rightarrow-\ln [T]/2$. We note that for input coherent states this scheme provides an alternative method of squeezing to electro-optic protocols presented in [@Fil05].
![ (Color online) Experimental fidelity for varying amplitudes $|\gamma^{+}|$ of the input coherent state, for $R=0.75$ and $x_0= 0.01$ Dark grey region: classical fidelity limit for an ancilla vacuum state; Light grey region: classical fidelity limit. Dot-dashed line: calculated theoretical prediction of experiment, with experimental losses and inefficiencies. []{data-label="ExptFidelity"}](Figure4_Lance_PRA.eps){width="\columnwidth"}
[*Experiment -* ]{} We experimentally demonstrated the principle of the post-selection protocol using input displaced coherent states for a realizable ancilla state squeezing. For the experiment, the quantum states we considered reside at the sideband frequency $(\omega)$ of the electromagnetic field. We denote the quadratures of these quantum states as $\hat{X}^{\pm}=\langle\hat{X}^{\pm}\rangle+\delta\hat{X}^{\pm}$, where $\langle\hat{X}^{\pm}\rangle$ are the mean quadrature displacements, and where the quadrature variances are expressed by $V^{\pm}=\langle(\delta\hat{X}^{\pm})^{2}\rangle$.
Fig. \[ExptSetup\] shows the experimental setup. We used a hemilithic MgO:LiNbO$_3$ below-threshold optical parametric amplifier, to produce an amplitude squeezed field at 1064 nm with squeezing of $s=0.52\pm0.03$, corresponding to a quadrature variance of $V^{+}_{\rm anc}=-4.5\pm 0.2$ dB with respect to the quantum noise limit. More detail of this experimental production of squeezing is given in [@Bow03]. The displaced coherent states were produced at the sideband frequency of 6.81 MHz of a laser field at 1064 nm, using standard electro-optic modulation techniques [@Bow03]. The post-selection protocol goes as follows: the amplitude squeezed ancilla field $\hat{X}^{\pm}_{\rm anc}$ was converted to a phase squeezed field by interfering it with the input coherent state $\hat{X}^{\pm}_{\rm in}$ with a much larger coherent amplitude on the beam splitter with a relative optical phase shift of $\pi/2$. This optical interference yielded two output states that were phase squeezed. The optical fringe visibility between the two fields was $\eta_{\rm vis}=0.96\pm 0.01$.
We directly detected the amplitude quadrature of the reflected state $\hat{X}^{+}_{\rm r}$ using a gate-detector, which had a quantum efficiency of $\eta_{\rm det}=0.92$ and an electronic noise of $6.5$ dB below the quantum noise limit. The post-selection could, in principle, be achieved using an all optical setup, but here we post-selected [*a posteriori*]{} the quadrature measurements of the transmitted state, $\hat{X}^{\pm}_{\rm t}$, which were measured using a balanced homodyne detector. The total homodyne detector efficiency was $\eta_{\rm hom}=0.89$, with the electronic noise of each detector $8.5$ dB below the quantum noise limit. To characterize the protocol, we also measured the quadratures of the input coherent state, $\hat{X}^{\pm}_{\rm in}$, using the same homodyne detector. To ensure accurate results, the total homodyne detector inefficiency was inferred out of all quadrature measurements [@Bow03].
The electronic photocurrents of the detected quantum states (at a sideband frequency of 6.81 MHz) from the gate and homodyne detectors were electronically filtered, amplified and demodulated down to 25 kHz using an electronic local oscillator at 6.785 MHz. The resulting photocurrents were digitally recorded using a NI PXI 5112 data acquisition system at a sample rate of 100 kS/s. We used computational methods to filter, demodulate and down-sample the data, so that it could be directly analyzed in the temporal domain. From this data, we post-selected the quadrature measurements of the transmitted state, $\hat{X}^{\pm}_{\rm t}$, which satisfied the threshold criteria $|X^{+}_{\rm r}|<x_0$. This post-selection threshold was independent of the input state and was experimentally optimized depending on the beam splitter reflectivity.
We characterized the efficacy of our protocol as an ideal single mode squeezer, by determining the fidelity of the post-selected output state with a target state that is an ideal squeezed operation of the input state \[Eq. (\[eq:ppt\])\]. The Wigner function of this ideal squeezed input state is given by $W_{\rm out}(\gamma;s^\prime)$, where $s\rightarrow\infty$ and $s^\prime\rightarrow-\ln [T]/2$. In this case, the fidelity is given by ${\cal F}(X^+_{\rm r})=
\pi\int^\infty_{-\infty} d^2\gamma W_{\rm expt}(\gamma;X^+_{\rm r})
W_{\rm out}(\gamma;s^\prime)$, where $W_{\rm expt}(\gamma;X^+_{\rm r})$ is the Wigner function of the post-selected output state. From this expression the average fidelity ${\cal F}_{\rm ave}$ for a post-selection threshold $x_0$ can be calculated. This corresponds to unity fidelity $\mathcal{F}_{\rm ave}=1$ in the limit of ideal ancilla state squeezing and $X^+_{\rm r}=0$. In the experiment, the input state was a slightly mixed state due to inherent low-frequency classical noise on the laser beam, with quadrature variances of $V^{+}_{\rm in}=1.13\pm 0.02$ and $V^{-}_{\rm in}=1.05\pm 0.02$, with respect to the quantum noise limit. Hence, we calculated the fidelity of the post-selected output state with an ideal squeezed transform of the experimental input state. Fig. \[ExptFidelity\] shows the classical fidelity limit $\mathcal{F}_{\rm clas}$, which signifies the highest fidelity achievable when the interaction of the ancilla and input coherent state yields no entanglement. Exceeding this classical fidelity limit can only be achieved using quantum resources.
Fig. \[ExptFidelity\] shows the experimental fidelity for varying input states $|\gamma^{+}| \equiv |\langle\hat{X}^+_{\rm in}\rangle|$. For a beam splitter reflectivity of $R=0.75$, we achieved a best fidelity of $\mathcal{F}_{\rm ave}=0.90\pm0.02$ for an input state $|\gamma^{+}|=0.18\pm0.01$, which exceeds the maximum classical fidelity of $\mathcal{F}_{\rm clas}=4/5=0.8$. This post-selected output state had quadrature variances of $V^{+}_{\rm out}=4.70\pm0.11$ and $V^{-}_{\rm out}=0.51\pm0.01$. The mean quadrature displacement gains, $g^{\pm}=\langle\hat{X}^{\pm}_{\rm out}\rangle/
\langle\hat{X}^{\pm}_{\rm in}\rangle$, were measured to be $g^{+}=0.71\pm0.16$ and $g^{-}=0.50\pm0.06$. This is compared with the ideal case of perfect ancilla state squeezing, where the ideal theoretical gains are $g^{+}_{\rm ideal}=2$ and $g^{-}_{\rm ideal}=1/2$. The phase gain was controlled by the beam splitter transmittivity, whilst the amplitude gain was less the ideal case due to finite ancilla state squeezing, finite post-selection threshold and experimental losses.
The quantum nature of the post-selection protocol is demonstrated by the experimental fidelity results that exceed the classical fidelity limit in Fig. \[ExptFidelity\]. For large input states $|\gamma^{+}|$, the experimental fidelity was less than the theoretical prediction due to electronic detector noise and the finite resolution of the data acquisition system, resulting in a smaller post-selected output state $|\gamma^{+}|$ and a corresponding decrease in the experimental fidelity. Fig. \[ExptFidelityPurity\] (a) illustrates how the experimental fidelity of a post-selected state transitions to the quantum fidelity region by decreasing the post-selection threshold (and corresponding probability of success).
![(Color online) (a) Experimental fidelity ($\mathcal{F}_{\rm ave}$) for varying post-selection success probability, for an amplitude $|\gamma^{+}|=1.07$ of the input state and $R=0.75$. (b) Experimental purity ($\mathcal{P}_{\rm norm}$) for varying input state $|\gamma^{+}|$, for $x_0= 0.01$ Dash arrows show purity prior to (diamonds) and after (circles) post-selection. Dot-dashed line: calculated theoretical prediction of the experiment.[]{data-label="ExptFidelityPurity"}](Figure5_Lance_PRA.eps){width="\columnwidth"}
We also characterized the experiment in terms of the purity of the post-selected output state, defined as $\mathcal{P}={\rm tr}(\rho^{2}_{\rm out})$. In the case of Gaussian states, the purity of the output state can be expressed as $\mathcal{P}=(V^{+}_{\rm out}V^{-}_{\rm out})^{-1/2}$. In the ideal case of a lossless experiment and a post-selection threshold $X^+_{\rm r}=0$, the protocol is a purity preserving transform, independent of the input state and the amount of squeezing of the ancilla state. In the experiment, as the input states are slightly mixed, we calculate the purity of the post-selected output state, normalized to the purity of the input state, which is given by $\mathcal{P}_{\rm norm}=
(V^{+}_{\rm out}V^{-}_{\rm out})^{-1/2}/(V^{+}_{\rm in}V^{-}_{\rm in})^{-1/2}$. Fig. \[ExptFidelityPurity\] (b) shows the experimental purity of the post-selected output state for varying input states, which illustrates how the purity is improved via the post-selection process. For a beam splitter reflectivity of $R=0.75$ we achieved a best purity of $\mathcal{P}_{\rm norm}=0.81\pm0.04$ for an input state of $|\gamma^{+}|=2.03\pm0.02$. Fig. \[ExptFidelityPurity\] (b) shows that the purity of the post-selected output states were approximately independent of the input states, for a large range of input states.
We also implemented our scheme for a beam splitter reflectivity of $R=0.5$. In this case, we measured a best fidelity of $\mathcal{F}_{\rm ave}=0.96\pm0.01$, which exceeded the maximum classical fidelity of $\mathcal{F}_{\rm clas}=\sqrt{8}/3\approx0.94$, and measured a best purity of $\mathcal{P}_{\rm norm}=0.80\pm0.04$.
In summary, we have investigated a continuous-variable conditioning scheme based on a beam splitter interaction, homodyne detection and an ancilla squeezed vacuum state. The conditional evolution of quantum systems based on continuous-variable measurements of the ancilla state are of particular interest as they can yield from input Fock states, non-Gaussian states, which have applications in the field of quantum information. Further, for Gaussian states, this technique provides an alternative to continuous electro-optic feed-forward schemes. We theoretically showed that our conditional post-selection scheme yields high fidelity squeezed single photon and superposition of coherent states from input one and two photon Fock states respectively, for realizable squeezing of the ancilla state. We experimentally demonstrated the principles of this scheme using coherent states, and measured experimental fidelities that were only achievable using quantum resources.
We thank the Australian Research Council for financial support through the Discovery Program.
[99]{}
M. Nielsen and I. Chuang, [*Quantum computation and quantum information*]{} (Cambridge University Press, Cambridge, UK 2000).
J. L. O’Brien [*et al.*]{}, [*Nature*]{} [**426**]{}, 264 (2003).
E. Knill, R. Laflamme, G. J. Milburn, [*Nature*]{} 409, [**46**]{} (2001).
M. Dakna, J. Clausen, L. Kn" oll, D.-G. Welsch, [*Phys. Rev. A*]{} [**59**]{}, 1658 (1999); J. Clausen, M. Dakna, L. Kn" oll, D.-G. Welsch, [*Opt. Commun.*]{} [**179**]{}, 189 (2000) .
J. Laurat [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**91**]{}, 213601 (2003).
S. A. Babichev, B. Brezger, A. I. Lvovsky, [*Phys. Rev. Lett.*]{} [**92**]{}, 047903 (2004).
T. C. Ralph, A. P. Lund, H. M. Wiseman, [*J. Opt. B: Quantum Semiclass. Opt.*]{} 7 S245ÐS249 (2005).
H. Jeong, M. S. Kim, J. Lee, [*Phys. Rev. A*]{} [**64**]{}, 052308 (2001); T. C. Ralph [*et al.*]{}, [*Phys. Rev. A*]{} [**68**]{}, 042319 (2003).
A. P. Lund, H. Jeong, T. C. Ralph, M. S. Kim, [*Phys. Rev. A*]{} [**70**]{}, 020101(R) (2004).
R. Filip, P. Marek, U. L. Andersen, [*Phys. Rev. A*]{} [**71**]{}, 042308 (2005); P. K. Lam, T. C. Ralph, E. H. Huntington, H.-A. Bachor, [*Phys. Rev. Lett.*]{} [**79**]{}, 1471 (1997)
W. P. Bowen [*et al.*]{}, [*Phys. Rev. A*]{} [**67**]{}, 032302 (2003); A. M. Lance [*et al.*]{}, [*Phys. Rev. Lett*]{} [**92**]{}, 177903 (2004).
|
---
abstract: 'We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$.'
author:
- Gregg Musiker and Ralf Schiffler
---
[Introduction]{}\[section intro\] Cluster algebras, introduced in [@FZ1], are commutative algebras equipped with a distinguished set of generators, the *cluster variables*. The cluster variables are grouped into sets of constant cardinality $n$, the *clusters*, and the integer $n$ is called the *rank* of the cluster algebra. Starting with an initial cluster $\mathbf{x}$ (together with a skew symmetrizable integer $n\times n$ matrix $B=(b_{ij})$ and a coefficient vector $\mathbf{y}=(y_i)$ whose entries are elements of a torsion-free abelian group $\mathbb{P}$) the set of cluster variables is obtained by repeated application of so called *mutations*. To be more precise, let $\mathcal{F}$ be the field of rational functions in the indeterminates $x_1,x_2,\ldots,x_n$ over the quotient field of the integer group ring $\mathbb{ZP}$. Thus $\mathbf{x}=\{x_1,x_2,\ldots,x_n\}$ is a transcendence basis for $\mathcal{F}$. For every $k=1,2,\ldots,n$, the mutation $\mu_k(\mathbf{x})$ of the cluster $\mathbf{x}=\{x_1,x_2,\ldots,x_n\}$ is a new cluster $\mu_k(\mathbf{x})=\mathbf{x}\setminus \{x_k\}\cup\{x_k'\}$ obtained from $\mathbf{x}$ by replacing the cluster variable $x_k$ by the new cluster variable $$\label{intro 1}
x_k'= \frac{1}{x_k}\,\left(y_k^+\,\prod_{b_{ki}>0} x_i^{b_{ki}} +
y_k^-\,\prod_{b_{ki}<0} x_i^{-b_{ki}}\right)$$ in $\mathcal{F}$, where $y_k^+,y_k^-$ are certain monomials in $y_1,y_2,\ldots,y_n$. Mutations also change the attached matrix $B$ as well as the coefficient vector $\mathbf{y}$, see [@FZ1].
The set of all cluster variables is the union of all clusters obtained from an initial cluster $\mathbf{x}$ by repeated mutations. Note that this set may be infinite.
It is clear from the construction that every cluster variable is a rational function in the initial cluster variables $x_1,x_2,\ldots,x_n$. In [@FZ1] it is shown that every cluster variable $u$ is actually a Laurent polynomial in the $x_i$, that is, $u$ can be written as a reduced fraction $$\label{intro 2}
u=\frac{f(x_1,x_2,\ldots,x_n)}{\prod_{i=1}^n x_i^{d_i}},$$ where $f\in\mathbb{ZP}[x_1,x_2,\ldots,x_n]$ and $d_i\ge 0$. The right hand side of equation (\[intro 2\]) is called the *cluster expansion* of $u$ in $\mathbf{x}$.
The cluster algebra is determined by the initial matrix $B$ and the choice of the coefficient system. A canonical choice of coefficients is the [*principal*]{} coefficient system, introduced in [@FZ4], which means that the coefficient group $\mathbb{P}$ is the free abelian group on $n$ generators $y_1,y_2,\ldots,y_n$, and the initial coefficient tuple $\mathbf{y}=\{y_1,y_2,\ldots,y_n\}$ consists of these $n$ generators. In [@FZ4], the authors show that knowing the expansion formulas in the case where the cluster algebra has principal coefficients allows one to compute the expansion formulas for arbitrary coefficient systems.
Inspired by the work of Fock and Goncharov [@FG1; @FG2; @FG3] and Gekhtman, Shapiro and Vainshtein [@GSV1; @GSV2] which discovered cluster structures in the context of Teichmüller theory, Fomin, Shapiro and Thurston [@FST; @FT] initiated a systematic study of the cluster algebras arising from triangulations of a surface with boundary and marked points. In this approach, cluster variables in the cluster algebra correspond to arcs in the surface, and clusters correspond to triangulations. In [@S2], building on earlier results in [@S1; @ST], this model was used to give a direct expansion formula for cluster variables in cluster algebras associated to unpunctured surfaces, with arbitrary coefficients, in terms of certain paths on the triangulation.
Our first main result in this paper is a new parametrization of this formula in terms of perfect matchings of a certain weighted graph that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. To be more precise, let $x_\gamma$ be a cluster variable corresponding to an arc ${\gamma}$ in the unpunctured surface and let $d$ be the number of crossings between ${\gamma}$ and the triangulation $T$ of the surface. Then ${\gamma}$ runs through $d+1$ triangles of $T$ and each pair of consecutive triangles forms a quadrilateral which we call a tile. So we obtain $d$ tiles, each of which is a weighted graph, whose weights are given by the cluster variables $x_\tau$ associated to the arcs $\tau$ of the triangulation $T$.
We obtain a weighted graph $G_{T,{\gamma}}$ by glueing the $d$ tiles in a specific way and then deleting the diagonal in each tile. To any perfect matching $M$ of this graph we associate its weight $w(M)$ which is the product of the weights of its edges, hence a product of cluster variables. We prove the following cluster expansion formula:
$$x_{\gamma}= \sum_M \frac{w(M)\,y(M)}{x_{i_1}x_{i_2}\ldots x_{i_d}},$$ where the sum is over all perfect matchings $M$ of $G_{T,{\gamma}}$, $w(M)$ is the weight of $M$, and $y(M)$ is a monomial in $\mathbf{y}$.
We also give a formula for the coefficients $y(M)$ in terms of perfect matchings as follows. The $F$-polynomial $F_{\gamma}$, introduced in [@FZ4] is obtained from the Laurent polynomial $x_{\gamma}$ (with principal coefficients) by substituting $1$ for each of the cluster variables $x_1,x_2,\ldots,x_n$. By [@S2 Theorem 6.2, Corollary 6.4], the $F$-polynomial has constant term $1$ and a unique term of maximal degree that is divisible by all the other occurring monomials. The two corresponding matchings are the unique two matchings that have all their edges on the boundary of the graph $G_{T,{\gamma}}$. We denote by $M_-$ the one with $y(M_-)=1$ and the other by $M_+$. Now, for an arbitrary perfect matching $M$, the coefficient $y(M)$ is determined by the set of edges of the symmetric difference $M_-\ominus M =(M_-\cup M)\setminus (M_-\cap M)$ as follows.
The set $M_-\ominus M$ is the set of boundary edges of a (possibly disconnected) subgraph $G_M$ of $G_{T,{\gamma}}$ which is a union of tiles $ G_M =\cup_{j\in J} S_j. $ Moreover, $$y(M)=\prod_{j\in J} y_{i_j}$$
As an immediate corollary, we see that the corresponding $g$-vector, introduced in [@FZ4], is $$g_{\gamma}= \deg\left(\frac{w(M_-)}{x_{i_1}\cdots x_{i_d}}\right).$$
Our third main result is yet another description of the formula of Theorem \[thm main\] in terms of the graph $G_{T,{\gamma}}$ only. In order to state this result, we need some notation. If $H$ is a graph, let $c(H)$ be the number of connected components of $H$, let $E(H)$ be the set of edges of $H$, and denote by $\partial H$ the set of boundary edges of $H$. Define $\mathcal{H}_k$ to be the set of all subgraphs $H$ of $G_{T,{\gamma}}$ such that $H$ is a union of $k$ tiles $H=S_{j_1}\cup\cdots\cup S_{j_k}$ and such that the number of edges of $M_-$ that are contained in $H$ is equal to $k+c(H)$. For $H\in \mathcal{H}_k$, let $$y(H)=\prod_{S_{i_j} \textup{\,tile\,in\,}H} y_{i_j}.$$
The cluster expansion of the cluster variable $x_{\gamma}$ is given by $$x_{\gamma}=\sum_{k=0}^d \ \sum_{H\in \mathcal{H}_k} \frac{w(\partial H\ominus M_-)\,y(H)}{x_{i_1} x_{i_2}\cdots x_{i_d}}.$$
Theorem \[thm main\] has interesting intersections with work of other people. In [@CCS2], the authors obtained a formula for the denominators of the cluster expansion in types $A,D$ and $E$, see also [@BMR]. In [@CC; @CK; @CK2] an expansion formula was given in the case where the cluster algebra is acyclic and the cluster lies in an acyclic seed. Palu generalized this formula to arbitrary clusters in an acyclic cluster algebra [@Palu]. These formulas use the cluster category introduced in [@BMRRT], and in [@CCS1] for type $A$, and do not give information about the coefficients.
Recently, Fu and Keller generalized this formula further to cluster algebras with principal coefficients that admit a categorification by a 2-Calabi-Yau category [@FK], and, combining results of [@A] and [@ABCP; @LF], such a categorification exists in the case of cluster algebras associated to unpunctured surfaces.
In [@SZ; @CZ; @Z; @MP] cluster expansions for cluster algebras of rank 2 are given, in [@Propp; @CP; @FZ3] the case $A$ is considered. In section 4 of [@Propp], Propp describes two constructions of snake graphs, the latter of which are unweighted analogues for the case A of the graphs $G_{T,\gamma}$ that we present in this paper. Propp assigns a snake graph to each arc in the triangulation of an $n$-gon and shows that the numbers of matchings in these graphs satisfy the Conway-Coxeter frieze pattern induced by the Ptolemy relations on the $n$-gon. In [@M] a cluster expansion for cluster algebras of classical type is given for clusters that lie in a bipartite seed, and the forthcoming work of [@MW] will concern cluster expansions for cluster algebras of classical type with principal coefficients, for an arbitary seed.
The formula for $y(M)$ given in Theorem \[thm y\] also can be formulated in terms of height functions, as found in literature such as [@EKLP] or [@ProppLattice]. We discuss this connection in Remark \[height\] of section \[sect y\].
The paper is organized as follows. In section \[sect FST\], we recall the construction of cluster algebras from surfaces of [@FST]. Section \[sect main\] contains the construction of the graph $G_{T,{\gamma}}$ and the statement of the cluster expansion formula. Section \[sect proof\] is devoted to the proof of the expansion formula. The formula for $y(M)$ and the formula for the $g$-vectors is given in section \[sect y\]. In section \[sect F-polynomial\], we present the expansion formula in terms of subgraphs and deduce a formula for the $F$-polynomials. We give an example in section \[sect example\].
The authors would like to thank Jim Propp for useful conversations related to this work.
[Cluster algebras from surfaces]{}\[sect FST\] In this section, we recall the construction of [@FST] in the case of surfaces without punctures.
Let $S$ be a connected oriented 2-dimensional Riemann surface with boundary and $M$ a non-empty set of marked points in the closure of $S$ with at least one marked point on each boundary component. The pair $(S,M)$ is called *bordered surface with marked points*. Marked points in the interior of $S$ are called *punctures*.
In this paper we will only consider surfaces $(S,M)$ such that all marked points lie on the boundary of $S$, and we will refer to $(S,M)$ simply by *unpunctured surface*.
We say that two curves in $S$ *do not cross* if they do not intersect each other except that endpoints may coincide.
An *arc* ${\gamma}$ in $(S,M)$ is a curve in $S$ such that
- the endpoints are in $M$,
- ${\gamma}$ does not cross itself,
- the relative interior of ${\gamma}$ is disjoint from $M$ and from the boundary of $S$,
- ${\gamma}$ does not cut out a monogon or a digon.
Curves that connect two marked points and lie entirely on the boundary of $S$ without passing through a third marked point are called *boundary arcs*. Hence an arc is a curve between two marked points, which does not intersect itself nor the boundary except possibly at its endpoints and which is not homotopic to a point or a boundary arc.
Each arc is considered up to isotopy inside the class of such curves. Moreover, each arc is considered up to orientation, so if an arc has endpoints $a,b\in M$ then it can be represented by a curve that runs from $a$ to $b$, as well as by a curve that runs from $b$ to $a$.
For any two arcs ${\gamma},{\gamma}'$ in $S$, let $e({\gamma},{\gamma}')$ be the minimal number of crossings of ${\gamma}$ and ${\gamma}'$, that is, $e({\gamma},{\gamma}')$ is the minimum of the numbers of crossings of arcs ${\alpha}$ and ${\alpha}'$, where ${\alpha}$ is isotopic to ${\gamma}$ and ${\alpha}'$ is isotopic to ${\gamma}'$. Two arcs ${\gamma},{\gamma}'$ are called *compatible* if $e({\gamma},{\gamma}')=0$. A *triangulation* is a maximal collection of compatible arcs together with all boundary arcs. The arcs of a triangulation cut the surface into *triangles*. Since $(S,M)$ is an unpunctured surface, the three sides of each triangle are distinct (in contrast to the case of surfaces with punctures). Any triangulation has $n+m$ elements, $n$ of which are arcs in $S$, and the remaining $m$ elements are boundary arcs. Note that the number of boundary arcs is equal to the number of marked points.
\[prop rank\] The number $n$ of arcs in any triangulation is given by the formula $n=6g+3b+m-6$, where $g$ is the genus of $S$, $b$ is the number of boundary components and $m=|M|$ is the number of marked points. The number $n$ is called the *rank* of $(S,M)$.
[@FST 2.10]
Note that $b> 0$ since the set $M$ is not empty. Table \[table 1\] gives some examples of unpunctured surfaces.
b g m surface
------- --------- --------- ---------------------------------------
1 0 n+3 polygon
1 1 n-3 torus with disk removed
1 2 n-9 genus 2 surface with disk removed
2 0 n annulus
2 1 n-6 torus with 2 disks removed
2 2 n-12 genus 2 surface with 2 disks removed
3 0 n-3 pair of pants
: Examples of unpunctured surfaces[]{data-label="table 1"}
Following [@FST], we associate a cluster algebra to the unpunctured surface $(S,M)$ as follows. Choose any triangulation $T$, let $\tau_1,\tau_2,\ldots,\tau_n$ be the $n$ interior arcs of $T$ and denote the $m$ boundary arcs of the surface by $\tau_{n+1},\tau_{n+2},\ldots,\tau_{n+m}$. For any triangle $\Delta$ in $T$ define a matrix $B^\Delta=(b^\Delta_{ij})_{1\le i\le n, 1\le j\le n}$ by $$b_{ij}^\Delta=\left\{
\begin{array}{ll}
1 & \textup{if $\tau_i$ and $\tau_j$ are sides of
$\Delta$ with $\tau_j$ following $\tau_i$ in the }
\\ &\textup{ counter-clockwise order;}\\
-1 & \textup{if $\tau_i$ and $\tau_j$ are sides of
$\Delta$ with $\tau_j$ following $\tau_i$ in the }\\
&\textup{ clockwise order;}\\
0& \textup{otherwise.}
\end{array} \right.$$ Then define the matrix $ B_{T}=(b_{ij})_{1\le i\le n, 1\le j\le n}$ by $b_{ij}=\sum_\Delta b_{ij}^\Delta$, where the sum is taken over all triangles in $T$. Note that the boundary arcs of the triangulation are ignored in the definition of $B_{T}$. Let $\tilde B_{T}=(b_{ij})_{1\le i\le 2n, 1\le j\le n}$ be the $2n\times n$ matrix whose upper $n\times n$ part is $B_{T}$ and whose lower $n\times n$ part is the identity matrix. The matrix $B_{T}$ is skew-symmetric and each of its entries $b_{ij}$ is either $0,1,-1,2$, or $-2$, since every arc $\tau$ can be in at most two triangles. An example where $b_{ij}=2 $ is given in Figure \[fig bij=2\].
Let $\mathcal{A}(\mathbf{x}_{T},\mathbf{y}_{T},B_{T})$ be the cluster algebra with principal coefficients in the triangulation $T$, that is, $\mathcal{A}(\mathbf{x}_{T},\mathbf{y}_{T},B_{T})$ is given by the seed $(\mathbf{x}_{T},\mathbf{y}_{T},B_{T})$ where $\mathbf{x}_{T}=\{x_{\tau_1},x_{\tau_2},\ldots,x_{\tau_n}\}$ is the cluster associated to the triangulation $T$, and the initial coefficient vector $\mathbf{y}_{T}=(y_1,y_2,\ldots,y_n)$ is the vector of generators of $\mathbb{P}=\textup{Trop}(y_1,y_2,\ldots,y_n)$.
For the boundary arcs we define $x_{\tau_k}=1$, $k=n+1,n+2,\ldots,n+m$.
For each $k=1,2,\ldots,n$, there is a unique quadrilateral in $T\setminus \{\tau_k\}$ in which $\tau_k$ is one of the diagonals. Let $\tau_k'$ denote the other diagonal in that quadrilateral. Define the *flip* $\mu_k T$ to be the triangulation $T\setminus\{\tau_k\}\cup\{\tau_k'\}$. The mutation $\mu_k$ of the seed ${\Sigma}_T$ in the cluster algebra $\mathcal{A}$ corresponds to the flip $\mu_k$ of the triangulation $T$ in the following sense. The matrix $\mu_k(B_T)$ is the matrix corresponding to the triangulation $\mu_k T$, the cluster $\mu_k(\mathbf{x}_T)$ is $\mathbf{x}_T\setminus\{x_{\tau_k}\}\cup
\{x_{\tau_k'}\}$, and the corresponding exchange relation is given by $$x_{\tau_k} x_{\tau_k'} = x_{\rho_1} x_{\rho_2} y^+ + x_{{\sigma}_1}
x_{{\sigma}_2} y^-,$$ where $y^+,y^-$ are some coefficients, and $\rho_1,{\sigma}_1,\rho_2,{\sigma}_2$ are the sides of the quadrilateral in which $\tau_k$ and $\tau_k'$ are the diagonals, such that $\rho_1,\rho_2$ are opposite sides and ${\sigma}_1,{\sigma}_2$ are opposite sides too.
[Expansion formula]{}\[sect main\]
In this section, we will present an expansion formula for the cluster variables in terms of perfect matchings of a graph that is constructed recursively using so-called *tiles*.
[Tiles]{}\[sect tiles\] For the purpose of this paper, a *tile* ${\overline{S}}_k$ is a planar four vertex graph with five weighted edges having the shape of two equilateral triangles that share one edge, see Figure \[figtile\].
The weight on each edge of the tile ${\overline{S}}_k$ is a single variable. The unique interior edge is called *diagonal* and the four exterior edges are called *sides* of ${\overline{S}}_k$. We shall use $S_k$ to denote the graph obtained from ${\overline{S}}_k$ by removing the diagonal.
Now let $T$ be a triangulation of the unpunctured surface $(S,M)$. If $\tau_k\in T$ is an interior arc, then $\tau_k $ lies in precisely two triangles in $T$, hence $\tau_k$ is the diagonal of a unique quadrilateral $Q_{\tau_k}$ in $T$. We associate to this quadrilateral a tile ${\overline{S}}_k$ by assigning the weight $x_k$ to the diagonal and the weights $x_a,x_b,x_c,x_d$ to the sides of ${\overline{S}}_k$ in such a way that there is a homeomorphism ${\overline{S}}_k\to
Q_{\tau_k}$ which sends the edge with weight $x_i$ to the arc labeled $\tau_i$, $i=a,b,c,d,k$, see Figure \[figtile\].
[The graph $\overline{G}_{T,{\gamma}}$]{}\[sect 1.2\] Let $T$ be a triangulation of an unpunctured surface $(S,M)$ and let ${\gamma}$ be an arc in $(S,M)$ which is not in $T$. Choose an orientation on ${\gamma}$ and let $s\in M$ be its starting point, and let $t\in M$ be its endpoint. We denote by $$p_0=s, p_1, p_2, \ldots, p_{d+1}=t$$ the points of intersection of ${\gamma}$ and $T$ in order. Let $i_1,
i_2, \ldots, i_d$ be such that $p_k$ lies on the arc $\tau_{i_k}\in T$. Note that $i_k$ may be equal to $i_j$ even if $k\ne j$. Let $\tilde S_1,\tilde S_2,\ldots,\tilde S_d $ be a sequence of tiles so that $\tilde S_k$ is isomorphic to the tile ${\overline{S}}_{i_k}$, for $k=1,2,\ldots,d$.
For $k$ from $0$ to $d$, let ${\gamma}_k$ denote the segment of the path ${\gamma}$ from the point $p_k $ to the point $p_{k+1}$. Each ${\gamma}_k$ lies in exactly one triangle ${\Delta}_k$ in $T$, and if $1\le k\le d-1$ then ${\Delta}_k$ is formed by the arcs $\tau_{i_k},
\tau_{i_{k+1}}$, and a third arc that we denote by $\tau_{[{\gamma}_k]}$.
We will define a graph $\overline{G}_{T,{\gamma}}$ by recursive glueing of tiles. Start with $\overline{G}_{T,{\gamma},1}\cong \tilde S_1$, where we orient the tile $\tilde
S_1$ so that the diagonal goes from northwest to southeast, and the starting point $p_0$ of ${\gamma}$ is in the southwest corner of $\tilde
S_1$. For all $k=1,2,\ldots,d-1$ let $\overline{G}_{T,{\gamma},k+1}$ be the graph obtained by adjoining the tile $\tilde
S_{k+1} $ to the tile $\tilde S_k$ of the graph $\overline{G}_{T,{\gamma},k}$ along the edge weighted $x_{[{\gamma}_k]}$, see Figure \[figglue\]. We always orient the tiles so that the diagonals go from northwest to southeast. Note that the edge weighted $x_{[{\gamma}_k]}$ is either the northern or the eastern edge of the tile $\tilde S_k$.
Finally, we define $\overline{G}_{T,{\gamma}}$ to be $\overline{G}_{T,{\gamma},d}$.
Let $G_{T,{\gamma}}$ be the graph obtained from $\overline{G}_{T,{\gamma}}$ by removing the diagonal in each tile, that is, $G_{T,{\gamma}} $ is constructed in the same way as $\overline{G}_{T,{\gamma}}$ but using tiles $S_{i_k}$ instead of ${\overline{S}}_{i_k}$.
A *perfect matching* of a graph is a subset of the edges so that each vertex is covered exactly once. We define the weight $w(M)$ of a perfect matching $M$ to be the product of the weights of all edges in $M$.
[Cluster expansion formula]{}\[sect cluster expansion formula\] Let $(S,M)$ be an unpunctured surface with triangulation $T$, and let $\mathcal{A}=\mathcal{A}(\mathbf{x}_T,\mathbf{y}_T,B)$ be the cluster algebra with principal coefficients in the initial seed $(\mathbf{x}_T,\mathbf{y}_T,B)$ defined in section \[sect FST\]. Each cluster variable in $\mathcal{A}$ corresponds to an arc in $(S,M)$. Let $x_{\gamma}$ be an arbitrary cluster variable corresponding to an arc ${\gamma}$. Choose an orientation of ${\gamma}$, and let $\tau_{i_1}$, $\tau_{i_2} \dots,
\tau_{i_d}$ be the arcs of the triangulation that are crossed by ${\gamma}$ in this order, with multiplicities possible. Let $G_{T,{\gamma}}$ be the graph constructed in section \[sect 1.2\].
\[thm main\] With the above notation $$x_{\gamma}= \sum_M \frac{w(M)\,y(M)}{x_{i_1}x_{i_2}\ldots x_{i_d}},$$ where the sum is over all perfect matchings $M$ of $G_{T,{\gamma}}$, $w(M)$ is the weight of $M$, and $y(M)$ is a monomial in $\mathbf{y}_T$.
The proof of Theorem \[thm main\] will be given in Section \[sect proof\].
[Proof of Theorem \[thm main\]]{}\[sect proof\] Throughout this section, $T$ is a triangulation of an unpunctured surface $(S,M)$, ${\gamma}$ is an arc in $S$ with a fixed orientation, and $s\in M$ is its starting point and $t\in M$ is its endpoint. Moreover, $ p_0=s, p_1, p_2, \ldots, p_{d+1}=t$ are the points of intersection of ${\gamma}$ and $T$ in order, and $i_1,
i_2, \ldots, i_d$ are such that $p_k$ lies on the arc $\tau_{i_k}\in T$.
[Complete $(T,{\gamma})$-paths]{} Following [@ST], we will consider paths ${\alpha}$ in $S$ that are concatenations of arcs in the triangulation $T$, more precisely, ${\alpha}=
({\alpha}_1,{\alpha}_2,\ldots,{\alpha}_{\ell({\alpha})})$ with ${\alpha}_i \in T$, for $i=1,2,\ldots, \ell({\alpha})$ and the starting point of ${\alpha}_i$ is the endpoint of ${\alpha}_{i-1}$. Such a path is called a *$T$-path*.
We call a $T$-path ${\alpha}=({\alpha}_1,{\alpha}_2,\dots ,
{\alpha}_{\ell({\alpha})}) $ a *complete $(T,\gamma)$-path* if the following axioms hold:
- The even arcs are precisely the arcs crossed by ${\gamma}$ in order, that is, ${\alpha}_{2k}=\tau_{i_k}$.
- For all $k=0,1,2,\ldots,d$, the segment ${\gamma}_k$ is homotopic to the segment of the path ${\alpha}$ starting at the point $p_k$ following ${\alpha}_{2k},{\alpha}_{2k+1}$ and ${\alpha}_{2k+2}$ until the point $p_{k+1}$.
We define the Laurent monomial $x({\alpha})$ of the complete $(T,{\gamma})$-path ${\alpha}$ by $$x({\alpha})=\prod_{i \textup{ odd}} x_{{\alpha}_i}\prod_{i \textup{ even}} x_{{\alpha}_i}^{-1}.$$
- Every complete $(T,{\gamma})$-path starts and ends at the same point as ${\gamma}$, because of .
- Every complete $(T,{\gamma})$-path has length $2d+1$.
- For all arcs $\tau$ in the triangulation $T$, the number of times that $\tau$ occurs as $\alpha_{2k}$ is exactly the number of crossings between ${\gamma}$ and $\tau$.
- In contrast to the ordinary $(T,{\gamma})$-paths defined in [@ST], complete $(T,{\gamma})$-paths allow backtracking.
- The denominator of the Laurent monomial $x({\alpha})$ is equal to $x_{i_1}x_{i_2}\cdots x_{i_d}$.
[Universal cover]{}\[sect universal cover\] Let $\pi:\tilde S \to S$ be a universal cover of the surface $S$, and let $\tilde M=\pi^{-1}(M)$ and $\tilde T=\pi^{-1}(T)$.
Choose $\tilde s \in \pi^{-1}(s)$. There exists a unique lift $\tilde
{\gamma}$ of ${\gamma}$ starting at $\tilde s$. Then $\tilde {\gamma}$ is the concatenation of subpaths $\tilde {\gamma}_0,\tilde {\gamma}_1,\ldots,\tilde
{\gamma}_{d+1}$ where $\tilde {\gamma}_k$ is a path from a point $\tilde p_k$ to a point $\tilde p_{k+1}$ such that $\tilde {\gamma}_k$ is a lift of ${\gamma}_k$ and $\tilde p_k\in\pi^{-1}(p_k)$, for $k=0,1,\ldots, d+1$. Let $\tilde t=\tilde p_{d+1}\in \pi^{-1}(t)$.
For $k$ from $1$ to $d$, let $\tilde \tau_{i_k}$ be the unique lift of $\tau_{i_k}$ running through $\tilde p_k$ and let $\tilde
\tau_{[{\gamma}_k]}$ be the unique lift of $\tau_{[{\gamma}_k]}$ that is bounding a triangle in $\tilde S$ with $\tilde \tau_{i_k}$ and $\tilde
\tau_{i_{k+1}}$. Each $\tilde {\gamma}_k$ lies in exactly one triangle $\tilde {\Delta}_k$ in $\tilde T$. Let $\tilde S({\gamma})\subset \tilde S$ be the union of the triangles $\tilde {\Delta}_0, \tilde {\Delta}_1,\ldots,\tilde {\Delta}_{d+1}$ and let $\tilde
M({\gamma})=\tilde M\cap \tilde S({\gamma})$ and $\tilde T({\gamma})=\tilde T\cap
\tilde S({\gamma})$. Then $(\tilde S({\gamma}),\tilde M({\gamma}))$ is a simply connected unpunctured surface of which $\tilde T({\gamma}) $ is a triangulation. This triangulation $\tilde T({\gamma})$ consists of arcs $\tilde
\tau_{i_k},\,\tilde\tau_{[{\gamma}_k]}$ with $k=1,2,\ldots,d,$ and two arcs incident to $\tilde s$ and two arcs incident to $\tilde t$.
The underlying graph of $\tilde T({\gamma})$ is the graph with vertex set $\tilde M({\gamma})$ and whose set of edges consists of the (unoriented) arcs in $\tilde T({\gamma})$.
By [@S2 Section 5.5], we can compute the Laurent expansion of $x_{{\gamma}}$ using complete $(\tilde T({\gamma}),\tilde {\gamma})$-paths in $(\tilde S({\gamma}),\tilde M({\gamma}))$.
[Folding]{}\[sect folding\] The graph $\overline{G}_{T,{\gamma}}$ was constructed by glueing tiles $\tilde S_{k+1}$ to tiles $\tilde S_k$ along edges with weight $x_{[{\gamma}_k]}$, see figure \[figglue\]. Now we will fold the graph along the edges weighted $x_{[{\gamma}_k]}$, thereby identifying the two triangles incident to $x_{[{\gamma}_k]}$, $k=1,2,\ldots,d-1$.
To be more precise, the edge with weight $x_{[{\gamma}_k]}$, that lies in the two tiles $\tilde S_{k+1}$ and $\tilde S_k$, is contained in precisely two triangles ${\Delta}_{k}$ and ${\Delta}_k'$ in $\overline{G}_{T,{\gamma}}$: ${\Delta}_{k}$ lying inside the tile $\tilde S_{k}$ and ${\Delta}_k'$ lying inside the tile $\tilde S_{k+1}$. Both ${\Delta}_{k}$ and ${\Delta}_k'$ have weights $x_{[{\gamma}_k]}$, $x_k$, $x_{k+1}$, but opposite orientations. Cutting $\overline{G}_{T,{\gamma}}$ along the edge with weight $x_{[{\gamma}_k]}$, one obtains two connected components. Let $R_k$ be the component that contains the tile $\tilde S_k$ and $R_{k+1}$ the component that contains $\tilde S_{k+1}$.
The *folding of the graph $\overline{G}_{T,{\gamma}}$ along $x_{[{\gamma}_k]}$* is the graph obtained by flipping $R_{k+1}$ and then glueing it to $R_k$ by identifying the two triangles ${\Delta}_k $ and ${\Delta}_k'$.
The graph obtained by consecutive folding of $\overline{G}_{T,{\gamma}}$ along all edges with weight $x_{[{\gamma}_k]}$ for $k=1,2,\ldots,d-1$, is isomorphic to the underlying graph of the triangulation $\tilde T({\gamma})$ of the unpunctured surface $(\tilde S({\gamma}),\tilde M({\gamma}))$. Indeed, there clearly is a bijection between the triangles in both graphs, and, in both graphs the way the triangles are glued together is uniquely determined by ${\gamma}$. Note that the two graphs may have opposite orientations.
We obtain a map that we call the *folding map* $$\begin{array}{rcccccc}
\phi & : & \left\{
\genfrac{}{}{0pt}{}{\textup{perfect matchings} }
{\textup{in $G_{T,{\gamma}}$}}
\right\}
&\to & \left\{ \genfrac{}{}{0pt}{}{\textup{complete $(\tilde T({\gamma}),\tilde
{\gamma})$-paths}} {\textup{in $(\tilde S({\gamma}),\tilde M({\gamma}))$}}\right\}
\\
\\
&& M&\mapsto &\tilde {\alpha}_M
\end{array}$$ as follows. First we associate a path ${\alpha}_M$ in $\overline G_{T,{\gamma}}$ to the matching $M$, by inserting a diagonal between any two consecutive edges of the perfect matching. More precisely, ${\alpha}_M$ is the path starting at $s$ going along the unique edge of $M$ that is incident to $s$, then going along the diagonal of the first tile $\tilde S_1$, then along the unique edge of $M$ that is incident to the endpoint of that diagonal, and so forth.
Since $M$ has cardinality $d+1$, the path ${\alpha}_M$ consists of $2d+1$ edges, thus ${\alpha}=({\alpha}_1 ,{\alpha}_2,\ldots,{\alpha}_{2d+1})$. Now we define $\tilde{\alpha}_M=(\tilde{\alpha}_1,\tilde{\alpha}_2,\ldots,\tilde{\alpha}_{2d+1})$ by folding the path ${\alpha}_M$. Thus, if $M=\{{\beta}_1,{\beta}_3,\ldots,{\beta}_{2d-1},{\beta}_{2d+1}\}$, where the edges are ordered according to ${\gamma}$, then $\phi(M) = (\tilde
{\alpha}_1,\tilde {\alpha}_2,\ldots,\tilde {\alpha}_{2d+1})$, where $\tilde
{\alpha}_{2k+1}$ is the image of ${\beta}_{2k+1}$ under the folding and $\tilde
{\alpha}_{2k}=\tau_{i_k}$ is the arc crossing ${\gamma}$ at $p_k$. Then $\phi(M)$ satisfies the axiom (T1) by construction. Moreover, $\phi(M)$ satisfies the axiom (T2), because, for each $k=0,1,\ldots,d$, the segment of the path $\phi(M) $, which starts at the point $p_k$ following $\tilde {\alpha}_{2k}, \tilde {\alpha}_{2k+1}$ and $\tilde{\alpha}_{2k+2}$ until the point $p_{k+1}$, is homotopic to the segment ${\gamma}_k$, since both segments lie in the simply connected triangle ${\Delta}_k$ formed by $\tau_{i_k},\tau_{i_{k+1}}$ and $\tau_{[{\gamma}_k]}$. Therefore, the folding map $\phi$ is well defined.
Note that it is possible that $\tilde {\alpha}_k, \tilde {\alpha}_{k+1}$ is backtracking, that is, $\tilde {\alpha}_k$ and $\tilde
{\alpha}_{k+1}$ run along the same arc $\tilde\tau \in \tilde T({\gamma})$.
[Unfolding the surface]{} Let ${\alpha}$ be a boundary arc in $(\tilde S({\gamma}),\tilde M({\gamma}))$ that is not adjacent to $\tilde s$ and not adjacent to $\tilde t$. Then there is a unique triangle $\Delta$ in $\tilde T({\gamma})$ in which ${\alpha}$ is a side. The other two sides of $\Delta$ are two consecutive diagonals, which we denote by $\tilde\tau_j$ and $\tilde\tau_{j+1}$, see Figure \[figcomplete\].
By cutting the underlying graph of $\tilde T({\gamma})$ along $\tilde\tau_j$, we obtain two pieces. Let $R_{j+1}$ denote the piece that contains ${\alpha},
\tilde\tau_{j+1}$ and $t$. Similarly, cutting $(\tilde S({\gamma}),\tilde
M({\gamma}))$ along $\tilde\tau_{j+1}$, we obtain two pieces, and we denote by $R_{j}$ the piece that contains $s,\tilde\tau_{j}$ and ${\alpha}$.
The *graph obtained by unfolding along ${\alpha}$* is the graph obtained by flipping $R_{j}$ and then glueing it to $R_{j+1}$ along ${\alpha}$. In this new graph, we label the edge of $R_j$ that had the label $\tilde\tau_{j+1}$ by $\tilde\tau_{j+1}^b$ and the edge of $R_{j+1}$ that had the label $\tilde\tau_{j}$ by $\tilde\tau_{j}^b$, indicating that these edges are on the boundary of the new graph, see Figure \[figcomplete\].
\[lem unfold\] The graph obtained by repeated unfolding of the underlying graph of $\tilde T({\gamma})$ along all boundary edges not adjacent to $s$ or $t$ is isomorphic to the graph $\overline{G}_{T,{\gamma}}$. Moreover, for each unfolding along an edge ${\alpha}$, the edges labeled $\tilde\tau_j^b, \tilde\tau_{j+1}^b$ are on the boundary of $\overline{G}_{T,{\gamma}}$ and carry the weights $x_j, x_{j+1}$ respectively, the edges labeled $\tilde\tau_j,\tilde\tau_{j+1}$ are diagonals in $\overline{G}_{T,{\gamma}}$ and carry the weights $x_j, x_{j+1}$ respectively, and ${\alpha}$ is an interior edge of $\overline{G}_{\gamma}$ that is not a diagonal and carries the weight $x_{[{\gamma}_j]}$.
This follows from the construction.
[Unfolding map]{} We define a map $$\begin{array}{rcl}
\{\textup{complete } (\tilde T({\gamma}),\tilde {\gamma})-\textup{paths}\} &\to &
\{\textup{perfect matchings of } G_{T,{\gamma}}\}\\
\tilde{\alpha}=(\tilde{\alpha}_1,\tilde{\alpha}_2,\ldots,\tilde{\alpha}_{2d+1}) &\mapsto &
M_{\tilde {\alpha}}
=\{{\beta}_1,{\beta}_3,{\beta}_5,\ldots,{\beta}_{2d+1}\}
\end{array}$$
where ${\beta}_1=\tilde{\alpha}_1$, ${\beta}_{2d+1}=\tilde{\alpha}_{2d+1}$ and $${\beta}_{2k+1}=\left\{
\begin{array}{ll}
\tilde{\alpha}_{2k+1}&\textup{if $\tilde{\alpha}_{2k+1}$ is a boundary arc in $\tilde T({\gamma})$,}\\
\tilde\tau_j^b &\textup{if $\tilde{\alpha}_{2k+1}=\tilde\tau_j$ is a diagonal in $\tilde T({\gamma})$.}
\end{array} \right.$$
We will show that this map is well-defined. Suppose ${\beta}_{2k+1}$ and ${\beta}_{2\ell+1}$ have a common endpoint $x$. Then ${\alpha}_{2k+1}$ and ${\alpha}_{2\ell+1}$ have a common endpoint $y$ in $(\tilde S({\gamma}),\tilde
M({\gamma}))$ and the two edges are not separated in the unfolding described in Lemma \[lem unfold\]. Consequently, there is no triangle in $\tilde T({\gamma})$ that is contained in the subpolygon spanned by ${\alpha}_{2k+1}$ and ${\alpha}_{2\ell+1}$, hence ${\alpha}_{2k+1}$ is equal to ${\alpha}_{2l+1}$. This implies that every arc in the subpath $({\alpha}_{2k+1},{\alpha}_{2k+2}\ldots {\alpha}_{2\ell+1})$ is equal to the same diagonal $\tilde\tau_j$, and the only way this can happen is when $\ell=k+1$ and $({\alpha}_{2k+1},{\alpha}_{2k+2}\ldots
{\alpha}_{2\ell+1})=(\tilde\tau_j,\tilde\tau_j,\tilde\tau_j)$ and both endpoints of $\tilde \tau_j$ are incident to an interior arc other than $\tilde \tau_j$. In this case, $\tilde\tau_j$ bounds the two triangles $\tilde\tau_{j-1},\tilde\tau_{j},\tilde\tau_{[{\gamma}_{j-1}]}$ and $\tilde\tau_{j},\tilde\tau_{j+1},\tilde\tau_{[{\gamma}_{j}]}$ in $\tilde T({\gamma})$. Unfolding along $\tilde\tau_{[{\gamma}_{j-1}]}$ and $\tilde\tau_{[{\gamma}_{j}]}$ will produce edges ${\beta}_{2k+1}$ and ${\beta}_{2\ell+1}$ that are not adjacent, see Figure \[fig 2\].
This shows that no vertex of ${G}_{T,{\gamma}}$ is covered twice in $M_{\alpha}$.
To show that every vertex of $G_{T,{\gamma}}$ is covered in $M_{\alpha}$, we use a counting argument. Indeed, the number of vertices of $G_{T,{\gamma}}$ is $2(d+1)$, and, on the other hand, $2d+1$ is the length of ${\alpha}$, since ${\alpha}$ is complete, and thus $M_{\tilde {\alpha}}$ has $d+1$ edges. The statement follows since every ${\beta}_j\in M_{\tilde{\alpha}}$ has two distinct endpoints. This shows that $M_{\tilde{\alpha}} $ is a perfect matching and our map is well-defined.
\[lem bijections\] The unfolding map $\tilde{\alpha}\mapsto M_{\tilde{\alpha}}$ is the inverse of the folding map $M\mapsto \tilde{\alpha}_M$. In particular, both maps are bijections.
Let $\tilde {\alpha}=(\tilde {\alpha}_1,\tilde {\alpha}_2,\ldots,\tilde
{\alpha}_{2d+1})$ be a complete $(\tilde T({\gamma}),\tilde {\gamma})$-path. Then $\tilde {\alpha}_{M_{\tilde {\alpha}}}=({\alpha}_1,{\alpha}_2,\ldots,{\alpha}_{2d+1})$ where ${\alpha}_{2k+1}$ is the image under folding of the arc $\tilde\tau_j^b$ if $\tilde{\alpha}_{2k+1}=\tilde\tau_j$ is a diagonal in $\tilde T({\gamma})$ or, otherwise, the image under the folding of the arc $\tilde
{\alpha}_{2k+1}$. Thus ${\alpha}_{2k+1}=\tilde {\alpha}_{2k+1}$. Moreover, ${\alpha}_{2k}=\tau_{i_k}=\tilde{\alpha}_{2k}$, and thus $\tilde {\alpha}_{M_{\tilde {\alpha}}}=\tilde {\alpha}$.
Conversely, let $M=\{{\beta}_1,{\beta}_3,\ldots,{\beta}_{2d-1},{\beta}_{2d+1}\} $ be a perfect matching of $G_{T,{\gamma}}$. Then $M_{\tilde
{\alpha}_M}=
\{\tilde{\beta}_1,\tilde{\beta}_3,\ldots,\tilde{\beta}_{2d-1},\tilde{\beta}_{2d+1}\} $ where $$\begin{array}{rcl}
\tilde{\beta}_{2k+1}&=&\left\{
\begin{array}{rl}
\tilde{\alpha}_{2k+1} &\textup{if $\tilde{\alpha}_{2k+1} $ is a boundary arc,}\\ \\
\tilde\tau^b_{j} &\textup{if $\tilde{\alpha}_{2k+1}=\tilde\tau_j $ is a diagonal}
\end{array} \right. \\
\\
&=&\left\{
\begin{array}{rl}
\tilde\tau_{[{\gamma}_j]} &\textup{if ${\beta}_{2k+1}=\tilde\tau_{[{\gamma}_j]} $,}\\ \\
\tilde\tau^b_{j} &\textup{if ${\beta}_{2k+1}=\tilde\tau_j^b .$}
\end{array} \right.\end{array}$$ Hence $M_{\tilde {\alpha}_M}=M$.
Combining Lemma \[lem bijections\] with the results of [@S2], we obtain the following Theorem.
\[thm bijections\] There is a bijection between the set of perfect matchings of the graph $G_{T,{\gamma}}$ and the set of complete $(T,{\gamma})$-paths in $(S,M)$ given by $M\mapsto \pi(\tilde {\alpha}_M)$, where $\tilde {\alpha}_M$ is the image of $M$ under the folding map and $\pi$ is induced by the universal cover $\pi:\tilde S\to S$. Moreover, the numerator of the Laurent monomial $x(\pi(\tilde {\alpha}_M))$ of the complete $(T,{\gamma})$-path $\pi(\tilde {\alpha}_M)$ is equal to the weight $ w(M)$ of the matching $M$.
The map in the Theorem is a bijection, because it is the composition of the folding map, which is a bijection, by Lemma \[lem bijections\], and the map $\pi$, which is a bijection, by [@S2 Lemma 5.8]. The last statement of the Theorem follows from the construction of the graph $G_{T,{\gamma}}$.
[Proof of Theorem \[thm main\]]{}\[ssect proof\] It has been shown in [@S2 Theorem 3.2] that
$$\label{eq 91}
x_{\gamma}= \sum_{{\alpha}} x({\alpha})\,y({\alpha}),$$
where the sum is over all complete $(T,{\gamma})$-paths ${\alpha}$ in $(S,M)$, $y({\alpha})$ is a monomial in $\mathbf{y}_T$, and
$$\label{eq 92}
x({\alpha})=\prod_{k \ \textup{odd}} x_{{\alpha}_k} \prod_{k \ \textup{even}} x_{{\alpha}_k}^{-1}.$$
Applying Theorem \[thm bijections\] to equation (\[eq 91\]) yields
$$\label{eq 93}
x_{\gamma}= \sum_M w(M)\,y(M) (x_{i_1}x_{i_2}\cdots x_{i_d})^{-1},$$
where the sum is over all perfect matchings $M$ of $G_{T,{\gamma}}$, $w(M)$ is the weight of the matching and $y(M)=y(\pi(\tilde {\alpha}_M))$. This completes the proof of Theorem \[thm main\].
[A formula for $y(M)$]{}\[sect y\] In this section, we give a description of the coefficients $y(M)$ in terms of the matching $M$. First, we need to recall some results from [@S2].
Recall that $T$ is a triangulation of the unpunctured surface $(S,M)$, ${\gamma}$ is an arc in $(S,M)$ that crosses $T$ exactly $d$ times, we have fixed an orientation for ${\gamma}$ and denote by $s=p_0,p_1,\ldots,p_d,p_{d+1}=t$ the intersection points of ${\gamma}$ and $T$ in order of occurrence on ${\gamma}$. Let $i_1,i_2,\ldots,i_d$ be such that $p_k$ lies on the arc $\tau_{i_k}\in T$, for $k=1,2,\ldots,d$.
For $k=0,1,\ldots,d$, let ${\gamma}_k$ denote the segment of the path ${\gamma}$ from the point $p_k$ to the point $p_{k+1}$. Each ${\gamma}_k$ lies in exactly one triangle ${\Delta}_k$ in $T$. If $1\le k\le d-1$, the triangle ${\Delta}_k$ is formed by the arcs $\tau_{i_k},
\tau_{i_{k+1}}$ and a third arc that we denote by $\tau_{[{\gamma}_k]}$.
The orientation of the surface $S$ induces an orientation on each of these triangles in such a way that, whenever two triangles ${\Delta},{\Delta}'$ share an edge $\tau$, then the orientation of $\tau $ in ${\Delta}$ is opposite to the orientation of $\tau $ in ${\Delta}'$, There are precisely two such orientations, we assume without loss of generality that we have the “clockwise orientation”, that is, in each triangle ${\Delta}$, going around the boundary of ${\Delta}$ according to the orientation of ${\Delta}$ is clockwise when looking at it from outside the surface.
Let ${\alpha}$ be a complete $(T,{\gamma})$-path. Then ${\alpha}_{2k}=\tau_{i_k}$ is a common edge of the two triangles ${\Delta}_{k-1}$ and ${\Delta}_k$. We say that ${\alpha}_{2k}$ is ${\gamma}$-*oriented* if the orientation of ${\alpha}_{2k}$ in the path ${\alpha}$ is the same as the orientation of $\tau_{i_k}$ in the triangle ${\Delta}_k$, see Figure \[fig gammaoriented\].
It is shown in [@S2 Theorem 3.2] that $$\label{eq 28}
y({\alpha})= \prod_{k:
{\alpha}_{2k}\textup{ is ${\gamma}$-oriented}} y_{i_k}.$$
Each perfect matching $M$ of $G_{T,{\gamma}}$ induces a path ${\alpha}_M$ in $\overline{G}_{T,{\gamma}}$ as in the construction of the folding map in section \[sect folding\]. The even arcs of ${\alpha}_M$ are the diagonals of the graph $\overline{G}_{T,{\gamma}}$. We say that an even arc of ${\alpha}_M$ has upward orientation if ${\alpha}_M$ is directed from southeast to northwest on that even arc, otherwise we say that the arc has downward orientation. If going upward on the first even arc of ${\alpha}_M$ is ${\gamma}$-oriented then we have that the $(2k)$-th arc of $\pi(\tilde{{\alpha}}_M)$ is ${\gamma}$-oriented if and only if the $2k$-th arc of ${\alpha}_M$ is upward if $k$ is odd, and downward if $k$ is even. If, on the other hand, going downward on the first even arc of ${\alpha}_M$ is ${\gamma}$-oriented then we have that the $(2k)$-th arc of $\pi(\tilde{{\alpha}}_M)$ is ${\gamma}$-oriented if and only if the $2k$-th arc of ${\alpha}_M$ is downward if $k$ is odd, and upward if $k$ is even.
There are precisely two perfect matchings $M_+$ and $M_-$ of $G_{T,{\gamma}}$ that contain only boundary edges of $G_{T,{\gamma}}$. The orientations of the even arcs in both of the induced $(T,{\gamma})$-paths ${\alpha}_{M_+}$ and ${\alpha}_{M_-}$ are alternatingly upward and downward, thus for one of the two paths, say $M_+$, each even arc of $\pi(\tilde{\alpha}_{M_+})$ is ${\gamma}$-oriented, whereas for $M_-$ none of the even arcs of $\pi(\tilde{\alpha}_{M_-})$ is ${\gamma}$-oriented. That is, $y(M_-)=1$ and $y(M_+)=y_{i_1}y_{i_2}\cdots y_{i_d}$.
For an arbitrary perfect matching $M$, the coefficient $y(M)$ is determined by the set of edges of the symmetric difference $M_-\ominus M =(M_-\cup M)\setminus (M_-\cap M)$ as follows.
\[thm y\] The set $M_-\ominus M$ is the set of boundary edges of a (possibly disconnected) subgraph $G_M$ of $G_{T,{\gamma}}$ which is a union of tiles $$G_M =\cup_{j\in J} S_j.$$ Moreover, $$y(M)=\prod_{j\in J} y_{i_j}$$
Choose any edge $e_1$ and either endpoint in $M_-\setminus(M_-\cap M)$, and walk along that edge until its other endpoint. Since $M$ is a perfect matching, this endpoint is incident to an edge $e_2$ in in $M$, which is different from $e_1$ and, hence, not in $M_-$. Thus $e_2\in M\setminus(M_-\cap M)$. Now walk along $e_2$ until its other endpoint. This endpoint is incident to an edge $e_3$ in $M_-$ which is different from $e_2$, and, hence, not in $M$. Thus $e_3\in M_-\setminus(M_-\cap M)$. Continuing this way, we construct a sequence of edges in $M_-\ominus M$. Since $G_{T,{\gamma}}$ has only finitely many edges, this sequence must become periodic after a certain number of steps; thus there exist $p,N$ such that $e_k=e_{k+p}$ for all $k\ge N$.
We will show that one can take $N=1$. Suppose to the contrary that $N\ge 2$ is the smallest integer such that $e_k=e_{k+p}$ for all $k\ge N$. Then $e_{N-1},e_N$ and $e_{N+p-1}$ share a common endpoint. But $e_{N-1},e_N$ and $e_{N+p-1}$ are elements of the union of two perfect matchings, hence $e_{N-1}=e_{N+p-1}$, contradicting the minimality of $N$.
Therefore the sequence $e_1, e_2,\ldots, e_p$ in $M\ominus M_-$ is the set of boundary edges of a connected subgraph of $G_{T,{\gamma}}$ which is a union of tiles.
The graph $G_M$ is the union of these connected subgraphs and, hence, it is a union of tiles. Let $H$ be a connected component of $G_M$. There are precisely two perfect matchings $M_-(H)$ and $M_+(H)$ of $H$ that consist only of boundary edges of $H$. Clearly, these two matchings are $M_-\cap E(H)$ and $M\cap E(H)$, where $E(H)$ is the set of edges of the graph $H$. Therefore, in each tile of $H$, the orientation of the diagonal in ${\alpha}_{M_-}$ and ${\alpha}_M$ are opposite. The restrictions of $M_-$ and $M$ to $E(G_{T,{\gamma}})\setminus E(G_M)$ are identical, hence in each tile of $G_{T,{\gamma}} \setminus G_M$, the orientations of the diagonal in ${\alpha}_{M_-}$ and ${\alpha}_M$ are equal. It follows from equation (\[eq 28\]) that $y(M)=\prod_{j\in J} y_{i_j}$.
It has been shown in [@FZ4] that, for any cluster variable $x_{\gamma}$ in $\mathcal{A}$, its Laurent expansion in the initial seed $(\mathbf{x}_T,\mathbf{y}_T,B_T)$ is homogeneous with respect to the grading given by $\textup{deg}(x_i)=\mathbf{e}_i$ and $\textup{deg}(y_i)=B_T\mathbf{e}_i$, where $\mathbf{e}_i=(0,\ldots,0,1,0,\ldots,0)^T
\in\mathbb{Z}^n$ with $1$ at position $i$. By definition, the *$g$-vector* $g_{\gamma}$ of a cluster variable $x_{\gamma}$ is the degree of its Laurent expansion with respect to this grading.
\[cor g\] The $g$-vector $g_{\gamma}$ of $x_{\gamma}$ is given by $$g_{\gamma}= \deg \frac{w(M_-)}{x_{i_1}x_{i_2}\cdots x_{i_d}}.$$
This follows from the fact that $y(M_-)=1$.
\[height\] The formula for $y(M)$ can also be phrased in terms of height functions. As described in section 3 of [@ProppLattice], one way to define the height function on the faces of a bipartite planar graph $G$, covered by a perfect matching $M$, is to superimpose each matching with the fixed matching $M_{\hat{0}}$ (the unique matching of minimal height). In the case where $G$ is a snake graph, we take $M_{\hat{0}}$ to be $M_-$, one of the two matchings of $G$ only involving edges on the boundary. Color the vertices of $G$ black and white so that no two adjacents vertices have the same color. In this superposition, we orient edges of $M$ from black to white, and edges of $M_-$ from white to black. We thereby obtain a spanning set of cycles, and removing the cycles of length two exactly corresponds to taking the symmetric difference $M \ominus M_-$. We can read the resulting graph as a relief-map, in which the altitude changes by $+1$ or $-1$ as one crosses over a contour line, according to whether the counter-line is directed clockwise or counter-clockwise. By this procedure, we obtain a height function $h_M : F(G) \rightarrow \mathbb{Z}$ which assigns integers to the faces of graph $G$. When $G$ is a snake graph, the set of faces $F(G)$ is simply the set of tiles $\{S_j\}$ of $G$. Comparing with the defintion of $y(M)$ in Theorem \[thm y\], we see that $$y(M) = \prod_{S_j \in F(G)} y_j^{h_M(j)}.$$ An alternative defintion of height functions comes from [@EKLP] by translating the matching problem into a domino tiling problem on a region colored as a checkerboard. We imagine an ant starting at an arbitary vertex at height $0$, walking along the boundary of each domino, and changing its height by $+1$ or $-1$ as it traverses the boundary of a black or white square, respectively. The values of the height function under these two formulations agree up to scaling by four.
[Cluster expansion without matchings]{}\[sect F-polynomial\] In this section, we give a formula for the cluster expansion of $x_{\gamma}$ in terms of the graph $G_{T,{\gamma}}$ only.
For any graph $H$, let $c(H)$ be the number of connected components of $H$. Let $E(H)$ be the set of edges of $H$, and denote by $\partial H$ the set of boundary edges of $H$. Define $\mathcal{H}_k$ to be the set of all subgraphs $H$ of $G_{T,{\gamma}}$ such that $H$ is a union of $k$ tiles $H=S_{j_1}\cup\cdots\cup S_{j_k}$ and the number of edges of $M_-$ that are contained in $H$ is equal to $k+c(H)$. For $H\in \mathcal{H}_k$, let $$y(H)=\prod_{S_{i_j} \textup{\,tile\,in\,}H} y_{i_j}.$$
\[thm F\] The cluster expansion of the cluster variable $x_{\gamma}$ is given by $$x_{\gamma}=\sum_{k=0}^d \ \sum_{H\in \mathcal{H}_k} \frac{w(\partial H\ominus M_-)\,y(H)}{x_{i_1} x_{i_2}\cdots x_{i_d}},$$
It follows from the theorems \[thm main\] and \[thm y\] that $$x_{\gamma}=\sum_{k=1}^d\quad \sum_{M:\mid y(M)\mid =k}\quad \frac{w(M)\,y(G_M)}{x_{i_1} x_{i_2}\cdots x_{i_d}},$$ where $| y(M)| $ is the number of tiles in $G_M$. We will show that for all $k$, the map $M\mapsto G_M$ is a bijection between the set of perfect matchings $M$ of $G_{T,{\gamma}}$ such that $| y(M)| =k$ and the set $\mathcal{H}_k$.
- The map is well-defined. Clearly, $G_M$ is the union of $k$ tiles. Moreover, $E(G_M)\cap M_-$ is a perfect matching of $G_M$, since $M_-$ consists of every other boundary edge of $G_{T,{\gamma}}$. Thus the cardinality of $(E(G_M)\cap M_-)$ is half the number of vertices of $G_M$, which is equal to $2k+2c(G_M)$. Therefore, the cardinality of $(E(G_M)\cap M_-)$ is $k+c(G_M)$ and $G_M\in \mathcal{H}_k$.
- The map is injective, since two graphs $G_M,G_{M'}$ are equal if and only if their boundaries are.
- The map is surjective. Let $H=S_{j_1}\cup\cdots\cup S_{j_k}$ such that the cardinality of $E(H)\cap M_-$ equals $k+c(H)$. The boundary of $H$ consists of $2k+2+2c(H)$ edges, half of which lie in $M_-$. As in the proof of Theorem \[thm y\], let $M_-(H)=E(H)\cap M_-$ and $M_+(H)$ be the two perfect matchings of $H$ that consist of boundary edges only. Let $M = M_+(H)\cup (M_-\setminus M_-(H))$. Then $M$ is a perfect matching of $G_{T,{\gamma}}$ such that $G_M=H$, and moreover, $|y(M)|$ is equal to the number of tiles in $H$, which is $k$. Thus the map is surjective.
Now the boundary edges of $G_M$ are precisely the elements of $M\ominus M_-$, which implies that $\partial(G_M)\ominus M_- = (M\ominus M_-)\ominus M_- = M\ominus (M_-\ominus M_-) =M$. Therefore $w(M)=w(\partial(G_M)\ominus M_-)$, and this completes the proof.
The $F$-polynomial of ${\gamma}$ is given by $$F_{\gamma}=\sum_{k=0}^d \ \sum_{H\in\mathcal{H}_k} y(H).$$
[Example]{}\[sect example\]
We illustrate Theorem \[thm main\], Theorem \[thm y\] and Theorem \[thm F\] in an example. Let $(S,M)$ be the annulus with two marked points on each of the two boundary components, and let $T=\{\tau_1,\ldots,\tau_8\}$ be the triangulation shown in Figure \[figex1\]. Let ${\gamma}$ be the dotted arc in that figure. It has $d=6$ crossings with the triangulation. The sequence of crossed arcs $\tau_{i_1},\ldots,\tau_{i_6}$ is $\tau_1,\tau_2,\tau_3,\tau_4,\tau_1,\tau_2$, and the corresponding segments ${\gamma}_0,\ldots,{\gamma}_6$ of the arc ${\gamma}$ are labeled in the figure. Moreover, $\tau_{[{\gamma}_1]}=\tau_6$, $\tau_{[{\gamma}_2]}=\tau_8$, $\tau_{[{\gamma}_3]}=\tau_7$, $\tau_{[{\gamma}_4]}=\tau_5$ and $\tau_{[{\gamma}_5]}=\tau_6$.
The graph $G_{T,{\gamma}}$ is obtained by glueing the corresponding six tiles $\tilde S_1$, $\tilde S_2$, $\tilde S_3$, $\tilde S_4$, $\tilde S_1$, and $\tilde S_2$. The result is shown in Figure \[figex1a\].
Theorems \[thm main\] and \[thm y\] imply that $x_{\gamma}(x_{i_1}x_{i_2}\cdots x_{i_d})$ is equal to $$\begin{array}{cccccc}
&x_4x_6x_8x_4x_4x_6x_8 \,y_1y_3y_4y_1&+& x_4x_6x_8x_4x_4x_1x_3 \, y_1y_3y_4y_1y_2\\+&
x_4x_6x_8x_4x_5x_2x_8 \,y_1y_3y_4
&+&x_4x_6x_2x_3x_1x_2x_8 \,y_1
\\
+& x_4x_6x_2x_7x_5x_2x_8\,y_1y_4 &+& x_4x_6x_2x_7x_4x_6x_8\,y_1y_4y_1\\
+& x_4x_1x_3x_4x_4x_6x_8 \,y_1y_2y_3y_4y_1 &+& x_4x_1x_3x_4x_4x_1x_3\,y_1y_2y_3y_4y_1y_2
\\
+& x_4x_1x_3x_4x_5x_2x_8\,y_1y_2y_3y_4
&+& x_5x_2x_8x_4x_4x_6x_8\, y_3y_4y_1\\
+& x_5x_2x_8x_4x_4x_1x_3\, y_3y_4y_1y_2 &+&x_5x_2x_8x_4x_5x_2x_8\, y_3y_4\\
+& x_5x_2x_2x_3x_1x_2x_8\, &+& x_5x_2x_2x_7x_4x_6x_8 \, y_4y_1 \\
+& x_5x_2x_2x_7x_4x_1x_3\, y_4y_1y_2
&+& x_5x_2x_2x_7x_5x_2x_8\,y_4,
\end{array}$$which is equal to
$$\begin{array}{cccccc}
&x_4^3\,y_1^2y_3y_4&+& x_1x_3x_4^3 \, y_1^2y_2y_3y_4 \\
+& x_2x_4^2 \,y_1y_3y_4&+&x_1x_2^2x_3x_4\,y_1\\
+& x_2^2x_4\,y_1y_4 &+& x_2x_4^2\,y_1^2y_4\\
+& x_1x_3x_4^3\,y_1^2y_2y_3y_4 &+&x_1^2x_3^2x_4^3\,y_1^2y_2^2y_3y_4\\
+& x_1x_2x_3x_4^2\,y_1y_2y_3y_4
&+& x_2x_4^2 \,y_1 y_3y_4\\
+& x_1x_2x_3x_4^2\, y_1y_2y_3y_4
&+&x_2^2x_4\, y_3y_4 \\
+& x_1x_2^3x_3 &+&x_2^2x_4 \, y_1y_4\\
+& x_1x_2^2x_3x_4\, y_1y_2y_4
&+& x_2^3\,y_4.\end{array}$$ For example, the first term corresponds to the matching $M$ consisting of the horizontal edges of the first three tiles and the horizontal edges of the last two tiles. The matching $M_-$ consists in the boundary edges weighted $x_5$ and $x_2$ in the first tile, $x_2$ in the third tile, $x_1$ and $x_3 $ in the forth, $x_2$ in the fifth and $x_8$ in the sixth tile. Thus $M_-\ominus M =(M_-\cup M)\setminus (M_-\cap M)$ is the union of the first, third, forth and fifth tile, whence $y(M)=y_{i_1}y_{i_3}y_{i_4}y_{i_5}=y_1y_3y_4y_1$.
To illustrate Theorem \[thm F\], let $k=2$. Then $\mathcal{H}_k$ consists of the subgraphs $H$ of $G_{T,{\gamma}}$ which are unions of two tiles and such that $E(H)\cap M_-$ has three elements if $H$ is connected, respectively four elements if $H$ has two connected components. Thus $\mathcal{H}_2$ has three elements $$\mathcal{H}_2=\{S_{i_3}\cup S_{i_4}, S_{i_4}\cup S_{i_5}, S_{i_1}\cup S_{i_4}\}$$ corresponding to the three terms $$x_2^2x_4y_3y_4 , x_2^2x_4y_1y_4 \textup{ and }x_2^2x_4y_1y_4.$$
C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, preprint, [arXiv:0805.1035]{}. I. Assem, T. Brüstle, G. Charbonneau-Jodoin, P.G. Plamondon, Gentle algebras arising from surface triangulations, in preparation. A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics, [*Adv. Math.*]{} [**204**]{} (2006), 572-612, [ arXiv:math.RT/0402054]{}. , Denominators of cluster variables, preprint, [arXiv:0710.4335]{}. P. Caldero and F. Chapoton, Cluster algebras as Hall algebras of quiver representations, [ *Comment. Math. Helv.*]{} [**81**]{} (2006), 595-616, [ arXiv:math.RT/0410187]{}. P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations arising from clusters ($A_n$ case), [*Trans. Amer. Math. Soc.*]{} [**358**]{} (2006), no. 3, 1347-1364, [arXiv:math.RT/0401316]{}. , Quivers with relations and cluster tilted algebras, *Algebr. and Represent. Theory* [ 9]{}, (2006), no. 4, 359–376, [arXiv:math.RT/0411238]{}. , From triangulated categories to cluster algebras, *Invent. Math.* [**172**]{} (2008), 169–211, [arXiv:math.RT/0506018]{}. P. Caldero, B. Keller, From triangulated categories to cluster algebras II, *Ann. Sci. École Norm. Sup. (4)* [**39**]{} (2006), no. 6, 983–1009, [arXiv:math.RT/0510251]{}. P. Caldero and A. Zelevinsky, Laurent expansions in cluster algebras via quiver representations, *Mosc. Math. J.* [**6**]{} (2006), no. 3, 411–429, [arXiv:math.RT/0604054]{}. G. Carroll and G. Price, (unpublished result). N. Elkies, G. Kuperberg, M. Larsen, J. Propp, Alternating-Sign Matrices and Domino Tilings (Part I), *J. Algebraic Combin.* [**1**]{} (1992), no. 2, 11–132,[arXiv:math/9201305]{}. V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory. [*Publ. Math. Inst. Hautes Études Sci.*]{} No. 103 (2006), 1–211. V. Fock and A. Goncharov, Cluster ensembles, quantization and the dilogarithm, preprint (2003), [ arXiv:math.AG/0311149]{}. V. Fock and A. Goncharov, Dual Teichmüller and lamination spaces. Handbook of Teichmüller theory. Vol. I, 647–684, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Zürich, 2007. S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes, preprint (2006), [arXiv:math/0608367]{}. S. Fomin and D. Thurston, Cluster algebras and triangulated surfaces. Part II: Lambda lengths, preprint (2008). S. Fomin and A. Zelevinsky, Cluster algebras I. Foundations, [*J. Amer. Math. Soc.* ]{}[ **15**]{}(2), (2002), 497-529 (electronic), [arXiv:math.RT/0104151]{}. S. Fomin and A. Zelevinsky, Cluster Algebras IV: Coefficients, [*Comp. Math.*]{} [**143**]{} (2007), 112–164, [arXiv:math.RA/0602259]{}. S. Fomin and A. Zelevinsky, (unpublished result). C. Fu and B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, preprint (2008), [arXiv:0710.3152]{}. M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry. [*Mosc. Math. J.*]{} [ **3**]{} (2003), no. 3, 899–934, 1199, [arXiv:math.QA/0208033]{}. M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil-Petersson forms. [*Duke Math. J.*]{} [**127**]{} (2005), no. 2, 291–311, [arXiv:math.QA/0309138]{}. D. Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, preprint (2008), [arXiv:0803.1328]{}. G. Musiker and J. Propp, Combinatorial interpretations for rank-two cluster algebras of affine type. *Electron. J. Combin.* [**14**]{} (2007), no. 1, Research Paper 15, 23 pp. (electronic). [arXiv:math/0602408]{}. G. Musiker, A graph theoretic expansion formula for cluster algebras of classical type, to appear in *Ann. Comb.*, [arXiv:0710.3574]{}. G. Musiker and L. Williams, Combinatorial formulas for $F$-polynomials and $g$-vectors for cluster algebras of classical type, in preparation. Y. Palu, Cluster characters for triangulated 2-Calabi–Yau categories, to appear in [*Annales de l’Institut Fourier*]{}, [arXiv:math/0703540]{}. J. Propp, The combinatorics of frieze patterns and Markoff numbers, preprint (2005), [arXiv:math.CO/0511633]{}. J. Propp, Lattice structure for orientations of graphs, preprint (1993), [arXiv:math/0209.5005]{}. R. Schiffler, A cluster expansion formula ($A_n$ case), [*Electron. J. Combin.*]{} [**15**]{} (2008), \#R64 1, [arXiv:math.RT/0611956]{}. R. Schiffler, On cluster algebras arising from unpunctured surfaces II, preprint (2008), [arXiv:0809.2593]{}. R. Schiffler and H. Thomas, On cluster algebras arising from unpunctured surfaces, preprint (2007), [arXiv:0712.4131]{}. P. Sherman and A. Zelevinsky, Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. Mosc. Math. J. 4 (2004), no. 4, 947–974, 982, [arXiv:math.RT/0307082]{}. A. Zelevinsky, Semicanonical basis generators of the cluster algebra of type $A\sp {(1)}\sb 1$. *Electron. J. Combin.* [**14**]{} (2007), no. 1, Note 4, 5 pp. (electronic), [arXiv:math.RA/0606775]{}.
-------------------------------------------- -----------------------------------
[Gregg Musiker ]{} [Ralf Schiffler]{}
[Department of Mathematics, Room 2-332 ]{} [Department of Mathematics ]{}
[Massachusetts Institute of Technology]{} [University of Connecticut]{}
[77 Massachusetts Ave.]{} [196 Auditorium Road]{}
[Cambridge, MA 02139]{} [Storrs, CT 06269-3009]{}
[musiker (at) math.mit.edu ]{} [schiffler (at) math.uconn.edu]{}
-------------------------------------------- -----------------------------------
|
---
abstract: |
We consider sufficient conditions of local removability of coincidences of maps $f,g:N\rightarrow M,$ where $M,N$ are manifolds with dimensions $\dim
N\geq\dim M.$ The coincidence index is the only obstruction to the removability for maps with fibers either acyclic or homeomorphic to spheres of certain dimensions$.$ We also address the normalization property of the index and coincidence-producing maps.
address: 'Marshall University, Huntington, WV 25755, USA'
author:
- Peter Saveliev
title: 'Removing Coincidences of Maps Between Manifolds of Different Dimensions.'
---
[^1]
Introduction
============
Let $N^{n+m}$ and $M^{n}$ be orientable compact smooth manifolds (possibly with boundaries $\partial N,\partial M)$, $n\geq2,$ and suppose $f,g:N\rightarrow M$ are maps. We shall call $m$ the *codimension*. The *coincidence set* is a compact subset of $N$ defined by ** $Coin(f,g)=\{x\in N:f(x)=g(x)\}.$
The Coincidence Problem asks what can be said about the coincidence set. When $m=0,$ the main tools for studying the problem is the *Lefschetz number* $L(f,g)$ defined as the alternating sum of traces of certain endomorphism on the (co)homology group of $M$. The famous Lefschetz coincidence theorem provides a sufficient condition for the existence of coincidences: $L(f,g)\neq0\Longrightarrow Coin(f,g)\neq\emptyset.$ Under some circumstances the converse is also true (up to homotopy): $L(f,g)=0\Longrightarrow$ there are maps $f^{\prime},g^{\prime}$ homotopic to $f,g$ respectively such that $Coin(f^{\prime},g^{\prime})=\emptyset.$ Now the problem reads as follows: Can we remove coincidences by a homotopy of $f$ and $g$?
Let $K=Coin(f,g)$. By $H_{\ast}$ ($H^{\ast}$) we denote the integral singular (co)homology. For any space $Y$ we define the diagonal map $d:Y\rightarrow
Y\times Y$ by $d(x)=(x,x).$ Let $$Y^{\times}=(Y\times Y,Y\times Y\backslash d(Y)).$$ For codimension $m=0,$ the (cohomology) *coincidence index* $I_{fg}$ of $(f,g)$ is defined as follows. Since all coincidences lie in $K,$ the map $(f,g):(N,N\backslash K)\rightarrow M^{\times}$ is well defined. Let $\tau$ be the generator of $H^{n}(M^{\times})=\mathbf{Z}$ and $O_{N}$ the fundamental class of $N$ around $K,$ then let $$I_{fg}=<(f,g)^{\ast}(\tau),O_{N}>\in\mathbf{Z.}$$ The coincidence index satisfies the following natural properties. (1) Homotopy Invariance: the index is invariant under homotopies of $f,g$; (2) Additivity: The index over a union of disjoint sets is equal to the sum of the indices over these sets; (3) Existence of Coincidences: if the index is nonzero then there is a coincidence; (4) Normalization: the index is equal to the Lefschetz number; (5) Removability: if the index is zero then the coincidence set can be removed by a homotopy.
While the coincidence theory for codimension $m=0$ is well developed [@Bredon VI.14], [@BS], [@Naka], [@Vick Chapter 7], very little is known beyond this case. For $m>0,$ the vanishing of the coincidence index does not always guarantee removability. For codimension $m=$ $1,$ the secondary obstruction to removability was considered by Fuller [@Fuller-th], [@Fuller] for $M$ simply connected. In the context of Nielsen Theory the sufficient conditions of the local removability for $m=1$ were studied by Dimovski and Geoghegan [@DG], Dimovski [@Dim] for the projection $f:M\times[0,1]\rightarrow M,$ and by Jezierski [@Jez] for $M,N$ subsets of Euclidean spaces or $M$ parallelizable. Necessary conditions of the global removability for arbitrary codimension were considered by Gonçalves, Jezierski, and Wong [@GJW Section 5] with $N$ a torus and $M$ a nilmanifold (see also [@GW]).
The main purpose of this note is to provide sufficient conditions of removability of coincidences for some codimensions higher than $1$. Under a certain technical condition, the coincidence index defined below is the only obstruction to removability. This condition holds when (1) $M$ *is a surface;* (2) *fibers* $f^{-1}(y)$ *of* $f$ *are acyclic;* or (2) *fibers of* $f$ *are* $m$*-spheres for* $m=4,5,12$ *and* $n$ *large*. The main theorem partially complements the results listed above. The proof follows and extends the one of Brown and Schirmer [@BS Theorem 3.1] for codimension $0$ (see also Vick [@Vick p. 194]).
An area of possible applications is discrete dynamical systems. A *dynamical system* on a manifold $M$ is determined by a map $f:M\rightarrow M.$ Then the next position, or state, $f(x)$ depends only on the current one, $x\in M$. Suppose we have a fiber bundle $F\rightarrow
N^{\underrightarrow{~\ \ \ g\ \ \ ~~}}M$ and a map $f:N\rightarrow M.$ Then this is a *parametrized dynamical system*, where the next position $f(x,s)$ depends not only on the current one, $x\in M,$ but also the input, $s\in F.$ Then the Coincidence Problem asks whether there are a position and an input such that the former remains unchanged, $f(x,s)=x$. A parametrized dynamical system can also be a model for a non-autonomous ordinary differential equation: $M$ is the space, $F$ is the time, and $N$ is the space-time.
Normalization Property.
=======================
For nonzero codimension the homology coincidence index $I_{fg}^{\prime
}=(f,g)_{\ast}(O_{N})$ is replaced with the *homology coincidence homomorphism* [@Brooks1] $$I_{fg}^{\prime}=(f,g)_{\ast}:H_{\ast}(N,N\backslash V)\rightarrow H_{\ast
}(M^{\times}),$$ where $V$ is a neighborhood of $Coin(f,g)$. Let $\pi:M\times M\rightarrow M$ be the projection on the first factor, then $\zeta=(M,\pi,M\times M,d)$ is the tangent microbundle of $M$ and the Thom isomorphism $\varphi:H_{\ast
}(M^{\times})\rightarrow H_{\ast}(M)$ is given by $\varphi(x)=\pi_{\ast}(\tau\frown x),$ where $\tau\in H^{n}(M^{\times}\mathbf{)}$ is the Thom class of $\zeta.$ The Lefschetz number is replaced with the *Lefschetz homomorphism* [@Sav1] $\Lambda_{fg}:H_{\ast}(N,N\backslash V;\mathbf{Q})\rightarrow H_{\ast}(M;\mathbf{Q})$ of degree $(-n)$ defined as follows. Suppose $f(N\backslash V)\subset\partial M$. For each $z\in H_{\ast
}(N,N\backslash V),$ let$$f_{!}^{z}=(f^{\ast}D^{-1})\frown z,$$ where $D:H^{\ast}(M,\partial M;\mathbf{Q})\rightarrow H_{n-\ast}(M;\mathbf{Q})$ is the Poincaré-Lefschetz duality isomorphism $D(x)=x\frown O_{M}$. Now let $$\Lambda_{fg}(z)=\sum_{k}(-1)^{k(k+m)}\sum_{j}x_{j}^{k}\frown g_{\ast}f_{!}^{z}(a_{j}^{k}),$$ where $\{a_{1}^{k},...,a_{m_{k}}^{k}\}$ is a basis for $H_{k}(M)$ and $\{x_{1}^{k},...,x_{m_{k}}^{k}\}$ the corresponding dual basis for $H^{k}(M).$ Then the Lefschetz-type coincidence theorem [@Sav1 Theorem 6.1] states that $\varphi I_{fg}^{\prime}=\Lambda_{fg}.$ This is the Normalization Property, which makes the coincidence homomorphism computable by algebraic means.
Since obstructions to removability of coincidences lie in certain cohomology groups, we need a cohomological analogue of the theory outlined above. Just as in the homology case, the cohomology coincidence index can be replaced with the cohomology coincidence homomorphism.
Let $C$ be an isolated subset of $Coin(f,g),$ $W,V$ neighborhoods of $C,$ $C\subset V\subset\overline{V}\subset W\subset N,$ and $W\cap Coin(f,g)=C.$ Then let$$I_{fg}=(f,g)^{\ast}:H^{\ast}(M^{\times})\rightarrow H^{\ast}(W,W\backslash
V).$$
However in this paper we consider only the restriction of $I_{fg}$ to $H^{n}(M^{\times})=\mathbf{Z}$. Therefore the only thing that matters is the class $I_{fg}(\tau)\in H^{n}(W,W\backslash V),$ where $\tau$ is the generator of $H^{n}(M^{\times})=\mathbf{Z,}$ which will still be called the *(cohomology)* *coincidence index*. This index satisfies the properties of additivity, existence of coincidences and homotopy invariance proven similarly to Lemmas 7.1, 7.2, 7.4 in [@Vick p. 190-191] respectively.
We will state the Normalization Property under assumptions similar to the ones in [@Sav Section 2], [@Sav1 Section 5]. Assume that $f(W\backslash
V)\subset\partial M.$
For each $z\in H_{n}(W,W\backslash V;\mathbf{Q}),$ define homomorphisms $\Theta_{q}:H^{q}(M,\partial M;\mathbf{Q})\rightarrow H^{q}(M,\partial
M;\mathbf{Q})$ by$$\Theta_{q}=D^{-1}g_{\ast}(f^{\ast}\frown z).$$ Then$$L_{z}(f,g)=\sum_{q}(-1)^{q}Tr\Theta_{q}$$ is called the (cohomology) *Lefschetz number* with respect to $z$ of the pair $(f,g).$
\[Normalization\]\[Normalization\]Suppose that $f(W\backslash V)\subset
\partial M.$ Then for each $z\in H_{n}(W,W\backslash V;\mathbf{Q}),$$$<I_{fg}(\tau),z>=(-1)^{n}L_{z}(f,g).$$ Therefore, $L_{z}(f,g)\neq0\Longrightarrow Coin(f,g)\neq\emptyset.$
The proof repeats the computation in the proof of Theorem 7.12 in [@Vick p. 197] with Lemmas 7.10 and 7.11 replaced with their generalizations, Lemmas 3.1 and 3.2 in [@Sav].
The theorem is true even when $N$ is not a manifold.
Local Removability.
===================
Let $f:\mathbf{S}^{3}\rightarrow\mathbf{S}^{2}$ be the Hopf map. Then $f$ is onto, in other words, it has a coincidence with any constant map $c$. However the coincidence homomorphism $I_{fc}:H^{\ast}((\mathbf{S}^{2})^{\times
})\rightarrow H^{\ast}(\mathbf{S}^{3})$ is zero. Therefore Theorem \[Normalization\] fails to detect coincidences. In fact, $f$ has a coincidence with any map homotopic to $c$ [@Brooks], therefore the converse of the Lefschetz coincidence theorem for spaces of different dimensions fails in general. Our main result below is a partial converse.
\[Local Removability\]\[Remov\]Suppose $f(C)=g(C)=\{u\},u\in M\backslash
\partial M,$ and the following condition is satisfied:$$\text{(A) \ }H^{k+1}(W,W\backslash V;\pi_{k}(\mathbf{S}^{n-1}))=0\text{ for
}k\geq n+1.$$ Then $I_{fg}(\tau)=0$ implies that $C$ can be removed via a local homotopy of $f$; specifically, there exists a map $f^{\prime}:N\rightarrow M$ homotopic to $f$ relative $N\backslash W$ such that $$W\cap Coin(f^{\prime},g)=\emptyset.$$
The proof uses the classical obstruction theory. Condition (A) guarantees that only the primary obstruction to the local removability, i.e., the coincidence index, may be nonzero.
We can assume that $U=\mathbf{D}^{n}$ is a neighborhood of $u$ in $M$ such that $f(W)=U$ and $g(W)\subset U.$ Define $Q:\mathbf{D}^{n}\times
\mathbf{D}^{n}\backslash d(\mathbf{D}^{n})\rightarrow\mathbf{D}^{n}\backslash\{0\}$ by $Q(x,y)=1/2(y-x).$ Consider the following commutative diagram:$$\begin{array}
[c]{ccccc}H^{n-1}(\mathbf{S}^{n-1}) & & & & \\
^{\simeq}\downarrow^{p^{\ast}} & & & & \\
H^{n-1}(\mathbf{D}^{n}\backslash\{0\}) & ^{\underrightarrow{~\ \delta^{\ast
}~\simeq~}} & H^{n}(\mathbf{D}^{n},\mathbf{D}^{n}\backslash\{0\}) & & \\
\downarrow^{Q^{\ast}} & & ^{\simeq}\downarrow^{Q^{\ast}} & & \\
H^{n-1}(\mathbf{D}^{n}\times\mathbf{D}^{n}\backslash d(\mathbf{D}^{n})) &
^{\underrightarrow{~\ \ \ \delta^{\ast}~~}} & H^{n}((\mathbf{D}^{n})^{\times
}) & ^{\underleftarrow{\ ~k^{\ast}~\simeq\ }} & H^{n}(M^{\times})\\
\downarrow^{(f,g)^{\ast}} & & \downarrow^{(f,g)^{\ast}} & ^{I_{fg}}\swarrow &
\\
H^{n-1}(W\backslash V) & ^{\underrightarrow{~\ \ \ \delta^{\ast}~~}} &
H^{n}(W,W\backslash V). & &
\end{array}$$ Here $\delta^{\ast}$ is the connecting homomorphism, $k$ the inclusion, $p$ the radial projection. Let $q=pQ(f,g):W\backslash V\rightarrow\mathbf{S}^{n-1}.$ Then $q^{\ast}$ is given in the first column of the diagram.
Now we apply the Extension Theorem, Corollary VII.13.13 in [@Bredon p. 509]. Suppose $I_{fg}=0.$ Then from the commutativity of the diagram, $\delta^{\ast}q^{\ast}=0$. Thus the primary obstruction to extending $q$ to $q^{\prime}:W\rightarrow\mathbf{S}^{n-1},$ $c^{n+1}(q)=\delta^{\ast}q^{\ast},$ vanishes. By condition (A) the other obstructions $c^{k+1}(q),k\geq n,$ also vanish$.$
Next, $q$ has the form$$q(x)=\dfrac{g(x)-f(x)}{||g(x)-f(x)||}.$$ Define a map $f^{\prime}:W\rightarrow\mathbf{D}^{n}$ by $f^{\prime
}(x)=g(x)-a(x)q^{\prime}(x),$ where $a:W\rightarrow(0,\infty)$ satisfies the following: (1) $a$ is small enough so that $f^{\prime}(x)\in\mathbf{D}^{n}$ for all $x\in W,$ (2) $a(x)=||g(x)-f(x)||$ for all $x\in W\backslash V.$ Then $Coin(f^{\prime},g)=\emptyset$ since $q^{\prime}(x)\neq0.$
To complete the proof observe that $f^{\prime}$ is homotopic to $f|_{W}$ relative $W\backslash V$ because $\mathbf{D}^{n}$ is convex.
The implications of this result for Nielsen theory will be addressed in a forthcoming paper.
Further Results.
================
Suppose $C=f^{-1}(y),$ where $y\in M\backslash\partial M$ is a regular values for both $f$ and $f|_{\partial N}.$ Then $C$ is a neat submanifold of $N$ and it has a tubular neighborhood $T$. Now $T$ can be treated as a disk bundle $(\mathbf{D}^{m},\mathbf{S}^{m-1})\rightarrow(T,T^{\prime})\rightarrow C.$ Therefore condition (A) takes the form$$\text{(A}^{\prime}\text{) \ }H^{k+1}(T,T^{\prime};\pi_{k}(\mathbf{S}^{n-1}))=0\text{ for }k\geq n+1.$$
In case $C$ is a boundaryless $m$-submanifold of $N,$ we have $H^{n+m}(T,T^{\prime};G)=H^{n+m}(T,\partial T;G)=G\oplus...\oplus G.$ Therefore if we let $k=n+m-1,$ then condition (A$^{\prime}$) implies the following:$$\text{(A}^{\ast}\text{) }\pi_{n+m-1}(\mathbf{S}^{n-1})=0.$$ This restriction cannot be relaxed, in the following sense. Suppose $$\lbrack h]\in\pi_{n+m-1}(\mathbf{S}^{n-1})\backslash\{0\}.$$ Then $h$ can be extended to a map $f:\mathbf{D}^{n+m}\rightarrow\mathbf{D}^{n}\subset M$ by setting $f(0)=0$ and $f(x)=||x||h\left( \dfrac{x}{||x||}\right) $ for $x\in\mathbf{D}^{n+m}\backslash\{0\}.$ Hence any map homotopic to $f$ relative $\mathbf{S}^{n+m-1}$ is onto [@Bredon Theorem VII.5.8, p. 448]. Therefore coincidences of $f$ and $g,$ where $g$ is constant, cannot be locally removed.
Below we treat condition (A) as a restriction on an arbitrary fiber $C=f^{-1}(y),y\in M\backslash\partial M$ of $f.$
\[Lemma\]Suppose for $1\leq p\leq m,$ the submanifold $C$ satisfies $$\begin{array}
[c]{c}\text{(1) }H^{p}(C)\otimes\pi_{n+p-1}(\mathbf{S}^{n-1})=0,\text{ and}\\
\text{(2) }Tor(H^{p+1}(C),\pi_{n+p-1}(\mathbf{S}^{n-1}))=0.
\end{array}$$ Then $C$ satisfies condition (A).
By the Thom Isomorphism Theorem [@Bredon Section VI.11], we have $$H^{k+1}(T,T^{\prime};\pi_{k}(\mathbf{S}^{n-1}))=H^{k+1-n}(C;\pi_{k}(\mathbf{S}^{n-1})).$$ By condition (2) and the Universal Coefficient Theorem, Corollary 25.14 in [@Gray p. 263], we have also$$H^{p}(C;\pi_{k}(\mathbf{S}^{n-1}))=H^{p}(C)\otimes\pi_{k}(\mathbf{S}^{n-1}).$$
It is known [@Toda] that $\pi_{n+m-1}(\mathbf{S}^{n-1})=0,$ for the following values of $m$ and $n$:
\(1) $m=4$ and $n\geq6;$
\(2) $m=5$ and $n\geq7;$
\(3) $m=12$ and $n=7,8,9,14,15,16,....$
The conclusion of Theorem \[Remov\] holds when (a) $M$ is a surface; (b) fibers of $f$ are acyclic; or (c) fibers of $f$ are unions of homology $m$-spheres for the above values of $m$ and $n$.
\(a) $n=2$ and $\pi_{n+p-1}(\mathbf{S}^{n-1})=0$ for all $p>0.$ (b) $H^{p}(C)=0$ for all $p>0.$ (c) Either $\pi_{n+p-1}(\mathbf{S}^{n-1})=0$ or $H^{p}(C)=H^{p+1}(C)=0$ for all $p>0.$ Thus the two conditions of Lemma \[Lemma\] are satisfied. Now the conclusion follows from Theorem \[Remov\].
Let $F\rightarrow N^{n+m\underrightarrow{~\ \ \ g\ \ \ ~~}}M^{n}$ be an $m$-sphere bundle with the above values of $m$ and $n,$ or an $m$-disk bundle. Then the set $C$ of stationary points of the parametrized dynamical system $F\rightarrow N^{n+m\underrightarrow{~\ \ \ f,g\ \ \ ~~}}M^{n}$ can be removed via a local homotopy of $f$ provided $I_{fg}=0.$
Coincidence-Producing Maps.
===========================
A boundary preserving map $f:(N,\partial N)\rightarrow(M,\partial M)$ is called *coincidence-producing* if every map $g:N\longrightarrow M$ has a coincidence with $f$. Brown and Schirmer [@BS Theorem 7.1] showed that if $M$ is acyclic, $\dim N=\dim M=n\geq2,$ then $f$ is coincidence-producing if and only if $f_{\ast}:H_{n}(N,\partial N)\rightarrow H_{n}(M,\partial M)$ is nonzero. Based on the Normalization and Removability Properties we prove a generalization of this theorem. We call a map $f:(N,\partial N)\rightarrow
(M,\partial M)$ *weakly coincidence-producing* [@Sav Section 5] if every map $g:N\rightarrow M$ with $g_{\ast}=0$ (in reduced homology) has a coincidence with $f.$ In particular every weakly coincidence-producing map is onto.
A corollary to the Lefschetz type coincidence theorem [@Sav Corollary 5.1] states that if $f_{\ast}:H_{n}(N,\partial N)\rightarrow
H_{n}(M,\partial M)$ is nonzero then the appropriate Lefschetz homomorphism is nontrivial and, therefore, $f$ is weakly coincidence-producing. For the converse we need condition (A) as an additional assumption.
Suppose $f$ is boundary preserving and suppose that each fiber $C$ of $f$ satisfies condition (A). Then the following are equivalent:
\(1) $f$ is weakly coincidence-producing;
\(2) $f_{\ast}:H_{n}(N,\partial N)\rightarrow H_{n}(M,\partial M)$ is nonzero.
Suppose $f_{\ast}:H_{n}(N,\partial N)\rightarrow H_{n}(M,\partial M)$ is zero. Choose $g$ to be identically equal to $y\in M\backslash\partial M.$ Then $C=Coin(f,g)=f^{-1}(y)\subset N\backslash\partial N$. Recall $I_{fg}=(f,g)^{\ast}:H^{\ast}(M^{\times})\rightarrow H^{\ast}(N,\partial N).$ Then for all $z\in H_{n}(N,\partial N),$ we have the following. $$\begin{tabular}
[c]{lll}$<I_{fg}^{N}(\tau),z>$ & $=(-1)^{n}L_{z}(f,g),$ by Theorem \ref{Normalization}
& \\
& $=(-1)^{n}Tr\Theta_{n},$ because $g_{\ast}=0$ & \\
& $=(-1)^{n}<f^{\ast}(\overline{O}_{M}),z>,$ where $\overline{O}_{M}$ is the
dual of $O_{M}$ & \\
& $=(-1)^{n}<\overline{O}_{M},f_{\ast}(z)>$ & \\
& $=0.$ &
\end{tabular}
\ \ \ \ \ \ \ \ \ $$ Hence $I_{fg}(\tau)=0.$ Therefore by Theorem \[Remov\] the coincidence set can be removed. Thus $f$ is not weakly coincidence-producing
Condition (A) is clearly satisfied for $m=0$. Therefore Brown and Schirmer’s Theorem [@BS Theorem 7.1] follows. Our theorem also includes the well-known fact that *a map has degree* $0$ *if and only if it can be deformed into a map which is not onto*.
Examples of maps satisfying condition (1) of the theorem can be found in [@BS Section 7], see also [@Sav1 Section 6].
I would like to thank the referee for a number helpful suggestions.
[99]{}
G. E. Bredon, “Topology and Geometry”, Springer-Verlag, 1993.
R. Brooks, On removing coincidences of two maps when only one, rather that both, of them may be deformed by homotopy, *Pacific J. Math*. **40** (1972), 45-52.
R. Brooks, On the sharpness of the $\Delta_{2}$ and $\Delta_{1}$ Nielsen numbers, *J. Reine Angew. Math.* **259** (1973), 101–108.
R. F. Brown and H. Schirmer, Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary, *Topology Appl.* **46** (1992), 65-79.
D. Dimovski, One-parameter fixed point indices, *Pacific J. Math.* **164** (1994) no. 2, 263–297.
D. Dimovski and R. Geoghegan, One-parameter fixed point theory. *Forum Math*. **2** (1990) no. 2, 125–154.
F. B. Fuller, “The homotopy theory of coincidences”, Ph.D. Thesis, Princeton, 1952.
F. B. Fuller, The homotopy theory of coincidences, *Ann. of Math.* (2) **59**, (1954), 219–226.
D. Gonçalves, J. Jezierski, and P. Wong, Obstruction theory and coincidences in positive codimension, preprint.
D. Gonçalves and P. Wong, Nilmanifolds are Jiang-type spaces for coincidences. *Forum Math.* **13** (2001), no. 1, 133–141.
B. Gray, Homotopy theory, An introduction to algebraic topology, Pure and Applied Mathematics, Vol. 64. New York - San Francisco - London: Academic Press. XIII, (1975).
J. Jezierski, One codimensional Wecken type theorems, *Forum Math.* **5** (1993) no. 5, 421–439.
M. Nakaoka, Coincidence Lefschetz number for a pair of fibre preserving maps, *J. Math. Soc. Japan* **32** (1980) no. 4, 751-779.
P. Saveliev, A Lefschetz-type coincidence theorem, *Fund. Math.* **162** (1999), 65-89.
P. Saveliev, The Lefschetz coincidence theory for maps between spaces of different dimensions, *Topology Appl.* **116** (2001) no. 1, 137-152.
H. Toda, Composition methods in homotopy groups of spheres. Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J. 1962.
J. W. Vick, “Homology Theory, An Introduction to Algebraic Topology”, Springer-Verlag, 1994.
[^1]:
|
---
abstract: 'In this paper, we use entangled states to construct $9\times9$-matrix representations of Temperley-Lieb algebra (TLA), then a family of $9\times9$-matrix representations of Birman-Wenzl-Murakami algebra (BWMA) have been presented. Based on which, three topological basis states have been found. And we apply topological basis states to recast nine-dimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special case.'
address:
- 'School of Physics, Northeast Normal University, Changchun 130024, People’s Republic of China'
- 'School of Physics, Northeast Normal University, Changchun 130024, People’s Republic of China'
- 'School of Physics, Northeast Normal University, Changchun 130024, People’s Republic of China'
- 'School of Physics, Northeast Normal University, Changchun 130024, People’s Republic of China'
- 'School of Physics, Northeast Normal University, Changchun 130024, People’s Republic of China'
author:
- Zhou Chengcheng
- Xue Kang
- Wang Gangcheng
- Sun Chunfang
- Du Guijiao
title: 'Birman-Wenzl-Murakami Algebra and the Topological Basis'
---
Introduction
============
Quantum entanglement (QE) is the most surprising nonclassical property of quantum systems which plays a key role in quantum information and quantum computation processing[@ent1; @ent2; @ent3; @ent4]. Because of these applications, QE has become one of the most fascinating topics in quantum information and quantum computation. To the best of our knowledge, the Yang-Baxter equation (YBE) plays an important role in quantum integrable problem, which was originated in solving the one-dimensional $\delta$-interacting models[@ye] and the statistical models[@be]. Braid group representations (BGRs) can be obtained from YBE by giving a particular spectral parameter. BGRs of two and three eigenvalues have direct relationship with Temperley-Lieb algebra (TLA)[@tla] and Birman-Wenzl-Murakami algebra (BWMA)[@bwma] respectively. TLA and BWMA have been widely used to construct the solutions of YBE[@cgx; @ybr1; @ybr2; @ybr3].
The TLA first appeared in statistical mechanics as a tool to analyze various interrelated lattice models[@tla] and was related to link and knot invariants[@tla1]. In the subsequent developments TLA is related to knot theory, topological quantum field theory, statistical physics, quantum teleportation, entangle swapping and universal quantum computation[@kk; @tla2]. On the other hand, the BWMA[@bwma] including braid algebra and TLA was first defined and independently studied by Birman, Wenzl and Murakami. It was designed partially help to understand Kauffman’s polynomial in knot theory. Recently, Ref.[@hxg] applied topological basis states for spin-1/2 system to recast 4-dimensional YBE into its 2-dimensional counterpart. As we know, few studies have reported topological basis states for spin-1 system. The motivation for our works is to find topological basis states for spin-1 system and study the topological basis states.
The purpose of this paper is twofold: one is that we construct a family of $9\times 9$-matrix representations of BWMA; the other concerns topological basis states for spin-1 system. This paper is organized as follows. In Sec. 2, we use entangled states to construct the $9\times9$ matrix representations of TLA, then we present a family of $9\times 9$-matrix representations of BWMA, and study the entangled states. In Sec. 3, we obtain three topological basis states of BWMA, and we recast nine-dimensional BWMA into its three-dimensional counterpart. We end with a summary.
$9\times 9$-matrix representations of BWMA
==========================================
The $4\times 4$ Hermitian matrix $E$, which satisfies TLA and can construct the well-known six-vertex model[@ybei], takes the representation $$E=\left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & q & \eta & 0 \\
0 & \eta^{-1} & q^{-1} & 0 \\
0 & 0 & 0 & 0
\end{array}
\right),$$
where $\eta=e^{i\varphi}$ with $\varphi$ being any flux. We can rewrite $E$ as that
$$\label{}\left\{ \begin{aligned}
&E=d|\Psi\rangle\langle\Psi| ,\\
&|\Psi\rangle=d^{-1/2}(q^{1/2}|\uparrow\downarrow\rangle+q^{-1/2}e^{-i\varphi}|\downarrow\uparrow\rangle),
\end{aligned} \right.$$
where $d=q+q^{-1}$.
So like this symmetrical method, we found the $9\times 9$ Hermitian matrices $E$’s, which satisfies TLA, take the representations as follows $$\label{}\left\{ \begin{aligned}
&E=d|\Psi\rangle\langle\Psi| ,\\
&|\Psi\rangle=d^{-1/2}(q^{1/2}|\lambda\mu\rangle+e^{i\phi_{\nu}}|\nu\nu\rangle+q^{-1/2}e^{i\varphi_{\mu\lambda}}|\mu\lambda\rangle),\\
\end{aligned} \right.$$ where $d=q+1+q^{-1}$, $\lambda\neq\mu\neq\nu\in(1,0,-1)$ and $ (d,q,\phi_{\nu},\varphi_{\mu\lambda}) \in real$. Recently, a $9\times 9$–matrix representation of BWMA has been presented[@bwma1; @bwma2]. We notice that $E$ is the same as Gou *et.al.*[@bwma1; @bwma2] presented, when $\phi_{\nu}=\varphi_2-\varphi_1+\pi$, $\varphi_{\mu\lambda}=-2\varphi_1$, $\lambda=1$, $\mu=-1$ and $\nu=0$.
As we know the BWMA relations[@bwma; @cgx; @gx2; @j] including braid relations and TLA relations satisfy the following relations, $$\label{bway} \left\{ \begin{aligned}
&S_i-S_i^{-1}=\omega (I-E_i) ,\\
&S_iS_{i\pm 1}S_i=S_{i\pm 1}S_iS_{i\pm 1},\ S_iS_j=S_jS_i,|i-j|\geq 2 ,\\
&E_iE_{i\pm 1}E_i=E_i ,\ E_iE_j=E_jE_i ,\ |i-j|\geq 2 ,\\
&E_iS_i=S_iE_i=\sigma E_i ,\\
&S_{i\pm 1}S_iE_{i\pm 1}=E_iS_{i\pm 1}S_i=E_iE_{i\pm 1},\\
&S_{i\pm 1}E_iS_{i\pm 1}=S_i^{-1}E_{i\pm 1}S_i^{-1} ,\\
&E_{i\pm 1}E_iS_{i\pm 1}=E_{i\pm 1}S_i^{-1} ,\ S_{i\pm 1}E_iE_{i\pm 1}=S_i^{-1}E_{i\pm 1} ,\\
&E_iS_{i\pm 1}E_i=\sigma ^{-1}E_i ,\\
&E_i^2=\left(1-\frac{\sigma-\sigma^{-1}}{\omega}\right)E_i,
\end{aligned} \right.$$ where $S_i,S_{i\pm 1}$ satisfy the braid relations, $E_i,E_{i\pm 1}$ satisfy the TLA relations [@tla] $$\label{tla1}\left\{ \begin{aligned}
&E_iE_{i\pm 1}E_i=E_i ,\ E_iE_j=E_jE_i ,\ |i-j|\geq 2 ,\\
&E_i^2=d E_i,
\end{aligned} \right.$$ where $0\ne d\in \mathbb{C}$ is topological parameter in the knot theory which does not depend on the sites of lattices. We denote $\sigma=q^{-2}$ and $\omega=q-q^{-1}$ throughout the text. The notations $E_i\equiv E_{i,i+1}$ and $S_i\equiv S_{i,i+1}$ are used, $E_{i,i+1}$ and $S_{i,i+1}$ are abbreviation of $I_1\otimes...\otimes I_{i-1}\otimes E_{i,i+1}\otimes I_{i+2}\otimes...\otimes I_{N}$ and $I_1\otimes...\otimes I_{i-1}\otimes S_{i,i+1}\otimes I_{i+2}\otimes...\otimes I_{N}$ respectively, and $I_j$ represents the unit matrix of the $j$-th particle.
Following the matrix representation of TLA we obtain a family of $9\times 9$-matrix representations of BWMA as follows $$\label{}\left\{ \begin{aligned}
&E=d|\Psi\rangle\langle \Psi| ,\\
&|\Psi\rangle=d^{-1/2}(q^{1/2}|\lambda\mu\rangle+e^{i\phi_{\nu}}|\nu\nu\rangle+q^{-1/2}e^{i\varphi_{\mu\lambda}}|\mu\lambda\rangle),
\end{aligned} \right.$$
$$\label{} \begin{aligned}
S&=q(|\lambda\lambda\rangle\langle\lambda\lambda|+|\mu\mu\rangle\langle\mu\mu|)+|\nu\nu\rangle\langle\nu\nu|\\
&+(q-q^{-1})(|\nu\lambda\rangle\langle\nu\lambda|+|\mu\nu\rangle\langle\mu\nu|)+(q-1)^2(q+1)q^{-2}|\mu\lambda\rangle\langle\mu\lambda|\\
&+e^{-i\varphi_{\mu\lambda}/2}(|\lambda\nu\rangle\langle\nu\lambda|+|\nu\mu\rangle\langle\mu\nu|)
+e^{i\varphi_{\mu\lambda}/2}(|\nu\lambda\rangle\langle\lambda\nu|+|\mu\nu\rangle\langle\nu\mu|)\\
&+q^{-1}e^{-i\varphi_{\mu\lambda}}|\lambda\mu\rangle\langle\mu\lambda|+q^{-1}e^{i\varphi_{\mu\lambda}}|\mu\lambda\rangle\langle\lambda\mu|\\
&-q^{-3/2}(q^2-1)(e^{i(\phi_{\nu}-\varphi_{\mu\lambda})}|\nu\nu\rangle\langle\mu\lambda|+
e^{-i(\phi_{\nu}-\varphi_{\mu\lambda})}|\mu\lambda\rangle\langle\nu\nu|),
\end{aligned}$$
$$\label{} \begin{aligned}
S^{-1}&=q^{-1}(|\lambda\lambda\rangle\langle\lambda\lambda|+|\mu\mu\rangle\langle\mu\mu|)+|\nu\nu\rangle\langle\nu\nu|\\
&+(q^{-1}-q)(|\lambda\nu\rangle\langle\lambda\nu|+|\nu\mu\rangle\langle\nu\mu|)+(q-1)^2(q+1)q^{-1}|\lambda\mu\rangle\langle\lambda\mu|\\
&+e^{-i\varphi_{\mu\lambda}/2}(|\lambda\nu\rangle\langle\nu\lambda|+|\nu\mu\rangle\langle\mu\nu|)
+e^{i\varphi_{\mu\lambda}/2}(|\nu\lambda\rangle\langle\lambda\nu|+|\mu\nu\rangle\langle\nu\mu|)\\
&+qe^{-i\varphi_{\mu\lambda}}|\lambda\mu\rangle\langle\mu\lambda|+qe^{i\varphi_{\mu\lambda}}|\mu\lambda\rangle\langle\lambda\mu|\\
&+q^{-1/2}(q^2-1)(e^{-i\phi_{\nu}}|\lambda\mu\rangle\langle\nu\nu|+
e^{i\phi_{\nu}}|\nu\nu\rangle\langle\lambda\mu|),
\end{aligned}$$
where $d=q+1+q^{-1}$, $\lambda\neq\mu\neq\nu\in(1,0,-1)$ and $ (d,q,\phi_{\nu},\varphi_{\mu\lambda}) \in real$.
It is worth noticing that the states $|\Psi\rangle$’s are entangled states. By means of negativity, we study these entangled states. The negativity for two qutrits is given by, $$\label{}
\mathcal{N}(\rho)\equiv \frac{\parallel\rho^{T_A}\parallel-1}{2},$$ where $\parallel\rho^{T_A}\parallel$ denotes the trace norm of $\rho^{T_A}$, which denotes the partial transpose of the bipartite state $\rho$ [@neg]. In fact, $\mathcal{N}(\rho)$ corresponds to the absolute value of the sum of negative eigenvalues of $\rho^{T_A}$ , and negativity vanishes for unentangled states[@ent]. By calculation, we can obtain the negativity of states $|\Psi\rangle$’s as $$\label{}
\mathcal{N}(q)= \frac{q^{1/2}+1+q^{-1/2}}{d},$$ where $d=q+1+q^{-1}$. The Fig.\[fig:neg\] corresponds to the negativity $\mathcal{N}(q)$. One demonstrates that the states $|\Psi\rangle$’s become maximally entangled states of two qutrits as $|\Psi\rangle=(|\lambda\mu\rangle+e^{i\phi_{\nu}}|\nu\nu\rangle+e^{i\varphi_{\mu\lambda}}|\mu\lambda\rangle)/\sqrt{3}$ when $q=1$.
![(The negativity $\mathcal{N}(q)$ versus the parameter $q$.)[]{data-label="fig:neg"}](fig1.eps){width="10cm"}
Topological Basis states
========================
In the topological quantum computation theory, the two-dimensional (2D) braid behavior under the exchange of anyons[@fw] has been investigated based on the $\nu=5/2$ fractional quantum Hall effect (FQHE)[@as]. The orthogonal topological basis states read[@as] $$\label{TB1} \begin{aligned}
|e_1>&=\frac{1}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}},\\
|e_2>&=\frac{1}{\sqrt{d^2-1}}({\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}-\frac{1}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}),
\end{aligned}$$ where the parameter $d$ represents the values of a unknotted loop. In Eq. there are two topological graphics ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}$ and ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}$. For four lattices, we can easy find four graphics ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}$, ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}$, ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}$, ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usiu}}}$. If we use Skein relations ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{s}}}=q^{1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{ii}}}+q^{-1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{un}}}$ ($S=q^{1/2}I+q^{-1/2}E$) and ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{si}}}=q^{-1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{ii}}}+q^{1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{un}}}$ ($S^{-1}=q^{-1/2}I+q^{1/2}E$), where the unknotted loop $d={\raisebox{-0.4\height}{\includegraphics[height=0.3cm]{d}}}=-q-q^{-1}$, the third and the fourth graphics recast to ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}$ and ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}$ (${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}=q^{1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}+q^{-1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}$, ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usiu}}}=q^{-1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}+q^{1/2}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}$). So the topological basis states are self-consistent. But in this paper, we focus on BWMA, the braid group representations ($S$) is independent of TLA representations ($E$), and in BWMA $S-S^{-1}=\omega(I-E)$. So we know the graphics ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}$ and ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usiu}}}$ have one independent graphic. We choose three independent graphics as ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}$, ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}$ and ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}$.
We define $$\label{} \begin{aligned}
&{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{uij}}}=d^{1/2}|\Phi_{ij}\rangle=q^{1/2}|\lambda\mu\rangle+e^{i\phi_{\nu}}|\nu\nu\rangle+q^{-1/2}|\mu\lambda\rangle,\\
&{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{nij}}}=d^{1/2}\langle\Phi_{ij}|=q^{1/2}\langle\lambda\mu|+e^{-i\phi_{\nu}}\langle\nu\nu|+q^{-1/2}\langle\mu\lambda|,\\
&[~{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}~]^{\dag}={\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{nn}}},\\
&[~{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}~]^{\dag}={\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{ninn}}},\\
&[~{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}~]^{\dag}={\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{nsn}}}.
\end{aligned}$$ So $E$ recasts to $E_{ij}={\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{unij}}}$. Following the BWMA, we define the graphic rules $$\label{} \begin{aligned}
&{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{su}}}=\sigma {\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{u}}}, {\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{siu}}}=\sigma^{-1}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{u}}}\\
&{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{s}}}-{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{si}}}=\omega({\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{ii}}}-{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{un}}}),\\
&{\raisebox{-0.4\height}{\includegraphics[height=0.3cm]{d}}}=d ~(the~ unknotted ~loop).
\end{aligned}$$
The orthogonal basis states read $$\label{}\left\{ \begin{aligned}
&|e_1\rangle=\frac{q}{(1+q^2)\sqrt{d^2-d-1}}({\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}+q{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}-\frac{q(q+1)}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}) ,\\
&|e_2\rangle=\frac{1}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}},\\
&|e_3\rangle=\frac{q}{(1+q^2)\sqrt{d}}({\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}-q^{-1}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}-\frac{q^2-q^{-1})}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}) .
\end{aligned} \right.$$ Let’s introduce the reduced operators $E_{A},E_{B},A$ and $B$ $$\label{}\left\{ \begin{aligned}
&(E_{A})_{ij}=\langle e_i|E_{12}|e_j\rangle ,\\
&(E_{B})_{ij}=\langle e_i|E_{23}|e_j\rangle ,\\
&A_{ij}=\langle e_i|S_{12}|e_j\rangle ,\\
&B_{ij}=\langle e_i|S_{23}|e_j\rangle .
\end{aligned} \right.$$ Due to the limited length, we only show how $S_{23}$ acts on $|e_3\rangle$ in detail as follows $$\label{} \begin{aligned}
S_{23}|e_3\rangle&=\frac{q}{(1+q^2)\sqrt{d}}({\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{susu}}}-q^{-1}{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{suinu}}}-\frac{q^2-q^{-1}}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{suu}}})\\
&=\frac{q}{(1+q^2)\sqrt{d}}({\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{siusu}}}+\omega({\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{iiusu}}}-{\raisebox{-0.4\height}{\includegraphics[height=0.7cm]{unusu}}})-q^{-1}\sigma {\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}-\frac{q^2-q^{-1}}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}})\\
&=\frac{q}{(1+q^2)\sqrt{d}}({\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}}+\omega({\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}-\sigma{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}})-q^{-1}\sigma{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}-\frac{q^2-q^{-1}}{d}{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}})\\
&=\frac{q}{(1+q^2)\sqrt{d}}((\omega-\frac{q^2-q^{-1}}{d}){\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}-(\omega+q^{-1})\sigma{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uinu}}}+{\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{uu}}})\\
&=-\frac{\sqrt{d^2-d-1}}{q^2\sqrt{d}(d-1)}|e_1\rangle+\frac{q}{\sqrt{d}}|e_2\rangle+\frac{d-2}{d-1}|e_3\rangle.
\end{aligned}$$
Thus their matrix representations in the basis states ($|e_1\rangle,|e_2\rangle,|e_3\rangle$) are given by $$E_A=diag\{0,d,0\},$$
$$E_B=\left(
\begin{array}{>{\displaystyle}l>{\displaystyle}c>{\displaystyle}r}
\frac{d^2-d-1}{d } & \frac{\sqrt{d^2-d-1} }{d } & -\frac{\sqrt{d^2-d-1} }{\sqrt{d} } \\
\frac{\sqrt{d^2-d-1} }{d} & \frac{1}{d} & -\frac{1}{\sqrt{d}} \\
-\frac{\sqrt{d^2-d-1} }{\sqrt{d}} & -\frac{1}{\sqrt{d}} & 1
\end{array}
\right),$$
$$A=diag\{q,q^{-2},-q^{-1}\},$$
$$B=\left(
\begin{array}{>{\displaystyle}l>{\displaystyle}c>{\displaystyle}r}
\frac{1}{q^4(d-1)d} & \frac{\sqrt{d^2-d-1} }{d q} & -\frac{\sqrt{d^2-d-1} }{q^2(d-1) \sqrt{d}} \\
\frac{\sqrt{d^2-d-1} }{d q} & \frac{q^2}{d} & \frac{q}{\sqrt{d}} \\
-\frac{\sqrt{d^2-d-1} }{q^2(d-1) \sqrt{d}} & \frac{q}{\sqrt{d}} & \frac{d-2}{d-1}
\end{array}
\right),$$
where $E_A$, $E_B$, $A$ and $B$ are Hermitian matrices. It is worth noting that $E_B=UE_AU^{-1},B=UAU^{-1}$, $$U=\left(
\begin{array}{>{\displaystyle}l>{\displaystyle}c>{\displaystyle}r}
\frac{1}{(d-1)d } & -\frac{\sqrt{d^2-d-1}}{d} & -\frac{ \sqrt{d^2-d-1} }{\sqrt{d} (d-1)} \\
\frac{\sqrt{d^2-d-1} }{d } & -\frac{1}{d} & \frac{1}{\sqrt{d}} \\
\frac{ \sqrt{d^2-d-1} }{\sqrt{d} (d-1)} & \frac{1}{\sqrt{d}}& -\frac{d-2}{d-1}
\end{array}
\right),$$ and they satisfy the reduced BWMA relations $$\label{bwa} \left\{ \begin{aligned}
&A-A^{-1}=\omega (I-E_A) ,\ B-B^{-1}=\omega (I-E_B),\\
&ABA=BAB ,\\
&E_AE_BE_A=E_A ,\ E_BE_AE_B=E_B ,\\
&E_AA=AE_A=\sigma E_A ,\ E_BB=BE_B=\sigma E_B ,\\
&ABE_A=E_BAB=E_BE_A ,\ BAE_B=E_ABA=E_AE_B ,\\
&AE_BA=B^{-1}E_AB^{-1} ,\ BE_AB=A^{-1}E_BA^{-1} ,\\
&E_AE_BA=E_AB^{-1} ,\ E_BE_AB=E_BA^{-1} ,\\
&AE_BE_A=B^{-1}E_A ,\ BE_AE_B=A^{-1}E_B ,\\
&E_ABE_A=\sigma^{-1}E_A ,\ E_BAE_B=\sigma^{-1}E_B, \\
&E_A^2=(1-\frac{\sigma-\sigma^{-1}}{\omega})E_A ,\
E_B^2=(1-\frac{\sigma-\sigma^{-1}}{\omega})E_B.
\end{aligned} \right.$$ We emphasize that acts on the basis ($|e_1\rangle,|e_2\rangle,|e_3\rangle$).
It is worth noting that the topological basis states are singlet states, when $\phi_{\nu}=\pi$, $\lambda=1$, $\mu=-1$, $\nu=0$ and $q=1$. In other words, $S^2|e_i\rangle=0$ and $S_z|e_i\rangle=0$, where $S=\sum_1^4S_j$, $S_j$ are the operators of spin-1 angular momentum for the $j$-th particle, $i=1,2,3$.
Summary
=======
In this paper we construct $9\times9$-matrix representations of TLA, where we used the entangled states ($|\Psi\rangle=d^{-1/2}(q^{1/2}|\lambda\mu\rangle+e^{i\phi_{\nu}}|\nu\nu\rangle+q^{-1/2}e^{i\varphi_{\mu\lambda}}|\mu\lambda\rangle)$). Then we get a family of $9\times9$ representations of BWMA. We study the entangled states $|\Psi\rangle$’s, and find the negativity related parameter $q$. The negativity became the maximum value if $q=1$. In Sce. 3, we defined the third topological graphic ${\raisebox{-0.4\height}{\includegraphics[height=0.5cm]{usu}}}$ and find three orthogonal topological basis states of BWMA, based on the former researchers. It was mentioned that the Hermitian matrices $E_A$, $E_B$, $A$ and $B$ have an interesting similar transformation matrix $U$ which satisfies $B=UAU^{-1}$ and $E_B=UE_AU^{-1}$. Based on them, we obtain a three-dimensional representation of BWMA. Finally we find the topological basis states are the spin singlet states, if $\phi_{\nu}=\pi$, $\lambda=1$, $\mu=-1$, $\nu=0$ and $q=1$. Our next work will study how the topological basis states play a role in quantum theory.
Acknowledgments
===============
This work was supported by NSF of China (Grant No.10875026)
[99]{} C.H. Bennett and D.P. DiVincenzo, [*Nature*]{} [**404**]{} (2003) 247. C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, [*Phys. Rev. Lett.*]{} [**70**]{} (1993) 1895. C.H. Bennett, and S.J. Wiesner, [*Phys. Rev. Lett.*]{} [**69**]{} (1992) 2881. M. Murao, D. Jonathan, M.B. Plenio, and V. Vedral, [*Phys. Rev. A*]{} [**59**]{} (1999) 156. C.N. Yang, [*Phys. Rev. Lett.*]{} [**19**]{} (1967) 1312; C.N. Yang, [*Phys. Rev.*]{} [**168**]{} (1968) 1920. R.J. Baxter, [*Exactly Solved Models in Statistical Mechanics*]{} (New York: Academic) (1982);\
R.J. Baxter, [*Ann. Phys.*]{} [**70**]{} (1972) 193. H.N.V. Temperley and E.H. Lieb, [*Proc. Roy. Soc. London,*]{} [**A 322**]{} (1971) 25. J. Birman, H. Wenzl, [*Trans. A.M.S.*]{} [**313**]{} (1989) 249;\
J. Murakami, [*Osaka J. Math.*]{} [**24**]{} (1987) 745. Y. Cheng, M.L. Ge, K. Xue, [*Commun Math Phys.*]{} [**136**]{} (1991) 195. V. Jones, [*Commun. Math. Phys.*]{} [**125**]{} (1989) 459. M.L. Ge, Y.S. Wu and K. Xue, [*Inter. J. Mod. Phys.*]{} [**A6**]{} (1991) 3735. G.C. Wang, K. Xue, *et al.*, [*Quantum Information Processing*]{} [**9(6)**]{} (2009) 699. M. Wadati, T. Deguchi, and Y. Akutsu, [*Phys. Rep.*]{} [**180**]{} (1989) 247332. Y. Zhang, [*J. Phys. A: Math. Gen.*]{} [**39**]{} (2006) 11599. L.H. Kauffman, [*Knots and Physics* ]{} (Singapore: World Scientific Publ CoLtd.) (1991) S.W. Hu, K.xue, and M.L. Ge, [*Phys. Rev. A*]{} [**78**]{} (2008) 022319. M. Jimbo, [*Yang-Baxter Equations on Integrable Systems*]{} (World Scientific, Singapore) (1990) L.D. Gou, *et al.*, [*Commun. Theor. Phys.*]{} [**55**]{} (2011) 263. L.D. Gou, *et al.*, [*International Journal of Quantum Information* ]{} [**8**]{} (2010) 1187. M.L. Ge and K. Xue, [*J. Phys. A: Math.*]{} [**26**]{} (1993) 281. V.F.R. Jones, [*Commun Math Phys.*]{} [**125**]{} (1987) 459. K. Zyczkowski, P. Horodecki, A. sanpera, M. lewenstein, [*Phys. Rev. A.*]{} [**58**]{} (1998) 883. G. Vidal, R.F. Werner, [*Phys. Rev. A*]{} [**65**]{} (2001) 032314. F. Wilczek, [*Phys. Rev. Lett.*]{} [**48**]{} (1982) 1144; G. Moore and N. Read, [*Nucl. Phys. B*]{} [**360**]{} (1991) 362. E. Ardonne, and K. Schoutens, [*Ann. Phys.*]{} [**322**]{} (2007) 21; A. Feiguin, S. Trebst, A.W.W. Ludwig, M. Troyer, A. Kitaev, Z.H. Wang, and M.H. Freedman, [*Phys. Rev. Lett.*]{} [**98**]{} (2007) 160409. for a review, see, C. Nayak, S.H. Simon, A. Stern, M. Freedman, and S.D. Sarma, [*e-print arXiv:0707.1889*]{}; K. Hikami, [*Ann. Phys.*]{} [**323**]{} (2008) 1729.
|
---
abstract: |
A Vehicular Ad-hoc NETwork (VANET) is a special form of Mobile Ad-hoc Network designed to provide communications among nearby vehicles and between vehicles and nearby fixed roadside equipment. Its main goal is to improve safety and comfort for passengers, but it can also be used for commercial applications. In this latter case, it will be necessary to motivate drivers to cooperate and contribute to packet forwarding in Vehicle-to-Vehicle and Vehicle-to-Roadside communications. This paper examines the problem, analyzes the drawbacks of known schemes and proposes a new secure incentive scheme to stimulate cooperation in VANETs, taking into account factors such as time and distance.
[**Keywords.**]{} Cooperation, Vehicular Ad-Hoc Network, VANET
author:
-
title: 'Stimulating Cooperation in Self-Organized Vehicular Networks'
---
Introduction
============
Vehicular ad hoc networks (VANETs) are important components of Intelligent Transportation Systems. The main benefit of VANET communication is seen in active safety systems that increase passenger safety by exchanging warning messages between vehicles. Other promising commercial applications are Added-Value Services such as: advertising support [@LPPGL07], request/provide information about nearby companies, access to Internet, etc.
A VANET may be seen as a special type of ad-hoc network used to provide communications between On-Board Units (OBUs) in nearby vehicles, and between OBUs in vehicles and Road-Side Units (RSUs), which are fixed equipment located on the road. In particular, this paper deals with the topic of Inter-Vehicle Communication when the systems in a VANET do not rely on RSUs, and consequently constitute a Mobile Ad-hoc Network (MANET).
The main advantage of VANETs is that they do not need an expensive infrastructure. However, their major drawback is the comparatively complex networking management system and security protocols that are required. This difficulty is mainly due to some specific characteristics of VANETs that allow differentiating them from the rest of MANETs such as their hybrid architecture, high mobility, dynamic topology, scalability problems, and intermittent and unpredictable communications. Consequently, these features have to be taken into account when designing any management service or security protocol.
In order to bring VANETs to their full potential, appropriate schemes to stimulate cooperation need to be developed according to the specific properties and potential applications of VANETs. Many incentive schemes to stimulate cooperation in ad-hoc networks may be found in the bibliography [@HHHH08] [@LZ03] [@SA06] [@SNR03]. Some authors have made first approaches to the topic of cooperation in VANETs [@DFM05] [@FF06] [@WC07a] [@WC07b]. Related to the proposal here described, Buttyan and Hubaux proposed in [@BH03] and [@BH07] the use of virtual credit in incentive schemes to stimulate packet forwarding. Also, Li et al. discussed some unique characteristics of the incentive schemes for VANETs in [@LW08] and proposed a receipt counting reward scheme that focuses on the incentive for spraying. However, the receipt counting scheme proposed there has a serious overspending problem. Based on the specific characteristics of VANETs, a more comprehensive weighted rewarding method is proposed here.
In particular, the proposed scheme is based on incentives where the behavior of a node is rewarded depending on its level of involvement in the routing process. Schemes based on reputation were here discarded due to the high mobility of nodes in VANETs, which makes infeasible to maintain historical information about peers behavior.
Note that an important problem that must be dealt with in rewarding incentive schemes is the possibility for selfish or malicious users in the vehicles to exaggerate their contribution in order to get more rewards. In our proposal, we assign different possible incentives to vehicles according to their contribution in packet forwarding, in an effort to achieve fairness and provide stimulation for participation. Our scheme utilizes a weighted rewarding component to decide the specific incentive in each case so they help to keep the packet forwarding attractive to the potential intermediate vehicles.
Background
==========
A VANET may be seen as a variation of a MANET where the nodes are vehicles. In both types of networks, cooperation between nodes is required for the adequate performance, so there might be thought that cooperation tools for MANETs can be also used for VANETs. In MANETs we can find two main approaches: Reputation-based schemes where packets are forwarded through the most reliable nodes, and Credit-based schemes where packet-forwarding is dealt with as a service that can be evaluated and charged. In our work we have analyzed both schemes in order to find out whether they are suitable for VANETs.
Why reputation based methods are not suitable?
----------------------------------------------
An important characteristic of VANETs is the high mobility of nodes. Taking into account such a parameter, it is impossible to establish a reputation-based scheme because it is infeasible to maintain historical information about peers’ behavior in a VANET. This is because it is possible that two vehicles meet just once in a long period of time, and it is very difficult to listen whether a neighbor node actually forwards a packet.
Why classical credit based schemes are not suitable?
----------------------------------------------------
Incentive schemes have been proposed in order to solve cooperation problems in MANETs. The so-called [*Packet Purse Model*]{} where every source node puts a sum of money that it considers enough to reach the destination has an overspending problem because the source vehicle can not predict accurately the global size of the required reward. Other known model is the so-called [*Packet Trade Model*]{} where the destination node pays the reward. In this case the model has problems because source nodes can send all the packets they want as they have not to pay for them. This model produces a network overload. Consequently, we can conclude that none of both schemes are a good solution for MANETs or for VANETs.
Main Scheme
===========
Figure \[tree\] shows a typical packet forwarding process in VANETs, here called [*Forwarding Tree*]{}. In such a figure several important features of routing in VANETs are represented:
1. The root node corresponds to the source vehicle that first sprays the message.
2. Each intermediate vehicle corresponds to one node in the tree.
3. Each node ignores those packets that it had previously received. Consequently, every vehicle is present just once in every forwarding tree.
4. Each link in the tree corresponds to an encounter in the vehicular network, which is associated with a timestamp and the spatial coordinates indicating the position of the vehicles.
![Forwarding tree[]{data-label="tree"}](Figure1.png)
According to the store-and-carry paradigm [@LW08] [@HCM09], if an intermediate vehicle stores a packet for a long time or actively sprays the packet to other vehicles, the packet will be either more likely to reach the intended destination, or to arrive to more destinations, depending on the specific goal of the routing. Therefore, by simply combining storage time and number of sprays, we can define a useful contribution metric for the intermediate vehicles. In order to stimulate intermediate vehicles to contribute more, the source vehicle should reward each intermediate vehicle according to its contribution.
Initially, the contribution $C_i$ to packet forwarding of a node $i$ during the forwarding process may be modeled as a linear convex combination balancing numbers of forwarding $f_i$ and the period the packet is stored $t_i$:
$C_i=\alpha t_i + (1 - \alpha) f_i$.
However, this basic model implies a constant share reward $R$ which is promised for the source node to each intermediate node. This model may cause an overspending problem because the source vehicle cannot guess in advance the total reward since the number of nodes in the tree cannot be predicted easily. Such a problem might be solved maintaining constant the total reward and calculating the reward associated to each intermediate node $R_i$ after the packet reaches the destination according to the following formula:
$R_i = \frac{R \cdot C_i }{C}$ where $C= \sum_i C_i$
When the packet reaches the destination, each node $i$ that participated in the forwarding should report its contribution $C_i$ to the source. The final contribution $C$ is calculated through the sum of the partial contribution of each node in the forwarding tree. Each intermediate node will receive $R_i$ as reward for forwarding. This model cannot be considered neither a good solution because selfish nodes might prefer keeping the packet rather than retransmitting it since they do not know in advance how much they can earn for forwarding and/or they might prefer not to share the reward. It happens when an intermediate node forwards the packet to a non final node because its proportional reward might decrease.
In our first proposal we incorporate several parameters to be considered when dealing with rewarding. They are related to information such as packet delivery deadline and number of forwardings. In order to avoid that nodes prefer keeping the packet rather than retransmitting it, we consider that a packet should have a deadline depending on the characteristics of the information contained. If the sent information is added-value information, then the deadline will be longer, whilst if the information is related to traffic safety, the deadline must be very short due to the urgency of transferring the information. The following notation is used to describe the parameters for the computation of rewards:
- Packet delivery deadline $T_{j}$.
- Period $t_{ij}$ that packet $j$ is stored by node $i$.
- Number of forwardings $f_{ij}$ of packet $j$ by node $i$ before the deadline $T_{j}$.
- Balancing factor $\alpha$
With these parameters we present our first proposal of contribution function: $
C_{ij} =\alpha$($t_{ij}T_{j})+ (1-\alpha)$$f_{ij}$ where $\alpha \in (0,1).
$
In this first approach to the solution, the contribution of node $i$ for spreading packet $j$ is proportional to the time $t_{ij}$ the packet is transported by the packet deadline $T_{j}$, and to the number of the forwardings $f_{ij}$. By counting the number of forwardings in this function, the objective is to encourage nodes to forward packets and not to keep them without forwarding them.
If the rate between time and deadline $t_{ij}/T_{j}$ were considered as factor, when the message is urgent and the deadline is short, the contribution would be higher. However in such a case, once the time $t_{ij}$ overpasses the deadline $T_{j}$, the contribution still goes on increasing and even faster because the proportional factor is greater than 1. This effect can be corrected by using the inverse of the deadline so that the more urgent the message, the greater the value $1/T_{j}$. Therefore, when the time that the packet is stored passes the deadline the user contribution is no more increased. However, this is neither a good solution because although the deadline has been reached, the forwarding node continues getting a reward although it is a small amount.
Our second proposal tries to solve these problems. We propose a new contribution function in which three parameters are used, which can be interesting both for the source node and/or for the forwarding node. In particular we consider the following additional notation to describe the parameters for the computation of rewards:
- Distance $d_{ij}$ between source and destination nodes when the packet $j$ is relayed by node $i$.
- Maximum distance $D_{j}$ where the information in the packet $j$ is considered interesting by the receivers.
Each of the parameters considered in this convex function has a balancing factor, represented by $\alpha_1$, $\alpha_2$ and $\alpha_3$. The value that is assigned to each $\alpha_i$ depends on the relevance that the source node prefers to assign to each component represented in the contribution function:
$
C_{ij} =\alpha_1$ $T_{j}(1-e^{-t_{ij}})+ \alpha_2$ $f_{ij}+ \alpha_3$ $(-D_{j}(1-e^{-d_{ij}})+D_j)$ where $\sum\limits_{k=1}^3 \alpha_k =1.
$
In the next subsections each part of this function is detailed, and both the justification why they are used and the repercussion they have in the contribution function are given.
Time
----
As discussed above, time is one of the most important parameters when trying to assure that a packet reaches the intended destination. If a vehicle stores a packet for a long time, it could forward the package to more vehicles. However, this parameter could produce a selfish behavior because a node could prefer not to forward it and in this way not to share the final reward with potential forwarding nodes. This effect is avoided by considering in the proposed metric the component associated to the following formula:
$T_{j}(1-e^{-t_{ij}})$.
This function corresponds to the Stokes formula, which has a characteristic asymptotical behavior. This function is intended to set a maximum time $T_{j}$ that a node should store one packet. Note that the value of contribution increases when time increases. When $t_{ij}$ reaches the threshold $T_{j}$, the growth of contribution stops. In this way, a selfish behavior can be avoided because if the time threshold is properly set, those vehicles that retransmit the packet before the deadline will have increased their contribution.
![Contribution versus time and distance[]{data-label="cvt"}](Ct-Cd.png)
Note in the example of Figure \[cvt\] that the value of contribution increases when the time increases, and that when $t_{ij}$ reaches the threshold $T_{j}$ the contribution increase stops. In this way, both selfish behavior and forwarding after deadline are discouraged because vehicles that retransmit the packet before deadline will have their contribution increased.
Forwarding
----------
The second term in the proposed contribution metric is related to the ultimate goal of our work. It deals with measuring the forwarding of packets by each intermediate node. This process is quite simple. It has not any restriction such as maximum or minimum possible values. It consists of increasing the contribution of node $i$ to relay the packet $j$ by:
$f_{ij}$.
According to this factor, the more the vehicles collaborate in forwarding a packet, the bigger their final contribution is. In the proposed function, this parameter is the one that increases the contribution faster before the deadline. Consequently, the balancing factor $\alpha_2$ must be higher than the other two factors in order to encourage the forwarding of packets.
Distance
--------
The evaluation of the effect of distance in the share rewarding process is the goal of the third term of the contribution function. This term has been incorporated thinking that in many cases information generated at a certain location is not interesting out of a radius distance from than point. With this idea in mind, when the vehicles go too far from the source of the original packet, this value decreases.
For example, if we talk about an accident in a city center, it has not sense that the message reaches a neighbor city. Other possible situations where the same idea is applicable is where the information is sent by a commercial centre, hotel or restaurant, for instance.
This term is similar to the one related to time commented in subsection \[time\]. The goal is to obtain a function with asymptotical behavior that tends to zero when distance is near to $D_{j}$. The value $D_j$ is established by the source node. The expression that models this behavior is:
$-D_{j}(1-e^{-d_{ij}})+D_j$.
Figure \[cvt\] shows an example where as the vehicle moves away from the source its contribution decreases, and when it reaches certain point it nulls. In this way, the vehicle does not get any benefit if it retransmits the packet outside the radio.
Simulation Analysis
===================
In order to make a study of the proposal, several VANETs simulations have been implemented in NS-2. The NS-2 simulation parameters are the following: 15 nodes placed at random in an area of 800m x 800m. The range of action of each node is 100m. In each simulation, a node is randomly chosen and it starts sending a packet to its neigbours, who send it to all the nodes they meet inside their range of action. In Figure \[Simulation\] we analyze the relationship among the rewards and the different parameters of the contribution function. According to the time plot, the scheme seems to send bigger rewards to those nodes who store packets for longer. However, note that these rewards are influenced by the number of forwards and the distance between the source node and the nodes forwarding the packet. In the forwards graph, the reward average increases according to the number of forwards. Finally, in the distance plot the scheme seems to give lower rewards to those nodes whose distance increases according to the source node initial position. For some nodes at a large distance, the reward average increased due to that their spray was bigger than the spray of the nodes in the same distance. This scheme provides more reward to the nodes that effectively sprayed the packet for a long time. Also according to Figure \[descents\], the proposal gives more reward to those nodes with higher contributions, which are usually those nodes that have more descendents. Consequently, cooperation among nodes is guaranteed thanks to the proposal.
Conclusion
==========
In this paper we have seen that a simple adaptation of known cooperation enforcement schemes defined originally for MANETs is not adequate to incentivize cooperation in VANETs. Consequently, we have proposed a new scheme where incentives are defined by a convex function that depends on different parameters. We have designed a metric for contribution according to the characteristics of VANETs and to parameters that are important both for source node and for enforcing cooperation among nodes. We conclude from our study that when designing these methods for distributing a reward, the parameters to be taken into account should be carefully assessed according to the network conditions.
Since this is a work in progress, many open questions exist such as the the analysis of how can data associated to traffic and weather conditions can be used in order to improve the efficiency of the proposal.
![Simulation[]{data-label="Simulation"}](Simulacion.png)
![Reward versus number of descents[]{data-label="descents"}](descents.png)
Acknowledgment {#acknowledgment .unnumbered}
==============
Research supported by the Spanish Ministry of Education and Science and the European FEDER Fund under TIN2008-02236/TSI Project, and by the Agencia Canaria de Investigación, Innovación y Sociedad de la Información under PI2007/005 Project.
[10]{}
Buttyan, L. and Hubaux, J.P.: Stimulating Cooperation in Self-Organizing Mobile Ad Hoc Networks. ACM Mobile Networks and Applications, 8(5), October (2003)
Buttyan, L. and Hubaux, J.P.: Security and Cooperation in Wireless Networks. Cambridge Univ. Press (2007)
Dotzer, F., Fischer, L. and Magiera, P.: VARS: A Vehicle Ad-Hoc Network Reputation System. Sixth IEEE International Symposium on a World of Wireless Mobile and Multimedia Networks. WoWMoM 2005. 13-16 June 2005 pp. 454-456 (2005)
Fonseca, E. and Festag, A.: A Survey of Existing Approaches for Secure Ad Hoc Routing and Their Applicability to VANETS, Technical Report NLE-PR-2006-19, NEC Network Laboratories, March (2006)
Ho, Y.H., Ho, A.H., Hamza-Lup, G.L. and Hua, K.A.: Cooperation Enforcement in Vehicular Networks. International Conference on Communication Theory, Reliability, and Quality of Service, 2008. CTRQ’08. June 29-July 5 pp. 7-12 (2008)
Lee, S., Pan, G., Park, J., Gerla, M. and Lu, S.: Secure Incentives for Commercial Ad Dissemination in Vehicular Networks, MobiHoc’07, Canada, Sep 9-14 (2007)
Li, F. and Wu, J.: FRAME: An Innovative Incentive Scheme in Vehicular Networks. Proc. of IEEE International Conference on Communications (ICC) (2009)
Hernandez-Goya, C., Caballero-Gil, P., Molina-Gil, J. and Caballero-Gil P.:Cooperation Enforcement Schemes in Vehicular Ad-Hoc Networks. Lecture Notes in Computer Science. EUROCAST. Vol: No. 5717, Spain Feb 15-20, (2009).
Liu, P. and Zang, W.: Incentive-based modeling and inference of attacker intent, objectives, and strategies, Proc. of the 10th ACM Computer and Communications Security Conference (CCS’03), Washington, DC, October pp. 179-189 (2003)
Shastry, N. and Adve, R.S.: Stimulating cooperative diversity in wireless ad hoc networks through pricing", IEEE Int. Conf. on Communications, June (2006)
Srinivasan, V., Nuggehalli, P. and Rao, R.R.: Cooperation in Ad Hoc Networks, Proc. of Infocom, San Francisco, CA (2003)
Wang, Z. and Chigan, C.: Countermeasure Uncooperative Behaviors with Dynamic Trust-Token in VANETs. IEEE International Conference on Communications, 2007. ICC’07. 24-28 June pp. 3959-3964 (2007)
Wang, Z. and Chigan, C.: Cooperation Enhancement for Message Transmission in VANETs. Wireless Personal Communications: An International Journal Volume 43, Issue 1 October pp. 141-156 (2007)
|
---
abstract: 'We have studied the underlying algebraic structure of the anharmonic oscillator by using the variational perturbation theory. To the first order of the variational perturbation, the Hamiltonian is found to be factorized into a supersymmetric form in terms of the annihilation and creation operators, which satisfy a $q$-deformed algebra. This algebraic structure is used to construct all the eigenstates of the Hamiltonian.'
address: |
$^1$ Department of Physics, University of Seoul, Seoul 130-743, Korea\
$^2$ Department of Physics, Kunsan National University, Kunsan 573-701, Korea\
$^3$ Department of Physics and Research Institute for Basic Sciences, Ewha Womans University, Seoul 120-750, Korea\
$^4$ Department of Physics Education, Seoul National University, Seoul 151-742, Korea\
$^5$ Department of Physics and Institute for Mathematical Sciences, Yonsei University, Seoul 120-749, Korea
author:
- 'Dongsu Bak, $^{1,}$[^1] Sang Pyo Kim,$^{2,}$[^2] Sung Ku Kim, $^{3,}$[^3] Kwang-Sup Soh, $^{4,}$[^4] and Jae Hyung Yee $^{5,}$[^5]'
title: 'Factorization and q-Deformed Algebra of Quantum Anharmonic Oscillator'
---
Quantum anharmonic oscillator has been frequently studied as a toy model for developing various approximation methods in quantum mechanics and quantum field theory [@dineykhan; @justin; @cea; @lee; @bak1]. Recently it has been used to develop various approaches to the variational perturbation theory [@cea; @lee], which enables one to compute the order by order correction terms to the well known variational approximation. More recently the model has been utilized to establish the Liouville-Neumann approach to the variational perturbation theory [@bak1], where one constructs the annihilation and creation operators as perturbation series in the coupling constant whose zeroth order terms constitute those of the Gaussian approximation.
However, the underlying algebraic structure of the anharmonic oscillator for its own sake has rarely been studied. In ref.[@bak1], we have shown that to the first order of the variational perturbation the Hamiltonian is factorized as in the case of the simple harmonic oscillator, while the annihilation and creation operators satisfy the $q$-deformed algebra rather than the usual commutation relations. This is an interesting algebraic structure of the theory which may enable one to obtain more information on the theory. The connection between the $q$-deformed algebra and the quasi-exactly solvability has been found for certain type of potentials [@bonatsos], and the possibility of a $q$-deformed quartic oscillator has also been suggested from the study of the energy spectra obtained by the standard perturbation method [@chung].
It is the purpose of this letter to study the algebraic structure of the anharmonic oscillator to the first-order variational perturbation, and to utilize this structure to obtain the general energy eigenstates of the system. It is the $q$-deformed algebraic structure of the theory that enables us to find the $q$-deformed Fock space [@biedenharn].
We now consider the anharmonic oscillator described by the Hamiltonian, $$\hat{H} = \frac{1}{2}{\hat{p}}^2 + \frac{1}{2} \omega^2 {\hat{x}}^2
+ \frac{1}{4} \lambda {\hat{x}}^4,
\label{anhar}$$ where the mass is scaled to unity for simplicity. In the variational Gaussian approximation one searches for a simple harmonic oscillator whose energy eigenstates minimize the expectation value of the Hamiltonian (\[anhar\]). For this purpose, we introduce a set of operators, $\hat{a}$ and $\hat{a}^{\dagger}$, as linear functions of the dynamical variables $\hat{x}$ and $\hat{p}$: $$\begin{aligned}
\hat{a} &=& \sqrt{\frac{\Omega_G}{2 \hbar}} \hat{x}
+ i \frac{1}{\sqrt{2 \Omega_G \hbar}} \hat{p},
\nonumber\\
\hat{a}^{\dagger} &=& \sqrt{\frac{\Omega_G}{2 \hbar}} \hat{x}
- i \frac{1}{\sqrt{2 \Omega_G \hbar}} \hat{p},
\label{ann-cre-operators}\end{aligned}$$ which are the annihilation and creation operators for the simple harmonic oscillator described by the Hamiltonian, $$\hat{H}_G = \frac{1}{2} {\hat{p}}^2 + \frac{1}{2} \Omega^2_G {\hat{x}}^2.
\label{har}$$ In terms of the annihilation and creation operators (\[ann-cre-operators\]), the Hamiltonian (\[anhar\]) is represented as $$\begin{aligned}
\hat{H} &=& \frac{\hbar}{2} \Bigl(\frac{\omega^2}{\Omega_G}
+ \Omega_G + \frac{3 \lambda \hbar}{2 \Omega_G^2} \Bigr) \Bigl( \hat{a}^{\dagger} \hat{a} + \frac{1}{2}\Bigr)
- \frac{3 \lambda \hbar}{16 \Omega_G^2}
\nonumber \\
&+& \frac{\hbar}{4} \Bigl(\frac{\omega^2}{\Omega_G}
- \Omega_G + \frac{3 \lambda \hbar}{2 \Omega_G^2} \Bigr) \Bigl( {\hat{a}}^2 + \hat{a}^{\dagger 2} \Bigr)
+ \frac{\lambda {\hbar}^2 }{16 \Omega_G^2 }
\sum_{k = 0}^{4} {4 \choose k} \hat{a}^{\dagger k} \hat{a}^{4-k}.
\label{oscillator1}\end{aligned}$$ In the variational Gaussian approximation one evaluates the expectation value of the Hamiltonian (\[oscillator1\]) with respect to the state annihilated by the annihilation operator $\hat{a}$: $$\hat{a} \vert 0 \rangle_{[0]} = 0,
\label{vacuum1}$$ and minimizes the expectation value, which leads to the gap equation, $$\Omega_G^2 = \omega^2 + \frac{3 \lambda \hbar}{2 \Omega_G}.
\label{freq}$$ The gap equation (\[freq\]) completely determines the operator $\hat{a}$ and $\hat{a}^{\dagger}$ of (\[ann-cre-operators\]), which gives the Gaussian approximation of the ground state of the system through Eq. (\[vacuum1\]).
By choosing the frequency (\[freq\]), the Hamiltonian of the anharmonic oscillator (\[oscillator1\]) can now be expressed as, $$\hat{H} = \frac{\hbar}{2} \Omega_G
\Bigl( \hat{a}^{\dagger} \hat{a} + \hat{a} \hat{a}^{\dagger}
\Bigr) - \frac{3 \lambda {\hbar}^2 }{16 \Omega_G^2}
+ \frac{\lambda {\hbar}^2 }{16 \Omega_G^2}
\sum_{k = 0}^{4} {4 \choose k} \hat{a}^{\dagger k} \hat{a}^{4-k}.
\label{anhar rep}$$ It is convenient to rewrite the Hamiltonian (\[anhar rep\]) as, $$\hat{H} = \hbar \Biggl\{ \frac{\Omega_{[1]}}{2}
\Bigl( \hat{a}^{\dagger} \hat{a} + \hat{a} \hat{a}^{\dagger}
\Bigr) + \frac{3 \lambda \hbar}{8 \Omega_G^2}
\hat{a}^{\dagger} \hat{a}
+ \frac{\lambda \hbar}{16 \Omega_G^2}
\sum_{k = 0}^{4} {4 \choose k} \hat{a}^{\dagger k} \hat{a}^{4-k}
\Biggr\},
\label{rep 2}$$ where $$\Omega_{[1]} = \Omega_G - \frac{3 \lambda \hbar}{8 \Omega_G^2},$$ and we have used the relation, $\bigl{[} \hat{a}, \hat{a}^{\dagger} \bigr{]} =
\hat{a} \hat{a}^{\dagger} + \hat{a}^{\dagger} \hat{a} - 2 \hat{a}^{\dagger} \hat{a} $. This representation can be used to construct the annihilation operator of the anharmonic oscillator as a perturbation series in the coupling constant, which leads to the Liouville-Neumann approach to the variational perturbation theory [@bak1].
We now turn to the main issue of this paper: the factorization of the Hamiltonian. The solvability of the simple harmonic oscillator (\[har\]) is rooted in the fact that the Hamiltonian is factorized into a supersymmetric form, $$\hat{H}_G = \frac{\hbar}{2} \Omega_G
\Bigl( \hat{a}^{\dagger} \hat{a} +
\hat{a} \hat{a}^{\dagger} \Bigr),
\label{supersymmetric form1}$$ and the fact that the annihilation and creation operators satisfy the standard commutation relation, $$[ \hat{a}, \hat{a}^{\dagger}] = 1.
\label{commutation relation}$$
We now ask whether the same kind of factorization occurs in the case of the anharmonic oscillator (\[anhar\]): $$\hat{H} = \frac{\hbar}{2} \Omega \Bigl( \hat{A}^{\dagger} \hat{A} +
\hat{A}\hat{A}^{\dagger} \Bigr),
\label{fact}$$ in a perturbative sense, and if it does, what kind of commutation relation does the operators $\hat{A}$ and $\hat{A}^{\dagger}$ satisfy. The recent result of ref.[@bak1] suggests that one can find the annihilation and creation operators $\hat{A}$ and $\hat{A}^{\dagger}$ for the anharmonic oscillator and their commutation relation to each order of the variational perturbation in $\lambda$. We have found that the operators $\hat{A}$ and $\hat{A}^{\dagger}$ are given by, to the order $\lambda \hbar$, $$\begin{aligned}
\hat{A}_{[1]} &=& \hat{a} + (\lambda \hbar) \sum_{k = 0}^{3}
c_k \hat{a}^{\dagger (3-k)} \hat{a}^{k},
\nonumber \\
\hat{A}^{\dagger}_{[1]} &=& {\Bigl( \hat{A}_{[1]} \Bigr)}^{\dagger} ,\end{aligned}$$ where $c_k$ are constants to be determined [@bak1]. The requirement that the Hamiltonian (\[rep 2\]) be of the factorized form (\[fact\]) determines the frequency $\Omega$ and the constants $c_k$’s to this order of the variational perturbation: $$\Omega = \Omega_{[1]} = \Omega_G - \frac{3 \lambda \hbar}{8 \Omega_G^2 },
\label{Omega equation1}$$ and $$c_0 = \frac{1}{16 \Omega_G^3}, ~~
c_1 = \frac{6}{16 \Omega_G^3}, ~~
c_2 = \frac{3}{16 \Omega_G^3},~~
c_3 = - \frac{2}{16 \Omega_G^3}.
\label{cons 1}$$ We thus find that the Hamiltonian of the anharmonic oscillator factorizes as, $$\hat{H} = \frac{\hbar}{2} \Omega_{[1]} \Bigl(
\hat{A}^{\dagger}_{[1]} \hat{A}_{[1]} +
\hat{A}_{[1]}\hat{A}^{\dagger}_{[1]} \Bigr) + {\cal O} (\lambda^2),
\label{fact 2}$$ and that the annihilation and creation operators $\hat{A}$ and $\hat{A}^{\dagger}$ satisfy the commutation relation, $$[\hat{A}_{[1]}, \hat{A}_{[1]}^{\dagger}] = 1
+ \frac{3 \lambda \hbar}{4 \Omega_G^3}
\hat{A}_{[1]}^{\dagger} \hat{A}_{[1]},
\label{com}$$ to this order of the variational perturbation.
Note that the commutation relation (\[com\]) defines a $q$-deformed algebra in the form [@kulish], $$\hat{A}_{[1]}
- q^2_{[1]} \hat{A}^{\dagger}_{[1]} \hat{A}_{[1]} = 1,$$ with the deformation parameter, $$q^2_{[1]} = 1 + \frac{3 \lambda \hbar}{4 \Omega_G^3}.
\label{q2-equation1}$$ This fact can be used to find all the eigenstates of the anharmonic oscillator (\[anhar\]) by using the Fock space structure of a $q$-deformed oscillator [@biedenharn]. The vacuum state is defined by, $$\hat{A}_{[1]} \vert 0 \rangle_{[1]} = 0,$$ and the number states by $$\vert n \rangle_{[1]} = \frac{1}{\sqrt{[n]!}} \Bigl( \hat{A}^{\dagger}_{[1]} \Bigr)^n \vert 0 \rangle_{[1]},$$ where $$[n] \equiv \frac{q^{2n}_{[1]} - 1}{q^2_{[1]} -1}.
\label{n-equation1}$$ These number states are the eigenstates of the following operators: $$\begin{aligned}
\hat{A}^{\dagger}_{[1]} \hat{A}_{[1]}
\vert n \rangle_{[1]} &=& [n] \vert n \rangle_{[1]}, \nonumber \\
\hat{A}_{[1]} \hat{A}^{\dagger}_{[1]}
\vert n \rangle_{[1]} &=& [n+1] \vert n \rangle_{[1]}.\end{aligned}$$ We thus find that the number states $\vert n \rangle_{[1]}$ are the eigenstates of the Hamiltonian corresponding to the eigenvalue, $$E_{[1] n} = \frac{\hbar}{2} \Omega_{[1]}
\Bigl([n] + [n+1] \Bigr),
\label{energy eigenvalue1}$$ to the first order of the variational perturbation. Using Eqs. (\[q2-equation1\]), (\[n-equation1\]) and (\[energy eigenvalue1\]), we finally obtain the energy eigenvalues of the anharmonic oscillator, to the order $\lambda \hbar $: $$E_{[1] n} = \frac{\hbar}{2} \Omega_{[1]} \Bigl(
2n + 1 + \frac{3 \lambda \hbar}{4 \Omega_G^3 } n^2 \Bigr),$$ which agrees with those obtained by You, et al. [@cea].
In summary, we have studied the algebriac structure of the anharmonic oscillator to the first-order in the variational perturbation theory. It has been found that the Hamiltonian is factorized into a supersymmetric form in terms of the annihilation and creation operators, which are expanded around those of the Gaussian variational approximation. It has also been shown that the anharmonic oscillator has the algebra of a $q$-deformed oscillator with $SU_q (1,1)$ symmetry, which was used to construct all the eigenstates of the Hamiltonian to this order of the variational perturbation. Further study is needed to see if this interesting algebraic structure is also respected by the higher order correction terms.
This work was supported by the Basic Science Research Institute Program, Korea Ministry of Education under Project No. BSRI-98-2418, BSRI-98-2425, BSRI-98-2427, and by the Center for Theoretical Physics, Seoul National University. SPK was also supported by the Non-Directed Research Fund, the Korea Research Foundation, 1997, and JHY by the Korea Science and Engineering Foundation under Project No. 97-07-02-02-01-3.
[99]{} M. Dineykhan, G. V. Effimov, G. Ganbold, and S. N. Nedelko, [*Oscillator Representation in Quantum Physics*]{} (Springer, Berlin, 1995). J. Zinn-Justin, [*Quantum Field Theory and Critical Phenomena*]{}(Claredon Press, Oxford, 1996). H. Kleinert, Phys. Lett. A [**173**]{}, 332 (1993); [*Path Integrals, $2^{nd}$ ed.*]{} (World Scientific, Singapore, 1995); P. Cea and L. Tedesco, Phys. Rev. D [**55**]{}, 4967 (1997); S. K. You, K. J. Jeon, C. K. Kim, and K. Nahm, Eur. J. Phys. [**19**]{}, 179 (1998); see references of these papers for earlier development. G. H. Lee and J. H. Yee, Phys. Rev. D [**56**]{}, 6573 (1997); G. H. Lee, T. H. Lee, and J. H. Yee, “Perturbative Expansion around the Gaussian Effective Potential of the Fermion Field Theory”, preprint hep-th/9804094. D. Bak, S. P. Kim, S. K. Kim, K.-S. Soh, and J. H. Yee, “Perturbation Method beyond the Variational Gaussian Approximation: The Liouville-Neumann Approach”, preprint SNUTP-98-074, hep-th/9808001. D. Bonatsos, C. Daskaloyannis and K. Kokkotas, J. Math. Phys. [**33**]{}, 2958 (1992); D. Bonatsos, C. Daskaloyannis and H. A. Mavromatis, Phys. Lett. A [**199**]{}, 1 (1995). W.-S. Chung, K.-S. Chung, S.-T. Nam, and C.-I. Um, J. Korean Phys. Soc. [**27**]{}, 117 (1994). L. C. Biedenharn, J. Phys. A [**22**]{}, L873 (1989); A. J. Macfarlane, J. Phys. A [**22**]{}, 4581 (1989). P. Kulish and E. Damaskinsky, J. Phys. A [**23**]{}, L415 (1990); M. Chaichian, R. G. Felipe, and C. Montonen, J. Phys. A [**26**]{}, 4017 (1993).
[^1]: Electronic address: dsbak@mach.uos.ac.kr
[^2]: Electronic address: sangkim@knusun1.kunsan.ac.kr
[^3]: Electronic address: skkim@theory.ewha.ac.kr
[^4]: Electronic address: kssoh@phya.snu.ac.kr
[^5]: Electronic address: jhyee@phya.yonsei.ac.kr
|
During recent years, low-temperature heat transport experiments on electrical insulators have been extended to mesoscopic systems [@Roukes], where the wavelength of thermal phonons can be comparable to the geometrical size of the device. In this regime, phonon transport through a thermal conductor such as an electrically insulating solid wire, formed from an undoped semiconductor may exhibit ballistic waveguide propagation [@Blencowe; @Kirczenow]. This possibility has stimulated interest in the guided-wave, phonon-mediated heat conductance, $\varkappa (T)$ of ballistic wires (with a width $W$ much smaller than the length $L$) connecting a heat reservoir to a thermal bath [@Blencowe; @Kirczenow].
In the present paper, we analyse the low-temperature ($k_{B}T\sim \hbar c\pi
/W$, where c is the sound velocity) heat transport in relatively long ($%
L/W\sim 100$) free-standing insulating wires by taking into account the effect of surface roughness. The idea behind this analysis is based on the assumption that, in a long wire, the wire edge or surface roughness may result in strong scattering, and even in the localisation of acoustic waves in the intermediate-frequency range, whereas the low-frequency part of the phonon spectrum would always have ballistic properties due to the specifics of sound waves. In the high-frequency part of the spectrum, phonons would have quasiballistic properties, too. This may result in a non-monotonic temperature dependence of the thermal conductance of such a system.
To verify the possibility of the existence of such a regime, in principle, we investigate the dependence on frequency $\omega $ of the transmission coefficient $\left\langle \Gamma (\omega )\right\rangle $ using a simplified model of a solid waveguide which has been chosen to reflect two features of this problem: the influence of roughness on the propagation of vibrations and the suppression of scattering on the roughness upon the decrease of the excitation frequency. We approach the problem numerically, by studying the transmission coefficent averaged over many realizations of a wire characterized by a given distribution of length scales in the surface roughness [@trans] and using this to find the heat conductance, $%
\varkappa (T)$. Phonon-mediated heat transport in quasi-one-dimensional systems can be studied using the same theoretical techniques as electron transport[@Ball; @Buttiker]. However, in contrast to electrical conductance, which at low temperatures is determined by transport properties of electron waves at the Fermi energy, the thermal conductance of a phonon waveguide is determined by all phonon energies $\omega $ up to $\hbar \omega \sim k_{B} T$. This smears out effects of confinement in the transverse direction in the temperature dependence of $\varkappa (T)$ in ballistic systems [@Blencowe], but results in a pronounced feature in $\varkappa (T)$ for strongly disordered free-standing wires. The latter has the form of an intermediate saturation regime in $\varkappa (T)$ following a linear $T$dependence at the lowest temperatures, with an anomalous length dependence of the saturation value, that scales as $\varkappa _{sat}\propto L^{-1/2}$ for wires with a white-noise spectrum of roughness.
[*Transmission coefficient analysis and localization of acoustic modes*]{}. Below, we classify phonon modes in a wire by the number $n$ of nodes in the displacement amplitude. The $n=0$ lowest-frequency vibrational mode ($\omega
<\omega _{1}=c\pi /W$, where $c$ is the velocity of sound) corresponds to equal displacements over the cross section of a free-standing wire and has a linear dispersion; others have frequency gaps, $\omega _{n}(q)=\sqrt{\left(
cq\right) ^{2}+\left( \pi nc/W\right) ^{2}}$. The aforementioned feature originates from the fact that edge disorder suppresses the transmission of all coherent phonon modes at high frequencies, but has a little influence on the $n=0$ mode at $\omega \rightarrow 0$, where $\Gamma (\omega \rightarrow
0)\rightarrow 1$. The transmission coefficient of this mode is mainly determined by direct backscattering, whose rate depends on the intensity of the surface roughness harmonic $\left\langle \delta W_{q}^{2}\right\rangle $ with wave number $q\sim \omega /c$. According to Rayleigh scattering theory in one dimension, the mean free path of this mode diverges at $\omega
\rightarrow 0$ as
$$l_{0}(\omega )\sim \left( \omega _{1}/\omega \right) ^{2}
\left( W^2 / \left\langle \left(
\delta W_{q=\omega /c}\right) ^{2}\right\rangle \right)W,$$
even if long-wavelength Fourier components $\delta W_{q}$ are equally represented in the surface roughness $\delta W(x)$. In an infinitely long wire with white-noise randomness on the surface, this results in a localization length $L_{\ast }$ for the lowest phonon mode which diverges at $\omega \rightarrow 0$ as
$$L_{\ast }(\omega )\sim l_{0}(\omega )\propto \omega ^{-2}. \label{eq:three}$$
The latter statement [@Fractal] is based on the equivalence between the localization problem for various types of waves [@Elattari]. For a wire shorter than $L_{\ast }$ scattering yields
$$1-\left\langle \Gamma (\omega )\right\rangle =\alpha \omega ^{2}\;{\rm at}%
\;\omega <\omega _{1}. \label{lomega}$$
In contrast, at higher frequencies, all modes in a wire with a white-noise randomness backscatter (either via intra- or inter-mode process) typically at the length scale of $l\sim W/\left\langle \left( \delta W/W\right)
^{2}\right\rangle $. Hence, for frequencies $\omega >\omega _{1}$, the transmission coefficient tends to follow a linear frequency dependence, $%
\left\langle \Gamma \left( \omega \right) \right\rangle \sim \left( \omega
W/c\right) \left( l/L\right) \ln (W\omega /c)$, which is typical for diffusion in quasiballistic systems [@Pippard; @Tesanovich; @Leadbeater]. The crossover from the low-frequency regime to the intermediate-frequency range in a long enough wire can, therefore, be nonmonotonic, with a pronounced fall towards zero at $\omega \lesssim \omega _{1}$, similar to that discussed by Blencowe [@Blencoweloc] in relation to the phonon propagation in thin films. As a result, an irregular wire may exhibit ballistic phonon propagation at low frequencies, whereas, at higher frequencies $\omega _{1}\lesssim \omega $, surface roughness would yield diffusive phonon propagation, or even localization.
The numerical simulations reported below confirm the above naive expectations. In these simulations, we model the phonons in a crystalline wire cut from a thin film (with the thickness much less than the wire width) as longitudinal waves in a two-dimensional strip whose width $W(x)$ fluctuates with rms value $\left\langle \left( \delta W/W\right)
^{2}\right\rangle ^{1/2}=0.1$ on a length scale $\xi $ longer or of order $W$. The effect of the width fluctuations consists of the scattering of acoustic waves propagating along the wire. The model that we adopt here gives a very simplified representation of a real system, since we ignore the existence of torsional and transverse bending modes of the wire excitations, which are known to transfer heat in adiabatic ballistic constrictions [@Kirczenow]. However, it takes into account two features of sound waves: their scattering and the possible localization by the surface roughness, and the almost ballistic properties at both ultra-low and high frequencies. In a continuum model, these lattice vibrations are described by a displacement field $u(x,y)$, which obeys 2D wave equation inside a wire
$$\omega ^{2}u+c^{2}\nabla ^{2}u=0, \label{eq:one}$$
Displacements obey free boundary conditions ${\bf n}\nabla u=0$ at $y=\pm
(W/2+\delta W_{s}(x))$, where $s=1,2$ indicates the upper and lower edge of the wire, and [**n**]{} stands for the local normal direction to the wire edge. In the simulations, we discretize Eq. (\[eq:one\]) on a square lattice with about 200 sites across the wire cross section and, then, compute $\left\langle \Gamma (\omega )\right\rangle $ numerically using the transfer matrix method for 100 disorder realisations [@RemNum]. Our numerical code overcomes problems of instability by $QL$ factorising the transfer matrices at each step [@Robinson]. Furthermore, at the end of each calculation, the S-matrix is checked for unitarity.
Fig. 1 shows the results of such simulations obtained for wires with a white-noise spectrum of roughness and an aspect ratio $L/W=30$. Both the low- and high-frequency asymptotic behaviour of transmission coefficient confirm the expected non-monotonic dependence of $\left\langle \Gamma
(\omega )\right\rangle $ for a wire of length $L\lesssim L_{\ast }$. Calculations using longer wires (with $L\gg L_{\ast }$) shows a deeper fall in $\left\langle \Gamma (\omega )\right\rangle $ over a broader range of frequencies, since phonons with frequencies $\omega \sim \omega _{1}$ behave as localized vibrations, and their transmission coefficient becomes exponentialy small. The inset of Fig. 1 shows the corresponding frequency dependence of the inverse localization length of very long wires (with $%
L/W\sim 1000$), which is approximately in agreement with the result of Eq. (\[eq:three\]). Therefore, the transmission of phonons through rough wires with white-noise roughness of the edges can be characterised by the following regimes. At low frequencies, $\omega <\omega _{\ast }$, where $%
\omega _{\ast }(L)\sim \omega _{1}\left( W^{2}/\left\langle \left(%
\delta W_{q\rightarrow 0}\right) ^{2}\right\rangle \right)^{1/2}\sqrt{W/L}$, $\Gamma \sim 1$ so that phonons pass through the wire almost ballistically. At intermediate frequencies, $\omega _{\ast }<\omega \lesssim \omega _{1}$, phonons in the lowest mode are localized on a length scale shorter than the wire length, and $\Gamma \rightarrow 0$. Upon a further increase of $\omega $, modes with $n\neq 0$ take part in the scattering, the multi-mode localization length increases, and at $\omega \gtrsim cL/Wl$ the localization length becomes longer than the sample length, thus restoring a quasi-ballistic character to phonon transport [@Tesanovich; @Leadbeater]. The first two regimes of low-frequency phonon propagation ($\omega <\omega _{1}$) through a wire with $L\gg l$ can be jointly described as a function of a single parameter, $%
L/l_{0}(\omega )$ $$\left\langle \Gamma (\omega <\omega _{1})\right\rangle =p(\omega /\omega
_{\ast });\;p(0)=1,{\rm \;}p(x\gg 1)\sim e^{-x}. \label{localized}$$ Note that the decline of $\Gamma $ in the vicinity of $\omega \sim \omega
_{1}$ strongly depends on the Fourier spectrum of the roughness. To illustrate this, we analyzed the effect of roughness composed of harmonics with wave numbers $q$ restricted to two intervals: (a) $0<q<\pi /W$ and (b)$%
\ \frac{3}{2}\pi /W<q<\frac{7}{2}\pi /W$. The result is shown in Fig. 2 (a) and (b), respectively. The spectral form of the randomness is relevant, since it determines the intensity of Bragg-type backscattering processes. Such processes are the most efficient in forming localization [@AltPrig], in which an incident phonon in mode $n$ with wave number $k$ along the wire axis scatters elastically to mode $n^{\prime }$ with wave number $%
-k^{\prime }$, $k^{\prime }=\sqrt{k^{2}+(n^{2}-n^{\prime 2})(\pi /W)^{2}}$. Therefore, values of $q=k+k^{\prime }$ represented in the spectrum of $%
\delta W_{q}$ identify the regions of frequencies for which the intra- and inter-mode Bragg-type scattering is allowed. In Fig. 2, the shaded frequency intervals indicate the corresponding conditions for two lowest modes, $n=0,1$.
[*Thermal conductance.* ]{}In the regime of elastic phonon propagation through the wire, the heat flow $\dot{Q}$ can be related to the transmission coefficient of phonons through the wire as
$$\dot{Q}=\sum_{n,m}\int_{0}^{\infty }\frac{dk}{2\pi }\hbar \omega
_{n}(k)v_{n}(k)(f_{1}(n,k)-f_{2}(n,k))\left| t_{nm}\right| ^{2},$$ where $v_{n}=\frac{d\omega _{n}}{dk}$ is the 1D velocity of a phonon in the mode $\omega _{n}(k)$, and $f_{1(2)}$ are equilibrium distributions of phonons at left (right) reservoirs. When the temperature difference $\Delta
T $ between the reservoirs is small [@RemTime], $\Delta T\ll T$, the thermal conductance, ${\bf \varkappa }=\dot{Q}/\Delta T$, has the form
$$\varkappa =\int_{0}^{\infty }\frac{d\omega }{2\pi }\frac{(\hbar \omega )^{2}%
}{k_{B}T^{2}}\frac{\exp (\hbar \omega /k_{B}T)}{\left[ \exp (\hbar \omega
/k_{B}T)-1\right] ^{2}}\left\langle \Gamma (\omega )\right\rangle \nonumber$$
For a wire with the transmission coefficient shown in Fig. 2(a), where $%
\left\langle \Gamma (\omega )\right\rangle \approx 1+\omega /\omega _{1}$, $%
\varkappa (T)$ is plotted by the dashed-line (1) in Fig. 3, which shows the crossover from linear to quadratic temperature dependence (at $T\sim
\vartheta _{1}=6\hbar \omega _{1}/k_{B}\pi ^{2}$) discussed in Ref. [@Blencowe]
$$\varkappa \approx \left( k_{B}^{2}\pi /6\hbar \right) T+\left(
0.7k_{B}^{2}/\hbar \right) T^{2}/\vartheta _{1}. \label{kappaball}$$
The ballistic character of heat transport in Eq. (\[kappaball\]) is reflected by the independence of $\varkappa $ on the sample length.
In a wire, where the transmission coefficient sufficiently drops at $\omega
\sim \omega _{1}$, as in Fig. 1, we approximate the low-frequency behavior of the transmission coefficient by a step function, $\left\langle \Gamma
(\omega )\right\rangle =\theta (\omega _{1}-\omega )$, which yields an intermediate saturation of the thermal conductance at the temperature $T\sim
\vartheta _{1}$,
$$\varkappa (T)\approx \frac{k_{B}\omega _{1}}{2\pi }\left\{
\begin{array}{c}
2T/\vartheta _{1},\;T\ll \vartheta _{1}; \\
1,\;\vartheta _{1}<T<\vartheta _{1}L/W.
\end{array}
\right. \label{kappadis}$$
The numerical result shown in Fig. 3 by a solid line is in a qualitative agreement with such an expectation. The horizontal arrow indicates the saturation value expected from equation (\[kappadis\]). The upper limit in the saturation interval mentioned in Eq. (\[kappadis\]) indicates the restoration of ballistic conditions for phonon propagation at wavelengths short enough to avoid wave diffraction at corrugated surfaces [@Tesanovich; @Leadbeater].
Theoretically, the intermediate saturation $\varkappa (T)\approx \varkappa
_{sat}$ at low temperatures is a more robust feature in longer wires of length $L\gg l\sim W/\left\langle \left( \delta W/W\right) ^{2}\right\rangle
$, where even the lowest mode, $n=0$ is localized at frequencies $\omega
_{\ast }(L)<\omega <\omega _{1}$. Here $\omega _{\ast }(L)\sim \omega
_{1}\left( W^{2}/\left\langle \left( \delta W_{q\rightarrow 0}\right)
^{2}\right\rangle\right)
^{1/2}\sqrt{W/L}$ is the frequency at which the localization length of $n=0$ acoustic mode is comparable to the wire length. In this case, the saturation takes place at a lower temperature $\vartheta _{\ast }\sim \hbar \omega
_{\ast }/k_{B}$, and we find that the saturation value of the thermal conductance within temperature interval $\vartheta _{1}\gtrsim T>\vartheta
_{\ast }$ has an anomalous dependence on the sample length, $$\varkappa _{sat}\approx \int_{0}^{\infty }\frac{d\omega }{2\pi }p\left(
\frac{\omega }{\omega _{\ast }}\right) \sim \frac{k_{B}\omega _{1}}{2\pi }%
\left( \frac{l}{L}\right) ^{1/2}. \label{lockappa}$$ This is an example of a more general scaling law for white-noise roughness; for a wire with a fractally rough edge, $\left\langle \left( \delta
W_{q}\right) ^{2}\right\rangle \propto q^{z}$, one obtains $\varkappa
_{sat}\propto L^{-1/\left( 2+z\right) }$.
[*In summary*]{}, our analysis of phonon propagation through long free-standing insulating wires with rough surfaces has highlighted a feature in the temperature dependence of the heat conductance $\varkappa (T)$, which results from the crossover from ballistic propagation of the lowest-frequency phonon mode at $\omega \ll \omega _{1}$ to diffusive (or even localized) behavior, with a re-entrance to the quasi-ballistic regime. Although the model used in this calculation has been restricted to only one (longitudinal) excitation branch in the wire spectrum, we believe that this feature persists also in more realistic multi-mode models (which take into account torsional modes and the wire vibrations of other polarizations), since all lowest sound modes are scattered by the surface roughness with the rate decreasing upon the decrease of the frequency. A drastic difference between phonon transport properties in different frequency intervals results in a tendency of the heat conductance of a wire to saturate provisionally at the temperature range of $T\sim hc/Wk_{B}$. An intermediate saturation value of the wire heat conductance depends on the length of a wire, and, in wires with length larger than the scattering length of phonons with frequencies $%
\omega \sim \omega _{1}$ has an anomalous length dependence, $\varkappa
_{sat}\propto L^{-1/2}$.
The authors thank M.Roukes and J.Worlock for attracting our attention to this problem. This work has been funded in parts by EPSRC and a European Union TMR programme.
T.S. Tighe [*et al*]{}, Appl. Phys. Lett. [**70**]{},2687 (1997); M.L. Roukes, Physica B [**263-264**]{}, 1 (1999) and refs. therein
M.P. Blencowe, Phys. Rev. B [**59**]{}, 4992 (1999)
L.G.C. Rego and G. Kirczenow, Phys. Rev. Lett. [**81,** ]{}232 (1998)
For a multichannel waveguide, $\Gamma
=\sum_{n,m}|t_{nm}(\omega )|^{2}$
L.I. Glazman [*et al*]{}, JETP Lett. [**48**]{}, 238 (1988); G. Kirczenow, Solid State Commun. [**68**]{}, 715 (1988)
M. Büttiker [*et al*]{}, Phys. Rev. B [**31**]{} 6207 (1985); C.W.J. Beenakker, Rev. Mod. Phys. [**69**]{}, 731 (1997)
For a fractally rough edge, $\left\langle \left( \delta
W_{q}\right) ^{2}\right\rangle \propto q^{z}$, $L_{\ast }(\omega )\propto
\omega ^{-\left( 2+z\right) }$.
B. Elattari, V. Kagalovsky and H.A. Weidenmuller, Phys. Rev. E [**57**]{}, 2733 (1998)
A.B. Pippard, [*Magnetoresistance in Metals*]{}, Cambridge Univ. Press 1989
Z. Tesanovich, M.C. Jaric and S. Maekawa, Phys. Rev. Lett. [**57**]{}, 2760 (1986)
M. Leadbeater, V.I. Falko and C.J. Lambert, Phys. Rev. Lett. [**81**]{}, 1274 (1998)
M.P. Blencowe, J. Phys. Cond. Matt. [**7**]{}, 5177 (1995)
After discretization, equation of motion takes the form of $\varepsilon _{ij}u_{ij}-\gamma \sum_{i^{\prime },j^{\prime
}=n.n.ofi,j}u_{i^{\prime }j^{\prime }}=\omega ^{2}u_{ij}$, where $\gamma
=c^{2}$ and $\varepsilon _{ij}=N_{ij}\gamma $, and $N_{ij}$ is the number of nearest neighbours of site $j$ from cross section $i$. By representing $%
u_{ij}=\sum^{n_{max}}_{n=0}\Phi _{i}^{n}\chi _{ij}^{n}$, with $\chi
_{ij}^{n}=\sqrt{\frac{2-\delta _{n0}}{W_{i}}}\cos (\frac{n\pi (j-\frac{1}{2})%
}{W_{i}})$ we prepare the equation of motion for the treatment by the discrete transfer-matrix method, $\sum_{m}I_{nm}^{ii}\Phi _{i}^{m}-\gamma
\sum_{m,\pm }I_{nm}^{ii\pm 1}\Phi _{i\pm 1}^{m}=\omega ^{2}\Phi _{i}^{m}$. The overlap integrals, $I_{nm}^{ii}=(2\gamma -2\gamma \cos (\frac{n\pi }{%
W_{i}}))\delta _{nm}+\gamma \sum_{j}\sum_{\pm }\beta _{j}^{ii\pm 1}\chi
_{ij}^{m}\chi_{ij}^{n}$, $I_{nm}^{ii\pm 1}=\gamma \sum_{j}\beta _{j}^{ii\pm 1}\chi
_{ij}^{n}\chi _{i\pm 1j}^{m}$ carry information about surface roughness through the factors $\beta _{j}^{ii^{\prime }}$, which indicate the existence ($\beta _{j}^{ii^{\prime }}=1$) or absence ($\beta
_{j}^{ii^{\prime }}=0$) of neighbouring sites. The width of each wire cross-section, $W_{i}=W+\delta W_{1}+\delta W_{2}$ is randomly generated by introducing a random function $r_{i}^{s}=\sum_{N_{min}}^{N_{max}}a_{m}^{s}%
\sin (\frac{2\pi mi}{L})$, where $N_{min}$ and $N_{max}$ are chosen to yield the desired harmonic content, $a_{m}$ are randomly taken from the interval $%
[-w/2,w/2]$, and $i$ is the coordinate along the wire axis. Finally, we normalise the fluctuation of the width to a desired r.m.s. value, so that $%
\delta W_{s}(i)=\sqrt{\left\langle \delta W^{2}\right\rangle /2}\left(
r^{s}(i)-<r^{s}(i)>\right) /\sqrt{<r^{s}(i)^{2}>}$.
S.J. Robinson, Ph.D. thesis, Lancaster University (1992); W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes (FORTRAN), Cambridge University Press (1989)
B. Altshuler and V. Prigodin, JETP Lett. [**47**]{}, 43 (1988) \[Pisma ZhETP [**47**]{}, 36 (1988)\]
One can also speak of a [*cooling time*]{} of an overheated specimen due to the phonon-mediated cooling through the wire into a bath, with$\ T_{bath}\ll T_{0}$. Then, the temporal evolution of the temperature can be described by equation $C(T)dT=-\dot{Q}(T)dt$, where $C(T)=%
\frac{12}{5}\pi ^{4}\rho Vk_{B}(\frac{T}{\Theta })^{3}$ is the Debye heat capacity of a specimen, and $V$ is its volume. The cooling time can be found by integrating this equation with respect to the temperature, which yeilds $%
\ \tau (T_{0})=\int_{0}^{T_{0}}dTC(T)/\dot{Q}(T).$ We estimate $\tau (T_{0})$ analytically in two limits: i) $\left\langle \Gamma (\omega )\right\rangle
\approx a+b\omega $ (with $a\sim 1$, $b\sim \omega _{1}^{-1}$) and ii) $%
\left\langle \Gamma (\omega )\right\rangle \approx \theta (\omega _{\ast
}-\omega )$ of a weakly disordered wire and a strongly disordered wire with the length $L\gg l$, respectively. In case i), we find approximately equal to $$\tau (T_{0})\approx 2\pi ^{6}V\rho \frac{\hbar ^{2}\omega _{1}}{%
k_{B}^{2}\Theta ^{3}}{T_{0}}\left( 1+\frac{0.7\hbar \omega _{1}}{k_{B}T_{0}}%
\ln \frac{\hbar \omega _{1}}{1.5k_{B}T_{0}+\hbar \omega _{1}}\right)$$ In case ii) of a long wire, we interpolate $\dot{Q}$ using $\int_{0}^{\infty
}z{\Gamma }(k_{B}Tz)\left[ e^{z}-1\right] ^{-1}dz\approx \left[ (T/\vartheta
_{\ast })/(1+T/\vartheta _{\ast })\right]\frac{k_{B}\omega _{\ast }}{2\pi }T$, and arrive at $\tau (T_{0})\approx 245V\rho \omega _{\ast }^{-1}(3\vartheta
_{\ast }T_{0}^{2}+2T_{0}^{3})/\Theta ^{3}$.
|
---
abstract: |
Projection pursuit is used to find interesting low-dimensional projections of high-dimensional data by optimizing an index over all possible projections. Most indexes have been developed to detect departure from known distributions, such as normality, or to find separations between known groups. Here, we are interested in finding projections revealing potentially complex bivariate patterns, using new indexes constructed from scagnostics and a maximum information coefficient, with a purpose to detect unusual relationships between model parameters describing physics phenomena. The performance of these indexes is examined with respect to ideal behaviour, using simulated data, and then applied to problems from gravitational wave astronomy. The implementation builds upon the projection pursuit tools available in the R package, tourr, with indexes constructed from code in the R packages, binostics, minerva and mbgraphic.\
author:
- Ursula Laa
- Dianne Cook
bibliography:
- 'bibliography.bib'
title: Using tours to visually investigate properties of new projection pursuit indexes with application to problems in physics
---
\#1
The term “projection pursuit” (PP) was coined by Friedman and Tukey (1974) to describe a procedure for searching high (say $p-$)dimensional data for “interesting” low-dimensional projections ($d=1$ or $2$ usually). The procedure, originally suggested by Kruskal (1969), involves defining a criterion function, or index, that measures the “interestingness” of each $d$-dimensional projection of $p$-dimensional data. This criterion function is optimized over the space of all $d$-dimensional projections of $p$-space, searching for both global and local maxima. It is hoped that the resulting solutions reveal low-dimensional structure in the data not found by methods such as principal component analysis. Projection pursuit is primarily used for visualization, with the projected data always reported as plots.
A large number of projection pursuit indexes have been developed, primarily based on departure from normality, which includes clusters, outliers and skewness, and also for finding separations between known groups (e.g. Friedman (1987), Hall (1989) Cook, Buja, and Cabrera (1992), Naito (1997), Lee et al. (2005), Ahn, Hofmann, and Cook (2003), Hou and Wentzell (2014), Jones and Sibson (1987), Rodriguez-Martinez et al. (2010), Pan, Fung, and Fang (2000), Ferraty et al. (2013), Loperfido (2018)). Less work has been done on indexes to find nonlinear dependence between variables, focused on $d=2$, which motivates this research.
The driving application is from physics, to aid the interpretation of model fits on experimental results. A physical model can be considered to be a set of $p$ free parameters, that cannot be measured directly and are determined by fitting a set of $q~ (p<q)$ experimental observations, for which predictions can be made once the $p$ parameters are estimated. (Note here, that while we may have analytic expressions for the predictions, this is not always the case and we often have to rely on numerical computation.) Different sets of model parameters ($n$) found to be compatible with the experimental results within a selected level of confidence yield the data to be examined using projection pursuit. A single prediction can be a complicated function of all of the free parameters, and typically $q \in [100,1000]$ and $p \sim 10$. Current practice is to examine pairs of parameters, or combinations produced by intuition or prior knowledge. This begs the question, whether important nonlinear associations are missed because they are hidden in linear combinations of more than two variables.
PP can be combined with other dimension reduction methods when $p$ is very high. For example, it can be beneficial to first do principal component analysis prior to PP, especially to remove linear dependencies before searching for other types of association. This is the approach used in Cook, Laa, and Valencia (2018), which explores a 56-dimensional parameter space, by first reducing the number of dimensions to the first six principal components, before applying projection pursuit. PCA was appropriate for this problem because reducing to principal component space removed the linear dependencies while preserving the nonlinear relationships that were interesting to discover. Some projection pursuit indexes do incorporate penalty terms to automate removing noise dimensions. It can also be important to have an efficient PP optimizer, particularly when working with high dimensions, because the search space increases exponentially with dimension.
To find appropriate projections pursuit indexes for detecting nonlinear dependencies, the literature on variable selection was a starting point. With high-dimensional data, even plotting all pairs of variables can lead to too many plots, which is what “scagnostics” (Wilkinson, Anand, and Grossman (2005), Wilkinson and Wills (2008)) were developed to address by providing metrics from which to select the most interesting variable pairs. There are eight scagnostics, of which three (“convex”, “skinny” and “stringy”) are used here. The question is whether these can be adapted into projection pursuit indexes, to search for unusual features in two-dimensional projections of high-dimensional data. Recent PhD research by Grimm (2016) explored the behavior of scagnostics for selecting variables, and proposed two more that have nicer properties, based on smoothing splines and distance correlation. In addition, two more indexes for measuring dependence have been proposed in the machine learning literature, based on information criteria, maximal and total information coefficient (MIC and TIC) (Reshef et al. 2011), with computationally more efficient versions (MIC\_e, TIC\_e) (Reshef et al. 2016). These are related to original 1D projection pursuit indexes based on entropy (e.g. Huber (1985), Jones and Sibson (1987)). This provides seven current indexes for measuring dependence between two variables, and each is available in an R (R Core Team 2018) package: binostics (Hofmann et al. 2019), mbgraphic (Grimm 2017) and minerva (Albanese et al. 2012).
PP index behavior can be understood and investigated more substantially when combined with a tour. A tour (Asimov (1985), Buja et al. (2005)) displays a smooth sequence of low dimensional projections from high dimensions to explore multivariate data for structure such as clusters, outliers, and nonlinear relationships. Cook et al. (1995) provided an approach combining the tour algorithm with PP, to interactively both search for interesting projections, and examine the behavior of the indexes. The projection pursuit guided tour is available in the R package, tourr (Wickham et al. 2011), and provides optimization routines, and visualization.
This paper is structured as follows. Section \[sec:construct\] discusses index construction, and how they can be used in the guided tour. Section \[sec:investigate\] investigates the behavior of the indexes, explored primarily using tour methods. The new guided tour with these indexes is applied to two examples from gravitational wave astronomy (Section \[sec:phys\]). The latter two parts are connected in that the application of the new indexes to these problems is the main motivation for the paper, and the simulation study, in the first part, was conducted to better understand the behavior of the indexes in general. The techniques in Section \[sec:investigate\] define procedures that will be generally useful for researchers developing new projection pursuit indexes to visually assess their behavior. Visual methods to diagnose the index behavior is important because PP is primarily used for visualization. The paper finishes with a discussion about the limitations of this work, and the potential future directions.
\[sec:construct\]
A projection pursuit index (PPI) is a scalar function $f$ defined on an $d$-dimensional data set, computed by taking a $d$-dimensional projection of an $n\times p$ data matrix. Typically the definition is such that larger values of $f$ indicate a more interesting distribution of observations, and therefore maximizing $f$ over all possible $d$ dimensional projections of a data set with $p>d$ variables will find the most interesting projections. This section describes the seven indexes that are to be used to explore bivariate association. Some data pre-processing, including standardization, is advisable, prior to optimizing the PPI.
Making a plot always involves some choice of scaling. When a scatterplot is made, effectively, albeit under the hood, the data is scaled into a range of $[a, b]$ (often $a=0, b=1$) on both axes to print it on a page or display in a graphics device window. The range deliminates page space within which to draw. The upshot is that the original data scale is standardized to the range and aspect ratio on the display space. It may be that the original range of one variable is $[1, 1000]$ and the other is $[1, 1.6]$ but the display linearly warps this to $[0, 1]$ and $[0, 1]$, say, giving both variables equal visual weight.
With high dimensional data, and particularly projections, it is also necessary to re-scale the original range, and it is important to pay attention to what is conventional, or possible, and the effects. The PPIs also may require specific scaling for them to be effectively computed. Both of these are addressed here. The common pre-processing include:
- Standardizing each variable, to have mean 0 and variance 1, so that individual variable scales do not affect the result. Different variable scales are examinable without resorting to projection pursuit, so can be handled prior to searching through high dimensions.
- Sphering the high-dimensional data is often done to remove linear dependence. This is typically done using principal component analysis, and using the principal components as the variables passed to PP. If linear dependence is the only structure PP is not needed, and thus this is removed before PP so that the PPIs are not distracted by simple structure.
- Transform single variables to reduce skewness. It is marginal structure, visible in a single variable, which doesn’t need a multivariate technique to reveal. Skewed distributions will inadvertently affect the PPIs, distracting the search for dependence.
- Remove outliers, which may be an iterative process, to discover, identify and delete. Extreme values will likely affect PPI performance. Outliers can be examined on a case by case basis later.
- Possibly remove noise dimensions, which is also likely to be an iterative process. Directions where the distribution is purely noise make optimization of a PPI more difficult. If a variable is suspected to have little structure and relationship with other variables, conducting PP on the subset of variables without them may improve the efficiency of the search.
- Centering and scaling of the projected data, can be helpful visually. If the data has a small amount of non-normal distribution in some directions, the projected data can appear to wander around the plot window during a tour. It doesn’t matter what the center of the projected data is, so centering removes a wandering scatterplot. Less commonly, it may be useful to scale the projected data to standard values, which would be done to remove any linear dependence remaining in the data.
------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$p$ be the number of variables in the data. For physics models the observable space is higher dimension, say $q$, and the data examined is the fitted model space, typically less than 10.
$d (=2) $ projection dimension. For studying physics models, typically $d=2$ and this is the focus for the index definition.
$n$ Number of observations, which for the physics models is the number of fitted models being examined and compared.
$\bm{X}_{n\times p} = (\bm{X}_1, ..., \bm{X}_p)$ $n\times p$-dimensional data matrix, where variables $\bm{X}_j$ may be scaled or standardized, and $\bm{X}$ may be sphered
$\bm{Y}_{n\times 2} = (\bm{Y}_1, ..., \bm{Y}_2)$ projected data matrix. where $Y_j = (\alpha_1 X_1, ..., \alpha_p X_p)$
$\bm{F}_{p\times 2} = (\bm{\alpha}_1, ..., \bm{\alpha}_2)$ orthonomal projection matrix
$H$ Convex hull
$A$ Alpha hull
------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Summary of notation
\[notation\]
\[sec:indexDef\]
Table \[notation\] summarizes the notation used for this section. Here we give an overview of the functions that are converted into projection pursuit indexes. Full details of the functions can be found in the original sources.
- **scagnostics**: The first step to computing the scagnostics is that the bivariate data is binned, and scaled between [\[]{}0,1[\]]{} for calculations. The convex (Eddy 1977) and alpha hulls (Edelsbrunner, Kirkpatrick, and Seidel 1983), and the minimal spanning tree (MST) (Kruskal 1956), are computed.
- *convex*: The ratio of the area of alpha to convex hull, $I_{\bm{F}, convex}= \frac{area(A(Y))}{area(H(Y))}$. This is the only measure where interesting projections will take low values, with a maximum of 1 if both areas are the same. Thus $1-c_{convex}$ is used.
- *skinny*: The ratio of the perimeter to the area of the alpha hull, $I_{\bm{F}, skinny} = 1 - \frac{\sqrt{4\pi area(A)}}{perimeter(A)}$, where the normalization is chosen such that $I_{\bm{F}, skinny} = 0$ for a full circle. Values close to 1 indicate a skinny polygon.
- *stringy*: Based on the MST, $I_{\bm{F}, stringy} = \frac{diameter(MST)}{length(MST)}$ where the diameter is the longest connected path, and the length is the total length (sum of all edges). If the MST contains no branches $I_{\bm{F}, skinny} = 1$.
- **association**: The index functions are available in Grimm (2017), and are defined to range in [\[]{}0,1[\]]{}. Both functions in the mbgraphic package can bin the data before computing the index, for computational performance.
- *dcor2D*: This function is based on distance correlation (Székely, Rizzo, and Bakirov 2007), which is designed to find both linear and nonlinear dependencies between variables. It involves computing the distances between pairs of observations, conducting an analysis of variance type breakdown of the distances relative to each variable, and the result is then passed to the usual co variance and hence correlation formula. The function wdcor, in the R package, extracat (Pilhöfer and Unwin 2013), computes the statistic, and the mbgraphic package utilises this function.
- *splines2D*: Measures nonlinear dependence by fitting a spline model (Wahba 1990) of $\bm{Y}_2$ on $\bm{Y}_1$ and also $\bm{Y}_1$ on $\bm{Y}_2$, using the `gam` function in the R package, mgcv (Wood et al. 2016). The index compares the variance of the residuals: $$I_{\bm{F}, splines2d} = max(1- \frac{Var(res_{\bm{Y}_1\sim \bm{Y}_2})}{Var(\bm{Y}_1)}, 1-\frac{Var(res_{\bm{Y}_2\sim {\bm{Y}_1}})}{Var(\bm{Y}_2)}),$$ which takes large values if functional dependence is strong.
- **information**: The index functions (Reshef et al. 2011) nonparametricly measure nonlinear association by computing the mutual information, $$\bm{I} = \sum_{by_1}\sum_{by_2} p(by_1, by_2) log(p(by_1, by_2)/(p(by_1)p(by_2))),$$ where $by_1, by_2$ are binned values of the projected data, and $p(by_1, by_2)$is the relative bin count in each cell, and $p(by_1), p(by_2)$ are the row and column relative counts, on a range of bin resolutions of the data. It is strictly a 2D measure. For a fixed binning, e.g. $2\times 2$ or $10\times 4$, the optimal binning is found by maximizing $I$. The values of I range between $[0,1]$ because they are normalized across bins by dividing by $log(min(\# bins_{y_1}, \# bins_{y_2}))$.
- *Maximum Information Coefficient (MIC)*: uses the maximum normalized $I$ across all bin resolutions.
- *Total Information Coefficient (TIC)*: sums the normalized $I$ for all bin resolutions computed. This creates a problem of scaling - there is no upper limit, although it is related to number of bins, and number of bin resolutions used. In the work below we have made empirical estimates of the maximum and scaled the TIC index using this to get it in the range $[0,1]$. This index should be more stable than MIC.
A comparison between these indexes for the **purpose of variable selection**, but not projection pursuit, was discussed in Grimm (2016). It is only available in German, we we summarize the main findings here. The scagnostics measures are flexible, and calculating the full set of measures provides useful guidance in variable selection. However, they are found to be highly sensitive to outlying points and sample size (as a consequence of the binning). Both splines2D and dcor2D are found to be robust in this respect, but splines2D is limited to functional dependence, while dcor2D is found to take large values only in scenarios with large linear correlation. The mutual information based index functions (MIC, TIC) are found to be flexible, but are sensitive to the sample size and often take relatively large values even when no association is present. A brief comparison of MIC and dcor2D was also provided in Simon and Tibshirani (2014).
In addition to the seven indexes described above, we will also include the holes index available in the `tourr` package, see (Cook, Buja, and Cabrera (1993), Cook and Swayne (2007)). This serves as a benchmark, demonstrating some desired behavior. The index takes maximum values for a central hole in the distribution of the projected data.
Given a PPI we are confronted with the task of finding the maximum over all possible $d$ dimensional projections. One challenge is to avoid getting trapped in local maxima that are only a result of sampling fluctuations or a consequence of a noisy index function. Posse (1995a) discusses the optimization, in particular that for most index functions and optimizers results are too local, largely dependent on starting point. Friedman (1987) suggested a two-step procedure: the first step is using a large step size to find the approximate global maximum while stepping over pseudomaxima. A second step is then starting from the projection corresponding to the approximate maximum and employing a gradient directed optimization for the identification of the maximum. For exploring high-dimensional data, it can be interesting to observe local maxima as well as a global maximum, and thus a hybrid algorithm that still allows lingering but not being trapped by local maxima is ideal. In addition, being able to visually monitor the optimization and see the optimal projection in the context of neighboring projections is useful. This is provided by combining projection pursuit with the grand tour (Cook et al. 1995). The properties of a suitable optimization algorithm include monotonicity of the index value, a variable step-size to avoid overshooting and to increase the chance of reaching the nearest maximum, and a stopping criterion allowing to move out of a local maximum and into a new search region (Wickham et al. 2011). A possible algorithm is inspired by simulated annealing and has been described in Lee et al. (2005), this has been implemented in the `search_better` and `search_better_random` search functions in the tourr package. The tourr package also provides the `search_geodesic` function, which first selects a promising direction by comparing index values between a selected number of small random steps, and then optimizes the function over the line along the geodesic in that direction considering projections up to $\pi/4$ away from the current view.
\[sec:investigate\]
A useful projection pursuit index needs to have several properties. This has been discussed in several seminal papers, e.g. Diaconis and Freedman (1984), Huber (1985), Jones and Sibson (1987), Posse (1995a), Hall (1989). The PPI should be minimized by the normal distribution, because this is not interesting from a data exploration perspective. If all projections are normally distributed, good modelling tools already exist. A PPI should be approximately affine invariant, regardless how the projection is rotated the index value should be the same, and the scale of each variable shouldn’t affect the index value. Interestingly the original index proposed by Friedman and Tukey (1974) was not rotationally invariant. A consistent index means that small perturbations to the sample do not dramatically change the index value. This is particularly important to making optimization feasible, small angles between projections correspond to small perturbations of the sample, and thus should be small changes to index value. Posse (1995a) suggests that indexes should be resistant to features in a tail of the distribution, but this is debatable because one departure from normality that is interesting to detect are anomalous observations. Some PPI are designed precisely for these reasons. Lastly, because we need to compute the PPI over many projections, it needs to be fast to compute. These form the basis of the criteria upon which the scagnostic indexes, and the several alternative indexes are examined, as explained below.
- **smoothness**: This is the consistency property mentioned above. The index function values are examined over interpolated tour paths, that is, the value is plotted against time, where time indexes the sequence of projections produced by the tour path. The signature of a good PPI is that the plotted function is relatively smooth. The interpolation path corresponds to small angle changes between projections, so the value should be very similar.
- **squintability**: Tukey and Tukey (1981) introduced the idea of squint angle to indicate resolution of structure in the data. Fine structure like the parallel planes in the infamous RANDU data (Marsaglia 1968) has a small squint angle because you have to be very close to the optimal projection plane to be able to see the structure. Structures with small squint angle are difficult to find, because the optimization algorithm needs to get very close to begin hill-climbing to the optimum. The analyst doesn’t have control over the data structure, but does have control over the PPI. Squintability is about the shape of the PPI over all projections. It should have smooth low values for noise projections and a clearly larger value, ideally with a big squint angle, for structured projections. The optimizer should be able to clearly see the optimal projections as different from noise. To examine squintability, the PPI values are examined on interpolated tour paths between a noise projection and a distant structured projection.
- **flexibility**: An analyst can have a toolbox of indices that may cover the range of fine and broad structure, which underlies the scagnostics suite. Early indexes, based on density estimation could be programmed to detect fine or large structure by varying the binwidth. This is examined by using a range of structure in the simulated data examples.
- **rotation invariance**: The orientation of structure within a projection plane should not change the index value. This is especially important when using the projection pursuit guided tour, because the tour path is defined between planes, along a geodesic path, not bases within planes. If a particular orientation is more optimal, this will get lost as the projection shown pays no attention to orientation. Buja et al. (2005) describes alternative interpolation paths based on Givens and Householder rotations which progress from basis to basis. It may be possible to ignore rotation invariance with these interpolations but there isn’t a current implementation, primarily because the within-plane spin that is generated is distracting from a visualization perspective. Rotation invariance is checked for the proposed PPIs by rotating the structured projection, within the plane.
- **speed**: Being fast to compute allows the index to be used in real-time in a guided tour, where the optimization can be watched. When the computations are shifted off-line, to watch in replay, computation times matter less. This is checked by comparing times for benchmark scenarios with varying sample size.
\[sec:dataOv\]
Three families of data simulations are used for examining the behavior of the index functions. Each generates structure in two variables, with the remaining variables containing various types of noise. This is a very simple construction, because there is no need for projection pursuit to find the structure, one could simply use the PPIs on pairs of variables. However, it serves the purpose to also evaluate the PPIs. The three data families are explained below. In each set, $n$ is used for the number of points, $p$ is the number of dimensions, and $d=2$ is the projection dimension. The three structures were selected to cover both functional and non-functional dependence, different types of nuisance distributions and different structure size and squintability properties.
- **pipe**: nuisance directions are generated by sampling independently from a uniform distribution between $[-1,1]$, and the circle is generated by sampling randomly on a 2D circle, and adding a small radial noise. The circle should be easy to see by some indices because it is large structure, but the nonlinearity creates a complication.
- **sine**: nuisance directions are generated by sampling independently from a standard normal distribution, and the sine curve is generated by $x_p = \sin(x_{p-1}) + \mathrm{jittering}$. The sine is a medium nonlinear structure, which should be visible to multiple indices.
- **spiral**: nuisance directions are generated by sampling independently from a normal distribution, and the structure directions are sampled from an Archimedean spiral, i.e. $r = a + b \theta$, with $a=b=0.1$ and we sample angles $\theta$ from a normal distribution with mean 0 and variance $2\pi$, giving a spiral with higher densities at lower radii. The absolute value of $\theta$ fixes the direction of the spiral shape. This is fine structure which is only visible close to the optimal projection.
![Scatterplots of pairs of variables from samples of each family, showing the nuisance variables and structured variables.[]{data-label="fig:data"}](figure/data-1){width="80.00000%"}
[llrrrrrr]{} & & & &\
(l[2pt]{}r[2pt]{})[3-4]{} (l[2pt]{}r[2pt]{})[5-6]{} (l[2pt]{}r[2pt]{})[7-8]{} Index & & lower & upper & lower & upper & lower & upper\
\
holes & noise & 0.37 & 0.46 & 0.00 & 0.01 & 0.00 & 0.01\
& structure & 0.98 & 0.99 & 0.00 & 0.00 & 0.00 & 0.00\
\
convex & noise & 0.70 & 0.73 & 0.57 & 0.69 & 0.54 & 0.68\
& structure & 0.06 & 0.07 & 0.00 & 0.00 & 0.00 & 0.00\
\
skinny & noise & 0.12 & 0.18 & 0.13 & 0.28 & 0.13 & 0.31\
& structure & 0.84 & 0.86 & 0.76 & 0.87 & 0.84 & 0.90\
\
stringy & noise & 0.31 & 0.48 & 0.20 & 0.42 & 0.22 & 0.43\
& structure & 0.57 & 0.74 & 1.00 & 1.00 & 0.88 & 0.98\
\
dcor2D & noise & 0.03 & 0.07 & 0.04 & 0.07 & 0.04 & 0.07\
& structure & 0.17 & 0.18 & 0.96 & 0.98 & 0.14 & 0.17\
\
splines2D & noise & 0.00 & 0.02 & 0.00 & 0.02 & 0.00 & 0.02\
& structure & 0.00 & 0.02 & 1.00 & 1.00 & 0.01 & 0.05\
\
MIC & noise & 0.03 & 0.04 & 0.03 & 0.04 & 0.03 & 0.05\
& structure & 0.56 & 0.58 & 0.98 & 1.00 & 0.40 & 0.45\
\
TIC & noise & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02\
& structure & 0.42 & 0.43 & 0.95 & 0.98 & 0.26 & 0.30\
For simplicity, in the investigations of the index behavior, we fix $p=6$, which corresponds to two independent 2D planes containing nuisance distributions, and one 2D plane containing the structured distribution. The structured projection is in variables 5 and 6 ($x_5, x_6$). Two samples sizes are used: $n=(100, 1000)$. All variables are standardized to have mean 0 and standard deviation 1. Figure \[fig:data\] shows samples from each of the families, of the nuisance and structured pairs of variables. Table \[tab:indexTable\] compares the PPIs for structured projections against those for nuisance variables, based on 100 simulated data sets of each type, using sample size 1000. The lower and upper show the 5th and 95th percentile of values. The holes index is sensitive only to the pipe distribution. All other indexes, except convex show distinctly higher values for the structured projections. The convex index shows the inverse scale to other indices, thus (1-convex) will be used in the assessment of performance of PPIs. The scale for the holes index in its original implementation is smaller than the others ranging from about 0.7 through 1, so it is re-scaled in the performance assessment so that all indices can be plotted on a common scale of 0-1 (details are given in the Appendix). Similarly, the TIC index is re-scaled depending on sample size.
The procedures for assessing the PPI properties of smoothness, squintability, flexibility, rotation invariance, and speed examined for samples from the family of data sets are:
1. Compute the PPI values on the tour path along an interpolation between pairs of nuisance variables, $x_1 - x_2$ to $x_3 - x_4$. The result is ideally a smooth change in low values. This checks the smoothness property.
2. Change to a tour path between a pair of nuisance variables $x_1 - x_2$ and the structured pair of variables $x_5 - x_6$ via the intermediate projection onto $x_1 - x_5$, and compute the PPI along this. This examines the squintability, and smoothness. If the function is smooth and slowly increases towards the structured projection, then the structure is visible from a distance.
3. Use the guided tour to examine the ease of optimization. This depends on having a relatively smooth function, with structure visible from a distance. One index is optimized to show how effectively the maximum is attained, and the values for other PPIs is examined along the same path, to examine the similarity between PPIs.
4. Rotation invariance is checked by computing PPIs on rotations of the structured projection.
5. Computational speed for the selected indexes is examined on a range of sample sizes.
\[sec:smooth\]
Figure \[fig:plotV1V2toV3V4\] shows the traces representing the index values calculated across a tour between a pair of nuisance projections. The tour path is generated by interpolating between the two independent nuisance planes, i.e. from the projection onto $x_1$-$x_2$ to one onto $x_3$-$x_4$. The range of each axis is set to be the limits of the index, as might be expected over many different data sets, 0 to 1. Each projection in the interpolation will also be noise. Two different sample sizes are show, $n=100$ as a dashed line, and $n=1000$ as a solid line. The ideal trace is a smooth function, with relatively low values, and no difference between the sample sizes. A major feature to notice is that the scagnostics produce noisy functions, which is problematic, because small changes in the projection result in big jumps in the value. This will make them difficult to optimize. On the other hand holes, dcor2d, splines2d, MIC and TIC are relatively smooth functions.
Several of the indexes are sensitive to sample size also, the same structured projection with differing numbers of points, produces different values.
![PPIs for projections along a geodesic interpolation between two nuisance projections. All projections would be nuisance so the PPI are ideally low and smooth, with little difference between sample sizes (solid lines: $n=1000$; dashed: $n=100$). The scagnostic PPIs are noisy. Some indexes have distinct differences in values between sample sizes. (This is not an optimization path, but an interpolation containing 41 projections between two known projections.)[]{data-label="fig:plotV1V2toV3V4"}](figure/plotV1V2toV3V4-1){width="\textwidth"}
\[sec:squintability\]
Figure \[fig:wIntermediate\] shows the PPIs for a tour sequence between a nuisance and structured projection. A long sequence is generated where the path interpolates between projections onto $x_1$-$x_2$, $x_1$-$x_5$, $x_5$-$x_6$, in order to see some of the intricacies of holes index. Sample size is indicated by line type: dashed being $n=100$ and solid is $n=1000$. The beginning of the sequence is the nuisance projection and the end is the structured projection. The index values for most PPIs increases substantially nearing the structured projection, indicating that they “see” the structure. Some indexes see all three structures: scagnostics, MIC and TIC, which means that they are flexible indexes capable of detecting a range of structure. (Grimm 2016)’s indexes, dcor2d and splines2d, are excellent for detecting the sine, and they can see it from far away, indicated by the long slow increase in index value. The holes index easily detects the pipe, and can see it from a distance, but also has local maxima along the tour path. The scagnostic index, stringy, can see the structure but is myopic, only when it is very close. Interestingly the scagnostic, skinny, sees the spiral from a distance.
![PPIs for projections along an interpolation between nuisance and structured projections, following $x_1$-$x_2$ to $x_1$-$x_5$ to $x_5$-$x_6$ (solid lines: $n=1000$; dashed: $n=100$). The vertical blue line indicates the position of the projection onto $x_1$-$x_5$ in the sequence. Peaks at the end of the sequence indicate the index sees the structure. The scagnostics, MIC and TIC see all three structures, so are more flexible for general pattern detection. Holes only responds to the pipe, and is a multimodal function for this data with a local maximum at $x_1$-$x_5$. (This is not an optimization path, but an interpolation containing 59 projections between three known projections.) []{data-label="fig:wIntermediate"}](figure/wIntermediate-1){width="\textwidth"}
\[sec:guided\]
Before applying the new index functions, with the guided tour on real examples, we test them on the simulated dataset to understand the performance of the optimization. The guided tour combines optimization with interpolation between pairs of planes. Target planes of the path are chosen to maximize the PPI. There are three derivative-free optimization methods available in the guided tour: ` search_better_random` (1), ` search_better` (2), and ` search_geodesic` (3). Method 1 casts a wide net randomly generating projection planes, computing the PPIs and keeping the best projection, and method 2 conducts a localized maximum search. Method 3 is quite different: a local search is conducted to determine a promising direction, and then this direction is followed until the maximum in that direction is found. For all methods the optimization is iterative, the best projections form target planes in the tour, the tour path is the interpolation to this target, and then a new search for a better projection is made, followed by the interpolation. For each projection during the interpolation steps, the PPI is recorded.
The stopping rule is that no better projections are found after a fixed number of tries, given a fixed tolerance value measuring difference. For method 1 and 2 two additional parameters control the optimization: the search window $\alpha$, giving the maximum distance from the current plane in which projections are sampled, and the cooling factor, giving the incremental decrease in search window size. Method 3 in principle also has two free parameters, which are however fixed in the current implementation. The first is the small step size used when evaluating the most promising direction, it is fixed to 0.01, and the second parameter being the window over which the line search is performed, fixed to $\pm \pi/4$ away from the current plane.
For distributions and indexes with smooth behavior and good squintability, method 3 is the most effective method for optimization. If these two criteria are not met the method may still be useful, but only given an informed starting projection. In such cases we can follow a method similar to that proposed by Friedman (1987): we break the optimization in two distinct steps. A first step (“scouting”) uses method 2 with large search window and no cooling as a way of stepping over fluctuations and local maxima and yielding an approximation of the global maximum. Note that this likely requires large number of tries, especially as dimension increases, since most randomly picked planes will not be interesting. The second step uses method 3 starting from the approximate maximum, which will take small steps to refine the result to be closer to the global maximum.
Despite the simple structure, the pipe is relatively difficult for the PPIs to find. For the TIC index, there is a fairly small squint angle. For the holes index, there are several local maxima, that divert the optimizer. There is a hint of this from Figure \[fig:wIntermediate\] because the initial projection (left side of trace) of purely noise variables has a higher index value than the linear combinations of noise and structured variables along the path. The uniform distribution was used to generate the noise variables, which has a higher PPI value than a normal distribution, yielding the higher initial value. In addition, a local maximum is observed whenever the pair of variables is one structured variable and one noise variable, because there is a lighter density in the center of the projection.
The optimization is done in two stages, a scouting phase using method 2, and a refinement stage using method 3. For the scouting we use $\alpha = 0.5$ and stopping condition of maximum 5000 tries, and we optimized the TIC index.
Figure \[fig:pipeFirstRun\] shows the target projections (points) selected during the scouting with method 2 on the TIC index. The focus is on the target projections rather than the interpolation between them, because the optimization is done off-line, and only the targets are used for the next step. The horizontal distance between the points in the plot reflects the relative geodesic distance between the planes in the 6D space. All of the other indexes are shown for interest. The TIC index value is generally low for this data, although it successfully detects the pipe. The holes, convex, skinny, and to some extent MIC, mirror the TIC performance. The holes differs in that it has some intermediate high values which are likely the indication of multi-modality of this index on this data.
The final views obtained in each of the two stages are compared in the Appendix.
![PPIs for a sequence of projections produced by scouting for the pipe using optimization method 2 on the TIC index. Other PPI values are shown for interest. Only the values on the target planes are shown. Despite the small maximum value of TIC for this data, it identifies the pipe.[]{data-label="fig:pipeFirstRun"}](figure/pipeFirstRun-1){width="80.00000%"}
Given the patterns in Figure \[fig:wIntermediate\] it would be expected that the sine could be found easily, using only optimization method 3 with the splines2d, dcor2d, MIC or TIC indexes. This is examined in Figure \[fig:findsine\]. Optimization is conducted using the splines2d index, and the trace of the PPI over the optimization is shown, along with the PPI values for the other indexes over that path. The vertical blue lines indicate anchor bases, where the optimizer stops, and does a new search. The distance between anchor planes is smaller as the maximum is neared.
The only complications arise from a lack of rotation invariance of the splines2d index. It is not easily visible here, but it is possible that the best projection will have a higher PPI. The index changes depending on the basis used to define the plane, but the geodesic interpolation conducted by the tour uses any suitable basis to describe the plane, ignoring that which optimizes the PPI. This is discussed in section \[sec:rot\].
![PPIs for sequence of projections produced by a guided tour optimizing the splines2d index, using optimization method 3, for the sine data, with $n=1000$. Anchor planes are marked by the blue vertical lines, and are closer to each other approaching the maxima. The sine is found relatively easily, by splines2d, and it is indicated that MIC, TIC, dcor2d and convex would also likely find this structure.[]{data-label="fig:findsine"}](figure/findsine-1){width="80.00000%"}
The spiral is the most challenging structure to detect because it has a small squint angle (Posse 1995b), especially as the ratio of noise to structure dimensions increases. This is explored using optimization method 2 to scout the space for approximate maxima. The skinny scagnostic index is used because it was observed (Figure \[fig:wIntermediate\]) to be sensitive to this structure, although the noisiness of the index might be problematic. The stringy appears to be more sensitive to the spiral, but it has a much smaller squint angle.
The search is conducted for $p=4,5,6$ which would correspond to 2, 3 and 4 noise dimensions respectively. In addition we examine the distance between planes, using a Frobenius norm, as defined by Equation 2 of Buja et al. (2005), and available in the `proj_dist` function in the tourr package, to compare searches across dimensions. The distance between planes is related to squint angle, how far away from the ideal projection can the structure be glimpsed. We estimate the squint angle depending on the number of noise dimensions in the appendix. In order for the optimizer to find the spiral, the distance between planes would need to be smaller than the squint angle. Figure \[fig:findspiral\] summarizes the results. When $p=4$ the scouting method effectively finds the spiral. Plot (a) shows the side-by-side boxplots of pairwise distances between planes examined during the optimization, for $p=4,5,6$. These are on average smaller for the lower dimension, and gradually increase as dimension increases. This is an indication of the extra computation needed to brute force find the spiral as noise dimensions increase. Plot (b) shows the distance of the plane in each iteration of the optimization to the ideal plane, where it can be seen that only when $p=4$ does it converge to the ideal. Its likely that expanding the search space should result in uncovering the spiral in higher dimensions, which however requires tuning of the stopping conditions and long run times.
![Guided tour optimizing the skinny index for the Sprial dataset with 1000 datapoints, with p = 4, 5, 6. The left plot shows the distribution of pairwise distances between planes obtained via the guided tour, the right shows the evolution of distance to the ideal plane as the index is being optimized.[]{data-label="fig:findspiral"}](figure/findspiral-1){width="90.00000%"}
\[sec:rot\]
Rotational invariance is examined using the sine data ($x_5$-$x_6$), computing PPI for different rotations within the 2D plane, parameterized by angle. Results are shown in Figure \[fig:rotationDep\]. Several indexes are invariant, holes, convex and MIC, because their value is constant around rotations. The dcor2d, splines2d and TIC index are clearly not rotationally invariant because the value changes depending on the rotation. The scagnostics indexes are approximately rotationally invariant, but particularly the skinny index has some random variation depending on rotation.
![PPI for rotations of the sine 1000 data, to examine rotation invariance. Most are close to rotation invariant, except for skinny, dcor2d, splines2d and TIC.[]{data-label="fig:rotationDep"}](figure/rotationDep-1){width=".6\textwidth"}
Examining the computing time as a function of sample size we find that scagnostics and splines2d are fast even for large samples, while all other index functions slow rapidly with increasing sample size. Detailed results are shown in the Appendix.
Some PPIs have a choice of parameters, and the choice can have an effect on function smoothness, and sensitivity to structure. In the Appendix we examine the dependence of the scagnostics indexes on the binning, showing that even with small number of bins the indexes are noisy, while they lose ability to see structure. Sensitivity to the number of bins in the MIC index is also examined, showing that tuning the parameter can improve the final result.
We identified two potential improvements. First, the issue of noisy index functions may be addressed via smoothing, and we explore different smoothing options for the examples of the skinny and stringy index in the Appendix. In addition, rotation dependent indexes may be enhanced by redefining them in a rotation invariant way.
Our results can be summarized by evaluating and comparing the advantages and disadvantages of each index function according to the criteria presented above. Such an overview is given in Table \[tab:summary\], listing if the criteria is fully met (), there are some shortcomings ($\cdot$) or failure ($\times$). (The holes index does not appear in the summary because its performance understood, and is not being examined here.) We find that none of the indexes considered meet all criteria, and in particular rotation invariance is often not fulfilled. In addition the limited flexibility of most indexes highlights the importance of index selection in the projection pursuit setup. Table \[tab:summary\] further suggests that there is much room for the improvement of index functions detecting unusual association between model parameters.
Index smooth squintability flexible rotation invariant speed
----------- ---------- --------------- ---------- -------------------- --------- --
convex $\cdot$ $\cdot$ $\cdot$
skinny $\cdot$ $\cdot$ $\cdot$ $\times$
stringy $\times$ $\times$ $\cdot$ $\cdot$
splines2D $\cdot$ $\times$
dcor2D $\cdot$ $\times$ $\cdot$
MIC $\cdot$ $\cdot$ $\cdot$
TIC $\cdot$ $\cdot$ $\cdot$
: Summary of findings, showing to what extend the considered new index functions pass the criteria for a good PPI. “" symbols good behavior, “$\cdot$" symbols some issues while “$\times$" symbols failure on the corresponding criteria. Each index has particular strengths and drawbacks and selection must be guided by the considered example, see text for details.[]{data-label="tab:summary"}
\[sec:phys\]
This section describes the application of these projection pursuit indices to find two-dimensional structure in two multidimensional gravitational waves problems.
The first example contains 2538 posterior samples obtained by fitting source parameters to the observed gravitational wave signal GW170817 from a neutron star merger (Abbott and others 2017). Data has been downloaded from (“LIGO” 2018). The fitting procedures are described in detail in Abbott and others (2018). We consider six parameters of physical interest (6-D) with some known relationships. Projection pursuit is used to find the known relationships.
The second example contains data generated from a simulation study of a binary black hole (BBH) merger event, as described in Smith et al. (2016). There are 12 parameters (12-D), with multiple nuisance parameters. Projection pursuit uncovers new relationships between parameters.
A scatterplot matrix (with transparency) of the six parameters is shown in the Appendix. (In astrophysics, scatterplot matrices are often called “corner plots” (Foreman-Mackey 2016).) The diagonal shows a univariate density plot of each parameter, and the upper triangle of cells displays the correlation between variables. From this it can be seen that m1 and m2 are strongly, and slightly, nonlinearly associated. Between the other variables we observe some linear association (R1, R2), some nonlinear association (L1, L2, R1, R2), heteroskedastic variance in most variables and some bimodality (R1, L1, L2, m1, m2).
The model describes a neutron star merger and contains 6 free parameters, with each neutron star described by its mass $m$ (m1, m2) and radius $R$ (R1, R2), and a so-called tidal deformability parameter $\Lambda$ (L1, L2) which is a function of the mass and radius, approximately proportional to $(m/R)^{-5}$.
Because m1 and m2 are very strongly associated, m2 is dropped before doing PP. This relationship is obvious from the scatterplots of pairs of variables and does not need to be re-discovered by PP.
All variables are scaled to range between 0 and 1. The purpose is that range differences in individual variables should not affect the detection of relationships between multiple variables. Standardizing the range will still leave differences between the standard deviations of the variables, and for this problem this is preferred. Differences in the standard deviations can be important for keeping the non-linear relationships visible to PP.
With only five parameters, a reasonable start is to examine the 5D space using a grand tour. This quickly shows the strong nonlinear relationships between the parameters. PP is then used to extract these relationships. The best index for this sort of problem is the splines2d, and it is fast to compute.
Figure \[fig:nsePlotOrig\] shows the optimal projection found by splines2d, a reconstructed view obtained by manually combining parameters, and a plot of the known relationship between parameters.
![Comparison of guided tour final view (left), approximation based on original parameters (middle) and expected relation based on analysis setup (right).[]{data-label="fig:nsePlotOrig"}](figure/nsePlotOrig-1){width="\textwidth"}
To further investigate relationships between parameters, $L1$ is removed and PP with the splines2D is applied to the remaining four parameters. The dependence of $L2$ on the mass and radius of the lighter neutron star, is revealed (Figure \[fig:nseRemL1\] left plot). A manual reconstruction shows this is a relationship between L2, R1, R2 and m1 (middle plot), but it is effectively the known relationship between L2, R2 and m2 (right plot) – m2 is latently in the relationship though m1.
![Removing L1 and optimize again of the remaining parameters, where m2 remains removed from the set. Because of parameter correlations we can recover clear description of L2 as a function of the other parameters, despite m2 missing.[]{data-label="fig:nseRemL1"}](figure/nseRemL1-1){width="\textwidth"}
This data contains posterior samples from simulation from a model describing a binary black hole (BBH) merger event. There are twelve model parameters. Flat priors are used for most model parameters.
A scatterplot matrix, of nine of the twelve parameters, is shown in the Appendix. (Parameter m2 is not shown because it is strongly linearly associated with m1, phi$\_$jl and psi are not shown because they are uniform, and not associated with other parameters.) Among the nine plotted parameters, strong nonlinear relationships can be seen between the parameters ra, dec and time. The first two describe the position of the event in the sky, and time is the merging time (in GPS units). Because of the elliptical relationship between dec and time, the TIC index is used for PP, even though it is slow to compute. Between the other parameters, the main structure seen is multimodality and some skewness. These patterns are representative of the likelihood function, since most priors are flat, or built to capture growth with volume rather than distance.
The analysis is conducted on 11 of the twelve parameters. One variable is removed, m2, because it is so strongly associated with m1. All parameters are scaled into the range 0 to 1.
All seven PP indexes are applied to the data. Figure \[fig:bbhGuided\] shows the projections that maximize three of the indexes. TIC and splines2d indexes identify very similar projections, that are based on the three parameters, dec, time and ra. This is to be expected based on the pairwise scatterplots. On the other hand, the 1-convex index finds a very different view, but this is because the optimization doesn’t adequately reach a maximum for this index.
![Projections corresponding to the maxima of three indices: TIC, splines2D and 1-convex. Projections a, b found by TIC and splines2d are very similar, and involve the same three parameters, ra, dec and time. The 1-convex index finds a very different view.[]{data-label="fig:bbhGuided"}](figure/bbhGuided-1){width="\textwidth"}
The variables time, dec and ra are dropped from the data, and PP is applied to the remaining 8D space. Figure \[fig:bbhGuided2\] shows the projections which maximize the TIC, splines2D and 1-convex indices. The results provide similar information as already learned from the scatterplot matrix. The parameters chi$\_$tot and chi$\_$p are linearly related (TIC maxima), and theta$\_$jn has a bimodal distribution yielding the figure 8 shape found by the splines2d index. The 1-convex index finds nothing interesting.
![Projections of the reduced 8D space corresponding to the maxima of three indices: TIC, splines2d and 1-convex.[]{data-label="fig:bbhGuided2"}](figure/bbhGuided2-1){width="\textwidth"}
The initial conditions for the optimization, and the subset of variables used, can have a large effect on the projections returned. We illustrate this using only the splines2d index, and find that there is one more association that can be learned that was masked earlier.
Figure \[fig:bbhGuided3\] shows six maxima obtained by different starts, for two types of parameters: first, spin related parameters (i.e. alpha, theta\_jn, chi\_tot and chi\_p), and second position related parameters (i.e. ra, dec and distance). Four of the six (a-d) are almost identical, but not interesting projections. Projection f has the highest PP index value but it is primarily the view seen in the bivariate plot of dec and ra. While none of these projections are particularly interesting on their own, moving between them can be revealing.
Choosing a different subset of variables reveals something new. The subspace of m1, ra, chi\_tot, alpha, distance, dec produces a more refined view of Figure \[fig:bbhGuided3\] projection f. When alpha contributes in contrast to dec, the relationship between the points is almost perfectly on a curve. This is shown in Figure \[fig:constructedExample\]. Manually reconstructing the optimal projection (left plot) can be done by differencing the two parameters, in their original units. This highlights the importance of improved optimization, that would use tiny final steps to polish the view to a finer optimal projection and possibly remove noise induced by small contributions of many variables.
![Final views identified in the dataset considering the seven dimensional parameter space (alpha, theta\_jn, chi\_tot, chi\_p, ra, dec, distance), differing only by randomly selected starting plane.[]{data-label="fig:bbhGuided3"}](figure/bbhGuided3-1){width="\textwidth"}
![Manual reconstruction of an optimal projection (left), constructed by differencing alpha from dec in the original units against ra (middle), compared with the two main variables (right).[]{data-label="fig:constructedExample"}](figure/constructedExample-1){width="\textwidth"}
Applying these procedures to new datasets can be done using the guided tour available in the `tourr` package. The typical steps required are:
1. Scale, standardize or sphere (principal components) the data.
2. Index function selection matching the type of structure that is interesting to detect. Any new function can be used as long as it takes a matrix as input and returns a single index value.
3. Call the guided tour with the data and index function:
- for exploration this can be done via the `tourr::animate` function
- for recording the results `tourr::save_history` should be used
4. Explore how the results depend on choices made (index function, starting planes, optimization method, prior dimension reduction).
These are steps followed in the above two applications and are documented in the comments of the source code (U. Laa and Cook 2019a). For simple usage examples, see documentation of the `tourr::guided_tour` function.
The motivation for this work was to discover dependencies between estimated parameters in multiple model fits in physics problems. This paper shows how projection pursuit with the new indexes can help address this problem. The results are encouraging, showing large potential for discovering unanticipated relations between multiple variables.
All of the indexes fall short against some aspect of the ideal properties of smoothness, squintability, flexibility, rotation invariance and speed. The paper describes how these properties can be assessed using tour methodology. Some potential fixes for the indexes are discussed but there is scope for further developing the new indexes. We recommend to use the spinebil (U. Laa and Cook 2019b) package when developing new indexes. It includes the functionalities needed to reproduce the assessments presented in this paper. While the current focus is on two-dimensional index functions, indexes in the tourr package apply to arbitrary projection dimension, and the methodology introduced here could be applied to the assessment of index functions where $d>2$.
The work also reveals inadequacies in the tour optimization algorithm, that may benefit from newly developed techniques and software tools. Exploring this area would help improve the guided tours. As new optimization techniques become available, adapting these to the guided tour would extend the technique to a broader range of problems. The current recommended approach is to first reduce the dimensionality, for example by using PCA, taking care to preserve nonlinear structure, prior to applying PP.
To apply the existing index functions in practice, we recommend to either use the tourr package directly, or if interaction is required to call the guided tour via the graphical interface available in the galahr (U. Laa and Cook 2019c) package. This package supersedes the now archived tourrGui (Huang, Cook, and Wickham 2012). Both packages contain examples to show how the guided tour can be used with different index functions.
- This article was created with R Markdown (Xie, Allaire, and Grolemund 2018), the code for the paper is available at (U. Laa and Cook 2019a).
- Methods for testing new index functions as presented in this work are implemented in the R package spinebil (U. Laa and Cook 2019b).
- The R package galahr (U. Laa and Cook 2019c) provides a graphical interface to the tourr package allowing for interactive exploration using the guided tour.
- Additional explanations are available in the Appendix, covering details of
- how the holes index was rescaled,
- estimations of the squint angle,
- a comparison of the computational performance of the index functions,
- testing the effect of index parameters on the results and
- suggestions how the index functions may be refined.
The authors gratefully acknowledge the support of the Australian Research Council. We thank Rory Smith for help with the gravitational wave examples, and German Valencia for useful comments. This article was created with knitr (Xie 2015), R Markdown (Xie, Allaire, and Grolemund 2018) and bookdown (Xie 2016) with embedded code.
Abbott, B. P., and others. 2017. “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral.” *Phys. Rev. Lett.* 119 (16): 161101. <https://doi.org/10.1103/PhysRevLett.119.161101>.
———. 2018. “GW170817: Measurements of neutron star radii and equation of state.” <http://arxiv.org/abs/1805.11581>.
Ahn, J. S., H. Hofmann, and D. Cook. 2003. “A Projection Pursuit Method on the Multidimensional Squared Contingency Table.” *Computational Statistics* 18 (3): 605–26.
Albanese, Davide, Michele Filosi, Roberto Visintainer, Samantha Riccadonna, Giuseppe Jurman, and Cesare Furlanello. 2012. “Minerva and Minepy: A c Engine for the Mine Suite and Its R, Python and Matlab Wrappers.” *Bioinformatics*, bts707.
Asimov, D. 1985. “The Grand Tour: A Tool for Viewing Multidimensional Data.” *SIAM Journal of Scientific and Statistical Computing* 6 (1): 128–43.
Buja, A., D. Cook, D. Asimov, and C. Hurley. 2005. “Computational Methods for High-Dimensional Rotations in Data Visualization.” In *Handbook of Statistics: Data Mining and Visualization*, edited by C. R. Rao, E. J. Wegman, and J. L. Solka, 391–413. http://www.elsevier.com: Elsevier/North Holland.
Cook, D., A. Buja, and J. Cabrera. 1992. “An Analysis of Polynomial-Based Projection Pursuit.” *Computing Science and Statistics* 24: 478–82.
Cook, D., A. Buja, J. Cabrera, and C. Hurley. 1995. “Grand Tour and Projection Pursuit.” *Journal of Computational and Graphical Statistics* 4 (3): 155–72.
Cook, Dianne, Andreas Buja, and Javier Cabrera. 1993. “Projection Pursuit Indexes Based on Orthonormal Function Expansions.” *Journal of Computational and Graphical Statistics* 2 (3): 225–50. <http://www.jstor.org/stable/1390644>.
Cook, Dianne, Ursula Laa, and German Valencia. 2018. “Dynamical projections for the visualization of PDFSense data.” *Eur. Phys. J.* C78 (9): 742. <https://doi.org/10.1140/epjc/s10052-018-6205-2>.
Cook, Dianne, and Deborah F. Swayne. 2007. *Interactive and Dynamic Graphics for Data Analysis with R and Ggobi*. 1st ed. Springer Publishing Company, Incorporated.
Diaconis, Persi, and David Freedman. 1984. “Asymptotics of Graphical Projection Pursuit.” *Ann. Statist.* 12 (3): 793–815. <https://doi.org/10.1214/aos/1176346703>.
Eddy, William F. 1977. “A New Convex Hull Algorithm for Planar Sets.” *ACM Trans. Math. Softw.* 3 (4): 398–403. <https://doi.org/10.1145/355759.355766>.
Edelsbrunner, H., D. Kirkpatrick, and R. Seidel. 1983. “On the Shape of a Set of Points in the Plane.” *IEEE Transactions on Information Theory* 29 (4): 551–59. <https://doi.org/10.1109/TIT.1983.1056714>.
Ferraty, F., A. Goia, E. Salinelli, and P. Vieu. 2013. “Functional Projection Pursuit Regression.” *Test* 22 (2): 293–320. <https://search-proquest-com.ezproxy.lib.monash.edu.au/docview/1354331474?accountid=12528>.
Foreman-Mackey, Daniel. 2016. “Corner.py: Scatterplot Matrices in Python.” *The Journal of Open Source Software* 24. <https://doi.org/10.21105/joss.00024>.
Friedman, J. H. 1987. “Exploratory Projection Pursuit.” *Journal of the American Statistical Association* 82 (1): 249–66.
Friedman, J. H., and J. W. Tukey. 1974. “A Projection Pursuit Algorithm for Exploratory Data Analysis.” *IEEE Transactionson Computers* 23: 881–89.
Grimm, Katrin. 2016. “Kennzahlenbasierte Grafikauswahl.” Doctoral thesis, Universität Augsburg.
———. 2017. *Mbgraphic: Measure Based Graphic Selection*. <https://CRAN.R-project.org/package=mbgraphic>.
Hall, Peter. 1989. “On Polynomial-Based Projection Indices for Exploratory Projection Pursuit.” *Ann. Statist.* 17 (2): 589–605. <https://doi.org/10.1214/aos/1176347127>.
Hofmann, Heike, Lee Wilkinson, Hadley Wickham, Duncan Temple Lang, and Anushka Anand. 2019. *Binostics: Compute Scagnostics*.
Hou, Siyuan, and Peter D. Wentzell. 2014. “Re-Centered Kurtosis as a Projection Pursuit Index for Multivariate Data Analysis.” *Journal of Chemometrics* 28 (5): 370–84. <https://doi.org/10.1002/cem.2568>.
Huang, Bei, Dianne Cook, and Hadley Wickham. 2012. “TourrGui: A gWidgets Gui for the Tour to Explore High-Dimensional Data Using Low-Dimensional Projections.” *Journal of Statistical Software* 49 (6): 1–12.
Huber, Peter J. 1985. “Projection Pursuit.” *Ann. Statist.* 13 (2): 435–75. <https://doi.org/10.1214/aos/1176349519>.
Jones, M. C., and R. Sibson. 1987. “What Is Projection Pursuit?” *J. Roy. Statist. Soc., Ser. A* 150: 1–36.
Kruskal, J. B. 1969. “Toward a Practical Method Which Helps Uncover the Structure of a Set of Observations by Finding the Line Transformation Which Optimizes a New ‘Index of Condensation’.” In *Statistical Computation*, edited by R. C. Milton and J. A. Nelder, 427–40. New York: Academic Press.
Kruskal, Joseph B. 1956. “On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem.” *Proceedings of the American Mathematical Society* 7 (1): 48–50. <http://www.jstor.org/stable/2033241>.
Laa, U., and D. Cook. 2019a. <https://github.com/uschiLaa/paper-ppi>.
———. 2019b. <https://github.com/uschiLaa/spinebil>.
———. 2019c. <https://github.com/uschiLaa/galahr>.
Lee, Eun-Kyung, Dianne Cook, Sigbert Klinke, and Thomas Lumley. 2005. “Projection Pursuit for Exploratory Supervised Classification.” *Journal of Computational and Graphical Statistics* 14 (4): 831–46. <https://doi.org/10.1198/106186005X77702>.
“LIGO.” 2018. 2018. [{https://dcc.ligo.org/public/0152/P1800115/005}](%7Bhttps://dcc.ligo.org/public/0152/P1800115/005%7D).
Loperfido, Nicola. 2018. “Skewness-Based Projection Pursuit: A Computational Approach.” *Computational Statistics & Data Analysis* 120: 42–57. <https://doi.org/https://doi.org/10.1016/j.csda.2017.11.001>.
Marsaglia, G. 1968. “Random Numbers Fall Mainly in the Planes.” *Proceedings of the National Academy of Science*.
Naito, Kanta. 1997. “A Generalized Projection Pursuit Procedure and Its Significance Level.” *Hiroshima Math. J.* 27 (3): 513–54. <https://doi.org/10.32917/hmj/1206126967>.
Pan, Jian-Xin, Wing-Kam Fung, and Kai-Tai Fang. 2000. “Multiple Outlier Detection in Multivariate Data Using Projection Pursuit Techniques.” *Journal of Statistical Planning and Inference* 83 (1): 153–67. <https://doi.org/https://doi.org/10.1016/S0378-3758(99)00091-9>.
Pilhöfer, Alexander, and Antony Unwin. 2013. “New Approaches in Visualization of Categorical Data: R Package extracat.” *Journal of Statistical Software* 53 (7): 1–25. <http://www.jstatsoft.org/v53/i07/>.
Posse, Christian. 1995a. “Projection Pursuit Exploratory Data Analysis.” *Computational Statistics & Data Analysis* 20 (6): 669–87. <https://doi.org/https://doi.org/10.1016/0167-9473(95)00002-8>.
———. 1995b. “Tools for Two-Dimensional Exploratory Projection Pursuit.” *Journal of Computational and Graphical Statistics* 4 (2): 83–100. <http://www.jstor.org/stable/1390759>.
R Core Team. 2018. *R: A Language and Environment for Statistical Computing*. Vienna, Austria: R Foundation for Statistical Computing. <https://www.R-project.org/>.
Reshef, David N., Yakir A. Reshef, Hilary K. Finucane, Sharon R. Grossman, Gilean McVean, Peter J. Turnbaugh, Eric S. Lander, Michael Mitzenmacher, and Pardis C. Sabeti. 2011. “Detecting Novel Associations in Large Data Sets.” *Science* 334 (6062): 1518–24. <https://doi.org/10.1126/science.1205438>.
Reshef, Yakir A., David N. Reshef, Hilary K. Finucane, Pardis C. Sabeti, and Michael Mitzenmacher. 2016. “Measuring Dependence Powerfully and Equitably.” *Journal of Machine Learning Research* 17 (212): 1–63. <http://jmlr.org/papers/v17/15-308.html>.
Rodriguez-Martinez, E., J. Yannis Goulermas, T. Mu, and J. F. Ralph. 2010. “Automatic Induction of Projection Pursuit Indices.” *IEEE Transactions on Neural Networks* 21 (8): 1281–95. <https://doi.org/10.1109/TNN.2010.2051161>.
Simon, Noah, and Robert Tibshirani. 2014. “Comment on ‘Detecting Novel Associations In Large Data Sets’ by Reshef Et Al, Science Dec 16, 2011.” *arXiv E-Prints*, January, arXiv:1401.7645. <http://arxiv.org/abs/1401.7645>.
Smith, Rory, Scott E. Field, Kent Blackburn, Carl-Johan Haster, Michael Pürrer, Vivien Raymond, and Patricia Schmidt. 2016. “Fast and accurate inference on gravitational waves from precessing compact binaries.” *Phys. Rev.* D94 (4): 044031. <https://doi.org/10.1103/PhysRevD.94.044031>.
Székely, Gábor J., Maria L. Rizzo, and Nail K. Bakirov. 2007. “Measuring and Testing Dependence by Correlation of Distances.” *Ann. Statist.* 35 (6): 2769–94. <https://doi.org/10.1214/009053607000000505>.
Tukey, P.A., and J.W. Tukey. 1981. *Graphical Display of Data in Three and Higher Dimensions*. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley. <https://books.google.com.au/books?id=WBzvAAAAMAAJ>.
Wahba, G. 1990. *Spline Models for Observational Data*. CBMS-Nsf Regional Conference Series in Applied Mathematics. Society for Industrial; Applied Mathematics. <https://books.google.com.au/books?id=ScRQJEETs0EC>.
Wickham, Hadley, Dianne Cook, Heike Hofmann, and Andreas Buja. 2011. “tourr: An R Package for Exploring Multivariate Data with Projections.” *Journal of Statistical Software* 40 (2): 1–18. <http://www.jstatsoft.org/v40/i02/>.
Wilkinson, L., A. Anand, and R. Grossman. 2005. “Graph-Theoretic Scagnostics.” In *IEEE Symposium on Information Visualization, 2005. INFOVIS 2005.*, 157–64. <https://doi.org/10.1109/INFVIS.2005.1532142>.
Wilkinson, Leland, and Graham Wills. 2008. “Scagnostics Distributions.” *Journal of Computational and Graphical Statistics* 17 (2): 473–91. <https://doi.org/10.1198/106186008X320465>.
Wood, S.N., N., Pya, and B. Säfken. 2016. “Smoothing Parameter and Model Selection for General Smooth Models (with Discussion).” *Journal of the American Statistical Association* 111: 1548–75.
Xie, Yihui. 2015. *Dynamic Documents with R and Knitr*. 2nd ed. Boca Raton, Florida: Chapman; Hall/CRC. <https://yihui.name/knitr/>.
———. 2016. *Bookdown: Authoring Books and Technical Documents with R Markdown*. Boca Raton, Florida: Chapman; Hall/CRC. <https://github.com/rstudio/bookdown>.
Xie, Yihui, J.J. Allaire, and Garrett Grolemund. 2018. *R Markdown: The Definitive Guide*. Boca Raton, Florida: Chapman; Hall/CRC. <https://bookdown.org/yihui/rmarkdown>.
|
---
abstract: '[We provide the detailed proof of a strengthened version of the M. Artin Approximation Theorem.]{}'
author:
- by Arkadiusz Płoski
title: Formal and convergent solutions of analytic equations
---
Impressed by the power of the Preparation Theorem – indeed, it prepares us so well! – I considered “Weierstrass Preparation Theorem and its immediate consequences” as a possible title for the entire book.
Sheeram S. Abhyankar, Preface to [@Abh1964]
Introduction
============
The famous Approximation Theorem of M. Artin [@Art1968] asserts that any formal solution of a system of analytic equations can be approximated by convergent solutions up to a given order. In my PhD thesis [@Pl1973] I was able by analysis of the argument used in [@Art1968] to sharpen the Approximation Theorem: any formal solution can be obtained by specializing parameters in a convergent parametric solution. The theorem was announced with a sketch of proof in [@Pl1974]. The aim of theses notes is to present the detailed proof of this result. It is based on the Weierstrass Preparation Theorem. The other tools are: a Jacobian Lemma which is an elementary version of the Regularity Jacobian Criterion used in [@Art1968], the trick of Kronecker (introducing and specializing variables) and a generalization of the Implicit Function Theorem due to Bourbaki [@Bou1961] and Tougeron [@Tou1972]. All theses ingredients are vital in the proofs of some other results of this type (see [@Art1969], [@Wav1975]). For more information on approximation theorems in local analytic geometry we refer the reader to Teissier’s article [@Te1993-94] and to Chapter 8 of the book [@deJong-Pf2000].
Let ${{\mathbb K}}$ be a field of characteristic zero with a non-trivial valuation. We put ${{\mathbb K}}[[x]]={{\mathbb K}}[[x_1,\dots,x_n]]$ the ring of formal power series in variables $x=(x_1,\dots,x_n)$ with coefficients in ${{\mathbb K}}$. If $f=\sum_{k\geq p}f_k$ is a nonzero power series represented as the sum of homogeneous forms with $f_p\neq 0$ then we write $\ord1\,f=p$. Additionally we put $\ord1\,0=+\infty$ and use the usual conventions on the symbol $+\infty$. The constant term of any series $f\in{{\mathbb K}}[[x]]$ we denote by $f(0)$. A power series $u\in{{\mathbb K}}[[x]]$ is a unit if $uv=1$ for a power series $v\in{{\mathbb K}}[[x]]$. Note that $u$ is a unit if and only if $u(0)\neq 0$. The non-units of ${{\mathbb K}}[[x]]$ form the unique maximal ideal ${{\mathbf m}}_{{{\mathbf x}}}$ of the ring ${{\mathbb K}}[[x]]$. The ideal ${{\mathbf m}}_{{{\mathbf x}}}$ is generated by the variables $x_1,\dots,x_n$. One has $f\in{{\mathbf m}}_{{{\mathbf x}}}^c$, where $c>0$ is an integer, if and only if $\ord1\,f\geq c$. Recall that if $g_1,\dots,g_n\in{{\mathbb K}}[[y]]$, $y=(y_1,\dots,y_n)$ are without constant term then the series $f(g_1,\dots,g_n)\in{{\mathbb K}}[[y]]$ is well-defined. The mapping which associates with $f\in{{\mathbb K}}[[x]]$ the power series $f(g_1,\dots,g_n)$ is the unique homomorphism sending $x_i$ for $g_i$ for $i=1,\dots,n$. Let ${{\mathbb K}}\{x\}$ be the subring of ${{\mathbb K}}[[x]]$ of all convergent power series. Then ${{\mathbb K}}\{x\}$ is a local ring. If $g_1,\dots,g_n\in{{\mathbb K}}\{y\}$ then $f(g_1,\dots,g_n)\in{{\mathbb K}}\{y\}$ for any $f\in{{\mathbb K}}\{x\}$.
In what follows we use intensively the Weierstrass Preparation and Division Theorems. The reader will find the basic facts concerning the rings of formal and convergent power series in [@Abh1964], [@Lef1953] and [@Zar-Sam1960].
Let $f(x,y)=(f_1(x,y),\dots,f_m(x,y))\in{{\mathbb K}}\{x,y\}^m$ be convergent power series in the variables $x=x_1,\dots,x_n)$ and $y=(y_1,\dots,y_N)$ where $m,n,N$ are arbitrary non-negative integers. The theorem quoted below is the main result of [@Art1968].
[THE ARTIN APPROXIMATION THEOREM]{}\
[*Suppose that there exists a sequence of formal power series $\bar y(x)=(\bar y_1(x),\dots,\bar y_N(x))$ without constant term such that $$f(x,\bar y(x))=0\;.$$ Then for any integer $c>0$ there exists a sequence of convergent power series $y(x)=(y_1(x),\dots,y_N(x))$ such that $$f(x,y(x))=0\mbox{ and }y(x)\equiv\bar y(x)\;(\mod\,{{\mathbf m}}_{{{\mathbf x}}}^c)\;.$$* ]{} The congruence condition means that the power series $y_\nu(x)-\bar y_\nu(x)$ are of order$\mbox{}\geq c$ i.e. the coefficients of monomials of degree$\mbox{}<c$ agree in $y_\nu(x)$ and $\bar y_\nu(x)$. We will deduce the Artin Approximation Theorem from the following result stated with a sketch of proof in [@Pl1974].
[THEOREM]{} [*With the notation and assumptions of the Artin theorem there exists a sequence of convergent power series $y(x,t)=(y_1(x,t),\dots,y_N(x,t))\in{{\mathbb K}}\{x,t\}^N$, $y(0,0)=0$, where $t=(t_1,\dots,t_s)$ are new variables, $s\geq 0$, and a sequence of formal power series $\bar t(x)=(\bar t_1(x),\dots,\bar t_s(x))\in{{\mathbb K}}[[x]]^s$, $\bar t(0)=0$ such that $$f(x,y(x,t))=0\mbox{ and }\bar y(x)=y(x,\bar t(x))\;.$$* ]{} The construction of the parametric solution $y(x,t)$ depends on the given formal solution $\bar y(x)$. To get the Artin Approximation Theorem from the stated above result fix an integer $c>0$. Let $y(x,t)$ and $\bar t(x)$ be series such as in the theorem and let $t(x)=(t_1(x),\dots,t_s(x))\in{{\mathbb K}}\{x\}^s$ be convergent power series such that $t(x)\equiv\bar t(x)\;\mod\,{{\mathbf m}}_{{{\mathbf x}}}^c$. Therefore $y(x,t(x))\equiv y(x,\bar t(x))\;\mod\,{{\mathbf m}}_{{{\mathbf x}}}^c$ and it suffices to set $y(x)=y(x,t(x))$
------------------------------------------------------------------------
Before beginning the proof of the theorem let us indicate two corollaries of it.
[COROLLARY 1]{}. [*Assume that $m=N$, $f(x,\bar y(x))=0$ and $$\det\frac{J(f_1,\dots,f_N)}{J(y_1,\dots,y_N)}(x,\bar y(x))\neq 0\;.$$ Then the power series $\bar y(x)$ are convergent.* ]{} Proof. Let $y(x,t)$ and $\bar t(x)$ be power series without constant term such that $f(x,y(x,t))=0$ and $\bar y(x)=y(x,\bar t(x))$. It is easy to check by differentiation of equalities $f(x,y(x,t))=0$ that $(\P{y_\nu}/\P{t_\sigma})(x,t)=0$ for $\nu=1,\dots,N$ and $\sigma=1,\dots,s$. Therefore the series $y(x,t)$ are independent of $t$ and the series $\bar y(x)$ are convergent
------------------------------------------------------------------------
[COROLLARY 2]{}. [*If $f(x,y)\in{{\mathbb K}}\{x,y\}$ is a nonzero power series of $n+1$ variables $(x,y)=(x_1,\dots,x_n,y)$ and $\bar y(x)$ is a formal power series without constant term such that $f(x,\bar y(x))=0$ then $\bar y(x)$ is a convergent power series.* ]{} Proof. By Corollary 1 it suffices to check that there exists a power series $g(x,y)\in{{\mathbb K}}\{x,y\}$ such that $g(x,\bar y(x))=0$ and $(\P{g}/\P{y})(x,\bar y(x))\neq 0$. Let $I=\{g(x,y)\in{{\mathbb K}}\{x,y\}:\,g(x,\bar y(x))=0\}$. Then $I\neq{{\mathbb K}}\{x,y\}$ is a prime ideal of ${{\mathbb K}}\{x,y\}$. Assume the contrary, that is, that for every $g\in I$: $(\P{g}/\P{y})\in I$. Then we get by differentiating the equality $g(x,\bar y(x))=0$ that $(\P{g}/\P{x_i})\in I$ for $i=1,\dots,n$ and, by induction, all partial derivatives of $g$ lie in $I$. Consequently $g=0$ for every $g\in I$ i.e. $I=(0)$. A contradiction since $0\neq f\in I$
------------------------------------------------------------------------
Reduction to the case of simple solutions
=========================================
We keep the notation introduced in Introduction. We will call a sequence of formal power series $\bar y(x)\in{{\mathbb K}}[[x]]$, $\bar y(0)=0$ a [*simple solution*]{} of the system of analytic equations $f(x,y)=0$ if $f(x,\bar y(x))=0$ and $$\rank\frac{J(f_1,\dots,f_m)}{J(y_1,\dots,y_N)}(x,\bar y(x))=m\;.$$ Thus, in this case, $m\leq N$.
In what follows we need
[THE JACOBIAN LEMMA]{}.
*Let $I$ be a nonzero prime ideal of the ring ${{\mathbb K}}\{x\}$, $x=(x_1,\dots,x_n)$. Then there exist covergent power series $h_1,\dots,h_r\in I$ such that*
- $\displaystyle
\rank\frac{J(h_1,\dots,h_r)}{J(x_1,\dots,x_n)}(\mod\,I)=r\;,$
- $\forall h\in I$, $\exists a\notin I$ such that $ah\in(h_1,\dots,h_r){{\mathbb K}}\{x\}$.
Before proving the above lemma let us note that it is invariant with respect to ${{\mathbb K}}$-linear nonsingular transformations. If $\Phi$ is an authomorphism of ${{\mathbb K}}\{x\}$ defined by $$\Phi(f(x_1,\dots,x_n))=
f\left(\sum_{j=1}^nc_{1j}x_j,\dots,\sum_{j=1}^nc_{nj}x_j\right)$$ with $\det(c_{ij})\neq 0$ then the Jacobian Lemma is true for $I$ if and only if it is true for $\Phi(I)$.
Proof of the Jacobian Lemma (by induction on the number $n$ of variables $x_i$).\
If $n=1$ then $I=(x_1){{\mathbb K}}\{x_1\}$ and $h_1=x_1$. Suppose that $n>1$ and that the lemma is true for prime ideals of the ring of power series in $n-1$ variables. Using a ${{\mathbb K}}$-linear nonsingular transformation we may assume that the ideal $I$ contains a power series $x_n$-regular of order $k>0$ i.e. such that the term $x_n^k$ appears in the power series with a non-zero coefficients. Therefore, by the Weierstrass Preparation Theorem $I$ contains a distinguished polynomial $$w(x',x_n)=x_n^k+a_1(x')x_n^{k-1}+\dots+a_k(x'),\mbox{ where }
x'=(x_1,\dots,x_{n-1})\;.$$ By the Weierstrass Division Theorem every power series $h=h(x)$ is of the form $h(x)=q(x)w(x',x_n)+r(x',x_n)$ where $r(x',x_n)$ is an $x_n$-polynomial (of degree$\mbox{}<k$). Therefore, the ideal $I$ is generated by the power series which are polynomials in $x_n$ and to prove the Jacobian Lemma it suffices to find power series $h_1,\dots,h_r$ such that (i) holds and (ii) is satisfied for $h\in I\cap{{\mathbb K}}\{x'\}[x_n]$.
Let $I'=I\cap{{\mathbb K}}\{x'\}$ and consider the set $I\setminus I'[x_n]$. Clearly $w(x',x_n)\in I\setminus I'[x_n]$. Let $$h_1(x',x_n)=c_0(x')x_n^l+c_1(x')x_n^{l-1}+\dots+c_l(x')$$ be a polynomial in $x_n$ of the minimal degree $l$, $l\geq 0$, which belongs to $I\setminus I'[x_n]$. Since the degree $l\geq 0$ is minimal, we have $$\begin{array}{ccc}
l & > & 0\;,\\
c_0(x') & \notin & I'\;,\\
\frac{\P{h_1}}{\P{x_n}} & \notin & I'\;.\\
\end{array}$$ Let $h(x',x_n)\in I$ be a polynomial in $x_n$. Dividing $h(x',x_n)$ by $h_1(x',x_n)$ (Euklid’s division) we get
- $c_0(x')^p\,h(x',x_n)=q(x',x_n)\,h_1(x',x_n)+r_1(x',x_n)$
where $x_n$-degree of $r_1(x',x_n)$ is less than $l$ and $p\geq 0$ is an integer. Since the $x_n$-degree of $r_1(x',x_n)$ is$\mbox{}<l$ then all coefficients of $r_1(x',x_n)$ lie in $I'$. If $I'=(0)$ then $r_1(x',x_n)=0$ and (E) proves the Jacobian Lemma.
If $I'\neq (0)$ then by the induction hypothesis there exists series $h_2,\dots,h_r\in I'$ such that
- $\displaystyle
\rank\frac{J(h_2,\dots,h_r)}{J(x_1,\dots,x_{n-1})}(\mod\,I')=r-1\;,$
- $\forall h'\in I'$, $\exists a'\notin I'$ such that $a'h'\in(h_2,\dots,h_r){{\mathbb K}}\{x'\}$.
We claim that $h_1,\dots,h_r$ satisfy (i) and (ii) of the Jacobian Lemma. To check (i) observe that $$\det\frac{J(h_1,\dots,h_r)}{J(x_{i_1},\dots,x_{i_{r-1}},x_n)}
=\det\frac{J(h_2,\dots,h_r)}{J(x_{i_1},\dots,x_{i_{r-1}})}\cdot
\frac{\P{h_1}}{\P{x_n}}\;,$$ where $i_1,\dots,i_{r-1}\in\{1,\dots,n-1\}$ and use (i$'$). Applying (ii$'$) to the coefficients of $r_1(x',x_n)$ we find a power series $a'(x')$ such that $a'(x')\,r_1(x',x_n)\in (h_2,\dots,h_r){{\mathbb K}}\{x\}$. By (E) we get $a(x')\,h(x',x_n)\in(h_1,\dots,h_r){{\mathbb K}}\{x\}$ where $a(x')=a'(x')\,c_0(x')^p\notin I$ which proves (ii)
------------------------------------------------------------------------
Now, we can check
[PROPOSITION 2.1.]{}
*Let $f(x,y)=(f_1(x,y),\dots,f_m(x,y))\in{{\mathbb K}}\{x,y\}^m$, $f(x,y)\neq 0$, $\bar y(x)=(\bar y_1(x),\dots,\bar y_N(x))\in{{\mathbb K}}[[x]]$, $\bar y(0)=0$, be formal power series such that $f(x,\bar y(x))=0$. Then there exist convergent power series $h(x,y)=(h_1(x,y),\dots,h_r(x,y))\in{{\mathbb K}}\{x,y\}^r$ such that*
- $h(x,\bar y(x))=0$,
- $\displaystyle
\rank\frac{J(h_1,\dots,h_r)}{J(y_1,\dots,y_{N})}(x,\bar y(x))=r\;,$
- suppose that there exist power series $y(x,t)=(y_1(x,t),\dots,y_N(x,t))$, $y(0,0)=0$ and $\bar t(x)=(\bar t_1(x),\dots,\bar t_N(x))$, $\bar t(0)=0$, such that $h(x,y(x,t))=0$ and $\bar y(x)=y(x,\bar t(x))$. Then $f(x,\bar y(x))=0$.
Proof. Consider the prime ideal $$I=\{g(x,y)\in{{\mathbb K}}\{x,y\}:\,g(x,\bar y(x))=0\}\;.$$ Clearly $f_1(x,y),\dots,f_m(x,y)\in I$ and $I\neq(0)$. By the Jacobian Lemma there exist series $h_1(x,y),\dots,h_r(x,y)\in I$ such that
- $\displaystyle
\rank\frac{J(h_1,\dots\dots\dots,h_r)}{J(x_1,\dots,x_n,y_1,\dots,y_N)}
(x,\bar y(x))=r\;,$
- $\forall g\in I$, $\exists a\notin I$ such that $a(x,y)\,g(x,y)\in(h_1,\dots,h_r){{\mathbb K}}\{x,y\}$.
We claim that $h_1,\dots,h_r$ satisfy the conditions (i), (ii), (iii). Condition (i) holds since $h_1,\dots,h_r\in I$. To check (ii) it suffices to observe that
- $\displaystyle
\rank\frac{J(h_1,\dots\dots\dots,h_r)}{J(x_1,\dots,x_n,y_1,\dots,y_N)}
(x,\bar y(x))=
\rank\frac{J(h_1,\dots,h_r)}{J(y_1,\dots,y_N)}(x,\bar y(x))\;.$
Indeed, differentiating the equations $h_i(x,\bar y(x))=0$, $i=1,\dots,r$, we get $$\frac{\P{h_i}}{\P{x_j}}(x,\bar y(x))+\sum_{\nu=1}^N
\frac{\P{h_i}}{\P{y_\nu}}(x,\bar y(x))\frac{\P{\bar y_\nu}}{\P{x_j}}=0
\mbox{ for }j=1,\dots,n$$ and (J) follows. To check (iii) let us write $$a_i(x,y)f_i(x,y)=\sum_{k=1}^ra_{i,k}(x,y)h_k(x,y)\mbox{ in }{{\mathbb K}}\{x,y\}\;,$$ where $a_i(x,y)\notin I$ for $i=1,\dots,m$. Thus $a_i(x,\bar y(x))\neq 0$ and $a_i(x,\bar y(x,t))\neq 0$ since $\bar y(x)=y(x,\bar t(x))$ and (iii) follows
------------------------------------------------------------------------
The Bourbaki–Tougeron implicit function theorem
===============================================
Let $f(x,y)=(f_1(x,y),\dots,f_m(x,y))\in{{\mathbb K}}\{x,y\}^m$ be convergent power series in the variables $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_N)$. Suppose that $m\leq N$ and put $$J(x,y)=\frac{J(f_1,\dots,f_m)}{J(y_{N-m+1},\dots,y_N)}\mbox{ and }
\delta(x,y)=\det\,J(x,y)\;.$$ Let $M(x,y)$ be the adjoint of the matrix $J(x,y)$. Thus we have $$M(x,y)J(x,y)=J(x,y)M(x,y)=\delta(x,y)I_m\;$$ where $I_m$ is the identity matrix of $m$ rows and $m$ columns. Let $g(x,y)=(g_1(x,y),\dots,g_m(x,y))\in{{\mathbb K}}\{x,y\}^m$ be convergent power series defined by $$\left[\begin{array}{c}g_1(x,y)\\\vdots\\g_m(x,y)\end{array}\right]=
M(x,y)\left[\begin{array}{c}f_1(x,y)\\\vdots\\f_m(x,y)\end{array}\right]\;.$$ It is easy to see that
- $g_i(x,y)\in(f_1(x,y),\dots,f_m(x,y)){{\mathbb K}}\{x,y\}$ for $i=1,\dots,m$
and
- $\delta(x,y)f_i(x,y)\in(g_1(x,y),\dots,g_m(x,y)){{\mathbb K}}\{x,y\}$ for $i=1,\dots,m$.
Now, we can state
[THE BOURBAKI-TOUGERON IMPLICIT FUNCTION THEOREM]{}\
*Suppose that there exists a sequence of formal power series $y^0(x)=(y_1^0(x),\dots,y_N^0(x))$, $y^0(0)=0$, such that $$g(x,y^0(x))\equiv 0\;\mod\,\delta(x,y^0(x))^2{{\mathbf m}}_{{{\mathbf x}}}\;.$$ Then*
- Let $y_\nu(x,t)=y^0_\nu(x)+\delta(x,y^0(x))^2t_\nu$ for $\nu=1,\dots,N-m$ where $t=(t_1,\dots,t_{N-m})$ are new variables. Then there exists a unique sequence of formal power series $u(x,t)=(u_{N-m+1}(x,t),\dots,u_N(x,t))\in{{\mathbb K}}[[x,t]]^m$, $u(0,0)=0$, such that if we let $y_\nu(x,t)=y_\nu^0(x)+\delta(x,y^0(x))u_\nu(x,t)$ for $\nu=N-m+1,\dots,N$ and $y(x,t)=(y_1(x,t),\dots,y_N(x,t))$ then $$f(x,y(x,t))=0\mbox{ in }{{\mathbb K}}[[x,t]]\;.$$ If the series $y^0(x)$ are convergent then $u(x,t)$ and $y(x,t)$ are covergent as well.
- For every sequence of formal power series $\bar y(x)=(\bar y_1(x),\dots,\bar y_N(x))$, $\bar y(0)=0$, the following two conditions are equivalent
- there exists a sequence of formal power series $\bar t(x)=(\bar t_1(x),\dots,\bar t_{N-m}(x))$, $\bar t(0)=0$, such that $\bar y(x)=y(x,\bar t(x))$,
- $f(x,\bar y(x))=0$ and $$\begin{aligned}
\bar y_\nu(x) & \equiv & y^0_\nu(x)\;\mod\,\delta(x,y^0(x))^2{{\mathbf m}}_{{{\mathbf x}}}^2
\mbox{ for }\nu=1,\dots,N-m\\
\bar y_\nu(x) & \equiv & y^0_\nu(x)\;\mod\,\delta(x,y^0(x)){{\mathbf m}}_{{{\mathbf x}}}^2
\mbox{ for }\nu=N-m+1,\dots,N\;.\end{aligned}$$
[Remark.]{} In what follows we call
------------- -----------------------------------------------------------------------------------
$y^0(x)$ an approximate solution of the system $f(x,y)=0$,
$y(x,t)$ a parametric solution determined by the approximate solution $y^0(x)$
$\bar y(x)$ satifying (i) or (ii) a subordinate solution to the approximate solution $y^0(x)$
------------- -----------------------------------------------------------------------------------
Proof. Let $v=(v_1,\dots,v_N)$ and $h=(h_1,\dots,h_n)$ be variables. Taylor’s formula reads
- $$\begin{aligned}
\left[\begin{array}{c}f_1(x,v+h)\\\vdots\\f_m(x,v+h)\end{array}\right] & = &
\left[\begin{array}{c}f_1(x,v)\\\vdots\\f_m(x,v)\end{array}\right]+
\frac{J(f_1,\dots,f_m)}{J(y_1,\dots,y_{N-m})}(x,v)
\left[\begin{array}{c}h_1\\\vdots\\h_{N-m}\end{array}\right]\\
& + & J(x,v)\left[\begin{array}{c}h_{N-m+1}\\\vdots\\h_{N}\end{array}\right]+
\left[\begin{array}{c}P_{1}(u,v,h)\\\vdots\\P_{m}(u,v,h)\end{array}\right]\end{aligned}$$
where $P_i(x,v,h)\in(h_1,\dots,h_{N})^2{{\mathbb K}}\{x,v,h\}$ for $i=1,\dots,m$. Let $u=(u_{N-m+1},\dots,u_N)$ be variables and put [$$F_i(x,t,u)=f_i(x,y_1(x,t),\dots,y_{N-m}(x,t),
y^0_{N-m+1}(x)+\delta(x,y^0(x))u_{N-m+1},\dots,
y^0_{N}(x)+\delta(x,y^0(x))u_{N})\;.$$ ]{} Substituting in Taylor’s formula (T) $v_i=y^0_i(x)$ for $i=1,\dots,N$, $h_i=\delta(x,y^0(x))^2t_i$ for $i=1,\dots,N-m$ and $h_i=\delta(x,y^0(x))u_i$ for $i=N-m+1,\dots,N$ we get $$\begin{aligned}
\left[\begin{array}{c}F_1(x,t,u)\\\vdots\\F_m(x,t,u)\end{array}\right] & = &
\left[\begin{array}{c}f_1(x,y^0(x))\\\vdots\\f_m(x,y^0(x))\end{array}\right]+
\delta(x,y^0(x))^2\frac{J(f_1,\dots,f_m)}{J(y_1,\dots,y_{N-m})}(x,y^0(x))
\left[\begin{array}{c}t_1\\\vdots\\t_{N-m}\end{array}\right]\\
& + & \delta(x,y^0(x))J(x,y^0(x))\left[\begin{array}{c}u_{N-m+1}\\\vdots\\u_{N}\end{array}\right]+
\delta(x,y^0(x))^2\left[\begin{array}{c}Q_{1}(x,t,u)\\\vdots\\Q_{m}(x,t,u)\end{array}\right]\end{aligned}$$ where $Q_i(x,t,u)\in(t,u)^2{{\mathbb K}}\{x,t,u\}$ for $i=1,\dots,m$. Multiplying the above identity by the matrix $M(x,y^0(x))$ and taking into account that $M(x,y^0(x))J(x,y^0(x))=\delta(x,y^0(x))I_m$ and $g_i(x,y^0(x))\equiv 0$ $(\mod\,\delta(x,y_0(x))^2{{\mathbf m}}_{{{\mathbf x}}})$ for $i=1,\dots,m$, we get
- $\displaystyle
M(x,y^0(x))\left[\begin{array}{c}F_1(x,t,u)\\\vdots\\F_m(x,t,u)\end{array}\right]=
\delta(x,y^0(x))^2\left[\begin{array}{c}G_1(x,t,u)\\\vdots\\G_m(x,t,u)\end{array}\right]
$
where $G_i(0,0,0)=0$ for $i=1,\dots,m$. Differentiating (\*) we obtain $$M(x,y^0(x))\frac{J(F_1,\dots,F_m)}{J(u_{N-m+1},\dots,u_N)}(x,t,u)=
\delta(x,y^0(x))^2\frac{J(G_1,\dots,G_m)}{J(u_{N-m+1},\dots,u_N)}(x,t,u)$$ which implies
- $\displaystyle
\det\frac{J(F_1,\dots,F_m)}{J(u_{N-m+1},\dots,u_N)}(x,t,u)=
\delta(x,y^0(x))^{m+1}det\frac{J(G_1,\dots,G_m)}{J(u_{N-m+1},\dots,u_N)}(x,t,u)
$
since $\det\,M(x,y^0(x))=\delta(x,y^0(x))^{m-1}$. On the other hand $$\frac{J(F_1,\dots,F_m)}{J(u_{N-m+1},\dots,u_N)}(x,0,0)=
\delta(x,y^0(x))^{m}J(x,y^0(x))$$ and $$\det\frac{J(F_1,\dots,F_m)}{J(u_{N-m+1},\dots,u_N)}(x,0,0)=
\delta(x,y^0(x))^{m+1}\;.$$ Therefore we get from (\*\*) $$\det\frac{J(G_1,\dots,G_m)}{J(u_{N-m+1},\dots,u_N)}(x,0,0)=1\;,$$ in particular $$\det\frac{J(G_1,\dots,G_m)}{J(u_{N-m+1},\dots,u_N)}(0,0,0)=1\;.$$ By the Implicit Function Theorem there exist formal power series $$u(x,t)=(u_{N-m+1}(x,t),\dots,u_{N}(x,t))$$ such that [$$(G_1(x,t,u),\dots,G_m(x,t,u)){{\mathbb K}}[[x,t,u]]=
(u_{N-m+1}-u_{N-m+1}(x,t),\dots,u_{N}-u_{N}(x,t)){{\mathbb K}}[[x,t,u]]\;.$$ ]{} If $y^0(x)$ are convergent then $G(x,t,u)$ and $u(x,t)$ are convergent as well. In particular $G(x,t,u(x,t))=0$ and by (\*) $F(x,t,u(x,t))=0$ which implies $f(x,y(x,t))=0$ where $y_\nu(x,t)=y^0_\nu(x)+\delta(x,y^0(x))u_\nu(x,t)$ for $\nu=N-m+1,\dots,N$. Let $\tilde u(x,t)=(\tilde u_{N-m+1}(x,t),\dots,\tilde u_{N}(x,t))$, $\tilde u(0,0)=0$, be power series such that $f(x,\tilde y(x,t))=0$ where [$$\tilde y(x,t)=(y_1(x,t),\dots,y_{N-m}(x,t),y^0_{N-m+1}(x)+\delta(x,y^0(x))\tilde u_{N-m+1}(x,t),\dots,y^0_{N}(x)+\delta(x,y^0(x))\tilde u_{N}(x,t))\;.$$ ]{} Then $F(x,t,\tilde u(x,t))=0$ and by (\*) $G(x,t,\tilde u(x,t))=0$. Thus we get $\tilde u(x,t)=u(x,t)$. This proves the first part of the Bourbaki-Tougeron Implicit Function Theorem. To check the second part it suffices to observe that for any formal power series $\bar t(x)=(\bar t_1(x),\dots,\bar t_{N-m}(x))$ and $\bar u(x)=(\bar u_{N-m+1}(x),\dots,\bar u_{N}(x))$ without constant term $G(x,\bar t(x),\bar u(x))=0$ if and only if $\bar u(x)=u(x,\bar t(x))$
------------------------------------------------------------------------
Approximate solutions
=====================
We keep the notions and assumptions of Section 3.
[PROPOSITION 4.1]{}. [*Let $\bar y(x)=(\bar y_1(x),\dots,\bar y_N(x))$, $\bar y(0)=0$, be a formal solution of the system of analytic equations $f(x,y)=0$, $m\leq N$, such that the power series $\delta(x,\bar y(x))$ is $x_n$-regular of strictly positive order $p>0$. Then there exists an approximate solution $\bar v(x)\in{{\mathbb K}}\{x'\}[x_n]^N$ of the system $f(x,y)=0$ and such that $\bar y(x)$ is a solution of $f(x,y)=0$ subordinate to $\bar v(x)$.* ]{} Proof. By the Weierstrass Preparation Theorem $\delta(x,\bar y(x))=\bar a(x)\cdot\mbox{unit}$ where $$\bar a(x)=x_n^p+\sum_{j=1}^p\bar a_j(x')x_n^{p-j}$$ is a distinguished polynomial. Using the Weierstrass Division Theorem we get $$\bar y_\nu(x)=\sum_{j=0}^{2p-1}\bar v_{\nu,j}(x')x_n^j+
\bar a(x)^2(c_\nu+\bar t_\nu(x))\mbox{ for }\nu=1,\dots,N-m$$ and $$\bar y_\nu(x)=\sum_{j=0}^{p-1}\bar v_{\nu,j}(x')x_n^j+
\bar a(x)(c_\nu+\bar u_\nu(x))\mbox{ for }\nu=N-m+1,\dots,N$$ where $c_\nu\in{{\mathbb K}}$ for $\nu=1,\dots,N$, while $$\bar t(x)=(\bar t_1(x),\dots,\bar t_{N-m}(x))\mbox{ and }
\bar u(x)=(\bar u_{N-m+1}(x),\dots,\bar u_{N}(x))$$ are formal power series without constant term. Let $$\bar v_\nu(x)=\sum_{j=0}^{2p-1}\bar v_{\nu,j}(x')x_n^j+
\bar a(x)^2 c_\nu\mbox{ for }\nu=1,\dots,N-m$$ and $$\bar v_\nu(x)=\sum_{j=0}^{p-1}\bar v_{\nu,j}(x')x_n^j+
\bar a(x)c_\nu\mbox{ for }\nu=N-m+1,\dots,N\;.$$ Clearly $\bar v(x)=(\bar v_1(x),\dots,\bar v_N(x))\in{{\mathbb K}}[[x']][x_n]^N\;.$
[Propery 1:]{} [*$\delta(x,\bar v(x))=\bar a(x)\cdot\mbox{unit}$* ]{} Proof. From $\bar y(x)\equiv\bar v(x)$ $(\mod\,\bar a(x){{\mathbf m}}_{{{\mathbf x}}})$ we get $\delta(x,\bar y(x))\equiv\delta(x,\bar v(x))$ $(\mod\,\bar a(x){{\mathbf m}}_{{{\mathbf x}}})$ and Property 1 follows since $\delta(x,\bar y(x))=\bar a(x)\cdot\mbox{unit}$
------------------------------------------------------------------------
[Propery 2:]{} [*$g(x,\bar v(x))\equiv 0$ $(\mod\,\bar a(x)^2{{\mathbf m}}_{{{\mathbf x}}})$*]{} Proof. Substituting in Taylor’s formula (T) $v=\bar v(x)$, $h_\nu=\bar a(x)^2\bar t_\nu(x)$ for $\nu=1,\dots,N-m$ and $h_\nu=\bar a(x)\bar u_\nu(x)$ for $\nu=N-m+1,\dots,N$ we get $$\begin{aligned}
\left[\begin{array}{c}0\\\vdots\\0\end{array}\right] & = &
\left[\begin{array}{c}f_1(x,\bar v(x))\\\vdots\\f_m(x,\bar v(x))\end{array}\right]+
\bar a(x)^2\frac{J(f_1,\dots,f_m)}{J(y_1,\dots,y_{N-m})}(x,\bar v(x))
\left[\begin{array}{c}\bar t_1(x)\\\vdots\\\bar t_{N-m}(x)\end{array}\right]\\
& + & \bar a(x)J(x,\bar v(x))\left[\begin{array}{c}\bar u_{N-m+1}(x)\\\vdots\\\bar u_{N}(x)\end{array}\right]+
\bar a(x)^2\left[\begin{array}{c}\bar Q_{1}(x)\\\vdots\\\bar Q_{m}(x)\end{array}\right]\;.\end{aligned}$$ Multiplying the above identity by $M(x,\bar v(x))$ and taking into account the formula $$M(x,\bar v(x))J(x,\bar v(x))=\delta(x,\bar v(x))I_m$$ we get Property 2
------------------------------------------------------------------------
[PROPOSITION 4.2]{}
*Let $(c^0_{\nu,j})$, $\nu=1,\dots,N$, $j=0,1,\dots,D$, be a family of constants such that $c^0_{\nu,0}=0$ for $\nu=1,\dots,N$. Suppose that $$\left(\sum_{j=0}^Dc_{1,j}^0x_n^j,\dots,\sum_{j=0}^Dc_{N,j}^0x_n^j\right)$$ is an approximate solution of the system of equations $f(0,x_n,y)=0$ such that $$\ord1\,\delta\left(0,x_n,\sum_{j=0}^Dc_{1,j}^0x_n^j,\dots,\sum_{j=0}^Dc_{N,j}^0x_n^j\right)
=p\,,\quad 0<p<+\infty\;.$$ Let $V^0=(V^0_{\nu,j})$, $\nu=1,\dots,N$, $j=0,1,\dots,D$ be variables. Then there exists a sequence $$F(x',V^0)=(F_1(x',V^0),\dots,F_M(x',V^0))\in{{\mathbb K}}\{x',V^0\}^M$$ such that for any family $(\bar v^0_{\nu,j}(x'))$ of formal power series without constant term the following two conditions are equivalent*
- $\displaystyle
\left(\sum_{j=0}^D(c_{1,j}^0+\bar v^0_{1,j}(x'))x_n^j,\dots,
\sum_{j=0}^D(c_{N,j}^0+\bar v^0_{N,j}(x'))x_n^j\right)$\
is an approximate solution of the system $f(x,y)=0$,
- $F(x',(\bar v^0_{\nu,j}(x')))=0$ in ${{\mathbb K}}[[x']]$.
Proof. Let $$v_{\bar\nu}(x_n)=\sum_{j=0}^D(c^0_{\nu,j}+V^0_{\nu,j})x_n^j,\quad
v(x_n)=(v_1(x_n),\dots,v_N(x_n))\;.$$ It is easy to check that $\delta(x,v(x_n))$ is $x_n$-regular of order $p$. By the Weierstrass Division Theorem $$g_i(x,v(x_n))=Q_i(x,V^0)\delta(x,v(x_n))^2+\sum_{j=0}^{2p-1}R_{i,j}(x',V^0)x_n^j$$ for $i=1,\dots,m$. Let $$\bar v_{\nu}(x)=\sum_{j=0}^D(c^0_{\nu,j}+\bar v^0_{\nu,j})x_n^j,\quad
\bar v(x)=(\bar v_1(x),\dots,\bar v_N(x))$$ where $(\bar v^0_{\nu,j}(x'))$ is a family of formal power series without constant term. Thus we get $$g_i(x,\bar v(x))=Q_i(x,\bar v(x))\delta(x,\bar v(x))^2+
\sum_{j=0}^{2p-1}R_{i,j}(x',\bar v^0_{\nu,j}(x'))x_n^j\mbox{ for }i=1,\dots,m\;.$$ By the uniqueness of the remainder in the Weierstrass Division Theorem we have that $\bar v(x)$ is an approximate solution of the system of analytic equations $f(x,y)=0$ if and only if $R_{i,j}(x',(\bar v^0_{\nu,j}(x')))=0$ for $i=1,\dots,m$ and $j=0,1,\dots,2p-1$ in ${{\mathbb K}}[[x']]$. This proves the proposition
------------------------------------------------------------------------
Proof of the theorem (by induction on the number $n$ of variables $x$)
======================================================================
The theorem is trivial for $n=0$. Suppose that $n>0$ and that the theorem is true for $n-1$. By PROPOSITION 2.1 we may suppose that $\bar y(x)$ is a simple solution of the system $f(x,y)=0$. Let $$\delta(x,y)=\det\frac{J(f_1,\dots,f_m)}{J(y_{N-m+1},\dots,y_N)}\;.$$ Without diminishing the generality we may suppose that $\delta(x,\bar y(x))\neq 0$. If $\delta(0,0)\neq 0$ then the theorem follows from the Implicit Function Theorem. Suppose that $\delta(0,0)=0$. After a linear change of the variables $x_1,\dots,x_n$ we may assume that $\delta(x,\bar y(x))$ is $x_n$-regular of order $p>0$. By PROPOSITION 4.1 the system of equations $f(x,y)=0$ has an approximate solution $\bar v(x)=(\bar v_1(x),\dots,\bar v_N(x))\in{{\mathbb K}}[[x']][x_n]$ such that the solution $\bar y(x)$ is subordinate to $\bar v(x)$. Write $$\bar v_{\nu}(x)=\sum_{j=0}^D(c^0_{\nu,j}+\bar v^0_{\nu,j}(x'))x_n^j,\quad
D\geq 0\mbox{ an integer}$$ where $(\bar v_{\nu,j}(x'))$ is a family of formal power series without constant term. It is easy to check that $$\left(\sum_{j=0}^Dc_{1,j}^0x_n^j,\dots,
\sum_{j=0}^Dc_{N,j}^0x_n^j\right)$$ is an approximate solution of the system $f(0,x_n,y)=0$ such that $$\ord1\,\delta
\left(0,x_n,\sum_{j=0}^Dc_{1,j}^0x_n^j,\dots,
\sum_{j=0}^Dc_{N,j}^0x_n^j\right)=p\;.$$ By PROPOSITION 4.2 there exist convergent power series $F(x',V^0)\in{{\mathbb K}}\{x',V^0\}^M$ such that $F(x',(\bar v^0_{\nu,j}(x')))=0$. By induction hypothesis there exist convergent power series $(V^0_{\nu,j}(x',s))$ in ${{\mathbb K}}\{x',s\}$ where $s=(s_1,\dots,s_q)$ are new variables and formal power series $\bar s(x')=(\bar s_1(x'),\dots,\bar s_q(x'))$ without constant term such that $$F(x',(V^0_{\nu,j}(x',s)))=0,\quad V^0_{\nu,j}(x',\bar s(x'))=
\bar v^0_{\nu,j}(x')\;.$$ Let $$v_{\nu}(x,s)=\sum_{j=0}^D(c^0_{\nu,j}+V^0_{\nu,j}(x',s))x_n^j
\mbox{ for }\nu=1,\dots,N$$ and $v(x,s)=(v_1(x,s),\dots,v_N(x,s))$. Thus $\bar v_\nu(x)=v_\nu(x,\bar s(x'))$ for $\nu=1,\dots,N$. Again by Proposition 4.2 $v(x,s)$ is an approximate solution of the system $f(x,y)=0$. By the Bourbaki-Tougeron Implicit Function Theorem the system $f(x,y)=0$ has the parametric solution determined by $v(x,s)$: $$\begin{aligned}
y_\nu(x,s,t) & = & v_\nu(x,s)+\delta(x,v(x,s))^2t_\nu\mbox{ for }
\nu=1,\dots,N-m\\
y_\nu(x,s,t) & = & v_\nu(x,s)+\delta(x,v(x,s))u_\nu(x,s,t)\mbox{ for }
\nu=N-m+1,\dots,N\;.\end{aligned}$$ On the other hand $$\begin{aligned}
\bar y_\nu(x,t) & = & \bar v_\nu(x)+\delta(x,\bar v(x))^2t_\nu\mbox{ for }
\nu=1,\dots,N-m\\
\bar y_\nu(x,t) & = & \bar v_\nu(x)+\delta(x,\bar v(x))\bar u_\nu(x,t)\mbox{ for }
\nu=N-m+1,\dots,N\end{aligned}$$ is the parametric solution determined by $\bar v(x)$. Since the formal solution $\bar y(x)$ is subordinate to the approximate solution $\bar v(x)$ there exist formal power series $\bar t(x)=(\bar t_1(x),\dots,\bar t_{N-m}(x))$, $\bar t(0)=0$, such that $\bar y(x)=\bar y(x,\bar t(x))$. We have $$\begin{aligned}
y_\nu(x,,\bar s(x'),t) & = & \bar v_\nu(x)+\delta(x,\bar v(x))^2t_\nu\mbox{ for }
\nu=1,\dots,N-m\\
y_\nu(x,\bar s(x'),t) & = & \bar v_\nu(x)+\delta(x,\bar v(x))u_\nu(x,\bar s(x'),t)\mbox{ for }
\nu=N-m+1,\dots,N\end{aligned}$$ By the uniqueness of the parametric solution determined by the approximate solution $\bar v(x)$ we get $$y(x,\bar s(x'),\bar t(x))=\bar y(x,\bar t(x))=\bar y(x)\;~\rule{1ex}{1ex}$$
[GGGGGGGG]{}
S. S. Abhyankar, [*Local analytic geometry*]{}, Academic Press 1964
M. Artin, [*On the solutions of analytic equations*]{}, Invent. math. 5 (1968), 277–291
M. Artin, [*Algebraic approximations of structures over complete local rings*]{}, Inst. Hautes Etudes Sci. Publ. Math. 36 (1969), 23–58
N. Bourbaki, [*Algèbre commutative*]{}, Fasc. XXVIII, Chap. III, §[4]{}, No. 5 (1961)
T. de Jong, G. Pfister, [*Local Analytic Geometry*]{}, vieweg 2000
S. Lefschetz, [*Algebraic Geometry*]{}, Princeton NJ, 1953
A. Płoski, [*Rozwizania formalne i zbiene røwna analitycznych*]{}, Ph D thesis, Institute of Mathematics, Polish Academy of Sciences (1973) (in Polish).
A. Płoski, Note on a theorem of M. Artin, Bull. Acad. Polonaise Sci. Ser. Math. 22 (1974), 1107–1109
B. Teissier, [*Résultats récents sur l’approximation des morphismes en algèbre commutative*]{} (d’après André, Artin, Popescu et Spivakovsky). Séminaire BOURBAKI, 16 ème année (1993-94), n$^\circ$784, 259–281
J. C. Tougeron, [*Idéaux de fonctions differentiables*]{}, New York: Springer 1972
John J. Wavrik, [*A Theorem on Solutions of Analytic Equations with Applications to Deformations of Complex Structures*]{}, Math. Ann. 216 (1975), 127–142
O. Zariski, P. Samuel, [*Commutative Algebra*]{}, vol.II, Van Nostrand, Princeton, New Jersey, 1960
[Department of Mathematics\
Kielce University of Technology\
AL. 1000 L PP 7\
25-314 Kielce, Poland]{}
|
---
abstract: |
In this paper we study Green measures of certain classes of Markov processes. In particular Brownian motion and processes with jump generators with different tails. The Green measures are represented as a sum of a singular and a regular part given in terms of the jump generator. The main technical question is to find a bound for the regular part.
[*Keywords*]{}: Markov processes, Green measures, compound Poisson process, Brownian motion.
[*AMS Subject Classification 2010*]{}: 47D07, 37P30, 60J65, 60G55.
author:
- |
**Yuri Kondratiev**\
Department of Mathematics, University of Bielefeld,\
D-33615 Bielefeld, Germany,\
Dragomanov University, Kiev, Ukraine\
Email: kondrat@mathematik.uni-bielefeld.de\
Email: kondrat@math.uni-bielefeld.de
- |
**Jos[é]{} L. da Silva**,\
CIMA, University of Madeira, Campus da Penteada,\
9020-105 Funchal, Portugal.\
Email: joses@staff.uma.pt
title: Green Measures for Markov Processes
---
Introduction
============
Let $X(t), t\geq 0$ be a time homogeneous Markov process in ${{{{\mathbb R}}^d}}$ starting from the point $x\in{{{{\mathbb R}}^d}}$. For a function $f:{{{{\mathbb R}}^d}}\to {{\mathbb R}}$ we consider the following heuristic object $$V(f,x)= \int_0^\infty E^x[f(X(t)] dt.$$ If this quantity exists then $V(f,x)$ is called the potential for the function $f$. The notions of potentials is well known in probability theory, see e.g., [@Blumenthal1968; @Revuz-Yor-94].
The existence of the potential $V(f,x) $ is a difficult question and the class of admissible $f$ shall be analyzed for each process separately. An alternative approach is based on the use of the generator $L$ of the process $X$. Namely, the potential $V(f,x) $ may be constructed as the solution to the following equation: $$-LV=f.$$ Of course, there appear a technical problem of the characterization of the domain for the inverse generator $L^{-1}$. For a general Markov process we can not characterize this domain.
In the analogy with the PDE framework, we would like to have a representation $$V(t,x) =\int_{{{{{\mathbb R}}^d}}} f(y) {\mathcal G}(x,\mathrm{d}y),$$ where ${\mathcal G}(x,dy)$ is a measure on ${{{{\mathbb R}}^d}}$. This measure is nothing but the fundamental solution to the considered equation and traditionally may be called the Green measure for the operator $L$. Our aim is to study Green measures for certain classes of Markov processes. We stress again that the concepts of potentials and Green measures are related in the same manner as the notions of solutions and fundamental solutions to the corresponding equations.
In this paper we discuss general notion of Green measures for Markov processes and consider particular examples of such processes for which the existence and properties of Green measures may be analyzed.
General Framework
=================
We will consider time homogeneous Markov processes $X(t)$, $t\ge0$ in ${{{{\mathbb R}}^d}}$, see for example [@Blumenthal1968; @Dynkin1965; @Meyer1967; @Revuz-Yor-94]. A standard way to define a Markov process is to give the probability $P_{t}(x,B)$ of the transition from the point $x\in{{{{\mathbb R}}^d}}$ to the Borel set $B\subset{{{{\mathbb R}}^d}}$ in the time $t>0$. In some cases we may have $$P_{t}(x,B)=\int_{B}p_{t}(x,y)\,\mathrm{d}y,$$ where $p_{t}(x,y)$ is the density of the transition probability (heat kernel), that is, $P_{t}(x,\mathrm{d}y)=p_{t}(x,y)\,\mathrm{d}y$. The function $$g(x,y):=\int_{0}^{\infty}p_{t}(x,y)\,\mathrm{d}t$$ is called the Green function, although the integral here may diverge. The existence of the Green function for a given process or transition probability, even for simple classes of Markov processes, is not always guaranteed, see examples below. Nevertheless, Green functions for different classes of Markov processes are well known in probability theory, see, e.g., [@CGL20; @Grigoryan2018a] and references therein.
As an alternative we introduce the Green measure by $${\mathcal G}(x,\mathrm{d}y):=\int_{0}^{\infty}P_{t}(x,\mathrm{d}y)\,\mathrm{d}t,\quad x\in{{{{\mathbb R}}^d}},$$ assuming the existence of this object as a Radon measure on ${{{{\mathbb R}}^d}}$. Then we would have ${\mathcal G}(x,\mathrm{d}y)=g(x,y)\,\mathrm{d}y$. The aim of this paper is to show how to define and study Green measures for certain particular Markov processes in ${{{{\mathbb R}}^d}}$.
Let us be more precise. We can start with a Markov semigroup $T(t),t\geq0$, that is, a family of linear operators in a Banach space $E$. As $E$, we may use bounded measurable functions $B({{{{\mathbb R}}^d}})$, bounded continuous functions $C_{b}({{{{\mathbb R}}^d}})$ or Lebesgue spaces $L^{p}({{{{\mathbb R}}^d}})$, $p\ge1$ depending on each particular case. This family of operators satisfy the following properties:
1. $T(t)\in{{\mathcal L}}(E),\;\;t\geq0$,
2. $T(0)=1$,
3. ${\displaystyle \lim_{t\to0^{+}}}T(t)f=f,\quad f\in E$,
4. $T(t+s)=T(t)T(s)$,
5. $\forall f\geq0\;\;\;T(t)f\geq0$.
The semigroup is conservative if
1. $T(t)1=1$.
The semigroup $T(t)$, $t\ge0$ is associated with a Markov process $\{X(t),t\geq0\;|\;P_{x},x\in{{{{\mathbb R}}^d}}\}$ if
$$(T(t)f)(x)=E^{x}[f(X(t))]=\int_{{{{{\mathbb R}}^d}}}f(y)P_{t}(x,\mathrm{d}y),\quad f\in E.$$ The transition probabilities may be constructed from the semigroup by choosing $f=\mathbbm{1}_{A}$, $A\in{{\mathcal B}}({{{{\mathbb R}}^d}})$, that is, $$P_{t}(x,A)=(T(t)\mathbbm{1}_{A})(x).$$ Now we introduce the resolvent of the Markov semigroup $T(t)$, $t\ge0$. Let $\lambda>0$ be given. The $\lambda$-resolvent of the Markov semigroup is the linear operator $R_{\lambda}:E\longrightarrow E$ defined by $$(R_{\lambda}f)(x):=\int_{0}^{\infty}e^{-\lambda t}(T(t)f)(x)\,\mathrm{d}t=\int_{0}^{\infty}e^{-\lambda t}\int_{{{{{\mathbb R}}^d}}}f(y)P_{t}(x,\mathrm{d}y)\,\mathrm{d}t,$$ for any $f\in E$ and $x\in{{{{\mathbb R}}^d}}$.
Denote by ${{\mathcal B}}_{b}({{{{\mathbb R}}^d}})$ the family of bounded Borel sets in ${{{{\mathbb R}}^d}}$ and $C_{0}({{{{\mathbb R}}^d}})$ the space of continuous functions with compact support.
The Green measure for a Markov process $X(t)$, $t\ge0$ with transition probability $P_{t}(x,B)$ is defined by $${\mathcal G}(x,B):=\int_{0}^{\infty}P_{t}(x,B)\,\mathrm{d}t,\;\;B\in{{\mathcal B}}_{b}({{{{\mathbb R}}^d}}),$$ or $$\int_{{{{{\mathbb R}}^d}}}f(y){\mathcal G}(x,dy)=\int_{0}^{\infty}f(y)P_{t}(x,dy)\,\mathrm{d}t,\;\;f\in C_{0}({{{{\mathbb R}}^d}})$$ whenever these integrals exist.
The Markov generator is a characteristic of a Markov semigroup. More precisely, we have the following definition.
We set $$D(L):=\left\{ f\in E\,\middle|\,\frac{T(t)f-f}{t}\;\mathrm{converges\;in}\;E\;\mathrm{when}\;t\to0^{+}\right\}$$ and for every $f\in D(L)$, $x\in{{{{\mathbb R}}^d}}$ $$(Lf)(x):=\lim_{t\to0^{+}}\frac{(T(t)f)(x)-f(x)}{t}.$$ Then $D(L)$ (the domain of $L$) is a linear subspace of $E$ and ${L:D(L)\longrightarrow E}$ is a linear operator called the generator of the semigroup $T(t)$, $t\ge0$.
There are several known properties of the generator $L$ and and we have the full description of the Markov generators via so-called maximum principle, see, e.g., [@CGL20; @Grigoryan2018a].
From the relation between semigroup and generator we have $$E^{x}\left[\int_{0}^{\infty}f(X(t))\,\mathrm{d}t\right]=\int_{{{{{\mathbb R}}^d}}}f(y){\mathcal G}(x,\mathrm{d}y)=-(L^{-1}f)(x)=\int_{0}^{\infty}(T(t)f)(x)\,\mathrm{d}t\label{FS}$$ for every $f\in C_{0}({{{{\mathbb R}}^d}})$. Because any Radon measure defines a generalized function, then we may write $${\mathcal G}(x,\mathrm{d}y)=g(x,y)\,\mathrm{d}y,$$ where $g(x,\cdot)\in D'({{{{\mathbb R}}^d}})$ is a positive generalized function for all $x\in{{{{\mathbb R}}^d}}$. In view of (\[FS\]) the Green measure is the fundamental solution corresponding to the operator $L$. Note that the existence and regularity of this fundamental solution produces a description of admissible Markov processes for which the Green measure exists. In Section \[sec:Particular-Models\] we present some examples and show the existence of the Green measure under the assumption $d\geq3$. This moment is one more demonstration on the essential influence of the dimension of the phase space on the properties of Markov processes.
Jump Generators and Green Measures
==================================
Let $a:{{{{\mathbb R}}^d}}\to{{\mathbb R}}$ be a fixed kernel with the following properties:
1. Symmetric, $a(-x)=a(x)$, for every $x\in{{{{\mathbb R}}^d}}$.
2. Positive and bounded, $a\geq0$, $a\in C_{b}({{{{\mathbb R}}^d}})$.
3. Integrable $$\int_{{{{{\mathbb R}}^d}}}a(y)\,\mathrm{d}y=1.$$
Consider the generator $L$ defined on $E$ (as mentioned above) by $$(Lf)(x)=\int_{{{{{\mathbb R}}^d}}}a(x-y)[f(y)-f(x)]\,\mathrm{d}y=(a*f)(x)-f(x),\quad x\in{{{{\mathbb R}}^d}}.$$ In particular, $L^{\ast}=L$ in $L^{2}({{{{\mathbb R}}^d}})$ and $L$ is a bounded linear operator in all $L^{p}({{{{\mathbb R}}^d}})$, $p\ge1$. We call this operator the jump generator with jump kernel $a$. The corresponding Markov process is of a pure jump type and is known in stochastic as compound Poisson process, see [@Skorohod1991].
Several analytic properties of the jump generator $L$ were studied recently, see for example [@Grigoryan2018; @Kondratiev2017; @Kondratiev2018]. Here we recall some of these properties necessary in what follows.
Because $L$ is a convolution operator, it is natural to apply Fourier transform to study it.
At first notice that, due to the symmetry of the kernel $a$, its Fourier image is given by
$$\hat{a}(k)=\int_{{{{{\mathbb R}}^d}}}e^{-i(k,y)}a(y)\,\mathrm{d}y=\int_{{{{{\mathbb R}}^d}}}\cos((k,y))a(y)\,\mathrm{d}y.$$ Then, it is easy to see that $$\hat{a}(0)=1,\quad\hat{a}(k)<1,\;k\neq0,$$ $$\hat{a}(k)\to0,k\to\infty.$$ On the other hand, the Fourier image $L$ is the multiplication operator by $$\hat{L}(k)=\hat{a}(k)-1$$ that is the symbol of $L$.
We make the following assumptions on the kernel $a$.
[(H)]{}
: The jump kernel $a$ is such that $\hat{a}\in L^{1}({{{{\mathbb R}}^d}})$ and has finite second moment, that is, $$\int_{{{{{\mathbb R}}^d}}}|x|^{2}a(x)\,\mathrm{d}x<\infty.$$
Denote by ${\mathcal G}_{{\lambda}}(x,y)$, $x,y\in{{{{\mathbb R}}^d}}$, ${\lambda}\in(0,\infty)$ the resolvent kernel of $R_{{\lambda}}(L)=({\lambda}-L)^{-1}$. This kernel admits the representation $${\mathcal G}_{{\lambda}}(x,y)=\frac{1}{1+{\lambda}}\big(\delta(x-y)+G_{{\lambda}}(x-y)\big),\quad{\lambda}\in(0,\infty),$$ with $$G_{{\lambda}}(x)=\sum_{k=1}^{\infty}\frac{a_{k}(x)}{(1+{\lambda})^{k}},\label{eq:Green-kernel}$$ where $a_{k}(x)=a^{\ast k}(x)$ is the $k$-fold convolution of the kernel $a$. Notice that the resolvent kernel ${\mathcal G}_{{\lambda}}(x,y)$ has a singular part, $\delta(x-y)$ and a regular part $G_{{\lambda}}(x-y)$. The Green function, as a generalized function, has the form $${\mathcal G}_{0}(x)=\delta(x)+G_{0}(x).$$
The transition probability density $p(t,x)$ in terms of Fourier transform has representation $$p(t,x)=\frac{1}{(2\pi)^{d}}\int_{{{{{\mathbb R}}^d}}}e^{i(k,x)+t(\hat{a}(k)-1)}\,\mathrm{d}k.$$ and for the resolvent kernel
$${\mathcal G}_{{\lambda}}(x,y)=-(L-{\lambda})^{-1}(x,y),$$ holds
$${\mathcal G}_{{\lambda}}(x-y)=\frac{1}{(2\pi)^{d}}\int_{{{{{\mathbb R}}^d}}}\frac{e^{i(k,x-y)}}{1-\hat{a}(k)+{\lambda}}\,\mathrm{d}k.$$ For a regularization of the last expression we write $$\frac{1}{1-\hat{a}(k)+{\lambda}}=\frac{1}{1+{\lambda}}+\frac{\hat{a}(k)}{(1+{\lambda})(1-\hat{a}(k)+{\lambda})}.$$ Then for operators we have $${\mathcal G}_{{\lambda}}=\frac{1}{1+{\lambda}}+G_{{\lambda}}$$ or in terms of kernels $$G_{{\lambda}}(x-y)=\frac{1}{(2\pi)^{d}}\int_{{{{{\mathbb R}}^d}}}e^{i(k,x-y)}\frac{\hat{a}(k)}{(1+{\lambda})(1-\hat{a}(k)+{\lambda})}\,\mathrm{d}k.$$
We summarize our considerations. The study the resolvent kernel (Green kernel) is reduced to the analysis of the regular part $G_{0}(x)$, that is, $$G_{0}(x)=\sum_{k=1}^{\infty}{a_{k}(x)},\qquad a_{k}(x)=a^{\ast k}(x),$$ where $a_{k}(x)$ is the $k$-fold convolution of the kernel $a$. The Fourier representation for $G_{0}$ is given by
$$G_{0}(x)=\frac{1}{(2\pi)^{d}}\int_{{{{{\mathbb R}}^d}}}e^{i(k,x)}\frac{\hat{a}(k)}{1-\hat{a}(k)}\,\mathrm{d}k.$$ For $d\geq3$ this integral exists for all $x\in{{{{\mathbb R}}^d}}$ that follows from the integrable singularity of $(1-\hat{a}(k))^{-1}$ at $k=0$. The latter is the consequence of our assumptions on $a(x)$.
Particular Models {#sec:Particular-Models}
=================
The main technical question is to obtain a bound for the $k$-fold convolution $a_{k}(x)$ in $k$ and $x$ together for the analysis of the properties of $G_{0}(x)$. From stochastic point of view, $a_{k}(x)$ is the density of sum of $k$ i.i.d. random variables with distribution density $a(x)$. Unfortunately, we can not find in the literature any general result in this direction. There are several particular classes of jump kernels for which we shall expect such kind of results, see [@Kondratiev2018].
1. *Exponential tails* or *light tail*s. That is, the kernel $a(x)$ satisfies the upper bound $$a(x)\leq Ce^{-\delta|x|},\quad\delta>0.$$
2. *Moderate tails*. In this case the asymptotic of $a(x)$ and $x\to\infty$ is given by $$a(x)\sim\frac{C}{|x|^{d+\gamma}},\quad\gamma>2.\label{eq:moderate-tails}$$
3. Heavy tails. The kernel $a(x)$ has an asymptotic similar to with $\gamma\in(0,2)$, that is, $$a(x)\sim\frac{C}{|x|^{d+\gamma}},\quad\gamma\in(0,2).$$
In both cases the exponential tails and moderate tails, the kernel $a(x)$ has second moment. On the other hand, the case of heavy tails the second moment of $a(x)$ does not exists.
Below we consider two examples of kernels $a(x)$ and show the bound for the regular part of the resolvent kernel $G_{0}(x)$.
Gauss Kernels
-------------
Assume that the jump kernels $a(x)$ has the following form:
$$a(x)=C\exp\left(-\frac{b|x|^{2}}{2}\right),\quad C,b>0.\label{G}$$
If the jump kernel $a(x)$ be given by (\[G\]) and $d\geq3$, then holds $$G_{0}(x)\leq C_{1}\exp\left(-\frac{b|x|^{2}}{4}\right).$$
By a direct calculation we find $$a_{k}(x)=\frac{C}{k^{d/2}}\exp\left(-\frac{b|x|^{2}}{2k}\right)$$ with $C=C(b,d)$. Therefore for $d\geq3$ we obtain $$\begin{aligned}
G_{0}(x) & = & \sum_{k=1}^{\infty}{a_{k}(x)}=C\sum_{k=1}^{\infty}\frac{1}{k^{d/2}}\exp\left(-\frac{b|x|^{2}}{2k}\right)\\
& = & C\sum_{k=1}^{\infty}\sum_{n=0}^{\infty}\frac{1}{k^{d/2}}\frac{1}{n!}\left(-\frac{b|x|^{2}}{2k}\right)^{n}\\
& = & C\sum_{n=0}^{\infty}\left(\sum_{k=1}^{\infty}\frac{1}{k^{d/2+n}}\right)\frac{1}{n!}\left(-\frac{b|x|^{2}}{2}\right)^{n}.\end{aligned}$$ As the series $\sum_{k=1}^{\infty}\frac{1}{k^{d/2+n}}=\zeta(d/2+n)\le\zeta(3/2)$ for $d/2+n>1\Leftrightarrow d\ge3$, where $\zeta(s)$, $s>1$ is the Riemann zeta function, then we obtain $$G_{0}(x)\le C\zeta(3/2)\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{b|x|^{2}}{2}\right)^{n}=C_{1}\exp\left(-\frac{b|x|^{2}}{2}\right).$$
Exponential Tails
-----------------
Now we investigate the case when the jump kernel $a(x)$ has exponential tails, that is, $$a(x)\leq C\exp(-\delta|x|),\quad\delta>0.\label{exp}$$
If the jump kernel $a(x)$ satisfies (\[exp\]) and $d\geq3$, then there exist $A,B>0$ such that the bound for $G_{0}(x)$ holds $$G_{0}(x)\leq A\exp(-B|x|).$$
It was shown in [@Kondratiev2018] that $$a_{n}(x)\leq Cn^{-d/2}\exp(-c\min(|x|,|x|^{2}/n)).$$ This implies the following bound for $a_{n}(x)$ $$a_{n}(x)\leq Cn^{-d/2}\big(\exp(-c|x|)+\exp(-c|x|^{2}/n)\big).$$ Hence, it is simple to obtain the bound for $G_{0}(x)$, namely for $d\ge3$ $$\begin{aligned}
G_{0}(x) & = & \sum_{n=1}^{\infty}{a_{n}(x)}\le C_{1}\exp(-c_{1}|x|)+C_{2}\exp(-c_{2}|x|^{2})\\
& \le & A\exp(-B|x|).\end{aligned}$$
Brownian Motion
---------------
Let us consider another concrete example of a Markov process. Namely, denote $B(t)$, $t\ge0$ the Brownian motion in ${{{{\mathbb R}}^d}}$. The generator of this process is the Laplace operator $\Delta$ considered in a proper Banach space $E$. As above we are interested in studying the expectation of the random variable $$Y(f)=\int_{0}^{\infty}f(B(t))\,\mathrm{d}t$$ for certain class of functions $f:{{{{\mathbb R}}^d}}\to{{\mathbb R}}$. To this end, we introduce the following class of functions $$CL({{{{\mathbb R}}^d}})=\{f:{{{{\mathbb R}}^d}}\rightarrow{{\mathbb R}}:f\;\text{is continuous, bounded and belongs to}\;L_{1}({{{{\mathbb R}}^d}})\}.$$ It is a Banach space with the norm $\|f\|_{CL}:=\|f\|_{\infty}+\|f\|_{1}$, where $\|\cdot\|_{\infty}$ is the supremum norm and $\|\cdot\|_{1}$ is the norm in $L_{1}({{{{\mathbb R}}^d}})$.
Let $d\geq3$ be given. The Green measure of Brownian motion is $${\mathcal G}(x,dy)=G_{0}(x-y)\,\mathrm{d}y=\frac{C(d)}{|x-y|^{d-2}}\,\mathrm{d}y.$$
Note that due to (\[FS\]) we have $$E^{x}[Y(f)]=-\Delta^{-1}f(x)=\int_{{{{{\mathbb R}}^d}}}C(d)\frac{f(y)}{|x-y|^{d-2}}\,\mathrm{d}y.$$ Then $$\begin{aligned}
\left|\int_{{{{{\mathbb R}}^d}}}\frac{f(y)}{|x-y|^{d-2}}\,\mathrm{d}y\right| & \leq\left|\,\int_{|x-y|\leq1}\frac{f(y)}{|x-y|^{d-2}}\,\mathrm{d}y\right|+\left|\,\int_{|x-y|>1}\frac{f(y)}{|x-y|^{d-2}}\,\mathrm{d}y\right|\\
& \le C_{1}\|f\|_{\infty}+C_{2}\|f\|_{L_{1}({{{{\mathbb R}}^d}})}\\
& \leq C\|f\|_{CL},\end{aligned}$$ where we have used the local integrability in $y$ of $|x-y|^{2-d}$. It means that every function from $CL({{{{\mathbb R}}^d}})$ is integrable with respect to the Green measure.
In a forthcoming paper we will investigate the additive functionals for time change Markov processes. More precisely, let $X(t)$, $t\ge0$ be a Markov process in ${{{{\mathbb R}}^d}}$ with generator $L$ and denote by $\mu_{t}(\mathrm{d}x)$ the marginal distribution of $X(t)$. That is, $\mu_{t}$ is the solution of the Fokker-Planck equation $$\frac{\partial\mu_{t}}{\partial t}=L^{\ast}\mu_{t}.$$ In addition, Assume that an inverse subordinator $E(t),$ $t\ge0$ is given and consider random time change $$Y(t):=X(E(t)),\quad t\ge0.$$ It is known that the marginal distribution $\nu_{t}$ of $Y(t)$ holds a subordination formula, see [@KKdS19] $$\nu_{t}(\mathrm{d}x)=\int_{0}^{\infty}D_{t}(\tau)\mu_{\tau}(\mathrm{d}x)\,\mathrm{d}\tau,$$ where $D_{t}(\tau)$ is the density distribution of $E(t)$. If $P_{t}(x,\mathrm{d}y)$ is the transition probability of $X(t)$, then the Green measure of $Y(t)$ is (heuristically) given by $$G(x,\mathrm{d}y)=\int_{0}^{\infty}\int_{0}^{\infty}P_{\tau}(x,\mathrm{d}y)G_{t}(\tau)\,\mathrm{d}\tau\,\mathrm{d}t.$$ But it is not hard to see that this definition leads to a contradiction and it has to be modified. More precisely, it has to e renormalized in such a way that then we are able to study the integral functionals $$\int_{0}^{\infty}f(Y(t))\,\mathrm{d}t$$ for a proper class of functions $f:{{{{\mathbb R}}^d}}\longrightarrow{{\mathbb R}}$. For the details, see [@KdS20].
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work has been partially supported by Center for Research in Mathematics and Applications (CIMA) related with the Statistics, Stochastic Processes and Applications (SSPA) group, through the grant UIDB/MAT/04674/2020 of FCT-Funda[çã]{}o para a Ci[ê]{}ncia e a Tecnologia, Portugal.
[12]{}
R. M. Blumenthal and R. K. Getoor. . Academic Press, 1968.
J. Cao, A. Grigor’yan, and L. Liu. . Preprint, University of Bielefeld, 2020.
E. B. Dynkin. . Springer, 1965.
A. Grigor’yan, E. Hu, and J. Hu. Two-sided estimates of heat kernels of jump type dirichlet forms. , 330:433–515, 2018.
A. Grigor’yan, Yu. G. Kondratiev, A. Piatnitski, and E. Zhizhina. Pointwise estimates for heat kernels of convolution-type operators. , 114(4):849–880, 2018.
Yu. G. Kondratiev and J. L. da Silva. . In preparation, 2020.
A. Kochubei, Yu. G. Kondratiev, and J. L. da Silva. Random time change and related evolution equations. [T]{}ime asymptotic behavior. , 4:2050034–1–24, 2019.
Yu. G. Kondratiev, S. Molchanov, A. Piatnitski, and E. Zhizhina. Resolvent bounds for jump generators. , 97(3):323–336, 2018.
Yu. G. Kondratiev, S. Molchanov, and B. Vainberg. Spectral analysis of non-local [S]{}chr[ö]{}dinger operators. , 273(3):1020–1048, 2017.
P.-A. Meyer. , volume 26 of [*Lecture Notes in Mathamatics*]{}. Springer, 1967.
D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]. Springer-Verlag, Berlin, 3rd edition, 1999.
A. V. Skorohod. , volume 47 of [*Mathematics and its applications (Soviet series)*]{}. Springer, 1991.
|
---
abstract: |
The irregularity strength of a graph $G$, $s(G)$, is the least $k$ admitting a $\{1,2,\ldots,k\}$-weighting of the edges of $G$ assuring distinct weighted degrees of all vertices, or equivalently the least possible maximal edge multiplicity in an irregular multigraph obtained of $G$ via multiplying some of its edges. The most well-known open problem concerning this graph invariant is the conjecture posed in 1987 by Faudree and Lehel that there exists a constant $C$ such that $s(G)\leq \frac{n}{d}+C$ for each $d$-regular graph $G$ with $n$ vertices and $d\geq 2$ (while a straightforward counting argument yields $s(G)\geq \frac{n+d-1}{d}$). The best known results towards this imply that $s(G)\leq 6\lceil\frac{n}{d}\rceil$ for every $d$-regular graph $G$ with $n$ vertices and $d\geq 2$, while $s(G)\leq (4+o(1))\frac{n}{d}+4$ if $d\geq n^{0.5}\ln n$.
We show that the conjecture of Faudree and Lehel holds asymptotically in the cases when $d$ is neither very small nor very close to $n$. We in particular prove that for large enough $n$ and $d\in [\ln^8n,\frac{n}{\ln^3 n}]$, $s(G)\leq \frac{n}{d}(1+\frac{8}{\ln n})$, and thereby we show that $s(G) = \frac{n}{d}(1+o(1))$ then. We moreover prove the latter to hold already when $d\in [\ln^{1+\varepsilon}n,\frac{n}{\ln^\varepsilon n}]$ where $\varepsilon$ is an arbitrary positive constant.
address: 'AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland'
author:
- 'Jakub Przyby[ł]{}o'
title: 'Asymptotic confirmation of the Faudree-Lehel Conjecture on irregularity strength for all but extreme degrees'
---
irregularity strength of a graph ,Faudree-Lehel Conjecture ,irregular edge labeling
Introduction
============
One of the most basic arguments in graph theory is the pigeonhole principle based observation that the vertices of a simple graph cannot all have pairwise distinct degrees, in other words no graph is in this sense *irregular* except the trivial one-vertex case. This exclusion considered, Chartrand, Erdős and Oellermann proposed and investigated in [@ChartrandErdosOellermann] possible alternative definitions of irregularity in graphs. In reference to this research, Chartrand et al. gave rise in [@Chartrand] to a related concept, designed to measure in some sense the level of irregularity of a given graph, exploiting for this aim the fact that irregular *multigraphs* are quite common and relatively easily constructed. Namely, they defined the *irregularity strength* of a graph $G=(V,E)$ as the least positive integer $k$ such that one may obtain an irregular multigraph (i.e. a multigraph with pairwise different vertex degrees) by blowing each edge of $G$ up to at most $k$ parallel edges. This graph invariant is usually denoted by $s(G)$ and is undefined for graphs with an isolated edge or two isolated vertices. An equivalent and often more convenient definition of $s(G)$ relates with so-called *weighted degrees*, defined for a given edge weighting function $\omega: E\to \mathbb{R}$ and a vertex $v\in V$ as: $$\sigma_{\omega}(v):=\sum_{e\ni v} \omega(e).$$ The irregularity strength of $G$ is in this setting equal to the minimum $k$ such that there exists a weighting $\omega:E\to\{1,2,\ldots,k\}$ with $\sigma_{\omega}(u)\neq \sigma_{\omega}(v)$ for every pair of distinct vertices $u,v$ of $G$.
A sharp upper bound $s(G)\leq n-1$ for all graphs with $n$ vertices for which the parameter is well defined except $K_3$ was settled in two papers: [@Aigner] and [@Nierhoff], devoted to connected and non-connected cases, respectively, with a $K_{1,n-1}$ star exemplifying its tightness. Nevertheless, much better bound was expected to hold in the case of graphs with larger minimum degree – a significantly smaller value of the irregularity strength was in particular anticipated for regular graphs. A straightforward counting argument implies a lower bound: $s(G)\geq \lceil\frac{n+d-1}{d}\rceil$ for any $d$-regular graph $G$ with $d\geq 2$. It was conjectured already in 1987 by Faudree and Lehel that this lower bound is not far from optimal.
\[FaudreeAndLehelConjecture\] There exists an absolute constant $C$ such that for every $d$-regular graph $G$ with $n$ vertices and $d\geq 2$, $$s(G)\leq\frac{n}{d}+C.$$
In fact this conjecture was first posed in the form of question by Jacobson (as mentioned by Lehel in [@Lehel]). It is this question that triggered extensive studies of the irregularity strength of graphs within the combinatorial community, and resulted in many papers devoted to this graph invariant, see e.g. [@Aigner; @Amar; @Amar_Togni; @Bohman_Kravitz; @Lazebnik; @Dinitz; @Ebert; @Ebert2; @Faudree2; @Faudree; @Ferrara; @Frieze; @Gyarfas; @Jendrol_Tkac; @KalKarPf; @MajerskiPrzybylo2; @Nierhoff; @Przybylo; @irreg_str2; @Togni] (or [@Lehel] for a survey), giving also rise to many related concepts [@Louigi30; @Louigi2; @Louigi; @AnhKalPrz; @Baca; @BarGrNiw; @LocalIrreg_1; @BensmailMerkerThomassen; @BonamyPrzybylo; @FlandrinMPSW; @Kalkowski12; @KalKarPf_123; @123KLT; @Majerski_Przybylo; @Przybylo_asym_optim2; @Przybylo_asym_optim; @LocalIrreg_2; @Przybylo_CN_1; @1234Reg123; @Przybylo_CN_2; @12Conjecture; @PrzybyloWozniakChoos; @Seamon123survey; @ThoWuZha; @Vuckovic_3-multisets; @WongZhu23Choos; @WongZhuChoos], to mention just a few out of the vastness of problems of this type. After over 30 years from its formulation, Conjecture \[FaudreeAndLehelConjecture\] still remains open. The first progress towards its solving was accomplished by Faudree and Lehel themselves, who already in [@Faudree] managed to push the upper bound of $n-1$ down to $n/2+9$ in the case of regular graphs. In 2002 Frieze, Gould, Karoński and Pfender applied the probabilistic method to make another big step forward and show among others that $s(G)\leq 10n/d+1$ when $d\leq\lfloor (n\ln n)^{1/4}\rfloor$ and $s(G)\leq 48n/d+1$ for $d\leq\lfloor n^{1/2}\rfloor$. They also achieved similar results but with slightly larger constants in the case of general graphs (with $d$ replaced by $\delta$). Later Cuckler and Lazebnik again exploited probabilistic approach to prove in particular that $s(G)\leq 48n/d+6$ for a $d$-regular graph with $d\geq 10^{4/3}n^{2/3}\ln^{1/3}n$ (and $s(G)\leq 48n/\delta+6$ for graphs with minimum degree $\delta\geq 10n^{3/4}\ln^{1/4}n$). No linear bounds in $n/d$ (or $n/\delta$) valid for entire spectrum of possible degrees were known at that point. The first such upper bounds were provided by Przyby[ł]{}o in [@Przybylo] and [@irreg_str2], where it was proved, resp., that $s(G)\leq 16\frac{n}{d}+6$ for $d$-regular graphs and $s(G)\leq 112\frac{n}{\delta}+28$ in the genaral case. Currently the best result of this type is due to Kalkowski, Karoński and Pfender, who made use of a relatively simple deterministic algorithm, refining and adapting its previous versions designed to tackle related subjects (see e.g. [@Kalkowski12] and [@AnhKalPrz]), and proved in [@KalKarPf] the general upper bound: $s(G)\leq 6\lceil n/\delta\rceil$ for all graphs with $n$ vertices and minimum degree $\delta\geq 1$ which do not contain isolated edges. Their approach was further developed with enhancement of the probabilistic method by Majerski and Przyby[ł]{}o, who showed in [@MajerskiPrzybylo2] that $s(G)\leq (4+o(1)) n/\delta+4$ for graphs with $\delta\geq n^{0.5}\ln n$. No better results have been achieved with these techniques for regular graphs.
In this paper we prove that the conjecture of Faudree and Lehel holds asymptotically in the cases when $d$ is neither very small nor very close to n. We in particular show that for large enough $n$ and $d\in [\ln^8n,\frac{n}{\ln^3 n}]$, $s(G)\leq \frac{n}{d}(1+\frac{8}{\ln n})$, thereby proving that $s(G) = \frac{n}{d}(1+o(1))$ then. We moreover prove the latter to hold already when $d\in [\ln^{1+\varepsilon}n,\frac{n}{\ln^\varepsilon n}]$ where $\varepsilon$ is an arbitrary positive constant, see Corollaries \[CorollaryLnNice\] and \[CorollaryLnBest\] in the last section. In fact the both results are implied by the following theorem with a slightly less self-evident form.
\[MainTheoremIrregStrAsympt\] For any positive real numbers $b,\varepsilon$, there exists a constant $N$ such that for every $d$-regular graph $G$ with order $n\geq N$ and $d\in\left[\ln^{1+6b+12\varepsilon}n,\frac{n}{\ln^{2b+5\varepsilon}n}\right]$, $$s(G)\leq \frac{n}{d}\left(1+\frac{8}{\ln^b n}\right).$$
Idea of Proof
=============
Given a $d$-regular graph $G=(V,E)$ with $n=|V|$ large enough and $d\in[\ln^{1+6b+12\varepsilon}n,$ $\frac{n}{\ln^{2b+5\varepsilon}n}]$, we first randomly choose some special sufficiently large subset $U\subseteq V$, yet still containing a small fraction of all vertices.
We then randomly and independently assign a real number $x_v\in[0,1]$ to every vertex $v$ in $V_0:=V\setminus U$ and associate weight $\lceil\frac{n}{d}\rceil$ to each edge $uv\in E(G[V_0])$ with $x_u+x_v\geq 1$ and weight $0$ otherwise. This more or less yields a desired distribution of weighted degrees in $V_0$, but only roughly.
In order to obtain actual precise distinction of the weighted degrees in $V_0$ we carefully choose weights for edges between $U$ and $V_0$. All of these weights are at the same time chosen relatively large (all appropriately larger than $\lceil\frac{n}{d}\rceil$), so that we are certain that the weighted degrees in $U$ are already larger than all those in $V_0$ (this shall be feasible in particular due to the fact that the randomly chosen $U$ shall be relatively small and most of the edges incident with any vertex in $U$ shall be joining it with $V_0$).
It is then sufficient to distinguish weighted degrees in $U$ via appropriate choice of weights for the edges of $G[U]$. For this goal we shall use an adaptation of the algorithm of Kalkowski, Karoński and Pfender from [@KalKarPf] (applicable to prove the general $6\lceil n/\delta\rceil$ upper bound). However, as in our randomly chosen $G[U]$ the proportion of the number of vertices to its minimum degree shall still be close to $\frac{n}{d}$ (as in $G$ itself), we shall have to be extra careful priory while choosing the weights for the edges between $U$ and $V_0$. Namely, we shall guarantee within that process that $U$ shall partition into $7$ subsets $U_1,U_2,\ldots,U_7$ with increasing weighted degrees and no possible conflicts between vertices from distinct subsets. This shall admit an adaptation of the mentioned algorithm with weights reduced up to at most $\frac{n}{d}+1$.
Within analysis of our random process we shall in particular use the Chernoff Bound. The following its version can be found e.g. in [@JansonLuczakRucinski] (Th. 2.1, page 26).
\[ChernofBoundTh\] For any $0\leq t\leq np$, $$\mathbf{Pr}({\rm BIN}(n,p)>np+t)<e^{-\frac{t^2}{3np}}~~~~{and}~~~~\mathbf{Pr}({\rm BIN}(n,p)<np-t)<e^{-\frac{t^2}{2np}}$$ where ${\rm BIN}(n,p)$ is the sum of $n$ independent Bernoulli variables, each equal to $1$ with probability $p$ and $0$ otherwise.
Proof of Theorem \[MainTheoremIrregStrAsympt\]
==============================================
Let $b,\varepsilon$ be two arbitrarily chosen and fixed positive real numbers.
Let $G=(V,E)$ be a $d$-regular graph with $n=|V|$ and $d\in\left[\ln^{1+6b+12\varepsilon}n,\frac{n}{\ln^{2b+5\varepsilon}n}\right]$. We shall not specify $N$, assuming whenever needed that $n$ is sufficiently large so that explicit inequalities below hold. We shall prove that then $$s(G)\leq \frac{n}{d}\left(1+\frac{8}{\ln^b n}\right).$$
Random Vertex Partition
-----------------------
We first observe that we may fix a partition of $V$ into a smaller part $U$ and a larger part $V_0=V\setminus U$ and a subpartition of $U$ into roughly equal seven parts with proportional distributions of neighbours of every vertex of $G$ between these sets.
\[UV-decomposition\] There is a subset $U$ of $V$ and its partition $U=U_1\cup U_2\cup \ldots \cup U_7$ such that for every $v\in V$ and $i\in\{1,2,\ldots,7\}$,
- $~~\left||U_i|-\frac{1}{7}\frac{n}{\ln^{b+\varepsilon} n}\right| \leq \frac{1}{7} \frac{n}{\ln^{2b+4\varepsilon}n}$, hence $~~~\left||U|-\frac{n}{\ln^{b+\varepsilon} n}\right| \leq \frac{n}{\ln^{2b+4\varepsilon}n}$;
- $~~\left|d_{U_i}(v) - \frac{1}{7}\frac{d}{\ln^{b+\varepsilon} n}\right| \leq \frac{1}{7}\frac{d}{\ln^{2b+4\varepsilon} n}$, hence $~~~\left|d_{U}(v) - \frac{d}{\ln^{b+\varepsilon} n}\right| \leq \frac{d}{\ln^{2b+4\varepsilon} n}$.
We place every vertex $v$ of $G$ in $U$ independently with probability $\frac{1}{\ln^{b+\varepsilon} n}$ and independently we randomly and equiprobably assign an integer $i_v\in\{1,2,\ldots,7\}$ to every vertex $v\in V$. For every $i=1,2,\ldots,7$, we define the set $U_i$ as the set of vertices $v\in U$ with $i_v=i$. Then $E(|U_i|)=\frac{1}{7}\frac{n}{\ln^{b+\varepsilon} n}$ and $E(d_{U_i}(v)) = \frac{1}{7}\frac{d}{\ln^{b+\varepsilon} n}$, and by the Chernoff Bound: $$\mathbf{Pr}\left(\left||U_i|-\frac{1}{7}\frac{n}{\ln^{b+\varepsilon} n}\right| > \frac{1}{7} \frac{n}{\ln^{2b+4\varepsilon}n}\right) < 2e^{-\frac{n}{3\cdot 7\ln^{3b+7\varepsilon}n}} < \frac{1}{14},$$ $$\textbf{Pr}\left( \left|d_{U_i}(v) - \frac{1}{7}\frac{d}{\ln^{b+\varepsilon} n}\right| > \frac{1}{7}\frac{d}{\ln^{2b+4\varepsilon} n}\right)
< 2e^{-\frac{d}{3\cdot 7\ln^{3b+7\varepsilon}n}}
\leq 2\left(e^{\ln n}\right)^{-\frac{\ln^{3b+5\varepsilon}n}{21}}
< \frac{1}{14n}.$$ Therefore, the probability that $(1^\circ)$ does not hold for some $i\in\{1,2,\ldots,7\}$ or $(2^\circ)$ does not hold for some $v\in V$ and $i\in\{1,2,\ldots,7\}$ is less than $$7\cdot \frac{1}{14} + n\cdot 7\cdot \frac{1}{14n} = 1,$$ hence there is a choice of $U,U_1,U_2,\ldots,U_7$ fulfilling all our requirements.
Denote $$V_0:=V\setminus U, ~~~~~~~~n_0:=|V_0|, ~~~~~~~~G_0:=G[V_0]$$ and $$d_0(v):=d_{V_0}(v)$$ for every $v\in V$. Then by $(1^\circ)$, $$\label{n_0Bounds}
n\left(1 - \frac{1}{\ln^{b+\varepsilon} n} - \frac{1}{\ln^{2b+4\varepsilon}n}\right)
\leq n_0 \leq
n\left(1 - \frac{1}{\ln^{b+\varepsilon} n} + \frac{1}{\ln^{2b+4\varepsilon}n}\right)$$ and by $(2^\circ)$ for each $v\in V$: $$\label{d_0Bounds}
d\left(1 - \frac{1}{\ln^{b+\varepsilon} n} - \frac{1}{\ln^{2b+4\varepsilon}n}\right)
\leq d_0(v) \leq
d\left(1 - \frac{1}{\ln^{b+\varepsilon} n} + \frac{1}{\ln^{2b+4\varepsilon}n}\right).$$
Random Labeling of the Vertices in $V_0$
----------------------------------------
We now aim at assigning weights to all edges with at least one end in $V_0$ so that the obtained weighted degrees in $V_0$ form an arithmetic progression with step size $1$ (i.e., they are consecutive integers). For this goal we first randomly and independently choose for every vertex $v$ of $V_0$ a real number $x_v\in[0,1]$ with uniform probability distribution (that is a realization of the random variable $X_v\sim U(0,1)$ associated to $v$). These $x_v$’s shall roughly refer to the positions of vertices in the arithmetic progression and define a natural ordering of the vertices (provided that all $x_v$’s are distinct).
Denote $$L_v:=\{u\in V_0: x_u < x_v\},~~~~~R_v:=\{u\in N_{G_0}(v): x_u\geq 1-x_v\}.$$ In fact $|L_v|$ shall correspond to the number of vertices preceding $v$ in the ordering (and thus to its position within it), while $R_v$ shall define the ends of the edges incident with $v$ in $G_0$ which shall receive weight $\lceil\frac{n}{d}\rceil$ (the remaining ones shall temporarily be weighted $0$).
\[GoodX\_vDistribution\] With positive probability all $x_v$’s are pairwise distinct and for every vertex $v\in V_0$ with $x_v=x$:
- if $x\geq \frac{1}{\ln^{2b+3\varepsilon}n}$, then $\left||L_v|-x(n_0-1)\right|\leq \frac{x(n_0-1)}{\ln^{2b+4\varepsilon} n}$;
- if $x < \frac{1}{\ln^{2b+3\varepsilon}n}$, then $|L_v| \leq \frac{n_0-1}{\ln^{2b+3\varepsilon}n} + \frac{n_0-1}{\ln^{4b+7\varepsilon}n}$;
- if $x\geq \frac{1}{\ln^{2b+3\varepsilon}n}$, then $\left||R_v|-xd_{0}(v)\right|\leq \frac{xd_{0}(v)}{\ln^{2b+4\varepsilon} n}$;
- if $x < \frac{1}{\ln^{2b+3\varepsilon}n}$, then $|R_v|\leq
\frac{d_{0}(v)}{\ln^{2b+3\varepsilon} n} + \frac{d_{0}(v)}{\ln^{4b+7\varepsilon} n}$.
For every $v\in V_0$ with $x_v=x$ we denote the following events: $$\begin{aligned}
A_v: && \left(x\geq \frac{1}{\ln^{2b+3\varepsilon}n}~~\wedge~~\left||L_v|-x(n_0-1)\right| > \frac{x(n_0-1)}{\ln^{2b+4\varepsilon} n}\right) \nonumber\\
&&~~\vee~~
\left(x < \frac{1}{\ln^{2b+3\varepsilon}n}~\wedge~|L_v| > \frac{n_0-1}{\ln^{2b+3\varepsilon}n} + \frac{n_0-1}{\ln^{4b+7\varepsilon}n}\right);\nonumber\end{aligned}$$ $$\begin{aligned}
B_v: && \left(x\geq \frac{1}{\ln^{2b+3\varepsilon}n}~\wedge~\left||R_v|-xd_{0}(v)\right| > \frac{xd_{0}(v)}{\ln^{2b+4\varepsilon} n}\right) \nonumber\\
&&~~\vee~~\left(x < \frac{1}{\ln^{2b+3\varepsilon}n}~\wedge~|R_v| >
\frac{d_{0}(v)}{\ln^{2b+3\varepsilon} n} + \frac{d_{0}(v)}{\ln^{4b+7\varepsilon} n}\right).\nonumber\end{aligned}$$
We note that for any $x\geq \frac{1}{\ln^{2b+3\varepsilon}n}$ (as $v\notin L_v$, while each remaining $u\in V_0$ lands in $L_v$ independently with probability $x$), by the Chernoff Bound and (\[n\_0Bounds\]): $$\begin{aligned}
\mathbf{Pr} \left(A_v|X_v=x\right) &=&
\mathbf{Pr} \left(\left|{\rm BIN}\left(n_0-1,x\right)-x(n_0-1)\right| > \frac{x(n_0-1)}{\ln^{2b+4\varepsilon} n}\right) \nonumber\\
&<& 2e^{-\frac{x(n_0-1)}{3\ln^{4b+8\varepsilon}n}} \leq 2e^{-\frac{n_0-1}{3\ln^{6b+11\varepsilon}n}} < 2e^{-\frac{n}{4\ln^{6b+11\varepsilon}n}} < \frac{1}{2n}. \label{PrAv1}\end{aligned}$$ By inequalities in (\[PrAv1\]) above (with $x = \frac{1}{\ln^{2b+3\varepsilon}n}$), for any $x < \frac{1}{\ln^{2b+3\varepsilon}n}$ we similarly have: $$\begin{aligned}
\mathbf{Pr} \left(A_v|X_v=x\right)
&=& \mathbf{Pr} \left({\rm BIN}\left(n_0-1,x\right) > \frac{n_0-1}{\ln^{2b+3\varepsilon}n} + \frac{n_0-1}{\ln^{4b+7\varepsilon}n}\right) \nonumber\\
&\leq& \mathbf{Pr} \left(\left|{\rm BIN}\left(n_0-1,\frac{1}{\ln^{2b+3\varepsilon}n}\right)-\frac{n_0-1}{\ln^{2b+3\varepsilon}n}\right| > \frac{\frac{n_0-1}{\ln^{2b+3\varepsilon}n}}{\ln^{2b+4\varepsilon} n}\right) < \frac{1}{2n}. \label{PrAv2}\end{aligned}$$
Analogously, for any $x\geq \frac{1}{\ln^{2b+3\varepsilon}n}$, by (\[d\_0Bounds\]) and the assumption that $d\geq\ln^{1+6b+12\varepsilon}n$: $$\begin{aligned}
\mathbf{Pr} \left(B_v|X_v=x\right) &=&
\mathbf{Pr} \left(\left|{\rm BIN}\left(d_0(v),x\right)-xd_0(v)\right| > \frac{xd_0(v)}{\ln^{2b+4\varepsilon} n}\right) \nonumber\\
&<& 2e^{-\frac{xd_0(v)}{3\ln^{4b+8\varepsilon}n}} \leq 2e^{-\frac{d_0(v)}{3\ln^{6b+11\varepsilon}n}} < 2e^{-\frac{d}{4\ln^{6b+11\varepsilon}n}} < \frac{1}{2n}. \label{PrBv1}\end{aligned}$$ By inequalities in (\[PrBv1\]) above, for any $x < \frac{1}{\ln^{2b+3\varepsilon}n}$ we in turn have: $$\label{PrBv2}
\mathbf{Pr} \left(B_v|X_v=x\right) \leq \mathbf{Pr} \left(\left|{\rm BIN}\left(d_0(v),\frac{1}{\ln^{2b+3\varepsilon}n}\right)-
\frac{1}{\ln^{2b+3\varepsilon}n}d_0(v)\right| > \frac{\frac{1}{\ln^{2b+3\varepsilon}n}d_0(v)}{\ln^{2b+4\varepsilon} n}\right) < \frac{1}{2n}.$$
By (\[PrAv1\]) and (\[PrAv2\]), $$\mathbf{Pr}(A_v) < \int_0^1\frac{1}{2n} dx = \frac{1}{2n},$$ and by (\[PrBv1\]) and (\[PrBv2\]), $$\mathbf{Pr}(B_v) < \int_0^1\frac{1}{2n} dx = \frac{1}{2n}.$$ Therefore, $$\mathbf{Pr} \left(\bigcap_{v\in V_0}\left(\overline{A_v}\cap \overline{B_v}\right)\right) = 1 - \mathbf{Pr}\left(\bigcup_{v\in V_0} A_v\cup B_v\right)> 1 - 2n\cdot \frac{1}{2n} = 0,$$ and the thesis follows (as the probability that all $x_v$ are pairwise distinct is obviously $1$).
Weighted Degrees in $V_0$
-------------------------
We fix any choice of $x_v$, $v\in V_0$ consistent with Observation \[GoodX\_vDistribution\] above.
Now to every edge $uv\in E(G_0)$ we assign an initial weight: $$\omega_0(uv)=\left\{\begin{array}{rcl}
0~~, & {\rm if} & x_u+x_v< 1;\\
\left\lceil\frac{n}{d}\right\rceil, & {\rm if} & x_u+x_v\geq 1.
\end{array}\right.$$ For every $e\in E[V_0,U_i]$ (the set of edges between $V_0$ and $U_i$) with $i\in\{1,2,\ldots,7\}$ we set: $$\label{Omega0V0U}
\omega_0(e)=\left\lceil\frac{n}{d}\right\rceil + i \left\lceil\frac{n}{d\ln^b n}\right\rceil,$$ and for every $e\subset U$ we temporarily set $$\omega_0(e)=0.$$
Therefore, $\omega_0(uv) = \left\lceil\frac{n}{d}\right\rceil$ if and only if $x_v \in R_u$ (or equivalently $x_u \in R_v$).
Note that for every $v\in V_0$ we thus have: $$\label{Omega_0FirstEstimation}
\sigma_{\omega_0} (v) = |R_v|\left\lceil\frac{n}{d}\right\rceil + \sum_{i=1}^7 d_{U_i}(v)\left(\left\lceil\frac{n}{d}\right\rceil + i \left\lceil\frac{n}{d\ln^b n}\right\rceil\right).$$
Set $$\label{B_0Definition}
B_0 = \left\lceil\frac{n}{\ln^{b+\varepsilon} n}\right\rceil + 4\left\lceil\frac{n}{\ln^{2b+\varepsilon} n}\right\rceil
+ 2\left\lceil\frac{n}{\ln^{2b+3\varepsilon} n}\right\rceil,$$
$$\label{NDefinition}
N=\left\lceil\frac{n}{\ln^{2b+2\varepsilon} n}\right\rceil - 2\left\lceil\frac{n}{\ln^{3b+5\varepsilon} n}\right\rceil.$$
By (\[Omega\_0FirstEstimation\]) and $(2^\circ)$, (\[B\_0Definition\]), (\[NDefinition\]), we thus have for every given $v\in V_0$ (and $n$ sufficiently large): $$\begin{aligned}
\sigma_{\omega_0} (v) &\geq& |R_v|\frac{n}{d} +
\sum_{i=1}^7 \frac{1}{7} \left(\frac{d}{\ln^{b+\varepsilon}n} - \frac{d}{\ln^{2b+4\varepsilon}n}\right)
\left(\frac{n}{d} + i \frac{n}{d\ln^b n}\right) \nonumber\\
&=& |R_v|\frac{n}{d} +
\frac{n}{\ln^{b+\varepsilon}n} + 4\frac{n}{\ln^{2b+\varepsilon}n} - \frac{n}{\ln^{2b+4\varepsilon}n} - 4\frac{n}{\ln^{3b+4\varepsilon}n}
\nonumber\\
&=& |R_v|\frac{n}{d} +
\left(\frac{n}{\ln^{b+\varepsilon}n} + 1\right) + 4\left(\frac{n}{\ln^{2b+\varepsilon}n} + 1\right) + 2\left(\frac{n}{\ln^{2b+3\varepsilon}n} + 1\right) \nonumber\\
&& - \frac{n}{\ln^{2b+2\varepsilon}n} + 2\left(\frac{n}{\ln^{3b+5\varepsilon}n} + 1\right)
+ \left(\frac{n}{\ln^{2b+3\varepsilon} n} + \frac{n}{\ln^{4b+7\varepsilon}n}\right) \nonumber\\
&&
+ \left( \frac{n}{\ln^{2b+2\varepsilon}n} - 3\frac{n}{\ln^{2b+3\varepsilon}n}
- \frac{n}{\ln^{2b+4\varepsilon}n} - 4\frac{n}{\ln^{3b+4\varepsilon}n} - 2\frac{n}{\ln^{3b+5\varepsilon}n} - \frac{n}{\ln^{4b+7\varepsilon}n}\ - 9\right)
\nonumber\\
&>& |R_v|\frac{n}{d} + B_0 - N
+ \left(\frac{n}{\ln^{2b+3\varepsilon} n} + \frac{n}{\ln^{4b+7\varepsilon}n}\right). \label{Sigma0Larger1}\end{aligned}$$ Thus if $x_v=x < \frac{1}{\ln^{2b+3\varepsilon}n}$, by (\[Sigma0Larger1\]) and $(4^\circ)$: $$\begin{aligned}
\sigma_{\omega_0} (v) &>&
B_0 - N
+ \left(\frac{n}{\ln^{2b+3\varepsilon} n} + \frac{n}{\ln^{4b+7\varepsilon}n}\right) \geq B_0 + |L_v| - N. \label{Sigma0Larger2}\end{aligned}$$ Analogously, if $x_v=x \geq \frac{1}{\ln^{2b+3\varepsilon}n}$, by (\[Sigma0Larger1\]) and $(5^\circ)$, (\[d\_0Bounds\]), (\[n\_0Bounds\]), $(3^\circ)$: $$\begin{aligned}
\sigma_{\omega_0} (v) &>&
x d_0(v)
\left(1 - \frac{1}{\ln^{2b+4\varepsilon} n}\right)\frac{n}{d}
+ B_0 - N
+ \left(\frac{n}{\ln^{2b+3\varepsilon} n} + \frac{n}{\ln^{4b+7\varepsilon}n}\right) \nonumber\\
&\geq&
x d\left(1 - \frac{1}{\ln^{b+\varepsilon} n} - \frac{1}{\ln^{2b+4\varepsilon}n}\right)
\left(1 - \frac{1}{\ln^{2b+4\varepsilon} n}\right)\frac{n}{d}
+ B_0 - N
+ \left(\frac{n}{\ln^{2b+3\varepsilon} n} + \frac{n}{\ln^{4b+7\varepsilon}n}\right) \nonumber\\
&>& xn - \frac{xn}{\ln^{b+\varepsilon}n} +
\frac{1}{2} \frac{n}{\ln^{2b+3\varepsilon}n} + B_0 - N \nonumber\\
&>& x n \left(1 - \frac{1}{\ln^{b+\varepsilon} n} + \frac{1}{\ln^{2b+4\varepsilon} n}\right)
\left(1 + \frac{1}{\ln^{2b+4\varepsilon} n}\right) + B_0 - N \nonumber\\
&\geq& x n_0
\left(1 + \frac{1}{\ln^{2b+4\varepsilon} n}\right) + B_0 - N \nonumber\\
&>& B_0 + |L_v| - N. \label{Sigma0Larger3}\end{aligned}$$
On the other hand, by (\[Omega\_0FirstEstimation\]) and $(2^\circ)$, (\[B\_0Definition\]), for every given $v\in V_0$ (and $n$ sufficiently large): $$\begin{aligned}
\sigma_{\omega_0} (v) &\leq& |R_v|\left(\frac{n}{d} + 1\right)
+ \sum_{i=1}^7 \frac{1}{7} \left(\frac{d}{\ln^{b+\varepsilon}n} + \frac{d}{\ln^{2b+4\varepsilon}n}\right)
\left(\frac{n}{d} + i \frac{n}{d\ln^b n} + 8\right) \nonumber\\
&<& |R_v|\frac{n}{d} + d
+ \sum_{i=1}^7 \frac{1}{7}\frac{d}{\ln^{b+\varepsilon}n}\left(\frac{n}{d} + i \frac{n}{d\ln^b n}\right)
+ \sum_{i=1}^7 \frac{1}{7}\frac{d}{\ln^{2b+4\varepsilon}n} \cdot 2\frac{n}{d} + 9 \frac{d}{\ln^{b+\varepsilon} n} \nonumber\\
&=& |R_v|\frac{n}{d}
+ \frac{n}{\ln^{b+\varepsilon}n} + 4\frac{n}{\ln^{2b+\varepsilon}n} + 2\frac{n}{\ln^{2b+4\varepsilon}n}
+ d + 9 \frac{d}{\ln^{b+\varepsilon} n}\nonumber\\
&<& |R_v|\frac{n}{d}
+ \frac{n}{\ln^{b+\varepsilon}n} + 4\frac{n}{\ln^{2b+\varepsilon}n} + 3\frac{n}{\ln^{2b+4\varepsilon}n} \nonumber\\
&<& |R_v|\frac{n}{d} + B_0
- \frac{3}{2} \frac{n}{\ln^{2b+3\varepsilon}n}. \label{Sigma0Smaller1}\end{aligned}$$ Thus if $x_v=x < \frac{1}{\ln^{2b+3\varepsilon}n}$, by (\[Sigma0Smaller1\]) and $(6^\circ)$: $$\begin{aligned}
\sigma_{\omega_0} (v) &<&
\left( \frac{d}{\ln^{2b+3\varepsilon} n} + \frac{d}{\ln^{4b+7\varepsilon} n}\right)\frac{n}{d}
+B_0
- \frac{3}{2} \frac{n}{\ln^{2b+3\varepsilon}n} <
B_0 \leq B_0 + |L_v|. \label{Sigma0Smaller2}\end{aligned}$$ Analogously, if $x_v=x \geq \frac{1}{\ln^{2b+3\varepsilon}n}$, by (\[Sigma0Smaller1\]) and $(5^\circ)$, (\[d\_0Bounds\]), (\[n\_0Bounds\]), $(3^\circ)$: $$\begin{aligned}
\sigma_{\omega_0} (v) &<&
x d_0(v)
\left(1 + \frac{1}{\ln^{2b+4\varepsilon} n}\right)\frac{n}{d}
+ B_0
- \frac{3}{2} \frac{n}{\ln^{2b+3\varepsilon}n} \nonumber\\
&\leq& xd \left(1 - \frac{1}{\ln^{b+\varepsilon} n} + \frac{1}{\ln^{2b+4\varepsilon}n}\right) \left(1 + \frac{1}{\ln^{2b+4\varepsilon} n}\right)\frac{n}{d}
+ B_0
- \frac{3}{2} \frac{n}{\ln^{2b+3\varepsilon}n} \nonumber\\
&<& xn - \frac{xn}{\ln^{b+\varepsilon} n} + B_0 -
\frac{n}{\ln^{2b+3\varepsilon}n} \nonumber\\
&<& x n \left(1 - \frac{1}{\ln^{b+\varepsilon} n} - \frac{1}{\ln^{2b+4\varepsilon} n} - \frac{1}{n}\right)
\left(1 - \frac{1}{\ln^{2b+4\varepsilon} n}\right) + B_0 \nonumber\\
&\leq& x (n_0-1)
\left(1 - \frac{1}{\ln^{2b+4\varepsilon} n}\right) + B_0 \nonumber\\
&\leq& B_0 + |L_v|. \label{Sigma0Smaller3}\end{aligned}$$
Fix the (unique) ordering $v_1,v_2,\ldots,v_{n_0}$ of the vertices in $V_0$ so that $$x_{v_i}\leq x_{v_j}~~ {\rm whenever}~~ i\leq j.$$
Now we admit some changes on the edges $e\in E[V_0,U]$ – namely for each of these we admit adding any integer $$\label{OmegaPrimInterval}
\omega'(e)\in \left[0,\left\lceil\frac{n}{d\ln^{b+\varepsilon} n}\right\rceil\right]$$ to its current weight. For definiteness, we set $\omega'(e)=0$ for the remaining edges of $G$.
As a result we may add any integer in $$\left[0,d_U(v)\left\lceil\frac{n}{d\ln^{b+\varepsilon} n}\right\rceil\right]$$ to $\sigma_{\omega_0}(v)$ of any $v\in V_0$. Thus if we set $$\omega_1:=\omega_0+\omega',$$ as $(2^\circ)$ implies that $$d_U(v)\left\lceil\frac{n}{d\ln^{b+\varepsilon} n}\right\rceil
\geq d\left(\frac{1}{\ln^{b+\varepsilon} n}-\frac{1}{\ln^{2b+4\varepsilon} n}\right) \frac{n}{d\ln^{b+\varepsilon} n} > N,$$ by (\[Sigma0Larger2\]), (\[Sigma0Larger3\]), (\[Sigma0Smaller2\]) and (\[Sigma0Smaller3\]), we can make our choices of $\omega'(e)$ so that (\[OmegaPrimInterval\]) holds and $$\sigma_{\omega_1}(v_j)=B_0+j$$ for every $j\in\{1,2,\ldots,n_0\}$.
Distinguishing Vertices in $U$
------------------------------
We shall not change weighted degrees of vertices in $V_0$ further on, but we shall increase the weights of (some of) the edges in $U$ in order to adjust the weighted degrees of vertices in $U$. We first observe that their weighted degrees are already larger than those in $V_0$. Note for this aim that for every $v\in V_0$ and $u\in U$, by (\[n\_0Bounds\]), (\[B\_0Definition\]), (\[d\_0Bounds\]) and (\[Omega0V0U\]): $$\label{sigma1vu}
\sigma_{\omega_1}(v)\leq B_0+n_0 <
n + 5\frac{n}{\ln^{2b+\varepsilon} n}
<
d_0(u) \left(\left\lceil\frac{n}{d}\right\rceil + \left\lceil\frac{n}{d\ln^b n}\right\rceil\right)
\leq \sigma_{\omega_1}(u).$$
In order to distinguish vertices in $U$ we shall now use an adaptation of the algorithm of Kalkowski, Karoński and Pfender from [@KalKarPf]. Within this we shall be admitting adding integers: $$\label{Omega''Limits}
\omega''(e)\in\left[0,\frac{n}{d}\right]$$ to the weight of every edge $e\subset U$. The almost final weighting of $G$ shall be defined as $$\label{Omega2asSumOmega1Omega''}
\omega_2=\omega_1+\omega''$$ (where we set $w''(e)=0$ for every edge $e\in E\setminus E(G[U])$). Note in particular that by (\[sigma1vu\]), for every $v\in V_0$ and $u\in U$ we shall (still) have: $$\label{DistinctionBetweenUV0}
\sigma_{\omega_2}(v) = \sigma_{\omega_1}(v) < \sigma_{\omega_1}(u) \leq \sigma_{\omega_2}(u).$$
Moreover, so that the algorithm could work effectively, we observe now that we shall be able to focus within its execution on distinguishing merely the vertices within the same $U_i$’s, as by our construction, for every $u\in U_i$ and $v\in U_{i+1}$ with $i\in\{1,2,3,4,5,6\}$, by (\[Omega0V0U\]), (\[OmegaPrimInterval\]), (\[Omega”Limits\]) and (\[d\_0Bounds\]) we shall have: $$\begin{aligned}
\sigma_{\omega_2}(u)
&\leq& d_0(u) \left(\left\lceil\frac{n}{d}\right\rceil + i \left\lceil\frac{n}{d\ln^b n}\right\rceil + \left\lceil\frac{n}{d\ln^{b+\varepsilon} n}\right\rceil\right) + d_U(u)\left\lceil\frac{n}{d}\right\rceil \nonumber\\
&=& d\left\lceil\frac{n}{d}\right\rceil + d_0(u) \left(i \left\lceil\frac{n}{d\ln^b n}\right\rceil + \left\lceil\frac{n}{d\ln^{b+\varepsilon} n}\right\rceil\right) \nonumber\\
&<& n + i\frac{n}{\ln^b n} + \frac{n}{\ln^{b+\varepsilon} n} \nonumber\\
&<& n + i\frac{n}{\ln^b n} + \frac{n}{\ln^b n} - 2\frac{n}{\ln^{b+\varepsilon} n} \nonumber\\
&<& d_0(v) \left(\left\lceil\frac{n}{d}\right\rceil + (i+1) \left\lceil\frac{n}{d\ln^b n}\right\rceil\right) \nonumber\\
&\leq& \sigma_{\omega_2}(v). \label{DistinctionBetweenUis}\end{aligned}$$
Initially we set $$\omega''(e)=\left\lfloor\frac{n}{3d}\right\rfloor$$ for every edge $e$ of $G[U]$. This shall be modified by the algorithm (while $\omega_2$ shall always refer to an up-to-date value of $\omega''$ below; cf. (\[Omega2asSumOmega1Omega”\])).
Within the algorithm we then analyze one component of $G[U]$ after another in arbitrary order, and in each consecutive component of $G[U]$, say $G'=(V',E')$, we arrange its vertices into a sequence $u_1,u_2,\ldots,u_{n'}$ so that each of these vertices, except the last one has a *forward edge*, i.e. an edge joining it in $G'$ with a vertex later in the ordering, which we shall call a *forward neighbour* of this vertex (we may use e.g. a reversed BFS ordering for this goal). We define *backward edges* and *neighbours* of a vertex correspondingly.
We analyze all vertices in the sequence one by one consistently with the fixed ordering in $G'$, and for any currently analyzed vertex $v$, we admit adding any integer in $$\label{ForwardChanges}
\left[0,\left\lfloor\frac{n}{3d}\right\rfloor\right]$$ to the weight of its every forward edge and adding one of the three numbers from the set $$\label{ThreeModifOptions}
\left\{-\left\lfloor\frac{n}{3d}\right\rfloor,0,\left\lfloor\frac{n}{3d}\right\rfloor\right\}$$ to the weights of its backward edges (thus every edge of $G[U]$ shall be modified at most twice within the algorithm). However, not all the mentioned three options in (\[ThreeModifOptions\]) shall be available for a given backward edge. Namely, the moment a given vertex $v$ is analyzed, we assign to it a suitable two-element set $\Sigma_v$ belonging to the family: $$S=\left\{\left\{2\lambda \left\lfloor\frac{n}{3d}\right\rfloor +a,(2\lambda+1)\left\lfloor\frac{n}{3d}\right\rfloor+a\right\}: \lambda\in\mathbb{Z}, a\in\left\{0,1,2,\ldots,\left\lfloor\frac{n}{3d}\right\rfloor-1\right\}\right\}$$ (note $S$ *partitions* $\mathbb{Z}$ into two-element sets) and perform admitted modifications of the weights of its forward and backward edges so that the obtained weighted degree $\sigma_{\omega_2}(v)$ of $v$ belongs to $\Sigma_v$. Once $\Sigma_v$ is fixed, $\sigma_{\omega_2}(v)$ is required to be its element (i.e. it may take only two distinct values from this point till the end of the construction). Therefore, if $uv$ is a backward edge of $v$ and $u=\min\Sigma_u$, then we may increase the weight of $uv$ by $\left\lfloor\frac{n}{3d}\right\rfloor$ (or do not alter it at all), while otherwise we may decrease it by $\left\lfloor\frac{n}{3d}\right\rfloor$. Such admitted changes of the weights of the backward edges of $v$ along with the admitted alterations concerning forward edges (cf. (\[ForwardChanges\])) guarantee by $(2^\circ)$ at least $$\label{NoOptionSima2}
d_U(v)\cdot \left\lfloor\frac{n}{3d}\right\rfloor+1 > 2 \cdot\frac{n}{7\ln^{b+\varepsilon}n}\left(1+\frac{1}{\ln^{b+3\varepsilon}n}\right)$$ options for $\sigma_{\omega_2}(v)$ for every currently analyzed vertex $v$ except for the last vertex of the component (which is the only vertex of $G'$ without a forward edge). Suppose $v\in U_i$ for some $i\in\{1,2,\ldots,7\}$ and $v\notin\{u_{n'-1},u_{n'}\}$. Then we may perform the admissible changes so that $\sigma_{\omega_2}(v)$ does not belong to $\Sigma_u$ for every vertex $u\in U_i$ (which has $\Sigma_u$ already fixed), as by (\[NoOptionSima2\]) and $(1^\circ)$, $2|U_i|<
d_U(v)\cdot \left\lfloor\frac{n}{3d}\right\rfloor+1$. Then we fix as $\Sigma_v$ the only set in $S$ which contains the obtained $\sigma_{\omega_2}(v)$.
Now suppose we have already analyzed $u_1,u_2,\ldots,u_{n'-2}$, hence we are left with just two vertices to by analyzed in the given component $G'$. Note that by our construction and (\[DistinctionBetweenUis\]), each $\Sigma_{u_j}$ with $j=1,2,\ldots,n'-2$ is disjoint with $\Sigma_u$ for every vertex $u\neq u_j$ which has already been analyzed. Let $L$ be the set consisting of all last and last but one vertices in the components of $G[U]$, hence $L\cap V'=\{u_{n'-1},u_{n'}\}$. The vertices in $L$ shall be admitted to have weighted degrees belonging to some previously fixed $\Sigma_u$ (yet different from $\sigma_{\omega_2}(u)$). Note that as this consent concerns only vertices in $L$ (and the corresponding sets $\Sigma_u$), the number of sets $\Sigma\in S$ assigned to more than one vertex equals at most twice the number of components of $G[U]$, i.e., by $(1^\circ)$ and $(2^\circ)$, less than $$2\cdot \frac{|U|}{\delta(G[U])} < 3\frac{n}{d}$$ sets. Denote temporarily the family of all such sets by $S_t$. Therefore, we may change (choose) the current value $\omega''(u_{n'-1}u_{n'}) = \left\lfloor\frac{n}{3d}\right\rfloor$ to some quantity in $\left\{0,1,\ldots,\left\lfloor\frac{n}{3d}\right\rfloor\right\}$ so that at most $$\left\lfloor \frac{2\cdot\frac{3n}{d}}{\left\lfloor\frac{n}{3d}\right\rfloor} \right\rfloor < 20$$ sets in $S_t$ contain elements congruent to the resulting $\sigma_{\omega_2}(u_{n'-1})$ and at the same time at most the same number (i.e. 20) of these sets contain elements congruent to the resulting $\sigma_{\omega_2}(u_{n'})$ modulo $\left\lfloor\frac{n}{3d}\right\rfloor$. Then the admitted alterations on the backward edges of $u_{n'-1}$ provide $d_U(u_{n'-1})\geq 45$ options (cf. $(2^\circ)$) for the weighted degree of $u_{n'-1}$ which form an arithmetic progression with step size $\left\lfloor\frac{n}{3d}\right\rfloor$. Thus at least one of these, say $\sigma'\in \Sigma'\in S$, does not belong to any set in $S_t$ and is neither the largest nor the smallest element of this arithmetic progression and moreover is not the current weighted degree of any already analyzed vertex – we then perform admissible modifications of the weights of backward edges of $u_{n'-1}$ to obtain $\sigma_{\omega_2}(u_{n'-1})=\sigma'$ not using for this aim a potential at most one edge $u'u_{n'-1}$ with $\sigma_{\omega_2}(u')\in \Sigma'$ (note it is always possible, as $\sigma'$ was chosen not to be the larges nor the smallest element of the mentioned arithmetic progression). We then set $\Sigma_{u_{n'-1}}=\Sigma'$ and note that the obtained $\sigma_{\omega_2}(u_{n'-1})$ is distinct from weighted degrees of all already analyzed vertices of $G'$.
Finally we analyze $u_{n'}$. The admitted alterations on the backward edges of $u_{n'}$ distinct from $u_{n'}u_{n'-1}$ and from a possible single edge $u_{n'}u'$ with $\sigma_{\omega_2}(u')\in \Sigma'$ provide $d_U(u_{n'})-1\geq 47$ options (cf. $(2^\circ)$) for the weighted degree of $u_{n'}$ which form an arithmetic progression with step size $\left\lfloor\frac{n}{3d}\right\rfloor$. Thus at least one of these, say $\sigma''\in \Sigma''\in S$, does not belong to any set in $S_t\cup\{\Sigma'\}$ and is neither the largest nor the smallest element of this arithmetic progression, and additionally is not the current weighted degree of any already analyzed vertex – we then perform admissible modifications of the weights of backward edges of $u_{n'}$ distinct from $u_{n'}u_{n'-1}$ and the possible edge $u_{n'}u'$ to obtain $\sigma_{\omega_2}(u_{n'})=\sigma''$ not using for this aim (analogously as previously) a potential at most one edge $u''u_{n'}$ with $\sigma_{\omega_2}(u'')\in \Sigma''$. We then set $\Sigma_{u_{n'}}=\Sigma''$ and note that all the already analyzed vertices of $G'$ have distinct current weighted degrees, which shall not be altered in the further part of the construction.
After analyzing all vertices of $G[U]$ we obtain a weighting $\omega_2$ such that $\sigma_{\omega_2}(u)\neq \sigma_{\omega_2}(v)$ for every $uv\in E$, as by our construction we have for every $e\in E(G[U])$: $$\omega''(e)\in\left[0,3\left\lfloor\frac{n}{3d}\right\rfloor\right],$$ hence (\[Omega”Limits\]) indeed holds, and thus in particular also (\[DistinctionBetweenUV0\]) and (\[DistinctionBetweenUis\]) are fulfilled. We then finally set: $$\sigma_3:=1+\sigma_2,$$ and as $G$ is regular, we obtain that for every $uv\in E$: $$\sigma_{\omega_3}(u)\neq \sigma_{\omega_3}(v)$$ and $$1\leq \sigma_3(uv) \leq \left\lceil\frac{n}{d}\right\rceil + 7 \left\lceil\frac{n}{d\ln^b n}\right\rceil + \left\lceil\frac{n}{d\ln^{b+\varepsilon} n}\right\rceil +1 \leq \frac{n}{d} + 8\frac{n}{d\ln^b n}.$$
Concluding Remarks
==================
By substituting e.g. $b=1$, $\varepsilon=\frac{1}{12}$ in Theorem \[MainTheoremIrregStrAsympt\] we obtain the following:
\[CorollaryLnNice\] For every $d$-regular graph with $n$ vertices and $d\in\left[\ln^{8}n,\frac{n}{\ln^3n}\right]$: $$s(G)\leq \frac{n}{d}\left(1+\frac{8}{\ln n}\right)$$ for $n$ sufficiently large.
So that our probabilistic argument works effectively, a poly-logarithmic in $n$ lower bound on $d$ is unfortunately unavoidable. We may however still conclude that $s(G)=(1+o(1))n/d$ for a wider domain of $d$ e.g. by fixing any small yet positive $\varepsilon_0$ and substituting $b=\frac{\varepsilon_0}{18}$ (i.e., $b>\frac{\varepsilon_0}{19}$) and $\varepsilon=\frac{\varepsilon_0}{18}$ in Theorem \[MainTheoremIrregStrAsympt\] to obtain:
\[CorollaryLnBest\] For each fixed $\epsilon_0>0$, for every $d$-regular graph with $n$ vertices and $d\in\left[\ln^{1+\varepsilon_0}n,\frac{n}{\ln^{\varepsilon_0}n}\right]$, if $n$ is sufficiently large, $$s(G)\leq \frac{n}{d}\left(1+\frac{1}{\ln^{\frac{\varepsilon_0}{19}} n}\right).$$ Hence, $$s(G)=(1+o(1))\frac{n}{d}.$$
We also note that on the other hand the second order term in our upper bounds above could also be greatly improved, but at the cost of narrowing down the interval for $d$. In particular, using the same technique as in the proof of Theorem \[MainTheoremIrregStrAsympt\] one may show that $s(G)\leq\frac{n}{d}(1+\frac{1}{n^\gamma})$ for any $d$-regular graph $G$ with sufficiently large order $n$ and $d\in[n^{\alpha},n^{\beta}]$ for appropriately selected constants $\alpha,\beta,\gamma\in (0,1)$, $\alpha<\beta$. We omit derails here, as the major goal of this paper was to settle an asymptotic confirmation of Conjecture \[FaudreeAndLehelConjecture\], namely that $s(G)=(1+o(1))\frac{n}{d}$, for possibly widest spectrum of degrees $d$.
[99]{}
L. Addario-Berry, K. Dalal, C. McDiarmid, B.A. Reed, A. Thomason, *Vertex-Colouring Edge-Weightings*, Combinatorica 27(1) (2007) 1–12.
L. Addario-Berry, R.E.L. Aldred, K. Dalal, B.A. Reed, *Vertex colouring edge partitions*, J. Combin. Theory Ser. B, 94(2) (2005) 237–244.
L. Addario-Berry, K. Dalal, B.A. Reed, *Degree Constrained Subgraphs*, Discrete Appl. Math. 156(7) (2008) 1168–1174.
M. Aigner, E. Triesch, *Irregular assignments of trees and forests*, SIAM J. Discrete Math. 3(4) (1990) 439–449.
D. Amar, Irregularity strength of regular graphs of large degree, Discrete Math. 114 (1993) 9–17.
D. Amar, O. Togni, Irregularity strength of trees, Discrete Math. 190 (1998) 15–38.
M. Anholcer, M. Kalkowski, J. Przyby[ł]{}o, A new upper bound for the total vertex irregularity strength of graphs, Discrete Math. 309 (2009) 6316–6317.
M. Bača, S. Jendroľ, M. Miller, J. Ryan, On irregular total labellings, Discrete Math. 307 (2007) 1378–1388.
T. Bartnicki, J. Grytczuk, S. Niwczyk, *Weight Choosability of Graphs*, J. Graph Theory 60(3) (2009) 242–256.
O. Baudon, J. Bensmail, J. Przyby[ł]{}o, M. Woźniak, *On decomposing regular graphs into locally irregular subgraphs*, European J. Combin. 49 (2015) 90–104.
J. Bensmail, M. Merker, C. Thomassen, *Decomposing graphs into a constant number of locally irregular subgraphs*, European J. Combin. 60 (2017) 124–134.
T. Bohman, D. Kravitz, On the irregularity strength of trees, J. Graph Theory 45 (2004) 241–254.
M. Bonamy, J. Przyby[ł]{}o, *On the neighbor sum distinguishing index of planar graphs*, J. Graph Theory 85(3) (2017) 669–690.
G. Chartrand, P. Erdős, O.R. Oellermann, *How to Define an Irregular Graph*, College Math. J. 19(1) (1988) 36–42.
G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz, F. Saba, *Irregular networks*, Congr. Numer. 64 (1988) 197–210.
B. Cuckler, F. Lazebnik, *Irregularity Strength of Dense Graphs*, J. Graph Theory 58(4) (2008) 299–313.
J.H. Dinitz, D.K. Garnick, A. Gyárfás, On the irregularity strength of the $m \times n$ grid, J. Graph Theory 16 (1992) 355–374.
G. Ebert, J. Hemmeter, F. Lazebnik, A.J. Woldar, On the irregularity strength of some graphs, Congr. Numer. 71 (1990) 39–52.
G. Ebert, J. Hemmeter, F. Lazebnik, A.J. Woldar, On the number of irregular assignments on a graph, Discrete Math. 93 (1991) 131–142.
R.J. Faudree, M.S. Jacobson, J. Lehel, R. Schelp, *Irregular networks, regular graphs and integer matrices with distinct row and column sums*, Discrete Math. 76 (1989) 223–240.
R.J. Faudree, J. Lehel, *Bound on the irregularity strength of regular graphs*, Colloq Math Soc Jańos Bolyai, 52, Combinatorics, Eger North Holland, Amsterdam, (1987), 247–256.
M. Ferrara, R.J. Gould, M. Karoński, F. Pfender, An iterative approach to graph irregularity strength, Discr. Appl. Math. 158 (2010) 1189–1194.
E. Flandrin, A. Marczyk, J. Przyby[ł]{}o, J-F. Saclé, M. Wo[ź]{}niak, *Neighbor sum distinguishing index*, Graphs Combin. 29(5) (2013) 1329–1336.
A. Frieze, R.J. Gould, M. Karoński, F. Pfender, *On Graph Irregularity Strength*, J. Graph Theory 41(2) (2002) 120–137.
A. Gyárfás, The irregularity strength of $K_{m,m}$ is $4$ for odd $m$, Discrete Math. 71 (1998) 273–274.
S. Janson, T. [Ł]{}uczak, A. Ruciński, *Random Graphs*, Wiley, New York, 2000.
S. Jendrol’, M. Tkač, Z. Tuza, The irregularity strength and cost of the union of cliques, Selected papers in honour of Paul Erdős on the occasion of his 80th birthday (Keszthely, 1993), Discrete Math. 150 (1996) 179–186.
M. Kalkowski, *A note on 1,2-Conjecture*, in Ph.D. Thesis, 2009. Available at: https://repozytorium.amu.edu.pl/bitstream/10593/445/1\
/Rozprawa%20Doktorska%202010%20Maciej%20Kalkowski.pdf
M. Kalkowski, M. Karoński, F. Pfender, *A new upper bound for the irregularity strength of graphs*, SIAM J. Discrete Math. 25 (2011) 1319–1321.
M. Kalkowski, M. Karoński, F. Pfender, *Vertex-coloring edge-weightings: Towards the 1-2-3 conjecture*, J. Combin. Theory Ser. B 100 (2010) 347–349.
M. Karoński, T. Łuczak, A. Thomason, *Edge weights and vertex colours*, J. Combin. Theory Ser. B 91 (2004) 151–157.
J. Lehel, *Facts and quests on degree irregular assignments*, Graph Theory, Combinatorics and Applications, Willey, New York, 1991, 765–782.
P. Majerski, J. Przyby[ł]{}o, *On the irregularity strength of dense graphs*, SIAM J. Discrete Math. 28(1) (2014) 197–205.
P. Majerski, J. Przyby[ł]{}o, Total vertex irregularity strength of dense graphs. [*J. Graph Theory*]{} 76(1) (2014) 34–41.
T. Nierhoff, *A tight bound on the irregularity strength of graphs*, SIAM J. Discrete Math. 13(3) (2000) 313–323.
J. Przyby[ł]{}o, *A note on asymptotically optimal neighbour sum distinguishing colourings*, European J. Combin. 77 (2019) 49–56.
J. Przyby[ł]{}o, *Asymptotically optimal neighbour sum distinguishing colourings of graphs*, Random Structures Algorithms 47 (2015) 776–791.
J. Przyby[ł]{}o, *On decomposing graphs of large minimum degree into locally irregular subgraphs*, Electron. J. Combin. 23(2) (2016) $\sharp$P2.31.
J. Przyby[ł]{}o, *Irregularity strength of regular graphs*, Electron. J. Combin. 15(1) (2008) $\sharp$R82.
J. Przyby[ł]{}o, *Linear bound on the irregularity strength and the total vertex irregularity strength of graphs*, SIAM J. Discrete Math. 23(1) (2009) 511–516.
J. Przyby[ł]{}o, *Neighbor distinguishing edge colorings via the Combinatorial Nullstellensatz*, SIAM J. Discrete Math. 27(3) (2013) 1313–1322.
J. Przyby[ł]{}o, *The 1–2–3 Conjecture almost holds for regular graphs*, submitted.
J. Przyby[ł]{}o, T-L. Wong, *Neighbor distinguishing edge colorings via the Combinatorial Nullstellensatz revisited*, J. Graph Theory 80(4) (2015) 299–312.
J. Przyby[ł]{}o, M. Woźniak, *On a 1,2 Conjecture*, Discrete Math. Theor. Comput. Sci. 12(1) (2010) 101–108.
J. Przyby[ł]{}o, M. Woźniak, *Total weight choosability of graphs*, Electron. J. Combin. 18(1) (2011) $\sharp$P112.
B. Seamone, *The 1-2-3 Conjecture and related problems: a survey*, Technical report, available online at http://arxiv.org/abs/1211.5122, 2012.
C. Thomassen, Y. Wu, C.Q. Zhang, *The $3$-flow conjecture, factors modulo $k$, and the 1–2–3 conjecture*, J. Combin. Theory Ser. B 121 (2016) 308–325.
O. Togni, Irregularity strength and compound graphs, Discrete Math. 218 (2000) 235–243.
B. Vučković, *Multi-set neighbor distinguishing 3-edge coloring*, Discrete Math. 341(3) (2018) 820–824.
T. Wong, X. Zhu, *Every graph is (2,3)-choosable*, Combinatorica 36(1) (2016) 121–127.
T. Wong, X. Zhu, *Total weight choosability of graphs*, J. Graph Theory 66 (2011) 198–212.
|
---
author:
- 'Yvonne Choquet-Bruhat$^{\ast}$ and James Isenberg$^{\ast\ast}$.'
title: 'Half polarized $U(1)$ symmetric vacuum spacetimes with AVTD behavior'
---
**Abstract.** In a previous work, we used a polarization condition to show that there is a family of $U(1)$ symmetric solutions of the vacuum Einstein equations on $\Sigma\times S^{1}\times R$ ($\Sigma$ any 2 dimensional manifold) such that each exhibits AVTD[^1] behavior in the neighborhood of its singularity. Here we consider the general case of $S^{1}$ bundles over the base $\Sigma\times R$ and determine a condition, called the half polarization condition, necessary and sufficient in our context, for AVTD behavior near the singularity.
Introduction.
=============
A rigorous study of the singularities in cosmological solutions of the vacuum Einstein equations has been hampered by the fact that the generic such solution is expected to have a singularity of the oscillatory type predicted by Belinsky, Lifshitz and Khalatnikov \[BLK\]. There is currently no satisfactory mathematical method for treating such oscillating singularities, at least when the spacetimes under study are spatially inhomogeneous. For this reason much effort has gone into the study of families of solutions which have milder cosmological singularities such as those of AVTD (asymptotically velocity term dominated) type, for which rigorous, so called Fuchsian, methods are available.
To suppress the oscillatory behavior expected for the generic solution, one can
- \(i) introduce suitable matter sources such as scalar fields and study the solutions of the associated non vacuum field equations \[AR\],
- \(ii) study higher dimensional models motivated by string or supergravity theories wherein (for sufficiently high dimensions at least) the oscillations are naturally suppressed \[DHRW\]; or
- \(iii) remain in 3+1 dimensions but impose a combination of symmetry and polarization conditions in order to achieve the desired AVTD behavior.
For the case of $U(1)$ symmetric vacuum solutions on the trivial $S^{1}$ bundle $T^{2}\times R\times S^{1}\rightarrow T^{2}\times R$ (with $U(1)$ symmetry imposed on the circular fibers) Isenberg and Moncrief \[IM\] have showed, using Fuchsian methods, that AVTD behavior is achieved provided the solutions considered are at least half polarized in a certain well defined sense. The half polarization condition includes, as a special case, the fully polarized solutions wherein the 3 planes orthogonal to orbits of the $U(1)$ isometry action are integrable and the vacuum 3+1 field equations reduce to a system of 2+1 dimensional Einstein equations coupled to a massless scalar field on the quotient manifold $T^{2}\times R.$ The more general (half polarized) solutions admit, in addition, half the extra (asymptotic) Cauchy data expected for a fully general, non polarized solution of the same ($U(1)$ symmetric) type. On the basis of numerical studies due to Berger and Moncrief the fully general, non polarized $U(1)$ symmetric vacuum solution on $T^{2}\times R\times S^{1}\rightarrow T^{2}\times R$ is expected to have an oscillatory singularity and hence not to be amenable to Fuchsian analysis \[BM\].
Choquet-Bruhat, Isenberg and Moncrief have extended the analysis given in \[IM\] to cover the case of polarized $U(1)$ symmetric vacuum solutions on manifolds of the more general type $\Sigma^{2}\times R\times S^{1}\rightarrow\Sigma
^{2}\times R,$ where $\Sigma^{2}$ is an arbitrary compact surface and the bundle (in view of the assumed polarization condition) is necessarily trivial. In the present paper the polarization restriction is eliminated in favor of an appropriate half polarization condition and the limitation to trivial $S^{1}$ bundles over the base $\Sigma^{2}\times R$ is also removed. The present work thus demontrates the existence of a large family of vacuum $U(1)$ symmetric solutions of half polarized type defined on trivial and non trivial bundles over $\Sigma^{2}\times R$ (with $\Sigma^{2}$ an arbitrary compact surface) and having AVTD singularity behavior. The half polarization condition used in \[IM\] involved requiring one of the asymptotic functions to vanish. The half polarization condition which we find here necessary and sufficient for possible AVTD behavior can be understood in terms of the behavior of the VTD solutions to which our solutions converge as one approaches the singularity. Specifically a VTD solution is half polarized if and only if the set of geodesics in the Poincaré plane which represent it (at different spatial points) all tend to the same point as $t$ appoaches the singularity.
Einstein equations.
===================
A spacetime metric on a manifold $V_{4}\equiv M\times R,$ with $M$ an $S^{1}$ principal fiber bundle over a surface $\Sigma,$ reads, if it is invariant under the $S^{1}$ action on $V_{4},$ $$^{(4)}g\equiv e^{-2\phi}\text{ }^{(3)}g+e^{2\phi}(d\theta+A)^{2}$$ with $\theta$ a parameter on the (spacelike) circular orbit, $\phi$ a scalar, $A$ a locally defined 1 - form and $^{(3)}g$ a lorentzian metric, all on $V_{3}:=\Sigma\times R.$
The vacuum 3+1 Einstein equations $Ricci(^{(4)}g)=0$ for such an $S^{1}$ symmetric metric on $V_{4}$ are known \[M86\], \[CB-M96\] to be equivalent[^2] to the wave map equation from $(V_{3},^{(3)}g)$ into the Poincaré plane $P=:(R^{2},G),$ $\Phi\equiv
(\phi,\omega):$ $V_{3}\rightarrow R^{2},$ where $$G\equiv2(d\phi)^{2}+\frac{1}{2}e^{-4\gamma}(d\omega)^{2},$$ coupled to the 2+1 Einstein equations for $^{(3)}g$ on $V_{3}$ with the wave map as the source field. The scalar function $\omega$ on $V_{3}$ is linked to the differential $F$ of $A$ by the relation $$d\omega=e^{4\phi}\ast F,\text{ \ \ with \ \ }F=dA.$$ Thus in local coordinates $x^{\alpha}$, $\alpha=0,1,2,$ on $V_{3},$ with $\eta$ the volume form of $^{(3)}g=^{(3)}g_{\alpha\beta}dx^{\alpha}dx^{\beta
},$ one has $$F_{\alpha\beta}\equiv\frac{1}{2}e^{-4\phi}\eta_{\alpha\beta\lambda}\partial^{\lambda}\omega.$$
The wave map equations are, with $^{(3)}\nabla$ the covariant derivative in the metric $^{(3)}g$ $$g^{\alpha\beta}(^{(3)}\nabla_{\alpha}\partial_{\beta}\phi+\frac{1}{2}e^{-4\phi}\partial_{\alpha}\omega\partial_{\beta}\omega)=0$$$$g^{\alpha\beta}(^{(3)}\nabla_{\alpha}\partial_{\beta}\omega-4\partial_{\alpha
}\omega\partial_{\beta}\phi)=0.$$
The $2+1$ Einstein equations are, with ”.” indicating a scalar product in the metric $G$$$^{(3)}R_{\alpha\beta}=\partial_{\alpha}\Phi.\partial_{\beta}\Phi.$$
To solve these equations we choose for $^{(3)}g$ a zero shift, we denote the lapse by $e^{\lambda}$ and we weigh by $e^{\lambda},$ without restricting the generality, the $t$ dependent space metric $g=g_{ab}dx^{a}dx^{b},$ $a,b=1,2.$ That is, we set $$^{(3)}g\equiv-N^{2}dt^{2}+g_{ab}dx^{a}dx^{b}\text{ \ \ with \ \ }N\equiv
e^{\lambda},\text{ \ \ }g_{ab}\equiv e^{\lambda}\sigma_{ab},\text{\ \ }$$ We denote by $\sigma^{ab}$ the contravariant form of $\sigma.$ The extrinsic curvature of $\Sigma_{t}$ in $(V_{3},^{(3)}g)$ is $$k_{ab}:=-\frac{1}{2N}\partial_{t}g_{ab}\equiv-\frac{1}{2}(\sigma_{ab}\partial_{t}\lambda+\partial_{t}\sigma_{ab}).$$ The mean extrinsic curvature $\tau$ is therefore $$\tau:=g^{ab}k_{ab}\equiv-e^{-\lambda}(\partial_{t}\lambda+\frac{1}{2}\psi),$$ where we have defined $$\psi:=\sigma^{ab}\partial_{t}\sigma_{ab}.$$
The connection coefficients (Christoffel symbols) of $^{(3)}g$ are found to be (note that $^{(3)}g^{00}=-e^{-2\lambda},$ $^{(3)}g^{ab}=g^{ab}=e^{-\lambda
}\sigma^{ab})$ $$^{(3)}\Gamma_{ab}^{c}=\Gamma_{ab}^{c}(g)=\Gamma_{ab}^{c}(\sigma)+\frac{1}{2}(\delta_{b}^{c}\partial_{a}\lambda+\delta_{a}^{c}\partial_{b}\lambda
-\sigma^{cd}\sigma_{ab}\partial_{d}\lambda)$$$$^{(3)}\Gamma_{00}^{0}=\partial_{t}\lambda\text{, \ \ }^{(3)}\Gamma_{0a}^{0}=\partial_{a}\lambda,\text{ \ \ }^{(3)}\Gamma_{00}^{a}=\sigma
^{ab}e^{\lambda}\partial_{a}\lambda,$$$$^{(3)}\Gamma_{ab}^{0}=-e^{-\lambda}k_{ab},\text{ \ }^{(3)}\Gamma_{a0}^{b}=-e^{\lambda}k_{a}^{b}.$$ In particular it holds that $$^{(3)}g^{\alpha\beta}{}^{(3)}\Gamma_{\alpha\beta}^{0}=\frac{1}{2}\psi
e^{-2\lambda}.$$ We see that the metric $^{(3)}g$ is in harmonic time gauge if and only if $\psi=0.$
The Einstein equations split into constraints and evolution equations. We denote by $S_{\beta}^{\alpha}\equiv^{(3)}R_{\beta}^{\alpha}-\frac{1}{2}\delta_{\beta}^{\alpha}{}^{(3)}R$ the Einstein tensor of $^{(3)}g,$ by $T_{\beta}^{\alpha}$ the stress energy tensor of $\Phi,$ and we set $\Sigma_{\beta}^{\alpha}\equiv S_{\beta}^{\alpha}-T_{\beta}^{\alpha}.$ The constraints are: $$C_{0}\equiv\Sigma_{0}^{0}\equiv-\frac{1}{2}\{R(g)-k.k+\tau^{2}-e^{-2\lambda
}\partial_{t}\Phi.\partial_{t}\Phi-g^{ab}\partial_{a}\Phi.\partial_{b}\Phi\}=0$$ and (indices raised with $g^{ab},$ $\nabla$ the covariant derivative in the metric $g)$$$C_{a}\equiv e^{\lambda}\Sigma_{a}^{0}\equiv-\{\nabla_{b}k_{a}^{b}-\partial
_{a}\tau+e^{-\lambda}\partial_{a}\Phi.\partial_{t}\Phi\}=0.$$ The evolution equations are, with $N=e^{\lambda},$ $$N(^{(3)}R_{a}^{b}-\rho_{a}^{b})\equiv-\partial_{t}k_{a}^{b}+N\tau k_{a}^{b}-\nabla^{b}\partial_{a}N+NR_{a}^{b}-N\partial_{a}\Phi.\partial^{b}\Phi=0.$$
In order to obtain a first order system in the Fuchsian analysis that we will make, we introduce auxiliary unknowns $\Phi_{t},\Phi_{a},\sigma_{c}^{ab} $ which are identified with the first partial derivatives of $\Phi$ and the covariant derivative of $\sigma$ with respect to a given $t$ independent metric $\tilde{\sigma}.$ These new unknowns satisfy the evolution equations $$\partial_{t}\Phi=\Phi_{t},$$$$\text{ \ \ }\partial_{t}\Phi_{a}=\partial_{a}\Phi_{t},$$$$\partial_{t}\sigma_{c}^{ab}=\tilde{\nabla}_{c}\partial_{t}\sigma^{ab}\text{
\ \ }$$ where, by the definitions of $\sigma$ and $k,$$$\partial_{t}\sigma^{ab}=2e^{2\lambda}k^{ab}+\sigma^{ab}\partial_{t}\lambda.$$ The function $\lambda$ is not left unknown, but rather is determined by a gauge condition from its VTD value.
VTD equations and solutions.
============================
The Velocity Terms Dominated equations are obtained by dropping the space derivatives in the equations.
We denote by a tilde quantities which are independent of $t,$ and we denote VTD solutions using a hat.
Einstein evolution VTD solutions.
---------------------------------
In order to obtain a global (on $\Sigma$) formulation we choose a VTD metric which remains in a fixed conformal class over $\Sigma$ as $t$ evolves; we set $$\hat{\sigma}_{ab}=\tilde{\sigma}_{ab}\text{ \ and \ \ }\hat{g}_{ab}=e^{\hat{\lambda}}\tilde{\sigma}_{ab}.$$ Then $$\hat{\psi}=0,\text{ \ \ }\hat{g}^{ab}=e^{-\hat{\lambda}}\tilde{\sigma}^{ab},\text{ \ \ }\partial_{t}\hat{g}_{ab}=e^{\hat{\lambda}}\tilde{\sigma
}_{ab}\partial_{t}\hat{\lambda},$$ and the definition of $k$ gives that $$\hat{k}_{ab}=-\frac{1}{2}\tilde{\sigma}_{ab}\partial_{t}\hat{\lambda},\text{
\ }\hat{k}_{a}^{b}=-\frac{1}{2}e^{-\hat{\lambda}}\delta_{a}^{b}\partial
_{t}\hat{\lambda},\text{ \ \ }\hat{\tau}:=\hat{k}_{a}^{a}=-e^{-\hat{\lambda}}\partial_{t}\hat{\lambda}.$$
Requiring that these VTD quantities satisfy the VTD evolution equations, we obtain $$\partial_{t}\hat{k}_{a}^{b}=\hat{N}\hat{\tau}\hat{k}_{a}^{b}.$$ Therefore, by straightforward computation $$\partial_{tt}^{2}\hat{\lambda}=0;\text{ \ \ \ hence \ \ \ }\hat{\lambda
}=\tilde{\lambda}-\tilde{v}t$$ with $\tilde{\lambda}$ and $\tilde{v}$ arbitrary functions on $\Sigma,$ independent of $t.$ Then we have $$\hat{k}_{ab}=\frac{1}{2}\tilde{v}\tilde{\sigma}_{ab},\text{ \ }\hat{k}_{a}^{b}=\frac{1}{2}\tilde{v}e^{-\hat{\lambda}}\delta_{a}^{b},\text{ \ \ }\hat{\tau}=e^{-\hat{\lambda}}\tilde{v}.$$
Wave map VTD solutions.
-----------------------
The results for a VTD wave map are very different from the results obtained for a scalar function \[CBIM\]. If we drop space derivatives in the wave map equations we obtain geodesic equations in the target manifold, with $t$ the length parameter on these geodesics so long as the 2+1 metric is in harmonic time gauge. If we make the change of coordinates $Y=e^{2\phi}$ in the target (which defines a diffeomorphism from $R^{2}$ onto the upper half plane $Y>0),$ the metric $G$ takes a standard form for the metric of a Poincaré half plane; namely $$G\equiv\frac{1}{2}\{\frac{d\omega^{2}+dY^{2}}{Y^{2}}\},\text{ \ \ }Y=e^{2\phi
}.$$ The VTD, geodesic, equations written in this metric read, with a prime denoting the derivative with respect to $t$$$\omega^{\prime\prime}-2Y^{-1}\omega^{\prime}Y^{\prime}=0,$$$$Y^{\prime\prime}+Y^{-1}\omega^{\prime}X^{\prime}=0.$$ The general solution of these geodesic equations is represented in these coordinates, as is well known, by half circles[^3] centered on the line $Y=0;$ specifically, with $A$ and $B$ arbitrary constants (that is, independent of $t)$, the solution takes the form $$\hat{\omega}=B+A\cos\theta,\text{ \ }\hat{Y}=A\sin\theta,\text{ \ \ }0<\theta<\pi.$$ These functions $\omega$ and $Y$ satisfy the differential equations 3.8, 3.9 if and only if it holds that: $$\frac{\theta^{\prime\prime}}{\theta^{\prime}}=\frac{\cos\theta}{\sin\theta
}\theta^{\prime}.$$ Integrating this equation we have that, with $\tilde{w}$ independent of $t$ $$\theta^{\prime}=-\tilde{w}\sin\theta.$$ Another integration gives that, with $\tilde{\Theta}$ independent of $t$ $$\tan\frac{\theta}{2}=\tilde{\Theta}e^{-\tilde{w}t}.$$ If we now make the substitution $A=e^{2\tilde{\phi}}$ and $B=\tilde{\omega},$ then 3.10 reads $$\hat{\phi}=\tilde{\phi}+\frac{1}{2}log(sin\theta),\text{ \ \ \ }\hat{\omega
}=\tilde{\omega}+e^{2\tilde{\phi}}cos\theta\text{\ }.\text{ \ \ }$$
The set of above formulas is identical to the following one $$\hat{\omega}\equiv\tilde{\omega}+e^{2\tilde{\phi}}\frac{1-\tilde{\Theta}^{2}e^{-2\tilde{w}t}}{1+\tilde{\Theta}^{2}e^{-2\tilde{w}t}},\text{
\ \ }e^{2\hat{\phi}}\equiv\hat{Y}=e^{2\tilde{\phi}}\frac{2\tilde{\Theta
}e^{-\tilde{w}t}}{1+\tilde{\Theta}^{2}e^{-2\tilde{w}t}}.$$ $\hat{Y}$ tends to zero when $t$ tends to $\infty,$ but $\hat{\omega}$ tends to $\tilde{\omega}+e^{2\tilde{\phi}}.$
Einstein constraint VTD solutions.
----------------------------------
We deduce from 3.14 and 3.12 that $$\partial_{t}\hat{\phi}\equiv\frac{1}{2}\hat{Y}^{-1}\hat{Y}^{\prime}=\frac
{1}{2}\frac{cos\theta\theta^{\prime}}{sin\theta}=-\frac{1}{2}\tilde
{w}cos\theta$$ and $$e^{-2\hat{\phi}}\partial_{t}\hat{\omega}=-\theta^{\prime}=\tilde{w}sin\theta.$$
The Einstein VTD constraints reduce to the following equation $$2\hat{C}_{0}\equiv-\hat{k}.\hat{k}+\hat{\tau}^{2}-e^{-2\hat{\lambda}}\{2(\partial_{t}\hat{\phi})^{2}+\frac{1}{2}e^{-4\hat{\phi}}(\partial_{t}\hat{\omega})^{2}\}=0.$$ We have, using 3.16 and 3.17, $$2(\partial_{t}\hat{\phi})^{2}+\frac{1}{2}e^{-4\hat{\phi}}(\partial_{t}\hat{\omega})^{2}=\frac{1}{2}\tilde{w}^{2}.$$ We deduce therefore from 3.6 that the VTD constraint 3.18 is satisfied if and only if $$\tilde{v}^{2}=\tilde{w}^{2}$$
Fuchsian expansion.
===================
2+1 metric expansions.
----------------------
For the unknowns $\sigma$ and $k$ we choose the following expansions, with the various $\varepsilon^{\prime}s$ being positive numbers to be chosen later $$\sigma^{ab}=\tilde{\sigma}^{ab}+e^{-\varepsilon_{\sigma}t}u_{\sigma}^{ab}$$$$k_{a}^{b}=e^{-\lambda}(\frac{1}{2}\tilde{v}\delta_{a}^{b}+e^{-\varepsilon
_{k}t}u_{k,a}^{b}).$$ Then $$\tau\equiv k_{a}^{a}=e^{-\lambda}(\tilde{v}+e^{-\varepsilon_{k}t}u_{k,a}^{a}).$$ We take as a gauge condition $$\lambda=\hat{\lambda};\text{ \ \ \ hence \ \ }\partial_{t}\lambda=-\tilde{v}.$$ Comparing the expressions 4.4 and 2.10 for $\tau,$ we find that this condition is equivalent to the gauge fixing requirement $$e^{-\varepsilon_{k}t}u_{k,a}^{a}+\frac{1}{2}\psi=0.$$
Wave map expansion.
-------------------
We expand $\Phi$ near its VTD value; that is we set $$\phi=\hat{\phi}+e^{-\varepsilon_{\phi}t}u_{\phi}\text{ \ \ \ with
\ \ }\ \ \hat{\phi}=\tilde{\phi}+\frac{1}{2}log(sin\theta),$$ while for $\omega,$ for convenience of computation, we choose to set $$\omega=\hat{\omega}+e^{2\phi}e^{-\varepsilon_{\omega}t}u_{\omega}\text{
\ \ with \ \ \ }\hat{\omega}=\tilde{\omega}+e^{2\tilde{\phi}}cos\theta,$$
Expansion for first derivatives.
--------------------------------
We expand the auxiliary unknowns near the values of the derivatives of the VTD solution. That is we set (see 3.16, 3.17) $$\phi_{t}=\partial_{t}\hat{\phi}+e^{-\varepsilon_{\phi_{t}}t}u_{\phi_{t}}\equiv-\frac{1}{2}\tilde{w}cos\theta+e^{-\varepsilon_{\phi_{t}}t}u_{\phi_{t}},$$$$\omega_{t}=\partial_{t}\hat{\omega}+e^{2\phi}e^{-\varepsilon_{\omega_{t}}}u_{\omega_{t}}\equiv e^{2\tilde{\phi}}\tilde{w}sin^{2}\theta+e^{2\phi
}e^{-\varepsilon_{\omega_{t}}}u_{\omega_{t}}$$ The expansions of $\phi_{a}$ and $\omega_{a}$ are defined similarly by setting $$\phi_{a}=\partial_{a}\hat{\phi}+e^{-\varepsilon_{\phi^{\prime}}}u_{\phi_{a}},\text{ \ \ \ }\omega_{a}=\partial_{a}\hat{\omega}+e^{2\phi}e^{-\varepsilon
_{\omega^{\prime}}t}u_{\omega_{a}}.$$ We next compute $\partial_{a}\hat{\phi}$ and $\partial_{a}\hat{\omega}.$ It follows from 3.14 that $$\partial_{a}\hat{\phi}=\partial_{a}\tilde{\phi}+\frac{cos\theta}{2sin\theta
}\partial_{a}\theta,\text{ \ \ \ \ }\partial_{a}\hat{\omega}=\partial
_{a}\tilde{\omega}+e^{2\tilde{\phi}}(2cos\theta\partial_{a}\tilde{\phi
}-sin\theta\partial_{a}\theta).$$ We compute $\partial_{a}\theta$ using 3.13 and elementary properties of sine and cosine. We find that $$\partial_{a}\theta=\tilde{\Theta}^{-1}sin\theta\partial_{a}(\tilde{\Theta
}-t\tilde{w})$$ Therefore it holds that $$\partial_{a}\hat{\phi}=\partial_{a}\tilde{\phi}+\tilde{\Theta}^{-1}\frac{cos\theta}{2}\partial_{a}(\tilde{\Theta}-\tilde{w}t).$$ Then, writing $\partial_{a}\hat{\omega}$ as sum of a term independent of $t$ plus terms tending to zero when $t$ tends to infinity, we have $$\partial_{a}\hat{\omega}\equiv\partial_{a}(\tilde{\omega}+e^{2\tilde{\phi}})-e^{2\tilde{\phi}}[2(1-cos\theta)\partial_{a}\tilde{\phi}-\tilde{\Theta
}^{-1}sin^{2}\theta\partial_{a}(\tilde{\Theta}-\tilde{w}t)]$$
For $\sigma_{c}^{ab},$ since $\tilde{\nabla}_{c}\tilde{\sigma}^{ab}=0,$ we set $$\sigma_{c}^{ab}\equiv e^{-\varepsilon_{\sigma^{\prime}}t}u_{\sigma^{\prime},c}^{ab}.$$
Fuchsian system for the evolution equations.
============================================
Given the Fuchsian expansions of the previous section, the Einstein - wave map evolution system reads as a first order system for the set of unknowns $U\equiv(u_{\sigma},$ $u_{k},$ $u_{\Phi},$ $u_{\Phi_{t}},$ $u_{\Phi^{\prime
},\text{ }}u_{\sigma^{\prime}}).$
The differential system for $U$ is Fuchsian in a neighbourhood of $t=+\infty$ if it takes the form $$\partial_{t}U-LU=e^{-\mu t}F(t,x,U,\tilde{\partial}U)$$ with $L$ a linear operator independent of $t$ with non negative eigenvalues, $\mu$ a positive number and $F$ a set of tensor fields linear in $\tilde{\partial}U$, continuous in $t,$ analytic in $x$ and $U$ and uniformly Lipshitzian in all its arguments in a neighbourhood of $U=0$, for $t$ large enough.
Einstein evolution equations.
-----------------------------
### Equation for $u_{\sigma}.$
The Fuchsian expansion 4.2 for $k$ yields the following equation: $$\partial_{t}g^{ab}\equiv2Nk^{ab}\equiv2e^{\lambda}g^{ac}k_{c}^{b}\equiv
e^{-\lambda}(\tilde{v}\sigma^{ab}+2e^{-\varepsilon_{k}t}\sigma^{ac}u_{k,c}^{b}).$$ Using $g^{ab}\equiv e^{-\lambda}\sigma^{ab}$ and $\partial_{t}\lambda
=-\tilde{v},$ we have $$\partial_{t}g^{ab}\equiv e^{-\lambda}(\tilde{v}\sigma^{ab}+\partial_{t}\sigma^{ab}).$$ Combining these equations together with the Fuchsian expansion of $\sigma$ results in the equation: $$\partial_{t}u_{\sigma}^{ab}-\varepsilon_{\sigma}u_{\sigma}^{ab}=2e^{(\varepsilon_{\sigma}-\varepsilon_{k})t}\sigma^{ac}u_{k,c}^{b},$$ which is of Fuchsian type if $\varepsilon_{k}>\varepsilon_{\sigma}>0.$
### Equation for $u_{k}.$
The Fuchsian expansion of $k$ together with $N=e^{\lambda}$ and $\partial
_{t}\lambda=-\tilde{v}$ imply by straightforward computation that $$\partial_{t}k_{a}^{b}\equiv e^{-\lambda}\{\frac{1}{2}\tilde{v}^{2}\delta
_{a}^{b}+e^{-\varepsilon_{k}t}(\tilde{v}-\varepsilon_{k})u_{k,a}^{b}+e^{-\varepsilon_{k}t}\partial_{t}u_{k,a}^{b}\},$$ and $$N\tau k_{a}^{b}\equiv e^{-\lambda}\{\frac{1}{2}\tilde{v}^{2}\delta_{a}^{b}+\tilde{v}e^{-\varepsilon_{k}t}u_{k,a}^{b})+e^{-\varepsilon_{k}t}u_{k,c}^{c}(\frac{1}{2}\tilde{v}\delta_{a}^{b}+e^{-\varepsilon_{k}t}u_{k,a}^{b})\}.$$ We see that $e^{-\lambda}\tilde{v}^{2}$ disappears from the difference $\partial_{t}k_{a}^{b}-N\tau k_{a}^{b},$ which motivates the choice of the Fuchsian expansion.
To write the evolution equation 2.18 for $k$ we now compute $$\nabla^{b}\partial_{a}N\equiv e^{-\lambda}\sigma^{bc}\nabla_{c}\partial
_{a}e^{\lambda}\equiv\sigma^{bc}[\partial_{c}\lambda\partial_{a}\lambda+\partial_{a}\partial_{c}\lambda-\Gamma_{ac}^{d}(g)\partial_{d}\lambda].$$ On the other hand, since $\Sigma$ is 2 dimensional and $g$ is conformal to $\sigma$ with a factor $e^{\lambda},$ we have that $$NR_{a}^{b}\equiv e^{\lambda}R_{a}^{b}\equiv\frac{1}{2}e^{\lambda}\delta
_{a}^{b}R(g)=\frac{1}{2}\delta_{a}^{b}\{R(\sigma)-\Delta_{\sigma}\lambda\}.$$ From these results, if we define $$f_{a}^{b}(t,u,u_{x}):=-\nabla^{b}\partial_{a}N+NR_{a}^{b}-N\partial_{a}\Phi.\partial^{b}\Phi$$ then we calculate $$f_{a}^{b}\equiv\sigma^{bc}[\partial_{c}\lambda\partial_{a}\lambda+\partial
_{a}\partial_{c}\lambda-\Gamma_{ac}^{d}(g)\partial_{d}\lambda]+\frac{1}{2}\delta_{a}^{b}[R(\sigma)-\Delta_{\sigma}\lambda]-\sigma^{bc}\Phi_{a}.\Phi_{c}$$ We see that $f_{a}^{b}$ is at most a second order polynomial in $t,$ is analytic in $x$ when $\tilde{v},\tilde{w},\tilde{\lambda},\tilde{\sigma}$ are analytic; is linear in $\partial u;$ and is analytic, bounded and Lipshitzian in $u$ for $u$ bounded and for large[^4] $t$, except eventually for the last term which reads $$\sigma^{bc}\Phi_{a}.\Phi_{c}=2\sigma^{bc}(\phi_{a}\phi_{c}+\frac{1}{2}e^{-4\phi}\omega_{a}\omega_{c}).$$ The expansion 4.10 of $\phi_{a}$ shows that it does not cause problems for the boundedness of $f_{a}^{b}$. However the expansion of $\omega_{a}$ gives $$e^{-2\phi}\omega_{a}=e^{-2\phi}\partial_{a}(\tilde{\omega}+e^{2\tilde{\phi}})-e^{-2\phi+2\tilde{\phi}}[2(1-cos\theta)\partial_{a}\tilde{\phi}+\tilde{\Theta}^{-1}sin^{2}\theta\partial_{a}(\tilde{\Theta}-\tilde{w}t)]$$$$+e^{-\varepsilon_{\omega^{\prime}}t}u_{\omega_{a}}.$$ It follows from 4.6 that $$e^{2(\tilde{\phi}-\phi)}=\frac{e^{-2\delta\phi}}{sin\theta},\text{ \ \ with
\ \ }\delta\phi\equiv e^{-\varepsilon_{\phi}t}u_{\phi}.$$ Therefore, using ($1-cos\theta)/sin\theta=tan(\theta/2)$ we have: $$e^{-2\phi}\omega_{a}=e^{2\tilde{\phi}}\frac{e^{-2\delta\phi}}{sin\theta
}\partial_{a}(\tilde{\omega}+e^{2\tilde{\phi}})-e^{-2\delta\phi}[2tan\frac{\theta}{2}\partial_{a}\tilde{\phi}+\tilde{\Theta}^{-1}sin\theta\partial_{a}(\tilde{\Theta}-\tilde{w}t)]+e^{-\varepsilon
_{\omega^{\prime}}t}u_{\omega_{a}}.$$ We see that $e^{-2\phi}\omega_{a}$ will increase like ($sin\theta)^{-1}$ - that is, like $e^{\tilde{w}t}$ - as $t$ tends to infinity, except if $$\tilde{\omega}+e^{2\tilde{\phi}}=constant.$$ Condition 5.13 is a generalization of the condition imposed on the fields in \[IM\], with other notations, to obtain AVTD behaviour, in the case that $\Sigma$ is a torus. Following the terminology of \[IM\] we call equation 5.13 the ”half polarization” condition. Its geometric meaning is that **the set of geodesics in the Poincaré plane representing the VTD solution all tend to the same point of the axis** $Y=0$ **as** $t$ **tends to infinity.**
After inserting the Fuchsian expansions and multiplying by $e^{\lambda
+\varepsilon_{k}:t}$ we find that the equation 2.18 takes the form $$\partial_{t}u_{k,a}^{b}-\varepsilon_{k}u_{k,a}^{b}-\frac{1}{2}v\delta_{a}^{b}u_{k,c}^{c}=e^{-\varepsilon_{k}t}u_{k,c}^{c}u_{k,a}^{b}+e^{\lambda
+\varepsilon_{k}t}f_{a}^{b}(t,u,u_{x}).$$ Since $\lambda=\tilde{\lambda}-\tilde{v}t$ and $\tilde{v}=\tilde{w}$ this system can take a Fuchsian form only if the functions $\tilde{\omega}$ and $\tilde{\phi}$ satisfy the half polarization condition 5.13. To obtain the system in obviously Fuchsian form in that case, we split 5.14 into its trace and its traceless parts. For the trace part we have $$\partial_{t}u_{k,a}^{a}-\varepsilon_{k}u_{k,a}^{a}-\tilde{v}u_{k,a}^{a}=e^{-\varepsilon_{k}t}u_{k,c}^{c}u_{k,a}^{a}+e^{\lambda+\varepsilon_{k}t}f_{a}^{a}(t,u,u_{x}).$$ This equation takes Fuchsian form if and only if 5.13 is satisfied and $\tilde{v}>\varepsilon_{k}.$ The same is verified for the traceless part $^{T}u_{k,a}^{b},$ which satisfies an equation with left hand side $$\partial_{t}\text{ }^{T}u_{k,a}^{b}-\varepsilon_{k}{}^{T}u_{k,a}^{b}.$$
### Equation for $u_{\sigma^{\prime}}.$
Using the expansion of $k$ and the relation $\partial_{t}\lambda=-\tilde{v},$ we find that $$\partial_{t}\sigma^{ab}=2e^{2\lambda}k^{ab}+\sigma^{ab}\partial_{t}\lambda=2e^{-\varepsilon_{k}t}\sigma^{ac}u_{k,c}^{b}$$ The equation for $\sigma_{c}^{ab}$ gives therefore the following equation for $u_{\sigma^{\prime}\text{ }}:$$$\partial_{t}u_{\sigma^{\prime},c}^{ab}-\varepsilon_{\sigma^{\prime}}u_{\sigma^{\prime},c}^{ab}=2e^{(\varepsilon_{\sigma^{\prime}}-\varepsilon
_{k})t}\tilde{\nabla}_{c}[\sigma^{ac}u_{k,c}^{b}.]\text{ }$$ which is of Fuchsian type so long as $\varepsilon_{\sigma^{\prime}}<\varepsilon_{k}.$
Wave map equations.
-------------------
### Equations for auxiliary variables.
The equations resulting from the introduction of the new variables $\phi
_{t},\omega_{t}$ are $$\partial_{t}\phi-\phi_{t}=0,\text{ \ \ \ \ }\partial_{t}\omega-\omega_{t}=0.$$ The first equation is of Fuchsian type for $u_{\phi}$ if $\varepsilon
_{\Phi_{t}}>\varepsilon_{\Phi\text{ }},$ since it reads $$\partial_{t}u_{\phi}-\varepsilon_{\phi}u_{\phi}=e^{(-\varepsilon_{\Phi_{t}}+\varepsilon_{\Phi})t}u_{\phi_{t}}.$$ The second equation reads $$\lbrack\partial_{t}u_{\omega}+(2\phi_{t}-\varepsilon_{\omega})u_{\omega
}]\text{ }-e^{(\varepsilon_{\Phi}-\varepsilon_{\Phi_{t}})t}u_{\omega_{t}}=0.$$ We replace $\phi_{t}$ by its value given in 4.8, which we write as follows $$\phi_{t}=-\frac{1}{2}\tilde{w}+\frac{1}{2}\tilde{w}(1-cos\theta
)+e^{-\varepsilon_{\Phi_{t}}t}u_{\phi_{t}}.$$ Since $1-cos\theta$ falls off to zero as $e^{-2\tilde{w}t},$ the equation 5.21 is of Fuchsian type for $u_{\omega}$ if $\tilde{w}>0$ and $\varepsilon
_{\Phi_{t}}>\varepsilon_{\Phi}.$
In the equations 2.20 to be satisfied by $\phi_{a}$ and $\omega_{a},$ the derivatives of the VTD terms disappear, due to the commutation of partial derivatives. The equation for $\phi_{a}$ reads $$\partial_{t}u_{\phi_{a}}-\varepsilon_{\phi^{\prime}}u_{\phi_{a}}=e^{-(\varepsilon_{\Phi_{t}}-\varepsilon_{\Phi^{\prime}})t}(\partial
_{a}u_{\phi_{t}}-t\partial_{a}w),$$ while the equation for $\omega_{a}$ becomes, using the expressions for $\omega_{t}$ and $\omega_{a}$ $$\partial_{t}u_{\omega_{a}}+(2\phi_{t}-\varepsilon_{\omega^{\prime}})u_{\omega_{a}}=e^{-(\varepsilon_{\Phi_{t}}-\varepsilon_{\Phi^{\prime}})t}(\partial_{a}u_{\omega_{t}}+2\phi_{a}u_{\omega_{t}}).$$ These equations are of Fuchsian type so long as $\tilde{w}>0$ and $\varepsilon_{\Phi_{t}}>\varepsilon_{\Phi^{\prime}}.$
### Equation for $u_{\Phi_{t}}.$
The first equation, 2.5, for the wave map reads $$g^{\alpha\beta}(\nabla_{\alpha}\partial_{\beta}\phi+\frac{1}{2}e^{-4\phi
}\partial_{\alpha}\omega\partial_{\beta}\omega)\equiv$$$$-e^{-2\lambda}(\partial_{t}\phi_{t}+\frac{1}{2}e^{-4\phi}\omega_{t}\omega
_{t})+e^{-\lambda}\sigma^{ab}(\nabla_{a}\phi_{b}+\frac{1}{2}e^{-4\phi}\omega_{a}\omega_{b})+g^{\alpha\beta}\Gamma_{\alpha\beta}^{0}\phi_{t}=0$$
Using the Fuchsian expansions for $\phi_{t}$ and $\omega_{t}$ together with $\theta^{\prime}=-\tilde{w}\sin\theta$ and the value given in section 5.1.2 for $e^{2(\tilde{\phi}-\phi)}$we find that: $$\partial_{t}\phi_{t}+\frac{1}{2}e^{-4\phi}\omega_{t}\omega_{t}\equiv$$$$e^{-\varepsilon_{\Phi_{t}}t}(\partial_{t}u_{\phi_{t}}-\varepsilon_{\phi_{t}}u_{\phi_{t}})-\frac{1}{2}\tilde{w}^{2}\sin^{2}\theta+\frac{1}{2}(e^{-2\delta\phi}\tilde{w}sin\theta+e^{-\varepsilon_{\omega_{t}}t}u_{\omega_{t}})^{2}$$
On the other hand, using the expansions for $\sigma^{ab},\phi_{a}$ and $\omega_{a}$ we find that: $$e^{-\lambda}\sigma^{ab}(\nabla_{a}\phi_{b}+\frac{1}{2}e^{-4\phi}\omega
_{a}\omega_{b})\equiv$$$$e^{-\lambda}(\tilde{\sigma}^{ab}+\delta\sigma^{ab})\{\nabla_{b}\partial
_{a}\hat{\phi}+e^{-\varepsilon_{\Phi^{\prime}}t}\nabla_{b}u_{\phi_{a}}+\frac{1}{2}(e^{-2\phi}\partial_{a}\hat{\omega}+e^{-\varepsilon_{\omega
^{\prime}}t}u_{\omega_{a}})(e^{-2\phi}\partial_{b}\hat{\omega}+e^{-\varepsilon
_{\omega^{\prime}}t}u_{\omega_{b}})\}$$ We recall that $$\nabla_{b}\partial_{a}\hat{\phi}\equiv\nabla_{b}[\partial_{a}\tilde{\phi
}+\frac{cos\theta}{2}\tilde{\Theta}^{-1}\partial_{a}(\tilde{\Theta}-\tilde
{w}t)],$$ while under the half polarization assumption $$\tilde{\omega}+e^{2\tilde{\phi}}=constant,$$ the product $e^{-2\phi}\partial_{a}\hat{\omega}$ is given by $$e^{-2\phi}\partial_{a}\hat{\omega}=-e^{-2\delta\phi}\tilde{\Theta}^{-1}sin\theta\partial_{a}(\tilde{\Theta}-\tilde{w}).$$ Finally we calculate $$g^{\alpha\beta}\Gamma_{\alpha\beta}^{0}\equiv\frac{1}{2}\psi e^{-2\lambda
}=-e^{-2\lambda-\varepsilon_{k}t}u_{k,a}^{a}.$$
Inserting these computations into the first wave map equation produces an equation of the form $$\partial_{t}u_{\phi_{t}}-\varepsilon_{\Phi_{t}}u_{\phi_{t}}=e^{-\mu t}f_{\phi_{t}}(x,t,u,\partial u)$$ which is of the Fuchsian type 5.1 (with $\mu>0)$ so long as $\tilde
{v}>\varepsilon_{\Phi_{t}},$ and $\varepsilon_{k}>\varepsilon_{\Phi_{t}}.$
Analogous computations show that the equation for $u_{\omega_{t}}$ is Fuchsian presuming these same inequalities hold.
Results for evolution.
----------------------
As a consequence of the calculations above we have proven the following theorem.
There exist a collection of positive numbers {$\varepsilon_{\sigma
},\varepsilon_{\sigma^{\prime}},\varepsilon_{k},\varepsilon_{\Phi},\varepsilon_{\Phi_{t}},\varepsilon_{\Phi^{\prime}}\}$ such that, given analytic asymptotic data on $\Sigma,$ $\tilde{A}=\{\tilde{v}=\tilde{w},$ $\tilde{\lambda},$ $\tilde{\sigma},\tilde{\Theta},\tilde{\phi},\tilde{\omega
}\},$ the Einstein - wave map evolution system written in first order form for the unknown $U,$ which defines $g,$ $k,$ $\Phi$ and auxiliary variables by the Fuchsian expansions of section 4, is a Fuchsian system for $U$ if and only if $\tilde{\phi}$ and $\tilde{\omega}$ satisfy the half polarisation condition 5.13 and $\tilde{v}>0.$ It admits then one and only one analytic solution tending to zero at infinity.
To show that this result implies that we have a family of solutions of the Einstein - wave map evolution system which decays to solutions of the VTD equations, we need to verify that for a large enough $t$ we have $\Phi
_{t}=\partial_{t}\Phi,$ $\Phi_{a}=\partial_{a}\Phi$ and the like. To show that $\Phi_{a}=\partial_{a}\Phi$ we use the equations 2.20 together with commutation of partial derivatives to show that: $$\partial_{t}(\phi_{a}-\partial_{a}\phi)=\partial_{a}\phi_{t}-\partial
_{a}\partial_{t}\phi=0;$$ hence $\phi_{a}-\partial_{a}\phi$ is independent of $t.$ As $t$ tends to $\infty$ it tends to zero because $$\phi_{a}-\partial_{a}\phi=e^{-\varepsilon_{\phi^{\prime}}t}u_{\phi_{a}}-e^{-\varepsilon_{\phi}t}(\partial_{a}u_{\phi}-\varepsilon_{\phi}u_{\phi}).$$ It must therefore always be zero. Analogous arguments can be used to show that $\omega_{a}=\partial_{a}\omega$ and $\sigma_{c}^{ab}=\tilde{\nabla}_{c}\sigma^{ab}.$
Constraints.
============
The solution of the evolution system satisfies the full Einstein equations so long as it satisfies also the Einstein constraints, that is $$C_{0}:=\Sigma_{0}^{0}\equiv-\frac{1}{2}\{R(g)-k.k+\tau^{2}-e^{-2\lambda
}\partial_{t}\Phi.\partial_{t}\Phi\}=0$$$$C_{a}:=e^{\lambda}\Sigma_{a}^{0}\equiv-\{\nabla_{b}k_{a}^{b}-\partial_{a}\tau+e^{-\lambda}\partial_{t}\Phi.\partial_{a}\Phi\}=0.$$
As usual we will rely on the Bianchi identities, here to construct a Fuchsian system satisfied by the constraints. Together with the wave equation satisfied by $\Phi,$ the Bianchi identities imply that $$^{(3)}\nabla_{\alpha}\Sigma_{\beta}^{\alpha}=0.\text{ }$$ Modulo the evolution equations $^{(3)}R_{a}^{b}-\rho_{a}^{b}=0$ that we have solved, with $\rho_{a}^{b}\equiv\Phi_{a}.\Phi^{b},$ it holds that $$^{(3)}R-\rho=R_{0}^{0}-\rho_{0}^{0};$$ hence $$\Sigma_{0}^{0}\equiv R_{0}^{0}-\rho_{0}^{0}-\frac{1}{2}\delta_{0}^{0}(^{(3)}R-\rho)=\frac{1}{2}\delta_{0}^{0}(^{(3)}R-\rho)$$ and $$\Sigma_{a}^{b}=-\frac{1}{2}\delta_{a}^{b}(^{(3)}R-\rho)=-\delta_{a}^{b}\Sigma_{0}^{0}$$ We use these equations and the identities $$\Sigma_{a}^{0}\equiv e^{-\lambda}C_{a},\text{ \ }\Sigma_{0}^{a}\equiv
-N^{2}\Sigma^{a0}\equiv-g^{ab}N^{2}\Sigma_{b}^{0}\equiv-e^{\lambda}g^{ab}C_{b}$$ together with the expressions for the Christoffel symbols of the metric $^{(3)}g.$ We find that the equations 6.1 can be written in the form $$\partial_{t}C_{0}-2e^{\lambda}\tau C_{0}=g^{ab}\nabla_{a}(e^{\lambda}C_{b})+g^{ab}e^{\lambda}\partial_{a}\lambda C_{a}$$ and (after some simplifications and multiplying by $e^{\lambda})$$$\partial_{t}C_{a}-e^{\lambda}\tau C_{a}=e^{\lambda}\nabla_{a}C_{0}+2e^{\lambda}\partial_{a}\lambda C_{0}.$$ Equivalently, we have $$\partial_{t}(e^{2\lambda}C_{0})-2\tilde{v}e^{2\lambda}C_{0}-2e^{\lambda}\tau
e^{2\lambda}C_{0}=e^{\lambda}\sigma^{ab}\nabla_{a}(e^{\lambda}C_{b})+\sigma^{ab}e^{\lambda}\partial_{a}\lambda e^{\lambda}C_{a}$$ and $$\partial_{t}(e^{\lambda}C_{a})-\tilde{v}e^{\lambda}C_{a}-e^{2\lambda}\tau
C_{a}=\nabla_{a}(e^{2\lambda}C_{0}).$$ We see that $e^{2\lambda}C_{0}$ and $e^{\lambda}C_{a}$ satisfy a linear homogeneous system, which admits zero as a solution. This solution is the unique one tending to zero at infinity, so long as the system is Fuchsian.
The system 6.8, 6.9 is Fuchsian, for a solution of the evolution system, if the VTD solution satisfies $\hat{C}_{0}=0$ (i.e. $\tilde{v}^{2}=\tilde{w}^{2}).$
Since the coefficients of the equations 6.8, 6.9 are constructed from solutions of the evolution system we may use the expansions and estimates derived in previous sections. In particular we calculate $$e^{\lambda}\tau\equiv\tilde{v}+e^{-\varepsilon_{k}t}u_{k,a}^{a}.$$ Equation 6.8 can therefore be written as the following equation of Fuchsian type: $$\partial_{t}(e^{2\lambda}C_{0})-4\tilde{v}e^{2\lambda}C_{0}e^{2\lambda}C_{0}=e^{-\varepsilon_{k}t}u_{k,a}^{a}e^{2\lambda}C_{0}+e^{\lambda}\sigma
^{ab}\nabla_{a}(e^{\lambda}C_{b})+\sigma^{ab}e^{\lambda}\partial_{a}\lambda
e^{\lambda}C_{a}.$$
Equation 6.9 is not a priori in Fuchsian form for the pair ($e^{\lambda}C_{a},e^{2\lambda}C_{0})$ in spite of the identity 6.10. However if we use the identity $$e^{2\lambda}C_{0}\equiv-\frac{1}{2}\{e^{2\lambda}R(g)-e^{2\lambda
}k.k+e^{2\lambda}\tau^{2}-\partial_{t}\Phi.\partial_{t}\Phi\}$$ and the property $\hat{C}_{0}=0$ together with the expression for $R(g)$ given in 5.9 we can show that there exists a number $\mu>0$ and a bounded function $F(x,t)$ such that we have $$|e^{2\lambda}C_{0}|\leq e^{-\mu t}F(x,t)\text{ \ and \ }|\partial
_{a}(e^{2\lambda}C_{0})|\leq e^{-\mu t}F(x,t).$$ It follows that 6.9 takes Fuchsian form.
A solution of the evolution system satisfies the full Einstein wave map equations if and only if the half polarized asymptotic data satisfies the condition $\tilde{w}=\tilde{v},$ and also $$\tilde{\Theta}=1\text{ \ \ and \ }\tilde{v}e^{-\tilde{\lambda}+2\tilde{\phi}}=constant.$$
To complete the proof that $C_{0}=C_{a}=0$ it suffices to show that $e^{2\lambda}C_{0}$ and $e^{\lambda}C_{a}$ tend to zero at infinity. We have already checked that this is true for $e^{2\lambda}C_{0},$ as long as $\tilde{w}=\tilde{v};$ i.e. $\hat{C}_{0}=0.$
We now study the asymptotic behaviour of $e^{\lambda}C_{a}.$ If we denote by $\delta u$ the difference between a field $u$ and its VTD value, we calculate (recall that $\lambda=\hat{\lambda},\hat{\sigma}=\tilde{\sigma})$ $$e^{\lambda}(C_{a}-\hat{C}_{a})\equiv e^{\lambda}\{(\nabla_{b}-\tilde{\nabla
}_{b})k_{a}^{b}+\tilde{\nabla}_{b}\delta k_{a}^{b}-\partial_{a}\delta
\tau\}+\delta(\Phi_{t}.\Phi_{a})$$ with $$\delta(\Phi_{t}.\Phi_{a})\equiv2\phi_{t}\delta\phi_{a}+2\hat{\phi}_{a}\delta\phi_{t}+\frac{1}{2}e^{-2\phi}\omega_{t}\delta(e^{-2\phi}\omega
_{a})+e^{-2\hat{\phi}}\hat{\omega}_{a}\delta(e^{-2\phi}\omega_{t}).$$ We see that, in the half polarized case, the Fuchsian expansions imply that $e^{\lambda}(C_{a}-\hat{C}_{a})$ tends to zero as $t$ tends to infinity. Using the expressions for $\ \hat{k}_{a}^{b}$ and $\hat{\lambda},$ we see that $e^{\hat{\lambda}}\hat{C}_{a}$ reads: $$e^{\hat{\lambda}}\hat{C}_{a}\equiv\frac{1}{2}e^{\hat{\lambda}}\partial
_{a}(e^{-\hat{\lambda}}\tilde{v})-\hat{\Phi}_{t}.\hat{\Phi}_{a}\equiv\frac
{1}{2}(\partial_{a}\tilde{v}-\tilde{v}\partial_{a}\tilde{\lambda}+\tilde
{v}\partial_{a}\tilde{v}t)-\hat{\Phi}_{t}.\hat{\Phi}_{a}$$ Using the expressions of $\hat{\lambda},\hat{\Phi}_{t},\hat{\Phi}_{a}$ and the half polarization condition, we find after some computation that $$\hat{\Phi}_{t}.\hat{\Phi}_{a}=-\tilde{w}\{cos\theta\partial_{a}\tilde{\phi
}+\frac{1}{2}\tilde{\Theta}^{-1}\partial_{a}(\tilde{\Theta}-\tilde{w}t)\}.$$ Thus we find that the terms containing $t$ disappear from $e^{\hat{\lambda}}\hat{C}_{a}$ if $\tilde{v}=\tilde{w}$ and $\tilde{\Theta}=1.$ It follows that $e^{\hat{\lambda}}\hat{C}_{a}$ tends to zero as $t$ tends to infinity (recall that $cos\theta$ tends then to $1)$ if and only if $$\frac{1}{2}[\partial_{a}\tilde{v}-\tilde{v}\partial_{a}\tilde{\lambda}]-\tilde{v}\partial_{a}\tilde{\phi}=0,$$ a condition equivalent to the hypothesis 6.14 of the theorem.
In the half polarized case the VTD solution only satisfies asymptotically the VTD momentum constraint, and only after being multiplied $e^{\hat{\lambda}}.$
**Aknowledgements.**
We are grateful to Vincent Moncrief for interesting discussions about this paper.
We thank the Kavli Institute for Theoretical Physics at Santa Barbara, L’Institut des Hautes Etudes Scientifiques at Bures-sur-Yvette and the department of mathematics of the University of Washington for providing very pleasant and stimulating environments for our collaboration on this work. This work was partially supported by the NSF, under grants PHY-0099373 and PHY-0354659 at Oregon.
**References.**
\[BM\] B. K. Berger and V. Moncrief ”Numerical evidence for an oscillatory singularity in generic $U(1)$ symmetric cosmologies on $T^{3}\times R"$ Phys. Rev. D 58 064023-1-8 (1998).
\[M86\] V. Moncrief Reduction of Einstein equations for vacuum spacetimes with U(1) spacelike isometry group, Annals of Physics 167 (1986), 118-142
\[CB-M 96\] Y. Choquet-Bruhat and V. Moncrief Existence theorem for solutions of Einstein equations with 1 parameter spacelike isometry group, Proc. Symposia in Pure Math, 59, 1996, H. Brezis and I.E. Segal ed. 67-80.
\[CBIM\] Y. Choquet-Bruhat, J. Isenberg and V.Moncrief ”Topologically general $U(1)$ symmetric Einsteinian spacetimes with AVTD behaviour” Il Nuovo Cimento B, Vol. 119, issue no. 7-9, 2005.
\[IM\] J. Isenberg and V. Moncrief, “Asymptotic behavior in polarized and half-polarized $U(1)$ symmetric spacetimes”, Class. Qtm. Grav. 19, 5361-5386 (2002).
\[KR\] S. Kichenassamy and A.D. Rendall, “Analytical description of singularities in Gowdy spacetimes”, Class. Qtm.Grav.15, 1339-1355 (1998).
\[AR\] L. Andersson and A. Rendall, “Quiescent cosmological singularities”, Comm. Math. Phys. 218, 479-511 (2001).
\[DHRW\] T. Damour, M Henneaux, A. Rendall, and M. Weaver, “Kasner-like behavior for subcritical Einstein-matter systems”, Ann. H. Poin. 3, 1049-1111 (2002).
\[IM92\] J. Isenberg and V, Moncrief, “Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes”, Ann. Phys. 99, 84-122 (1992).
\[CBM\] Y. Choquet-Bruhat and V.Moncrief, “Future complete $U(1)$ symmetric einsteinian spacetimes”, Ann. Henri Poincare, 2, 1007-1064 (2001) See also “Non linear stability of einsteinian spacetimes with $U(1)$ isometry group”, gr-qc/0302021.
\[CB\] Y. Choquet-Bruhat, “Future complete $U(1)$ symmetric einsteinian spacetimes, the unpolarized case”, in ”50 Years of the Cauchy Problem”, eds. P. Chrusciel and H. Friedrich (2004).
\[K\] S. Kichenassamy, ”Nonlinear Wave Equations” (Dekker, NY) (1996).
\*Académie des Sciences, 23 quai Conti 755270 Paris cedex 06, France
\*\*Department of Mathematics and Institute of Theoretical Science, University of Oregon, Eugene, OR 97403-5203, USA
[^1]: [Asymptotic Velocity Term Dominated]{}
[^2]: [If we choose an arbitrary harmonic 1 - form appearing in the solution to be zero.]{}
[^3]: [We discard here the special case which corresponds to the polarized case, treated elsewhere, where these circles are centered at infinity. The geodesics are then the half lines X]{}$\equiv\omega=$[constant,.]{}
[^4]: [This restriction on t comes from the covariant components of ]{}$\sigma$[ which remain bounded as long as ]{}$\sigma^{ab}$[ remains positive definite. ]{}
|
---
author:
- 'M. Revnivtsev , E. Churazov , K. Postnov , S. Tsygankov'
title: Quenching of the accretion disk strong aperiodic variability at the magnetospheric boundary
---
Introduction
============
It has been recognized since the beginning of X-ray astronomy that the flux of accreting X-ray binaries demonstrates strong aperiodic variability [see e.g. @rappaport71; @oda74]. Almost immediately after the discovery, the noise in the X-ray light curves of accreting binaries (like e.g. Cyg X-1) was explained as a superposition of randomly occurring X-ray emission flashes (shots) of similar duration (the shot noise model, @terrell72). This provided an explanation to the shape of the power density spectra (PDS, the Fourier transform of the autocorrelation function of the lightcurve of a source) of different X-ray sources.
However, the accumulation of more data posed serious questions to this paradigm. In particular, it was very hard to explain a huge range of the X-ray variability time scales observed in some sources [see e.g. @churazov01] and the linear correlation between the variability amplitude and the average flux of sources [@uttley01]. Indeed, to explain observed large variability amplitude, the individual flashes/shots in the shot noise model should be very powerful. Therefore these flashes must come from the innermost region of the accretion flow, where most of the energy is released. The characteristic time scales in this region are very short – milliseconds or tens of milliseconds for stellar-mass compact objects. However, very often, e.g. in the soft/high state of accreting binaries, the observed power spectra have a power law shape extending down to frequencies as low as $10^{-5}-10^{-6}$ Hz, i.e. 5-7 orders of magnitude longer time scales than all time scales characteristic for the region of the main energy release [see @churazov01; @gilfanov05].
A very promising model for the aperiodic X-ray variability of accreting sources is the “perturbation propagation” model [@lyubarskii97; @churazov01; @kotov01; @arevalo06]. In this model, the X-ray flux variability is caused by the variations of the instantaneous value of the mass accretion rate in the inner accretion flow. In turn, the variations of the mass accretion rate are due to the perturbations introduced to the accretion flow by the stochastic variations of the disk viscous stresses. In this model the observed variability is a multiplicative superposition of perturbations introduced at different radii. Assuming that the fractional amplitudes of the mass accretion rate perturbations are the same at all radii, the PDS of the emerging lightcurve will naturally appear as a self-similar power-law with slope $-1...-1.5$ up to the maximal frequencies that can be generated in the disk [@lyubarskii97]. Direct magneto hydrodynamic simulations of accretion flows [se e.g. @brandenburg95; @balbus99; @hirose06] provide further support to this semi-phenomenological model. In particular, these simulations show that perturbations in the instantaneous mass accretion rate generated at any given radius of the disk have characteristic time scale proportional to the local dynamical time.
This model of the aperiodic X-ray flux variability implies that the presence of the accretion disk edges, both outer and inner, should be reflected in the noise properties of the X-ray light curve. Signatures of outer edges of accretion disks were found by [@gilfanov05] in the low frequency parts of the noise power spectra of low mass X-ray binaries.
Accretion disks around compact stars in X-ray binaries should also have inner edges, which should manifest itself in the power spectrum of their X-ray light curves. Specifically, a definite break is expected to be present in the PDS at the characteristic frequency of variability generated at the inner edge of the disk. At frequencies below this break the power spectrum is expected to be produced in the accretion disk and have a self-similar slope about -1.0...-1.5 [@lyubarskii97; @churazov01; @gilfanov05; @revnivtsev06], while at higher frequencies the character of the flow changes and the PDS slope may be different.
In accreting X-ray pulsars and intermediate polars the central compact object (a neutron star or a white dwarf) has a strong magnetic field which can disrupt the disk-like accretion flow at the magnetospheric boundary, or even prevent the formation of accretion disk at all (like in polars), dividing the flow into two distinct parts – the accretion disk and the magnetospheric flow. The noise properties of these flows may be very different.
In the present paper we compare PDS of different types of accreting X-ray binaries and discuss the observational support for the qualitative picture of aperiodic variability outlined above.
Truncated accretion disks in different classes of sources
=========================================================
We shall consider several classes of accretion X-ray binaries:
- X-ray binaries with compact objects, which have magnetospheres powerful enough to disrupt the accretion disk at large distances (accreting X-ray pulsars, intermediate polars)
- X-ray binaries with large magnetospheres in which the accretion disk does not form and the accretion proceeds along the magnetosphere from the very beginning (polars)
In a fair fraction of persistent accreting X-ray pulsars the spin period of neutron star is observed to be close to synchronization (corotation) with the Keplerian rotation of the accretion disk at the magnetospheric boundary, which is explained by the standard description of the interaction of accretion disk with a magnetized neutron star [see e.g. @davidson73; @shakura75; @lipunov76; @gl79; @corbet84; @ziolkowski85].
On the contrary, due to much larger moments of inertia of white dwarfs, it takes a much longer time to bring accreting magnetic white dwarfs in cataclysmic variables into corotation with surrounding accretion disks, and their magnetospheres rotate with periods much larger than the Keplerian ones at the inner disk edge.
The perturbation propagation model makes distinct predictions for the noise properties of accretion flows in these classes of objects. Namely:
1. The noise power spectra of accreting sources with large magnetospheres and without accretion disk (polars, e.g. AM Her), in which matter is transferred directly from the companion star to the compact star via magnetospheric accretion, should be different from those of sources with accretion disks and without magnetospheres (e.g. accreting black holes like Cyg X-1 in their soft state).
2. The noise power spectra of sources with accretion disk truncated at the magnetospheric boundary should have a break corresponding to the characteristic frequency in the disk near its inner boundary.
3. If the size of the magnetosphere changes (e.g. as a response to the change in mass accretion rate like in transient pulsars), the break frequency must change correspondingly.
4. For persistent X-ray pulsars which are close to corotation, the comparison of the PDS break frequency with the compact object spin frequency can be used to determine the ratio of the characteristic frequency of perturbations generated in the disk to the Keplerian frequency.
![Power density spectra of persistent accretion powered X-ray pulsars. The frequency shown along the X-axis is expressed in units of the compact object spin frequency. The power density spectra are multiplied by frequency to show the square of the fractional RMS per decade of frequency. For comparison with the observed profiles, the thick dashed curve shows an analytical model $P\propto f^{-1}\,(1+(f/f_0)^2)^{-0.5}$.[]{data-label="powers"}](./powers_scaled_depulse.ps){width="\columnwidth"}
Breaks in the power spectra spectra of magnetized accretors
-----------------------------------------------------------
It has been noticed in earlier studies [e.g. @hoshino93] that noise power spectra of accreting X-ray pulsars typically have breaks around the pulse frequency. This is shown more clearly in Fig.\[powers\], where we plot the noise power spectra of several persistent X-ray pulsars, including Cen X-3, 4U 1626-67, Vela X-1, 4U 1538-52, GX 301-2, X Persei, GX 1+4, and of the magnetic white dwarf (polar) AM Her. The frequency scale of their power spectra is normalized through multiplication by the pulse period of the sources. The variability, associated with regular pulsation was removed from the original X-ray light curves by subtracting folded segments of light curves with a duration of 10-20 spin periods . Strictly speaking this procedure does not remove the contribution of regular pulsations completely since often the periodic and aperiodic variabilities show signs of nonlinear interactions (see e.g. the discussion in @tsygankov07); however, it is good enough as the first approximation (see e.g. @finger96).
It is seen that [*all power spectra of accreting X-ray pulsars show clear breaks approximately at their spin frequency*]{}. PDSs of all sources have a similar power-law slope above the break frequency irrespective of the PDS form below it.
In the framework of the perturbation propagation model (see e.g. @lyubarskii97 [@churazov01]), we can interpret this observational fact as a signature of the truncation of the accretion disk flow and its conversion into a magnetospheric flow.
The proximity of the PDS break frequency to the spin frequency (which in the case of corotating systems is close to the frequency of the Keplerian rotation around the compact object at the inner edge of the accretion disk) allows us to conclude that [*the characteristic time scale of variability produced at some distance from the central compact objects are close to the local Keplerian time scale.*]{}
Fig. \[powers\] shows that above the break frequency the power spectrum of the flux variability typically is a power law with the slope close to $\sim -2$. We can not exclude that such slope of the flux variability is a property of the magnetospheric accretion, however, we note that similar slopes of power spectra sometimes observed in accreting systems without magnetospheric accretion, for example unmagnetized cataclysmic variables [see e.g. @kraicheva99; @pandel03].
Change of the magnetospheric size with mass accretion rate
----------------------------------------------------------
If the break frequency in the noise power spectra of accreting X-ray pulsars indeed reflects the time scale of the noise generation at the inner boundary of the accretion disk/flow, its value should depend on the mass accretion rate in the binary system. Increase in the mass accretion rate decreases the size of the magnetosphere (and hence the inner radius of the disk) and brings the system off the corotation, so the characteristic frequency at the inner edge of the disk/flow increase. A similar situation should be observed in the case of luminous intermediate polars (accreting magnetized white dwarfs with moderate magnetospheres) which typically are always out of the corotation.
We can verify this hypothesis by examining power spectra of bright transient X-ray pulsars (like A0535+26, 4U0115+63, V0332+53, KS 1947+300), which demonstrate wide range of X-ray luminosities during the outbursts, and power spectra of luminous intermediate polars (e.g. V1223 Sgr).
In Fig.\[dif0535\] we show examples of power spectra of the pulsar A0535+26 during its bright outburst in 2005 and the power spectrum of V1223 Sgr. The power spectra of A0535+26 were averaged over two time intervals: a) with small X-ray luminosity ($L_{\rm x}\sim10^{36}$ erg/s) (low mass accretion rate) and b) with higher luminosity ($L_{\rm x}\sim10^{37}$ erg/s) (higher mass accretion rate). The power spectrum of V1223 Sgr is averaged over all publicly available RXTE observations.
From Fig.\[dif0535\] it is clear that the power spectra of A0535+26 in different luminosity states differ significantly at the frequencies higher than the pulse frequency, while at lower frequencies the PDSs are almost identical. The difference at high frequencies is essentially an addition of an extra noise component, which is presumably generated in the ring of accretion disk between the radii corresponding to the magnetosphere size in the high accretion rate (small radius) and low accretion rate (large radius). This ring and associated variability were absent in the state with low accretion rate.
The noise power spectrum of V1223 Sgr closely resembles that of A0535+26 in the bright state. This similarity reflects the fact that both systems are out of corotation with their accretion disks – the magnetospheres are squeezed by the increased accretion flow, and the inner parts of the accretion disks rotate much faster than the central object.
![Power density spectra of accreting X-ray pulsar A0535+26 at low accretion rate (labeled as “faint”) and in the strong spin-up regime during the outburst (labeled as “bright”). The upper curves shows the power spectrum of luminous intermediate polar V1223 Sgr[]{data-label="dif0535"}](./p0535_dif_v1223.ps){width="\columnwidth"}
If the characteristic frequency $f_0$ of the noise at the magnetospheric boundary $R_{\rm m}$ is proportional to the frequency of the Keplerian rotation $\nu_{\rm K}$ of matter at the inner edge of the accretion disk $R_{\rm in}\approx R_{\rm m}$, we can relate the observed break frequency to the instantaneous value of the mass accretion rate $\dot{M}$ (see e.g. @pringle72 [@lamb73; @davidson73; @bildsten97]): $$2\pi \nu_{\rm K} = (GM)^{1/2} R_{\rm m}^{-3/2}$$ where $$R_{\rm m}\approx \mu^{4/7} (GM)^{-1/7} \dot{M}^{-2/7}$$ is the standard expression for the magnetospheric radius, $\mu$ is the dipole magnetic moment of the neutron star. Therefore we can anticipate that the break frequency will follow the dependence: $$f_{\rm b}\propto\nu_{\rm K}\propto (GM)^{10/14} \mu^{-6/7} \dot{M}^{3/7}
\label{fb}$$
![Dependence of the break frequency in the noise power spectrum of A0535+26 on the 3-20 keV X-ray flux (filled circles). The dependence of the QPO frequency observed by [@finger96] during the source outburst in 1994 is shown by open dotted circles. The solid circles show the results of [@finger96] when the QPO frequency is recalculated to the break frequency as $f_{break}=2.5\times f_{QPO}$ (see Fig. \[supportingfigures\], the upper panel). The dashed line shows the prediction of the simplest “magnetospheric” model of the break frequency $f_{\rm b}\propto {L_{\rm x}}^{3/7}$ described in the text. The dotted line shows the neutron star spin frequency.[]{data-label="correlation"}](./correlation_0535.ps){width="\columnwidth"}
It is exactly what we see during the evolution of the outburst of A0535+262 observed by RXTE in 2005 (we have substituted the X-ray luminosity $L_{\rm x}\simeq
0.1 \dot M c^2$ instead of the mass accretion rate $\dot{M}$ into Eq. (\[fb\])). The dependence of the break frequency ($f_b$ in Eq. (\[fb\])) on the X-ray flux of A0535+262 is shown in Fig. \[correlation\] (filled circles). For this this plot we have used all observations of RXTE of this outburst after $\sim$MJD 53613, when the stable regime of accretion was established (see @caballero08 [@postnov08]).
It is interesting to note that the dependence $f\propto L_{\rm x}^{3/7}$ was previously established for the centroid frequency of quasi-periodic X-ray oscillations in the power spectra of A0535+26 detected during its giant outburst in 1994 [@finger96]. The authors argued that the oscillations originate at the inner boundary of the accretion disk and are related either to the Keplerian rotation at the inner edge of the disk or to the beat frequency between the Keplerian rotation and that of the neutron star magnetosphere [e.g. @alpar85].
Actually we can try to combine our present RXTE measurements with QPO studies by [@finger96] making use of two facts:
- when low-frequency QPOs are observed in power spectra of accreting pulsars, their centroid frequency is related to the break frequency as $\sim$1:2.5 (Fig.\[supportingfigures\], upper panel, see also [@angelini89] for QPO in EXO 2030+375). The correlation of the QPO and the break frequencies is also observed in LMXBs, although in these objects the centroid of the QPO feature is usually above the break frequency [see e.g. @wijnands99].
- during RXTE observations of the source outburst in 2005, the X-ray flux of the A0535+26 in the energy band 20-100 keV (the range where measurements by @finger96 were done) is approximately factor of 0.55 of the X-ray flux in the energy band 3-20 keV (see the RXTE spectrum shown in the lower panel of Fig.\[supportingfigures\]).
The dependence of the QPO frequency on the X-ray flux during its 1994 outburst, renormalized using the above factors, is shown in Fig. \[correlation\] (solid open circles). The renormalized dependence (solid open circles) perfectly continues the observed break frequency-flux dependence (filled circles) and corresponds to Eq. (\[fb\]) (the dashed line).
Prominent QPO features similar to those detected by [@finger96] are not always observed in the power spectra of accreting X-ray pulsars. On the other hand a break in the PDS is more ubiquitous and therefore the diagnostics of the accretion flow based on the break frequency can be applied to larger datasets. For example, the break frequency in the noise power spectrum can be used as an estimate of the dipole magnetic moment of compact objects using Eq. (\[fb\]).
Conclusions
===========
We studied aperiodic variability of the X-ray flux from accreting binaries, in which the truncation of the disk-like accretion flow by the magnetosphere of the compact object is important. The results can be summarized as follows:
- There is a distinct break in the Power Density Spectra of accreting magnetized neutron stars and white dwarfs, apparently associated with the change of the disk-like accretion flow to the magnetospheric flow near the Alfvenic surface.
- In transient systems with variable X-ray luminosity the PDS break frequency $f_b$ changes with the X-ray luminosity (mass accretion rate) as $f_b\propto \L_x^{3/7}$, in agreement with the standard theory of accretion onto magnetized compact stars.
- This break can naturally be explained in the “perturbation propagation” model, which assumes that at any given radius in the accretion disk stochastic perturbations at frequencies characteristic for this radius are introduced to the flow. These perturbations are then advected by the flow to the region of main energy release leading to a self-similar form of the PDS $P\propto f^{-1...-1.5}$. The break in the PDS corresponds to a frequency characteristic for the accretion disk truncation radius (the magnetospheric radius).
- We suggest that the PDS break frequency is directly related to the magnetospheric radius for a given value of the mass accretion rate and can be used to estimate the magnetic moment of accreting compact stars.
- For systems which are close to corotation (accreting X-ray pulsars) the PDS break frequency is close to the spin frequency of the neutron star. This strongly suggests that the characteristic frequency of perturbations introduced to the accretion flow in the disk is of order of the local Keplerian frequency.
- In all studied objects the PDS above the break frequency follows the $P\sim f^{-2}$ law over a broad range of frequencies, suggesting that strong $f^{-1...-1.5}$ aperiodic variability which is ubiquitous in accretion disks is not characteristic for magnetospheric flows.
The authors thank the anonymous referee for useful comments. Authors thank Marat Gilfanov for useful discussions. This research made use of data obtained from the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. KP thanks the MPA for hospitality. This work was supported by DFG-Schwerpunktprogramme (SPP 1177), grants of Russian Foundation of Basic Research 07-02-01051, 07-02-00961-a, 08-08-13734, NSh-5579.2008.2 and the RAS program “The origin and evolution of stars and galaxies” (P04).
Alpar M. A., Shaham J., 1985, Natur, 316, 239 Angelini L., Stella L., Parmar A. N., 1989, ApJ, 346, 906 Ar[é]{}valo P., Uttley P., 2006, MNRAS, 367, 801 Balbus S. A., Papaloizou J. C. B., 1999, ApJ, 521, 650 Bildsten L., et al., 1997, ApJS, 113, 367 Bradt H. V., Rothschild R. E., Swank J. H., 1993, A&AS, 97, 355 Brandenburg A., Nordlund A., Stein R. F., Torkelsson U., 1995, ApJ, 446, 741 Caballero I., et al., 2008, A&A, 480, L17 Churazov E., Gilfanov M., Revnivtsev M., 2001, MNRAS, 321, 759 Corbet R. H. D., 1984, A&A, 141, 91 Davidson K., Ostriker J. P., 1973, ApJ, 179, 585 Finger M. H., Wilson R. B., Harmon B. A., 1996, ApJ, 459, 288 Ghosh P., Lamb F. K., 1979, ApJ, 234, 296 Gilfanov M., Arefiev V., 2005, astro, arXiv:astro-ph/0501215 Hirose S., Krolik J. H., Stone J. M., 2006, ApJ, 640, 901 Hoshino M., Takeshima T., 1993, ApJ, 411, L79 Kotov O., Churazov E., Gilfanov M., 2001, MNRAS, 327, 799 Kraicheva Z., Stanishev V., Genkov V., Iliev L., 1999, A&A, 351, 607 Lamb F. K., Pethick C. J., Pines D., 1973, ApJ, 184, 271 Lipunov V. M., Shakura N. I., 1976, SvAL, 2, 133 Lyubarskii Y. E., 1997, MNRAS, 292, 679 Matthaeus W.H, Klein L.W., Ghosh S., Brown M.R., 1991, J. Geophys. Res., 96, 5421 Oda M., Takagishi K., Matsuoka M., Miyamoto S., Ogawara Y., 1974, PASJ, 26, 303 Pandel D., C[ó]{}rdova F. A., Howell S. B., 2003, MNRAS, 346, 1231 Postnov K., Staubert R., Santangelo A., Klochkov D., Kretschmar P., Caballero I., 2008, A&A, 480, L21 Pringle J. E., Rees M. J., 1972, A&A, 21, 1 Rappaport S., Doxsey R., Zaumen W., 1971, ApJ, 168, L43 Revnivtsev M., et al., 2006, A&A, 447, 545 Shakura N. I., 1975, SvAL, 1, 223 Terrell N. J. J., 1972, ApJ, 174, L35 Tsygankov S. S., Lutovinov A. A., Churazov E. M., Sunyaev R. A., 2007, AstL, 33, 368 Uttley P., McHardy I. M., 2001, MNRAS, 323, L26 Wijnands R., van der Klis M., 1999, ApJ, 514, 939 Ziolkowski J., 1985, AcA, 35, 185
|
---
abstract: 'Drug repositioning (DR) refers to identification of novel indications for the approved drugs. The requirement of huge investment of time as well as money and risk of failure in clinical trials have led to surge in interest in drug repositioning. DR exploits two major aspects associated with drugs and diseases: existence of similarity among drugs and among diseases due to their shared involved genes or pathways or common biological effects. Existing methods of identifying drug-disease association majorly rely on the information available in the structured databases only. On the other hand, abundant information available in form of free texts in biomedical research articles are not being fully exploited. Word-embedding or obtaining vector representation of words from a large corpora of free texts using neural network methods have been shown to give significant performance for several natural language processing tasks. In this work we propose a novel way of representation learning to obtain features of drugs and diseases by combining complementary information available in unstructured texts and structured datasets. Next we use matrix completion approach on these feature vectors to learn projection matrix between drug and disease vector spaces. The proposed method has shown competitive performance with state-of-the-art methods. Further, the case studies on Alzheimer’s and Hypertension diseases have shown that the predicted associations are matching with the existing knowledge.'
author:
-
-
bibliography:
- 'references.bib'
title: Representation learning of drug and disease terms for drug repositioning
---
Representation Learning, Vector Representation, Drug repositioning, Word vector, Heterogeneous Inference
Introduction
============
Development of new drugs is associated with huge investment of time and money, and risk of failure in clinical trials. It has been estimated that on an average, drug development process takes 15 years [@dimasi2001new] and associated cost is approximately \$1 billion [@adams2006estimating]. Finding novel indications for approved drugs, referred as [*drug repositioning*]{} or [*drug repurposing*]{}(DR), has attracted researchers and pharmaceutical industry as a cost-effective and faster alternative to overcome this challenge [@hurle2013computational]. The candidates for drug repositioning are drugs which are already in market or which have been discontinued due to various reasons other than safety issues. As per the estimate in [@hurle2013computational], DR allows a significant reduction in time from 10-17 years to 3-12 years in novel drug discovery. According to [@jin2014toward], among all the drugs which have been approved by the US Food and Drug Administration (FDA), approximately $30\%$ of them were the result of drug repositioning. Significant examples of drug repositioning includes [*Aspirin*]{} (regular use as analgesic and now also being widely adapted to treat heart related disease [@wolff2009aspirin]), [*Plerixafor*]{} (initially developed to treat HIV but later being used as a drug to mobilize stem cells [@flomenberg2005use]), and [*Thalidomide*]{} (initially developed to treat nausea but after drug repositioning research, being used to treat dermatological issues and the myelome disease [@calabrese2000thalidomide]).
There have been significant number of methods developed for the drug repositioning problem including machine learning methods. We summarize the prominent methods in the section \[sec:relwork\]. Working principle of all methods rely on two important aspects related to drugs and diseases. First, drugs often bind to multiple targets resulting into various biological effects including side-effects [@pujol2010unveiling]. Second, a biological target of a drug which is relevant to a particular disease, may also be directly or indirectly associated with other diseases. In other words, overlapping pathways or common associated targets between various diseases are important factors and thereby making it possible that an approved drug for one disease may be useful in treating a similar disease [@sardana2011drug].
Existing methods of identifying drug-disease association majorly rely on the information available in the structured databases only. However these databases are unable to keep pace with the exponential growth of information appearing in research articles. In this paper, our primary aim is to develop a method which can exploit information present in free texts as well as in structured databases. In recent years, vector representation of words, learned using neural network based methods from a large corpora of free texts, have been shown to give significant performance for several natural language processing tasks. Word vectors thus obtained are also shown to capture syntactic and semantic properties. We employ a novel method to learn representation of each drug and disease terms which then are projected on a common vector space to obtain similarity between drugs and diseases. Towards this end, first we learn vector representation of drugs and diseases by using the knowledge present in literature. Next, these vectors are updated to accommodate various similarity measures of drugs and diseases respectively. The resultant drug and disease vector representation are not necessarily in the same vector-space. So we employ matrix completion approach [@yu2014largescal] to learn a projection matrix between drug and disease vector space. We evaluate the performance of our method using ten fold cross validation and top k rank threshold methods and compare it with 3 other competitive methods. We further perform case studies on Alzheimer’s disease and Hypertension and verify our predictions for these diseases from literature. Our study shows that all our top ten drugs predicted for Alzheimer’s disease are approved to treat neurodegenerative diseases. Similarly 7 out 10 drugs predicted for Hypertension are approved and 2 out of remaining 3 are used to treat Ocular Hypertension.
Materials and Methods
=====================
In this study we describe the datasets used. Later we explain our method which includes learning the feature vector and learning the projection matrix between drug and disease vector space.
Dataset {#sec:dataset}
-------
This section discusses all datasets and their sources used in this work.
1. [**Drug-Disease Association Data-**]{}
We use the same drug-disease association data as used in PREDICT [@gottlieb2011predict]. Data is made available as the supplementary material of the corresponding paper [@gottlieb2011predict]. It contains $1933$ drug-disease associations between $593$ drugs and $313$ diseases. All $593$ drugs are registered in DrugBank [@DrugBank] and all $313$ diseases are listed in OMIM [@Omim] database. As we have mentioned earlier that the proposed method rely on obtaining word embedding for each drug and disease from a huge corpus of biomedical articles, we discarded some drugs and diseases which we were not present in the corpus. Finally we consider $584$ drugs, $294$ diseases and $1854$ drug-disease associations known between them.
2. [**SIDER: Drug Side Effect Data-**]{} We obtain the list of side effects corresponding to all $584$ drugs from the SIDER [@sider] database.
3. [**Chemical Fingerprint of Drugs-**]{} Chemical fingerprint of a drug corresponds to the record of component fragment present in their chemical structure. For each drug their chemical fingerprint was obtained from the DrugBank [@DrugBank].
4. [**DrugBank-**]{} Similar to the chemical fingerprint of drugs, drug targets are obtained from the DrugBank. Drug targets one or more cellular molecules such as metabolites or proteins for desired effects. A list of targets corresponding to all $584$ drugs are obtained.
5. [**DisGeNet: Disease Associated Genes-**]{} We collect genes associated with disease from DisGeNET [@pinero2015disgenet].
Construction of similarity measures
-----------------------------------
We calculate three types of similarity for drugs which are based upon side-effects, chemical structure and target proteins. Two similarities are calculated for each disease pair based upon the disease phenotypes and associated genes.
### Drug Similarity measures
1. Side effect similarity : A side effect is an undesired consequence of a drug. Drugs cause side-effects when they bind to off-target apart from their desired on-targets. Under the assumption that if 2 drugs share side-effects and hence off-targets, there is a possibility that they might share on-targets which can be used to cure diseases. Studies [@campillos2008drug] show that drugs sharing off targets might also share on targets. For each drug pair ($d_i,d_j$), this similarity is : $$Sim(d_i,d_j) = \frac{|SE(d_i) \cap SE(d_j)|}{|SE(d_i) \cup SE(d_j)|}$$ where $SE(d)$ is the set of side-effects related to drug $d$.
2. Chemical Similarity : Similarity of two chemicals is based upon comparing their chemical fingerprint. A fingerprint is a record of component fragment present in a chemical structure. It has been shown in [@bajusz2015tanimoto] that [*Tanimoto coefficient*]{} can be an effective measure to calculate similarity between two chemicals based on their structures. Pairwise similarity between two drugs was calculated as Tanimoto score of their fingerprint using RDKit [@Rdkit] library of Python.
3. Drug-Target Similarity : A biological target is the protein in the body which is either up regulated or down regulated due to the action of a particular drug on it . If two drugs share same targets, the probability of them causing the similar effect may also increase. Pairwise drug-target similarity between drugs $d_i$ and $d_j$ is calculated as :
@size[8]{}@mathfonts @@@\#1$$\begin{aligned}
Sim(d_i,d_j)=\frac{1}{|P(d_i)||P(d_j)| } \sum_{i=1}^{P(d_i)}\sum_{j=1}^{P(d_j)} SW(P(d_i),P(d_j)) \end{aligned}$$ where $P(d)$ denotes the set of genes associated to drug $d$ and $SW$ is the Smith-Waterman Sequence alignment score [@smith1985statistical].
### Disease Similarity measures
1. Phenotypic similarity : A phenotypic feature is an observable biological or clinical characteristic of a disease. It is a amalgamation of gene expression as well as influence of external environmental factors. The similarity is collected from MIMMiner Tool [@van2006text]. The tool measures disease similarity by computing similarity between MeSH terms [@lipscomb2000medical] that appear in the medical description of diseases in the OMIM database.
2. Gene Similarity : Disease causing or associated genes are collected from DisGeNET [@pinero2015disgenet]. Pairwise gene similarity between disease $d_i$ and $d_j$ is calculated as: @size[8]{}@mathfonts @@@\#1$$\begin{aligned}
Sim(d_i,d_j) = \frac{1}{|P(d_i)||P(d_j)| }\sum_{i=1}^{P(d_i)}\sum_{j=1}^{P(d_j)}SW(P(d_i),P(d_j))\end{aligned}$$
where $P(d)$ denotes the set of genes associated to disease $d$ and $SW$ is the Smith-Waterman Sequence alignment score [@smith1985statistical].
Method {#sec:method}
------
The proposed method has three major steps. In the first step we obtain vector representation of drugs and diseases using neural embedding method [@mikolov2013distributed]. We update these representations using similarity scores calculated from the various structured datasets. And in the last step, we learn a projection matrix between the two vector-spaces so that a final association score between drug-disease pair can be obtained. It is noteworthy to mention here again that there is no requirement of negative datasets. Figure \[fig:method\] summarizes the proposed method.
![Diagram depicting the flow of our method. First, the drug and disease feature vector are learned. To do this, drug word vector and disease word vector are updated using the similarity measures. Second, the projection matrix is learned by using the well known associations and our drug/disease feature vectors. []{data-label="fig:method"}](figures/diagram)
### Word vectors for drugs and diseases
To capture the information present in literature, we obtain the word vector representation of drugs and diseases. We use Pubmed [@pubmed] open access set as our corpus. Each disease is mapped to its OMIM id. As diseases can appear under various names in Pubmed [@pubmed] corpus, each disease in the corpus is mapped to a Concept Unique Identifier(CUI) by using UMLS Meta thesaurus [@umls]. If a disease (OMIM indication) has multiple concept names associated to it, then the resultant vector is taken as the simple average of all the vectors associated to that OMIM indication. The concept names for each OMIM indication is obtained from Supplementary Information of PREDICT [@gottlieb2011predict]. Word vector representation of each drug and disease is obtained by training Pubmed Corpus using word2Vec [@word2vec] Python library. To train vectors, we set window size to 5. We have experimented using various vector dimensions ranging from 100 to 200.
### Learning vector representation by combining similarity measures
Let $N_d$ be the number of drugs and $N_s$ be the number of diseases. Each drug word vector is denoted as $d_i \in \mathcal{R}^N$, where $i$ ranges from $1$ to $N_d$. Each disease word vector is denoted as $s_i \in \mathcal{R}^N$, where $i$ ranges from $1$ to $N_s$. Let the updated drug vector (feature vector) for i^th^ drug be denoted as $\tilde{d_i}$, which is initialized to $d_i$. Let the updated disease vector (feature vector) for i^th^ disease be denoted as $\tilde{s_i}$, which is initialized to $s_i$. Let $Sim_k(i,j)$ denote the k^th^ similarity between drug i and drug j or disease i and disease j.
Let $M$ be the number of drug similarity measures and $L$ be the number of disease similarity measures. The motive is to obtain a feature vector for each drug and disease by combining the above mentioned similarities and updating the word vectors. For each drug $i$ ( $i$ varies from 1 to $N_d$), $\tilde{d_i}$ is updated when the below objective ( $J_1$ ) is minimized: $$J_1=\sum_{j=1}^{N_d}\sum_{k=1}^{M}(\frac{\tilde{d_i} . d_j} { |\tilde{d_i}||d_j| } - Sim_k(i,j))^2$$ where $|d_i|$ denote the length of the vector $d_i$ .
Each drug word vector is updated using all the other drug vectors and for each similarity measure. The updated set of drug vectors (called feature vectors) is denoted as $D=[\tilde{d_1},\tilde{d_2},...,\tilde{d_{N_d}}] $, where each $\tilde{d_i}\in \mathcal{R}^N$.
Similar kind of objective ( $J_2$ ) is minimized for all disease $\tilde{s_i}$, where $i$ varies from 1 to $N_s$ . $$J_2=\sum_{j=1}^{N_s}\sum_{k=1}^{L}(\frac{\tilde{s_i} . s_j} { |\tilde{s_i}||s_j| } - Sim_k(i,j))^2$$ where $|s_i|$ denote the length of the vector $s_i$.
Each disease word vector is updated using all the other disease vectors and for each similarity measure. The updated set of disease vectors (called feature vectors) is denoted as $S=[\tilde{s_1},\tilde{s_2},...,\tilde{s_{N_s}} ] $, where each $\tilde{s_i}\in \mathcal{R}^N$.
The optimization problem is solved using Theano [@theano] library of Python. We have obtained a drug vector space and a disease vector space where each vector is of dimension $\mathcal{R}^N$.
### Learning projection from drug vector space to disease vector space
Our motive is to learn a projection matrix from drug vector space to disease vector space which will help us in predicting drug-disease association scores. The projection matrix should be such that the projected drug vectors are geometrically close to vectors of their well known disease vectors. The drugs that are in proximity in the directions of their feature vectors may share diseases and vice-versa. Let $I \in \mathcal{R}^{N_d \times N_s}$ be called the association matrix where $I_{ij}$ is 1 if $drug_i$ treats $disease_j$ else 0. The projection matrix is denoted as Z $\in\mathcal{R}^{N\times N}$ .
To learn this projection matrix we use inductive matrix completion approach [@yu2014largescal][@natarajan2014inductive] which minimizes the following objective function: $$\min\limits_{G,H} \sum_{i,j} ||I_{ij} - \tilde{d_i}GH^T \tilde{s_j}^ T ||^2 + \frac{\lambda}{2} ( ||G||^2 + ||H||^2 )$$ where the projection matrix $Z = GH^T$, where $G\in \mathcal{R}^{N \times K}$ and $H \in \mathcal{R}^{N \times K}$. The score of a drug $i$ and disease $j$ pair is calculated as: $$score(i,j) = \tilde{d_i}Z\tilde{s_j}^T$$ Higher the score, greater is the possibility of drug i treating disease j .
Experiments
===========
We conduct 10-fold cross-validation experiments to evaluate the performance of all methods. We use AUC, ROC and top-rank thresholds as evaluation metrics. In the top-rank threshold measure, a well known drug-disease association is considered as correctly predicted if its rank based on the predicted score is within the specified rank threshold.
Baseline Methods
----------------
We compare the proposed method with three other methods, MBIRW [@luo2016drug], HGBI [@wang2013drugHET] and TP-NRWRH [@liu2016inferring]. We briefly summarize each of the three methods for the sake of completeness.
The HGBI method creates a heterogeneous network of two different type of nodes. One set of nodes are representing different drugs and another set of nodes represent targets. Edges exist between within same node types as well as between two different node types. Existence of edge depends on drug-drug similarities, target-target similarities and drug-target interactions. The edge weights of the network are updated in an iterative fashion by incorporating all the paths between the drug-target pair.
The MBIRW method constructs two separate networks on drugs and diseases. Both similarity networks were created using novel similarity measure which takes into account correlation between different similarities. Further MBIRW performs bi-directional random walk on these two networks to get scores for drug disease associations.
TP-NRWH again uses random walk method but on single heterogeneous drug-disease network. This network is similar to the network used in the the HGBI method and integrates all the similarity measures (drug-drug and disease-disease) and well known drug-disease associations. This is in contrast to the MBIRW method which creates two separate networks on drugs and diseases.
We use default parameter settings for all the three methods. Parameters of TP-NRWRH are set as ($\alpha=0.3,\lambda=0.8,\eta=0.4$). For MBIRW $\alpha$ is set to default 0.3 and max iterations for right and left random walk is set to 2. For HGBI, restart probability $\alpha$ is set to default 0.4 and cut off was set to 0.3.
Results
=======
Vector representation
---------------------
First we analyze the performance of our method with respect to varying length of feature vectors between 100 and 200. Fig \[fig:dim\_vector\] shows AUC obtained by using different size of drugs and disease vectors. Although increasing dimension generally led to improved AUC score but improvement was not really significant. Next we analyze the importance of updating the word vectors based on similarity scores. We obtain an AUC score of $0.77$ when word-vectors obtained using word2vec method on biomedical copora are not updated. On the other hand an AUC score of $0.86$ is obtained when updated word-vectors are used. The relative improvement of $10\%$ clearly indicates that the vectors learned through our method captured the similarity of drugs and diseases in better manner.
![Performance of our method in terms of AUC with respect to different dimensionality of feature vector of drugs and diseases. []{data-label="fig:dim_vector"}](figures/vectordimauc)
Comparison with existing methods
--------------------------------
Fig \[fig:roc\_comparison\] shows the AUC values and the ROC of 10-fold cross-validation experiments. Although the proposed method has obtained AUC value of $0.86$ which is better than the one obtained by HGBI ($0.79$) but the other two methods were best performing methods.
![Diagram depicting the Receiver operating characteristics of our method and 3 other competitive methods. The AUC values are also mentioned.[]{data-label="fig:roc_comparison"}](figures/ROC)
Similar observations are made based on the top-rank threshold metric. The number of correctly predicted associations by our method is greater than that of HGBI for every top rank thresholds as shown in the Fig \[fig:toprank\].
![Diagram depicting the number of correctly predicted associations with respect to five different top-ranked thresholds.[]{data-label="fig:toprank"}](figures/topkrank)
Case Studies
------------
After finding the performance of our model, we conducted leave-disease-out experiment. For this, first we select a disease and train our model only with the remaining data after excluding all known associations related to it. Then scores are calculated for the held out disease and top scoring drugs are reported. We perform the case studies on two diseases, Alzheimer and Hypertension.
1. Alzheimer’s Disease: Table \[tab:alzheimer\] shows the top scoring drugs predicted by the model for Alzheimer’s disease. Out of the top 10 drugs predicted by our method, 6 drugs namely [*Rivastigmine*]{}, [*Galantamine*]{}, [*Donepezil*]{}, [*Memantine*]{}, [*Tacrine*]{} and [*Valproic Acid*]{} have been approved for Alzheimer’s disease. Other drugs namely [*Ropinirole*]{}, [*Entacapone*]{}, [*Pramipexole*]{} and [*Carbidopa*]{} have been used to treat Parkinson’s disease. Although there is a difference in pathogenesis of Parkinson’s disease and Alzheimer’s disease, but both of them are neuro-degenerative disease associated with aging [@nussbaum2003alzheimer]. Pramipexole has been under Phase 2 of Clinical trials (NCT01388478) [@ClinicalTrials] for the treatment of Alzheimer’s disease. The above results have been verified from PREDICT [@gottlieb2011predict] and DrugBank [@DrugBank].
[ccccc]{} Rank & Drug Name &
-----------
Predicted
Score
-----------
: TOP 10 PREDICTED DRUGS FOR ALZHEIMER’S DISEASE BY OUR METHOD[]{data-label="tab:alzheimer"}
&
----------
Clinical
Evidence
----------
: TOP 10 PREDICTED DRUGS FOR ALZHEIMER’S DISEASE BY OUR METHOD[]{data-label="tab:alzheimer"}
& Mean Score\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
2. Hypertension: Another indication, Hypertension was analysed and its top scoring drugs are shown in Table \[tab:hypertension\]. Out of the top 10 drugs predicted by our method, 7 drugs namely [*Trandolapril*]{}, [*Prazosin*]{}, [*Mecamylamine*]{}, [*Labetalol*]{}, [*Captopril*]{}, [*Losartan*]{} and [*Valsartan*]{} are approved drugs for hypertension. [*Brinzolamide*]{} and [*Travoprost*]{} have been used for treating ocular hypertension. The scores of these top predicted drugs for treating Hypertension and Alzheimer’s disease are much higher than the mean score of these drugs for all diseases.
[ccclc]{} Rank & Drug Name &
-----------
Predicted
Score
-----------
: TOP 10 PREDICTED DRUGS FOR HYPERTENSION BY OUR METHOD[]{data-label="tab:hypertension"}
&
----------
Clinical
Evidence
----------
: TOP 10 PREDICTED DRUGS FOR HYPERTENSION BY OUR METHOD[]{data-label="tab:hypertension"}
& Mean Score\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
& & & &\
Related work {#sec:relwork}
============
Several computational methods have been developed to solve the drug repositioning problem including machine learning as well as literature mining based methods. Literature mining methods mainly rely on co-occurrence of drug, disease and targets within a context [@andronis2011literature][@napolitano2013drug]. These methods generally have poor performance as they do not use any contextual semantic information and treat all types of relations between two relevant terms equivalently. On the other hand, machine learning based models have been shown to perform relatively better than co-occurrence based methods. Gottlieb et al. [@gottlieb2011predict] developed a logistic regression based model and combined various similarity measures of drugs and diseases in order to predict drug-disease associations. Wang et al. [@wang2013drug] applied Support Vector Machine model on drug chemical structures, protein sequences and disease phenotypic data in order to identify new relations between drugs and diseases. One of the major issue faced by these methods is requirement of negative data, which is not available.
Network based models are good alternative as they do not require both positive and negatively labeled data. Chiang et al. developed a network based model which predict novel indications for drugs based upon the fact that if two disease share treatment profiles, then drugs approved for one of those disease may be used to treat the other disease too. The method relied upon “guilt by association” technique [@chiang2009systematic]. Wang et al. developed a novel method which consists of three-layer heterogeneous graph model to integrate relationships between drugs, diseases and targets. Based on this model, an iterative algorithm was used in order to rank drugs for each disease. The method measures the strength between unlinked drug-target pairs by using all the paths in the network [@wang2014drug]. Martinez et al. developed the DrugNet method, a network -based prioritization method, that utilizes the information of drugs, diseases and drug targets and performs prioritization of drug-disease and disease-drug [@martinez2015drugnet] associations. Lee et al. exploited the structural properties of biological networks. They developed shared neighborhood scoring algorithm and applied it on an integrated drug-protein-disease tripartite network in order to predict new indications for drugs [@lee2012rational]. Chen et al. utilized already existing recommendation systems ProbS and Heats to give recommendation score for disease with respect to drugs [@chen2015network]. Wu et al. constructed a weighted heterogeneous network of drugs and diseases and applied clustering technique using ClusterONE [@nepusz2012detecting] algorithm to predict new drug-disease associations [@wu2013computational]. Luo et al. improved drug-drug and disease-disease similarity measures by using concepts of clustering and by performing various analysis on similarity values. Further they performed a bi-random walk on the similarity network and well known drug-disease associations and learned drug-disease association matrix [@luo2016drug].
Conclusion
==========
In this paper we have presented a novel representation learning method to obtain vector representation of drugs and diseases. These representations are then utilized to obtain association score between drug-disease pairs. The main contribution of this work is combining complementary information available in unstructured texts and structured datasets. Heterogeneous information was combined and feature vectors were learned for drugs and diseases. Prediction using updated feature vectors gave better results than using the original word vectors. Case studies on Hypertension and Alzheimer’s disease indicate that predictions made by our method can be used for biomedical research. We compared our method with existing methods on drug repositioning. Our results are fairly comparable to those methods in terms of AUC and top k rank threshold scoring mechanism.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Shubhakar Reddy (Former bachelor’s student at Indian Institute of Technology Guwahati) for providing us the disease to CUI mapping tool developed by him.
[John Doe]{}
|
---
abstract: |
We study the online constrained ranking problem motivated by an application to web-traffic shaping: an online stream of sessions arrive in which, within each session, we are asked to rank items. The challenge involves optimizing the ranking in each session so that local vs. global objectives are controlled: within each session one wishes to maximize a reward (local) while satisfying certain constraints over the entire set of sessions (global). A typical application of this setup is that of page optimization in a web portal. We wish to rank items so that not only is user engagement maximized in each session, but also other business constraints (such as the number of views/clicks delivered to various publishing partners) are satisfied.
We describe an online algorithm for performing this optimization. A novel element of our approach is the use of linear programming duality and connections to the celebrated Hungarian algorithm. This framework enables us to determine a set of *shadow prices* for each traffic-shaping constraint that can then be used directly in the final ranking function to assign near-optimal rankings. The (dual) linear program can be solved off-line periodically to determine the prices. At serving time these prices are used as weights to compute weighted rank-scores for the items, and the simplicity of the approach facilitates scalability to web applications. We provide rigorous theoretical guarantees for the performance of our online algorithm and validate our approach using numerical experiments on real web-traffic data from a prominent internet portal.
author:
- Parikshit Shah Akshay Soni Troy Chevalier
bibliography:
- 'sigproc.bib'
title: 'Online Ranking with Constraints: A Primal-Dual Algorithm and Applications to Web Traffic-Shaping'
---
Introduction
============
This paper investigates the online constrained ranking problem — a collection of sessions arrive in an online manner. In each session we wish to make optimal decisions that balance local versus global trade-offs: locally we wish to maximize the engagement of that session, while globally we wish to satisfy certain constraints over the entire set of sessions. Each session involves a user interacting with a collection of items, and the decision-making task at hand is that of selecting a ranking of these items. The rankings chosen across all sessions influence the rewards collected in the sessions as well as whether or not the overall constraints are satisfied. While our main focus in this paper is to develop algorithms for the general online constrained ranking problem, the key motivating application involves webpage layout optimization and traffic-shaping. In this context, each session corresponds to a user interacting with a webpage (for example a web portal), and the decision-maker wishes to optimize the placement of content from an assortment of content (e.g. news articles, sports articles, ads, blogs, financial news, etc.).
The webpage is assumed to be divided into slots, and the problem of assigning content to slots may be viewed as an assignment or ranking problem. The natural locally optimal approach to optimizing a webpage involves computing an engagement score (e.g., a click probability or a predicted dwell-time) for each content, and displaying the content within slots on the page in descending order of the scores so that the highest (predicted) engaging content is in the most highly clicked slot, and so on.
However, in practice there are often other business constraints in place. For example we may wish to deliver a certain guaranteed number of clicks to different segments of the traffic. Business considerations may dictate that a certain number of clicks be delivered to a premium advertiser, or that a certain number of top-ranked page-views be delivered to certain premium news-agency partners over the entire collection of sessions in the traffic. These additional side constraints introduce trade-offs; performing a simple “locally optimal” ranking may violate these constraints. The challenge thus is to pick assignments in each session, online, in a way that satisfy the traffic-wide business constraints while maximizing user engagement. Since the traffic distribution is quite well-known in advance (and data is available for the same), the natural question to ask is whether we can optimize the pages in each session to maximize user engagement while satisfying the other traffic constraints. Our paper deals with developing a principled approach to deal with this traffic-shaping problem.
Note that our paper does not seek to address the learning-to-rank problem [@LTR; @LTR_online], i.e. the problem of learning how to predict user engagement from data. Indeed, in many practical systems, a natural separation of concerns is assumed between learning-to-rank (or machine learning for predicting other user-engagement signals such as click-through rate, dwell-time etc.) and the traffic-shaping problem. The machine learned models are used as input signals to a *federation* layer that performs traffic-shaping, i.e. ingesting the signals and optimizing the ranking to maximize engagement and satisfying the constraints. However, the algorithm for traffic-shaping itself is completely agnostic to the details of how these signals are learned.
**Comparison to related work:** The traffic-shaping and click-shaping problems were studied by Agarwal et al. [@Agarwal_cs1; @Agarwal_cs2; @Agarwal_cs3] using an optimization based framework that involved probabilistic sampling. Several important features distinguish our work. The aforementioned work deals only with the single slot case (i.e. each session involves choosing a single item from a collection and picking the best one that satisfies a constraint), and they defer the multi-slot problem as an open problem. While they also adopt a linear programming approach, their model is substantially different; their optimization variables are probabilities of sampling items that maximize reward and satisfy traffic constraints. Moreover, they do not present theoretical guarantees on the solution quality. In contrast, our approach directly addresses the multi-slot situation (i.e. ranking items), our optimization involves optimizing over the set of permutations (i.e. assignments), and we tackle the online aspect of the problem by incorporating a learning phase followed by an online decision-making phase. We evaluate our method empirically and provide rigorous theoretical guarantees.
Our work draws on ideas and tools from the online optimization literature, specifically the primal-dual method for online linear programming [@Ye] and its applications to online combinatorial optimization problems [@Buchbinder] such as the ad-words problem [@Mehta; @Devanur], whole-page optimization problem for ads [@Devanur2], combinatorial auctions [@comb_auc], and extensions to the nonlinear case [@Chen]. While our approach substantially uses ideas and tools from this literature, a key distinction is the nature of the decision-making problem. Each online session in the aforementioned lines of work involves a binary decision-making problem, i.e. whether or not to assign a unit of an item in the session. In contrast, in our problem, each session involves choosing a permutation (more specifically a perfect matching between the items and the slots).
Moreover, unlike [@Ye] where the decision-maker must satisfy budget (upper bound) constraints, in our paper we have allocation (lower bound) constraints wherein a certain number of units (for e.g. clicks) must be delivered. Finally, we mention that within the literature, different lines of work arise from different distributional assumptions made on the sessions; e.g. [@Mehta] analyze the problem in the setting where the ordering of sessions may be adversarial with respect to the online decision maker. Another line of work [@Devanur; @Ye] assumes the *random permutation model*, wherein the distribution order of sessions may be assumed to exchangeable. Different theoretical guarantees are possible under different scenarios, and our work employs the random permutation model also. Our main contributions in this paper are the following:
- **Solution Form:** We present a new algorithm for the online constrained ranking problem. In the initial learning phase, dual prices $\lambda_t$ corresponding to each traffic constraint $t$ are learned. In the subsequent online phase, in each session, one is provided matrices $C$ and $A_{t}$ (see Sec. \[sec:formulation\] for details) that capture the predictions for engagement and traffic-shaping units contributed for different possible rankings. The online ranking algorithm assumes the simple form: $$\label{eq:form}
\sigma = \texttt{MaxWeightMatching}\left(C + \sum_{t=1}^T \lambda_t A_t \right).$$ When $\lambda_t=0$, one recovers the “greedy” solution, i.e. that of optimizing each session individually and disregarding the constraints. In order to satisfy the constraints, the session rankings must be re-optimized (over the space of permutations) as per . As a consequence of the connection to maximum-weight matching, we intimately use the Hungarian algorithm [@Hungarian] and its analysis.
- **Scalability:** In the learning phase, data is collected and a single linear program needs to be solved offline to obtain the prices. Thereafter, the online aspect is extremely scalable as it involves computing a matching using the prices.
- **Evaluation and Guarantees:** We present detailed empirical validation of our algorithm on real data. We also theoretically analyze our algorithm using the primal-dual method. Specifically we show that our algorithm achieves a competitive ratio of $1-O(\epsilon)$, where $\epsilon$ is the fraction of samples that are used in the learning phase.
- **Flexibility:** Another desirable feature of our algorithm is that it is flexible; indeed constraints can change, performance of the machine learned input signals can vary, and we would like a solution that is robust. Robustness is achieved by simply repeating the learning phase (with the new constraints in place, for instance), and obtaining the new prices. Thereafter, the implementation of the online algorithm remains the same, only the new prices need be propagated.
**Notation:** Throughout the paper, we will use the notation $[m]:=\left\{1, \ldots, m \right\}$; this quantity will refer to the size of the list (of documents) to be ranked in each session. A permutation (or ranking), denoted by $\sigma$ refers to a bijection between sets $\sigma: [m] \rightarrow [m]$. Usually the sets of interest are the set of documents (in a fixed session), and the different slots, and a permutation $\sigma$ specifies the assignment of documents to slots. We will use the terms permutation, assignment, and ranking interchangeably throughout the paper. It will often be convenient to represent the ranking $\sigma$ by a permutation matrix $P \in {\mathbb{R}}^{m \times m}$ where $(P)_{ij}=1$ if $\sigma(i)=j$ and $0$ otherwise. Such a matrix has a single entry equal to one in each row and column, with the remaining entries equal to zero. We will refer to the set of $m!$ permutation matrices as $\mathcal{P}$. The convex hull of $\mathcal{P}$, denoted by $co(\mathcal{P})$ is the Birkhoff polytope [@Hungarian], which will play an important role.
Throughout the paper, the index $k$ is reserved for sessions, i.e. $k \in [n]$, the index $t$ is reserved for the $T$ different traffic-shaping constraints, i.e. $t \in [T]$. In a session $k$, if document $d$ is shown in position $p$, a user engagement of $\left(C_k \right)_{d,p}$ is assumed to be accrued. Given a permutation $\sigma$, the overall engagement reward is assumed to be additive, i.e. given by $\sum_{i=1}^m \left(C_k \right)_{i,\sigma(i)}$. The matrix $C_k$ is conveniently represented by a bipartite graph with $m$ nodes on the left and right and the edges having weights $\left(C_k \right)_{d,p}$. A matching $\sigma$ achieves a weight of $\sum_{i=1}^m \left(C_k \right)_{i,\sigma(i)}$. A maximum-weight perfect matching is a permutation $\sigma \in \mathcal{P}$ with maximum possible weight. We remind the reader that a maximum-weight perfect matching in a bipartite graph is obtained via the Hungarian algorithm [@Hungarian]. We will use the notation $P \geq 0$ to specify that the matrix $P$ is entry-wise non-negative, and $P^{\prime}$ denotes the matrix transpose. We will use $\langle P,Q \rangle := \sum_{i=1}^{m} \sum_{j=1}^m P_{ij}Q_{ij}$ to denote the inner-product between the matrices $P, Q \in {\mathbb{R}}^{m \times m}$. When a ranking $P$ is chosen, the engagement corresponding to that ranking is given by $\langle C_k, P \rangle$ due to the additive nature of the reward.
Formulation {#sec:formulation}
===========
Let $k \in [n]$ index a collection of sessions where different users interact with the content-delivery system of interest. In each session, a collection of $m$ documents are made available. The task of the ranking system is, in each session $k$, to rank the documents, i.e. produce a permutation $\sigma_k: [m] \rightarrow [m]$.
For each session, we assume access to predictions of how each document would perform with respect to user engagement (e.g. clicks or dwell time) if document $d$ was shown in the $p^{th}$ position. Let $C \in {\mathbb{R}}^{m \times m}$ be the matrix such that $C_{d,p}$ models the engagement when document $d$ is shown in the $p^{th}$ position. (Typically, this type of information is available from the output of a machine-learned model that predicts the engagement for each document, personalized for each user corresponding to the session in question). For each session, the engagement is given by $\sum_{p=1}^{m} C_{p,\sigma(p)} = \langle C, P \rangle$, where $P$ is the permutation matrix corresponding to $\sigma$. The total engagement across all sessions is therefore $\sum_{k=1}^{n} \langle C_k, P_k \rangle$, where $k$ is the user engagement for the $k^{th}$ session and $P_k$ is the corresponding permutation chosen.
Simultaneously, we assume that a number of traffic-shaping constraints are present. As an example, we may have constraints on the number of clicks that must be delivered to various publishing partners (in a fixed number of sessions). These can be captured as: $$\sum_{k=1}^{n} \langle A_{kt}, P_k \rangle \geq b_t,$$ where the index $k$ refers to the session, and $t$ is refers to $t^{th}$ traffic-shaping constraint. The matrix $A_{kt} \in {\mathbb{R}}^{m \times m}$ is the matrix whose $(d,p)$ entry captures the number of engagement units (e.g. clicks) delivered for the $t^{th}$ constraint in the $k^{th}$ session when document $d$ is shown in position $p$ (such estimates are also typically obtained as the output of a machine-learned click model)[^1] . The constant $b_t$ is the number of clicks that are committed to the publishing partner in $n$ sessions contractually. The objective of the ranking system is to maximize the overall user-engagement while satisfying certain constraints.
The above description suggests a natural optimization formulation:
$$\begin{split} \label{opt:P0}
\underset{P_1, \ldots, P_n}{\text{maximize}} & \qquad \sum_{k=1}^{n} \langle C_k, P_k \rangle \\
\text{subject to} & \qquad \sum_{k=1}^{n} \langle A_{kt}, P_k \rangle \geq b_t \; \; t=1, \ldots, T\\
& \qquad P_k \in \mathcal{P} \; \; k=1, \ldots, n.
\end{split}$$
While conceptually clean, working directly with this formulation has several disadvantages:
- Solving this formulation requires knowledge of the $C_k, A_{kt}$ before-hand, whereas the problem is online in nature (i.e. the sessions arrive in an online manner).
- The above optimization formulation is combinatorial and therefore intractable, i.e. it involves optimizing over the set of permutations.
To address the last point, we perform a *convex relaxation* [@Bubeck Chapter 6] of the problem, i.e. relax the optimization constraints $P_k \in \mathcal{P}$ to $P_k \in co(\mathcal{P})$, where the $co(\mathcal{P})$ represents the convex hull of $\mathcal{P}$ [@Hungarian Chapter 18]. The resulting relaxation converts the optimization formulation into a convex optimization problem (indeed, it is a linear programming problem), which can therefore be solved (in principle) in polynomial time. There are a number of difficulties pertaining to the relaxation:
- Linear programming is computationally intensive and solving one at serving-time is often infeasible due to the strict latency requirements. Hence we desire to avoid solving them at run-time (although solving offline on a periodic basis is acceptable).
- A problem associated with convex relaxations of combinatorial problems is that the resulting solution may be fractional (i.e. the solutions $P_k$ may not be permutation matrices). We then have to devise a scheme to convert the optimal solution to a permutation matrix via a rounding scheme. As we will show in our subsequent analysis, we will not need to round solutions. The special structure of our problem and well-known results from graph matching and the analysis of the Hungarian algorithm [@Hungarian Chapter 17] actually guarantee that we can always produce a valid extremal solution, rather than a fractional one.
- In order to solve the convex relaxation, one needs a tractable representation of the set $co(\mathcal{P})$ in terms of a small number of equations and inequalities. It turns out that the convex hull of the set of permutation matrices has a compact description via the Birkhoff-von Neumann Theorem [@Hungarian Chapter 18], it is simply the set of so-called doubly stochastic matrices (sometimes also called the Birkhoff polytope) given by: $$\mathcal{B}:=co(\mathcal{P})=\left\{ P \in {\mathbb{R}}^{m \times m} \; | \; P \mathbf{1} = \mathbf{1} \; P^{\prime} \mathbf{1} = \mathbf{1} \; P \geq 0 \right\}.$$
We will be able to resolve the above difficulties by computing and analyzing the dual; indeed the analysis will reveal a natural and simple online algorithm which we will describe below.
We first explicitly state the primal and dual linear programs that result from the convex relaxation of .
$$\begin{split} \label{opt:P1}
\underset{P_1, \ldots, P_n}{\text{maximize}} & \qquad \sum_{k=1}^{n} \langle C_k, P_k \rangle \\
\text{subject to} & \qquad \sum_{k=1}^{n} \langle A_{kt}, P_k \rangle \geq b_t \; \; \forall \; t=1, \ldots, T\\
& \qquad P_k \in \mathcal{B} \; \; k=1, \ldots, n
\end{split}$$
$$\begin{split} \label{opt:D1}
\underset{\lambda, \alpha_k,\beta_k}{\text{maximize}} & \qquad \sum_{t=1}^T \lambda_t b_t + \sum_{k=1}^n \alpha_k^{\prime} \mathbf{1} + \sum_{k=1}^n \beta_k^{\prime} \mathbf{1} \\
\text{subject to} & \qquad C_k + \sum_{t=1}^{T} \lambda_t A_{kt} + \mathbf{1} \alpha_k^{\prime}+ \beta_k \mathbf{1}^{\prime} \leq 0 \; \; \forall k=1, \ldots, n \\
& \qquad \lambda \geq 0
\end{split}$$
The dual linear program is of key interest in the paper — our algorithm will consist of solving on historical data, and computing the optimal dual variables $\lambda$ which have a natural price interpretation. We describe this in the next section.
The Algorithm {#sec:algorithm}
=============
Recall that in our setup, the $n$ sessions arrive in a sequential manner online, and for each session $k$ we have access to the matrices $C_k$ (the engagement), and $A_{tk}$ (corresponding to the traffic-shaping constraint). We denote the full set of sessions by $N$. Our approach involves using a subset of the sessions, $S$, initially to learn the dual prices. The corresponding primal/dual linear programs are called the *sample linear programs*. In these learning-phase sessions, we make arbitrary decisions — concretely we let $P_k = I$, the identity permutation. Let $\hat{n}:=|S|$ be chosen and denote $\epsilon = \frac{\hat{n}}{n}$. Since the constraints require that a budget of $b_t$ clicks are to be delivered over $n$ sessions, it follows that when $\epsilon n$ sessions are present, the corresponding commitment is only $\epsilon b_t$ clicks for each traffic-constraint $b_t$. In this way, the $b_t$ is rescaled. In order to be conservative, we instead require that for each constraint: $
\sum_{k=1}^{\hat{n}} \langle A_{kt}, P_k \rangle \geq \nu \epsilon b_t.
$ The additional $\nu$ multiplicative factor ensures that the solution obtained is robust to random variations (we will discuss this in Section \[sec:theory\]), and also to counter the conservative estimate that since the first $\epsilon$ fraction of sessions are used for learning, zero (or negligible) contributions to the constraints are made from it. Accordingly, the primal/dual sample linear programs are: $$\begin{split} \label{opt:P2}
\underset{P_1, \ldots, P_{\hat{n}}}{\text{maximize}} & \qquad \sum_{k=1}^{\hat{n}} \langle C_k, P_k \rangle \\
\text{subject to} & \qquad \sum_{k=1}^{\hat{n}} \langle A_{kt}, P_k \rangle \geq \nu \epsilon b_t \; \; \forall \; t=1, \ldots, T\\
& \qquad P_k \in \mathcal{B} \; \; k=1, \ldots, \hat{n}
\end{split}$$ $$\begin{split} \label{opt:D2}
\underset{\lambda, \alpha_k, \beta_k}{\text{maximize}} & \qquad \sum_{t=1}^T \nu\epsilon \lambda_t b_t + \sum_{k=1}^{\hat{n}} \alpha_k^{\prime} \mathbf{1} + \sum_{k=1}^{\hat{n}} \beta_k^{\prime} \mathbf{1} \\
\text{subject to} & \qquad C_k + \sum_{t=1}^{T} \lambda_t A_{kt} + \mathbf{1} \alpha_k^{\prime}+ \beta_k \mathbf{1}^{\prime} \leq 0 \; \; \forall k=1, \ldots, \hat{n} \\
& \qquad \lambda \geq 0
\end{split}$$ In the theoretical anaylsis section we prove that when $\nu = 1+ 4 \epsilon$, with high probability a feasible solution is returned. We also prove that when $\nu = 1 - \epsilon$, an almost feasible solution is returned, and that the objective is almost as large as the hindsight optimal feasible solution.
Data $\left\{ (C_k, A_{k1}, \ldots, A_{kT})\right\}$ for sessions $k=1, \ldots, n$ presented online for ranking, sample linear program parameters $\nu$, $\epsilon$. For the first $\hat{n}$ sessions, let $P_k = I$ (arbitrary decisions) and log the session data $\left\{ (C_k, A_{k1}, \ldots, A_{kT})\right\}$ corresponding to this sample set $k \in S$. Solve the sample linear program on $S$ to obtain the dual prices $\hat{\lambda}_1, \ldots, \hat{\lambda}_m$. Compute document score matrix $S_k = C_k + \sum_{t=1}^T \hat{\lambda}_t A_{kt}$ where $S_{d,p}$ represents the score when document $d$ is shown in position $p$. \[step:scoring\] Set $\sigma_k:=\texttt{MaxWeightMatching}(S_k)$. \[step:hungarian\] Ranking $\sigma_k$ for session $k$. Increment session counter $k:=k+1$
In words, the algorithm works as follows. First, using historical data, we offline solve the linear program to obtain the optimal dual prices $\hat{\lambda}_t$ for each traffic shaping constraint. These prices are then made available to the online serving system for ranking. Once the learning phase is over, upon the arrival of new sessions, the matrices $C_k, A_{kt}$ are computed, and the score matrix $S_k$ is computed as per step \[step:scoring\]. As mentioned before, the $(d,p)$ entry of the matrix $S_k$ represents the score when document $d$ is shown in position $p$.
Once the score matrix is computed, we call a subroutine to compute the maximum-weight bipartite perfect matching with respect the weight matrix $S_k$, this perfect matching assigns documents to positions, and thereby yields the desired ranking $\sigma_k$ for that session.
The run-time complexity is essentially the complexity of computing a perfect matching, for which a number of options are available. In order to compute a max-weight perfect matching exactly, one may use the celebrated Hungarian algorithm. The computational complexity of implementing the same is $O(m^3 \log m)$ [@Hungarian]. In scenarios of interest to us, we are typically only interested in ranking of the order of $20-200$ webpages in each session, and hence the algorithm is feasible to implement at run-time. When the scale is larger and the Hungarian algorithm is not a viable option, one can instead implement the naive greedy algorithm[^2] as an alternative. The naive greedy algorithm is known to be a $\frac{1}{2}$-optimal algorithm for maximum matching, with a worst-case running time of $O(m^2)$. In our experiments, we use a position effects model called the “reference CTR” model, in which the greedy algorithm is actually optimal. The offline computational complexity is polynomial in the size of the problem — it involves the solution of the linear program , which scales with the sample size of the sampled linear program and is hence high. However, since offline computation is not a bottleneck, our approach is scalable to web applications.
While Algorithm \[alg:traffic\_shaping\] advocates solving the sampled linear program once (in advance) on historical data, this is mostly to simplify the analysis of the algorithm. In practice, we recommend solving the dual linear program periodically (e.g. once daily) to refresh the prices to capture the random as well as seasonal variations in traffic. The analysis of periodically recomputing the solution is also possible (in the spirit of [@Ye]), but the analysis is more complicated with only marginally better bounds — we leave this for future work.
Numerical Experiments {#sec:expts}
=====================
Experimental Setup
------------------
In this section we validate our algorithm via numerical experiments. Our numerical experiments are conducted on real data collected from session history on a major web-portal. We describe the setup for our experiment below.
Our data consists of $2000$ sessions of web-traffic data. In each session, we are presented with $20$ documents to rank on the web-page. Each document has associated with it the following attributes:
1. The predicted dwell-time: This is a score generated by a machine-learned model on a large corpus of data that incorporates document features, user features, and document popularity signals. The predicted score is to be interpreted as a surrogate for the expected dwell-time when the user is shown that document.
2. The click-through rate: This is again generated via a machine learned model, and is an estimate for the click-probability for a particular user to click on the document in question when that document is shown in the first position.
3. Newsiness score: This is a normalized score that captures how newsy a particular document is. It is also obtained as the output of a machine-learned model. When a newsy article is shown in a highly clickable slot on the page, the overall page is deemed more newsy.
4. Publisher A: This is a binary $\left\{ 0,1 \right\}$ score that captures whether a document is curated by Publisher A or not. Showing a Publisher A-curated document in a more clickable slot delivers more clicks to that publisher, and is hence more desirable to it.
5. Publisher B: This is also a binary score (as above) corresponding to whether document is curated by Publisher B.
In our experiments, we implement Algorithm \[alg:traffic\_shaping\], i.e. use a subset of the data for learning the dual prices (making arbitrary decisions in the process), and in the remaining sessions making the optimal decisions with respect to the learned dual prices. In our experiments, the online optimization problem involves: (a) maximizing the dwell-time in each session, (b) delivering a fixed number of clicks to Publishers A and B, (c) achieving at least a certain average newsiness score over the entire set of sessions.
One of the key requirements of our framework is that we need the dwell-time, click probability, and newsiness score for each document when it is shown in any one of the $20$ slots. The machine learned models generate these scores with respect to the first slot only. In order to address this, we utilize the (somewhat standard) Reference CTR model, which is explained below.
The Reference CTR
-----------------
When users are presented with a list of content on a web-page, it is well-known [@Chapelle] that there is a strong *position effect* for clicks, dwell-time, and other engagement metrics (such as the newsiness score of interest to us in this paper). It is commonly assumed [@Chapelle], that a document $d$, when shown in slot $p$, has a CTR that depends on its CTR in position $1$ (denoted by $c_1(d)$) and a factor that depends only on the position (denoted by $\text{ref}_p$). It is normalized so that $\text{ref}_1=1$. Hence the predicted CTR of document $d$ in position $p$ is given by: $$c_p(d)=c_1(d)\times\text{ref}_p.$$
The quantity $\text{ref}_p$ is computed by computing the aggregate CTR for each position, and normalizing by the aggregate CTR for position $1$ [@Chapelle]. We will also assume (for this experiment) that the dwell-time and newsiness have a similar position effect as above.
For a list of items presented to a user according to a permutation $\sigma$, the total number of (predicted) clicks generated by the session, and the clicks generated for publisher $t$ (where $t \in \left\{A,B\right\})$ are respectively given by: $$\begin{aligned}
C(\sigma)&=\sum_{p=1}^{m}c_p(\sigma(p))\times\text{ref}_p \\
C_t(\sigma)&=\sum_{p=1}^{m}c_p(\sigma(p))\times\text{ref}_p \times \mathbf{1}\left(\sigma(p) \text{ of traffic type } t \right).\end{aligned}$$
Similarly, we assume that the total dwell-time and newsiness respectively are: $$D(\sigma)=\sum_{p=1}^{m}\texttt{dwell}(\sigma(p))\times\text{ref}_p,
\qquad
N(\sigma)=\sum_{p=1}^{m}\texttt{news}(\sigma(p))\times\text{ref}_p.$$
Figure \[fig:refctr\] shows the reference CTR distribution.
![The reference CTR distribution for the slots on the page. Interestingly, the reference CTR distribution is not monotonically decreasing with the slot position due to various page presentation effects.[]{data-label="fig:refctr"}](refCTR.eps)
Observations
------------
We first investigate the performance of sampled linear program when the size of the sample set increases to $2000$ sessions (entire dataset). Figure \[fig:costVSsessions\] shows how the average (normalized) dwell-time per session varies as we increase the size of the sampled linear program. Note that when all $2000$ sessions are used, we essentially obtain the hindsight-optimal solution. As Fig. \[fig:costVSsessions\] shows, this level of performance is approximately achieved by the sampled linear program withing just $800$ sessions, justifying our intuition that this quantity is distributionally stable with respect to the traffic.
[cc]{}
[0.45]{}
&
[0.45]{}
[cc]{}
[0.45]{}
&
[0.45]{}
Figure \[fig:pricesVSsessions\] shows how the dual prices change as a function of the number of sessions over which the sampled linear program is solved. Again, we see that the prices stabilize at approximately $800$ sessions. In Fig. \[fig:costVSconstraint\] we investigate the trade-offs that are inevitable as a consequence of traffic-shaping. We fix the number of sessions to be $2000$ and solve the corresponding linear program (i.e. the hindsight optimal solution). We fix the values of the constraints for clicks delivered to publisher $B$ and the newsiness constraint. We vary the number of clicks deliverable to publisher $A$ by changing a parameter $\theta$ (as $\theta$ increases, the corresponding click constraint commitment $b_t$ increases multiplicatively). We study how the dwell-time per session changes as $\theta$ increases — the corresponding trade-off curve is shown in Fig. \[fig:costVSconstraint\].
In Fig. \[fig:performanceVSepsilon\] we study the performance of the online algorithm. On the $x$-axis is the fraction of the $2000$ sessions which were used for the sampled linear program. On the $y$-axis is the performance ratio for different metrics. A value of $\nu=1.05$ (as required in ) was chosen for this experiment. The solid-blue curve is the competitive ratio (i.e. the ratio of the performance by the online algorithm to that of the hindsight optimal solution, see Sec. \[sec:theory\]). Note that in the first $\epsilon$ fraction of the sessions (the price-learning phase), arbitrary suboptimal decisions are made, and hence the competitive ratio decreases as a function of $\epsilon$ (see Theorem \[thm:main\]). The yellow, purple, and green plots are the ratio of the clicks/newsiness delivered versus the value committed by the constraints (i.e. the $b_t$), and values under one indicate constraint violation. Note that for small values of $\epsilon$, some of the constraints are actually violated — hence while the competitive ratio is high, training on that small subset of sessions yields infeasible solutions. The red dotted line indicates the online performance ratio as a function of $\epsilon$. The online performance ratio is the ratio of the reward of the online algorithm to that of the hindsight optimal sessions, *restricted to the segment of the sessions where online decisions are made*. While the online performance ratio is close to $1$ throughout, it attains this ratio by violating constraints at small values of $\epsilon$.
Another interesting feature that we found was that, under the RefCTR model, the greedy algorithm (i.e. sorting documents by the price-weighted scores, and then matching them to positions in sorted order of their RefCTR value) achieve the *same ranking* compared to the Hungarian algorithm. This is simply a structural consequence of the RefCTR model (we omit the proof here), however the scalability implications are substantial; instead of implementing the Hungarian algorithm, the serving implementation only needs a sorting procedure.
Theoretical Guarantees {#sec:theory}
======================
In order to describe the performance guarantees, we need to define a critical (and well-known) solution concept related to online algorithms, namely the notion of the competitive ratio of the algorithm.
Given online events $\left\{ (C_k, A_{k1}, \ldots, A_{km})\right\}$ for sessions $k \in [n]$, let OPT denote the solution obtained by solving the linear program on this data — we call the solution thus obtained the *optimal hindsight solution*. A hypothetical decision-maker with advance knowledge of the “future” sessions would make these optimal decisions, and this is the best performance achievable by any algorithm. Suppose Algorithm \[alg:traffic\_shaping\] (which is online) achieves a performance of $\widehat{\text{OPT}}$. We define the *competitive ratio* of the algorithm to be $\frac{\widehat{\text{OPT}}}{\text{OPT}}$. The competitive ratio measures how the performance of the online algorithm compares to the optimal hindsight solution. Our main result shows that, under certain assumptions, our algorithm achieves an $1-O\left( \epsilon \right)$ competitive ratio. We state our assumptions next:
1. **Assumption 1:** The sessions $\left\{ (C_k, A_{k1}, \ldots, A_{kT})\right\}$ arrive in a random order, i.e. the matrices $\left\{ (C_k, A_{k1}, \ldots, A_{kT})\right\}$ can be arbitrary, but every permutation of the indices $k$ have an equal probability.
2. **Assumption 2:** The optimization problems and are both strictly feasible.
3. **Assumption 3:** The total number of sessions $n$ is known a priori.
4. **Assumption 4:** For every permutation $P$ we have $\langle C_k, P \rangle \in [0,1]$ and $\langle A_{kt}, P \rangle \in [0,1]$ for all $k=1, \ldots, n$ and $t=1, \ldots, T$.
\[thm:main\] Consider the online traffic-shaping problem when the above assumptions hold true, and let $\epsilon \in (0,\frac{1}{3})$ be fixed. Define $B:=\min_{t=1, \ldots, T} b_t$. Suppose the constraints satisfy the requirement that $$\label{eq:sample_complexity}
\epsilon \geq \left( \frac{\log \left( \frac{2C_0T}{\epsilon} \right) + (T+1) \log \left( m^2n \right)}{B} \right)^{\frac{1}{3}}.$$
Let the dual prices computed by the sampled linear program be denoted by $\hat{\lambda}$. Then the solution constructed by Algorithm \[alg:traffic\_shaping\], denoted by $P_k(\hat{\lambda})$, has the following properties:
1. In each session the optimal hindsight solution and the online approach via Algorithm \[alg:traffic\_shaping\] produce valid assignments $P_k^*$ and $P_k^{\text{online}}(\hat{\lambda})$. Moreover, if the optimal hindsight prices, denoted $\lambda^*$, are given \[alg:traffic\_shaping\], then $P_k^* = P_k^{\text{online}}(\lambda^*)$.
2. When $\nu = 1 + 4 \epsilon$, With probability exceeding $1 - \epsilon$, the assignments satisfy the traffic-shaping constraints, i.e. $$\sum_{k=1}^{n} \langle A_{kt}, P_k(\hat{\lambda}) \rangle \geq b_t \; \; \forall t=1, \ldots, T$$
3. When $\nu = 1 - \epsilon$, with probability exceeding $1-\epsilon$ the online solution satisfies $$\sum_{k=1}^{n} \langle A_{kt}, P_k(\hat{\lambda}) \rangle \geq (1-2\epsilon)b_t \; \; \forall t=1, \ldots, T$$ and objective attained by the online assignments $P_k(\hat{\lambda})$ satisfies: $$\sum_{k=1}^{n} \langle C_k, P_k(\hat{\lambda}) \rangle \geq (1-\epsilon) \text{OPT},$$ (i.e. the competitive ratio is $1-O(\epsilon)$ with high probability).
We give a detailed proof in Sec \[sec:proof\]. We make the following remarks about the main result:\
**Remarks\
** (1) Note that the first part states that if the optimal prices are available to the online algorithm, then optimal decisions will be made in each session. Hence the prices $\lambda^*$ are sufficient to make good decisions online. The second and third parts of the theorem quantify the intuition that if we use a small set of samples initially to learn the prices using the sampled linear program, the online algorithm will then perform almost optimally.
\(2) Note that Algorithm \[alg:traffic\_shaping\] advocates using the Hungarian algorithm to compute the best matching in each session. Depending on the number of items to be ranked, this may or may not be feasible at run time — in the latter case a greedy algorithm may be used. The greedy algorithm, being a $\frac{1}{2}$-approximation in the worst case, can also be analyzed and bounds on the competitive ratio may be derived. In our experiments, we find that using the greedy algorithm does not lead to any loss of performance as a consequence of the RefCTR assumption (see Sec. \[sec:expts\]).
\(3) Assumption $4$ requires that the for each permutation, the rewards and the influence to each traffic shaping is non-negative and bounded by $1$. The non-negativity assumption is natural in our setting (since we are interested in quantities such as views and clicks). The upper bound of $1$ is somewhat arbitrary, we merely need the rewards and influences to be bounded by some uniform quantity (say $U$). Our results would be more or less unaffected (modulo certain logarithmic factors in the statement of Theorem \[thm:main\]) if $U > 1$).
\(4) We now comment on the qualitative relationship between $\epsilon$, $B$ and the competitive ratio. In order to make sense of the statement, we assume that $B$ increases linearly with the number of sessions $n$ (e.g. clicks committed increase linearly as the number of sessions increases); i.e. $B=\rho n$. (Note that $B$ growing with $n$ fixed would be problematic from a feasibility point of view - if we over-commit to a traffic-shaping constraint the optimization problem becomes infeasible violating Assumption $2$). Thus as $n$ increases, the right hand side of decreases to zero and we may make $\epsilon$ arbitrarily small, thus yielding (a) a small fraction of the samples used for learning the prices (b) a better competitive ratio, and (c) a high probability of success simultaneously. Indeed may be viewed as a “sample-complexity” type result.
\(5) The competitive ratio decreases as the fraction of sessions $\epsilon$ used for learning increases. While on the one hand the prices are learned more accurately as $\epsilon$ increases, on the other hand suboptimal decisions are made by the online algorithm in the learning phase. Indeed, we assume (somewhat pessimistically) that arbitrary decision are made in the learning phase with zero reward accrued, thus an $\epsilon$-fraction of the possible reward is lost in this phase.
Conclusion {#sec:conclusion}
===========
In this paper we investigated the problem of online constrained ranking and its application to the web traffic-shaping problem. We developed a linear-programming based primal-dual algorithm for online ranking and demonstrated it’s efficacy on real data. We also proved rigorous guarantees about the algorithm in terms of the achieved competitive ratio.
While the paper focuses on traffic-shaping as the motivating application, we believe this algorithm to have broad applicability to whole-page optimization and beyond. For instance, when optimizing the simultaneous placement of ads and organic content, additional constraints (or modifications to the objective) can be made to factor in the revenue considerations traffic-wide. More generally, this approach can be used for solving a variety of constrained online assignment problems such as matching customers to servers (e.g. matching drivers to passengers) subject to overall service-level constraints. We believe this approach will scale well to a variety of such application domains.
A number of further avenues for future work are worth mentioning:
- Our model involves optimizing (in the objective) subject to constraints over the predicted dwell-time and the predicted clicks over a period of sessions. These predictions are obtained as the output of a machine-learned model. The number of actual clicks obtained, and the actual dwell-time realized will of course be different. As a practical strategy, we advocate re-solving the linear program periodically (and suitably altering the constraints) to capture these variations. However, there is an unavoidable feedback loop between the rankings and the machine learned models — the decisions made in the ranking phase affect the learning of these online models. Consequently, there is a natural exploration-exploitation trade-off that is unavoidable in this setup. Ranking unknown items in a highly clickable slot will lead to more exploration, whereas ranking more certain items will lead to exploitation. How one navigates this will likely be of key practical importance.
- Our approach assumes that the objective and the constraints can be expressed as linear functions of the ranking; the expected number of clicks and expected dwell time are assumed to be a weighted sum of the components derived from each slot. Other types of constraints may not be linearly expressible (e.g. suppose one wishes to enforce a certain notion of diversity in each ranking and enforce a certain minimum amount of average diversity across sessions). Working with non-linear constraints will likely enable us to enrich the quality of ranking in our traffic.
Proofs {#sec:proof}
======
We begin by stating two lemmas (their proofs are omitted due to space constraints) which follow as a standard consequence of the analysis of the Hungarian algorithm (we refer the reader to [@Hungarian] for details).
\[lemma:hung\] Let $C \in {\mathbb{R}}^{m \times m}$ be an entry-wise non-negative matrix. Consider the following optimization problems: $$\begin{split} \label{opt:P3}
\underset{P}{\text{maximize}} & \qquad \langle C, P \rangle \\
\text{subject to}
& \qquad P \in \mathcal{B}
\end{split}$$
$$\begin{split} \label{opt:D3}
\underset{\alpha, \beta}{\text{maximize}} & \qquad \alpha^{\prime} \mathbf{1} + \beta^{\prime} \mathbf{1} \\
\text{subject to} & \qquad C + \mathbf{1} \alpha^{\prime}+ \beta \mathbf{1}^{\prime} \leq 0
\end{split}$$
Then the following hold:
1. The pair , is a primal-dual pair, and strong duality holds.
2. An optimal solution $P^*$ to is attained at a permutation matrix, and this solution can be found using the Hungarian algorithm.
The duality between and is a standard exercise - we leave it to the reader. To see that strong duality holds, we note that the matrix with entries $P_{ij}=\frac{1}{m}$ is in the relative interior of the feasible set of . By Slater’s condition [@Boyd Chap. 5.2.3] strong duality follows.
The second part follows from the fact that the permutation matrix corresponding to the maximum weight perfect matching is an optimal solution. To see this, first note that a perfect matching exists since the bipartite graph in question is complete (since $C \in {\mathbb{R}}^{m \times m}$ is square), and thus satisfies Hall’s theorem [@Hungarian Chap. 22] - guaranteeing existence. Furthermore, the Hungarian algorithm [@Hungarian] yields a permutation matrix $P^*$ such that dual feasibility is achieved (see e.g. [@Hungarian Chap. 17]), indeed dual feasibility of is one of the key invariants of the Hungarian algorithm. Furthermore, the corresponding permutation matrix $P^*$ satisfies complementary slackness, only edges $(i,j)$ corresponding to a perfect matching are chosen (primal feasibility) corresponding to edges where the dual constraint $\left( C + \mathbf{1} \alpha^{\prime}+ \mathbf{1} \beta^{\prime} \right)_{ij} = 0$ i.e. corresponding to tight constraints, and hence $\left( C + \mathbf{1} \alpha^{\prime}+ \mathbf{1} \beta^{\prime} \right)_{ij} P^*_{ij} = 0$. Hence $P^*$ must be a primal optimal solution.
\[lemma:wt\_hung\] Let $\lambda = \bar{\lambda} \in {\mathbb{R}}^{T}$ be fixed such that $\bar{\lambda} \geq 0$. Consider the solution to the optimization problem $$\begin{split} \label{opt:D4}
\underset{\alpha_k, \beta_k}{\text{maximize}} & \qquad \sum_{t=1}^T \bar{\lambda}_t b_t + \sum_{k=1}^n \alpha_k^{\prime} \mathbf{1} + \sum_{k=1}^n \beta_k^{\prime} \mathbf{1} \\
\text{subject to} & \qquad C_k + \sum_{t=1}^{T} \lambda_t A_{kt} + \mathbf{1} \alpha_k^{\prime}+ \beta_k \mathbf{1}^{\prime} \leq 0 \; \; \forall k=1, \ldots, n
\end{split}$$ A set of optimal dual solutions (w.r.t. the constraints of ) $P_k(\bar{\lambda})$ for $k=1, \ldots, n$ corresponding to are achieved at extreme points of the Birkhoff polytope, (i.e. permutation matrices). Moreover each of the $P_k(\bar{\lambda})$ may be computed by computing a maximum-weight perfect matching with respect to the bipartite graph with weights $C_k + \sum_{t=1}^T \bar{\lambda}_t A_{kt}$ (using the Hungarian algorithm).
Note that for each fixed $\lambda = \bar{\lambda}$ and the separability (with respect to $k$) of the objective function, decouples into the following optimization problems: $$\begin{split} \label{opt:D6}
\underset{\alpha_k, \beta_k}{\text{maximize}} & \qquad \alpha_k^{\prime} \mathbf{1} + \beta_k^{\prime} \mathbf{1} \\
\text{subject to} & \qquad C_k + \sum_{t=1}^{T} \bar{\lambda}_t A_{kt} + \mathbf{1} \alpha_k^{\prime}+ \beta_k \mathbf{1}^{\prime} \leq 0,
\end{split}$$ and the corresponding optimal solutions $\alpha_k^*(\bar{\lambda})$, $\beta_k^*(\bar{\lambda})$ are optimal with respect to . The strong dual of is precisely: $$\begin{split} \label{opt:P3}
\underset{P}{\text{maximize}} & \qquad \langle C_k + \sum_{t=1}^{T} \bar{\lambda}_t A_{kt} , P \rangle \\
\text{subject to}
& \qquad P^{\prime} \mathbf{1} =\mathbf{1} \\
& \qquad P \geq 0 \; \; k=1, \ldots, n
\end{split}$$ By Lemma \[lemma:hung\], the solutions $P_k(\bar{\lambda})$ to the above are yielded by the permutations corresponding to the maximum-weight perfect matchings yielded by running the Hungarian algorithm on bipartite graphs with weight matrices $C_k + \sum_{t=1}^{T} \bar{\lambda}_t A_{kt}$ for $k=1, \ldots, T$.
We will also need the following lemma concerning concentration of random variables. For a proof we refer the reader to [@Ye Lemma A.1], [@Devanur Lemma 3] and the references therein.
\[lemma:conc\] Let $u_1, \ldots, u_r$ be a uniformly random sample without replacement from real numbers $$\left\{ | \sum_{i=1}^r u_i - r \bar{c} | \geq t \right\} \leq \exp\left( -\frac{t^2}{2r\sigma_R^2 + t \Delta_R} \right),$$ where $\Delta_R= \max_i c_i - \min_i c_i$, $\bar{c}=\frac{1}{R} \sum_{i=1}^R c_i$, and $\sigma_R^2=\frac{1}{R}\sum_{i=1}^R(c_i -\bar{c})^2$.
We now provide a detailed proof of Theorem 1. Note that both our main result for online ranking, and our assumptions are of a similar flavor as existing literature in online algorithms [@Devanur; @Ye]. Indeed, it is known [@Devanur; @Ye], that in these related results, these assumptions are also tight, i.e. it is not possible to get $1-O\left( \epsilon \right)$ competitive ratio if a single one of these assumptions is removed. While we do not show the tightness of the assumptions, we believe it likely that they are needed to achieve the stated competitive ratio.
The main idea behind the proof, which mimics [@Ye] with suitable adaptations, will be to use the complementary slackness conditions for linear programming. Let $\hat{\lambda}$ be a dual price vector obtained as the solution of the sampled linear program. We denote by $P_k(\hat{\lambda})$ the permutation that is chosen as per steps \[step:scoring\], \[step:hungarian\] of the algorithm. **\
\
Part 1.** The online assignments $P_k^{\text{online}}(\hat{\lambda})$ (and similarly $P_k^{\text{online}}(\lambda^*)$) are valid assignments since they are constructed via the Hungarian algorithm. Note that they are both trivially primal-feasible (since assignments are extreme points of the Birkhoff polytope).\
Let $\lambda^*$ be an optimal dual solution of . Note that when $\bar{\lambda} = \lambda^*$, the corresponding optimal solutions of coincide with those of . By strong duality, a set of corresponding primal solutions also coincide. By Lemma \[lemma:wt\_hung\], a dual variable corresponding to the constraints of achieves optimality at permutation matrices $P_k(\lambda^*)$. Hence, $P_k(\lambda^*)=P_k^*$.\
**Part 2.** Our proof strategy will be as follows. We will take a hypothetical optimal solution $\hat{\lambda}$ to the sampled linear program. It will be assumed to satisfy that the complementary slackness conditions of the primal dual pair , - but otherwise arbitrary. We will show that as a consequence of complementary slackness and the assumptions, with high probability the solution yielded by $P_k(\hat{\lambda})$ are feasible for this hypothetical $\hat{\lambda}$. We will then take a union bound over all possible $\hat{\lambda}$ to obtain that the solution is feasible unconditionally.
Let us fix a constraint $t$ and note that for the first $\epsilon n$ sessions, arbitrary decisions are made and we thus assume that no contributions to the constraints are made from these. As a result we required that the sampled linear program satisfy: $$\sum_{k \in S} \langle A_{kt}, P_k \rangle \geq (1+4 \epsilon) \epsilon b_t,$$ where the additional $(1+4 \epsilon)$ factor ensures that the solution robustly satisfies the constraints.
The bad event of interest to us is the event that a constraint is violated. We will call the set $S$ bad when the solution $\lambda$ leads us to a situation of such a constraint violation. Concretely, we consider the following set relations corresponding to a set $S$ being bad:
$$\begin{aligned}
&\left\{\sum_{k \in N \setminus S } \langle A_{kt}, P_k(\lambda) \rangle < b_t, \sum_{k \in S} \langle A_{kt}, P_k(\lambda) \rangle \geq (1+4 \epsilon)\epsilon b_t \right\} \subseteq A_1 \cup A_2, \\
&A_1 = \left\{\sum_{k \in N \setminus S } \langle A_{kt}, P_k(\lambda) \rangle < b_t, \sum_{t \in S} \langle A_{kt}, P_k(\lambda) \rangle \geq (1+4 \epsilon)\epsilon b_t, \right. \\
& \qquad \left. \sum_{k \in N} \langle A_{kt}, P_k(\lambda) \rangle \geq (1+3 \epsilon) b_t \right\} \\
&A_2 = \left\{ \sum_{k \in S} \langle A_{kt}, P_k(\lambda) \rangle \geq (1+4 \epsilon)\epsilon b_t, \sum_{k \in N} \langle A_{kt}, P_k(\lambda) \rangle < (1+3 \epsilon) b_t \right\}\end{aligned}$$
Next we note that: $$\begin{aligned}
{\mathbb{P}}(A_1) & \leq {\mathbb{P}}\left(\sum_{k \in N \setminus S } \langle A_{kt}, P_k(\lambda) \rangle < b_t, \sum_{k \in N} \langle A_{kt}, P_k(\lambda) \rangle \geq (1+3 \epsilon) b_t \right) \\
& \hspace{-4mm} \leq {\mathbb{P}}\left(\sum_{k \in N \setminus S } \langle A_{kt}, P_k(\lambda) \rangle < b_t, \sum_{k \in N} \langle A_{kt}, P_k(\lambda) \rangle \geq \frac{1+\epsilon}{1-\epsilon} b_t \right) \\
&\hspace{-4mm} = {\mathbb{P}}\left(\sum_{k \in N \setminus S } \langle A_{kt}, P_k(\lambda) \rangle < b_t, (1-\epsilon)\sum_{k \in N} \langle A_{kt}, P_k(\lambda) \rangle \geq (1+\epsilon) b_t \right) \\
&\hspace{-4mm} \leq 2 \exp \left( -\frac{\epsilon^2 b_t}{2-\epsilon }\right)\end{aligned}$$ where the second inequality follows since $\frac{1+\epsilon}{1-\epsilon} < 1+3 \epsilon$ for all $\epsilon \in (0,\frac{1}{3})$, and the last inequality follows from Lemma \[lemma:conc\].
Defining $Y_k:=\langle A_{kt}, P_k(\lambda) \rangle$, we also note that $$\begin{aligned}
{\mathbb{P}}(A_2) & \leq {\mathbb{P}}\left( \sum_{t \in S} \langle A_{kt}, P_k(\lambda) \rangle \geq (1+4 \epsilon)\epsilon b_t, \sum_{t \in N} \langle A_{kt}, P_k(\lambda) \rangle < (1+3 \epsilon) b_t \right) \\
& \leq {\mathbb{P}}\left( |\sum_{t \in S} Y_t - \epsilon \sum_{t \in N} Y_t| \geq \epsilon^2 b_t\right) \\
& \leq 2 \exp \left( -\frac{\epsilon^3 b_t}{2+ \epsilon}\right)\end{aligned}$$ where the last inequality again follows from an application of Lemma \[lemma:conc\].
Hence, the probability of a bad sample is bounded above by $4 \exp \left( -\frac{\epsilon^3 b_t}{2+ \epsilon} \right) \leq \delta$, where $\delta:= \frac{\epsilon}{C_0 (m^2n)^{T+1}} $
Since $\lambda$ is the solution obtained from the sampled linear program, it follows that: $$\sum_{k \in S} Y_k \geq (1+\epsilon) \epsilon b_t.$$
where $\delta := \frac{\epsilon}{2C_0T(m^2n)^{T+2}}$ for some constant $C_0$. In the preceding chain of inequalities, the last one follows from Lemma \[lemma:conc\].
Since the solution $\lambda$ obtained from the sampled linear program can be arbitrary, we take a union bound over all the possible “distinct” $\lambda$. We call two solutions $\lambda_1$ and $\lambda_2$ as distinct if $P_k(\lambda_1) \neq P_k(\lambda_2)$ for some session $k \in N$. Consider a dual constraint with complementary slackness: $$\left( C_k + \sum_{t=1}^{T} \lambda_t A_{kt} + \mathbf{1} \alpha_k^{\prime}+ \mathbf{1} \beta_k^{\prime} \right)_{ij} \left(P_k \right)_{ij}=0.$$ and note that $\left( P_k \right)_{ij} > 0$ is only possible when $$\left( C_k + \sum_{t=1}^{T} \lambda_t A_{kt} + \mathbf{1} \alpha_k^{\prime}+ \mathbf{1} \beta_k^{\prime} \right)_{ij} \geq 0.$$ (Indeed, the set of possible solutions yielded by the Hungarian algorithm will be a subset of such possible $P_k$ - by design the Hungarian algorithm maintains the dual inequality as an invariant [@Hungarian]). The set of possible $P_k$ is thus bounded above by the number of unique separations of the points $$\left\{ \left( \left( C_k \right)_{ij}, \left( A_{k1}, \right)_{ij}, \ldots, \left( A_{kT}, \right)_{ij},1 \right) \right\}_{i,j,k \in [m] \times [m] \times [n]}$$ in $(T+1)$-dimensional space. By results from computational geometry [@comp_geom], the number of such distinct hyperplanes, is bounded above by $C_0 (m^2n)^{T+1}$ for some constant $C_0$. Applying a union bound over the possible different prices, and $t=1, \ldots, T$ we obtain the required result.\
\
**Part 3.** Given $\nu=1-\epsilon$, one can prove approximate feasibility in a similar manner as above by observing that $${\mathbb{P}}\left( \sum_{k \in N \setminus S } \langle A_{kt}, P_k(\lambda) \rangle <(1-2\epsilon) b_t, \sum_{k \in N} \langle A_{kt}, P_k(\lambda) \rangle \geq (1 - \epsilon) b_t \right) \leq \delta,$$ where $\delta$ is the quantity used in the preceding part.
To bound the competitive ratio, our proof strategy will be to construct an auxiliary linear program for which the online solutions are optimal. We will show that a point $P_k^*$ (the solution to ) is feasible with respect to this auxiliary linear program, and hence the objective function values between the online solution and that achieved by $P_k^*$ cannot be too different.
Let $\hat{\lambda}$ be the solution of the sampled linear program and $P_k(\hat{\lambda})$ the corresponding assignment. Consider the linear program: $$\begin{split} \label{opt:P_aux}
\underset{P_1, \ldots, P_{n}}{\text{maximize}} & \qquad \sum_{k=1}^{n} \langle C_k, P_k \rangle \\
\text{subject to} & \qquad \sum_{k=1}^{n} \langle A_{kt}, P_k \rangle \geq \hat{b}_t \; \; \forall \; t=1, \ldots, T\\
& \qquad P_k \in \mathcal{B} \; \; k=1, \ldots, n
\end{split}$$ where the quantity $\hat{b}_t$ is defined as $\hat{b}_t = \sum_{t \in N} \langle A_{kt}, P_k(\hat{\lambda})\rangle$ if $\hat{\lambda}_t>0$ and $b_t=\min \left\{ \sum_{t \in N} \langle A_{kt}, P_k(\hat{\lambda})\rangle, b_t\right\}$ if $\lambda_t =0$. Since $\hat{\lambda}, P_k(\hat{\lambda})$ satisfy the complementary slackness conditions for by construction, they are optimal solutions to it. Note that in general the solution of the sampled linear program could be fractional (i.e. $\hat{P}_k$ for some members of the Birkhoff polytope). However, as a consequence of Lemma \[lemma:wt\_hung\], it is in fact the case that $\hat{P}_k=P_k(\hat{\lambda})$ are optimal.
Suppose $\hat{\lambda}_t >0$, then the $t^{th}$ primal constraint is tight in the sampled linear program, i.e. $\sum_{k \in S} \langle A_{kt}, P_k(\hat{\lambda})\rangle = (1-\epsilon)\epsilon b_t$. By applying an argument along similar lines to part $2$ above and applying concentration, it follows that with probability at least $1-\epsilon$, $$\hat{b}_t = \sum_{t \in N} \langle A_{kt}, P_k(\hat{\lambda})\rangle \leq b_t.$$
Let $P_k^*$ be the optimal solution to . It follows that $P_k^*$ are also feasible solutions to . As a consequence, $\sum_{k \in N}\langle C_k, P_k(\hat{\lambda}) \rangle \geq \text{OPT}$, where OPT refers to the objective function value of the optimal solution to .
By again applying a concentration argument as above, we can argue that with probability exceeding $1-\epsilon$, $$\sum_{k \in N \setminus S}\langle C_k, P_k(\hat{\lambda}) \rangle \geq (1-\epsilon)\sum_{k \in N }\langle C_k, P_k(\hat{\lambda}) \geq (1-\epsilon)\text{OPT}.$$ Since the returns from the first $\hat{n}$ sessions are non-negative (under an arbitrary assignment), the online algorithm returns a reward of at least $(1-\epsilon)\text{OPT}$.
[^1]: As an aside, we note that in practice, $C_k$ and $A_{kt}$ are predictions obtained from a machine learned model, and the engagement (i.e., dwell-time and clicks) realized will likely be different from the predictions. We ignore this distinction in this paper, and make the simplifying assumption that the predictions are “perfect”. When the models are consistent and a sufficiently large number of sessions are involved, the expected performance (being optimized here) will be close to that of the realized one.
[^2]: The greedy algorithm is simply the following: sort the edges of the bipartite graph in decreasing order. We maintain a set of edges $M$ (initialized to be empty), and loop over the sorted edges. In each step, add the corresponding edge to $M$ if it is possible to do so while maintaining the requirement that $M$ is a (possibly partial) matching.
|
---
abstract: 'We present an analytical solution for the tunneling process in a piecewise linear and quadratic potential which does not make use of the thin-wall approximation. A quadratic potential allows for smooth attachment of various slopes exiting into the final minimum of a realistic potential. Our tunneling solution thus serves as a realistic approximation to situations such as populating a landscape of slow-roll inflationary regions by tunneling, and it is valid for all regimes of the barrier parameters. We shortly comment on the inclusion of gravity.'
author:
- Koushik Dutta
- 'Pascal M. Vaudrevange'
- Alexander Westphal
bibliography:
- 'tunneling\_linear\_quadratic\_v1.bib'
date: 'February 23, 2011'
title: An Exact Tunneling Solution in a Simple Realistic Landscape
---
Vacuum decay is one of the most drastic environmental changes in field theory, which often proceeds via nucleating a bubble of true vacuum inside a sea of false vacuum by a quantum mechanical tunneling event. Tunneling processes (in first order phase transitions) play a vital role in many aspects of high-energy theory and cosmology. The groundwork for the computation of tunneling amplitudes was laid many years ago by Coleman and de Luccia (CdL) [@Coleman:1977py; @Coleman:1980aw]. Their method uses the Euclidean path integral to calculate what is now known as the CdL instanton, often appearing to be the stationary point of minimal action. The CdL analysis consist of a single real scalar field in a potential with a false and a true vacuum located at the position of the corresponding minima of the scalar potential, $\phi_+$ and $\phi_-$ respectively, see Figure \[fig:potential\_shape\]. The tunneling probability per unit volume $\Gamma/V = A e^{-B}\,,$ can be conveniently expressed [^1] in terms of the Euclidean action $S_E$ of the O(4) symmetric so-called bounce solution $\phi_{{\text Bounce}}$ as $B =
S_E[\phi_{{\text Bounce}}] - S_E[\phi_+]\,$. In the interior of the nucleating bubble, the scalar field exits at some point on the slope towards the minimum, while outside of the bubble, beyond the bubble wall, it still sits in the false minimum.
In general, the equation of motion for the bounce is difficult to solve analytically. This lead to the development of the thin-wall approximation [@Coleman:1977py]. In this limit of small energy difference between the true and false vacuum, the bubble wall becomes infinitely thin, and an approximate solution can be found.
To the best of our knowledge, an exact tunneling solution is only known in the case of a piecewise linear potential [@Duncan:1992ai]. In this article, we will derive the analytic tunneling solution for the piecewise linear and quadratic potential $$\begin{aligned}
V(\phi)&=&\left\{\begin{array}{cl}
V_L\equiv \lambda_+ \phi+V_{-}+\frac{1}{2}m^2\phi_-^2\,,&\phi<0\\
V_R\equiv \frac{1}{2}m^2(\phi-\phi_-)^2+V_{-}\,,&\phi\ge0
\end{array}\right. \label{eq:potential}\,,\end{aligned}$$ (see Fig \[fig:potential\_shape\]) which does not make use of the thin-wall approximation, see also recent work by [@Gen:1999gi]. However, one (non-differential) equation remains to be solved either numerically or in the limits of a large and small bubble radius. We show that our general analytical results reduce to the thin-wall results and the results of [@Duncan:1992ai] in the appropriate limits. For the most part of this work, we exclude the effects of gravity and thus ignore the magnitude of the true vacuum energy $V_{-}$. We will only offer some qualitative arguments about the inclusion of gravity in the end, except for the thin-wall limit.
We expect this tunneling solution to be the simplest yet realistic approximation to an arbitrarily shaped barrier exiting smoothly into a true minimum. The reason is that a quadratic potential towards the true vacuum allows us to attach to its critical point smoothly a given shallowly sloped region of scalar potential containing a minimum. This may be particularly relevant for studying the dynamics of populating a landscape of slow-roll inflationary slopes via tunneling from some false vacuum. Any discussion of the relative prevalence of different classes of inflationary models in a candidate fundamental theory such as string theory will necessarily have to include the discussion of the bias incurred by population via tunneling.
In the absence of gravity, the Euclidean action for a single scalar field with potential energy $V(\phi)$ is given by $$\begin{aligned}
\label{eq:full:S_E}
S_E&=&2\pi^2\int_{0}^{\infty}\!dr\,r^3\left(\frac{1}{2}\phi^{\prime^2}+V(\phi)\right)\,,\end{aligned}$$ and the bounce solution $\phi(r)$ is determined by solving the Euclidean $O(4)$ symmetric equations of motion $$\begin{aligned}
\label{eq:eom}
\phi^{{\prime\prime}}(r)+\frac{3}{r}\phi^\prime(r)&=&\partial_\phi V\,,\end{aligned}$$ where $\phi^\prime\equiv\partial_r\phi$ and the potential for our case is given by Eq. (\[eq:potential\]). Initially, the field is sitting in the false vacuum at $\phi_+<0$, kept in place by a linear potential piece (shown as a dashed line in Fig \[fig:potential\_shape\]) attached to the left of the false minimum. Its slope is irrelevant in the following computation as long as it classically stabilizes the field in the false minimum.
![Schematic shape of the potential Eq. (\[eq:potential\])[]{data-label="fig:potential_shape"}](potential){width="0.8\linewidth"}
To start with, we first depict the numerically solved bounce solution schematically in Fig \[fig:BounceSolution\]. The field sits at some value $\phi_0$ (not necessarily close to the true vacuum $\phi_-=1$ in this case but to the right of the maximum of the potential) inside the bubble at $r=0$. At the bubble radius $R_T$, the field crosses through the maximum of the potential at $\phi(R_T)=0$. Well outside the bubble, the field sits in the false vacuum $\phi(R_+)=\phi_+=-0.0001$.
{width="0.8\linewidth"}
\[fig:BounceSolution\]
The solutions $\phi_L$ and $\phi_R$ in the left and right part of the potential, respectively, have to fulfill the following boundary conditions: The bubble nucleates at rest $\phi_R(0) = \phi_0>0, \phi_R^\prime(0)=0\,.$ Outside of the bubble wall at finite $r> R_{+}$, the field sits at rest in the false vacuum $i.e$ $\phi_L(R_+) = \phi_+,
\phi_L^\prime(R_+)=0$. Solving Eq. (\[eq:eom\]) with these boundary conditions for both parts of the potential, we find $$\begin{aligned}
\label{eq:full:phi_sols}
\phi_L&=&\phi_++\frac{\lambda_+}{8r^2}\left(r^2-R_+^2\right)^2\,,\\
\phi_R&=&\phi_-+2(\phi_0-\phi_-)\frac{I_1(mr)}{mr}\,,\end{aligned}$$ where $I_1(z)$ is the modified Bessel function of the first kind (see also equation (3.14) in [@Coleman:1977py]).
In order to determine the constants $\phi_0, R_+$, and $R_T$, we use that the field configurations need to match smoothly at the bubble radius $R_T$ with $\phi_L(R_T) = \phi_R(R_T) = 0$ and $\phi_L^\prime(R_T) = \phi_R^\prime(R_T)$. Thus we need to solve $$\begin{aligned}
\label{eq:full:matching_conditions}
(R_+^2-R_T^2)^2&=&-\frac{8R_T^2\phi_+}{\lambda_+}\;,\; I_1(mR_T)=\frac{\phi_-\,mR_T}{2(\phi_- - \phi_0)} \;,\nonumber\\
R_+^4-R_T^4&=&\frac{8R_T^2}{\lambda_+}(\phi_--\phi_0)I_2(m R_T)\,.\end{aligned}$$ Calculating the Euclidean action Eq. (\[eq:full:S\_E\]) for the solutions Eq. (\[eq:full:phi\_sols\]), we can express both $\phi_0$ and $R_+$ in terms of $R_T$ to obtain $$\begin{aligned}
\label{eq:full:B:exact}
B&=&2\pi^2\phi_-^2R_T^2[\alpha^2 +\frac{1}{2}\left(\frac{4}{3}\alpha\sqrt{\Delta}+\frac{I_2(mR_T)}{I_1(mR_T)}\right)mR_T\nonumber\\
&&-\frac{(1 - \Delta)}{8}m^2R_T^2 ],\end{aligned}$$ where we have introduced $\alpha = -\phi_+/\phi_- > 0$, and $\Delta =
(-2\lambda_+\phi_+)/(m^2\phi_-^2)$ as a measure of the height of the potential barrier with values $0<\Delta<1$. In order to find $R_T$, we combine Eq. (\[eq:full:matching\_conditions\]) to get $$\begin{aligned}
\label{eq:full:RtEq}
2\alpha+\sqrt{\Delta}m R_T&=&m R_T \frac{I_2(m R_T)}{I_1(m R_T)}\,,\end{aligned}$$ which can be solved numerically. However, it is much more instructive to examine it in the limits of large and small $mR_T$. But first we should briefly note that Eq. (\[eq:full:RtEq\]) can be used to remove the Bessel functions from Eq. (\[eq:full:B:exact\]). We emphasize that the above expression for $B$ now depends on the potential parameters and the only unknown quantity $R_T$, which is fully determined in terms of $\alpha,\Delta$ by an algebraic Eq. (\[eq:full:RtEq\]). This is one of our central results.
Now we turn towards the two approximate solutions of our results. We start by taking the limit $m R_T \gg 1$, and in this limit, Eq. (\[eq:full:RtEq\]) can be solved as $$\begin{aligned}
\label{eq:largemRt:RtFull}
m R_T&=&\frac{3+4\alpha}{2}\frac{1}{1-\sqrt{\Delta}}\,.\end{aligned}$$ We note that for all allowed values of $\alpha$ and $\Delta$, $m
R_T > \frac{3}{2}$. In particular, we find that $mR_T$ can be large either for the thin-wall limit $\Delta\approx1$, or $\Delta<1$, but $\alpha>\Delta$ (see Eq. (\[eq:separatrix\])). For the latter case, the potential minima are separated by sizable distances in the field space, as well as the potential energy between the false and true vacuum. Therefore, the large $m R_T$ limit encompasses more than the just thin-wall solution, see Table \[Tab.1\]. For small differences in the vacuum energy $\epsilon =(1-\Delta)\frac{1}{2}m^2\phi_-^2$, in leading order of $\epsilon$, $$\begin{aligned}
\label{eq:largemRt:RtApprox}
m R_T& \approx&\frac{(3+4\alpha)m^2\phi_-^2}{2\epsilon}\,,\end{aligned}$$ which we will find to be identical to the result from using the thin-wall formalism. This is in accord with expectations from the thin-wall formalism: The radius of the nucleating bubble grows toward $\infty$ as the minima become degenerate. Also, looking at the limit of the matching condition $\phi_R=0$ in Eq. (\[eq:full:matching\_conditions\]) $$\begin{aligned}
\phi_0&=&\phi_-\left(1- \sqrt{\frac{\pi}{2}} (mR_T)^{3/2}e^{-mR_T}\right)\,,\end{aligned}$$ it is clear that the true vacuum bubble nucleates close to the minimum of the quadratic part of the potential, $\phi_0\sim\phi_{-}$, for large values of $m R_T$. Plugging in the expression for $R_T$ from Eq. (\[eq:largemRt:RtFull\]) yields an unwieldy result for $B$. Thus we only display the expression for $B$ in the $\epsilon\rightarrow0$ limit $$\begin{aligned}
B&\approx&\frac{\pi^2}{96 \epsilon^3}m^4\phi_-^8\left(3+4\alpha\right)^4\,.\label{eq:largemRt:B}\end{aligned}$$ Finally, we give an expression of $B$ in terms of $\phi_0$ $$\begin{aligned}
B&\approx&\frac{\pi^2\phi_-^2}{12m^2} \,(4\alpha+3)\cdot\ln^3\hspace{-0.3ex}\Big(1-\frac{\phi_0}{\phi_-}\Big)\,.\label{eq:smallmRt2:B}\end{aligned}$$ In summary, in this limit, when either the thin-wall approximations are valid or the minima are separated far away from each other, a large bubble nucleates close to the true vacuum.
For the sake of comparison, we shall now give the essential results of the thin-wall calculation, where the nucleation rate is given by $B =
\frac{27\pi^2S_1^4}{2\epsilon^3}$ and for the potential Eq. (\[eq:potential\]) we can compute $S_1$ directly, and in the small $\epsilon$ limit it is $$S_1 = -\frac{m}{2}\left(1 + \frac{4}{3} \alpha \right) \phi_{-}^2.$$ Plugging this into the expression for $B$ we find the same expression as Eq. (\[eq:largemRt:B\]). The same way, we get $R_T=3S_1/\epsilon$ which, after plugging in $S_1$ and expanding in $\epsilon$, agrees with .
We will now proceed to the opposite case of taking the limit of small $m R_T \ll 1$. To second order in $mR_T$, Eq. (\[eq:full:RtEq\]) is solved by $$\begin{aligned}
\label{eq:smallmRt2:Rt}
mR_T&=&2\left(\sqrt{\Delta}+\sqrt{2\alpha+\Delta}\right)\,,\end{aligned}$$ where we discarded the negative solution. Now $mR_T<\mu_T$ can be equivalently casted as $ (0<\alpha < \mu_T^2/8) \wedge (0<\Delta < (\mu_T^2-8\alpha)^2/16\mu_T^2\,.$ Therefore small $m R_T$ limit corresponds to the small values of $\alpha$ and $\Delta$, where potential minima are closely spaced in the field space, but the potential difference between the false and the true minima is considerably large. In this limit, the exit point of the bubble can be conveniently expressed as $$\phi_{0} = \phi_{-}\left( 1 + \frac{8}{m R_T}\right)^{-1},$$ and it shows that the bubble nucleates close to the tip of potential barrier which is intuitively understandable from the form the potential in this particular limit. We can express $B$ for small $mR_T$ in terms of $\alpha, \Delta$ as $$\begin{aligned}
\label{eq:smallmRt2:B:Rt}
B&=&\frac{16\pi^2}{3}\frac{\phi_-^2}{m^2}\left(2\alpha+\Delta\right)\\
&&\times\left[(\alpha+\Delta)^2+\Delta^2\left(1+\frac{2\alpha}{\Delta}\right)\left(1+\sqrt{1+\frac{2\alpha}{\Delta}}\right)\right].\nonumber\end{aligned}$$ Again, we give an expression of $B$ in terms of $\phi_0$ $$\begin{aligned}
B\approx \frac{16\pi^2\phi_-^2}{m^2}\left[\alpha^2\frac{\phi_0}{\phi_-}+\frac{4\alpha\sqrt{2\Delta}}{3}\left(\frac{\phi_0}{\phi_-}\right)^{\hspace*{-0.5ex}3/2}\hspace*{-2ex}+\Delta\left(\frac{\phi_0}{\phi_-}\right)^{2}\right]\label{eq:smallmRt2:B:phi0}\end{aligned}$$
In the limit where the quadratic part of the potential can be regarded as almost flat, the solution we found for small $mR_T$ should agree with the results of a piecewise linear potential as studied by [@Duncan:1992ai]. This is the case for the limit $\Delta\ll1$. Taking the slope of the quadratic potential to be $\lambda_-=m^2\phi_-$, the parameter $c$ defined by [@Duncan:1992ai] becomes $c=2\alpha/\Delta$. The value of the tunneling radius $m R_T$ found by [@Duncan:1992ai] is given by $$\begin{aligned}
m R_T=\frac{4\alpha}{\sqrt{2\alpha+\Delta}-\sqrt{\Delta}}\end{aligned}$$ in agreement with the corresponding expansion of Eq. (\[eq:smallmRt2:Rt\]). Expanding the expression for $B$ in Eq. (\[eq:full:B:exact\]), we find perfect agreement with $B$ from [@Duncan:1992ai].
a){width="0.31\linewidth"} b){width="0.31\linewidth"} c){width="0.32\linewidth"}
\[fig:inclusion\_of\_gravity:BB0overRt\] \[fig:RelErrorB\] \[fig:RelErrormRt\]
$mR_T$ $B_{exact}$ $B_{DJ}$ $B_{thin-wall}$
--------------- -------- ------------- ---------- -----------------
$\alpha=0.01$ 0.6 0.0023 0.0022 72.4
$\alpha=0.1$ 1.2 0.3 0.04 113.3
$\alpha=0.5$ 2.5 23.9 0.6 529.8
: The mismatch of action: the linearized result $B_{DJ}$ of [@Duncan:1992ai] compared to the thin-wall result $B_{thin-wall}$ and the exact result found here. $B$ is given in units of $\phi_-^2/m^2$ and all values are quoted for $\Delta=0.01$. Clearly, there are regimes with $mR_T={\cal O}(1)$ where neither the linearized treatment of [@Duncan:1992ai] nor the thin-wall approximation are sufficient.[]{data-label="Tab.1"}
Let us recall what has been done so far. We reduced the computation of the tunneling amplitude for tunneling from the linear to the quadratic part of the potential Eq. (\[eq:potential\]) to solving an exact Eq. (\[eq:full:RtEq\]). We provided approximate solutions for the limits of large and small tunnel radius $mR_T^{L,S}$. Surprisingly, the smaller of the relative error between either the numerical result $R_T^N$ and $R_T^L$ or $R_T^S$ is always smaller than $10\%$, see Fig. \[fig:RelErrormRt\] (a). This then defines globally the approximate solution $mR_T$ $$\begin{aligned}
m R_T=\left\{\begin{array}{cl}
2\sqrt{\Delta}\left(1+\sqrt{1+\frac{2\alpha}{\Delta}}\right),& \Delta<(0.8\alpha-0.5)^2\\ & \\
\frac{3+4\alpha}{2}\frac{1}{1-\sqrt{\Delta}}\;,&(0.8\alpha-0.5)^2<\Delta
\end{array}
\right.\label{eq:separatrix}\end{aligned}$$ In other words, we succeeded in computing the tunneling amplitude analytically in a non thin-wall solution with error better than $50\%$, see Fig. \[fig:RelErrormRt\] (b).
Finally we shall comment on the effects of gravity on the tunneling process. In the thin-wall limit [@Coleman:1980aw] studied the case where either true or false vacuum are at zero energy, whereas [@Parke:1982pm] examined this issue for arbitrary values of the false and true vacuum energy. Among other things, their work suggests the following intuitive idea. When the true vacuum is also in de-Sitter space, due to the “pull” on the bubble both from the inside and the outside of the wall, the nucleation rate should be enhanced compared to the flat space limit. We can see this explicitly in the thin-wall approximation, [*i.e.*]{} large $mR_T$ and $1-\Delta\ll 1$, where the effect of gravity is well understood. Following [@Parke:1982pm], we compute the effect of gravity on the exponent $B$ of the tunnel rate, see Fig. \[fig:inclusion\_of\_gravity:BB0overRt\] (c).
For sufficiently sub-Planckian values of $\phi_{-}$ we always have a constant asymptotic value for $B/B_0$, where $B_{0}$ is without gravity. In the case of small $\eta_- \equiv m^2/V_{-}$ ($i.e$ a very flat quadratic part) the ratio $B/B_0$ is very small, giving a large gravitational enhancement of tunneling. Contrary, for a steep potential with large $\eta_-$ the tunneling probability is less enhanced. Increasing $\alpha$ suppresses $B$ further and thus enhances tunneling. This fits with the notion, that the gravitational correction is more important the thicker the barrier is in the field space, since for a given $\phi_-$ the barrier thickness increases with increasing $\alpha$. Note, that for $\alpha\gg 1$ and $\eta_-\ll1$ there is also Hawking-Moss tunneling possible [@Hawking:1981fz; @Starobinsky:1986fx; @Linde:1991sk].
In this work we have discussed quantum tunneling in field theory in a piecewise linear and quadratic scalar potential with a a false vacuum and a true vacuum. Such a potential is arguably the most simple yet realistic approximation to an arbitrarily shaped barrier exiting smoothly into a true minimum. The reason is that a quadratic potential allows us to attach to its critical point smoothly a given shallowly sloped region of scalar potential containing a minimum. Our result gives the tunneling rate in this situation exactly. Further, in the appropriate limits our result reduces to either the thin-wall result or the known result for piecewise linear potentials by [@Duncan:1992ai]. However, there is a large region of barrier shape parameter space where neither of them is a good approximation to the full solution given here. The inclusion of further effects from approximating a given potential to higher than quadratic order, as well as a detailed incorporation of gravity in the non-thin-wall regime we leave as a topic for future work. Let us note in passing that in the context of meta-stable supersymmetry breaking vacua in gauge theories [@Intriligator:2006dd], choosing $N_f=3 N_c/2$ flavors can imply a barrier shape with $\alpha\ll\Delta\simeq 2/3$. For $\alpha=0.1$ this gives $mR_T\simeq 9$ which shows the linearized result of [@Duncan:1992ai] with $B_{DJ}\simeq2.5\times 10^2$ overestimating the tunneling rate by an order of magnitude compared to our exact result $B\simeq 1.4\times 10^3$. Finally, we expect one relevant application of our results to be the dynamics of populating a landscape of slow-roll inflationary slopes via tunneling from some false vacuum. For this purpose, we have given the instanton action for the tunneling also as a function of $\phi_0$, as the exit from the tunneling $\phi=\phi_0,\dot\phi_0=0$ forms the initial conditions for slow-roll inflation.
[**Note added:**]{} While this paper was being finished, we became aware of [@Pastras:2011zr], which overlaps with our results.
[**Acknowledgements:**]{} We are grateful to A. Linde for enlightening comments. This work was supported by the Impuls und Vernetzungsfond of the Helmholtz Association of German Research Centres under grant HZ-NG-603, and German Science Foundation (DFG) within the Collaborative Research Center 676 ÔParticles, Strings and the Early UniverseÕ.
[^1]: $A$ is of order $m^4$ with $m$ being the characteristic scale of the problem [@Callan:1977pt], which we neglect here for simplicity.
|
---
abstract: 'Prediction of future states of the environment and interacting agents is a key competence required for autonomous agents to operate successfully in the real world. Prior work for structured sequence prediction based on latent variable models imposes a uni-modal standard Gaussian prior on the latent variables. This induces a strong model bias which makes it challenging to fully capture the multi-modality of the distribution of the future states. In this work, we introduce *Conditional Flow Variational Autoencoders (CF-VAE)* using our novel conditional normalizing flow based prior to capture complex multi-modal conditional distributions for effective structured sequence prediction. Moreover, we propose two novel regularization schemes which stabilizes training and deals with posterior collapse for stable training and better fit to the target data distribution. Our experiments on three multi-modal structured sequence prediction datasets – MNIST Sequences, Stanford Drone and HighD – show that the proposed method obtains state of art results across different evaluation metrics.'
author:
- |
Apratim Bhattacharyya [^1] Michael Hanselmann [^2] Mario Fritz [^3] Bernt Schiele\
Christoph-Nikolas Straehle
bibliography:
- 'iclr2020\_conference.bib'
title: Conditional Flow Variational Autoencoders for Structured Sequence Prediction
---
Introduction
============
Anticipating future states of the environment is a key competence necessary for the success of autonomous agents. In complex real world environments, the future is highly uncertain. Therefore, structured predictions, one to many mappings [@sohn2015learning; @bhattacharyya2018accurate] of the likely future states of the world, are important. In many scenarios, these tasks can be cast as sequence prediction problems. Particularly, Conditional Variational Autoencoders (CVAE) [@sohn2015learning] have been successful for such problems – from prediction of future pedestrians trajectories [@lee2017desire; @bhattacharyya2018accurate; @pajouheshgar2018back] to outcomes of robotic actions [@babaeizadeh2017stochastic]. The distribution of future sequences is diverse and highly multi-modal. CVAEs model diverse futures by factorizing the distribution of future states using a set of latent variables which are mapped to likely future states. However, CVAEs assume a standard Gaussian prior on the latent variables which induces a strong model bias [@hoffman2016elbo; @tomczak2017vae] which makes it challenging to capture multi-modal distributions. This also leads to missing modes due to posterior collapse [@bowman2015generating; @razavi2019preventing].
Recent work [@tomczak2017vae; @wang2017diverse; @gu2018dialogwae] has therefore focused on more expressive Gaussian mixture based priors. However, Gaussian mixtures still have limited expressiveness and optimization suffers from complications e.g. determining the number of mixture components. In contrast, normalizing flows are more expressive and enable the modelling of complex multi-modal priors. Recent work on flow based priors [@chen2016variational; @ziegler2019latent], have focused only on the unconditional (plain VAE) case. However, this not sufficient for CVAEs because in the conditional case the complexity of the distributions are highly dependent on the condition.
[Latent Prior]{} [Latent Prior]{}
------------------ -- ------------------ -- -- --
In this work,
We propose *Conditional Flow Variational Autoencoders (CF-VAE)* based on novel conditional normalizing flow based priors In order to model complex multi-modal conditional distributions over sequences. In , we show example predictions of MNIST handwriting stroke of our CF-VAE. We observe that, given a starting stroke, our CF-VAE model with data dependent normalizing flow based latent prior captures the two main modes of the conditional distribution – i.e. 1 and 8 – while CVAEs with fixed uni-modal Gaussian prior predictions have limited diversity.
We propose a novel regularization scheme that stabilizes the optimization of the evidence lower bound and leads to better fit to the target data distribution.
We leverage our conditional flow prior to deal with posterior collapse which causes standard CVAEs to ignore modes in sequence prediction tasks.
Finally, our method outperforms the state of the art on three structured sequence prediction tasks – handwriting stroke prediction on MNIST, trajectory prediction on Stanford Drone and HighD.
Related Work {#sec:relatedwork}
============
[**Normalizing Flows.**]{} Normalizing flows are a powerful class of density estimation methods with exact inference. [@dinh2014nice] introduced affine normalizing flows with triangular Jacobians. [@dinh2016density] extend flows with masked convolutions which allow for complex (non-autoregessive) dependence between the dimensions. In [@kingma2018glow], $1 \times 1$ convolutions were proposed for improved image generation compared to [@dinh2016density]. In [@huang2018neural] normalizing flows are auto-regressive and [@behrmann2018invertible] extend it to ResNet. [@lu2019structured] extended normalizing flows to model conditional distributions. Here, we propose conditional normalizing flows to learn conditional priors for variational latent models.
[**Variational Autoencoders.**]{} The original variational autoencoder [@kingma2013auto] used uni-modal Gaussian prior and posterior distributions. Thereafter, two lines of work have focused on developing either more expressive prior or posterior distributions. [@rezende2015variational] propose normalizing flows to model complex posterior distributions. [@kingma2016improved; @tomczak2016improving; @berg2018sylvester] present more complex inverse autoregessive flows, householder and Sylvester normalizing flow based posteriors. Here, we focus on the orthogonal direction of more expressive priors and the above approaches are compatible with our approach.
Recent work which focus more expressive priors include [@nalisnick2016stick] which proposes a Dirichlet process prior and [@goyal2017nonparametric] which proposes a nested Chinese restaurant process prior. However, these methods require sophisticated learning methods. In contrast, [@tomczak2017vae] proposes a mixture of Gaussians based prior (with fixed number of components) which is easier to train and shows promising results on some image generation tasks. [@chen2016variational], proposes a inverse autoregressive flow based prior which leads to improvements in complex image generation tasks like CIFAR-10. [@ziegler2019latent] proposes a prior for VAE based text generation using complex non-linear flows which allows for complex multi-modal priors. While these works focus on unconditional priors, we aim to develop more expressive conditional priors.
[**Posterior Collapse.**]{} Posterior collapse arises when the latent posterior does not encode useful information. Most prior work [@yang2017improved; @dieng2018avoiding; @higgins2017beta] concentrate on unconditional VAEs and modify the training objective – the KL divergence term is annealed to prevent collapse to the prior. [@liu2019cyclical] extends KL annealing to CVAEs. However, KL annealing does not optimize a true lower bound of the ELBO for most of training. [@zhao2017infovae] also modifies the objective to choose the model with the maximal rate. [@razavi2019preventing] propose anti-causal sequential priors for text modelling tasks. [@bowman2015generating; @gulrajani2016pixelvae] proposes to weaken the decoder so that the latent variables cannot be ignored, however only unconditional VAEs are considered. [@wang2019riemannian] shows the advantage of normalizing flow based posteriors for preventing posterior collapse. In contrast, we study for the first time posterior collapse in conditional models on datasets with minor modes.
[**Structured Sequence Prediction.**]{} [@helbing1995social; @robicquet2016learning; @alahi2016social; @gupta2018social; @zhao2019multi; @sadeghian2018sophie] consider the problem of traffic participant trajectory prediction in a social context. Notably, [@gupta2018social; @zhao2019multi; @sadeghian2018sophie] use generative adversarial networks to generate socially compliant trajectories. However, the predictions are uni-modal. [@lee2017desire; @bhattacharyya2018accurate; @rhinehart2018r2p2; @deo2019scene; @pajouheshgar2018back] considers structured (one to many) predictions using – a CVAE, improved CVAE training, pushforward policies for vehicle ego-motion prediction, motion planning, spatio-temporal convolutional network respectively. [@kumar2019videoflow] proposes a normalizing flow based model for video sequence prediction, however the sequences considered have very limited diversity compared to the trajectory prediction tasks considered here. Here, we focus on improving structured predictions using conditional normalizing flows based priors.
Conditional Flow Variational Autoencoder (CF-VAE) {#sec:cvae}
=================================================
Our Conditional Flow Variational Autoencoder is based on the conditional variational autoencoder [@sohn2015learning] which is a deep directed graphical model for modeling conditional data distributions $p_{\theta}(\text{y}|\text{x})$. Here, $\text{x}$ is the sequence up to time $t$, $x = \left[ x^{1}, \cdots, x^{t} \right]$ and $\text{y}$ is the sequence to be predicted up to time $T$, $y = \left[ y^{t+1}, \cdots, y^{T} \right]$. CVAEs factorize the conditional distribution using latent variables $\text{z}$ – $p_{\theta}(\text{y}|\text{x})$ is factorized as $p_{\theta}(\text{y}|\text{z}, \text{x}) p(\text{z}|\text{x})$, where $p(\text{z}|\text{x})$ is the prior on the latent variables. During training, amortized variational inference is used and the posterior distribution $q_{\phi}(\text{z}|\text{x},\text{y})$ is learnt using a recognition network. The ELBO is maximized, given by, $$\begin{aligned}
\label{eq:cvae}
\log(p_{\theta}(\text{y}|\text{x})) \geq \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})} \log(p_{\theta}(\text{y}|\text{z},\text{x})) - D_{\text{KL}}( q_{\phi}(\text{z}|\text{x},\text{y}) || p(\text{z}|\text{x})). \end{aligned}$$ In practice, to simplify learning, simple unconditional standard Gaussian priors are used [@sohn2015learning]. However, the complexity e.g. the number of modes of the target distributions $p_{\theta}(\text{y}|\text{x})$, is highly dependent upon the condition $x$. An unconditional prior demands identical latent distributions irrespective complexity of the target conditional distribution – a very strong constraint on the recognition network. Moreover, the latent variables cannot encode any conditioning information and this leaves the burden of learning the dependence on the condition completely on the decoder.
Furthermore, on complex conditional multi-modal data, Gaussian priors have been shown to induce a strong model bias [@tomczak2016improving; @ziegler2019latent]. It becomes increasingly difficult to map complex multi-modal distributions to uni-modal Gaussian distributions, further complicated by the sensitivity of the RNNs encoder/decoders to subtle variations in the hidden states [@bowman2015generating]. Moreover, the standard closed form estimate of the KL-divergence pushes the encoded latent distributions to the mean of the Gaussian leading to latent variable collapse [@wang2017diverse; @gu2018dialogwae] while discriminator based approaches [@tolstikhin2017wasserstein] lead to underestimates of the KL-divergence [@rosca2017variational].
Therefore, we propose conditional priors based on conditional normalizing flows to enable the latent variables to encode conditional information and allow for complex multi-modal latent representations. Next, we introduce our novel conditional non-linear normalizing flows followed by our novel regularized Conditional Flow Variational Autoencoder (CF-VAE) formulation.
Conditional Normalizing Flows
-----------------------------
Recently, normalizing flow [@tabak2010density; @dinh2014nice] based priors for VAEs have been proposed [@chen2016variational; @ziegler2019latent]. Normalizing flows allows for complex priors by transforming a simple base density e.g. standard Gaussian to a complex multi-modal density through a series of $n$ layers of invertible transformations $f_{i}$, $$\begin{aligned}
\label{eq:unflowtrans}
\epsilon \overset{f_{1}}{\longleftrightarrow} \text{h}_{1} \overset{f_{2}}{\longleftrightarrow} \text{h}_{2} \cdots \overset{f_{n}}{\longleftrightarrow} \text{z}.\end{aligned}$$
However, such flows cannot model conditional priors. In contrast to prior work, we utilize conditional normalizing flows to model complex conditional priors. Conditional normalizing flows also consists of a series of $n$ layers of invertible transformations $f_{i}$ (with parameters $\psi$), however we modify the transformations $f_{i}$ such that they are dependent on the condition $\text{x}$, $$\begin{aligned}
\label{eq:flowtrans}
\epsilon | \text{x} \overset{f_{1} | \text{x} }{\longleftrightarrow} \text{h}_{1} | \text{x} \overset{f_{2} | \text{x} }{\longleftrightarrow} \text{h}_{2} | \text{x} \cdots \overset{f_{n} | \text{x} }{\longleftrightarrow} \text{z} | \text{x}.\end{aligned}$$
Further, in contrast to prior work [@lu2019structured; @atanov2019semi; @ardizzone2018analyzing] which use affine flows ($f_{i}$), we build upon [@ziegler2019latent] and introduce conditional non-linear normalizing flows with split coupling. Split couplings ensure invertibility by applying a flow layer $f_{i}$ on only half of the dimensions at a time. To compute (\[eq:cflow\]), we split the dimensions $\text{z}^{D}$ of the latent variable into halfs, $\text{z}^{L} = \left\{ 1, \cdots, \nicefrac{D}{2} \right\}$ and $\text{z}^{R} = \left\{ \nicefrac{D}{2}, \cdots, d \right\}$ at each invertible layer $f_{i}$. Our transformation takes the following form for each dimension $\text{z}^{j}$ alternatively from $\text{z}^{L}$ or $\text{z}^{R}$, $$\begin{aligned}
\label{eq:cnlsq}
f_{i}^{-1}(\text{z}^{j} | \text{z}^{R}, \text{x}) = \epsilon^{j} = a(\text{z}^{R}, \text{x}) + b(\text{z}^{R}, \text{x}) \times \text{z}^{j} + \frac{c(\text{z}^{R}, \text{x})}{1 + (d(\text{z}^{R}, \text{x}) \times \text{z}^{j} + g\big(\text{z}^{R}, \text{x})\big)^2}.\end{aligned}$$ where, $\text{z}^{j} \in \text{z}^{L}$. Details of the forward (generating) operation $f_{i}$ are in Appendix A. To ensure that the generated prior distribution is conditioned on $\text{x}$, in (\[eq:cnlsq\]) and in the corresponding forward operation $f_{i}$, the coefficients $\left\{a,b,c,d,g\right\} \in \mathbb{R}$ are functions of both the other half of the dimensions of $\text{z}$ *and* the condition $\text{x}$ (unlike [@ziegler2019latent]). Finally, due to the expressive power of our conditional non-linear normalizing flows, simple spherical Gaussians base distributions were sufficient.
Variational Inference using Conditional Normalizing Flows based Priors
----------------------------------------------------------------------
Here, we derive the ELBO (\[eq:cvae\]) for our novel regularized CF-VAE with our conditional flow based prior. In case of the standard CVAE with the Gaussian prior, the KL divergence term in the ELBO has a simple closed form expression. In case of our conditional flow based prior, we can use the change of variables formula to compute the KL divergence. In detail, given the base density $p(\epsilon|\text{x})$ and the Jacobian $J_{i}$ of each layer $i$ of the transformation, the log-likelihood of the latent variable $\text{z}$ under the prior can be expressed using the change of variables formula, $$\begin{aligned}
\label{eq:cflow}
\log(p_{\psi}(\text{z}|\text{x})) = \log(p(\epsilon|\text{x})) + \sum\limits_{i=1}^{n} \log( \lvert \det J_{i} \lvert ).\end{aligned}$$
This change of variables allows us to evaluate the likelihood of latent variable $\text{z}$ over the base distribution instead of the complex conditional prior and to express the KL divergence as, $$\begin{aligned}
\label{eq:cfkl}
\begin{split}
- D_{\text{KL}}( q_{\phi}(\text{z}|\text{x},\text{y}) || p_{\psi}(\text{z}|\text{x})) &= - \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})} \log(q_{\phi}(\text{z}|\text{x},\text{y})) + \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})} \log(p_{\psi}(\text{z}|\text{x}))\\
&= \mathcal{H}(q_{\phi}) + \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})}\log(p(\epsilon|\text{x})) + \sum\limits_{i=1}^{n} \log( \lvert \det J_{i} \lvert) .
\end{split}\end{aligned}$$
where, $\mathcal{H}(q_{\phi})$ is the entropy of the variational distribution. Therefore, the ELBO can be expressed as, $$\begin{aligned}
\label{eq:cfcvae}
\log(p_{\theta}(\text{y}|\text{x})) \geq \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})} \log(p_{\theta}(\text{y}|\text{z},\text{x})) + \mathcal{H}(q_{\phi}) + \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})}\log(p(\epsilon|\text{x})) + \sum\limits_{i=1}^{n} \log( \lvert \det J_{i} \lvert)\end{aligned}$$
[r]{}[0.4]{}
{width="\textwidth"}
To learn complex conditional priors, we alternately optimize both the variational posterior distribution $q_{\phi}(\text{z}|\text{x},\text{y})$ and the conditional prior $p_{\psi}(\text{z}|\text{x})$ in (\[eq:cfcvae\]). This would allow the variational posterior $q_{\theta}$ to match the conditional prior and vice-versa so that the ELBO (\[eq:cfcvae\]) is maximized. However, in practice we observe instabilities during training and posterior collapse. Next, we introduce our novel regularization schemes to deal with both these problems.
[**Posterior Regularization for Stability (pR).**]{} The entropy and the $\log$-Jacobian of the CF-VAE objective (\[eq:cfcvae\]) are at odds with each other. The $\log$-Jacobian favours the contraction of the base density. Therefore, $\log$-Jacobian at the right of (\[eq:cfcvae\]) is maximized when the conditional flow maps the base distribution ($\epsilon \leftrightarrow \text{z}$ in ) to a low entropy conditional prior and thus a low entropy variational distribution $q_{\phi}(\text{z}|\text{x},\text{y})$. Therefore, in practice we observe instabilities during training. We observe that either the entropy or the $\log$-Jacobian term dominates and the data log-likelihood is fully or partially ignored. Therefore, we regularize the posterior $q_{\phi}(\text{z}|\text{x},\text{y})$ by fixing the variance to $\text{C}$. This leads to a constant entropy term which in turn bounds the maximum possible amount of contraction, thus upper bounding the $\log$-Jacobian. This encourages our model to concentrate on explaining the data and leads better fit to the target data distribution. Note that, although $q_{\phi}(\text{z}|\text{x},\text{y})$ has fixed variance, this does not significantly effect expressivity as the marginal $q_{\phi}(\text{z}|\text{x})$ can be arbitrarily complex due to our conditional flow prior. Moreover, we observe that the LSTM based decoders employed demonstrate robust performance across a wide range of values $\text{C}=\left[ 0.05, 0.25\right]$.
[**Condition Regularization for Posterior Collapse (cR).**]{} We observe missing modes when the target conditional data distribution has a major mode(s) and one or more minor modes (corresponding to rare events). This is because the condition $\text{x}$ on the decoder is already enough to model the main mode(s). If the cost of ignoring the minor modes is out-weighed by the cost of encoding a more complex latent distribution reflecting all modes, the minor modes and the latent variables are ignored. We propose a novel regularization scheme by removing the additional conditioning $\text{x}$ on the decoder, when the dataset in question has a dominating mode(s). This enabled by our novel conditional flow prior, which already ensures that conditioning information can be encoded in the latent space. This assumes a simpler factorization of the conditional distribution $p_{\theta}(\text{y}|\text{x}) = p_{\theta}(\text{y}|\text{z}) p_{\psi}(\text{z}|\text{x})$. This ensures that the latent variable $\text{z}$ cannot be ignored by the CF-VAE and thus must encode useful information. Note that this regularization scheme is only possible due to our conditional prior, the unconditional Gaussian prior of CVAE would always need to condition the decoder. Finally, we discuss the integration of diverse sources of contextual information into the conditional prior $p_{\psi}(\text{z}|\text{x})$ for even richer conditional latent distributions of our regularized CF-VAE.
Conditioning Priors on Contextual Information {#sec:context}
---------------------------------------------
For prediction tasks, it is often crucial to integrate sources of contextual information e.g. past trajectories or environmental information for accurate predictions. As these sources are heterogeneous, we employ source specific networks to extract fixed length vectors from each source.
[**Past Trajectory.**]{} We encode the past trajectories using a LSTM to an fixed length vector $\text{x}_{t}$. For efficiency we share the condition encoder between the conditional flow and the CF-VAE decoder.
[**Environmental Map.**]{} We use a CNN to encode environmental information to a set of region specific feature vectors. We apply attention conditioned on the past trajectory to extract a fixed length conditioning vector $\text{x}_{m}$, such that $\text{x}_{m}$ contains information relevant to the future trajectory.
[**Interacting Agents.**]{} To encode information of interacting traffic participants/agents, we build on [@deo2018convolutional] and propose a fully convolutional social pooling layer. We aggregate information of interacting agents using a grid overlayed on the environment. This grid is represented using a tensor, where the past trajectory information of traffic participants are aggregated into the tensor indexed corresponding to the grid in the environment. In [@deo2018convolutional] past trajectory information is aggregated using a LSTM. We aggregate the past trajectory information into the tensor using $1\times 1$ convolutions as it allows for stable learning and is computationally efficient. Finally, we apply several layers of $k\times k$ convolutions to capture interaction aware contextual features $\text{x}_{p}$ of traffic participants in the scene.
Due to the expressive power of our conditional non-linear normalizing flows, simple concatenation into a single vector $\text{x} = \left\{ \text{x}_{t}, \text{x}_{m}, \text{x}_{t} \right\}$ was sufficient to learn powerful conditional priors.
Experiments {#sec:experiments}
===========
We evaluate our CF-VAE on three popular and highly multi-modal sequence prediction datasets. We begin with a description of our evaluation metrics and model architecture.
[**Evaluation Metrics.**]{} In line with prior work [@lee2017desire; @bhattacharyya2018accurate; @pajouheshgar2018back; @deo2019scene; @bhattacharyya2018bayesian], we use the negative conditional $\log$-likelihood (-CLL) and mean Euclidean distances of the oracle Top $n$% of $N$ predictions. The oracle Top $n$% metric measures not only the coverage of all modes but also discourages random guessing for a reasonably large value of $n$ (e.g. $n=10$%). This is because, a model can only improve this metric by moving randomly guessed samples from an overestimated mode to the correct modes (detailed analysis in Appendix F).
[**Conditional Flow Model Architecture.**]{} Our conditional flow prior consists of 16 layers of conditional non-linear flows with split coupling. Increasing the number of conditional non-linear flows generally led to “over-fitting” on the training latent distribution.
MNIST Sequences
---------------
The MNIST Sequence dataset [@mnist_seq] consists of sequences of handwriting strokes of the MNIST digits. The state-of-the-art approach is the “Best-of-Many”-CVAE [@bhattacharyya2018accurate] with a Gaussian prior. We follow the evaluation protocol of [@bhattacharyya2018accurate] and predict the complete stroke given the first ten steps. We also compare with,
A standard CVAE with uni-modal Gaussian prior;
A CVAE with a data dependent conditional mixture of Gaussians (MoG) prior;
A CF-VAE without any regularization ;
A CF-VAE without the conditional non-linear flow layers (CF-VAE**-***Affine*, replaced with affine flows [@lu2019structured; @atanov2019semi]).
We also experiment with a conditional MoG prior (see Appendix D and E). We use the same model architecture [@bhattacharyya2018accurate] across all baselines.
[r]{}[6cm]{}
\[tab:mnistseq\]
We report the results in . We see that our CF-VAE with posterior regularization (pR) performs best. It has a performance advantage of over 20% against the state of the art BMS-CVAE. We see that without regularization (pR) ($\text{C}=0.2$) there is a 40% drop in performance, highlighting the effectiveness of our novel regularization scheme. We further illustrate the modes captured and the learnt multi-modal conditional flow priors in . We do not use condition regularization here (cR) as we do not observe posterior collapse. In contrast, the BMS-CVAE is unable to fully capture all modes – its predictions are pushed to the mean due to the strong model bias induced by the Gaussian prior. The results improve considerably with the multi-modal MoG prior ($M=3$ components work best). We also experiment with optimizing the standard CVAE architecture. This improves performance only slightly (after increasing LSTM encoder/decoder units to 256 from 48, increasing the number of layers did not help). Moreover, our experiments with a conditional (MoG) AAE/WAE [@gu2018dialogwae] based baseline did not improve performance beyond the standard (MoG) CVAE, because the discriminator based KL estimate tends to be an underestimate [@rosca2017variational]. This illustrates that in practice it is difficult to map highly multi-modal sequences to a Gaussian prior and highlights the need of a data-dependent multi-modal priors. Our CF-VAE still significantly outperforms the MoG-CVAE as normalizing flows are better at learning complex multi-modal distributions [@kingma2018glow]. We also see that affine conditional flow based priors leads to a drop in performance (77.2 vs 74.9 CLL) illustrating the advantage of our non-linear conditional flows.
Stanford Drone
--------------
\[tab:stanford\_drone\_cross\]
The Stanford Drone dataset [@robicquet2016learning] consists of multi-model trajectories of traffic participant e.g. pedestrians, bicyclists, cars captured from a drone. Prior works follow two different evaluation protocols,
[@lee2017desire; @bhattacharyya2018accurate; @pajouheshgar2018back] use 5 fold cross validation,
[@robicquet2016learning; @sadeghian2018car; @sadeghian2018sophie; @deo2019scene] use a single split
. We evaluate using the first protocol in and the second in .
[r]{}[6cm]{}
Additionally, [@pajouheshgar2018back] suggest a “Shotgun” baseline. This baseline extrapolates the trajectory from the last known position and orientation in 10 different ways – 5 orientations: $(0^{\circ},\, \pm 8^{\circ},\, \pm 15^{\circ})$ and 5 velocities: None or exponentially weighted over the past with coefficients $(0, \, 0.3, \, 0.7, \, 1.0)$. This baseline obtains results at par with the state-of-the-art because it a good template which covers the most likely possible futures (modes) for traffic participant motion in this dataset. We report the results using 5 fold cross validation in . We additionally compare to a mixture of Gaussians prior (Appendix D). We use the same model architecture as in [@bhattacharyya2018accurate] and a CNN encoder with attention to extract features from the last observed RGB image (Appendix C). These visual features serve as additional conditioning ($\text{x}_{m}$) to our Conditional Flow model. We see that our CF-VAE model with RGB input and posterior regularization (pR) performs best – outperforming the state-of-art “Shotgun” and BMS-CVAE by over 20% (Error $@$ 4sec). We see that our conditional flows are able to utilize visual scene (RGB) information to improve performance (3.5 vs 3.6 Error $@$ 4sec). We also see that the MoG-CVAE and our CF-VAE + pR outperforms the BMS-CVAE, even without visual scene information. This again reinforces our claim that the standard Gaussian prior induces a strong model bias and data dependent multi-modal priors are needed for best performance. The performance advantage of CF-VAE over the MoG-CVAE again illustrates the advantage of normalizing flows at learning complex conditional multi-modal distributions. The performance advantage over the “Shotgun” baseline shows that our CF-VAE + pR not only learns to capture the correct modes but also generates more fine-grained predictions. The qualitative examples in shows that our CF-VAE is better able to capture complex trajectories with sharp turns.
We report results using the single train/test split of [@robicquet2016learning; @sadeghian2018car; @sadeghian2018sophie; @deo2019scene] in . We use the minimum Average Displacement Error (mADE) and minimum Final Displacement Error (mFDE) metrics as in [@deo2019scene]. The minimum is over as set of predictions of size $N$. Although this metric is less robust to random guessing compared to the Top $n$% metric, it avoids rewarding random guessing for a small enough value of $N$. We choose $N=20$ as in [@deo2019scene]. Similar to the results with 5 fold cross validation, we observe 20% improvement over the state-of-the-art.
HighD
-----
The HighD dataset [@highDdataset] consists of vehicle trajectories recorded using a drone over highways. In contrast to other vehicle trajectory datasets e.g. NGSIM it contains minimal false positive trajectory collisions or physically improvable velocities.
[r]{}[7.5cm]{}
The HighD dataset is challenging because lane changes or interactions are rare $\sim$ 10% of all trajectories. The distribution of future trajectories contain a single main mode (linear continuations) along with several minor modes. Thus, approaches which predict a single mean trajectory (targeting the main mode) are challenging to outperform. In , we see that the simple Feed Forward (FF) model performs well and the Graph Convolutional GAT model of [@diehl2019graph], which captures interactions, only narrowly outperforms the FF model. This dataset is challenging for CVAE based models as they frequently suffer from posterior collapse when a single mode dominates. This is clearly observed with our CVAE baseline in . To prevent posterior collapse, we use the cyclic KL annealing scheme proposed in [@liu2019cyclical] (using a MoG prior did not help). This already leads to significant improvement over the deterministic FF and GAT baselines. We also observe posterior collapse with our CF-VAE model. Therefore, we regularize by removing additional conditioning (cR). Our CF-VAE **+** $\left\{\text{pR,cR}\right\}$ with condition regularization significantly outperforms the CF-VAE **+** pR and CVAE baselines (with cyclic KL annealing), demonstrating the effectiveness of our condition regularization scheme (cR) in preventing posterior collapse. The addition of contextual information of interacting traffic participants using our convolutional social pooling network with 1$\times$1 convolutions significantly improves performance (also see Appendix G), demonstrating the effectiveness of our conditional normalizing flow based priors.
Conclusion {#sec:conclusion}
==========
In this work, we presented the first variational model for learning multi-modal conditional data distributions with Conditional Flow based priors – the Conditional Flow Variational Autoencoder (CF-VAE). Furthermore, we propose two novel regularization techniques – posterior regularization (pR) and condition regularization (cR) – which stabilizes training solutions and prevents posterior collapse leading to better fit to the target distribution. This techniques lead to better match to the target distribution. Our experiments on diverse sequence prediction datasets show that our CF-VAE achieves state-of-the-art results across different performance metrics.
Appendix A. Conditional Non-Linear Normalizing Flows {#appendix-a.-conditional-non-linear-normalizing-flows .unnumbered}
====================================================
In Subsection 3.1 of the main paper, we describe the inverse operation $f_{i}^{-1}$ of our non-linear conditional normalizing flows. Here, we describe the forward operation. Note that while the forward operation is necessary to compute the likelihood (3) (in the main paper) during training, the forward operation is necessary to sample from the latent prior distribution of our CF-VAE. The forward operation consists of solving for the roots of the following equation (more details in [@ziegler2019latent]), $$\begin{aligned}
\label{eq:cnlsq_f}
\begin{split}
& - b d^2 (\epsilon^{j})^3 + ((\text{z}^{j} - a) d^{2} - 2 d g b) (\epsilon^{j})^{2} \\
& +(2 d g (\text{z}^{j} - a) - b(g^{2} + 1)) \epsilon^{j} + ((\text{z}^{j} - a)(g^{2} + 1) - c) = 0
\end{split}\end{aligned}$$
This equation has one real root which can be found analytically [@holmes2002use]. As mentioned in the main paper, note that the coefficients $\left\{a,b,c,d,g\right\}$ are also functions of the condition $\text{x}$ (unlike [@ziegler2019latent]).
Appendix B. Additional Evaluation of Conditional Non-Linear Flows {#appendix-b.-additional-evaluation-of-conditional-non-linear-flows .unnumbered}
=================================================================
We compare conditional affine flows of [@atanov2019semi; @lu2019structured] and our conditional non-linear (Cond NL) flows in and . We plot the conditional distribution $p(\text{y} | \text{x})$ and the corresponding condition $\text{x}$ in the second and first columns. We use 8 and 16 layers of flow in case of the densities in and respectively. We see that the estimated density by the conditional affine flows of [@atanov2019semi; @lu2019structured] contains distinctive “tails” in case of and discontinuities in case of . In comparison our conditional non-linear flows does not have distinctive “tails” or discontinuities and is able to complex capture the multi-modal distributions better. Note, the “ring”-like distributions in cannot be well captured by more traditional methods like Mixture of Gaussians. We see in that even with 64 mixture components, the learnt density is not smooth in comparison to our conditional non-linear flows. This again demonstrates the advantage of our conditional non-linear flows.
Appendix C. Additional Details of Our Model Architectures {#appendix-c.-additional-details-of-our-model-architectures .unnumbered}
=========================================================
Here, we provide details of the model architectures used across the three datasets used in the main paper.
[**MNIST Sequences.**]{} We use the same model architecture as in [@bhattacharyya2018accurate]. The LSTM condition encoder on the input sequence $\text{x}$, the LSTM recognition network $q_{\theta}$ and the decoder LSTM network has 48 hidden neurons each. Also as in [@bhattacharyya2018accurate], we use a 64 dimensional latent space.
[**Stanford Drone.**]{} Again, we use the same model architecture as in [@bhattacharyya2018accurate] except for the CNN encoder. The LSTM condition encoder on the input sequence $\text{x}$ and the decoder LSTM network has 64 hidden neurons each. The LSTM recognition network $q_{\theta}$ has 128 hidden neurons. Also as in [@bhattacharyya2018accurate], we use a 64 dimensional latent space. Our CNN encoder has 6 convolutional layers of size 32, 64, 128, 256, 512 and 512. We predict the attention weights on the final feature vectors using the encoding of the LSTM condition encoder. The attention weighted feature vectors are passed through a final fully connected layer to obtain the final CNN encoding. Furthermore, we found it helpful to additionally encode the past trajectory as an image (as in [@pajouheshgar2018back]) as provide this as an additional channel to the CNN encoder.
[**HighD.**]{} We use the same model architecture with both the CVAE and CF-VAE models. As in the Stanford drone dataset, we use LSTM condition encoder on the input sequence $\text{x}$ and the decoder LSTM network with 64 hidden neurons each and the LSTM recognition network $q_{\theta}$ with 128 hidden neurons. The contextual information of interacting traffic participants are encoded into a spatial grid tensor of size 13$\times$3 (see Section 3.2 of the main paper). We use a CNN with 5 layers of sizes 64, 128, 256, 256 and 256 to extract contextual features.
Appendix D. Details of the mixture of Gaussians (MoG) baseline {#appendix-d.-details-of-the-mixture-of-gaussians-mog-baseline .unnumbered}
==============================================================
In the main paper, we include results on the MNIST Sequence and Stanford Drone dataset with a Mixture of Gaussians (MoG) prior. In detail, instead of a normalizing flow, we set the prior to a MoG form, $$\begin{aligned}
\label{eq:mog}
p_{\xi}(\text{z}|\text{x}) = \sum\limits_{i=1}^{M} p(\text{c}_{i} | \text{x}) \mathcal{N}(\text{z}; \mu_{i}, \sigma_{i} | \text{x}).\end{aligned}$$
We use a simple feed forward neural network that takes in the condition $\text{x}$ (see Section 3.4 of the main paper) and predicts the parameters of the MoG, $\xi = \left\{ \text{c}_{1}, \mu_{1}, \sigma_{1}, \cdots, \text{c}_{M}, \mu_{M}, \sigma_{M} \right\}$. Note, to ensure a reasonable number of parameters, we consider spherical Gaussians. Similar to (5) in the main paper, the ELBO can be expressed as,
$$\begin{aligned}
\label{eq:mog_elbo}
\log(p_{\theta}(\text{y}|\text{x})) \geq \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})} \log(p_{\theta}(\text{y}|\text{z},\text{x})) + \mathcal{H}(q_{\phi}) + \mathbb{E}_{q_{\phi}(\text{z}|\text{x},\text{y})}\log(p_{\xi}(\text{z}|\text{x})).\end{aligned}$$
Note that we fix the entropy of the posterior distribution $q_{\phi}$ for stability
Appendix E. Additional Evaluation on the MNIST Sequence Dataset {#appendix-e.-additional-evaluation-on-the-mnist-sequence-dataset .unnumbered}
===============================================================
Here, we perform a comprehensive evaluation using the MoG prior with varying mixture components. Moreover, we experiment with a CVAE with unconditional non-linear flow based prior (NL-CVAE). We report the results in .
Method -CLL $\downarrow$
---------------------------------- -------------------
NL-CVAE 107.6
CVAE ($M=1$) [@sohn2015learning] 96.4
MoG-CVAE, $M=2$ 85.3
MoG-CVAE, $M=3$ 84.6
MoG-CVAE, $M=4$ 85.7
MoG-CVAE, $M=5$ 86.3
CF-VAE + pR **74.9**
: Evaluation on MNIST Sequences (CLL: lower is better).[]{data-label="tab:mnistseq_mog"}
As mentioned in the main paper, we see that the MoG-CVAE outperforms the plain CVAE. This again reinforces our claim that the standard Gaussian prior induces a strong model bias. We see that using $M=3$ components with the variance of the posterior distribution fixed to $\text{C}=0.2$ leads to the best performance. This is expected as 3 is the most frequent number of possible strokes in the MNIST Sequence dataset. Also note that the results with the MoG prior are also relatively robust across $\text{C}=\left[ 0.05, 0.2 \right]$ as we learn the variance of the prior (see the section above). Finally, our CF-VAE + pR still significantly outperforms the MoG-CVAE (74.9 vs 84.6). This is expected as normalizing flows are more powerful compared to MoG at learning complex multi-modal distributions [@kingma2018glow] (also see ).
We also see that using an unconditional non-linear flow based prior actually harms performance (107.6 vs 96.4). This is because the latent distribution is highly dependent upon the condition. Therefore, without conditioning information the non-linear conditional flow learns a global representation of the latent space which leads to out-of-distribution samples at prediction time.
Appendix F. Evaluation of the Robustness of the Top n% Metric {#appendix-f.-evaluation-of-the-robustness-of-the-top-n-metric .unnumbered}
=============================================================
We use two simpler uniform “Shotgun” baselines to study the robustness of the Top n% metric against random guessing. In particular, we consider the “Shotgun”-u$90^{\circ}$ and “Shotgun”-u$135^{\circ}$ baselines which: given a budget of N predictions, it uniformly distributes the predictions between $(-90^{\circ},90^{\circ})$ and $(-135^{\circ},135^{\circ})$ respectively of the original orientation and using the velocity of the last time-step. In we compare the Top 1 (best guess) to Top 10% metric with N$={50,100,500}$ predictions.
[lccccc]{} Method & K & Error $@$ 1sec & Error $@$ 2sec & Error $@$ 3sec & Error $@$ 4sec\
&&\
------------------------------------------------------------------------
“Shotgun”-u$90^{\circ}$ & 50 & 0.9 & 1.9 & 3.1 & 4.4\
“Shotgun”-u$90^{\circ}$ & 100 & 0.9 & 1.9 & 3.0 & 4.3\
“Shotgun”-u$90^{\circ}$ & 500 & 0.9 & 1.9 & 3.0 & 4.3\
&&\
------------------------------------------------------------------------
“Shotgun”-u$90^{\circ}$ & 50 & 1.2 & 2.5 & 3.9 & 5.4\
“Shotgun”-u$90^{\circ}$ & 100 & 1.2 & 2.5 & 3.9 & 5.4\
“Shotgun”-u$90^{\circ}$ & 500 & 1.2 & 2.5 & 3.9 & 5.4\
------------------------------------------------------------------------
&&\
------------------------------------------------------------------------
“Shotgun”-u$135^{\circ}$ & 50 & 0.9 & 2.0 & 3.1 & 4.5\
“Shotgun”-u$135^{\circ}$ & 100 & 0.9 & 1.9 & 3.0 & 4.3\
“Shotgun”-u$135^{\circ}$ & 500 & 0.9 & 1.9 & 3.0 & 4.2\
&&\
------------------------------------------------------------------------
“Shotgun”-u$135^{\circ}$ & 50 & 1.4 & 2.9 & 4.5 & 6.2\
“Shotgun”-u$135^{\circ}$ & 100 & 1.4 & 2.9 & 4.5 & 6.2\
“Shotgun”-u$135^{\circ}$ & 500 & 1.4 & 2.9 & 4.5 & 6.2\
We see that in case of both the “Shotgun”-u$90^{\circ}$ and “Shotgun”-u$135^{\circ}$ baselines, the Top 1 (best guess) metric improves with increasing number of guesses. This effect is even more pronounced in case of the “Shotgun”-u$135^{\circ}$ baseline as the random guesses are distributed over a larger spatial range. In contrast, the Top 10% metric remains remarkably stable. This is because, in order to improve the Top 10% metric, random guessing is not enough – the predictions have to be on the correct modes. In other words, the only way to improve the Top 10% metric is move random predictions to any of the correct modes.
Appendix G. Qualitative Examples on the HighD Dataset {#appendix-g.-qualitative-examples-on-the-highd-dataset .unnumbered}
=====================================================
We show qualitative examples on the HighD dataset in . In the left of we show 128 random samples from the HighD test set. In the middle we show predictions on these samples by the CVAE (with cyclic Kl annealing [@liu2019cyclical]). We see that even with cyclic KL annealing, we observe posterior collapse. All samples have been pushed towards the mean and the variance in the 5 samples per test set example is minimal. E.g. note the top most sample track from the test set in (left). All CVAE sample predictions are a linear continuation of the trajectory (continuing on the same lane), while there is in fact a turn (change of lanes). In contrast, our CF-VAE + $\left\{\text{pR,cR}\right\}$ sample predictions are much more diverse and cover such eventualities. This also shows that our CF-VAE + $\left\{\text{pR,cR}\right\}$ does not suffer from such posterior variable collapse.
[^1]:
[^2]:
[^3]:
|
---
abstract: 'In this paper we analyse the propagation of warps in protostellar circumbinary discs. We use these systems as a test environment in which to study warp propagation in the bending-wave regime, with the addition of an external torque due to the binary gravitational potential. In particular, we want to test the linear regime, for which an analytic theory has been developed. In order to do so, we first compute analytically the steady state shape of an inviscid disc subject to the binary torques. The steady state tilt is a monotonically increasing function of radius, but misalignment is found at the disc inner edge. In the absence of viscosity, the disc does not present any twist. Then, we compare the time-dependent evolution of the warped disc calculated via the known linearised equations both with the analytic solutions and with full 3D numerical simulations. The simulations have been performed with the <span style="font-variant:small-caps;">phantom</span> SPH code using $2$ million particles. We find a good agreement both in the tilt and in the phase evolution for small inclinations, even at very low viscosities. Moreover, we have verified that the linearised equations are able to reproduce the diffusive behaviour when $\alpha>H/R$, where $\alpha$ is the disc viscosity parameter. Finally, we have used the 3D simulations to explore the non-linear regime. We observe a strongly non-linear behaviour, which leads to the breaking of the disc. Then, the inner disc starts precessing with its own precessional frequency. This behaviour has already been observed with numerical simulations in accretion discs around spinning black holes. The evolution of circumstellar accretion discs strongly depends on the warp evolution. Therefore the issue explored in this paper could be of fundamental importance in order to understand the evolution of accretion discs in crowded environments, when the gravitational interaction with other stars is highly likely, and in multiple systems. Moreover, the evolution of the angular momentum of the disc will affect the history of the angular momentum of forming planets.'
author:
- |
Stefano Facchini$^{1,2}$[^1], Giuseppe Lodato$^{1}$ and Daniel J. Price$^3$\
$^1$Dipartimento di Fisica, Università Degli Studi di Milano, Via Celoria, 16, Milano, 20133, Italy\
$^2$Institute of Astronomy, Madingley Road, Cambridge CB3 OHA\
$^3$Centre for Stellar and Planetary Astrophysics, School of Mathematical Sciences, Monash University, Clayton 3800, Australia.
bibliography:
- 'warp\_bib.bib'
date: Submission date
title: |
Wave-like warp propagation in circumbinary discs I.\
Analytic theory and numerical simulations
---
\[firstpage\]
accretion, accretion discs — protoplanetary discs — hydrodynamics.
Introduction {#sec:intro}
============
It is now well known that the majority of stars in star forming regions are in binary or higher order multiple systems [see e.g. the review by @mckee_ostriker07]. Moreover, in the last decade the possibility of detecting discs around multiple systems (and both stellar components in close binaries) has dramatically improved. These observations have shown that many stars of this kind do have circumstellar discs and evidence of accretion [e.g. @mathieu97]. Therefore, the probability of circumstellar or circumbinary discs around young stars is quite high.
If we focus on the circumbinary case, only a few circumbinary discs have been detected [@dutrey94; @CM04; @beust_dutrey05]. Indirect evidence of their past presence is the recent measure of circumbinary planets [@deeg08; @lee09; @beuermann10], some of which have been measured by Kepler [e.g. @doyle11; @welsh12; @orosz12].
In star forming regions, accretion discs are affected by gravitational interactions with the surroundings [@bate10]. These perturbations will strongly affect the evolution of the systems. In particular, the discs are likely to gain a warp, and in the case of multiple systems, to misalign with respect to the stars’ orbital plane. This fact has been invoked as a possible explanation of the misalignment between the stellar rotation axis and planets’ orbits [@bate10], measurable via the Rossiter McLaughlin effect [@triaud10; @albrecht12]. In this context, it is of fundamental importance to study the propagation of such interactions into the disc. In particular, warps can be produced by tidal torques due to the binary stars, whenever the binary is misaligned with respect to the disc plane. In this paper we focus on warp propagation in protostellar circumbinary systems in order to take into account these external torques.
Tilted discs have been observed in many other astrophysical environments, such as other galactic binaries as the X-ray binary Her X-1 [@tananbaum72; @wijers_pringle99], or the microquasar GRO J1655-40 [@hjellming95; @martin08]. Warps have also been found in thin discs around AGN, such as NGC 4258 [@herrnstein96]. In this last case, additional forcing torques can arise from the general relativistic Lense–Thirring precession around a spinning black hole. The warp evolution is strongly connected with the spin history of the SMBH (Supermassive Black Hole) via the @bardeen_petterson75 effect, and has been studied in depth in recent years [e.g. @king08]. Finally, recent studies have been made on spinning SMBH binaries, where the tidal binary torque and the Lense–Thirring effect could coexist [@dotti10; @nixon2012; @lodato_gerosa12].
Analytic and semi-analytic theories have been developed in order to study warp propagation in two different regimes (see section \[sec:waves\_theory\]): one where the warp evolves diffusively in a thin accretion disc with a diffusive coefficient inversely proportional to disc viscosity [@papaloizou_pringle83; @pringle92; @ogilvie99], and one where the warp propagates via bending waves in thick or inviscid discs [@papaloizou_lin95; @lubow_ogilvie2000].
The diffusive case has been explored in the last years both analytically [@scheuer_feiler96; @pringle96; @LOP02] and numerically [@lodato_pringle07; @nixon_king12]. In particular, high resolution numerical simulations have been performed, obtaining a good agreement with the analytic theory in the linear and mildly non-linear regime [@lodato_price10].
The bending–wave regime has been less analysed. Analytic studies have given a description of the wave propagation in the linear regime [@lubow_ogilvie2000; @lubow_ogilvie01; @LOP02], and other studies have been accomplished on the non-linear case [@gammie00; @ogilvie06]. Only a few poorly resolved numerical simulations have been performed to date via SPH codes [@larwood_pap97; @nelson_pap99; @nelson_pap00]. More recently, @fragner10 performed 3D simulations using a 3D grid code.
In this work we focus on warp propagation via bending waves in protostellar circumbinary discs. After obtaining an analytic solution for the steady state of the warp in a disc extending to infinity, we test the linear regime with numerical simulations. In order to address this issue, we use a 1D ring code [as in @LOP02 hereafter ] and full 3D SPH simulations with a much higher resolution than the ones by @nelson_pap99 [@nelson_pap00]. We find that the agreement between the linear theory and the simulations is good, even at very low viscosities. Finally, 3D simulations allow us to explore the non-linear regime, which has been poorly addressed so far.
While working on this this paper, we found out that a similar problem had been recently studied by @foucart13. In that paper, they focus on the steady state solutions for circumbinary discs in the linear regime, and analyse the alignment timescale between the disc and the binary. We will compare our respective results when needed.
The paper is organised as follows. In section \[sec:potential\] we derive a time independent approximation of the gravitational potential generated by the central binary and we extrapolate the torques to which the disc is subjected. In section \[sec:waves\_theory\] we describe warp propagation in the linear regime. In section \[sec:analyt\] we obtain an analytic solution of the disc steady state in the inviscid limit, and in section \[sec:1D\] we compare it with 1D time-dependent calculations. In section \[sec:res\] we describe the numerical setup used to perform the 3D SPH simulations and we report our main results both in linear and in non-linear regime. Finally, in section \[sec:concl\] we compare our results with the paper by @foucart13 and draw our conclusions.
In a companion paper [@facchini13_2], we analyse the warp evolution in a specific circumbinary protostellar disc, surrounding the binary system KH 15D [@CM04].
The binary - disc torque {#sec:potential}
========================
In this section we determine the gravitational potential generated by the binary. A similar analysis, restricted to binaries with extreme mass ratios, has been done by @nixon2011. Here, we generalise their results to arbitrary mass ratios and correct a few typos (compare equation 4 by @nixon2011 with equation \[eq:phi\_fin\]). Note that similar derivations are present in a few other papers, such as @ivanov99 and @naya05.
We consider two stars with masses $M_1$ and $M_2$, that rotate in two circular orbits, on a plane described by the cylindrical polar coordinates $(R,\phi)$ and perpendicular to the $z-$axis. We place the origin of our coordinate system in the centre of mass. The geometry of the system, as well as the definition of various quantities of interest, is shown in Fig. \[fig:grafico\_pot\]. We denote with $r_1$ the distance between $M_1$ and the centre of mass, and similarly for $r_2$ and $M_2$. The distance between the two stars is $a=r_1+r_2$. We finally place a test particle $m$ in a generic position with coordinates $(R,\phi,z)$. We call $s_1$ and $s_2$ the distance between $m$ and $M_1$ and $m$ and $M_2$. We are focusing on a restricted three body problem in 3D space.
![The binary system in the corotating frame of reference centred in the centre of mass. $M_1$ and $M_2$ are the two stars, and $m$ is the test particle. For simplicity in this figure we locate $m$ in the $(R,\phi)$ plane. The quantity $r_1$ indicates the distance between $M_1$ and the centre of mass, and analogously for $r_2$ and $M_2$. The distances between $m-M_1$ and $m-M_2$ are called $s_1$ and $s_2$, respectively.[]{data-label="fig:grafico_pot"}](grafico_pot.eps){width=".8\columnwidth"}
We now consider a reference frame $S$ corotating with the two stars. In this particular system they are both at rest at $(r_1,\pi,0)$ and $(r_2,0,0)$, respectively. The potential in this reference frame is: $$\label{eq:phi_init}
\Phi(R,\phi,z)=-G\left(\frac{M_1}{s_1} + \frac{M_2}{s_2} \right) - \frac{1}{2}\Omega_{\rm b}^2R^2,$$ where $\Omega_{\rm b}$ is the angular velocity of the binary, given by $$\Omega_{\rm b}^2=\frac{G(M_1+M_2)}{a^3}.$$ In equation \[eq:phi\_init\] the second term is due to the non-inertial nature of the reference frame, and it represents the centrifugal potential. The orbital frequency $\Omega_{\rm b}$ is simply calculated from Kepler’s third law. Finally it can be easily shown that: $$\begin{aligned}
\nonumber s_1^2 = r_1^2 + R^2 + 2r_1R\cos{\phi} + z^2;\\
s_2^2 = r_2^2 + R^2 - 2r_2R\cos{\phi} + z^2.\end{aligned}$$ We now move to an inertial reference frame $S'$, with the origin coincident with that of $S$. We do not have the centrifugal term anymore, and the $\phi$ angle undergoes the simple transformation $\phi\rightarrow\phi '=\Omega_{\rm b} t$. In this frame we obtain the following gravitational potential: $$\begin{aligned}
\nonumber\Phi(R,\phi'=\Omega_{\rm b} t,z) =& -G \displaystyle \frac{M_1}{(R^2 + r_1^2 + 2r_1R\cos{\Omega_bt} + z^2)^{1/2}}\\
& - G \displaystyle \frac{M_2}{(R^2 + r_2^2 - 2r_2R\cos{\Omega_b t} + z^2)^{1/2}}.\end{aligned}$$ This is the most general form of the gravitational potential of a circular binary. We could now expand the above relation in a Fourier series with azimuthal wavenumber $\sigma$ to distinguish the various contributions to the potential. In order to avoid it, we make the assumption, as it has been made by @nixon2011 and earlier by e.g. @lubow_ogilvie01, that the perturbations to the potential of the $\sigma\geq1$ modes are oscillatory, and if we are far enough from resonances, they will have no long-term secular effect. Long-term effects on the orbit of the test particle, and hence eventually on the disc, come from the only zero-frequency term. Since we are interested in the secular dynamics of the disc, we can just consider this $\sigma=0$ mode [@bate00].
In order to calculate this time-independent term we use the fact that physically this $\sigma=0$ term is given by replacing the two masses $M_1$ and $M_2$ with the same masses spread uniformly over their orbit, i.e. two rings of mass $M_1$ and $M_2$ and radius $r_1$ and $r_2$ in the $(R,\phi)$ plane [see @nixon2011]. The gravitational potential for the generic test particle is then: $$\label{eq:phi_m1_m2}
\Phi(R,z)=-\frac{GM_1}{2\pi}\int_0^{2\pi}{\frac{d\phi}{\tilde{s}_1}}-\frac{GM_2}{2\pi}\int_0^{2\pi}{\frac{d\phi}{\tilde{s}_2}},$$ where $$\begin{aligned}
\nonumber\tilde{s}_1^2=R^2+r_1^2+z^2+2Rr_1\cos{\phi}, \\
\tilde{s}_2^2=R^2+r_2^2+z^2-2Rr_2\cos{\phi}.\end{aligned}$$ The new $\tilde{s}_1$ and $\tilde{s}_2$ define the distances between a generic point of the two massive annuli and the test particle positioned at $(R,0,z)$ (we place it at $\phi=0$ because of the rotational symmetry about the $z$-axis of the problem, see Fig. \[fig:grafico\_pot\_2\]), while $s_1$ and $s_2$ indicate the distances between the two stars at rest in the corotating reference frame and a generic particle.
![The binary system in an inertial frame of reference centred in the centre of mass. $M_1$ and $M_2$ are now spread in two massive annuli with radii $r_1$ and $r_2$, respectively, and $m$ is the test particle, positioned at $(R,0,z)$ (we place it at $\phi=0$ because of the rotational symmetry about the $z$-axis of the problem). The new $\tilde{s}_1$ and $\tilde{s}_2$ define the distances between a generic point of the two massive annuli and the test particle.[]{data-label="fig:grafico_pot_2"}](grafico_pot_2.eps){width=".8\columnwidth"}
We introduce the factor $\eta=M_1M_2/M^2$ and the total mass $M=M_1+M_2$. We recall that $a=r_1+r_2$. If we now expand equation \[eq:phi\_m1\_m2\] in powers of $r_1/R$, $r_2/R$ and $z/R$, keeping terms only up to second order we find: $$\label{eq:phi_fin}
\nonumber\Phi(R,z)=-\frac{GM}{R}-\frac{GM\eta a^2}{4R^3} + \frac{GMz^2}{2R^3}+\frac{9}{8}\frac{GM\eta a^2z^2}{R^5}.$$ We notice that by taking the limits $\eta\rightarrow M_2/M_1$ and $M\rightarrow M_1$ we obtain the equivalent of equation $4$ of @nixon2011, in which they calculated the same gravitational potential but in the simplified case where $M_2\ll M_1$.
The perturbations affecting the disc particles can be expressed in terms of the orbital frequency $\Omega$, the vertical oscillation frequency $\Omega_z$, and the epicyclic frequency $\kappa$ (see section \[sec:waves\_theory\]). Their definition follows below: $$\Omega^2=\frac{1}{R}\frac{\partial\Phi}{\partial R}\bigg|_{z=0},$$ $$\Omega_z^2=\frac{\partial^2\Phi}{\partial z^2}\bigg|_{z=0},$$ $$\kappa^2=4\Omega^2+2R\Omega\frac{d \Omega}{d R}=4\Omega^2\left[1+\frac{1}{2}\frac{d\ln{\Omega}}{d\ln{R}}\right].$$
For the binary potential of equation \[eq:phi\_fin\] we obtain: $$\label{eq:omega}
\Omega^2=\frac{GM}{R^3}+\frac{3}{4}\frac{GM\eta a^2}{R^5},$$ $$\Omega_z^2=\frac{GM}{R^3}+\frac{9}{4}\frac{GM\eta a^2}{R^5},$$ $$\label{eq:kappa}
\kappa^2=\frac{GM}{R^3}-\frac{3}{4}\frac{GM\eta a^2}{R^5}.$$ To first order, thus, $$\label{eq:omegaz2}
\frac{\Omega_z^2-\Omega^2}{\Omega^2}=\frac{3}{2}\frac{\eta a^2}{R^2},$$ $$\label{eq:kappa2}
\frac{\kappa^2-\Omega^2}{\Omega^2}=-\frac{3}{2}\frac{\eta a^2}{R^2}.$$ We will use this approximation throughout the paper.
In summary, in this section we have deduced the time independent term of the gravitational potential of a generic binary system, formed by two different stars of mass $M_1$ and $M_2$, in the hypothesis that the disc particles rotate at a large radius ($R\gg r_1,\ r_2$) and low height ($z\ll R$).
Theory of wave-like warp propagation {#sec:waves_theory}
====================================
We consider here the propagation of warps in thin, almost Keplerian accretion discs. The quantities that define their dynamics are the angular velocity $\Omega(R)$, the surface density $\Sigma(R)$ and the angular momentum per unit area ${\bf L}(R)$. $H$ describes the scale height of the disc, and it is related to the sound speed $c_{\rm s}$ via $H=c_{\rm s}/\Omega$. We assume here that the disc is composed of a series of flat, infinitesimally thin rings, each of which can be oriented arbitrarily in space. A single ring at radius $R$ is thus described by two angles: the tilt angle $\beta$ with respect to the $z$ axis, and the azimuthal angle $\gamma$ that defines the orientation of the tilt with respect to an arbitrary axis, perpendicular to $z$. If $\beta$ varies with $R$, we will have a warped disc. If $\gamma$ varies with radius the disc is additionally twisted. Here, $R$ should be intended as a spherical coordinate, even though at each radius the disc is thin in the direction perpendicular to the local rotation plane. Therefore we define in complex notation the tilt of the disc $W(R,t)$ at each radius as $W(R,t)=\beta(R,t)\exp{[i\gamma(R,t)]}$ [@pringle96]. These quantities are related to the specific angular momentum through ${\bf l}(R) = {\bf L}(R)/L(R) = (\cos\gamma\sin\beta,\sin\gamma\sin\beta,\cos\beta)$.
We consider a standard $\alpha$-prescription for the viscosity: $\nu=\alpha c_{\rm s}H$ [@shakura73].
Warp propagation can be described in two different regimes. @papaloizou_pringle83 had already suggested that whenever $\alpha < H/R < 1$ warps would probably propagate via bending waves, whereas when $H/R < \alpha < 1$ the equations describing the evolution would be diffusive [@pringle92]. These results have been confirmed analytically; @papaloizou_lin95 derived the equations describing the evolution in the case $\alpha < H/R < 1$, and they confirmed that they evolve via wave equations. Equivalent formulations have been derived later by @demianski_ivanov97 and @lubow_ogilvie2000 (hereafter, ). Throughout this work we use the formulation by .
have shown that when the disc is nearly Keplerian and non self-gravitating, the linearised equations for bending waves (with azimuthal wavenumber $m=1$) may be written as: $$\label{eq:wave_l_real}
\Sigma R^2\Omega\frac{\partial {\bf l}}{\partial t}=\frac{1}{R}\frac{\partial {\bf G}}{\partial R}+{\bf T},$$ and $$\label{eq:wave_g_real}
\frac{\partial{\bf G}}{\partial t}+\left(\frac{\kappa^2-\Omega^2}{\Omega^2}\right)\frac{\Omega}{2}{\bf e}_z\times{\bf G} +\alpha\Omega{\bf G}=\Sigma R^3\Omega\frac{c_{\mathrm{s}}^2}{4}\frac{\partial{\bf l}}{\partial R},$$ where $$\label{eq:external_torque}
{\bf T}=-\Sigma R^2\Omega\left(\frac{\Omega_z^2-\Omega^2}{\Omega^2}\right)\frac{\Omega}{2}{\bf e}_z\times{\bf l},$$ where ${\bf e}_z$ is the unit vector perpendicular to the binary orbit.
In our assumptions the warp is small, therefore $l_x,l_y \ll 1$ and $l_z \approx 1$. Thus, by considering $l_z = 1$, we can consider the equations on the $xy$-plane only. The term $2\pi {\bf G}$ is the internal torque and ${\bf T}$ the external torque density. The external torque is due to the lack of spherical symmetry in the potential, and is proportional to the term $\Omega_z^2 - \Omega^2$ (equation \[eq:external\_torque\]). Instead, equation \[eq:wave\_g\_real\] shows that the internal torque is mediated by horizontal epicyclic motions. The term proportional to $\alpha$ tends to dissipate the waves through an exponential factor. Finally, note that the external torque is related to the precession frequency of the ring ${\bf \Omega}_{\rm p}$, as we know it should be from simple mechanics [@lodato_pringle06]. By knowing that ${\bf L}=\Sigma R^2 \Omega {\bf l}$ we can rewrite equation \[eq:external\_torque\] as: $${\bf T}=\frac{-(\Omega_z-\Omega)(\Omega_z+\Omega)}{\Omega^2}\frac{\Omega}{2}{\bf e}_z \times {\bf L} \approx {\bf \Omega}_{\rm p}\times{\bf L}.$$ If $\Omega_z\approx\Omega$, then ${\bf \Omega}_{\rm p}=(\Omega-\Omega_z){\bf e}_z$ [@nixon2011].
A different but equivalent set of equations can be used by defining the dimensionless complex variable $W(R,t)=l_x+il_y$ and the complex variable $G(R,t)=G_x+iG_y$. We can thus rewrite equation \[eq:wave\_l\_real\] and \[eq:wave\_g\_real\] as:
$$\label{eq:wave_l_complex}
\Sigma R^2\Omega\left[\frac{\partial W}{\partial t}+\left(\frac{\Omega_z^2-\Omega^2}{\Omega^2}\right)\frac{i\Omega}{2}W \right]=\frac{1}{R}\frac{\partial G}{\partial R},$$
and
$$\label{eq:wave_g_complex}
\frac{\partial G}{\partial t}+\left(\frac{\kappa^2-\Omega^2}{\Omega^2}\right)\frac{i\Omega}{2}G+\alpha\Omega G=\Sigma R^3\Omega\frac{c_{\mathrm{s}}^2}{4}\frac{\partial W}{\partial R}.$$
Let us consider the propagation velocity of the waves. By neglecting the external torque and the non-Keplerian term we can obtain a first order approximated dispersion relation [@nelson_pap99]:
$$\omega=\frac{1}{2}[i\alpha\Omega \pm (c_{\rm s}^2k^2 - \alpha^2\Omega^2)^{\frac{1}{2}}],$$
where $k$ is the radial wavenumber and $\omega$ is the wave frequency. Therefore if the disc is inviscid ($\alpha=0$) the warp propagates as a non-dispersive wave with wave speed $c_{\rm s}/2$. Moreover, propagation becomes purely diffusive in the limit $|\omega| \ll \alpha\Omega$. Note that in the Keplerian limit ($\Omega=\Omega_z=\kappa$) equations \[eq:wave\_l\_real\] and \[eq:wave\_g\_real\] explicitly tend to a diffusive equation when this condition is verified (i.e. when the third term of the l.h.s. of equation \[eq:wave\_g\_real\] dominates over the first two).
Finally, have shown that in the inviscid case the dispersion relation associated to equation \[eq:wave\_l\_real\] and \[eq:wave\_g\_real\] is given by: $$\left[ \omega - \left( \frac{\Omega^2-\Omega_z^2}{2\Omega}\right)\right] \left[ \omega - \left( \frac{\Omega^2-\kappa^2}{2\Omega}\right)\right]=\frac{c_{\rm s}^2}{4}k^2.$$ In the case of $\omega=0$, whenever $\kappa^2-2\Omega^2+\Omega_z^2=0$, which is the case for our binary potential, the spatial configuration that the disc will reach is an evanescent wave. We shall see that this theoretical prediction made by is verified by our results.
Finally, note that in this section we have considered the linear case only. Some efforts have been spent in the last decade to cover the non-linear case in Keplerian and nearly Keplerian discs, both in the diffusive regime [@ogilvie99; @lodato_price10] and in the wave-like one [@ogilvie06].
Analytic considerations for the steady state {#sec:analyt}
============================================
In this section we deduce an analytic solution for the warped disc shape in its steady state. We follow the procedure by , who considered the solution for an external torque due to a spinning black hole. We consider a circumbinary disc extending from $R_{\rm in}$ up to $R_{\rm out}\rightarrow\infty$, under all the approximations made in section \[sec:potential\]. Therefore we can implement equations (\[eq:omega\]-\[eq:kappa\]) in equations (\[eq:wave\_l\_complex\]) and \[eq:wave\_g\_complex\]. By setting the time-derivatives to $0$, we obtain equation $17$ of : $$\frac{d}{dR}\left[\left(\frac{\Sigma c_{\mathrm{s}}^2R^3\Omega^2}{\Omega^2-\kappa^2+2i\alpha\Omega}\right)\frac{dW}{dR}\right] +\Sigma R^3(\Omega^2-\Omega_z^2)W=0.$$ We focus on the inviscid case. In order to explicitly write the above equation, we consider $\Sigma\propto R^{-p}$ and $c_{\rm s}\propto R^{-q}$. We use the parametrisation $R=R_{\rm in}x$. We obtain $$\label{eq:analyt}
\frac{d}{dx}\left[x^{5-2q-p}\frac{dW}{dx}\right]=4\chi^2x^{-2-p}W,$$ where $$\chi=\frac{3}{4}\eta\frac{(a/R_{\rm in})^2}{H_{\rm in}/R_{\rm in}}.
\label{eq:defchi}$$ Note that equation \[eq:analyt\] can be projected to the real domain only. The real and the imaginary parts of $W$ are not entangled anymore. This has the important consequence that the steady state shape of an inviscid disc will have no twist.
In the absence of viscosity we have just two phenomena that contrast each other: gravity and pressure. The parameter $\chi$ indicates which one dominates. Note that the amplitude of the warp will be regulated by the $\chi$ parameter only (cf. equation 20 of @foucart13).
After some algebra, we can express the solutions of equation \[eq:analyt\] in terms of modified Bessel functions of the first and second kind:
$$W(x)=x^{-\zeta}[c_1 I_{\xi}(y(x))+c_2 K_{\xi}(y(x))],$$
where $$y(x)=\left|\frac{2\chi x^{\psi}}{\psi}\right|,$$ and $$\zeta=-\frac{1}{2}(2q+p-4),\ \ \ \ \psi=\frac{1}{2}(2q-5),\ \ \ \ \xi = \frac{\zeta}{\psi}.$$ This is the most general solution for the steady shape of an inviscid disc around a circular binary system. The modified Bessel functions $I$ and $K$ are exponentially growing and decaying functions and are not oscillatory. This confirms the theoretical prediction highlighted by and reported at the end of section \[sec:waves\_theory\] in the case of $\omega=0$. This is a relevant difference between the binary case and the steady state of a disc under Lense-Thirring torques, where the solutions are oscillatory.
Finally, we need to specify the boundary conditions. We set $\lim_{x\rightarrow\infty}{W(x)}=W_{\infty}$, which comes from the fact that $W$ has an horizontal asymptote (at large radii the disc is not affected by the external torque, which decreases as $R^{-7/2}$) and $W'(x=1)=0$ (zero-torque boundary condition at the inner edge). In this way we constrain the two constants $c_1$ and $c_2$ [^2]. Note that $W_{\infty}$ is an arbitrary number, that in real cases will depend on the initial condition of the disc tilt. All solutions are valid to within a constant scale amplitude, since we deal with a linear model. To compute the limits, we use the fact that when $r\rightarrow 0$: $$I_{\xi}(y)\sim\frac{1}{\Gamma(\xi+1)}\left(\frac{y}{2}\right)^{\xi},$$ and, in general: $$K_{\xi}(y)=\frac{\pi[I_{-\xi}(y)-I_{\xi}(y)]}{2\sin{\xi\pi}},$$ where $\Gamma$ is the gamma function. We do not need any other condition, in particular we do not need to fix a value for $W$ at the inner edge of the disc. In general the tilt at the inner edge of the disc will not be very small. This is an importance difference between the wave-like and the diffusive regime, which tends to have a very small tilt in the inner regions of the disc.
![Analytic solution of $W(x)$ with $H_{\mathrm{in}}/R_{\mathrm{in}}=0.1$, $p=0.5$ and $q=0.75$. The bottom panel shows the same plot as in the top one with a logarithmically scaled $x-$axis. The three coloured lines (blue, green, red) illustrate the case in which $R_{\rm in}=2a$, and $\eta=0.0475$, $0.16$ and $0.25$, respectively. Note that the amplitude of the warp increases with the binary mass ratio $M_2/M_1<1$. The black line shows the case $\eta=0.25$ and $R_{\rm in}=a$. In all the solutions $W_{\infty}$ has been set to $1$. The general solution is an evanescent wave, without any oscillation.[]{data-label="fig:analytical"}](fig3a "fig:"){width=".85\columnwidth"} ![Analytic solution of $W(x)$ with $H_{\mathrm{in}}/R_{\mathrm{in}}=0.1$, $p=0.5$ and $q=0.75$. The bottom panel shows the same plot as in the top one with a logarithmically scaled $x-$axis. The three coloured lines (blue, green, red) illustrate the case in which $R_{\rm in}=2a$, and $\eta=0.0475$, $0.16$ and $0.25$, respectively. Note that the amplitude of the warp increases with the binary mass ratio $M_2/M_1<1$. The black line shows the case $\eta=0.25$ and $R_{\rm in}=a$. In all the solutions $W_{\infty}$ has been set to $1$. The general solution is an evanescent wave, without any oscillation.[]{data-label="fig:analytical"}](fig3b_modif "fig:"){width=".85\columnwidth"}
So far we have used two generic values for $q$ and $p$. Henceforth we set the two values to $3/4$ and $1/2$, respectively. In Fig. \[fig:analytical\] we report the solution obtained with a few sets of typical physical parameters for protostellar circumbinary discs. The scale height is equal in all the portrayed solutions: $H_{\mathrm{in}}/R_{\mathrm{in}}=0.1$, and scales as $H/R\propto R^{-1/4}$. From @art_lubow94 we know that the inner radius of the disc is equal to the tidal truncation radius, that in the case of circular orbits is $R_{\rm t}\sim 2a$ (the dependence on $\eta$ is very weak). In Fig. \[fig:analytical\] we set $R_{\rm in}=2a$ for the three coloured (blue, green, red) lines, which illustrate the analytic solution for the following values of $\eta$: $0.0475$, $0.16$ and $0.25$ (corresponding to the mass ratio $M_2/M_1 = 0.052, 0.25$ and 1, respectively).The black line shows the case in which $\eta=0.25$, but $R_{\rm in}=a$ so that the disc extends to an inner radius that is slightly smaller than the tidal truncation radius. Note that, for the same inner radius, the warp becomes more prominent as the mass ratio between the two stars becomes closer to 1 ($\eta\rightarrow 0.25$), but still maintains a significant misalignment even for equal masses. For the same mass ratio, obviously, the warp increases as the disc moves closer in towards the binary orbit. We show the $R_{\rm in}=a$ case to underline the strong dependence of the warping on the inner radius of the disc. Finally, we have verified that the order of magnitude of the amplitude of the warp agrees with the estimate given by equation 20 by @foucart13 [compare the case $\eta=0.25$, $R_{\rm in}=2a$, i.e. the red line in Fig. \[fig:analytical\], with the tilting showed in fig. 1 by @foucart13].
Time-dependent evolution: a 1D model {#sec:1D}
====================================
In this section we describe and use a 1D model for warp propagation via bending waves in a disc subject to a binary torque. We consider $R$ as the only spatial variable of the system, as described at the beginning of section \[sec:analyt\]. The disc is discretised into a set of thin annuli that can be tilted and interact with one another via pressure and viscous forces. In this dynamical evolution, we neglect the dependence of $\Sigma$ on time. In fact, from the dispersion relation of the wave equations we know that for low viscosity discs, bending waves propagate on a timescale $t_{\rm dyn}=2R/c_{\rm s}$, whereas the viscous evolution of $\Sigma$ occurs on a timescale $t_{\nu}=R^2/\nu$. Therefore: $$\frac{t_{\rm dyn}}{t_{\nu}}=\frac{2R\nu}{c_{\rm s}R^2}=2\alpha\frac{H}{R}.$$ Since we know that $\alpha<H/R\ll 1$ we can neglect the evolution of $\Sigma$.
Moreover, we neglect the angular momentum variations of the binary, which is affected by the gravitational potential of the disc. We do not consider the back-reaction of the disc onto the binary angular momentum because we focus on low mass discs, where their angular momentum is negligible compared to the binary one (this back-reaction is considered in @foucart13).
In order to compute the evolution we move to four dimensionless differential equations from equations \[eq:wave\_l\_real\] and \[eq:wave\_g\_real\]. We use the following parametrisation: $R=R_{\rm in}x$, $\Omega=\Omega_{\rm in}x^{-3/2}$, $\Sigma=\Sigma_{\rm in}x^{-p}$, $t=\Omega_{\rm in}^{-1}(H_{\rm in}/R_{\rm in})^{-1}\tau$, ${\bf l}=W_{\infty}\boldsymbol{\lambda}$ and ${\bf G}=G_{\rm in}{\bf \Gamma}$. Then, we set $G_{\rm in}=\Sigma_{\rm in}R_{\rm in}^4\Omega_{\rm in}^2(H_{\rm in}/R_{\rm in})W_{\infty}$. With these definitions, we obtain the following set of 4 equations: $$\label{eq:lambda_1}
\frac{\partial\lambda_x}{\partial \tau}=x^{p-3/2}\frac{\partial\Gamma_x}{\partial x}+\chi x^{-7/2}\lambda_y,$$ $$\frac{\partial\lambda_y}{\partial \tau}=x^{p-3/2}\frac{\partial\Gamma_y}{\partial x}-\chi x^{-7/2}\lambda_y,$$ $$\frac{\partial\Gamma_x}{\partial \tau}+\alpha \frac{R_{\mathrm{in}}}{H_{\mathrm{in}}} x^{-3/2}\Gamma_x +\chi x^{-7/2}\Gamma_y=x^{3/2-p}\left(\frac{c}{2}\right)^2\frac{\partial\lambda_x}{\partial x},$$ $$\label{eq:gamma_2}
\frac{\partial\Gamma_y}{\partial \tau}+\alpha \frac{R_{\mathrm{in}}}{H_{\mathrm{in}}} x^{-3/2}\Gamma_y -\chi x^{-7/2}\Gamma_x=x^{3/2-p}\left(\frac{c}{2}\right)^2\frac{\partial\lambda_y}{\partial x},$$ where $c$ is the dimensionless sound speed ($c=x^{-3/4}$ in this paper), and $\chi$ has been previously defined in equation \[eq:defchi\]. All the physics is included in two parameters: $\alpha/(H_{\rm in}/R_{\rm in})$ and $\chi$. The first one is the measure of the importance of viscous and pressure effects, the second one determines the magnitude of the external torque due to the binary potential with respect to pressure forces. The parameter $H_{\rm in}/R_{\rm in}$ is a scale parameter determining the speed of the temporal evolution.
solved the same problem for the Lense-Thirring case with a different but equivalent set of equations in the complex domain. We prefer to use our equations, because they show the dependence of the evolution on the physical parameters more transparently. However, we use their result as an important comparison. By implementing the Lense-Thirring torque, and by setting the exact same set of parameters as they did, we obtain their same result for the disc tilt shape. This confirms the equivalence of the two sets of equations.
Code and boundary conditions
----------------------------
The numerical code we use to solve equations \[eq:lambda\_1\]-\[eq:gamma\_2\] implements the same numerical algorithm used by . We consider ${\bf \Gamma}$ to be defined at $N$ logarithmically distributed grid points (typically $N=1001$, for the inviscid simulations we used $N=4001$), and $\boldsymbol{\lambda}$ to be defined at the half grid points. We opt for a logarithmically distributed spatial grid because the torque is much stronger at the inner edge. In this way we can proceed with a leapfrog algorithm. The tracking of the evolution can be read in section $4.1$ of .
Results {#sec:res_1D}
-------
Henceforth in the whole paper we use $p=1/2$. We perform a first generic simulation with the following set of parameters: $(H_{\rm in}/R_{\rm in})=0.1$, $\alpha=0.05$, $\eta=0.25$ and $(a/R_{\rm in})=0.5$. We recall that $\eta=0.25$ corresponds to the case $M_1=M_2$. We make these choices because $H_{\mathrm{in}}/R_{\mathrm{in}}=0.1$ is the typical value for protostellar discs, when $\alpha=0.05$ we expect a low-viscosity behaviour, $M_1 = M_2$ gives a sizable torque, and finally $a/R_{\mathrm{in}}=0.5$ makes the approximation of a time independent gravitational potential reasonable. In the whole section we will then use: $x_{\rm in}=1$, $x_{\rm out}=90$ and $W_{\infty}=1$. The time unit will naturally be $\Omega_{\rm in}^{-1}$.
We consider a disc initially aligned with the binary plane in the inner parts, and misaligned in the outer parts. Thus as an initial condition we take $\lambda_x=0$ for $x\leq18$, $\lambda_x=\frac{1}{2}\{1+\sin{[\pi(x-20)/4]}\}$ for $18\leq x\leq22$, and $\lambda_x=1$ for $x\geq22$ . $\lambda_y$ is initially set to $0$. In this way at $t=0$ we just have a tilt, with no twist. In other terms, $\gamma(x,t=0)=0$ at each radius.
The evolution of the warp in this case is shown in Fig. \[fig:wave\_example\_tstop4000\]. It is apparent that the discontinuity does propagate inwards and outwards as a bending wave. As the inwardly propagating wave reaches the inner edge, it bounces back and reacts to the strong external torque due to the binary, until it forms a stationary wave reaching a steady state on a sound crossing timescale. The outwardly travelling discontinuity keeps on propagating throughout the whole simulation. In the top panel of Fig. \[fig:wave\_example\_tstop4000\] we report the tilt evolution, where the tilt is defined as $\sqrt{\lambda_x^2+\lambda_y^2}$ (normalised at $1$ at infinity) up to a computational time $t=4000$. The bottom panel of Fig. \[fig:wave\_example\_tstop4000\] shows the same simulation up to $t=20000$. Note that the steady state has a tilt shape with the same features as the inviscid analytic solution: it is an evanescent wave, and the tilt does not tend to $0$ at the inner edge. These figures do not show the information about the twist, but we will analyse it later in this section. Finally, note that the waves propagate with a velocity $\approx c_{\rm s}/2$, as predicted by the dispersion relation in the low-viscosity case. From the figures we can observe small ripples propagating behind the outwardly travelling wave front. These are given by small numerical instabilities and are damped away by viscous interactions in almost a simulation time. We have verified that they can be reduced by increasing the spatial resolution. We have performed simulations with a set of initial conditions (e.g. an initially tilted untwisted disc), and the final state of the disc shape we obtain does not depend on them.
The warp in the presence of viscosity is much larger in this case with respect to the inviscid case (compare fig. \[fig:analytical\] and \[fig:wave\_example\_tstop4000\]). Also in this case, the warp amplitude agrees to order of magnitude with that predicted by @foucart13 (for the same set of parameters, their equation 16 would predict $\Delta\beta/\beta_{\infty}\approx 0.4$, and we observe $\Delta\beta/\beta_{\infty}\approx 0.6$, where $\beta_{\infty}$ is the tilt at large radii).
![Tilt evolution as a function of radius of an initially warped protostellar circumbinary disc in the linear low-viscosity regime, given the following set of parameters: $(H_{\rm in}/R_{\rm in})=0.1$, $\alpha=0.05$, $\eta=0.25$ and $(a/R_{\rm in})=0.5$. We recall that $\eta=0.25$ corresponds to $M_1=M_2$. Top panel: early evolution of the tilt, shown at 11 equally spaced times $t=0$, $400$, $800$,...,$4000$. Bottom panel: late evolution of the the tilt at 11 equally spaced times $t=0$, $2000$, $4000$,...,$20000$. As expected the discontinuity in the tilt initially propagates inwards and outwards in a wave-like fashion. The inward propagation of the tilt interacts with the strong external torque, and by the end of the simulation the steady state shape of the disc close to the binary is established. At $t=20000$ the initial outwardly propagating warp wave is approaching the outer edge of the grid, followed closely by the reflection of the initially inwardly propagating warp wave. By this time, the steady warped disc solution has been established over about half of the grid.[]{data-label="fig:wave_example_tstop4000"}](a05_alpha005_t4000 "fig:"){width=".9\columnwidth"} ![Tilt evolution as a function of radius of an initially warped protostellar circumbinary disc in the linear low-viscosity regime, given the following set of parameters: $(H_{\rm in}/R_{\rm in})=0.1$, $\alpha=0.05$, $\eta=0.25$ and $(a/R_{\rm in})=0.5$. We recall that $\eta=0.25$ corresponds to $M_1=M_2$. Top panel: early evolution of the tilt, shown at 11 equally spaced times $t=0$, $400$, $800$,...,$4000$. Bottom panel: late evolution of the the tilt at 11 equally spaced times $t=0$, $2000$, $4000$,...,$20000$. As expected the discontinuity in the tilt initially propagates inwards and outwards in a wave-like fashion. The inward propagation of the tilt interacts with the strong external torque, and by the end of the simulation the steady state shape of the disc close to the binary is established. At $t=20000$ the initial outwardly propagating warp wave is approaching the outer edge of the grid, followed closely by the reflection of the initially inwardly propagating warp wave. By this time, the steady warped disc solution has been established over about half of the grid.[]{data-label="fig:wave_example_tstop4000"}](a05_alpha005_t20000 "fig:"){width=".9\columnwidth"}
![Tilt evolution as a function of radius at 11 equally spaced times $t=0$, $2000$, $4000$,...,$20000$ with $(H_{\rm in}/R_{\rm in})=0.1$, $\alpha=0.05$, $\eta=0.25$ and $(a/R_{\rm in})=0.1$. At equal viscosity, when the external torque is less significant the disc tends to stay misaligned, since the torque is not sufficiently strong to induce a warp against internal pressure stresses.[]{data-label="fig:gravity"}](a01_alpha005_t20000){width=".9\columnwidth"}
We have then explored the parameters space by modifying the two parameters governing the physics of the system: $\chi$ and $\alpha/(H_{\rm in}/R_{\rm in})$. By modifying $\chi$, we change the physical parameters of the binary system, either the distance between the two stars or the ratio between the two masses. Fig. \[fig:gravity\] shows the tilt evolution when $a/R_{\rm in}=0.1$, $\eta=0.25$, $H_{\rm in}/R_{\rm in}=0.1$ and $\alpha=0.05$. The disc edge lies further away from the binary (the inner radius is larger than the tidal truncation radius). This choice is equivalent to reducing the binary mass ratio and keeping the inner radius fixed. In this case, the external torque due to the non spherical symmetry of the potential is reduced, and therefore the disc tends to stay misaligned with respect to the binary plane (see also Fig. \[fig:analytical\]). In Fig. \[fig:visc\] we portray two simulations with different values of viscosity. The top panel has $\alpha=0.01$, the bottom one $\alpha=0.6$, while the other parameters are $a/R_{\rm in}=0.5$, $\eta=0.25$ and $H_{\rm in}/R_{\rm in}=0.1$. The second one lies in the diffusive regime ($H/R<\alpha<1$), and it shows its characteristic behaviour. We can note two relevant differences between the two panels. First, in the diffusive simulation the disc tends to align with the binary plane in the inner parts, as already illustrated in previous works [@lodato_pringle06; @lodato_pringle07]. Secondly, in this case the evolution of the shape of the disc slows down significantly, since the dispersion relation does heavily depend on $\alpha$.
So far we have not specified how the final solutions differ from one another with respect to the phase. In Fig. \[fig:visc\_phase\] we illustrate the phase (as a function of radius) for the two cases reported in Fig. \[fig:visc\]. We can observe that both discs are twisted in the inner regions. Moreover, the magnitude of the twist increases as the viscosity increases. The twisting of the disc is therefore strongly correlated to viscosity (in fact viscosity is the only physical quantity that can induce shear forces in this case). The limit case is $\alpha=0$; in section \[sec:inviscid\] we will see that inviscid discs present no twist at all for $t\rightarrow\infty$, independently of the initial condition.
We can compare these results with the ones obtained by @foucart13. We agree with the fact that the steady state solutions depend on two parameters only: $\alpha/(H_{\rm in}/R_{\rm in})$ and $\chi$, as they show in their equations 15, 16 and 20 when they estimate the amplitude of both the warping and the twisting. Moreover, we confirm that equation 20 gives a right estimate of the warping of the disc when the disc is (nearly) inviscid (see Fig. \[fig:analytical\]). When $\alpha/(H_{\rm in}/R_{\rm in})$ is not negligible, we confirm that the warping is dominated by the term given by their equation 16.
The inviscid case {#sec:inviscid}
-----------------
![Top panel: tilt evolution for an initially warped disc with $(H_{\rm in}/R_{\rm in})=0.1$, $\alpha=0.01$, $\eta=0.25$ and $(a/R_{\rm in})=0.5$ at 11 equally spaced times $t=0$, $2000$, $4000$,...,$20000$. Bottom panel: tilt evolution for the same disc with $\alpha=0.6$. In the less viscous disc the warp propagates at a half of the sound speed, and the shape of the tilt tends to a final steady state which is comparable to the analytic solution for completely inviscid discs. In the figure reported in the bottom panel we are in a diffusive regime ($\alpha>H/R$); the diffusive behaviour is apparent.[]{data-label="fig:visc"}](a05_alpha001_t20000_tilt "fig:"){width=".9\columnwidth"} ![Top panel: tilt evolution for an initially warped disc with $(H_{\rm in}/R_{\rm in})=0.1$, $\alpha=0.01$, $\eta=0.25$ and $(a/R_{\rm in})=0.5$ at 11 equally spaced times $t=0$, $2000$, $4000$,...,$20000$. Bottom panel: tilt evolution for the same disc with $\alpha=0.6$. In the less viscous disc the warp propagates at a half of the sound speed, and the shape of the tilt tends to a final steady state which is comparable to the analytic solution for completely inviscid discs. In the figure reported in the bottom panel we are in a diffusive regime ($\alpha>H/R$); the diffusive behaviour is apparent.[]{data-label="fig:visc"}](a05_alpha060_t20000_tilt "fig:"){width=".9\columnwidth"}
![Phase evolution of the same discs portrayed in Fig. \[fig:visc\] (top panel: $\alpha=0.01$; bottom panel: $\alpha=0.6$). The twist of the disc correlates with the viscosity: the more viscous the disc is, the more twisted it gets. In the bottom panel the twisting is so strong that $\Delta\gamma/\Delta R\approx2\pi$.[]{data-label="fig:visc_phase"}](a05_alpha001_t20000_phase "fig:"){width=".9\columnwidth"} ![Phase evolution of the same discs portrayed in Fig. \[fig:visc\] (top panel: $\alpha=0.01$; bottom panel: $\alpha=0.6$). The twist of the disc correlates with the viscosity: the more viscous the disc is, the more twisted it gets. In the bottom panel the twisting is so strong that $\Delta\gamma/\Delta R\approx2\pi$.[]{data-label="fig:visc_phase"}](a05_alpha060_t20000_phase "fig:"){width=".9\columnwidth"}
We now compare the analytic solution for the steady state shape (section \[sec:analyt\]) with the results of the 1D time-dependent simulations. We recall that we deduced the analytic solution for inviscid discs only, therefore we have to set $\alpha=0$ in order to be able to compare our results. We consider an initially warped disc with $R_{\mathrm{in}}=2a$, $H_{\mathrm{in}}/R_{\mathrm{in}}=0.1$ and $\eta=0.25$. Fig. \[fig:alpha\_0\] shows the comparison with the analytic solution (highlighted with the red line). For this simulations we have used $N=4001$ (number of grid points) in order to reduce the ripples due to low resolution. The agreement is good: the time-dependent calculation leads to a steady state that is well described by the analytic solution. The disc rearranges in such a way that it reaches a steady state with a constant phase. As predicted by the analytic model, time-dependent inviscid simulations tend to an untwisted steady state, which is uniformly rotated with respect to the initial condition (see Fig. \[fig:alpha\_0\_phase\]). Such rotation is needed because in order to reach the steady warped configuration, the disc has to mix the $x$- and $y$- components of ${\bf l}$, thus producing a transient twist that eventually settles in an untwisted, but rotated, configuration. These results have been confirmed by using different initial conditions. Moreover, we ran a simulation with the analytic solution as initial condition. We see little evolution in this case, mostly driven by small numerical noise.
![Top panel: tilt evolution of an inviscid initially warped disc with $(H_{\rm in}/R_{\rm in})=0.1$, $\eta=0.25$ and $(a/R_{\rm in})=0.5$ at 11 equally spaced times $t=0$, $2000$, $4000$,...,$20000$. The red line depicts the analytic solution of the steady state’s shape. Bottom panel: blow up of the inner region.[]{data-label="fig:alpha_0"}](a05_alpha000_t20000_tilt "fig:"){width=".9\columnwidth"} ![Top panel: tilt evolution of an inviscid initially warped disc with $(H_{\rm in}/R_{\rm in})=0.1$, $\eta=0.25$ and $(a/R_{\rm in})=0.5$ at 11 equally spaced times $t=0$, $2000$, $4000$,...,$20000$. The red line depicts the analytic solution of the steady state’s shape. Bottom panel: blow up of the inner region.[]{data-label="fig:alpha_0"}](a05_alpha000_t20000_tilt_blown "fig:"){width=".9\columnwidth"}
![Phase evolution of the disc depicted in Fig. \[fig:alpha\_0\]. The initial condition is $\gamma(R)=0$. The disc tends to a steady state with $\gamma=$const. The disc rearranges towards the analytic solution in the tilt; in order to do that, it initially gains a twist, that then propagates outwards while the disc reaches an untwisted shape. This fact confirms that an inviscid disc tends to a steady state that is well described by the analytic solution both in the tilt shape and in the untwisted behaviour.[]{data-label="fig:alpha_0_phase"}](a05_alpha000_t20000_phase){width=".9\columnwidth"}
Full 3D simulations {#sec:res}
===================
Hitherto in order to describe warp propagation in protostellar circumbinary discs, we have made many approximations. We recall them here. First of all, we have considered the gravitational potential generated by the two central stars as time independent. In order to do so, we assumed that the inner edge of the disc was quite far from the binary, $r_1/R_{\mathrm{in}} < 1$ and $r_2/R_{\mathrm{in}} < 1$. Secondly, in the thin-disc approximation, we have considered $\partial_t \Sigma = 0$, because the viscous time scale is much longer than the sound crossing time, which is the warp evolution time in the wave-like regime. Then, we have considered the linear (small amplitude) waves propagating with the single $m=1$ mode.
3D hydrodynamical simulations allow us to model our physical systems without making any of the assumptions listed above. First of all, we can implement the rotation of the two central stars directly without estimating the time-independent contribution of their gravitational potential. The particles describing the disc flow will be subject to the full potential generated by the binary. Secondly, we do not have to make any assumption about the warp dynamics, and especially the assumption of linear perturbations. By comparing the results of the 3D simulations with the results obtained by the 1D code, we will be able to check whether our assumptions were reasonable, and in which parameters range we can consider them valid.
SPH and viscosity {#sec:SPH}
-----------------
We perform our 3D simulations by using a smoothed particle hydrodynamics (SPH) code [see @price12 for a recent review]. We have used the <span style="font-variant:small-caps;">phantom</span> code by Daniel Price [see e.g. @lodato_price10; @price_fed10], which has been shown to perform well in dealing with warp propagation in the diffusive (non self-gravitating) regime [see @lodato_price10; @nixon_al12 for two recent applications]. As already mentioned in the introduction, SPH codes have already been used to simulate warp propagation in the bending-wave (thick disc) regime, but with very low resolution [@nelson_pap99; @nelson_pap00].
In this section we shall not go through SPH theory. We will just describe how we implemented an isotropic viscosity in the code. We know that SPH codes implement an artificial viscosity in order to spread discontinuities over a few smoothing lengths. It has been known for some time [@murray96] that the artificial terms in SPH can be understood straightforwardly as numerical representations of second derivatives of the velocity, and this makes viscosity act also when there is a purely shear flow, as in an accretion disc. The $\alpha$ parameter for the shear viscosity is related to the artificial one ($\alpha_{\rm art}$) by: $$\alpha_{\rm SS}=\frac{1}{10}\alpha_{\rm art} \frac{\bar{h}}{H},
\label{eq:visc_sph}$$ where $\bar{h}$ is the averaged smoothing length at radius $R$ and $H$ the scale-height of the disc at the same radius. @lodato_price10 showed excellent agreement between this relation and the outcome of their simulations. The notation $\alpha_{\rm SS}$ is used to discriminate between the directly implemented physical viscosity (see below) and the physical viscosity due to the artificial one. Thus, we could simulate the physical viscosity of our discs by using the artificial one. However, in order to keep a uniform value for $\alpha_{\rm SS}$, we would need a constant ratio $\bar{h}/H$ in the disc. In the case of power-law density and sound speed profiles, this requires the two exponents to be related by $p+2q=3$ (e.g. $p=1.5$, $q=0.75$). Our choice for these two parameters is different though, since we preferred to use $p=0.5$ in order to compare our 1D code with . To mimic viscosity, in this work we used an alternative formulation proposed by @flebbe94, where we evaluate directly the stress tensor in the Navier-Stokes equation. We preferred this formulation to the similar one by @espanol_revenga07 because the former has been well tested by @lodato_price10, and the latter does not conserve angular momentum. By this method, we have direct control of the viscous terms, since we set the shear viscosity by hand: $$\nu=\alpha c_{\rm s}^2(R)/\Omega(R),$$ where $\alpha$ is the chosen value of the viscosity parameter, and the profiles of the sound speed $c_{\rm s}$ and of the (Keplerian) angular velocity $\Omega$ are prescribed functions of $R$. Moreover, we can set the bulk viscosity to $0$. In this work, however, we still keep a small amount of artificial viscosity in order to correctly dissipate shocks if they are present and prevent particle interpenetration using the @morris_mon97 switch. Moreover, when the physical $\alpha$ is set to $0$ in order to simulate an inviscid motion, we will need an artificial viscosity in order to prevent chaotic motions of SPH particles [@price_fed10] that would increase the effective viscosity instead of reducing it. In the simulations presented in this section, we have used $\alpha_{\rm art,max}=0.5$ and $\alpha_{\rm art,min}=0.01$ (unless specified otherwise), where the value of $\alpha_{\rm art}$ between this two values is estimated via the Morris and Monaghan switch. The @von50 $\beta_{\rm art}$ parameter has been set equal to $2$.
Finally, note that in our simulations we did not compute the energy equation. We adopted a locally isothermal equation of state (set by the $q$ parameter assigned to the local sound-speed).
{width=".9\columnwidth"} {width=".9\columnwidth"}
{width=".95\columnwidth"}
{width=".9\columnwidth"} {width=".9\columnwidth"}
Numerical setup and initial conditions
--------------------------------------
We model the two central stars as sink particles. i.e. non-gaseous particles with appropriate boundary conditions [@bate95]. We assign an accretion radius to each of the two sink particles, i.e. the radius within which we can consider the fluid as accreted onto the star. This feature allows us not to follow the dynamics of gas particles too close to the binary system, since it would be computationally expensive. We assign an accretion radius of $0.05 R_{\rm in}$ to each star.
We have run simulations with different resolutions. We go from $N=10^5$ to $N=2\cdot10^6$, where $N$ is the number of particles used. Once we have assigned the values of the physical parameters of the system (see below), we distribute the particles so that the disc attains a prescribed initial density profile. We assign each particle a radius dependent sound-speed $c_{\rm s}$, where $c_{\rm s}\propto R^{-q}$ as usual. We then distribute the particles in the vertical direction, in order for the density to have a Gaussian profile in the $z$-direction. The scale-height is set to $H=c_{\rm s}/\Omega$. Finally, the initial angular velocity is assigned by taking into account pressure contributions:
$$v_{\phi}=v_{\rm K}\left[1-(p+2q)\left(\frac{c_{\rm s}}{v_{\rm K}}\right)^2\right]^{1/2},$$
where $v_{\rm K}=\sqrt{GM/R}$ is the usual Keplerian velocity.
As initial condition, we implement a surface density equal to:
$$\label{eq:sigma_in}
\Sigma(R)=\Sigma_0 R^{-p} \left( 1-\sqrt{\frac{R_{\mathrm{in}}}{R}} \right),$$
where $p$ is the usual coefficient introduced in section \[sec:analyt\]. Note that this is slightly different from the pure power law used above in section \[sec:analyt\]. We use the above relation because otherwise the inner gas would be pushed inwardly by the strong pressure gradient at the inner edge. Moreover, with this setup it is reasonable to consider the $x$ and $y$ components of the torque equal to $0$. Note that $\Sigma_0$ does not play a key role in our simulations, as long as $M_d \ll M_1+M_2$, where $M_d$ is the mass of the disc. In fact, self-gravity is not implemented in the SPH code. However, sink particles do feel the gravitational force generated by the gas particles. Since $M_d \ll M_1+M_2$, the back-reaction of the disc on the angular momentum of the binary is negligible ($M_d=0.01M$ in all the simulations). We do not impose any condition at the outer edge. We just consider wide discs, in such a way that $t_{\nu} = R_{\rm out}^2/\nu \gg t_{\mathrm{sound}}$, where $t_{\rm sound}$ is the sound crossing time, so that the external boundary condition does not affect the evolution in the inner regions, which are the ones we are interested in. However, we do not simulate too wide discs because most of the mass lies in the outer regions ($M(R)\propto R^{2-q}$), but we need a high resolution at the very inner edge where the external torque is stronger. The dynamic range of our simulations is $R_{\rm out}/R_{\rm in}=35$.
The other parameters have been set to: $p=0.5$, $q=0.75$, $H_{\rm in}/R_{\rm in}=0.1$, $\eta=0.25$ and $R_{\rm in}=a$. This set of parameters is slightly different from our standard runs of section \[sec:1D\]. In particular, here we consider a disc that is initially closer to the central binary. We made this choice in order to have a more prominent warp. Otherwise, since the resolution at small radii is relatively poor (because $\Sigma\rightarrow 0$), the features of the warp would be masked by the low signal to noise ratio. Note however that the inner radius of the disc will be pushed further from the central binary due to tidal forcing. We will discuss this issue in detail in section \[subsec:results\]. The time variable $t$ is expressed in terms of $\Omega_{\mathrm{in}}^{-1}$ in the whole section.
As initial condition we used an untwisted disc uniformly tilted with respect to the binary plane. Therefore, at $t=0$, $\beta(R)={\rm const}$ and $\gamma(R)=0$. The initial inclination angle and the physical viscosity will be specified later while presenting the results. The simulations are run in the same dimensionless units as in section \[sec:analyt\].
![Tilt of two differently resolved discs, with $\alpha=0.05$ and $\beta_{\infty}=5^\circ$, at $t=4000$. The black line portrays a disc with $N=1$ million particles, and the red line a disc with $N=100000$ particles. For less well resolved discs, the tidal torques at the inner disc edge are less effective, leading to a smaller warp. The number of shells to compute the averaged quantities is $200$ in order to reduce the scatter in the inner regions.[]{data-label="fig:res_resolution"}](resolution_alpha005_200_rin17){width=".9\columnwidth"}
Results {#subsec:results}
-------
In this section we compare the results of the 3D SPH simulations with the ones obtained by the 1D code. New 1D simulations are reported here, with a surface density profile equal to the one reported in equation \[eq:sigma\_in\]. In order to compare the results we had to compute azimuthally averaged disc quantities of the the SPH simulations in a number of thin shells. The procedure is the one described in section $3.2.6$ of @lodato_price10. The number of shells has been set to $300$.
Before comparing the results, we have verified that $\partial_t \Sigma\approx0$ in the SPH simulations. We looked at the evolution of $\Sigma$ in a simulation time ($t_{\mathrm{stop}} = 4000$) for two cases: $\alpha = 0.05$ and $\alpha = 0.2$. Note that with our setup $t_{\nu}\approx10^5(0.2/\alpha)\gg t_{\rm stop}$. For these two simulations $N = 1$ million. In Fig. \[fig:sigma\_evol\] we show $\Sigma$ at the beginning of the two simulations and at their very end. From the results illustrated in the figure we can conclude that the assumption $\partial_t \Sigma \approx 0$ is satisfied at least in the bulk of the disc, even in the diffusive regime ($\alpha=0.2$). However, we can make the following observations.
Firstly, at the outer edge the surface density profile smooths towards a continuous configuration. The initial condition presents a discontinuity at the outer radius, which is damped out quite quickly by pressure forces. Secondly, and more importantly, in both cases we observe an evolution at the inner edge: the inner radius is pushed further from the binary by tidal forcing, as we expect since the initial inner radius is smaller than the tidal truncation one. From @art_lubow94 we expect it to be at a radius $R_{\rm t}\approx 1.7 a$ when $\eta = 0.25$. Indeed, in section \[sec:analyt\] we had set this value equal to $2a$ for simplicity. Note that the location of the inner disc edge has a strong effect on the warp, given the strong radial dependence of the binary torques. In the two cases shown in Fig. \[fig:sigma\_evol\] $R_{\rm t}$ lies around $1.5-1.9 a$, in good agreement with the predicted $1.7 a$.
Finally, we have verified that the binary angular momentum variations are negligible over the simulation time.
{width=".9\columnwidth"} {width=".9\columnwidth"}
{width=".9\columnwidth"} {width=".9\columnwidth"}
### Linear regime
We analyse the warp dynamics in three cases: $\alpha = 0.05$, $\alpha = 0.2$ and $\alpha \approx 0$. In the first case the disc falls in the wave-like regime, in the second one it is in the diffusive regime, and the third one is the closest possible value to the inviscid case. Initially, at $t=0$, the disc is tilted with respect to the binary plane by an angle $\beta_{\infty} = 5^\circ$. Both @nelson_pap99 and @ogilvie06 have shown that in absence of an external torque (which does not modify the regime the waves propagate with, anyway) such a small inclination ensures a linear regime for the wave propagation. We have run other simulations with lower values of the initial $\beta_{\infty}$, but since we are dealing with full 3D simulations, we have to take the finite thickness into account. When $\beta_{\infty}$ is too small (i.e. $\tan{\beta_{\infty}} \lesssim H/R$), we obtain a very noisy measure of the tilt. By increasing $\beta_{\infty}$ up to $5^\circ$, we reduce the noise-to-signal ratio. Therefore we report the results of the $5^\circ$ simulations only.
Let us start with the case of $\alpha = 0.05$. In Fig. \[fig:res\_005\] we report the evolution of the tilt and the phase at $t = 2000$. The number of particles used in this simulation is $N=1$ million. We illustrate the results of the 3D simulation with the black lines, and the ones of the corresponding 1D simulation with the red lines. In the 1D simulations the inner edge of the disc has been set equal to $R_{\rm t}=1.7 a$. The general trend of the 3D simulations is in very good agreement with the 1D ones. Both the tilt and the twist tend to a steady state. Moreover, the tilt tends to a steady state that has a shape described by an evanescent wave, as discussed in section \[sec:waves\_theory\] theoretically, and in sections \[sec:analyt\] and \[sec:1D\] both analytically and numerically. There is a good agreement also in the propagation velocity. Overall, the evolution of both the tilt and the phase throughout the simulation is in very good agreement with that obtained by solving the linearised warp equations in 1D.
An additional effect (not apparent in these plots) is that the evolution shows a small periodic oscillation with time. This is due to the imperfect rotational symmetry about the $z$-axis of the gravitational potential, which enforces wobbling modes into the disc [@bate00], especially with $\sigma=2$.
We have also tested the effects of limited resolution in SPH by running a simulation with the same parameters as above but with 10 times fewer particles. The tilt evolution in this case is shown in Fig. \[fig:res\_resolution\]. We see that in this case the disc develops a smaller warp (i.e., the inner disc tends to stay more aligned with the outer disc). This is due to the fact that decreasing the resolution, we do not resolve equally well the inner disc edge. The disc thus appears truncated by tidal torques at a slightly larger radius ($\sim 2a$ in this case), thus decreasing the warp amplitude. We find a very good agreement in the evolution of the tilt by comparing this poorly resolved simulation with a 1D simulation with an inner radius equal to $2a$.
We have just described a disc in which the wave-like regime is expected in the whole disc. Now we use the same setup (simulations with $N = 1$ million particles) as above, but we set $\alpha = 0.2$. By knowing that $H/R = 0.1\ x^{-1/4}$, the condition $\alpha \gtrsim H/R$ is verified in the whole disc. Therefore with $\alpha = 0.2$ the warp evolves diffusively. We compare the 3D results with the ones obtained in section \[sec:1D\] for the same value of $\alpha$. In Fig. \[fig:res\_02\] we report the results for the tilt and the phase evolution, respectively. The inner edge of the disc has been set to $R_{\rm t}=1.7 a$ for the 1D simulations. We observe a good agreement. This confirms that the equations do describe the evolution even in a diffusive regime (see section \[sec:waves\_theory\]). A small discrepancy is still present at the very inner edge because of the low resolution when $\Sigma$ tends to $0$ (and the associated error in the estimate of $R_{\rm t}$), but this does not affect the shape in the outer regions of the disc.
Finally, we try to simulate the dynamics of an inviscid disc. In order to do it, we consider a disc with physical viscosity equal to 0. However, as we have already mentioned in section \[sec:SPH\], we cannot remove the artificial viscosity from the simulations completely. In this section we use the following viscosity parameters: $\alpha_{\mathrm{art,max}} = 0.5$, $\alpha_{\mathrm{art,min}} = 10^{-5}$ and $\alpha = 0$. We used such a low value for $\alpha_{\mathrm{art,min}}$ because when ${\bf \nabla} \cdot {\bf v} < 0$ the Morris and Monaghan switch ensures that discontinuities are smoothed by a higher value of $\alpha_{\mathrm{art}}$. The artificial viscosity grows to its maximum value when . We tried to use lower values of $\alpha_{\mathrm{art,max}}$, but, as predicted, the simulations become noisy. In Fig. \[fig:res\_00\] we report the tilt and the phase evolution of such a disc.
In the tilt we see a good agreement between the 3D simulation and the 1D solution (where, as usual, the inner radius of the disc has been set equal to $R_{\rm t}=1.7a$). In the 3D tilt we note a bump on the wavefront. This is due to the fact that in the 3D simulation $N$ is not large enough to resolve the tilt discontinuity at the wavefront of the 1D code: such a discontinuity is smoothed over some smoothing lengths.
The phase evolution is very different. As described in section \[sec:inviscid\], in the 1D case when $\alpha = 0$ the disc rotates and reaches a steady state that is untwisted. Instead, in the 3D case the disc does present a twist. This fact emphasises that our simulations are not completely inviscid, since the artificial viscosity is acting as a small effective viscosity in the disc (see equation \[eq:visc\_sph\]). The small amount of this effective viscosity produces the twist observable in the figure. Moreover, the resolution is not high enough to resolve the strong phase discontinuity of the 1D simulation. SPH is smoothing the discontinuity over some smoothing lengths.
Still, we note that the agreement between theory and simulations is remarkable, even for almost inviscid discs.
![Tilt and phase evolution of an initially untwisted tilted disc, with $\beta_{\infty} = 60^\circ$, $\eta=0.25$, $(H_{\rm in}/R_{\rm in})=0.1$ and $N = 2$ million at $t=2000$. A sharp break occurs in the 3D simulations (black line). The linear theory (red line) fails to describe the warp evolution for such high inclinations.[]{data-label="fig:nonlinear_tilt"}](tilt_non_linear_17){width=".9\columnwidth"}
![3D structure of the disc shown in Fig. \[fig:nonlinear\_tilt\] at $t=1760$. The disc breaks in two almost separated discs. The inner one starts precessing, since its width is very narrow.[]{data-label="fig:nonlinear"}](nonlinear_blue){width=".95\columnwidth"}
### Non-linear regime
If we enhance the initial inclination angle, the linear theory fails. As reported above, some efforts have been made in order to describe the non-linear regime, both numerically [@nelson_pap99] and analytically [@ogilvie06], the latter in the absence of external torques. In this section we simulate the non-linear regime of wave-like warp propagation via SPH simulations with a much higher resolution than in previous works. Moreover, we focus on the case where the disc is subject to external torques, due to the central misaligned binary. We use $2$ million particles, and the following set of parameters: $\alpha_{\rm art,max}=0.5$, $\alpha_{\rm art,min}=0.01$, $\alpha=0.05$, $H_{\rm in}/R_{\rm in}=0.1$, $M_1=M_2=0.5$ and $r_1/R_{\rm in}=r_2/R_{\rm in}=0.5$. This is the same set used for the simulation portrayed in Fig. \[fig:res\_005\]. We have used the usual formulation for the viscosity, since we know that in non-linear cases shocks are much more likely to occur.
We have performed simulations with three different initial inclinations of the disc plane with respect to the binary plane: $\beta_{\infty}=20$, $40$ and $60^\circ$. The $20^\circ$ inclined disc starts showing relevant discrepancies from the linear regime, but it is in the $40^\circ$ and even more in the $60^\circ$ inclined disc that the evolution is completely different from the one predicted by the linear theory. In this section we report the results of the most inclined disc only as an example. Further studies on this issue are required.
Let us analyse the case with $\beta_{\infty}=60^\circ$. In Fig. \[fig:nonlinear\_tilt\] we show the evolution of the tilt at $t=2000$. As above, we illustrate the results of the 3D simulation with the black line, and the ones corresponding to the linear 1D simulation with the red line. For the 1D simulation the inner edge has been set equal to $R_{\rm t}=1.7 a$ We can immediately observe that the discrepancy is very remarkable, and that the linear theory fails to describe the warp propagation in such an extreme case. From Fig. \[fig:nonlinear\] we note that the disc breaks sharply. Moreover, the 3D simulation shows that the inner ring starts precessing.
This kind of behaviour has already been observed by @lodato_price10 and @nixon_al12, the latter focusing on discs subject to Lense-Thirring precession. Their simulations focus on the diffusive regime ($\alpha>H/R$), but they both obtain the breaking of the disc with low values of $\alpha$. Moreover, @fragner10 have seen the same result via a grid code, and @larwood_pap97 via SPH simulations, but with a much lower resolution than ours. Note that we have performed the same kind of simulation, but with a lower viscosity ($\alpha=0.01$), and the general result is equivalent to the more viscous case.
Conclusions {#sec:concl}
===========
In this paper we have analysed the bending-wave regime of protostellar circumbinary warped accretion discs. Analytically, we have found the general solution for the shape of an inclined disc around a binary of arbitrary mass ratio in the inviscid limit and in the linear approximation. We have verified that the solution for the steady state tilt of the disc is an evanescent wave, as predicted by (where they obtained the same kind of solution for a retrograde rotating disc around a spinning black hole). In the inviscid limit, the disc does not present any twist, and the inner parts do not become aligned with the binary. This is different to the diffusive limit, where the disc tends to align in the inner regions.
Then, we have performed 1D time-dependent calculations for low viscosity discs affected by the binary torque. We can summarise our results as follows: firstly, we have verified that the warp does evolve as a bending wave with a wave front moving at $c_{\rm s}/2$. The disc reaches a steady state described by a stationary wave, and its shape does not depend on the initial condition. Internal torques due to a warped structure in the inner region compensate the differential precession that would occur if the rings forming the disc were disconnected. Secondly, we have explored the parameter space in terms of viscosity and amplitude of the external torque. We have found that the tilt shape of the steady state depends strongly on these two parameters. The amplitude of the external torque affects the amplitude of the warp in the inner regions. The smaller the mass ratio of the two stars (or the farther away from the disc they are), the smaller the amplitude of the warp. This fact is intuitive: as the mass ratio gets lower, the gravitational potential becomes more spherically symmetric. Therefore, the disc is less affected by an external torque. Instead, viscosity affects the disc in two ways. If the disc is viscous enough ($\alpha>H/R$), the warp propagates diffusively, as predicted by the theory, and aligns with the binary plane at the inner edge. Moreover, the amount of viscosity regulates the amplitude of the twist in the the disc. The more viscous the disc is, the more twisted it gets. Thirdly, we have run simulations with $\alpha=0$. In this last case, the disc reaches a steady state in very good agreement with the analytic solution. It tends to the same shape in the tilt, and to a constant phase that depends on the initial condition. However, we recall that the tilt normalisation and the constant phase angle are two degrees of freedom of the system, since we deal with a linear regime and rotational symmetry.
We can compare these results with the ones obtained by @foucart13, who also give approximate analytic estimates of the amplitude of the warp in the steady state shape of misaligned circumbinary discs. In our paper we have added significant contributions to their results. Firstly, we have explicitly derived an analytic solution for the inviscid case, in the form of modified Bessel functions. Secondly, we have analysed the temporal evolution of the shape of the discs, whereas @foucart13 only focus on the steady state solutions, and we have explored the parameters space with time-dependent simulations. Thirdly, as they do, we show that the equations (and therefore the solutions) depend on two dimensionless parameters only, $\chi$ and $\alpha/(H_{\rm in}/R_{\rm in})$, and we broadly confirm their results on the amplitude of both the warping and the twisting.
We have then performed 3D SPH simulations, in order to explore both the linear and the non-linear regime. We have compared these results with the ones obtained with the 1D ring code. By focusing on the linear case, we have firstly verified the validity of the assumptions made in the 1D model. Secondly, we have tested that the agreement is good, both in the tilt and in the twist, in the wave-like regime. Small discrepancies are due to the low resolution in the inner regions of the disc. By increasing the value of viscosity, we have verified that the wave equations do succeed in simulating the viscous regime (the agreement between 3D and 1D simulation is remarkable), with the *caveat* that this is valid for short enough timescales. Finally, we have performed simulations at very low viscosities, close to the inviscid case. By comparing the SPH results to the 1D inviscid ones, we have obtained a good agreement, at least in the tilt evolution. This fact emphasises how well SPH is able to reproduce the warp evolution of discs, even for the case of extremely low viscosities. We have shown that standard tools, such as the Morris & Monaghan switch, are indeed effective at reducing artificial viscosity and ensure the possibility of running almost inviscid warped disc SPH simulations.
In the non-linear regime, we have shown that for high inclination angles of the disc plane with respect to the binary one the disc breaks, and the inner ring precesses almost completely disconnected from the outer regions of the disc. Additional studies are required to further explore this last issue.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Chris Nixon, Cathie Clarke and Jim Pringle for stimulating discussion. We thank the anonymous referee for useful advice and for pointing to us the paper by @foucart13. SF thanks the Science and Technology Facility Council and the Isaac Newton Trust for the award of a studentship. Figs \[fig:analytical\]-\[fig:nonlinear\] were produced using <span style="font-variant:small-caps;">splash</span> [@price07], a visualisation tool for SPH data.
\[lastpage\]
[^1]: facchini@ast.cam.ac.uk
[^2]: Note that in order to have a finite tilt angle for $R\rightarrow\infty$, we need to require that $\zeta>0$. This is generally satisfied, unless either the sound speed or the surface density is a very steeply decaying function of $R$. In these cases, the warp cannot be communicated effectively through the disc, which will behave essentially as a single precessing ring, and will thus not reach a steady (non-precessing) shape.
|
---
abstract: 'The closed time path (CTP) formalism is applied, in the framework of open quantum systems, to study the time evolution of the expectation value of the energy-momentum tensor of a scalar field in the presence of real materials. We analyze quantum (Casimir) fluctuations in a fully non-equilibrium scenario, when the scalar field is interacting with the polarization degrees of freedom of matter, described as quantum Brownian particles (harmonic oscillators coupled to a bath) in each point of space. A generalized analysis was done for two types of couplings between the field and the polarization degrees of freedom. On the one hand, we considered a bilinear coupling between the field and the polarization degrees of freedom, and on the other hand, a (more realistic) current-type coupling as in the case of the electromagnetic field interacting with matter. We successfully computed the CTP generating functional for the field, through calculating the corresponding influence functionals. We considered the high temperature limit for the field, keeping arbitrary temperatures for each part of the material’s volume elements. We obtained a closed form for the Hadamard propagator, which let us study the dynamical evolution of the expectations values of the energy-momentum tensor components from the initial time, when the interactions are turned on. We showed that two contributions always take place in the transient evolution: one of these is associated to the material and the other one is only associated to the field. Transient features were studied and the long-time limit was derived in several cases. We proved that in the steady situation of a field in $n+1$ dimensions, the material always contribute unless is non-dissipative. Conversely, the proper field contribution vanishes unless the material is non-dissipative or, moreover, at least for the $1+1$ case, if there are regions without material. We finally conclude that any steady quantization scheme in $1+1$ dimensions must consider both contributions and, on the other hand, we argue why these results are physically expected from a dynamical point of view, and also could be valid for higher dimensions based on the expected continuity between the non-dissipative and real material cases'
author:
- 'Adrián E. Rubio López[^1] and Fernando C. Lombardo[^2]'
date: today
title: Closed time path approach to the Casimir energy in real media
---
Introduction
============
The study of the Casimir forces in the framework of open quantum systems is the possibility of analyzing non-equilibrium effects, such as the Casimir force between objects at different temperatures [@Antezza], the power of heat transfer between bodies [@Bimonte], and the inclusion of time dependent evolutions until reaching a stationary situation. Even though, the celebrated Lifshitz formula [@Lifshitz] describes the forces between dielectrics at steady situation in terms of their macroscopic electromagnetic properties, this is not derived from a first principle quantum framework. The original derivation of this very general formula is based on a macroscopic approach, starting from stochastic Maxwell equations and using thermodynamical properties for the stochastic fields. As pointed out in several papers, the connection between this approach and an approach based on a fully quantized model is not completely clear. Moreover, some doubts have been raised about the applicability of the Lifshitz formula to lossy dielectrics [@Barton2010; @Philbin2010; @daRosaetal].
Moreover, from a conceptual point of view, the theoretical calculations for mirrors with general electromagnetic properties, including absorption, is not a completely settled issue [@Barton2010; @Philbin2010; @daRosaetal]. Since dissipative effects imply the possibility of energy interchanges between different parts of the full system (mirrors, vacuum field and environment), the theory of open quantum systems [@BreuerPett] is the natural approach to clarify the role of dissipation in Casimir physics. Indeed, in this framework, dissipation and noise appear in the effective theory of the relevant degrees of freedom (the electromagnetic field) after integration of the matter and other environmental degrees of freedom.
The quantization at the steady situation (steady quantization scheme) can be performed starting from the macroscopic Maxwell equations, and including noise terms to account for absorption [@Buhmann2007]. In this approach a canonical quantization scheme is not possible, unless one couples the electromagnetic field to a reservoir (see [@Philbin2010]), following the standard route to include dissipation in simple quantum mechanical systems. Another possibility is to establish a first principles model in which the slabs are described through their microscopic degrees of freedom, which are coupled to the electromagnetic field. In this kind of models, losses are also incorporated by considering a thermal bath, to allow for the possibility of absorption of light. There is a large body of literature on the quantization of the electromagnetic field in dielectrics. Regarding microscopic models, the fully canonical quantization of the electromagnetic field in dispersive and lossy dielectrics has been performed by Huttner and Barnett (HB) [@HB]. In the HB model, the electromagnetic field is coupled to matter (the polarization field), and the matter is coupled to a reservoir that is included into the model to describe the losses. In the context of the theory of quantum open systems, one can think the HB model as a composite system in which the relevant degrees of freedom belong to two subsystems (the electromagnetic field and the matter), and the matter degrees of freedom are in turn coupled to an environment (the thermal reservoir). The indirect coupling between the electromagnetic field and the thermal reservoir is responsible for the losses. As we will comment below, this will be our starting point to compute the Casimir force between absorbing media.
In a previous work [@LombiMazziRL], we have followed a steady canonical quantization program similar to that of Ref.[@Dorota1992], generalizing it by considering a general and well defined open quantum system. In this work, we will work with two simplified models analogous to the one of HB, both assuming that the dielectric atoms in the slabs are quantum Brownian particles, and that they are subjected to fluctuations (noise) and dissipation, due to the coupling to an external thermal environment. We will keep generality in the type of spectral density to specify the bath to which the atoms are coupled, generalizing the constant dissipation model as in Ref.[@LombiMazziRL]. Indeed, after integration of the environmental degrees of freedom, it will be possible to obtain the dissipation and noise kernels that modify the unitary equation of motion of the dielectric atoms.
The difference between both models relies in their couplings to the field. On the one hand, the first model, which we will call bilinear coupling model, consists in a direct coupling between the field and the atom’s polarization degree of freedom in each point of space. On the other hand, the current-type coupling model consists in a coupling between the field time derivative and the atom’s polarization degree of freedom. The former model is more suitable for the development of the calculations, while the latter is more realistic in the sense that is closer to the real coupling between the electromagnetic (EM) field and the matter. However, both models are of interest and can be studied in a compact way altogether to obtain general conclusions about the non-equilibrium thermodynamics and transient time evolution of quantum fields in the framework of quantum open system.
With this aim, we used the Schwinger-Keldysh formalism (or closed time path (CTP) - in-in - formalism) to provide the theoretical framework, which is based on the original papers by Schwinger [@Schwinger] and Keldysh [@Keldish] and is particularly useful for non-equilibrium quantum field theory (also see [@Chou; @Jordan; @Weinberg]). According to the CTP formalism, the expectation value of an operator and its correlation functions can be derived from an in-in generating functional in a path integral representation [@CalHu], in a similar way as it happens in the well-known in-out formalism [@GreiRein] but by doubling the fields and connecting them by a CTP boundary condition, which ensures that the functional derivatives of the generating functional give expectation values of the field operator. In this scheme, the open quantum systems framework is totally integrated through the concept of the influence action [@FeynHibbs], resulting from partial trace over the environment degrees of freedom, giving the effective dynamics for the system through a coarse graining of the environments. Influence actions have been calculated in different context in the Literature, for example, specific models assume that during cosmological inflation the UV (or sub-Hubble) modes of a field, once integrated out, decohere the IR (or super-Hubble) modes because the former modes are inaccessible observationally. In these models, the CTP formalism applied to cosmological perturbations aims to describing the transition between the quantum nature of the initial density inhomogeneities as a consequence of inflation and the classical stochastic behaviour [@lombardo].
This paper is organized as follows: In the next Section we introduce the bilinear model. In Sec. \[GIF\], we fully develop the CTP formalism for the open quantum system to obtain the generating functional for the field and the influence actions that result after each functional integration, identifying the dissipation and noise kernels in each influence action. In Sec. \[CC\], we extend, by a few number of modifications, the calculation of the generating functional done in Sec. \[GIF\] to the current-type model, calculating the new dissipation and noise kernels. In Sec. \[EMTFC\], we derive a closed form for the expectation values of the energy-momentum tensor components in terms of the Hadamard propagators for each coupling model. Then, in Sec. \[NEBFDCOTCS\] we study different scenarios of interest where our general results give different transient time behaviors and different conclusions about the steady situations in each coupling case. Finally, Sec. \[FR\] summarize our findings. The Appendix contains some details of intermediate calculations.
Bilinear Coupling
=================
In order to include effects of dissipation and noise (fluctuations) in the calculation of the energy density of the electromagnetic field in interaction with real media, we will develop a full CTP approach to the problem.
Therefore, we will consider a composite system consisting in two parts: the field, which we will consider a real massless scalar field and the real media, which in turn are modeled by continuous sets of quantum Brownian particles localized in certain regions of space. With this, we represent the polarization density degrees of freedom. These degrees of freedom are basically harmonic oscillators coupled to the field at each point, that can be associated with the material’s atoms. The composite system (field and material atoms) is also coupled to an external bath of harmonic oscillators throught the interaction between the atoms in the material and the thermal environment.
Then, the total action for the whole system is given by
$$S[\phi,r,q_{n}]=S_{0}[\phi]+S_{0}[r]+\sum_{n}S_{0}[q_{n}]+S_{\rm int}[\phi,r]+\sum_{n}S_{\rm int}[r,q_{n}],$$
where each term given by
$$S_{0}[\phi]=\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~\frac{1}{2}~\partial_{\mu}\phi~\partial^{\mu}\phi,
\label{FreeFieldAction}$$
$$S_{0}[r]=\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\frac{m_{\mathbf{x}}}{2}\left(\dot{r}_{\mathbf{x}}^{2}(\tau)-\omega_{\mathbf{x}}^{2}~r_{\mathbf{x}}^{2}(\tau)\right),
\label{FreePolAction}$$
$$S_{0}[q_{n}]=\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\frac{m_{n,\mathbf{x}}}{2}\left(\dot{q}_{n,\mathbf{x}}^{2}(\tau)-\omega_{n,\mathbf{x}}^{2}~q_{n,\mathbf{x}}^{2}(\tau)\right),
\label{FreeBathHOAction}$$
$$S_{\rm int}[\phi,r]=\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\lambda_{0,\mathbf{x}}~\phi(\mathbf{x},\tau)~r_{\mathbf{x}}(\tau),
\label{IntFieldPolAction}$$
$$S_{\rm int}[r,q_{n}]=\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\frac{\lambda_{n,\mathbf{x}}}{\sqrt{2m_{n,\mathbf{x}}~\omega_{n,\mathbf{x}}}}~r_{\mathbf{x}}(\tau)~q_{n,\mathbf{x}}(\tau),
\label{IntPolBathHOAction}$$
where the subindex $\mathbf{x}$ denotes the fact that the oscillators (and its properties) in each point of the space are independent of each other. In other words, we have to conceive the set of oscillators $r$ associated to the polarization density that form the material more like a continuous of independent quantum degrees of freedom (with density $\eta_{\mathbf{x}}$), where each polarization degree of freedom has its own material properties (masses $m_{\mathbf{x}}$, frequency $\omega_{\mathbf{x}}$ and coupling $\lambda_{0,\mathbf{x}}$), being $\mathbf{x}$ only a label when appearing as a subindex (we are assuming that the material can be inhomogeneous). Analogously, we consider for the respective properties of each thermal bath interacting with the polarization degrees of freedom represented by the sets of oscillators $\{q_{n,\mathbf{x}}\}$ in each spatial point.
On the other hand, the matter distribution $g(\mathbf{x})$ defines the regions of material and is $g= 1$ for this regions and $g=0$ outside them.
It is also worth noting that the scalar field seems to be one of the electromagnetic field components interacting with matter. In this first model we consider, for simplicity, a bilinear coupling between the field and the polarization degree of freedom.
Finally, we will assume that the total system is initially uncorrelated, thus the initial density matrix is written as a direct product of each part, which we also suppose to be initially in a thermal equilibrium at a proper characteristic temperatures ($\beta_{\phi},\beta_{r_{\mathbf{x}}},\beta_{B,\mathbf{x}}$ -the material can also be thermally inhomogeneous-),
$$\widehat{\rho}(t_{0})=\widehat{\rho}_{\phi}(t_{0})\otimes\widehat{\rho}_{r_{\mathbf{x}}}(t_{0})\otimes\widehat{\rho}_{\{q_{n,\mathbf{x}}\}}(t_{0}).
\label{InitialState}$$
Generating and Influence Functionals {#GIF}
====================================
Our goal in this Section is to compute the expectation value of the field quantum correlation function. We will employ the in-in formalism by means of a closed time path (CTP) to write the field’s generating functional, after integrating out the environment by generalizing the procedure known from, for example, Refs. [@CalHu; @CalRouVer],
$$\begin{aligned}
Z[J,J']=\int d\phi_{\rm f}\int d\phi_{0}~d\phi'_{0}\int_{\phi(\mathbf{x},t_{0})=\phi_{0}(\mathbf{x})}^{\phi(\mathbf{x},t_{\rm f})=\phi_{\rm f}(\mathbf{x})}\mathcal{D}\phi\int_{\phi'(\mathbf{x},t_{0})=\phi'_{0}(\mathbf{x})}^{\phi'(\mathbf{x},t_{\rm f})=\phi_{\rm f}(\mathbf{x})}\mathcal{D}\phi'&&\rho_{\phi}(\phi_{0},\phi'_{0},t_{0})~e^{i\left(S_{0}[\phi]-S_{0}[\phi']\right)}~\mathcal{F}[\phi,\phi']\nonumber\\
&&\times~ e^{i\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau\left(J(\mathbf{x},\tau)~\phi(\mathbf{x},\tau)-J'(\mathbf{x},\tau)~\phi'(\mathbf{x},\tau)\right)},
\label{GeneratingFunctional}\end{aligned}$$
where the field’s functional $\mathcal{F}$ is known as the influence functional [@FeynHibbs] which is related to the field’s influence action $S_{\rm IF}[\phi,\phi']$ generated by the material degrees of freedom (atoms plus baths).
Since the material is modeled as a continuum of spatially-independent oscillators each one interacting with its own bath, the influence functional clearly factorizes in the spatial label resulting
$$\begin{aligned}
\mathcal{F}[\phi,\phi']=e^{iS_{\rm IF}[\phi,\phi']}&=&\prod_{\mathbf{x}}\int dr_{\rm f,\mathbf{x}}\int dr_{0,\mathbf{x}}~dr'_{0,\mathbf{x}}\int_{r_{\mathbf{x}}(t_{0})=r_{0,\mathbf{x}}}^{r_{\mathbf{x}}(t_{\rm f})=r_{\rm f,\mathbf{x}}}\mathcal{D}r_{\mathbf{x}}\int_{r'_{\mathbf{x}}(t_{0})=r'_{0,\mathbf{x}}}^{r'_{\mathbf{x}}(t_{\rm f})=r_{\rm f,\mathbf{x}}}\mathcal{D}r'_{\mathbf{x}} ~ \rho_{r_{\mathbf{x}}}\left(r_{0,\mathbf{x}},r'_{0,\mathbf{x}},t_{0}\right)~\nonumber\\
&\times & e^{i4\pi\eta_{\mathbf{x}}g(\mathbf{x})\left(S_{0}[r_{\mathbf{x}}]-S_{0}[r'_{\mathbf{x}}]\right)}~e^{i4\pi\eta_{\mathbf{x}}g(\mathbf{x})~S_{\rm QBM}[r_{\mathbf{x}},r'_{\mathbf{x}}]}~e^{i4\pi\eta_{\mathbf{x}}g(\mathbf{x})\left(S_{\rm int}[\phi,r_{\mathbf{x}}]-S_{\rm int}[\phi',r'_{\mathbf{x}}]\right)}.
\label{InfluenceFuntionalField}\end{aligned}$$
where $S_{\rm QBM}[r_{\mathbf{x}},r'_{\mathbf{x}}]=\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~\Delta r_{\mathbf{x}}(\tau)\left(-2~D_{\rm QBM,\mathbf{x}}(\tau-\tau')~\Sigma r_{\mathbf{x}}(\tau')+\frac{i}{2}~N_{\rm QBM,\mathbf{x}}(\tau-\tau')~\Delta r_{\mathbf{x}}(\tau')\right)$ is the well-known influence action for the QBM theory [@CaldeLegg; @HuPazZhang], which represents the influence of a bath at $\mathbf{x}$ (given by the set $\{q_{n,\mathbf{x}}\}$) over the polarization degrees of freedom $r_{\mathbf{x}}$ at the same spatial point. It is worth noting that in this expression the scalar fields $\phi$ and $\phi'$ appear as additional external sources as $J$ and $J'$ were so for the field. It is worth noting that we have set $\Delta r_{\mathbf{x}}=r'_{\mathbf{x}}-r_{\mathbf{x}}$ and $\Sigma r_{\mathbf{x}}=\left(r_{\mathbf{x}}+r'_{\mathbf{x}}\right)/2$, and that the QBM’s influence action is clearly the analogous result of the CTP expression for the influence functional of the field of Eq.(\[InfluenceFuntionalField\]), where the trace has been taken over the bath’s degree of freedom $\{q_{n,\mathbf{x}}\}$ considering them in a thermal state.
The kernels $N_{\rm QBM,\mathbf{x}}$ and $D_{\rm QBM,\mathbf{x}}$ in $S_{\rm QBM}$ are nothing more than the QBM noise and dissipation kernels respectively [@CalHu; @HuPazZhang]. It is clear that the expression of the influence action is quite general and applies to all type of baths (characterized by the spectral density being subohmic, ohmic or supraohmic [@HuPazZhang; @BreuerPett]) characterized by a particular temperature. In the same way, it turns out that the noise kernel $N_{\rm QBM,\mathbf{x}}$ corresponds to the sum of the Hadamard propagators for the bath oscillators at the point $\mathbf{x}$, while the dissipation kernel $D_{\rm QBM,\mathbf{x}}$ corresponds to the sum of the retarded propagators at the same point, which clearly shows a causal behavior ($D_{\rm QBM,\mathbf{x}}(\tau,\tau')\propto\Theta(\tau-\tau')$).
Field’s Influence Functional
----------------------------
At this point, we have to compute the influence functional for the field, $\mathcal{F}$ of Eq.(\[InfluenceFuntionalField\]). For this purpose, we have to evaluate each factor in the product. The result of this type of CTP integrals can be found in Ref. [@CalRouVer]. We present a generalization due to consider the degrees of freedom of the polarization density which is straightforward (polarization or bath degrees of freedom must contain a dimensional normalization factor $\frac{1}{4\pi\eta_{\mathbf{x}}}$ -see Ref. [@LombiMazziRL]- and we also have to take in account that the matter distribution satisfies $g^{2}(\mathbf{x})=g(\mathbf{x})$), therefore we obtain,
$$\begin{aligned}
\mathcal{F}[\phi,\phi']&=&\prod_{\mathbf{x}}\Big\langle e^{-i4\pi\eta_{\mathbf{x}}g(\mathbf{x})~\lambda_{0,\mathbf{x}}\int_{t_{0}}^{t_{\rm f}}d\tau~\Delta\phi(\mathbf{x},\tau)~\mathcal{R}_{0,\mathbf{x}}(\tau)}\Big\rangle_{r_{0,\mathbf{x}},p_{0,\mathbf{x}}}e^{-\frac{1}{2}~4\pi\eta_{\mathbf{x}}g(\mathbf{x})\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~\Delta\phi(\mathbf{x},\tau)~\mathcal{N}_{B,\mathbf{x}}(\tau,\tau')~\Delta\phi(\mathbf{x},\tau')}\nonumber\\
&&\times~e^{-i4\pi\eta_{\mathbf{x}}g(\mathbf{x})\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~\Delta\phi(\mathbf{x},\tau)~2~\mathcal{D}_{\mathbf{x}}(\tau,\tau')~\Sigma\phi(\mathbf{x},\tau')},
\label{CTPIntPol}\end{aligned}$$
where $\Delta\phi=\phi'-\phi$, $\Sigma\phi=(\phi+\phi')/2$ and $\mathcal{D}_{\mathbf{x}}(\tau,\tau')\equiv\mathcal{D}_{\mathbf{x}}(\tau-\tau')=\frac{\lambda_{0,\mathbf{x}}^{2}}{2}~G_{\rm Ret,\mathbf{x}}(\tau-\tau')$ is the dissipation kernel over the field, being $G_{\rm Ret,\mathbf{x}}$ the retarded Green function and $\mathcal{R}_{0,\mathbf{x}}$ the solution with initial conditions $\{r_{0,\mathbf{x}},p_{0,\mathbf{x}}\}$ associated to the semiclassical equation of motion, that results from the homogeneous equation
$$\begin{aligned}
\frac{\delta S_{\rm CTP}[r_{\mathbf{x}},r'_{\mathbf{x}}]}{\delta r_{\mathbf{x}}}\Big|_{r_{\mathbf{x}}=r'_{\mathbf{x}}}=\frac{\delta S_{\rm CTP}[\Delta r_{\mathbf{x}},\Sigma r_{\mathbf{x}}]}{\delta \Delta r_{\mathbf{x}}}\Big|_{\Delta r_{\mathbf{x}}=0}&=&0,\nonumber\\
\ddot{r}_{\mathbf{x}}+\omega_{\mathbf{x}}^{2}~r_{\mathbf{x}}-\frac{2}{m_{\mathbf{x}}}\int_{t_{0}}^{t}d\tau~D_{\rm QBM,\mathbf{x}}(t-\tau)~r_{\mathbf{x}}(\tau)&=&0,
\label{EqMotionR}\end{aligned}$$
where $S_{\rm CTP}[r_{\mathbf{x}},r'_{\mathbf{x}}]=S_{0}[r_{\mathbf{x}}]-S_{0}[r'_{\mathbf{x}}]+S_{\rm QBM}[r_{\mathbf{x}},r'_{\mathbf{x}}]$ and,
$$\mathcal{R}_{0,\mathbf{x}}(\tau)=r_{0,\mathbf{x}}~\dot{G}_{\rm Ret,\mathbf{x}}(\tau-t_{0})+\frac{p_{0,\mathbf{x}}}{m_{\mathbf{x}}}~G_{\rm Ret,\mathbf{x}}(\tau-t_{0}).
\label{HomoSolQBM}$$
On the other hand, the kernel $\mathcal{N}_{B,\mathbf{x}}$ is the part of the noise kernel associated to the baths that acts on the field (there is another part associated to the first factor in the right hand side of Eq.(\[CTPIntPol\])),
$$\mathcal{N}_{B,\mathbf{x}}(\tau,\tau')=\lambda_{0,\mathbf{x}}^{2}\int_{t_{0}}^{t_{\rm f}}ds\int_{t_{0}}^{t_{\rm f}}ds'~G_{\rm Ret,\mathbf{x}}(\tau-s)~N_{\rm QBM,\mathbf{x}}\left(s-s'\right)~G_{\rm Ret,\mathbf{x}}(\tau'-s').
\label{PhiNoiseKernelB}$$
Finally, the first factor in the right hand side of Eq.(\[CTPIntPol\]) is given by (see Ref. [@CalRouVer])
$$\begin{aligned}
\Big\langle e^{-i4\pi\eta_{\mathbf{x}}g(\mathbf{x})~\lambda_{0,\mathbf{x}}\int_{t_{0}}^{t_{\rm f}}d\tau~\Delta\phi(\mathbf{x},\tau)~\mathcal{R}_{0,\mathbf{x}}(\tau)}\Big\rangle_{r_{0,\mathbf{x}},p_{0,\mathbf{x}}} &=& \int dr_{0,\mathbf{x}}\int dp_{0,\mathbf{x}} e^{-i4\pi\eta_{\mathbf{x}}g(\mathbf{x})~\lambda_{0,\mathbf{x}}\int_{t_{0}}^{t_{\rm f}}d\tau~\Delta\phi(\mathbf{x},\tau)~\mathcal{R}_{0,\mathbf{x}}(\tau)} \nonumber \\
&\times& W_{r_{\mathbf{x}}}\left(r_{0,\mathbf{x}},p_{0,\mathbf{x}},t_{0}\right),\label{eq14}\end{aligned}$$
where $W_{r_{\mathbf{x}}}\left(r_{0,\mathbf{x}},p_{0,\mathbf{x}},t_{0}\right)$ is the Wigner functional associated to the density matrix of the polarization degrees of freedom $\widehat{\rho}_{r_{\mathbf{x}}}(t_{0})$. This functional can be written by generalizing the expression found in Ref. [@CalRouVer],
$$W_{r_{\mathbf{x}}}\left(r_{0,\mathbf{x}},p_{0,\mathbf{x}},t_{0}\right)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\Gamma~e^{i4\pi\eta_{\mathbf{x}}g(\mathbf{x})~p_{0,\mathbf{x}}\Gamma}~\rho_{r_{\mathbf{x}}}\left(r_{0,\mathbf{x}}-\frac{\Gamma}{2},r_{0,\mathbf{x}}+\frac{\Gamma}{2},t_{0}\right).$$
Considering thermal initial states for each part of the total composite system, we take the density matrices for the polarization degrees of freedom to be Gaussian functions. Therefore, Eq.(\[eq14\]) also Gaussian since the Wigner function is Gaussian in $r_{0,\mathbf{x}}$ and $p_{0,\mathbf{x}}$. This way, by considering Eq.(\[HomoSolQBM\]), we can easily calculate the first factor on the right hand of Eq.(\[CTPIntPol\]) as
$$\begin{aligned}
\Big\langle e^{-i4\pi\eta_{\mathbf{x}}g(\mathbf{x})~\lambda_{0,\mathbf{x}}\int_{t_{0}}^{t_{\rm f}}d\tau~\Delta\phi(\mathbf{x},\tau)~\mathcal{R}_{0,\mathbf{x}}(\tau)}\Big\rangle_{r_{0,\mathbf{x}},p_{0,\mathbf{x}}} &=& \frac{1}{4\pi\eta_{\mathbf{x}}g(\mathbf{x})~2\sinh\left(\frac{\beta_{r_{\mathbf{x}}}\omega_{\mathbf{x}}}{2}\right)} \nonumber \\
&\times& e^{-\frac{1}{2}~4\pi\eta_{\mathbf{x}}g(\mathbf{x})\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~\Delta\phi(\mathbf{x},\tau)~\mathcal{N}_{r,\mathbf{x}}(\tau,\tau')~\Delta\phi(\mathbf{x},\tau')},
\label{FirstFactorPol}\end{aligned}$$
with
$$\begin{aligned}
\mathcal{N}_{r,\mathbf{x}}(\tau,\tau')=\frac{\lambda_{0,\mathbf{x}}^{2}}{2m_{\mathbf{x}}\omega_{\mathbf{x}}}~\coth\left(\frac{\beta_{r_{\mathbf{x}}}\omega_{\mathbf{x}}}{2}\right)\left[\dot{G}_{\rm Ret,\mathbf{x}}(\tau-t_{0})~\dot{G}_{\rm Ret,\mathbf{x}}(\tau'-t_{0})+\omega_{\mathbf{x}}^{2}~G_{\rm Ret,\mathbf{x}}(\tau-t_{0})~G_{\rm Ret,\mathbf{x}}(\tau'-t_{0})\right],
\label{PhiNoiseKernelR}\end{aligned}$$
which is the other part of the noise kernel that acts on the field. This is associated to the influence generated by the polarization degrees of freedom (it carries a global thermal factor containing the temperature of the polarization degrees of freedom $\beta_{r_{\mathbf{x}}}$).
Hence, after the normalization procedure of $Z[J,J']$, Eq.(\[CTPIntPol\]) finally reads
$$\mathcal{F}[\phi,\phi']=e^{iS_{\rm IF}[\phi,\phi']},$$
with
$$\begin{aligned}
S_{\rm IF}[\phi,\phi']&=&\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\Delta\phi(\mathbf{x},\tau)\left[-2~\mathcal{D}_{\mathbf{x}}(\tau-\tau')~\Sigma\phi(\mathbf{x},\tau')+\frac{i}{2}~\mathcal{N}_{\mathbf{x}}(\tau,\tau')~\Delta\phi(\mathbf{x},\tau')\right]\nonumber\\
&=&\int d^{4}x\int d^{4}x'~\Delta\phi(x)\left[-2~\mathcal{D}(x,x')~\Sigma\phi(x')+\frac{i}{2}~\mathcal{N}(x,x')~\Delta\phi(x')\right],
\label{FieldSIF}\end{aligned}$$
where in the last line $\mathcal{D}(x,x')\equiv 4\pi\eta_{\mathbf{x}}g(\mathbf{x})~\delta(\mathbf{x}-\mathbf{x}')~\mathcal{D}_{\mathbf{x}}(\tau-\tau')$ and $\mathcal{N}(x,x')\equiv 4\pi\eta_{\mathbf{x}}g(\mathbf{x})~\delta(\mathbf{x}-\mathbf{x}')~\mathcal{N}_{\mathbf{x}}(\tau,\tau')$ (with $\mathcal{N}_{\mathbf{x}}(\tau,\tau')=\mathcal{N}_{r,\mathbf{x}}(\tau,\tau')+\mathcal{N}_{B,\mathbf{x}}(\tau,\tau')$) for the dissipation and noise kernels respectively. The four-dimensional translational symmetry is broken by the spatial coordinates, because the $n+1$ field is interacting with $0+1$ fields, the polarization degrees of freedom. This causes that the temporal and spatial coordinates are not in equal footing.
As expected for linear couplings, the influence action for the field have the same form as $S_{QBM}$ obtained after bath’s integration but for a field in four dimensions all over the space (this is not only true for bilinear couplings between the coordinates, it is also true for bilinear couplings between a coordinate and a momenta, but logically the kernels change as we will see in next sections).
CTP Generating Functional
-------------------------
We have achieved an exact result for the influence functional and, consequently, for the influence action. Thus, going back to Eq.(\[GeneratingFunctional\]) we can note that this CTP integrals are of the form of Eq.(\[CTPIntPol\]), replacing the degree of freedom by a scalar field. Generalizing the result found in Ref. [@CalRouVer] for fields, we get
$$\begin{aligned}
Z[J,J']&=&\Big\langle e^{-i\int d^{4}x~J_{\Delta}(x)~\Phi_{0}(x)}\Big\rangle_{\phi_{0}(\mathbf{x}),\Pi_{0}(\mathbf{x})}~e^{-\frac{1}{2}\int d^{4}x\int d^{4}x'\int d^{4}y'\int d^{4}y~J_{\Delta}(x)~\mathcal{G}_{\rm Ret}(x,x')~\mathcal{N}(x',y')~\mathcal{G}_{\rm Ret}(y,y')~J_{\Delta}(y)}\nonumber\\
&\times & ~e^{-i\int d^{4}x\int d^{4}y~J_{\Delta}(x)~\mathcal{G}_{\rm Ret}(x,y)~J_{\Sigma}(y)},
\label{GeneratingFunctionalNOFinal}\end{aligned}$$
where $\phi_{0}(\mathbf{x})=\phi(\mathbf{x},t_{0})$ and $\Pi_{0}(\mathbf{x})=\dot{\phi}(\mathbf{x},t_{0})$ are the initial conditions for the field, while $J_{\Delta}=J'-J$ and $J_{\Sigma}=(J+J')/2$.
Analogously to the integration performed in the last section, $\mathcal{G}_{\rm Ret}$ is a retarded Green function, this time associated to the field’s semiclassical equation that results from the homogeneous equation of motion for the CTP effective action for the field: $S_{\rm CTP}[\phi,\phi']=S_{0}[\phi]-S_{0}[\phi']+S_{IF}[\phi,\phi']$,
$$\begin{aligned}
\frac{\delta S_{\rm CTP}[\phi,\phi']}{\delta\phi}\Big|_{\phi=\phi'}=\frac{\delta S_{\rm CTP}[\Delta\phi,\Sigma \phi]}{\delta \Delta\phi}\Big|_{\Delta\phi=0}&=&0,\nonumber\\
\partial_{\mu}\partial^{\mu}\phi-2\int d^{4}x'~\mathcal{D}(x,x')~\phi(x')&=&0.
\label{EqMotionPhi}\end{aligned}$$
In the same way, $\Phi_{0}(x)$ is the solution of last equation that satisfies the initial condition $\{\phi_{0}(\mathbf{x}),\Pi_{0}(\mathbf{x})\}$, i. e.:
$$\Phi_{0}(x)=\int d\mathbf{x}'~\dot{\mathcal{G}}_{\rm Ret}(\mathbf{x},\mathbf{x}',t-t_{0})~\phi_{0}(\mathbf{x}')+\int d\mathbf{x}'~\mathcal{G}_{\rm Ret}(\mathbf{x},\mathbf{x}',t-t_{0})~\Pi_{0}(\mathbf{x}').$$
To calculate the first factor involving the average over the initial conditions, we use
$$\begin{aligned}
\Big\langle e^{-i\int d^{4}x~J_{\Delta}(x)~\Phi_{0}(x)}\Big\rangle_{\phi_{0}(\mathbf{x}),\Pi_{0}(\mathbf{x})}&=&\int \mathcal{D}\phi_{0}(\mathbf{x}')\int \mathcal{D}\Pi_{0}(\mathbf{x}')~W_{\phi}\left[\phi_{0}(\mathbf{x}'),\Pi_{0}(\mathbf{x}'),t_{0}\right]\nonumber\\
&&\times~e^{-i\int d\mathbf{x}'\int d^{4}x~J_{\Delta}(x)\left[\dot{\mathcal{G}}_{\rm Ret}(\mathbf{x},\mathbf{x}',\tau-t_{0})~\phi_{0}(\mathbf{x}')+\mathcal{G}_{\rm Ret}(\mathbf{x},\mathbf{x}',\tau-t_{0})~\Pi_{0}(\mathbf{x}')\right]},
\label{InitialCalzettaFactor}\end{aligned}$$
where $W_{\phi}\left[\phi_{0}(\mathbf{x}'),\Pi_{0}(\mathbf{x}'),t_{0}\right]$ plays the same role that the Wigner function in Ref.[@MrowMull].
### Initial State Contribution of the field
Once we have calculated the Wigner functional for the field in a thermal state (Eq.(\[FieldWignerCoordinate\])), we go back to Eq.(\[InitialCalzettaFactor\]) to finally calculate the first factor in the right hand of Eq.(\[GeneratingFunctionalNOFinal\]).
For an arbitrary value for the field’s temperature, the factor, which in principle are functional integrals over the field $\phi_{0}(\mathbf{x})$ and its associated momentum $\Pi_{0}(\mathbf{x})$, splits in each functional integration because the exponent also separated in each variables, therefore
$$\begin{aligned}
\Big\langle e^{-i\int d^{4}x~J_{\Delta}(x)~\Phi_{0}(x)}\Big\rangle_{\phi_{0}(\mathbf{x}),\Pi_{0}(\mathbf{x})}&=&\int\mathcal{D}\phi_{0}(\mathbf{x})~e^{-\frac{\beta_{\phi}}{2}\int d\mathbf{x}\int d\mathbf{x}'~\Delta_{\beta_{\phi}}(\mathbf{x}-\mathbf{x}')~\nabla\phi_{0}(\mathbf{x})\cdot\nabla\phi_{0}(\mathbf{x}')}~e^{\beta_{\phi}\int d\mathbf{x}~\mathcal{J}_{\phi}(\mathbf{x})~\phi_{0}(\mathbf{x})}\nonumber\\
&&\times\int\mathcal{D}\Pi_{0}(\mathbf{x})~e^{-\frac{\beta_{\phi}}{2}\int d\mathbf{x}\int d\mathbf{x}'~\Delta_{\beta_{\phi}}(\mathbf{x}-\mathbf{x}')~\Pi_{0}(\mathbf{x})~\Pi_{0}(\mathbf{x}')}~e^{\beta_{\phi}\int d\mathbf{x}~\mathcal{J}_{\Pi}(\mathbf{x})~\Pi_{0}(\mathbf{x})},
\label{InitialFactorFunctional}\end{aligned}$$
where:
$$\begin{aligned}
\mathcal{J}_{\phi}(\mathbf{x})\equiv-\frac{i}{\beta_{\phi}}\int d^{4}x'~J_{\Delta}(x')~\dot{\mathcal{G}}_{\rm Ret}\left(\mathbf{x}',\mathbf{x},t'-t_{0}\right),\end{aligned}$$
$$\begin{aligned}
\mathcal{J}_{\Pi}(\mathbf{x})\equiv-\frac{i}{\beta_{\phi}}\int d^{4}x'~J_{\Delta}(x')~\mathcal{G}_{\rm Ret}\left(\mathbf{x}',\mathbf{x},t'-t_{0}\right).\end{aligned}$$
Both functional integrals will define the contribution of the first factor to the generating functional of Eq.(\[FieldWignerCoordinate\]). In fact, it will define the contribution of the initial state of the field to the dynamical evolution, relaxation and steady situation of the system.
### High Temperature Limit
First of all, to continue the calculation, we can explore the high temperature limit for the field, which seems to be the easier case to solve the functional integrals in Eq.(\[InitialFactorFunctional\]). The high temperature approximation is given by $\frac{\beta_{\phi}|\mathbf{p}|}{2}<<1$ on the thermal weight in momentum space (Eq.(\[ThermalWeightMomentum\])). Then, $\tanh\left(\frac{\beta_{\phi}|\mathbf{p}|}{2}\right)\approx\frac{\beta_{\phi}|\mathbf{p}|}{2}$, and the thermal weight in momentum space is approximately $1$. Thus, in the coordinate space:
$$\begin{aligned}
\Delta_{\beta_{\phi}}(\mathbf{x}'-\mathbf{x}'')\approx\int\frac{d\mathbf{p}}{(2\pi)^{3}}~e^{-i\mathbf{p}\cdot\left(\mathbf{x}'-\mathbf{x}''\right)}\equiv\delta\left(\mathbf{x}'-\mathbf{x}''\right).\end{aligned}$$
In this approximation, Eq.(\[InitialFactorFunctional\]) simplifies because one integral in the exponents is straightforwardly evaluated. In this limit, both functional integrals are easily calculated, and in fact the integration over the momentum $\Pi_{0}(\mathbf{x})$, is simply a Gaussian
$$\begin{aligned}
&&\int\mathcal{D}\Pi_{0}(\mathbf{x})~e^{-\frac{\beta_{\phi}}{2}\int d\mathbf{x}~\Pi_{0}(\mathbf{x})~\Pi_{0}(\mathbf{x})}~e^{\beta_{\phi}\int d\mathbf{x}~\mathcal{J}_{\Pi}(\mathbf{x})~\Pi_{0}(\mathbf{x})}\nonumber\\
&=&e^{-\frac{1}{2\beta_{\phi}}\int d^{4}y\int d^{4}y'~J_{\Delta}(y)\left[\int d\mathbf{x}~\mathcal{G}_{\rm Ret}\left(\mathbf{y},\mathbf{x},\tau-t_{0}\right)~\mathcal{G}_{\rm Ret}\left(\mathbf{y}',\mathbf{x},\tau'-t_{0}\right)\right]J_{\Delta}(y')},\end{aligned}$$
where in this notation $y=(\tau,\mathbf{y}),y'=(\tau',\mathbf{y}')$ and we are discarding any normalization constant that will eventually go away, at the end, in the normalization of the generating functional.
At this point, it is interesting that the high temperature approximation seems to erase all the differences between the result obtained for a single problem due to the number of spatial dimensions as it was remarked at the end of the last section. This can be noted in the fact that the thermal weight, in this limit, turns out to be the Dirac delta in all the coordinates in question, independently of the spatial dimensionality, and thus all the possible differences due to the different functional forms of the thermal weight on a given number of dimensions, seems to disappear. However, the dimensionality appears again to make differences when the functional integral over $\phi_{0}(\mathbf{x})$ has to be solved. That integral is a Gaussian functional integral too. Then, we can proceed by integrating by parts the exponent involving gradients and discard terms involving the vanishing asymptotic decay of the field at infinity ($\phi_{0}(x_{i}=\pm\infty)\rightarrow 0$). Therefore, the functional integral over $\phi_{0}(\mathbf{x})$, in the high temperature limit, is a simple Gaussian functional integral, finally obtaining
$$\begin{aligned}
\int\mathcal{D}\phi_{0}(\mathbf{x})~e^{-\frac{\beta_{\phi}}{2}\int d\mathbf{x}~\nabla\phi_{0}(\mathbf{x})\cdot\nabla\phi_{0}(\mathbf{x})}&&e^{\beta_{\phi}\int d\mathbf{x}~\mathcal{J}_{\phi}(\mathbf{x})~\phi_{0}(\mathbf{x})}\propto e^{\frac{1}{2}\int d\mathbf{x}\int d\mathbf{x}'~\beta_{\phi}^{2}~\mathcal{J}_{\phi}(\mathbf{x}')~K\left(\mathbf{x},\mathbf{x}'\right)~\mathcal{J}_{\phi}(\mathbf{x}')}\nonumber\\
&=&e^{-\frac{1}{2}\int d^{4}y\int d^{4}y'~J_{\Delta}(y)\left[\int d\mathbf{x}\int d\mathbf{x}'~\dot{\mathcal{G}}_{\rm Ret}(\mathbf{y},\mathbf{x},\tau-t_{0})~K(\mathbf{x}-\mathbf{x}')~\dot{\mathcal{G}}_{\rm Ret}(\mathbf{y}',\mathbf{x}',\tau'-t_{0})\right]J_{\Delta}(y')},\end{aligned}$$
where $K(\mathbf{x},\mathbf{x}')=\left(-\beta_{\phi}\nabla^{2}\right)^{-1}$ is the inverse of the Laplace operator, i. e., is the Green function defined by
$$-\beta_{\phi}~\nabla^{2}K(\mathbf{x},\mathbf{x}')=\delta(\mathbf{x}-\mathbf{x}').$$
It is clear the kernel has to depend on $\mathbf{x}-\mathbf{x}'$.
Since the equation is analogous to the one for the Green function of a point charge in free space (although the thermal factor appears to be as a constant permittivity), we can solve the equation taking the Fourier transform
$$K(\mathbf{x}-\mathbf{x}')=\int\frac{d\mathbf{p}}{(2\pi)^{3}}~e^{-i\mathbf{p}\cdot\left(\mathbf{x}-\mathbf{x}'\right)}~\overline{K}(\mathbf{p}),
\label{KernelKHighTCoordinate}$$
where
$$\overline{K}(\mathbf{p})=\frac{1}{\beta_{\phi}~|\mathbf{p}|^{2}}.
\label{KernelKHighTMomentum}$$
It is worth noting that the kernel $K(\mathbf{x}-\mathbf{x}')$ strongly depends on the dimensionality of the problem, so the number of dimensions in the problem could modify the final results.
Finally, the first factor on the generating functional in the high temperature limit results
$$\begin{aligned}
\Big\langle e^{-i\int d^{4}x~J_{\Delta}(x)~\Phi_{0}(x)}\Big\rangle_{\phi_{0}(\mathbf{x}),\Pi_{0}(\mathbf{x})}&=&e^{-\frac{1}{2}\int d^{4}y\int d^{4}y'~J_{\Delta}(y)\left[\mathcal{A}(y,y')+\mathcal{B}(y,y')\right]J_{\Delta}(y')},
\label{InitialFactorFunctionalHighTFINAL}\end{aligned}$$
where the kernels are:
$$\mathcal{A}(y,y')\equiv\frac{1}{\beta_{\phi}}\int d\mathbf{x}~\mathcal{G}_{\rm Ret}\left(\mathbf{y},\mathbf{x},\tau-t_{0}\right)~\mathcal{G}_{\rm Ret}\left(\mathbf{y}',\mathbf{x},\tau'-t_{0}\right),
\label{KernelAHighT}$$
$$\mathcal{B}(y,y')\equiv\int d\mathbf{x}\int d\mathbf{x}'~\dot{\mathcal{G}}_{\rm Ret}(\mathbf{y},\mathbf{x},\tau-t_{0})~K(\mathbf{x}-\mathbf{x}')~\dot{\mathcal{G}}_{\rm Ret}(\mathbf{y}',\mathbf{x}',\tau'-t_{0}).
\label{KernelBHighT}$$
The result is symmetric, i. e., $\mathcal{A}(y,y')=\mathcal{A}(y',y)$ and $\mathcal{B}(y,y')=\mathcal{B}(y',y)$ and we can clearly note that both kernels depend linearly with the field initial temperature as we expected from the high temperature approximation. The presence of kernels $\mathcal{A}(y,y')$ and $\mathcal{B}(y,y')$ is one of the main results of this article. We will remark their role in the Casimir energy density, and in the contribution to the energy in the long time regime.
All in all, we now can finally write the normalized generating functional for the field in the high temperature limit, by inserting Eq.(\[InitialFactorFunctionalHighTFINAL\]) in Eq.(\[GeneratingFunctionalNOFinal\]),
$$\begin{aligned}
Z[J,J']&=&e^{-\frac{1}{2}\int d^{4}y\int d^{4}y'~J_{\Delta}(y)\left[\mathcal{A}(y,y')+\mathcal{B}(y,y')+\int d^{4}x\int d^{4}x'~\mathcal{G}_{\rm Ret}(y,x)~\mathcal{N}(x,x')~\mathcal{G}_{\rm Ret}(y',x')\right]J_{\Delta}(y')}\nonumber\\
&&\times~e^{-i\int d^{4}y\int d^{4}y'~J_{\Delta}(y)~\mathcal{G}_{\rm Ret}(y,y')~J_{\Sigma}(y')},
\label{GeneratingFunctionalHighTFINAL}\end{aligned}$$
where it is worth noting that the first factor on the right hand side is accompanied by two $J_{\Delta}$, whereas that the second one is accompanied by one $J_{\Delta}$ and one $J_{\Sigma}$. This difference will make that the first and third exponents will contribute to the energy while the second one will not.
Finally, we have calculated the field generating functional in a fully dynamical scenario in the high temperature limit for the field. This was done keeping the polarization degrees of freedom volume elements and its baths, with their own properties and temperatures. However, the model contains a bilinear interaction between the matter and the field. In the next section, we will see how to obtain, straightforwardly, the generating functional for the case of a more realistic model, i. e., a current-type interaction.
Current-type Coupling {#CC}
=====================
At this point, we have calculated the generating functional for a massless scalar field interacting with matter as Brownian particles. It is clear that in the calculation done in the previous Sections, the field and the polarization degrees of freedom are coupled linearly, i. e., the coupling is directly on the quantum degrees of freedom. Therefore, that model is not a scalar version for one of the electromagnetic field components interacting with matter, since the interaction is not a current-type interaction. Therefore, in this Section, we will show how to extend the calculation to the case of a current-type interaction between the matter and the field, getting closer to a realistic electromagnetic model.
Then, we have to start by replacing the interaction action $S_{\rm int}[\phi,r]$ between the field and the matter in Eq.(\[IntFieldPolAction\]) by a current-type interaction term as
$$\widetilde{S}_{\rm int}[\phi,r]=\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\lambda_{0,\mathbf{x}}~\dot{\phi}(\mathbf{x},\tau)~r_{\mathbf{x}}(\tau)\equiv S_{\rm int}[\dot{\phi},r],
\label{InteractionFieldPolCurrent}$$
where $\lambda_{0,\mathbf{x}}$ effectively plays the role of the electric charge in the electromagnetic model. It is also worth noting that we write the time derivative acting on the field, instead on the polarization degree of freedom. Both choices lead to the same equations of motion for the composite system so they are physically equivalent. In fact, all the calculations of the last Section, devoted to calculate the field influence action, are formally the same, and we can obtain it, in principle, by simply replacing $\phi$ by $\dot{\phi}$. Therefore, the influence action on the field, in this case, reads
$$\begin{aligned}
\widetilde{S}_{\rm IF}[\phi,\phi']\equiv S_{\rm IF}[\dot{\phi},\dot{\phi}']&=&\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\Delta\dot{\phi}(\mathbf{x},\tau)\left[-2~\mathcal{D}_{\mathbf{x}}(\tau-\tau')~\Sigma\dot{\phi}(\mathbf{x},\tau')+\frac{i}{2}~\mathcal{N}_{\mathbf{x}}(\tau,\tau')~\Delta\dot{\phi}(\mathbf{x},\tau')\right]\nonumber\\
&=&\int d^{4}x\int d^{4}x'~\Delta\dot{\phi}(x)\left[-2~\mathcal{D}(x,x')~\Sigma\dot{\phi}(x')+\frac{i}{2}~\mathcal{N}(x,x')~\Delta\dot{\phi}(x')\right].
\label{FieldSIFCurrent}\end{aligned}$$
Now, to continue the calculation as in the last Section, and to identify the noise and dissipation kernels of the present model, we integrate by parts in both time variables to obtain an influence action depending on the sum and difference of the fields, instead of their time derivatives. Therefore, as in Ref. [@ArteagaBarrielTesis], we obtain
$$\begin{aligned}
\widetilde{S}_{\rm IF}[\phi,\phi']&=&\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'4\pi\eta_{\mathbf{x}} g(\mathbf{x})~\Delta\phi(\mathbf{x},\tau)\left[-2\partial_{\tau\tau'}^{2}\mathcal{D}_{\mathbf{x}}(\tau-\tau')\Sigma\phi(\mathbf{x},\tau')+\frac{i}{2}\partial_{\tau\tau'}^{2}\mathcal{N}_{\mathbf{x}}(\tau,\tau') \Delta\phi(\mathbf{x},\tau')\right].
\label{FieldSIFCurrentIntParts}\end{aligned}$$
Since the dissipation kernel $\mathcal{D}$ involves the product of two distributions (because $\mathcal{D}(\tau-\tau')$ contains $\Theta(\tau-\tau')$ times an accompanying function of the times difference), the kernel is not well-defined [@ArteagaBarrielTesis]. Differentiating twice the kernel, firstly with respect $\tau'$ and secondly respect to $\tau$, causes that
$$\partial_{\tau\tau'}^{2}\mathcal{D}_{\mathbf{x}}(\tau-\tau')=-\delta(\tau-\tau')~\dot{\mathcal{D}}_{\mathbf{x}}(\tau-\tau')-\ddot{\mathcal{D}}_{\mathbf{x}}(\tau-\tau'),
\label{DobleDerD}$$
where dots over the kernels represent time derivatives involving differentiation over the accompanying function of the times difference, avoiding the differentiation of the Heavyside function contained in the kernel. Without confusion, it is remarkable that in the first term the Dirac delta function comes from the differentiation of the Heavyside function, but the actual notation makes that the Heavyside function contained in $\dot{\mathcal{D}}_{\mathbf{x}}$ is superfluous and meaningless. On the other hand, we shall also note that we have exploited the fact that the dissipation kernel $\mathcal{D}$ depends on the times difference $\tau-\tau'$, which gives $\partial_{\tau'}\mathcal{D}=-\partial_{\tau}\mathcal{D}=-\dot{\mathcal{D}}$. On the other hand, this is unnecessary for the kernel $\mathcal{N}_{\mathbf{x}}$.
Inserting Eq.(\[DobleDerD\]) into the influence action Eq.(\[FieldSIFCurrentIntParts\]), by considering that from its definition $\dot{\mathcal{D}}_{\mathbf{x}}(0^{+})=\lambda_{0,\mathbf{x}}^{2}/2$ for the first term of Eq.(\[DobleDerD\]), we clearly obtain
$$\begin{aligned}
\widetilde{S}_{\rm IF}[\phi,\phi']&=&\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}~g(\mathbf{x})~\Delta\phi(\mathbf{x},\tau)~\Sigma\phi(\mathbf{x},\tau)\nonumber\\
&&+\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~4\pi\eta_{\mathbf{x}}~g(\mathbf{x})~\Delta\phi(\mathbf{x},\tau)\left[2~\ddot{\mathcal{D}}_{\mathbf{x}}(\tau-\tau')~\Sigma\phi(\mathbf{x},\tau')+\frac{i}{2}~\partial_{\tau\tau'}^{2}\mathcal{N}_{\mathbf{x}}(\tau,\tau')~\Delta\phi(\mathbf{x},\tau')\right]
\label{FieldSIFCurrentIntPartsFINAL}\end{aligned}$$
where the first term is a finite renormalization position-dependent mass term for the scalar field which will be meaningless in the determination of the Green function as will see in next Sections. These renormalization mass terms also appears in the QBM theory, but in general they are divergent due to the fact that the bath is a set of infinite harmonic oscillators, each one contributing to the mass renormalization. In our case, the field is coupled in each space point $\mathbf{x}$ to an unique harmonic oscillator represented by the polarization degree of freedom located at $\mathbf{x}$, so the renormalization term is only one, and then, it is finite.
It is worth noting that from the second term, we shall call current-dissipation kernel and current-noise kernel, to the derivatives of the dissipation and noise kernels of the bilinear model, i. e., the current-dissipation kernel is $-\ddot{\mathcal{D}}_{\mathbf{x}}$ while the current-noise kernel is $\partial_{\tau\tau'}^{2}\mathcal{N}_{\mathbf{x}}$. From this Section and so on, and to avoid confusion, we shall use the prefixed ’current’ for the kernels referring to the current-type model, keeping the terms dissipation and noise kernels for $\mathcal{D}_{\mathbf{x}}$ and $\mathcal{N}_{\mathbf{x}}$ respectively.
All in all, and having written the influence action of Eq.(\[FieldSIFCurrentIntPartsFINAL\]) formally as a renormalization mass term plus a non-local term (identical to Eq.(\[FieldSIF\]) but with different kernels), we can go back and resume the procedure done for the bilinear coupling in the last Section.
Despite the renormalization mass term, the CTP functional integral over the field variables can be done as in the last Section. Therefore, the generating functional results formally identical to Eq.(\[GeneratingFunctionalHighTFINAL\]). However, in the present case, the current-noise and current-dissipation kernels are different, so the first one will define the contribution due to the matter fluctuations, while the second one will contribute to the definition of the retarded Green function by appearing in the field’s semiclassical equation obtained from the CTP effective action for the current-type model. This equation can be easily derived as Eq.(\[EqMotionPhi\]), obtaining
$$\partial_{\mu}\partial^{\mu}\phi+4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}~g(\mathbf{x})~\phi(\mathbf{x},t)+8\pi\eta_{\mathbf{x}}~g(\mathbf{x})\int_{t_{0}}^{t} d\tau~\ddot{\mathcal{D}}_{\mathbf{x}}(t-\tau)~\phi(\mathbf{x},\tau)=0,
\label{EqMotionPhiCurrent}$$
where the scalar field has a well-defined (positive) position-dependent mass $2\sqrt{\pi\eta_{\mathbf{x}}}~|\lambda_{0,\mathbf{x}}|$ in every point $\mathbf{x}$ where there is material (so $g(\mathbf{x})=1$), while it is massless in the free regions. This last equation is in agreement with the one obtained from a canonical quantization scheme (see Ref. [@LombiMazziRL] for example) and it is in fact its generalization.
Energy-Momentum Tensor and Field Correlation {#EMTFC}
============================================
At this point, we have obtained the field CTP generating functional for both coupling models after tracing out all the material degrees of freedom (polarization plus thermal baths). Then, we are interested in evaluating the expectation value of the symmetric energy-momentum tensor operator $\langle\widehat{T}_{\mu\nu}\rangle$, which gives the energy density and radiation pressure associated to the field, it is defined by [@GreiRein; @Ramond]:
$$\widehat{T}_{\mu\nu}(x_{1})\equiv-\eta_{\mu\nu}~\frac{1}{2}~\partial_{\gamma}\widehat{\phi}(x_{1})~\partial^{\gamma}\widehat{\phi}(x_{1})+\partial_{\mu}\widehat{\phi}(x_{1})~\partial_{\nu}\widehat{\phi}(x_{1}),$$
where $\eta_{\mu\nu}$ is the Minkowski metric ($\eta_{00}=-\eta_{ii}=1$ for the non-vanishing elements).
We can proceed through the point splitting technique, employing the field correlation function as
$$\Big\langle\widehat{T}_{\mu\nu}(x_{1})\Big\rangle=\lim_{x_{2}\rightarrow x_{1}}\left(-\eta_{\mu\nu}~\frac{1}{2}~\partial_{\gamma_{1}}\partial^{\gamma_{2}}+\partial_{\mu_{1}}\partial_{\nu_{2}}\right)\Big\langle\widehat{\phi}(x_{1})\widehat{\phi}(x_{2})\Big\rangle,$$
where the notation implies $\partial_{\gamma_{1}}\partial^{\gamma_{2}}\equiv\partial_{t_{1}}\partial_{t_{2}}-\nabla_{1}\cdot\nabla_{2}$ and so on for $\partial_{\mu_{1}}\partial_{\nu_{2}}$.
Therefore, we need the field correlation function to know the expectation value of every energy-momentum tensor component. In fact, we need the correlation to be finite, so we have to insert a regularized expression of the correlation function. From the generating functional in Eq.(\[GeneratingFunctionalHighTFINAL\]), this is straightforward [@CalHu]. We will evaluate the field correlation in two different points $x_{1}$ and $x_{2}$ (having no specific relation between the points because they are in different branches of the CTP). Then, we have four alternatives depending on the relation between $x_{1}$ and $x_{2}$, however in the coincidence limit this is not relevant,
$$\Big\langle\widehat{\phi}(x_{1})\widehat{\phi}(x_{2})\Big\rangle=\frac{\delta^{2}Z}{\delta J'(x_{1})\delta J(x_{2})}\Big|_{J=J'=0}.$$
Because the generating functional has a simple form in Eq.(\[GeneratingFunctionalHighTFINAL\]), we can easily compute its functional derivatives, taking advantage of the symmetry kernel’s properties, to obtain:
$$\Big\langle\widehat{\phi}(x_{1})\widehat{\phi}(x_{2})\Big\rangle=\mathcal{A}(x_{1},x_{2})+\mathcal{B}(x_{1},x_{2})+\int d^{4}x\int d^{4}x'~\mathcal{G}_{\rm Ret}(x_{1},x)~\mathcal{N}(x,x')~\mathcal{G}_{\rm Ret}(x_{2},x')+\frac{1}{2}~\mathcal{G}_{\rm Jordan}(x_{1},x_{2}),
\label{FieldCorrelationHighT}$$
where $\mathcal{G}_{\rm Jordan}(x_{1},x_{2})\equiv i\left(\mathcal{G}_{\rm Ret}(x_{2},x_{1})-\mathcal{G}_{\rm Ret}(x_{1},x_{2})\right)$ is the Jordan propagator [@CalHu]. Then, the kernels are the ones in Eqs.(\[FieldSIF\]), (\[KernelAHighT\]) and (\[KernelBHighT\]) for the case of the bilinear model, being the retarded Green function defined from semiclassical equation of motion on Eq.(\[EqMotionPhi\]). On the other hand, to obtain the result for the current-type model we have to take into account that the retarded Green function is defined from the corresponding semiclassical equation of motion for the field in this model, given by Eq.(\[EqMotionPhiCurrent\]), but the formal expressions for the kernels $\mathcal{A}$ and $\mathcal{B}$ are unchanged. To finish, we have to replace the noise kernel $\mathcal{N}$ on Eq.(\[FieldCorrelationHighT\]) by the current-noise kernel $\partial_{\tau\tau'}^{2}\mathcal{N}$ associated to the field’s influence action for this model of Eq.(\[FieldSIFCurrentIntPartsFINAL\]). All in all, a smart and compact notation can be achieved by including a parameter $\alpha$ encompassing both models, therefore, we can write the generalized noise kernel as $\partial_{\tau\tau'}^{2\alpha}\mathcal{N}$, with $\alpha=0,1$ for the bilinear and current-type model respectively.
It is worth noting that the correlation function in Eq.(\[FieldCorrelationHighT\]) corresponds to the Whightman function for the field in this open system. In fact, considering that $\mathcal{G}_{\rm Ret}$ is real, as it is written is clear that the correlation is a complex quantity, and its imaginary part is given by $\mathcal{G}_{\rm Jordan}$ whereas that the real part is formed by the others three terms. If we want to match the Whightman propagator with the typical relations for propagators, the Hadamard propagator is given by
$$\mathcal{G}_{\rm H}(x_{1},x_{2})\equiv 2\left[\mathcal{A}(x_{1},x_{2})+\mathcal{B}(x_{1},x_{2})+\int d^{4}x\int d^{4}x'~\mathcal{G}_{\rm Ret}(x_{1},x)~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}(x,x')\right]~\mathcal{G}_{\rm Ret}(x_{2},x')\right].
\label{HadamardPropagator}$$
On the other hand, we want to calculate energy-momentum tensor expectation values, so the result must be real. This apparently seems not to be the case because the correlation function is complex and its imaginary part is given by $\mathcal{G}_{\rm Jordan}$. However, to compute the expectation values, we will have to derive in a symmetric way in both coordinates $x_{1,2}$ and then we will have to calculate the coincidence limit when $x_{2}\rightarrow x_{1}$. Due to the definition of the Jordan propagator, this operation (symmetric derivation plus the coincidence limit) makes that the contribution vanishes. Therefore, the expectation values turns out to be, effectively, real numbers, as is expected.
Finally, the expectation value of the energy-momentum tensor can be written as
$$\Big\langle\widehat{T}_{\mu\nu}(x_{1})\Big\rangle=\frac{1}{2}\lim_{x_{2}\rightarrow x_{1}}\left(-\eta_{\mu\nu}~\frac{1}{2}~\partial_{\gamma_{1}}\partial^{\gamma_{2}}+\partial_{\mu_{1}}\partial_{\nu_{2}}\right)\mathcal{G}_{\rm H}(x_{1},x_{2}),
\label{TmunuExpValue}$$
where the Hadamard propagator must be a well-defined (non-divergent) propagator.
It is important to note that all the full non-equilibrium dynamics, both time evolution and thermodynamical non-equilibrium, is contained in this result.
Due to the structure of the noise kernel $\partial_{\tau\tau'}^{2\alpha}\mathcal{N}$ for each model, this term accounts for the influence generated by the material (polarization degrees of freedom plus thermal baths), from the initial time when the interaction with the field is turn on, describing the relaxation process of the material forming the contours. Both parts, polarization degrees of freedom and thermal baths, can have different initial temperature, having thermal non-equilibrium. In fact, each volume element on the material can have its own properties.
On the other hand, there are two terms proportionals to the field initial temperature, which are the kernels $\mathcal{A}$ and $\mathcal{B}$ (and their derivatives in the contribution of the expectation values). Those terms account for the dynamical evolution and change of the field in the presence of the material contours, when the interaction is turned on. Therefore, those terms must be entirely related to the modified normal modes that appear in a (steady situation) canonical quantization scheme as a vacuum contribution [@LombiMazziRL; @Dorota1992].
Non-equilibrium behaviour for different configurations of the composite system {#NEBFDCOTCS}
==============================================================================
$0+1$ Field {#0+1F}
-----------
As a first example, we consider the case of a scalar field in $0+1$ dimensions, i. e., we take the field $\phi$ as a quantum harmonic oscillator degree of freedom of unit mass. Therefore, to adapt our results to this situation, a few changes are needed. In this case the spatial notion is erased, and the volume element concept is meaningless, so the composite system is an harmonic oscillator ($0+1$ field) coupled to another one (polarization degree of freedom) which is also coupled to a set of harmonic oscillators (thermal bath).
The spatial label $\mathbf{x}$ will be unnecessary and the quantum degree of freedom will be characterized by a frequency $\Omega$ (which plays the role that the spatial derivative has on the field in $n+1$ dimensions, with $n>0$). The initial action for the field (Eq.(\[FreeFieldAction\])) must be replaced by the straightforward expression
$$S_{0}[\phi]=\int d\mathbf{x}\int_{t_{0}}^{t_{\rm f}}d\tau~\frac{1}{2}~\partial_{\mu}\phi~\partial^{\mu}\phi\longrightarrow\int_{t_{0}}^{t_{\rm f}}
d\tau~\frac{1}{2}~\left[\left(\frac{d\phi}{d\tau}\right)^{2}-\Omega^{2}~\phi(\tau)\right].$$
Eqs.(\[FreePolAction\]) - (\[IntPolBathHOAction\]) can also be simply adapted by discarding the spatial integrals, labels, the density $\eta$ (together with the factor $4\pi$) and the distribution function $g$ in all the actions. All the integrations and traces can be performed in the same way without further modifications until the functional integration over the field. The influence action in Eq.(\[FieldSIF\]) still remains valid. In this way, the generating functional from Eq.(\[GeneratingFunctional\]) to Eq.(\[GeneratingFunctionalNOFinal\]) is formally the same. However, in this case, the calculation of the first factor, that involves the initial state of the $0+1$ field $\phi(t)$, implies a Wigner function and not a functional, so the factor results from the same formal calculation done for the polarization degree of freedom $r$ in Eq.(\[FirstFactorPol\]), but the kernel obtained is clearly different. Therefore, we have for the both models ($\alpha=0,1$):
$$\begin{aligned}
Z[J,J']&=&e^{-\frac{1}{2}\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}ds~J_{\Delta}(\tau)\left[\mathcal{A}(\tau,s)+\mathcal{B}(\tau,s)\right]J_{\Delta}(s)}~e^{-\frac{1}{2}\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'\int_{t_{0}}^{t_{\rm f}}ds'\int_{t_{0}}^{t_{\rm f}}ds~J_{\Delta}(\tau)~\mathcal{G}_{\rm Ret}^{\Omega}(\tau,\tau')~\partial_{\tau's'}^{2\alpha}\left[\mathcal{N}(\tau',s')\right]~\mathcal{G}_{\rm Ret}^{\Omega}(s,s')~J_{\Delta}(s)}\nonumber\\
&\times &~e^{-i\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}ds~J_{\Delta}(\tau)~\mathcal{G}_{\rm Ret}^{\Omega}(\tau,s)~J_{\Sigma}(s)},
\label{GeneratingFunctional0+1}\end{aligned}$$
where the sum of the kernels $\mathcal{A}$ and $\mathcal{B}$ results from the ordinary integration over the initial values of the field $\phi_{0}\equiv\phi(t_{0})$ and $\Pi_{0}\equiv\Pi(t_{0})$ and it is of the form of Eq.(\[PhiNoiseKernelR\]) (indeed, the high temperature limit of this expression has exactly the form of Eq.(\[PhiNoiseKernelR\]) by discarding it spatial features),
$$\mathcal{A}(\tau,s)+\mathcal{B}(\tau,s)\equiv\frac{1}{2\Omega}~\coth\left(\frac{\beta_{\phi}\Omega}{2}\right)\left[\Omega^{2}~\mathcal{G}_{\rm Ret}^{\Omega}(\tau-t_{0})~\mathcal{G}_{\rm Ret}^{\Omega}(s-t_{0})+\dot{\mathcal{G}}_{\rm Ret}^{\Omega}(\tau-t_{0})~\dot{\mathcal{G}}_{\rm Ret}^{\Omega}(s-t_{0})\right],$$
where $\mathcal{G}_{\rm Ret}^{\Omega}$ is the retarded Green function (which is a function of the time difference, i. e., $\mathcal{G}_{\rm Ret}^{\Omega}(t,s)\equiv\mathcal{G}_{\rm Ret}^{\Omega}(t-s)$, as we can infer from its equation of motion), associated to Eqs. (\[EqMotionPhi\]) and (\[EqMotionPhiCurrent\]) for each model respectively in the $0+1$ case, which can be written together as
$$\begin{aligned}
\frac{d^{2}\phi}{dt^{2}}+(\Omega^{2}+\alpha~\lambda_{0}^{2})~\phi(t)-(-1)^{\alpha}~2\int_{t_{0}}^{t}d\tau~\partial_{tt}^{2\alpha}\left[\mathcal{D}(t-\tau)\right]~\phi(\tau)=0,
\label{EqMotionPhi0+1}\end{aligned}$$
where in this case, the (finite) mass term presents as a frequency renormalization term, and $\partial_{tt}^{2\alpha}\left[\mathcal{D}(t-\tau)\right]$ is the generalized dissipation kernel.
Therefore, the $0+1$ counterpart of the Hadamard propagator of Eq.(\[HadamardPropagator\]) is
$$\begin{aligned}
\mathcal{G}_{\rm H}^{\Omega}(t_{1},t_{2})&\equiv& \frac{1}{\Omega}~\coth\left(\frac{\beta_{\phi}\Omega}{2}\right)\left[\Omega^{2}~\mathcal{G}_{\rm Ret}^{\Omega}(t_{1}-t_{0})~\mathcal{G}_{\rm Ret}^{\Omega}(t_{2}-t_{0})+\dot{\mathcal{G}}_{\rm Ret}^{\Omega}(t_{1}-t_{0})~\dot{\mathcal{G}}_{\rm Ret}^{\Omega}(t_{2}-t_{0})\right]\nonumber\\
&&+~2\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~\mathcal{G}_{\rm Ret}^{\Omega}(t_{1}-\tau)~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}(\tau,\tau')\right]~\mathcal{G}_{\rm Ret}^{\Omega}(t_{2}-\tau'),
\label{HadamardPropagator0+1}\end{aligned}$$
with the noise kernel in two contributions $\mathcal{N}(\tau,\tau')=\mathcal{N}_{B}(\tau,\tau')+\mathcal{N}_{r}(\tau,\tau')$, each one characterized by its own temperature $\beta_{r,B}$, given in Eqs.(\[PhiNoiseKernelB\]) and (\[PhiNoiseKernelR\]) respectively. In fact, we can correspond the temperature value associated to the term to the contribution of that part of the total system, i. e., the terms carrying the field’s temperature $\beta_{\phi}$ are associated to the proper (influenced) system contribution, while each part of the noise kernel $\mathcal{N}$ has one term associated to the polarization degree of freedom (denoted by containing the temperature $\beta_{r}$) and another one associated to the bath (denoted by containing the temperature $\beta_{B}$).
It is clear now, that the energy-momentum tensor is simply the energy of the $0+1$ field, where the evolution of the expectation value can be easily written in terms of the Hadamard propagator, as it happens in Eq.(\[TmunuExpValue\]):
$$\Big\langle E(t_{1})\Big\rangle\equiv\frac{1}{2}\lim_{t_{2}\rightarrow t_{1}}\left(\frac{\partial}{\partial t_{1}}\frac{\partial}{\partial t_{2}}+\Omega^{2}\right)\mathcal{G}_{\rm H}^{\Omega}(t_{1},t_{2}).
\label{EnergyExpValue}$$
Finally, we have written the mean value of the energy as a function of time, from the initial conditions for the composite system. It is clear that the dynamic depends on the retarded Green functions $\mathcal{G}_{\rm Ret}^{\Omega}, G_{\rm Ret}$ (where $G_{\rm Ret}$ is contained in the field’s noise kernels of each model through Eqs. (\[PhiNoiseKernelB\]) and (\[PhiNoiseKernelR\])), from each part of the system and the QBM noise kernel $N_{\rm QBM}$ (which depends on the type of bath we are considering) after tracing out the degrees of freedom that influences its dynamics. Since we are interested in the field dynamics, the traces are performed taking the field $\phi$ as the system and doing them in sequential steps: the partial traces over each part of the complex environment formed by the polarization degree of freedom $r$ and the bath $\{q_{n}\}$ [@FeynHibbs].
Therefore, the transient time behavior of the energy expectation value and its relaxation to a steady state, will depend on the fluctuations of each part of the environment, through the noise kernels, and how the system evolves to the steady situation, depends on its own Green function $\mathcal{G}_{\rm Ret}^{\Omega}$, as it is clear from Eq.(\[HadamardPropagator0+1\]).
Then, for the long-time limit ($t_{0}\rightarrow-\infty$), we need to know how is the long-time behavior of each retarded Green function $\mathcal{G}_{\rm Ret}^{\Omega}, G_{\rm Ret}$. Thus, we must focus on the specific Green functions that we have in our system, which are determined by each equation of motion we have obtained at each stage of the tracing.
The retarded Green function for the polarization degree of freedom $r$ is determined by the equation of motion of the polarization degree of freedom, Eq.(\[EqMotionR\]). The associated equation for the Green function $G_{\rm Ret}$ can be solved by Laplace transforming the equation subjected to the initial conditions $G_{\rm Ret}(0)=0,\dot{G}_{\rm Ret}(0)=1$ (see Ref. [@BreuerPett]). It is straightforward to prove that, for every type of bath, the Laplace transform of the retarded Green function is given by
$$\widetilde{G}_{\rm Ret}(z)=\frac{1}{\left(z^{2}+\omega^{2}-2~\widetilde{D}_{\rm QBM}(z)\right)},
\label{RetGreenFunctionQBM}$$
where $\widetilde{D}_{\rm QBM}$ is the Laplace transform of the QBM’s dissipation kernel contained in $S_{QBM}$ [@CaldeLegg; @HuPazZhang].
The analyticity properties of the the Laplace transform $\widetilde{G}_{\rm Ret}$ and the location of its poles define the time evolution and the asymptotic behavior of the Green function $G_{\rm Ret}$. In this direction, causality implies, by Cauchy’s theorem, that the poles of $\widetilde{G}_{\rm Ret}$ should be located in the left-half of the complex $z$-plane, i. e., the poles’ real parts must be negative or zero. Assuming that $\omega\neq 0$ and that the bath modeled includes a cutoff function in frequencies (see [@BreuerPett]), considering the discussion given in Ref. [@LombiMazziRL], which results that all the poles are simple and have negative real parts. Through the Mellin’s formula and the Residue theorem to retransform to the time dependent function [@Schiff], we easily obtain that, formally, the Green function reads
$$G_{\rm Ret}(t)=\Theta(t)\sum_{j}Res\left[\widetilde{G}_{\rm Ret}(z),z_{j}\right]~e^{z_{j}t}.$$
Since $Re[z_{j}]<0$, it is clear that in the long-time limit, when $t_{0}\rightarrow-\infty$, we have $G_{\rm Ret}(t-t_{0})\rightarrow 0$ and also the same for its time derivatives.
Indeed, this asymptotic behavior defines the long-time contribution of the polarization degree of freedom to the field’s energy density at the steady situation. Since the Green function goes to zero, we have also that the part of the field’s noise kernels, directly associated to the polarization degree of freedom $\mathcal{N}_{r}$, goes to zero. This means that the polarization degree of freedom do not contribute through its thermal state to the energy at the steady situation in none of the two coupling models. Although the dependence on the temperature $\beta_{r}$ is erased in the long-time regime (due to the asymptotic decay of the retarded Green function $G_{\rm Ret}(t-t_{0})$), this function also appears in the bath’s contribution $\mathcal{N}_{B}$. That term is characterized, of course, by the bath’s temperature $\beta_{B}$.
All in all, in the long-time limit ($t_{0}\rightarrow-\infty$) we have that the (generalized) noise kernel contribution (polarization degree of freedom plus bath) in Eq.(\[HadamardPropagator0+1\]) results
$$\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~\mathcal{G}_{\rm Ret}^{\Omega}(t_{1}-\tau)~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}(\tau,\tau')\right]~\mathcal{G}_{\rm Ret}^{\Omega}(t_{2}-\tau')\longrightarrow\int_{-\infty}^{t_{\rm f}}d\tau\int_{-\infty}^{t_{\rm f}}d\tau'~\mathcal{G}_{\rm Ret}^{\Omega}(t_{1}-\tau)~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}_{B}(\tau,\tau')\right]~\mathcal{G}_{\rm Ret}^{\Omega}(t_{2}-\tau')$$
where the QBM noise kernel does not depends on $t_{0}$ so it makes that the bath contribution do not vanish in the steady situation.
Finally, we have to analyze the behavior of the contribution associated to the proper field-system. Thus we have to study the retarded Green function $\mathcal{G}_{\rm Ret}^{\Omega}$. We then proceed as in the polarization degree of freedom case, for studying $G_{\rm Ret}$ by considering the same initial conditions ($\mathcal{G}_{\rm Ret}^{\Omega}(0)=0,\dot{\mathcal{G}}_{\rm Ret}^{\Omega}(0)=-1$). From the equation of motion for the Green function $\mathcal{G}_{\rm Ret}^{\Omega}$, associated to the field in both models, Eq.(\[EqMotionPhi0+1\]), we can easily obtain an analogous expression as in the first case for the Laplace transform
$$\begin{aligned}
\widetilde{\mathcal{G}}_{\rm Ret}^{\Omega}(z)&=&\frac{-1}{\left(z^{2}+\Omega^{2}-\lambda_{0}^{2}~(-z^{2})^{\alpha}~\widetilde{G}_{\rm Ret}(z)\right)},\end{aligned}$$
where it is worth noting that this compact expression is due to the fact that the renormalization (mass) frequency term cancels out with a term coming from the derivative of the dissipation kernel $\mathcal{D}$ at the initial time.
Analyticity properties of this Laplace transform define the asymptotic behavior of the proper contribution of the field. For $\lambda_{0},\Omega,\omega\neq 0$ and an Ohmic bath, it is easy to show that the Laplace transform for both models has four simple poles with negative real parts, verifying the causality requirement. We assume that the general case gives the same features and the poles are simple and have negative real parts. From this, in the time domain, it follows that
$$\mathcal{G}_{\rm Ret}^{\Omega}(t)=\Theta(t)\sum_{l}Res\left[\widetilde{\mathcal{G}}_{\rm Ret}^{\Omega}(z),z_{l}\right]~e^{z_{l}t}.$$
Therefore, since $Re[z_{l}]<0$, we clearly have in the long-time limit ($t_{0}\rightarrow-\infty$) that $\mathcal{G}_{\rm Ret}^{\Omega}(t-t_{0})\rightarrow 0$ and also the same for its time derivatives.
The long-time limit of the Hadamard propagator $\mathcal{G}_{\rm H}^{\Omega}$ is given only by the bath’s long-time contribution:
$$\begin{aligned}
\mathcal{G}_{\rm H}^{\Omega}(t_{1},t_{2})\rightarrow 2\int_{-\infty}^{t_{\rm f}}d\tau\int_{-\infty}^{t_{\rm f}}d\tau'~\mathcal{G}_{\rm Ret}^{\Omega}(t_{1}-\tau)~\mathcal{N}_{B}(\tau,\tau')~\mathcal{G}_{\rm Ret}^{\Omega}(t_{2}-\tau'),
\label{HadamardPropagator0+1}\end{aligned}$$
corresponding to the steady situation with the bath’s fluctuation at temperature $\beta_{B}$, as the fluctuation-dissipation theorem asserts.
Finally, summarizing thus section, for a $0+1$ field in both types of coupling models, the energy density at the steady situation, have only contributions from the bath, while the polarization degree of freedom and the proper field contributions go to zero through the time evolution.
Now, let see how these calculations apply for the case of a field in $n+1$ dimensions with an homogeneous material all over the space.
Field In Infinite Material {#FIIM}
--------------------------
Let us now consider a scalar field in $n+1$ dimensions (with $n\neq 0$) with no boundaries, i. e., this is the case of an homogeneous material that appears all over the space at the initial time $t_{0}$. In this situation, $g(\mathbf{x})\equiv 1$ for every $\mathbf{x}$ and we have to eliminate the spatial label due to the homogeneity of the problem. Then, Eqs.(\[EqMotionPhi\]) and (\[EqMotionPhiCurrent\]) can be written together through its generalized form as
$$\begin{aligned}
\partial_{\mu}\partial^{\mu}\phi+4\pi\eta\lambda_{0}^{2}~\alpha~\phi-(-1)^{\alpha}8\pi\eta\int_{t_{0}}^{t}d\tau~\partial_{tt}^{2\alpha}\left[\mathcal{D}(t-\tau)\right]~\phi(\mathbf{x},\tau)&=&0,
\label{EqMotionPhiHomogeneousAllSpace}\end{aligned}$$
which is basically a wave-type equation for the field in a dissipative media.
Therefore, the associated equation for the retarded Green function $\mathcal{G}_{\rm Ret}$ is straightforward and it is subjected to the typical wave equation initial conditions
$$\mathcal{G}_{\rm Ret}(\mathbf{x},\mathbf{x}',0)=0~~~~~,~~~~~\dot{\mathcal{G}}_{\rm Ret}(\mathbf{x},\mathbf{x}',0)=-\delta(\mathbf{x}-\mathbf{x}').
\label{InitialConditionsHomogeneous}$$
Due to the translational symmetry of the problem, $\mathcal{G}_{\rm Ret}(\mathbf{x},\mathbf{x}',t)=\mathcal{G}_{\rm Ret}(\mathbf{x}-\mathbf{x}',t)$, the Fourier transform satisfies
$$\begin{aligned}
\partial_{tt}^{2}\overline{\mathcal{G}}_{\rm Ret}(\mathbf{k},t)+(k^{2}+4\pi\eta\lambda_{0}^{2}~\alpha)~\overline{\mathcal{G}}_{\rm Ret}(\mathbf{k},t)-(-1)^{\alpha}8\pi\eta\int_{0}^{t}d\tau~\partial_{tt}^{2\alpha}\left[\mathcal{D}(t-\tau)\right]~\overline{\mathcal{G}}_{\rm Ret}(\mathbf{k},\tau)=0,
\label{EqMotionPhiHomogeneousAllSpaceFourier}\end{aligned}$$
where, as in the last section, $\mathcal{D}(t-\tau)=\lambda_{0}^{2}~G_{\rm Ret}(t-\tau)$, $k\equiv|\mathbf{k}|$, and the initial conditions are
$$\overline{\mathcal{G}}_{\rm Ret}(\mathbf{k},0)=0~~~~~,~~~~~\dot{\overline{\mathcal{G}}}_{\rm Ret}(\mathbf{k},0)=-1.
\label{InitialConditionsHomogeneousFourier}$$
Eqs.(\[EqMotionPhiHomogeneousAllSpaceFourier\]) and (\[InitialConditionsHomogeneousFourier\]) are equivalent to the field equation and initial conditions for the retarded Green function for the $0+1$ field, i.e., each field mode behaves as a $0+1$ field of natural frequency $k$ and the dynamics are equivalent. Then, the Fourier transform of the retarded Green function is closely related to the retarded Green function in the last Section, in fact, we have,
$$\overline{\mathcal{G}}_{\rm Ret}(\mathbf{k},t)\equiv\overline{\mathcal{G}}_{\rm Ret}^{k}(t),$$
where $\mathcal{G}_{\rm Ret}^{k}$ is the retarded function of a $0+1$ field of frequency $k$.
We can write
$$\mathcal{G}_{\rm Ret}(\mathbf{x}-\mathbf{x}',t)=\int\frac{d\mathbf{k}}{(2\pi)^{3}}~e^{-i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x}')}~\mathcal{G}_{\rm Ret}^{k}(t).$$
In order to study the behavior of the contributions to the expectation value of the energy-momentum tensor $\langle\widehat{T}_{\mu\nu}\rangle$ in Eq.(\[TmunuExpValue\]), let us firstly consider the contributions of the polarization degrees of freedom and the thermal baths in each point $\mathbf{x}$ in the last term of Eq.(\[HadamardPropagator\]). Since we are considering an homogeneous material with all the polarization degrees of freedom having the same temperature $\beta_{r}$ and the same for the baths in each point with $\beta_{B}$ (note that this does not means thermal equilibrium because each part of the material can have different temperatures, i. e., we can still have the situation in which $\beta_{r}\neq\beta_{B}$). In the present case $\mathcal{N}(x,x')=4\pi\eta~\delta(\mathbf{x}-\mathbf{x}')~\mathcal{N}(\tau,\tau')$.
If we use the Fourier representation of $\mathcal{G}_{\rm Ret}$ to write the last term of Eq.(\[HadamardPropagator\]), it is straightforward that
$$\begin{aligned}
\int d^{4}x\int d^{4}x'&\mathcal{G}_{\rm Ret}(x_{1},x)&~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}(x,x')\right]\mathcal{G}_{\rm Ret}(x_{2},x')\nonumber\\
&=4\pi\eta&\int\frac{d\mathbf{k}}{(2\pi)^{3}}e^{-i\mathbf{k}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}\int_{t_{0}}^{t_{\rm f}}d\tau\int_{t_{0}}^{t_{\rm f}}d\tau'~\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{1}-\tau)~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}(\tau,\tau')\right]\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{2}-\tau'),\end{aligned}$$
where it is remarkable that both integrals over $\tau$ and $\tau'$, and the integrand are exactly one half of the last term in Eq.(\[HadamardPropagator0+1\]), the contribution of the polarization degree of freedom and the bath in the last Section, i. e., for the $0+1$ field of frequency $\Omega$. This is clear because, as we have inferred from the equation for the Fourier transformed Green function, each field $\mathbf{k}$-mode is matched to a $0+1$ field of frequency $k=|\mathbf{k}|$.
Then, we have for each field mode the same time evolution as for a $0+1$ field of natural frequency $k$ in any coupling model. Considering the analysis done in the last Section about the Green function $G_{\rm Ret}$, we can easily conclude that the long-time regime ($t_{0}\rightarrow-\infty$) of this contribution is given by
$$\begin{aligned}
\int d^{4}x\int d^{4}x'~&\mathcal{G}_{\rm Ret}(x_{1},x)&~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}(x,x')\right]~\mathcal{G}_{\rm Ret}(x_{2},x')\longrightarrow\nonumber\\
&\longrightarrow 4\pi\eta&\int\frac{d\mathbf{k}}{(2\pi)^{3}}e^{-i\mathbf{k}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}\int_{-\infty}^{t_{\rm f}}d\tau\int_{-\infty}^{t_{\rm f}}d\tau'~\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{1}-\tau)~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}_{B}(\tau,\tau')\right]~\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{2}-\tau'),\end{aligned}$$
where, as in the last Section, we have that the polarization degrees of freedom do not contribute to the steady situation of the $n+1$ field in an homogeneous material.
On the other hand, for the proper contribution of the field, contained in the kernels $\mathcal{A}$ and $\mathcal{B}$ of Eqs. (\[KernelAHighT\]) and (\[KernelBHighT\]), we can again exploit the Fourier representation
$$\mathcal{A}(x_{1},x_{2})+\mathcal{B}(x_{1},x_{2})=\int\frac{d\mathbf{k}}{(2\pi)^{3}}~e^{-i\mathbf{k}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}\left[\frac{1}{\beta_{\phi}}~\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{1}-t_{0})~\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{2}-t_{0})+\overline{K}(k)~\dot{\overline{\mathcal{G}}}_{\rm Ret}^{k}(t_{1}-t_{0})~\dot{\overline{\mathcal{G}}}_{\rm Ret}^{k}(t_{2}-t_{0})\right].$$
Therefore, considering the analysis done in the last Section for the retarded Green function $\mathcal{G}_{\rm Ret}^{\Omega}$, in the long-time limit ($t_{0}\rightarrow-\infty$) the Fourier transform of the retarded Green function vanishes, i. e., $\overline{\mathcal{G}}_{\rm Ret}^{k}(t-t_{0})\rightarrow 0$; and this makes that also the proper contribution vanishes at the steady situation.
All in all, as in the $0+1$ field, the long-time regime is defined by the bath contribution to the Hadamard propagator, and it is expected to satisfy the fluctuation-dissipation theorem in the steady situation by both coupling models
$$\begin{aligned}
\mathcal{G}_{\rm H}(x_{1},x_{2})\rightarrow 8\pi\eta\int\frac{d\mathbf{k}}{(2\pi)^{3}}~e^{-i\mathbf{k}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}\int_{-\infty}^{t_{\rm f}}d\tau\int_{-\infty}^{t_{\rm f}}d\tau'~\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{1}-\tau)~\partial_{\tau\tau'}^{2\alpha}\left[\mathcal{N}_{B}(\tau,\tau')\right]~\overline{\mathcal{G}}_{\rm Ret}^{k}(t_{2}-\tau').\end{aligned}$$
Finally, the energy density at the steady situation will also depend only on the baths fluctuations in the long-time regime for anyone of the coupling models. This conclusion is not necessarily true if the material is not homogeneous or if there are temperature gradients, whether between the polarization degrees of freedom or between the baths. In fact, in the next Sections we will show that the conclusion could be different if, on the one hand, we consider non-dissipative (constant permittivity) media or, in the other hand, there are regions where the field fluctuates freely, i. e., regions where there is no material ($g(\mathbf{x})=0$) and the field is subjected to the presence of boundaries.
Constant Dielectric Permittivity Limit {#CDPL}
--------------------------------------
In previous Sections we have analyzed two situations (a field in $0+1$ dimensions and a field in $n+1$ dimensions in the presence if an infinite material) where we have shown that, beyond the transient time evolution of the system, the steady regime is described only by the fluctuations of the thermal baths which are in contact with the polarization degrees of freedom of the material, as it is expected from a formalism only based on the fluctuation-dissipation theorem. This result can be seen from the final temperature dependence of the Hadamard propagator, which in the analyzed cases, was $\beta_{B}$. On the other hand, we have shown that the kernels $\mathcal{A}$ and $\mathcal{B}$, associated to the proper contribution of the field, and the contribution from the polarization degrees of freedom vanish at the steady situation (due to the dissipative dynamics of the field in every point of the space and of the polarization degrees of freedom as Brownian particles).
It is clear that these conclusions are due, physically, to the dissipative dynamics of the field in contact to reservoirs conforming the real material, which generates the damping and the absorption dominating the steady situation through its fluctuations.
We will assume now that the material is a non-dissipative dielectric, i. e., a constant permittivity material which presents no absorption and it is no dispersive because the permittivity function in the complex frequency domain is real and it is not a smooth function over the imaginary frequency axis. It is worth noting that this verifies Kramers-Kronig relations for the complex permittivity function in the frequency domain although the function is real. In fact, Kramers-Kronig relations are not satisfied by dispersive and real permittivity functions in the imaginary frequency axis. Therefore, our calculations must include this scenario as a limiting case.
As a first step, if we clearly turn off the dissipation provided by the baths in each point of the material, we have to set $D_{\rm QBM}\equiv 0$. From the fluctuation-dissipation theorem is straightforward that $N_{\rm QBM}\equiv 0$. Therefore, this directly implies that the noise kernel also vanishes, i. e., $\mathcal{N}_{B,\mathbf{x}}\equiv 0$. This way, the bath contribution is erased from the result.
However, this is not enough because it leaves a material formed by harmonic oscillators without damping, i. e., which do not relax to a steady situation. This can be seen from the Laplace transform of the retarded Green function of the polarization degrees of freedom, which, through Eq. (\[RetGreenFunctionQBM\]), turns out to be $\widetilde{G}_{\rm Ret,\mathbf{x}}(z)=1/(z^{2}+\omega_{\mathbf{x}}^{2})$, which presents purely imaginary poles at $z=\pm i\omega_{\mathbf{x}}$, so the retarded Green function in the time domain will be sinusoidal functions. This causes, in principle, that the contribution coming from the polarization degrees of freedom do not vanishes, i. e., $\mathcal{N}_{r,\mathbf{x}}$ not necesarily vanishes.
Nevertheless, since the dissipation kernel is $\mathcal{D}_{\mathbf{x}}=\frac{\lambda_{0,\mathbf{x}}^{2}}{2}~G_{\rm Ret,\mathbf{x}}$, and the generalized dissipation kernel $\partial_{tt}^{2\alpha}\left[\mathcal{D}_{\mathbf{x}}(t-\tau)\right]$ that acts over the field and forms the dielectric function through its Laplace transform, will give a dispersive and real permittivity function for purely imaginary frequencies. Thus, it does not verify the Kramers-Kronig relations. Then, vanishing bath dissipation is not enough to achieve the constant dielectric limit and, in fact, it is a non-physical model.
To get a clue about how this limit can be taken, we can use the equation of motion for the field, for arbitrary shapes of material boundaries, which in both coupling models can be written as
$$\begin{aligned}
\partial_{\mu}\partial^{\mu}\phi+4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}~g(\mathbf{x})~\alpha~\phi-(-1)^{\alpha}8\pi\eta_{\mathbf{x}}~g(\mathbf{x})\int_{t_{0}}^{t}d\tau~\partial_{tt}^{2\alpha}\left[\mathcal{D}_{\mathbf{x}}(t-\tau)\right]~\phi(\mathbf{x},\tau)&=&0.
\label{EqMotionPhiGeneralized}\end{aligned}$$
Going to the complex frequency domain, we Laplace-transform the associated equation for the Green function, imposing the same initial conditions as in the last Section, easily obtaining for each model
$$\begin{aligned}
\nabla^{2}\widetilde{\mathcal{G}}_{\rm Ret}-z^{2}\left[1-(-1)^{\alpha}~4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}~g(\mathbf{x})~\frac{z^{2(\alpha-1)}}{(z^{2}+\omega_{\mathbf{x}}^{2})}\right]\widetilde{\mathcal{G}}_{\rm Ret}=\delta(\mathbf{x}-\mathbf{x}').
\label{EqGreenLaplaceGeneralizedNoDiss}\end{aligned}$$
If we now consider the equation of motion of retarded Green function, corresponding to a field subjected to the same initial conditions and with boundaries of constant dielectric permittivity $\epsilon(\mathbf{x})$ from the very beginning, we would have obtained
$$\nabla^{2}\widetilde{\mathcal{G}}_{\rm Ret}-z^{2}~\epsilon(\mathbf{x})~\widetilde{\mathcal{G}}_{\rm Ret}=\delta(\mathbf{x}-\mathbf{x}').$$
which is analogous to what is found from a steady canonical quantization scheme of a field with constant permittivity dielectric boundaries [@Dorota1990].
Comparing the equations it is clear that in our case we must achieve that the permittivity function given by the expression in brackets should not depend on $z$, i. e., we have to replace it by a constant. So, we can try by replacing it by its zeroth order. On the one hand, this is not possible in a simple way for the bilinear model ($\alpha=0$) because it diverges for $z=0$. On the other hand, the current-type model ($\alpha=1$) gives a finite zeroth order, allowing us to find a feasible replacement, obtaining:
$$\begin{aligned}
\nabla^{2}\widetilde{\mathcal{G}}_{\rm Ret}-z^{2}\left[1+\frac{4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}}{\omega_{\mathbf{x}}^{2}}~g(\mathbf{x})\right]\widetilde{\mathcal{G}}_{\rm Ret}=\delta(\mathbf{x}-\mathbf{x}'),\end{aligned}$$
where it is clear that the permittivity function results $\epsilon(\mathbf{x})\equiv 1+\frac{4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}}{\omega_{\mathbf{x}}^{2}}~g(\mathbf{x})$, which correctly satisfies Kramers-Kronig relations and is constant in time.
In fact, with this replacement from the very beginning, we are removing all the dynamics of the polarization degrees of freedom and setting then equal to the steady situation in the scenario including dissipation by the evaluation at $z=0$. We clearly have that $G_{\rm Ret}\equiv 0$. All in all, it gives that the terms corresponding to the contribution of the material (polarization degrees of freedom and baths) vanishes since $\mathcal{N}\equiv 0$.
Therefore, the Hadamard propagator in Eq.(\[HadamardPropagator\]) is up to the kernels $\mathcal{A}$ and $\mathcal{B}$, which in this case, clearly do not vanish. In fact, as long as there is Casimir force between constant dielectric boundaries due to the modification of the vacuum modes, these kernels should not vanish at the steady situation. This is in complete agreement with many results, that can be found for non-dissipative media boundaries, obtained from a steady canonical quantization scheme (see for example Ref. [@Dorota1990]), where quantization is carried out only by considering a Hilbert space associated to the field, and developing the Heisenberg’s canonical operator method in terms of creation and annihilation field mode operators. Thus, we can write that the long-time limit ($t_{0}\rightarrow -\infty$) is given by
$$\mathcal{G}_{\rm H}(x_{1},x_{2})\longrightarrow 2\Big(\mathcal{A}(x_{1},x_{2})+\mathcal{B}(x_{1},x_{2})\Big).$$
It is worth noting that, as it follows from steady canonical quantization schemes, the field’s state must be taken as a thermal one, being this an additional requirement of consistency, which results in the correct thermal global factors for the correlation and Green functions in the steady situation. However, our approach naturally gives the correct thermal dependence at least when an initial high-temperature state is considered for the field, which is in agreement with the high-temperature limit of the canonical quantization or in-out formalism schemes [@LombiMazziRL].
Finally, we have shown a first and simplest example where the kernels $\mathcal{A}$ and $\mathcal{B}$ do not vanish at the steady situation and in fact, in this case, they define the long-time regime. However, this is not totally new because we clearly know that there exist Casimir force between constant dielectric boundaries due to the modified vacuum modes. Anyway, we have just proved that our approach correctly reproduce that situation as a limiting case. In the next Section, we will study another situation where these terms do not vanish but neither define completely the long-time regime.
Field and Material Boundaries {#FAMB}
-----------------------------
In this Section, let us study a particular situation of the presence of boundaries. At this point, we have already seen that for the case of a $0+1$ dimensions field of frequency $\Omega$, the long-time regime is defined by the bath’s contribution, while the polarization degree of freedom and the proper field contributions vanish at the steady situation. Then, we have also seen that a $n+1$ dimensions scalar field, interacting with homogeneous material all over the space, can be reduced to an infinite set of $0+1$ fields with frequency $k$, representing the field modes that evolve in time due to the sudden appearance of the material. We have shown that in the long-time limit, as in the $0+1$ case, the only contribution to the energy-momentum tensor that survives is also the one associated to the baths. The polarization degrees of freedom and the own field have vanishing contributions at the steady situation.
However, although we were tempted to assume that the result of the last two Sections is quite general and always valid, we have presented a limiting case where the reverse is true and the annulation of the dissipation makes that the kernels $\mathcal{A}$ and $\mathcal{B}$ become the responsible of the Casimir force between non-dissipative boundaries in the long-time regime.
As we pointed out before, this is not the only case where these kernels contribute to the steady situation. If the material is inhomogeneous or there exist regions without material (i.e., vacuum regions that define material boundaries) the same could be true. Therefore, this Section gives a simple example of the presence of boundaries and the analysis of the steady situation.
### The Retarded Green Function
Back to the field equation for general boundaries given in Eq.(\[EqMotionPhiGeneralized\]) for both coupling models, we can again Laplace transform the associated equation for the retarded Green function with appropriate initial conditions to obtain
$$\begin{aligned}
\nabla^{2}\widetilde{\mathcal{G}}_{\rm Ret}-z^{2}\left[1-(-1)^{\alpha}~4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}~g(\mathbf{x})~z^{2(\alpha-1)}~\widetilde{G}_{\rm{Ret},\mathbf{x}}(z)\right]\widetilde{\mathcal{G}}_{\rm Ret}=\delta(\mathbf{x}-\mathbf{x}'),
\label{EqGreenLaplaceGeneralized}\end{aligned}$$
where $\widetilde{\mathcal{G}}_{\rm Ret}$ results to be the inverse of the differential operator $\nabla^{2}-z^{2}\left[1-(-1)^{\alpha}~4\pi\eta_{\mathbf{x}}\lambda_{0,\mathbf{x}}^{2}~g(\mathbf{x})~z^{2(\alpha-1)}~\widetilde{G}_{\rm{Ret},\mathbf{x}}(z)\right]$, i.e., it is directly the Green function associated to this operator.
We will consider a single homogeneous Dirac delta plate located at $x_{\perp}=0$ ($x_{\perp},\mathbf{x}_{\parallel}$ are the orthogonal and parallel coordinates to the plate of a given space point $\mathbf{x}$), which is described by the material distribution $g(\mathbf{x})\equiv\delta(x_{\perp})$. Thus, Eq.(\[EqGreenLaplaceGeneralized\]) results
$$\begin{aligned}
\nabla^{2}\widetilde{\mathcal{G}}_{\rm Ret}-z^{2}\left[1-(-1)^{\alpha}~4\pi\eta\lambda_{0}^{2}~\delta(x_{\perp})~z^{2(\alpha-1)}~\widetilde{G}_{\rm{Ret}}(z)\right]\widetilde{\mathcal{G}}_{\rm Ret}=\delta(\mathbf{x}-\mathbf{x}').
\label{EqGreenLaplaceGeneralizedDiracHomogenea}\end{aligned}$$
It is clear that the last equation presents translational invariance on the parallel coordinates $\mathbf{x}_{\parallel}$, so the Green function must depends on $\mathbf{x}_{\parallel}-\mathbf{x}'_{\parallel}$. Then,
$$\begin{aligned}
\frac{\partial^{2}\widetilde{\mathcal{G}}_{\rm Ret}}{\partial x_{\perp}^{2}}-(z^{2}+k_{\parallel}^{2})~\widetilde{\mathcal{G}}_{\rm Ret}+(-1)^{\alpha}~4\pi\eta\lambda_{0}^{2}~\delta(x_{\perp})~z^{2\alpha}~\widetilde{G}_{\rm{Ret}}(z)~\widetilde{\mathcal{G}}_{\rm Ret}=\delta(x_{\perp}-x'_{\perp}).
\label{EqGreenLaplaceGeneralizedDiracHomogeneaFourierParallel}\end{aligned}$$
where $k_{\parallel}=|\mathbf{k}_{\parallel}|$ and $\widetilde{\mathcal{G}}_{\rm Ret}\equiv\widetilde{\mathcal{G}}_{\rm Ret}(x_{\perp},x'_{\perp},k_{\parallel},z)$.
It is worth noting that the last equation turns out to be a Sturn-Liouville equation for the Green function, so it can be calculated by the technique described in Ref. [@Collin], where it is constructed as
$$\widetilde{\mathcal{G}}_{\rm Ret}(x_{\perp},x'_{\perp},k_{\parallel},z)=\frac{\Phi^{(L)}(x_{<})~\Phi^{(R)}(x_{>})}{W(x'_{\perp})},
\label{GreenFunctionCollin}$$
where $x_{<}$ ($x_{>}$) is the smaller (bigger) between $x_{\perp}$ and $x'_{\perp}$, $W(x)=\Phi^{(L)}(x)~\frac{d\Phi^{(R)}}{dx}-\frac{d\Phi^{(L)}}{dx}~\Phi^{(R)}(x)$ is the Wronskian (which has to be a constant function) of the solutions $\{\Phi^{(L)},\Phi^{(R)}\}$, which are two homogeneous solutions $\Phi^{(L,R)}$ that satisfy the associated homogeneous equation
$$\begin{aligned}
\frac{\partial^{2}\Phi}{\partial x_{\perp}^{2}}-(z^{2}+k_{\parallel}^{2})~\Phi+(-1)^{\alpha}~4\pi\eta\lambda_{0}^{2}~\delta(x_{\perp})~z^{2\alpha}~\widetilde{G}_{\rm{Ret}}(z)~\Phi=0.
\label{EqSolutionsPhi}\end{aligned}$$
and the boundary condition in one of the two range endpoints, i.e., $\Phi^{L}$ ($\Phi^{R}$) satisfies the boundary condition in the left (right) endpoint of the interval. In our case, that boundary condition is to have outgoing waves in the corresponding region including the respective endpoint.
The presence of a Dirac delta function in one of the terms of the equation makes that we will obtain the solution in two regions, each one with positive and negative coordinates $x_{\perp}$ respectively; and on the other hand, it gives a boundary condition with a jolt on the derivative, which can be obtained from the equation itself by integrating over an interval containing the root of the delta function and then take its length to zero around the root, clearly obtaining
$$\frac{\partial\Phi}{\partial x_{\perp}}\Big|_{x_{\perp}=0^{+}}-\frac{\partial\Phi}{\partial x_{\perp}}\Big|_{x_{\perp}=0^{-}}=(-1)^{\alpha}~4\pi\eta\lambda_{0}^{2}~z^{2\alpha}~\widetilde{G}_{\rm Ret}(z)~\Phi(0),
\label{BoundaryConditionDerDirac}$$
which goes together with the continuity of the solution.
Therefore, in each region, the solutions are plane waves so, after imposing the boundary conditions, both solutions result
$$\begin{aligned}
\Phi^{(L)}(x_{\perp})= \left\{
\begin{array}{lr rl}
t~e^{\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}, &&& \text{for}~x_{\perp}<0\\
e^{\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}+r~e^{-\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}, &&&\text{for}~0<x_{\perp}\\
\end{array}
\right.
\label{PhiSolutionL}\end{aligned}$$
$$\begin{aligned}
\Phi^{(R)}(x_{\perp})= \left\{
\begin{array}{lr rl}
e^{-\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}+r~e^{\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}, &&& \text{for}~x_{\perp}<0\\
t~e^{-\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}, &&&\text{for}~0<x_{\perp}\\
\end{array}
\right.
\label{PhiSolutionR}\end{aligned}$$
where $r$ and $t$ are the reflection and transmission coefficients for one plate respectively and given by:
$$r=-(-1)^{\alpha}~2\pi\eta\lambda_{0}^{2}~\frac{z^{2\alpha}}{\sqrt{z^{2}+k_{\parallel}^{2}}}~\widetilde{G}_{\rm Ret}(z)~t~~~~~~~,~~~~~~~t=\frac{1}{\left(1+(-1)^{\alpha}~2\pi\eta\lambda_{0}^{2}~\frac{z^{2\alpha}}{\sqrt{z^{2}+k_{\parallel}^{2}}}~\widetilde{G}_{\rm Ret}(z)\right)},
\label{CoefficientsRTnD}$$
where it is clear that $t=1+r$.
Then, the Laplace-Fourier-transform of the retarded Green function for a field point $x_{\perp}<0$ follows:
$$\begin{aligned}
\widetilde{\mathcal{G}}_{\rm Ret}(x_{\perp},x'_{\perp},k_{\parallel},z)=-\frac{1}{2\sqrt{z^{2}+k_{\parallel}^{2}}}\left\{
\begin{array}{lr rl}
e^{\sqrt{z^{2}+k_{\parallel}^{2}}~x'_{\perp}}\left(e^{-\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}+r~e^{\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}\right), &&& \text{for}~x'_{\perp}<x_{\perp}<0\\
\left(e^{-\sqrt{z^{2}+k_{\parallel}^{2}}~x'_{\perp}}+r~e^{\sqrt{z^{2}+k_{\parallel}^{2}}~x'_{\perp}}\right)e^{\sqrt{z^{2}+k_{\parallel}^{2}}~x_{\perp}}, &&&\text{for}~x_{\perp}<x'_{\perp}<0\\
t~e^{\sqrt{z^{2}+k_{\parallel}^{2}}~(x_{\perp}-x'_{\perp})}. &&&\text{for}~x_{\perp}<0<x'_{\perp}\\
\end{array}
\right.
\label{LaplaceFourierGreenFunctionDirac}\end{aligned}$$
For simplicity in the calculations, we continue with the one-dimensional version of the problem, i. e., the case of a $1+1$ field where the only dimension of interest clearly is the one associated to the perpendicular coordinate $x_{\perp}$, which we call now $x$. Therefore, to obtain the results for this case we also have to discard everything related to the parallel dimensions. We can do this simply by setting $k_{\parallel}$ equal to $0$ in all the results. This simplifies all the expressions and the Laplace transform of the retarded Green function in Eq.(\[LaplaceFourierGreenFunctionDirac\]) is
$$\begin{aligned}
\widetilde{\mathcal{G}}_{\rm Ret}(x,x',z)=-\frac{1}{2z}\left\{
\begin{array}{lr rl}
e^{zx'}\left(e^{-zx}+r~e^{zx}\right), &&& \text{for}~x'<x<0\\
\left(e^{-zx'}+r~e^{zx'}\right)e^{zx}, &&&\text{for}~x<x'<0\\
t~e^{z(x-x')}, &&&\text{for}~x<0<x'\\
\end{array}
\right.
\label{LaplaceFourierGreenFunctionDirac1D}\end{aligned}$$
where the reflection and transmission coefficients are now given by:
$$r=-(-1)^{\alpha}~2\pi\eta\lambda_{0}^{2}~z^{2\alpha-1}~\widetilde{G}_{\rm Ret}(z)~t~~~~~~~,~~~~~~~t=\frac{1}{\left(1+(-1)^{\alpha}~2\pi\eta\lambda_{0}^{2}~z^{2\alpha-1}~\widetilde{G}_{\rm Ret}(z)\right)}.
\label{CoefficientsRT1D}$$
We can transform Laplace back, by Mellin’s formula and the Residue theorem [@Schiff], assuming that the poles of the Laplace transform of the retarded Green function have non-positive real parts. It is important to remark that, following the discussion done in Ref.[@LombiMazziRL], for both coupling models, this can always be ensured by introducing an appropriate cut-off function in the Laplace transform of the dissipation kernel $\widetilde{D}_{\rm QBM}$ (in fact, any spectral density that characterizes the environment has a physical cutoff function). Moreover, assuming that the dissipation (represented by $D_{\rm QBM}$), the frequency $\omega$ and coupling constant $\lambda_{0}$ are not zero, the only pole with vanishing real part is the one at $z=0$, which appears (for each coupling case) in different terms of the Laplace transform of retarded Green function, resulting in different behaviors of the retarded Green function. This can be seen working out the last expression of the reflection coefficient $r$ in each model. However, this pole do not change the conclusions of the present Section, so we will continue the analysis without losing generality.
Therefore, the Green function can be formally written as:
$$\begin{aligned}
\mathcal{G}_{\rm Ret}(x,x',t)=-\frac{1}{2}\left\{
\begin{array}{lr rl}
\Theta(x'-x+t)+\Theta(x+x'+t)\left[\alpha-1+\sum_{z_{j}}R_{j}~e^{z_{j}(x+x'+t)}\right], &&& \text{for}~x'<x<0\\
\Theta(x-x'+t)+\Theta(x+x'+t)\left[\alpha-1+\sum_{z_{j}}R_{j}~e^{z_{j}(x+x'+t)}\right], &&&\text{for}~x<x'<0\\
\Theta(x-x'+t)\left[\alpha+\sum_{z_{j}}R_{j}~e^{z_{j}(x-x'+t)}\right], &&&\text{for}~x<0<x'\\
\end{array}
\right.
\label{GreenFunctionDirac1DDisplay}\end{aligned}$$
where $z_{j}$ are all the poles of $r$ with negative real part, i.e., the pole at $z=0$ is calculated explicitly in each model. For the others poles we have $R_{j}\equiv Res\left[\frac{r}{z},z_{j}\right]=Res\left[\frac{t}{z},z_{j}\right]$. Given that the retarded Green function must be real, its poles must come in pairs (i.e., if $z_{j}$ is a pole then its conjugate $z_{j}^{*}$ is a pole too) unless $z_{j}$ is real.
This expression, however, can be worked out by re-arranging the terms and combining their Heavyside functions to obtain a suitable closed form for the retarded Green function in each model for a field point $x<0$
$$\begin{aligned}
\mathcal{G}_{\rm Ret}(x,x',t)&=&\mathcal{G}_{\rm Ret}^{0}(x,x',t)+\frac{(1-\alpha)}{2}~\Theta(-x)~\Theta(x+x'+t)~\Theta(x-x'+t)\nonumber\\
&&-\frac{\Theta(-x)}{2}\sum_{z_{j}}R_{j}~e^{z_{j}(x+t)}\left[e^{z_{j}x'}~\Theta(-x')~\Theta(x+x'+t)+e^{-z_{j}x'}~\Theta(x')~\Theta(x-x'+t)\right],
\label{GreenFunctionDirac1DClosedForm}\end{aligned}$$
where $\mathcal{G}_{\rm Ret}^{0}(x,x',t)\equiv-\frac{\Theta(-x)}{2}~\Theta(x'-x+t)~\Theta(x-x'+t)$ is the retarded Green function in free space for a field point $x<0$.
It is worth noting, on the one hand, that the second term is an extra term only for the bilinear model due to the presence of the plate but independent of the material properties. On the other hand, the third term is directly and entirely related to the presence of the plate and it contains all the information about the material contribution to the transient evolution (i.e., relaxation) and the new steady situation that the field will achieve. It is clear that it implicitly depends on the coupling model because the poles $z_{j}$ depend on it.
From Eq.(\[GreenFunctionDirac1DClosedForm\]) it can be easily proved, by looking carefully the products of distributions, that the time derivative of the retarded Green function has a simple form given by:
$$\begin{aligned}
\dot{\mathcal{G}}_{\rm Ret}(x,x',t)&=&\dot{\mathcal{G}}_{\rm Ret}^{0}(x,x',t)\nonumber \\
&-&\frac{\Theta(-x)}{2}\sum_{z_{j}}z_{j}~R_{j}~e^{z_{j}(x+t)}\left[e^{z_{j}x'}~\Theta(-x')~\Theta(x+x'+t)+e^{-z_{j}x'}~\Theta(x')~\Theta(x-x'+t)\right],
\label{DerivativeGreenFunctionDirac1DClosedForm}\end{aligned}$$
where in this expression the only difference between the coupling models relies on the poles $z_{j}$.
### The Long-Time Regime
With the retarded Green function for the present problem, we can proceed to study some dynamical aspects and features about the steady situation.
As we just obtained in previous sections, our interest is the Hadamard propagator given in Eq.(\[HadamardPropagator\]), which we can use to calculate the expectation value of the energy-momentum components through Eq.(\[TmunuExpValue\]). As is stated by Eq.(\[HadamardPropagator\]), the Hadamard propagator has several contributions that can be divided in two parts, one coming from the field generated by all the components of the material (polarization degrees of freedom and baths) which is represented by the noise kernel $\mathcal{N}$, and another one coming from the field generated by the vacuum fluctuations subjected to the actual boundary conditions, which is represented by the kernels $\mathcal{A}$ and $\mathcal{B}$ and will imply a modification of the field modes through a transient evolution from the initial free field to the new steady field.
Let’s study firstly the material contribution. As we have proved in Sec.\[FIIM\], when the material is modeled as Brownian particles interacting with the field by both coupling models, the material contribution at the steady situation have only the contributions coming from the baths, while the particles merely act as a bridge connecting the field with the baths, but having no contribution in the long-time regime due to their dissipative Brownian dynamics. This was basically contained in the fact that in the long-time limit ($t_{0}\rightarrow-\infty$), we clearly have that $\mathcal{N}\rightarrow\mathcal{N}_{B}$. In the present case, although there are regions without material, the result is still valid. It is clear that in this case the Green function is given by Eq.(\[GreenFunctionDirac1DClosedForm\]) but the formal expression is the same. In fact, it is worth noting also that the material distribution $g$ will define the range of integration, having no contribution from the points outside the material.
On the other hand, we have the contribution to the field generated by the vacuum fluctuations represented by $\mathcal{A}$ and $\mathcal{B}$. We are tempted to assume that, as in Sec.\[FIIM\], these contributions vanish at the steady situation giving only, a transient behavior. However, as we just pointed out at the end of that Section, this could not be true when there are vacuum regions where the field fluctuates freely.
Therefore, considering the kernel $\mathcal{A}$ in Eq.(\[KernelAHighT\]) and the expression for the retarded Green function given in Eq.(\[GreenFunctionDirac1DClosedForm\]), is clear that the product $\mathcal{G}_{\rm Ret}(x_{1},x,t_{1}-t_{0})~\mathcal{G}_{\rm Ret}(x_{2},x,t_{2}-t_{0})$ will have at most nine terms (depending which coupling model we are considering) due to all the possible combinations of the three separated terms which then have to be integrated over $x$. Using the symmetry of the kernel, the number of integrals to calculate is six at most. The complication in the full calculation is due to the fact that each integral involves products of distributions having the integration variable and both field points $(x_{1},t_{1})$ and $(x_{2},t_{2})$; results will depend on multiple relations between the coordinates of the field points.
On the other hand, the full calculation of kernel $\mathcal{B}$ is so complicated as for the previous kernel. In the one dimensional case, it is easy to calculate the kernel $K$ in Eq.(\[KernelKHighTCoordinate\]) through the Residue theorem, obtaining that $K(x-x')=-\frac{|x-x'|}{2\beta_{\phi}}$, which can be written as two terms. Then, kernel $\mathcal{B}$ involves a double integration (over $x$ and $x'$) of the triple product $\dot{\mathcal{G}}_{\rm Ret}(x_{1},x,t_{1}-t_{0})~K(x-x')~\dot{\mathcal{G}}_{\rm Ret}(x_{2},x',t_{2}-t_{0})$. From Eq.(\[DerivativeGreenFunctionDirac1DClosedForm\]) it is clear that the derivative of the retarded Green function has two terms, so to obtain $\mathcal{B}$ the number of integrals to perform is in principle eight. Due to the symmetry, the final number of double integrals reduce to six for this kernel too.
As we are interested in general features about the transient time evolution and the steady situation, we will not proceed to a complete calculation of the terms, but we will show that there are steady terms associated to the contribution of these kernels.
We should note that the terms in the kernels $\mathcal{A}$ and $\mathcal{B}$ associated to the products of the free field retarded Green function $\mathcal{G}_{\rm Ret}^{0}$ and its derivative, i.e., the terms $\mathcal{G}_{\rm Ret}^{0}(x_{1},x,t_{1}-t_{0})~\mathcal{G}_{\rm Ret}^{0}(x_{2},x,t_{2}-t_{0})$ in $\mathcal{A}$ and $\dot{\mathcal{G}}_{\rm Ret}^{0}(x_{1},x,t_{1}-t_{0})~K(x-x')~\dot{\mathcal{G}}_{\rm Ret}^{0}(x_{2},x',t_{2}-t_{0})$ in $\mathcal{B}$ are the ones that will be removed by the Casimir prescription (subtraction with the free field case), so we do not have to calculate them.
We should note also that the crossed terms (i.e. terms combining different terms of the Green function) will be transient terms since those integrations will generate constant terms that will vanish in the derivatives and through the limit needed to calculate the expectation values of the energy-momentum tensor, or terms that will exponentially decay at the long-time limit, or divergent terms that must be subtracted to define a correct (non-divergent) Hadamard propagator. As we are interested now in the steady situation, we will not calculate them.
Independently of which model we are considering, to study the long-time regime, the products involving two sums over poles will be the ones that result in steady contributions. As a first case, we consider the corresponding term found in the kernel $\mathcal{A}$ for field point $x_{1,2}<0$:
$$\begin{aligned}
\mathcal{A}(x_{1},x_{2},t_{1},t_{2})&=&(\text{Free Field Terms})+(\text{Crossed Terms})+\frac{\Theta(-x_{1})~\Theta(-x_{2})}{4\beta_{\phi}}\sum_{j,l}R_{j}~R_{l}~e^{z_{j}(x_{1}+t_{1}-t_{0})}~e^{z_{l}(x_{2}+t_{2}-t_{0})}\nonumber\\
&&\times\int_{-\infty}^{+\infty}dx\left[e^{z_{j}x}~\Theta(-x)~\Theta(x_{1}+x+t_{1}-t_{0})+e^{-z_{j}x}~\Theta(x)~\Theta(x_{1}-x+t_{1}-t_{0})\right]\nonumber\\
&&~~~~~~~~~~~\times\left[e^{z_{l}x}~\Theta(-x)~\Theta(x_{2}+x+t_{2}-t_{0})+e^{-z_{l}x}~\Theta(x)~\Theta(x_{2}-x+t_{2}-t_{0})\right].
\label{KernelAHighTDirac1DSteadyTerms}\end{aligned}$$
Considering that $\Theta(x)~\Theta(-x)\equiv 0$ and $\Theta(\pm x)~\Theta(\pm x)\equiv\Theta(\pm x)$, there are vanishing integrals in the expression. Then, making a substitution $x\rightarrow -x$ on one of the two resulting terms, all the integrals show to be the same, clearly obtaining:
$$\begin{aligned}
\mathcal{A}(x_{1},x_{2},t_{1},t_{2})&=&(\text{Free Field Terms})+(\text{Crossed Terms})\nonumber\\
&+&\frac{\Theta(-x_{1})\Theta(-x_{2})}{2\beta_{\phi}}\sum_{j,l}R_{j}~R_{l}~e^{z_{j}(x_{1}+t_{1}-t_{0})}~e^{z_{l}(x_{2}+t_{2}-t_{0})}\nonumber \\
&\times & \int_{-\infty}^{+\infty}dx~e^{(z_{j}+z_{l})x}~\Theta(-x)~\Theta(x_{1}+x+t_{1}-t_{0})~\Theta(x_{2}+x+t_{2}-t_{0}).\end{aligned}$$
Considering that $\Theta(x_{1}+x+t_{1}-t_{0})~\Theta(x_{2}+x+t_{2}-t_{0})=\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+x+t_{2}-t_{0})+\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+x+t_{1}-t_{0})$, the last integral can be easily calculated
$$\begin{aligned}
&&\mathcal{A}(x_{1},x_{2},t_{1},t_{2})=(\text{Free Field Terms})+(\text{Crossed Terms})+\frac{\Theta(-x_{1})~\Theta(-x_{2})}{2\beta_{\phi}}\sum_{j,l}\frac{R_{j}~R_{l}}{(z_{j}+z_{l})}\nonumber\\
&&\times\left[\Big(\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})+\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})\Big)~e^{z_{j}(x_{1}+t_{1}-t_{0})}~e^{z_{l}(x_{2}+t_{2}-t_{0})}\right. \\
&&\left.-~\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})~e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}-\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})~e^{z_{l}(x_{2}-x_{1}+t_{2}-t_{1})}\right].\nonumber
\label{KernelAHighTDirac1DSteadyTermsFinal}\end{aligned}$$
On the other hand, the kernel $\mathcal{B}$ presents a more complicated structure because it involves two integrations (one over $x$ and other one over $x'$) and an extra kernel $K(x-x')$ which couples both integrations preventing a separate calculation. Following the same train of thought, we focus in the terms involving two sums over poles. Therefore, kernel $\mathcal{B}$ reads:
$$\begin{aligned}
\mathcal{B}(x_{1},x_{2},t_{1},t_{2})&=&(\text{Free Field Terms})+(\text{Crossed Terms})\nonumber \\ &-&\frac{\Theta(-x_{1})~\Theta(-x_{2})}{8\beta_{\phi}}\sum_{j,l}z_{j}~z_{l}~R_{j}~R_{l}~e^{z_{j}(x_{1}+t_{1}-t_{0})}~e^{z_{l}(x_{2}+t_{2}-t_{0})}\nonumber\\
&&\times\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}dx'~|x-x'|\left[e^{z_{j}x}~\Theta(-x)~\Theta(x_{1}+x+t_{1}-t_{0})+e^{-z_{j}x}~\Theta(x)~\Theta(x_{1}-x+t_{1}-t_{0})\right]\nonumber\\
&& \times\left[e^{z_{l}x'}~\Theta(-x')~\Theta(x_{2}+x'+t_{2}-t_{0})+e^{-z_{l}x'}~\Theta(x')~\Theta(x_{2}-x'+t_{2}-t_{0})\right].\end{aligned}$$
The integration over $x'$ can be done first by writing $|x-x'|=\Theta(x-x')~(x-x')+\Theta(x'-x)~(x'-x)$. Working out the integral, we obtain that the result can be separated again in terms that will be part of the transient evolution and will vanish at the long-time regime and terms that will give steady results. In fact, the integral can be written as:
$$\begin{aligned}
\int_{-\infty}^{+\infty}&dx'&~|x-x'|\left[e^{z_{l}x'}~\Theta(-x')~\Theta(x_{2}+x'+t_{2}-t_{0})+e^{-z_{l}x'}~\Theta(x')~\Theta(x_{2}-x'+t_{2}-t_{0})\right]=\nonumber\\
&=&\frac{2}{z_{l}^{2}}\left[\Theta(-x)~\Theta(x_{2}+x+t_{2}-t_{0})~e^{z_{l}x}+\Theta(x)~\Theta(x_{2}-x+t_{2}-t_{0})~e^{-z_{l}x}\right]+(\text{Transient Terms}).\end{aligned}$$
Thus, kernel $\mathcal{B}$ reads:
$$\begin{aligned}
&&\mathcal{B}(x_{1},x_{2},t_{1},t_{2})=(\text{Free Field Terms})+(\text{Crossed Terms})+(\text{Transient Terms}) \\ &-&\frac{\Theta(-x_{1})~\Theta(-x_{2})}{4\beta_{\phi}}\sum_{j,l}\frac{z_{j}}{z_{l}}~R_{j}~R_{l}\nonumber\\
&&\times~e^{z_{j}(x_{1}+t_{1}-t_{0})}~e^{z_{l}(x_{2}+t_{2}-t_{0})}\int_{-\infty}^{+\infty}dx\left[e^{z_{j}x}~\Theta(-x)~\Theta(x_{1}+x+t_{1}-t_{0})+e^{-z_{j}x}~\Theta(x)~\Theta(x_{1}-x+t_{1}-t_{0})\right]\nonumber\\
&&\times\left[\Theta(-x)~\Theta(x_{2}+x+t_{2}-t_{0})~e^{z_{l}x}+\Theta(x)~\Theta(x_{2}-x+t_{2}-t_{0})~e^{-z_{l}x}\right],\nonumber\end{aligned}$$
where it is worth noting that the resulting integral is the same as the one in kernel $\mathcal{A}$ in Eq.(\[KernelAHighTDirac1DSteadyTerms\]).
Then, the result is the same and the kernel can be written as:
$$\begin{aligned}
&&\mathcal{B}(x_{1},x_{2},t_{1},t_{2})=(\text{Free Field Terms})+(\text{Crossed Terms})+(\text{Transient Terms}) \\ &-& \frac{\Theta(-x_{1})~\Theta(-x_{2})}{2\beta_{\phi}}\sum_{j,l}\frac{z_{j}}{z_{l}}~\frac{R_{j}~R_{l}}{(z_{j}+z_{l})}\nonumber\\
&&\times\left[\Big(\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})+\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})\Big)~e^{z_{j}(x_{1}+t_{1}-t_{0})}~e^{z_{l}(x_{2}+t_{2}-t_{0})}\right.\nonumber\\
&&\left.-~\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})~e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}-\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})~e^{z_{l}(x_{2}-x_{1}+t_{2}-t_{1})}\right].\nonumber\end{aligned}$$
By considering this last equation and Eq.(\[KernelAHighTDirac1DSteadyTermsFinal\]), it is now straightforward that the proper field contribution, given by the sum of the kernels $\mathcal{A}$ and $\mathcal{B}$ can be written as:
$$\begin{aligned}
&&\mathcal{A}(x_{1},x_{2},t_{1},t_{2})+\mathcal{B}(x_{1},x_{2},t_{1},t_{2})=(\text{Free Field Terms})+(\text{Crossed Terms})+(\text{Transient Terms}) \\
&+&\frac{\Theta(-x_{1})~\Theta(-x_{2})}{2\beta_{\phi}}\sum_{j,l}\left(1-\frac{z_{j}}{z_{l}}\right)~\frac{R_{j}~R_{l}}{(z_{j}+z_{l})}\nonumber\\
&&\times\left[\Big(\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})+\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})\Big)~e^{z_{j}(x_{1}+t_{1}-t_{0})}~e^{z_{l}(x_{2}+t_{2}-t_{0})}\right.\nonumber\\
&&\left.-~\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})~e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}-\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})~e^{z_{l}(x_{2}-x_{1}+t_{2}-t_{1})}\right]. \nonumber\end{aligned}$$
The last two terms in the brackets contain exponentials whose exponents do not depend on the initial time $t_{0}$. Therefore those terms will not vanish at the long-time limit ($t_{0}\rightarrow-\infty$). This shows that a part of the proper field contribution has not only transient but also steady terms that contributes to the long-time regime. Note, in fact, that these terms in the Hadamard propagator will result as constant terms in the expectation values of the energy-momentum tensor components of Eq.(\[TmunuExpValue\]) after differentiating and calculating the coincidence limit.
Moreover, we can work out these terms and write them in a more familiar way connecting with previous works. It should be noted that in the first term (associated to $e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}$) the sum over $l$ can be worked out through the Residue theorem, while in the second term (associated to $e^{z_{l}(x_{2}-x_{1}+t_{2}-t_{1})}$) the sum over $j$ can be done.
Then, let us take firstly this last sum over $j$. Considering that all the poles are simple and $R_{j}\equiv Res\left[\frac{r}{z},z_{j}\right]$, we can write:
$$\sum_{j}\frac{(z_{l}-z_{j})}{(z_{j}+z_{l})}~R_{j}=\sum_{j}Res\left[\frac{(z_{l}-z)}{(z+z_{l})}\frac{r}{z},z_{j}\right].$$
From Eq.(\[CoefficientsRT1D\]) and given that $Re(z_{l})<0$ for every pole $z_{l}$ (so $z_{l}+z_{j}\neq 0$), we can show that the complex function $\frac{(z_{l}-z)}{(z+z_{l})}\frac{r}{z}$ goes to $0$ when $|z|\rightarrow+\infty$, independently of the direction in the complex plane, and that its set of poles is given by all the poles $z_{j}$, the pole $-z_{l}$ (which depend on the term on the sum over $l$ that we are considering) and the pole $0$ only in the bilinear coupling model. Therefore, through the Residue theorem, for a circle $\mathcal{C}_{R}^{+}$ of radius $R$ in the complex plane that contains all the poles, when $R\rightarrow+\infty$, we can write:
$$\begin{aligned}
&&0=\int_{\mathcal{C}_{\infty}}\frac{dz}{2\pi i}~\frac{(z_{l}-z)}{(z+z_{l})}\frac{r}{z}=\sum_{j}Res\left[\frac{(z_{l}-z)}{(z+z_{l})}\frac{r}{z},z_{j}\right]-2~r(-z_{l})+\alpha-1,\end{aligned}$$
where the last two terms are the results of calculating explicitly the poles at $-z_{l}$ and at $0$.
Therefore, the whole term associated to $e^{z_{l}(x_{2}-x_{1}+t_{2}-t_{1})}$ reads:
$$\begin{aligned}
\sum_{j,l}\left(1-\frac{z_{j}}{z_{l}}\right)~\frac{R_{j}~R_{l}}{(z_{j}+z_{l})}~e^{z_{l}(x_{2}-x_{1}+t_{2}-t_{1})}=\sum_{l}\frac{R_{l}}{z_{l}}\Big(2~r(-z_{l})+1-\alpha\Big)~e^{z_{l}(x_{2}-x_{1}+t_{2}-t_{1})}.
\label{SumaL}\end{aligned}$$
Analogously, we can proceed for the other term, associated to $e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}$, by starting with the sum over $l$. The calculation is the same but over the complex function $\frac{(z-z_{j})}{(z+z_{j})}\frac{r}{z^{2}}$ and except that the pole at $z=0$ is simple for the current-type model, while is of second order in the bilinear model. Then we finally have:
$$\begin{aligned}
\sum_{j,l}\left(1-\frac{z_{j}}{z_{l}}\right)~\frac{R_{j}~R_{l}}{(z_{j}+z_{l})}~e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}=\sum_{j}R_{j}\Bigg(2~\frac{r(-z_{j})}{z_{j}}+\frac{2\pi\eta\lambda_{0}^{2}}{\omega^{2}}~\alpha+(1-\alpha)\frac{\omega^{2}}{2\pi\eta\lambda_{0}^{2}}\Bigg)~e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}
\label{SumaJ}\end{aligned}$$
where the difference in units between the last two terms in the brackets is due to the fact that the coupling constant $\lambda_{0}$ change its units depending on the coupling model.
The last terms involves differences between both coupling models in Eqs.(\[SumaL\]) and (\[SumaJ\]). It can be shown that are divergent terms in the coincidence limit, so we discard them by regularizing the expression. This way, we can write the proper field contribution as:
$$\begin{aligned}
\mathcal{A}(x_{1},x_{2},t_{1},t_{2})&+&\mathcal{B}(x_{1},x_{2},t_{1},t_{2})=(\text{Free Field Terms})+(\text{Crossed Terms})+(\text{Transient Terms})\nonumber\\
&&-\frac{\Theta(-x_{1})~\Theta(-x_{2})}{\beta_{\phi}}\sum_{j}R_{j}~\frac{r(-z_{j})}{z_{j}}\left[\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})~e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}\right.\nonumber\\
&&\left.+~\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})~e^{-z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}\right].\end{aligned}$$
At this point, we can exploit one more time the Residue theorem to obtain a final closed form for these terms. It is straightforward to show that first the sum over $j$, as we did for the others sums and taking into account the convergence requirements, can be written as an integral in the complex plane over a curve $\mathcal{C}^{+}=\mathcal{C}_{L}^{+}\bigcup\mathcal{R}^{+}$ where $\mathcal{C}_{L}^{+}$ is a half-infinite circle enclosing the left half of the complex plane and $\mathcal{R}^{+}$ is a straight path over the imaginary axis from the bottom to the top. Therefore, since the integrand function $\frac{r(z)~r(-z)}{(-z^{2})}~e^{z(x_{1}-x_{2}+t_{1}-t_{2})}$ vanishes for $|z|\rightarrow\infty$ with $Re(z)<0$, the integral over $\mathcal{C}_{L}^{+}$ is null and the integral is directly over the imaginary axis, which can be parametrized as $z=-i\Omega$, finally obtaining:
$$\begin{aligned}
-\sum_{j}R_{j}~\frac{r(-z_{j})}{z_{j}}~e^{z_{j}(x_{1}-x_{2}+t_{1}-t_{2})}&=&\int_{\mathcal{C}^{+}}\frac{dz}{2\pi i}~\frac{r(z)~r(-z)}{(-z^{2})}~e^{z(x_{1}-x_{2}+t_{1}-t_{2})}\nonumber \\
&=&\int_{-\infty}^{+\infty}\frac{d\Omega}{2\pi}~\frac{|r(-i\Omega)|^{2}}{(\Omega^{2})}~e^{-i\Omega(x_{1}-x_{2}+t_{1}-t_{2})},\end{aligned}$$
where we have used that $r(i\Omega)=r^{*}(-i\Omega)$ for real $\Omega$.
Finally, the proper field contribution reads:
$$\begin{aligned}
\mathcal{A}(x_{1},x_{2},t_{1},t_{2})&+&\mathcal{B}(x_{1},x_{2},t_{1},t_{2})=(\text{Free Field Terms})+(\text{Crossed Terms})+(\text{Transient Terms})\nonumber\\
&&+\frac{\Theta(-x_{1})~\Theta(-x_{2})}{\beta_{\phi}}\int_{-\infty}^{+\infty}\frac{d\Omega}{2\pi}~\frac{|r(-i\Omega)|^{2}}{\Omega^{2}}\left[\Theta(x_{1}-x_{2}+t_{1}-t_{2})~\Theta(x_{2}+t_{2}-t_{0})~e^{-i\Omega(x_{1}-x_{2}+t_{1}-t_{2})}\right.\nonumber\\
&&\left.+~\Theta(x_{2}-x_{1}+t_{2}-t_{1})~\Theta(x_{1}+t_{1}-t_{0})~e^{i\Omega(x_{1}-x_{2}+t_{1}-t_{2})}\right],\end{aligned}$$
where it is remarkable that it has the form of the long-time contributions considered without demonstration in Refs.[@LombiMazziRL; @Dorota1992] for the proper field contribution in a steady canonical quantization scheme in the case of the force between two plates, the Casimir-Lifshitz problem.
All in all, we have showed that the long-time limit of a Dirac delta plate of real material have contributions both from the bath as from the field by itself. The result can be extended to other configurations in one dimension ($1+1$) also but the calculations are more complicated. The conclusion seems to be general, i.e., we have showed that a situation including boundaries or, analogously, including vacuum regions, will present not only the contributions from the baths, but also the proper field contribution in the long-time regime. Therefore, this situation shows a new type of scenario, where the long-time regime is steady but it has contributions from two parts of the composite system. Note that the behavior and the steady situation in the case with vacuum regions is radically different from the case of material in the entire space. However, the material contribution, considered separately, behaves in the same way, i.e., the contribution associated to the polarization degrees of freedom is transient and vanish at the steady situation, while the baths’ contribution survives and it is part of the long-time regime. The great difference of including boundaries is that it is not the only one that survives. This is due to the fact that while the field tends to vanish inside the material due to dissipation, outside it is fluctuating freely without damping. This makes that the fluctuations outside propagate inside the material and finally reach a steady situation at the long-time regime, when the material has relaxed and the dynamics are reduced to the steady ones, then allowing us to describe the proper contribution effectively by modified vacuum modes as Refs.[@LombiMazziRL; @Dorota1992] for the Casimir problem.
Therefore, any quantization procedure in the long-time regime, i.e., any quantization scheme at the steady situation, at least for this models in $1+1$, must consider this contribution to obtain the correct results.
Following this train of thought, as a final comment we should note that we have proved this fact in the one dimensional ($1+1$) case, but the conclusion for higher dimensions could change. Although we have not made here the calculation for the $n+1$ case in this scenario including vacuum regions, from the comparison between the reflection and transmission coefficients in Eqs.(\[CoefficientsRTnD\]) and (\[CoefficientsRT1D\]), and the Laplace transform of the retarded Green functions in Eqs. (\[LaplaceFourierGreenFunctionDirac\]) and (\[LaplaceFourierGreenFunctionDirac1D\]) for both cases, we can note that for the higher dimensions problem we have two branch cuts $\sqrt{z^{2}+k_{\parallel}}$ involving the parallel momentum $k_{\parallel}$ instead of a simple $z$. Therefore, the analytical properties of the Laplace transform are different and the time behavior of the retarded Green function will change critically. It could happen that the proper field contribution vanishes for this higher dimensions case, but the continuity between the actual case of real material and the result obtained in the $n+1$ case for a constant dielectric and arbitrary boundaries in Sec.\[CDPL\] suggest that the result of this Section is quite general even for the higher dimensions case.
Final remarks {#FR}
=============
In this article we have extensively used the CTP approach to calculate a general expression for the time evolution of the expectation value of the energy-momentum tensor components, in a completely general non-equilibrium scenario, for a scalar field in the presence of real materials. The interaction is turned on at an initial time $t_{0}$, coupling the field to the polarization degrees of freedom of a volume element of the material, which is also linearly coupled to thermal baths in each point of the space. Throughout the work, we studied two coupling models between the field and the polarization degrees of freedom. One is the bilinear model, analogous to the one considered in the QBM theory [@BreuerPett; @CaldeLegg; @HuPazZhang], and the other one is (a more realistic) current-type model, where the polarization degrees of freedom couples to the field’s time derivative (as in the EM case interacting with matter).
It is remarkable that the material is free to be inhomogeneous, i.e., its properties (density $\eta_{\mathbf{x}}$, coupling constant to the field $\lambda_{0,\mathbf{x}}$, mass $m_{\mathbf{x}}$, and frecuency $\omega_{\mathbf{x}}$ of the volume elements polarization degrees of freedom) can change with the position. The baths’ properties (coupling constant to the polarization degrees of freedom $\lambda_{n,\mathbf{x}}$, mass $m_{n,\mathbf{x}}$, and frecuency $\omega_{n,\mathbf{x}}$ of each bath oscillator) also can change with position, resulting in an effective-position-dependent properties over the volume elements of the material; which are represented by the dissipation and noise kernels $D_{\rm QBM,\mathbf{x}}$ and $N_{\rm QBM,\mathbf{x}}$ after the first integration over the baths’ degrees of freedom.
On the other hand, thermodynamical non-equilibrium is included by letting both each volume element and thermal bath to have their own temperature ($\beta_{r_{\mathbf{x}}},\beta_{B,\mathbf{x}}$) by choosing the initial density operators of each part to be thermal states. The field has also its own temperature $\beta_{\phi}$, which analogously comes from the field initial state, but for simplicity in the calculations, we have taken the high temperature approximation, except for the $0+1$ field case, in Sec.\[0+1F\], where the calculation can be done for arbitrary temperatures of the field.
It is worth noting that the approach also consider that the material bodies can be of finite extension and have arbitrary shape, i.e., vacuum regions were the field is free are included. All these features are concentrated in the matter distribution function $g(\mathbf{x})$, which takes binary values $1$ or $0$ whether there is material at $\mathbf{x}$ or not.
In the field’s high temperature limit, the expectation value of the energy-momentum tensor components are given by Eq.(\[TmunuExpValue\]), where the Hadamard propagator is defined in Eq.(\[HadamardPropagator\]). That equation contains the full dynamics of the field correlation, with one contribution clearly associated to the material, which is contained in the third term of Eq.(\[HadamardPropagator\]); and the other one clearly associated to the field by itself, which is contained in the first two terms.
All in all, the third term is directly associated to the material (polarization degrees of freedom plus thermal baths), represented by the field’s noise kernel, which is also separated in two contributions, $\mathcal{N}(x,x')=4\pi\eta_{\mathbf{x}}g(\mathbf{x})~\delta(\mathbf{x}-\mathbf{x}')\left[\mathcal{N}_{r,\mathbf{x}}(\tau,\tau')+\mathcal{N}_{B,\mathbf{x}}(\tau,\tau')\right]$, one associated to the polarization degrees of freedom that define the material and has an effective dissipative (damping) dynamics due to the interaction with the baths (QBM), and other one associated to the baths’ fluctuations that, thanks to its (non-damping) dynamics, acts like source of constant temperatures in each point of space and results in an influence over the field, although they are not in direct interaction, but the polarization degrees connect them as a bridge in a non-direct interaction. Both contributions define the field’s transient dynamics, while the steady situation seems to be determined only by the baths’, which has non-damping dynamics. The relaxation dynamics of the polarization degrees of freedom will make that they will not have any contribution to the long-time regime.
We have shown that the polarization degrees of freedom never contribute to the steady situation, while the bath always do, independently of the situation considered. However, there is a case (Sec.\[CDPL\]) that both contributions were absent. It is the constant dielectric case, where it is trivially expected that there is no contribution because the material dynamics is suppressed.
On the other hand, the first two terms of Eq.(\[HadamardPropagator\]) are directly associated to the field initial state. As we have considered the high temperature limit, both terms are linear in the field’s initial temperature $\beta_{\phi}$, as is expected. These terms give the transient field evolution from the initial free field over the whole space to the field in interaction with the polarization degrees of freedom of the material in certain regions defined by the distribution function $g(\mathbf{x})$. This evolution involves two dynamical aspects. One is related to the fact that the properties of the boundaries are time-dependent. This adds extra dynamical features associated to the relaxation process of the material, which enter the field dynamics encoded in the field’s dissipation kernel $\partial_{tt}^{2\alpha}\mathcal{D}$ for each coupling model that defines the form of the field’s retarded Green function $\mathcal{G}_{\rm Ret}$ through Eq.(\[EqMotionPhiGeneralized\]). In Sec.\[CDPL\], this aspect is turned off by suppressing the material’s relaxation.
The other dynamical aspect is the adaptation of the free field to the condition of being bounded by the sudden appearance of boundaries. We have shown that this clearly takes place in two scenarios, one was studied in Sec.\[FAMB\] when the distribution function $g(\mathbf{x})$ has null values for some space point $\mathbf{x}$, i.e., when there are vacuum regions in the particular problem, while the other one is the aforementioned lossless case (Sec.\[CDPL\]). In both, the aspect is basically the conversion of the field modes from free field modes to interacting (modified) field modes.
This is completely related to the long-time regime of these terms and it is not so easy to analyze. In Secs.\[0+1F\] and \[FIIM\], the proper field contribution vanishes for both coupling models, so there is no interacting (modified) field modes associated to the field’s initial state. Tempted to consider that this result applies to all the cases involving material boundaries, in Secs.\[CDPL\] and \[FAMB\], we have shown that the proper field contribution does not vanish in the long-time regime. The result is trivially expected for the constant dielectric permittivity, as there exists steady Casimir force between bodies of constant dielectric permittivity (see Refs.[@LombiMazziRL; @Dorota1990]) involving a $n+1$ field. Therefore, since this case and the real material including losses and absorption are expected to be continuously connected, as occurs in Sec.\[FAMB\] for the one dimensional ($1+1$) case, it is suggested that the result found in $1+1$ is quite general and holds for the $n+1$ case with material boundaries (with $n>1$).
Physically, for the case without boundaries, the dissipative (damping) dynamics of the field vanishes at the steady situation as in the QBM case.
For the case with vacuum regions, the field inside the material tends to behave as a damped field, but in the vacuum regions, the field fluctuates without damping. So, there is a competition between the behavior in both regions which will make the field evolves to a modified modes steady field in the long-time regime, beyond the material relaxation, analogously to the case of the constant dielectric properties steady situation, but with frequency-dependent effective properties. The free fluctuations propagate inside the material regions and keep the field in a continuous situation that survives to the long-time regime. In other words, the transient dynamics of these terms will be different from the constant dielectric case. The steady situation will be also different because the frequency dependence of the properties. But, taking into account the considerations made above, the formal expression must be the same as the constant dielectric case by replacing it by the corresponding actual frequency-dependent permittivity (as it happens for the Casimir force for a $1+1$ field in absorbing media [@LombiMazziRL]).
Therefore, at least in the $1+1$ case, any quantization scheme of these models at a steady scenario, although involving thermodynamical non-equilibrium, must take into account the proper field contribution besides the expected baths’ contribution.
All in all, this is in fact a natural physical conclusion, all the parts of the system having no damping dynamics contribute to the long-time regime. Following this train of thought, is naturally expected that the bath contributes and, since there are regions where the field has no damping dynamics, modified field modes takes place in the long-time proper field contribution.
As future work, for completeness, it should be possible to extend the calculation to the case of arbitrary field’s temperature without many complications. More interesting would be investigate the implications of this analysis over the vacuum fluctuations in the real Casimir problem, extending the complete study in $1+1$ scalar field to the $3+1$ EM field. Finally, it would be also interesting to study the heat transfer and other thermodynamical features in situations where the material be thermally inhomogeneous.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank A. J. Roncaglia and F. D. Mazzitelli for useful discussions that stimulated this research. This work was supported by UBA, CONICET and ANPCyT.
The field Wigner functional {#A}
===========================
The Wigner functional for a quantum field can be defined as in Ref. [@MrowMull]:
$$\begin{aligned}
W_{\phi}\left[\phi_{0}(\mathbf{x}),\Pi_{0}(\mathbf{x}),t_{0}\right]=\int\mathcal{D}\varphi(\mathbf{x})~e^{-i\int d\mathbf{x}~\Pi_{0}(\mathbf{x})~\varphi(\mathbf{x})}~\Big\langle\phi_{0}(\mathbf{x})+\frac{1}{2}~\varphi(\mathbf{x})\Big|\widehat{\rho}_{\phi}(t_{0})\Big|\phi_{0}(\mathbf{x})-\frac{1}{2}~\varphi(\mathbf{x})\Big\rangle.
\label{FieldWignerCoordinate}\end{aligned}$$
It is worth noting that sometimes seems easier to compute the Wigner functional in momentum space. However, it is not so easy. Even though the field $\phi(\mathbf{x})$ is real, it Fourier transform $\phi(\mathbf{p})$ is complex but its real and imaginary parts are not independent, because to have a real field, $\phi(-\mathbf{p})=\phi^{*}(\mathbf{p})$. As in Ref. [@MrowMull], for the Fourier transform, we will treat the real and imaginary parts of $\phi(\mathbf{p})$ as independent variables, but considering $p_{i}\in(0,+\infty)$ for each momentum component instead of $p_{i}\in(-\infty,+\infty)$. This way, the Wigner functional in the momentum space can be defined as:
$$\begin{aligned}
\widetilde{W}_{\phi}\left[\phi_{0}(\mathbf{p}),\Pi_{0}(\mathbf{p}),t_{0}\right]=\int\mathcal{D}\varphi(\mathbf{p})~e^{-i\int_{0}^{+\infty} d\mathbf{p}~\left[\Pi_{0}^{*}(\mathbf{p})~\varphi(\mathbf{p})+\Pi_{0}(\mathbf{p})~\varphi^{*}(\mathbf{p})\right]}~\Big\langle\phi_{0}(\mathbf{p})+\frac{1}{2}~\varphi(\mathbf{p})\Big|\widehat{\rho}_{\phi}(t_{0})\Big|\phi_{0}(\mathbf{p})-\frac{1}{2}~\varphi(\mathbf{p})\Big\rangle,
\label{FieldWignerMomentum}\end{aligned}$$
with the functional integrations running over real and imaginary components of $\phi(\mathbf{p})$ [@MrowMull]. Going from Eq.(\[FieldWignerCoordinate\]) to (\[FieldWignerMomentum\]) implies a nontrivial Jacobian $\det\left[\frac{\delta\varphi(\mathbf{x})}{\delta\varphi(\mathbf{p})}\right]$, which does not depends on the fields because the Fourier transformation is a linear mapping, consequently, it appears merely as a new normalization factor of the Wigner functional.
Now, we consider the scalar field initially in thermodynamical equilibrium. Then, the density matrix operator $\widehat{\rho}_{\phi}(t_{0})$ is given by:
$$\widehat{\rho}_{\phi}(t_{0})=\frac{1}{Z}~e^{-\beta_{\phi}\widehat{H}_{0}},$$
where $Z$ is the partition function associated to the initial field’s hamiltonian $\widehat{H}_{0}$, which can be written as:
$$\widehat{H}_{0}=\int_{0}^{+\infty}d\mathbf{p}\left(\widehat{\Pi}^{\dag}(\mathbf{p})~\widehat{\Pi}(\mathbf{p})+p^{2}~\widehat{\Phi}^{\dag}(\mathbf{p})~\widehat{\Phi}(\mathbf{p})\right).$$
In order to check that the hamiltonian is a sum of two harmonic oscillator hamiltonians for each component at fixed $\mathbf{p}$. Thus, taking $\mathbf{p}$ as a label for each pair of oscillators, we can introduce a complete set of energy eigenstates of the two dimensional (isotropic) oscillator $|n_{1},n_{2}\rangle$, writing Eq.(\[FieldWignerMomentum\]) as:
$$\begin{aligned}
\widetilde{W}_{\phi}\left[\phi_{0}(\mathbf{p}),\Pi_{0}(\mathbf{p}),t_{0}\right]=\sum_{n_{1},n_{2}}\int\mathcal{D}\varphi(\mathbf{p})&&e^{-\int_{0}^{+\infty}d\mathbf{p}\left[i\left(\Pi_{0}^{*}(\mathbf{p})~\varphi(\mathbf{p})+\Pi_{0}(\mathbf{p})~\varphi^{*}(\mathbf{p})\right)+\beta_{\phi}|\mathbf{p}|\right]}\times\nonumber\\
&&\times\Big\langle\phi_{0}(\mathbf{p})+\frac{1}{2}\varphi(\mathbf{p})\Big|n_{1},n_{2}\Big\rangle~\Big\langle n_{2},n_{1}\Big|\phi_{0}(\mathbf{p})-\frac{1}{2}\varphi(\mathbf{p})\Big\rangle.\end{aligned}$$
The eigenfunctions for the two dimensional (isotropic) harmonic oscillator are given by:
$$\begin{aligned}
\Big\langle\Phi_{R},\Phi_{I}\Big|n_{1},n_{2}\Big\rangle=\left(\frac{\alpha^{2}}{\pi~2^{n_{1}}~n_{1}!~2^{n_{2}}~n_{2}!}\right)^{1/2}H_{n_{1}}\left(\alpha~\Phi_{R}\right)~H_{n_{2}}\left(\alpha~\Phi_{I}\right)~e^{-\frac{\alpha^{2}}{2}\left(\Phi_{R}^{2}+\Phi_{I}^{2}\right)},\end{aligned}$$
where $\Phi_{R,I}$ is the real or imaginary part of the field, respectively, $H_{n}$ are the Hermite polynomials and $\alpha\equiv\left(\frac{|\mathbf{p}|}{2}\right)^{1/2}$.
Inserting this into the last expression for the Wigner functional in momentum space and using the identity for the Hermite polynomials:
$$\sum_{n}^{\infty}\frac{a^{n}}{n!}~H_{n}(x)~H_{n}(y)=\frac{1}{\sqrt{1-4a^{2}}}~e^{\frac{4axy-4a^{2}(x^{2}+y^{2})}{1-4a^{2}}},$$
which holds for $a<1/2$, condition which is satisfied in our case because $a=e^{-\beta_{\phi}|\mathbf{p}|}/2$; one finds that:
$$\begin{aligned}
\widetilde{W}_{\phi}\left[\phi_{0}(\mathbf{p}),\Pi_{0}(\mathbf{p}),t_{0}\right]&=&\int\mathcal{D}\varphi(\mathbf{p})~e^{-i\int_{0}^{+\infty}d\mathbf{p}\left(\Pi_{0}^{*}(\mathbf{p})~\varphi(\mathbf{p})+\Pi_{0}(\mathbf{p})~\varphi^{*}(\mathbf{p})-i\frac{\alpha^{2}}{4}~\varphi^{*}(\mathbf{p})~\varphi(\mathbf{p})-i\alpha^{2}~\phi_{0}^{*}(\mathbf{p})~\phi_{0}(\mathbf{p})-i\beta_{\phi}|\mathbf{p}|\right)}\nonumber\\
&&\times\prod_{\mathbf{p}}\frac{\alpha^{2}}{\pi\left(1-e^{-2\beta_{\phi}|\mathbf{p}|}\right)}~e^{\frac{\alpha^{2}~e^{-\beta_{\phi}|\mathbf{p}|}}{2\left(1-e^{-2\beta_{\phi}|\mathbf{p}|}\right)}\left[4\left(1-e^{-\beta_{\phi}|\mathbf{p}|}\right)~\phi_{0}^{*}(\mathbf{p})~\phi_{0}(\mathbf{p})-\left(1+e^{-\beta_{\phi}|\mathbf{p}|}\right)~\varphi^{*}(\mathbf{p})~\varphi(\mathbf{p})\right]}\nonumber\\
&=&C~e^{\int_{0}^{+\infty}d\mathbf{p}~\alpha^{2}~\tanh\left(\frac{\beta_{\phi}|\mathbf{p}|}{2}\right)\phi_{0}^{*}(\mathbf{p})~\phi_{0}(\mathbf{p})}\int\mathcal{D}\varphi(\mathbf{p})~e^{-i\int_{0}^{+\infty}d\mathbf{p}\left(\Pi_{0}^{*}(\mathbf{p})~\varphi(\mathbf{p})+\Pi_{0}(\mathbf{p})~\varphi^{*}(\mathbf{p})\right)}\nonumber\\
&&\times~e^{\int_{0}^{+\infty}d\mathbf{p}~\frac{\alpha^{2}}{4}~\coth\left(\frac{\beta_{\phi}|\mathbf{p}|}{2}\right)\varphi^{*}(\mathbf{p})~\varphi(\mathbf{p})}.\end{aligned}$$
where in the coefficient $C$ we have included all the terms which are not functionals of the fields and momenta.
Integrating trivially over the real and imaginary components of $\varphi(\mathbf{p})$, we arrive to the three-dimensional generalization of the Wigner functional in momentum space found in Ref. [@MrowMull] for the one-dimensional case:
$$\begin{aligned}
\widetilde{W}_{\phi}\left[\phi_{0}(\mathbf{p}),\Pi_{0}(\mathbf{p}),t_{0}\right]=C~e^{-\frac{\beta_{\phi}}{2}\int d\mathbf{p}~\widetilde{\Delta}_{\beta_{\phi}}(|\mathbf{p}|)\left[\Pi_{0}^{*}(\mathbf{p})~\Pi_{0}(\mathbf{p})+|\mathbf{p}|^{2}~\phi_{0}^{*}(\mathbf{p})~\phi_{0}(\mathbf{p})\right]},
\label{FieldWignerP3DThermal}\end{aligned}$$
where the thermal weight factor is given by:
$$\widetilde{\Delta}_{\beta_{\phi}}(|\mathbf{p}|)=\frac{2}{\beta_{\phi}|\mathbf{p}|}~\tanh\left(\frac{\beta_{\phi}|\mathbf{p}|}{2}\right).
\label{ThermalWeightMomentum}$$
This is an even function of the momentum’s absolute value, so in Eq.(\[FieldWignerP3DThermal\]) the integrals over the momentum components were extended to all the real values.
Finally, knowing the Wigner functional in momentum space, one can easily finds the Wigner functional in coordinate space by writing all the momentum’s functions as Fourier transforms of the function in coordinate space. This way, we can write an extension of the result found in [@MrowMull]:
$$\begin{aligned}
W_{\phi}\left[\phi_{0}(\mathbf{x}),\Pi_{0}(\mathbf{x}),t_{0}\right]=C'~e^{-\beta\int d\mathbf{x}\int d\mathbf{x}'~\mathcal{H}(\mathbf{x},\mathbf{x}')},
\label{FieldWignerCoordinate}\end{aligned}$$
where $C'$ is the normalization constant in coordinate space and the integrand $\mathcal{H}$ is given by
$$\begin{aligned}
\mathcal{H}(\mathbf{x},\mathbf{x}')\equiv\frac{1}{2}~\Delta_{\beta_{\phi}}(\mathbf{x}-\mathbf{x}')\left[\Pi_{0}(\mathbf{x})~\Pi_{0}(\mathbf{x}')+\nabla\phi_{0}(\mathbf{x})\cdot\nabla\phi_{0}(\mathbf{x}')\right],\end{aligned}$$
where the thermal weight factor in coordinate space is given by:
$$\begin{aligned}
\Delta_{\beta_{\phi}}(\mathbf{x}-\mathbf{x}')=\int\frac{d\mathbf{p}}{(2\pi)^{3}}~e^{-i\mathbf{p}\cdot\left(\mathbf{x}-\mathbf{x}'\right)}~\widetilde{\Delta}_{\beta_{\phi}}\left(|\mathbf{p}|\right).\end{aligned}$$
It is worth noting that, due to the interchange symmetry of the integrand, the thermal weight factor in coordinate space must be symmetric, i. e., $\Delta_{\beta_{\phi}}(\mathbf{x}'-\mathbf{x})=\Delta_{\beta_{\phi}}(\mathbf{x}-\mathbf{x}')$.
It is remarkable that although the expression of the thermal weight factor in momentum space do not change formally with the number of dimensions we are considering, on the other hand, the thermal factor in coordinate space clearly does.
[5]{} M. Antezza, L. P. Pitaevskii, S. Stringari, V. B. Svetovoy, *Phys. Rev. A* **77**, 022901 (2008). G. Bimonte, *Phys. Rev. A* **80**, 042102 (2009). E. M. Lifshitz, *Sov. Phys. JETP* **2**, 73 (1956). G. Barton, *New J. Phys.* **12**, 113045 (2010). T.G. Philbin, *New J. Phys.* **12**, 123008 (2010). F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, *Phys. Rev. A* **81**, 033812 (2010). H.P. Breuer and F. Petruccione, *The Theory of Open Quantum Systems* (Oxford University Press, New York, USA, 2002). S. Y. Buhmann and D. G. Welsch, *Prog. Quantum Electron.* **31**, 51 (2007). B. Huttner and S. M. Barnett, *Phys. Rev. A* **46**, 4306 (1992). F.C. Lombardo, F.D. Mazzitelli, A.E. Rubio Lopez, *Phys. Rev. A* **84**, 052517 (2011). D. Kupiszewska, *Phys. Rev. A* **46**, 2286 (1992). J. S. Schwinger, *J. Math. Phys.* **2**, 407 (1961). L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 \[Sov. Phys. JETP 20 (1965) 1018\]. K. c. Chou, Z. b. Su, B. l. Hao and L. Yu, *Phys. Rept.* **118**, 1 (1985). R. D. Jordan, *Phys. Rev. D* **33**, 444 (1986). S. Weinberg, *Phys. Rev. D* **72**, 043514 (2005). E.A. Calzetta and B.L. Hu, *Nonequilibrium Quantum Field Theory* (Cambridge University Press, Cambridge, 2008). W. Greiner and J. Reinhardt, *Field Quantization* (Springer-Verlag, Berlin, 1996). R.P. Feynman and A.R. Hibbs, *Quantum mechanics and path integrals* (McGraw-Hill, New York, 1965). F. Lombardo and F. D. Mazzitelli, Phys. Rev. D 53, 2001 (1996); F. C. Lombardo, F. D. Mazzitelli and R. J. Rivers, Nucl. Phys. B 672 (2003) 462; F. C. Lombardo and D. Lopez Nacir, Phys. Rev. D 72, 063506 (2005). E.A. Calzetta, A. Roura and E. Verdaguer, *Physica* **319A**, 188-212 (2003). A.O. Caldeira and A.J. Leggett, *Physica* **121A**, 587-616 (1983). B.L. Hu, J.P. Paz and Y. Zhang, *Phys. Rev. D* **45**, 2843 (1992). S. Mrówczyński and B. Müller, *Phys. Rev. D* **50**, 12, 7542-7552 (1994). D. Arteaga Barriel, *Particle Propagation in Non-trivial Backgrounds: A Quantum Field Theory Approach*, Phd. Thesis, arXiv:0707.3899 (2007). P. Ramond, *Field Theory: A Modern Primer* (Westview Press, 1990). J.L. Schiff, *The Laplace Transform: Theory and Applications* (Springer-Verlag, New York, USA, 1999). D. Kupiszewska and J. Mostowski, *Phys. Rev. A* **41**, 4636 (1990). R.E. Collins, *Field Theory of Guided Waves*, 2nd Edition (IEEE Press, New York, USA, 1990).
[^1]: arubio@df.uba.ar
[^2]: lombardo@df.uba.ar
|
---
abstract: 'The contribution of a chiral three-nucleon force to the strength of an effective spin-orbit coupling is estimated. We first construct a reduced two-body interaction by folding one-nucleon degrees of freedom of the three-nucleon force in nuclear matter. The spin-orbit strength is evaluated by a Scheerbaum factor obtained by the $G$-matrix calculation in nuclear matter with the two-nucleon interaction plus the reduced two-nucleon interaction. The problem of the insufficiency of modern realistic two-nucleon interactions to account for the empirical spin-orbit strength is resolved. It is also indicated that the spin-orbit coupling is weaker in the neutron-rich environment. Because the spin-orbit component from the three-nucleon force is determined by the low-energy constants fixed in the two-nucleon sector, there is little uncertainty in the present estimation.'
author:
- 'M. Kohno'
title: 'Strength of reduced two-body spin-orbit interaction from chiral three-nucleon force'
---
Spin-orbit field in atomic nuclei is essential to reproduce well-established single-particle shell structure. The empirical strength of the spin-orbit potential, however, has not been fully understood on the basis of the realistic nucleon-nucleon force. The possible role of intermediate isobar $\Delta$-excitation to the nuclear spin-orbit field was considered in parallel with the construction of the two-pion-exchange three-nucleon force (3NF) by Fujita and Miyazawa [@FM57]. The problem was reinvestigated in the early 1980s [@OTT80; @AB81] to search for the additional spin-orbit strength. Later, the Illinois group showed [@PP93] that their 3NF makes a substantial contribution to the spin-orbit splitting in $^{15}$N.
Kaiser and his collaborators investigated, in their several papers [@KFW; @NK03; @NK04], the nuclear spin-orbit coupling in the framework of chiral perturbation theory. The large contributions generated by iterated one-pion exchange and the 3NF almost cancel each other [@KFW; @NK03], and the short-range spin-orbit strength in the form of the effective four-nucleon contact-coupling deduced from realistic nucleon-nucleon interactions accounts well [@NK04] for the empirical one. Because the contact-interaction in the chiral perturbation, however, is still needed to be regulated for the application to low-energy nuclear structure calculations, and their arguments for various contributions seem not to be fully unified, it is worthwhile to analyze the effective strength of the spin-orbit coupling by applying the established microscopic theory, namely the lowest-order Brueckner theory, to the two-nucleon and three-nucleon interactions in the chiral effective field theory (Ch-EFT).
The Thomas form of an average single-particle spin-orbit potential has been used to describe nucleon spin-orbit coupling: $$U_{\ell s}^0 \frac{1}{r}\frac{d\rho(r)}{dr}\bfell\cdot \bfsigma,\label{eq:tf}$$ where the radial function $\rho(r)$ is a nucleon total density distribution. The relation of the strength $U_{\ell s}^0$ to a two-body effective spin-orbit interaction was derived by Scheerbaum [@SCH76b]. By defining the constant $B_S(\bar{q})$ for the triplet odd component of the effective two-body spin-orbit interaction $v_{\ell s}^{3O}(r)$ $$B_S(\bar{q})=-\frac{2\pi}{\bar{q}}\int_0^{\infty} dr r^3
j_1(\bar{q}r)v_{\ell s}^{3O}(r),$$ with $j_1$ being a spherical Bessel function, the single-particle spin-orbit potential for spin-saturated nuclei may be written as $$U_{\ell s,\tau}(r)=\frac{1}{2}B_S(\bar{q}) \frac{1}{r}
\frac{d\{\rho(r)+\rho_\tau(r)\}}{dr}\bfell\cdot \bfsigma,$$ where $\tau$ specifies either a proton or neutron. We refer to $B_S(\bar{q})$ as a Scheerbaum factor, which is different from the original constant in Ref. [@SCH76b] by a factor of $-\frac{2\pi}{3}$. Scheerbaum prescribed $\bar{q}\approx 0.7$ fm$^{-1}$ on the basis of the wavelength of the density distribution. We employ this prescription. If we assume a naive relation $\rho_p(r)=\rho_n(r)=\frac{1}{2}\rho(r)$, we recover the Thomas form, Eq. (\[eq:tf\]), with $U_{\ell s}^0=\frac{3}{4}B_S(\bar{q})$. It has also been customary to use a $\delta$-type two-body spin-orbit interaction $$iW(\bfsigma_1+\bfsigma_2)\cdot (\nabla_r\times \delta(\br)\nabla_r)$$ in nuclear Hartree-Fock calculations using $\delta$-type Skyrme interactions [@SKY75; @SKY80] and even with finite range effective forces, e.g., the Gogny force [@GP77]. This two-body force provides a single-particle spin-orbit potential: $$\frac{1}{2}W \frac{1}{r}\frac{d\{\rho(r)+\rho_\tau(r)\}}{dr}\bfell\cdot\bfsigma.$$ Therefore, the strength $W$ may be identified as the Scheerbaum factor $B_S(\bar{q})$. The empirical value of $W$ is around $120$ MeV$\cdot$fm$^5$ in various nuclear Hartree-Fock calculations. As will be shown below, the modern nucleon-nucleon interactions underestimate the spin-orbit strength by about 25 %.
Applying Scheerbaum’s formulation to the momentum-space $G$-matrix calculation in nuclear matter with the Fermi momentum $k_F$, we obtain the corresponding spin-orbit strength as follows [@FK00]: $$\begin{aligned}
B_S(\bar{q})=\frac{1}{k_F^3}\sum_{JT}(2J+1)(2T+1)\int_0^{q_{max}} dq \nonumber\\
\times W(\bar{q},q)\{ (J+2)G_{1J+1,1J+1}^{JT}(q)+G_{1J,1J}^{JT}(q) \nonumber \\
-(J-1)G_{1J-1,1J-1}^{JT}(q) \}.\end{aligned}$$ Here, $q_{max}=\frac{1}{2}(k_F+\bar{q})$ and the weight factor $W(\bar{q},q)$ is $$W(\bar{q},q)=\left\{ \begin{array}{l}
\theta (k_F-\bar{q})\;\;\mbox{for} \;\; 0\leq q\leq \frac{|k_F-\bar{q}|}{2} \\
\frac{k_F^2-(\bar{q}-2q)^2}{8\bar{q}q}\;\;\mbox{for}\;\;
\frac{|k_F-\bar{q}|}{2} \leq q\leq \frac{k_F+\bar{q}}{2},
\end{array} \right.$$ where $\theta (k_F-\bar{q})$ is a step function. In Eq. (6), $G_{1\ell',1\ell}^{JT}$ is the abbreviation of the momentum-space diagonal $G$-matrix element in the spin-triplet channel with the total isospin $T$, total spin $J$, and orbital momenta $\ell'$ and $\ell$.
Calculating $B_S(\bar{q})$ in the lowest-order Brueckner theory with the continuous prescription for intermediate spectra, as presented below explicitly in Table 1, modern two-body nucleon-nucleon potentials are found to give smaller values of around 90 Mev$\cdot$fm$^5$ compared with the empirical one. As has been well known that LOBT calculations in symmetric nuclear matter with realistic two-nucleon force do not reproduce correct saturation property. However, in most case, calculated energies at the empirical saturation point $k_F=1.35$ fm$^{-1}$ are close to the empirical energy of about $-16$ MeV. This suggests that $G$ matrices provide basic information on the effective nucleon-nucleon interaction in the nuclear medium, by incorporating important short-range correlations, Pauli effects and dispersion effects.
Now we consider the contribution of the 3NF. In this article, we estimate it in a two-step procedure. First, the 3NF $v_{123}$ defined in momentum space is reduced to an effective two-nucleon interaction $v_{12(3)}$ by folding one-nucleon degrees of freedom:
$$\langle \bk_1' \sigma_1' \tau_1', \bk_2'\sigma_2'\tau_2'|v_{12(3)}
|\bk_1 \sigma_1 \tau_1, \bk_2\sigma_2\tau_2\rangle_A
=\frac{1}{3}\sum_{\bk_3\sigma_3\tau_3} \langle \bk_1'\sigma_1'\tau_1',
\bk_2'\sigma_2'\tau_2', \bk_3\sigma_3\tau_3|v_{123}|\bk_1\sigma_1\tau_1,
\bk_2\sigma_2\tau_2, \bk_3\sigma_3\tau_3\rangle_A.\label{eq:efv}$$
Here, we have to assume that remaining two nucleons are in the center-of-mass frame, namely $\bk_1'+\bk_2'=\bk_1+\bk_2=0$. The density-dependent effective two-nucleon interaction as the effect of the 3NF has been commonly introduced in the literature [@KAT74; @FP81; @HKW10]. Note that the suffix $A$ means an antisymmetrized matrix element; namely $|ab\rangle_A\equiv |ab-ba\rangle$ and $|abc\rangle_A\equiv |abc-acb+bca-bac+cab-cba\rangle$, and the factor $\frac{1}{3}$ in Eq. (\[eq:efv\]) is an additional statistical one. This statistical factor has been often slipped in the literature. The recent derivation of the effective two-body interaction from the Ch-EFT 3NF by Holt, Kaiser and Weise [@HKW10] also seems not to be an exception. If an adjustable strength is introduced, the statistical factor may be hidden in the fitting procedure. In our case of using the Ch-EFT 3NF, the low-energy constants except for $c_D$ and $c_E$ are fixed. Although there may be a room to adjust $c_D$ and $c_E$, the contributions to the energy from these terms are rather small, if they are in a reasonable range. In addition, $c_D$ and $c_E$ do not contribute to the reduce two-nucleon spin-orbit interaction. By comparing the nuclear matter energy directly calculated from $v_{123}$ and that by the reduced $v_{12(3)}$, the error due to this approximation can be checked to be less than 10 %, if we calculate Born energy without including a form factor.
To explain the procedure of obtaining $v_{12(3)}$ more explicitly, we write the reduced spin-orbit component originating from the $c_1$ term of the Ch-EFT 3NF: $$-\frac{c_1 g_A^2 m_\pi^2}{f_\pi^4}\sum_{1\leq i<j\leq 3}
\frac{(\bfsigma_i\cdot \bq_i)(\bfsigma_j\cdot\bq_j)}
{(\bq_i^2+m_\pi^2)(\bq_j^2+m_\pi^2)}(\bftau_i\cdot\bftau_j),$$ where $g_A=1.29$, $f_\pi =92.4$ MeV, $m_\pi$ is a pion mass, and $\bq_i$ is a momentum transfer of the $i$-th nucleon. The momentum transfer of the third nucleon $k$ is dictated by the relation $\bq_k=-\bq_i-\bq_j$. The folding of the 3NF by one nucleon is carried out without incorporating a three-body form factor. A form factor is later introduced on the two-body level. The folding in symmetric nuclear matter with the Fermi momentum $k_F$ gives, besides the central and tensor components, the following spin-orbit term: $$\begin{aligned}
& & \frac{c_1 g_A^2 m_\pi^2}{f_\pi^4}\frac{1}{(2\pi)^3}\iiint_{|\bk_3|\le k_F}
d\bk_3 \nonumber\\
& & \times \frac{i(\bfsigma_1+\bfsigma_2)\cdot
(-\bk_1'\times \bk_1+(\bk_1'-\bk_1)\times\bk_3)}
{((\bk_1'-\bk_3)^2+m_\pi^2)((\bk_1-\bk_3)^2+m_\pi^2)}.\label{eq:nmc1}\end{aligned}$$ When carrying out the folding in pure neutron matter, the restriction of the isotopic spin brings about an additional factor of $\frac{1}{3}$.
The partial-wave decomposition of the above spin-orbit term becomes $$\begin{aligned}
& & -\delta_{S 1} \frac{c_1 g_A^2 m_\pi^2}{f_\pi^4}
\frac{\ell(\ell+1)+2-J(J+1)}{2\ell+1} \nonumber \\
& & \left\{ Q_{W,0}^{\ell-1}(k_1',k_1)-Q_{W,0}^{\ell+1}(k_1',k_1)
-W_{\ell s,0}^\ell (k_1',k_1)\right\}\end{aligned}$$ for the orbital and total angular momenta $\ell$ and $J$. The functions $Q_{W,0}^\ell$ and $W_{\ell s,0}^\ell$ are defined by $$\begin{aligned}
Q_{W,0}^\ell (k_1',k_1)\!&\!\equiv\!& \!\frac{2\pi}{(2\pi)^3}\frac{1}{2}
\int_0^{k_F}\!dk_3 Q_\ell (x')Q_\ell (x),\\
W_{\ell s,0}^\ell (k_1',k_1)\!&\!\equiv\!&\!\frac{2\pi}{(2\pi)^3}
\frac{1}{2k_1'k_1}\int_0^{k_F}\!dk_3k_3 \nonumber \\
& & \times\left\{ k_1'Q_\ell (x)(Q_{\ell-1}(x')-Q_{\ell+1}(x'))\right.\nonumber\\
& &\left. \!\!+ k_1Q_\ell (x')(Q_{\ell-1}(x)-Q_{\ell+1}(x))\right\},\end{aligned}$$ where $Q_\ell(x)$ is a Legendre function of the second kind, and $x'\equiv \frac{k_3^2+k_1'^2+m_\pi^2}{2k_1'k_3}$ and $x\equiv \frac{k_3^2+k_1^2+m_\pi^2}{2k_1k_3}$, respectively.
The spin-orbit component arises also from the $c_3$ term of the Ch-EFT 3NF. This case, in addition to the replacement of the coupling constant, an additional factor $(\bk_1'-\bk_3)\cdot (\bk_3-\bk_1)$ appears in the denominator in Eq. (\[eq:nmc1\]). The partial-wave decomposition reads
$$\begin{aligned}
\delta_{S 1}\frac{c_3g_A^2}{2f_\pi^4} \frac{\ell(\ell+1)+2-J(J+1)}{2\ell+1}
\left[ (m_\pi^2+\frac{1}{2}(k_1'^2+k_1^2))\{ Q_{W,0}^{\ell-1}(k_1',k_1)-
Q_{W,0}^{\ell+1}(k_1',k_1)-W_{\ell s,0}^{\ell}(k_1',k_1)\}\right.\nonumber\\
+3k_1'k_1 \left\{ Q_{W,0}^\ell (k_1',k_1)-(\ell-1)Q_{W,0}^{\ell-2}(k_1',k_1)
+(\ell +2)Q_{W,0}^{\ell+2}(k_1',k_1) +\frac{\ell-1}{2\ell-1}
W_{\ell s,0}^{\ell-1}(k_1',k_1)
+\frac{\ell+2}{2\ell+3} W_{\ell s,0}^{\ell+1}(k_1',k_1)\right\}\nonumber\\
\left. -\delta_{\ell 1} \frac{k_1'k_1}{2}
(F_0(k_1')+F_0(k_1)-F_1(k_1')-F_1(k_1)) \right],
\end{aligned}$$
where the new functions $F_0(k)$ and $F_1(k)$ are defined by $$\begin{aligned}
F_0(k) \equiv \frac{1}{(2\pi)^3}\iiint_{|\bk_3|\leq k_F}d\bk_3
\frac{1}{(\bk-\bk_3)^2+m_\pi^2},\\
F_1(k) \equiv \frac{1}{(2\pi)^3}\frac{1}{k^2}\iiint_{|\bk_3|\leq k_F}d\bk_3
\frac{\bk\cdot \bk_3}{(\bk-\bk_3)^2+m_\pi^2}.\end{aligned}$$
Adding the reduced two-nucleon interaction to the Ch-EFT two-nucleon interaction, we repeat the LOBT $G$-matrix calculation. Although explicit expressions are not shown in this Letter except for the spin-orbit part, we include all central, tensor and spin-orbit components of the reduced interaction $v_{12(3)}$. The form factor in a functional form of $f(k_1',k_1)=\exp\{-[(k_1'/\Lambda)^4+(k_1/\Lambda)^4]\}$ is introduced for $v_{12(3)}$ with the cut-off mass $\Lambda=$550 MeV. We use the low-energy constants fixed for the Jülich Ch-EFT potential by Hebeler [*et al.*]{} [@HEB]; $c_D=-4.381$, and $c_E=-1.126$. Other constants are $c_1=-0.81$ GeV$^{-1}$, $c_3=-3.4$ GeV$^{-1}$, and $c_4=3.4$ GeV$^{-1}$. Because the reduction of the 3NF to the two-nucleon force was carried out in nuclear matter, $v_{12(3)}$ may not be directly applied to very light nuclei, such as $^3$H and $^4$He.
First, we comment on calculated saturation curves, which are given in Fig. 1. Without the contribution of the 3NF, the saturation curve attains its minimum at larger $k_F$ as a function of the Fermi momentum $k_F$ than the empirical saturation momentum, as has been known. Nucleon-nucleon interactions, AV18 [@AV18], NSC97 [@NSC], and J[ü]{}lich N$^3$LO with the cutoff mass of 550 MeV [@EHM09] give similar saturation curves, and the CD-Bonn potential [@CDB] predicts somewhat deeper binding. For the reference of what saturation curve is preferable for nuclear mean filed calculations, we also show the result with the Gogny D1S interaction [@GP77].
The thin dotted curve shows the result in which the plane wave expectation value of the 3NF $v_{123}$ is added to the result of the two-nucleon N$^3$LO. The thick dotted curve alongside the thin dotted curve is the result with the plane wave expectation value of the reduced two-nucleon interaction $v_{12(3)}$. The difference between the thin and thick curves is due to the difference of the form-factors and the necessary approximation $\bk_1'+\bk_2'=\bk_1+\bk_2=0$ in Eq. (8).
The solid curve is the result of the $G$-matrix calculation with including the reduced two-nucleon interaction, $v_{12(3)}$. Although the energy is seen to be underestimated by a few MeV, the saturation property is largely improved by the repulsive contribution from the three-nucleon force. It is not necessary at present to expect a perfect agreement with the empirical properties in the LOBT calculation in nuclear matter.
{width="75mm"}
$k_F=1.35$ fm$^{-1}$ AV18 NSC97 CD-B N$^3$LO N$^3$LO+3NF
---------------------- ------ ------- ------ --------- -------------
$B_S(T=0)$ 2.09 1.9 3.1 2.5 7.0
$B_S(T=1)$ 86.4 86.7 90.2 84.6 116.2
$B_S(\bar{q})$ 88.4 88.6 93.3 87.1 123.2
$k_F=1.07$ fm$^{-1}$ AV18 NSC97 CD-B N$^3$LO N$^3$LO+3NF
$B_S(T=0)$ 1.4 1.3 2.3 1.6 4.1
$B_S(T=1)$ 88.1 88.7 92.2 86.5 106.7
$B_S(\bar{q})$ 89.5 90.0 94.5 88.1 110.8
: $B_S(\bar{q})$ in the unit of MeV$\cdot$fm$^5$ given by Eq. (6) with $\bar{q}=0.7$ fm$^{-1}$ for modern nucleon-nucleon interaction: AV18 [@AV18], NSC97 [@NSC], CD-Bonn [@CDB], and J[ü]{}lich N$^3$LO [@EHM09]. The last entry is the result with including the reduced two-body interaction from the Ch-EFT 3NF.
Now we examine the spin-orbit strength. We tabulate values for $B_S(\bar{q})$ of Eq. (6) at $\bar{q}=0.7$ fm$^{-1}$ calculated in the LOBT with modern nucleon-nucleon interactions: AV18 [@AV18], NSC97 [@NSC], CD-Bonn [@CDB], and J[ü]{}lich N$^3$LO [@EHM09]. The Scheerbaum factors obtained by realistic two-nucleon forces are seen to be similar but insufficient to explain the strength needed in nuclear mean field calculations. Namely only about three-fourths of the empirically needed strength is accounted for. The two-body part of the Ch-EFT, N$^3$LO, shows little difference with other realistic two-nucleon force. It is also noticed that values at $k_F=1.07$ fm$^{-1}$, namely at the half of the normal density, change little from those at the normal density with $k_F=1.35$ fm$^{-1}$. It turns out, as the last column of Table I shows, that the addition of the reduced two-body interaction from the Ch-EFT 3NF bring about a good effect to fill the gap, though the 3NF contribution is smaller at $k_F=1.07$ fm$^{-1}$. This is in accord with the important role of the 3NF to the spin-orbit splitting demonstrated in quantum Monte Carlo calculations of low-energy neutron-alpha scattering [@NOL]. Although there are ambiguities from the form factor and uncertainties inherent in the folding procedure without taking into account nucleon-nucleon correlations, no additional adjustable parameter exists, because low-energy constants $c_1$ and $c_3$ which contribute solely to the spin-orbit strength are determined on the two-nucleon sector.
As noted after Eq. (\[eq:nmc1\]), the reduced two-body spin-orbit term in neutron matter is one-third of that in symmetric nuclear matter. Actual $G$-matrix calculations using the Ch-EFT N$^3$LO plus $v_{12(3)}$ in pure neutron matter with $k_F^n=1.35$ fm$^{-1}$ tell that $B_S(\bar{q})$ values at $\bar{q}=0.7$ fm$^{-1}$ are 84.7 and 93.5 MeV$\cdot$fm$^5$ without and with the reduced two-nucleon interaction $v_{12(3)}$, respectively. If $k_F^n=1.07$ fm$^{-1}$ is assumed, the corresponding values are 87.0 and 94.6 MeV$\cdot$fm$^5$, respectively. Again, the $k_F^n$-dependence is weak. While the spin-orbit strength from the two-nucleon force is scarcely different from that in symmetric nuclear matter, the additional contribution from the three-nucleon force is in fact almost one-third of that in symmetric nuclear matter. Thus, the spin-orbit strength is expected to be smaller in the neutron-rich environment. This seems to be consistent with the trend observed in the shell structure near the neutron drip line [@SCH04] that a decreasing spin-orbit interaction is preferable with increasing neutron excess.
In summary, we have estimated quantitatively the contribution of the three-nucleon force of the chiral effective field theory to the single-particle spin-orbit strength, using the formulation by Scheerbaum [@SCH76b]. We first introduced the reduced two-body interaction by folding one-nucleon degrees of freedom of the 3NF in nuclear matter. Making partial-wave expansion of the resulting two-body interaction and adding it to the genuine two-nucleon interaction with including the necessary statistical factor of $\frac{1}{3}$, we carried out LOBT $G$-matrix calculations in infinite matter and evaluated the Scheerbaum factor corresponding to the spin-orbit strength. Because the spin-orbit field in the atomic nuclei is fundamentally important as the nuclear magic numbers exhibit, it is important to learn that the inclusion of the 3NF in the chiral effective field theory can account for the spin-orbit strength empirically required for nuclear mean filed calculations. Because the relevant low-energy constants $c_1$ and $c_3$ are determined in the two-nucleon interaction sector, there should be little uncertainty for the additional spin-orbit strength except for the treatment of the two-body form factor. We have also noted that the additional spin-orbit strength from the 3NF should be weaker in neutron-excess nuclei.
This work is supported by Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (Grant No. 22540288). The author thanks H. Kamada for valuable comments concerning the Ch-EFT interaction. He is also grateful to M. Yahiro for his interest in this work.
[20]{} J. Fujita and H. Miyazawa, Prog. Theor. Phys [**17**]{}, 366 (1957). K. Ohta, T. Terasawa, and M. Tohyama, Phys. Rev. C [**22**]{}, 2233 (1980). K. Andō and H. Bandō, Prog. Theor. Phys [**66**]{}, 227 (1981). S.C. Pieper and V.R. Pandharipande, Phys. Rev. Lett. [**70**]{}, 2541 (1993). N. Kaiser, S. Fritsch, and W. Weise, Nucl. Phys. [**A724**]{}, 47 (2003). N. Kaiser, Phys. Rev. [**C68**]{}, 054001 (2003). N. Kaiser, Phys. Rev. [**C70**]{}, 034307 (2004). R.R. Scheerbaum, Nucl. Phys. [**A257**]{}, 77 (1976). M. Beiner, H. Flocard, N. Van Giai, and P. Quentin, Nucl. Phys. [**A238**]{}, 29 (1975). H. Krivine, J. Treiner, and O. Bohigas, Nucl. Phys. [**A336**]{}, 155 (1980). D. Gogny and R. Padjen, Nucl. Phys. [**293**]{}, 365 (1977). Y. Fujiwara, M. Kohno, T. Fujita, C. Nakamoto, and Y. Suzuki, Nucl. Phys. [**A674**]{}, 493 (2000). T. Kasahara, Y. Akaishi, and H. Tanaka, Prog. Theor. Phys. Suppl. [**56**]{}, 96 (1974). B. Friedman and V.R. Pandharipande, Nucl. Phys. [**A361**]{}, 502 (1981). J.W. Holt, N. Kaiser, and W. Weise, Phys. Rev. C [**81**]{}, 024002 (2010). K. Hebeler, S.K. Bogner, R.J. Furnstahl, A. Nogga, and A. Schwenk, Phys. Rev. [**C83**]{}, 031301(R) (2011). R.B. Wiringa, V.G.J.Stoks, and R. Schiavilla, Phys. Rev. [**C51**]{}, 38 (1995). T. A. Rijken, V. G. J. Stoks, and Y. Yamamoto, Phys. Rev. C [**59**]{}, 21 (1999). E. Epelbaum, W. Göckle, and U.-G. Mei[ß]{}ner, Nucl. Phys. [**A747**]{}, 362 (2005). R. Machleidt, Phys. Rev. [**C63**]{}, 024001 (2001). K.M. Nollett, S.C. Pieper, R.B. Wiringa, J. Carlson, and G.M. Hale, Phys. Rev. Lett. [**99**]{}, 022502 (2007). J.P. Schiffer [*et al.*]{}, Phys. Rev. Lett. [**92**]{}, 162501 (2004).
|
---
abstract: 'The cops and robbers game has been extensively studied under the assumption of optimal play by both the cops and the robbers. In this paper we study the problem in which cops are chasing a *drunk* robber (that is, a robber who performs a random walk) on a graph. Our main goal is to characterize the “cost of drunkenness.” Specifically, we study the ratio of expected capture times for the optimal version and the drunk robber one. We also examine the algorithmic side of the problem; that is, how to compute near-optimal search schedules for the cops. Finally, we present a preliminary investigation of the *invisible* robber game and point out differences between this game and *graph search*.'
address:
- 'Department of Mathematics, Physics and Computer Sciences, Aristotle University of Thessaloniki, Thessaloniki GR54124, Greece'
- 'Department of Mathematics, Ryerson University, Toronto, ON, Canada, M5B 2K3'
author:
- Athanasios Kehagias
- 'Pawe[ł]{} Pra[ł]{}at'
title: Some Remarks on Cops and Drunk Robbers
---
Introduction {#sec01}
============
The game of *Cops and Robbers*, introduced independently by Nowakowski and Winkler [@nw] and Quilliot [@q] almost thirty years ago, is played on a fixed undirected, simple, and finite graph $G$. There are two players, a team of $k$ *cops*, where $k\geq1$ is a fixed integer, and the *robber*. In the first round of the game, the cops occupy any set of $k$ vertices and then the robber chooses a vertex to start from; in the following rounds, first the cops and then the robber move from vertex to vertex, following the edges of $G$. More than one cop is allowed to occupy a vertex, and the players may remain on their current positions. At every step of the game, both players know the positions of all cops and the robber. The cops win if they capture the robber; that is, if at least one of cop eventually occupies the same vertex as the robber; the robber wins if he can avoid being captured indefinitely. The players are *adversarial*; that is, they play optimally against each other. Since placing a cop on each vertex guarantees that the cops win, we may define the *cop number*, written $c(G)$, to be the minimum number of cops needed to win on $G$. The cop number was introduced by Aigner and Fromme in [@af].
In this paper we study a new version of the game, in which the robber is *drunk*; that is, he performs a random walk on $G$. The cops are assumed to follow a strategy which is optimal with respect to the robber’s random behavior. This version was proposed by D. Thilikos during the 4th Workshop on GRAph Searching, Theory and Applications (GRASTA 2011) and he specifically asked the following question: “what is the *cost of drunkenness*?” In other words, how much faster than the adversarial robber is the drunk one captured? We try to answer various versions of this question. In addition, we study some algorithmic questions; for example, how to compute the expected capture time for an optimal strategy of cops.
There is a large bibliography on pursuit games on graphs. The reader interested in cops and robbers can start by perusing the surveys [@al; @ft; @h] and the recent book [@NowBook]. To the best of our knowledge, the problem of a drunk robber has not been previously studied in the cops and robbers literature. However there is a strong connection to the *Markov Decision Processes* (MDP) literature; we will comment on this connection (and use it) in Section \[sec06\]. The reader can refer to [@MDPOnline; @MDP03; @MDP02] for MDP surveys.
While the emphasis of the current paper is on cops chasing the visible robber, we also touch briefly the case of *invisible* robber, both adversarial and drunk. Not much has been written on this problem, but a related problem which has been extensively studied is the *Graph Search* problem, where a team of searchers try to locate in a graph an *invisible* fugitive, who is also assumed to be *arbitrarily fast* and *omniscient* (he always knows the searchers’ locations as well as their strategy). A recent comprehensive review of graph search appears in [@ft]. We emphasize that the graph search problem is similar but *not* identical to cops chasing an invisible robber.
The paper is structured as follows. In Section \[sec02\] we present definitions and our notation; the formulation is, naturally, probabilistic. In particular, we define the cost of drunkenness to be the ratio of the capture time for the adversarial robber and the *expected* capture time for the drunk robber. We also present a number of lemmas which we will repeatedly use in the following sections. In Section \[sec03\] we obtain bounds on the cost of drunkenness for various special families of graphs; for example, paths, cycles, grids, and complete $d$-ary trees. In Section \[sec04\] we look at the problem more generally and show that, for any $c\in[1,\infty)$, there is a graph for which the cost of drunkenness is arbitrarily close to $c$. In Section \[sec06\] we connect the cops and drunk robber problem to *Markov Decision Processes* (MDP); that is, Markov chains with a *control input* which can modify the transition probabilities. MDP’s provide a natural language for the problem; in particular they are useful in the computation of optimal cop strategies; that is, strategies which minimize the expected robber capture time. We then use the MDP machinery to present algorithms which compute the optimal cop strategy for a given graph and a drunk robber. In Section \[sec07\] we give a brief, preliminary discussion of the cost of drunkenness for an *invisible* robber. Finally, in Section \[sec08\] we list possible future research directions.
Preliminaries {#sec02}
=============
Definitions
-----------
Let $G=(V,E)$ be a fixed undirected, simple, and finite graph. Since the game played on a disconnected graph can be analyzed by investigating each component separately, we assume that $G$ is connected. We will use the following notation and assumptions.
1. There are $k$ cops (for the time being we assume $k\geq c(G)$ but this assumption will be relaxed in later sections).
2. $X_{t}^{i}$ denotes the position of the $i$-th cop at time $t$ ($i \in\{1,2,\ldots,k\}$, $t \in\{0,1,2, \ldots\}$); $X_{t}=(X_{t}^{1},\ldots,X_{t}^{k})$ denotes the vector of all cop positions at time $t$; $\mathbf{X}=( X_{0},X_{1},X_{2},\ldots)$ denotes the positions of all cops during the game ($\mathbf{X}$ may have finite or infinite length).
3. $Y_{t}$ denotes the position of the robber at time $t$ and $\mathbf{Y}=(Y_{0},Y_{1},Y_{2}, \ldots)$ the positions of the robber during the game. (Let us note that there is a correlation between $\mathbf{X}$ and $\mathbf{Y}$; that is, players adjust their strategies observing moves of the opponent.)
4. The moving sequence is as follows: first the cops choose initial positions $X_{0} \in V$, then the robber chooses $Y_{0} \in V$. For $t\in\{1,2,\ldots\}$ first the cops choose $X_{t}$ and then the robber chooses $Y_{t}$. Players use edges of the graph $G$ to move from vertex to another one; that is, $\{X_t^i, X_{t+1}^i\} \in E$ for $i \in\{1,2,\ldots,k\}$ and $t \in\{0,1,2, \ldots\}$, and $\{Y_t, Y_{t+1}\} \in E$ for $t \in\{0,1,2, \ldots\}$.
5. The *capture time* is denoted by $T$ and defined as follows $$T=\min\{ t : \exists i \text{ such that }X_{t}^{i}=Y_{t} \};$$ that is, it is the first time a cop is located at the same vertex as the robber (note that this can happen either after the cops move or after the evader moves). Note that $T<\infty$, since $k\geq c(G)$ and $c(G)$ cops can capture the adversarial robber (and so, of course, the drunk one too).
Assuming for the moment adversarial cops *and* robber, and given initial cop positions $x\in V^{k}$ and robber position $y\in V$, we let $\mathrm{ct}_{x,y}(G,k)=T$. The *$k$-capture time* is defined as follows: $$\mathrm{ct}(G,k) =\min_{x\in V^{k}}\max_{y\in V}\mathrm{ct}_{x,y}(G,k).$$ In other words, we allow our perfect players to choose their initial positions in order to achieve the best outcome. Finally, when $k=c(G)$ we simply write $\mathrm{ct}(G)$ instead of $\mathrm{ct}(G,c(G))$, and call it the *capture time* instead of $c(G)$-capture time. Let us stress one more time that the above quantities are defined under the assumption of *optimal play by both players*.
Next let us assume that the cops are adversarial but the robber is *drunk*. More specifically, we assume the robber performs a random walk on $G$. Given that he is at vertex $v\in V$ at time $t$, he moves to $u\in N(v)$ at time $(t+1)$ with probability equal to $1/|N(v)|$. Note that we do *not* include $v$ in $N(v)$; that is, we consider open, not closed, neighbourhoods. Moreover, the robber probability distribution does not depend on current position of cops; in particular, it can happen that the robber moves to a vertex occupied by a cop (something the adversarial robber would never do).
Under the above assumptions, the drunk robber game is actually a one-player game and, for given initial configuration and cops strategy, the capture time $T$ is a random variable. For any $x \in V^k$ and $y \in V$, let $$\mathrm{dct}_{x,y}\left( G,k\right) = \mathbb{E}\left( T~~|~~X_{0}=x,Y_{0}=y,\text{$k$ cops are used optimally}\right);$$ in other words, it is the expected capture time given initial cops and robber configurations $x$, $y$ and optimal play by the $k$ cops.
Since the robber is drunk, we cannot expect him to choose the most suitable vertex to start with—instead, he chooses an initial vertex uniformly at random. Cops are, of course, aware of this and so they try to choose an initial configuration so that the expected length of the game is as small as possible. Hence, we define the expected $k$-capture time as follows: $$\mathrm{dct}\left( G,k\right) =\min_{x\in V^{k}} \sum_{y\in V} \frac{\mathrm{dct}_{x,y}\left( G,k\right) }{\left\vert V\right\vert }.$$ As before, $\mathrm{dct}(G)=\mathrm{dct}(G,c(G))$. We define the *cost of drunkenness* as follows $$F(G)=\frac{\mathrm{ct}(G)}{\mathrm{dct}(G)}$$ and we obviously have $F(G)\geq1$.
While we concentrate on the case $k=c(G)$, it is also natural to consider expected capture time $\mathrm{dct}(G,k)$ for $k\neq c(G)$. The next theorem shows that this is well defined for any $k\geq1$ (in particular, even for $k<c(G)$).
\[thm:dct\_finite\] $\mathrm{dct}(G,k)<\infty$ for any connected graph $G$ and $k\geq1$.
Let $G=(V,E)$ be any connected graph, $D=D(G)$ be the diameter of $G$, and $\Delta=\Delta(G)$ be the maximum degree of $G$. Fix any vertex $v\in V$, place $k$ cops on $v$, and let $X_{t}^{i}=v$ for all $i$ and $t$ (that is, cops never move; this is clearly a suboptimal strategy). For a given vertex $y\in V$ occupied by the drunk robber, the probability that he uses a shortest path from $y$ to $v$ to move straight to $v$ is at least $(1/\Delta)^{D}$. This implies that, regardless of the current position of the robber at time $t$, the probability that he will be caught after at most $D$ further rounds is at least $\varepsilon= (1/\Delta)^{D}$. Moreover, corresponding events for times $t+iD$, $i\in{\mathbb{N}}\cup\{0\}$ are mutually independent. Thus, we get immediately that $$\begin{aligned}
\mathbb{E}T & =\sum_{t\geq0}\mathbb{P}(T>t)~~\leq~~\sum_{t\geq0} \mathbb{P} \left( T> \left\lfloor \frac{t}{D} \right\rfloor D \right) \nonumber\\
& =\sum_{i\geq0} D \cdot\mathbb{P}(T>iD)~~\leq~~D \sum_{i\geq0} (1-\varepsilon)^{i}~~=~~\frac{D}{\varepsilon}~~=~~D \Delta^{D}~~<~~\infty,\end{aligned}$$ and we are done.
Let us remark that sharper bounds can be obtained for the capture time of a drunk robber, even in the case that the cops are also drunk; for example see [@Winkler]. However, Theorem \[thm:dct\_finite\] will be sufficient for our needs.
Some Useful Lemmas
------------------
We will be using the following version of a well-known Chernoff bound many times so let us state it explicitly.
\[lem:Chernoff\] Let $X$ be a random variable that can be expressed as a sum $X=\sum_{i=1}^{n} X_{i}$ of independent random indicator variables where $X_{i}\in\mathrm{Be}(p_{i})$ with (possibly) different $p_{i}=\mathbb{P} (X_{i} = 1)= \mathbb{E}X_{i}$. Then the following holds for $t \ge0$: $$\begin{aligned}
\mathbb{P} (X \ge\mathbb{E }X + t) & \le\exp\left( - \frac{t^{2}}{2(\mathbb{E }X+t/3)} \right) ,\\
\mathbb{P} (X \le\mathbb{E }X - t) & \le\exp\left( - \frac{t^{2}}{2\mathbb{E }X} \right) .\end{aligned}$$ In particular, if $\varepsilon\le3/2$, then $$\begin{aligned}
\mathbb{P} (|X - \mathbb{E }X| \ge\varepsilon\mathbb{E }X) & \le2 \exp\left( - \frac{\varepsilon^{2} \mathbb{E }X}{3} \right) .\end{aligned}$$
Let us now consider the following (simple) random walk on ${\mathbb{Z}}$. Understanding the behaviour of this Markov chain will be important in investigating simple families of graphs later. Let $X_{0}=0$, and for a given $t\geq0$, let $$X_{t+1}=
\begin{cases}
X_{t}+1 & \text{with probability }1/2\\
X_{t}-1 & \text{otherwise.}
\end{cases}$$ It is known that with high probability, random variable $X_{t}$ stays relatively close to zero. We make this precise below using the Chernoff bound.
\[lem:walk\] Let $n \in{\mathbb{N}}$ and $c \in(2, \infty)$. For a simple random walk $(X_{t})$ on ${\mathbb{Z}}$ with $X_{0} = 0$ we have that $|X_{t}|\le c \sqrt{n \log n}$ for every $t \in\{0, 1, \dots, n\}$ with probability at least $1-2n^{1-c^{2}/4}$.
Fix $n\in{\mathbb{N}}$ and $c\in(2,\infty)$. Let us perform $n$ steps of a simple random walk on ${\mathbb{Z}}$ starting with $X_{0}=0$. Let $Y_{t}$ ($1\leq t\leq n$) denote the number of times the process goes ‘up’ until time $t$. It is clear that $\mathbb{E}Y_{t}=t/2$ and $$X_{t}=Y_{t}-(t-Y_{t})=2(Y_{t}-t/2).$$ For a given $t$, it follows from Chernoff bound (Lemma \[lem:Chernoff\]) that $$\begin{aligned}
\mathbb{P}\left( X_{t}<-c\sqrt{n\log n}\right) & =\mathbb{P}\left( Y_{t}\leq\frac{t}{2}-\frac{c}{2}\sqrt{n\log n}\right) \\
& \leq\exp\left( -\frac{(c\sqrt{n\log n}/2)^{2}}{2(t/2)}\right) \\
& \leq\exp\left( -\frac{c^{2}}{4}\log n\right) =n^{-c^{2}/4}.\end{aligned}$$ A symmetric argument can be used to get that $X_{t}>c\sqrt{n\log n}$ with probability at most $n^{-c^{2}/4}$. Finally, from a union bound we get that the probability that there exists $t$ ($1\leq t\leq n$) with $|X_{t}|>c\sqrt{n\log n}$ is at most $n\cdot2n^{-c^{2}/4}=2n^{1-c^{2}/4}$.
Bounds on the Cost of Drunkenness {#sec03}
=================================
In this section we place upper and lower bounds on the cost of drunkenness $F(G)$ when $k$ cops are available. We emphasize the case $k=c(G)$ but also consider values of $k \neq c(G)$. We start with simple graphs (namely: paths, cycles, trees, and grids) in order to prepare for slightly more complicated families in the next section.
Paths and a Suboptimal Strategy
-------------------------------
In this subsection we play the game on $P_{n}$, a path on $n$ vertices ($V(P_{n})=\{0,1,\dots,n-1\}$, $E(P_{n})=\{\{i-1,i\}:i\in\{1,2,\dots ,n-1\}\}$). Clearly, $c(P_{n})=1$; that is, one cop can catch the adversarial robber. Since the drunk robber is easier to catch than the adversarial one, let us study the drunk robber playing against a single cop.
In this subsection we will compute the expected capture time using a *suboptimal* strategy, namely starting the cop at $X_0=0$ and moving him to the other end until he reaches $n-1$ (or until capture takes place). It is clear that this strategy achieves capture; furthermore (as will become apparent in the following sections) many optimal strategies can be analyzed using this suboptimal one.
Let $Z_{t}=Y_{t}-X_{t}$ be the distance between players at time $t$. If the drunk robber starts at vertex $k\in\{0,1,\dots,n-1\}$, we have $Z_{0}=Y_{0}=k$. (In order to simplify the argument, we allow players to “pass each other” which is never the case in the real game; that is, $Z_{t}$ can be negative.) We can redefine the capture time as $$T_n=T_n(k)=\min\{t:Z_{t}\leq0\}.$$ Now, it is not so difficult to see the behaviour of the sequence $(Z_{t})_{t \ge 0}$. Note that at time t, the maximum distance between players is $n-1-t$ which implies that the robber will be caught in at most $n-1$ steps. We have the following Markov chain to investigate: for $t\in\{0,1,\dots,n-2\}$, if $Z_{t}<n-1-t$, then $$Z_{t+1}=
\begin{cases}
Z_{t}-2 & \text{with probability 1/2 (the robber goes toward the cop)}\\
Z_{t} & \text{with probability 1/2 (the robber goes away from the cop)}.
\end{cases}$$ If $Z_{t}=n-1-t$ (that is, the robber occupies the end of the path), then $Z_{t+1}=Z_{t}-2$ (deterministically).
Consider another Markov chain $Z_{t}^{\prime}$, which has the following simple behaviour: $Z_{0}^{\prime}=k$ and for every $t\geq0$, $Z_{t+1}^{\prime}=Z_{t}^{\prime}-2$ with probability 1/2; otherwise $Z_{t+1}^{\prime}=Z_{t}^{\prime}$. Define $T^{\prime}=\min\{t:Z_{t}\leq0\}$. In other words, we will be chasing the robber on the infinite ray $R$ ($V(R)={\mathbb{N}} \cup\{0\}$, $E(R)=\{\{i-1,i\}:i\in{\mathbb{N}}\}$), which is slightly more difficult for the cop. Hence, it is easy to prove that $\mathbb{E}(T_n~~|~~Z_{0}=k) \leq\mathbb{E}(T^{\prime}~~|~~Z_{0}^{\prime}=k) $. Moreover, it is also easy (using a recursive argument) to show that $\mathbb{E}(T^{\prime}~~|~~Z_{0}^{\prime}=k)=k$, and so $\mathbb{E}(T_n(k)) \leq k$. Now we are ready to show the following.
\[thm:dst\_pn\] Consider that the cop starts on one end of the path $P_{n}$ and moves toward the other end. Let $T_{n}$ be the capture time, provided that the robber is drunk. Then, $$\frac{n}{2} \left( 1-O\left( \frac{\log n}{n}\right) \right)~~\leq~~\mathbb{E}T_{n}~~\leq~~\frac{n-1}{2}.$$
Before we move to the proof of this theorem let us mention that, in fact, with a slightly more sophisticated argument, it is possible to show that $\mathbb{E}T_n=n/2-O(1)$.
Let $n \in{\mathbb{N}}$ and fix any $c>2$. The robber starts his walk on a vertex $k \in\{0, 1, \dots, n-1\}$. Let us note that he is captured after at most $n-1$ steps of the process (deterministically); that is, $T_n(k) \le n-1$. As we already mentioned $\mathbb{E }T_n(k) \le k$. Since the starting vertex for the robber is chosen uniformly at random, we get that $\mathbb{E }T_n \le \sum_{k=0}^{n-1} k/n = (n-1)/2$, so it remains to investigate a lower bound.
Suppose first that $k \le(n-1) - c \sqrt{n \log n}$. It follows from Lemma \[lem:walk\] that the robber reaches the other end of the path with probability at most $2n^{1-c^{2}/4}$. If this is the case, we apply a trivial lower bound for $T_n(k)$, namely, $T_n(k) \ge0$; otherwise we get that the (conditional) expectation for $T_n(k)$ is equal to $k$. Hence, $\mathbb{E }T_n(k) \ge k (1-2n^{1-c^{2}/4})$. Suppose now that $k > (n-1) - c \sqrt{n \log n}$. Using Lemma \[lem:walk\] one more time, we get that with probability at least $1-2n^{1-c^{2}/4}$ the robber is not caught before time $k - c \sqrt{n \log n}$.
Since the starting vertex for the robber is chosen uniformly at random, we get that $$\begin{aligned}
\mathbb{E }T_n & \ge\frac{1}{n} \sum_{k=0}^{n-1} \mathbb{E }T_n(k)\\
& \ge\frac{1}{n} \left( \sum_{k=0}^{n-1-c \sqrt{n \log n}} k + \sum_{k=n-c \sqrt{n \log n}}^{n-1} (k - c \sqrt{n \log n}) \right) (1-2n^{1-c^{2}/4})\\
& \ge\left( \frac{n-1}{2} - c^{2} \log n \right) (1-2n^{1-c^{2}/4}).\end{aligned}$$ For a given $n$, the parameter $c$ can be adjusted for the best outcome. To get an asymptotic behaviour, we can use, say, $c=3$ to get that $$\mathbb{E }T_n \ge\frac{n}{2} \left( 1 - O \left( \frac{\log n}{n} \right) \right) ,$$ and the proof is complete.
The proof of the theorem actually gives us more. We get that with probability tending to 1 as $n\rightarrow\infty$, for all starting points for the robber ($k \in\{0, 1, \dots, n-1\}$), the cop needs $k+O(\sqrt{n\log n})$ moves to catch the robber.
Paths
-----
We continue studying a visible robber on $P_{n}$ but we now apply the optimal capture strategy (it is optimal for both adversarial and drunk robber). If $n$ is odd, we start by placing a cop on vertex $(n-1)/2$; if $n$ is even we have two optimal strategies, the cop can start on $n/2$ or $n/2-1$. In any case, after selecting an initial vertex the strategy is the same: the cop keeps moving toward the robber. Except for initial placement, this is the strategy examined in the previous subsection and we have $\mathrm{ct}(P_{n})=\lfloor n/2\rfloor$. We easily get the following result.
$$\frac{n}{4}\left( 1-O\left( \frac{\log n}{n}\right) \right) ~~\leq~~\mathrm{dct(P_{n})~~\leq~~\frac{n}{4}.}$$ In particular, $\mathrm{dct}(P_{n})=(1+o(1))n/4$ and the cost of drunkenness is $$F (P_{n}) = \frac{\mathrm{ct}(P_{n})}{\mathrm{dct}(P_{n})}=2+o(1).$$
As we already mentioned, after the robber selects his initial vertex to start from, the game is played essentially on a path of length at most $\lfloor n/2\rfloor+1$. From Theorem \[thm:dst\_pn\], we get immediately that $$\mathrm{dct}(P_{n})\leq\mathbb{E} T_{\lfloor n/2\rfloor+1} \leq n/4.$$ For a lower bound, we notice that the length of each subpath is at least $\lfloor n/2\rfloor$. By Theorem \[thm:dst\_pn\], $$\mathrm{dct}(P_{n}) \geq\mathbb{E} T_{\lfloor n/2\rfloor} \geq\frac{n}{4}\left( 1-O\left( \frac{\log n}{n}\right) \right) ,$$ and the proof is complete.
In the general case when $k \in\mathbb{N}$ cops are available, we need to ‘slice’ a path into $k$ shorter paths and place a cop on their centers. We get that $\mathrm{dct}(P_{n},k)=(1+o(1))n/(4k)$.
Cycles
------
Let us play the game on a cycle $C_{n}$ for $n\geq4$ ($V(C_{n})=\{1,2,\dots,n\}$, $E(C_{n})=\{\{i,i+1\}:i\in\{1,2,\dots,n-1\}\}\cup\{\{1,n\}\}$). It is not difficult to see that $c(C_{n})=2$; we use two cops to chase the robber. They start by occupying two vertices at the distance $\lfloor(n+1)/2\rfloor$, the maximum possible distance on the cycle. When the robber selects his vertex to start with, they move toward him and capture occurs at time $\mathrm{ct}(C_{n})=\lfloor(n+1)/4\rfloor$. The same strategy is used when the robber is drunk.
As for paths, one can introduce a random variable $Z_{t}$ to measure the distance between the robber and cops at time $t$. The problem (almost) reduces to the problem on a path. We mention briefly the difference below but the formal proof is omitted. If $n$ is odd, then $Z_{t}$ has exactly the same behaviour as before. However, $Z_{0}=\lfloor(n+1)/2\rfloor$ with probability two times smaller than any other legal starting value (note that a uniform distribution on $V(C_{n})$ is used but there is just one vertex at the distance $\lfloor(n+1)/2\rfloor$). If $n$ is even, then we get a uniform distribution for starting values but the transition from $Z_{t}$ to $Z_{t+1}$ is slightly different, namely, there is a chance for $Z_{t}$ to stay at the same value, provided that the robber occupies the vertex which is at the maximum distance from cops. In any case, it is straightforward to show that both upper and lower bounds still hold so we get the following.
\[thm:dct\_cn\] $$\frac{n}{8}\left( 1-O\left( \frac{\log n}{n}\right) \right) ~~\leq~~\mathrm{dct}(C_{n})~~\leq~~\frac{n+1}{8}.$$ In particular, $\mathrm{dct}(C_{n})=(1+o(1))n/8$ and the cost of drunkenness is $$F(C_{n}) = \frac{\mathrm{ct}(C_{n})}{\mathrm{dct}(C_{n})}=2+o(1).$$
In the general case when $k \in\mathbb{N}$ cops are available, we spread them as evenly as possible. We get that $\mathrm{dct}(C_{n},k)=(1+o(1))n/(4k)$.
Trees
-----
All families of graphs we discussed so far have a very nice property, namely, it is clear what the optimal strategy for the cops is. Once players fix their initial positions (that is, $X_{0}$ and $Y_{0}$), cops must move toward the robber in order to decrease the expected capture time. As we mentioned before, it is natural to measure the distance $Z_{t}$ between players at time $t$; $Z_{t}$ decreases by 2 if the robber makes a bad move or is occupying a leaf; otherwise the distance remains the same. This applies to the family of trees as well (note that $c(T)=1$ for any tree $T$). However, this time it is not clear which vertex should be used for the cop to start with in order to optimize the expected capture time. For this family, the random variable $Z_{t}$ decreases with probability $1/\deg(v)$, provided that the robber occupies vertex $v$, and the behaviour of the sequence $(Z_{t})_{t \ge 0}$ highly depends not only on the degree distribution but on the structure of a tree as well. It is non-trivial to estimate the cost of drunkenness for a particular tree without performing extensive calculations for every vertex as a starting point (these calculations can be performed by computer, using the algorithms of Section \[sec0602\]). However, some sub-families of trees are still relatively easy to deal with.
Let us consider $d$ regular, rooted tree $T(d,k)$ of depth $k$. The root vertex on the level 0 has $d$ neighbours (children), vertices on levels 1 to $k-1$ have degree $d+1$ (one parent and $d$ children), leaves on the level $k$ have degree 1 (just one parent). There are $d^{i}$ vertices on level $i$ for a total of $(d^{k+1}-1)/(d-1)$ vertices. Due to the symmetry, the cop must start the game on the root. Since the drunk robber prefers to move toward leaves, it is natural to expect that his behaviour is similar to the one of the adversarial robber. Moreover, almost all vertices are located on levels $k-o(k)$ so the robber almost always starts on these vertices which is clearly a good move. We show that the cost of drunkenness is as best as possible; that is, $\mathrm{dct}(T(d,k))$ is tending to $\mathrm{ct}(T(d,k))=k$ as $k \to\infty$.
\[thm:dct\_tdk\] $$k - O(\sqrt{k \log k}) ~~\leq~~\mathrm{dct}(T(d,k))~~\leq~~k.$$ In particular, $\mathrm{dct}(T(d,k))=(1+o(1))k$ and the cost of drunkenness is $$F(T(d,k)) = \frac{\mathrm{ct}(T(d,k))}{\mathrm{dct}(T(d,k))}=1+o(1).$$
Suppose that the drunk robber starts on level $i \ge k - \sqrt{k \log k}$. It follows from Lemma \[lem:Chernoff\] that with probability $1-O(k^{-1})$ he will be caught on level $k - O(\sqrt{k \log k})$. (In fact, it is also true for $i \ge k/d$, since the robber moves toward leaves with higher rate, namely, with probability $(d-1)/d$. However, an error following from this part is negligible comparing to the other error, so we stay with this obvious bound for $i$.) Therefore, $$\begin{aligned}
\mathrm{dct}(T(d,k)) & \ge\sum_{i=k - \sqrt{k \log k}}^{k} \frac{d^{i}}{(d^{k+1}-1)/(d-1)} (k - O(\sqrt{k \log k})) (1 - O(k^{-1})) \\
& = ( 1 - O(d^{- \sqrt{k \log k}}) ) (k - O(\sqrt{k \log k})) (1 - O(k^{-1})) \\
& = k - O(\sqrt{k \log k}),\end{aligned}$$ which finishes the proof.
Grids
-----
The *Cartesian product* of two graphs $G$ and $H$ is a graph with vertex set $V(G) \times V(H)$ and with the vertices $(u_{1}, v_{1})$ and $(u_{2}, v_{2})$ adjacent if either $u_{1} = u_{2}$ and $v_{1}, v_{2}$ are adjacent in $H$, or $v_{1} = v_{2}$ and $u_{1}, u_{2}$ are adjacent in $G$. We denote the Cartesian product of $G$ and $H$ by $G \square H$. In this subsection, we will study a square grid $P_{n} \square P_{n}$.
It is known that for any two trees $T_{1},T_{2}$, we have $c(T_{1} \square T_{2}) = 2$ [@NN]. The capture time of the Cartesian product of trees was recently studied in [@Mehrabian]. It was shown that for any two trees $T_{1}, T_{2}$ we have $$\mathrm{ct}(T_{1}\square T_{2}) = \left\lfloor \frac{D(T_{1} \square T_{2})}{2} \right\rfloor = \left\lfloor \frac{D(T_{1}) + D(T_{2})}{2} \right\rfloor,$$ where $D=D(G)$ is the diameter of $G$. In particular, for a square grid we have that $\mathrm{ct}(P_{n} \square P_{n}) = n-1$.
We will show that the cost of drunkenness for a grid is asymptotic to $8/3$.
\[thm:dct\_grid\] $$\mathrm{dct}(P_{n} \square P_{n})= (1+o(1)) \frac38 n,$$ and the cost of drunkenness is $F(P_{n} \square P_{n}) = 8/3 + o(1)$.
Suppose that the drunk robber occupies an internal vertex $(u,v)$. The decision where to go from there can be made in the following way: toss a coin to decide whether modify the first coordinate ($u$) or the second one ($v$); independently, another coin is tossed to decide whether we increase or decrease the value. Hence the robber will move with probability 1/4 to one of the four neighbors of $(u,v)$. Note that, if we restrict ourselves to look at one dimension only (for example, let us call it North/South direction) we see the robber going North with probability 1/4, going South with the same probability and staying in place with probability 1/2. In other words the robber performs a *lazy* random walk on the path. Hence, both coordinates behave similarly to the lazy random walk on integers (move with probability $1/2$; do nothing, otherwise). The same argument as in the previous proofs can be used to show that with probability, say, $1-o(n^{-1})$, the robber stays within the distance $O(\sqrt{n \log n})=o(n)$ from the initial vertex. Hence, if we look at the grid from the ‘large distance’ the drunk robber is not moving at all.
Therefore, since we would like to investigate an asymptotic behaviour, the problem reduces to finding a set $S$ consisting of two vertices such that the average distance to $S$ is as small as possible. Cops should start on $S$ to achieve the best outcome. It is clear that, due to the symmetry of $P_{n} \square P_{n}$, there are two symmetric optimal configurations for set $S$: $$\begin{aligned}
S = \{ (n/2+O(1),n/4+O(1)), (n/2+O(1),3n/4+O(1))\}, \\
S = \{ (n/4+O(1),n/2+O(1)), (3n/4+O(1),n/2+O(1))\}.\end{aligned}$$ In any case, the average distance is $$\sum_{u=0}^{n-1} \sum_{v=0}^{n-1} dist( (u,v), S) = (1+o(1)) 8n \int_{x = 0}^{1/2} \int_{y=0}^{1/4} (x+y) dy dx = (1+o(1)) \frac38 n.$$ The result follows.
The cost of drunkenness {#sec04}
=======================
In this section we show that the cost of drunkenness can be arbitrarily close to any real number $c \in[1,\infty)$. In order to do it, we introduce two families of graphs, barbells and lollipops.
Barbell
-------
Let $n \in\mathbb{N}$ and $c \ge0$. The *barbell* $B(n,c)$ is a graph that is obtained from two complete graphs $K_{\lfloor cn \rfloor}$ connected by a path $P_{n}$ (that is, one end of the path belongs to the first clique whereas the other end belongs to the second one). The number of vertices of $B(n,c)$ is $(1+2c)n + O(1)$, $c(B(n,c))=1$. In order to catch (either the adversarial or the drunk) robber, the cop should start at the center of the path and move toward the robber; $\mathrm{ct}(B(n,c)) = n/2 + O(1)$. This family can be used to get any ratio from $(1,2]$.
\[thm:dct\_cn1\] Let $c \ge0$. Then, $$\mathrm{dct}(B(n,c))=(1+o(1)) \frac{n}{2} \cdot\frac{1+4c}{2+4c},$$ and the cost of drunkenness is $$F(B(n,c))=\frac{\mathrm{ct}(B(n,c))}{\mathrm{dct}(B(n,c))}=1 + \frac{1}{1+4c}+o(1).$$
The drunk robber starts on a clique with probability $(2c)/(1+2c) + o(1)$. If this is the case, the capture occurs at time $n/2 + O(\sqrt{n \log n})$ with probability, say, $1-o(n^{-1})$ by Lemma \[lem:walk\]. If the robber chooses a vertex at the distance $k$ from the robber to start with, he is captured after $k+O(\sqrt{n \log n})$ steps, again with probability $1-o(n^{-1})$. Hence the expected capture time is $$(1+o(1)) \left( \frac{2c}{1+2c} \cdot\frac{n}{2} + \frac{1}{1+2c} \cdot
\frac{n}{4} \right) = (1+o(1)) \frac{n}{2} \cdot\frac{1+4c}{2+4c}.$$ The theorem holds.
Lollipop
--------
Let $n \in\mathbb{N}$ and $c \ge0$. The *lollipop* $L(n,c)$ is a graph that is obtained from a complete graph $K_{\lfloor cn \rfloor}$ connected to a path $P_{n}$ (that is, one end of the path belongs to the clique). The number of vertices of $L(n,c)$ is $(1+c)n + O(1)$, and the cop number $c(L(n,c))$ is $1$. In order to catch the perfect robber, the cop should start at the center of the path and move toward the robber; $\mathrm{ct}(L(n,c)) = n/2 + O(1)$. However, it is not clear what the optimal strategy for the drunk robber is. The larger the clique is, the closer to the clique the cop should start the game.
\[thm:dct\_cn2\] Let $c \ge0$. Then, $$\mathrm{dct}(L(n,c))=
\begin{cases}
(1+o(1)) \frac{n}{4} \cdot\frac{(\sqrt{2}-1+c)(\sqrt{2}+1-c)}{1+c}, \mbox{ for } c \in[0,1]\\
(1+o(1)) \frac{n}{2(1+c)}, \mbox{ for } c > 1.
\end{cases}$$ and the cost of drunkenness is $$F(L(n,c))=\frac{\mathrm{ct}(L(n,c))}{\mathrm{dct}(L(n,c))}=
\begin{cases}
\frac{2(1+c)}{(\sqrt{2}-1+c)(\sqrt{2}+1-c)} + o(1), \mbox{ for } c \in[0,1]\\
(1+c) + o(1), \mbox{ for } c > 1.
\end{cases}$$
Before we move to the proof of this result, let us mention that the cost of drunkenness (as a function of the parameter $c$) has an interesting behaviour. For $c=0$ it is $2$ (we play on the path), but then it is decreasing to hit its minimum of $1+\sqrt{2}/2$ for $c=\sqrt{2}-1$. After that it is increasing back to $2$ for $c=1$, and goes to infinity together with $c$. Therefore, this family can be used to get any ratio at least $1+\sqrt{2}/2 \approx1.71$.
Let the cop start on vertex $v$ at the distance $(1+o(1))bn$ from the clique ($b \in[0,1]$ will be chosen to obtain the minimum expected capture time). The drunk robber starts on a clique with probability $c/(1+c)+o(1)$. If this is the case, the capture occurs at time $bn + O(\sqrt{n \log n})$ with probability, say, $1-o(n^{-1})$ by Lemma \[lem:walk\]. If the robber chooses vertex at the distance $k$ from the cop, then he is captured, again with probability $1-o(n^{-1})$, after $k+O(\sqrt{n \log n})$ rounds. The robber starts between the cop and the clique with probability $b/(1+c)+o(1)$ and on the other side with remaining probability. Hence the expected capture time is equal to $$\begin{aligned}
(1+o(1)) & \left( \frac{c}{1+c} \cdot bn + \frac{b}{1+c} \cdot\frac{bn}{2} + \frac{1-b}{1+c} \cdot\frac{(1-b)n}{2} \right) \\
& = (1+o(1)) \frac{n}{1+c} \left( b^{2} + (c-1)b + 1/2 \right) .\end{aligned}$$ The above expression is a function of $b$ (that is, a function of the starting vertex $v$ for the cop) and is minimized at $$b = \min\left\{ \frac{1-c}{2} , 0 \right\} .$$ The theorem holds.
It follows immediately from Theorems \[thm:dct\_tdk\], \[thm:dct\_cn1\], and \[thm:dct\_cn2\] that the cost of drunkenness can be arbitrarily close to any constant $c \ge1$.
For every real constant $c\geq1$, there exists a sequence of graphs $(G_{n})_{n \ge 1}$ such that $$\lim_{n\rightarrow\infty}F(G_{n})=\lim_{n\rightarrow\infty}\frac{\mathrm{ct}(G_{n})}{\mathrm{dct}(G_{n})}=c.$$
Computational Aspects {#sec06}
=====================
In this section we deal with computational aspects of the cop against drunk robber problem. Our analysis holds for any number of cops, that is, we no longer assume that $k=c\left(G\right)$.
Computing expected capture time for a given strategy\[sec0601\]
---------------------------------------------------------------
Suppose that we are given a graph and we fix a strategy before the game actually starts. We will now show how to explicitly compute the probability of capture at time $t\in\{0,1,2,\ldots\}$ as well as the expected capture time.
Fixing a strategy in advance is the best one can do for the invisible robber case (see Section \[sec07\]) but for a visible one, cops should adjust their strategy based on the behaviour of the opponent; this will be treated in the next subsection \[sec0602\]. However, the approach presented here is less demanding computationally and can be used to provide an upper bound for the optimal expected capture time.
Let $G=(V,E)$ be a connected graph with $V=\{0,1,\ldots,n-1\}$. Letting $$P_{i,j}=\Pr\left( Y_{t}=j~~|~~Y_{t-1}=i\right)$$ we have $$P_{i,j}=
\left\{
\begin{array}
[c]{clc}
\frac{1}{|N(i)|} & \text{for }j\in N(i) & \\
0 & \text{otherwise.} &
\end{array}
\right.$$ Note that $P$ is the $n\times n$ transition probability matrix governing the robber’s random walk in $G$ *in the absence of cops*. To account for capture by the cops, define a new state space $\overline{V}=V\cup\left\{n\right\}$, that is, the old state space augmented by the *capture state* $n$. The corresponding $(n+1)\times(n+1)$ transition matrix is $$\overline{P}=
\left(
\begin{array}
[c]{cc}
P & \mathbf{0}\\
\mathbf{0} & 1
\end{array}
\right).$$ In the absence of cops, the robber performs a standard random walk on $G$ and never enters the capture state; if however he starts in the capture state, he remains there forever: $\overline{P}_{n,n}=1$. In other words, the Markov chain governed by $\overline{P}$ contains two noncommunicating equivalence classes: $\{0,1,\ldots,n-1\}$ and $\{n\}$.
Suppose now that a single cop is located in vertex $x$. We will denote the corresponding transition probability matrix by $\overline{P} \left(x\right)$. Obviously, $\overline{P}\left(x\right)\neq\overline{P}$. The difference is caused by the possibility of capture, which can occur in two ways.
1. At the $(t-1)$-th round the robber is located at $x$ and, in the first phase of the $t$-th round, the cop moves into $x$. Then the robber is captured, so $\overline{P}_{x,n}\left(x\right)=1$ and $\overline{P}_{x,y}\left(x\right)=0$ for $y \in V$.
2. At the $(t-1)$-th round the robber is located at $y\neq x$ and, in the second phase of the $t$-th round, he moves from $y$ to $x$. Hence the robber is captured with probability $P_{y,x}$. So, for all $y\in V-\left\{x\right\}$, $\overline{P}_{y,n}\left(x\right)=P_{y,x}$, $\overline{P}_{y,x}\left(x\right)=0$.
We can summarize the above by writing $$\overline{P}\left(x\right)=\left(
\begin{array}
[c]{cc}
P\left(x\right) & \mathbf{p}(x)\\
\mathbf{0} & 1
\end{array}
\right),$$ where $P\left(x\right)$ has 0’s in the $x$-th row and column and the corresponding probabilities have been moved into the $\mathbf{p}(x)$ vector. For example, letting $G$ be the path with 5 nodes, the matrices $\overline{P}$ and $\overline{P}\left(2\right)$ are: [$$\overline{P}=
\left(
\begin{array}
[c]{cccccc}
0 & 1 & 0 & 0 & 0 & 0\\
1/2 & 0 & 1/2 & 0 & 0 & 0\\
0 & 1/2 & 0 & 1/2 & 0 & 0\\
0 & 0 & 1/2 & 0 & 1/2 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right),
\quad
\overline{P}\left(2\right)=
\left(
\begin{array}
[c]{cccccc}
0 & 1 & 0 & 0 & 0 & 0\\
1/2 & 0 & 0 & 0 & 0 & 1/2\\
0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 1/2 & 1/2\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right).$$ ]{} Especially for the placement round of the game ($t=0$) we need a different matrix, because the robber does not perform a random-walk, but simply chooses an initial position uniformly at random; if he chooses the one already occupied by the cop, then he is captured immediately. Hence, for this round the appropriate transition matrix is $\widehat{P}\left(x\right)$, which is the unit matrix with the one of the $x$-th row moved to the $\left(n+1\right)$-th column.
Let $\pi_{i}(t)=\mathbb{P}(Y_{t}=i)$ for $i\in\overline{V}$ and $t\in\left\{0,1,\ldots,s\right\}$ and $\pi(t)=\left(\pi_{0}(t),\pi_{1}(t), \ldots,\pi_{n}(t)\right)$; also let $\widehat{\pi}\left(0\right) =\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n},0\right)$. Then, given a strategy $\mathbf{X}=\left(x_{0},x_{1},\ldots,x_{s}\right)$, the above formulation yields $$\pi\left(0\right)=\widehat{\pi}\left(0\right)\widehat{P}\left(x_{0}\right)$$ and, for $t\in\left\{1,2, \ldots\right\}$, $$\pi\left(t\right)=\pi\left(t-1\right)\overline{P}\left(x_{t-1}\right).$$ This implies that $\pi\left(t\right)= \widehat{\pi}\left(0\right)\widehat{P}\left(x_{0}\right)\overline{P}\left(x_{1}\right)\overline{P}\left(x_{2}\right)\ldots\overline{P}\left(x_{t}\right)$. To illustrate this, let us continue the example. Suppose a single cop enters the path and follows the strategy $\mathbf{X}=(0,1,2,3,4)$ (start on one end of the path and move to the other one). Then we have
[ $\pi\left(0\right)=\widehat{\pi}\left(0\right)\widehat{P}\left(x_{0}\right)=
\left(
\begin{array}
[c]{cccccc}
1/5 & 1/5 & 1/5 & 1/5 & 1/5 & 0
\end{array}
\right)
\left(
\begin{array}
[c]{cccccc}
0 & 0 & 0 & 0 & 0 & 1\\
0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right)
=\allowbreak
\left(
\begin{array}
[c]{cccccc}
0 & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5}
\end{array}
\right)$ ]{}
[ $\pi\left(1\right)=\pi\left( 0\right) \overline{P}\left(x_{1}\right)=
\left(
\begin{array}
[c]{cccccc}
0 & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5}
\end{array}
\right)
\left(
\begin{array}
[c]{cccccc}
0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 1/2 & 0 & 1/2\\
0 & 0 & 1/2 & 0 & 1/2 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right)
=\allowbreak
\left(
\begin{array}
[c]{cccccc}
0 & 0 & \frac{1}{10} & \frac{3}{10} & \frac{1}{10} & \frac{1}{2}
\end{array}
\right)$ ]{}
[ $\pi\left(2\right)=\pi\left(1\right)\overline{P}\left(x_{2}\right)
=\allowbreak
\left(
\begin{array}
[c]{cccccc}
0 & 0 & \frac{1}{10} & \frac{3}{10} & \frac{1}{10} & \frac{1}{2}
\end{array}
\right)
\left(
\begin{array}
[c]{cccccc}
0 & 1 & 0 & 0 & 0 & 0\\
1/2 & 0 & 0 & 0 & 0 & 1/2\\
0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 1/2 & 1/2\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right)
=\allowbreak
\left(
\begin{array}
[c]{cccccc}
0 & 0 & 0 & \frac{1}{10} & \frac{3}{20} & \frac{3}{4}
\end{array}
\right)$ ]{}
[ $\pi\left(3\right)=\pi\left(2\right)\overline{P}\left(x_{3}\right)
=\allowbreak
\left(
\begin{array}
[c]{cccccc}
0 & 0 & 0 & \frac{1}{10} & \frac{3}{20} & \frac{3}{4}
\end{array}
\right)
\left(
\begin{array}
[c]{cccccc}
0 & 1 & 0 & 0 & 0 & 0\\
1/2 & 0 & 1/2 & 0 & 0 & 0\\
0 & 1/2 & 0 & 0 & 0 & 1/2\\
0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right)
=\allowbreak
\left(
\begin{array}
[c]{cccccc}
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right)$ ]{}
The elements $\pi_{n}(t)$ give the probabilities $P(X_{t}=n)$ at time $t$, that is, the probabilities of capture in *at most* $t$ steps. The probabilities of capture *exactly* at time $t$ are then given by $\pi_{n}(t)-\pi_{n}(t-1)$. The expected capture time (conditional on strategy $\mathbf{X}$ being used) is $$\mathbb{E}T=\sum_{t=1}^{\infty}t\cdot\left(\pi_{n}(t)-\pi_{n}(t-1)\right).$$ In the above example we have $$\mathbb{E}T=1\cdot\left(\frac{1}{2}-\frac{1}{5}\right)+2\cdot\left(\frac{3}{4}-\frac{1}{2}\right)+3\cdot\left(1-\frac{3}{4}\right)=\frac{31}{20}.$$
The approach can be generalized to more than one cop, by letting $\mathbf{x}=(x_{1},x_{2},\ldots,x_{k})$ be a configuration of cops and defining $\overline{P}(\mathbf{x})$, $P(\mathbf{x})$ analogously to the one cop case. Given that the cops follow the strategy $\mathbf{X}=(X_{1},X_{2},\ldots,X_{s})$, the transition probabilities of $Y$ satisfy $$\mathbb{P}(Y_{t}=j~~|~~Y_{t-1}=i)=P_{ij}(X_{t})$$ for $t\leq s$. So the robber process is an inhomogeneous Markov chain, with the transitions controlled by the cops’ actions. Markov chains of this type are called *Markov Decision Processes* (MDP) or *Controlled Markov Processes*, where the control function is $X_{t}$; it is a (stochastic) control in the sense that it allows us to change the transition probabilities of $Y_{t}$. We can use the MDP formulation to compute $\mathbb{E}T$ for any given strategy $\mathbf{X}$ in reasonable time. Computing the *optimal* strategy is not computationally viable; for example, with $|V|=n$ and $k$ cops there may exist up to $\Theta((n^{k})^{t})$ strategies of length $t$ (and the same number of corresponding $\mathbb{E}T$’s) to evaluate. In the Section \[sec0602\] we will present a computationally viable approach to compute the strategy that is arbitrarily close to the optimal one.
MDP’s were introduced in the book [@MDP01]; book-length treatments are [@BerTsi; @laBarriere; @MDP03; @MDP02]; an online tutorial is [@MDPOnline]. They have been applied to a version of the cops-robber problem in [@Zadeh].
Computing near-optimal strategies and minimum expected capture time\[sec0602\]
------------------------------------------------------------------------------
Let us now present and algorithm to compute $F(G)=\frac{\textrm{ct}(G)}{\textrm{dct}(G)}$ with arbitrarily good precision. Basically this reduces to computing $\textrm{ct}(G)$ and a good approximation of $\textrm{dct}(G)$, which can be done independently. To this end we present two algorithms, both of which have previously appeared in the literature. To improve the presentation we assign a name to each algorithm and make a few notational modifications; also we point out the similarity between the two algorithms (which apparently has not been noticed before).
1. The *CAAR* (*C*op *A*gainst *A*dversarial *R*obber) algorithm computes $\text{ct}_{x,y}(G)$ for every initial cop/robber configuration $(x,y)$. In addition, CAAR computes the optimal cop and robber play for every $(x,y)$. Capture time $\text{ct}\left(G\right)$ is easily computed from $\text{ct}(G)=\min_{x}\max_{y}\text{ct}_{x,y}(G)$.
2. Similarly, the *CADR*(*C*op *A*gainst *D*runk *R*obber) algorithm computes (an arbitrarily good approximation of) $\text{dct}_{x,y}(G)$ and the (near-)optimal cop play for every $(x,y)$; drunken capture time $\text{dct}(G)$ is computed from $\text{dct}(G)=\min_{x}\frac{\sum_{y}\text{dct}_{x,y}(G)}{n}$.
CAAR was introduced by Hahn and MacGillivray in [@Hahn2006]. We present the algorithm for the case of a single cop (the generalization for more than one cops is straightforward). Slightly changing notation, we will use $C_{x,y}$ to denote the game duration when the cop is located at $x$, the robber at $y$ and it is the cop’s turn to move (in other words, $C_{x,y}$ equals $\text{ct}_{x,y}(G)$). Similarly $R_{x,y}$ denotes game duration when it is the robber’s turn to move. For both $C_{x,y}$ and $R_{x,y}$ we assume optimal play by both cop and robber. Let us also define $$\widehat{V}^{2}=V\times V-\left\{\left(x,x\right):x\in V\right\},$$ (that is, $V^{2}$ excluding the diagonal) and for all $x\in V$, let $N^{+}\left(x\right)=N\left(x\right)\cup\left\{x\right\}$ be the closed neighbourhood of $x$. CAAR consists of the following recursion (for $i=1,2,\ldots$): $$\begin{aligned}
\forall\left(x,y\right) &
\in\widehat{V}^{2}:R_{x,y}^{\left(i\right)}=\max_{y^{\prime}\in N^{+}\left(y\right)}C_{x,y^{\prime}}^{\left(i-1\right)},
\label{eq0601}\\
\forall\left(x,y\right) &
\in\widehat{V}^{2}:C_{x,y}^{\left(i\right)}=1+\min_{x^{\prime}\in N^{+}\left(x\right)}R_{x^{\prime},y}^{\left(i\right)}.
\label{eq0602}\end{aligned}$$ $C$ and $R$ are initialized with $C_{x,y}^{(0)}=R_{x,y}^{(0)}=\infty$ for all $x\neq y$. We take $C_{x,x}^{(i)}=R_{x,x}^{(i)}=0$ for $i=0,1,2,\ldots$. Then (\[eq0601\])-(\[eq0602\]) is essentially equivalent to the version presented by Hahn and MacGillivray in [@Hahn2006], with just one difference which we will now discuss.
In (\[eq0601\])-(\[eq0602\]) the matrix $C$ is computed iteratively: the ($i-1$)-th matrix $C^{(i-1)}$ is stored and used in the $i$-th iteration to compute $C^{\left(i\right)}$. In numerical analysis this is known as a *Jacobi* iteration. It is well known that an alternative approach to computations of this type is the *Gauss-Seidel* iteration. In this iteration a single copy of $C$ is stored and its elements are updated “in place.” In [@Hahn2006], Hahn and MacGillivray present the Jacobi version of CAAR and prove that the algorithm converges (in a finite number of steps) if and only if $c\left(G\right)=1$. Hence CAAR computes the solution of the equations $$\begin{aligned}
\forall\left(x,y\right) &
\in\widehat{V}^{2}:R_{x,y}=\max_{y^{\prime}\in N^{+}\left(y\right)}C_{x,y^{\prime}},
\label{eq0603}\\
\forall\left(x,y\right)&
\in\widehat{V}^{2}:C_{x,y}=1+\min_{x^{\prime}\in N^{+}\left(x\right)}R_{x^{\prime},y},
\label{eq0604}\\
\forall x &
\in V:C_{x,x}=R_{x,x}=0.
\label{eq0604a}\end{aligned}$$ The interpretation of the equations is the following. Equation (\[eq0603\]) captures the property that from configuration $\left(x,y\right)$ the robber moves so as to maximize the length of the game; similarly, (\[eq0604\]) describes the cop’s goal to minimize the game duration (since the cop moves in the first phase of each round, 1 time unit must be added to $\min R_{x^{\prime},y}$); finally (\[eq0604a\]) says that the game ends when cop and robber occupy the same vertex.
Extending the CAAR idea to the *drunk* robber game, let us now use $C_{x,y}$ to denote $\text{dct}_{x,y}(G)$. In other words $C_{x,y}$ (respectively, $R_{x,y}$) is the *expected* game duration after the cop’s (respectively, robber’s) move. Recall (see Subsection \[sec0601\]) that $P_{y,y^{\prime}}(x)$ is the probability of the robber transiting from $y$ to $y^{\prime}$, given that the cop is at $x$; note that $P(x)$ is a *substochastic* matrix. The analog of (\[eq0601\])-(\[eq0602\]) is $$\begin{aligned}
\forall\left( x,y\right) & \in\widehat{V}^{2}:R_{x,y}^{\left( i\right)
}=\sum_{y^{\prime}\in N\left( y\right) }P_{y,y^{\prime}}\left( x\right)
C_{x,y^{\prime}}^{\left( i-1\right) },\label{eq0605}\\
\forall\left( x,y\right) & \in\widehat{V}^{2}:C_{x,y}^{\left( i\right)
}=1+\min_{x^{\prime}\in N^{+}\left( x\right) }R_{x^{\prime},y}^{\left(
i\right) }\label{eq0606}$$ and the analog of (\[eq0603\])-(\[eq0604a\]) is $$\begin{aligned}
\forall\left(x,y\right) &
\in\widehat{V}^{2}:R\left(x,y\right)=\sum_{y^{\prime}\in N\left(y\right)}P_{y,y^{\prime}}\left(x\right)C_{x,y^{\prime}},
\label{eq0607}\\
\forall\left(x,y\right) &
\in\widehat{V}^{2}:C_{x,y}=1+\min_{x^{\prime}\in N^{+}\left(x\right)}R_{x^{\prime},y}.
\label{eq0608}\\
\forall x &
\in V:C_{x,x}=R_{x,x}=0.
\label{eq0608a}\end{aligned}$$ We want (\[eq0605\])-(\[eq0606\]) to converge to the solution of (\[eq0607\])-(\[eq0608a\]). We will discuss convergence conditions (and initialization) presently.
Actually (\[eq0605\])-(\[eq0606\]) can be simplified. Since the drunk robber does not choose his moves, we can eliminate $R_{x,y}^{(i)}$ from (\[eq0605\])-(\[eq0606\]) and obtain the CADR algorithm recursion: $$\forall\left(x,y\right)\in\widehat{V}^{2}:C_{x,y}^{\left(i\right)}
=1+\min_{x^{\prime}\in N^{+}\left(x\right)}
\left(\sum_{y^{\prime}\in N\left(y\right)}P_{y,y^{\prime}}\left(x^{\prime}\right) C_{x^{\prime},y^{\prime}}^{\left(i-1\right)}\right).
\label{eq0609}$$
We have derived (\[eq0609\]) from (\[eq0605\])-(\[eq0606\]), which we see as an analog of (\[eq0601\])-(\[eq0602\]). However, we will now show that (\[eq0609\]) is a version of the *value iteration* algorithm, introduced and studied in the MDP literature [@BerTsi; @laBarriere; @MDP03; @MDP02]. Consider a general MDP process with state space $S$, action space $A$, transition matrix $Q$ and cost matrix $G(a)$ (that is, $G_{s,s^{\prime}}\left(a\right)$ is the cost of transition $s\rightarrow s^{\prime}$ using action $a$). The state space satisfies $S=S_{T}\cup S_{A}$, where $S_{T}$ are the transient states and $S_{A}$ the absorbing ones; it is assumed that transitions after absorption have zero cost: $G_{s,s^{\prime}}\left(a\right)=0$ for $s,s^{\prime}\in S_{A}$. Let $C_s$ be the expected total cost of the process starting from state $s$ and continuing until absorption. Then [@MDP03] $C$ satisfies the equations $$\forall s\in S_{T}:C_{s}=
\min_{a\in A}\left(G_{s,s^{\prime}}\left(a\right)+\sum_{s^{\prime}\in S_{T}}Q_{s,s^{\prime}}\left(a\right)C_{s^{\prime}}\right)
\label{eq0611}$$ and the solutions to (\[eq0611\]) can be obtained by the following value iteration: $$\forall s\in S_{T}:C_{s}^{\left(i\right)}=
\min_{a\in A}\left(G_{s,s^{\prime}}\left(a\right)+
\sum_{s^{\prime}\in S_{T}}Q_{s,s^{\prime}}\left(a\right) C_{s^{\prime}}^{\left(i-1\right)}\right).
\label{eq0610}$$ To show that (\[eq0610\]) can be reduced to (\[eq0609\]) let us take $S_{T}=\widehat{V}^{2}$ and $A=V$; in other words, states $s=(x,y)$ are cop/robber configurations and actions $a=x^{\prime}$ are new cop positions. Regarding move costs: (a) before capture every move has unit cost, (b) after capture only moves of the form $(x,x) \rightarrow (x,x)$ are possible and these have zero cost; in short $$G_{\left(x,y\right),\left(x^{\prime},y^{\prime}\right)}\left(x^{\prime}\right) =
\left\{
\begin{array}
[c]{ll}
1 & \text{if and only if }x\neq y\\
0 & \text{otherwise.}\end{array}
\right.$$ Finally, $$Q_{\left(x,y\right),\left(x^{\prime},y^{\prime}\right)}\left(a\right)=
\left\{
\begin{array}
[c]{ll}
P_{y,y^{\prime}}\left(x^{\prime}\right) & \text{if }a=x^{\prime}\in N^{+}\left(x\right)\text{ and }y^{\prime}\in N\left(y\right) \\
0 & \text{otherwise.}\end{array}
\right.$$ Using the above, it is easy to reduce (\[eq0610\]) to (\[eq0609\]).
The convergence of the CADR algorithm has been studied by several authors, in various degrees of generality [@Zadeh; @MDP01; @MDP02]. A simple yet strong result, derived in [@Zadeh], uses the concept of *proper strategy*: a strategy is called proper if it yields finite expected capture time. It is proved in [@Zadeh] that: if a proper strategy exists for graph $G$, then the Gauss-Seidel version of CADR converges to the true $C$ for arbitrary $C^{(0)}$ provided $C_{x,y}^{(0)}\geq0$ for all $\left(x,y\right)\in\widehat{V}^{2}$. As we have seen in Theorem \[thm:dct\_finite\], the cop has a proper strategy for every $G$. It can be proved that the Jacobi version of CADR also converges under the same conditions.
Now, $F(G)$ can be computed, easily. For every pair $(x,y)$, one can obtain a desired approximation of $\text{ct}_{x,y}(G)$ and $\text{dct}_{xy}(G)$ by performing CAAR and CADR, respectively. Then $$F\left(G\right)=
\frac{\text{ct}\left(G\right)}{\text{dct}\left(G\right)}=
\frac{\min_{x\in V}\max_{y\in V}\text{ct}_{xy}\left(G\right)}{\min_{x\in V}\frac{1}{\left\vert V\right\vert }\sum_{y\in V}\text{dct}_{xy}\left(G\right)}.$$ Both CAAR and CADR can be generalized for the case of $k$ cops, replacing $x$ by a $k$-tuple $\mathbf{x}=(x_{1},x_{2},\ldots,x_{k})$; however, execution time of both algorithms increases exponentially with $k$, hence the algorithms are computationally viable only for small $k$’s. Also CADR will work for any transition probability matrix $P$, not just for random walks. Hence, if desired, we can compute the cost of drunkenness for any number of cops (not just for $k=c(G)$) and for non-uniform random walks (i.e., discrete time birth-and-death processes) and other kinds of Markovian robbers.
Both CAAR and CADR can easily provide an optimal and near-optimal cop strategy in *feedback* form $U_{x,y}$, that is, the optimal cop move when the cop/robber configuration is $(x,y)$. This is achieved by recording a minimizing $x^{\prime}$ in (\[eq0604\]) / (\[eq0609\]). The optimal robber strategy $W_{x,y}$ (for the adversarial robber) can be similarly obtained by CAAR. For every $(x,y)$ configuration we can have more than one optimal moves, but they all yield the same (optimal) game duration.
We have implemented the CAAR and CADR algorithms in the Matlab package `CopsRobber`, which can be downloaded from [@web-page]. We have used this package to perform a number of numerical experiments, some of which are presented in the technical report [@KehPraTR]. This report also contains presentation of the algorithms in pseudo-code and a discussion of various computational issues.
The Invisible Robber\[sec07\]
=============================
In this section we present an introductory discussion of the cops and robber game when the robber is *invisible*; in other words, the cops do not know the robber’s location unless he is occupying the same vertex as one of the cops. All the other rules of the game remain the same. This version raises several interesting questions, a full study of which will be undertaken in a future paper.
Since the cops never see the robber until capture, they cannot use feedback strategies. In other words, the cop strategy is determined before the game starts. This does *not* mean that every cop move is predetermined because in certain cases it makes sense for the cops to randomize their moves. Hence capture time will in general be a random variable, even in the case of adversarial robber (who may also benefit from a randomized strategy).
Let us first examine the case of adversarial invisible robber. It is clear that, given enough cops, expected capture time will be finite. This is obviously true for $| V |$ cops, but in fact $c(G)$ cops suffice, as seen by the following theorem.
\[thm:random\_cops\] Suppose that $c(G)$ cops perform a random walk on a connected graph $G$, starting from any initial position. The robber, playing perfectly, is trying to avoid being captured. Let random variable $T$ be the capture time. Then, $$\mathbb{E }T < \infty.$$
Let $G=(V,E)$ be any connected graph, and let $\Delta=\Delta(G)$ be the maximum degree of $G$. Put $k=c(G)$. For any configuration of cops $x\in V^{k}$ and any vertex occupied by the robber $y\in V$, there exists a winning strategy $S_{x,y}$ that guarantees that the robber is caught after at most $t_{x,y}$ rounds. It is clear that cops will follow $S_{x,y}$ with probability at least $(1/\Delta)^{kt_{x,y}}$. Now, let us define $$\varepsilon=\min_{x\in V^{k},y\in V}(1/\Delta)^{kt_{x,y}}=(1/\Delta)^{kT_{0}}>0\text{, where }T_{0}=\max_{x\in V^{k},y\in V}t_{x,y}.$$ This implies that, regardless of the current position of players at time $t$, the probability that the robber will be caught after at most $T_{0}$ further rounds is at least $\varepsilon$. Moreover, corresponding events for times $t,t+T_{0},t+2T_{0},\dots$ are mutually independent. Thus, we get immediately that $$\begin{aligned}
\mathbb{E}T & =\sum_{t\geq0}\mathbb{P}(T>t)~~\leq~~\sum_{t\geq0}
\mathbb{P}\left(T> \left\lfloor \frac{t}{T_{0}} \right\rfloor T_{0}\right)\nonumber\label{eq:1}\\
& =\sum_{i\geq0}T_{0}\mathbb{P}(T>iT_{0})~~\leq~~T_{0}\sum_{i\geq
0}(1-\varepsilon)^{i}~~=~~\frac{T_{0}}{\varepsilon}~~<~~\infty,\end{aligned}$$ and we are done.
Hence $c(G)$ is the *minimum* number of cops required to capture the adversarial invisible robber in finite expected time, since this task is at least as hard as capturing the adversarial *visible* robber. Of course, generally it will take longer, comparing to the visible robber case, to capture the invisible robber. Let us define $\mathrm{ict}_{x,y}(G,k)$ to be the expected capture time when the initial cops/robber configuration is $(x,y)$ and both the $k$ cops and the robber play optimally; we also define $$\mathrm{ict}(G,k)=\min_{x\in V^{k}} \max_{y\in V}\mathrm{ict}_{x,y}(G,k)$$ and, finally, $\mathrm{ict}(G)=\mathrm{ict}(G,c(G))$.
We now turn to the *drunk* invisible robber. He chooses his starting vertex uniformly at random and performs a random walk, as before. For a given starting position $x\in V^{k}$ for $k$ cops, there is a strategy that yields the smallest expected capture time $\mathrm{idct}_{x}(G,k)$. Cops have to minimize this by selecting a good starting position: $$\mathrm{idct}(G,k)=\min_{x\in V^{k}}\mathrm{idct}_{x}(G,k).$$ As usual, $\mathrm{idct(G)=idct(G,c(G))}$ but it makes sense to consider any value of $k\geq1$. The proof of the next theorem is exactly the same as Theorem \[thm:dct\_finite\] and so is omitted.
$\mathrm{idct(G,k)<\infty}$ for any connected graph $G$ and $k\geq1$.
Finally, the cost of drunkenness for the invisible robber game is $F_i(G)=\frac{\mathrm{ict}(G)}{\mathrm{idct}(G)}$. It follows from last theorem that this graph parameter is well defined (that is, finite).
Let us make a few remarks regarding the invisible robber with “*infinite*” speed (actually, what we mean by this is an arbitrarily high speed). Let us define the cop number for this case by $c^{\infty}(G)$; it is the minimum number of cops that have a strategy to obtain a finite expected capture time. It is clear that $c(G)\le c^{\infty}(G)\le s(G)$, where $s(G)$ is the *search number* of $G$, that is, the minimum number of cops required to *clean* the graph in the *Graph Search* (GS) game (mentioned in Section \[sec01\]). We want to emphasize that the cops and robber game (with invisible, infinite speed robber) is *different* from the GS game and, in particular, there are graphs for which $c^{\infty}(G)< s(G)$. For example, for the $C_3$ cycle, $s(C_3)$=2 but $c^{\infty}(G)=1$, namely one cop using a *randomized* strategy, can capture the invisible, adversary, infinite speed robber in $T$ with $\mathbb{E}T=2$. Similarly, one cop on $K_{1,3}$, the star with 3 rays, can achieve $\mathbb{E}T=11/3$. Many other examples can be found. The main reason for the discrepancy between $c^{\infty}(G)$ and $s(G)$ is that, in the GS game, the fugitive is assumed *omniscient* and (under one interpretation) this means he knows *in advance* all the cop moves (until the end of the game). In the cops and robber family of games, on the other hand, omniscience is not assumed, either explicitly or implicitly. We can summarize in one phrase: *clearing is harder than capturing* even an infinite speed robber. We intend to further explore this issue, as well the computation of optimal strategies for cops chasing an invisible adversarial robber in a future publication.
We will finish this section with the computation of the cost of drunkenness for two examples (path and cycle) involving an invisible (unit speed) robber. In both cases the computation is possible because the optimal strategy (for both the cops and the adversarial robber) is “obvious.” Our examples are similar to the ones we have considered for the visible robber and proofs are omitted, since they are almost identical to those of Section \[sec03\].
Consider the path $P_{n}$ again, with a single cop and an invisible robber. It is clear that the best strategy for the cop (regardless of whether he is playing against a perfect robber or a drunk one) is to start from one end of the path (say, from vertex $0$) and move along the path until the robber is captured. We have $\mathrm{ict}(P_{n})=n-1$. When cops are playing agains a drunk robber, the expected capture time is roughly two times smaller.
$$\frac{n}{2} \left( 1 - O \left( \frac{\log n}{n} \right) \right) ~~\le~~
\mathrm{idct}(P_{n}) ~~\le~~ \frac{n-1}{2}.$$ In particular, $\mathrm{idct}(P_{n}) = (1+o(1)) n/2$ and the cost of drunkenness is $$F_i(G)=\frac{\mathrm{ict}(P_{n})}{\mathrm{idct}(P_{n})} = 2+o(1).$$
Let us now play the game with two cops and an invisible robber on the cycle $C_{n}$ for $n\geq4$. It is not difficult to see that $s(C_{n})=2=c(C_{n})$. The best cop strategy is to start on vertices $1$ and $n$; the cop occupying vertex $1$ will move toward higher values, the other one will move in the opposite direction. The game ends after $\mathrm{ict}(C_{n})=\lfloor(n-1)/2\rfloor$ steps. When cops are playing against a drunk robber, the expected capture time is roughly two times smaller.
We have $$\frac{n}{4}\left( 1-O\left( \frac{\log n}{n}\right) \right) \leq
\mathrm{idct}(C_{n})\leq\frac{n-1}{4}.$$ In particular, $\mathrm{idct}(C_{n})=(1+o(1))n/4$ and the cost of drunkenness is $$F_i(G)=\frac{\mathrm{ict}(C_{n})}{\mathrm{idct}(C_{n})}=2+o(1).$$
Conclusion\[sec08\]
===================
Most of the results in the paper pertain to the case of a visible (adversarial / drunk) robber, pursued by $k=c(G)$ cops. The cases of arbitrary $k$ and invisible robber have been briefly touched. We conclude the current paper by listing additional questions regarding the cost of drunkennes. We begin by listing several questions related to the visible robber.
1. Our analysis can be expanded to strategies which use an arbitrary number of cops. As shown in Theorem \[thm:dct\_finite\], even a single cop can catch a drunk robber in finite expected time. Hence, for a given $G$ we can study $\mathrm{dct}(G,k)$ as a function of $k$. Obviously this is a decreasing function; what more can be said about it? As a first step in this direction, the numerical approach of Section \[sec06\] can be used to explore the properties of $\mathrm{dct}(G,k)$ for a given graph $G$.
2. Let us define $\mathrm{dct}(G,\mathbf{X})$ to be the expected capture time in graph $G$ using strategy $\mathbf{X}$. It is no longer assumed that $\mathbf{X}$ is an optimal strategy. Under what conditions on $\mathbf{X}$ and/or $G$ will $\mathrm{dct}(G,\mathbf{X})$ be finite? Can we use the approach of Section \[sec06\] to obtain non-trivial bounds on $\mathrm{dct}(G,\mathbf{X})$?
3. A related question is whether (for a specific $G$ and either optimal or general strategies) expected capture time can be connected to some graph parameter such as treewidth, pathwidth etc.
4. How robust are our results to slight (natural) modifications of the cops/robber game rules? For example, would the cost of drunkenness change if we allowed the robber to loop into its current location (that is, to perform lazy random walk)? What about a “general” random walk (that is, with nonuniform transition probabilities). What about *directed* graphs? Finally, does the situation change significantly if the cops and the robber move simultaneously rather than the cops moving first? The algorithm of Section \[sec06\] can be easily modified to handle these cases and numerical experiments may be useful for an initial exploration.
One can try to obtain similar results for the *invisible* robber. In Section \[sec07\] we showed how our approach can be extended (at least for certain families of graphs) to this case. In the examples we examined (paths, cycles) the optimal cop strategy is obvious. For general graphs, finding the search strategy optimal for the invisible (adversarial / drunk) robber will be more complicated. Is there a (computationally viable, perhaps approximate) algorithm to achieve this?
Finally, let us note that all of the above analyses adopt the cops’ point of view. It will be interesting to study the cost of drunkenness for the cops. In other worlds, assuming an adversarial evader and $k$ drunk cops, can we place bounds on the increase of expected capture time as compared to the case of adversarial cops? Theorem \[thm:random\_cops\] may be used as a starting point to achieve this goal.
[99]{}
M. Aigner and M. Fromme, A game of cops and robbers, *Discrete Applied Mathematics* **8** (1984) 1–12.
B. Alspach, Sweeping and searching in graphs: a brief survey, *Matematiche* **59** (2006) 5–37.
J. Bertsekas and J. Tsitsiklis. *Parallel and Distributed Computation*. Addison-Wesley, 1989.
A. Bonato and R. Nowakowski. *The Game of Cops and Robbers on Graphs*. AMS, 2011.
D. Coppersmith, P. Tetali and P. Winkler, Collisions among random walks on a graph, *SIAM J. Disc. Math.* **6** (1993) 363–374.
J.H. Eaton and L.A. Zadeh, Optimal pursuit strategies in discrete-state probabilistic systems, *Trans. ASME Ser. D, J. Basic Eng* **84** (1962) 23–29.
F.V. Fomin and D. Thilikos, An annotated bibliography on guaranteed graph searching, *Theoretical Computer Science* **399** (2008) 236–245.
G. Hahn, Cops, robbers and graphs, *Tatra Mountain Mathematical Publications* **36** (2007) 163–176.
G. Hahn and G. MacGillivray, A note on $k$-cop, $l$-robber games on graphs, *Discrete Mathematics*, **306** (2006) 2492–2497.
R.A. Howard, *Dynamic programming and Markov process*, MIT Press, 1960.
S. Janson, T. [Ł]{}uczak, and A. Ruciński, *Random Graphs*, Wiley, New York, 2000.
L. Kallenberg, Markov Decision Processes, http://www.math.leidenuniv.nl/kallenberg/Survey%20MDP.pdf
Ath. Kehagias and P. Prałat, Cops and visible robbers, Technical Report available at `http://users.auth.gr/kehagiat/GraphSearch/TRCODvis.pdf`
A. Mehrabian, The capture time of grids, *Discrete Math.* **311** (2011), 102–105.
S. Neufeld and R. Nowakowski, A game of cops and robbers played on products of graphs, *Discrete Mathematics* **186** (1998), 253–268.
R. Nowakowski and P. Winkler, Vertex to vertex pursuit in a graph, *Discrete Mathematics* **43** (1983) 230–239.
R. Pallu de la Barriere. *Optimal Control Theory*. Dover, 1980.
M.L. Puterman. *Markov Decision Processes*, Wiley, 1994.
A. Quilliot, Jeux et pointes fixes sur les graphes, Ph.D. Dissertation, Université de Paris VI, 1978.
D.J. White, *Markov decision processes*, Wiley, 1993.
`http://users.auth.gr/kehagiat/GraphSearch/`.
|
---
abstract: 'This paper proposes a reliable approach for human gait symmetry assessment using a depth camera and two mirrors. The input of our system is a sequence of 3D point clouds which are formed from a setup including a Time-of-Flight (ToF) depth camera and two mirrors. A cylindrical histogram is estimated for describing the posture in each point cloud. The sequence of such histograms is then separated into two sequences of sub-histograms representing two half-bodies. A cross-correlation technique is finally applied to provide values describing gait symmetry indices. The evaluation was performed on 9 different gait types to demonstrate the ability of our approach in assessing gait symmetry. A comparison between our system and related methods, that employ different input data types, is also provided.'
author:
- |
Trong-Nguyen Nguyen\
DIRO, University of Montreal\
Montreal, QC, Canada\
`nguyetn@iro.umontreal.ca`\
Huu-Hung Huynh\
University of Science and Technology\
Danang, Vietnam\
`hhhung@dut.udn.vn`\
Jean Meunier\
DIRO, University of Montreal\
Montreal, QC, Canada\
`meunier@iro.umontreal.ca`\
bibliography:
- 'references.bib'
title: Human Gait Symmetry Assessment using a Depth Camera and Mirrors
---
Introduction {#sec:introduction}
============
The problem of assessing human gait has received a great attention in the literature since gait analysis is a key component of health diagnosis. Marker-based and multi-camera systems are widely employed to deal with this problem. However, such systems usually require specific equipments with high price tag and sometimes have high computational cost. In order to reduce the cost of devices, we focus on a system of gait analysis which employs only one depth sensor. The principle is similar to a multi-camera system, but the collection of cameras are replaced by one depth sensor and mirrors. Each mirror in our setup plays the role of a camera which captures the scene at a different viewpoint. Since we use only one camera, synchronization can thus be avoided, and the cost of devices is reduced.
In order to simplify the setup, recent studies used a color or depth camera to perform gait analysis. The input of such systems is thus either the subject’s silhouette or depth map. Many gait signatures have been proposed based on the former input type such as Gait Energy Image (GEI) [@Han2006], Motion History Image (MHI) [@Davis2001], or Active Energy Image (AEI) [@Zhuowen2015]. Typically they are computed based on a side view camera and are usually applied on the problem of human identification. In order to deal with pathological gaits, the input sequence of silhouettes needs more elaborate processing. In the work [@Nguyen2014extracting], the input sequence of silhouettes was separated into consecutive sub-sequences corresponding to gait cycles. The feature extraction was applied on each individual silhouette and the gait assessment was performed based on a combination of such features in each sub-sequence. Instead of capturing a side view of the subject, the authors in [@Bauckhage2005; @Bauckhage2009] put the camera in front of a walking person and tried to detect unusual movement. The movement of the subject was encoded based on a sequence of lattices applied on the captured silhouettes. A feature vector was then estimated for each lattice according to a predefined set of points, and the characteristic representing the whole motion was formed by concatenating such vectors. This step of concatenation is to incorporate the temporal context into the classification with a Support Vector Machine (SVM). A common limitation of such silhouette-based approaches is the reduction of data dimension since the 3D scene is represented by 2D images. In order to overcome this drawback, a depth camera is often employed. One of the devices that are widely used is the Microsoft Kinect. Beside its low price, this camera provides a built-in functionality of human skeleton localization, estimated in each single depth frame [@Shotton2011realtime; @Shotton2013efficient]. Such skeletal information is useful for gait-related problems such as abnormal gait detection [@Nguyen2016], gait-based recognition [@Jiang2015], and pathological gait analysis [@Bigy2015]. A limitation of skeleton-based approaches is that the skeleton may be deformed due to self-occlusions in the depth map. Unfortunately, such problem usually occurs in pathological gaits [@Pfister2014; @Auvinet2015].
For that reason, other researchers have used depth images without skeleton fitting to assess gait. For instance, @Auvinet2015 [@Auvinet2015] have proposed an asymmetry index based on spatial differences between the lower limb positions recorded by a depth camera placed in front of the subject walking on a treadmill. They have demonstrated that this approach was more reliable than a similar index based on skeleton information. However these differences were computed only in a small predefined leg zone, required averaging depth maps over several gait cycles (to be determined) and limited to a single depth map representing only a partial view of the whole body. @Nguyen2018BHI [@Nguyen2018BHI] have also employed successfully (enhanced) depth maps for gait assessment using a weighted combination of a PoI-score, based on depth map key points, and a LoPS-score describing a measurement of body balance from the body silhouette. However, their method was still limited to a partial view of the body and basic features.
Taking all this into account, we present an original approach that estimates an index of human gait symmetry without requiring skeleton extraction or gait cycle detection. To improve the performance, the input of our system is a sequence of 3D point clouds of the whole body obtained with a combination of a depth camera and two mirrors. Cylindrical histograms of point cloud are then computed and analysed for left-right symmetry for subjects walking on a treadmill to obtain their symmetry index. The remaining of this paper is organized as follow: Section \[sec:method\] describes details of our method including the setup, point cloud formation, feature extraction, and gait symmetry assessment; our experiments and evaluation are presented in Section \[sec:experiment\], and Section \[sec:conclusion\] gives the conclusion.
Proposed method {#sec:method}
===============
In order to give a visual understanding, an overview of the proposed approach is shown in Fig. \[fig:overview\].
Point cloud formation
---------------------
Beside a ToF depth camera and two mirrors, our setup also employs a treadmill where each subject performs his/her walking gait. The ToF camera is put in front of the subject and the two mirrors are behind so that the walking person nearly stands at the center (see Fig. \[fig:setupsketch\]). An example of such captured depth map is presented in Fig. \[fig:depthmap\].
(250,240) (20,0)[![Basic principle of the depth camera system with mirrors. The depth information visible by the depth camera (blue surface of the object) is complemented by the reflected depth information from the two mirrors (red and green surfaces) to obtain the full 3D reconstruction of the object. Notice that in practice some unreliable points must be removed due to multiple reflections with ToF camera (see [@NguyenKinect2; @NguyenReportKinect2]).[]{data-label="fig:setupsketch"}](images/setupsketch.png "fig:")]{} (59,149) (160,170) (94,5)[depth camera]{}(110,102)[object]{}
![A depth map captured by our system, in which there are 3 collections of subject’s pixels (highlighted by cyan ellipses). The two mirrors and the treadmill are highlighted with yellow rectangles.[]{data-label="fig:depthmap"}](images/graydepthmap2.png){width="64.00000%"}
There are two popular types of depth sensor that are distinguished based on the scheme of depth estimation: structured light (SL) and Time-of-Flight (ToF). In our work, the second type was employed because it is more accurate [@Wasenmuller2017] and consequently its point cloud has a higher level of details compared with the first one.
As shown in Fig. \[fig:depthmap\], each captured depth map provides subject’s images from 3 different view points. The 3D reconstruction could also be performed when the depth camera is replaced by a color one. However, the process of reconstruction based on such data produces an object (visual hull) that is bigger, less accurate and contains redundancies as described in [@Auvinet2012]. Therefore employing a depth camera in our setup is advantageous to provide a better model of 3D information.
Let us briefly describe the formation of a 3D point cloud from each depth map captured by a depth camera in our work. According to the example shown in Fig. \[fig:depthmap\], a depth map contains 3 partial surfaces of the subject. A point cloud representing the walking person can thus be formed by combining (a) the direct cloud (highlighted by the middle ellipse) and (b) reflections of two indirect ones (smaller ellipses), which are behind the mirrors [@Nguyen2018SPIE; @NguyenKinect2]. We used the method described in [@NguyenKinect2; @NguyenReportKinect2] because it was specifically designed for ToF camera and is robust to unreliable points caused by unwanted multiple reflections. Figure \[fig:cloudexample\] illustrates an example of a 3D point cloud obtained with the setup of Fig. \[fig:depthmap\].
![A point cloud obtained in our setup seen from different view points.[]{data-label="fig:cloudexample"}](images/cloud.png){width="70.00000%"}
Feature extraction
------------------
In order to perform gait symmetry assessment, we separate the entire movement into two non-overlapping sub-movements corresponding to the left and right half-bodies. In more details, each individual point cloud is processed to obtain a cylindrical histogram, and then the histogram is split into two sub-histograms representing two half-bodies.
### Coordinate system transformation
Let us notice that the point cloud is initially computed in the camera space $(x_c, y_c, z_c)$. Therefore, to facilitate the computation of the cylindrical histogram, we need a rigid transformation from the camera coordinate system to the object (body) coordinate system. The latter is defined by its origin assigned to the centroid of the body 3D point cloud, the $y$-axis normal to the ground (treadmill), the $x$-axis along the walking direction and the $z$-axis in the left to right direction (see Fig. \[fig:rigidtransform\]). The $y$-axis is easily estimated as the normal to the treadmill plane obtained during calibration using a few markers (4 in our work). The walking direction ($x$-axis) is determined from the vector between two appropriate markers on the treadmill. The remaining dimension ($z$-axis) is estimated by performing a cross product.
(250,314) (0,167)[![Visualizations of our scene from two different view points that show the camera coordinate system and the body coordinate system used for matching a cylinder with a point cloud. They are right-handed.[]{data-label="fig:rigidtransform"}](images/rigid4x.png "fig:")]{} (0,0)[![Visualizations of our scene from two different view points that show the camera coordinate system and the body coordinate system used for matching a cylinder with a point cloud. They are right-handed.[]{data-label="fig:rigidtransform"}](images/rigid3x.png "fig:")]{} (22,310)[$y_c$]{}(59,253)[$z_c$]{} (184,306)[$y$]{}(115,236)[$x$]{} (-3,98)[$x_c$]{}(27,119)[$y_c$]{}(62,74)[$z_c$]{} (112,75)[$x$]{}(175,150)[$y$]{}(142,120)[$z$]{}
### Cylindrical histogram estimation
Once the subject’s point cloud corresponding to each depth frame has been transformed, its symmetrical characteristic is then extracted with a cylindrical histogram. Concretely, a cylinder is estimated with the main axis coinciding with the $y$-axis of the body coordinate system, and the top and bottom surfaces going through the highest and lowest points along this dimension. The cylinder’s radius is long enough to guarantee that the entire point cloud is within the cylinder.
Given a cloud $P$ of $n$ 3D points and the size $h \times w$ of a target cylindrical histogram (see Fig. \[fig:correspondance\]), the sector’s zero-based index of each point $P^{(i)}$ is determined as $$\begin{cases}h^{(i)}=min\big(\floor*{h(max_y-P^{(i)}_y)(max_y-min_y)^{-1}},h-1\big)\\w^{(i)}=\floor*{\frac{w}{2\pi}\{[2\pi+sgn(\vec{v}^{(i)}_z)cos^{-1}(\frac{\vec{v}^{(i)}_x}{\|\vec{v}^{(i)}\|})]\bmod (2\pi)\}}\end{cases}
\label{eq:sectorindex}$$ where $max_y$ and $min_y$ respectively indicates the $y$-coordinate of highest and lowest points in the cloud $P$ along the $y$-axis, $\lfloor \circ \rfloor$ is the floor function, $P^{(i)}_y$ is the $y$ value of point $P^{(i)}$, $sgn(\circ)$ is the sign function, and $\vec{v}^{(i)}$ is a 2D vector computed from the $y$-axis to the point $P^{(i)}$. Notice that the notation $\vec{v}^{(i)}_z$ in eq. (\[eq:sectorindex\]) is the $z$ coordinate of $\vec{v}^{(i)}$. The subscript $z$ is to indicate the axis used in this calculation. The $min$ function in (\[eq:sectorindex\]) is to guarantee that the output index is in the range $[0, h-1]$.
Although a cylinder is employed to estimate a histogram for each point cloud, the representation of such histogram is flat, i.e. a matrix of size $h \times w$. The correspondence between a histogram’s bin and its original cylinder’s sector is illustrated in Fig. \[fig:correspondance\]. As illustrated in Fig. \[fig:histseparation\], the head is aligned at the center of the cylindrical histogram after performing the estimation. Notice that a slight *rotation* of the cylinder might be necessary to ensure that the body is well centered in the cylindrical histogram depending on the camera-to-body rigid transformation accuracy (see above).
(250,105) (0,0)[![Mapping from cylindrical sectors to histogram’s bins. The sub-figure (a) shows a 3D visualization. The histogram can be considered as a flattened cylinder seen from a specific view point as the sub-figure (b). In this simplified representation, the histogram’s size is $4 \times 4$ corresponding to 16 sectors.[]{data-label="fig:correspondance"}](images/sectorsx2.png "fig:")]{} (154,0)[![Mapping from cylindrical sectors to histogram’s bins. The sub-figure (a) shows a 3D visualization. The histogram can be considered as a flattened cylinder seen from a specific view point as the sub-figure (b). In this simplified representation, the histogram’s size is $4 \times 4$ corresponding to 16 sectors.[]{data-label="fig:correspondance"}](images/sectorsx3.png "fig:")]{} (128,54)[$x$]{}(150,96)[$y$]{}(128,87)[$z$]{} (218,41)[$x$]{}(245,67)[$z$]{} (60,90)[(a)]{}(195,90)[(b)]{} (5,75)(18,75)(31,75)(44,75) (54,63)(54,51)(54,39)(54,27)
(250,128) (0,0)[![Example of flattened cylindrical histogram. The original histogram (gray image) of size $8 \times 8$ is scaled and is represented as a heat map for a better visualization. We can explicitly see the posture’s self-symmetry since the head is at the center of the histogram.[]{data-label="fig:histseparation"}](images/gait32a.png "fig:")]{} (45,0)[![Example of flattened cylindrical histogram. The original histogram (gray image) of size $8 \times 8$ is scaled and is represented as a heat map for a better visualization. We can explicitly see the posture’s self-symmetry since the head is at the center of the histogram.[]{data-label="fig:histseparation"}](images/gait32b.png "fig:")]{} (100,69.7)[![Example of flattened cylindrical histogram. The original histogram (gray image) of size $8 \times 8$ is scaled and is represented as a heat map for a better visualization. We can explicitly see the posture’s self-symmetry since the head is at the center of the histogram.[]{data-label="fig:histseparation"}](images/hist_rearranged_8by8_original.png "fig:")]{} (142,0)[![Example of flattened cylindrical histogram. The original histogram (gray image) of size $8 \times 8$ is scaled and is represented as a heat map for a better visualization. We can explicitly see the posture’s self-symmetry since the head is at the center of the histogram.[]{data-label="fig:histseparation"}](images/hist_rearranged_8by8.png "fig:")]{} (186,120)[head]{}(196,118)[(0,-1)[15]{}]{}
Gait symmetry assessment {#sec:assessment}
------------------------
Similarly to related studies on gait analysis (e.g. [@Bauckhage2009; @Nguyen2016; @Auvinet2015; @Nguyen2018BHI]), the assessment of gait symmetry in our system also considers the temporal factor. Concretely, the value measuring the gait symmetry is estimated on a sequence of consecutive histograms. Symmetry can be measured by separating a histogram into two sub-histograms corresponding to two half-bodies. In other words, a sequence of histograms of size $h \times w$ becomes two sequences of sub-histograms of size $h \times 0.5w$. According to the nature of normal walking gait, there is a shifting along the time axis between a left sub-histogram and its corresponding symmetric right one. Therefore our method employs a cross-correlation technique [@Stoica2005] to measure the gait symmetry index. A good symmetry occurs if the left sub-histogram is similar to the horizontal flip version of the (shifted) right sub-histogram (Fig. \[fig:shifting\]).
(404,150) (0,10)[{width="\textwidth"}]{} (291,65) (195,87)[$h$]{}(180,102)[$w$]{} (262,87)[$h$]{}(247,102)[$w/2$]{}(294,87)[$h$]{}(279,102)[$w/2$]{} (143.5,40)[$i^{th}$]{}(167,64)[$j^{th}$]{} (53,0)[(a) Sequence of histograms]{} (158,0)[(b) Half-body sequences]{} (255,0)[(c) Best matching of different shiftings]{} (58,46)[]{} (3,46)[]{} (27,46)[]{} (58,105)[]{} (3,105)[]{} (27,105)[]{}
$$S \big(L,R,D\big) =min \big(\big\{\frac{1}{l-|d|} \sum_{i=0}^{l-|d|-1} Diff \big(L_{max(0,d)+i},R^f_{max(0,-d)+i}\big) \mid d \in D \big\} \big)
\label{eq:crosscorrelation}$$
(250,126) (0,0)[![Correlation between two sequences corresponding to positive and negative shifting values $d$, and indices of beginning positions. The notation $Ref$ indicates the reference (left sequence in our work). In these two examples, the lengths of each input sequence and the common one are 8 and 6, respectively.[]{data-label="fig:shiftdirection"}](images/shiftdistance.png "fig:"){width="64.00000%"}]{} (215,38)[$Ref$]{}(215,75)[$Ref$]{} (3,25)[$d=-2$]{}(51,94)[$d=2$]{} (55,58)[$max(0,d)$]{} (5,115)[$max(0,-d)$]{}
The processing of this stage is as follows. The input is a sequence of histograms. Although many related studies tried to process on gait cycles, our assessment is performed on consecutive (i.e. non-overlapping) sub-sequences (or segments) that have the same length. There are several reasons leading to our choice: (1) gait cycle determination would be difficult to perform when working on pathological gaits, (2) the symmetry can be measured well by dealing with the mentioned shifting on an arbitrary (long enough) sequence of histograms, and (3) sub-sequences do not need to have common properties (e.g. similar beginning and ending postures as in [@Nguyen2016] or [@Auvinet2015]) because we do not focus on training a model representing the gait. Each sub-sequence is then separated into two sequences of left and right sub-histograms. We can expect that by assigning a sequence as the reference and shifting the other with an appropriate delay, the two registered sub-sequences would have a good symmetry (see Fig. \[fig:shifting\]). Because such suitable delay is various with different subjects, we perform the shifting with a set of delays and choose the best match. Given two sequences of sub-histograms $L$ and $R^f$ ($R$ horizontally flipped) of length $l$ representing two half-bodies, a set of shifting delays $D$, the symmetry index $S$ is measured as in eq. (\[eq:crosscorrelation\]). The $Diff$ function estimates the distance between two sub-histograms, the $min$ function thus provides the best matching. Notice that the left segment is assigned as the reference, and the set $D$ contains both negative and positive values indicating the shifting direction of the other segment (see Fig. \[fig:shiftdirection\]). At the end of this stage, the system provides a sequence of scores measuring the gait symmetry corresponding to consecutive segments.
Experimental results {#sec:experiment}
====================
Data acquisition {#sec:acquisition}
----------------
Our experiments were performed on 9 different gait types consisting of normal walking gaits and 8 simulated asymmetrical (so-called abnormal) ones. These abnormal gaits were simulated by either padding a sole with a thickness of 5/10/15-centimeters under one foot or attaching a weight (4 kilograms) to one ankle. We use the notations L$|$5cm, L$|$10cm, L$|$15cm, and L$|$4kg to indicate these abnormal gaits with left leg, and so on for the other leg. Such set up can provide gaits having a higher level of asymmetry compared with normal walking ones. A Kinect 2 was employed for data acquisition since it uses ToF for depth measurement and had a low price. There were 9 volunteers that performed the 9 mentioned walking gaits, in which each motion was captured as 1200 continuous frames with a frame rate of 13 fps. The treadmill speed was set at 1.28 km/h. In order to provide a comparison with related approaches, we also captured other data types including skeleton and silhouette using built-in functionalities of the Kinect 2. Therefore each walking gait of a volunteer is represented by 1200 point clouds, 1200 skeletons, and 1200 silhouettes [@NguyenReportDataset][^1]^,^[^2]. These experimental procedures involving human subjects were approved by the Institutional Review Board (IRB).
System parameters
-----------------
As mentioned in Section \[sec:assessment\], the input sequence of point clouds is segmented into non-overlapping segments. In our experiments, each input sequence was separated into 10 segments of length 120 (about 9 seconds), the corresponding output was thus a vector of 10 elements measuring the gait symmetry. The size of cylindrical histogram was $16 \times 16$, so each half-body volume in Fig. \[fig:shifting\] had a size of $[16 \times 8 \times 120]$. The $L_1$ norm was used for measuring the distance \[the term $Diff$ in eq. (\[eq:crosscorrelation\])\] between two normalized histograms, i.e. dividing each bin value by the sum. The shifting delays were in the range $[-50, 50]$ to guarantee that the length of the common sub-sequence would be greater than a half of input length. Let us notice that $16 \times 16$ is *not* the optimal size of cylindrical histograms. This is just an arbitrarily selected value for our experiments. The effect of that hyperparameter will be discussed in Section \[sec:sizeaffection\].
Testing results {#sec:ourresult}
---------------
Since our system returned 10 measurement values (corresponding to 10 segments of length 120) for each input sequence of point clouds, their mean can be used as an index of gait symmetry. The experimental results are shown in Fig. \[fig:overallresult\].
(250,235) (19,3)[![Mean values of 10 measurements provided by our system for each gait of each volunteer. The notation N indicates normal gaits, L and R respectively represent left and right legs, and $v_i$ is the $i^{th}$ volunteer.[]{data-label="fig:overallresult"}](images/meanMatrix.png "fig:")]{} (238,0)[0]{} (238,20)[0.1]{} (238,39.5)[0.2]{} (238,59)[0.3]{} (238,78.5)[0.4]{} (238,98)[0.5]{} (238,118)[0.6]{} (238,136.5)[0.7]{} (238,156)[0.8]{} (238,175)[0.9]{} (238,195)[1]{} (5,11)[$v_9$]{}(5,33)[$v_8$]{}(5,55)[$v_7$]{} (5,78)[$v_6$]{}(5,100)[$v_5$]{}(5,122)[$v_4$]{} (5,143)[$v_3$]{}(5,165)[$v_2$]{}(5,187)[$v_1$]{} (40,206)[![Mean values of 10 measurements provided by our system for each gait of each volunteer. The notation N indicates normal gaits, L and R respectively represent left and right legs, and $v_i$ is the $i^{th}$ volunteer.[]{data-label="fig:overallresult"}](images/bracket.png "fig:")]{} (83,206)[![Mean values of 10 measurements provided by our system for each gait of each volunteer. The notation N indicates normal gaits, L and R respectively represent left and right legs, and $v_i$ is the $i^{th}$ volunteer.[]{data-label="fig:overallresult"}](images/bracket.png "fig:")]{} (126,206)[![Mean values of 10 measurements provided by our system for each gait of each volunteer. The notation N indicates normal gaits, L and R respectively represent left and right legs, and $v_i$ is the $i^{th}$ volunteer.[]{data-label="fig:overallresult"}](images/bracket.png "fig:")]{} (169,206)[![Mean values of 10 measurements provided by our system for each gait of each volunteer. The notation N indicates normal gaits, L and R respectively represent left and right legs, and $v_i$ is the $i^{th}$ volunteer.[]{data-label="fig:overallresult"}](images/bracket.png "fig:")]{} (52,225)[5cm]{} (93,225)[10cm]{} (136,225)[15cm]{} (183,225)[4kg]{} (25,200)[N]{} (47,200)[L]{}(69,200)[R]{}(91,200)[L]{}(113,200)[R]{} (135,200)[L]{}(156,200)[R]{}(177,200)[L]{}(198,200)[R]{} (22,185)[.35]{}(22,163)[.35]{}(22,141)[.36]{} (22,120)[.37]{}(22,98)[.41]{}(22,76)[.44]{} (22,53)[.39]{}(22,31)[.30]{}(22,9)[.39]{} (44,185)[.43]{}(44,163)[.48]{}(44,141)[.53]{} (44,120)[.45]{}(44,98)[.63]{}(44,76)[.77]{} (44,53)[.60]{}(44,31)[.55]{}(44,9)[.71]{} (66,185)[.53]{}(66,163)[.55]{}(66,141)[.60]{} (66,120)[.42]{}(66,98)[.58]{}(66,76)[.64]{} (66,53)[.61]{}(66,31)[.58]{}(66,9)[.71]{} (88,185)[.53]{}(88,163)[.54]{}(88,141)[.58]{} (88,120)[.69]{}(88,98)[.61]{}(88,76)[.69]{} (88,53)[.70]{}(88,31)[.56]{}(88,9)[.77]{} (110,185)[.69]{}(110,163)[.65]{}(110,141)[.82]{} (110,120)[.67]{}(110,98)[.69]{}(110,76)[.72]{} (110,53)[.74]{}(110,31)[.72]{}(110,9)[.71]{} (132,185)[.64]{}(132,163)[.63]{}(132,141)[.66]{} (132,120)[.75]{}(132,98)[.66]{}(132,76)[.78]{} (132,53)[.69]{}(132,31)[.60]{}(132,9)[.77]{} (153,185)[.74]{}(153,163)[.74]{}(153,141)[.70]{} (153,120)[.77]{}(153,98)[.70]{}(153,76)[.74]{} (153,53)[.82]{}(153,31)[.78]{}(153,9)[.77]{} (174,185)[.64]{}(174,163)[.59]{}(174,141)[.83]{} (174,120)[.58]{}(174,98)[.50]{}(174,76)[.81]{} (174,53)[.59]{}(174,31)[.79]{}(174,9)[.53]{} (195,185)[.71]{}(195,163)[.69]{}(195,141)[.95]{} (195,120)[.58]{}(195,98)[.61]{}(195,76)[.82]{} (195,53)[.65]{}(195,31)[.62]{}(195,9)[.65]{}
The mean values were in the range between 0.30 and 0.44 for normal gaits, and higher measures for the asymmetrical ones. Therefore, considering the returned estimation of an arbitrary gait and that range may allow gait symmetry assessment. However, that range is formed from a set of volunteers, an asymmetrical gait of a subject may thus have an estimation falling inside the normal range of other subjects though this value is still higher than the measure of normal gait with the same subject. This case happened for the R$|$5cm gait of the $4^{th}$ volunteer which was lower than the normal gait of the $6^{th}$ volunteer. Therefore, within-subject analysis should be considered to increase the confidence of the symmetry assessment. Let us see more details of our experimental results in Fig. \[fig:details\] instead of only mean values. With most subjects, the measured values tended to decrease when the asymmetry reduces (e.g. L$|$10cm compared with L$|$15cm). This means that our system could be used to assess the recovery of patients after a (knee, hip, etc.) surgery, during a musculoskeletal treatment or after a stroke for instance. In summary, the assessment of gait symmetry can be performed by checking estimated measures with a specific range and confirming the decision based on recent changes of these values (e.g. day by day). Let us notice again that considering only the normal range may not be sufficient since the actual gait symmetry depends on various factors such as health, physical body, and even walking habit. Therefore checking the convergence of symmetry measurements helps us to confirm the normality of patient’s gaits.
Comparison with other related methods
-------------------------------------
**Test subjects** **Evaluation** **Our method** **HMM [@Nguyen2016]** **One-class SVM [@Bauckhage2009]** **Binary SVM [@Bauckhage2009]**
------------------- ---------------- --------------------- ----------------------- ------------------------------------ ---------------------------------
short-term 0.042 0.335 0.227 0.157
full sequence 0.000 0.250 0.139 0.139
short-term 0.025 ($\pm$ 0.038) 0.396 ($\pm$ 0.117) 0.274 ($\pm$ 0.183) 0.152 ($\pm$ 0.058)
full sequence 0.000 ($\pm$ 0.000) 0.198 ($\pm$ 0.250) 0.136 ($\pm$ 0.070) 0.111 ($\pm$ 0.000)
short-term 0.051 - - -
full sequence 0.037 - - -
-- ----- --------------- ---------------- ---------------- ---------------- --------------- ---------------- ---------------- ----------------
$16 \times 8$ $16 \times 16$ $16 \times 24$ $16 \times 32$ $8 \times 16$ $16 \times 16$ $24 \times 16$ $32 \times 16$
AUC 0.989 **0.989** 0.988 0.987 0.989 0.989 0.989 **0.989**
EER **0.043** 0.050 0.044 0.044 **0.046** 0.050 0.050 0.050
AUC **0.998** 0.997 0.995 0.995 0.997 0.997 0.997 0.997
EER **0.014** 0.028 0.028 0.028 **0.014** 0.028 0.028 0.028
-- ----- --------------- ---------------- ---------------- ---------------- --------------- ---------------- ---------------- ----------------
In order to compare the gait-related information gained when exploiting 3D point clouds with other data types, we also performed experiments on the skeletons and silhouettes mentioned in Section \[sec:acquisition\]. Method [@Nguyen2016] was employed to deal with the former data type. That study separated an input sequence of skeletons into consecutive gait cycles detected using the distance between two foot joints. A hidden Markov model (HMM) with a specific structure was employed to build a model of normal walking gait cycles as well as to provide a likelihood for each input cycle. The categorization was finally performed by comparing such log-likelihoods with a predefined threshold. For the silhouette input, we used the approach [@Bauckhage2009], in which the feature extraction was performed on each frame, the temporal context was embedded by vector concatenation, and a support vector machine (SVM) was employed for the task of classification. Both methods aim to classify each input sequence into two categories: normal and abnormal gaits. Their ability was evaluated based on different measures: the Area Under Curve of a Receiver Operating Characteristic (ROC) curve for [@Nguyen2016] and typical classification accuracy for [@Bauckhage2009]. We decided to use the Equal Error Rate (EER) as the measure for comparison because this is estimated according to the ROC curve and its meaning is related to the classification accuracy. Such ROC-based measures have been employed in many problems of binary classification.
The HMM in [@Nguyen2016] was built with only normal gaits. Therefore beside the typical binary SVM, we also modified the model in approach [@Bauckhage2009] to have a one-class SVM. That unsupervised learning is reasonable in practical situations because there are numerous walking gaits that have abnormality, collecting a dataset of such gaits with a high generality is thus difficult. In our experiments, the HMM and one-class SVM were trained with the same dataset consisting of normal gaits of 5 (over 9) subjects ($v_1, v_3, v_5, v_6, v_9$ in Fig. \[fig:overallresult\] as suggested in [@NguyenReportDataset]), and the (normal and abnormal) gaits of the remaining subjects were the test set. The binary SVM was also trained on all gaits of those 5 volunteers, and the test set included all gaits of the other 4 volunteers. In order to have a more general evaluation, we also performed the experiments using leave-one-out, i.e. 9-fold cross-validation where each fold contains all 9 gaits of a subject. The assessment was thus represented as mean ($\pm$ std) of the evaluation quantity. The experimental results are presented in Table \[table:comparison\]. The notation *short-term* has different meanings: a segment of 120 point clouds in our method, an automatically detected gait cycle in [@Nguyen2016], and a temporal context of $\Delta = 20$ in [@Bauckhage2009] (i.e. per-frame classification based on vector concatenation of features in 21 recent frames). The notation *full sequence* indicates the classification based on mean values in our work (as shown in Fig. \[fig:overallresult\]), lowest averages of log-likelihoods computed on three consecutive cycles in each sequence in [@Nguyen2016], and alarm triggers on whole input sequences in [@Bauckhage2009].
According to Table \[table:comparison\], the classification errors resulting from our method are much lower compared with the others. Table \[table:comparison\] also shows that in all the 3 methods, the decision provided based on the whole input sequence had a higher confidence compared with short segments. In other words, the mean values in Fig. \[fig:overallresult\] were better than individual segment measures in indicating the gait symmetry embedded inside a sequence of point clouds. During our experiments, we observed that the binary SVM [@Bauckhage2009] always classified sequences of normal gaits (according to alarm triggers) into the category of anomaly. This property was clearly showed in the leave-one-out cross validation where the error was 0.111 for all 9 folds. This problem might be due to the large ratio between abnormal and normal gaits (8:1), and a binary (i.e. supervised) SVM was thus not really appropriate for the task of detecting abnormal gaits where there are numerous types of abnormal walking. It was also noticeable that the approach [@Nguyen2016] could be improved to get better results by modifying the width of sliding window since the frame rate of our data acquisition was lower than the system in [@Nguyen2016].
Sensitivity to size of cylindrical histogram {#sec:sizeaffection}
--------------------------------------------
The cylindrical histogram plays the main role in our approach and also affects the gait symmetry assessment. By changing the histogram’s size, i.e. number of sectors, the range of mean values in Section \[sec:ourresult\] would be different. The ability of distinguishing two gait types would also change. We can guess that a histogram with small resolution can reduce the computational cost of the entire system but may not have enough details for describing body postures. On the contrary, using a histogram formed from a large number of cylindrical sectors may also reduce the system’s efficiency. In that case, each sector covers a small volume with low numbers of 3D points, the result of eq. (\[eq:crosscorrelation\]) is thus sensitive to noise in the input 3D point clouds. In summary, the system accuracy can be improved by a careful selection of histogram size. Table \[table:sizecomparison\] shows the abilities (according to AUCs and EERs of ROC curves) of our system for various histogram resolutions in distinguishing symmetrical and asymmetrical walking gaits. In this table, we focus on the mean-based measurement because it describes the gait symmetry better than segments (according to Table \[table:comparison\]). The ability of our method tended to reduce, i.e. increasing of EER and decreasing of AUC, when we set a high value for the histogram width. The height of cylindrical histograms had a lower effect since the AUC and EER (for both segments and means) were almost unchanged when the height exceeded a particular threshold.
Conclusion {#sec:conclusion}
==========
In this paper, we have presented an original and efficient low-cost system for assessing gait symmetry using a ToF depth camera together with two mirrors. The input of the proposed method is a sequence of 3D point clouds representing the subject’s postures when walking on a treadmill. By fitting a cylinder on each point cloud, a cylindrical histogram is formed to describe the corresponding gait in the manner of self-symmetry. Cross-correlation is then applied on each pair of sequences of half-body sub-histograms to measure the gait symmetry along the movement. The ability of our method has been demonstrated via a dataset of 9 subjects and 9 gait types. Our approach also outperforms some related works, that employ skeletons and frontal view silhouettes as the input, in the task of distinguishing normal (symmetric) and abnormal (asymmetric) walking gaits. The resulting system is thus a promising tool for a wide range of clinical applications by providing relevant gait symmetry information. Patient screening, follow-up after surgery, treatment or assessing recovery after a stroke are obvious applications that come to mind. As future work, the proposed method will be modified focusing on particular pathological gaits such as diplegic, hemiplegic, choreiform, and Parkinsonian [@Neurologic] in order to support the gait diagnosis on patients.
### Acknowledgment {#acknowledgment .unnumbered}
The authors would like to thank the NSERC (Natural Sciences and Engineering Research Council of Canada) for having supported this work (Discovery Grant RGPIN-2015-05671).
[^1]: <http://www.iro.umontreal.ca/~labimage/GaitDataset>
[^2]: Prepared cylindrical histograms of size $16 \times 16$ are available at <http://www.iro.umontreal.ca/~labimage/GaitDataset/rawhists.zip> (unzip password: **8sxDaUEmUG**)
|
---
abstract: 'Comet 17P/Holmes underwent a massive outburst in 2007 Oct., brightening by a factor of almost a million in under 48 hours. We used infrared images taken by the Wide-Field Survey Explorer mission to characterize the comet as it appeared at a heliocentric distance of 5.1 AU almost 3 years after the outburst. The comet appeared to be active with a coma and dust trail along the orbital plane. We constrained the diameter, albedo, and beaming parameter of the nucleus to 4.135 $\pm$ 0.610 km, 0.03 $\pm$ 0.01 and 1.03 $\pm$ 0.21, respectively. The properties of the nucleus are consistent with those of other Jupiter Family comets. The best-fit temperature of the coma was 134 $\pm$ 11 K, slightly higher than the blackbody temperature at that heliocentric distance. Using Finson-Probstein modeling we found that the morphology of the trail was consistent with ejection during the 2007 outburst and was made up of dust grains between 250 $\mu$m and a few cm in radius. The trail mass was $\sim$ 1.2 - 5.3 $\times$ 10$^{10}$ kg.'
author:
- 'R. Stevenson\*, J. M. Bauer, E. A. Kramer, T. Grav, A. K. Mainzer, J. R. Masiero'
title: 'Lingering grains of truth around comet 17P/Holmes'
---
Introduction
============
Comet 17P/Holmes (hereafter 17P) has undergone 3 massive outbursts since its discovery in 1892 [@1892Obs....15..441H], most recently brightening by a factor of almost a million and becoming visible to the naked eye in 2007 Oct. The outbursts are likely thermally-driven since all three occurred 6-9 months after 17P passed through perihelion. Dynamically, 17P appears to be a typical Jupiter family comet (JFC) with a semi-major axis of 3.62 AU, eccentricity of 0.43, and inclination of 19$^{\circ}$.1. The comet is enigmatic given its propensity for unusually large outbursts but dynamical and physical properties similar to other JFCs.
The material ejected during the 2007 outburst included gas species, dust grains, and macroscopic fragments (e.g. ). Much of the smaller dust expanded in an almost spherical shell around the nucleus, while larger dust grains were observed to separate as a “blob” at a slower rate of $\sim$ 120 - 135 m s$^{-1}$ (e.g. ). [@2010Icar..208..276R] and detected a slower moving core-component of the largest grains that separated from the nucleus at a relative velocity of $\sim$ 7-9 m s$^{-1}$. Large dust grains may persist in the vicinity or along the trail of a comet for years after ejection from the nucleus [@1990Icar...86..236S; @1998ApJ...496..971L; @2011ApJ...738..171B]. In this work we used infrared (IR) images obtained with the Wide-Field Infrared Survey Explorer (WISE) to examine the evolution of 17P several years after the 2007 outburst.
WISE Observations and Reduction {#sec:wobs}
===============================
The WISE telescope launched in 2009 Dec. and conducted an all-sky survey over the following year. The 40 cm telescope covered a 47$^{\prime}$ $\times$ 47$^{\prime}$ field of view (FOV) in four IR bands simultaneously. The bands had central wavelengths of 3.4 $\mu$m, 4.6 $\mu$m, 12 $\mu$m, and 22 $\mu$m, and are referred to as W1, W2, W3, and W4, respectively. The median pixel scale in bands W1, W2, and W3 was 2$^{\prime\prime}$.8 pixel$^{-1}$, while 2 $\times$ 2 binned W4 images had a pixel scale of 5$^{\prime\prime}$.5 pixel$^{-1}$ [@2010AJ....140.1868W]. The effective exposure times were 7.7 s for W1 and W2 images, and 8.8 s for W3 and W4 images. The individual exposures and extracted sources from each frame were archived and searched using tools developed as part of the NEOWISE project [@2011ApJ...736..100M].
The data were initially processed by the WISE Science Data System, which removed the instrumental signatures and provided astrometric and photometric calibration. Astrometric accuracy was $\sim$ 0$^{\prime\prime}$.2, while absolute photometric accuracy was $\sim$ 5-10% [@2011ApJ...736..100M; @2012wise.rept....1C].
WISE pointed towards 17P a total of 14 times during the mission. Three of these sets of images were obtained after the cryogen had been depleted and thus only produced images at the two shortest wavelengths. Here we use the 11 sets of images in all 4 bands that were obtained within 24 hours between UT 2010 May 14 and 15 (Table \[table:obs\]). At this time 17P was at a heliocentric distance of 5.1 AU and a true anomaly of $\sim$ 170$^{\circ}$, approximately 5 months prior to reaching aphelion. These frames were aligned using the predicted orbital motion of the comet as calculated by the JPL Horizons ephemeris service and combined using the AWAIC (A WISE Astronomical Image Co-Adder) stacking algorithm [@2009ASPC..411...67M], which includes an outlier-rejection algorithm. By using the stacked images for this work, we increased the signal to noise ratio, and average over rotational variations, which may amount to 0.3 mag in R-band [@2006MNRAS.373.1590S]. The stacked images were resampled to have pixel scales of 1$^{\prime\prime}$ pixel$^{-1}$, corresponding to a projected on-sky distance of 3600 km pixel$^{-1}$, with PSFs having average full-width half-maxima (FWHM) of 6$^{\prime\prime}$.1, 6$^{\prime\prime}$.4, 6$^{\prime\prime}$.5, and 12$^{\prime\prime}$.0 in bands W1, W2, W3, and W4, respectively [@2010AJ....140.1868W]. Though the spacecraft did not track the comet’s motion, trailing is not a concern since the maximum motion of the comet during an exposure was 0.03$^{\prime\prime}$, significantly less than the FWHM or pixel scale of any image.
[cccccc]{} 17P/Holmes & 2010 May 14-15 & 11 & 5.13 & 4.93 & 11.3\
\[table:obs\]
To convert counts to fluxes we used the instrumental zero points given in [@2010AJ....140.1868W]. We revised the zero points by -8% in W3 and +4% in W4 to account for the observed discrepancy between red and blue calibrators. We corrected for the loss of light outside of fixed apertures by using the aperture corrections given in [@2011wise.rept....1C]. These amounted to -0.34 and -0.65 mag for W3 and W4 for apertures of radius 11$^{\prime\prime}$. In this work we chose to use apertures with radii of 11$^{\prime\prime}$ as a compromise between limiting the intrusion of background signal into the aperture and still capturing the majority of the PSF. Finally, we performed a color correction, which was necessary due to the wide band pass of the filters. We calculated the correction by interpolating the color corrections given for a range of temperatures in [@2010AJ....140.1868W] to the estimated blackbody temperature of 17P.
Results
=======
17P was detected in the longer wavelength W3 and W4 bands (Figure \[fig:holmes\]) but was not detected in bands W1 or W2. A dust trail was seen in both W3 and W4 images, though it was considerably brighter in W4. The modal average of the background was calculated $\sim$ 5$^{\prime}$ from the nucleus over an area of $\sim$ 2.2 square arcminutes and subtracted from the stacked images. We set 5 $\sigma$ upper limits on the signal from 17P in the W1 and W2 bands of 0.03 mJy and 0.10 mJy, respectively, using an 11$^{\prime\prime}$ radius aperture. The total fluxes within 11$^{\prime\prime}$ radius apertures were 0.77 $\pm$ 0.15 mJy and 8.97 $\pm$ 1.91 mJy in bands W3 and W4, respectively. Table \[table:fluxes\] shows the fluxes measured for 17P and the fluxes obtained by best-fit thermal models to the nucleus and coma signals (discussed in sections \[sec:nuc\] and \[sec:coma\], respectively).
[ccccc]{} 17P/Holmes, total flux & $<$ 0.03 & $<$ 0.10 & 0.77 $\pm$ 0.15 & 8.97 $\pm$ 1.91\
Nucleus (measured) & $<$ 0.03 & $<$ 0.10 & 0.33 $\pm$ 0.05 & 1.56 $\pm$ 0.31\
Nucleus (best-fit) & 1.92 $\times$ 10$^{-4}$ & 2.04 $\times$ 10$^{-4}$ & 0.27 & 1.68\
Coma (measured) & $<$ 0.03 & $<$ 0.10 & 0.45 $\pm$ 0.09 & 7.41 $\pm$ 1.58\
Coma (best-fit) & 5.91 $\times$ 10$^{-3}$ & 3.59 $\times$ 10$^{-3}$ & 0.45 & 7.41\
\[table:fluxes\]
Nucleus {#sec:nuc}
-------
The total signal was a mix of contributions from the nucleus and dust particles in the coma and trail around the nucleus. We separated those signals by fitting the non-PSF-like signal with an analytical function of the form F $\rho^{-n}$, where F is a scalar, $\rho$ is the distance from the nucleus, and $n$ is a power law index [@1999PhDT.........8F; @1999Icar..140..189L]. This model was centered on the nucleus and fitted for 120 azimuthal slices, each 3$^{\circ}$ in azimuth. We assumed that the coma behavior is constant near the nucleus and is well-modeled by a power law. We experimented with a range of annuli of varying positions and varying widths. The best fit was identified by examining the remaining “nucleus” signal and comparing its shape to a model PSF for the WISE images using a least-squares minimization technique. We elected to use annuli fitted between nucleo-centric distances of 11$^{\prime\prime}$ and 14$^{\prime\prime}$ for the W3 image, and 13$^{\prime\prime}$ and 29$^{\prime\prime}$ for the W4 image. The model of the coma was then subtracted, leaving the nucleus signal behind. The remaining nucleus signal is compared to the model PSF in Figure \[fig:PSFcomp\].
We used aperture photometry to investigate the extracted nucleus signal. The aperture was centered on the position of the nucleus as predicted by the JPL Horizons service. The signal from the nucleus was 0.33 $\pm$ 0.05 mJy and 1.56 $\pm$ 0.31 mJy in W3 and W4, respectively.
The W3 and W4 signals were fit to a Near-Earth Asteroid Thermal Model (NEATM; @1998Icar..131..291H). The model assumes that the nucleus is spherical and acts as a smooth Lambertian surface [@1977Icar...31..427C; @1985Icar...64...53B]. We assumed that the emissivity of the surface was 0.9, consistent with refractory materials [@1986Icar...68..239L]. The free-fit parameters were diameter ($D$) and beaming parameter ($\eta$). The best fit results were $D$ = 4.135 $\pm$ 0.610 km, $\eta$ = 1.03 $\pm$ 0.21. We adopted the absolute magnitude of the nucleus to be $H$ = 16.24 $\pm$ 0.02 as determined by [@2006MNRAS.373.1590S] when the comet appeared to be inactive in 2005. By coupling the absolute magnitude to the thermal fit we derive the albedo to be $p_{v}$ = 0.03 $\pm$ 0.01.
Coma {#sec:coma}
----
We used aperture photometry to investigate the dust coma. We subtracted the extracted nucleus signal reported in section \[sec:nuc\] from the results, giving coma fluxes of 0.45 $\pm$ 0.09 mJy in W3 and 7.41 $\pm$ 1.58 mJy in W4. We assumed that the signal is thermal emission, rather than reflected light, and fitted the data points at thermal wavelengths with a simple blackbody curve to determine the temperature. The best fit is shown in Figure \[fig:comfit\] and corresponds to a temperature of 134 $\pm$ 11 K, $\sim$ 10% higher than the local blackbody temperature.
![Coma temperature fit to signal from the dust coma of 17P/Holmes. The best-fit temperature is 134 $\pm$ 11 K, which is $\sim$ 10% higher than the local blackbody temperature.[]{data-label="fig:comfit"}](17Pcomatemp.eps)
Dust Trail {#sec:trail}
----------
We observed a broad dust trail lagging the nucleus of 17P in the W3 and W4 images. Given the extremely low surface brightness of the trail in W3, we restricted our analyses to the W4 data. The trail lay in the orbital plane of 17P and was observed to stretch over 3.1 $\times$ 10$^{6}$ km ($\sim$ 14$^{\prime}$.7) as projected on the sky.
In order to determine the best model, the trail shape was first characterized using the following method: (1) Using a 40$^{\prime\prime}$ annulus centered on the comet, unwrap the image in $r$ - $\theta$ space, (2) fit a Gaussian to the unwrapped data across radial bins of 2$^{\circ}$ width, giving a location that can be converted back to x-y space, (3) repeat the process along the length of the trail. This process compresses the trail into a series of 21 discrete points that can then be analytically compared to the models described below. Several combinations of annulus width (20$^{\prime\prime}$ to 50$^{\prime\prime}$ in steps of 10$^{\prime\prime}$) and radial bin size (1$^{\circ}$, 2$^{\circ}$, and 3$^{\circ}$) were considered, with 40$^{\prime\prime}$ annuli and 2$^{\circ}$ radial bins giving the least amount of scatter in the positions.
The dust trail was modeled using the Finson-Probstein method [@1968ApJ...154..327F], which assumes that the motion of cometary dust particles is controlled by Solar gravity, $F_{grav}$, and Solar radiation, $F_{rad}$. The motion can be parameterized using the ratio of the two forces, $\beta$:
$$\beta = \frac{F_{rad}}{F_{grav}} = \frac{5.76 \times 10^{-4}~Q_{pr}}{\rho_{d}~a_{d}}
\label{eq:betaratio}$$
where $Q_{pr}$ is the scattering efficiency, $\rho_{d}$ is the density of the particle \[kg m$^{-3}$\], and $a_{r}$ is the particle radius \[m\]. For grains with radii larger than the wavelength of observation $Q_{pr} \sim$ 1 [@1979Icar...40....1B]. Thus, $\beta$ depends on the inverse of particle diameter, i.e. for smaller grains, $\beta$ is larger, meaning the radiation pressure pushing the particles outwards has a larger effect than the gravitational force pulling them inwards. We integrated the motion of the dust particles over a period of 5 years. This generated a set of points that can be shown as curves of constant radius particles released at a range of times (syndynes) or curve of constant release date with a range of particle radii (synchrones).
For both the syndynes and synchrones, we calculated the RMS between each model and the fitted tail points. The lowest RMS value for any of the syndynes is higher than for any of the synchrones, thus we proceeded to fit the trail with a synchrone. The synchrone with the lowest RMS to the fitted trail points was determined to correspond to the best-fit particle ejection date. To constrain the error on the best-fit date, we computed the best-fit synchrone for each fitted point along the trail and computed the RMS between those synchrones and the overall best-fit synchrone. This analysis yielded a best-fit ejection date of 2007 Oct. 27 $\pm$ 221 days. The large error bars are due to some of the fitted trail points deviating significantly from the best-fit synchrone, and thus giving dramatically different best-fit ejection dates. Example syndynes and synchrones, as well as the best fit, are plotted in Figure \[fig:synchrones\]. It is possible that some of the largest grains were released on previous perihelion passages as [@2010Icar..208..276R] observed an old debris trail along the orbit of 17P using the Spitzer space telescope in 2008. However, the contribution is likely negligible given the good fit by the synchrone analysis.
The surface brightness of the trail was too low to estimate the size distribution of particles along its length, although we can set the minimum radius of observed particles to $\sim$ 250 $\mu$m. This is done by finding the syndyne that crosses the trail near the observed edge (14.3$^{\prime}$ from the nucleus). Using the same method, we estimate that particles near ($\sim$ 11$^{\prime\prime}$ from) the nucleus are on the order of a few cm in diameter. Smaller particles likely existed beyond the trail observed by WISE but are below the detection limit.
Discussion
==========
Physical properties of the nucleus
----------------------------------
The effective nucleus diameter of 4.135 $\pm$ 0.610 km calculated here is larger than previous estimates of 3.24 $\pm$ 0.02 km [@2006MNRAS.373.1590S] and 3.42 $\pm$ 0.14 km , derived when 17P appeared to be inactive or only weakly active prior to its 2007 outburst. Both of the previously mentioned results were determined from optical observations using an assumed albedo of 0.04. The diameter reported here is consistent with previously reported results when they are corrected using the NEOWISE-derived albedo.
We derived an albedo of 0.03 $\pm$ 0.01. The albedo of the nucleus is consistent with those measured for other comets, which generally occupy a narrow range between 0.02 and 0.06 [@2004come.book..223L]. We note that we are unable to derive an albedo for material in the trail as we do not have simultaneous high signal-to-noise observations at optical wavelengths. [@2010ApJ...714.1324I] used optical, near-IR, and mid-IR observations to constrain the albedo of the ejecta within a few days of the outburst. They found that the albedo of the material (as observed at a phase angle of 16$^{\circ}$) decreased during their observations from 0.12 $\pm$ 0.04 to 0.032 $\pm$ 0.014 and suggested that sublimating volatiles would lower the albedo. [@2012ApJ...760L...2L] estimated the geometric albedo of the dust in the coma as 0.006 $\pm$ 0.002, also at a phase angle of 16$^{\circ}$. Such a low value is not unheard-of [@2002Sci...296.1087S; @2004Icar..167...37N], though it does not match well with results from [@2010ApJ...714.1324I]. The discrepancy may be due to different populations of grains dominating the thermal emission in the IR and the stellar extinction at optical wavelengths. Our result for the albedo of the nucleus is generally consistent with the albedo of the ejecta observed in 2007 when the albedo was determined from combined optical and mid-IR wavelengths.
The beaming parameter of 1.03 $\pm$ 0.21 is consistent with the average value of 1.03 $\pm$ 0.11 reported for 57 JFCs by the SePPCoN survey, which used thermal infrared measurements by the Spitzer Space Telescope [@2013Icar..226.1138F]. They found that beaming parameters for 57 JFC nuclei were approximately normally distributed and suggested that there appears to be little variation among bulk thermal properties of JFCs. As discussed in [@2013Icar..226.1138F], a beaming parameter close to 1.0 implies low thermal inertia and little nightside emission. 17P appears typical in this regard.
Thermal emission in the coma
----------------------------
The best-fit coma temperature of 134 $\pm$ 11 K is $\sim$ 10% warmer than the temperature expected for an ideal blackbody ($T_{BB}$) at a heliocentric distance of 5.13 AU (123 K assuming $T_{BB}$ $\propto$ 278 K $r_{H}^{-0.5}$; @1992Icar..100..162G). Previous results from observations taken close to the time of outburst ($r_{H}$ $\sim$ 2.4 AU) have also suggested that the dust temperature of the ejecta exceeded the local blackbody temperature of $\sim$ 180 K. [@2009AJ....137.4538Y] reported a dust temperature near the nucleus of 360 $\pm$ 40 K using near-IR observations obtained with the NASA Infrared Telescope Facility several days after the outburst, while mid-IR results obtained around the same time suggested cooler temperatures between 172 K and 200 K [@2010ApJ...714.1324I; @2009PASJ...61..679W]. Spitzer Space Telescope observations obtained on 2007 Nov. 10 resulted in an estimated temperature of 260 K for the near-nucleus dust [@2010Icar..208..276R]. Most of these results are higher than the estimated blackbody temperature to varying degrees, matching well with our results here.
Numerous IR observations of comets have shown that it is common, perhaps even the norm, for comae and dust tail and trail temperatures to exceed the temperature expected for a co-located blackbody (e.g. @1988AJ.....96.1971T [@1992Icar..100..162G; @1998ApJ...496..971L; @2000ApJ...538..428H]). Generally, excess emission at IR wavelengths is attributed to either small grains ($\lesssim$ 1$\mu$m) that are unable to radiate efficiently at IR wavelengths, rough surfaced grains that are more emissive than the smooth spherical grains modeled by the blackbody temperature, or larger grains that maintain a thermal gradient across their surface [@1982come.coll..341C; @1990Icar...86..236S; @2001indu.book...95S]. In the case of 17P, all of these effects may be present. The nucleus was seen to remain active in the months and years following the outburst, likely releasing small dust grains from the surface (@2012AJ....144..138S; Snodgrass, private communication). Based on results from the best-fit synchrone determined in section \[sec:trail\], particles larger than a few cm in diameter would still be close enough to the nucleus to contribute to the excess thermal emission observed here.
An old trail
------------
The morphology of the trail is consistent with being debris ejected during the 2007 outburst and observations by [@2013ApJ...778...19I] that showed large dust grains following the comet around aphelion in Oct. 2010. We constrained the range of particle diameters observed between $\sim$ 250 $\mu$m and a few cm. The larger grain diameters are consistent with the sizes of grains observed in a slow-moving “core” near the nucleus just a few days after the outburst, which were determined to be $\gtrsim$ 200 $\mu$m .
We measured the flux along the trail using a box aperture that extends between 11$^{\prime\prime}$ and 880$^{\prime\prime}$ from the nucleus and has a width perpendicular to the length of the trail of 52$^{\prime\prime}$. The flux was calibrated and color-corrected as described in section \[sec:wobs\]. To correct for light potentially lost outside of the large aperture, we applied an aperture correction of -0.03 mag derived by [@2013AJ....145....6J]. To estimate the cross-section of material present we used the following relation from [@2005Icar..179..158M]:
$$\sigma_{\lambda} = \frac{F_{\lambda} ~ \Delta^{2}}{B_{\lambda}(T)}
\label{eq:cross sec}$$
where $\sigma_{\lambda}$ is the cross-section of material observed at wavelength $\lambda$ (in this case, 22 $\mu$m, or W4), F$_{\lambda}$ is the observed flux, $\Delta$ is the geocentric distance, and $B_{\lambda}(T)$ is the Planck function at temperature $T$. We were unable to constrain the temperature of the dust along the trail as the signal in W3 is too low to fit a Planck function to. We therefore assumed that the temperature is between the expected blackbody temperature of 123 K and the measured coma temperature of 134 K. This is consistent with the previously-discussed finding that many comet trails are at or exceed local blackbody temperatures. We also assumed that the temperature is constant along the trail and is not dependent on the size of the dust grains present. The cross-section of material was 1.5 $\times$ 10$^{9}$ m$^{2}$ in the case of the local blackbody temperature or 10$^{9}$ m$^{2}$ in the higher temperature case. We used previously measured minimum and maximum particle sizes ($a-$, $a+$) of 250 $\mu$m and 3 cm, and assumed that the differential size distribution of particles follows a power law of the form $n(a)~da \propto a^{-q}~da$, with the value of $q$ set between 2.2 and 3.4, as measured by [@2010Icar..208..276R] and , respectively. The mean particle size within the trail was given by:
$$\bar{a} = \frac{\int_{a-}^{a+} \pi a^{3} n(a) da}{\int_{a-}^{a+} \pi a^{2} n(a) da}
\label{eq:area}$$
The mass within the observed trail was then given by:
$$M = \frac{4~\rho~\bar{a}~\sigma_{\lambda}}{3}
\label{eq:mass}$$
where $\rho$ is the bulk density of the grains and was assumed to be 1000 kg m$^{-3}$ [@1991ASSL..167...19J]. The mass in the trail was $\sim$ 1.2 - 5.3 $\times$ 10$^{10}$ kg. This represented approximately 1 - 100% of the total ejected mass .
Assuming an average grain size of 200 $\mu$m, @2010Icar..208..276R estimated the mass of the slow-moving core seen in 2007 to be $\sim$ 4 $\times$ 10$^{9}$ kg. estimated the mass to be significantly higher at $\sim$ 0.7-4 $\times$ 10$^{11}$ kg by summing over an estimated particle size distribution with $a_{-}$ = 0.1 $\mu$m, 10 $< a_{+} <$ 1000 mm, and -3.3 $< q <$ -3.0. Thus, the dust trail observed by WISE represented 3 - 75% of the core modeled by in 2007.
Why did 17P outburst?
---------------------
The overarching question remains to be answered: why does 17P undergo massive outbursts when most JFCs experience only mild mass loss? The diameter, beaming parameter, and albedo of the nucleus are similar to those of other JFC nuclei. The volatile species observed shortly after the outburst in 2007 similarly fail to provide any obvious clues about the cause of the outburst. Relative abundances of CN, C$_{2}$, C$_{3}$ and NH and the isotopic ratios of $^{12}$C/$^{13}$C and $^{14}$N/$^{15}$N in CN and HCN were similar to those observed for other comets [@2008ApJ...679L..49B; @2009AJ....138.1062S]. Several species, including C$_{2}$H$_{6}$, HCN, CH$_{3}$OH, and C$_{2}$H$_{2}$, were enhanced with respect to H$_{2}$O although only by a factor of a few [@2008ApJ...680..793D].
The perihelion distance of 17P changed from 2.17 AU in 2000 to 2.05 AU in 2007 following a close encounter with Jupiter. The change resulted in a $\sim$ 10% increase in solar insolation at the surface. The small difference may have caused the thermal wave to propagate deeper than on previous perihelion passages, reaching previously unheated pockets of volatiles. A runaway exothermic phase transition of amorphous water ice to crystalline is probably insufficient to cause the outburst [@2010Icar..207..320K]. If supervolatiles such as CO and/or CO$_{2}$ are trapped within the amorphous ice and heated sufficiently, the resulting gas production may be able to drive such activity, if the gas can build up sufficient internal pressure [@2009AJ....138.1062S; @2009ICQ....31...99S; @2011Icar..212..847K; @2012Icar..221..147H].
It is possible that the nucleus of 17P has unusually high tensile strength that allows gas pressure to build up in the interior before releasing the energy in a sudden outburst upon surface failure. [@2010Icar..208..276R] suggested that the nucleus must have a strength between 10 - 100 kPa in order to have survived the 2007 outburst. However, previous studies of other comets suggest much lower strengths for JFCs. 16P/Brooks 2 and D/1993 F2 (Shoemaker-Levy 9) both underwent tidal splitting during close encounters with Jupiter leading to estimates of 0.1 kPa and $\sim$ 0.38 kPa for the tensile strengths of the nuclei . Based on observations of mini-outbursts, [@2008Icar..198..189B] estimated the strength of the sub-surface material of 9P/Tempel 1 to be not much more than 0.01 - 0.1 kPa, while [@2005Sci...310..258A] found that the strength of the surface must also be extremely low ($<$ 0.065 kPa). Only a few comets have estimated tensile strengths and are not necessarily representative of all JFCs. We note simply that the estimated tensile strength required of 17P is an order of magnitude higher than those estimated for other JFCs.
Summary
=======
We used wide-field IR images obtained by the WISE mission in 2010 May to characterize 17P. Years later, 17P still exhibited evidence of the 2007 outburst. Our results suggest that 17P is a JFC with a typical diameter, albedo, and beaming parameter, but atypical outgassing behavior.
1. [The diameter, albedo, and beaming parameter of the nucleus of 17P were constrained to values of 4.135 $\pm$ 0.610 km, 0.03 $\pm$ 0.01, and 1.03 $\pm$ 0.21, respectively. The physical and bulk thermal properties of the nucleus appear to be consistent with those of other JFCs.]{}
2. [The temperature of dust near the nucleus was 134 $\pm$ 11 K, slightly higher than the local blackbody temperature. Possible explanations for the elevated temperature include emission from small sub-micron grains that cannot effectively radiate at IR wavelengths, or contributions from larger dust grains that maintain a temperature gradient across their surface. Both effects may have been present at the time of observation.]{}
3. [17P was observed to have a debris trail in 2010 May. Dynamical modeling of the dust suggests that this was leftover from the massive 2007 outburst. The range of grain diameters observed is 800 $\mu$m to a few cm. The mass of trail was estimated at 1.2 - 5.3 $\times$ 10$^{10}$ kg, which represents $\sim$ 1 - 100% of the total mass ejected during the 2007 outburst.]{}
Acknowledgments
===============
This publication makes use of data products from the Wide-field Infrared Survey Explorer and NEOWISE, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. RS acknowledges support from the NASA Postdoctoral Fellowship Program. EK was supported by the JPL Graduate Fellowship Program and the NASA Earth and Space Sciences Fellowship program. This research was funded in part by a grant from NASA through the Near Earth Object Observations Program for the NEOWISE project, and was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
natexlab\#1[\#1]{}
A’Hearn, M. F., Belton, M. J. S., Delamere, W. A., et al. 2005, Science, 310, 258
Asphaug, E., & Benz, W. 1996, , 121, 225
Bauer, J. M., Walker, R. G., Mainzer, A. K., et al. 2011, , 738, 171
Bockel[é]{}e-Morvan, D., Biver, N., Jehin, E., et al. 2008, , 679, L49
Boissier, J., Bockel[é]{}e-Morvan, D., Biver, N., et al. 2012, , 542, A73
Belton, M. J. S., Feldman, P. D., A’Hearn, M. F., & Carcich, B. 2008, , 198, 189
Brown, R. H. 1985, , 64, 53
, J. A., [Lamy]{}, P. L., & [Soter]{}, S. 1979, , 40, 1
Campins, H., & Hanner, M. S. 1982, IAU Colloq. 61: Comet Discoveries, Statistics, and Observational Selection, 341
Crovisier, J., Biver, N., Bockel[é]{}e-Morvan, D., & Colom, P. 2009, , 57, 1162
Cruikshank, D. P., & Jones, T. J. 1977, , 31, 427
Cutri, R. M., Wright, E. L., Conrow, T., et al. 2011, Explanatory Supplement to the WISE Preliminary Data Release Products, 1
Cutri, R. M., Wright, E. L., Conrow, T., et al. 2012, Explanatory Supplement to the WISE All-Sky Data Release Products, 1
Dello Russo, N., Vervack, R. J., Jr., Weaver, H. A., et al. 2008, , 680, 793
Fern[á]{}ndez, Y. R. 1999, Ph.D. Thesis
Fern[á]{}ndez, Y. R., Kelley, M. S., Lamy, P. L., et al. 2013, , 226, 1138
Finson, M. J., & Probstein, R. F. 1968, , 154, 327
Gehrz, R. D., & Ney, E. P. 1992, , 100, 162
Harris, A. W. 1998, , 131, 291
Hayward, T. L., Hanner, M. S., & Sekanina, Z. 2000, , 538, 428
Hillman, Y., & Prialnik, D. 2012, , 221, 147
Hsieh, H. H., Fitzsimmons, A., Joshi, Y., Christian, D., & Pollacco, D. L. 2010, , 407, 1784
Holmes, E. 1892, The Observatory, 15, 441
Ishiguro, M., Watanabe, J.-i., Sarugaku, Y., et al. 2010, , 714, 1324
Ishiguro, M., Kim, Y., Kim, J., et al. 2013, , 778, 19
Jarrett, T. H., Masci, F., Tsai, C. W., et al. 2013, , 145, 6
Jewitt, D. 1991, IAU Colloq. 116: Comets in the post-Halley era, 167, 19
Kossacki, K. J., & Szutowicz, S. 2010, , 207, 320
Kossacki, K. J., & Szutowicz, S. 2011, , 212, 847
Lacerda, P., & Jewitt, D. 2012, , 760, L2
Lamy, P. L., Toth, I., Fern[á]{}ndez, Y. R., & Weaver, H. A. 2004, Comets II, 223
Lamy, P. L., Toth, I., Weaver, H. A., A’Hearn, M. F., & Jorda, L. 2009, , 508, 1045
Lebofsky, L. A., Sykes, M. V., Tedesco, E. F., et al. 1986, , 68, 239
Li, J., Jewitt, D., Clover, J. M., & Jackson, B. V. 2011, , 728, 31
Lin, Z.-Y., Lin, C.-S., Ip, W.-H., & Lara, L. M. 2009, , 138, 625
Lisse, C. M., A’Hearn, M. F., Hauser, M. G., et al. 1998, , 496, 971
Lisse, C. M., Fern[á]{}ndez, Y. R., Kundu, A., et al. 1999, , 140, 189
Mainzer, A., Grav, T., Masiero, J., et al. 2011, , 736, 100
Masci, F. J., & Fowler, J. W. 2009, Astronomical Data Analysis Software and Systems XVIII, 411, 67
Min, M., Hovenier, J. W., de Koter, A., Waters, L. B. F. M., & Dominik, C. 2005, , 179, 158
Montalto, M., Riffeser, A., Hopp, U., Wilke, S., & Carraro, G. 2008, , 479, L45
Nelson, R. M., Soderblom, L. A., & Hapke, B. W. 2004, , 167, 37
Reach, W. T., Vaubaillon, J., Lisse, C. M., Holloway, M., & Rho, J. 2010, , 208, 276
Schleicher, D. G. 2009, , 138, 1062
Sekanina, Z., & Yeomans, D. K. 1985, , 90, 2335
Sekanina, Z., Chodas, P. W., & Yeomans, D. K. 1998, , 46, 21
Sekanina, Z., Hanner, M. S., Jessberger, E. K., & Fomenkova, M. N. 2001, Interplanetary Dust, Edited by E. Gr[ü]{}n, B.A.S. Gustafson, S. Dermott, and H. Fechtig. Astronomy and Astrophysics Library. 2001, 804 p., ISBN: 3-540-42067-3. Berlin: Springer, 2001, p. 95, 95
Sekanina, Z. 2008, International Comet Quarterly, 30, 3
Sekanina, Z. 2009, International Comet Quarterly, 31, 99
Snodgrass, C., Lowry, S. C., & Fitzsimmons, A. 2006, , 373, 1590
Soderblom, L. A., Becker, T. L., Bennett, G., et al. 2002, Science, 296, 1087
Stevenson, R., Kleyna, J., & Jewitt, D. 2010, , 139, 2230
Stevenson, R., & Jewitt, D. 2012, , 144, 138
Sykes, M. V., Lien, D. J., & Walker, R. G. 1990, , 86, 236
Tokunaga, A. T., Golisch, W. F., Griep, D. M., Kaminski, C. D., & Hanner, M. S. 1988, , 96, 1971
Watanabe, J.-I., Honda, M., Ishiguro, M., et al. 2009, , 61, 679
Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, , 140, 1868
Yang, B., Jewitt, D., & Bus, S. J. 2009, , 137, 4538
|
---
abstract: |
We compute the class of arithmetic genus two Teichmüller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants.
The main technical tool is the computation of divisor classes of Hilbert Jacobi forms on the universal abelian surface over the pseudo-Hilbert modular surface.
address: 'Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany'
author:
- André Kappes and Martin Möller
bibliography:
- 'bib\_irred.bib'
title: Cutting out arithmetic Teichmüller curves in genus two via Theta functions
---
|
---
abstract: 'We study the relation between the scale of chiral symmetry spontaneously breaking and constituent quark mass. We argue that this relation partly reveals strong interaction origination of chiral symmetry breaking. We show that the relation can be obtained via checking unitarity region of low-energy effective field theory of QCD. This effective field theory must manifestly include consistent quark mass as parameter. Thus we derive this effective field theory from naive chiral constituent quark model. The phenomenological value obtained by this method agree with usual one determined by pion decay constant.'
author:
- 'Mu-Lin Yan[^1], Yi-Bin Huang[^2]'
- 'Xiao-Jun Wang'
title: The Scale on Chiral Symmetry Breaking
---
The typical difficulty on studies on QCD is from its dramatic properties when dynamics of QCD lies in non-perturbative region. The analysis of renormalization group shows that QCD is asymptotic freedom at high energy scale, but should lie in confinement phase at very low energy. Consequently, a phase transition must occur when energy scale varies from higher to lower one. The phase transition are dynamically characterized by well-known fermion (quark) condensation phenomena. A dynamical scale (it is usually referred as $\Lambda_{\rm QCD}$ in QCD) is consequently generated by the quark condensation. It is just order parameter associating the phase transition. Sometimes this scale is also transferred to another effective parameter: so-called constituent quark mass $m$ and treated it as order parameter. Focusing on dynamics of QCD with light flavor quarks only, however, the story is more complicated: The quark condensation also breaks the (approximate) global chiral symmetry of QCD. Accompanying with the chiral symmetry spontaneously breaking (CSSB), another scale $\Lambda_{\rm CSSB}$ must be dynamically generated and Goldstone bosons appear as dynamical degrees of freedom. CSSB is one of the most important features for the hadron physics. It together with color confinement governs full low-energy dynamics of QCD. A typical example is success of chiral perturbation theory (ChPT)[@GL85a].
An interesting issue is that global chiral symmetry is broken due to pure strong interaction. To localize the global chiral symmetry one has to introduce electroweak interactions. Because the quark condensation breaks both of local and global chiral symmetry, CSSB actually involves both of strong and electroweak interactions (in contrast to color confinement caused by pure strong interaction). The fact of electroweak relevance of CSSB has been shown in determination on CSSB scale via weak decay constant of pion [@MG84], i.e., $\Lambda_{\rm CSSB}\sim 2\pi F_{\pi}\sim
1.2$GeV. The studies on role of strong interaction in CSSB, however, seems to be more difficult, since complete understanding on this issue requires underlying knowledge on dynamical mechanics of color confinement. During the past decades, CSSB has been extensively studied along this way, i.e., so called the formalism of the gap equations (or Schwinger-Dyson equations, see refs.[@Farhi; @Roberts; @Ripka; @Hatsuda] and the references within, and [@Cheng]). This method is rigorous and achieves some successes, but still far from our final expectation so far. Alternately, it should be also possible to explore CSSB by starting within confinement phase. The key point is to find relation between $\Lambda_{\rm CSSB}$ and constituent quark mass $m$ (or $\Lambda_{\rm QCD}$). It will partly reflect the role of strong interaction in CSSB. This is just purpose of this letter. In this phase, the dynamical description is replaced by effective one with constituent quarks and Goldstone bosons. In such effective description a critic-like energy scale must exist. Above this energy scale, this effective description on the system collapses and below it, the description works. This critic-like energy scale should be just the scale of CSSB, $\Lambda_{\rm CSSB}$.
The natural criterion on whether an quantum effective field theory (QEFT) description collapses or not is to check unitarity of the QEFT. This claim bases on the fact that QEFT does not describe full degrees of freedom of fundamental theory. When energy is higher than characteristic scale of the QEFT, some new degrees of freedoms will be excited consequently the unitarity of the QEFT is lost. In this letter, we will derive a low-energy QEFT of pure meson interaction from naive chiral constituent quark model, and to obtain $\Lambda_{\rm CSSB}$ via checking unitarity region of that QEFT.
It is well-known that a low energy effective meson theory should be a well-defined perturbative theory in $N_c^{-1}$ expansion[@tH74]. Therefore, unitarity condition of $S$-matrix, or optical theorem, has to satisfied order by order in powers of $N_c^{-1}$ expansion, $$\begin{aligned}
\label{1}
{\rm Im}({\cal T}_{\beta,\alpha})_n=\frac{1}{2}\sum_{\rm
all\;\gamma}\sum_{m\leq n}({\cal T}_{\gamma,\alpha})_m
({\cal T}_{\gamma,\beta}^*)_{n-m}.\end{aligned}$$ where the ${\cal T}_{\beta,\alpha}$ is transition amplitude from state initial $\alpha$ to final state $\beta$, and $\gamma$ denotes all possible intermediate states on mass shells, and ${\cal T}_n\sim O((1/\sqrt{N_c})^n)$. According to standard power counting law on large $N_c$ expansion in meson interaction[@tH74; @Dono90], any transition amplitudes with $n_V$ vertices, $n_e$ external meson lines, $n_i$ internal meson lines and $n_l$ loops of mesons are of order $$\begin{aligned}
\label{2}
N_c^{n_V-n_i-n_e/2}=(N_c^{-\frac{1}{2}})^{2n_l+n_e-2},\end{aligned}$$ where topological relation $n_l=n_i-n_V+1$ has been used. We focus on transition amplitude from single meson initial state $\alpha$ to $k$ mesons final state $\beta=\{\beta_1,\beta_2,\cdots,\beta_k\}$. Assuming intermediate state $\gamma$ includes $s$ mesons $\{\gamma_1,\gamma_2,\cdots,\gamma_s\}$, then using the power counting rule (\[2\]), eq. (\[1\]) can be written as $$\begin{aligned}
\label{3}
{\rm Im}({\cal
T}_{\beta,\alpha})_{(2n_l+k-1)}=\frac{1}{2}\sum_{{\rm
all}\;\gamma(s)}\sum_{n'}
({\cal T}_{\gamma,\alpha})_{(2n'_l+s-1)}
({\cal T}_{\gamma,\beta}^*)_{(2n''_l+s+k-2)},\end{aligned}$$ where $n_l$, $n'_l$ and $n''_l$ are meson loop numbers in transition amplitude ${\cal T}_{\beta,\alpha}$, ${\cal
T}_{\gamma,\alpha}$ and ${\cal T}_{\gamma,\beta}$ respectively. Both side of eq. (\[3\]) should be of the same order, thus $$\begin{aligned}
\label{4}
n'_l+n''_l+s=1+n_l.\end{aligned}$$ For the case of leading order of transition amplitude ${\cal
T}_{\beta,\alpha}$, i.e., $n_l=0$, we have $n'_l=n''_l=0$ and $s=1$ according to eq. (\[4\]). Consequently only $\gamma=\alpha$ is allowed at the leading order. Since meson fields are free point-particle at limit $N_c\rightarrow\infty$[@tH74], we have $({\cal
T}_{\alpha,\alpha})_0\equiv 0$. Therefore, it can be claimed that, if any effective meson theory is unitary below its characteristic scale, the on-shell transition amplitude from any meson state to any multi-meson state must be real at leading order of $N_c^{-1}$ expansion, $$\begin{aligned}
\label{5}
{\rm Im}({\cal T}^{(0)}_{\beta,\alpha})_{k-1}=0.\end{aligned}$$ where the superscript $(0)$ denotes the leading order of $N_c^{-1}$ expansion. This claim will serve as equivalent description of unitarity for any QEFTs on meson interaction.
A convenient effective description on the low energy QCD is naive chiral constituent quark model (ChQM) proposed in ref.[@MG84]. The constituent quark mass as order parameter associating to phase transition is manifestly appear in this model. Thus this model provides a possible framework to explore relation between $\Lambda_{\rm CSSB}$ and order parameter. The simplest ChQM is parameterized by the following SU(3)$_{V}$ invariant Lagrangian $$\begin{aligned}
\label{6}
{\cal L}_{\rm ChQM}&=&i\bar{q}({\slash\!\!\! {\partial}}+{\slash\!\!\! \Gamma}+
g_{A}{\slash\!\!\!\!\Delta}\gamma_5-i{\slash\!\!\! V})q-m\bar{q}q
-\bar{q}Sq-\kappa\bar{q}P\gamma_5q \nonumber \\
&&+\frac{F^2}{16}<\nabla_\mu U\nabla^\mu U^{\dag}>
+\frac{1}{4}m_0^2<V_\mu V^{\mu}>.\end{aligned}$$ Here $V_\mu$ are vector meson octet, $<\cdots>$ denotes trace in SU(3) flavor space, $\bar{q}=(\bar{u},\bar{d},\bar{s})$ are constituent quark fields, $g_{A}=0.75$ is fitted by beta decay of neutron, and $$\begin{aligned}
\label{7}
\Delta_\mu&=&\frac{1}{2}[\xi^{\dag}({\partial}_\mu-ir_\mu)\xi
-\xi({\partial}_\mu-il_\mu)\xi^{\dag}], \nonumber \\
\Gamma_\mu&=&\frac{1}{2}[\xi^{\dag}({\partial}_\mu-ir_\mu)\xi
+\xi({\partial}_\mu-il_\mu)\xi^{\dag}], \nonumber \\
\nabla_\mu U&=&{\partial}_\mu U-ir_\mu U+iUl_\mu=2\xi\Delta_\mu\xi,
\\
\nabla_\mu U^{\dag}&=&{\partial}_\mu U^{\dag}-il_\mu
U^{\dag}+iU^{\dag}r_\mu
=-2\xi^{\dag}\Delta_\mu\xi^{\dag}, \nonumber\\
S&=&\frac{1}{2}(\xi^{\dag}{\tilde}{\chi}\xi^{\dag}+\xi{\tilde}{\chi}^{\dag}\xi),
\hspace{1in}
P=\frac{1}{2}(\xi^{\dag}{\tilde}{\chi}\xi^{\dag}-\xi{\tilde}{\chi}^{\dag}\xi),
\nonumber\end{aligned}$$ where $l_\mu=v_\mu+a_\mu$ and $r_\mu=v_\mu-a_\mu$, ${\tilde}{\chi}=s_{\rm ext}+{\cal M}+ip$ with external fields $v_\mu$ (vector), $a_\mu$ (axial-vector), $s_{\rm ext}$ (scalar), $p$ (pseudoscalar), and current quark mass matrix ${\cal M}={\rm
diag}\{m_u,m_d,m_s\}$ respectively. $\xi$ associates with non-linear realization of spontaneously broken global chiral symmetry $G=SU(3)_L\times SU(3)_R$ introduced by Weinberg [@Wein68], $$\label{8}
\xi(\Phi)\rightarrow
g_R\xi(\Phi)h^{\dag}(\Phi)=h(\Phi)\xi(\Phi)g_L^{\dag},\hspace{0.5in}
g_L, g_R\in G,\;\;h(\Phi)\in H=SU(3)_{V}.$$ Explicit form of $\xi(\Phi)$ is usually taken as $$\label{9}
\xi(\Phi)=\exp{\{i\lambda^a \Phi^a(x)/2\}},\hspace{1in}
U(\Phi)=\xi^2(\Phi),$$ where $\lambda^1,\cdots,\lambda^8$ are SU(3) Gell-Mann matrices in flavor space, and the Goldstone bosons $\Phi^a$ are identified to pseudoscalar meson octet.
The transformation law under SU(3)$_{V}$ for any quantities defined in eqs. (\[6\]) and (\[8\]) are $$\begin{aligned}
\label{10}
q&{\longrightarrow}& h(\Phi)q, \hspace{0.6in}
\Delta_\mu{\longrightarrow}h(\Phi)\Delta_\mu h^{\dag}(\Phi), \hspace{0.6in}
V_\mu\rightarrow h(\Phi)V_\mu h^{\dag}(\Phi), \nonumber \\
\Gamma_\mu & {\longrightarrow}& h(\Phi)\Gamma_\mu
h^{\dag}(\Phi)+h(\Phi){\partial}_\mu
h^{\dag}(\Phi).\end{aligned}$$ The homogenous transformation law on vector meson field is usually referred as WCCWZ realization on vector meson.[@Wein68; @WCCWZ]
Because there is no kinetic term for vector fields in ${\cal
L}_{\rm ChQM}$, they serve as auxiliary fields in this formalism. From the equation of motion $\delta{\cal L}_{\rm ChQM}/\delta
V_{\mu}=0$, we can see the vector fields in ${\cal L}_{\rm ChQM}$ are the composite fields of constituent quarks, Therefore, WCCWZ method is actually a way to catch the effects of constituent quark bound states in the ChQM. $F,\;g_{A},\;m,\;\kappa$ and $m_0$ in eq. (\[6\]) are free parameters of the model.
The effective action on meson interaction, $S_{\rm eff}[U,V]$, can be obtained via integrating out quark fields, $$\begin{aligned}
\label{11}
S_{\rm eff}[U,V]=\ln \det ({\cal D})+\int d^4x
\{\frac{F^2}{16}<\nabla_\mu U\nabla^\mu U^{\dag}>
+\frac{1}{4}m_0^2<V_\mu V^{\mu}>\},\end{aligned}$$ where ${\cal D}={\slash\!\!\! {\partial}}+{\slash\!\!\! \Gamma}+
g_{A}{\slash\!\!\!\!\Delta}\gamma_5-i{\slash\!\!\! V}-m
-S-\kappa P\gamma_5$, $F$ and $m_0$ will receive quark loop effects and then are renormalized into $F_\pi=186{\rm MeV}$ and the physical masses $m_V$ of vector mesons respectively. Then $S_{\rm eff}[U,V]$ parameterizes an QEFT on pure meson interaction.
Now let us consider unitarity of this QEFT. In particular, we focus on $V\to\Phi\Phi$ decay amplitude and impose eq. (\[5\]) to find unitarity region of the QEFT. To separate relevant effective action from $S_{\rm eff}[U,V]$ and rewrite it into appropriate form $$\begin{aligned}
\label{12}
S_{\rm eff}^{V\Phi\Phi}= \sum\limits_{abc}\int\frac{d^4p d^4q_1
d^4q_2}{(2\pi2\pi)^4}\delta(p+q_1+q_2)
V_\mu^{ab}(p)\Phi^{bc}(q_1)\Phi^{ca}(q_2)q_2^\mu
f_{abc}(p^2,q_1^2,q_2^2),\end{aligned}$$ we have $$\begin{aligned}
\label{13}
{\cal T}_{\Phi\Phi, V}^{(0)}\equiv
<\Phi^{bc}(q_1)\Phi^{ca}(q_2)|{\cal
T}^{(0)}|V^{ab}(p,\lambda)>=(2\pi)^4\delta^4(p-q_1-q_2)q_2^\mu
\epsilon_\mu^\lambda f_{abc}(p^2,q_1^2,q_2^2).\end{aligned}$$ where $\epsilon_\mu^\lambda $ is the polarization vector of the vector meson $V^{ab}(p,\lambda)$. Consequently $$\begin{aligned}
\label{14}
{\rm Im}{\cal T}^{(0)}_{\Phi\Phi,V}\propto {\rm Im}
f_{abc}(p^2,q_1^2,q_2^2).\end{aligned}$$ The form factor $f_{abc}(p^2,q_1^2,q_2^2)$ can be rewritten as $f_{abc}(p^2,q_1^2,q_2^2)=f_2(p^2)+f_3(p^2,q_1^2,q_2^2)$, where $f_2$ and $f_3$, with subscript $abc$ suppressed, are two-point Green function (Fig. 1-a) and three-point Green function (Fig.1-b) of constituent quark fields respectively, and are linearly independent.
{width="5in"}
Explicitly, the calculations on form factors $f_2(p^2)$ and $f_3(p^2,q_1^2,q_2^2)$ are straightforward, $$\begin{aligned}
\label{15}
f_2(p^2)&=&
i\int\frac{d^4k}{(2\pi)^4}\frac{g(k,p)}{[(k-p)^2-M_b^2+i{\epsilon}]
(k^2-M_a^2+i{\epsilon})} =\int_0^1 dx\int_0^\infty du\frac{{{\tilde}g}(x,u,p^2)}{(u+D_2-i{\epsilon})^2},
\nonumber \\
f_3(p^2,q_1^2,q_2^2)&=&
i\int\frac{d^4k}{(2\pi)^4}\frac{h(k,p,q_1,q_2)}{[(k+q_1)^2-M_b^2+i{\epsilon}]
(k^2-M_c^2+i{\epsilon})[(k-q_2)^2-M_a^2+i{\epsilon}]}\nonumber \\
&=&\int_0^1 xdx\int_0^1 dy\int_0^\infty du\frac{{{\tilde}h}(x,y,u,p^2,q_1^2,q_2^2)}{(u+D_3-i{\epsilon})^3},\end{aligned}$$ where $g,\;h$ and ${{\tilde}g},\; {\tilde}{h}$ are definite real and polynomial functions of $u=k_E^2$ ($k_E^\mu=-ik_0,k_x,k_y,k_z$), $M_{a}=m+m_{a}\;\;(a=u,d,s)$ and $$\begin{aligned}
\label{16}
D_2&=&M_a^2(1-x)+M_b^2x-p^2x(1-x), \nonumber \\
D_3&=&M_a^2x(1-y)+M_b^2(1-x)+M_c^2xy-p^2x(1-x)(1-y)
-q_1^2xy(1-x)-q_2^2x^2y(1-y).\end{aligned}$$ Using principle value formula $$\begin{aligned}
\label{17}
\frac{1}{z\pm i{\epsilon}}=\frac{{\cal P}}{z}\mp i\pi\delta(z),\end{aligned}$$ where $\frac{{\cal P}}{z}$ is the principle value and is real, we can express $f_2$ and $f_3$ as $$\begin{aligned}
\label{18}
f_2(p^2)&=&-\int_0^1 dx\int_0^\infty du{{\tilde}g}\frac{{\partial}}{{\partial}u}\Big(\frac{{\cal P}}{u+D_1}+i\pi\delta(u+D_1)\Big), \nonumber \\
f_3(p^2,q_1^2,q_2^2)&=&\frac{1}{2}\int_0^1 xdx\int_0^1
dy\int_0^\infty du{{\tilde}h}\frac{{\partial}^2}{{\partial}u^2}\Big(\frac{{\cal
P}}{u+D_2}+i\pi\delta(u+D_2)\Big).\end{aligned}$$ Then we obtain $$\begin{aligned}
\label{19}
{\rm Im}f_2(p^2)&\propto &\int_0^1 dx\int_0^\infty du\frac{{\partial}{{\tilde}g(x,u,p^2)}}{{\partial}u}\delta(u+D_2), \nonumber \\
{\rm Im}f_3(p^2,q_1^2,q_2^2)&\propto &\int_0^1 xdx\int_0^1
dy\int_0^\infty du\frac{{\partial}^2{{\tilde}h(x,y,u,p^2,q_1^2,q_2^2)}}{{\partial}u^2}\delta(u+D_3).\end{aligned}$$ Finally we have $$\begin{aligned}
\label{20}
{\rm Im}f_i(p^2)=0\Longleftrightarrow
u+D_i\neq0\Longleftrightarrow D_i>0\hspace{1in}i=2,3,\end{aligned}$$ where $u>0,0<x,y<1$ have been considered. More precisely, $$\begin{aligned}
\label{21}
{\rm Im}{\cal T}_{\Phi\Phi,V}^{(0)}=0\Longleftrightarrow
\left\{\begin{array}{cc}D_1>0&(0<x<1)\\
D_2>0&\;\;\;\;(0<x,y<1).\end{array}\right.\end{aligned}$$ This will lead to a restriction on the range of $p^2$. The former inequality will hold in domain $0\leq x\leq1$ if and only if $$M_{V^{ab}}=\sqrt{p^2}\leq M_a+M_b.$$ As to the latter, the right side of it has no stationary point in $x-y$ plane, therefore this inequality holding in the square domain is equivalent to it holding at boundary of the square, which gives $$\left\{\begin{array}{c}
M_{V^{ab}}=\sqrt{p^2}\leq M_a+M_b,\\
M_{\Phi^{ab}}=\sqrt{q^2}\leq M_a+M_b.
\end{array}
\right.$$ Because $M_{\Phi^{ab}}<M_{V^{ab}}$, we see that the second condition is satisfied if the first one does. Therefore we conclude that the necessary condition for the effective theory to be unitary is $$\begin{aligned}
\label{22}
M_{V^{ab}}=\sqrt{p^2}\leq \Lambda^{ab}\equiv 2m+m_a+m_b.\end{aligned}$$
In the beginning of this letter, we have actually argued an important fact that $\Lambda^{ab} \equiv 2m+m_a+m_b$ is a critical energy scale in the meson QEFT parameterized by $S_{\rm
eff}[U,V]$. As $\sqrt{p^2}$ is below $ \Lambda^{ab}$, the $S$-matrices yielded from the Feynman rules of that QEFT are unitary, while as $\sqrt{p^2}$ is above this scale, the unitarity of that QEFT will be violated. This fact indicates that the well-defined QEFT describing the meson physics in the framework of ChQM exists only as the characteristic energy is below $\Lambda^{ab}$. When energy is above $\Lambda^{ab}$, the effective meson Lagrangian description of the dynamics is illegal in principle because the unitarity fails. This is precisely a critical phenomenon, or quantum phase transition in quantum field theory, which is caused by quantum fluctuations in the system[@Sachdev]. Recalling the meaning of the scale $\Lambda_{\rm CSSB}$ of chiral symmetry spontaneously breaking in QCD, we can see that $\Lambda^{ab}$ play the same role as $\Lambda_{\rm CSSB}$. Then, in the framework of ChQM, we identify $$\label{23}
\Lambda_{\rm CSSB}=\Lambda^{ab}\equiv 2m+m_a+m_b.$$
The above equation just explores a simple relation between $\Lambda_{\rm CSSB}$ and constituent quark mass, and is the main result of this letter. It should be notice that $\Lambda_{\rm CSSB}$ is flavor-dependent as we expect. This reflects the fact that although the vacuum (quark condensation) is $SU(3)_{\rm V}$ invariant at chiral limit, it is explicitly broken to Abelian subgroup of $SU(3)_{\rm V}$ when current quark masses are turned on. As discussed at the beginning of this letter, $\Lambda_{\rm QCD}$ should be unique scale of QCD at low energy. In other words, other dimensional quantities, even including $\Lambda_{\rm CSSB}$, should be related to $\Lambda_{\rm QCD}$. Thus more fundamental task is to find relation between $\Lambda_{\rm CSSB}$ and $\Lambda_{\rm QCD}$ from eq. (\[23\]). Roughly we can expect $m\simeq\Lambda_{\rm
QCD}$, at some definite low energy limits at least. However, the precise coefficient is no longer 1. A direct evidence is that if interaction between gluons and constituent quarks are turned on (this coupling is usually expected to be weak, but should not vanish exactly), the self-energy diagram of constituent quarks will contribute to mass term of constituent quarks. To explore relation between $\Lambda_{\rm CSSB}$ and $\Lambda_{\rm QCD}$ means that we should explore exactly relation between $\Lambda_{\rm QCD}$ and constituent quark mass. It actually requires that we should know underlying dynamical mechanism of low energy QCD and thus will be great challenge.
Phenomenologically, it is also interesting to fix numerical value of $\Lambda_{\rm CSSB}$. The low-energy limit of the QEFT can obtained via integrating out vector meson fields[@Wang98; @Eck89]. It means that, at very low energy, the dynamics of vector mesons are replaced by pseudoscalar meson fields. Expanding the resulted Lagrangian up to ${\cal O}(p^4)$ in terms of Schwinger’s proper time method[@Sch54; @Ball89], we get ${\cal O}(p^4)$ ChPT-coefficients as follows $$\begin{aligned}
\label{24}
L_1&=&\frac{1}{2}L_2=\frac{1}{128\pi^2}, \hspace{0.8in}
L_3=-\frac{3}{64\pi^2}+\frac{1}{64\pi^2}g_A^4, \nonumber \\
L_4&=&L_6=0, \hspace{1.2in}L_5=\frac{3m}{32\pi^2B_0}g_A^2,\nonumber \\
L_8&=&\frac{F_\pi^2}{128B_0m}(3-\kappa^2)+\frac{3m}{64\pi^2B_0}
(\frac{m}{B_0}-\kappa g_A-\frac{g_A^2}{2}-\frac{B_0}{6m}g_A^2)
+\frac{L_5}{2},\nonumber \\
L_9&=&\frac{1}{16\pi^2}, \hspace{1.35in}
L_{10}=-\frac{1}{16\pi^2}+\frac{1}{32\pi^2}g_A^2.\end{aligned}$$ The above expressions of $L_i$ have been obtained in some previous refs.[@Esp90; @Wang98; @Bijnens93] (except $L_8$). Then inputting experimental values of $L_5$ and $L_8$ and takeing $g_{_A}=0.75$ (fitted by $n\rightarrow pe^-\bar{\nu}_e$ decay[@MG84]) and $m_u+m_d\simeq 11$MeV, we can fix phenomenological values of other free parameters as $B_0\simeq 1.8$GeV, $m\simeq 460$MeV and $\kappa\simeq0.5$. The numerical results for those low energy constants are listed in table II.
[cccccccccc]{}\
&$L_1$&$L_2$&$L_3$&$L_4$&$L_5$&$L_6$&$L_8$&$L_9$&$L_{10}$\
ChPT&$0.7\pm 0.3$&$1.3\pm 0.7$&$-4.4\pm 2.5$&$-0.3\pm 0.5$&$1.4\pm
0.5$&$-0.2\pm 0.3$&$0.9\pm 0.3$&$6.9\pm 0.7$&$-5.2\pm 0.3$\
[ChQM]{}&0.79&1.58&-4.25&0&$1.4^{a)}$&0&$0.9^{a)}$&6.33&-4.55\
Numerically, for $ud$-flavor system (e.g., $\pi-\rho-\omega$ physics), $$\label{ud}
\Lambda_{\rm CSSB}(ud)\simeq 2m=920{\rm MeV}.$$ For $u(d)s$-flavor system (e.g., $K-K^*$ physics), $$\label{ud-s}
\Lambda_{\rm CSSB}(u(d)s)\simeq 2m+m_s=1090{\rm MeV}.$$ For $\bar{s}s$ case (e.g., $\phi$-physics), $$\label{ss}
\Lambda_{\rm CSSB}(ss)\simeq 2(m+m_s)=1260{\rm MeV}.$$ Since $m_\rho<\Lambda_{\rm CSSB}(ud)$, $m_{K^*}<\Lambda_{\rm
CSSB}(u(d)s)$ and $m_\phi<\Lambda_{\rm CSSB}(ss)$, the effective meson field theory derived by resummation derivation in ChQM in this paper is unitary. And the low energy expansions in powers of $p$ are legitimate and convergent due to $p^2/\Lambda_{\rm
CSSB}^2<1$. It means that all light flavor vector meson resonances can be included in ChQM consistently. It is remarkable that the quantum phase transitions in ChQM can be explored successfully in resummation derivation method, and the corresponding critical scales are determined analytically.
To conclude, it is shown that the scale of CSSB is not independent of the scale of color confinement. The relation between two scales reveals strong interaction origination of CSSB phenomena. However, to explore this relation precisely is very difficult due to lack of underlying knowledge on color confinement. Instead we argued that this relation can be partly replaced by one between $\Lambda_{\rm CSSB}$ and constituent quark mass $m$. We used naive chiral constituent quark model to find this relation via checking unitarity region of induced QEFT of meson interaction. Phenomenologically, we determined numerical value of $\Lambda_{\rm CSSB}$ in terms of consistent fit on values of free parameters of ChQM. The result agree with usual value of $\Lambda_{\rm CSSB}$ determined by pion decay constant. Our evaluation also shows that lowest order vector meson resonances can be consistently included in naive ChQM.
[**ACKNOWLEDGMENTS**]{}
This work is partially supported by NSF of China 90103002 and the Grant of the Chinese Academy of Sciences. The authors wish to thank Yong-Shi Wu (Utah U) for his stimulating discussions on quantum phase transitions.
[99]{} J.Gasser and H.Leutwyler, Ann. Phys. [**158**]{}(1984) 142; Nucl. Phys. [**B250**]{}(1985) 465. A. Manohar and H. Georgi, Nucl. Phys. [**B234**]{} (1984) 189; H. Georgi, [*Weak Interactions and Modern Particle Theory*]{} (Benjamin/Cimmings, Menlo Park, CA, 1984) sect. 6. , ed. E.Farhi and R.Jackiw, Word Scientific, (1982); [*Proceedings of the 1991 Nagoya Spring School on Dynamical Symmetry Breaking,*]{} ed. K.Yamawaki, Word Scientific, Singapore, (1992). C. D. Roberts, nucl-th/0007054. G. Ripka, [*Quarks Bound by Chiral Fields*]{}, Clarendon Press, Oxford, (1997). T. Hatsuda and T. Kunihiro, Phys. Rep. [**247**]{} (1994) 221. G. Cheng and T.K.Kuo, J. Math Phys. [**38**]{}, (1997) 6119. G. ’t Hooft, Nucl. Phys. [**B75**]{} (1974) 461. J.F. Donoghue, E.Golowich and B.R.Holstein, [*Dynamics of the Standard Model*]{}, pp258-272, Cambridge Univ Press, (1992). S. Weinberg, Phys. Rev. [**166**]{} (1968) 1568. S. Coleman, J. Wess and B. Zumino, Phys. Rev. [**177**]{} (1969) 2239;C. G. Callan, S. Coleman, J. Wess and B. Zumino, [*ibid*]{} 2247. e.g., see, S. Sachdev, [*Quantum Phase Transitions*]{}, Cambridge Univ. Press, 1999. X. J. Wang and M. L. Yan, Jour. Phys. [**G24**]{} (1998) 1077. G. Ecker, J.Gasser, A. Pich and E.de Rafel, Nucl. Phys. [**B321**]{} (1989) 311; G. Ecker, H. Leutwyer, J. Gasser, A. Pich and E.de Rafel, Phys. Lett. [**B223**]{} (1989) 425. J. Schwinger, Phys. Rev. [**82**]{} (1951) 664; J. Schwinger, Phys, Rev. [**93**]{} (1954) 615. R. D. Ball, Phys. Rep. [**182**]{} (1989) 1. B. A. Li, Phys. Rev. [**D52**]{} (1995) 5165, 5184. D. Espriu, E. de Rafael and J. Taron, Nucl. Phys. [**B345**]{} (1990) 22. J. Bijnens, C. Bruno and E. de Rafael, Nucl. Phys. [**B390**]{} (1993) 501.
[^1]: mlyan@ustc.edu.cn
[^2]: huangyb@mail.ustc.edu.cn
|
---
abstract: 'We extend our combinatorial approach of decomposing the partition function of the Potts model on finite two-dimensional lattices of size $L \times N$ to the case of toroidal boundary conditions. The elementary quantities in this decomposition are characters $K_{l,D}$ labelled by a number of bridges $l=0,1,\ldots,L$ and an irreducible representation $D$ of the symmetric group $S_l$. We develop an operational method of determining the amplitudes of the eigenvalues as well as some of their degeneracies.'
author:
- |
[Jean-François Richard${}^{1,2}$ and Jesper Lykke Jacobsen${}^{1,3}$]{}\
[*${}^1$Laboratoire de Physique Théorique et Modèles Statistiques*]{}\
[*Université Paris-Sud, Bât. 100, 91405 Orsay, France*]{}\
[*${}^2$Laboratoire de Physique Théorique et Hautes Energies*]{}\
[*Université Paris VI, Boîte 126, Tour 24, 5${}^{\mbox{\`eme}}$ [é]{}tage*]{}\
[*4 place Jussieu, 75252 Paris cedex 05, France*]{}\
[*${}^3$Service de Physique Théorique*]{}\
[*CEA Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France*]{}\
\
title: |
Character decomposition of Potts model partition functions.\
II. Toroidal geometry
---
Introduction
============
The $Q$-state Potts model on a graph $G=(V,E)$ with vertices $V$ and edges $E$ can be defined geometrically through the cluster expansion of the partition function [@FK] $$Z=\sum_{E' \subseteq E} Q^{n(E')} ({\rm e}^J-1)^{b(E')} \,,
\label{Zcluster}$$ where $n(E')$ and $b(E')=|E'|$ are respectively the number of connected components (clusters) and the cardinality (number of links) of the edge subsets $E'$. We are interested in the case where $G$ is a finite regular two-dimensional lattice of width $L$ and length $N$, so that $Z$ can be constructed by a transfer matrix ${\rm T}_L$ propagating in the $N$-direction.
In a companion paper [@cyclic], we studied the case of cyclic boundary conditions (periodic in the $N$-direction and non-periodic in the $L$-direction). We decomposed $Z$ into linear combinations of certain restricted partition functions (characters) $K_l$ (with $l=0,1,\ldots,L$) in which $l$ [*bridges*]{} (that is, marked non-contractible clusters) wound around the periodic lattice direction. We shall often refer to $l$ as the [*level*]{}. Unlike $Z$ itself, the $K_l$ could be written as (restricted) traces of the transfer matrix, and hence be directly related to its eigenvalues. It was thus straightforward to deduce from this decomposition the amplitudes in $Z$ of the eigenvalues of ${\rm T}_L$.
The goal of this second part of our work is to repeat this procedure in the case of toroidal boundary conditions. This case has been a lot less studied than the cyclic case (a noticeable exception is [@shrock]). Indeed, when the boundary conditions are toroidal, the transfer matrix (of the related six-vertex model, to be precise) does no longer commute with the generators of the quantum group $U_q(sl(2))$. Therefore, there is no simple algebraic way of obtaining the amplitudes of eigenvalues, although some progress has been made by considering representations of the periodic Temperley-Lieb algebra (see for instance [@nichols]). But the representations of this algebra are not all known, and therefore we choose to pursue here another approach than the algebraic one.
We use instead the combinatorial approach we developed in [@cyclic], as it is for now the only approach which can be easily extended to the toroidal case. There are however several complications due to the boundary conditions, the first of which is that the bridges can now be permuted (by exploiting the periodic $L$-direction). In the following this leads us to consider decomposition of $Z$ into more elementary quantities than $K_l$, namely characters $K_{l,C}$ labeled by $l$ [*and*]{} a class $C$ of permuations of the symmetric group $S_l$. However, $K_{l,C}$ is not simply linked to the eigenvalues of $T$, and thus we will further consider its expansion over related quantities $K_{l,D}$, where $D$ labels an irreducible representation (irrep) of $S_l$. It is $K_{l,D}$ which are the elementary quantities in the case of toroidal boundary conditions.
The second complication comes from the fact that, due to the planarity of the lattice, not all the permutations between bridges can be realised. It follows that the $K_{l,D}$ are not all independent, and so there are eigenvalue degeneracies inside and between levels. Finally, there can be additional degeneracies because of the particular symmetry of the lattice, and even accidental degeneracies[^1]. We have therefore not been able to go as far as in the cyclic case, where the amplitude of any eigenvalue in $K_l$ was given by a simple expression, depending only on $l$. We do however establish an operational method of determining, for any fixed (but in practice small) $L$, the amplitudes and degeneracies of eigenvalues in the case of a generic lattice[^2].
The structure of the article is as follows. In section \[sec2\], we define appropriate generalisations of the quantities we used in the cyclic case [@cyclic]. Then, in section \[sec3\], we decompose restricted partition functions—and as a byproduct the total partition function—into characters $K_l$ and $K_{l,C}$. Finally, in section \[sec4\], we expose a method of determining the amplitudes of eigenvalues.
Preliminaries {#sec2}
=============
Definition of the $Z_{j,n1,P}$
------------------------------
As in the cyclic case, the existence of a periodic boundary condition allows for non-trivial clusters (henceforth abbreviated NTC), i.e., clusters which are not homotopic to a point. However, the fact that the torus has [*two*]{} periodic directions means that the topology of the NTC is more complicated that in the cyclic case. Indeed, each NTC belongs to a given homotopy class, which can be characterised by two coprime numbers $(n_1,n_2)$, where $n_1$ (resp. $n_2$) denotes the number of times the cluster percolates horizontally (resp. vertically) [@zuber]. The fact that all clusters (non-trivial or not) are still constrained by planarity to be non-intersecting induces a convenient simplification: all NTC in a given configuration belong to the same homotopy class. For comparison, we recall that in the cyclic case the only possible homotopy class for a NTC was $(n_1,n_2)=(1,0)$.
It is a well-known fact [@pasquier; @RJ2] that the difficulty in decomposing the Potts model partition function—or relating it to partition functions of locally equivalent models (of the six-vertex or RSOS type)—is due solely to the weighing of the NTC. Although a typical cluster configuration will of course contain trivial clusters (i.e., clusters that are homotopic to a point) with seemingly complicated topologies (e.g., trivial clusters can surround other trivial clusters, or be surrounded by trivial clusters or by NTC), we shall therefore tacitly disregard such clusters in most of the arguments that follow. Note also that the so-called degenerate clusters of Ref. [@RJ2] in the present context correspond to $n_1=1$.
![Cluster configuration with $j=2$ non-trivial clusters (NTC), here represented in red and blue colours. Each NTC is characterised by its number of branches, $n_1=2$, and by the permutation it realises, $P=(12)$. Within a given configuration, all NTC have the same topology.[]{data-label="fig1"}](torconfig.eps){width="150pt"}
Consider therefore first the case of a configuration having a single NTC. For the purpose of studying its topology, we can imagine that is has been shrunk to a line that winds the two periodic directions $(n_1,n_2)$ times. In our approach we focus on the the properties of the NTC along the direction of propagation of the transfer matrix ${\rm T}_L$, henceforth taken as the horizontal direction. If we imagine cutting the lattice along a vertical line, the NTC will be cut into $n_1$ horizontally percolating parts, which we shall call the $n_1$ [*branches*]{} of the NTC. Seen horizontally, a given NTC realises a permutation $P$ between the vertical coordinates of its $n_1$ branches, as shown in Fig. \[fig1\]. Up to a trivial relabelling of the vertical coordinate, the permutation $P$ is independent of the horizontal coordinate of the (imaginary) vertical cut, and so, forms part of the topological description of the NTC. We thus describe totally the topology along the horizontal direction of a NTC by $n_1$ and the permutation $P \in
S_{n_1}$.
Note that there are restrictions on the admissible permutations $P$. Firstly, $P$ cannot have any proper invariant subspace, or else the corresponding NTC would in fact correspond to several distinct NTC, each having a smaller value of $n_1$. For example, the case $n_1=4$ and $P=(13)(24)$ is not admissible, as $P$ corresponds in fact to two distinct NTC with $n_1=2$. In general, therefore, the admissible permutations $P$ for a given $n_1$ are simply cyclic permutations of $n_1$ coordinates. Secondly, planarity implies that the different branches of a NTC cannot intersect, and so not all cyclic permutations are admissible $P$. For example, the case $n_1=4$ and $P=(1324)$ is not admissible. In general the admissible cyclic permutations are characterised by having a constant coordinate difference between two consecutive branches, i.e., they are of the form $(k,2k,3k,\ldots)$ for some constant $k$, with all coordinates considered modulo $n_1$. For example, for $n_1=4$, the only admissible permutations are then finally $(1234)$ and $(1432)$.
Consider now the case of a configuration with several NTC. Recalling that all NTC belong to the same homotopy class, they must all be characterised by the same $n_1$ and $P$. Alternatively one can say that the branches of the different NTC are entangled. Henceforth we denote by $j$ the number of NTC with $n_1\geq 1$ in a given configuration. Note in particular that, seen along the horizontal direction, configurations with no NTC and configurations with one or more NTC percolating only vertically are topologically equivalent. This is an important limitation of our approach.
Let us denote by $Z_{j,n_1,P}$ the partition function of the Potts model on an $L \times N$ torus, restricted to configurations with exactly $j$ NTC characterised by the index $n_1\geq 1$ and the permutation $P \in S_{n_1}$; if $P$ is not admissible, or if $n_1 j > L$, we set $Z_{j,n_1,P}=0$. Further, let $Z_{j,n_1}$ be the partition function restricted to configurations with $j$ NTC of index $n_1$, let $Z_j$ be the partition function restricted to configurations with $j$ NTC [*percolating horizontally*]{}, and let $Z$ be the total partition function. Obviously, we have $Z_{j,n_1}=\sum_{P\in
S_{n_1}} Z_{j,n_1,P}$, and $Z_{j}=\sum_{n_1=1}^{L} Z_{j,n_1}$, and $Z=\sum_{j=0}^{L} Z_{j}$. In particular, $Z_0$ corresponds to the partition function restricted to configurations with no NTC, or with NTC percolating only vertically.
In the case of a generic lattice all the $Z_{j,n_1,P}$ are non-zero, provided that $P$ is an admissible cyclic permutation of length $n_1$, and that $n_1 j
\leq L$. The triangular lattice is a simple example of a generic lattice. Note however that other regular lattices may be unable to realise certain admissible $P$. For example, in the case of a square lattice or a honeycomb lattice, all $Z_{j,n_1,P}$ with $n_1 j =L$ and $n_1 > 1$ are zero, since there is not enough “space” on the lattice to permit all NTC branches to percolate horizontally while realising a non-trivial permutation. Such non-generic lattices introduce additional difficulties in the analysis which have to be considered on a case-to-case basis. In the following, except when explicitly stated, we consider therefore the case of a generic lattice.
Structure of the transfer matrix
--------------------------------
The construction and structure of the transfer matrix ${\rm T}$ can be taken over from the cyclic case [@cyclic]. In particular, we recall that ${\rm
T}$ acts towards the right on states of connectivities between two time slices (left and right) and has a block-trigonal structure with respect to the number of [*bridges*]{} (connectivity components linking left and right) and a block-diagonal structure with respect to the residual connectivity among the non-bridged points on the left time slice. As before, we denote by ${\rm T}_l$ the diagonal block with a fixed number of bridges $l$ and a trivial residual connectivity. Each eigenvalue of ${\rm T}$ is also an eigenvalue of one or more ${\rm T}_l$. In analogy with [@shrock] we shall sometimes call ${\rm
T}_l$ the transfer matrix at level $l$. It acts on connectivity states which can be represented graphically as a partition of the $L$ points in the right time slice with a special marking (represented as a [*black point*]{}) of precisely $l$ distinct components of the partition (i.e., the components that are linked to the left time slice via a bridge).
A crucial difference with the cyclic case is that for a given partition of the right time slice, there are more possibilities for attributing the black points. Namely, a connectivity component which is not [*apparently*]{} accessible from the left (and thus markable) may in fact be so due to the periodic boundary conditions identifying the top and the bottom rows. This will obviously increase the dimension of the level $l$ subspace of connectivities (for $0 < l < L$). Considering for the moment the black points to be indistinguishable, we denote the corresponding dimension as $n_{\rm
tor}(L,l)$. It can be shown [@shrock] that n\_[tor]{}(L,l) =
[ll]{} & l=0\
[2L-1 L-1]{} & l=1\
[2L L-l]{} & 2 l L
. and clearly $n_{\rm tor}(L,l)=0$ for $l>L$.
Suppose now that a connectivity state at level $l$ is time evolved by a cluster configuration of index $n_1$ and corresponding to a permutation $P$. This can be represented graphically by adjoining the initial connectivity state to the left rim of the cluster configuration, as represented in Fig. \[fig1\], and reading off the final connectivity state as seen from the right rim of the cluster configuration. Evidently, the positions of the black points in the final state will be permuted with respect to their positions in the intial state, according to the permutation $P$. As we have seen, not all $P$ are admissible, but it turns out to be advantageous to consider formally also the action of non-admissible permutations. This is permissible since in any case ${\rm T}_l$ will have only zero matrix elements between states which are related by a non-admissible permutation. Since $n_{\rm tor}(L,l)$ was just defined as the number of possible connectivity states without taking into account the possible permutations between black points, the dimension of $T_l$ is $l! \; n_{\rm tor}(L,l)$.
Let us denote by $|v_{l,i} \rangle$ (where $1\leq i \leq n_{\rm tor}(L,l)$) the $n_{\rm tor}(L,l)$ standard connectivity states at level $l$. The full space of connectivities at level $l$, i.e., with $l$ distinguishable black points, can then be obtained by subjecting the $|v_{l,i} \rangle$ to permutations of the black points. It is obvious that ${\rm T}_l$ commutes with the permutations between black points (the physical reason being that ${\rm
T}_l$ cannot “see” to which positions on the left time slice each bridge is attached). Therefore ${\rm T}_l$ itself has a block structure in a appropriate basis. Indeed, ${\rm T}_l$ can be decomposed into ${\rm T}_{l,D}$ where ${\rm
T}_{l,D}$ is the restriction of ${\rm T}_l$ to the states transforming according to the irreducible representation (irrep) of $S_l$ corresponding to the Young diagram $D$. One can obtain the corresponding basis by applying the projectors $p_D$ on all the connectivity states at level $l$, where $p_D$ is given by $$p_D=\frac{{\rm dim}(D)}{l!} \sum_{P} \chi_D(P) \, P \;.
\label{projpD}$$ Here ${\rm dim}(D)$ is the dimension of the irrep $D$ and $\chi_D(P)$ the character of $P$ in this irrep. We have used the fact that all characters of $S_l$ are real. The application of all possible permutations on any given standard vector $|v_{l,i} \rangle$ generates a regular representation of $S_l$, which contains therefore ${\rm dim}(D)$ representations $D$ (each of dimension ${\rm dim}(D)$). As there are $n_{\rm tor}(L,l)$ standard vectors, the dimension of ${\rm T}_{l,D}$ is thus $\left[ {\rm dim}(D)\right]^2 n_{\rm
tor}(L,l)$. Furthermore, using Schur’s lemma, we deduce that each of its eigenvalues is (at least) ${\rm dim}(D)$ times degenerate. Therefore ${\rm
T}_{l,D}$ has (at most) ${\rm dim}(D) \, n_{\rm tor}(L,l)$ different eigenvalues, which we shall denote $\lambda_{l,D,k}$.[^3]
Definition of the $K_{l,D}$ {#sec:defKlD}
---------------------------
We now define, as in the cyclic case [@cyclic], $K_l$ as the trace of $\left({\rm T}_l\right)^N$. Since ${\rm T}_l$ commutes with $S_l$, we can write $$K_l=l! \sum_{i=1}^{n_{\rm tor}(L,l)} \langle v_{l,i}| \left({\rm T}_l\right)^N
|v_{l,i} \rangle \; .
\label{defKltor}$$ In distinction with the cyclic case, we cannot decompose the partition function $Z$ over $K_l$ because of the possible permutations of black points (see below). We shall therefore resort to more elementary quantities, the $K_{l,D}$, which we define as the trace of $\left({\rm T}_{l,D}\right)^N$. Since both ${\rm T}_l$ and the projectors $p_D$ commute with $S_l$, we have $$K_{l,D}=l! \sum_{i=1}^{n_{\rm tor}(L,l)} \langle v_{l,i}| p_D \left({\rm T}_l\right)^N |v_{l,i} \rangle \; .
\label{defKlDtor}$$ Obviously one has $$K_l = \sum_D K_{l,D} \;,$$ the sum being over all the irreps $D$ of $S_l$. Recall that in the cyclic case the amplitudes of the eigenvalues at level $l$ are all identical. This is no longer the case, since the amplitudes depend on $D$ as well. Indeed $$K_{l,D}=\sum_{k=1}^{{\rm dim}(D) n_{\rm tor}(L,l)} {\rm dim}(D)
\left(\lambda_{l,D,k}\right)^N \; .
\label{defastrace}$$ In order to decompose $Z$ over $K_{l,D}$ we first introduce the auxiliary quantities $$K_{l,C_l}=\sum_{P_l \in C_l} K_{l,P_l} \;,
\label{defKlCtor}$$ the sum being over permutations $P_l \in S_l$ belonging to the class $C_l$. We then have $$K_{l,P_l}=\sum_{i=1}^{n_{\rm tor}(L,l)} \langle v_{l,i}|\left(P_l\right)^{-1} \left(T_l\right)^N |v_{l,i} \rangle \;.
\label{defKlPtor}$$ So $K_{l,P_l}$ (resp. $K_{l,C_l}$) can be thought of as modified traces in which the final state differs from the initial state by the application of the permutation $P_l$ (resp. the class $C_l$). Note that $K_{l,{\rm Id}}$ is simply equal to $\frac{K_l}{l!}$. Since the character is the same for all permutations belonging to the same class, Eqs. (\[defKlDtor\]) and (\[projpD\]) yield a relation between $K_{l,D}$ and $K_{l,P_l}$: $$K_{l,D}={\rm dim}(D) \sum_{C_l} \chi_D(C_l) K_{l,C_l} \; .
\label{KlDfC}$$ These relations can be inverted so as to obtain $K_{l,C_l}$ in terms of $K_{l,D}$, since the number of classes equals the number of irreps $D$: $$K_{l,C_l}=\sum_D \frac{c(D,C_l)}{l!} K_{l,D}
\label{KlCfD}$$ With the chosen normalisation, the coefficients $c(D,C_l)$ are integer. Multiplying Eq. (\[KlDfC\]) by $\chi_D(C'_l)$ and summing over $D$, and using the orthogonality relation $\sum_D \chi_D(C_l) \chi_D(C'_l) =
\frac{l!}{|C_l|} \delta_{C_l,C'_l}$ one easily deduces that $$c(D,C_l) = \frac{|C_l| \; \chi_D(C_l)}{{\rm dim}(D)} \;.$$ We also note that $$\sum_D \left[ {\rm dim}(D) \right]^2 c(D,C_l) = l! \, \delta_{C_l,{\rm Id}}
\label{cDClsum}
\label{relcDC}$$
Decomposition of the partition function {#sec3}
=======================================
The characters $K_l$ {#sec:charKl}
--------------------
By generalising the working for the cyclic case, we can now obtain a decomposition of the $K_l$ in terms of the $Z_{j,n_1}$. To that end, we first determine the number of states $|v_{l,i}\rangle$ which are [*compatible*]{} with a given configuration of $Z_{j,n_1}$, i.e., the number of initial states $|v_{l,i}\rangle$ which are thus that the action by the given configuration produces an identical final state. The notion of compatability is illustrated in Fig. \[fig2\].
![Standard connectivity states at level $l=1$ which are compatible with a given cluster configuration contributing to $Z_{2,1}$.[]{data-label="fig2"}](torcomp1.eps){width="300pt"}
We consider first the case $n_1=1$ and suppose that the $k$’th NTC connects onto the points $\{y_k\}$. The rules for constructing the compatible $|v_{l,i}\rangle$ are identical to those of the cyclic case:
1. The points $y \notin \cup_{k=1}^j \{ y_k \}$ must be connected in the same way in $|v_{l,i}\rangle$ as in the cluster configuration.
2. The points $\{y_k\}$ within the same bridge must be connected in $|v_{l,i}\rangle$.
3. One can independently choose to associate or not a black point to each of the sets $\{y_k\}$. One is free to connect or not two distinct sets $\{y_k\}$ and $\{y_{k'}\}$.
The choices mentioned in rule 3 leave $n_{\rm tor}(j,l)$ possibilities for constructing a compatible $|v_{l,i}\rangle$. The coefficient of $Z_{j,1}$ in the decomposition of $K_l$ is therefore $\frac{l! \; n_{\rm tor}(j,l)}{Q^j}$, since the permutation of black points in a standard vector $|v_{l,i}\rangle$ allows for the construction of $l!$ distinct states, and since the weight of the $j$ NTC in $K_l$ is $1$ instead of $Q^j$. It follows that K\_l = \_[j=l]{}\^L l! n\_[tor]{}(j,l) n\_1=1.
![Standard connectivity states at level $l=1$ which are compatible with a given cluster configuration contributing to $Z_{2,2}$.[]{data-label="fig3"}](torcomp2.eps){width="300pt"}
We next consider the case $n_1>1$. Let us denote by $\{y_{k,m}\}$ the points that connect onto the $m$’th branch of the $k$’th NTC (with $1 \le m \le n_1$ and $1 \le k \le j$), and by $\{y_k\}=\cup_{m=1}^{n_1}\{y_{k,m}\}$ all the points that connect onto the $k$’th NTC. As shown in Fig. \[fig3\], the $|v_{l,i}\rangle$ which are compatible with this configuration are such that
1. The connectivities of the points $y\notin\cup_{k=1}^j\{y_k\}$ are identical to those appearing in the cluster configuration.
2. All points $\{y_{k,m}\}$ corresponding to the branch of a NTC must be connected.
3. For each of the $k$ NTC there are two possibilities. A) Either one connects all $\{y_{k,m}\}$ (with $1\leq m \leq n_1$) corresponding to all $n_1$ branches of the NTC, obtaining what we shall henceforth call a [*big block*]{}. B) Or alternatively one connects none of the $n_1$ branches.
4. Because of the constraint of planarity and the fact that the NTC are entangled, all the different big blocks are automatically connected among themselves. One can therefore attribute at most one black point to the collection of big blocks.
To obtain rule 3 we have used the fact that the permutations $P$ characterising the NTC do not have any proper invariant subspace. Note that rule 4 implies that the decomposition of $K_l$ with $l\geq 2$ does not contain any of the $Z_{j,n_1}$ with $n_1>1$. We therefore have simply $$K_l=\sum_{j=l}^L l! \, n_{\rm tor}(j,l) \frac{Z_{j,1}}{Q^j} \qquad
\mbox{for }l \geq 2 \;.
\label{expKltor}$$
It remains to obtain the decomposition of $K_1$ and $K_0$. The number of standard connectivities $|v_{l,i}\rangle$ compatible with $r$ big blocks is $0$ for $l\geq 2$ (because of rule 4); ${j \choose r}$ for $l=1$ and $r\geq 1$ (by rule 3 we independently choose to link up $r$ of the $j$ NTC, and by rule 4 the resulting big block must carry the black point); $0$ for $l=1$ and $r=0$ (since one needs a big block to attribute the black point); and ${j \choose
r}$ for $l=0$. Summing over $r$, we finally obtain the number of compatible $|v_{l,i}\rangle$: $0$ for $l\geq 2$; $\sum_{r=1}^j {j \choose r}=2^j - 1$ for $l=1$; and $\sum_{r=0}^j {j \choose r}=2^j$ for $l=0$. The decomposition of $K_1$ reads therefore $$K_1=\sum_{j=1}^L n_{\rm tor}(j,1)\frac{Z_{j,1}}{Q^j} + \sum_{j=1}^{\left
\lfloor \frac{L}{2}\right \rfloor} (2^j-1) \frac{Z_{j,n_1>1}}{Q^j}
\label{expK1tor}$$ and that of $K_0$ is $$K_0=\sum_{j=0}^L n_{\rm tor}(j,1)\frac{Z_{j,1}}{Q^j} + \sum_{j=1}^{\left\lfloor\frac{L}{2}\right\rfloor} 2^j\,
\frac{Z_{j,n_1>1}}{Q^j} \; .
\label{expK0tor}$$ Note that the coefficients in front of $Z_{j,n_1}$ do not depend on the precise value of $n_1$ when $n_1>1$. To simplify the notation we have defined $Z_{0,1}=Z_0$.
The coefficients $b^{(l)}$ {#sec:coef_bl}
--------------------------
Since the coefficients in front of $Z_{j,1}$ and $Z_{j,n_1>1}$ in Eqs. (\[expK1tor\])–(\[expK0tor\]) are different, we cannot invert the system of relations (\[expKltor\])–(\[expK0tor\]) so as to obtain $Z_j \equiv Z_{j,1}+Z_{j,n_1>1}$ in terms of the $K_l$. It is thus precisely because of NTC with several branches contributing to $Z_{j,n_1>1}$ that the problem is more complicated than in the cyclic case.
In order to appreciate this effect, and compare with the precise results that we shall find later, let us for a moment assume that Eq. (\[expKltor\]) were valid also for $l=0,1$. We would then obtain $$Z_{j,1}=\sum_{l=j}^L b_j^{(l)} \frac{K_l}{l!}
\label{expZj1i}$$ where $$b^{(l)} \equiv \sum_{j=0}^l b_j^{(l)}
= \left \lbrace \begin{array}{ll}
\sum_{j=0}^l (-1)^{l-j} \frac{2l}{l+j} {l+j \choose l-j} Q^j + (-1)^l (Q-1)
& \mbox{for }l \ge 2 \\
\sum_{j=0}^l (-1)^{l-j} {l+j \choose l-j } Q^j
& \mbox{for }l \le 2 \\
\end{array} \right.
\label{defbltor}$$ The coefficients $b^{(l)}$ play a role analogous to those denoted $c^{(l)}$ in the cyclic case [@cyclic]; note also that $b^{(l)}= c^{(l)}$ for $l \leq
2$. Chang and Schrock have developed a diagrammatic technique for obtaining the $b^{(l)}$ [@shrock].
Supposing still the unconditional validity of Eq. (\[expKltor\]), one would obtain for the full partition function $$Z=\sum_{l=0}^L b^{(l)} \frac{K_l}{l!} \; .
\label{devZtorosim}$$ This relation will be modified due to the terms $Z_{j,n_1>1}$ realising permutations of the black points, which we have here disregarded. To get things right we shall introduce Young diagram dependent coefficients $b^{(l,D)}$ and write $Z=\sum_{l=0}^L \sum_D b^{(l,D)}K_{l,D}$. Neglecting $Z_{j,n_1>1}$ terms would lead, according to Eq. (\[devZtorosim\]), to $b^{(l,D)}=\frac{b^{(l)}}{l!}$ independently of $D$. We shall see that the $Z_{j,n_1>1}$ will lift this degeneracy of amplitudes in a particular way, since there exists certain relations between the $b^{(l,D)}$ and the $b^{(l)}$.
Decomposition of the $K_{l,C_l}$ {#sec:decKlCl}
--------------------------------
The relations (\[expKltor\])–(\[expK0tor\]) were not invertible due to an insufficient number of elementary quantities $K_l$. Let us now show how to produce a development in terms of $K_{l,C_l}$, i.e., taking into account the possible permutations of black points. This development turns out to be invertible.
A standard connectivity state with $l$ black points is said to be [*$C_l$-compatible*]{} with a given cluster configuration if the action of that cluster configuration on the connectivity state produces a final state that differs from the initial one just by a permutation $C_l$ of the black points. This generalises the notion of compatibility used in Sec. \[sec:charKl\] to take into account the permutations of black points.
Let us first count the number of standard connectivities $|v_{l,i}\rangle$ which are $C_l$-compatible with a cluster configuration contributing to $Z_{j,n_1,P}$. For $n_1=1$, $S_{n_1}$ contains only the identity element ${\rm
Id}$, and so the results of Sec. \[sec:charKl\] apply: the $Z_{j,1}$ contribute only to $K_{l,{\rm Id}}$. We consider next a configuration contributing to $Z_{j,n_1,P}$ with $n_1>1$. The $|v_{l,i}\rangle$ which are $C_l$-compatible with this configuration satisfy the same four rules as given in Sec. \[sec:charKl\] for the case $n_1>1$, with the slight modification of rule 4 that the black points must be attributed to the big blocks in such a way that [*the final state differs from the initial one by a permutation $C_l$*]{}.
This modification makes the attribution of black points considerably more involved than was the case in Sec. \[sec:charKl\]. First note that not all $C_l$ are allowed. To be precise, the cycle decomposition of the allowed permutations can only contain ${\rm id}$ (the identity acting on a single black point) or $P$ (recall that $P$ is the permutation of coordinates realised by the branches of a single NTC). Indeed, if one attributes a black point to a big block its position remains unchanged by action of the cluster configuration, whereas if one attributes $n_1$ black points to the $n_1$ branches of one same NTC these points will be permuted by $P$. Furthermore, since the big blocks are automatically connected among themselves, one can at most attribute to them a single black point, and so ${\rm id}$ is contained in the cycle decomposition $0$ or $1$ times. Note also that the entanglement of the NTC will imply the entanglement of the structure of the allowed permutations, but this fact is of no importance here since we are only interested in $C_l$, i.e., the [*classes*]{} of allowed permutations.
Denoting by $n_P$ the number of times the permutations of class $C_l$ contains $P$, the two types of allowed $C_l$ are: 1) those associated with permutations that only contain $P$, i.e., such that $l=n_Pn_1$, and 2) those associated with permutations that contain ${\rm id}$ once, i.e., such that $l=n_Pn_1+1$. In the following we denote these two types as $(n_P,n_1)$ and $(n_P,n_1)'$, respectively, and the corresponding $K_{l,C_l}$ will be denoted $K_{(n_P,n_1)}$ and $K_{(n_P,n_1)'}$ respectively.
![Standard connectivity states at level $l=2$ which are $(12)$-compatible with a given cluster configuration contributing to $Z_{2,2}$. The action of the cluster configuration on the connectivity states permutes the positions of the two black points.[]{data-label="fig4"}](torccomp.eps){width="300pt"}
Let us consider the first case, which is depicted in Fig. \[fig4\]. If the $|v_{l,i}\rangle$ have $r$ big blocks, there are ${j \choose r}$ ways of choosing them among the $j$ NTC, and ${j-r \choose n_P}$ ways to attribute the black points. Indeed one needs to distribute $l=n_P n_1$ black points among $n_P$ groups of $n_1$ non-connected blocks corresponding to the same NTC, out of a total of $j-r$. Since the $|v_{l,i}\rangle$ can contain at most $j-n_P$ big blocks, the number of $C_l$-compatible standard connectivities is $$\sum_{r=0}^{j-n_P} {j \choose r} {j-r \choose n_P}=\sum_{r=0}^{j-n_P} {j \choose n_P} {j-n_P \choose r}=
{j \choose n_P} 2^{j-n_P} \; .
\label{sommer}$$ From this we infer the decomposition of $K_{(n_P,n_1)}$: $$K_{(n_P,n_1)}=\sum_{j=n_P}^{\left \lfloor \frac{L}{n_1} \right \rfloor} {j
\choose n_P} 2^{j-n_P} \frac{Z_{j,n_1}}{Q^j} \; .
\label{KnPn1}$$ Consider next the second case. The $|v_{l,i}\rangle$ can still contain at most $j-n_P$ big blocks, but they are now required to contain at least one, as one black point needs to be attributed. Therefore, the sums in Eq. (\[sommer\]) start from $r=1$, leading to the following result for the decomposition of $K_{(n_P,n_1)'}$: $$K_{(n_P,n_1)'}=\sum_{j=n_P+1}^{\left \lfloor \frac{L}{n_1} \right \rfloor} {j
\choose n_P} (2^{j-n_P}-1) \frac{Z_{j,n_1}}{Q^j} \; .
\label{KnPn1'}$$ It remains to study the special case of $n_P=0$, i.e., the case of $C_l = {\rm
Id}$. This is in fact trivial. Indeed, in that case, the value of $n_1$ in $Z_{j,n_1}$ is no longer fixed, and one must sum over all possible values of $n_1$, taking into account that the case of $n_1=1$ is particular (absense of big blocks). Since $K_{l,{\rm Id}}=\frac{K_l}{l!}$, one obtains simply Eqs. (\[expKltor\])–(\[expK0tor\]) of Sec. \[sec:charKl\] up to a global factor.
Decomposition of $Z_j$ over the $K_{l,C_l}$ {#sec:expZj_KlCl}
-------------------------------------------
To obtain the decomposition of $Z_j$ in terms of the $K_{l,C_l}$, one would need to invert Eqs. (\[KnPn1\])–(\[KnPn1’\]) obtained above. But we now encounter the opposite problem of that announced in the beginning of Sec. \[sec:decKlCl\]: there are too many $K_{l,C_l}$. Indeed, the elementary quantities $K_{l,C_l}$ are not independent, since a given cluster configuration can realise different permutations depending on the way in which the black points are attributed. We must therefore select an independent set of $K_{l,C_l}$, and we make the choice of selecting the $K_{(n_P,n_1)}$, i.e., the $C_l$ of the first type. Inverting Eq. (\[KnPn1\]) for varying $n_P$ and fixed $n_1>1$ one obtains: $$Z_{j,n_1}=Q^j \sum_{n_P=j}^{\left \lfloor \frac{L}{n_1} \right \rfloor}
{n_P \choose j}
(-2)^{n_P-j} K_{(n_P,n_1)} \qquad \mbox{for }n_1>1 \; .
\label{expZjn1}$$ Since the coefficients in this sum do not depend on $n_1$ (provided that $n_1>1$), we can sum this relation over $n_1$ and write it as $$Z_{j,n_1>1}=Q^j \sum_{n_P=j}^{\left \lfloor \frac{L}{2} \right\rfloor}
{n_P \choose j}
(-2)^{n_P-j} K_{(n_P,n_1>1)}
\label{expZjn1>1}$$ where we recall the notations $Z_{j,n_1>1}=\sum_{n_1=2}^L Z_{j,n_1}$ and $K_{(n_P,n_1>1)}=\sum_{n_1=2}^L K_{(n_P,n_1)}$, corresponding to permutations composed of $n_P$ cycles of the same length $>1$.
Consider next the case $n_1=1$. For $j\geq 2$ one has simply $$Z_{j,1}=\sum_{l=j}^{L} \frac{b^{(l)}_{j}}{l!} K_l \;,
\label{expZj12}$$ recalling Eq. (\[expZj1i\]) and the fact that for $l\ge 2$ the $Z_{j,n_1>1}$ do not appear in the decomposition of $K_l$. However, according to Eqs. (\[expK1tor\])–(\[expK0tor\]), the $Z_{j,n_1>1}$ do appear for $l=0$ and $l=1$, and one obtains $$Z_{1,1} = \left( Q K_1-Q\sum_{j=1}^L (2^j-1)\frac{Z_{j,n_1>1}}{Q^j} \right)
+ \sum_{l=2}^L \frac{b^{(l)}_{j}}{l!} K_l \; .
\label{expZ11i}$$ Inserting the decomposition (\[expZjn1>1\]) of $Z_{j,n_1>1}$ into Eq. (\[expZ11i\]) one obtains the decomposition of $Z_{1,1}$ over $K_{l}$ and $K_{(n_P,n_1)}$: $$Z_{1,1}=\sum_{l=1}^L \frac{b^{(l)}_j}{l!} K_{l}+\sum_{n_P=1}^L Q (-1)^{n_P}
K_{(n_P,n_1>1)} \; .
\label{expZ11}$$
We proceed in the same fashion for the decomposition of $Z_0 \equiv Z_{0,1}$, finding $$Z_{0}=\sum_{l=0}^L \frac{b^{(l)}_{j}}{l!} K_l - \sum_{j=1}^L
\frac{Z_{j,n_1>1}}{Q^j} \; .
\label{expZ0i}$$ Upon insertion of the decomposition (\[expZjn1>1\]) of $Z_{j,n_1>1}$, one arrives at $$Z_{0}=\sum_{l=0}^L \frac{b^{(l)}_{j}}{l!} K_l +
\sum_{n_P=1}^L \left[ (-1)^{n_P+1}+(-2)^{n_P} \right]
K_{(n_P,n_1>1)} \; .
\label{expZ0}$$ Since $Z_j=Z_{j,1}+Z_{j,n_1>1}$, we conclude from Eqs. (\[expZj12\])–(\[expZjn1>1\]) that, for any $j$, $$Z_j=\sum_{l=j}^L \frac{b^{(l)}_j}{l!} K_{l}
+ \sum_{n_P=j}^L b^{(n_P,n_1>1)}_{j} K_{(n_P,n_1>1)} \;,
\label{expZj}$$ with the coefficients $$b^{(n_P,n_1>1)}_{j} = \left \lbrace \begin{array}{ll}
Q^j {n_P \choose j} (-2)^{n_P-j} &
\mbox{for }j\ge 2 \\
Q \left[ n_P (-2)^{n_P-1} + (-1)^{n_P} \right] &
\mbox{for }j=1 \\
(-1)^{n_P+1}+(-2)^{n_P} &
\mbox{for }j=0 \\
\end{array} \right.
\label{defbnPn1>10}$$ The decomposition of $Z \equiv \sum_{0 \leq j \leq L} Z_j$ is therefore $$Z=\sum_{l=0}^L \frac{b^{(l)}}{l!} K_{l} +
\sum_{n_P=1}^L b^{(n_P,n_1>1)} K_{(n_P,n_1>1)}
\label{expZ}$$ with $$\begin{aligned}
b^{(l)}&=&\sum_{0\leq j \leq l} b^{(l)}_j \;, \nonumber \\
b^{(n_P,n_1>1)}&=&\sum_{0\leq j \leq n_P} b^{(n_P,n_1>1)}_j \; .
\label{deffbnPn1>1i}\end{aligned}$$ Note that $b^{(l)}_j$ (resp. $b^{(n_P,n_1>1)}_j$) is just the term multiplying $Q^j$ in $b^{(l)}$ (resp. $b^{(n_P,n_1>1)}$). Computing the sum over $j$, we obtain the simple result $$b^{(n_P,n_1>1)}=(Q-2)^{n_P}+(-1)^{n_P}(Q-1) \; .
\label{defbnPn1>1}$$
Amplitudes of the eigenvalues {#sec4}
=============================
Decomposition of $Z$ over the $K_{l,D}$
---------------------------------------
The culmination of the preceeding section was the decomposition (\[expZj\]) of $Z_j$ in terms of $K_{l,C_l}$. However, it is the $K_{l,D}$ which are directly related to the eigenvalues of the transfer matrix ${\rm T}$. For that reason, we now use the relation (\[KlCfD\]) between $K_{l,C_l}$ and $K_{l,D}$ to obtain the decomposition of $Z_j$ in terms of $K_{l,D}$. The result is: $$Z_j=\sum_{l,D} b^{(l,D)}_j K_{l,D}
\label{expZj2}$$ where the coefficients $b^{(l,D)}_j$ are given by $$b^{(l,D)}_j=\frac{b^{l}_j}{l!} +
\sum_{(n_1>1) | l} \frac{b^{\left(\frac{l}{n_1},n_1>1\right)}_j}{l!} \;
c\left(D,\left(\frac{l}{n_1},n_1\right)\right) \; .
\label{defblDj}$$ Indeed, $K_{l}=\sum_{D} K_{l,D}$, and since $K_{(n_P,n_1)}$ corresponds to the level $l=n_P n_1$, we have $K_{(n_P,n_1)}=\sum_{D \in S_{n_Pn_1}}
\frac{c(D,(n_P,n_1))}{l!} K_{n_P n_1,D}$. (Recall that $(n_P,n_1)$ is the class of permutations composed of $n_P$ cycles of the same length $n_1$.) As explained in Sec. \[sec:coef\_bl\], the $b^{(l,D)}_j$ are not simply equal to $\frac{b^{l}_j}{l!}$ because of the $n_1>1$ terms. Using Eq. (\[relcDC\]) we find that they nevertheless obey the following relation $$\sum_{D \in S_l} \left[{\rm dim}(D)\right]^2 b^{(l,D)}_j=b^{(l)}_j \; .
\label{relblDj}$$ But from Eq. (\[defblDj\]) the $b^{(l,D)}_j$ with $l<2j$ are trivial, i.e., equal to $\frac{b^{l}_j}{l!}$ independently of $D$. This could have been shown directly by considering the decomposition (\[expKltor\]) of $K_l$. Finally, since $b^{(1,n_1>1)}_1=0$ from Eq. (\[defbnPn1>10\]), only $b^{(l,D)}_0$ is non-trivial for $l=2$ or $l=3$.
The decomposition of $Z$ over $K_{l,D}$ is obviously given by $$Z=\sum_{l,D} b^{(l,D)} K_{l,D}
\label{expZKlD}$$ where $$b^{(l,D)}=\sum_{j=1}^l b^{(l,D)}_j \; .
\label{defblD}$$ The $b^{(l,D)}$ then satisfy $$\sum_{D \in S_l} \left[{\rm dim}(D)\right]^2 b^{(l,D)}=b^{(l)} \; .
\label{relblD2}$$
Relations among the $K_{l,D}$
-----------------------------
Just like the $K_{l,C_l}$, the $K_{l,D}$ are not independent, and for the same reasons. Indeed, the number of $K_{l,D}$ which are independent among themselves, and independent of $K_{l',D'}$ at higher levels $l'>l$, equals the number of independent $K_{l,C_l}$. This number in turn equals the number of integers dividing $l$, since the independent $K_{l,C_l}$ are $K_{l,{\rm Id}}$ and the $K_{(n_P,n_1)}$ with $l=n_P n_1$.
Therefore one can write relations between the $K_{l,D}$, by selecting an independent number of $K_{l,D}$ and expressing the others in terms of those selected. This produces relations of the form $$K_{l,D_l}=\sum_{D_{l'}} e(D_l,D_{l'}) K_{l',D_{l'}}
\label{defe}$$ where the $K_{l,D_l}$ are now those not selected, and the sum of $K_{l',D_{l'}}$ is over the $D_{l'}$ selected with $l'\geq l$. The expressions of the coefficients $e(D_l,D_{l'})$ depend of the choice of $K_{l,D}$ made. Note in particular that to obtain the $e(D_l,D_{l'})$, the $K_{(n_P,n_1)'}$ must be expressed in terms of the $K_{(n_P,n_1)}$. By combining Eqs. (\[KnPn1’\]) and (\[expZjn1\]) we obtain $$K_{(n_P,n_1)'}=\sum_{n_P'=n_P+1}^{\left \lfloor \frac{L}{n_1} \right \rfloor}
\sum_{j=n_P+1}^{n_P'}
{j \choose n_P} {n_P' \choose j}
(2^{j-n_P}-1)(-2)^{n_P'-j}
K_{(n_P',n_1)} \;,$$ and performing the sum over $j$ this becomes $$K_{(n_P,n_1)'}=\sum_{n_P'=n_P+1}^{\left \lfloor \frac{L}{n_1} \right \rfloor}
{n_P' \choose n_P}
(-1)^{n_P'-n_P+1}
K_{(n_P',n_1)} \;.
\label{relKnPn1}$$ Let us give an example of this relation: for $L=4$ we find that $K_{3,(1,2)'}=2K_{4,(2,2)}$.
The coefficients $e(D_l,D_{l'})$ have the following properties: $$\begin{aligned}
\sum_{D_{l'} \in S_{l'=l}} \left[ {\rm dim}(D_{l'}) \right]^2 e(D_l,D_{l'})
&=& \left[{\rm dim}(D_{l})\right]^2 \\
\sum_{D_{l'} \in S_{l'>l}} \left[ {\rm dim}(D_{l'}) \right]^2 e(D_l,D_{l'})
&=& 0
\label{prope}\end{aligned}$$ which can be proved using the fact that the $e(D_l,D_{l'})$ are independent of $L$ and that the number of eigenvalues, including degeneracies, corresponding to $K_{l,D_l}$ is $[{\rm dim}(D_{l})]^2 \, n_{\rm tor}(L,l)$. These relations between $K_{l,D_l}$ have strong physical implications: additional degeneracies inside a level and between different levels. We shall give in the next subsection a method to determine these degeneracies, but note for now that they depend of $L$ although the $e(D_l,D_{l'})$ are independent of $L$.
We can now repeat the decompositions of the preceeding subsection, but expanding only over the selected independent $K_{l,D}$. To that end, we define the coefficients $\tilde{c}(D,C_l)$ by $$K_{l,C_l}= \sum_{{\rm indep.}\ D \in S_{l'\geq l}}
\frac{\tilde{c}(D,C_l)}{(l')!} K_{l',D} \;.
\label{relKlCDind}$$ Note that contrary to Eq. (\[KlCfD\]), the sum carries over all independent (selected) $D$ at levels $l'\geq l$. Because of Eq. (\[prope\]), the $\tilde{c}(D,C_l)$ have the following properties: if $C_l \neq {\rm Id}$ $$\sum_{{\rm indep.}\ D \in S_{l'}} [{\rm dim}(D)]^2 \tilde{c}(D,C_l) = 0 \;,
\label{propctilde}$$ whereas if $C_l={\rm Id}$ $$\begin{aligned}
\sum_{{\rm indep.}\ D \in S_{l'=l}} [{\rm dim}(D)]^2 \tilde{c}(D,Id) &=&
|C_l| \;, \nonumber \\
\sum_{{\rm indep.}\ D \in S_{l'>l}} [{\rm dim}(D)]^2 \tilde{c}(D,Id) &=& 0 \;.
\label{propctilde2}\end{aligned}$$ Inserting the decomposition (\[relKlCDind\]) of $K_{l,C_l}$ into Eq. (\[expZ\]), we obtain the decomposition of $Z$ over independent $K_{l,D}$: $$Z=\sum_{l,D} \frac{\tilde{b}^{(l,D)}}{l!} K_{l,D}
\label{expZind}$$ where the $\tilde{b}$ can be obtained using the $\tilde{c}$.
We do not have any general closed-form expression[^4] for $\tilde{b}$, but in the next subsection we show how they can be determined in practice by a straightforward, though somewhat lengthy, procedure. More precisely, we determine all the $\tilde{b}^{(l,D)}$ up to $l=4$, with a given convention for the choice of independent $K_{l,D}$. As the $b^{(l,D)}$, the $\tilde{b}^{(l,D)}$ verify $$\sum_{{\rm indep.}\ D \in S_l} [{\rm dim}(D)]^2 \tilde{b}^{(l,D)} = b^{(l)}$$ except that now the sum is over independent $D$. This is a consequence of the properties of the $\tilde{c}$.
Method to obtain the amplitudes of the eigenvalues {#sec:amplitudes}
--------------------------------------------------
Because of the additional degeneracies between the $K_{l,D}$, we have not been able to find a general formula giving the total degeneracies of the eigenvalues. But, using Eq. (\[expZ\]) and the fact that the $c(D,C_l)$ defined by Eq. (\[KlCfD\]) are integers, we deduce that the amplitudes of the eigenvalues are integer combinations of the $\frac{b^{(l)}}{l!}$ and the $\frac{b^{(n_P,n_1>1)}}{(n_Pn_1)!}$. Determining precisely with which integers is not an easy task, and we give here a method which is operational for all values of $L$, though in practice it will probably become quite cumbersome for large $L$.
One must begin at the highest possible level, $l=L$. Since not all permutations are admissible, one can write relations between the $K_{L,D}$ and deduce which eigenvalues are shared by several different $K_{L,D}$. One then proceeds to the next lower-lying level, $l=L-1$. Since not all permutations are admissible, and as some permutations are not independent of those at level $l+1$, one can write relations between the $K_{l,D}$ and the $K_{l+1,D}$. These relations permit to deduce which eigenvalues appearing at level $l$ are new and what are their degeneracies. This method is then iterated until one attains level $l=3$. Considering $l\leq 2$ is not necessary: all the eigenvalues at these levels are new as there are no relations between the corresponding $K_{l,D}$. Finally, using Eq. (\[expZ\]) where all $K_{l,D}$ have been expressed in terms of an independent number of $K_{l,D}$, we deduce the amplitudes of the eigenvalues.
Let us consider in detail the case $L=4$. At level $4$, the possible $K_{4,D}$ are $K_{\yng(4)}$, $K_{\yng(1,1,1,1)}$, $K_{\yng(3,1)}$, $K_{\yng(2,1,1)}$ and $K_{\yng(2,2)}$, while the admissible $K_{l,C_l}$ are $K_{4,{\rm Id}}$, $K_{4,(2,2)}$ and $K_{4,(1,4)}$. Using the table of characters of $S_4$, we can write: $$\begin{aligned}
K_{\yng(4)} &=& K_{4,{\rm Id}}+K_{4,(2,2)}+K_{4,(1,4)} \label{chtabK4} \\
K_{\yng(1,1,1,1)} &=& K_{4,{\rm Id}}+K_{4,(2,2)}-K_{4,(1,4)} \label{chtabK1111} \\
K_{\yng(3,1)} &=& 9K_{4,{\rm Id}}-3K_{4,(2,2)}-3K_{4,(1,4)} \\
K_{\yng(2,1,1)} &=& 9K_{4,{\rm Id}}-3K_{4,(2,2)}+3K_{4,(1,4)} \\
K_{\yng(2,2)} &=& 4K_{4,{\rm Id}}+4K_{4,(2,2)}\end{aligned}$$ We choose $K_{\yng(4)}$, $K_{\yng(3,1)}$ and $K_{\yng(2,2)}$ as independent $K_{4,D}$, and we express the $K_{4,C_l}$ in terms of those $K_{4,D}$: $$\begin{aligned}
K_{4,{\rm Id}}&=&\frac{K_{\yng(4)}}{4}+\frac{K_{\yng(3,1)}}{12} \\
K_{4,(2,2)}&=&-\frac{K_{\yng(4)}}{4}-\frac{K_{\yng(3,1)}}{12}+\frac{K_{\yng(2,2)}}{4} \\
K_{4,(1,4)}&=&K_{\yng(4)}-\frac{K_{\yng(2,2)}}{4} \end{aligned}$$ Next, using these expressions, we obtain $K_{\yng(1,1,1,1)}$ and $K_{\yng(2,1,1)}$ in terms of the independent $K_{4,D}$ chosen: $$\begin{aligned}
K_{\yng(1,1,1,1)}&=&-K_{\yng(4)}+\frac{K_{\yng(2,2)}}{2} \label{relK1111} \\
K_{\yng(2,1,1)}&=&6K_{\yng(4)}+K_{\yng(3,1)}-\frac{3}{2}K_{\yng(2,2)}
\label{relK211}\end{aligned}$$ With these two relations we can determine the eigenvalue degeneracies between the chosen $K_{4,D}$. Recall first that according to Eq. (\[defastrace\]) the number of eigenvalues contributing to $K_{l,D}$ is $n_{\rm tor}(L,l) \,
{\rm dim}(D)$ and that each eigenvalue has multiplicity ${\rm dim}(D)$. Further, ${\rm dim}(\yng(4))=1$, ${\rm dim}(\yng(3,1))=3$ and ${\rm
dim}(\yng(2,2))=2$. Consider now Eq. (\[relK1111\]), recalling that the $K_{l,D}$ have been defined in Eq. (\[defastrace\]) as a trace. We deduce that the corresponding eigenvalues must satisfy ( \_[(1,1,1,1)]{} )\^N = - (\_[(4)]{})\^N + \_[i=1]{}\^2 (\_[(2,2),i]{})\^N for any positive integer $N$. This implies that $\lambda_{\yng(2,2),1} =
\lambda_{\yng(4)}$ and that $\lambda_{\yng(2,2),2} = \lambda_{\yng(1,1,1,1)}$. Using this, Eq. (\[relK211\]) then yields \_[i=1]{}\^3 (\_[(2,1,1),i]{})\^N = (\_[(4)]{})\^N - (\_[(1,1,1,1)]{})\^N + \_[i=1]{}\^3 (\_[(3,1),i]{})\^N for any $N$. This is possible provided that either $\lambda_{\yng(1,1,1,1)}=\lambda_{\yng(4)}$ or $\lambda_{\yng(1,1,1,1)}=\lambda_{\yng(3,1),1}$. But the first possibility can be excluded since, by Eqs. (\[chtabK4\])–(\[chtabK1111\]), it would imply $K_{4,(1,4)}=0$ which is inconsistent with our hypothesis that we work on a generic lattice where all admissible $K$ are non-zero. We conclude that $\lambda_{\yng(1,1,1,1)}=\lambda_{\yng(3,1),1}$ and hence that $\lambda_{\yng(4)}=\lambda_{\yng(2,1,1),1}$ and $\lambda_{\yng(3,1),i} =
\lambda_{\yng(2,1,1),i}$ for $i=2,3$. There are therefore only $4$ different eigenvalues at level $4$ instead of $10$.
Consider now the level $l=3$. From the character table of $S_3$ we obtain: $$\begin{aligned}
K_{\yng(3)} &=& K_{3,{\rm Id}}+K_{3,(1,2)'}+K_{3,(1,3)} \\
K_{\yng(1,1,1)} &=& K_{3,{\rm Id}}-K_{3,(1,2)'}+K_{3,(1,3)} \\
K_{\yng(2,1)} &=& 4K_{3,{\rm Id}}-2K_{3,(1,3)}\end{aligned}$$ $K_{3,(1,2)'}$ must be expressed terms of the independent $K_{4,D_l}$ chosen: $$K_{3,(1,2)'}=2K_{4,(2,2)}=-\frac{K_{\yng(4)}}{2}-\frac{K_{\yng(3,1)}}{6}+\frac{K_{\yng(2,2)}}{2}$$ We choose $K_{\yng(3)}$ and $K_{\yng(2,1)}$ as independent $K_{3,D}$, and we express the $K_{3,C_l}$ in terms of the independent $K_{3,D}$ and $K_{4,D}$ chosen: $$\begin{aligned}
K_{3,{\rm Id}}&=&\frac{K_{\yng(3)}}{3}+\frac{K_{\yng(2,1)}}{6}+\frac{K_{\yng(4)}}{6}+\frac{K_{\yng(3,1)}}{18}-\frac{K_{\yng(2,2)}}{6} \\
K_{3,(1,3)}&=&\frac{2}{3}K_{\yng(3)}-\frac{K_{\yng(2,1)}}{6}+\frac{K_{\yng(4)}}{3}+\frac{K_{\yng(3,1)}}{9}-\frac{K_{\yng(2,2)}}{3}\end{aligned}$$ We obtain then the expression of $K_{\yng(1,1,1)}$: $$K_{\yng(1,1,1)}=K_{\yng(3)}+K_{\yng(4)}+\frac{K_{\yng(3,1)}}{3}-K_{\yng(2,2)}
\label{Kidlevel3}$$ Using again Eq. (\[defastrace\]) and the eigenvalue identities obtained at level $l=4$, this becomes \_[i=1]{}\^8 (\_[(1,1,1),i]{})\^N = \_[i=1]{}\^8 (\_[(3),i]{})\^N - ( \_[(4)]{} )\^N - ( \_[(3,1),1]{} )\^N + ( \_[(3,1),2]{} )\^N + ( \_[(3,1),3]{} )\^N from which we deduce that $\lambda_{\yng(3),1} = \lambda_{\yng(4)}$, and that $\lambda_{\yng(3),2} = \lambda_{\yng(3,1),1}$. (Note that we cannot have, for example, $\lambda_{\yng(4)} = \lambda_{\yng(3,1),2}$ since these eigenvalues were shown to be independent in the preceeding analysis at level $l=4$.) We can then further deduce that $\lambda_{\yng(1,1,1),1} =
\lambda_{\yng(3,1),2}$, that $\lambda_{\yng(1,1,1),2} =
\lambda_{\yng(3,1),3}$, and that $\lambda_{\yng(1,1,1),i} =
\lambda_{\yng(3),i}$ for $i=3,4,\ldots,8$. So among the 8 eigenvalues participating in $K_{\yng(3)}$ and the 8 participating in $K_{\yng(1,1,1)}$ only a total of 6 are new. On the other hand, all $16$ eigenvalues participating in $K_{\yng(2,1)}$ are new, since $K_{\yng(2,1)}$ did not appear in an identity such as Eq. (\[Kidlevel3\]).
The eigenvalues for $l\leq 2$ are all new, as there are no relations between the $K_{l,D}$. Therefore, there are $28$ new eigenvalues associated to $K_{\yng(2)}$, $28$ to $K_{\yng(1,1)}$, $35$ to $K_1$ and $14$ to $K_0$.
To obtain the amplitudes associated to the eigenvalues, we use Eq. (\[expZ\]): $$\begin{aligned}
Z&=&K_0 + b^{(1)} K_1 + b^{(2)} K_{2,{\rm Id}} + b^{(1,n_1>1)} K_{2,(1,2)} +
b^{(3)} K_{3,{\rm Id}} + b^{(1,n_1>1)} K_{3,(1,3)} \nonumber \\
&+& b^{(4)} K_{4,{\rm Id}} + b^{(1,n_1>1)} K_{4,(1,4)} + b^{(2,n_1>1)}
K_{4,(2,2)}\end{aligned}$$ and insert the expressions of the $K_{l,C_l}$ in terms of the independent $K_{l,D}$ chosen above: $$\begin{aligned}
Z&=&K_0 + b^{(1)} K_1 + \tilde{b}^{(\yng(2))}K_{\yng(2)}+ \tilde{b}^{(\yng(1,1))}K_{\yng(1,1)} + \tilde{b}^{(\yng(3))}K_{\yng(3)}+ \tilde{b}^{(\yng(2,1))}K_{\yng(2,1)} \nonumber \\
&+&\tilde{b}^{(\yng(4))}K_{\yng(4)} + \tilde{b}^{(\yng(3,1))}K_{\yng(3,1)} + \tilde{b}^{(\yng(2,2))}K_{\yng(2,2)} \label{expZforL4}\end{aligned}$$ where the amplitudes associated to the independent $K_{l,D}$ are: $$\begin{aligned}
\tilde{b}^{(\yng(2))}&=&\frac{b^{(2)}}{2}+\frac{b^{(1,n_1>1)}}{2} \\
\tilde{b}^{(\yng(1,1))}&=&\frac{b^{(2)}}{2}-\frac{b^{(1,n_1>1)}}{2} \\
\tilde{b}^{(\yng(3))}&=&\frac{b^{(3)}}{3}+\frac{2 b^{(1,n_1>1)}}{3} \\
\tilde{b}^{(\yng(2,1))}&=&\frac{b^{(3)}}{6}-\frac{b^{(1,n_1>1)}}{6} \\
\tilde{b}^{(\yng(4))}&=&\frac{b^{(3)}}{6}+\frac{4b^{(1,n_1>1)}}{3}+\frac{b^{(4)}}{4}-\frac{b^{(2,n_1>1)}}{4}\\
\tilde{b}^{(\yng(3,1))}&=&\frac{b^{(3)}}{18}+\frac{b^{(1,n_1>1)}}{9}+\frac{b^{(4)}}{12}-\frac{b^{(2,n_1>1)}}{12} \\
\tilde{b}^{(\yng(2,2))}&=&-\frac{b^{(3)}}{6}-\frac{7b^{(1,n_1>1)}}{12}+\frac{b^{(2,n_1>1)}}{4}\end{aligned}$$ These can now be calculated explicitly from Eqs. (\[defbltor\]) and (\[defbnPn1>1\]): $$\begin{aligned}
\tilde{b}^{(\yng(2))}&=&\frac{Q^2}{2}-\frac{3Q}{2} \\
\tilde{b}^{(\yng(1,1))}&=&\frac{Q^2}{2}-\frac{3Q}{2}+1 \\
\tilde{b}^{(\yng(3))}&=&\frac{Q^3}{3}-2Q^2+\frac{8Q}{3}-1 \\
\tilde{b}^{(\yng(2,1))}&=&\frac{Q^3}{6}-Q^2+\frac{4Q}{3} \\
\tilde{b}^{(\yng(4))}&=&\frac{Q^4}{4}-\frac{11Q^3}{6}+\frac{15Q^2}{4}-\frac{5Q}{3}-2 \\
\tilde{b}^{(\yng(3,1))}&=&\frac{Q^4}{12}-\frac{11Q^3}{18}+\frac{15Q^2}{12}-\frac{5Q}{9}-\frac{1}{3} \\
\tilde{b}^{(\yng(2,2))}&=&-\frac{Q^3}{6}+\frac{5Q^2}{4}-\frac{25Q}{12}+\frac{3}{2}\end{aligned}$$ Note that the four first amplitudes in this list have been obtained by Chang and Schrock [@shrock] using a different method.
We can finally give the amplitudes of the eigenvalues themselves. The $14$ eigenvalues at level $0$ have amplitude $1$. The $35$ eigenvalues at level $1$ have amplitude $\tilde{b}^{(1)}$. At level $2$, the $24$ eigenvalues contributing to $K_{\yng(2)}$ have amplitude $\tilde{b}^{\yng(2)}$, and the $24$ eigenvalues contributing to $K_{\yng(1,1)}$ have amplitude $\tilde{b}^{\yng(1,1)}$. At level $3$, the $6$ new eigenvalues contributing to $K_{\yng(3)}$ have amplitude $\tilde{b}^{\yng(3)}$, and the $16$ eigenvalues contributing to $K_{\yng(2,1)}$ have amplitude $2\tilde{b}^{\yng(2,1)}$. At level $4$, $\lambda_{\yng(4)}$ has amplitude $\tilde{b}^{\yng(4)}+2\tilde{b}^{\yng(2,2)}+\tilde{b}^{\yng(3)}$, $\lambda_{\yng(1,1,1,1)}$ has amplitude $3\tilde{b}^{\yng(3,1)}+2\tilde{b}^{\yng(2,2)}+\tilde{b}^{\yng(3)}$, and $\lambda_{\yng(3,1),1}$ and $\lambda_{\yng(3,1),2}$ both have the same amplitude $3\tilde{b}^{\yng(3,1)}$.
Note that when we know the amplitudes of the $K_{l,D}$ in the expansion of $Z$ and the relations between $K_{l,D}$ for a given width $L$, we know it for all widths smaller than $L$, as they do not change. The only difference is that for smaller widths some of the $K_{l,D}$ vanish, so the equations must be truncated. For example, the result (\[Kidlevel3\]) obtained here for $L=4$, implies by truncation of the $K_{4,D}$ terms that for $L=3$: $$K_{\yng(1,1,1)}=K_{\yng(3)}$$ Likewise, the expansion (\[expZforL4\]) of $Z$ obtained here for $L=4$, implies by truncation that for $L=3$: $$Z=K_0 + b^{(1)} K_1 + \tilde{b}^{(\yng(2))}K_{\yng(2)}+ \tilde{b}^{(\yng(1,1))}K_{\yng(1,1)}+ \tilde{b}^{(\yng(3))}K_{\yng(3)}+ \tilde{b}^{(\yng(2,1))}K_{\yng(2,1)}$$ From these equations, it is then simple to obtain the amplitudes of eigenvalues for $L=3$.
Particular non-generic lattices
-------------------------------
The degeneracies we have obtained apply to the case of a generic lattice. In the case of a specific lattice, i.e., one having extra non-generic symmetries, there might be additional degeneracies. An example is the case of a square or a honeycomb lattice, because of the invariance of the lattice under reflection by its symmetry axis.
Specifically, the transfer matrix ${\rm T}_L$ of the square lattice commutes with the dihedral group $D_L$, since the lattice enjoys both translational and reflectional symmetries in the space perpendicular to the transfer direction. Likewise, for the honeycomb lattice, the symmetry group is $D_{\frac{L}{2}}$. Those groups act on the right time slice, not on the left one (i.e., the black points), and thus commute with the symmetry group $S_l$ at level $l$ of the bridges. There are therefore additional degeneracies inside a given $K_{l,D}$.
In the case of the dihedral group, since all its irreps are of dimension $1$ or $2$, there are additional degeneracies between pairs of eigenvalues, as observed in Ref. [@shrock]. The method to determine precisely these degeneracies is to decompose the space at level $l$ of symmetry $D$ into irreps of the dihedral group, and to count the number of irreps of dimension $2$.
Furthermore, in the case of a square or a hexagonal lattice, there are yet additional degeneracies (which do not exist for a generic lattice with a dihedral symmetry). Indeed, for these two lattices, the $Z_{L,n_1}$ with $n_1>1$ are found to be zero: at level $l=L$ no permutation between bridges is allowed. At this level there is thus only one eigenvalue of total degeneracy $\sum_{D \in S_l} \left[ {\rm dim}(D)\right]^2 \tilde{b}^{(L,D)}=b^{(L)}$. Finally, in the case of a square lattice, there appears to be additional “accidental” degeneracies: for example, an eigenvalue at level $1$ coincides with an eigenvalue at level $2$, as observed in Ref. [@shrock].
Conclusion
==========
To summarise, we have generalised the combinatorial approach developed in Ref. [@cyclic] for cyclic boundary conditions to the case of toroidal boundary conditions. In particular, we have obtained the decomposition of the partition function for the Potts model on finite tori in terms of the generalised characters $K_{l,D}$. This decomposition is considerably more difficult to interpret than in the cyclic case, as some eigenvalues coincide between different levels $l$ for all values of $Q$. We have nevertheless succeeded in giving an operational method of determining the amplitudes of the eigenvalues as well as their [*generic*]{} degeneracies.
The eigenvalue amplitudes are instrumental in determining the physics of the Potts model, in particular in the antiferromagnetic regime [@af]. Generically, this regime belongs to a so-called Berker-Kadanoff (BK) phase in which the temperature variable is irrelevant in the renormalisation group sense, and whose properties can be obtained by analytic continuation of the well-known ferromagnetic phase transition [@hs]. Due to the Beraha-Kahane-Weiss (BKW) theorem [@bkw], partition function zeros accumulate at the values of $Q$ where either the amplitude of the dominant eigenvalue vanishes, or where the two dominant eigenvalues become equimodular. When this happens, the BK phase disappears, and the system undergoes a phase transition with control parameter $Q$. Determining analytically the eigenvalue amplitudes is thus directly relevant for the first of the hypotheses in the BKW theorem.
For the cyclic geometry, the amplitudes are very simple, and the values of $Q$ satisfying the hypothesis of the BKW theorem are simply the so-called Beraha numbers, $Q=B_n=(2 \cos(\pi/n))^2$ with $n=2,3,\ldots$, independently of the width $L$. For the toroidal case, we have no general formula for the amplitudes, valid for any $L$. It is however clear from the amplitudes given for $L \le 4$ in Sec. \[sec:amplitudes\] that many of them vanish at $Q=2$, and yet other differ just by a sign by virtue of Eq. (\[defbnPn1>1\]). Indeed, it is consistent with simple physical arguments, that a phase transition in the antiferromagnetic regime must take place at $Q=2$. However, it remains to elucidate whether the BK phase exists for all other $Q \in
(0,4)$, and whether the Beraha numbers play any special role in the toroidal case.
[**Acknowledgments.**]{}
The authors are grateful to P. Zinn-Justin and J.-B. Zuber for some useful discussions. JLJ further thanks the members of the SPhT, where part of this work was done, for their kind hospitality.
[99]{}
P.W. Kasteleyn and C.M. Fortuin, J. Phys. Soc. Jap. Suppl. [**26**]{}, 11 (1969); C.M. Fortuin and P.W. Kasteleyn, Physica [**57**]{}, 536 (1972).
J.-F. Richard and J. Jacobsen, [*Character decomposition of Potts model partition functions. I. Cyclic geometry*]{}, math-ph/0605xxx.
S.-C. Chang and R. Shrock, [*Transfer matrices for the partition function of the Potts model on toroidal lattice strips*]{}, cond-mat/0506274.
A. Nichols, J. Stat. Mech. [**0601**]{}, 3 (2006); hep-th/0509069.
P. Di Francesco, H. Saleur and J.B. Zuber, J. Stat. Phys. [**49**]{}, 57 (1987).
V. Pasquier, J. Phys. A [**20**]{}, L1229 (1987).
J.-F. Richard and J.L. Jacobsen, Nucl. Phys. B [**731**]{}, 335 (2005); math-ph/0507048.
J.L. Jacobsen and H. Saleur, Nucl. Phys. B [**743**]{}, 207 (2006); cond-mat/0512056.
H. Saleur, Commun. Math. Phys. [**132**]{}, 657 (1990); Nucl. Phys. B [**360**]{}, 219 (1991).
S. Beraha, J. Kahane and N.J. Weiss, Proc. Natl. Acad. Sci. USA [**72**]{}, 4209 (1975).
[^1]: An example occurs for the square lattice of width $L=4$, where an eigenvalue at level $1$ coincides with an eigenvalue at level $2$ [@shrock], without any apparent reason.
[^2]: By a generic lattice we understand one without mirror symmetry with respect to the transfer axis, i.e., without any accidental degeneracies. An example of a generic lattice is the triangular lattice, drawn as a square lattice with diagonals added.
[^3]: It turns out that there are more degeneracies than warranted by this argument. The reason is that the cluster configurations cannot realise all the permutations of $S_l$ (recall our preceeding discussion), and thus some $\lambda_{l,D,k}$ with different $l$ and/or $D$ are in fact equal. We shall come back to this point later.
[^4]: The best one could hope for would be an explicit formula relating $\tilde{b}$ to the characters of the symmetric group.
|
---
abstract: 'We present the latest results on the search for bottom baryon states using $ \sim1\invfb $ of CDF data. The study is performed with the world’s largest sample of fully reconstructed decays collected by CDF II detector at $ \sqrt{s}=1.96\tev $ in the hadronic trigger path. We observe 4 new states consistent with bottom baryons.'
address: |
Department of Physics and Astronomy,\
University of New Mexico,\
800 Yale Blvd. NE, Albuquerque, NM 87131, USA
author:
- |
Igor V. Gorelov \
(For the CDF Collaboration)
title: |
First Observation of Bottom Baryon\
$ \mathbf{\Sigb} $ States at CDF
---
[FERMILAB-CONF-07-027-E]{}\
\[sec:Introduction\] Introduction
=================================
High energy particle colliders provide a wealth of experimental data on bottom mesons. However, only one bottom baryon, the , has been directly observed [@exp:lb-delphi; @exp:lb-aleph; @exp:lb-cdf1; @exp:lb-cdf2].
Heavy baryons containing one heavy quark and a light diquark became a nice 3-body laboratory to test QCD models. In the limit of heavy quark mass $ m_{Q}\to\infty $, heavy baryons’ properties are governed by the dynamics of the light diquark in a gluon field created by the heavy quark acting as a static source. The heavy baryon like can be considered as a “helium atom” of QCD. In this Heavy Quark Symmetry (HQS) approach [@th:isgur-wise] at the heavy quark limit a heavy quark spin does not interact with the gluon field, the spin decouples from the degrees of freedom of the light quark and the quantum numbers of the heavy quark and the light diquark are separately conserved by the strong interaction. Consequently the light diquark momentum $\mathbf{j_{qq}=s_{qq}+L_{qq}}$, the heavy quark spin $\mathbf{s_{Q}}$ and total momentum $\mathbf{J_{Qqq}=s_{Q}+j_{qq}}$ are considered as good quantum numbers. Based on the HQS principles an effective field theory was constructed where $\mathbf{\frac{1}{m_{Q}}}$ corrections can be systematically included in the perturbative expansions. The theory was named as Heavy Quark Effective Theory (HQET) (see [@th:hqet] and references therein).
For the bottom $Q\equiv\,b$ baryons with a single heavy quark and two light ones (see Table \[tab:bbaryons\]) the bottom quark spin, $ \mathbf{s_Q={\frac{1}{2}}^{+}} $, is combined with the light diquark momentum $\mathbf{j_{qq}}$ comprised of spin $\mathbf{s_{qq}=0^{+}\oplus 1^{+}}$ and its angular momentum $\mathbf{L_{qq}}$. The baryons with a diquark having $\mathbf{s_{qq}=0^{+}}$ and isospin $\mathbf{I=0}$ are called $\Lambda$- type, while the states with $\mathbf{s_{qq}=1^{+}}$ and isospin $\mathbf{I=1}$ are called $\Sigma$- type. The ground state baryon has $\mathbf{I=0,\,J^{P}={\frac{1}{2}}^{+}}$. A doublet of ground $\Sigma$- like bottom baryons comprises with $\mathbf{I=1,\,J^{P}={\frac{1}{2}}^{+}}$ and with $\mathbf{I=1,\,J^{P}={\frac{3}{2}}^{+}}$.
The combination of an orbital momentum $\mathbf{L_{qq}=1^{-}}$ of a diquark with its spin of $\mathbf{s_{qq}=0^{+}}$ adds to the spectroscopy a number of excited $P$- wave bottom $\Lambda$ -states. The lowest lying orbital excitations are with $\mathbf{J^{P}={\frac{1}{2}}^{-},\,{\frac{3}{2}}^{-}}$ [@Chow:1994sz; @Chow:1995nw; @Isgur:1999rf].
[llll]{} & [**Quarks**]{} & $\mathbf{J^{P}}$ & $\mathbf{(I,I_{3})}$\
$\Lb$ & $b[ud]$ & $(1/2)^{+}$ & $(0,0)$\
$\Sigbp$ & $buu$ & $(1/2)^{+}$ & $(1,+1)$\
$\Sigbz$ & $b\{ud\}$ & $(1/2)^{+}$ & $(1,0)$\
$\Sigbm$ & $bdd$ & $(1/2)^{+}$ & $(1,-1)$\
$\Sigbstp$ & $buu$ & $(3/2)^{+}$ & $(1,+1)$\
$\Sigbstz$ & $b\{ud\}$ & $(3/2)^{+}$ & $(1,0)$\
$\Sigbstm$ & $bdd$ & $(3/2)^{+}$ & $(1,-1)$\
$\Lbst$ & $b[ud]$ & $(1/2)^{-}$ & $(0,0)$\
$\Lbst$ & $b[ud]$ & $(3/2)^{-}$ & $(0,0)$\
Theoretical expectations for ground bottom baryon states are summarized in Table \[tab:theory\]. The calculations have been done with non-relativistic and relativistic potential quark models, $1/N_{c}$ expansion, quark models in the HQET approximation, sum rules, and finally with lattice QCD models [@th:mass-pred].
In a physics reality $\mathbf{m_{Q}} $ is finite and a degeneration of a $\{\Sigb,\,\Sigbst\}$ doublet is resolved by a hyperfine mass splitting between its states. There is also an isospin mass splitting (see Table \[tab:theory\]) between members of and isotriplets [@th:lichtenberg; @th:capstick; @th:jrosner1998; @th:jrosner]. As it was pointed out [@th:jrosner] the value of the isospin splitting within triplet does differ from triplet, namely $(\Sigbstp\,-\,\Sigbstm)-(\Sigbp\,-\,\Sigbm)=0.40\pm0.07\mevcc$. This number contributes to systematic uncertainty of our experimental results (see Section \[sec:Candidates\]).
[ll]{} & $ \mathbf{\mevcc} $\
$ {m(\Sigb) - m(\Lb)} $ & ${180 - 210}$\
$ {m(\Sigbst) - m(\Sigb)} $ & ${10 - 40}$\
$ {m(\Sigbm) - m(\Sigbp)}$ & ${5 - 7}$\
$ {m(\Lb)} $,[** fixed**]{} & [${5619.7}$]{}\
[**from CDF II**]{} & [$ \pm1.2\pm1.2 $]{}\
$ {\Gamma(\Sigb),\Gamma(\Sigbst)}$ & [$ \sim8, \sim15 $]{}\
see below &\
According to HQS the physics of pion transitions between heavy baryons is governed by the light diquark. The one or two pions are emitted from the light diquark while the heavy quark propagates unaffected by the pion emission process. Various pion transitions of bottom baryons into the lower ground states are summarized at the Figure \[fig:exc-pions\]. The mass predictions for the $S$- wave (i.e. ground state) $\Sigma$- like baryons show that there is enough phase space for the both and to decay into via single-pion emission. The two excited ($P$-wave) states might decay into via two-pion transitions provided a sufficient phase space.
![ \[fig:exc-pions\] Transitions of ground states , into a low lying ground state via a single-pion emission in a $P$-wave as the diquark with $j_{qq}=1^{+}$ is converted to one with $j_{qq}=0^{+}$. If states have masses sufficiently higher than the decay thresholds, the two-pion transitions into or a single-pion decays into or states are possible (see also a discussion [@Chow:1994sz; @Chow:1995nw; @Isgur:1999rf]). single-pion transitions to are forbidden due to isospin conservation and also by parity (for the higher $J^{P}={\frac{3}{2}}^{-}$ state). ](./LamBPiTrans.1.eps){width="22pc"}
It is important for our experimental expectations to understand the natural width of baryons. As we expect that masses lie within $(180 - 210)\mevcc$ above and well above a threshold for a single-pion mode, we would expect that the single-pion $P$-wave transition will dominate the total width [@th:peskin; @th:koerner]. The authors [@th:koerner] find $$\Gamma_{\Sigma_Q\to\Lambda_Q\pi}
= \frac{1}{6\pi}\frac{M_{\Lambda_Q}}{M_{\Sigma_Q}}
\left|f_p\right|^2\left|\vec{p}_\pi\right|^{3},$$ where $\vec{p}_\pi$ is the three-momentum of soft $ \pi_{\Sigma_Q} $, $f_p\equiv\,g_A/f_\pi $, $f_\pi=92\mev $ and $g_A$ is the axial vector coupling of the constituent quark for the nucleon. A fit of this formula to the known PDG width measurements [@pdg] of charm states and yields $g_A=0.75\pm0.05$ which is in excellent agreement with $g_A=0.75$ numerical theoretical value for the nucleon. Using the fit results we have estimated $\Gamma(\Sgbst)$, see the Table \[tab:theory\]. The error of the fitted $g_A$ contributes as a systematic uncertainty to our experimental measurements (see Section \[sec:Candidates\]).
\[sec:Principle\] Principle of the Analysis
===========================================
The topology of the event with state produced in Tevatron collisions is demonstrated in Figure \[fig:topology\]. The candidates are searched in the decay chain:
- Strong decay $\Sgbstpm\to\LambB\pi_{\Sigb}^{\pm}$ with both $\LambB\to\Lcpim$ and its daughter $\Lc\to\pKpi$ in weak decay modes.
To remove a contribution due to a mass resolution of each candidate and to avoid absolute mass scale systematic uncertainties, the candidates are reconstructed in the mass difference [*Q-value*]{} spectra defined as $$Q =
M(\LambB\pi_{\Sigb}^{\pm})-M(\Lb)-\,M_{\rm PDG}(\pipm).$$ The narrow signatures are searched for in the $Q$-spectrum constructed separately for every charge state of candidates. The subsample of contains $ \Lb\pim $ and $
\overline{\Lb}\pip $ combinations from the decays of the particles and the antiparticles $ \overline{\Sigb}^{(*)+} $, respectively. The subsample of contains $ \Lb\pip $ and $ \overline{\Lb}\pim $ combinations from the decays of the particles and the antiparticles $ \overline{\Sigb}^{(*)-} $, respectively.
![ \[fig:topology\] Sketch of the event topology of a produced in the CDF detector [@cdf:det]. The bottom baryon is produced at the collision origin, i.e. primary vertex of the event. The decays strongly into with soft $\pi_{\Sigb}^{\pm}$ emitted. Due to a fast nature of the strong decay the soft pion originates from the primary vertex. The and its daughter decay weakly with a secondary decay vertex measured in SVX II[@cdf:svx2]. The tracks with a common secondary vertex are displaced relative to a primary vertex and the proton from decay and from decay most likely contribute to the CDF hadron Two Track Trigger (see Section \[sec:Triggers\] and [@cdf:svt]). ](./sigmab.xfig.eps){width="18pc"}
The signal region at $Q$- value spectrum is defined as $30\mevcc\lsim Q\lsim100\mevcc$, based on the theoretical expectation (see Table \[tab:theory\]). We pursued a [*blind analysis*]{} and developed the selection criteria using [*only*]{} the pure background sample in the upper and lower sideband regions of $0\mevcc\lsim Q\lsim30\mevcc$ and $100\mevcc\lsim Q\lsim500\mevcc$. The signal was modeled by a [PYTHIA]{} [@th:pythia] Monte Carlo.
\[sec:Triggers\] Triggers and Datasets
======================================
Our results are based on data collected with the 2 detector [@cdf:det] and corresponding to an integrated luminosity of $\sim1.1\invfb$. As $\pap$ collisions at 1.96TeV have an enormous inelastic total cross-section of $
\sim\,60\,\mbarn $, while - hadron events comprise only $
\approx\,20\,\mub\,\,(|\eta|<1.0) $, triggers selecting - hadron events are of vital importance. Our analysis is based on a data sample collected by a three-level Two displaced Track Trigger. It reconstructs a pair of high $ \pt>2.0\gevc $ tracks at Level 1 with the CDF central tracker and enables secondary vertex selection at Level 2. This requires each of the tracks to have an impact parameter measured by the CDF silicon detector SVX II [@cdf:svx2; @cdf:svt] to be larger than 120. The excellent impact parameter resolution of SVX II makes this challenging task possible. The trigger proceeds with a full event reconstruction at Level 3. The Two displaced Track Trigger is efficient for heavy quark hadron decay modes (see Figure \[fig:topology\] and its caption).
\[sec:Selection\] Event Selection
=================================
The candidates of the basic state in our analysis, , have been reconstructed in the mode $ \Lb\to\Lcpim $ with $ \Lc\to\pKpi
$. The Two displaced Track Trigger requirements (see Section \[sec:Triggers\]) are confirmed offline for each candidate. The Charm and bottom candidates have both been subjected to 3-dimensional vertex fits. The collection of the fitted candidates has been confined to a mass range of $m_{\Lc}^{PDG}\pm 16\mevcc $ [@pdg]. To suppress a prompt background we apply a cut on the proper decay time $ \ct(\Lb)>250\mum
$ with its significance $ \ct(\Lb)/\sigma_{\ct}>10 $. The proper decay time of with respect to the vertex is required to be $
-70\mum<\ct(\Lc\from\Lb)<200\mum $. We define the topological quantities as $ \ct\equiv L_{xy}\cdot m_{\Lambda_{Q}}/\pt $ and $
L_{xy}=\vec{D_{xy}}\cdot\vec{\pt}/\pt $ where $
\vec{D_{xy}},\,\vec{\pt} $ are the vectors of corresponding distance or momentum in a transverse plane. To reduce combinatorial background and contribution from partially reconstructed modes the impact parameter $ d_0(\Lb) $ is also restricted to be below of 80, where $ d_{0}=|\vec{D_{xy}}\times\vec{\pt}/\pt| $. The kinematic cuts for and candidates, $ \pt(\Lb)>6.0\gevc $ and $ \pt(\Lc)>4.5\gevc $ are applied as well.
The powerful $ \Lb\to\Lcpim $ signal is shown in Figure \[fig:lb\_mass\] with a binned maximum likelihood fit superimposed. The background from physical states contributing to the left side of the signal is analyzed with Monte Carlo simulations. The fit to the invariant mass distribution yields $ 3125\pm62\stat $ of $ \Lb\to\Lcpim $ candidates. We posses the world’s largest sample.
****\
![ \[fig:lb\_mass\] Fit to the invariant mass of $ \Lb\to\Lcpim $ candidates. Fully reconstructed decays such as $ \Lb\to\Lcpim $ and $\Lb\to\Lc{K^{-}}$ are not indicated on the figure. The signal region, $5.565\gevcc < m(\Lcpim) < 5.670\gevcc$, consists primarily of baryons, with some contamination from mesons and combinatorial events. The left side band is enriched by partially reconstructed decays like $\Lb\to\Lambda_c^{*+}\pi^-$ and by fully reconstructed 4-prong - meson decays like $\Bd\to\Dp\pim,\,\Dp\to\Km\pip\pip $. The right side band consists from a combinatorial background. ](./BaselineBlindedData_massFit.eps "fig:"){width="25pc"}
\[sec:Candidates\] $ \mathbf{\Sigb} $ Candidates and Signals
============================================================
Following a method outlined in a Section \[sec:Principle\] the candidates (see a Section \[sec:Selection\]) from a signal region of $ [5.565, 5.670]\gevcc $ have been coupled with the pion tracks $
\pi_{\Sigb}^{\pm} $ to create candidates and the pairs $\Lb
\pi_{\Sigb}^{\pm} $ have been subjected to a common vertex fit.
The analysis cuts are optimized according to a [*blind analysis*]{} technique using the upper and lower sideband regions (see a Section \[sec:Selection\]) of experimental $Q$- value spectrum while the signal is modeled by a [PYTHIA]{} [@th:pythia]. The following kinematic and topological variables are used : $
\pt(\Sigb) $, the soft pion track impact parameter significance $
\IPsig\,(\pi_{\Sigb})$, and the polar angle of the soft pion in a - rest frame, ${\costhst\,(\pi_{\Sigb})\,=}$ ${\vec{p}_{\Sigb}\cdot\vec{p}_{\pi}^{*}/(|\vec{p}_{\Sigb}|\cdot|\vec{p}_{\pi}^{*}|)}$. A figure of merit is defined as $\epsilon(S_{\rm MC})/\sqrt{B}$, where $\epsilon(S_{\rm MC})$ is the signal efficiency measured in the Monte Carlo sample and $B$ is the background in the signal region estimated from the upper and lower sidebands. The maximum of the figure of merit is reached for $
\pt(\Sigb)>9.5\gevc $, $ \IPsig\,(\pi_{\Sigb})< 3.0 $, and $
\costhst >-0.35 $.
![ \[fig:Qblinded\] **a)** The three background sources described in the text and their sum are shown superimposed on the $Q$ distributions, with the signal region blinded. The top plot shows the $
\LambB\pi_{\Sigb}^{-} $ distribution, which contains all $\Sigbm$. The bottom plot shows the $ \LambB\pi_{\Sigb}^{+}
$ distribution, with all $\Sigbp$. The sideband regions are parameterized with a power law multiplied by an exponential. The percentage of each background component in the signal region is computed from the mass fit (see Figure \[fig:lb\_mass\]), and is: $N(\Lb)\approx90.1\%$, $N(B)\approx6.3\%$ and $N({\rm
combinatorial})\approx3.6\%$; **b)** The excess in the $ \LambB\pi_{\Sigb}^{-} $ (top plot) subsample is $148$ candidates over $268$ expected background candidates, whereas in the $ \LambB\pi_{\Sigb}^{+} $ (bottom plot) subsample the excess is $108$ over $298$ expected background candidates. ](./Blinded_Points_FullRange.eps "fig:"){width="18pc"} ![ \[fig:Qblinded\] **a)** The three background sources described in the text and their sum are shown superimposed on the $Q$ distributions, with the signal region blinded. The top plot shows the $
\LambB\pi_{\Sigb}^{-} $ distribution, which contains all $\Sigbm$. The bottom plot shows the $ \LambB\pi_{\Sigb}^{+}
$ distribution, with all $\Sigbp$. The sideband regions are parameterized with a power law multiplied by an exponential. The percentage of each background component in the signal region is computed from the mass fit (see Figure \[fig:lb\_mass\]), and is: $N(\Lb)\approx90.1\%$, $N(B)\approx6.3\%$ and $N({\rm
combinatorial})\approx3.6\%$; **b)** The excess in the $ \LambB\pi_{\Sigb}^{-} $ (top plot) subsample is $148$ candidates over $268$ expected background candidates, whereas in the $ \LambB\pi_{\Sigb}^{+} $ (bottom plot) subsample the excess is $108$ over $298$ expected background candidates. ](./UnblindedBkgs_Points_FullRange.eps "fig:"){width="18pc"}\
****
The $Q$- value spectra with blinded signal region are shown in Figure \[fig:Qblinded\] with its detailed caption. In the search, the dominant background is from the combination of prompt baryons with extra tracks produced in the hadronization process. The remaining backgrounds are from the combination of hadronization tracks with $B$ mesons reconstructed as baryons, and from combinatorial background events.
Upon unblinding the $Q$ signal region in both spectra we observe an excess of events over the background as shown in Figure \[fig:Qblinded\] with the details explained in the caption.
Next we perform a simultaneous unbinned maximum likelihood fit to the $ \LambB\pi_{\Sigb}^{-} $ and $\LambB\pi_{\Sigb}^{+} $ subsamples for a signal from each expected state plus the background, referred to as the “four signal hypothesis.” Each signal consists of a Breit-Wigner distribution convoluted with two Gaussian distributions describing the detector resolution, with a dominant narrow core and a small broad component for the tails. The natural width of each Breit-Wigner distribution is computed from the central $Q$- value (see a Section \[sec:Introduction\] and [@th:koerner]). The fit shown in Figure 5 results in the yields $N_{\Sigbp} = 33^{+13}_{-12}$, $N_{\Sigbm} = 62^{+15}_{-14}$, $N_{\Sigbstp}= 82\pm17$, and $N_{\Sigbstm}= 79\pm18$ candidates, with the signals located at $Q_{\Sigbp} = 48.2^{+1.9}_{-2.2}\mevcc$, $Q_{\Sigbm} = 55.9^{+1.0}_{-0.9}\mevcc$, and $\Delta_{\Sigbst} = 21.5^{+2.0}_{-1.8}\mevcc$, where $\Delta_{\Sigbst}\equiv(Q_{\Sigbst}\,-\,Q_{\Sigb})$.
+:---------------------------------:+:---------------------------------:+
| {width="18pc"}\ | ------------------------------- |
| | **Table 3.** Likelihood ratios |
| ------------------------------- | calculated for the |
| --------------------------------- | alternative signal hypothesis w |
| --------------- | ith respect to |
| **Figure 5.** Simultaneous fit | the one of four states. The str |
| to the $ \LambB\pi_{\Sigb}^{-} $ | ength of |
| (top) and $ \LambB\pi_{\Sigb}^{ | the signal hypothesis is furthe |
| +} $ (bottom) spectra for $ {\Sig | r given by the |
| b}^{(*)\mp} $ | likelihood ratio, $LR\equiv L/L |
| candidates shown on a range of | _{\rm alt}$, where $L$ is the |
| $Q \in$ \[0, 200\] $\mevcc$. | likelihood of the four signal h |
| ------------------------------- | ypothesis and |
| --------------------------------- | $L_{\rm alt}$ is the likelihood |
| --------------- | of an alternate |
| | hypothesis. |
| \ | ------------------------------- |
| | ------------------------------- |
| | |
| | \ |
| | |
| | [ll]{} Hypothesis & $LR$\ |
| | Null & $2.6\times 10^{19}$\ |
| | Two States & $1.6\times 10^{6}$\ |
| | No Signal & $3.3\times 10^{4}$\ |
| | No Signal & $3$\ |
| | No Signal & $2.4\times 10^{4}$\ |
| | No Signal & $1.8\times 10^{4}$\ |
| | |
| | \ |
| | |
| | ------------------------------- |
| | ---------------------------- |
| | The alternative hypothesis is e |
| | stimated using |
| | all systematic variations of th |
| | e background |
| | and signal functions. The varia |
| | tion with the |
| | largest value of $L_{\rm alt}$ |
| | corresponding to the least |
| | favorable hypothesis is taken. |
| | ------------------------------- |
| | ---------------------------- |
+-----------------------------------+-----------------------------------+
Systematic uncertainties on the mass difference and yield measurements fall into three categories: mass scale, background model, and signal parameterization. The mass scale is determined from the difference in the mean of the narrow resonances $m(\Dstarp)-m(\Dz)$, $m(\Sigczpp)-m(\Lc)$, $m(\Lambda_{c}(2625)^{+})-m(\Lc)$ between data and PDG[@pdg]. The uncertainties on the background come from the assumption on the sample composition of the signal region, the normalization and functional form of the hadronization background. The systematic effects related to assumptions made on the signal parameterization are: underestimation of the detector resolution in Monte Carlo, the accuracy of the natural width prediction from [@th:koerner], and the fit constraint that $(Q_{\Sigbstp}\,-\,Q_{\Sigbp})=(Q_{\Sigbstm}\,-\,Q_{\Sigbm})$ [@th:jrosner]. All systematic uncertainties on the $Q_{\Sgbstpm}$ mass difference measurements are small compared to the statistical uncertainties. The final results [@cdf:sigbweb] for the signal yields, including systematic errors, are $ N_{\Sigbp} = 33^{+13}_{-12}~\mbox{(stat.)} ^{+5}_{-3}~\mbox{(syst.)}$, $ N_{\Sigbm} = 62^{+15}_{-14}~\mbox{(stat.)} ^{+9}_{-4}~\mbox{(syst.)}$, $ N_{\Sigbstp} = 82^{+17}_{-17}~\mbox{(stat.)} ^{+10}_{-6}~\mbox{(syst.)}$, and $ N_{\Sigbstm} = 79^{+18}_{-18}~\mbox{(stat.)} ^{+16}_{-5}~\mbox{(syst.)}$. The final results [@cdf:sigbweb] for the masses are $ Q_{\Sigbp} = 48.2^{+1.9}_{-2.2}~\mbox{(stat.)}^{+0.1}_{-0.2}~\mbox{(syst.)}~\mevcc $, $ Q_{\Sigbm} = 55.9^{+1.0}_{-0.9}~\mbox{(stat.)} ^{+0.1}_{-0.1}~\mbox{(syst.)}~\mevcc $, and $ \Delta_{\Sigbst} = 21.5^{+2.0}_{-1.9}~\mbox{(stat.)} ^{+0.4}_{-0.3}~\mbox{(syst.)}~\mevcc$. Using the CDF II measurement of $m_{\Lb} = 5619.7 \pm 1.2~\mbox{(stat.)} \pm 1.2~\mbox{(syst.)}\mevcc$ [@Acosta:2005mq], the masses of the four states are [@cdf:sigbweb]:
$ m_{\Sigbp} = 5807.5^{+1.9}_{-2.2}~\mbox{(stat.)} \pm 1.7~\mbox{(syst.)}~\mevcc$, $ m_{\Sigbm} = 5815.2^{+1.0}_{-0.9}~\mbox{(stat.)} \pm 1.7~\mbox{(syst.)}~\mevcc$, $ m_{\Sigbstp} = 5829.0^{+1.6}_{-1.7}~\mbox{(stat.)} \pm 1.7~\mbox{(syst.)}~\mevcc$, $ m_{\Sigbstm} = 5836.7^{+2.0}_{-1.8}~\mbox{(stat.)} \pm 1.7~\mbox{(syst.)}~\mevcc$,
where the systematic uncertainties are now dominated by the total mass uncertainty.
The significance of the signal is evaluated with two methods: using statistical Monte-Carlo pseudo-experiments and comparing the likelihoods of the default four signal hypothesis with pessimistic alternate ones. The randomly generated background samples are fit with the four signal hypothesis. The probability for background to produce the observed experimental number of signal events or more is found to be less than $8.5\times 10^{-8}$, corresponding to a significance of greater than $5.2~\sigma$. The results on study of likelihood ratios are summarized in a Table 3 and its detailed caption.
Conclusions
===========
In summary, using a sample of $ \sim3100~\Lb\to\Lcpim $ candidates reconstructed in $1.1\invfb$ of CDF II data, we search for resonant $ \Lb\pipm $ states. We observe a significant signal of four states whose masses and widths are consistent with those expected for the lowest-lying charged $ \Sgbst $ baryons: , , and . This result represents the first observation of the $ \Sgbst $ baryons. The author is grateful to his colleagues from the CDF $B$-Physics Working Group for useful suggestions and comments made during preparation of this talk. The author thanks J. Rosner for useful discussions. The author thanks S. C. Seidel for support of this work and J. E. Metcalfe for reading the manuscript.
References {#references .unnumbered}
==========
[999]{} Abreu P 1996 [**]{} 351 Buskulic D 1996 [**]{} 442 Abe F 1997 [**]{} 1142 Acosta D 2006 [**]{} 202001 De Rujula A, Georgi H and Glashow S L 1976 [**]{} 785; Godfrey S and Isgur N 1985 [**]{} 189; Rosner J L 1986 [**]{} 109; Isgur N and Wise M B 1989 [**]{} 113; Isgur N and Wise M B 1991 [**]{} 1130; Godfrey S and Kokoski R 1991 [**]{} 1679; Neubert M 1994 [*Phys. Rept. *]{}[**245**]{} 259 and references herein Manohar A V and Wise M B 1994 [**]{} 1310; Manohar A V and Wise M B 2000 [*Heavy Quark Physics*]{} (Cambridge: Cambridge University Press) Chow C K and Wise M B 1994 [**]{} 2135 Chow C K 1996 [**]{} 3374 Isgur N 2000 [**]{} 014025 Kwong W, Rosner J L and Quigg C 1987 [**]{} 325; Bowler K C 1996 \[UKQCD Collaboration\] [**]{} 3619; Jenkins E 1996 [**]{} 4515; Jenkins E 1997 [**]{} 10; Karliner M and Lipkin H J [*Preprint*]{} ; Mathur N, Lewis R and Woloshyn R M 2002 [**]{} 014502; Albertus C, Amaro J E, Hernandez E and Nieves J 2004 333 Ebert D, Faustov R N and Galkin V O 2005 [**]{} 034026 Hwang W-Y P and Lichtenberg D B 1987 [**]{} 3526 Capstick S 1987 [**]{} 2800 Rosner J L 1998 [**]{} 4310 Rosner J L [*Preprint*]{} , submitted to [**]{} Falk A F and Peskin M E 1994 [**]{} 3320 Körner J G, Krämer M and Pirjol D 1994 [**]{} 787 and references herein Yao W-M 2006 [**]{} 1 Acosta D 2005 [**]{} 032001 Sill A 2000, [**]{} 1 Ashmanskas W 2004, [**]{} 532 Sjostrand T, Eden P, Friberg C, Lonnblad L, Miu G, Mrenna S, Norrbin E\
2001 238 CDF Collaboration, [*Observation of New Heavy Baryon and* ]{},\
`http://www-cdf.fnal.gov/physics/new/bottom/060921.blessed-sigmab/` Acosta D 2006 [**]{} 202001
|
---
abstract: 'We report on the experimental and theoretical study of the spatial fluctuations of the local density of states (EM-LDOS) and of the fluorescence intensity in the near-field of a gold nanoantenna. EM-LDOS, fluorescence intensity and topography maps are acquired simultaneously by scanning a fluorescent nanosource grafted on the tip of an atomic force microscope at the surface of the sample. The results are in good quantitative agreement with numerical simulations. This work paves the way for a full near-field characterization of an optical nanoantenna.'
address: '${}^1$Institut Langevin, ESPCI ParisTech & CNRS UMR 7587, 1 rue Jussieu, 75005 Paris, France ${}^2$Laboratoire de Photonique et Nanostructures (LPN-CNRS), Route de Nozay, 91460 Marcoussis, France'
author:
- 'V. Krachmalnicoff$^1$, D. Cao$^1$, A. Cazé$^1$, E. Castanié$^1$, R. Pierrat$^1$, N. Bardou$^2$, S. Collin$^2$, R. Carminati$^1$, and Y. De Wilde$^1$'
title: 'Towards a full characterization of a plasmonic nanostructure with a fluorescent near-field probe'
---
[99]{}
L. Novotny, and N. van Hulst, “Antennas for light", Nat. Phot. [**5**]{}, 83 (2011).
V. Krachmalnicoff, E. Castanié, Y. De Wilde, and R. Carminati, “Fluctuations of the Local Density of States Probe Localized Surface Plasmons on Disordered Metal Films", Phys. Rev. Lett. [**105**]{}, 183901 (2010).
E. Castanié, V. Krachmalnicoff, A. Cazé, R. Pierrat, Y. De Wilde, and R. Carminati, “Distance dependence of the local density of states in the near field of a disordered plasmonic film", Opt. Lett. [**37**]{}, 3006 (2012).
D. Canneson, I. Mallek-Zouari, S. Buil, X. Quélin, C. Javaux, B. Mahler, B. Dubertret, and J.-P. Hermier, “Strong Purcell effect observed in single thick-shell CdSe/CdS nanocrystals coupled to localized surface plasmons", Phys. Rev. B [**84**]{}, 245423 (2011).
K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface", Phys. Rev. B [**68**]{}, 245405 (2003).
L. Novotny and B. Hecht, [*Principles of Nano-Optics*]{} (Cambridge University, 2006).
K.J. Vahala, “Optical microcavities", Nature [**424**]{}, 839 (2003).
M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of Spatial Fluctuations of the Local Density of States in Random Photonic Media", Phys. Rev. Lett. [**105**]{}, 013904 (2010).
R. Sapienza, P. Bondareff, R. Pierrat, B. Habert, R. Carminati, and N. F. van Hulst, “Long-Tail Statistics of the Purcell Factor in Disordered Media Driven by Near-Field Interactions", Phys. Rev. Lett. [**106**]{}, 163902 (2011).
L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, P. Lodahl, “Cavity Quantum Electrodynamics with Anderson-Localized Modes", Science [**327**]{}, 1352 (2010).
A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna", Science [**329**]{}, 930-933 (2010).
J.-J. Greffet, “Nanoantennas for Light Emission", Science [**308**]{}, 1561 (2005).
R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical Patch Antennas for Single Photon Emission Using Surface Plasmon Resonances", Phys. Rev. Lett. [**104**]{}, 026802 (2010).
M. Agio, “Optical antennas as nanoscale resonators", Nanoscale [**4**]{}, 692 (2012).
C. Vandenbem, L. S. Froufe-Pérez, and R. Carminati, “Fluorescence signal of a single emitter coupled to a nanoparticle through a plasmonic film", J. Opt. A: Pure Appl. Opt. [**11**]{}, 114007 (2009).
C. Vandenbem, D. Brayer, L. S. Froufe-Pérez, and R. Carminati, “Controlling the quantum yield of a dipole emitter with coupled plasmonic modes, Phys. Rev. B [**81**]{}, 085444 (2010).
Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy", Nature [**444**]{}, 740 (2012).
C. Chicanne, T. David, R. Quidant, J. C. Weeber, Y. Lacroute, E. Bourillot, and A. Dereux, “Imaging the Local Density of States of Optical Corrals", Phys. Rev. Lett. [**88**]{}, 097402 (2002).
R. Sapienza, T. Coenen, J. Renger, M. Kuttge, N. F. van Hulst, and A. Polman, “Deep-subwavelength imaging of the modal dispersion of light", Nat. Mat., [**11**]{}, 781 (2012).
S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, “Enhancement of Single-Molecule Fluorescence Using a Gold Nanoparticle as an Optical Nanoantenna", Phys. Rev. Lett. [**97**]{}, 017402 (2006).
P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and Quenching of Single-Molecule Fluorescence", Phys. Rev. Lett. [**96**]{}, 113002 (2006).
M. Frimmer, Y. Chen, and A.F. Koenderink, “Scanning Emitter Lifetime Imaging Microscopy for Spontaneous Emission Control", Phys. Rev. Lett. [**107**]{}, 123602 (2011).
J. Wenger, D. Gérard, J. Dintinger, O. Mahboub, N. Bonod, E. Popov, T. W. Ebbesen, and H. Rigneault, “Emission and excitation contributions to enhanced single molecule fluorescence by gold nanometric apertures", Opt. Exp. [**16**]{}, 3008 (2008).
K. Lieberman, S. Harush, A. Lewis, and R. Kopelman, “A light source smaller than the optical wavelength", Science [**247**]{}, 59 (1990).
J. Michaelis, C. Hettich, J. Mlynek, and V. Sandoghdar, “Optical microscopy using a single- molecule light source", Nature [**405**]{}, 325 (2000).
S. Kühn, C. Hettich, C. Schmitt, J. P. Poizat, and V. Sandoghdar, “Diamond colour centres as a nanoscopic light source for scanning near-field optical microscopy", J. Microsc. [**202**]{}, 2 (2001).
L. Aigouy, Y. De Wilde, and M. Mortier, “Local optical imaging of nanoholes using a single fluorescent rare-earth-doped glass particle as a probe", Appl. Phys. Lett. [**83**]{}, 147 (2003).
A. Cuche, O. Mollet, A. Drezet, and S. Huant, “Deterministic quantum plasmonics", Nano Lett. [**10**]{}, 4566 (2010).
G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup, “Nanoscale imaging magnetometry with diamond spins under ambient conditions", Nature [**455**]{}, 648 (2008).
L. Rondin, J.-P. Tetienne, P. Spinicelli, C. Dal Savio, K. Karrai, G. Dantelle, A. Thiaville, S. Rohart, J.-F. Roch, and V. Jacques, “Nanoscale magnetic field mapping with a single spin scanning probe magnetometer", Appl. Phys. Lett. [**100**]{}, 153118 (2012).
K. Karrai, and I. Tiemann, “Interfacial shear force microscopy", Phys. Rev. B [**62**]{}, 13174 (2000).
A. Cuche, O. Mollet, A. Drezet, and S. Huant, “Deterministic Quantum Plasmonics", Nano Lett. [**10**]{}, 4566 (2010).
K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field", Surf. Sci. Rep. [**57**]{}, 59 (2005).
S. Y. Suck, S. Collin, N. Bardou, Y. De Wilde, and G. Tessier, “Imaging the three-dimensional scattering pattern of plasmonic nanodisk chains by digital heterodyne holography", Opt. Lett. [**36**]{}, 849 (2011).
E.D. Palik, [*Handbook of Optical constants of solids*]{} (Academic Press, San Diego, 1998).
P.C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity", Phys. Rev. E [**70**]{}, 036606 (2004).
A. Cazé, R. Pierrat, and R. Carminati, “Radiative and non-radiative local density of states on disordered plasmonic films", Phot. Nano. Fund. Appl. [**10**]{}, 339 (2012).
A.F. Koenderink, M. Kafesaki, C.M. Soukoulis, and V. Sandoghdar, “Spontaneous emission in the near field of two-dimensional photonic crystals", Opt. Lett. [**30**]{}, 3210 (2005).
Introduction
============
Much interest has recently been paid to plasmonic nanostructures due to the capacity they offer to enhance light-matter interaction of elementary dipoles such as fluorescent molecules and quantum dots. This encompasses three different mechanisms which are often hard to disentangle. (1) Light absorption can be enhanced by an optical antenna, leading to an increased effective absorption cross section. This can be advantageously used in photodetection and photovoltaics [@Novotny_NatPhot2011]. (2) Plasmonic nanostructures may also induce significant changes in the spontaneous emission dynamics by the Purcell effect. As an example, large Purcell factors associated to electromagnetic modes strongly localized in subwavelength volumes have been reported at the surface of disordered thin metallic films, which opens new perspectives such as the investigation of strong coupling in a regime where high Purcell factors coexist with high absorption [@Krachmalnicoff_PRL2010; @Castanie2012; @Canneson_PRB2011]. (3) A change of fluorescence intensity is also expected when a dipole is in near-field interaction with a plasmonic nanostructure. In an optical antenna, a field enhancement occurs, but it is often obscured by non-radiative processes leading to fluorescence quenching.
The local density of electromagnetic states (EM-LDOS) is the basic quantity which governs these three mechanisms. As it rules the energy stored in all available modes, it provides a direct measurement of the probability of spontaneous light emission. The decay rate of a fluorescent dipole is proportional to the EM-LDOS. While in vacuum the EM-LDOS is homogeneous, it is known that it can be significantly affected by the proximity of an interface or a nanostructure [@Joulain_PRB2003; @NovotnyBook]. Nanostructures made of dielectric materials such as microcavities have been shown to induce an increase or a decrease of the fluorescence decay rate [@Vahala_Nature2003]. Several groups have also reported spatial variations of the EM-LDOS in disordered photonic media, based on statistical measurements of the fluorescent emission rate from a large collection of dipole emitters [@Krachmalnicoff_PRL2010; @Birowosuto_PRL2010; @Sapienza_PRL2011]. In such systems, the direct observation of modes strongly localized in subwavelength volumes is crucial as it can provide an important signature of Anderson localization and opens the route to cavity quantum electrodynamics in systems which are inherently disordered [@Sapienza_Science2010]. Optical nanoantennas constitute another example of systems in which electromagnetic field can be controlled on a nanometer scale, producing an environment which can modify the amount of energy emitted by molecules and their direction of emission [@Novotny_NatPhot2011; @Curto_Quidant_Science2010; @Greffet_Science2005; @Greffet_PRL2010; @Agio_NanoScale2012; @Vandembem2009; @Vandembem2010].
A key issue to probe the EM-LDOS experimentally is that the detection process should have the same response for all modes. Several methods have been proposed to map the spatial variations of the EM-LDOS on photonic nanostructures, among which measuring the thermal emission in the near-field [@DeWilde_Nature2006], measuring the “forbidden light" signals from the aperture of a near-field scanning optical microscope [@Dereux], or using a scanning electron beam as a point dipole source [@Sapienza_NatMat2012]. The EM-LDOS can be directly inferred from measurements of the spontaneous fluorescence decay rate of a single nanoemitter in its local environment, $\Gamma=1/\tau$, where $\tau$ is the fluorescence lifetime of the emitter [@Krachmalnicoff_PRL2010; @NovotnyBook]. An emitter constituted by several randomly oriented dipoles probes simultaneously the EM-LDOS in the three spatial directions. One dimensional maps of the decay rate were initially obtained by scanning a gold bead near single isolated fluorescent molecules fixed on a substrate [@Kuhn_PRL2006; @Anger_PRL2006]. Recently, a decrease of the fluorescence lifetime was measured in the reverse situation, when scanning a fluorescent bead across a 250 nm diameter silver rod, pointing to an increased EM-LDOS due to the existence of plasmonic modes on the rod [@Frimmer_PRL2011].
While a detailed knowledge of the EM-LDOS is clearly required, it is not enough to provide a full characterization of a system involving dipoles coupled with plasmonic nanostructures. Local changes of fluorescence intensity depend on other parameters such as the radiative and non-radiative part of the EM-LDOS and the local field enhancement factor [@Wenger2008]. To characterize a plasmonic antenna, one needs at least to measure both the EM-LDOS and the fluorescence enhancement factor at the nanometer scale in the near-field of the antenna.
In this paper, we present the use of a fluorescent 100-nanometers bead grafted at the apex of a scanning probe for the near-field investigation of an optical antenna. Two dimensional (2D) maps of the decay rate and intensity of the fluorescent bead are obtained with subwavelength resolution. The experimental near-field images of fluorescence intensity and EM-LDOS provide complementary information that are both needed to characterize the optical response of a nanoantenna. A good understanding of these maps is achieved by comparison with numerical simulations, which also allows one to analyze the optical response of the fluorescent near-field probe used to perform the experiments.
The decay rate averaged over the orientations of the transition dipole ${{\bf p}}$ reads $$\Gamma = \frac{\pi \omega}{3 \epsilon_0 \hbar} |{{\bf p}}|^2 \, \rho({{\bf r}},\omega),$$\[eq:Gamma\] where $\rho({{\bf r}},\omega)$ is the EM-LDOS. Hence, measuring the fluorescence lifetime $\tau=1/\Gamma$ is a direct way to probe the EM-LDOS.
If the molecules are far from saturation, the fluorescence signal reads $$\label{fluorescence_signal}
S = C \left [ \int_{\Omega} \eta({{\bf u}},\omega_{fluo}) {{\mathrm{d}}}\Omega \right ] \, \sigma(\omega_{exc}) \, K^2(\omega_{exc}) \, I_{inc}.$$ In this equation, $\eta({{\bf u}},\omega_{fluo})$ is the apparent quantum yield for a detection in direction ${{\bf u}}$, and $\Omega$ is the solid angle of the detection objective. The apparent quantum yield is defined as $\eta({{\bf u}},\omega_{fluo}) = \Gamma_R({{\bf u}})/\Gamma$ , with $\Gamma_R({{\bf u}})$ the directional radiative decay rate and $\Gamma=1/\tau$ the full decay rate. $\omega_{exc}$ and $\omega_{fluo}$ are the absorption and emission frequencies of the molecules. The constant $C$ is a calibration parameter of the detection (that accounts for transmissivity of filters, detector efficiency, etc), $\sigma(\omega_{exc})$ is the absorption cross-section of the bare fluorescent beads, $I_{inc}$ is the incident laser intensity and $K^2(\omega_{exc})$ is the local-intensity enhancement factor. In terms of the fluorescence signal, the product $F=\left [ \int_{\Omega} \eta({{\bf u}},\omega_{fluo}) {{\mathrm{d}}}\Omega \right ] \, K^2(\omega_{exc})$ is the fluorescence enhancement factor and drives the contrast of the images.
Experimental results
====================
Active fluorescent probes which allow near-field intensity measurements have been reported [@Lewis1990; @Michaelis2000; @Kuhn2001; @Aigouy2003; @Huant2010; @Wrachtrup2008; @Rondin2012], but a scanning probe based on a fluorescent nanobead in which [*both*]{} the fluorescence decay rate and the fluorescence intensity are simultaneously measured with subwavelength resolution is still very uncommon [@Frimmer_PRL2011]. Such a probe constitutes a major advance in the field of nano-optics as it provides a direct access to a map of the EM-LDOS and the intensity. Its practical realization firstly requires manipulating the bead to graft it at the apex of the tip of the scanning probe, and then recording its fluorescence lifetime and intensity at a rate compatible with the acquisition of 2D images. The experimental setup which we have designed to this aim is sketched in Fig.\[fig:experimental\_setup\]a. It consists of a homebuilt atomic force microscope (AFM) at the tip of which a fluorescent nano-bead is fixed. The AFM is mounted on the stage of an inverted confocal fluorescence microscope equipped with a white pulsed laser (Fianum SC450), an avalanche photodiode (Micro Photon Devices), and a time correlated single photon counting (TCSPC) electronics for fluorescence lifetime measurements (Picoquant HydraHarp 400) [@Krachmalnicoff_PRL2010]. The scans are performed on the sample stage using a three axes (xyz) piezoelectric assembly (Piezosystem Jena). The tip is a pulled silica fiber with an apex of $\approx 100$ nm diameter, mounted at the extremity of one arm of a quartz tuning fork excited by a dither piezoceramic. Stepper motors (Smaract GmbH) allow three dimensional displacements of the tip to perform the coarse approach and the fine positioning of the tip within the tightly focused laser spot produced by a high numerical aperture (NA=1.4) microscope objective. The tip oscillates laterally in shear mode at a height of approximately $10$ nm above the surface of the sample thanks to a feedback electronics [@Karrai_PRB2000]. The optical probe consists of a $100$ nm diameter polymer bead filled with fluorescent dye molecules (Invitrogen Red Fluospheres) which is grafted at the extremity of the silica tip. Fluorescent beads with a diameter of $100$ nm are initially spin-coated on a glass coverslip with a separation of $5$ to $10$ $\mu \mathrm{m}$ between them. Wide-field illumination of the coverslip is initially used to excite simultaneously the fluorescence of several beads (see Fig.\[fig:experimental\_setup\]b), which are imaged with an EM-CCD camera (Andor iXon 897) mounted on an extra port of the inverted microscope. The silica tip is brought at the glass surface. The grafting of a single fluorescent bead on its apex is achieved by moving the bead in contact with the tip while monitoring the process in real time with the CCD camera (see movie in supplemental data). Fig.\[fig:experimental\_setup\]b shows an optical fluorescence image of the beads spread on the glass coverslip, one of them being grafted on the silica tip visible on the right side of the image. Remarkably, there is no glue involved in the grafting which is successful in about $50$% of the attempts and can last up to several days. Once the grafting procedure is achieved, the nano-bead is brought close to the structure under study, the laser ($\lambda_{exc} =560$ nm) is tightly focused on the bead in order to excite its fluorescence and the sample is scanned. Note that, as the grafting is achieved by lateral displacements of the bead towards the tip (see supplemental data), the bead is expected to be laterally displaced with respect to the tip apex, as schematically sketched in Fig.\[fig:experimental\_setup\]c.
As a first example of the investigation of a nanoantenna, we have studied a linear chain of three $150$ nm-diameter gold nano-disks separated by $50$ nm gaps on a glass substrate [@Suck_OL2011]. According to numerical simulations discussed later in the paper, such a nanostructure is expected to produce significant variations of the EM-LDOS and of the fluorescence enhancement factor on a scale of several tens of nanometers, which we have observed with the fluorescence scanning probe. The disk chains are fabricated by electron beam lithography on a glass microscope coverslip. Each disk is made of a $2$ nm-thick wetting layer of chromium and of a $30$ nm-thick layer of gold.
Three signals are simultaneously recorded during the scans as a function of the position of the fluorescent probe at the surface of the sample, producing three different images of the nanoantenna: (1) the topography (AFM image, see Fig.\[fig:experimental\_maps\]a); (2) the fluorescence intensity, which corresponds to the integral of the decay rate histogram (Fig.\[fig:experimental\_maps\]b); and (3) the spontaneous decay rate (Fig.\[fig:experimental\_maps\]c), obtained after measuring, at each point of the scan during $1$ s, the histogram of the arrival time of the fluorescence photons with respect to the excitation laser pulse [@Krachmalnicoff_PRL2010]. This long acquisition time is a major difficulty to record 2D maps of the decay rate on a very small area with a scanning probe, because the system has to be stable with nanometric precision against thermal and mechanical drifts. This explains the low number of pixels ($32\times22$) of the experimental images.
![\[fig:experimental\_setup\] a) Sketch of the experimental setup. The active AFM tip is mounted on a piezoelectric system allowing the positioning of the tip within the laser diffraction limited spot. The excitation and fluorescence photons are respectively focused and collected from the same high NA objective. The sample can be moved on the XY plane to perform the fluorescence intensity, EM-LDOS and topography maps. b) Wide-field fluorescence image of the beads spread on the microscope coverslip. The tip can be seen on the right side of the image. One bead is grafted on the apex of the tip (see text and movie in supplemental data). c) Artist view of the active AFM tip scanning the near-field of the gold nanoantenna.](fig_1.eps){width="7cm"}
![\[fig:experimental\_maps\] Experimental results: a) Topography of the sample. b) Fluorescence intensity map. c) Decay rate (EM-LDOS) map. The contour of the topographic relief (dashed line), as measured by the active AFM asymmetric probe (see text), is reported on the three maps to guide the eye.](fig_2.eps){width="7cm"}
The topography of the sample shows three distinct objects with a height of $30$ nm each. The measured structures seem not to be circularly shaped as expected, but rather elliptical. This is because the measured topography is given by the convolution of the topography of the real object (three disks) with the shape of the probe used to make the scan, here the AFM tip on which a fluorescent bead is grafted sideways. Due to the lateral shift of the bead with respect to the tip apex (Fig.\[fig:experimental\_setup\]c), the topography probe presents an anisotropic shape. The size of the short axis in the AFM image of a disk is ruled by the larger object which constitutes the probe, while the size of the longer axis is related to the sum of the sizes of the bead and the tip. Hence, the topographic image confirms the lateral displacement of the bead with respect to the tip apex and allows one to affirm that the size of the tip is of the same order of the size of the bead.
The geometry of the scanning probe as sketched in Fig.\[fig:experimental\_setup\]c, is confirmed by looking at the fluorescence intensity map shown in Fig.\[fig:experimental\_maps\]b. The contour of the measured topography (Fig.\[fig:experimental\_maps\]a) is reported on this map to guide the eye (dashed lines). The intensity signal decreases in three regions, circularly shaped, situated on the upper half of the elliptical contour. Since the fluorescence signal only comes from the $100$ nm diameter dye doped bead, this confirms that the three golden disks are scanned twice, once by the bead and then by the tip. The decrease of the fluorescence intensity in correspondence of the gold structure is confirmed by the numerical simulations shown in Fig.\[fig:numerical\_maps\]. Remarkably, the numerical and experimental intensity maps present an almost quantitative agreement, showing both a contrast of about a factor $3$ of the fluorescence intensity. Note also that the combination of the topography and the fluorescence intensity signals allows us to intuit quite precisely what is the position of the bead with respect to the tip apex.
Beyond the fluorescence intensity map, a deeper insight of the properties of the electromagnetic field at the surface of this nanoantenna is obtained by mapping the decay rate of the fluorescent bead, which is proportional to the EM-LDOS (see Eq.\[eq:Gamma\]). Fig.\[fig:experimental\_maps\]c shows a decay rate map of the scanned area. It is observed that the EM-LDOS increases by about $30\%$ in three regions presenting an extension of about $60$ nm each and separated by $100$ nm. Two of these regions are situated between the gold disks, as confirmed by numerical simulations reported in Fig.\[fig:numerical\_maps\]c and discussed below. As in the case of the fluorescence intensity map, numerical and experimental data are in almost quantitative agreement regarding the expected change of the decay rate in the region between the disks with respect to a region far away from the nanoantenna. Note that the numerical simulations also show the presence of two other regions with an enhanced EM-LDOS, on the external sides of the nanoantenna. Experimentally, we are only able to see one of these regions, situated on the right side of the antenna. A possible explanation for this is an asymmetry of the gold structure, caused for example by a defect of the lift-off process, that would translate in an asymmetry of the structure of the electromagnetic field on the surface of the nanoantenna. Numerical calculations with asymmetric shaped nanoantennas have been done and produce similar asymmetries in the EM-LDOS images. However, since the exact shape of the nanoantenna is not accessible at the required level of resolution, having an exact matching between theory and experiment is a very speculative task and the discussion is therefore limited here to a comparison between the experimental results with numerical simulations made on an ideal antenna formed by three regularly spaced circular disks.
Numerical simulations: comparison with the experiment
=====================================================
In order to analyze the experimental results precisely, we have performed exact 3D numerical simulations. We use a volume integral method based on the Lippmann-Schwinger equation $$\label{lippmann_schwinger}
{{\bf E}}({{\bf r}},\omega) = {{\bf E}}_0({{\bf r}},\omega) + \frac{\omega^2}{c^2} \int \left[\epsilon({{\bf r}}',\omega)-1\right]{{\bf G}}_0({{\bf r}},{{\bf r}}',\omega){{\bf E}}({{\bf r}}',\omega){{\mathrm{d}}^3}{{\bf r}}',$$
where ${{\bf E}}_0$ is the incident field, ${{\bf G}}_0$ the dyadic Green function of the background (vacuum in our simulations) and $\epsilon(\omega)$ is the dielectric function of gold [@PalikBook]. The numerical computation is done by a moment method, discretizing the trimer into 5 nm cells (see Fig.\[fig:numerical\_maps\]a). The Green function ${{\bf G}}_0$ is integrated over the cell to improve convergence [@Chaumet2004]. Computing the electric field under the illumination of a source dipole ${{\bf E}}_0({{\bf r}}, \omega) = \mu_0\omega^2{{\bf G}}_0({{\bf r}},{{\bf r}}_0,\omega){{\bf p}}$, we can deduce the total Green function ${{\bf G}}$ from ${{\bf E}}({{\bf r}}, \omega) = \mu_0\omega^2{{\bf G}}({{\bf r}},{{\bf r}}_0,\omega){{\bf p}}$. The EM-LDOS is then given by $$\label{ldos}
\frac{\rho({{\bf r}}_0,\omega)}{\rho_0} = \frac{2\pi}{k_0}\, {\mathrm{Im} \, }\,{\mathrm{Tr} \, }\,\left[ {{\bf G}}({{\bf r}}_0,{{\bf r}}_0,\omega) \right],$$ where $\rho_0$ is the EM-LDOS in vacuum, given by $\rho_0=\omega^2/(\pi^2c^3)$, $k_0 = \omega/c$ and ${\mathrm{Tr} \, }$ denotes the trace of a tensor. The fluorescence enhancement factor, given by Eq.\[fluorescence\_signal\], is driven by the product $$F = \left [ \int_{\Omega} \eta({{\bf u}},\omega_{fluo}) {{\mathrm{d}}}\Omega \right ] \, K^2(\omega_{exc}).$$ The apparent quantum yield of the molecules can be deduced from the total Green function ${{\bf G}}$ [@Caze2012]. For sake of simplicity, we do not take into account any intrinsic non-radiative decay rate that would account for internal losses. Moreover, in our simulation, we do not integrate over the directions ${{\bf u}}$ of collection but we only consider the direction orthogonal to the plane of the antenna. To compute $K(\omega_{exc})$ we solve Eq.\[lippmann\_schwinger\] under the illumination of a plane-wave with normal incidence. We average on two orthogonal polarizations to mimic the fact that the incident laser beam in the experiment is not polarized.
In Fig.\[fig:numerical\_maps\], we present the results of our calculations. Every point of each map is averaged over 100 positions of the emitter randomly chosen inside a $100$ nm diameter sphere, to mimic the experimental fluorescent beads. The distance between the top of the trimer and the bottom of the bead is fixed at $d=20$ nm. The excitation and emission wavelengths have respectively been set to $\lambda_{exc}=560\,\mathrm{nm}$ and $\lambda_{fluo}=605\,\mathrm{nm}$. The dimensions of the trimer are the same as the experimental ones.
The general trends observed in the experimental maps are in excellent agreement with the simulations. The decay rate exhibits two major hot-spots in the two gaps between the disks, and two minor ones on both sides. The fluorescence signal is significantly reduced when the molecules are on top of a disk. The contrast of both maps are in nearly quantitative agreement with the experimental results, which confirms us in the idea that the order of magnitude of the distance between the bead and the trimer is sound.
![\[fig:numerical\_maps\] (a) Top view of the topography of the discretized trimer. Note that the trimer is $30\,\mathrm{nm}$ thick, like the one used in the experiment; (b) Numerical fluorescence signal map (expressed in arbitrary units); (c) Numerical EM-LDOS map normalized to its value in vacuum.](fig_3.eps){width="7cm"}
Numerical study of the resolution of the EM-LDOS map
====================================================
One interesting feature of both the experimental and numerical EM-LDOS maps is that both seem to exhibit variations on scales well below 100$\,\mathrm{nm}$, the size of the fluorescent bead. To explain this phenomenon, already observed in [@Frimmer_PRL2011], we compare the contribution to the decay rate of the emitters located in the lower half and the upper half of the bead. Fig.\[fig:smallest\_detail\]a-b show the EM-LDOS maps averaged over 100 emitters located at different positions inside a $100{\,\mathrm{nm}}$ bead for two distances $d$ between the bottom of the bead and the top of the trimer. Fig.\[fig:smallest\_detail\]c-d and Fig.\[fig:smallest\_detail\]e-f show the maps averaged over the emitters located respectively in the lower and the upper half of the bead. Each map is normalized by the value of the EM-LDOS in vacuum to allow for the comparison between the maps.
A comparison between Fig.\[fig:smallest\_detail\]a and b shows that the smallest details present on the EM-LDOS map (such as for example the two EM-LDOS hot spots visible on the right and left sides of the nanoantenna) are washed out when the distance of the bead to the sample surface increases. This is a known feature in near-field optics. In fact, subwavelength details are evanescent and can only be probed by emitters in the very near-field of the system [@Castanie2012]. Since these details are visible on the experimental map, this study confirms that the real distance between the bottom of the probe and the sample surface is of the order of $20$ nm.
Furthermore, a detailed observation of Fig.\[fig:smallest\_detail\]a, b and c allows us to affirm that the resolution of the EM-LDOS map is not limited by the size of the bead. Indeed, the similarity between Fig.\[fig:smallest\_detail\]a and c clearly shows that the EM-LDOS map is driven by the emitters situated on the lower half of the bead. The two EM-LDOS hot spots which are visible on the right and on the left side of the nanoantenna are smeared out when considering only the contribution of the emitters populating the upper part of the sphere. More insight can be given by plotting the section of the EM-LDOS maps along the lines drawn in Fig.\[fig:smallest\_detail\]a, c, e. The obtained profiles are normalized by the maximum value of the corresponding map $\rho_{max}$, in order to quantify the contrast of each hotspot. They are reported in Fig.\[fig:smallest\_detail\]g and in Fig.\[fig:smallest\_detail\]h for $d=20$ nm and $d=50$ nm respectively. For $d=20{\,\mathrm{nm}}$, the lateral hot-spot is clearly resolved when the EM-LDOS signal is averaged on the emitters located on the bottom of the sphere or over all the sphere, while it is washed out when the signal is averaged over the top of the sphere. Therefore the resolution of this detail is clearly due to the bottom emitters. Consequently, the effective resolution is not limited by the size of the bead but is smaller and in the case presented in this paper is of the order of $50$ nm. Interestingly, at $d=50{\,\mathrm{nm}}$, even if a non-monotonic behavior is observed in the profiles reported on Fig.\[fig:smallest\_detail\]h), the bottom emitters are too far away from the sample surface and the smallest details are washed out.
![\[fig:smallest\_detail\] Computed normalized EM-LDOS maps for two distances $d$ between the *bottom* of the bead and the top of the trimer. (a-b) Average over 100 emitters randomly located in the bead; (c-d) Contribution of the 48 emitters located in the lower half of the bead; (e-f) Contribution of the 52 emitters located in the upper half of the bead. (g-h) Section view of the maps (a,c,e) and (b,d,f) respectively along the line shown on the maps. Note that in this case the EM-LDOS has been normalized by the maximum value of each map $\rho_{max}$ to quantify the contrast of the image. $\lambda_{exc} = 560\,\mathrm{nm}$; $\lambda_{fluo} = 605\,\mathrm{nm}$. Diameter of the bead: $100\,\mathrm{nm}$.](fig_4.eps){width="10cm"}
Conclusion
==========
In conclusion, the scanning probe described in this paper has demonstrated its ability to map both the fluorescence intensity and the EM-LDOS simultaneously with nanometer accuracy. While its use was devoted here to the investigation of an optical antenna, it can be used to the investigation of the electromagnetic modes on any type of photonic nanostructure. The manipulation of a fluorescent nano-object with nanometer accuracy at the surface of a photonic nanostructure opens interesting new perspectives for studies of quantum electrodynamics, such as the investigation of the coupling of quantum emitters with plasmonic (or dielectric) devices, the characterization of electromagnetic modes on photonic nanostructures, or the search for Anderson localized modes in disordered systems.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge Abdel Souilah for technical support. This work is supported by the French National Research Agency (ANR-11-BS10-0015,“3DBROM”), by the Region Ile-de-France in the framework of DIM Nano-K and by LABEX WIFI (Laboratory of Excellence within the French Program “Investments for the Future”) under reference ANR-10-IDEX-0001-02 PSL\*.
|
\#1\#2
Ł Ł
The resent generalization of AdS/CFT correspondence due to Witten and Yau [@W-Y] is based on manifolds with a positive scalar curvature. This result renew an interest to alternative theories of classical gravity. Indeed, all solutions of the vacuum Einstein field equations yield the metric with zero scalar curvature.\
Einstein’s gravitational theory is in a very good accordance with all known experimental data and this fact makes a natural obstacle for modification the foundation of GR. Indeed, almost all alternative theories of gravity are constructed in such a way that they preserve the classical Schwarzschild solution for a static spherical-symmetric massive particle $$\label{1}
ds^2=(1-\frac m{\r})dt^2-\frac 1{1-\frac m{\r}}d{\r}^2-{\r}^2d\Omega^2.$$ On the other hand the most reliable corroborations of GR (such as gravitational red-shift, perihelion procession, bending of light and time delay of radar signals) are based only on two theoretical facts:\
1) Geodesic Postulate,\
2) Long distance behavior of Schwarzschild metric.\
It is known a rather different metric $$\label{2}
ds^2=e^{-2\frac mr}dt^2-e^{2\frac mr}(dx^2+dy^2+dz^2),$$ with $r=(x^2+y^2+z^2)^{1/2}$, which is also compatible with the classical macroscopic observation data.\
The metric (\[2\]) appeared for the first time in the framework of Yilmaz’s scalar theory of gravity and hardly later in the bi-metric theory of Rosen (see [@yilmaz76], [@rose74] and the reference therein). We will refer to (\[2\]) as Yilmaz-Rosen metric (see, for instance, [@Hehl]). Recently the metric (\[2\]) was obtained in the Kaniel-Itin model based on a wave-type field equation for a coframe field [@K-I], [@it2] and in the Watt and Misner scalar model of gravity [@WM].\
Thus there are two different metrics that are in a good accordance with the macroscopic ($r>>m$) observation data. On a small distance $r\simeq m$ these metrics are rather different: the Schwarzschild metric describes the black hole while the Yilmaz-Rosen metric are regular in all points on the manifold (except of the origin).\
The question in turn is: [*What is similar in a long-distance analytical behavior of these two metrics and which other metrics have the same behavior?*]{}\
We work backwards. We do not specify the field equations, but simply postulate a jet-space of solutions which looks promising.\
Consider the general spherical symmetric static metric in spatial conformal (isotropic) coordinates $$\label{3}
ds^2=e^{f(r)}dt^2-e^{g(r)}(dx^2+dy^2+dz^2).$$ We are interested in asymptotically flat metrics, thus require the functions $f$ and $g$ to have Taylor expansions of the form \[4\] f(r)&=&r++O(1[r\^3]{}),\
g(r)&=&r++O(1[r\^3]{}). (the zeroth order terms can be vanishing by rescaling the coordinates). Write the Schwarzschild line element, in the same isotropic coordinates $$\label{5}
ds^2=\Big(\frac{1-\frac{m}{2r}} {1+\frac{m}{2r}}
\Big)^2dt^2-\Big({1+\frac{m}{2r}}\Big)^4(dx^2+dy^2+dz^2).$$ The Taylor expansions of the functions $f$ and $g$ for this metric are \[6\] f(r)&=&2=-r+O(1[r\^3]{}),\
g(r)&=&4([1+]{})=r-+O(1[r\^3]{}) and the coefficients are $$\label{7}
a_1=-2m, \quad b_1=2m, \quad a_2=0, \quad b_2=-\frac {m^2}2, \ ...$$ The corresponding coefficients for the Yilmaz-Rosen metric (\[2\]) are $$\label{8}
a_1=-2m,\quad b_1=2m,\quad a_i=b_i=0\quad \text{for} \ i>1$$ Comparing (\[7\]) and (\[8\]) we obtain *[ a sufficient condition]{} for a general static spherical-symmetric asymptotic flat metric in isotropic coordinates (\[3\]) to be compatible with the macroscopic classical tests: $$\label{9}
a_1+b_1=0, \qquad a_2=0.$$ In order to describe by the metric (\[3\]) the field of a point with arbitrary mass, the actual value of the coefficient $a_1$ should, also, be arbitrary. Thus the conditions (\[9\]) define a 2-jet space of functions $f(r)$ and a 1-jet space of functions $g(r)$. These spaces correspond to the $\frac 32$-order approximation of GR.\
As an example of functions containing in these jet-spaces consider a family of metrics $$\label{10}
ds^2=\Big(\frac{1-\frac{m}{kr}} {1+\frac{m}{kr}}
\Big)^kdt^2-\Big({1+\frac{m}{kr}}\Big)^{2k}(dx^2+dy^2+dz^2),$$ where $k$ is a dimensionless parameter. It is easy to see that the metric (\[10\]) satisfies the conditions (\[10\]) for an arbitrary choice of the parameter $k$.\
In order to have analytically correct functions in (\[10\]) we require the parameter $k$ to be integer. Note that this restriction is taken only for simplification. We can also consider the parameter $k$ as an arbitrary real number, but in this case we have made an analytical redefinition of the metric on the small distances $r \le \frac mk$.\
For $k=2$ one obtain, certainly, Schwarzschild metric while in the limits $k\to \pm\infty$ (\[10\]) approaches Yilmaz-Rosen metric.\
In order to obtain the the metric (\[10\]) in Schwarzschild coordinates we have to use the new radial coordinate $r=r(\r)$, which is implicit defined by the equation $$\label{11}
\Big({1+\frac{m}{kr}}\Big)^{k}r=\r.$$ For $k<0$ the metric (\[10\]) is singular at a distance $$\label{12}
r=-\frac mk \quad ==>\quad \r=0$$ i.e. in the origin of the Schwarzschild coordinates.\
As for $k>0$ the metric (\[8\]) is singular on a sphere $$\label{13}
r=\frac mk \quad ==>\quad \r=m\frac {2^k}k.$$ Note that the physical radius of singular sphere $\r$ increases very fast with growth of the parameter $k$. In order to clarify the nature of the singularities compute the scalar curvature of the metric (\[10\]) $$\label{14}
R=\frac{2-k}{k}\bigg(\frac{m^2}{r^4}\cdot\frac{1+(1-\frac{m}{kr})^2}{(1-\frac{m^2}{k^2r^2})^2}\Big(1+\frac{m}{kr}\Big)^{-2k}\bigg).$$ Note that the expression in the brackets are positive thus the sign of the scalar curvature depends only on the value of the parameter $k$. Thus all the manifolds endowed the metric (\[10\]) contain in a class of pseudo-Riemannian manifolds with a scalar curvature of a fixed sign.\
The scalar curvature is zero only in the case of Schwarzschild metric - $k=2$.\
Consider the different regions for the values of the parameter $k$:\
${\mathbf 1)\qquad k> 2}$\
The scalar curvature negative in every final point on the manifold. Near the singular value of coordinates $r=\frac mk$ the scalar curvature is singular $R\to -\infty$. Thus this coordinate singularity is physical. The scalar curvature inside of the spherical envelope decreases very fast. This singularity can be interpreted as a rigid sphere - close 2-brane.*
\[fig:1\]
=2.25in ![ The scalar curvature (\[14\]) plotted as a function $R/m^2$ of a radial distance $r/m$ for $k=3$. ](k=3.ps "fig:"){width="8cm"}
For $k\to \infty$ we obtain the expression for the scalar curvature of Yilmaz-Rosen metric [@K-I] $$\label{15}
R= -2\frac{m^2}{r^4}e^{-2\frac mr}.$$\
${\mathbf 2)\qquad k=1}$\
The scalar curvature is positive. We have the metric in isotropic coordinates $$\label{16}
ds^2=\Big(\frac{1-\frac{m}{r}} {1+\frac{m}{r}}
\Big)dt^2-\Big({1+\frac{m}{r}}\Big)^{2}(dx^2+dy^2+dz^2).$$ From the relation (\[11\]) we obtain the transform to the Schwarzschild radial coordinate $$\r=r+m.$$ The metric in these coordinates takes the form $$\label{17}
ds^2=(1-2\frac m{\r})dt^2-\frac 1{(1-\frac m{\r})^2}d{\r}^2-{\r}^2d\Omega^2.$$ The scalar curvature of the metric (\[16\]) is $$\label{18}
R=\frac{m^2}{r^4}\cdot\frac{1+(1-\frac{m}{r})^2}{(1-\frac{m}{r})^2(1+\frac{m}{r})^{4}}.$$ This expression is positive in every point of the manifold. It is singular for $r=m$ or equivalently for $\r=2m$. Thus the coordinate singularity for $\r=2m$ is physical.
\[fig:2\]
=2.25in ![ The scalar curvature (\[14\]) plotted as a function $R/m^2$ of $r/m$ for $k=1$. ](k=1.ps "fig:"){width="8cm"}
This singularity can be also interpreted as a rigid sphere - close 2-brane.\
${\mathbf 3)\qquad k\le -1}$\
The scalar curvature is negative. Consider for instance $k=-1$. The metric takes the form $$\label{19}
ds^2=\Big(\frac{1-\frac{m}{r}} {1+\frac{m}{r}}
\Big)dt^2-\Big({1-\frac{m}{r}}\Big)^{-2}(dx^2+dy^2+dz^2).$$ The metric is singular at the Schwarzschild radius $r=m$ The scalar curvature takes a form $$\label{20}
R=-{3}\frac{m^2}{r^4}\cdot\frac{1+(1+\frac{m}{r})^2}{(1+\frac{m}{r})^2}.$$ This value is regular for every $r$ (except of the origin).
\[fig:3\]
=2.25in ![ The scalar curvature (\[20\]) plotted as a function $R/m^2$ of $r/m$ for $k=1$. ](k=-1.ps "fig:"){width="8cm"}
In order to describe by the metric (\[19\]) a black hole one should locate it’s surface at a distance $r=m$ where $g_{00}=0$. In fact this surface cannot be reached by any material object. The proper radial distance from the surface $r=m$ to a point $r_0>m$ is $$l=\int^{r_0}_m{\frac {dr}{(1-\frac mr)^2}}\to \infty$$ The proper time for a radial null geodesic is also infinite $$T=\int^{r_0}_m \frac {(1+\frac mr)^\frac 12} {(1-\frac mr)^\frac 32}dr\to \infty.$$ Thus the surface of a star cannot never reach the Schwarzschild radius and a realistic physical system can be modeled by the metric (\[19\]) only for a distance $r>m$. The behavior of the metric is similar to the spherical-symmetric solution in the gravity model of Lee and Lightman. [@L-L].
[999]{}
|
LPTENS-08/27
[**FOLLOWING THE PATH OF CHARM:\
NEW PHYSICS AT THE LHC**]{}
[JOHN ILIOPOULOS]{}
Laboratoire de Physique Théorique\
de L’Ecole Normale Supérieure\
75231 Paris Cedex 05, France
Talk presented at the ICTP on the occasion of the Dirac medal award ceremony Trieste, 27/03/08
It is a great honour for me to speak on this occasion and I want to express my gratitude to the Abdus Salam International Centre for Theoretical Physics as well as the Selection Committee of the Dirac Medal. It is also a great pleasure to be here with Luciano Maiani and discuss the consequences of our common work with Sheldon Glashow on charmed particles [@GIM]. I will argue in this talk that the same kind of reasoning, which led us to predict the opening of a new chapter in hadron physics, may shed some light on the existence of new physics at the as yet unexplored energy scales of LHC.
The argument is based on the observation that precision measurements at a given energy scale allow us to make predictions concerning the next energy scale. It is remarkable that the origin of this observation can be traced back to 1927, the two fundamental papers on the interaction of atoms with the electromagnetic field written by Dirac, which are among the cornerstones of quantum field theory. In the second of these papers [@Dirac] Dirac computes the scattering of light quanta by an atom $\gamma(k_1)+A_i \rightarrow \gamma(k_2)+A_f$, where $A_i$ and $A_f$ are the initial and final atomic states, respectively. He obtains the perturbation theory result:
\[Diracpert\] H\_[fi]{}=\_j where $H$ are the amplitudes. For the significance of the rhs, Dirac notes: “...The scattered radiation thus appears as a result of the two processes $i
\rightarrow j$ and $j\rightarrow f$, one of which must be an absorption, the other an emission, in neither of which the total proper energy is even approximately conserved.” This is the crux of the matter: In the calculation of a transition amplitude we find contributions from states whose energy may put them beyond our reach. The size of their contribution decreases with their energy, see (\[Diracpert\]), so, the highest the precision of our measurements, the further away we can see.
Let me illustrate the argument with two examples, one with a non-renormalisable theory and one with a renormalisable one. A quantum field theory, whether renormalisable or not, should be viewed as an effective theory valid up to a given scale $\Lambda$. It makes no sense to assume a theory for all energies, because we know already that at very high energies entirely new physical phenomena appear (example: quantum gravity at the Planck scale). The first example is the Fermi four-fermion theory with a coupling constant $G_F \sim 10^{-5}GeV^{-2}$. It is a non-renormalisable theory and, at the $n$th order of perturbation, the $\Lambda$ dependence of a given quantity $A$ is given by:
$$A^{(n)}=C_0^{(n)} (G_F \Lambda^2)^n+ C_1^{(n)}G_F (G_F \Lambda^2)^{n-1}+
C_2^{(n)}G_F^2 (G_F \Lambda^2)^{n-2}+....
\label{Fermiexp}$$
where the $C_i$’s are functions of the masses and external momenta, but their dependence on $\Lambda$ is, at most, logarithmic. Perturbation theory breaks down obviously when $A^{(n)} \sim A^{(n+1)}$ and this happens when $G_F
\Lambda^2 \sim 1$. This gives a scale of $\Lambda \sim 300$GeV as an upper bound for the validity of the Fermi theory. Indeed, we know today that at 100GeV the $W$ and $Z$ bosons change the structure of the theory. But, in fact, we can do much better than that [@JOFSHAB]. Weak interactions violate some of the conservation laws of strong interactions, such as parity and strangeness. The absence of such violations in precision measurements will tell us that $G_F
\Lambda^2 \sim \epsilon$ with $\epsilon$ being the experimental precision. The resulting limit depends on the value of the $C$ coefficient for the quantity under consideration. In this particular case it turned out that, under the assumption that the chiral symmetry of strong interactions is broken only by terms transforming like the quark mass terms, the coefficient $C_0^{(n)}$ for parity and/or strangeness violating amplitudes vanishes and no new limit is obtained [@BoucIlPrent]. However, the second order coefficient $C_1^{(n)}$ contributes to flavour changing neutral current transitions and the smallness of the $K_1-K_2$ mass difference, or the $K^0_L \rightarrow \mu^+ +\mu^-$ decay amplitude, give a limit of $\Lambda
\sim 3$GeV before new physics should appear. The new physics in this case turned out to be the charmed particles [@GIM]. We see in this example that the scale $\Lambda$ turned out to be rather low and this is due to the non-renormalisable nature of the effective theory which implies a power-law behaviour of the radiative corrections on $\Lambda$.
The second example in which new physics has been discovered through its effects in radiative corrections is the well-known “discovery” of the $t$ quark at LEP, before its actual production at Fermilab. The effective theory is now the Standard Model, which is renormalisable. In this case the dependence of the radiative corrections on the scale $\Lambda$ is, generically, logarithmic and the sensitivity of the low energy effective theory on the high scale is weak (there is an important exception to this rule for the Standard Model which we shall see presently). In spite of that, the discovery was made possible because of the special property of the Yukawa coupling constants in the Standard Model to be proportional to the fermion mass. Therefore, the effects of the top quark in the radiative corrections are quadratic in $m_t$. The LEP precision measurements were able to extract a very accurate prediction for the top mass.
I claim that we are in a similar situation with the precision measurements of the Standard Model. Our confidence in this model is amply justified on the basis of its ability to accurately describe the bulk of our present day data and, especially, of its enormous success in predicting new phenomena. All these spectacular successes are in fact successes of renormalised perturbation theory. Indeed what we have learnt was how to apply the methods which had been proven so powerful in quantum electrodynamics, to other elementary particle interactions. The remarkable quality of modern High Energy Physics experiments, mostly at LEP, but also elsewhere, has provided us with a large amount of data of unprecedented accuracy. All can be fit using the Standard Model with the Higgs mass as the only free parameter. Let me show some examples: Figure \[globfit\] indicates the overall quality of such a fit. There are a couple of measurements which lay between 2 and 3 standard deviations away from the theoretical predictions, but it is too early to say whether this is accidental, a manifestation of new physics, or the result of incorrectly combining incompatible experiments.
Another impressive fit concerns the strong interaction effective coupling constant as a function of the momentum scale (Figure \[alphas\]). This fit already shows the importance of taking into account the radiative corrections, since, in the tree approximation, $\alpha_s$ is obviously a constant. Similarly, Figure \[epsilon\] shows the importance of the weak radiative corrections in the framework of the Standard Model. Because of the special Yukawa couplings, the dependence of these corrections on the fermion masses is quadratic, while it is only logarithmic in the Higgs mass. The $\epsilon$ parameters are designed to disentangle the two. The ones we use in Figure \[epsilon\] are defined by:
$$\label{epspar1}
\epsilon_1=\frac{3G_Fm_t^2}{8\sqrt{2}\pi^2}-\frac{3G_Fm_W^2}{4\sqrt{2}\pi^2}\tan^2\theta_W \ln \frac{m_H}{m_Z}+...$$
$$\label{epspar3}
\epsilon_3=\frac{G_Fm_W^2}{12\sqrt{2}\pi^2}\ln \frac{m_H}{m_Z}-\frac{G_Fm_W^2}{6\sqrt{2}\pi^2} \ln \frac{m_t}{m_Z}+...$$
where the dots stand for subleading corrections. As you can see, the $\epsilon$ s vanish in the absence of weak interaction radiative corrections, in other words, $\epsilon_1=\epsilon_3=0$ are the values we get in the tree approximation of the Standard Model but after having included the purely QED and QCD radiative corrections. We see clearly in Figure \[epsilon\] that this point is excluded by the data. The latest values for these parameters are $\epsilon_1=5.4 \pm 1.0$ and $\epsilon_3=5.34 \pm 0.94$ [@Altar].
Using all combined data we can extract the predicted values for the Standard Model Higgs mass which are given in Figure \[limhiggs\]. The data clearly favour a low mass ($\leq 200$ GeV) Higgs, although, this prediction may be less solid than what Figure \[limhiggs\] seems to indicate.
The main conclusion I want to draw from this comparison can be stated as follows:
[*Looking at all the data, from low energies to the Tevatron, we have learnt that perturbation theory is remarkably successful, outside the specific regions where strong interactions are important.*]{}
Let me explain this point better: At any given model with a coupling constant $g$ we expect to have a weak coupling region $g\ll 1$, in which weak coupling expansions, such as perturbation theory, are reliable, a strong coupling region with $g\gg 1$, in which strong coupling expansions may be relevant, and a more or less large gray region $g\sim 1$, in which no expansion is applicable. The remarkable conclusion is that this gray area appears to be extremely narrow. And this is achieved by an enlargement of the area in which weak coupling expansion applies. The perturbation expansion is reliable, not only for very small couplings, such as $\alpha_{em}\sim 1/137$, but also for moderate QCD couplings $\alpha_s \sim 1/3$, as shown in Figure \[alphas\]. This is extremely important because without this property no calculation would have been possible. If we had to wait until $\alpha_s$ drops to values as low as $\alpha_{em}$ we could not use any available accelerator. Uncalculable QCD backgrounds would have washed out any signal. And this applies, not only to the Tevatron and LHC, but also to LEP. We can illustrate this observation using a qualitative argument first introduced by F. Dyson. He noted that in a field theory like QED, the contribution of the $2n$-th order perturbation term to a physical amplitude $A^{(2n)}$ grows with $n$ roughly as[^1]
A\^[(2n)]{} \~\^[n]{} (2n-1)!! where $\alpha$ is (the square of) the coupling constant. Again, perturbation theory will break down when $A^{(2n)}\sim A^{(2n+2)}$ which gives
2n+1 \~\^[-1]{}
This leaves a comfortable margin for QED but leaves totally unexplained the successes of QCD at moderate energies.
A global view of the weak and strong coupling regions is given in Figure \[R\] which shows the $R$-ratio, [*i.e.*]{} the $e^+ +e^-$ total cross section to hadrons normalised to that of $e^+ +e^- \rightarrow \mu^+ +\mu^-$ as a function of the centre-of-mass energy. The lowest order perturbation value for this ratio is a constant, equal to $\Sigma Q_i^2$, the sum of the squares of the quark charges accessible at this energy. We see clearly in this Figure the areas of applicability of perturbation theory: At very low energies, below 1 GeV, we are in the strong coupling regime characterised by resonance production. The strong interaction effective coupling constant becomes of order one (we can extrapolate from Figure \[alphas\]), and perturbation breaks down. However, as soon as we go slightly above 1 GeV, $R$ settles to a constant value and it remains such except for very narrow regions when new thresholds open. In these regions the cross section is again dominated by resonances and perturbation breaks down. But these areas are extremely well localised and threshold effects do not spread outside these small regions.
In this talk I want to exploit this observational fact and argue that the available precision tests of the Standard Model allow us to claim with confidence that new physics will be unravelled at the LHC, although we have no unique answer on the nature of this new physics. The argument assumes the validity of perturbation theory and it will fail if the latter fails. But, as we just saw, perturbation theory breaks down only when strong interactions become important. But new strong interactions do imply new physics.
The key is again the Higgs boson. As we explained above, the data favour a low mass Higgs. However, the opposite cannot be excluded, first because it depends on the subset of the data one is looking at[^2], and, second, because the analysis is done taking the minimal Standard Model.
Given this result, let us see what, if any, are the theoretical constraints. The Standard Model Higgs mass is given, at the classical level, by $m_H^2=2\lambda v^2$, with $v$ the vacuum expectation value of the Higgs field. $v$ is fixed by the value of the Fermi coupling constant $G_F/\sqrt 2 =1/(2v^2)$ which implies $v\approx$246 GeV. Therefore, any constraints will come from the allowed values of $\lambda$. A first set of such constraints is given by the classical requirement:
$$\label{clconstr}
1>\lambda >0 ~~~~\Rightarrow ~~~~ m_H<400-500 GeV$$
The lower limit for $\lambda$ comes from the classical stability of the theory. If $\lambda$ is negative the Higgs potential is unbounded from below and there is no ground state. The upper limit comes from the requirement of keeping the theory in the weak coupling regime. If $\lambda \geq 1$ the Higgs sector of the theory becomes strongly interacting and we expect to see plenty of resonances and bound states rather than a single elementary particle.
Going to higher orders is straightforward, using the renormalisation group equations. The running of the effective mass is determined by that of $\lambda$. Keeping only the dominant terms and assuming $t=\log (v^2/\mu^2)$ is small ($\mu \sim v$), we find
$$\label{quconstr1}
\frac{d\lambda}{dt}=\frac{3}{4\pi^2}[\lambda^2+3\lambda h_t^2-9h_t^4+...]$$
where $h_t$ is the coupling of the Higgs boson to the top quark. The dots stand for less important terms, such as the other Yukawa couplings to the fermions and the couplings with the gauge bosons. This equation is correct as long as all couplings remain smaller than one, so that perturbation theory is valid, and no new physics beyond the standard model becomes important. Now we can repeat the argument on the upper and lower bounds for $\lambda$ but this time taking into account the full scale dependence $\lambda (\mu)$. We thus obtain for the Higgs mass an upper bound given by the requirement of weak coupling regime ($\lambda (\mu)<1$) all the way up to the scale $\mu$, and a lower bound by the requirement of vacuum stability ($\lambda (\mu)>0$), again up to $\mu$. Obviously, the bounds will be more stringent the larger the assumed value of $\mu$. Figure \[bounds\] gives the allowed region for the Higgs mass as a function of the scale for scales up to the Planck mass. We see that for small $\mu \sim 1$TeV, the limits are, essentially, those of the tree approximation equation (\[clconstr\]), while for $\mu \sim M_P$ we obtain only a narrow window of allowed masses 130GeV$<m_H<$200GeV, remarkably similar to the experimental results.
This analysis gives the first conclusion: If perturbation theory remains valid, in other words, if we have no new strong interactions, there exists at least one, relatively light, Higgs boson: 0.5cm
[*Conclusion 1: The absence of a light Higgs boson implies New Physics.*]{} 0.5cm
Here “heavy Higgs” is not clearly distinguished from “no-Higgs”, because a very heavy Higgs, above 1 TeV, is not expected to appear as an elementary particle. As we explained above, this will be accompanied by new strong interactions. A particular version of this possibility is the “Technicolor” model, which assumes the existence of a new type of fermions with strong interactions at the multi-hundred-GeV scale. The role of the Higgs is played by a fermion-antifermion bound state. “New Physics” is precisely the discovery of a completely new sector of elementary particles. Other strongly interacting models can and have been constructed. The general conclusion is that a heavy Higgs always implies new forces whose effects are expected to be visible at the LHC.[^3]
The possibility which seems to be favoured by the data is the presence of a “light” Higgs particle. In this case new strong interactions are not needed and, therefore, we can assume that perturbation theory remains valid. But then we are faced with a new problem. The Standard Model is a renormalisable theory and the dependence on the high energy scale is expected to be only logarithmic. This is almost true, but with one notable exception: The radiative corrections to the Higgs mass are quadratic in whichever scale $\Lambda$ we are using. The technical reason is that $m_H$ is the only parameter of the Standard Model which requires, by power counting, a quadratically divergent counterterm. The gauge bosons require no mass counterterm at all because they are protected by gauge invariance and the fermions need only a logarithmic one. The physical reason is that, if we put a fermion mass to zero we increase the symmetry of the model because now we can perform chiral transformations on this fermion field. Therefore the massless theory will require no counterterm, so the one needed for the massive theory will be proportional to the fermion mass and not the cut-off. In contrast, putting $m_H=0$ does not increase the symmetry of the model.[^4] As a result the effective mass of the Higgs boson will be given by
$$\label{mhiggseff}
(m_H^2)_{eff}=m_H^2+C\alpha_{eff}\Lambda^2$$
where $C$ is a calculable numerical coefficient of order one and $\alpha_{eff}$ some effective coupling constant. In practice it is dominated by the large coupling to the top quark. The moral of the story is that the Higgs particle cannot remain light unless there is a precise mechanism to cancel this quadratic dependence on the high scale. This is a particular aspect of a general problem called “scale hierarchy”. The only known mechanism which reconciles a light Higgs and a high value of the scale $\Lambda$ with the validity of perturbation theory is supersymmetry. In this case the Higgs mass is protected against the quadratic corrections of eq. (\[mhiggseff\]) because it behaves like the mass of the companion fermion which, as we just said, receives only logarithmic corrections. It is closest in spirit to the charm mechanism, in the sense that a heavy effective cut-off is made compatible with the low energy data by the presence of new particles. The alternative is to have a low value of $\Lambda$, [*i.e.*]{} new physics, at a low scale. The models with large compact extra dimensions enter into this category. This brings us to our second conclusion: 0.5cm
[*Conclusion 2: A light Higgs boson is unstable without new physics.*]{} 0.5cm
Both conclusions are good news for LHC. But the time for speculations is coming to an end. The LHC is coming. Never before a new experimental facility had such a rich discovery potential and never before was it loaded with so high expectations.
[99]{} S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. P.A.M. Dirac, Proc. Roy. Soc. A114 (1927) 710. B.L. Joffe and E.P. Shabalin, Yadern Fiz. 6 (1967) 828 (Soviet J. Nucl. Phys. 6 (1968) 603); Z. Eksp. i Teor. Fiz. Pis’ma U Red. 6 (1967) 978 (Soviet. Phys. JETP Lett. 6 (1967) 390). C. Bouchiat, J. Iliopoulos and J. Prentki, Nuov. Cim. 56A (1968) 1150; J. Iliopoulos, Nuov. Cim. 62A (1969) 209. G. Altarelli, Private Communication.
[^1]: The estimation is only heuristic. It is based on a rough counting of the number of diagrams and assumes that they all contribute equally and have the same sign, neither of which is exact. The estimation can be improved but the result remains the same.
[^2]: This prediction is, in fact, an average between a much lower value, around 50 GeV, given by the data from leptonic asymmetries, and a much higher one, of 400 GeV, obtained from the hadronic asymmetries. Although the difference sounds dramatic, the two are still mutually consistent at the level of 2-3 standard deviations.
[^3]: We can build specific models in which the effects are well hidden and pushed above the LHC discovery potential, at least with the kind of accuracy one can hope to achieve in a hadron machine. In this case one would need very high precision measurements, probably with a multi-TeV $e^+-e^-$ collider.
[^4]: At the classical level, the Standard Model with a massless Higgs does acquire a new symmetry, namely scale invariance, but this symmetry is always broken for the quantum theory and offers no protection against the appearance of quadratic counterterms.
|
---
abstract: 'Nonlinear Schrödinger equations and corresponding quantum hydrodynamic (QHD) equations are widely used in studying ultracold boson-fermion mixtures and superconductors. In this article, we show that a more exact account of interaction in Bose-Einstein condensate (BEC), in comparison with the Gross-Pitaevskii (GP) approximation, leads to the existence of a new type of solitons. We use a set of QHD equations in the third order by the interaction radius (TOIR), which corresponds to the GP equation in a first order by the interaction radius. The solution for the soliton in a form of expression for the particle concentration is obtained analytically. The conditions of existence of the soliton are studied. It is shown what solution exists if the interaction between the particles is repulsive. Particle concentration of order of $10^{12}$-$10^{14}$ $cm^{-3}$ has been achieved experimentally for the BEC, the solution exists if the scattering length is of the order of 1 $\mu$m, which can be reached using the Feshbach resonance. It is one of the limit case of existence of new solution. The corresponding scattering length decrease with the increasing of concentration of particles. The investigation of effects in the TOIR approximation gives a more detail information on interaction potentials between the atoms and can be used for a more detail investigation into the potential structure.'
author:
- 'P. A. Andreev'
- 'L. S. Kuzmenkov'
title: 'Bright-like soliton solution in quasi-one-dimensional BEC in third order on interaction radius '
---
\[sec:level1\]I. Introduction
=============================
Nonlinear structures, solitons, and vortices have been actively studied in atomic Bose-Einstein condensate (BEC) and boson-fermion mixtures situated in magnetic and optical traps [@Young-S; @JP; @B; @11]- [@Wen; @JP; @B; @10]. Solitons exist in inquest of nonlinearity of interactionally conditioned terms. These nonlinear terms compensate the dispersion emerging particularly as a consequence of a free motion of quantum particles. Solitons in the BEC are described via the Gross-Pitaevskii (GP) equation [@L.P.Pitaevskii; @RMP; @99] which has form of one-particle nonlinear Schrödinger equation (NLSE). The NLSE plays an important role when describing the dynamics of various physical systems; name just a few degenerated chargeless bosons and fermions as well as superconductors. A more detailed account of the interaction in comparison with the GP equation leads to the appearance of additional terms both in the GP equation and in the corresponding quantum hydrodynamics (QHD) equations. A similar generalization is acquired for ultracold boson-fermion mixtures [@Andreev; @PRA08]. Such generalization leads to that the term depending on the spatial derivatives of the concentration (modulus quadrate of wave-function in the medium) appears in the NLSE. Different authors [@Rosanov], [@Braaten] suggested the NLSEs for the description of the BEC dynamic taking into account the terms depending on high degrees of concentration. The occurrence of additional terms in the GP equation leads to varying the characteristics of wave perturbations [@Andreev; @Izv.Vuzov.; @09; @1], solitons [@Andreev; @Izv.Vuzov.; @10], vortices, dispersion shock waves, and can also lead to new types of solutions. The last point is considered in this paper.
The possibility to derive the GP equation from a microscopic many-particle Schrödinger equation (MPSE) is substantiated in [@Erdos; @PRL; @07]. The method of direct derivation of the GP equation from the MPSE was suggested in Ref. [@Andreev; @PRA08]. It was made by means of QHD method [@Maksimov; @QHM; @99-01], [@Andreev; @PRB; @11]. It is well-known that the GP equation can be present in the form of hydrodynamic equations [@L.P.Pitaevskii; @RMP; @99] $$\label{TOIR Sol cont from GP}\partial_{t}n(\textbf{r},t)+\partial^{\alpha}(n(\textbf{r},t)v^{\alpha}(\textbf{r},t))=0$$ and $$mn(\textbf{r},t)\partial_{t}v^{\alpha}(\textbf{r},t)+\frac{1}{2}mn(\textbf{r},t)\partial^{\alpha}v^{2}(\textbf{r},t)$$ $$-\frac{\hbar^{2}}{4m}\partial^{\alpha}\triangle n(\textbf{r},t)+\frac{\hbar^{2}}{4m}\partial^{\beta}\biggl(\frac{\partial^{\alpha}n(\textbf{r},t)\cdot\partial^{\beta}n(\textbf{r},t)}{n(\textbf{r},t)}\biggr)$$ $$\label{TOIR Sol eiler from GP}+gn(\textbf{r},t)\partial^{\alpha}n(\textbf{r},t)=-n(\textbf{r},t)\partial^{\alpha}V_{ext}(\textbf{r},t),$$ where $$g=\int d\textbf{r}U(r),$$ and $n(\textbf{r},t)$ is the concentration of particles and $v^{\alpha}(\textbf{r},t)$ is the velocity field. The quantity $\triangle$ is the Laplace operator.
For dilute gases the quantity $g$ can be express via scattering length by formula $$g=\frac{4\pi\hbar^{2}a}{m},$$ where $a$ is the scattering length.
In this paper we use set of equations derived with the QHD method. There are different methods for obtaining equations describing the BEC evolution. For example, in Ref. [@Nakamura; @AnnPh; @11], equation for BEC evolution was derived in the framework of nonequilibrium Thermo Field Dynamics. In the QHD method the system of equation is appeared directly from many-particle Schrödinger equation. The first step of derivation is the definition of concentration of particles in three dimensional physical space. Differentiation of concentration with respect to time and applying of the Schrödinger equation leads to continuity equation, a current of density is arisen there. Next step of derivation is differentiation of the current density. In this way we obtain a momentum balance equation, in another terms the Euler equation. A force field exists in the obtained Euler equation. For neutral particles with the short-range interaction the force field could be present in the form of a expansion in a series. In this case, the GP equation emerges when we take into account the first member of decomposition of the force field by the interaction radius. The next nonzero term appears in the third order by the interaction radius (TOIR).
Different types of solitons occur in the BEC and boson-fermion mixtures. If the interaction between the bosons is repulsive $a>0$, dark solitons that are the regions with a lowered concentration of the particles can propagate in the BEC [@Burger; @et.al.], [@Denschlag; @Sc; @00], [@Anderson; @PRL; @01]. Bright solitons, i.e., solitons of compression, can exist in the system of Bose particles coupled by attractive forces in quasi-one-dimensional (1D) traps [@Khaykovich; @et.al.], [@Strecker; @Nat; @02]. Gap solitons manifest themselves in periodic structures, particularly, the existens of gap bright solitons are found experimentally in the system of bosons with $a>0$ [@Eiermann; @cond-mat]. Solitons of compression occur in boson-fermion mixtures if repulsive forces $a_{bb}>0$ act between the bosons, while the interaction between bosons and fermions is attractive with force $a_{bf}<0$ [@Karpiuk; @PRL; @04]. In the work [@Andreev; @Izv.Vuzov.; @10] authors obtain a change of the form of the well-known bright soliton due to TOIR terms. The bright soliton solution arise from GP equation. More detailed account of interaction with accuracy to TOIR leads to change of form of bright soliton.
In this paper, we report about the existence of a new soliton solution in a one dimensional (1D) BEC. This solution appears when we account the interaction accurate to the TOIR. To obtain this soliton we solve the set of the QHD equations by the perturbative method suggested Washimi et al. [@Washimi; @PRL; @66], which is widely used in the plasma physics, see for example [@perturb; @meth], [@Infeld; @book]. The obtained solution is the soliton of compression; it exists under the condition $a>0$, i.e., in the case of repulsion between the particles. The existence of such a solution can be conditioned by higher spatial concentration derivatives in the term for interaction in the TOIR. We consider a two cases it are 1D configuration and quasi-1D trap. Let us notice the limiting cases of existence of the solution. One of the limiting cases in the region of parameters when the scattering length (SL) $a$ is of the order $10^{-8} cm$, and the corresponding equilibrium concentration is $10^{18} cm^{-3}$. That could be actual in connection with the development of cooling methods for dense gases [@Vogl; @Nat; @09], [@Sheik; @NatP; @09]. Another limiting case is the region of parameters with the SL of the order $10^{-4} cm$. This case corresponds to concentrations about of $10^{12}-10^{14} cm^{-3}$, which are usually dealt with in experiments with BEC. Thus, in order to form the conditions for soliton occurrence, the Feshbach resonance (FR) phenomenon should be used [@Chin; @RMP; @10], [@Bloch; @RMP; @08]. They attains the wide-limit SL change in FR experiments, particularly the values $10^{3}-10^{4} a_{0}$ ($a_{0}$-Bohr radius) can be reached for magnetically trapped $^{85}Rb$ [@Cornish; @PRL; @00].
We use in this paper short-range interaction potential quantum hydrodynamic equation derived for the system of ultracold neutral particles. In connection with this, our attention should be paid to the fact that an increase in the SL can be caused both by a decrease and an increase in the depth or width of the interaction potential. Assuming that an increase in the SL is caused by a decrease in the interaction potential depth, the conditions of existence of equations could be considered as fulfilled.
In a general case, the fact that under the FR condition larger values of SL $a$ are attained, can point to the fact that a more successive account for the interaction should be necessary.
The processes and effects in the TOIR, along with the effects in the spinor BEC [@Szankowski; @PRL; @10], magnetically [@Gligoric; @PRA; @10], [@Wilson; @PRL; @10], [@Bijnen; @PRA; @10] and electrical [@Fischer; @PRA; @06], [@Ticknor; @PRL; @11] polarized BEC, can play an important role when investigating BEC and interatomic interaction.
Our paper is organized as follows. In Sect. 2 we present basic equation and describe using model. In Sect. 3 we consider a solitons in 1D BEC and describe a method of getting of solution. We show that with solution is a new solution and receive a condition of existence of this solution. In Sect. 4 we obtain system of QHD equations for the quasi-one dimensional case. In Sect. 5 we investigate soliton solution obtained in sect. 3 for quasi-one dimensional case. In Sect. 6 brief summary of obtained results is presented.
\[sec:level1\] II. Model
========================
To investigate solitons in BEC, we use the set of QHD equations up to the TOIR approximation [@Andreev; @PRA08]. The calculation of the first member in a quantum stress tensor that corresponds to the GP equation is fulfilled in [@Andreev; @PRA08] under the condition that the particles do not interact. A more complete investigation into the conditions of derivation of the GP equation from the MPSE shows that the GP equation appears in the first order by the interaction radius (FOIR), if the particles are in an arbitrary state that can be simulated by a single-particle wave function. Such a state can particularly appears as a result of strong interaction between the particles that takes place in quantum fluids.
The QHD equations set for the atoms with a two-particle interaction with the potential $U(r)$ and located in external field $V_{ext}(\textbf{r},t)$ in the TOIR approximation has the form [@Andreev; @PRA08] $$\label{TOIR Sol cont boze TOIR}\partial_{t}n(\textbf{r},t)+\partial^{\alpha}(n(\textbf{r},t)v^{\alpha}(\textbf{r},t))=0$$ and $$mn(\textbf{r},t)\partial_{t}v^{\alpha}(\textbf{r},t)+\frac{1}{2}mn(\textbf{r},t)\partial^{\alpha}v^{2}(\textbf{r},t)$$ $$-\frac{\hbar^{2}}{4m}\partial^{\alpha}\triangle n(\textbf{r},t)+\frac{\hbar^{2}}{4m}\partial^{\beta}\biggl(\frac{\partial^{\alpha}n(\textbf{r},t)\cdot\partial^{\beta}n(\textbf{r},t)}{n(\textbf{r},t)}\biggr)$$ $$-\Upsilon n(\textbf{r},t)\partial_{\alpha}n(\textbf{r},t)-\frac{1}{16}\Upsilon_{2}\partial_{\alpha}\triangle n^{2}(\textbf{r},t)$$ $$\label{TOIR Sol eiler boze TOIR}=-n(\textbf{r},t)\partial^{\alpha}V_{ext}(\textbf{r},t),$$ where $$\label{TOIR Sol Upsilon} \Upsilon=\frac{4\pi}{3}\int
dr(r)^{3}\frac{\partial U(r)}{\partial r}$$and $$\label{TOIR Sol Upsilon 2}\Upsilon_{2}=\frac{4\pi}{15}\int dr
(r)^{5}\frac{\partial U(r)}{\partial r}.$$ We also have $\Upsilon=-g$. Equations (\[TOIR Sol cont boze TOIR\]) and (\[TOIR Sol eiler boze TOIR\]) determine the dynamic of concentration of particles $n(\textbf{r},t)$ and velocity field $v^{\alpha}(\textbf{r},t)$. From equation (\[TOIR Sol eiler boze TOIR\]) we see that dynamics of BEC depends on different moments of interaction potential $\Upsilon$, $\Upsilon_{2}$. The system of equations (\[TOIR Sol cont boze TOIR\]) and (\[TOIR Sol eiler boze TOIR\]) is differ from (\[TOIR Sol cont from GP\]) and (\[TOIR Sol eiler from GP\]) by existence of one new term. It is a last term in left hand side of equation (\[TOIR Sol eiler boze TOIR\]). This term appears at interaction account up to TOIR approximation.
In diluted alkali gases, the interaction between particles can be considered as scattering. In this case, the FOIR interaction constant can be expressed in terms of SL $\Upsilon=-4\pi\hbar^{2}a/m$ [@L.P.Pitaevskii; @RMP; @99]. The second interaction constant $\Upsilon_{2}$ emerges in the TOIR. In a general case, parameter $\Upsilon_{2}$ is independent of $\Upsilon$. In [@Andreev; @PRA08], the approximate expression $\Upsilon_{2}$ via $\Upsilon$ is considered. We use this expression in our work when we investigate the existence region of the soliton solution.
Considering the dispersion equation for elementary excitations in BEC accurate to TOIR $$\omega^{2}(k)=\biggl(\frac{\hbar^{2}}{4m^{2}}+\frac{n_{0}\Upsilon_{2}}{8m}\biggr)k^{4}-\frac{\Upsilon n_{0}}{m_{}}k^{2},$$ which obtained in [@Andreev; @PRA08] we can see that coefficient at $k^{4}$ could be negative. It is realized at condition $\Upsilon_{2}<-2\hbar^{2}/mn_{0}$. Consequently, we can expect that value $\Upsilon_{2}=-2\hbar^{2}/mn_{0}$ could play important role at investigation of nonlinear processes.
\[sec:level1\] III. Bright-like soliton in 1D BEC
=================================================
In this section, we consider the solitons in the 1D BEC. For this purpose, we use the perturbative method [@Washimi; @PRL; @66], [@perturb; @meth]. Here we present some detail of calculations and describe the perturbative method.
We investigate the case when the stretched variables include the expansion parameter in follows combination: $$\label{TOIR masht of var 1D} \begin{array}{ccc}\xi=\varepsilon^{1/2}(z-ut),&\tau=\varepsilon^{3/2}ut &\end{array}$$ where $u$ is the phase velocity of the wave, $\varepsilon$- is a small nondimension parameter.
An operational relations are arisen from (\[TOIR masht of var 1D\]) $$\label{TOIR repres. of var. 1D}\begin{array}{ccc} \partial_{x}=\varepsilon^{1/2}\partial_{\xi},&\partial_{t}=u\biggl(\varepsilon^{3/2}\partial_{\tau}-\varepsilon^{1/2}\partial_{\xi}\biggr)&\end{array}$$
The decomposition of the concentration and velocity field involves a small parameter $\varepsilon$ in the following form: $$\label{TOIR Sol expansion 1D of conc}n=n_{0}+\varepsilon n_{1}
+\varepsilon^{2}n_{2}+...$$ $$\label{TOIR Sol expansion 1D of vel}v=\varepsilon v_{1}
+\varepsilon^{2}v_{2}+...$$ Presented in (\[TOIR Sol expansion 1D of conc\]) equilibrium concentration $n_{0}$ is a constant. We put expansions (\[TOIR repres. of var. 1D\])-(\[TOIR Sol expansion 1D of vel\]) in equations (\[TOIR Sol cont boze TOIR\]) and (\[TOIR Sol eiler boze TOIR\]). Then, the system of equation is divided into systems of equations in different orders on $\varepsilon$.
Equations emerging in the first order by $\varepsilon$ from the system of equations (\[TOIR Sol eiler boze TOIR QODim\]) have form $$-u\partial_{\xi}n_{1}+n_{0}\partial_{\xi}v_{1}=0,$$ $$\label{TOIR Sol 1 order syst}-mun_{0}\partial_{\xi}v_{1}=\Upsilon n_{0}\partial_{\xi}n_{1}$$ and lead to the following expression for the phase velocity $u$: $$\label{TOIR Sol rel for U 1D}u^{2}=-\frac{\Upsilon n_{0}}{m}.$$ Square of phase velocity $u^{2}$ must be positive. Consequently $\Upsilon$ is negative, i.e. $$\label{TOIR Sol cond of ex 1D}\Upsilon<0.$$ It corresponds to the repulsive SRI.
Also, from (\[TOIR Sol 1 order syst\]) we obtain relation between $n_{1}$ and $v_{1}$ and their derivatives $$\partial_{\xi}n_{1}=\frac{n_{0}}{u}\partial_{\xi}v_{1}.$$ Integrating this equation and using a boundary conditions $$\label{TOIR boundari cond the first} \begin{array}{cccc} n_{1} ,& v_{1}\rightarrow 0 & at &
x\rightarrow\pm\infty\end{array}$$ we have $$\label{TOIR 1e connection of var}n_{1}=\frac{n_{0}}{u}v_{1}.$$
In the second order by $\varepsilon$, from equations (\[TOIR Sol cont boze TOIR\]) and (\[TOIR Sol eiler boze TOIR\]), we derive $$\label{TOIR 2e cont eq}-u\partial_{\xi}n_{2}+u\partial_{\tau}n_{1}+\partial_{\xi}(n_{0}v_{2}+n_{1}v_{1})=0$$ and $$-mu(n_{0}\partial_{\xi}v_{2}+n_{1}\partial_{\xi}v_{1})+mun_{0}\partial_{\tau}v_{1}+mn_{0}v_{1}\partial_{\xi}v_{1}$$ $$\label{TOIR Sol 2 order syst}-\frac{\hbar^{2}}{4m}\partial_{\xi}^{3}n_{1}=\Upsilon n_{0}\partial_{\xi}n_{2}+\Upsilon n_{1}\partial_{\xi}n_{1}+\frac{1}{8}\Upsilon_{2}n_{0}\partial_{\xi}^{3}n_{1}.$$
In (\[TOIR 2e cont eq\]) we can express $n_{2}$ via $v_{2}$ and $n_{1}$, $v_{1}$ *and* put it in equation (\[TOIR Sol 2 order syst\]). Using (\[TOIR Sol rel for U 1D\]), we exclude $v_{2}$ from the obtained equation (\[TOIR Sol 2 order syst\]). Thus, we obtain an equation which contain $n_{1}$ and $v_{1}$, only. Using (\[TOIR 1e connection of var\]), expressing $v_{1}$ via $n_{1}$ we get a Korteweg-de Vries equation for $n_{1}$ $$\label{TOIR Sol KdV eq sos 1D}\partial_{\tau}n_{1}
+p_{1D}n_{1}\partial_{\xi}n_{1}+q_{1D}\partial_{\xi}^{3}n_{1}=0.$$ In this equation the coefficients $p_{1D}$ and $q_{1D}$ arise in the form $$\label{TOIR Sol KdV koef 1D 1}p_{1D}=\frac{3}{2n_{0}},$$ and $$\label{TOIR Sol KdV koef 1D 2}q_{1D}=\frac{\frac{\hbar^{2}}{2m}+\frac{1}{8}n_{0}\Upsilon_{2}}{2n_{0}\Upsilon}.$$
From equation (\[TOIR Sol KdV eq sos 1D\]) we can find the solution in the form of a solitary wave using transformation $\eta=\xi-V\tau$ *and* taking into account boundary condition $n_{1}=0$ and $\partial_{\eta}^{2}n_{1}=0$ at $\eta\rightarrow\pm\infty$, we get $$\label{TOIR Sol solution of KdV 1D}n_{1}=\frac{3V}{p_{1D}}\frac{1}{\cosh^{2}\biggl(\frac{1}{2}\sqrt{\frac{V}{q_{1D}}}\eta\biggr)},$$ where $V$ is the velocity of solition propagation to the right. From expression $p_{1D}=3/2n_{0}$ and solution (\[TOIR Sol solution of KdV 1D\]) we can find that a perturbation of concentration is positive. Consequently, obtained solution is the bright like soliton (BLS). A width of the soliton is given with formula $d=2\sqrt{q_{1D}/V}$. BLS exists in the case $q_{1D}$ is positive. From condition $q_{1D}>0$ (\[TOIR Sol cond of ex 1D\]) we have $$\label{TOIR Sol solution existence cond}\frac{\hbar^{2}}{2m}+\frac{1}{8}n_{0}\Upsilon_{2}<0.$$ Relation (\[TOIR Sol solution existence cond\]) is fulfil only in the case when $\Upsilon_{2}$ is negative. In the absence of the second interaction constant $\Upsilon_{2}$ (i. e. in the Gross-Pitaevskii approximation) the relation (\[TOIR Sol solution existence cond\]) does not fulfil and, consequently, BLS does not exist. From (\[TOIR Sol solution existence cond\]) we receive that the second interaction constant $\Upsilon_{2}$ must be negative and it’s module must be more than $4\hbar^{2}/mn_{0}$ $$\label{TOIR Sol solution existence cond sv}|\Upsilon_{2}|>\frac{4\hbar^{2}}{mn_{0}}.$$
Using representation $\Upsilon_{2}$ via the s-wave SL $a$ [@Andreev; @PRA08] we get $$\label{TOIR Sol appr for G2}\Upsilon_{2}=\theta a^{2}\Upsilon=-4\pi\theta\hbar^{2}a^{3}/m ,$$ where $\theta$- is a constant, which is determined by an explicit form of the interaction potential, $\theta>0$, $\theta\sim 1$ [@Andreev; @PRA08].
From (\[TOIR Sol solution existence cond\]) and (\[TOIR Sol appr for G2\]) we obtain $$\label{TOIR Sol solution existence cond SL}\pi a^{3}n_{0}>0.$$ It is the condition of BLS existence.
Due to used method perturbation $n_{1}$ must be smaller than equilibrium concentration $n_{0}$: $n_{0}>>n_{1}$. Here we consider the rate $n_{1}/n_{0}$ at the centre of soliton at $\cosh(\sqrt{V/q_{1D}}/2\eta)=1$: $$\frac{n_{1}(centre)}{n_{0}}=\frac{3V}{p_{1}n_{0}}=2V.$$ Correspondingly, dimensionless velocity $V$ must be much smaller than one.
At equality in formula (\[TOIR Sol solution existence cond SL\]) the BLS has infinite width, with the increasing of interaction solitons width becomes finite. Formula (\[TOIR Sol solution existence cond SL\]) shows the bound condition for existence of soliton.
Below we consider the same problem for cigar-shaped trap.
\[sec:level1\]IV. The quantum hydrodynamics equation in the cigar-shaped traps
==============================================================================
Let us to consider the variation of the form of equations of QHD in the case of cigar-shaped magnetic traps: $$V_{ext}=\frac{m\omega_{0}^2}{2}(\rho^2+\lambda^2z^2),$$ where $\omega_{0}$ and $\lambda\omega_{0}$ are angular frequencies in radial and axial directions and $\lambda$ is the anisotropy parameter. In a quasi-1D geometry, the anisotropy parameter in the axially-free motion approximation becomes zero $\lambda=0$. Thus, the solution for the radial wave-function appears in the form $$\label{TOIR Sol Rad WF}\mid\Phi_{0}(\rho)\mid^{2}=n(\rho)=\frac{m\omega_{0}}{\pi\hbar}exp\biggl(-\frac{m\omega_{0}\rho^{2}}{\hbar}\biggr).$$
In the TOIR during the quasi-1D motion of bosons in magnetic traps, the GP equation preserves the form but the interaction constant [@Adhikari; @PRA05] changes. A complete 3D particle concentration $n_{w}(\rho,z,t)$ can be presented as product of one-dimensional time dependent concentration $n(z,t)$ and static radial two-dimensional concentration $n(\rho)$ $$\label{TOIR Sol concentr for rho and z}n_{w}(\rho,z,t)=n(z,t)n(\rho),$$ where the value $n(\rho)$ is presented by the formula (\[TOIR Sol Rad WF\]).
Applying the procedure described in [@Adhikari; @PRA05], from the set of equations (\[TOIR Sol eiler boze TOIR\]) and using the corresponding NLSE, we can acquire the system of QHD equations for cigar-shaped trap. Here, we describe basic steps of this procedure. Starting from equation (\[TOIR Sol eiler boze TOIR\]) we get a equation of evolution of a following function, which sometimes called wave function in medium or order parameter, $$\Phi(\textbf{r},t)=\sqrt{n(\textbf{r},t)}\exp(\imath m\theta(\textbf{r},t)/\hbar),$$ where $\theta$ is the potential of velocity field, i.e. $\textbf{v}=\nabla\theta$. Equation for $\Phi(\textbf{r},t)$ is the NLSE corresponding to system of equations (\[TOIR Sol eiler boze TOIR\]). Approximately we can present $\Phi(\textbf{r},t)$ in the form $\Phi(\textbf{r},t)=\Phi(\rho,z,t)=\Phi(z,t)\Phi(\rho)$, where $\Phi(\rho)$ is the wave function of the ground state of harmonic oscillator and the square of module of $\Phi(\rho)$ presented by formula (\[TOIR Sol Rad WF\]). Since, we get a NLSE for $\Phi(z,t)$. This equation describes the evolution of BEC in quasi-one dimensional trap. From obtained NLSE we derive the system of QHD equations for quasi-1D trap. In the results we have $$\partial_{t}n(z,t)+\partial_{z}(n(z,t)v(z,t))=0$$ and $$mn(z,t)\partial_{t}v(z,t)+\frac{1}{2}mn(z,t)\partial_{z}v^{2}(z,t)$$ $$-\frac{\hbar^{2}}{4m}\partial_{z}^{3}n(z,t)+\frac{\hbar^{2}}{4m}\partial_{z}\frac{(\partial_{z}n(z,t))^{2}}{n(z,t)}$$ $$+\alpha_{1} n(z,t)\partial_{z}n(z,t)+\alpha_{2} n(z,t)\partial_{z}^{3}n(z,t)$$ $$\label{TOIR Sol eiler boze TOIR QODim}-\frac{7}{2}\alpha_{2}(\partial_{z} n(z,t))\partial_{z} n^{2}(z,t)
-\alpha_{2}\frac{(\partial_{z}n(z,t))^2}{n(z,t)}=0.$$
The following parameters appear in equation (\[TOIR Sol eiler boze TOIR QODim\]): $$\alpha_{1}=-\Upsilon\frac{1}{2}\frac{m\omega_{0}}{\pi\hbar}+\frac{5\pi}{2}\Upsilon_{2}\Biggl(\frac{m\omega_{0}}{\pi\hbar}\Biggr)^{2}$$ and $$\alpha_{2}=-\frac{3}{16}\Upsilon_{2}\frac{m\omega_{0}}{\pi\hbar}.$$
The form of nonlinear terms that describe the interaction in equation (\[TOIR Sol eiler boze TOIR QODim\]) differs from corresponding terms in (\[TOIR Sol eiler boze TOIR\]). This leads to varying the form of solutions and conditions of their existence in a quasi-1D geometry compared with a 1D case.
\[sec:level1\] V. The small amplitude solitons in quasi-1D BEC
==============================================================
In this section, we consider the solitons in the BEC for the case of small nonlinearity taking into account the TOIR. For this purpose, we use the perturbative method [@Washimi; @PRL; @66], [@perturb; @meth].
We investigate the case when the stretched variables include the expansion parameter in follows combination: $$\label{TOIR masht of var} \begin{array}{ccc}\xi=\varepsilon^{1/2}(z-ut),&\tau=\varepsilon^{3/2}ut &\end{array}$$ where $u$ is the phase velocity of the wave.
The decomposition of the concentration and velocity field involves a small parameter $\varepsilon$ in the following form: $$\label{TOIR Sol expansion of conc}n=n_{0}+\varepsilon n_{1}
+\varepsilon^{2}n_{2}+...$$ $$\label{TOIR Sol expansion of vel}v=\varepsilon v_{1}
+\varepsilon^{2}v_{2}+...$$ Presented in (\[TOIR Sol expansion of conc\]) equilibrium concentration $n_{0}$ is a constant. Equations emerging in the first order by $\varepsilon$ from the set of equations (\[TOIR Sol eiler boze TOIR QODim\]) lead to the following expression for the phase velocity $u$: $$u^{2}=\frac{n_{0}\alpha_{1}}{m}$$ $$\label{TOIR Sol phase vel in sos}=\frac{n_{0}}{m}\biggl(-\Upsilon\frac{1}{2}\frac{m\omega_{0}}{\pi\hbar}+\frac{5\pi}{2}\Upsilon_{2}\biggl(\frac{m\omega_{0}}{\pi\hbar}\biggr)^{2}\biggr)$$ It is evident from (\[TOIR Sol phase vel in sos\]) that the wave can exist under the condition $\alpha_{1}>0$. The obtained condition means that in the repulsive forces should act in the case under consideration under the condition that the contribution of terms in FOIR prevails over the TOIR terms.
Relationship (\[TOIR Sol phase vel in sos\]) is the analog of the dispersion dependence. The use of scaling (\[TOIR masht of var\]) leads to simplifying the dispersion relationship compared with the case when we consider small perturbations proportional to $exp(-\imath\omega t+\imath kz)$. In the latter case, we obtained the dispersion relation $\omega(k)$ form the set of equations (\[TOIR Sol eiler boze TOIR QODim\]) in the form: $$\omega^{2}=\biggl(\frac{\hbar^{2}}{4m^{2}}
+\frac{3n_{0}}{16m}\Upsilon_{2}\frac{m\omega_{0}}{\pi\hbar}\biggr)k^{4}$$ $$\label{TOIR disp rel gen QOD}
+n_{0}k^{2}\biggl(-\frac{1}{2}\Upsilon\frac{m\omega_{0}}{\pi\hbar}
+\frac{5\pi}{2}\Upsilon_{2}\biggl(\frac{m\omega_{0}}
{\pi\hbar}\biggr)^{2}\biggr).$$ Thus, relationship (\[TOIR Sol phase vel in sos\]) corresponds to the phonon part of the dispersion dependence (\[TOIR disp rel gen QOD\]).
From the second-order set of equations (\[TOIR Sol eiler boze TOIR QODim\]) by $\varepsilon$ we find that the concentration $n_{1}$ satisfies the Korteweg-de Vries equation: $$\label{TOIR Sol KdV eq sos}\partial_{\tau}n_{1}
+pn_{1}\partial_{\xi}n_{1}+q\partial_{\xi}^{3}n_{1}=0,$$ where $$\label{TOIR Sol p-q gen 1}p=\frac{3}{2n_{0}},$$ $$\label{TOIR Sol p-q gen 2} q=\frac{4mn_{0}\alpha_{2}-\hbar^{2}}{8mn_{0}\alpha_{1}}.$$ When we derive this equation, we used boundary conditions $n_{1}=0$ and $v_{1}=0$ at $\xi\rightarrow\pm\infty$. Using transformation $\eta=\xi-V\tau$, *and* taking into account boundary condition $n_{1}=0$ and $\partial_{\eta}^{2}n_{1}=0$ at $\eta\rightarrow\pm\infty$, we can obtain the solution in the form of a solitary wave from equation (\[TOIR Sol KdV eq sos\]) $$\label{TOIR Sol solution of KdV}n_{1}=\frac{3V}{p}\frac{1}{\cosh^{2}\biggl(\frac{1}{2}\sqrt{\frac{V}{q}}\eta\biggr)},$$ where $V$ is the velocity of solition propagation to the right. Sign of perturbation is determined by the sign of $p$. From formulas (\[TOIR Sol p-q gen 1\]), (\[TOIR Sol p-q gen 2\]) we can see the quantity $p$ is positive. Consequently, obtained solution is the soliton of compression or bright like soliton solution, by analogy with well-known bright soliton in BEC [@Andreev; @Izv.Vuzov.; @10], [@Khaykovich; @et.al.], [@Strecker; @Nat; @02]. As it will be shown below, this solution exists only when taking into account the TOIR.
Let us pass on to a detail consideration of the conditions of existence of the solution (\[TOIR Sol solution of KdV\]). The solution (\[TOIR Sol solution of KdV\]) of the equation (\[TOIR Sol KdV eq sos\]) exists as the conditions $q>0$ and $\alpha_{1}>0$ are fulfilled. We start with consideration the condition $q>0$. As $\alpha_{1}>0$, then to fulfill the condition $q>0$ we need $-\hbar^{2}+4mn_{0}\alpha_{2}>0$.
In the case when $\alpha_{2}$ is vanish (i.e. in FOIR approximation) the solution (\[TOIR Sol solution of KdV\]) is not exist. It means, that solution arises in TOIR approximation which developed in [@Andreev; @PRA08]. One of the condition of existence of the solution (\[TOIR Sol solution of KdV\]) is: $$\label{TOIR Sol cond of ex sol 1}\alpha_{2}>\frac{\hbar^{2}}{4mn_{0}}.$$ Consequently, for the second interaction constant $\Upsilon_{2}$ we obtain: $$\label{TOIR Sol cond of ex sol 1f2}\Upsilon_{2}<-\frac{4\pi\hbar^{3}}{3m^{2}n_{0}\omega_{0}}.$$ Using relation (\[TOIR Sol appr for G2\]) we can make estimation for corresponding SL. It is useful to present the value of possible SL in the terms of space parameter of the trap $a_{\perp}=\sqrt{\hbar/m\omega_{0}}$. From conditions (\[TOIR Sol cond of ex sol 1f2\]), (\[TOIR Sol appr for G2\]), the conditions for SL $a$ appear: $$\label{TOIR Sol cond of ex sol 1f2 bbbb}a>\sqrt[3]{\frac{a_{\perp}^{2}}{3\theta n_{0}}}.$$ In addition, from $\alpha_{1}>0$ and (\[TOIR Sol appr for G2\]), we obtain $$\label{TOIR sec cond ex of sol} a<\frac{a_{\perp}}{\sqrt{5\theta}}.$$ Using equation (\[TOIR Sol appr for G2\]), the particle concentration $n$ can be presented in the form $$\label{TOIR Sol wiev sol}n=n_{0}+2Vn_{0}\varepsilon\cdot sech^{2}\biggl(\frac{1}{2}\sqrt{\frac{V}{q'}}\eta\biggr),$$ where $$\label{TOIR Sol q'}q'=\frac{3\theta mn_{0}\omega_{0}a^{3}\hbar-\hbar^{2}}{16mn_{0}\omega_{0}a(\hbar-5a^{2}\theta m\omega_{0})}$$
The soliton width $d$ arises in the form $d=2\sqrt{q'/V}$. The numerical analysis of formula (\[TOIR Sol q’\]) is presented in Fig. \[fig1:epsart\]- \[fig2:epsart\]. It is evident from Fig. \[fig1:epsart\]- \[fig2:epsart\] that there is a narrow interval of the SL values, for which the solution (\[TOIR Sol wiev sol\]), (\[TOIR Sol q’\]) exists. The dependence of the soliton width on the SL is resonant-shaped. The resonant value of the SL $a_{r}$ depends on the particle concentration $n_{0}$ and trap parameter. The values of $a_{r}$ become lower at increasing of equilibrium concentration $n_{0}$. The SL $a_{r}$ reaches 0.1 nm at the concentration of the order $10^{18}$ $cm^{-3}$. As the concentration decreases to values $10^{12}$-$10^{14}$ $cm^{-3}$, which are usually used in BEC experiments, the SL increases to the values of the order of 1 $\mu$m. Such values of the SL can be attained when using the FR.
![\[fig1:epsart\] The dependence of the soliton width $d$ on the scattering length $a$ at fixed parameter of the trap $a_{\perp}=\sqrt{\hbar/m\omega_{0}}=10^{-5}cm$ and equilibrium concentration $n_{0}=10^{6}cm^{-1}$ *and* at $V=1$, $\theta=1$. On Fig.1a we can see what width of soliton is the positive in small range of the values of the SL. On Fig.1b the area of the resonance is presented more detailed than on Fig.1a.](BLSS01.eps "fig:"){width="8cm"} ![\[fig1:epsart\] The dependence of the soliton width $d$ on the scattering length $a$ at fixed parameter of the trap $a_{\perp}=\sqrt{\hbar/m\omega_{0}}=10^{-5}cm$ and equilibrium concentration $n_{0}=10^{6}cm^{-1}$ *and* at $V=1$, $\theta=1$. On Fig.1a we can see what width of soliton is the positive in small range of the values of the SL. On Fig.1b the area of the resonance is presented more detailed than on Fig.1a.](BLSS02.eps "fig:"){width="8cm"}
![\[fig2:epsart\] (Color online) The dependence of soliton width $d$ on radial parameter of the trap $a_{\perp}$ and the nonperturbative concentration of the particles $n_{0}$ at the fixed scattering length $a=10^{-6}cm$, and at $V=1$, $\theta=1$. The soliton width $d$ is positive in two area. But in area with smaller value of the particles concentration $n_{0}$ the square of phase velocity is negative. Thus, the solution exist in area of bigger concentrations.](BLSS03.eps){width="8cm"}
\[sec:level1\] VI. Conclusion
=============================
In this article, we showed that at a more exact accounting of the interaction, specifically, taking into account the TOIR, a new type of solitons emerges in the BEC. We also studied the conditions of existence of such a solution. For this problem solving we used the set of QHD equations where the interactions included up to TOIR approximation. The TOIR approximation is an example of the nonlocal interaction. The GP approximation gives us the force density in the right hand side of the Euler equation $\textbf{F}=-g\nabla n^{2}/2=\Upsilon\nabla n^{2}/2$. It is corresponds to the first order on interaction radius. The interaction including up to TOIR approximation gives us the second tern in the force field $\textbf{F}=\Upsilon\nabla
n^{2}/2+\Upsilon_{2}\nabla\triangle n^{2}/16$. The new term contain the third spatial derivative of the concentration square and the new interaction constant. For obtained results analysis and estimation we used approximate estimation of $\Upsilon_{2}$ *and* its approximate connection with the $\Upsilon$ or SL $a$.
We found that BLS (soliton of compression) exists in 1D case (one dimensional propagation in three dimensional medium) in the case of strong enough repulsive interaction. BLS appearance strongly connects with the second interaction constant $\Upsilon_{2}$. If we consider the interaction in the first order by the interaction radius there is no BLS. We also studied the BLS behavior in the case of quasi-1D trap and describe contribution of external fields on BLS amplitude and width.
In a general case, the second interaction constant $\Upsilon_{2}$, which appears in the TOIR, is independent of $\Upsilon$ and, consequently, of SL $a$. Thus, the relationships obtained in this article (\[TOIR Sol p-q gen 1\]), (\[TOIR Sol p-q gen 2\]), (\[TOIR Sol solution of KdV\]) can be used for an independent experimental determination of $\Upsilon_{2}$. In this case, parameter $\hbar^{2}/mn_{w0}$ can be used for the qualitative evaluation of the second interaction constant $\Upsilon_{2}$.
Thus, in this paper we showed that new physical effects appear at account of interaction up to TOIR approximation. The processes and effects in the TOIR approximation, along with the effects in the spinor and polarized BEC, can play an important role at investigation of BEC and interatomic interaction.
[99]{}
Luis E. Young-S, P. Muruganandam and S. K. Adhikari, J. Phys. B: At. Mol. Opt. Phys. **44**, 101001 (2011).
R. G. Scott, F. Dalfovo, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. **106**, 185301 (2011).
S. Tsuchiya, F. Dalfovo, and L. Pitaevskii, Phys. Rev. A **77**, 045601 (2008).
Zhen-Jun Yang, et. al., Chin. Phys. Lett. **28**, 074213 (2011).
L. Wen, Y. Zhang and J. Feng J. Phys. B: At. Mol. Opt. Phys. **43**, 225302 (2010).
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari, Rev. Mod. Phys. **71**, 463 (1999).
P. A. Andreev, L. S. Kuz’menkov, Phys. Rev. A. **78**, 053624 (2008).
N. N. Rosanov, A. G. Vladimirov, D. V. Skryabin, W. J. Firth, Phys. Lett. A. **293**, 45 (2002).
E. Braaten, H.-W. Hammer, and Shawn Hermans, Phys. Rev. A. **63**, 063609 (2001).
P. A. Andreev, L. S. Kuz’menkov, Russian Physics Journal **52**, 912 (2009).
P. A. Andreev, M. I. Trukhanova, Russian Physics Journal **53**, 1196 (2011).
L. Erdős, B. Schlein, H.-T. Yau, Phys. Rev. Lett. **98**, 040404 (2007).
L. S. Kuz’menkov, S. G. Maksimov, Theoretical and Mathematical Physics **118**, 227 (1999).
P. A. Andreev, L. S. Kuzmenkov and M. I. Trukhanova, Phys. Rev. B **84**, 245401 (2011).
Y. Nakamura, Y. Yamanaka, Annals Of Phys. **326**, 1070 (2011).
S. Burger, et.al., Phys. Rev. Lett. **83**, 5198 (1999).
J. Denschlag et.al., Science **287**, 97 (2000).
B. P. Anderson et.al., Phys. Rev. Lett. **86**, 2926 (2001).
L. Khaykovich et.al., Science **296**, 1290 (2002).
K. E. Strecker et al., Nature (London) **417**, 150 (2002).
B. Eiermann et al., Phys. Rev. Lett. **92**, 230401 (2004).
T. Karpiuk, et. al., Phys. Rev. Lett. **93**, 100401 (2004).
H. Washimi, T. Taniuti, Phys. Rev. Lett. **17**, 996 (1966).
B. C. Kalita, R. Das, Phys. Plasmas **5**, 3588 (1998); H. Leblond, J. Phys. B **41**, 043001 (2008).
E. Infeld, G. Rowlands, *Nonlinear waves, Solitons and Chaos*, Cambridge University press, (2000).
U. Vogl, M. Weitz, Nature **461**, 70 (2009). M. Sheik-Bahae, D. Seletskiy, Nature Photonics **3**, 680 (2009).
Cheng Chin, R. Grimm, P. Julienne and E. Tiesinga, Rev. Mod. Phys. **82**, 1225 (2010).
I. Bloch, J. Dalibard, W. Zwerger, Rew. Mod. Phys. **80**, 885 (2008).
S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. **85**, 1795 (2000).
P. Szankowski, M. Trippenbach, E. Infeld, and G. Rowlands, Phys. Rev. Lett. **105**, 125302 (2010). G. Gligoric, A. Maluckov, M. Stepic, L. Hadzievski, and B. A. Malomed, Phys. Rev. A **81**, 013633 (2010).
R. M. Wilson, S. Ronen, and J. L. Bohn , Phys. Rev. Lett. **104**, 094501 (2010).
R. M. W. van Bijnen, N. G. Parker, S. J. J. M. F. Kokkelmans, A. M. Martin, and D. H. J. O’Dell, Phys. Rev. A **82**, 033612 (2010).
U. R. Fischer, Phys. Rev. A **73**, 031602(R) (2006).
C. Ticknor, R. M. Wilson, and J. L. Bohn, Phys. Rev. Lett. **106**, 065301 (2011).
S. K. Adhikari, Phys. Rev. A **72**, 053608 (2005).
|
---
abstract: '[We give positive answers for questions by Berestovskiǐ. Namely, we prove that every bijection of locally compact geodesically complete and connected at infinity $CAT(0)$-space $X\/$ onto itself preserving some fixed distance or satellite relations is an isometry of this space. The proof of this theorem is based on another result stated by Berestovskiǐ as a problem: the metric of the space $X\/$ may be recovered from its diagonal tube corresponding to an arbitrary number $r \, >\, 0$.]{}'
address: 'Pomor State University, Arkhangelsk, Russia'
author:
- 'Pavel D. Andreev'
date: 'Received August 11, 2003; received in final form January 11, 2004'
title: 'A characterization of isometries of $CAT(0)$-space as maps preserving diagonal tube'
---
Introduction {#introduction .unnumbered}
============
V.N.Berestovskiĭ in [@Berestovskii] has established following characterisation for isometries of Aleksandrov spaces of curvature negatively bounded above:
\[isom\] Let $X\/$ be a locally compact geodesically complete $CAT(\kappa)$-space, $\kappa \, <\, 0$, in which all spheres are path connected. Then every bijection $f\/$ of $X\/$ onto itself such that both $f\/$ and $f^{-1}\/$ map any closed ball of some fixed radius $r\, >\,0$ onto some ball of radius $r\/$ is isometry of $X$.
The key point of the proof is
Let $X\/$ be as in Theorem \[isom\] and $V \subset X \times X\/$ be its diagonal tube corresponding to a number $r\, >\,0$. Then the metric of $X\/$ is uniquely determined by $V$.
A *diagonal tube* $V\/$ of a metric space $X\/$ corresponding to $r\, >\,0$ is by definition a set $$V := \{ (x,y) \in X \times X |\quad | xy| \, \le\, r\} \subset X \times X\, ,$$ where $|xy|\/$ is a distance between points $x,y \in X$. For a set $V\/$ we put $$\partial V := \{ (x,y) \in X \times X |\quad | xy|\, =\, r\} \subset X \times X\, ,$$ and $${\rm Int}V := \{ (x,y) \in X \times X |\quad | xy| \, <\, r\} \subset X \times X\, .$$
A question whether analogous theorems are true in the case $\kappa\, =\, 0\/$ is raised in [@Berestovskii]. We give an affirmative answer to this question. Namely we prove
\[main\] Let $(X, d)\/$ be a locally compact geodesically complete and connected at infinity $CAT(0)$-space, $f\,\colon X \, \to\, X$ — bijection and $V \subset X \times X\/$ be diagonal tube of space $X\/$ corresponding to a number $r\, >\,0$. Set a map $\phi := f\times f \,\colon X\times X \, \to\, X \times X\/$ by $\phi(x_1, x_2) := (f(x_1), f(x_2))$. Then following statements are equivalent:
1. $\phi(V)\, =\, V$,
2. $\phi(\partial V)\, =\, \partial V$,
3. $\phi({\operatorname{Int}}V)\, =\, {\operatorname{Int}}V$,
4. $f\/$ is an isometry of $X\/$ onto itself.
When we say that a map $\phi\/$ moves $V\/$ onto $V\/$ this means that for every pair $x,y \in X\/$ with $d(x,y) \, \le\, r\/$ inequality $d(\phi(x), \phi(y)) \, \, \le\, r\/$ holds, but not necessarily $d(\phi(x), \phi(y))\, =\, d(x, y)$. The last equality is the target of the theorem.
Connectedness at infinity of space $X\/$ means that the complement $X\setminus B\/$ of every metric ball $B \subset X\/$ is path connected. It was shown in [@Berestovskii] that connectedness at infinity is equivalent to the condition of path connectedness for every sphere in $X$.
Following [@Berestovskii] we base the proof on the uniqueness of the metric on $X\/$ with prescribed properties and given diagonal tube. We formulate this statement as following.
\[tube\] Let the space $(X, d)\/$ be as in theorem \[main\] and the number $r\, >\,0\/$ is fixed. Then every metric $d'\/$ on $X\/$ such that $(X, d')\/$ is locally compact, geodesically complete $CAT(0)$-space, coincides with $d$: for any $x,y\in X\, d'(x,y)=d(x,y)$, iff any one of following equivalent conditions hold:
1. $\forall x,y\in X\quad d(x,y) \, \le\, r \, \Leftrightarrow\, d'(x,y)\, \le\, r$;
2. $\forall x,y\in X\quad d(x,y)\, =\, r\, \Leftrightarrow\, d'(x,y)\, =\, r$;
3. $\forall x,y\in X\quad d(x,y) \, <\, r\, \Leftrightarrow\, d'(x,y) \, <\, r$.
Theorem \[main\] is direct consequence of theorem \[tube\]: it sufficies to take the metric $d'\, =\, f^\ast d$: $$d'(x,y)\, =\, d(f(x), f(y))\, ,$$ and apply equality $f^\ast d (x,y) := d(x,y)\/$ as claim of theorem \[tube\].
For simplicity, from now on we will assume $r=1$, but conveniently, the notation $r\/$ will be kept in some notions, such as *$r$-sequences* etc.
The author have considered a case of Busemann spaces ([@Busemann]), i.e. locally compact complete metric spaces with intrinsic metric and properties of nonbranching and local extendability of minimizing segments in earlier paper ([@andreev]). It was shown in [@berestov-2] that every $CAT(0)$-space which is a Busemann space is really a Riemannian manifold with continuous metric tensor relatively distance coordinates. Thus the connectedness at infinity of a space $X\/$ is equivalent to the estimation $n\, >\,1\/$ for its topological dimension $n := \, \operatorname{TopDim}(X)\/$ in this case. The proof of the theorem \[tube\] in the present paper is based on methods developed in [@andreev] with necessary modification.
Problems resolved in the paper are conjugated with the question suggested by A.D.Aleksandrov in 1960-s:
- Under what conditions is a map of a metric space into itself preserving a fixed distance (for example, distance 1) an isometry of this metric space?
A number of similar questions are partially resolved by Aleksandrov himself. For example, several theorems presenting sufficient conditions for map of classical spaces of constant curvature preserving congruence to be isometry are proved in [@A.D.1973]. Beckman and Quarles have proved the version of theorem \[main\] for maps of Euclidean spaces in [@Beck-Quart], Kuzminykh — for Lobachevskii space in [@Kuzm].
The paper consists of three sections. The first section is introductory and contains a list of basic notions and facts used by author as preliminaries and the concept of parallel-equivalence for $r$-sequences. We introduse two types of $r$-sequences according to behavior of corresponding classes of parallel-equivalence. In the rest sections we show that the metric of any geodesic in $X\/$ can be recovered from $V$. In the Section 2 we study geodesics bounding flat strip, and in Section 3 geodesics of strictly rank one. Combining two mentioned situations we get whole proof of Theorem \[tube\].
[**Acknowledgment**]{} I’m very grateful to V.N.Berestovskiǐ for support and useful consultations given during my work on the paper. I would like to thank members of geometrical seminar of Steklov PDMI RAS (St. Petersburg) for attention and number of important remarks. Especially I want to thank S.V.Buyalo, for suggesting an idea simplifying the proof of existence theorem for scissors.
Preliminaries
==============
$CAT(\kappa)$-spaces {#catk}
--------------------
Main definitions and properties of Aleksandrov spaces with curvature bounded above and so called $CAT(\kappa)$-spaces may be found in [@B], [@BH] or [@Bu].
Let $(X,d)\/$ be a metric space. The distance $d\/$ between points $x,y \in X\/$ conveniently will be denoted as $|xy| := d(x,y)$. An open ball of radius $\rho\/$ centered at point $x\in X\/$ is denoted as $B(x, \rho)$, corresponding closed ball — as $\overline{B}(x, \rho)$, the boundary sphere — as $S(x, \rho)$. For any subset $A\subset X\/$ and any $\epsilon\, >\,0$, the $\epsilon$-neighborhood of $A\/$ is $$N_\epsilon(A):=\{x\in X|\, d(x, a) \, <\, \epsilon\quad \text{for some } a\in A\}\, .$$ For any two closed subsets $A, B\subset X$, the *Hausdorff distance* between $A\/$ and $B\/$ is $$d_H(A, B):=\inf \{\,\epsilon\,|\, A\subset N_\epsilon(B),\, B\subset N_\epsilon(A)\};$$ $d_H(A, B)\/$ is defined to be $\infty\/$ if there is no $\epsilon\, >\,0\/$ with $A\subset N_\epsilon(B)\/$ nor $B\subset N_\epsilon(A)$.
A *geodesic* in $X\/$ is a continuous map $c\,\colon I\, \rightarrow\, X\/$ from an interval $I\subset {\mathbb R}\/$ into $X$, such that for any point $t\in I\/$ there exists a neighborhood $U\/$ of $t\/$ with $d(c(s_1),c(s_2))=|s_1-s_2|\/$ for all $s_1,s_2 \in U$. If one can take $U\, =\, I$, then such geodesic is said to be *minimizing*. The image of a geodesic or a minimizing will also be called a geodesic or a minimizing. When $I\/$ is a closed interval $[a,b]\subset {\mathbb R}$, we say that $c\/$ is a *geodesic segment* of length $b-a\/$ and $c\/$ *connects* $c(a)\/$ and $c(b)$. If $I\, =\, {\mathbb R}$, we say that $c\/$ is a complete geodesic. A metric space $X\/$ is called a *geodesic metric space* if for any two points $x,y\in X\/$ there is a minimal geodesic segment connecting them. A geodesic metric space is called *geodesically complete* if every geodesic segment is contained in some complete geodesic (not necessarily unique).
It follows from Hopf-Rinow theorem (cf. [@Bu Ch.1, Theorem 2.3]) that geodesically complete locally compact space $X\/$ is *proper* or *finitely compact*, i.e. every its closed ball is compact.
For $\kappa \in {\mathbb R}\/$ we let $M_\kappa\/$ be the *model* (i.e. complete simply connected) surface of constant curvature $\kappa$, $D(\kappa)\/$ be its *diameter*, that is, $D(\kappa)\, =\, \infty\/$ if $\kappa\, \le\, 0\/$ and $D(\kappa)\, =\, \pi/\sqrt{\kappa}\/$ if $\kappa \, >\, 0$. The metric of $M_\kappa\/$ is denoted as $d_\kappa$.
A *triangle* in $X\/$ is the union of three mimimizing segments $c_i\,\colon [a_i, b_i] \, \to\, X$, $i\, =\, \overline{1,3}$, called *sides* of triangle, pairwise connecting three points $x_i, i\, =\, \overline{1,3}\/$ which we call *vertices* of triangle.
For a collection $\Delta := (x_1, x_2, x_3)\/$ of points in $X\/$ the *comparison triangle* $\overline{\Delta}\subset M_\kappa\/$ has vertices $\overline{x}_1, \overline{x}_2\/$ and $\overline{x}_3\/$ such that $d_\kappa(\overline{x}_i, \overline{x}_j)\, =\, d(x_i, x_j)$, $i,j=\overline{1,3}$. A comparison triangle exists and is unique up to isometry if the perimeter $$P(\Delta) := d(x_1, x_2) \, +\, d(x_2, x_3) \, +\, d(x_1, x_3) \, <\, 2 D(\kappa)\, .$$ If $c_i\/$ are sides of $\Delta$, we denote $\overline{c}_i\/$ sides of $\overline{\Delta}$.
A point $\overline{x}\in \overline{\Delta}\/$ corresponds to a point $x\/$ of triangle $\Delta\/$ if there is some $i\/$ and some $t_i\in [a_i, b_i]\/$ with $x=c_i(t_i)\/$ and $\overline{x}=\overline{c}_i(t_i)$.
Let $\kappa\in {\mathbb R}$. A complete metric space $X\/$ is called a *$CAT(\kappa)$-space* if\
[(i)]{} Every two points $x, y\in X\/$ with $d(x, y)\, <\, D(\kappa)\/$ are connected by a minimizing; For any triangle $\Delta\/$ in $X\/$ with perimeter less than $2D(\kappa)\/$ and any two points $x,y\in \Delta$, the inequality $d(x,y)\, \le\, d(\overline{x},\overline{y})\/$ holds, where $\overline{x}\/$ and $\overline{y}\/$ are the points of the comparison triangle $\overline{\Delta}\subset M_\kappa\/$ corresponding to $x\/$ and $y\/$ respectively.
Every geodesic $c\,\colon {\mathbb R}\, \to\, X\/$ in $CAT(0)$-space $(X,d)\/$ has two opposite directions defined by its rays $c|_{(-\infty, 0]}\/$ and $c|_{[0, +\infty)}$. Geodesic rays $c_i\,\colon [0, \infty) \, \to\, X, i=\overline{1,2}\/$ are called *asymptotic* if Hausdorff distance between them is finite: $d_H(c_1, c_2) \, <\, \infty$. A relation *to be asymptotic* on a set of all rays in $X\/$ is equivalence. Equivalence classes called *ideal points* (or *points at infinity*) form *geometric boundary* (*boundary at infinity*) $\partial_\infty X\/$ of $X$. Complete geodesics $c_1\/$ and $c_2\/$ are *asymptotic* if they contain asymptotic rays.
Complete geodesics $c_1\/$ and $c_2\/$ are called *parallel* if their Hausdorff distance ${\rm Hd}(c_1, c_2)\/$ is finite. In this case they are asymptotic in both directions and bound a *flat strip*, i.e. subset isometric to a strip in Euclidean plane.
We say that a geodesic $c\/$ *virtually bounds a flat strip* if there is a finite sequence $c\, =\, c_0, c_1, \dots, c_n\/$ of geodesics such that for every $1\, \le\, i\, \le\, n\/$ geodesics $c_{i-1}\/$ and $c_i\/$ are asymptotic and $c_n\/$ is a boundary geodesic of some flat strip. In particular every line bounding the flat strip also bounds it virtually. Otherwize we say that $c\/$ *has strictly rank one*.
Spaces of directions
--------------------
A *pseudo-metric* on a set $\Sigma\/$ is a function $d\,\colon \Sigma\times \Sigma\, \rightarrow\, [0, \infty)\/$ that is symmetric and satisfies the triangle inequality. If $(\Sigma,d)\/$ is a pseudo-metric space, then we get a metric space $(\Sigma^\ast, d^\ast)\/$ by letting $\Sigma^\ast\/$ be the set of maximal zero diameter subsets and setting $d^\ast(S_1,S_2):=d(s_1,s_2)\/$ for any $s_i\in S_i$. $(X^\ast, d^\ast)\/$ is called the metric space *associated* to the pseudo-metric $d$.
Given two geodesic segments $c=[x\eta]\/$ and $d=[x\zeta]\/$ with common vertex $x\/$ in $CAT(\kappa)$-space $X$, $\kappa \in {\mathbb R}\/$ with lengthes less than $D(\kappa)$, we have a well-defined function $\angle_x(c,d)\, =\, \lim\limits_{t \, \to\, 0} \widetilde {\angle_x}(c(t),d(t))$, where $\widetilde {\angle_x}(y,z)\/$ denotes *the comparison angle*, i.e. angle at the vertex $\overline x\/$ of comparison triangle $\overline x\,\overline y\,\overline z\/$ at the Euclidean plane for triangle $xyz$. The function $\angle_x\/$ defines a pseudo-metric on a set of all geodesic segments begining from $x$. The metric space associated to the pseudo-metric $\angle_x\/$ is denoted by $\Sigma^\ast_x X$. Its metric completion $\Sigma_x X\/$ is called *the space of directions* at $x$. If a space $X\/$ is geodesically complete, then we have $\Sigma^\ast_x X\, =\, \Sigma_x X\/$ for any $x \in X$.
Let $X\/$ be a $CAT(\kappa)$-space and $p\in X$. Then $\Sigma_p X\/$ is a $CAT(1)$-space.
Diagonal tube and satellite relations {#satel}
-------------------------------------
From now on we assume that $(X,d)\/$ is a locally compact geodesically complete connected at infinity $CAT(0)$-space (*Hadamard space*) and $\overline{X}\, =\, X \cup \partial_\infty X\/$ be its geometric closure. The metric $d\/$ of $CAT(0)$-space is convex function in following sence. For every two geodesics $c_1,c_2\,\colon {\mathbb R}\, \to\, X\/$ the function $d(c_1(t), c_2(t))\/$ is convex.
For two points $y,z \in \overline{X}\/$ notation $[yz]\/$ means:
- connecting them geodesic segment, if both lies in $X$, or
- connecting them geodesic ray if one of them lies in $X$, and another (order is ignored) — in $\partial_\infty X$, or
- any complete geodesic connecting $y\/$ and $z\/$ in the case when $y,z \in \partial_\infty X$, if such geodesic does exist. When the geodesic is not unique, we will detect it by additional features.
We will identify every geodesic as a map of real interval into $X\/$ with its image and use the same notation.
For a number $r\, >\,0\/$ $V\/$ will be *the diagonal tube* corresponding it. Recall that we have assumed $r=1$, hence $$V := \{(x,y) \in X\times X|\quad |xy|\, \le\, 1\}\, .$$
When we say that diagonal tube $V\/$ defines metric $d$, this means that every metric $d'\/$ on $X\/$ satisfying conditions of theorem \[tube\] with the same diagonal tube $V\/$ coincides with $d$. We will consider $d\/$ as initial metric and $d'\/$ as trial metric for which we need to show that $d'\, =\, d$. Below $d'\/$ will always be trial metric satisfying conditions of theorem \[tube\] and with the same diagonal tube $V$.
Also, sometimes we will use the terminology, such as following: “diagonal tube defines ...” or “$V\/$ allows to recover...” etc. Opinions of such type will mean that some object, value or some property of object is the same for metric $d\/$ and any trial metric $d'$. In particular this holds when mensioned object admits description in terms of $V$.
First, we note that relations $nV$, ${\rm int}(nV)\/$ and $\partial(nV)\/$ are defined for all $n \in {\mathbb N}\/$ simultaneously with $V\/$ by equalities $$nV := \{(x,y)\in X\times X |\quad |xy|\, \le\, n\}$$ $$\partial(nV) := \{(x,y) \in X\times X |\quad |xy|\, =\, n \}$$ and $${\rm Int} (nV) := nV \setminus \partial (nV)\, .$$
\[poli\] If two metrics $d\/$ and $d'\/$ on $X\/$ as in theorem \[tube\] have common diagonal tube $V$, then they have also common relations $nV$, $\partial(nV)\/$ and ${\rm Int}(nV)\/$ for all $n \in {\mathbb N}$.
Pair $(x,y) \in 2V\/$ iff there exists a point $z \in X\/$ such that $(x,z), (z,y) \in V$. Inductively, $(x, y) \in nV\/$ iff there exists $z \in X\/$ such that $(x,z)\in(n-\nolinebreak 1)V\/$ and $(z,y) \in V$. We have $(x, y) \in \partial (2V)\/$ iff there exists unique point $z\/$ with $(x,z), (z,y) \in V$. In this case $(x,z)\/$ and $(z,y)\in \partial V\/$ and $z\/$ is a midpoint of segment $[xy]$. Moreover, pair $(x,z) \in \partial V\/$ iff there exists a pair $(x,y) \in \partial (2V)\/$ such that $z\/$ is a midpoint of $[xy]$. Inductively one may define relations $\partial (nV)$. Relations ${\operatorname{Int}}(nV)\/$ are by definition $${\operatorname{Int}}(nV)\, =\, nV \setminus \partial (nV)\, .$$
Consequently metrics $d\/$ and $d'\/$ with common $V\/$ have common open, closed balls and spheres with integer radii.
\[inter\] If two metrics $d\/$ and $d'\/$ on $X\/$ as in theorem \[tube\] have common any one of three relations $V$, $\partial V\/$ and ${\operatorname{Int}}V$, then they have common another two relations.
It was shown in Lemma \[poli\] that $V\/$ defines $\partial V\/$ and ${\rm Int}(V)$.
Assume that $d\/$ and $d'\/$ have common boundary relation $\partial V$. Arguments similar to those used in the proof of Lemma \[poli\] show that they have common diagonal tube $2V\/$ as well. In particular, metrics $d\/$ and $d'\/$ have common all balls $\overline{B}(x, 2n)\/$ of even radii and consequently all spheres $S(x, 1)\/$ and $S(x, 2n)\/$ with $x \in X$. Then by the connectedness at infinity of $X\/$ we have $(x, y) \in {\rm int} V\/$ iff $S(x,1) \cap S(y, 1) \ne \emptyset\/$ and $S(x,2) \cap S(y, 1)\, =\, \emptyset$. This defines ${\rm Int} V\/$ by already defined relations. $V\/$ is defined as $V\, =\, {\rm Int} V \cup \partial V$.
Let now $d\/$ and $d'\/$ have common ${\rm Int} V$. Then for all $n \in {\mathbb N}\/$ relations ${\rm Int}(nV)\/$ are also common for $d\/$ and $d'$. Hence from geodesic completeness of $X\/$ we have that $(x,y) \in \partial V\/$ iff $(x,y) \notin {\rm Int} V\/$ and $B(y,1) \subset B(x,2)$. Indeed, if $1 \, <\, d(x,y) \, <\, 2$, then we may include geodesic segment $[xy]\/$ into segment $[xz]\/$ of length $d(x,z)\, =\, \frac 12 (3 \, +\, d(x,y))$, and $z \in B(y,1) \setminus B(x,2)$. This defines $\partial V\/$ and hence $V\/$ in terms of ${\rm Int} V$.
Consequently for the proof of theorem \[tube\] we need only to prove that the metric $d\/$ may be recovered from diagonal tube $V\/$ itself.
$r$-sequences.
--------------
The main tool of [@Berestovskii], called $r$-sequence was defined in terms of relations above. We give another definition here. *$r$-sequence* in $X\/$ is by definition a homothety ${\mathbb Z}\, \to\, X\/$ with coefficient $r$, that is an integral parametrized sequence $\{ x_z\}_{z \in {\mathbb Z}}\subset X\/$ of points in $X\/$, such that for all $z_1, z_2 \in {\mathbb Z}\/$ equalities $|x_{z_{1}}\, x_{z_{2}}|\, =\, r |z_2\, -\, z_1|\/$ hold. When we put $r=1$, the last equation becomes written $|x_{z_{1}}\, x_{z_{2}}|\, =\, |z_2\, -\, z_1|\/$ and we consider $r$-sequence as isometric map ${\mathbb Z}\, \to\, X$.
It immediately follows from lemmas \[poli\] and \[inter\] and geometry of space $(X, d)\/$ that:
- for every $r$-sequence there is unique containing it geodesic in $(X,d)\/$ and geodesic in $(X, d')$. A priori these geodesics may be different. If they coincide, we will say that the incidence relation on this geodesic is detected by $V$.
- for any trial metric $d'$, sequence $\{x_z \}_{z \in {\mathbb Z}}\/$ is $r$-sequence with respect to the metric $d\/$ iff it is $r$-sequence with respect to $d'$, and
- $V\/$ allows us to reveal, given two $r$-sequences $\{x_z \}_{z \in {\mathbb Z}}\/$ and $\{y_z \}_{z \in {\mathbb Z}}\/$ whether geodesics in $(X,d)\/$ containing them are asymptotic in any direction.
The segment of $r$-sequence $\{ x_z\}_{z \in {\mathbb Z}}\subset X\/$ between points $x_z\/$ and $x_{z\, +\,k}\/$ will be denoted as $[x_z, \dots , x_{z\, +\,k}]_r$, ideal points, defined by $r$-sequence $\{x_z \}_{z \in {\mathbb Z}}$, — as $x_{+\infty}\/$ and $x_{-\infty}$.
Metric transfer {#trans}
---------------
We will use an effective trick proposed in [@Berestovskii], — a map $R_{c_{1}c_{2}}\/$ from a geodesic to its asymptotic one by means of *Busemann functions*. Let $c_i \,\colon {\mathbb R}\, \to\, X, i\, =\, \overline{1, 2}\/$ be two asymptotic geodesics such that $c_1(+\infty)\, =\, c_2(+\infty)$. Let $\beta_\xi\/$ be some Busemann function corresponding to an ideal point $\xi\, =\, c_i(+\infty)$: $$\label{hs}
\beta_\xi (x) := \lim\limits_{t \, \to\, \infty}(|x\, c(t)|\, -\, t).$$ Level sets of function $\beta_\xi\/$ are called *horospheres*, sublevels — *horoballs*. We will distinguish open horoballs defined by strict inequality $\beta_\xi(x) \, <\, b_0\/$ and horoballs $\beta_\xi(x)\, \le\, b_0$.
The horosphere centered in the end $\xi := c(+\infty)\in \partial_\infty X\/$ of geodesic $c\,\colon{\mathbb R}\, \to\, X\/$ containing a point $z \in X$, i.e. level set $$\{x \in X| \, \beta_\xi(x)\, =\, \beta_\xi(z)\}$$ of Busemann function $\beta_\xi(x)\/$ will be denoted as $\mathcal{HS}_{\xi, z}$, corresponding closed horoball as $\mathcal{HB}_{\xi,z}$. An open horoball is $hb_{\xi,z} := \mathcal{HB}_{\xi,z}\setminus \mathcal{HS}_{\xi,z}$.
Horoballs and horospheres admits following definitions in terms of ${\rm Int} V\/$ (and hence in terms of $V$). Let $\{x_z\}_{z \in {\mathbb Z}}\/$ be $r$-sequence with $\xi := x_{+\infty}\, =\, c_i(+\infty)\/$ and given point $x_0$. Then open horoball $hb_{\xi, x_0}\/$ is $$hb_{\xi, x_0}\, =\, \bigcup\limits_{n=1}^{\infty} B(x_n, n)\, .$$ For a point $y \in X\/$ we have $y \in \mathcal{HS}_{\xi,x_0}\/$ iff $y \notin hb_{\xi, x_0}\/$ and $B(y,1) \subset hb_{\xi,x_{-1}}$. Finally, $\mathcal{HB}_{\xi,x_0}\, =\, hb_{\xi,x_0} \cup \mathcal{HS}_{\xi,x_0}$.
There exist length parametrisations of geodesics $c_i$, such that $$\label{parametr}
\forall t \in {\mathbb R}\quad \beta_\xi (c_1(t))\, =\, \beta_\xi (c_2(t)).$$
We set $R_{c_{1}c_{2}} (c_1(t)) := c_2(t)$. A map $R_{c_{1}c_{2}}\/$ defined above is independent on a choise of Busemann function $\beta_\xi\/$ centered in $c_i(+\infty)\/$ and on parametrisations with the property .
It was shown in [@Berestovskii] that whenever the relation of incidence of points in lines $c_i\/$ is detected, then relation $V\/$ defines every map $R_{c_{1}c_{2}}\/$ in metric space $(X, d)\/$ itself. This means that whenever geodesics $c_1\/$ and $c_2\/$ are also geodesics with respect to trial metric $d'$, then the image of any point $c_1(t)\/$ under the map $R_{c_{1}c_{2}}\/$ is independent on the choise of metric $d\/$ or $d'\/$.
We say that two geodesics $a\/$ and $b\/$ are *connected by the asymptotic chain* if there is a finite sequence $$\label{a1n}
a\, =\, a_0, a_1, \dots, a_n=b$$ of geodesics such that for every $1\, \le\, i\, \le\, n\/$ geodesics $a_{i-1}\/$ and $a_i\/$ are asymptotic in some direction.
If geodesic $a\/$ virtually bounds flat strip, it is connected by asymptotic chain with some geodesic $b\/$ lying in the boundary of flat strip.
\[asymchain\] Let geodesics $a\/$ and $b\/$ are connected by the asymptotic chain such that for all $i\, =\, \overline{0,n}\/$ relation of incidence in geodesics $a_i\/$ is detected by $V$. Then if distances $d\/$ and $d'\/$ coincide along $a$, i.e. for every pair $t_1, t_2\in {\mathbb R}\/$ equality $d(a(t_1), a(t_2))\, =\, d'(a(t_1), a(t_2)\/$ holds, then metrics coincide along $b\/$ as well.
The claim may be proved by a multiple transfer of a metric from a geodesic to its asymptotic one in the asymptotic chain connecting $a\/$ and $b$.
Parallel equivalence {#paral}
--------------------
Let $(\{x_z\}_{z \in {\mathbb Z}},\, \{y_z\}_{z \in {\mathbb Z}})\/$ be a pair of $r$-sequences in $X$. We say that they are *parallel-equivalent*, iff Hausdorff distance $d_H\/$ between sets $\{x_z\}_{z \in {\mathbb Z}}\/$ and $\{y_z\}_{z \in {\mathbb Z}}\/$ is finite: $$d_H\left( \{x_z\}_{z \in {\mathbb Z}},\, \{y_z\}_{z \in {\mathbb Z}}\right) \, <\, +\infty\, .$$
We will use the notation $\{x_z\}_{z \in {\mathbb Z}}\, \parallel \{y_z\}_{z \in {\mathbb Z}}\/$ for this relation. It is obviously a really equivalence on a set of all $r$-sequences in $X$. In fact every $r$-sequence belongs to a unique geodesic in $X$, and $\{x_z\}_{z \in {\mathbb Z}} \parallel \{y_z\}_{z \in {\mathbb Z}}\/$ iff geodesics containing $\{x_z\}_{z \in {\mathbb Z}}\/$ and $\{y_z\}_{z \in {\mathbb Z}}\/$ are parallel or coinside.
Property “to be parallel-equivalent” can be revealed by $V$.
$\{x_z\}_{z \in {\mathbb Z}}\parallel\{y_z\}_{z \in {\mathbb Z}}\/$ iff there exists $k\in {\mathbb N}\/$ such that for all $z\in{\mathbb Z}\/$ $(x_z, \, y_z) \in kV$. This completes the proof.
$r$-sequence is called $r$-sequence *of rank* 1, if there exists unique geodesic $c\/$ in $X\/$ containing all $r$-sequences parallel-equivalent to it. $c\/$ has not different parallel geodesics in $X\/$ and does not bound a flat strip in this case. It was shown in [@Berestovskii] that the incidence relation is detected by the diagonal tube in this case. In opposite case we say that $r$-sequense is *of higher rank* or has *rank over* 1. Geodesic containing $r$-sequence of higher rank bounds some flat strip.
\[rank\] A property of an arbitrary $r$-sequence $(\{x_z\}_{z \in {\mathbb Z}}\/$ to be of rank 1 or of higher rank is determined by relation $V$.
$\{x_z\}_{z \in {\mathbb Z}}\/$ is of higher rank iff there exists $r$-sequence $\{ y_z \}_{z \in {\mathbb Z}}\/$ parallel-equivalent to it and not equal to $\{ x_{z\pm 1} \}_{z \in {\mathbb Z}}$, such that $| x_0\, y_0 |\, =\, 1$. Since the relation $\partial V\/$ is determined by $V\/$ (cf Lemma \[poli\]), the proposition is proved.
The plan of further consideration is following. We will examine every single geodesic in accordance with its rank. It will be shown that in any case the incidence relation and the metric of a geodesic may be recovered from a diagonal tube $V$. First we consider a case of geodesic bounding a flat strip.
The geodesic bounding flat strip
================================
Splitting of parallel-equivalence class {#split}
---------------------------------------
Let $c \,\colon (-\infty , +\infty) \, \to\, X\/$ be a geodesic connecting ideal points $\xi_- := c(-\infty)\/$ and $\xi_+ := c(+\infty)\/$ and passing through the point $x_0\, =\, c(0)$. For $x \in X\/$ and $\xi, \eta \in \partial_\infty X$, angle at point $x\/$ between rays $[x \, \xi]\/$ and $[x \, \eta]\/$ is denoted as $\angle_x (\xi, \, \eta)$.
The following lemma is well-known (cf. [@Okun Lemma 5] for example).
\[secheniya\] Set $$C := \{ y \in X |\, \angle_y (\xi_-,\, \xi_+)\, =\, \pi\}\, .$$ Then
1. $\mathcal{HS}_{\xi_-, \, x_0} \cap \mathcal{HS}_{\xi_+, \, x_0}\, =\, \mathcal{HB}_{\xi_-, \, x_0} \cap \mathcal{HB}_{\xi_+, \, x_0}$,
2. $C\/$ is a union of all geodesics parallel to $c$,
3. $C\/$ is a closed convex set in $X\/$ and
4. $C\/$ splits as a product $C\, =\, C\,' \times c$, where $C\,' := \mathcal{HS}_{\xi_-, \, x_0} \cap \mathcal{HS}_{\xi_+, \, x_0}\/$ is a closed convex subset.
Hence the set $C\/$ of all points of all $r$-sequences parallel-equivalent to given $r$-sequence $\{x_z\}_{z \in {\mathbb Z}} \subset X\/$ splits as $C\, =\, C\,' \times c$. The set $C\,'\/$ has more than one point iff the $r$-sequence $\{x_z\}_{z \in {\mathbb Z}} \subset X\/$ is of higher rank.
Given arbitrary $r$-sequence $\{x_z'\}_{z \in {\mathbb Z}}\/$ parallel-equivalent to $r$-sequence $\{x_z\}_{z \in {\mathbb Z}} \subset X$, denote as $C\,'(x_z')\/$ the fiber of a product $C=C\,'\times~c\/$ containing a point $x_z'$. Since horospheres $\mathcal{HS}_{\xi_-, \, x_0}\/$ and $\mathcal{HS}_{\xi_+, \, x_0}\/$ may be defined in terms of $V$, the *horisontal structure* of splitting is also determined by $V$. In other words any trial metric $d'\/$ gives rise to the same set of fibers of type $C\,'(x)$, which we will call *horisontal sections* of $C$.
As well, diagonal tube $V\/$ allows to restore the order of horisontal sections of $C\/$ as following. We say that a section $C\,'(x)\/$ of the set $C\/$ *lays below* of a section $C\,'(y)\/$ if a ray $[x \xi_+]\/$ intersects $C\,'(y)$. We have $\mathcal{HB}_{\xi_-, \, x} \cap \mathcal{HB}_{\xi_+, \, y}\, =\, \emptyset\/$ in this case. It is left to restore distances between horisontal sections of $C\/$ and incidence of points of $c\/$ for completing the case.
Tapes
-----
We say that $4p \quad (p \in {\mathbb N})\/$ pairwise parallel-equivalent $r$-sequences $$\label{baza}
\{x_{i,\, j;\, z}\}_{z \in {\mathbb Z}},\, i= \overline{0,\, 3}, j= \overline{1,\, p}$$ form *$p$-tape*, if following $4p\, +\,4\/$ points $$\begin{array}{l}
x_{i,\, 1,\, 0}, \dots, x_{i,\, p,\, 0}, \, i\, =\, \overline{0,\, 3} \\
x_{0,\, 1,\, 2p-1},\, x_{2,\, p,\, 1-2p},\, x_{3,\, p-1,\, 1-2p}, x_{3,\, p,\, 1-2p}
\end{array}$$ in addition satisfy the system of $2p\/$ relations $$\label{tesyomka}
\begin{cases}
[x_{0,\, 1,\, 0},\, x_{1,\, 1,\, 0},\, x_{2,\, 1,\, 0},\, x_{3,\, 1,\, 0}]_r \\
\qquad\qquad\qquad\dots \\
\left[x_{0,\, p,\, 0},\, x_{1,\, p,\, 0},\, x_{2,\, p,\, 0},\, x_{3,\, p,\, 0}\right]_r \\
\left[x_{0,\, 2,\, 0},\, x_{1,\, 1,\, 0},\, x_{2,\, p,\, 1-2p},\, x_{3,\, p-2,\, 1-2p}\right]_r\\
\left[x_{0,\, 3,\, 0},\, x_{1,\, 2,\, 0},\, x_{2,\, 1,\, 0},\, x_{3,\, p-1,\, 1-2p}\right]_r\\
\left[x_{0,\, 4,\, 0},\, x_{1,\, 3,\, 0},\, x_{2,\, 2,\, 0},\, x_{3,\, 1,\, 0}\right]_r\\
\qquad\qquad\qquad\dots\\
\left[x_{0,\, p,\, 0},\, x_{1,\, p-1,\, 0},\, x_{2,\, p-2,\, 0},\, x_{3,\, p-3,\, 0}\right]_r\\
\left[x_{0,\, 1,\, 2p-1},\, x_{1,\, p,\, 0},\, x_{2,\, p-1,\, 0},\, x_{3,\, p-2,\, 0}\right]_r
\end{cases}$$ presenting segments of $r$-sequences.
Since $r$-sequences admit definition in terms of $V$, notion of $p$-tape is independent on the choise of metric $d\/$ or trial metric $d'$. A fragment of a $p$-tape is shown at figure \[tesma\]. Here only a part of points of $r$-sequences forming $p$-tape is picked out. The main idea is that a segment between points $x_{0,\, 1,\, 0}\/$ and $x_{0,\, 1,\, 2p-1}\/$ of the same $r$-sequence contains $2(p-1)\/$ points of this $r$-sequence dividing corresponding geodesical segment by $2p-1\/$ equal parts, and simultaneously marked segment contains $p-1\/$ points of kind $x_{0,\, j, \, 0} \quad j\, =\, \overline{2,\, p}$, dividing it by $p\/$ equal segments. Hence all points of $r$-sequences $\{ x_{0, \, j,\, z} \}_{z \in {\mathbb Z}}\/$ at segment $[x_{0,\, 1,\, 0}\, x_{0,\, 1,\, 2p-1}]\/$ divide it by $p(2p-1)\/$ equal parts, and in particular segment $[x_{0, \, 1,\, 0}\, x_{0,\, 1,\, 1}]\/$ is divided by points $x_{0,\, p,\, 3-2p},\, x_{0,\, p-1,\, 5-2p}\,,\, \dots ,\, x_{0,\, 2,\, -1}\/$ by $p\/$ equal segments.
(320,60) (0,10)(80,0)[3]{}[(5,1)[120]{}]{} (40,34)(80,0)[3]{}[(5,-1)[120]{}]{} (0,26)[(5,-1)[80]{}]{} (0,26)[(5,1)[40]{}]{} (240,10)[(5,1)[80]{}]{} (280,34)[(5,-1)[40]{}]{} (0,10)(80,0)[5]{} (40,18)(80,0)[4]{} (0,26)(80,0)[5]{} (40,34)(80,0)[4]{} (0,0)[$x_{0,\, 1,\, 0}$]{} (110,40)[$x_{3,\, 1,\, 0}$]{} (225,0)[$x_{0,\, p,\, 0}$]{} (295,0)[$x_{0,\, 1,\, 2p-1}$]{} (18,40)[$x_{3,\, p,\, 1-2p}$]{} (70,0)[$x_{0,\, 2,\, 0}$]{} (70,35)[$x_{2,\, 1,\, 0}$]{} (30,5)[$x_{1,\, 1,\, 0}$]{} (260,40)[$x_{3,\, p-1,\, 0}$]{} (265,5)[$x_{1,\, p,\, 0}$]{}
Now the plan of restoring the metric of geodesic of higher rank is following. Given $r$-sequence $\{x_z \}_{z \in {\mathbb Z}}\/$ of higher rank, geodesic $c\,\colon(-\infty,+\infty)\, \to\,~X\/$ containing it spans a flat strip (not unique in general). For sufficiently great $P\/$ for any $p \, >\, P$, $p$-tape defined by a family of $r$-sequences of type with $x_{0,\, 1,\, z}\, =\, x_z\/$ for all $z\in {\mathbb Z}$, does exist. Points of all such $r$-sequences $x_{0,\, j,\, z}\/$ for every possible $p$-tapes with $p \, >\, P\/$ cover a set of rational points on considered geodesic.
As it was mensioned earlier, horospheres and horoballs of $CAT(0)$-space $X\/$ define horisontal sections of a set $C$. Hence the metric of a term ${\mathbb R}\/$ in the splitting $C\, =\, C' \times {\mathbb R}\/$ of the set $C\/$ above may be recovered from the relation $V$.
It remains to show, that the incidence of points on $c\/$ is restorable by $V$. It is already done for rational points. The limiting procedure using balls is possible for irrationals.
Recovery of the metric of geodesic bounding flat strip
------------------------------------------------------
Every $p$-tape in $X\/$ is contained in a flat strip.
Let $p$-tape be defined by $r$-sequences . Consider a flat strip $F\/$ spanned by parallel geodesics $c\/$ and $c'$, containing correspondingly $r$-sequences $\{ x_{0,\, 1,\, z}\}_{z \in {\mathbb Z}}\/$ and $\{ x_{3,\, 1,\, z}\}_{z \in {\mathbb Z}}$. $r$-sequences $\{ x_{1,\,1,\, z }\}_{z \in {\mathbb Z}}\/$ and $\{ x_{2,\,1,\, z }\}_{z \in {\mathbb Z}}\/$ are also contained in $F$.
We will show that $F\/$ really contains all $r$-sequences in . Consider points $$\label{6-points}
x_{0,\, 2,\, 0}, \quad x_{1,\, 1,\, 0}, \quad x_{2,\, 1,\, 0}, \quad x_{3,\, 1,\, 0}, \quad x_{2,\, 2,\, 0}, \quad x_{1,\, 2,\, 0}.$$
As a corollary of we have relations: $$[x_{0,\, 2,\, 0}, \, x_{1,\, 2,\, 0}, \, x_{2,\, 2,\, 0}]_r$$ and $$[x_{1,\, 1,\, 0}, \, x_{2,\, 1,\, 0},\, x_{3,\, 1,\, 0}]_r\, .$$
Because of convexity of metric in a space of nonpositive curvature the function $$d_{2,\, 1}(t)\, =\, |\, x_{t,\, 2,\, 0}\, x_{t + 1,\, 1,\, 0}\, |\, ,$$ where $x_{t, \, j,\, 0}\, =\, c_j(t)$, $c_j\,\colon [0, 2r] \, \to\, X, \quad j\, =\, \overline{1,2}\/$ — geodesic segments connecting $x_{0,\, j,\, 0}\/$ with $x_{1, \, j,\, 0}$, is a convex function. But $$d_{2,\, 1}(0)\, =\, d_{2,\, 1}(1)\, =\, d_{2,\, 1}(2)\, =\, 1\, ,$$ hence $d_{2,\, 1}(t)\, =\, 1\/$ for all $t \in [0,\, 2]$, and points belong to a flat parallelogram $$\label{parallelogramm-1}
x_{0,\, 2,\, 0} \, x_{1,\, 1,\, 0} \, x_{3,\, 1,\, 0}, \, x_{2,\, 2,\, 0}$$ isometrically embedded into $X$.
Similarly six points $$x_{2,\, 1,\, 0}, \quad x_{3,\, 1,\, 0}, \quad x_{2,\, 2,\, 0}, \quad x_{1,\, 3,\, 0}, \quad x_{0,\, 3,\, 0}, \quad x_{1,\, 2,\, 0}$$ belong to isometrically embedded into $X\/$ flat parallelogram $$\label{parallelogramm-2}
x_{2,\, 1,\, 0} \, x_{3,\, 1,\, 0} \, x_{1,\, 3,\, 0} \, x_{0,\, 3,\, 0}.$$
Parallelograms and have the common part — parallelogram $$x_{2,\, 1,\, 0} \, x_{3,\, 1,\, 0} \, x_{2,\, 2,\, 0} \, x_{1,\, 2,\, 0}\, ,$$ and hence their union is nonconvex flat hexagon $$x_{0, \, 2, \, 0} \, x_{1,\, 1,\, 0} \, x_{3,\, 1,\, 0} \, x_{1,\, 3,\, 0} \, x_{0,\, 3,\, 0} \, x_{1, \, 2, \, 0}\, .$$
It is isometric to a nonconvex hexagon in Euclidean plane, constructed by two equal parallelograms. Continuing so on we get a flat poligon $$\label{poligon}
\begin{array}{c}
P\, =\, x_{0, \, 2, \, 0} \, x_{1,\, 1,\, 0} \, x_{3,\, 1,\, 0} \, x_{2,\, 2,\, 0} \, x_{3, \, 2, \, 0} \, \dots \\
\dots x_{3, \, p-1, \, 0} \, x_{2, \, p, \, 0} \, x_{3, \, p, \, 0} \, x_{2, \, 1, \, 2p-1}\, x_{0, 1, 2p-1} \, x_{0, \, p, \, 0} \,
x_{1, \, p-1, \, 0} \, \dots \\
\dots x_{1,\, 3,\, 0} \, x_{0,\, 3,\, 0} \, x_{1, \, 2, \, 0},
\end{array}$$ (see Figure \[tesma\]) which is isometric to a $(4p-2)$-gon constructed from $2p-3\/$ parallelograms, consecutively intersecting each other.
$P\/$ containes whole geodesic segments $x_{1,\, 1, \, 0}\, x_{1, \, 1, \, 2p-1}\/$ and $x_{2,\, 1, \, 0}\, x_{2, \, 1, \, 2p-1}$. Hence intersection of $P\/$ with a flat strip $F\/$ contains parallelogram $$x_{1,\, 1, \, 0}\, x_{2,\, 1, \, 0}\, x_{2, \, 1, \, 2p-1} x_{1, \, 1, \, 2p-1}\, .$$ Furthemore $P\/$ is contained in parallelogram $x_{0,\, 1, \, 0}\, x_{3,\, 1, \, 0}\, x_{3, \, 1, \, 2p-1} x_{0, \, 1, \, 2p-1}\/$ and hence in $F$. Since there are points of all $r$-sequences in $P$, we may say that the whole tape is contained in $F$.
*The width* of $p$-tape in $X\/$ is a distance between parallel geodesics including $r$-sequences $\{x_{0, \, j,\, z}\}_{z \in {\mathbb Z}}\/$ and $\{x_{3, \, j,\, z}\}_{z \in {\mathbb Z}}$.
The width $s(p)\/$ of $p$-tape equals to $$s(p)\, =\, \frac{3\sqrt{4p-1}}{2p}\, .$$
Since every flat strip in $CAT(0)$-space is isometric to a strip in Euclidean plane, it suffuces to calculate a width of standard Euclidean $p$-tape.
For every flat strip containing a geodesic $c$, there exists a number $P \, >\, 0$, such that for all $p \, >\, P\/$ the strip contains a $p$-tape generated by a family of $r$-sequences of type with $\{ x_{0,\, j,\, z}\} \subset c$.
\[rational\] Let the geodesic $c\,\colon (-\infty,\, +\infty ) \, \to\, X\/$ contains $r$-sequence $\{ x_z \}_{z \in {\mathbb Z}}$, $x_0\, =\, c(0)\/$ of higher rank. Then for every rational $q := \frac mn\/$ and $p := kn$, multiple of $n$, every $p$-tape for which $x_{0, \, 1,\, z}=x_z\/$ contains a point $c(q)\, =\, x_{0,\, j,\, z'}\/$ with $$j := n \, +\, 1\, -\, km'$$ and $$z' := q' \, +\, 1\, -\, 2p \, +\, 2k(m' \, +\, 1)\, ,$$ where $q\,' := \left[q-\frac 1n\right]\/$ — integral and $\frac{m'}{n} := \left\{ q\, -\, \frac 1n \right\}\/$ — fractional parts of the number $q-\frac 1n$.
By direct calculation for a standard flat $p$-tape.
This yelds, that for all $p\/$ multiple to $n\/$ every $p$-tape constructed with base points $x_z\/$ of given $r$-sequence $\{x_z\}_{z \in {\mathbb Z}}$, contains points $c(\frac mn)\/$ as elements with defined by $p\/$ multiindex. We are ready now to prove the part of Theorem \[tube\] in the case we consider.
\[parabolic\] Let metric space $(X,d)\/$ and trial metric $d'\/$ on $X\/$ be as in Theorem \[tube\]. Assume that $c \,\colon (-\infty , +\infty) \, \to\, X\/$ is a geodesic in metric $d\/$ bounding flat strip in $X$. Then $c\/$ is geodesic bounding flat strip in metric $d'\/$ and for all $t_1, t_2 \in {\mathbb R}\/$ $$\label{dd1}
d'(c(t_1), c(t_2))\, =\, d(c(t_1), c(t_2)).$$
We may assume that $t_1\, =\,0$. Let $\{x_z\}_{z \in {\mathbb Z}}\/$ be $r$-sequence lying in $c\/$ such that $x_z\, =\, c(z)$. It is $r$-sequence of higher rank relatively both metrics $d\/$ and $d'\/$ and every $p$-tape in metric $d\/$ based on $\{x_z\}_{z \in {\mathbb Z}}\/$ is $p$-tape in metric $d'\/$ as well. When $t\/$ is rational, equality is a consequence of Lemma \[rational\].
Let the number $t\/$ be irrational and $t_n, t'_n \, \to\, t\/$ be a pair of sequences of its rational approximations from below and from above correspondingly. Without loss of generality we may assume that $t\, >\,0$. Then $$c(t)\, =\, \left(\bigcap\limits_{n\, =\, 1}^\infty B(c((t'_n\, -\, 1)), \, 1)\right) \cap \left(\bigcap\limits_{n\, =\, 1}^\infty B(c((t_n \, +\, 1)), \, 1)\right)\, ,$$ and for metric $D \in \{d, d'\}\/$ we have $$t_n \, <\, D(c(0), c(t)) \, <\, t'_n\, ,$$ implying the claim.
In other words, Theorem \[parabolic\] asserts that the diagonal tube $V\/$ allows to recover the metric of geodesic $c\/$ bounding a flat strip.
\[chain\] $V\/$ recovers a metric of any geodesic which virtually bounds a flat strip in $X$.
The claim is a partial case of lemma \[asymchain\], since every geodesic in metric $d\/$ does not bounding flat strip is also geodesic in trial metric $d'$.
Geodesics of rank one
=====================
Scissors
--------
We say that four complete geodesics $a, b, c, d\,\colon (- \infty , +\infty) \, \to\, X \/$ in $CAT(0)$-space $X\/$ form *scissors centered* in $x\in X\/$ if:
- $a(-\infty)=b(-\infty)$;
- $a(+\infty)=c(+\infty)$;
- $c(-\infty)=d(-\infty)$;
- $b(+\infty)=d(+\infty)$; and
- $b \cap c\, =\, x$.
Such configuration will be denoted as $\langle a,b,c,d ;x \rangle\/$ (figure \[scissors\]). Geodesics $a\/$ and $d\/$ above are called *bases* of scissors. One of them, for example $a$, will be labeled as *lowest base*. Writing $\langle a,b,c,d ;x \rangle\/$ we put the lowest base first. One or both of bases may in general pass throw $x_0$, but when $X\/$ has a property of nonbranching of geodesics, such a situation is excluded.
(200,50) (0,0)[(1,0)[200]{}]{} (0,5)[(6,1)[200]{}]{} (200,5)[(-6,1)[200]{}]{} (0,45)[(1,0)[200]{}]{} (100,22) (98,26)[$x$]{} (98,3)[$a$]{} (98,48)[$d$]{} (172,12)[$c$]{} (28,12)[$b$]{}
Note that when scissors are given, they are not uniquelly determined by a choice of their lowest base $a\/$ and a center $x\/$ in general: we must take into account the possibility of branching of geodesics $b\/$ and $c$. Highest bases $d_1\/$ and $d_2\/$ of scissors $\langle a, b, c, d_1, x \rangle\/$ and $\langle a, b, c, d_2, x \rangle\/$ with common lines $a$, $b$, $c\/$ may be parallel. Lines forming scissors may be partially or even totally attached. Some examples of scissors with branching of their lines are presented at figure \[scisdeg\].
(200,70) (0,0)[(5,1)[50]{}]{} (50,10)[(1,0)[100]{}]{} (150,10)[(5,-1)[50]{}]{} (50,10)[(2,1)[100]{}]{} (50,60)[(2,-1)[100]{}]{} (0,70)[(5,-1)[50]{}]{} (50,60)[(1,0)[100]{}]{} (150,60)[(5,1)[50]{}]{} (100,35) (98,38)[$x$]{} (98,13)[$a$]{} (98,63)[$d$]{} (130,22)[$c$]{} (65,22)[$b$]{}
(200,50) (0,0)[(3,1)[90]{}]{} (90,30)[(1,0)[20]{}]{} (110,30)[(3,-1)[90]{}]{} (0,60)[(3,-1)[90]{}]{} (110,30)[(3,1)[90]{}]{} (100,30) (98,33)[$x$]{} (25,3)[$a$]{} (170,3)[$a$]{} (25,55)[$d$]{} (170,55)[$d$]{} (135,12)[$c$]{} (65,42)[$c$]{} (135,42)[$b$]{} (65,12)[$b$]{}
The main advantage of using scissors consists in a scissors translation $T$, along their lowest base. It may be described as following. Assume that $R_{ac}\/$ is metric transfer from geodesic line $a\/$ to the line $c\/$ generated by Busemann function $\beta_{a(+\infty)}\/$ as in subsection \[trans\]: any point $m \in a\/$ moves to the unique point $m^\prime\, =\, R_{ac}(m) \in c\/$ with $\beta_{a(+\infty)}(m^\prime)\, =\, \beta_{a(+\infty)}(m)$. Similarly maps $R_{cd}$, $R_{db}\/$ and $R_{ba}\/$ are defined as metric transfers of corresponding lines. All maps above do not depend on the chois of Busemann functions $\beta\/$ in classes defined by corresponding ideal points.
*Scissors translation* $T\/$ is a composition $$T:=R_{ba}\circ R_{db} \circ R_{cd} \circ R_{ac} \,\colon a \, \to\, a\, .$$ Since all maps forming the translation $T\/$ above are isometric maps, $T\/$ is an isometry of a geodesic $a\/$ preserving its direction. *The displacement* of the tanslation $T\/$ i.e. difference $\beta_{a(-\infty)}(T(m))\, -\, \beta_{a(-\infty)}(m)\/$ does not depend on the choice of the point $m\in a$. We denote this value as $\delta T$.
The term $\delta T\/$ admits following approach. Let $\beta_{a-}$, $\beta_{a+}$, $\beta_{d-}\/$ and $\beta_{d+}\/$ be four Busemann functions such that there exists points $p \in a\/$ and $q \in d$, satisfying equalities $\beta_{a-}(p)\, =\, \beta_{a+}(p)\, =\, 0\/$ and $\beta_{d-}(q)\, =\, \beta_{d+}(q)\, =\, 0$.
$$\label{15.02.03-1}
\delta T\, =\, \beta_{a-}(x) \, +\, \beta_{a+}(x) \, +\, \beta_{d-}(x) \, +\, \beta_{d+}(x)\, \ge\, 0,$$
where $x\/$ is a center of scissors $\langle a, b, c, d; x \rangle\/$ with translation $T$. Moreover, if none of geodesics $a\/$ or $d\/$ bounds flat strip and if $a\cap d\, =\, \emptyset$, then $\delta T \, >\, 0$
First, note that sums $\beta_{a-}(x) \, +\, \beta_{a+}(x)\/$ and $\beta_{d-}(x) \, +\, \beta_{d+}(x)\/$ are independent on the choice of points $p \in a\/$ and $q \in d$, since, for example, substituting point $p\/$ by $p' \in a$, one adds constants $\beta_{a-}(p')\/$ and $\beta_{a+}(p')$, where $\beta_{a-}(p')\, =\,\, -\, \beta_{a+}(p')$, to functions $\beta_{a-}\/$ and $\beta_{a+}\/$ correspondingly. We have for $\delta T$: $$\delta T\, =\, t'\, -\, t\, ,$$ where $$a(t')\, =\, T(a(t))\, .$$
If $d(s)\, =\, R_{cd}\circ R_{ac} (a(0))$, then for all $t \in {\mathbb R}\/$ $$d(s\, +\,t)\, =\, R_{cd} \circ R_{ac}(a(t))\, .$$ Analogously, if $a(t)\, =\, R_{ba} \circ R_{db}(d(0))$, then for all $s \in {\mathbb R}\/$ $$a(t \, +\, s)\, =\, R_{ba} \circ R_{db}(d(s))\, .$$
Set $p := a(0)\, =\, R_{ac}^{-1}(x)\/$ and $q := d(0)\, =\, R_{db}^{-1}(x)$. Then $$\beta_{a+}(x)\, =\, \beta_{a+}(p)\, =\, 0\, ,$$ $$\beta_{d+}(x)\, =\, \beta_{d+}(q)\, =\, 0$$ and $$T(p)\, =\, R_{ba} \circ R_{db} \circ R_{cd} (x)\, =\, R_{ba} \circ R_{db} (d(s))\, ,$$ where $s= \beta_{d-}(x)$. We have: $$T(p)\, =\, a(\beta_{d-}(x) \, +\, t)\, ,$$ with $t := \beta_{a-}(x)$. Hence displacement of a point $p \in a$, and consequently of any point of $a\/$ equals to
$$\delta T\, =\, \beta_{a-}(x) \, +\, \beta_{d-}(x)\, -\, 0\, =\, \beta_{a-}(x) \, +\, \beta_{a+}(x) \, +\, \beta_{d-}(x) \, +\, \beta_{d+}(x)\, .$$
We have $\delta T \, \ge\, 0$, since $x\/$ lies in intersections $hb(a(+\infty), x) \cap hb(a(-\infty), x)\/$ and $hb(d(+\infty), x) \cap hb(d(-\infty), x)$ of horoballs. When additional conditions are cassumed, we may suppose without lost of generality, that $x\notin a$. In this case $\beta_{d-}(x) \, +\, \beta_{d+}(x) \, \ge\, 0\/$ and $\beta_{a-}(x) \, +\, \beta_{a+}(x) \, >\, 0$.
Shadows {#Sha}
-------
Fix a point $x_0$. *The complete shadow* of a point $x_0\/$ *with respect to point* $y \in \overline{X}\setminus \{x_0\}\/$ is by definition a set $${\operatorname{Shadow}}_y(x_0) := \{z \in \overline{X}|\quad \exists [yz] \quad x_0 \in [yz]\}\, .$$ Assuming of existence is necessary here only if both points $y,z\/$ are infinite: $y,z \in \partial_\infty X$. *Spherical shadow* of a point $x_0\/$ *of radius* $\rho \, >\, 0\/$ relatively point $y\in \overline{X}\/$ is intersection ${\operatorname{Shadow}}_y (x_0, \rho)\/$ of its complete shadow ${\operatorname{Shadow}}_y(x_0)\/$ with the sphere $S(x, \rho)$. In particular, when $\rho\, =\, +\infty\/$ $${\operatorname{Shadow}}_y (x_0, +\infty) := \partial_\infty({\operatorname{Shadow}}_y (x_0)) := {\operatorname{Shadow}}_y(x_0) \cap \partial_\infty X\, .$$
The horosphere $\mathcal{HS}_{y, z}\/$ where $z \in X\/$ is such a point that $\beta_y(z)\, -\, \beta_y(x_0)\, =\, \rho$, will be denoted as $\mathcal{HS}_{y, \rho}$. Here $\beta_y\/$ is arbitrary Busemann function centered in $y\in \partial_\infty X$. Following properties of shadows are obvious.
1. Sets ${\operatorname{Shadow}}_y(x_0)\cup \{x_0\}\/$ and ${\operatorname{Shadow}}_y (x_0, \rho)\/$ are closed in $\overline{X}\/$ for all $\rho \, >\, 0$;
2. ${\operatorname{Shadow}}_y (x_0)\, =\, \overline{\bigcup\limits_{\rho \, >\, 0}{\operatorname{Shadow}}_y (x_0, \rho)}\setminus \{x_0\}$;
3. If $y \in X$, then ${\operatorname{Shadow}}_y (x_0, \rho)\, =\, S(y, |xy|\, +\,\rho) \cap S(x, \rho)\/$ for all $\rho \, >\, 0$;
4. If $y \in \partial_\infty X$, then ${\operatorname{Shadow}}_y (x_0, \rho)\, =\, (\mathcal{HS}_{y, \rho}) \cap S(x, \rho)\/$ for all $\rho \, >\, 0$;
5. \[null\] If $\angle_{x_0}(y,z)\, =\, 0$, then ${\operatorname{Shadow}}_y(x_0)\, =\, {\operatorname{Shadow}}_z(x_0)$.
The statement \[null\] implies consequence:
If the direction of the ray $c\,\colon[0, \infty)\, \to\, X\/$ in a point $x_0\/$ with $c(0)= x_0\/$ and $c(|xy|)\, =\, y$, has unique inverse in $x_0$, then for any two $z', z'' \in {\operatorname{Shadow}}_y(x_0)\/$ $${\operatorname{Shadow}}_{z'}(x_0)\, =\, {\operatorname{Shadow}}_{z''}(x_0)\, .$$
For sufficiently small $\varepsilon \, >\, 0\/$ $\mathcal N_\varepsilon ({\operatorname{Shadow}}_y(x_0, \rho))\/$ will denote *$\varepsilon$-neighbourhood* of spherical shadow ${\operatorname{Shadow}}_y(x_0, \rho)\/$
- in the sphere $S(y, |xy|\, +\,\rho)$, if $y \in X$, or
- in the horosphere $\mathcal{HS}_{y, \rho}$, if $y \in \partial_\infty X$.
\[finite\] For all $x_0 \in X, y \in \overline{X}\setminus\{x_0\}, 0\, <\, \rho \, <\, +\infty\/$ and $\varepsilon \, >\, 0 \/$ there exists $\delta \, >\, 0\/$ such that for any $x_1 \in B(x_0, \delta)$, satisfying equality $|yx_1|\, =\, |yx_0|\/$ (or $\beta_y(x_1)\, =\, \beta_y(x_0)\/$ when $y \in \partial_\infty X$), inclusion $${\operatorname{Shadow}}_y(x_1, \rho) \subset \mathcal N_\varepsilon ({\operatorname{Shadow}}_y(x_0, \rho))$$ holds.
We prove the statement in the case $y \in X$. Situation when $y \in \partial_\infty X\/$ is similar. Assume that for any $\delta \, >\, 0\/$ there exists a point $x_\delta \in B(x_0, \delta) \cap S(y, \, |yx_{0}|)\/$ and a point $z_\delta \in S(y, |yx_0|\, +\,\rho) \setminus \mathcal N_\varepsilon ({\operatorname{Shadow}}_y(x_0, \rho))$, for which $x_\delta \in [yz_\delta]$. We take a sequense $\delta_n \, \to\, 0\/$ and corresponding sequences of points $x_{\delta_{n}}\/$ and $z_{\delta_{n}}$. Then $x_{\delta_{n}} \, \to\, x_0$. Since the space $X\/$ is finitely compact, one may choose converging subsequence from $z_{\delta_{n}}$. We may assume that sequence $z_{\delta_{n}}\/$ converges to a point $z \in S(y, |yx_0|\, +\,\rho)\/$ itself. Parametrizing the segment $\gamma\, =\, [yz]\/$ by arclength, we obtain $$\label{sered}
|x_0 \gamma(|yx_0|)|\, \le\, |x_0 x_{\delta_{n}}| \, +\, |x_{\delta_{n}} \gamma(|yx_0|)\, \le\, \delta_n \, +\, |z_{\delta_{n}} z|.$$ The value on the right hand of vanishes when $n \, \to\, \infty$, hence the constant on the left hand is 0. Consequently, the point $z \in {\operatorname{Shadow}}_y(x_0, \rho)$, and points $z_n\/$ belong to $\mathcal N_\varepsilon ({\operatorname{Shadow}}_y(x_0, \rho))\/$ when $n\/$ is sufficiently large. Contradiction.
Let a set $\mathcal V \subset \partial_\infty X\/$ and a number $K, \varepsilon \, >\, 0\/$ be given. *$(y, K, \varepsilon)$-neighbourhood* of $\mathcal V\/$ is by definition a set $$\mathcal N_{y,\, K,\, \varepsilon}(\mathcal V) := \{\zeta \in \partial_\infty X | \exists \xi \in \mathcal V \quad \zeta \in \mathcal U(\xi, y ,K, \varepsilon)\}\, ,$$ where $$\mathcal U(\xi,x_0,K, \delta) := \{\eta \in \partial_\infty X |\,\, |\,c(K)d(K)|\,\, <\,\varepsilon, \quad c=[y, \xi] ; d\, =\, [y, \eta] \}\, .$$
Following claim in fact is only reformulated statement of the theorem \[finite\] for a case $y \in X$.
\[infty\]
For any point $y \in X\/$ and any $(y, K, \varepsilon)$-neighbourhood $\mathcal N_{y,\, K,\, \varepsilon}(\partial_\infty({\operatorname{Shadow}}_y (x_0)))\/$ of shadow at infinity $\partial_\infty ({\operatorname{Shadow}}_y(x_0))\/$ there exists $\delta\, >\,0$, such that for any $x_1\in B(x_0, \delta)$, with $|yx_1|\, =\, |yx_0|$, we have $$\partial_\infty ({\operatorname{Shadow}}_y(x_1)) \subset \mathcal N_{y,\, K,\, \varepsilon} (\partial_\infty ({\operatorname{Shadow}}_y(x_0)))\, .$$
The geometry of ideal boundary of $CAT(0)$-space {#Z}
------------------------------------------------
\[existence\]
In this subsection we recall some well-known facts from asymptotic geometry of Hadamard spaces. We refer the reader to [@BH] for more detailed considerations.
First of all, we note that given a point $y_0 \in X$, an ideal point $\xi \in \partial_\infty X\/$ and a number $t \, >\, 0$, a family of sets $$\label{sphebaza}
\mathcal B_{y_{0},\, \xi} := \{ \mathcal U_{\delta, t}(y_0,\, \xi)|\quad \delta,t \, >\, 0\}$$ form a basis of neighbourhoods of point $\xi\/$ in the cone topology in $\partial_\infty X$. Here $$\mathcal U_{\delta, t}(y_0,\, \xi) := \{ \eta \in \partial_\infty X |\quad |c(t) d(t)|\, <\,\delta \}\, ,$$ and $c, d\,\colon [0, +\infty) \, \to\, X\/$ are rays emanating from $y_0\/$ in $\xi, \eta \in \partial_\infty X\/$ directions correspondingly.
The space $X\/$ instead of the cone topology on ideal boundary $\partial_\infty X\/$ has one induced by so called *angle metric*. For $\xi, \eta \in \partial_\infty X\/$ angle distance is by definition equal to $$\angle(\xi, \eta) := \sup \{\angle_x (\xi, \eta)| \quad x \in X\}\, .$$ The interior metric on $\partial_\infty X\/$ associated with angle metric is called *Tits metric* and denoted as $Td$. The two metrics on $\partial_\infty X\/$ are equivalent in the sence that they induce the same topology on $\partial_\infty X$. We will write $\partial_T X\/$ for denoting the ideal boundary equiped with Tits metric. We need following two propositions.
Angle metric considered as a function $(\xi, \eta) \, \to\, \angle (\xi, \eta)\/$ is lower semicontinuous with respect to cone topology: for all $\varepsilon \, >\, 0\/$ there exist neighbourhoods $\mathcal U\/$ of point $\xi\/$ and $\mathcal V\/$ of point $\eta$, such that for all $\xi\,' \in \mathcal U\/$ è $\eta\,' \in \mathcal V\/$ inequality $$\angle (\xi\,', \eta\,') \, >\, \angle (\xi, \eta)\, -\, \varepsilon$$ holds.
As a corollary, the Tits metric is also lower semicontinuous in the cone topology.
Let $X\/$ be a proper $CAT(0)$-space and let $\xi_0\/$ and $\xi_1\/$ be distinct points of $\partial_\infty X$.
1. If $Td(\xi_0, \xi_1 \, >\, \pi\/$ then there is a geodesic $c\,\colon {\mathbb R}\, \to\, X\/$ with $c(+\infty)\, =\, \xi_0\/$ and $c(-\infty)\, =\, \xi_1$,
2. If there is no geodesic $c\,\colon {\mathbb R}\, \to\, X\/$ with $c(+\infty)\, =\, \xi_0\/$ and $c(-\infty)\, =\, \xi_1$, then $Td(\xi_0, \xi_1)\, =\, \angle(\xi_0, \xi_1)\/$ and there is a geodesic segment in $\partial_T X\/$ joining $\xi_0\/$ and $\xi_1$,
3. If $c\,\colon {\mathbb R}\, \to\, X\/$ is a geodesic, then $Td(c(-\infty), c(+\infty)) \, \ge\, \pi\/$ with equality iff $c\/$ bounds a flat half-plane,
4. If the diameter of the Tits boundary $\partial_TX\/$ is $\pi$, then every geodesic line in $X\/$ bounds a flat half-plane.
Points with uniqueness of inverse direction
-------------------------------------------
Directions $\xi, \eta \in \Sigma_xX\/$ are called *mutually inverse* if $\angle_x(\xi, \eta)\, =\, \pi$. In the case of geodesically complete $CAT(0)$-space two directions $\xi, \eta\in \Sigma_xX\/$ are mutually inverse iff there exists a geodesic throw $x\/$ whose positive direction in $x\/$ is $\xi\/$ and negative one is $\eta$.
Given geodesic $a\/$ in $CAT(0)$-space $X\/$ we denote $\omega^+\,(a)\/$ a set of points $x \in a\/$ with the property that the positive direction $\xi \in \Sigma_xX:\, [xa(+\infty)] \in \xi\/$ of $a\/$ has more than one inverse direction. Similarly $\omega^-(a)\/$ be the set of points where the negative direction $\eta \in \Sigma_xX:\, [xa(-\infty)] \in \eta\/$ of $a\/$ has more than one inverse.
\[singpoints\] Let $a\/$ be a geodesic in a geodesically complete locally compact $CAT(0)$-space $X$. Then sets $\omega^+(a)\/$ and $\omega^-(a)\/$ are at most countable.
For $\phi \, >\, 0\/$ consider set $\Omega^+_\phi(a) \subset \omega^+(a)\/$ defined as following. $x \in \Omega^+_\phi(a)\/$ iff there exists a direction $\zeta \in \Sigma_xX\/$ inverse to direction of the ray $[xa(+\infty)]$, such that $\angle_x(\zeta, a(-\infty)) \, >\, \phi$. We will show that intersection of $\Omega^+_\phi(a)\/$ with any segment $[xy]\subset a\/$ is finite.
Really, assume that there exists a segment $[xy]\subset a\/$ containing infinite sequence $\{a(t_n)\}_{n=1}^{\infty}\subset \Omega^+_\phi(a)$. We may assume that $x\, =\, a(0)\/$ and $y\, =\, a(-L)$, where $L\, =\, |xy|$. For a point $a(t_n)\/$ we set $z_n \in S(x, 2L)\/$ a point such that $a(t_n) \in [xz_n]\/$ and $\angle_{a(t_n) }(a(-\infty), z_n) \, >\, \phi$. Then for $n \ne k\/$ we have $|z_n z_k| \, >\, 2L\sin\frac{\phi}{2}\/$ and the sequence $\{z_n\}_{n=1}^{\infty}\/$ does not contain any fundamental subsequence, contradicting finitely compactness of $X$.
Because of $\omega^+(a)\, =\, \bigcup\limits_{\phi \, >\, 0}\Omega^+_\phi(a)\/$ and $\Omega^+_{\phi}(a) \subset \Omega^+_{\psi}(a)\/$ for $\psi \, <\, \phi$, we have the claim for $\omega^+(a)$. Consideration of $\omega^-(a)\/$ is similar.
Otsu and Tanoue in [@OT] introduce the following notion. For $\delta \, >\, 0\/$ and $y \in X\/$ point $x\/$ is called *$\delta$-branched point of $y$* if the diameter of the set $\{v\in \Sigma_xX| \angle_x(v_{xy}v)=\pi\}\/$ is not smaller then $\delta$. Here $v_{xy}\in \Sigma_xX\/$ is the direction of the segment $[xy]$. Such a notion has evident extension on the case $y \in \overline{X}$. In fact, every set $\Omega^+_\phi(a)\/$ is contained in a set of $\phi$-branching points of $a(+\infty)\/$ lying in $a$, which also has finite intersection with every segment $[xy]\subset a$.
Every point of a geodesic $a\subset X\/$ may occur to be a branching point. For example, exclude from Euclidean plane the interior domain bounded by a parabola and attach a flat half-plane on its place. We get a $CAT(0)$-space $X$. The parabola $a\/$ dividing two flat domains is its geodesic and every its point is a point of branching of geodesics in both directions. However we have $\omega^+(a)\, =\, \omega^-(a)\, =\, \emptyset$, since angle between any two branches of $a\/$ in its opposite direction starting from the same point vanishes.
Sets $\omega^+(a)\/$ and $\omega^-(a)\/$ may occur to be dence in $a$. To see this one may take a convex continuous natural parametrised curve on which every point has a positive and negative semitangents, not opposite to each other in rational points, instead of the parabola in previous example.
Existence of scissors
---------------------
We prove the existence theorem for scissors in this subsection.
\[15.02.03-4\] Let $a\,\colon (-\infty , +\infty) \, \to\, X\/$ be a geodesic of strictly rank one. Let $x_0 \in a\setminus (\omega^+(a)\cup\omega^-(a))\/$ be a point where both directions of $a\/$ have unique inverse. Then there exists a geodesic $a'\/$ with $a'(0)= x_0\/$ and $\angle_{x_{0}}(a(+\infty), a'(+\infty))\, =\, 0\/$ with following property. For every neighbourhood $\mathcal U\/$ of a triple $$(a'(+\infty), a'(-\infty), x_0) \in \partial_\infty X \times \partial_\infty X \times X$$ there exists a triple $(\xi, \eta, x) \in \mathcal U\/$ with $x \ne x_0\/$ and geodesics $b'\, =\, [a'(-\infty) \xi], c'=[\eta a'(+\infty)]\/$ and $d'=[\eta\xi]$, forming scissors $\langle a',b',c',d'; x\rangle$.
We note that by the condition on the rank of $a$, any geodesic $a'\/$ with $a'(+\infty) \in \partial_\infty({\operatorname{Shadow}}_{a(-\infty)}(x_0))\/$ or $a'(-\infty) \in \partial_\infty({\operatorname{Shadow}}_{a(+\infty)}(x_0))\/$ has rank $1$. First we show that there exists scissors with lowest base $a\/$ and center $x\/$ arbitrary closed to $x_0$. In view of the first remark, the considerations will be applicable also for a geodesic $a'\/$ passing throw $x_0\/$ in the same direction.
Take points $y'\, =\, a(-\rho)\/$ and $y''\, =\, a(\rho)$, where $\rho \, >\, 0$. We have $$\partial_\infty({\operatorname{Shadow}}_{a(-\infty)}(x_0))\, =\, \partial_\infty({\operatorname{Shadow}}_{y'}(x_0))$$ and $$\partial_\infty({\operatorname{Shadow}}_{a(+\infty)}(x_0))\, =\, \partial_\infty({\operatorname{Shadow}}_{y''}(x_0))\, .$$
Low semicontinuous funtion $$Td\,\colon \partial_\infty X \times \partial_\infty X \, \to\, {\mathbb R}_+\cup \{+\infty\}$$ attains its minimum on the compact set $$Q=\partial_\infty({\operatorname{Shadow}}_{y'}(x_0)) \times \partial_\infty({\operatorname{Shadow}}_{y''}(x_0))\, ,$$ and inequality $$\label{mintd}
{\rm min}(Td)|_Q \, >\, \pi$$ holds as a consequence of the condition on the rank of $a$. Moreover, there exists neighbourhoods $$\mathcal N\,':=\mathcal N_{y', K, \varepsilon}(\partial_\infty({\operatorname{Shadow}}_{y'}(x_0)))$$ and $$\mathcal N\,'':=\mathcal N_{y'', K, \varepsilon}(\partial_\infty({\operatorname{Shadow}}_{y''}(x_0)))$$ with some $K \, >\, \rho\/$ and $\varepsilon \, >\, 0$, for which $$\label{inftd}
\inf \{Td(\xi, \eta)|\quad (\xi, \eta) \in \mathcal N'\times \mathcal N''\} \, >\, \pi.$$
Choose $\delta_1$-neighbourhood $B(x_0, \delta_1)\/$ of a point $x_0\/$ defined by Corollary \[infty\] relatively $\mathcal N_{y', K, \varepsilon/2}(\partial({\operatorname{Shadow}}_{y')}(x_0, \rho)\/$ and $\mathcal N_{y'', K, \varepsilon/2}(\partial({\operatorname{Shadow}}_{y'')}(x_0, \rho)$.
Let also $\mathcal N_{\varepsilon/2}(\partial({\operatorname{Shadow}}_{a(-\infty)}(x_0, \rho)))\/$ and $\mathcal N_{\varepsilon/2}(\partial({\operatorname{Shadow}}_{a(+\infty)}(x_0, \rho)))\/$ be $\varepsilon/2$-neighbourhoods of boundaries of shadows of point $x_0\/$ relatively points $a(-\infty)\/$ and $a(+\infty)$.
From Theorem \[finite\] there exists $\delta_2$-neighbourhood $B(x_0, \delta_2)\/$ of point $x_0$, such that for every $x' \in B(x_0, \delta_2)\/$ inclusions $${\operatorname{Shadow}}_{a(-\infty)}(x', \rho) \subset \mathcal N_{\varepsilon/2}(\partial({\operatorname{Shadow}}_{a(-\infty)}(x_0, \rho)))$$ and $${\operatorname{Shadow}}_{a(+\infty)}(x', \rho) \subset \mathcal N_{\varepsilon/2}(\partial({\operatorname{Shadow}}_{a(+\infty)}(x_0, \rho)))$$ hold.
Set $\delta_0 := \min \{\delta_1, \delta_2\}$. Then for any point $x \in \mathcal U_{\delta_{0}}(x_0)\/$ and geodesics $b\/$ and $c\/$ satisfying conditions
- $b(0)\, =\, c(0)\, =\, x$,
- $b(-\infty)\, =\, a(-\infty)\/$ and
- $c(+\infty)\, =\, a(+\infty)$
we get $$\label{bincl}
b(\rho) \in \mathcal N_{\varepsilon/2}(\partial({\operatorname{Shadow}}_{a(-\infty)}(x_0, \rho)))$$ and $$c(-\rho) \in \mathcal N_{\varepsilon/2}(\partial({\operatorname{Shadow}}_{a(+\infty)}(x_0, \rho)))$$
We will show that $$\label{b+infty}
b(+\infty) \in \mathcal N\,'$$ and $$\label{c-infty}
c(-\infty) \in \mathcal N\,''.$$
Given geodesic ray $\gamma\, =\, [y'b(+\infty)]\/$ with length parametrisation $$\gamma\,\colon [0, +\infty) \, \to\, X$$ and geodesic line $a'\/$ passing throw $x_0\/$ such that $a'(+\infty) \in \partial_\infty({\operatorname{Shadow}}_{y'}(x_0))\/$ we have $$|\gamma(2\rho)a'(\rho)|\, \le\, |\gamma(2\rho)b(\rho)| \, +\,|b(\rho)a'(\rho)|\, .$$
First item has estimation $$|\gamma(2\rho)b(\rho)|\, \le\,|\gamma(0)b(-\rho)|\, =\, |a(-\rho)b(-\rho)|\, \le\, |a(0)b(0)| \, <\, \frac\varepsilon2\, .$$
Because of one may choose the geodesic $a'\/$ such that second item satisfies to inequality $$|b(\rho)a'(\rho)|\, <\,\frac\varepsilon2\, .$$ Finally we get $$|\gamma(2\rho)a'(\rho)|\, <\,\varepsilon\, ,$$ proving the inclusion . Inclusion is similar.
So because of (\[inftd\]) there exists a geodesic $d$ in $X$ connecting points $c(-\infty)$ and $b(+\infty)$ such that we have scissors $\langle a,b,c,d; x\rangle$.
Now we take a sequences $\delta_n \, \to\, 0\/$ of scales and $\langle a_n, b_n, c_n, d_n; x_n \rangle\/$ of corresponding scissors for which $|x_0x_n| \, <\, \delta_n$. Choose as $\xi,\eta \in \partial_\infty X\/$ limit points of sequences $b_n(+\infty)\/$ and $c_n(-\infty)\/$ correspondingly. Then $\xi \in \partial_\infty({\operatorname{Shadow}}_{a(-\infty)}(x_0))\/$ and $\eta \in \partial_\infty({\operatorname{Shadow}}_{a(+\infty)}(x_0))$. The points above may be connected by a geodesic $a'=[\eta\xi]\/$ in $X\/$ such that $a'(0)=x_0\/$ (cf. inequality ). For any $\delta \, >\, 0\/$ there exists scissors $\langle a', b', c', d';x\rangle\/$ with lowest base $a'\/$ and $|xx_0| \, <\, \delta$. Note that the point $x\/$ may always be different of $x_0\/$ and not belong to $a'$.
It is left to show that such scissors may be chosen with $b'(+\infty)\/$ arbitrary closed to $\xi$, and $c'(-\infty)\/$ arbitrary closed to $\eta\/$ in the sence of cone topology on $\partial_\infty X$. It may be done with applying the same method as above but with neighborhoods of points $a'(\pm \infty)\/$ instead of neighborhoods of shadows at infinity of point $x_0$. Construction of a line $a'\/$ garantees that sets $$C_{-\infty}(B(a'(K),\varepsilon))\, =\, \bigcup\limits_{y\in B(a'(K),\varepsilon)}[a'(-\infty)y]$$ and $$C_{+\infty}(B(a'(-K),\varepsilon))\, =\, \bigcup\limits_{z\in B(a'(-K),\varepsilon)}[za'(+\infty)]$$ has nonempty intersection for any $\varepsilon, K\, >\,0$, and moreover $$\begin{aligned}
C_{-\infty}&(\mathcal N_\varepsilon (a'(K))) \cap C_{+\infty}(\mathcal N_\varepsilon (a'(-K)))\\
\cap (X \setminus ({\operatorname{Shadow}}_{y'} &(x_0) \cup {\operatorname{Shadow}}_{y''}(x_0)\cup \{ x_0\}) \cap B(x_0, \varepsilon)) \ne \emptyset.\end{aligned}$$ This fact provides for necessary construction.
\[aa’\] Note that geodesics $a\/$ and $a'\/$ are connected by the asymptotic chain $a\, =\, a_0, a_1, a_2\, =\, a'$, where the geodesic $a_1\/$ is obtained as union of rays $[x_0 a(-\infty)]\/$ and $[x_0 a'(+\infty)]$.
Continuity of the displacement function
---------------------------------------
The goal of this subsection is a theorem \[contin\] which is a theorem of continuity for a displacement function $\delta\/$ defined by the scissors translations along given geodesic $a\/$ as a function defined on appropriate subset of $\partial_\infty X \times \partial_\infty X \times X$. First we need some estimation of distance between projections.
We say that a point $x \in X\/$ *projects* to a point $x_0 \in a$, if $x_0\/$ is nearest to $x\/$ point of a geodesic $a$. It is called a *projection* of $x$. Every point $x\in X\/$ has unique projection on any given geodesic $a$.
\[vareps\] Let $d\,\colon~(-\infty,\,+\infty)~\, \to\,~X\/$ be a strongly rank one geodesic, $x \notin d\/$ be a point in $X\/$ and $x_1\/$ — its projection onto $d$. Then for any $\varepsilon \, >\, 0\/$ points $\xi\, =\, d(+\infty)\/$ and $\eta\, =\, d(-\infty) \in \partial_\infty X\/$ have neighbourhoods $\mathcal U_+\/$ and $\mathcal U_-\/$ correspondingly, such that if a geodesic $d'\/$ connects points $d'(+\infty) \in \mathcal U_+\/$ and $d'(-\infty) \in \mathcal U_-$, and $x'_1 \in d'\/$ is a projection of the point $x\/$ onto $d'$, then $|x_1 x'_1| \, <\, \varepsilon$.
The first step of its proof is the next simple lemma.
\[tech\] Let $d\/$ be a geodesic in the $CAT(0)$-space $X\/$ and $x'\/$ be the projection of the point $x\/$ onto $d$. Then for any point $y \in d\/$ $$|\, x'y|\, \le\, \sqrt{|\, xy|^2\, -\, |\, xx'|^2}\, .$$
Assume that $|\, x'y| \, >\, \sqrt{|\, xy|^2\, -\, |\, xx'|^2}.\/$ Consider a comparison triangle $\overline{x} \overline{x}' \overline{y}\/$ of a triangle $xx'y$. We have by assumption $\angle_{\overline{x}'}(\overline{x},\, \overline{y}) \, <\, \pi/2$. Hence there exists a point $\overline{m} \in \overline{x}' \overline{y}$, with $|\, \overline{x}\, \overline{m}| \, <\, |\, \overline{x}\, \overline{x}'|$. Its corresponding point $m\/$ of triangle $xx'y\/$ satisfies inequality $|\, xm|\, \le\, |\, \overline{x}\, \overline{m}| \, <\, |\, xx'|$. The contradiction proves the claim.
Let $d\,\colon~(-\infty,\,+\infty)~\, \to\,~X\/$ be given geodesic and $\sigma \, >\, 0\/$ an arbitrary number. We denote $HS_{\xi}(t, \rho)\/$ the intersection of a horosphere $\mathcal{HS}_{\xi, d(t)}\/$ centered at a point $\xi \in \partial_\infty X\/$ with $B(d(t), \rho)$.
We need the fact that the cone topology of $CAT(0)$-space $X\/$ has a base consisting of sets $$\mathcal U(\xi,x_0,K, \delta) := \{\eta \in \partial_\infty X |\,\, |\,c(K)c_*(K)|\,\, <\,\delta \quad c=[x_0, \xi] ; c_*\, =\, [x_0, \eta] \}$$
Choose points $x_{-K}\, =\, d(-K)\/$ and $x_K\, =\, d(K)$, where $K\, >\,0\/$ is a sufficiently large nubmer. In particular one may assume for $K\/$ to satisfy the following condition: if $y\in B(x_{-K},\, \sigma) $, then $[y d(+\infty)] \cap HS_{d(+\infty)}(K,\, \sigma/4) \ne \emptyset$, and if $z\in B(x_K, \, \sigma)$, then $[z d(-\infty)] \cap HS_{d(-\infty)}(-K,\, \sigma/4) \ne \emptyset$. This condition can be accomplished because $d\/$ has strictly rank one.
Denote as $$\mathcal U_{+\infty}(K, \sigma) := \left\{ \xi \in \partial_\infty X |\, \forall y \in B(x_{-K}, \sigma) | \quad [y \xi] \cap HS_{d(+\infty)}\left(K,\, \frac{\sigma}{2}\right) \ne \emptyset\right\}$$ and $$\mathcal U_{-\infty}(K, \sigma) := \left\{ \eta \in \partial_\infty X |\, \forall z \in B(x_K,\, \sigma) | \quad [z \eta] \cap HS_{d(-\infty)}\left(-K,\, \frac{\sigma}{2}\right) \ne \emptyset\right\}$$ sets of those ideal points which serve as centers of projections onto corresponding horospheres, such that a ball $B(x_{-K},\, \sigma)\/$ moves inside a ball $B(x_K,\,\sigma/2)$, and a ball $B(x_K, \sigma)\/$ moves inside a ball $B(x_{-K}, \sigma/2)$. Sets $\mathcal U_{\pm\infty}(K, \sigma)\/$ are nonempty, since $d(+\infty) \in \mathcal U_{+\infty}(K, \sigma)\/$ and $d(-\infty) \in \mathcal U_{-\infty}(K, \sigma)$. Furthermore, points $\gamma(\pm \infty)\/$ are interior points of sets $\mathcal U_{\pm\infty}(K, \sigma)$. This is a consequence of the definition of the cone topology on $\partial_\infty X\/$ and local compactness of $X$.
For an arbitrary point $x \notin d\/$ let $x_1\/$ be its projection onto $d$. Fix $\varepsilon \, >\, 0$. One may assume that $\varepsilon \, <\, \min \{|\, x \, x_1|, \, 1 \}$, besause it is sufficient to make all considerations for arbitrary small $\varepsilon$. Set $$\sigma := \frac{\varepsilon^2}{9|\, x\, x_1|}\, .$$ For $K\/$ as above we denote $\mathcal U_\pm\subset \partial_\infty X\/$ neighbourhoods of ideal points $d(\pm \infty)$, contained correspondingly in $\mathcal U_{\pm\infty}(K, \sigma)$.
We show that neighbourhoods $\mathcal U_\pm\/$ are the ones satisfying the claim of the theorem: if the geodesic $d'\/$ has ends $d'(\pm \infty) \in \mathcal U_\pm\/$ and $x'_1\in d'\/$ is a projection of a point $x\/$ onto $d'$, then $|\, x_1\, x'_1| \, <\, \varepsilon$. For this we show that $d' \cap \overline{HS}_{d(+\infty)}(K,\, \sigma/2) \ne \emptyset\/$ and $d' \cap \overline{HS}_{d(-\infty)}(-K,\, \sigma/2) \ne \emptyset\/$ at first.
Indeed, let $\pi_+ \,\colon HS_{d(-\infty)}(-K,\, \sigma) \, \to\, HS_{d(+\infty)}(K, \, \sigma/2)\/$ be the projection map centered in $\gamma'(+\infty)\/$ and $\pi_- \,\colon HS_{\gamma(+\infty)}(K,\, \sigma) \, \to\, HS_{\gamma(-\infty)}(-K, \, \sigma/2)\/$ be analogous projection centered in $d'(-\infty)$. Then by the condition on the rank of $d'$, the composition $\pi_- \circ \pi_+\, : HS_{d(-\infty)}(-K, \sigma) \, \to\, HS_{d(-\infty)}(-K, \sigma/2)\/$ is a continuous map which is contraction operator on $HS_{d(-\infty)}(-K, \sigma)$. Since the closure $\overline{HS}_{d(-\infty)}(-K, \, \sigma/2)\/$ is compact, the contraction operator $\pi_- \circ \pi_+\/$ has unique fixed point in $\overline{HS}_{d(-\infty)}(-K, \, \sigma/2)$. Let such a point $z\/$ have an image $y\, =\, \pi_+(z)$. Then $y \in [z \, d'(+\infty)]\/$ and $z \in [y \, d'(- \infty)]$. The union of rays $[z \, d'(+\infty)] \ni y\/$ and $[y \, d'(- \infty)] \ni z\/$ gives us precisely a geodesic $d'$, since it does not bound any flat strip.
It is only left to estimate the distance $|\, x'_1\, x_1|\/$ to complete the proof. Since $d'\/$ passes throw interior points of balls $B(-K,\, \sigma)\/$ and $B(K,\, \sigma)$, there exists a point $m \in d'$, with $|\, m\, x_1| \, <\, \sigma$. Then we have: $$\label{first}
|\, x'_1\, x_1 |\, \le\, |\, x'_1\, m| \, +\, |m \, x_1|\, \le\, \sqrt{|\, x\, m|^2\, -\, |\, x\, x'_1|^2} \, +\, \sigma.$$
By the triangle inequality we have for point $m\/$ $$\label{m}
|\, x\, m| \, <\, |\, x \, x_1| \, +\, \sigma,$$ and since $x_1\/$ is the nearest to $x\/$ point of $d\/$ and the distance of $x'_1\/$ to $d\/$ is less then $\sigma$, the estimate $$\label{x1}
|\, x \, x'_1| \, >\, |\, x \, x_1|\, -\, \sigma$$ holds.
Substituting inequalities and to , we get $$\begin{aligned}
|\, x'_1 \, x_1| \, <\, \sqrt{(|\, x\, x_1| \, +\, \sigma)^2\, -\, (|\, x\, x_1|\, -\, \sigma)^2} \, +\, \sigma\, &=\, 2\sqrt{\sigma|\, x \, x_1|} \, +\, \sigma\\
= \sqrt{\sigma}(2 \sqrt{|\, x \, x_1|} \, +\, \sqrt{\sigma}) \, <\, 3 \sqrt{\sigma |\, x x_1|}\, &=\, \varepsilon.\end{aligned}$$
Fix a strictly rank one geodesic $a\,\colon {\mathbb R}\, \to\, X$. Set $Z(a)\subset \partial_\infty X \times \partial_\infty X \times X\/$ to be a subset consisting of all triples $(\xi, \eta, x) \in \partial_\infty X \times \partial_\infty X \times X\/$ such that there exists scissors $\langle a,b,c,d; x\rangle\/$ with $b(+\infty)\, =\, \xi\/$ and $c(-\infty)\, =\, \eta$.
\[contin\] The displacement function $\delta\/$ is continuous on the set $Z(a)$.
We use the equality . Fix a triple $(\xi_0, \eta_0, x_0) \in Z(a)$. It means that $x_0\/$ is the center of scissors $\langle a, b_0, c_0, d_0; x_0 \rangle$, where $b_0(+\infty)\, =\, \xi_0\/$ and $c_0(-\infty)\, =\, \eta_0$.
Fix an arbitrary $\varepsilon \, >\, 0$. Then first, by continuity of Busemann functions $\beta_{a-}\/$ and $\beta_{a\, +\,}\/$ there exists $\sigma_1\/$ such that if for a point $x'\in X\/$ inequality $|\, x_0x' | \, <\, \sigma_1\/$ holds, then $$\label{15.02.03-2}
|\, \beta_{a\, +\,}(x') \, +\, \beta_{a-}(x')\, -\, \beta_{a\, +\,}(x_0)\, -\, \beta_{a-}(x_0) | \, <\,
\varepsilon/2.$$
Using the theorem \[vareps\] we choose neighbourhoods $\mathcal U_+\/$ and $\mathcal U_-\/$ of points $d_0(+\infty)=\xi_0\/$ and $d_0(-\infty)=\eta_0 \in \partial_\infty X\/$ with the condition: for any geodesic $d\,'\/$ such that $d\,'(-\infty) \in \mathcal U_-\/$ and $d\,'(+\infty) \in \mathcal U_+$, the projection $x_1\/$ of $x_0\/$ onto $d_0\/$ and projection $x'_1\/$ of $x_0\/$ onto $d\,'\/$ satisfy inequality $|\, x_1x'_1 | \, <\, \varepsilon/4$. For these neighbourhoods there exists $\sigma_2\/$ such that if the point $x'\/$ satisfy conditions
a\) there exists a ray $[x'\xi']$, inverse to the ray $[x' a(-\infty)]\/$ with $\xi' \in \mathcal U_+$,
b\) there exists a ray $[x'\eta']$, inverse to the ray $[x' a(+\infty)]\/$ with $\eta' \in \mathcal U_-$,
c\) $d\,'\,\colon (-\infty, +\infty) \, \to\, X\/$ is a geodesic connecting points $\xi'\/$ and $\eta'$, and
d\) $|\, x_0x' | \, <\, \sigma_2$,\
then $$|\, \beta_{d\,'-}(x')\, -\, \beta_{d_{0}-}(x_0)| \, <\, \varepsilon/4$$ and $$|\, \beta_{d\,'+}(x')\, -\, \beta_{d_{0}\, +\,}(x_0)| \, <\, \varepsilon/4\, .$$
Here $\beta_{d'-}\/$ and $\beta_{d'\, +\,}\/$ are Busemann functions corresponding to points $d'(-\infty)\/$ and $d'(+\infty)\/$ and satisfying the condition $\beta_{d'-}(x'_1)\, =\, \beta_{d'\, +\,}(x'_1)\, =\, 0$, and $\beta_{d_{0}-}$, $\beta_{d_{0}\, +\,}\/$ — analogous Busemann functions for the geodesic $d_0$. As a result, for $\sigma\, =\, \min \{\sigma_1 , \sigma_ 2\}\/$ and for all $x'\/$ such that $|\, x_0x' | \, <\, \sigma\/$ and conditions a), b), c) above holds, we have an estimation: $$\label{15.02.03-3}
|\, \beta_{d'-}(x') \, +\, \beta_{d'\, +\,}(x')\, -\, \beta_{d_{0}-}(x_0)\, -\, \beta_{d_{0}\, +\,}(x_0)|\, \le\, \varepsilon/2.$$
Taking $\mathcal N\, =\, \mathcal U_+ \times \mathcal U_- \times B(x_0, \sigma)\/$ as neighbourhood of triple $(\xi_0, \eta_0, x_0)\/$ and compairing inequalities and with representation , we get a condition of the continuity for the function $\delta\/$ as function of triple $(\xi, \eta, x)\/$ in the point $(\xi_0, \eta_0, x_0)\in Z(a)$.
Let $x_0\in a\/$ be an arbitrary point of a geodesic $a\/$ of strictly rank one. Then $\delta(\xi, \eta, x)\, \to\, 0\/$ when $(\xi, \eta, x)\, =\, (b(+\infty), c(-\infty), x)\/$ tends to $(a(+\infty), a(-\infty), x_0)\/$ in the sence of topology in the set $Z(a)\/$ inherited from $\partial_\infty X\times \partial_\infty X\times X$.
Let $A\/$ be a set of lines $a'\/$ which passes throw $x_0\/$ such that $a'(+\infty) \in {\operatorname{Shadow}}_{a(-\infty)} (x_0, +\infty)\/$ and $a'(-\infty) \in {\operatorname{Shadow}}_{a(+\infty)} (x_0, +\infty)$. For $a' \in A\/$ we define “*closed*” scissors $\langle a', a', a', a'; x_0 \rangle\/$ as a system of four items of a line $a'\/$ and a center $x_0$. They also have a transformation map $T\, =\, {\rm id}_{a'}$, for which $\delta T\, =\, 0\/$ and a function $\delta\/$ remains continuous when it is defined on $Z(a) \cup \{(a'(+\infty), a'(-\infty), x)|~x\in a',\, a'\in A\}\/$ by equality $\delta(a'(+\infty), a'(-\infty), x)\, =\, 0\/$ for $x \in a'$.
Following evident corollary will be the key point in remainder of the proof.
\[small\] Let $a\/$ be the geodesic of strictly rank one. Then there exists a point $x_0\in a$, a geodesic $a'$, connected with $a\/$ by the asymptotic chain, and $\Delta \, >\, 0$, such that for every $\varepsilon \in (0, \Delta)\/$ there exist scissors $\langle a', b, c, d; x \rangle$, with displacement of scissors translation $T\/$ equal to $\delta T\, =\, \varepsilon$.
Recovery of the metric on geodesic of strictly rank one
-------------------------------------------------------
The goal of this paragraph is to show that the metric of an arbitrary geodesic of strictly rank one is restorable from the diagonal tube $V$.
Let $x \notin a\/$ and $\xi, \eta \in \partial_\infty X$. Then the relation $V\/$ allows to detect whether the triple $(\xi, \eta, x)\/$ belongs to a set $Z(a)$.
$(\xi, \eta, x) \in Z(a)\/$ if and only if there exists $r$-sequences $\{x_z\}_{z\in {\mathbb Z}}\subset a$, $\{ u_z \}_{z \in {\mathbb Z}}$, $\{ v_z \}_{z \in {\mathbb Z}}\/$ and $\{ w_z \}_{z \in {\mathbb Z}}$, for which $u_0\, =\, v_0\, =\, x$, and such that their limiting ponts in $\partial_\infty X\/$ satisfy equalities $u_{-\infty}\, =\, x_{-\infty}$, $v_{+\infty}\, =\, x_{+\infty}$, $w_{-\infty}\, =\, v_{-\infty}=\eta\/$ and $w_{+\infty}\, =\, u_{+\infty}=\xi$. Given any four $r$-sequences $\{ x_z \}_{z \in {\mathbb Z}}$, $\{ u_z \}_{z \in {\mathbb Z}}$, $\{ v_z \}_{z \in {\mathbb Z}}\/$ and $\{ w_z \}_{z \in {\mathbb Z}}$, relation $V\/$ detects, whether marked equalities hold or not. Also, $V\/$ lets to reveal such four $r$-sequences if they exist.
Given scissors $\langle a,\, b,\, c,\, d;\, x \rangle$, relation $V\/$ defines an image $T(m)\in a\/$ of any point $m \in a\/$ in scissors translation $T$.
We use the fact that all horospheres are defined by $V\/$ and serves as level sets of Busemann functions. Hence $V\/$ defines a point $m_1\, =\, R_{ac}(m)\/$ as $\mathcal{HS}_{a(+\infty),\, m} \cap c$, a point $m_2\, =\, R_{cd}(m_1)\/$ as $\mathcal{HS}_{d(-\infty),\, m_1} \cap d$, a point $m_3\, =\, R_{db}(m_2)\/$ as $\mathcal{HS}_{d(+\infty),\, m_2} \cap b\/$ and finally, a point $T(m)\, =\, m_4\, =\, R_{ba}(m_3)\/$ as $\mathcal{HS}_{a(-\infty),\, m_3} \cap a$.
\[sdvig\] Let $\langle a,\, b,\, c,\, d;\, x \rangle\/$ be scissors with displacement of translation $\delta T$. Then $V\/$ allows to display the value $\delta T$.
We have: $$\delta T\, =\, \frac 1n d(x_0, T^n(x_0))\, =\, \lim\limits_{n \, \to\, +\infty} \frac 1n d(x_0, T^n(x_0))\, =\, \lim\limits_{n \, \to\, +\infty} \frac 1n [d(x_0, T^n(x_0))],$$ where $[\, t]\/$ is integral part of the number $t$. Since every item of the sequence $a_n\, =\, [d(\, x_0, T^n(x_0)\, )]\/$ is definable by $V$, hence the value $\delta T\/$ is.
We are ready now to prove the following theorem.
\[hyperbolic\] Let metric space $(X,d)\/$ and trial metric $d'\/$ on $X\/$ be as in Theorem \[tube\]. Assume that $a \,\colon (-\infty , \, +\,\infty) \, \to\, X\/$ is a geodesic of strictly rank one in metric $d$. Then $a\/$ is of strictly rank one in metric $d'\/$ as well, and for all $t_1, t_2 \in {\mathbb R}\/$ $$\label{dd2}
d'(c(t_1), c(t_2))\, =\, d(c(t_1), c(t_2)).$$
We may think the assertion on the rank of $a\/$ as already proved in Proposition \[rank\], since the property of geodesic to have rank one is accomplished when any its $r$-sequence is of rank one.
According Corollary \[small\] there exists $x_0 \in a$, geodesic $a'\/$ and $\Delta\, >\,0\/$ such that for any $q \in {\mathbb N}$, satisfying $1/q \, <\, \Delta$, there exist scissors $\langle a',\, b,\, c,\, d;\, x \rangle\/$ with displacement $\delta T\, =\, 1/q$. By lemma \[sdvig\] the relation $V\/$ lets to detect, whether the value of displacement equal $1/q$. Take scissors with this value of displacement, which is independent on the choise of metric $d\/$ or $d'$. We have $$d(a'(t), T^p(a'(t)))\, =\, d'(a'(t), T^p(a'(t)))\, =\, p/q$$ for all $t \in {\mathbb R}$. Described procedure defines all points of type $a'(t)\/$ with $t \in {\mathbb Q}\/$ in $a'\/$ from relation $V$, and we have $d(a'(0), a'(t))\, =\, d'(a'(0), a'(t))\/$ for them. When $t \in {\mathbb R}\setminus {\mathbb Q}\/$ we may use the fact that $V\/$ defines the incidence relation for $a'\/$ and its order of points.
It remains to apply remark \[aa’\] and Lemma \[asymchain\] for completing the proof.
Combining Theorem \[hyperbolic\] with Theorem \[parabolic\], one gets the whole proof of the Theorem \[tube\]. Consequently Theorem \[main\] is also proved.
[99]{}
<span style="font-variant:small-caps;">A.D.Aleksandrov</span> *On maps preserving congruence*, Dokl.Akad.Nauk SSSR, **211** (1973), 6, 1257–1260.
<span style="font-variant:small-caps;">P.D.Andreev</span> *Recovering the metric of $CAT(0)$-space by a diagonal tube*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), **299** (2003), 5–29.
<span style="font-variant:small-caps;">W.Ballmann</span> *Lectures on spaces of nonpositive curvature*, volume [**[25]{}**]{} of DMV seminar. Birkhaüser, 1995.
<span style="font-variant:small-caps;">F.S.Beckman and D.A.Quarles, Jr.</span> *On isometries of Euclidean spaces*, Proc. Amer. Math. Soc. **4** (1953), 810–815.
<span style="font-variant:small-caps;">V.N.Berestovskiǐ</span> *Isometries in Aleksandrov spaces of curvature bounded above*, Illinois J. Math. **46** (2002), 2, 645–656.
<span style="font-variant:small-caps;">V.N.Berestovskiǐ</span> *Busemann spaces with upper bounded Aleksandrov curvature*, Algebra and Analiz, **14** (2002), 5, 3–18.
<span style="font-variant:small-caps;">M.Bridson, A.Haefliger</span> *Metric spaces of nonpositive curvature,* Grundlehren der Math. Wiss. [**[319]{}**]{}, Springer-Verlag, Berlin(1999).
<span style="font-variant:small-caps;">H.Busemann</span> *Geometry of geodesics.* New York, Academic Press, 1955.
<span style="font-variant:small-caps;">S.V.Buyalo</span> *Lectures on spaces of curvature bounded above*, preprint, 1994.
<span style="font-variant:small-caps;">M.V.Davis, B.Okun, W.Zheng</span> *Piecewise Euclidean structures and Eberlein’s Rigidity theorem in the singular case*, Geom. Topol. **3** (1999), 303–330.
<span style="font-variant:small-caps;">A.V.Kuzminykh</span> *Mappings preserving the distance 1* , Sibirsk. Mat. Zh. [**20**]{} (1979), 3, 597–602.
<span style="font-variant:small-caps;">I.Nikolaev</span> *The tangent cone of an Aleksandrov space of curvature $\le\, K$,* Manuscripta Math. [**[86]{}**]{} no.2 (1995), 137-147.
<span style="font-variant:small-caps;">Y.Otsu, H.Tanoue</span> *The Riemannian structure of Alexandrov spaces with curvature bounded above*, preprint.
|
\
\
University of Göttingen - Institute for Theoretical Physics\
Friedrich-Hund-Platz 1 - D 37077 Göttingen
:\
A variational principle is suggested within Riemannian geometry, in which an auxiliary metric and the Levi Civita connection are varied independently. The auxiliary metric plays the role of a Lagrange multiplier and introduces non-minimal coupling of matter to the curvature scalar. The field equations are 2nd order PDEs and easier to handle than those following from the so-called Palatini method. Moreover, in contrast to the latter method, no gradients of the matter variables appear. In cosmological modeling, the physics resulting from the new variational principle will differ from the modeling using the Palatini method.\
PACS numbers: 04.20 Fy, 04.50 Kd, 98.80 Cq, 02.40 Ky
Introduction
============
For the derivation of the field equation of Einstein’s theory of gravitation and of alternative gravitational theories sometimes a method named, alternatively, “Palatini’s Principle”, “the Palatini method of variation” or “Palatini’s device” is used. Although the starting point is Riemannian geometry, besides the metric an independent affine connection forming the curvature tensor is imagined; in the Lagrangian, both metric and connection then are varied independently. An advantage of the method is that it leads to 2nd order field equation for Lagranians of higher order in curvature while a variation of the metric as the only variable results in 4th-order PDEs. On the other hand, a main conceptual difficulty of the method is that the variational procedure mixes Riemannian and metric-affine geometry. Authors either leave undetermined the space-time geometry as a frame for the new connection, or tacitely fix it mentally by introducing constraints (symmetric connection, no torsion etc) which do not show up in the formalism.
Since many years, warnings have been voiced that the method be working reliably only for the Hilbert-Einstein Lagrangian (plus the matter part) ${\cal L} = \sqrt{-g}~[R(g_{ij}) + 2\kappa L_{mat}(g_{ij}, u^{A})]$ with curvature scalar $R=g^{lm} R_{lm}(g_{ij})$ and matter variables $u^A$, but otherwise leads to under- and un-determinacies [@Buchdahl1960], [@Buchdahl1977], [@Stephen1977].[^1] Recently, Palatini’s method has been unearthed in attempts to build cosmological models thought to explain the accelerated expansion of the universe with its consequences for dark energy [@Voll03], [@Flana04], [@VaTaTsu07], [@SotLib07], [@Olmo08]. The method also has been applied to loop quantum cosmology [@OlmSin09]. Often, the starting point is a Lagrangian of the form ${\cal L}
= \sqrt{-g}~[~R(g_{ij}) + \tilde{f}(R)~] + \sqrt{-g}~ 2\kappa L_{mat}(g_{ij},
u^{A})$ with $\tilde{f}$ an arbitrary smooth function.[^2] In the following, we suggest another variational principle leading to 2nd order field equations and lacking the deficiencies of the Palatini method. After its introduction, it is applied to the class of $f(R)$-theories in section \[section:ftheory\] and compared with the Palatini method in section \[section:compalat\]. A recent particular choice for $f(R)$ in the framework of cosmological modeling then is used as an example for the working of the new principle.
The new variational principle {#section:newvar}
=============================
Whereas in the Palatini method the Levi Civita connection (represented by the Christoffel symbol) is replaced by a general affine connection, here we keep the geometry (pseudo-)Riemannian but introduce an auxiliary Lorentz metric. This is done by replacing, in an action integral set up within Riemannian geometry, the (Lorentz-)metric $g_{ab}$ by an auxiliary metric $\gamma_{ab}$ except in the Levi Civita connection which is left unchanged. The independent variables for the variation are $\gamma_{ab}$ and the Levi Civita connection formed from $g_{ab}$ $$\{ _{ij}^{k}\}_g = \frac{1}{2}g^{kl}(\frac{\partial
g_{il}}{\partial x^j}+\frac{\partial g_{jl}}{\partial x^i} -
\frac{\partial g_{ij}}{\partial x^l})~\label{christo}.$$ The equations following from the variation will give the dynamics of the gravitational field and link $\gamma_{ab}$ with $
g_{ab}$. We wish to emphasize that it is [*not*]{} a bi-metric theory which is aimed at.[^3] The auxiliary metric may be seen as playing the role of a Lagrange multiplier. This is analogous to the case of scalar-tensor theories replacing $f(R)$-theories of gravitation (cf. [@Soti06], [@Igle07]). The new variational principle for Einstein gravity starts from:[^4] $${\cal L} =
\sqrt{-\gamma}~[\gamma^{ab}~ R_{ab}(\{ _{ij}^{k}\}_{g}) + 2\kappa
L_{mat}(\gamma_{lm}, u^A)]~.\label{lagr1}$$ Variation with respect to $\gamma^{ab}$ leads to: $${\delta_{\gamma}\cal L} =\sqrt{-\gamma}~[~R_{ab}(\{
_{ij}^{k}\}_g) - \frac{1}{2}\gamma_{ab}~R_{\gamma}
+ 2\kappa~T_{ab}(\gamma_{lm}~, u^A)~]~\delta \gamma^{ab} \label{new1},$$ where $R_{\gamma}:= \gamma^{lm}R_{lm}(\{_{ij}^{k}\}_g)$ and $T_{ab}:= \frac{2}{\sqrt{-\gamma}}~\frac{\delta{\cal L}_{mat}}{\delta
\gamma^{ab}}$. Variation with respect to $\{ _{ij}^{k}\}_g$ gives: $${\delta_{\{ _{ij}^{k}\}_g }\cal L} =
[~-(\sqrt{-\gamma}~\gamma^{b(i})_{;b}\delta_k^{~j)}+
(\sqrt{-\gamma}~\gamma^{ij})_{;k}~]~\delta(\{ _{ij}^{k}\}_g) \label{new2}$$ up to divergence terms.[^5] From ${\delta_{\{ _{ij}^{k}\}_g
}\cal L}=0$, after a brief calculation using the trace of (\[new2\]), $$(\sqrt{-\gamma}~\gamma^{ij})_{;k}=0\label{det met}$$ follows, where the covariant derivative is formed with the Levi Civita connection. Thus, $\gamma^{ab} = const \cdot g^{ab}$ follows. ${\delta_{\gamma}\cal L}=0$ from (\[new1\]) reduces to Einstein’s field equations.\
The method is particularly well suited to a calculus with differential forms. Here, the usual basic 1-forms $\theta^{i}=
e^{i}_{~r}~ dx^r$ and the curvature 2-form $\Omega_{ij}= \frac{1}{2}
R_{ijkl}(g_{lm})\theta^{k} \wedge \theta^{l}$ are taken as the independent variables. In place of the auxiliary metric $\gamma_{ij}$, now an auxiliary 1-form is introduced and denoted by $\bar{\theta}^{i}= \bar{e}^{i}_{~r}~ dx^r$ where $$\bar{e}^{i}_{~r}~ \bar{e}^{j}_{~s}~\eta^{ij}
=\gamma^{rs},~ e^{i}_{~r}~e^{j}_{~s}~\eta^{ij} = g^{rs}~.$$ The Einstein-Hilbert Lagrangian is $ {\cal L}_E = \Omega_{ab}\wedge
*(\bar{\theta}^{a}\wedge \bar{\theta}^{b})$ with the Hodge-star operation: $*(\bar{\theta}^{a}\wedge \bar{\theta}^{b})=: \bar{\epsilon}^{ab}$ and $\bar{\epsilon}_{ab}:= \frac{1}{2!}\epsilon_{ablm}\bar{\theta}^{l}\wedge
\bar{\theta}^{m}$.[^6] Variation with regard to the fundamental 1-forms and curvature form leads to the field equations:$$\begin{aligned}
D(\frac{\partial{\cal
L}_E}{\partial \Omega_{ij}}) = 0,~~ \frac{\partial{\cal
L}_E}{\partial \bar{\theta}_{i}} = 0~ \end{aligned}$$ with the covariant external derivative $D$ using the Levi Civita connection (1-form). Because of $\frac{\partial{\cal L}_E}{\partial \Omega_{ij}}=
*(\bar{\theta}^{i}\wedge \bar{\theta}^{j})$ and of $ \frac{\partial{\cal
L}_E}{\partial \bar{\theta}_{i}}= \Omega_{lm}\wedge \bar{\epsilon}_{ilm}~,
$ the field equations are:$$D\bar{\epsilon}^{ij}=0~,~
\Omega_{lm}\wedge \bar{\epsilon}_{ilm}=0~,\label{fieldeqforms}$$ where $ \bar{\epsilon}_{ilm}:= \epsilon_{ilmp}\theta_{p}$ is a 1-form; $\bar{\epsilon}^{ilm}$ is dual to $ \bar{\theta}^{i}\wedge
\bar{\theta}^{l}\wedge \bar{\theta}^{m}.$ Standard manipulations with the forms show that the 1st equation (\[fieldeqforms\]) is satisfied identically due to the absence of torsion, i.e, $D \bar{\theta}^{m}=0$; and that the 2nd becomes: $2 G^{c}_{~a}(g)\bar{\epsilon}_{c}=0$ with the Einstein tensor $G^{c}_{~a}(g)$ and the 3-form $\bar{\epsilon}_{i}:=
\frac{1}{3!}\epsilon_{iklm}\bar{\theta}^{k}\wedge \bar{\theta}^{l} \wedge
\bar{\theta}^{m}~. $ An advantage of this formalism is that it may be adapted easily to gauge theories.
Extension to f(R)-theories {#section:ftheory}
==========================
The new variational principle easily applies to the Lagrangian $${\cal L} = \sqrt{-\gamma}~[f(\gamma^{lm}R_{lm}(\{
_{ij}^{k}\}_g)) + 2\kappa~ {\cal L}_{mat} (\gamma_{ij}, u^{A})]~.$$ The variations lead to: $${\delta_{\gamma}\cal L} =\sqrt{-\gamma}~[~f'(R_{\gamma})R_{ab}(\{
_{ij}^{k}\}_g) - \frac{1}{2}\gamma_{ab}~f(R_{\gamma})
+ 2\kappa~T_{ab}(\gamma_{lm}~, u^A, \partial u^A)~]~\delta
\gamma^{ab} \label{new1f},$$ whith $f':=
\frac{df}{dR}$ and to $$\delta_{\{ _{ij}^{k}\}_g \cal L} =
[~-(\sqrt{-\gamma}~f'(R_{\gamma})\gamma^{b(i})_{;b}\delta_k^{~j)}+
(\sqrt{-\gamma}~f'(R_{\gamma})\gamma^{ij})_{;k}~]~\delta(\{
_{ij}^{k}\}_g) \label{new2f}$$ up to divergence terms. As in section \[section:newvar\], from (\[new2f\]) $$(\sqrt{-\gamma}~f'(R_{\gamma})\gamma^{ij})_{;k}=0\label{detmet}~,$$ but from which now follows: $$\gamma^{ab}=f'(R_{\gamma}) g^{ab},~~\gamma_{ab}=(f'(R_{\gamma}))^{-1}
g_{ab}~.\label{metric2}$$ From (\[metric2\]), $R_{\gamma}= f'(R_{\gamma}) R_{g}$ with $R_{g}:=
g^{lm}R_{lm}(\{_{ij}^{k}\}_g)$, i.e., the curvature scalar in (pseudo-)Riemannian space-time. Writing $$R_{g}=
\frac{R_{\gamma}}{f'(R_{\gamma})}=:r(R_{\gamma})~, \label{g-gamma}$$ the relation $R_{\gamma} = r^{-1}(R_g)$ can be used to remove all entries of $\gamma_{ab}$ via the curvature scalar in the field equations following from (\[new1f\]). Expressed by $g_{ab}$, they read as: $$f'(r^{-1}(R_{g}))~R_{ab}(\{_{ij}^{k}\}_g) -
\frac{1}{2}~g_{ab}~\frac{f(r^{-1}(R_{g}))}{f'(r^{-1}(R_{g}))}
+ 2\kappa~T_{ab}((f')^{-1}(r^{-1}(R_{g}))~g_{lm}~, u^A) = 0
\label{fieldeq2}~.$$ Equation (\[fieldeq2\]) shows that, in contrast to f(R)-theories leading to 4th-order differential equations when derived by variation of only the metric $g_{ab}$, the new field equations are of 2nd order in the derivatives of $g_{ab}$. The auxiliary metric is fully determined: $\gamma^{ab}=
f'(r^{-1}(R_g))g^{ab}$; it is not an absolute object. Beyond acting as a Lagrange multiplier its main function is its appearance in the matter tensor causing non-minimal coupling to the curvature scalar. No further role in the description of the gravitational field is played.[^7] For a Lagrangian of the form $\sqrt{-g}~[~R(g_{ij}) + \tilde{f}(R)~]$, in the formalism given above $f$ is to be replaced by $R+\tilde{f}(R), f'$ by $1+\tilde{f}'$ while $f''=
\tilde{f}'', f'''= \tilde{f}'''$.\
A.\
First, a non-vanishing trace (with respect to the auxiliary metric $\gamma$) of the matter tensor will be assumed $T_{\gamma}:=
\gamma^{lm}T_{lm}(\gamma_{rs}~, u^A)\neq 0$. In this case, the curvature scalar is seen to be a functional of the trace of the matter tensor. Because of $$\begin{aligned}
T_{\gamma}= f'(R_{\gamma})~ g^{lm} T_{lm}(f'(r^{-1}(R_g))
g_{rs}, u^A)= f'(r^{-1}(R_g)) T_g(f'(r^{-1}(R_g)) g_{rs}, u^A),
\label{traceq} \end{aligned}$$ with $\tilde{T}_g := g^{lm}
T_{lm}(\gamma_{rs}, u^A)$ from the g-trace of (\[fieldeq2\]) follows:$$f'^2~R_g -2f
+ 2\kappa~f'~\tilde{T}_g= 0~,$$ or, precisely, $$(f'(r^{-1}(R_g))^2 R_g - 2f(r^{-1}(R_g)) + 2\kappa f'(r^{-1}(R_g))\tilde{T}_g
((f'(R_g))^{-1} g_{lm} u^A) = 0~. \label{traceT}$$ With a newly defined function $\omega$ this can be written as $$R_g
= \omega (2\kappa T_g)~,$$ where now $T_g:= g^{lm}T_{lm}
(g_{rs}, u^A)$. From (\[traceT\]) we conclude that (\[fieldeq2\]) can be cast into the form of Einstein’s equations with an effective matter tensor. The curvature scalar is coupled directly to the matter variables showing up in its trace; no derivatives are involved. In fact:$$R_{ab}(\{_{ij}^{k}\}_g)-\frac{1}{2}~g_{ab}(R_{g})= - \frac{2\kappa}{f'}~[~T_{ab} (\frac{1}{f'}~g_{lm}, u^A) - \frac{1}{2}T g_{ab}~] - \frac{1}{2}g_{ab}~ \frac{f}{(f')^2}~. \label{newEineq}$$
In the case of [*perfect fluid*]{} matter with energy density $\mu$ and pressure $p$ $$T_{ab}(\gamma_{rs}, u^A) = (\mu + p)~\gamma_{al}\gamma_{bm}\bar{u}^l
\bar{u}^m - p~ \gamma_{ab} \label{perfect}$$ with $\bar{u}^l := \frac{dx^l}{d\bar{s}}$ and $~d\bar{s}^2=\gamma_{lm}dx^l dx^m. $ Hence, $\bar{u}^l= (f')^{1/2}~ u^l~, u^l= \frac{dx^l}{ds} $ and $$T_{ab}(\gamma_{rs},
u^A) = (f')^{-1}(R_g)~ T_{lm}(g_{rs}, u^A)~.$$ In this case, from (\[traceq\]) a simple relationship for the $\gamma$- and $g$-traces of the matter tensor follows: $$T_{\gamma}(\gamma_{rs}~, u^A)= T_g(g_{rs}, u^A)= \mu-3 p~.$$ In place of $T^{ab}_{~~;~b}=0$ for the Einstein-Hilbert Lagrangian, in this theory a more general relationship with $T^{ab}_{~~;~b}\neq 0$ follows from general covariance. This is also seen by forming the divergence of the Einstein tensor in (\[newEineq\]).\
B.\
For vanishing trace of the matter tensor $T_{\gamma}=0$, (\[traceT\]) reduces to $$f'(R_{\gamma})R_{\gamma} - 2
f(R_{\gamma}) = 0~.\label{tracezero}$$ This implies two cases:\
i) $f=(f_0 R_{\gamma})^2$, and ii) $f \neq (f_0 R_{\gamma})^2$. The exceptional case i) is characterized by an additional scale invariance implying zero trace for the matter tensor. The field equations (\[fieldeq2\]) become $$\begin{aligned}
2(f_0)^2
(R_{\gamma})[R_{ab}(g)-\frac{1}{4}R_g~g_{ab}] + 2\kappa
T_{ab}(\frac{1}{2f_0^2R_{\gamma}}~ g_{lm}, u^A) =
0~,\nonumber \\ R_g = \frac{1}{2(f_0)^2}~,~ T_{\gamma}= T_g
=0~. \end{aligned}$$ If we take a sourceless Maxwell field as matter, then$$T_{ab}(\gamma_{lm}, F_{lm})= \gamma^{lm}F_{al}F_{bm} -
\frac{1}{4}\gamma_{ab}\gamma^{il}\gamma^{jm} F_{il}F{jm} = f'(R_{\gamma})
T_{ab}(g_{lm}, F_{lm})~.$$ $R_{\gamma}$ drops out and the field equations are: $$\begin{aligned}
R_{ab}(g)- \frac{1}{4}R_g~g_{ab} + \kappa~
T_{ab}(g_{lm}, u^A) = 0~,\nonumber \\ R_g =\frac{c_1}{f'(c_1)}~,~T_{\gamma}= T_g =0~. \end{aligned}$$
In case ii), from (\[tracezero\]) $R_{\gamma}=c_1=
const$ and we may proceed only if one real solution of $ f'(c_1)~c_1 - 2
f(c_1) = 0~$ does exist and if $f(c_1),~ f'(c_1)$ remain finite. The field equations then are $$\begin{aligned}
f'(c_1)R_{ab}(g)-\frac{c_1}{4}~g_{ab} + 2\kappa
T_{ab}(\frac{1}{f'(c_1)}~g_{lm}, u^A) = 0~,\nonumber \\ T_{\gamma}= T_g
=0~. \end{aligned}$$
In Einstein’s theory, $R=0$ follows if the trace of the matter tensor is vanishing. Here, the larger set of solutions $R = const$ is obtained.\
Above, it has been assumed that the matter tensor does not contain covariant derivatives; this covers most cases of physical interest. Otherwise, formidable complications result even when the Einstein-Hilbert Lagrangian is taken. E.g., if the additional term in the matter Lagrangian is $\sim \sqrt{-\gamma} \gamma^{il} \gamma^{km}u_{i;k}~u_{l;m}$ (\[det met\]) must be replaced by $(\sqrt{-\gamma}~\gamma^{ij})_{;k} = f^{ij}_{k}(\gamma^{lm},
u^A, \partial u^A)$ with a particular functional $ f^{ij}_{k}$. Hence, the elimination of the Lagrangian multiplier will requirec quite an effort.
Comparison with the Palatini method {#section:compalat}
===================================
For the Palatini method of variation with variables $g_{ij}$ and $\Gamma_{ij}^{k}$, the field equations of the f(R)-theory are: $$\begin{aligned}
f'(R) R_{ik}(\Gamma)-\frac{1}{2}
f(R)g_{ik} = - 2\kappa~T_{ik},~ \label{Palat1}\\ (\sqrt{-g}f'(R)g^{il})_{\parallel l} =0~,\label{Palat2}\end{aligned}$$ where the covariant derivative is formed with the connection $\Gamma$ and $R=g^{ik}R_{ik}(\Gamma)$. From (\[Palat2\]) we obtain a metric $\bar{g}_{ij}$ compatible with the connection $\Gamma$: $$\bar{g}_{ij}=f'(R)~ g_{ij}~,~
\bar{g}^{ij}=(f')^{-1}(R)~ g^{ij}~, \label{Palatmet}$$ and the relation between $\Gamma$ and the Levi Civita connection is: $$\Gamma_{ij}^k \equiv \{ _{ij}^{k}\}_{\bar{g}}=\{
_{ij}^{k}\}_{g} + \frac{1}{2}\frac{d}{dR}(ln
f'(R))~[2\delta_{(i}^kR_{,j)}-g_{ij}g^{kl}R_{,l}]~.\label{Palatcon}$$ A comparison of (\[metric2\]) and (\[Palatmet\]) shows the difference between $\gamma^{ij}$ and $\bar{g}^{ij}$. With the help of (\[Palatcon\]) and (\[Palatmet\]) we can rewrite the tracefree part of (\[Palat1\]) in terms of the conformally related metric $\bar{g}_{ij}$ $$\begin{aligned}
\bar{R}_{ab}(\bar{g})-\frac{1}{4}~ \bar{R}(\bar{g})~\bar{g}_{ab}=
R_{ab}(g)-\frac{1}{4}~R(g)~g_{ab}-
\frac{f''}{f'}[R_{,i;j}-\frac{1}{4}~g_{ij} \square R]- \nonumber\\ -
(\frac{f''}{f'}-\frac{3}{2} (\frac{f''}{f'})^2)~
[R_{,i}R_{,j}-\frac{1}{4}~g_{ij}R_{,l}R_{,m}g^{lm}]~. \end{aligned}$$ When bringing the field equations into the form of Einstein’s equations, the result is:$$\begin{aligned}
R_{ab}(g)-\frac{1}{2}~R(g)~g_{ab}=- 2\kappa
T_{ab}({g}) - \frac{f''}{f'}[R_{,i;j}- g_{ij} \square R]- \nonumber\\ -
[\frac{f'''}{f'}-\frac{3}{2} (\frac{f''}{f'})^2]~
R_{,i}R_{,j} + (\frac{f'''}{f'}-\frac{3}{4}
(\frac{f''}{f'})^2)~g_{ij}R_{,l}R_{,m}g^{lm} ~.\label{gradient2}\end{aligned}$$ Again, the trace equation of (\[Palat1\]), i.e., $$f'(R) R - 2 f(R) = - 2\kappa~T_g ~,$$ is used to eliminate the curvature scalar in favour of the trace of the matter tensor. This means that the non-minimal coupling to the curvature scalar and its derivatives will be replaced by a coupling to the [ *gradients*]{} of the matter variables contained in $g^{ik}T_{ik}$.
The remark at the end of section \[section:ftheory\] for the case of covariant derivatives in the matter tensor applies here as well.\
An example: Exponential gravity
===============================
New variational principle
-------------------------
As an example, we now take a recent model for $f(R)$-gravity [@Linder09] with:
$$f(R)= -c r (1- e^{-\frac{R}{r}})~, f'=-c
e^{-\frac{R}{r}}~,\label{expgrav}$$
where $r$ of dimension $(length)^2$ and $c$, dimensionless, are constants. From (\[g-gamma\]) $R_g
=-\frac{1}{c}R_{\gamma}e^{\frac{R_{\gamma}}{r}}$. Thus, the inverse $R_{\gamma}= r^{-1}(R_g)$ can be obtained only numerically. A series expansion for $ \frac{R_{\gamma}}{r} << 1$ leads to:$$R_{\gamma}=-c R_g -\frac{c^2}{r}R_g^2
-\frac{3}{2}\frac{c^3}{r^2}R_g^3~\pm \cdot
\cdot~, \label{expg-gamma}$$ and $$R_g=
\frac{2\kappa T_g~}{c^2}~[1 - \frac{2\kappa T_g}{rc}+ \frac{2}{3}~
(\frac{2\kappa T_g}{rc})^2 \pm \cdot \cdot ]$$ If the further calculations are restricted [*to the lowest order*]{} in the expansion (\[expg-gamma\]), with the Einstein tensor $G_{ab}= R_{ab}
-\frac{1}{2}R_g~g_{ab}$ the field equations (\[newEineq\]) become: $$G_{ab}(\{_{ij}^{k}\}_g)=
-\frac{2\kappa}{c^2}~[1- 4\frac{\kappa T_g}{rc}]~T_{ab}~- \frac{\kappa
T_g}{c^2} ~\frac{\kappa T_g}{rc}~g_{ab}~.$$ For a perfect fluid with pressure $p=0$, from (\[perfect\]) to lowest order the equations replacing Einstein’s are: $$R_{ab}(\{_{ij}^{k}\}_g)-\frac{1}{2}R_g~g_{ab}=
-\frac{2\kappa}{c^2}(1- 4\frac{\kappa \mu}{rc})~ \mu~ u_a u_b -
\frac{\kappa^2 \mu^2}{rc^3}~g_{ab} \label{newfieldperf}~.$$ The result is a variable coupling “constant” in the effective matter tensor and a variable cosmological term both depending on the energy density of matter.
For a homogeneous and isotropic cosmological model with scale factor $a(t)$ and flat space sections, (\[newfieldperf\]) leads to altered Friemann equations: $$\begin{aligned}
(\frac{\dot{a}}{a})^2 = \frac{2\kappa
\mu}{3c}~(1 -\frac{7}{2} \frac{\kappa \mu}{cr})~,\\ 2\frac{\ddot{a}}{a} +
(\frac{\dot{a}}{a})^2 = \frac{\kappa^2 \mu^2}{rc^3}~. \label{newFried}\end{aligned}$$ Keep in mind that $r,c$ are free constants of the model; the velocity of light has been put equal to 1 in (\[newFried\]). As numerical calculations would have to be done, and the main aim of this paper is the introduction of a new variational principle, we will not comment on this particular model (exponential gravity) and the physics following from it.
Palatini method
---------------
For exponential gravity as given by (\[expgrav\]) and for pressureless fluid matter, the field equations according to the first equation of (\[Palat1\]) turn out to be $$R_{ik}(\Gamma) = \frac{2\kappa}{c}~T_{ik}
(g,u^A) + g_{ik}~r(1-e^{-\frac{R}{r}}).$$ This does not look complicated; however the connection $\Gamma$ first must be expressed by the conformally related metric $\bar{g}_{ab}$. To the same order of approximation, the final field equation then can be written as:$$\begin{aligned}
R_{ab}(g)-\frac{1}{2}~ R(g)~g_{ab}=- 2\kappa \mu
u_{a} u_{b} - \frac{2\kappa}{rc}[\mu_{,i;j}- g_{ij} \square \mu]-
\nonumber\\ - \frac{2\kappa^2}{r^4c^2}~\mu_{,i}~\mu_{,j} +\frac{\kappa^2}{
r^4c^2}~g_{ij}~\mu_{,l}~\mu_{,m}~g^{lm} ~.\label{expgravgradient}\end{aligned}$$ The effective matter tensor in (\[expgravgradient\]) depends on 1st and 2nd [*gradients*]{} of the energy density of matter. This shows that the physics resulting from the two variational principles may be quite different. The same can be said with regard to the Einstein-Hilbert metric variation used in [@Linder09] and e.g., in [@SoHuSa07], [@HuSa07] and the new variational principle.
Concluding remarks
==================
For physics, a significant difference between the new variational method presented here and the Palatini method is that non-minimal coupling of matter and the curvature scalar $R$ occurs by multiplication with functions of $R$ or the trace of the matter tensor. In the Palatini method, non-minimal coupling happens via the [*gradients*]{} of the scalar curvature (trace of the matter tensor). A conceptual advantage of the new method is that it works within (pseudo)-Riemannian geometry; metric-affine geometry never does appear.[^8] When dealing with $R+f(R)$-Lagrangians, in both approaches a new dimensionful constant is needed whose physical meaning must be defined. Application to $f(R,~ R_{ab}R^{ab})$ is unproblematic; here, two new parameters will occur. In general, via the field equations both curvature invariants can be expressed as functionals of invariants of the matter tensor. The Einstein-Hilbert Lagrangian seems to be very robust: now there are at least [*three*]{} different methods for a derivation of the Einstein field equations. As the example treated shows, for more general Lagrangians the variation will lead to different physical theories. Whether the new variational principle introduced here, if applied to cosmological models, produces convincing physics will have to be shown by further studies.
[999]{}
H. Buchdahl, Proc. Cambr. Philos. Soc. [ **56**]{}, 369-400 (1960). H. Buchdahl, J. Phys. A [**12**]{}, 1229-1234 (1979). G. Stephenson, J. Phys. A [**10**]{}, 181-184 (1977). F. W. Hehl, pp. 5-62, in: Cosmology and Gravitation, Eds. P. G. Bergmann and V. De Sabbata (1980). M. Ferraris and M. Francaviglia, Gen. Rel. Grav. [**14**]{}, 243-254 (1982). M. Kohler, Z. Physik [**134**]{}, 286-305 (1952). D. N. Vollick, Phys. Rev. D [**68**]{}, 063510/1-3 (2003). É Flannagan, Phys. Rev. Lett. [**92**]{}, 071101 (2004) and response by D. N. Vollick, Class. Quant. Grav. [**21**]{}, 3813-3816 (2004). St. Fay, R. Tavakol, and S.Tsujikawa, Phys. Rev. D [**75**]{}, 063509 (2007). Th. Sotiriou and At. Liberati, Annals Phys. [ **322**]{}, 935-966 (2007). G. J. Olmo, Phys. Rev. D [**78**]{}, 104026 (2008). G. J. Olmo and P. Singh, J. Cosm. Astropart. Phys. 01 (2009) 030. G. J. Olmo et al., Phys. Rev D [**80**]{}, 024013 (2009). Th. Sotiriou, Class. Quant. Grav. [**23**]{}, 5117-5128 (2006). A. Iglesias et al., Phys. Rev. D [**76**]{}, 104001 (2007). E. V. Linder, Phys. Rev D [**80**]{}, 123528 (2009). Y.-S. Song, W. Hu, and I. Sawicki, Phys. Rev. D [**75**]{}, 044004 (2007). W. Hu and I. Sawicki, Phys. Rev. D [**76**]{}, 064004 (2007).
[^1]: For incorrectly relating Palatini’s name with what is ascibed to him cf. [@Hehl1980], footnote on p. 40 as well as the English translation of Palatini’s paper in the same volume on pp. 477-488 (1980). Cf. also [@Ferra1982].
[^2]: Recently, Lagrangians with two curvature invariants, i.e., $f(R,~
R_{ab}R_{lm}g^{al}g^{bm})$ have been considered [@OlmSanTri09].
[^3]: In bi-metric theories, one metric usually is fixed to be the flat Minkowskian metric and not varied. A formal variation of the second metric often is restricted to an infinitesimal coordinate change in order to derive conservation laws. Cf. [@Kohler1952].
[^4]: Latin indices a, b, i, j, ... run from 0 to 3; the summation convention is implied.
[^5]: ($\sqrt{-\gamma}A^k)_{;k}$ always may be written as $\sqrt{-g}~(\sqrt{\frac{\gamma}{g}}A^k)_{;k}$ and thus as ($\sqrt{-\gamma}A^k)_{,k}$.
[^6]: Notation here is somewhat ambiguous: e.g., the curvature form depends on both the Levi Civita connection and the auxiliary tetrad: $\Omega_{ij}= \frac{1}{2} R_{ijkl}(\{ _{ij}^{k}\}_g)~\bar{\theta}^{k}
\wedge \bar{\theta}^{l}$. Nevertheless, no bar will be put on $\Omega$. The notation $\Omega_{ij}(g,\bar{\theta})$ would be inconvenient.
[^7]: In particular, $\gamma^{ab}$ does [*not*]{} enter the Levi Civita connection, but only the matter tensor. As a metric $\gamma_{ab}$ is incompatible with the Levi Civita connection; its non-metricity tensor does not vanish.
[^8]: For metric-affine geometry, independent variation of metric and connection is mandatory anyway.
|
---
abstract: 'We explore further controllability problems through a standard least square approach as in [@pedregal]. By setting up a suitable error functional $E$, and putting $m(\ge0)$ for the infimum, we interpret approximate controllability by asking $m=0$, while exact controllability corresponds, in addition, to demanding that $m$ is attained. We also provide a condition, formulated entirely in terms of the error $E$, which turns out to be equivalent to the unique continuation property, and to approximate controllability. Though we restrict attention here to the 1D, homogeneous heat equation to explain the main ideas, they can be extended in a similar way to many other scenarios some of which have already been explored numerically, due to the flexibility of the procedure for the numerical approximation.'
address: 'ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain'
author:
- Pablo Pedregal
title: A different look at controllability
---
[^1]
Introduction
============
We would like to explore controllability problems through a least square approximation strategy. As a matter of fact, it has already been considered and proposed in [@pedregal].
As in a typical least-square approximation, we will set up a non-negative error functional $E$, defined in a suitable space, and let $m=\inf E\ge0$. Approximate controllability is then defined by demanding $m=0$, while exact controllability occurs when, in addition, $m$ is a minimum. The main benefit we have found concerning this viewpoint is that the controllability problem is translated, in an equivalent way, into minimizing such an error functional. As such, from the numerical point of view, one can then proceed to produce numerical approximation by utilizing typical descent strategies. Especially in linear situations, such an error functional is convex and quadratic (though not necessarily coercive), and this numerical procedure should work fine. This is indeed so. In addition, it is also true that $m=0$ is equivalent to the interesting property $E'=0$ implies $E=0$, so that the only possible critical value for $E$ is zero ([@pedregal]).
We only treat explicitly these ideas for the homogeneous heat equation in (spatial) dimension $N=1$, as it will be pretty clear how to extend this philosophy to many other situations. Specifically, we take $\Omega=(0,1)\subset\R$, and $T>0$, the time horizon.
> For initial and final data $u_0(x)$, and $u_T(x)$, respectively, we would like to find the right-point condition $f(t)$, so that the solution of the heat problem $$u_t-u_{xx}=0\hbox{ in }(0, T)\times(0, 1),\quad u(t, 0)=0, u(0, x)=u_0(x), u(t, 1)=f(t),$$ will comply with $u(T, x)=u_T(x)$.
This is the boundary controllability situation. There is also an inner controllability case in which we take a fixed subdomain $\omega\subset\Omega$.
> Determine the source term $f(t, x)$ supported in $\omega$ in such a way that the solution of the problem $$u_t-u_{xx}=f\chi_\omega\hbox{ in }(0, T)\times(0, 1),\quad u(t, 0)=0, u(0, x)=u_0(x), u(t, 1)=0,$$ will comply with $u(T, x)=u_T(x)$.
We refer to [@FR], and to [@Russell] for classic results on controllability, and to [@FurIma], [@LebRob] for a more recent analysis. [@carthel] also contains important ideas through duality arguments in the context of the Hilbert Uniqueness Method. See also [@Coron], [@glowinski08]. The numerical analysis of this kind of controllability problems has attracted a lot of work. Without pretending to be exhaustive, we would mention important contributions covering a whole range of methods and approaches in [@belgacem], [@carthel], [@EFC-AM-I], [@kindermann], [@micu-zuazua], [@AM-EZ].
Approximate controllability {#prime}
===========================
To keep the formalism to a minimum without compromising rigor, let us stick to the situation described in the Introduction by taking $Q_T=(0, T)\times(0, 1)$, $\overline u\in H^1(Q_T)$ furnishing the data for $t=0$ and $t=T$, $\overline u(0, x)=u_0(x)$ and $\overline u(T, x)=u_T(x)$, respectively, and assuming that $\overline u(t, 0)=0$ in the sense of traces. Let $$\begin{gathered}
\A_0=\{U\in H^1(Q_T): U(t, 0)=U(0, x)=U(T, x)=0 \hbox{ for }x\in(0, 1), t\in(0, T)\},\nonumber\\
\A=\overline u+\A_0.\nonumber\end{gathered}$$ For $u\in\A$, define its “corrector" $v\in H^1_0(Q_T)$ to be the unique solution of the variational equality $$\label{calor}
\int_{Q_T}[(u_t+v)\phi+(u_x+v_x)\phi_x+v_t\phi_t]\,dx\,dt=0$$ for all $\phi\in H^1_0(Q_T)$. Note that $v$ is the unique solution of the minimization problem $$\hbox{Minimize in }w\in H^1_0(Q_T):\quad \int_{Q_T}\left(\frac12[(u_x+w_x)^2+w_t^2+w^2]+u_tw\right)\,dx\,dt,$$ whose equilibrium equation is $$\label{eq}
-(u_x+v_x)_x-v_{tt}+v+u_t=0\hbox{ in }Q_T.$$ The weak formulation of (\[eq\]) is precisely (\[calor\]). The error functional $E_T:\A\to\R^+$ is taken to be the size of the corrector $$E_T(u)=\int_{Q_T}\frac12 (v_x^2+v_t^2+v^2)\,dx\,dt.$$ It is obvious to realize that if $E(u)=0$ because $u$ has a vanishing corrector, then $u$ is a solution of the heat equation (\[eq\]) with $v\equiv0$, complying with boundary, initial, and final conditions provided by $\overline u$, i.e. the trace of $u$ over $x=1$ is the boundary control sought. Let $\T:\A\mapsto H^1_0(Q_T)$ be the linear, continuous operator taking $u$ into its corrector $v$. Our setting is definitely a least-square approach in the spirit of [@bochevgunz], [@glowinski83].
\[defprin\]
1. We say that the datum $\overline u(T, x)$ is approximately controllable from $\overline u(0, x)$ through the subset $\{1\}$ of the boundary of $(0, 1)$, if for every $\epsilon>0$, there is $u_\epsilon\in\A$ such that $E(u_\epsilon)<\epsilon$.
2. We say that the datum $\overline u(T, x)$ is exactly controllable from $\overline u(0, x)$ through the subset $\{1\}$ of the boundary of $(0, 1)$, if there is $u\in\A$ such that $E(u)=0$.
3. The unique continuation property holds, given our framework, if the only function $v\in\im\T\subset H^1_0(Q_T)$ with $$\label{unique}
\int_{Q_T}(U_tv+U_xv_x)\,dx\,dt=0$$ for all $U\in\A_0$ is the trivial one $v\equiv0$.
Within the framework just introduced through this definition, we can now state our main result.
Let $T>0$, and $\overline u\in H^1(Q_T)$ be given. The following assertions are equivalent:
1. the trace $\overline u(T, x)$ for $t=T$ of $\overline u$ is approximately controllable from $\overline u(0, x)$ through the end-point $\{1\}$;
2. the unique continuation property holds;
3. $E'_T(u)=0$ implies $E_T(u)=0$ for $u\in\A$.
We will show $(2)\Longrightarrow (1)\Longrightarrow (3)\Longrightarrow(2)$.
Let $u\in\A$, and take $U\in\A_0$. Consider the variation $u+\eta U$, preserving boundary, initial, and final data. Let $v$ be the corrector associated with $u$, and put $v+\eta V$ for the corrector associated with $u+\eta U$. By linearity, it is elementary to argue that $$\label{basico}
\int_{Q_T}[(U_t+V)\phi+(U_x+V_x)\phi_x+V_t\phi_t]\,dx\,dt=0$$ for all $\phi\in H^1_0(Q_T)$, where $V$, as $v$ itself, belongs to $H^1_0(Q_T)$. On the other hand, it is also elementary to compute the derivative of $E_T(u+\eta U)$ with respect to $\eta$ at $\eta=0$. It is given by $$\int_{Q_T}(vV+v_xV_x+v_tV_t)\,dx\,dt.$$ By using (\[basico\]) for $\phi=v$, we also can write $$\langle E'_T(u), U\rangle=\left.\frac{d E_T(u+\eta U)}{d\eta}\right|_{\eta=0}=-\int_{Q_T}(U_tv+U_xv_x)\,dx\,dt.$$ Recall that $\T:\A=\overline u+\A_0\mapsto H^1_0(Q_T)$ can be regarded as a linear, continuous operator taking $u$ into its corrector $v$. Then, because of the unique continuation property, $\langle E'(u), U\rangle=0$ for all $U\in\A_0$ if and only if $\T u=0$. Therefore, it is a standard fact in Functional Analysis that over the quotient space $\A_0/\hbox{ker}\T$ both quantities $\|\T u\|$ and $$\sup_{U\in\A_0, \|U\|=1}\langle E'_T(u), U\rangle=\sup_{U\in\A_0, \|U\|=1}\int_{Q_T}(U_tv+U_xv_x)\,dx\,dt$$ should be equivalent norms. Hence, for some positive constant $C>0$, $$\|E'_T(u)\|\equiv\sup_{U\in\A_0, \|U\|=1}\langle E'_T(u), U\rangle\ge C\|\T u\|=CE_T(u)^{1/2}.$$ If, starting out at arbitrary $u^0\in\A$, we follow the flow of $-E'_T$, we would eventually reach a certain $u\in\A$ so that $\|E'_T(u^0+u)\|<\epsilon$. This, together with the previous inequality, yields the approximate controllability result.
Assume now that $\tilde u\in\A$ in a critical point of $E_T$. Under the approximate controlability property, we would like to conclude that $u$ is indeed a solution of the controlability problem. To this end, notice that:
- the infimum of $E_T$ over $\A$ vanishes: this is the approximate controlability property;
- $E_T$ is a non-negative, convex functional.
As a consequence of the convexity, if $\tilde u$ is a critical point of $E_T$, it has to be a minimizer as well. But then the infimum becomes a minimum, and it has to vanish, i.e. $E_T(\tilde u)=0$.
Finally, let $\tilde v\in \im\T$ be such that (\[unique\]) holds for all $U\in\A_0$. Let $\tilde u\in\A$ be such that $\T\tilde u=\tilde v$. By the computations performed above, (\[unique\]) implies $E'_T(\tilde u)=0$, and so, by hypothesis, $\|\tilde v\|^2=E_T(\tilde u)=0$, that is $\tilde v\equiv0$.
The equivalence stated in this theorem implies that the above concept of approximate controllability might be a bit more flexible than the classic one, at least for data sets which are traces for $t=0$ and $t=T$ of $H^1(Q_T)$-functions.
\[ucp\] For every positive time $T>0$, the unique continuation property in Definition \[defprin\] holds.
Simply notice that $H^1_0(Q_T)\subset\A_0$, and so we can take $U=v$ in (\[unique\]) to obtain $$\int_{Q_T}(v_tv+v_x^2)\,dx\,dt=0.$$ Because $v(0, x)=v(T, x)=0$ for all $x\in(0, 1)$, we conclude that $v_x\equiv0$ in $Q_T$. This together with the vanishing boundary conditions $v(t, 0)=v(t, 1)=0$ implies $v\equiv0$.
Extension
=========
The setting described in the previous section admits some straightforward variations. The choice of the space for the correctors $v\in H^1_0(Q_T)$ can be changed. For example, one can take $v\in H^1(Q_T)$ for a broader situation, and in this case the corrector will enjoy the natural boundary condition all around $Q_T$: $u_x=0$ for $x=0$ and $x=1$, while $u_t=0$ for $t=0$ and $t=T$. But intermediate alternatives are also possible: $v=0$ for $t=0$, and $t=T$, and so $u_x=0$ for $x=0$ and $x=1$, or $v\in\A_0$, as well. Another possibility is to define the corrector $v$ for a.e. time slice as a minimization problem only in space. This can again be easily set up in more or less the same terms (see [@pedregal]).
Rather than considering these various possibilities which are straightforward variations, we would like to explore the most general framework that this approach may allow for $u$ instead of for $v$. Our ambient space will now be $$\begin{aligned}
\A_0=\{ &U\in L^2(0, T; \tilde H^1_0(0, 1)): U_t\in L^2(0, T; \tilde H^{-1}(0, 1)), \nonumber\\
&U(0, x)=U(T, x)=0 \hbox{ for }x\in(0, 1), t\in(0, T)\}.\nonumber\end{aligned}$$ We are taking here $$\tilde H^1_0(0, 1))=\{U\in H^1(0, 1): U(0)=0\},$$ while $\tilde H^{-1}(0, 1)$ is its dual. Notice that every $U\in\A_0$ belongs to the space $\C([0, T]; L^2(0, 1))$ so that traces of $U$ are defined for every time $t\in[0, T]$ ([@EvansB]). If $\overline u\in L^2(0, T; \tilde H^1_0(0, 1))$ (and so $\overline u(t, 0)=0$ for a.e. $t\in(0, T)$), with $\overline u_t\in L^2(0, T; \tilde H^{-1}(0, 1))$, furnishes initial and final data, we will put as before $\A=\overline u+\A_0$. This time initial and final data, $\overline u(0, x)$, $\overline u(T, x)$ merely belong to $L^2(0, 1)$.
For $u\in\A$, define its “corrector" $v\in H^1_0(Q_T)$ to be the unique solution of the variational problem $$\int_{Q_T}[(u_x+v_x)\phi_x+(v_t-u)\phi_t+v\phi]\,dx\,dt=0$$ for all $\phi\in H^1_0(Q_T)$. Note that $v$ is the unique solution of the minimization problem $$\hbox{Minimize in }w\in H^1_0(Q_T):\quad \int_{Q_T}\left(\frac12[(u_x+w_x)^2+(w_t-u)^2+w^2]\right)\,dx\,dt.$$ The error functional $E:\A\to\R^+$ is taken to be the size of the corrector $$E(u)=\int_{Q_T}\frac12 (v_x^2+v_t^2+v^2)\,dx\,dt.$$ As above, we investigate the derivative of the error functional. To this end, put $u+\eta U$ for $U\in\A_0$, and $v+\eta V$, its corresponding corrector, with $v$ the corrector for $u$. Then $$\int_{Q_T}[(U_x+V_x)\phi_x+(V_t-U)\phi_t+V\phi]\,dx\,dt=0$$ for all $\phi\in H^1_0(Q_T)$. In the same way, $$\langle E'(u), U\rangle=\int_{Q_T}(v_xV_x+v_tV_t+vV)\,dx\,dt=\int_{Q_T}(Uv_t-U_xv_x)\,dx\,dt,$$ by taking $\phi=v$ in the last identity.
The unique continuation property, and the equivalence with approximate controllability are established in the same way as before.
Exact controllability
=====================
Within this framework, exact controllability can be deduced as a consequence of the fact that the range of the map $\T$ is closed, in addition to the unique continuation property. More precisely, recall the definition of the map $\T:\overline u+\A_0\mapsto H^1_0(Q_T)$, taking every feasible $u\in\overline u+\A_0$ into its corrector $v$, in the analytical framework of Section \[prime\]. The error functional corresponds to the least-square problem $$\hbox{Minimize in }u\in\overline u+\A_0:\quad \frac12\|\T u\|^2.$$ Exact controllability can then be achieved as a consequence of two facts:
1. the infimum $m$ is in fact a minimum;
2. $m$ does vanish.
The unique continuation property is related to the second issue, but the first is equivalent to the fact that the range of $\T$ is closed. Except for general remarks involving the adjoint operator $\T^*$, the exact controllability issue involves subtle and delicate ideas about Carleman inequalities and observability. This elegant theory is very well established (see some of the references indicated in the Introduction).
The inner controllability case
==============================
Let $\omega\subset(0, 1)$ be an interval. Put $Q_T=(0, T)\times(0, 1)$, $q_T=(0, T)\times\omega$. Let $\A_0$ be taken now as the space $$\A_0=\{U\in L^2(0, T; H^2(0, 1)\cap H^1_0(0, 1)): U_t\in L^2(Q_T), U=0\hbox{ on }\partial Q_T\},$$ and $\overline u\in L^2(0, T; H^2(0, 1)\cap H^1_0(0, 1))$ with $\overline u_t\in L^2(Q_T)$, carrying the boundary (around $\partial\Omega$), initial, and final data. For feasible functions $u\in\A\equiv\overline u+\A_0$, let $v$ be its associated corrector, the unique solution of the problem $$\label{lap}
v_{tt}+v_{xx}=\chi_{Q_T\setminus q_T}(t, x)\left(u_t-u_{xx}\right)\hbox{ in }Q_T,\quad v=0\hbox{ on }\partial Q_T,$$ and the error functional $E_T:\A\mapsto\R^+$ $$E_T(u)=\frac12\int_{Q_T}(v_t^2+v_x^2)\,dx\,dt.$$ We also put $\T:\A\mapsto H^1_0(Q_T)$ for the linear, continuous mapping taking $u$ into its corrector $v$.
Assume that $u\in\A$ is such that $E(u)=0$. Then $v\equiv0$, and $$\chi_{Q_T\setminus q_T}\left(u_t-u_{xx}\right)\equiv0\hbox{ in }Q_T.$$ Hence if we take $f=(u_t-u_{xx})\in L^2(Q_T)$, then $$u_t-u_{xx}=\chi_{q_T}f\hbox{ in }Q_T,$$ and the restriction $f$ becomes the sought control.
\[defprinseg\]
1. We say that the datum $\overline u(T, x)$ is approximately controllable from $\overline u(0, x)$ through the subset $\omega$ of $\Omega$, if for every $\epsilon>0$, there is $u_\epsilon\in\A$ such that $E(u_\epsilon)<\epsilon$.
2. We say that the datum $\overline u(T, x)$ is exactly controllable from $\overline u(0, x)$ through the subset $\omega$ of $\Omega$, if there is $u\in\A$ such that $E(u)=0$.
3. The unique continuation holds if the only $v\in\im\T\subset H^1_0(Q_T)$ with $v\equiv0$ in $q_T$ and $$\label{uniquein}
\int_{Q_T\setminus q_T}(-Uv_t+U_xv_x)\,dx\,dt=0$$ for all $U\in\A_0$ is the trivial one $v\equiv0$.
4. We say that $E_T$ is an error functional if $E'_T(u)=0$ implies $E_T(u)=0$.
Just as in the boundary situation, the integral occurring in the unique continuation property is precisely the integral that appears when computing the Gateaux derivative $$\left.\frac{d E_T(u+\epsilon U)}{d\epsilon}\right|_{\epsilon=0}=\langle E'_T(u), U\rangle.$$ Indeed, because of linearity, $$\label{linealint}
V_{tt}+V_{xx}=\chi_{Q_T\setminus q_T}(U_t-U_{xx})\hbox{ in }Q_T, \quad V=0\hbox{ on }\partial Q_T,$$ if $V$ is the variation produced in $v$ by $U\in\A_0$ in $u$. Then $$\langle E'_T(u), U\rangle=\int_{Q_T}(v_tV_t+v_xV_x)\,dx\,dt.$$ By using (\[linealint\]), we obtain $$\langle E'_T(u), U\rangle=-\int_{Q_T\setminus q_T}(U_t-U_{xx})v\,dx\,dt.$$ Let us focus on the second term $$\int_{Q_T\setminus q_T}U_{xx}v\,dx\,dt.$$ A first integration by parts yields $$-\int_{Q_T\setminus q_T}U_xv_x\,dx\,dt+\int_{\partial(Q_T\setminus q_T)}U_xv\,dS.$$ But since $v=0$ around $\partial Q_T$, we find that the boundary integral equals $$\int_{\partial q_T} U_xv\,dS=-\int_{q_T}U_{xx}v\,dx\,dt.$$ We can always take $U$ to be arbitrary in $q_T$, and independent of time, so that $v\equiv0$ in $q_T$. Hence, altogether, $$\label{derivada}
\langle E'_T(u), U\rangle=-\int_{Q_T\setminus q_T}(U_tv+U_xv_x)\,dx\,dt.$$ This is the basic computation for an equivalence as in the boundary situation. The proof follows exactly along the same lines as with the boundary counterpart.
Let $T>0$, and $\omega\subset\Omega$ be given. Let also $\overline u\in\A$ furnish initial and final data. The following are equivalent:
1. the trace $\overline u(T, x)$ for $t=T$ of $\overline u$ is approximately controllable from $\overline u(0, x)$ through $\omega$;
2. the corresponding unique continuation holds;
3. $E_T$ is an error functional in the sense of Definition \[defprinseg\].
In this setting, it is also immediate to check that the unique continuation condition holds, so that we have approximate controllability as well. Just notice that the corrector $v$, being the solution in (\[lap\]), is a feasible direction $U$ because $H^2(Q_T)\cap H^1_0(Q_T)\subset\A_0$. By taking $U=v$ in (\[derivada\]), we conclude immediately that $v\equiv0$.
Final comments
==============
The formalism introduced here, and described in detail for the linear, homogeneous heat equation in (spatial) dimension $N=1$ can be formally extended, in a rather direct way, to many other frameworks because of its flexibility. The specific treatment of the unique continuation property may however change from situation to situation. For instance, it is well-known that for the wave equation the unique continuation property requires a certain size of the horizon $T$ due to the finite speed of propagation. Some of those situations include, but are not limited to:
- higher dimension $N>1$;
- inhomogeneous heat equation;
- wave equation;
- systems of differential equations;
- situations for degenerate equations;
- non-linear problems.
Especially in linear cases, this viewpoint naturally leads to an iterative approximation scheme based on a standard descent method. It has already been tested in various scenarios and, at least numerically, it performs very well (see [@arandapedregal], [@AM], [@AM2], [@AM-PP], [@AM-PP2]).
[99]{}
Aranda, E., Pedregal, P., A variational method for the numerical simulation of boundary controllability problems for the linear and semi-linear 1D wave equation, (submitted).
F. Ben Belgacem and S.M. Kaber, *On the Dirichlet boundary controllability of the 1-D heat equation: semi-analytical calculations and ill-posedness degre,* Inverse Problems, 27 (2011).
Bochev, P., Gunzburger, Max D., [*Least-squares finite element methods*]{}. Applied Mathematical Sciences, 166. Springer, New York, 2009.
C. Carthel, R. Glowinski and J.-L. Lions, *On exact and approximate Boundary Controllability for the heat equation: A numerical approach,* J. Optimization, Theory and Applications 82(3), (1994) 429–484.
J.M. Coron, *Control and Nonlinearity*, Mathematical Surveys and Monographs, AMS, Vol. 136, 2007.
Evans, L. C., *Partial Differential Equation*, Grad. Studies Math., Volume 19, AMS, 1999, Providence.
H.O. Fattorini and D.L. Russel, *Exact controllability theorems for linear parabolic equation in one space dimension*, Arch. Rational Mech. 43 (1971) 272-292.
E. Fernández-Cara and A. Münch, *Numerical null controllability of the 1D heat equation: primal algorithms.* Séma Journal, 61(1) (2013), 49–78.
A.V. Fursikov and O.Yu. Imanuvilov, *Controllability of Evolution Equations,* Lecture Notes Series, number 34. Seoul National University, Korea, (1996) 1–163.
R. Glowinski, *Numerical Methods for Nonlinear Variational Problems* Springer series in computational physics 1983.
R. Glowinski, J.L. Lions and J. He, *Exact and approximate controllability for distributed parameter systems: a numerical approach* Encyclopedia of Mathematics and its Applications, 117. Cambridge University Press, Cambridge, 2008.
S. Kindermann, *Convergence Rates of the Hilbert Uniqueness Method via Tikhonov regularization*, J. of Optimization Theory and Applications 103(3), (1999) 657-673.
G. Lebeau and L. Robbiano, *Contrôle exact de l’équation de la chaleur,* Comm. Partial Differential Equations 20 (1995), no. 1–2, 335–356.
S. Micu and E. Zuazua, *On the regularity of null-controls of the linear 1-d heat equation,* C. R. Acad. Sci. Paris, Ser. I 349 (2011) 673-677.
A. Münch and E. Zuazua, *Numerical approximation of null controls for the heat equation: ill-posedness and remedies,* Inverse Problems 26 (2010) no. 8 085018, 39pp.
A. Münch., A variational approach to approximate controls for systems with essential spectrum: application to membranal arch, Evolt. Eq. Cont. Th., 2 (2013) no 1, 119-151.
A. Münch, A least-squares formulation for the approximation of controls for the Stokes system, submitted.
A. Münch and P. Pedregal, *Numerical null controllability of the heat equation through a least squares and variational approach,* Eurp. J. Appl. Math., (accepted).
A. Münch and P. Pedregal, A least-squares formulation for the approximation of null controls for the Stokes system, C.R. Acad. Sci. Paris, SŽrie. I 351, 545-550 (2013).
P. Pedregal, *A variational perspective on controllability,* Inverse Problems 26 (2010) no. 1, 015004, 17pp.
D. L. Russell, *Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions*, SIAM Review, 20 (1978), 639-739.
[^1]: E.T.S. Ingenieros Industriales. Universidad de Castilla La Mancha. Campus de Ciudad Real (Spain). Research supported by MTM2010-19739 of the MCyT (Spain). e-mail:[pablo.pedregal@uclm.es ]{}
|
---
abstract: 'We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches. As an example, we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions. Applications to nonlinear partial differential equations will be discussed in a subsequent paper.'
address: 'University of Trento, Italy'
author:
- Emanuele Bottazzi
title: Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations
---
Introduction
============
The theory of distributions, pioneered by Dirac in [@dirac] and developed in the first half of the XX Century, has become one of the fundamental tools of functional analysis. In particular, the possibility to define the weak derivative of a non-differentiable function has allowed the formulation and the study of a wide variety of nonsmooth phenomena by the theory of partial differential equations. However, the lack of a nonlinear theory of distributions is a limiting factor both for the applications and for the theoretical study of nonlinear PDEs. On the one hand, in the description of some physical phenomena such as shock waves and relativistic fields, it arises the need to have some mathematical objects which cannot be formalized in the sense of distributions (we refer to [@colombeau; @advances] for some examples). On the other hand, the absence of a nonlinear theory of distributions poses some limitations in the study of nonlinear partial differential equations: while some nonlinear problems can be solved by studying the limit of suitable regularized problems, other problems do not allow for solutions in the sense of distributions (see for instance the discussion in [@evans; @nonlinear]).
In 1954, L. Schwartz proved that the absence of a nonlinear theory for distributions is intrinsic: more formally, the main theorem of [@schwartz] entails that there is no differential algebra $(A, +, \otimes, D)$ in which the real distributions $\mathcal{D}'$ can be embedded and the following conditions are satisfied:
1. $\otimes$ extends the product over $C^0$ functions;
2. $D$ extends the distributional derivative $\partial$;
3. the product rule holds: $D(u\otimes v) = (Du)\otimes v + u\otimes(Dv)$.
Despite this negative result, there have been many attempts at defining some notions of product between distributions (see for instance [@neutrix; @survey]). Following this line of research, Colombeau in 1983 proposed an organic approach to a theory of generalized functions [@colombeau; @1983]: Colombeau’s idea is to embed the distributions in a differential algebra with a good nonlinear theory, but at the cost of sacrificing the coherence between the product of the differential algebra with the product over $C^0$ functions. This approach has been met with interest and has proved to be a prolific field of research. For a survey of the approach by Colombeau and for recent advances, we refer to [@colombeau; @advances].
Research about generalized functions beyond distributions is also being carried out within the setting of nonstandard analysis. Possibly the earliest result in this sense is the proof by Robinson that the distributions can be represented by smooth functions of nonstandard analysis and by polynomials of a hyperfinite degree [@nsa; @robinson]. Distributions have also been represented by functions defined on hyperfinite domains, for instance by Kinoshita in [@moto] and, with a different approach, by Sousa Pinto and Hoskins in [@hyperfinite; @pinto]. Another nonstandard approach to the theory of generalized functions has been proposed by Oberbuggenberg and Todorov in [@oberbuggenberg] and further studied by Todorov et al. [@todorov2; @todorov3]. In this approach, the distributions are embedded in an algebra of asymptotic functions defined over a Robinson field of asymptotic numbers. Moreover, this algebra of asymptotic functions can be seen as a generalized Colombeau algebra where the set of scalars is an algebraically closed field rather than a ring with zero divisors. In this setting, it is possible to study generalized solutions to differential equations, and in particular to those with nonsmooth coefficient and distributional initial data [@nonsmo2; @nonsmo1].
Another theory of generalized functions oriented towards the applications in the field of partial differential equations and of the calculus of variations has been developed by Benci and Luperi Baglini. In [@ultrafunctions1] and subsequent papers [@benci; @ultramodel; @ultraschwartz; @ultraapps], the authors developed a theory of ultrafunctions, i.e. nonstandard vector spaces of a hyperfinite dimension that extend the space of distributions. In particular, the space of distributions can be embedded in an algebra of ultrafunctions $V$ such that the following inclusions hold: ${\mathscr{D}}'({\mathbb{R}}) \subset V \subset {\,\!^\astC^1({\mathbb{R}})}$ [@ultraschwartz]. This can be seen as a variation on a result by Robinson and Bernstein, that in [@invariant] showed that any Hilbert space $H$ can be embedded in a hyperfinite dimensional subspace of ${\,\!^\astH}$. In the setting of ultrafunctions, some partial differential equations can be formulated coherently by a Galerkin approximation, while the problem of finding the minimum of a functional can be turned to a minimization problem over a formally finite vector space. For a discussion of the applications of ultrafunctions to functional analysis, we refer to [@ultrafunctions1; @ultramodel; @ultraapps].
The idea of studying the solutions to a partial differential equation via a hyperfinite Galerkin approximation is not new. For instance, Capińsky and Cutland in [@capicutland; @statistic] studied statistical solutions to parabolic differential equations by discretizing the equation in space by a Galerkin approximation in an hyperfinite dimension. The nonstandard model becomes then a hyperfinite system of ODEs that, by transfer, has a unique nonstandard solution. From this solution, the authors showed that it is possible to define a standard weak solution to the original problem. In the subsequent [@capicutland1], the authors proved the existence of weak and statistical solutions to the Navier-Stokes equations in 3-dimensions by modelling the equations with a similar hyperfinite Galerkin discretization in space. This approach has spanned a whole line of research on the Navier-Stokes equations, concerning both the proof of the existence of solutions (see for instance [@capicutland; @noise; @cutland; @n+1]) and the definition and the existence of attractors (see for instance [@capicutland; @attractors; @cutland; @attractors]). One of the advantages of this approach is that, by a hyperfinite discretization in space, the nonstandard models have a unique global solution, even when the original problem does not. For a discussion of the relation between the uniqueness of the solutions of the nonstandard formulation and the non-uniqueness of the weak solutions of the original problem in the case of the Navier-Stokes equations, we refer to [@capicutland1].
In the theoretical study of nonlinear partial differential equations, sometimes problems do not allow even for a weak solution. However, the development of the notion of Young measures, originally introduced by L. C. Young in the field of optimal control in [@young1], has allowed for a synthetic characterization of the behaviour of the weak-$\star$ limit of the composition between a nonlinear continuous function and a uniformly bounded sequence in $L^\infty$. By enlarging the class of admissible solutions to include Young measures, one can define generalized solutions for some class of nonlinear problems as the weak-$\star$ limit of the solutions to a sequence of regularized problems [@demoulini; @evans; @nonlinear; @matete; @matete2; @plotnikov; @slemrod; @smarrazzo]. A similar approach can be carried out in the field of optimal controls, where generalized controls in the sense of Young measures can be defined as the measure-valued limit points of a minimizing sequence of controls. For an in-depth discussion of the role of Young measures as generalized solution to PDEs and as generalized controls, we refer to [@balder; @evans; @nonlinear; @sychev; @webbym]. In [@cutland; @controls3; @cutland; @controls2; @cutland; @controls], Cutland showed that Young measures can be interpreted also as the standard part of internal controls of nonstandard analysis. The possibility to obtain a Young measure from a nonstandard control allows to study generalized solutions to nonlinear variational problems by means of nonstandard techniques: such an approach has been carried out for instance by Cutland in the aforementioned papers, and by Tuckey in [@tuckey]. For a discussion of this field of research, we refer to [@neves].
Structure of the paper {#structure-of-the-paper .unnumbered}
----------------------
In this paper, we will discuss another theory of generalized functions of nonstandard analysis, hereafter called grid functions (see Definition \[def grid functions\]), that provide a coherent generalization both of the space of distributions and of a space of parametrized measures that extends the space of Young measures. In Section \[prelim\], we will define the space of grid functions, and recall some well-established nonstandard results that will be used throughout the paper. In particular, we will formulate in the setting of grid functions some known results regarding the relations between the hyperfinite sum and the Riemann integral, and the finite difference operators of an infinitesimal step and the derivative of a $C^1$ function.
In Section \[distri\], we will study the relations between the grid functions and the distributions, with the aim of proving that every distribution can be obtained from a suitable grid function. In order to reach this result, we will introduce an algebra of nonstandard test functions that can be seen as the grid function counterpart to the space ${\mathscr{D}}(\Omega)$ of smooth functions with compact support over $\Omega \subseteq {\mathbb{R}}^k$. By duality with respect to the algebra of test functions, we will define a module of grid distributions, and an an equivalence relation between grid functions (see Definition \[def equiv\] and Definition \[def bounded grid functions\]). We will then prove that the set of equivalence classes of grid distributions with respect to this equivalence relation is a real vector space that is isomorphic to the space of distributions. Afterwards, we will discuss how the finite difference operators generalize not only the usual derivative for $C^1$ functions, but also the distributional derivative. After having shown that the finite difference operator generalizes the distributional derivative, our study of the relations between grid functions and distributions concludes with a discussion of the Schwartz impossibility theorem. In particular, we will show that the space of distributions can be embedded in the space of grid functions in a way that
1. the product over the grid functions generalizes the pointwise product between continuous functions;
2. the finite difference is coherent with the distributional derivative modulo the equivalence relation induced by duality with test functions;
3. a discrete chain rule for products holds.
This theorem supports our claim that the space of grid functions provides a nontrivial generalization of the space of distributions.
In Section \[sez young\], we will embed the space of grid functions in the spaces ${\,\!^\astL^p}$ with $1 \leq p \leq \infty$, and we will study some properties of grid functions through this embedding. Moreover, we will discuss a generalization of the embedding of $L^2(\Omega)$ in a hyperfinite subspace of ${\,\!^\astL^2}(\Omega)$ due to Robinson and Bernstein [@invariant]. This classic result will be generalized in two directions:
1. for every $1 \leq p \leq \infty$, we will embed the spaces $L^p(\Omega)$ in the space of grid functions, which is a subspace of ${\,\!^\astL^p(\Omega)}$ of a hyperfinite dimension;
2. the above embedding is actually an embedding of the bigger space ${\mathscr{D}}'(\Omega)$ into a hyperfinite subspace of ${\,\!^\astL^p(\Omega)}$ for all $1 \leq p \leq \infty$.
Moreover, this embedding is obtained with different techniques from the original result by Robinson and Bernstein.
In the second part of Section \[sez young\], we will establish a correspondence between grid functions and parametrized measures, in a way that is coherent with the isomorphism between equivalence classes of grid distributions and distributions discussed in Section \[distri\]. The results discussed in Section \[sez young\] will be used in Section \[solutions\], where we will discuss the grid function formulation of partial differential equations, in Section \[selected applications\], where we will show selected applications of grid functions from different fields of functional analysis, and in the paper [@illposed], where we will study in detail a grid function formulation of a class of ill-posed partial differential equations with variable parabolicity direction.
In Section \[solutions\], we will discuss how to formulate partial differential equations in the space of grid functions in a way that coherently generalizes the standard notions of solutions. In particular, stationary PDEs will be given a fully discrete formulation, while time-dependent PDEs will be given a continuous-in-time and discrete-in-space formulation, resulting in a hyperfinite system of ordinary differential equations, as in the nonstandard formulation of the Navier-Stokes equations by Capińsky and Cutland.
In Section \[selected applications\], we will use the theory of grid functions developed so far to study two problems in the nonlinear theory of distributions and in the calculus of variations. These problems are classically studied within different frameworks, but we will show that each of these problems can be formulated in the space of grid functions in a way that the nonstandard solutions generalize the respective standard solutions.
Terminology and preliminary notions {#prelim}
===================================
In this section, we will now fix some notation and recall some results from nonstandard analysis that will be useful throughout the paper.
If $A \subseteq {\mathbb{R}}^k$, then $\overline{A}$ is the closure of $A$ with respect to any norm in ${\mathbb{R}}^k$, $\partial A$ is the boundary of $A$, and $\chi_A$ is the characteristic function of $A$. If $x \in {\mathbb{R}}$, then $\chi_x = \chi_{\{x\}}$. If $f : A \rightarrow {\mathbb{R}}$, ${\mathrm{supp\,}}f$ is the closure of the set $\{x \in A : f(x)\not=0\}$. These definitions are generalized as expected also to nonstandard objects.
We consider the following norms over ${\mathbb{R}}^k$: if $x \in {\mathbb{R}}^k$ or $x \in {{\,\!^\ast{\mathbb{R}}}}^k$, then $|x| = \sqrt{\sum_{i=1}^k x_i^2}$ is the euclidean norm, and $|x|_\infty = \max_{i = 1, \ldots, k} |x_i|$ is the maximum norm.
We will denote by $e_1, \ldots, e_k$ the canonical basis of ${\mathbb{R}}^k$. If $f : A \subseteq {\mathbb{R}}^m \rightarrow {\mathbb{R}}^k$, we will denote by $f_1, \ldots, f_k$ the hyperreal valued functions that satisfy the equality $f(x) = (f_1(x), \ldots, f_k(x))$ for all $x \in {{\,\!^\ast{\mathbb{R}}}}$.
In the sequel, $\Omega \subseteq {\mathbb{R}}^k$ will be an open set.
If $f \in C^1(\Omega)$, we will denote the partial derivative of $f$ in the direction $e_i$ by $\frac{df}{dx_i}$ or $D_i f$. If $\Omega \subseteq {\mathbb{R}}$, we will also write $f'$ for the derivative of $f$. We adopt the multi-index notation for partial derivatives and, if $\alpha$ is a multi-index, we will denote by $D^\alpha f$ the function $$D^\alpha f = \frac{\partial^{|\alpha|} f}{\partial x_1^{\alpha_1} \partial x_2^{\alpha_2}\ldots \partial x_k^{\alpha_k}}.$$ If $\alpha = (\alpha_1, \ldots, \alpha_k)$ is a multi-index, then $\alpha-e_i = (\alpha_1, \ldots, \alpha_i -1, \ldots, \alpha_k)$. If $f : [0,T]\times \Omega \rightarrow {\mathbb{R}}$, we will think of the first variable of $f$ as the time variable, denoted by $t$, and we will write $f_t$ for the derivative $\frac{\partial f}{\partial t}$.
We will often reference the following real vector spaces:
- ${C^0_b}({\mathbb{R}}) = \{ f \in C^0({\mathbb{R}}) : f \text{ is bounded and } \lim_{|x|\rightarrow \infty} f(x) = 0 \}$.
- $C^0_c(\Omega) = \{ f \in C^0_b(\Omega) : {\mathrm{supp\,}}f \subset \subset \Omega\}$.
- ${\mathscr{D}}(\Omega) = \{ f \in C^\infty(\Omega) : {\mathrm{supp\,}}f \subset \subset \Omega\}$.
- A real distribution over $\Omega$ is an element of ${\mathscr{D}}'(\Omega)$, i.e. a continuous linear functional $T : {\mathscr{D}}(\Omega)\rightarrow{\mathbb{R}}$. If $T$ is a distribution and $\varphi$ is a test function, we will denote the action of $T$ over $\varphi$ by $ {\langle}T, \varphi{\rangle_{{\mathscr{D}}(\Omega)}}$. When $T$ can be identified with a $L^p$ function, we will sometimes write $
\int_{\Omega} T\varphi dx
$ instead of ${\langle}T, \varphi{\rangle_{{\mathscr{D}}(\Omega)}}$.
If $T \in {\mathscr{D}}'({\mathbb{R}})$, we will denote the derivative of $T$ by $T'$ or $D T$. Recall that $T'$ is defined by the formula $
{\langle}DT, \varphi{\rangle_{{\mathscr{D}}(\Omega)}}= - {\langle}T, D\varphi{\rangle_{{\mathscr{D}}(\Omega)}}.
$ If $T \in {\mathscr{D}}'(\Omega)$ and $\alpha$ is a multi-index, the distribution $D^\alpha T$ is defined in a similar way: $
{\langle}D^\alpha T, \varphi{\rangle_{{\mathscr{D}}(\Omega)}}= (-1)^{|\alpha|} {\langle}T, D^\alpha \varphi{\rangle_{{\mathscr{D}}(\Omega)}}.
$
- In the sequel, measurable will mean measurable with respect to ${\mu_L}$, the Lebesgue measure over ${\mathbb{R}}^n$. Consider the equivalence relation given by equality almost everywhere: two measurable functions $f$ and $g$ are equivalent if ${\mu_L}(\{x\in\Omega : f(x) \not = g(x)\})=0$. We will not distinguish between the function $f$ and its equivalence class, and we will say that $f = g$ whenever the functions $f$ and $g$ are equal almost everywhere.
For all $1 \leq p < \infty$, $L^p(\Omega)$ is the set of equivalence classes of measurable functions $f: \Omega \rightarrow {\mathbb{R}}$ that satisfy $$\int_{\Omega} |f|^p dx < \infty.$$ If $f \in L^p(\Omega)$, the $L^p$ norm of $f$ is defined by $${\Vertf\Vert}_p^p = \int_{\Omega} |f|^p dx.$$
$L^\infty(\Omega)$ is the set of equivalence classes of measurable functions that are essentially bounded: we will say that $f: \Omega \rightarrow {\mathbb{R}}$ belongs to $L^\infty(\Omega)$ if there exists $y \in {\mathbb{R}}$ such that ${\mu_L}(\{x\in\Omega:|f(x)| > y\}) = 0$. In this case, $${\Vertf\Vert}_\infty = \inf\{y \in {\mathbb{R}}: {\mu_L}(\{x\in\Omega:f(x) > y\}) = 0 \}.$$ If $1 < p < \infty$, we recall that $p'$ is defined as the unique solution to the equation $$\frac{1}{p}+\frac{1}{p'}=1,$$ while $1' = \infty$ and $\infty' = 1$.
- It is well-established that the distributional derivative allows to define a notion of weak derivative for $L^p$ functions. $L^2$ functions whose weak derivatives up to order $p < \infty$ are still $L^2$ functions are of a particular relevance in the study of partial differential equations. For $p \in {\mathbb{N}}$, $p \geq 1$, the space $H^p(\Omega)$ is defined as $$H^p(\Omega) = \{ f \in L^2(\Omega) : D^\alpha f \in L^2(\Omega) \text{ for every } \alpha \text{ with }|\alpha|\leq p \}.$$ We also consider the following norm over the space $H^p(\Omega)$: $${\Vertf\Vert}_{H^p} = \sum_{|\alpha|\leq p} {\VertD^\alpha f\Vert}_2,$$ and we will call it the $H^p$ norm. Recall also that $H_0^p(\Omega) \subset H^p(\Omega)$ is defined as the closure of ${\mathscr{D}}(\Omega)$ in $H^p(\Omega)$ with respect to the $H^p$ norm. For further properties of the weak derivative and of the spaces $H^p(\Omega)$ and $H_0^p(\Omega)$, we refer to [@strichartz; @tartar].
- ${\mathbb{M}}({\mathbb{R}}) = \{ \nu : \nu \text{ is a Radon measure over } {\mathbb{R}}\text{ satisfying } |\nu|({\mathbb{R}})<+\infty \}$.
- ${{\mathbb{M}}^{\mathbb{P}}}({\mathbb{R}}) = \{ \nu \in {\mathbb{M}}({\mathbb{R}}) : \nu \text{ is a probability measure}\}$.
- Following [@balder; @ball; @webbym] and others, measurable functions $\nu : \Omega \rightarrow {{\mathbb{M}}^{\mathbb{P}}}({\mathbb{R}})$ will be called Young measures. Measurable functions $\nu : \Omega \rightarrow {\mathbb{M}}({\mathbb{R}})$ will be called parametrized measures, even though in the literature the term parametrized measure is used as a synonym for Young measure. If $\nu$ is a parametrized measure and if $x \in \Omega$, we will write $\nu_x$ instead of $\nu(x)$.
Throughout the paper, we will work with a $|{\mathscr{D}_{\mathbb{X}}}'(\Omega)|$-saturated nonstandard model ${{\,\!^\ast{\mathbb{R}}}}$, and we will assume familiarity with the basics of nonstandard analysis. For an introduction on the subject, we refer for instance to Goldblatt [@go], but see also [@nsa; @theory; @apps; @da; @keisler; @nsa; @working; @math; @nsa; @robinson]. The definitions introduced so far are extended by transfer, as usual, so for instance ${{\,\!^\ast{\mathbb{R}}}}^k$ is the set of $k$-uples of hyperreal numbers, and $\{e_1, \ldots, e_k\}$ is a basis of ${{\,\!^\ast{\mathbb{R}}}}^k$.
We will denote by ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ the set of finite numbers in ${{\,\!^\ast{\mathbb{R}}}}$, i.e. ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}= \{ x \in {{\,\!^\ast{\mathbb{R}}}}: x$ is finite$\}$. For any $x, y \in {\,\!^\ast{\mathbb{R}}}$ we will write $x {\approx}y$ to denote that $x-y$ is infinitesimal, we will say that $x$ is finite if there exists a standard $M \in {\mathbb{R}}$ satisfying $|x| < M$, and we will say that $x$ is infinite whenever $x$ is not finite. The notion of finiteness can be extended componentwise to elements of ${\,\!^\ast{\mathbb{R}}^k}$ whenever $k\in{\mathbb{N}}$: we will say that $x \in {\,\!^\ast{\mathbb{R}}^k}$ is finite iff all of its components are finite, and we define ${\,\!^\circx} = ({\,\!^\circx_1}, {\,\!^\circx_2}, \ldots, {\,\!^\circx_k}) \in {\mathbb{R}}^k$. Similarly, if $x, y \in {\,\!^\ast{\mathbb{R}}^k}$, we will write $x {\approx}y$ if $|x-y|{\approx}0$ (notice that this is equivalent to $|x-y|_\infty {\approx}0$).
If $k$ is finite, for all finite $x \in {{\,\!^\ast{\mathbb{R}}}}^k$ we will denote by ${\,\!^\circx}\in{\mathbb{R}}^k$ the standard part of $x$, i.e. the unique vector in ${\mathbb{R}}^k$ closest to $x$. Similarly, for any $A \subseteq {{\,\!^\ast{\mathbb{R}}}}^k$, ${\,\!^\circX}$ will denote the set of the standard parts of the finite elements of $X$.
The space of grid functions is defined as the space of functions whose domain is a uniform hyperfinite grid.
\[def grid functions\] Let $N_0\in{\,\!^\ast{\mathbb{N}}}$ be an infinite hypernatural number. Set $N = N_0!$ and $\varepsilon = 1/N$, and define $${\mathbb{X}}= \{ n\varepsilon : n \in [-N^2, N^2] \cap {\,\!^\ast{\mathbb{Z}}}\}.$$ We will say that an internal function $f : {\mathbb{X}}^k \rightarrow {{\,\!^\ast{\mathbb{R}}}}$ is a grid function and, if $A \subseteq {\mathbb{X}}^k$ is internal, we denote by ${\mathbb{G}({A})}$ the space of grid functions defined over $A$: ${\mathbb{G}({A})} = \mathbf{Intl}({{\,\!^\ast{\mathbb{R}}}}^{A}) = \{ f : A \rightarrow {{\,\!^\ast{\mathbb{R}}}}\text{ and } f \text{ is internal}\}.$
Some elements of nonstandard topology {#ns topology}
-------------------------------------
In the next definition, we will give a canonical extension of subsets of the standard euclidean space ${\mathbb{R}}^k$ to internal subsets of the grid ${\mathbb{X}}^k$.
For any $A \subseteq {\mathbb{R}}^k$, we define $A_{\mathbb{X}}= {\,\!^\astA} \cap {\mathbb{X}}^k$. Notice that $A_{\mathbb{X}}$ is an internal subset of ${\mathbb{X}}^k$, and in particular it is hyperfinite.
In general, we expect that for a generic set $A \subseteq {{\,\!^\ast{\mathbb{R}}}}^k$, ${\,\!^\circA_{\mathbb{X}}} \not = \overline{A}$. For instance, if $A \cap {\mathbb{Q}}^k = \emptyset$, then $A_{\mathbb{X}}= {\,\!^\circA_{\mathbb{X}}} = \emptyset$. In this section, we will prove that if $A$ is an open set, then indeed $A_{\mathbb{X}}$ is a faithful extension of $A$, in the sense that ${\,\!^\circA_{\mathbb{X}}} = {\,\!^\circ\overline{A}_{\mathbb{X}}} = \overline{A}$. Moreover, there is a nice characterization of the boundary of $A_{\mathbb{X}}$ which is projected to the boundary of $A$ via the standard part map.
In order to prove these results, we need to show that for an open set $A$, $\mu(x) \cap {\,\!^\astA} \not = \emptyset$ is equivalent to $\mu(x) \cap A_{\mathbb{X}}\not = \emptyset$ for all $x\in \overline{A}$.
\[aperti\] If $A \subseteq {\mathbb{R}}^k$ is an open set, then for all $x \in \overline{A}$ it holds $$\label{closed3}
\mu(x) \cap {\,\!^\astA} \not= \emptyset \Longleftrightarrow \mu(x) \cap A_{\mathbb{X}}\not= \emptyset.$$
Let $x \in \overline{A}$. The hypothesis $N=N_0!$ for an infinite $N_0 \in {\,\!^\ast{\mathbb{N}}}$ ensures that for all $p \in {\mathbb{Q}}^k$, $p \in{\mathbb{X}}^k$. As a consequence, for all $n \in {\mathbb{N}}$ there exists $p \in A_{\mathbb{X}}$ with $|x-p|<1/n$. By overspill, for some infinite $M \in {\,\!^\ast{\mathbb{N}}}$ there exists $p \in A_{\mathbb{X}}$ that satisfies $|x-p|<1/M$.
We want to define a boundary for the set $A_{\mathbb{X}}$ that is coherent with the usual notion of boundary for $A$. The idea is to define the ${\mathbb{X}}$-boundary of $A_{\mathbb{X}}$ as the set of points of $A_{\mathbb{X}}$ that are within a step of length $\varepsilon$ from a point of ${\,\!^\astA}^c$.
Let $A \subseteq {\mathbb{R}}^k$. We define the ${\mathbb{X}}$-boundary of $A_{\mathbb{X}}$ as $$\partial_{\mathbb{X}}A_{\mathbb{X}}= \{ x \in A_{\mathbb{X}}: \exists y \in {\,\!^\astA}^c \text{ satisfying } |x-y|_\infty\leq \varepsilon \}.$$
This definition is coherent with the usual boundary of an open set.
\[topologia bella\] Let $A \subseteq {\mathbb{R}}^k$ be an open set. Then ${\,\!^\circA_{\mathbb{X}}} = \overline{A}$ and ${\,\!^\circ (\partial_{\mathbb{X}}A_{\mathbb{X}})} = \partial A$.
The equality ${\,\!^\circA_{\mathbb{X}}} = \overline{A}$ is a consequence of Lemma \[aperti\].
Recall the nonstandard characterization of the boundary of $A$: $x \in \partial A$ if and only if there exists $y \in {\,\!^\astA}$, $x \not = y$, and $z \in {\,\!^\astA}^c$ with $x {\approx}y {\approx}z$. This is sufficient to conclude that $\partial A \supseteq {\,\!^\circ (\partial_{\mathbb{X}}A_{\mathbb{X}})}$.
To prove that the other inclusion holds, we only need to show that if $x \in \partial A$, then there exists $y \in \partial_{\mathbb{X}}A_{\mathbb{X}}$ with $y {\approx}x$. Let $x \in \partial A$: since $A_{\mathbb{X}}$ is a hyperfinite set, we can pick $y \in A_{\mathbb{X}}$ satisfying $$|{\,\!^\astx}-y|_\infty = \min_{z \in A_{\mathbb{X}}}\{|{\,\!^\astx}-z|_\infty\}.$$ Our choice of $y$ and the hypothesis that $x \in \partial A$ ensure that $y \not = {\,\!^\astx}$ and $|{\,\!^\astx}-y|_\infty {\approx}0$. We claim that $y \in \partial_{\mathbb{X}}A_{\mathbb{X}}$. In fact, suppose towards a contradiction that $y \not\in \partial_{\mathbb{X}}A_{\mathbb{X}}$: in this case, for all $z \in {\,\!^\astA}^c$, $|y-z|_\infty>\varepsilon$ and, in particular, $|{\,\!^\astx}-y|_\infty > \varepsilon$. Let ${\,\!^\astx}-y = \sum_{i = 1}^k a_i e_i$, let $I = \{i \leq k : |a_i| = |{\,\!^\astx}-y|_\infty\}$, and define $$\tilde{y} = y+\sum_{i \in I} \frac{a_i}{|a_i|}\varepsilon e_i.$$ Since $|\tilde{y}-y|_\infty = \varepsilon$ and since $y \not\in \partial_{\mathbb{X}}A_{\mathbb{X}}$, then $\tilde{y} \in A_{\mathbb{X}}$. Moreover, $$|{\,\!^\astx}-\tilde{y}|_\infty = \max_{i\not\in I}\{|{\,\!^\astx}-y|-\varepsilon, |a_i|\} < |x-y|_\infty,$$ contradicting $|{\,\!^\astx}-y|_\infty = \min_{z \in A_{\mathbb{X}}}\{|{\,\!^\astx}-z|_\infty\}$.
Let $\Omega \subseteq {\mathbb{R}}^k$ be an open set. By Proposition \[topologia bella\], this hypothesis is sufficient to ensure the equalities ${\,\!^\circ{\Omega_{{\mathbb{X}}}}} = {\,\!^\circ\overline{\Omega}_{\mathbb{X}}} = \overline{\Omega}$ and ${\,\!^\circ(\partial_{\mathbb{X}}{\Omega_{{\mathbb{X}}}})} = {\,\!^\circ(\partial_{\mathbb{X}}\overline{\Omega}_{\mathbb{X}})} = \partial \Omega$.
Derivatives and integrals of grid functions
-------------------------------------------
Since grid functions are defined on a discrete set, there is no notion of derivative for grid functions. However, in nonstandard analysis it is fairly usual to replace the derivative by a finite difference operator with an infinitesimal step.
\[fd\] For an internal grid function $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, we define the $i$-th forward finite difference of step $\varepsilon$ as $${\mathbb{D}}_i f(x)= {\mathbb{D}}_i^+ f(x)= \frac{f(x+\varepsilon e_i)-f(x)}{\varepsilon }$$ and the $i$-th backward finite difference of step $\varepsilon$ as $${\mathbb{D}}_i^- f(x) = \frac{f(x)-f(x-\varepsilon e_i)}{\varepsilon }.$$ If $n \in {\,\!^\ast{\mathbb{N}}}$, ${\mathbb{D}}^n_i$ is recursively defined as ${\mathbb{D}}_i({\mathbb{D}}_i^{n-1})$ and, if $\alpha$ is a multi-index, then ${\mathbb{D}}^\alpha$ is defined as expected: $${\mathbb{D}}^\alpha f = {\mathbb{D}}_1^{\alpha_1} {\mathbb{D}}_2^{\alpha_2} \ldots {\mathbb{D}}_n^{\alpha_k} f.$$ These definitions can be extended to ${\mathbb{D}}^-$ by replacing every occurrence of ${\mathbb{D}}$ with ${\mathbb{D}}^-$.
For further details about the properties of the finite difference operators, we refer to Hanqiao, St. Mary and Wattenberg [@watt], to Keisler [@keisler] and to van den Berg [@imme2; @imme1].
Notice that if $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ and if $\alpha$ is a standard multi-index, then ${\mathbb{D}}^{\alpha} f$ is not defined on all of ${\Omega_{{\mathbb{X}}}}$. However, if we let $${\Omega_{{\mathbb{X}}}}^\alpha = \{ x \in {\Omega_{{\mathbb{X}}}}: {\mathbb{D}}^{\alpha} f \text{ is defined} \}
= \{ x \in {\Omega_{{\mathbb{X}}}}: x+\alpha\varepsilon\in{\Omega_{{\mathbb{X}}}}\}$$ then we have ${\,\!^\circ{\Omega_{{\mathbb{X}}}}^\alpha} = {\,\!^\circ{\Omega_{{\mathbb{X}}}}} = \overline{\Omega}$, since for every $x \in {\Omega_{{\mathbb{X}}}}^\alpha$ we have $x+\alpha\varepsilon\in{\Omega_{{\mathbb{X}}}}$ and $x+\alpha\varepsilon {\approx}x$ by the standardness of $\alpha$.
In a similar way, if we define $$\partial_{\mathbb{X}}^\alpha {\Omega_{{\mathbb{X}}}}= \{ x \in {\Omega_{{\mathbb{X}}}}: x+\alpha\varepsilon\in\partial_{\mathbb{X}}{\Omega_{{\mathbb{X}}}}\},$$ then, from the relation $x+\alpha\varepsilon {\approx}x$ and from Proposition \[topologia bella\], we deduce that it holds also the equality ${\,\!^\circ\partial_{\mathbb{X}}^\alpha {\Omega_{{\mathbb{X}}}}} = {\,\!^\circ\partial_{\mathbb{X}}{\Omega_{{\mathbb{X}}}}} = \partial\Omega$. In section \[linear pdes\], we will use this result in order to show show how Dirichlet boundary conditions can be expressed in the sense of grid functions. Since ${\Omega_{{\mathbb{X}}}}^\alpha$ is a faithful extension of $\Omega$ in the sense of proposition \[topologia bella\], we will often abuse notation and write ${\mathbb{D}}^{\alpha} f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ instead of the correct ${\mathbb{D}}^\alpha f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}^\alpha})}$.
In the setting of grid functions, integrals are replaced by hyperfinite sums.
\[def inner\] Let $f,g : {\,\!^\ast\Omega} \rightarrow {{\,\!^\ast{\mathbb{R}}}}$ and let $A \subseteq {\Omega_{{\mathbb{X}}}}\subseteq {\mathbb{X}}^k$ be an internal set. We define $$\int_{A} f(x) d{\mathbb{X}}^k = \varepsilon^k \cdot \sum_{x \in A} f(x)$$ and $$ \langle f, g \rangle
= \displaystyle \int_{{\mathbb{X}}^k} f(x) g(x) d{\mathbb{X}}^k = \displaystyle \varepsilon^k \cdot \sum_{x \in {\mathbb{X}}^k} f(x)g(x),
$$ with the convention that, if $x \not \in {\,\!^\ast\Omega}$, $f(x) = g(x) = 0$.
A simple calculation shows that the fundamental theorem of calculus holds. In particular, for all $f:{\mathbb{G}({{\mathbb{X}}})}\rightarrow {{\,\!^\ast{\mathbb{R}}}}$ and for all $a, b \in {\mathbb{X}}$, $b < N$, we have $$\varepsilon \sum_{x=a}^{b} {\mathbb{D}}f(x) = f(b+\varepsilon)-f(a) \text{ and } {\mathbb{D}}\left( \varepsilon \sum_{x=a}^b f(x) \right) = f(b+\varepsilon).$$ The next Lemma is a well-known compatibility result between the grid integral and the Riemann integral of continuous functions.
\[equivalenza integrali\] Let $A \subset {\mathbb{R}}^k$ be a compact set. If $f \in C^0(A)$, then $$\int_{A_{{\mathbb{X}}}} {\,\!^\astf}(x) d{\mathbb{X}}^k {\approx}\int_{A} f(x) dx.$$
See for instance Section 1.11 of [@nsa; @working; @math].
In order to introduce the grid functions that correspond to real distributions, we will use a notion of duality induced by the inner product \[def inner\].
For any $V \subseteq {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, we define $$V' = \{ f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} : \langle g, f \rangle \text{ is finite for all } g \in V\}.$$
For any $V \subseteq {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, $V'$ with pointwise sum and product is a module over ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$. Moreover, $V'/\equiv$ inherits a structure of real vector space from $V'$.
Notice that, contrary to what happened for the space $S^0({\Omega_{{\mathbb{X}}}})$, $V'$ is not an algebra, since in general the hypothesis $f, g \in V'$ is not sufficient to ensure that $fg \in V'$.
$S^\alpha$ functions and $C^\alpha$ functions {#S-continuous}
---------------------------------------------
We will now define functions of class $S^\alpha$, that will be the grid functions counterpart of functions of class $C^\alpha$. This definition is grounded upon the well-known notion of S-continuity, as S-continuity has been widely used as a bridge between discrete functions of nonstandard analysis and standard continuous functions.
We will say that $x \in {\Omega_{{\mathbb{X}}}}$ is nearstandard in $\Omega$ iff there exists $y \in \Omega$ such that $x {\approx}y$.
We say that a function $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ is of class $S^0$ iff $f(x)$ is finite for some nearstandard $x \in {\Omega_{{\mathbb{X}}}}$ and for every nearstandard $x, y \in {\Omega_{{\mathbb{X}}}}$, $x {\approx}y$ implies $f(x) {\approx}f(y)$.
We also define functions of class $S^\alpha$ for every multi-index $\alpha$:
- $f$ is of class $S^\alpha({\Omega_{{\mathbb{X}}}})$ if ${\mathbb{D}}^\alpha f \in S^0({\Omega_{{\mathbb{X}}}})$;
- $f$ is of class $S^{\infty}({\Omega_{{\mathbb{X}}}})$ if ${\mathbb{D}}^\alpha f \in S^0({\Omega_{{\mathbb{X}}}})$ for any standard multi-index $\alpha$.
Notice that if $f \in S^\alpha({\Omega_{{\mathbb{X}}}})$ for some standard multi-index $\alpha$, then $f(x)$ is finite at all nearstandard $x \in {\Omega_{{\mathbb{X}}}}$.
In the study of $S$-continuous functions, we find it useful to introduce the following equivalence relation.
\[equiv1\] Let $f, g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. We say that $f \equiv_S g$ iff $(f-g)(x){\approx}0$ for all nearstandard $x \in {\Omega_{{\mathbb{X}}}}$. From the properties of ${\approx}$, it can be proved that $\equiv_S$ is an equivalence relation. We will denote by $\pi_S$ the projection from ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ to the quotient space ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} / \equiv_S$, and will denote by $[f]_S$ the equivalence class of $f$ with respect to $\equiv_S$.
The rest of this section is devoted to the proof that the quotient $S^\alpha({\Omega_{{\mathbb{X}}}}) / \equiv_S$ is real algebra isomorphic to the algebra of $C^\alpha$ functions over $\Omega$. This result is a reformulation in the language of grid functions of some results by van den Berg [@imme2] and by Wattenberg, Hanqiao, and St. Mary [@watt].
\[lemma piccolo\] For every standard multi-index $\alpha$, $S^\alpha({\Omega_{{\mathbb{X}}}})$ with pointwise sum and product is an algebra over ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, and $S^\alpha({\Omega_{{\mathbb{X}}}}) / \equiv_S$ inherits a structure of real algebra from $S^\alpha({\Omega_{{\mathbb{X}}}})$.
The only non-trivial assertion that needs to be verified is closure of $S^\alpha({\Omega_{{\mathbb{X}}}})$ with respect to pointwise product. This property is a consequence of Proposition 2.6 of [@imme2].
\[teorema isomorfismo\] $S^0({\Omega_{{\mathbb{X}}}})/\equiv_S$ is a real algebra isomorphic to $C^0(\Omega)$. The isomorphism is given by $i[f]_S={\,\!^\circf}$. The inverse of $i$ is the function $i^{-1}(f) = [{\,\!^\astf}_{|{\mathbb{X}}}]_S$.
If $f \in S^0({\Omega_{{\mathbb{X}}}})$, then it is well-known that ${\,\!^\circf}$ is a well-defined function and that ${\,\!^\circf} \in C^0(\Omega)$. Surjectivity of ${\,\!^\circ}$ is a consequence of Lemma II.6 of [@watt]. Since $$\ker({\,\!^\circ}) = \left\{ f \in S^\alpha({\Omega_{{\mathbb{X}}}}) : f(x) {\approx}0 \text{ for all finite } x \in {\Omega_{{\mathbb{X}}}}\right\} = [0]_S,$$ we deduce that $i$ is injective and surjective. Since ${\,\!^\circ(x+y)} = {\,\!^\circx}+{\,\!^\circy}$ and ${\,\!^\circ(xy)} = {\,\!^\circx}{\,\!^\circy}$ for all $x, y \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, $i_\alpha$ is an isomorphism of real algebras.
We will now show that, for grid functions of class $S^\alpha$, the finite difference operators ${\mathbb{D}}_i^+$ and ${\mathbb{D}}_i^-$ assume the role of the usual partial derivative for $C^\alpha$ functions. In particular, these finite difference operators can be seen as generalized derivatives.
\[teorema equivalenza derivate\] For all $1 \leq i \leq k$ and for all standard multi-indices $\alpha$ with $\alpha_i\geq1$, the diagrams $$\begin{array}{ccc}
\begin{array}{ccc}
S^{\alpha}({\Omega_{{\mathbb{X}}}}) & \stackrel{{\mathbb{D}}^+_i}{\longrightarrow} & S^{\alpha-e_i}({\Omega_{{\mathbb{X}}}}) \\
i \circ \pi_S \downarrow & & \downarrow i \circ \pi_S\\
C^\alpha(\Omega) & \stackrel{D_i}{\longrightarrow} & C^{\alpha-e_i}(\Omega)
\end{array}
&
\text{ and }
&
\begin{array}{ccc}
S^\alpha({\Omega_{{\mathbb{X}}}}) & \stackrel{{\mathbb{D}}^-_i}{\longrightarrow} & S^{\alpha-e_i}({\Omega_{{\mathbb{X}}}}) \\
i \circ \pi_S \downarrow & & \downarrow i \circ \pi_S\\
C^\alpha(\Omega) & \stackrel{D_i}{\longrightarrow} & C^{\alpha-e_i}(\Omega)
\end{array}
\end{array}$$ commute.
By Theorem \[teorema isomorfismo\], if $f \in S^\alpha({\Omega_{{\mathbb{X}}}})\subseteq S^0({\Omega_{{\mathbb{X}}}})$ then $(i_\alpha \circ \pi_S)(f) = {\,\!^\circf}$ and, by Lemma II.7 of [@watt], ${\,\!^\circ({\mathbb{D}}^{\pm}_i f)} = D_i {\,\!^\circf}$.
By Theorem \[teorema equivalenza derivate\], the isomorphism $i$ defined in Theorem \[teorema isomorfismo\] induces an isomorphism between $S^\alpha({\Omega_{{\mathbb{X}}}}) / \equiv_S$ and $C^\alpha(\Omega)$ as real algebras.
\[corollario isomorfismo\] For any multi-index $\alpha$, the isomorphism $i$ restricted to $S^\alpha({\Omega_{{\mathbb{X}}}}) / \equiv_S$ induces an isomorphism between $S^\alpha({\Omega_{{\mathbb{X}}}}) / \equiv_S$ and $C^\alpha(\Omega)$ as real algebras.
Thanks to this isomorphism, if $f \in S^\alpha({\Omega_{{\mathbb{X}}}})$, we can identify the equivalence class $[f]_S$ with the standard function ${\,\!^\circf} \in C^\alpha(\Omega)$.
Grid functions as generalized distributions\[distri\]
=====================================================
In this section, we will study the relations between the space of grid functions and the space of distributions. In particular, we will prove that the space of grid functions can be seen as generalization of the space of distributions, and that the operators ${\mathbb{D}}^+$ and ${\mathbb{D}}^-$ coherently extend the distributional derivative to the space of grid functions.
In order to prove the above results, we start by defining a projection from an external ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$-submodule of ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ to the space of distributions. This projection is defined by duality with an external ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$-algebra of grid functions that is a counterpart to the space of test functions.
We define the algebra of test functions over ${\Omega_{{\mathbb{X}}}}$ as follows: $${\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}}) =
\left\{ f \in S^{\infty}({\Omega_{{\mathbb{X}}}}) : {\,\!^\circ{\mathrm{supp\,}}f} \subset \subset \Omega \right\}.$$
The above definition provides a nonstandard counterpart of the usual space of smooth functions with compact support.
\[lemma test\] The isomorphism $i$ defined in Theorem \[teorema isomorfismo\] induces an isomorphism between the real algebras ${\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}}) / \equiv_S$ and ${\mathscr{D}}(\Omega)$. The isomorphism preserves integrals, i.e. for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, it holds the equality $$\label{uguaglianza integrali}
{\,\!^\circ \int_{{\Omega_{{\mathbb{X}}}}} \varphi d{\mathbb{X}}^k } = \int_{\Omega} i[\varphi]_S dx.$$ Moreover, if $\varphi \in {\mathscr{D}}(\Omega)$, then ${\,\!^\ast\varphi}_{|{\mathbb{X}}} \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, so that $i^{-1}(\varphi) = \left[{\,\!^\ast\varphi}_{|{\mathbb{X}}}\right]_S \cap {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$.
From Theorem \[teorema isomorfismo\], from Theorem \[teorema equivalenza derivate\] and from the definition of ${\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, we can conclude that the hypothesis $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ ensures that $i[\varphi] \in {\mathscr{D}}(\Omega)$. Since ${\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}}) \subset S^0({\Omega_{{\mathbb{X}}}})$, injectivity of $i$ is a consequence of Theorem \[teorema isomorfismo\].
Similarly, surjectivity of $i$ can be deduced from Theorem \[teorema isomorfismo\] and from Theorem \[teorema equivalenza derivate\]. In fact, suppose towards a contradiction that there exists $\psi \in {\mathscr{D}}(\Omega)$ such that $\psi \not = i[\varphi]$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}(\Omega)$. Since $\psi \in C^0(\Omega)$, Theorem \[teorema isomorfismo\] ensures that there exists $\phi \in S^0({\Omega_{{\mathbb{X}}}})$ with $i[\phi]=\psi$. If $\phi \not \in S^\infty({\Omega_{{\mathbb{X}}}})$, then for some standard multi-index $\alpha$, ${\mathbb{D}}^\alpha \phi \not \in S^0({\Omega_{{\mathbb{X}}}})$, contradicting Theorem \[teorema equivalenza derivate\]. As a consequence, $i$ is an isomorphism between ${\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}}) / \equiv_S$ and ${\mathscr{D}}(\Omega)$.
Equality \[uguaglianza integrali\] is a consequence of the hypothesis ${\,\!^\circ{\mathrm{supp\,}}\varphi} \subset \subset \Omega$ and of Lemma \[equivalenza integrali\].
Now let $\varphi \in {\mathscr{D}}(\Omega)$: by Theorem \[teorema isomorfismo\] and by Theorem \[teorema equivalenza derivate\], ${\,\!^\ast\varphi}_{|{\mathbb{X}}} \in S^\infty({\Omega_{{\mathbb{X}}}})$. Let $A = {\mathrm{supp\,}}\varphi$: since $A$ is the closure of an open set, by Proposition \[topologia bella\] ${\,\!^\circA_{\mathbb{X}}} = A \subset \subset \Omega$, from which we deduce ${\,\!^\ast\varphi}_{|{\mathbb{X}}} \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$. As a consequence, $i^{-1}(\varphi) = \left[{\,\!^\ast\varphi}_{|{\mathbb{X}}}\right]_S \cap {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, as we claimed.
The duality with respect to the space of test functions can be used to define an equivalence relation on the space of grid functions. This equivalence relation plays the role of a weak equality.
\[def equiv\] Let $f, g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. We say that $f \equiv g$ iff for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ it holds $
\langle f, \varphi \rangle {\approx}\langle g, \varphi \rangle.
$ We will call $\pi$ the projection from ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ to the quotient $
{\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} / \equiv,
$ and we will denote by $[f]$ the equivalence class of $f$ with respect to $\equiv$.
The new equivalence relation $\equiv$ is coarser than $\equiv_S$.
For all $f, g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, $f \equiv_S g$ implies $f \equiv g$.
We will show that $f \equiv_S g$ implies $\langle f-g, \varphi\rangle {\approx}0$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$: by linearity of the hyperfinite sum, this result is equivalent to $f \equiv g$.
Let $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, and let $\eta = \max_{x \in {\mathrm{supp\,}}\varphi} \{|(f-g)(x)|\}$. The hypothesis that $f \equiv_S g$ and the hypothesis that ${\,\!^\circ{\mathrm{supp\,}}\varphi}$ is bounded are sufficient to ensure that $\eta {\approx}0$. As a consequence, we have the following inequalities $$\displaystyle
\left| \langle f-g,\varphi\rangle \right|
\leq
\displaystyle
\langle |f-g|,|\varphi|\rangle
\leq
\displaystyle
|\eta| \int_{{\Omega_{{\mathbb{X}}}}} |\varphi(x)| d{\mathbb{X}}^k {\approx}0,$$ that are sufficient to conclude the proof.
We can now define the grid distributions as the the dual of ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ with respect to the inner product introduced in Definition \[def inner\].
\[def bounded grid functions\] The ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$-module $${\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}})
=
\left\{ f\in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}\ |\ \langle f, \varphi \rangle \text{ is finite for all } \varphi\in{\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})\right\}$$ is called the module of grid distributions.
The rest of this section is devoted to the proof that the quotient ${\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}}) / \equiv$ is real vector space isomorphic to the space of distributions ${\mathscr{D}}'(\Omega)$. The following characterization of grid distributions will be used in the proof of this isomorphism.
\[character\] The following are equivalent:
1. $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$;
2. $\langle f, \varphi \rangle {\approx}0$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ satisfying $\varphi(x) {\approx}0$ for all $x \in {\Omega_{{\mathbb{X}}}}$;
3. $\langle f, {\,\!^\ast\varphi} \rangle \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ for all $\varphi \in {\mathscr{D}}(\Omega)$.
\(1) implies (2), by contrapositive. Suppose that $\langle f, \varphi \rangle \not {\approx}0$ for some $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ with $\varphi(x) {\approx}0$ for all $x \in {\Omega_{{\mathbb{X}}}}$. If $\varphi(x) \geq 0$ for all $x\in{\Omega_{{\mathbb{X}}}}$, take some $\psi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ with $\psi(x) \geq n\varphi(x)$ for all $x \in {\Omega_{{\mathbb{X}}}}$ and for all $n \in {\mathbb{N}}$. From the inequality $|\langle f, \psi \rangle| \geq n |\langle f, \varphi \rangle|$ for all $n \in {\mathbb{N}}$, we deduce that $\langle f, \psi \rangle$ is infinite, i.e. that $f \not \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. The remaining cases, namely if $\varphi(x) \leq 0$ for all $x \in {\Omega_{{\mathbb{X}}}}$ or if there exists $x, y \in {\Omega_{{\mathbb{X}}}}$ with $\varphi(x)\varphi(y)<0$, can be dealt in a similar way, thanks to the linearity of the hyperfinite sum.
\(2) implies (1), by contrapositive. Suppose that $\langle f, \varphi \rangle = M$ is infinite for some $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$. Since $\varphi/M \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ and $\varphi/M(x) {\approx}0$ for all $x \in {\Omega_{{\mathbb{X}}}}$, we deduce that (2) does not hold.
It is clear that (1) implies (3).
\(3) implies (1), by contradiction. Suppose that $\langle f, {\,\!^\ast\varphi} \rangle \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ for all $\varphi \in {\mathscr{D}}(\Omega)$, but that $f \not \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. Since (1) and (2) are equivalent, there exists $\psi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ with $\psi(x) {\approx}0$ for all $x \in {\Omega_{{\mathbb{X}}}}$ such that $\langle f, \psi\rangle \not {\approx}0$. By reasoning as in the first part of the proof, we deduce that there exists $\phi \in {\mathscr{D}}(\Omega)$ with $\langle f, {\,\!^\ast\phi} \rangle \not\in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, a contradiction.
From the above Lemma, we deduce that the action of a grid distribution over the space of test functions is continuous.
\[cor continuity\] If $\varphi, \psi \in {\mathscr{D}_{\mathbb{X}}}{({\Omega_{{\mathbb{X}}}})}$ and $\varphi \equiv_S \psi$, then $\langle f, \varphi\rangle {\approx}\langle f, \psi\rangle$ for all $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$.
The hypotheses $\varphi, \psi \in {\mathscr{D}_{\mathbb{X}}}{({\Omega_{{\mathbb{X}}}})}$ and $\varphi \equiv_S \psi$ imply $\varphi - \psi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ and $(\varphi - \psi)(x) {\approx}0$ for all $x \in {\Omega_{{\mathbb{X}}}}$. Then, by Lemma \[character\], we have $\langle f, \varphi - \psi\rangle {\approx}0$ for all $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, as we wanted.
We are now ready to prove that ${{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}/\equiv$ is isomorphic to the space of distributions over $\Omega$.
\[bello\] The function $\Phi:({{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}/\equiv) \rightarrow {\mathscr{D}}'(\Omega)$ defined by $${\langle}\Phi([f]), \varphi {\rangle_{{\mathscr{D}}(\Omega)}}=
{\,\!^\circ\langle f, {\,\!^\ast\varphi} \rangle}$$ is an isomorphism of real vector spaces.
At first, we will show that the definition of $\Phi$ does not depend upon the choice of the representative for $[f]$. Let $g, h \in [f]$: then, by definition of $\equiv$, ${\,\!^\circ\langle g, \varphi \rangle} = {\,\!^\circ\langle h, \varphi \rangle}$ for all $\varphi\in{\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$. By Lemma \[lemma test\], for all $\varphi \in {\mathscr{D}}'(\Omega)$, ${\,\!^\ast\varphi}_{|{\mathbb{X}}} \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, so that if $g,h \in [f]$, then ${\,\!^\circ\langle g, {\,\!^\ast\varphi} \rangle} = {\,\!^\circ\langle h, {\,\!^\ast\varphi} \rangle}$ so that the definition of $\Phi$ is independent on the choice of the representative for $[f]$.
Lemma \[cor continuity\] ensures that for all $[f]\in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}/\equiv$, $\Phi([f]) \in {\mathscr{D}_{\mathbb{X}}}'(\Omega)$, and in particular that $\Phi([f])$ is continuous.
We will prove by contradiction that $\Phi$ is injective. Suppose that $\langle \Phi([f]),\varphi\rangle = 0$ for all $\varphi \in {\mathscr{D}}(\Omega)$ and that $[f] \not = [0]$. The latter hypothesis implies that there exists $\psi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ such that $\langle f, \psi \rangle \not {\approx}0$. But, since ${\,\!^\ast({\,\!^\circ\psi})}_{|{\mathbb{X}}} \equiv_S \psi$, by Corollary \[cor continuity\] we deduce $${\langle}\Phi([f]),{\,\!^\circ\psi}{\rangle_{{\mathscr{D}}(\Omega)}}=
{\,\!^\circ\langle f, {\,\!^\ast({\,\!^\circ\psi})} \rangle}
=
{\,\!^\circ
\langle f, \psi \rangle}
\not = 0,$$ contradicting the hypothesis $\langle \Phi([f]),\varphi\rangle = 0$ for all $\varphi \in {\mathscr{D}}(\Omega)$. As a consequence, $\Phi$ is injective.
Surjectivity of $\Phi$ is a consequence of Theorem 1 of [@moto] and of Lemma \[character\].
Thanks to the previous theorem, from now on we will identify the equivalence class $[f]$ with the distribution $\Phi([f])$. Notice that if $f \in S^0({\Omega_{{\mathbb{X}}}})$, this identification is coherent with $[f]_S$.
\[corollario teorema equivalenza\] If $f \in S^0({\Omega_{{\mathbb{X}}}})$, then $[f] = [f]_S = {\,\!^\circf}$.
Since $f$ is S-continuous, by Lemma \[equivalenza integrali\] and by Lemma \[lemma test\] we have the equality $$\int_{\Omega} {\,\!^\circf} \varphi dx = {\,\!^\circ\langle f, {\,\!^\ast\varphi} \rangle}$$ for all $\varphi \in {\mathscr{D}}(\Omega)$, and this is sufficient to deduce the thesis.
\[remark vector valued\] If $k\in{\mathbb{N}}$, define $${\mathscr{D}_{\mathbb{X}}}'(\Omega, {{\,\!^\ast{\mathbb{R}}}}^k)= \left\{ f: {\Omega_{{\mathbb{X}}}}\rightarrow {\mathbb{R}}^k : f_i \in {\mathscr{D}_{\mathbb{X}}}'(\Omega) \text{ for all } 1 \leq i \leq k \right\}.$$ If $f \in {\mathscr{D}_{\mathbb{X}}}'(\Omega, {{\,\!^\ast{\mathbb{R}}}}^k)$, then we can define a functional $[f]$ over the dual of the space of vector-valued test functions $${\mathscr{D}}(\Omega, {\mathbb{R}}^k) = \left\{ \varphi: \Omega \rightarrow {\mathbb{R}}^k : \varphi_i \in {\mathscr{D}}(\Omega) \text{ for all } 1 \leq i \leq k \right\}$$ by posing $
\langle [f], \varphi \rangle_{{\mathscr{D}}(\Omega, {\mathbb{R}}^k)} = \sum_{i = 1}^k{\,\!^\circ\langle f_i, {\,\!^\ast\varphi_i}\rangle}
$ for all $\varphi \in {\mathscr{D}}(\Omega, {\mathbb{R}}^k)$. From Theorem \[bello\], we deduce that the quotient of the ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$-module $
{\mathscr{D}_{\mathbb{X}}}'(\Omega, {{\,\!^\ast{\mathbb{R}}}}^k) $ with respect to $\equiv$ is isomorphic to the real vector space of linear continuous functionals over ${\mathscr{D}}(\Omega, {\mathbb{R}}^k)$.
\[remark estensioni\] Theorem \[bello\] can be used to define more general projections of nonstandard functions. For instance, if $f \in {\,\!^\astC^0}({{\,\!^\ast{\mathbb{R}}}}, {\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}}))$, then for all $T \in {\mathbb{R}}$ $f$ induces a continuous linear functional $[f]$ over the space $C^0([0,T],{\mathscr{D}}'(\Omega))$ defined by the formula $$\int_0^T {\langle}[f], \varphi {\rangle_{{\mathscr{D}}(\Omega)}}dt = {\,\!^\circ\left({\,\!^\ast\int}_0^T \langle f(t), {\,\!^\ast\varphi}(t) \rangle dt \right)}$$ for all $\varphi \in C^0([0,T],{\mathscr{D}}'(\Omega))$. Moreover, if $f \in {\,\!^\astC^1}({{\,\!^\ast{\mathbb{R}}}}, {\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}}))$, then $[f]$ allows for a weak derivative with respect to time: for all $T \in {\mathbb{R}}$, $[f]_t$ is the distribution that satisfies $$\int_0^T {\langle}[f]_t, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}dt = - {\,\!^\circ\left({\,\!^\ast\int}_0^T \langle f(t), {\,\!^\ast\varphi}(t) \rangle dt \right)}$$ for all $\varphi \in C^1([0,T],{\mathscr{D}}'(\Omega))$.
Discrete derivative and distributional derivative {#distributonal derivative}
-------------------------------------------------
In Theorem \[teorema equivalenza derivate\], we have seen that the finite difference operators ${\mathbb{D}}_i^+$ and ${\mathbb{D}}_i^-$ generalize the derivative for smooth functions to the setting of grid functions. We will now see that it holds a more general result: the operators ${\mathbb{D}}_i^+$ and ${\mathbb{D}}_i^-$ generalize also the distributional derivative, in the sense that $[{\mathbb{D}}_i^{\pm} f] = D_i[f]$ for all $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. For a matter of commodity, we will suppose that ${\Omega_{{\mathbb{X}}}}\subseteq {\mathbb{X}}$: the generalization to an arbitrary dimension can be deduced from the proof of Theorem \[teorema equivalenza derivate2\] with an argument relying on Theorem \[teorema equivalenza derivate\].
Recall the discrete summation by parts formula: for all grid functions $f$ and $g$ and for all $a, b \in {\,\!^\ast{\mathbb{N}}}$ with $N^2 \leq a < b < N^2$ it holds the equality $$\begin{aligned}
\sum_{n = a}^{b} (f((n+1)\varepsilon)-f(n\varepsilon))g(n\varepsilon)
&=& f((b+1)\varepsilon)g((b+1)\varepsilon) - f(a\varepsilon)g(a\varepsilon) +\\
&&- \sum_{n = a}^{b} f((n+1)\varepsilon) (g((n+1)\varepsilon)-g(n\varepsilon))\end{aligned}$$ that, in particular, implies $$\label{party}
\left\langle {\mathbb{D}}f, \varphi \right\rangle = - \left\langle f(x+\varepsilon), {\mathbb{D}}\varphi \right\rangle$$ for all $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ and for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$.
Inspired by the above formula, we will now prove that if we shift a grid distribution by an infinitesimal displacement, we still obtain the same grid distribution.
\[shift2\] Let $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. Then $f(x) \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ if and only if $f(x+\varepsilon)\in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. If $f(x) \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ then, $[f(x)] = [f(x+\varepsilon)]$.
The hypothesis that for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ it holds ${\,\!^\circ{\mathrm{supp\,}}\varphi} \subset \subset \Omega$ ensures the equality $$\langle f(x), \varphi(x) \rangle = \langle f(x+\varepsilon), \varphi(x+\varepsilon)\rangle$$ from which we deduce the equivalence $f(x) \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ if and only if $f(x+\varepsilon)\in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$.
We will now prove that, $f(x) \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, then $[f(x)] = [f(x+\varepsilon)]$. By equation \[party\], we have $$\label{equation mai citata}
\langle f(x+\varepsilon)-f(x) , \varphi \rangle = - \langle f(x+\varepsilon), \varepsilon {\mathbb{D}}\varphi \rangle$$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$. Notice that $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ implies that $\varepsilon {\mathbb{D}}\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$ and $\varepsilon {\mathbb{D}}\varphi(x) {\approx}0$ for all $x \in {\Omega_{{\mathbb{X}}}}$. Hence, by the hypothesis $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ and by Lemma \[character\], we deduce that $\langle f(x+\varepsilon), \varepsilon {\mathbb{D}}\varphi \rangle {\approx}0$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$. By equation \[equation mai citata\], this is sufficient to deduce the equality $[f(x)] = [f(x+\varepsilon)]$.
As a consequence of the above Lemma, we can characterize a nonstandard counterpart of the shift operator.
\[corollario shift\] Let $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. For all $n$ such that $n\varepsilon$ is finite, $[f(x\pm n\varepsilon)] = [f](x\pm{\,\!^\circ(n\varepsilon)})$.
Thanks to the above results, we can now prove that the finite difference operators generalize the distributional derivative.
\[teorema equivalenza derivate2\] The diagrams $$\begin{array}{ccc}
\begin{array}{ccc}
{{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}& \stackrel{{\mathbb{D}}^+}{\longrightarrow} & {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}\\
\Phi \circ \pi \downarrow & & \downarrow \Phi \circ \pi\\
\mbox{}{\mathscr{D}}'({\Omega}) & \stackrel{D}{\longrightarrow} & {\mathscr{D}}'({\Omega}) \\
\end{array}
&
\text{and}
&
\begin{array}{ccc}
{{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}& \stackrel{{\mathbb{D}}^-}{\longrightarrow} & {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}\\
\Phi \circ \pi \downarrow & & \downarrow \Phi \circ \pi\\
\mbox{}{\mathscr{D}}'({\Omega}) & \stackrel{D}{\longrightarrow} & {\mathscr{D}}'({\Omega}) \\
\end{array}
\end{array}$$ commute.
We will prove that the first diagram commutes, as the proof for the second is similar.
Let $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$: we have the following equality chain $${\langle}D[f],\varphi {\rangle_{{\mathscr{D}}(\Omega)}}=
-{\langle}[f], D\varphi {\rangle_{{\mathscr{D}}(\Omega)}}=
-{\,\!^\circ\langle f, {\,\!^\astD}\varphi \rangle}.$$ By Theorem \[teorema equivalenza derivate\], ${\,\!^\astD}\varphi \equiv_S {\mathbb{D}}^{\pm} {\,\!^\ast\varphi}$ and, by Corollary \[cor continuity\], $$\langle f, {\,\!^\astD}\varphi\rangle
{\approx}\langle f, {\mathbb{D}}^{\pm} {\,\!^\ast\varphi} \rangle.$$ By the discrete summation by parts formula \[party\] and by Lemma \[shift2\] we have $$\langle f, {\mathbb{D}}^{\pm} {\,\!^\ast\varphi} \rangle
{\approx}- \langle {\mathbb{D}}^{\pm} f, {\,\!^\ast\varphi} \rangle$$ from which we deduce $${\langle}D[f], \varphi {\rangle_{{\mathscr{D}}(\Omega)}}=
{\,\!^\circ\langle [{\mathbb{D}}^{\pm} f], {\,\!^\ast\varphi} \rangle}$$ for all $\varphi \in {\mathscr{D}}(\Omega)$.
By composing finite difference operators, the theorem is easily extended to other differential operators. We will discuss an example of such a grid function formulation of a differential operator in Section \[section nonli\]. Moreover, in [@illposed] it is discussed a grid function formulation of the gradient and of the Laplacian.
Discrete product rule for generalized distributions and the Schwartz impossibility theorem {#section schwartz}
------------------------------------------------------------------------------------------
For the usual distributions, it is a consequence of the impossibility theorem by Schwartz that no extension of the distributional derivative satisfies a product rule. However, for the grid functions there are some discrete product rules that generalize the product rule for smooth functions. Indeed, the following identities can be established by a simple calculation.
\[chain\] Let $f, g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. Then $$\begin{aligned}
{\mathbb{D}}^+(f\cdot g)(x) & = & \frac{f(x+\varepsilon)g(x+\varepsilon) - f(x)g(x)}{\varepsilon} \\
& = & f(x+\varepsilon){\mathbb{D}}^+ g(x) + g(x){\mathbb{D}}^+ f(x) \\
& = & f(x){\mathbb{D}}^+ g(x) + g(x+\varepsilon){\mathbb{D}}^+ f(x)
\end{aligned}$$ and $$\begin{aligned}
{\mathbb{D}}^-(f\cdot g)(x) & = & \frac{f(x)g(x) - f(x-\varepsilon)g(x-\varepsilon)}{\varepsilon} \\
& = & f(x){\mathbb{D}}^- g(x) + g(x-\varepsilon){\mathbb{D}}^- f(x) \\
& = & f(x-\varepsilon){\mathbb{D}}^- g(x) + g(x){\mathbb{D}}^- f(x).
\end{aligned}$$
For an in-depth discussion of this example and of the limitations in the definition of a product rule for the distributional derivative, we refer to [@tartar]. Consider the following representative of the sign function $$f(x) = \left\{ \begin{array}{rl}
-1 & \text{if } x<0\\
1 & \text{if } x \geq 0.
\end{array}
\right.$$ For this function $f$, $f^2 = 1$ and $f^3 = f$, but the distributional derivative $f_x = 2\delta_{0}$ is different from $(f^3)_x = 3 f^2 f_x = 3 f_x= 6 \delta_0$. So, even if $f^2$ is smooth, the product rule does not hold.
If we regard $f$ as a grid function, however, the boundedness of $f$ ensures that $f \in {\mathscr{D}_{\mathbb{X}}}'({\mathbb{X}})$, and with a simple calculation we obtain: $$\label{direct calculation}
{\mathbb{D}}f(x) = \left\{ \begin{array}{rl}
2 \varepsilon^{-1} & \text{if } x=-\varepsilon\\
0 & \text{otherwise}.
\end{array}
\right.$$ Notice also that $[{\mathbb{D}}f] = 2 \delta_{0}$, as we expected from Theorem \[teorema equivalenza derivate2\]. Applying one of the chain rule formulas of Lemma \[chain\] and taking into account that $f^2(x) = 1$ for all $x \in {\mathbb{X}}$, we obtain $$\begin{aligned}
{\mathbb{D}}f^3(x)
&=& f(x) {\mathbb{D}}f^2(x) + f^2(x+\varepsilon) {\mathbb{D}}f(x)\\
&=& f(x) (f(x) {\mathbb{D}}f(x) + f(x+\varepsilon) {\mathbb{D}}f(x)) + {\mathbb{D}}f(x)\\
&=& {\mathbb{D}}f(x)(2+ f(x) f(x+\varepsilon))
\end{aligned}$$ so that $${\mathbb{D}}f^3(x) = \left\{ \begin{array}{ll}
{\mathbb{D}}f(-\varepsilon) = 2\varepsilon^{-1} & \text{if } x=-\varepsilon\\
0 & \text{otherwise},
\end{array}
\right.$$ in agreement with \[direct calculation\].
We can summarize the results obtained so far as follows: the space of grid functions
- is a vector space over ${{\,\!^\ast{\mathbb{R}}}}$ that extends the space of distributions in the sense of Theorem \[bello\];
- has a well-defined pointwise multiplication that extends the one defined for $S^0$ functions;
- has a derivative ${\mathbb{D}}$ that generalizes the distributional derivative and for which the discrete version of the chain rule established in Proposition \[chain\] holds.
These properties are the nonstandard, discrete counterparts to the ones itemized in the impossibility theorem by Schwartz [@schwartz]. As a consequence, the space of grid functions can be seen as a non-trivial generalization of the space of distributions, as we claimed at the beginning of this section.
We will complete our study of the relations between the space of grid functions and the space of distributions by showing that the space of distributions can be embedded, albeit in a non-canonical way, in the space of grid functions. Notice that we cannot ask to this embedding to be fully coherent with derivatives: in fact, there is already an infinitesimal discrepancy between the usual derivative and the discrete derivative in the set of polynomials: the derivative of $x^2$ is $2x$, but ${\mathbb{D}}x^2 = 2x+\varepsilon$. However, as shown in Theorem \[teorema equivalenza derivate\], for all $f \in C^n$, $D^n f = [{\mathbb{D}}^n ({\,\!^\astf}_{|{\mathbb{X}}})]$. In fact, the canonical linear embedding $l : C^0({\mathbb{R}}) \hookrightarrow S^0({\mathbb{X}})$ given by $l(f) = {\,\!^\astf}_{|{\mathbb{X}}}$ does not preserve derivatives, but it has the weaker property $$\label{weak agreement}
l(f') \equiv {\mathbb{D}}(l(f)).$$ This will be the weaker coherence request that we will impose on the embedding from the space of distributions to the space of grid functions.
\[schwartz\] Let $\{\psi_n\}_{n \in {\mathbb{N}}}$ be a partition of unity, and let $H$ be a Hamel basis for ${\mathscr{D}}'({\mathbb{R}})$. There is a linear embedding $l : {\mathscr{D}}'({\mathbb{R}}) \rightarrow {\mathscr{D}_{\mathbb{X}}}'({\mathbb{X}})$, that depends on $\{\psi_n\}_{n \in {\mathbb{N}}}$ and $H$, that satisfies the following properties:
1. $\Phi \circ l = id$;
2. the product over ${\mathscr{D}_{\mathbb{X}}}'({\mathbb{X}}) \times {\mathscr{D}_{\mathbb{X}}}'({\mathbb{X}})$ generalizes the pointwise product over $C^0({\mathbb{R}}) \times C^0({\mathbb{R}})$;
3. the derivative ${\mathbb{D}}$ over ${\mathscr{D}_{\mathbb{X}}}'({\mathbb{X}})$ extends the distributional derivative in the sense of equation \[weak agreement\];
4. the chain rule for products holds in the form established in Lemma \[chain\].
We will define $l$ over $H$ and extend it to all of ${\mathscr{D}}'({\mathbb{R}})$ by linearity. Let $T \in H$. From the representation theorem of distributions (see for instance [@strichartz]), we obtain $$\label{struttura distri standard}
T = \sum_{n \in {\mathbb{N}}} T\psi_n = \sum_{n \in {\mathbb{N}}} D^{a_n} f_n$$ with $f_n \in C^0({\mathbb{R}})$ and ${\mathrm{supp\,}}(D^{a_n} f_n) \subseteq {\mathrm{supp\,}}\psi_n$ for all $n \in {\mathbb{N}}$. Moreover, the sum is locally finite and for all $\varphi \in {\mathscr{D}}(\Omega)$ there exists a finite set $I_{\varphi} \subset {\mathbb{N}}$ such that $$\label{locally finite}
{\langle}T, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}=
{\langle}\sum_{i \in I_\varphi}D^{a_i} f_i, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}.$$
Let $\{\phi_n\}_{n\in{\,\!^\ast{\mathbb{N}}}}$ be the nonstandard extension of the sequence $\{\psi_n\}_{n \in {\mathbb{N}}}$, and let $\{b_n\}_{n \in {\,\!^\ast{\mathbb{N}}}}$ be the nonstandard extension of the sequence $\{a_n\}_{n \in {\mathbb{N}}}$. By transfer, from the representation \[struttura distri standard\] we obtain $$\label{struttura distri}
{\,\!^\astT} = \sum_{n \in {\,\!^\ast{\mathbb{N}}}} {\,\!^\astT}\phi_n = \sum_{n \in {\,\!^\ast{\mathbb{N}}}} {\,\!^\astD}^{b_n} g_n$$ with $g_i \in {\,\!^\astC^0({\mathbb{R}})}$ and ${\mathrm{supp\,}}(D^{b_n} g_n) \subseteq {\mathrm{supp\,}}\psi_n$ for all $n \in {\,\!^\ast{\mathbb{N}}}$. We may also assume that the representation \[struttura distri\] has the following properties:
1. $ b_n = \min \left\{
m \in {\,\!^\ast{\mathbb{N}}} : {\,\!^\astT}\phi_n = {\,\!^\astD}^{m} f \text{ with } f \in {\,\!^\astC^0({\mathbb{R}})}
\right\}$ for all $n \in {\,\!^\ast{\mathbb{N}}}$
2. if ${\,\!^\astT}\phi_n = {\,\!^\astD}^{b_n} g = {\,\!^\astD}^{b_n} h$ with $g, h \in {\,\!^\astC^0({\mathbb{R}})}$, then $g-h$ is a polynomial of a degree not greater than $b_n-1$;
3. if $n$ is finite and ${\,\!^\astT}\phi_n = {\,\!^\astD}^{b_n}g_n$, then $g_n = {\,\!^\astf}_n$ and $b_n = a_n$, where $f_n$ and $a_n$ satisfy $T\psi_n = D^{a_n}f_n$.
For $T\in H$, we define $$l(T) = \sum_{n \in {\,\!^\ast{\mathbb{N}}}:\ b_n \leq N} {\mathbb{D}}^{b_n} ({g_n}_{|{\mathbb{X}}}),$$ and we extend $l$ to ${\mathscr{D}}'({\mathbb{R}})$ by linearity. Notice that $l$ does not depend on the choice of the functions $\{g_n\}_{n \in {\,\!^\ast{\mathbb{N}}}}$. In fact, suppose that ${\,\!^\astT}\phi_n = {\,\!^\astD}^{b_n} g = {\,\!^\astD}^{b_n} h$ with $g, h \in {\,\!^\astC^0({\mathbb{R}})}$. By property (2) of the representation \[struttura distri\], $g-h$ is a polynomial of a degree not greater than $b_n-1$. Recall that, if $p \in {\mathbb{G}({{\mathbb{X}}})}$ is a polynomial of degree at most $b_n-1$, then ${\mathbb{D}}^{b_n} p = 0$. As a consequence, ${\mathbb{D}}^{b_n} ({g}_{|{\mathbb{X}}}) = {\mathbb{D}}^{b_n}({h}_{|{\mathbb{X}}})$, as we wanted.
We will now show that, for all $T \in H$, ${\langle}\Phi([l(T)]), \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= {\langle}T, \varphi{\rangle_{{\mathscr{D}}(\Omega)}}$ for all $\varphi \in {\mathscr{D}}'({\mathbb{R}})$. This equality and linearity of $l$ entail that $\Phi \circ l = id$. Let $\varphi \in {\mathscr{D}}({\mathbb{R}})$, and let $I_\varphi \subset {\mathbb{N}}$ a finite set such that equality \[locally finite\] holds. We claim that whenever $i \not \in I_\varphi$, then $\langle {\mathbb{D}}^{b_i} ({g_i}_{|{\mathbb{X}}}), {\,\!^\ast\varphi} \rangle = 0$. In fact, if $i \not \in I_\varphi$ is finite, then by formula \[locally finite\] and by property (3) of the representation \[struttura distri\] we have $${\,\!^\circ\langle {\mathbb{D}}^{b_i} ({g_i}_{|{\mathbb{X}}}), \varphi} \rangle = {\,\!^\circ\langle {\mathbb{D}}^{a_i} ({{\,\!^\astf}_i}_{|{\mathbb{X}}}), \varphi} \rangle = {\langle}D^{a_i} f_i, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= 0.$$
We want to show that $\langle {\mathbb{D}}^{b_i} ({g_i}_{|{\mathbb{X}}}), {\,\!^\ast\varphi} \rangle = 0$ also when $i$ is infinite. Notice that if $x \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, then for sufficiently large $n \in {\mathbb{N}}$ it holds $x \not \in {\mathrm{supp\,}}\phi_n$: otherwise, we would also have ${\,\!^\circx} \in {\mathrm{supp\,}}\psi_n$ for arbitrarily large $n$, against the fact that for all $x \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, ${\,\!^\circx}\in{\mathrm{supp\,}}\phi_n$ only for finitely many $n$. As a consequence, ${\mathrm{supp\,}}\phi_i \cap {{{\,\!^\ast{\mathbb{R}}}}_{fin}}= \emptyset$, and by the inclusion ${\mathrm{supp\,}}(D^{b_i} g_i) \subseteq {\mathrm{supp\,}}\phi_i$, then also ${\mathrm{supp\,}}(D^{b_i} g_i) \cap {{{\,\!^\ast{\mathbb{R}}}}_{fin}}= \emptyset$. Taking into account property (2) of the representation \[struttura distri\], we deduce that the restriction of $g_i$ to ${{\,\!^\ast{\mathbb{R}}}}\setminus {\mathrm{supp\,}}(D^{b_i} g_i)$ is a polynomial $p$ of degree at most $b_n -1$. We have already observed that ${\mathbb{D}}^{b_i} p = 0$ and, as a consequence, ${\,\!^\circ\langle {\mathbb{D}}^{b_i} ({g_i}_{|{\mathbb{X}}}), \varphi} \rangle = 0$.
We then have the following equality: $$\langle l(T), {\,\!^\ast\varphi} \rangle = \langle \sum_{i \in I_\varphi}{\mathbb{D}}^{a_i} ({{\,\!^\astf}_i}_{|{\mathbb{X}}}), {\,\!^\ast\varphi} \rangle.$$ By Theorem \[teorema equivalenza derivate2\], we obtain $$ \langle l(T), {\,\!^\ast\varphi} \rangle =
\langle \sum_{i \in I_\varphi}{\mathbb{D}}^{a_i} ({{\,\!^\astf}_i}_{|{\mathbb{X}}}), {\,\!^\ast\varphi} \rangle =
{\langle}\sum_{i \in I_\varphi}D^{a_i} f_i, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}=
{\langle}T, \varphi{\rangle_{{\mathscr{D}}(\Omega)}},
$$ that is sufficient to conclude that $\Phi([l(T)]) = T$.
Assertion (2) is a consequence of Lemma \[lemma piccolo\], assertion (3) is a consequence of Theorem \[teorema equivalenza derivate2\], and assertion (4) is a consequence of Proposition \[chain\].
Grid functions as ${\,\!^\astL}^p$ functions and as parametrized measures {#sez young}
=========================================================================
The main goal of this section is to show that there is an external ${{{\,\!^\ast{\mathbb{R}}}}_{fin}}$-submodule of the space of grid functions whose elements correspond to Young measures, and that this correspondence is coherent with the projection $\Phi$ defined in Theorem \[bello\]. Moreover, we will show how this correspondence can be generalized to arbitrary grid functions. Before we prove these results, we find it useful to discuss some properties of grid functions when they are interpreted as ${\,\!^\astL^p}$ functions. These properties are interesting on their own, and will also be used also in Section \[solutions\], when we will discuss the grid function formulation of partial differential equations.
Recall that for all $1 \leq p \leq \infty$, a function $g \in L^p(\Omega)$ induces a distribution $T_g \in {\mathscr{D}}'(\Omega)$ defined by $${\langle}T_g, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= \int_{\Omega} g \varphi dx$$ for all $\varphi \in {\mathscr{D}}(\Omega)$. As a consequence, by identifying $g$ with $T_g$ we have the inclusions $L^p(\Omega) \subset {\mathscr{D}}'(\Omega)$ for all $1 \leq p \leq \infty$. Since $\Phi$ is surjective, we expect that for all $g \in L^p(\Omega)$ there exists $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ satisfying $[f] = T_g$. In this case, we will often write $[f] = g$ and $[f] \in L^p(\Omega)$. If $[f] \in L^p(\Omega)$, thanks to the Riesz representation theorem, we can think of $[f]$ either as a functional acting on $L^{p'}(\Omega)$, or as a member of an equivalence class of $L^p(\Omega)$ functions. To our purposes, we find it more convenient to treat $[f]$ as a function. With this interpretation, if $g = [f]$ and $g\in L^p(\Omega)$, then it holds the equality $g(x) = [f](x)$ for almost every $x \in \Omega$.
Grid functions as ${\,\!^\astL}^p$ functions {#sub lp}
--------------------------------------------
We can identify every grid function with a piecewise constant function defined on all of ${{\,\!^\ast{\mathbb{R}}}}^k$. Among many different identifications, we choose the following: if $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, then ${\widehat{f}}$ is defined by $${\widehat{f}}(x) = \left\{
\begin{array}{ll}
f((n_1,n_2,\ldots,n_k)\varepsilon) & \text{if } n_i \varepsilon \leq x_i < (n_i+1)\varepsilon \text{ for all } 1 \leq i \leq k\\
0 & \text{if } |x_i| > N \text{ for some } 1 \leq i \leq k,
\end{array}
\right.$$ with the agreement that $f((n_1,n_2,\ldots,n_k)\varepsilon) = 0$ if $(n_1,n_2,\ldots,n_k)\varepsilon \not \in {\Omega_{{\mathbb{X}}}}$.
If $f$ is a grid function, the function ${\widehat{f}}$ is an internal ${\,\!^\ast}$simple function and, as such, it belongs to ${\,\!^\astL^p}({\mathbb{R}}^k)$ for all $1 \leq p \leq \infty$. The integral of ${\widehat{f}}$ is related with the grid integral of $f$ by the following formula: $${\,\!^\ast\int}_{{{\,\!^\ast{\mathbb{R}}}}^k} {\widehat{f}} dx = \int_{{\Omega_{{\mathbb{X}}}}} f(x) d{\mathbb{X}}^k = \varepsilon^k \sum_{x \in {\Omega_{{\mathbb{X}}}}} f(x).$$ As a consequence, the ${\,\!^\astL^p}$ norm of ${\widehat{f}}$ can be expressed by $${\Vert{\widehat{f}} \Vert}_p^p = \varepsilon^k \sum_{x \in {\Omega_{{\mathbb{X}}}}} |f(x)|^p \text{ if } 1 \leq p < \infty, \text{ and } {\Vert{\widehat{f}}\Vert}_\infty = \max_{x \in {\Omega_{{\mathbb{X}}}}} |f(x)|.$$
Notice that if $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, then ${\,\!^\circ{\mathrm{supp\,}}{\widehat{f}}} \subseteq {\,\!^\circ{\mathrm{supp\,}}{\widehat{\chi_{{\Omega_{{\mathbb{X}}}}}}}} = \overline{\Omega}$. If we define ${\widehat{\Omega}} = {\mathrm{supp\,}}{\widehat{\chi_{{\Omega_{{\mathbb{X}}}}}}}$, then from the above inclusion we can write ${\widehat{f}} \in {\,\!^\astL}^p({\widehat{\Omega}})$ for all $1 \leq p \leq \infty$. By identifying $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ with ${\widehat{f}}$, for all $1 \leq p \leq \infty$ the space of grid functions is identified with a subspace of ${\,\!^\astL^p({\widehat{\Omega}})}$ which is closed with respect to the ${\,\!^\astL}^p$ norm. Since ${\widehat{\Omega}}$ is ${\,\!^\ast}$bounded in ${{\,\!^\ast{\mathbb{R}}}}^k$, for $1 \leq p \leq \infty$ we have the usual relations between the ${\,\!^\astL^p}$ norms of $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$: if $1\leq p < q \leq +\infty$ and if $r$ satisfies the equality $1/p = 1/q + 1/r$, then ${\Vert{\widehat{f}}\Vert}_p \leq {\,\!^\ast{\mu_L}({\widehat{\Omega}})}^r{\Vert{\widehat{f}}\Vert}_q$.
From now on, when there is no risk of confusion, we will often abuse the notation and write $f$ instead of ${\widehat{f}}$.
We begin our study of grid functions as ${\,\!^\astL^p}$ functions by showing that if a grid function $f$ has finite ${\,\!^\astL}^p$ norm for some $1\leq p\leq \infty$, then $f\in{{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ and, as a consequence, $[f]$ is a well-defined distribution.
\[lemma sufficiente\] If ${\Vertf\Vert}_p \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ for some $1 \leq p \leq \infty$, then $f \in {\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}})$.
Notice that ${\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})\subset {\,\!^\astL^p}({\widehat{\Omega}})$ for all $1 \leq p \leq \infty$ and, for any $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, ${\Vert\varphi\Vert}_p \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ for all $1 \leq p \leq \infty$. By the discrete Hölder’s inequality $$|\langle f, \varphi \rangle| \leq {\Vertf\varphi\Vert}_1 \leq {\Vertf\Vert}_p {\Vert\varphi\Vert}_{p'}$$ so that if ${\Vertf\Vert}_p \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, then $\langle f, \varphi \rangle \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, as desired.
From the previous Lemma we deduce that, if the $L^p$ norm of the difference of two grid functions $f$ and $g$ is infinitesimal, then $f \equiv g$.
\[corollario lp\] Let $f, g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. If ${\Vertf-g\Vert}_p {\approx}0$ for some $1 \leq p \leq \infty$, then $f \equiv g$.
If ${\Vertf-g\Vert}_p {\approx}0$, then by Lemma \[lemma sufficiente\] $$\langle f-g, \varphi \rangle \leq {\Vertf-g\Vert}_p {\Vert\varphi\Vert}_{p'} {\approx}0$$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$. As a consequence, $f \equiv g$.
Notice that the other implication does not hold, in general. As an example, consider the grid function $f(n\varepsilon) = (-1)^n$. Since $\langle f, \varphi \rangle {\approx}0$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, we deduce that $[f] = 0$, but ${\Vertf\Vert}_{p} = 1$ for all $1 \leq p \leq \infty$. Notice also that ${\Vertf\Vert}_p$ is finite, but ${\Vert{\widehat{f}}-{\,\!^\astg}\Vert}_p \not {\approx}0$ for all $g \in L^p(\Omega)$ and for all $1 \leq p \leq \infty$.
In the next section, we will show that the hypothesis ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ is sufficient to ensure that $[f]\in L^\infty(\Omega)$. If $1 \leq p < \infty$, however, the hypothesis ${\Vertf\Vert}_p \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ is not sufficient to imply that $[f] \in L^p(\Omega)$. An example is given by $N\chi_0 \in{\mathbb{G}({{\mathbb{X}}})}$, a representative of the Dirac distribution centred at $0$. It can be calculated that ${\VertN\chi_0\Vert}_1 = \varepsilon N = 1,$ but $[N\chi_0] = \delta_0 \not\in L^p({\mathbb{R}})$ for any $p$. In general, whenever $[f]\in L^p(\Omega)$, it holds the inequality ${\Vertf\Vert}_p \geq {\Vert[f]\Vert}_p$.
\[norm inequality\] For all $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ and for all $1 \leq p \leq \infty$, if $[f] \in L^p(\Omega)$, then
1. if $[|f|] \in L^p(\Omega)$, then $[|f|]\geq|[f]|$ a.e. in $\Omega$;
2. ${\,\!^\circ{\Vertf\Vert}_p} \geq {\Vert[f]\Vert}_p$.
Define $f^+(x) = \max\{f(x),0\}$ and $f^-(x) = \min\{f(x),0\}$, so that $f = f^+ + f^-$ and $|f|^p = |f^+|^p + |f^-|^p$ for all $1 \leq p < \infty$. If $[|f|]\in L^p(\Omega)$, then $[f^+]$ and $[f^-] \in L^p(\Omega)$ and, by linearity of $\Phi$, $$[|f|](x) = [f^+](x)-[f^-](x) \geq [f^+](x)+[f^-](x) = [f](x)$$ for a.e. $x \in\Omega$.
Let $f\in{\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ and suppose that $[f]\in L^p(\Omega)$ with $p < \infty$. If either $|f^+| \not \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, $|f^+|^p \not \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, $|f^-| \not \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ or $|f^-| \not \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ then by Lemma \[lemma sufficiente\] we would have $|f|^p \not \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ and, as a consequence, $${\Vertf\Vert}_p^p = {\Vert|f|^p\Vert}_1 \not \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}},$$ so that inequality (2) would hold. Suppose then that $|f^+| \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, $|f^+|^p \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, $|f^-| \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ and $|f^-|^p \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. As a consequence, both $|f| \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ and $|f|^p \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. If $[|f|]\in L^p(\Omega)$, then (2) is a consequence of (1). The only case left is $[|f|]\not\in L^p(\Omega)$.
For a matter of commodity, let $g = [f]$, and let $g^+(x) = \max\{g(x),0\}$ and $g^-(x) = \min\{g(x),0\}$. Since $
[f^+]+[f^-] = [f] = g^++g^- \text{ in } {\mathscr{D}}'(\Omega),
$ we deduce that $
[f^+]-g^+ = -([f^-]-g^-).
$ The hypothesis $[f^+]\not \in L^p(\Omega)$ entails that also $[f^+]-g^+ \not \in L^p(\Omega)$. Let $K = {\mathrm{supp\,}}([f^+]-g^+)$: then for all $\varphi \in {\mathscr{D}}(\Omega)$ with ${\mathrm{supp\,}}\varphi \subset K$ and with $\varphi(x) \geq 0$ for all $x \in \Omega$, $$0 \leq {\langle}[f^+]-g^+, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= {\,\!^\circ\langle f^+, {\,\!^\ast\varphi}\rangle} - \int_{\Omega} g^+ \varphi dx.$$ Similarly, $$0 \leq -{\langle}[f^-]-g^-, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= {\,\!^\circ\langle |f^-|, {\,\!^\ast\varphi}\rangle} - \int_{\Omega} |g^-| \varphi dx.$$ From the arbitrariness of $\varphi$, we deduce $ {\Vertf\chi_{K_{\mathbb{X}}}\Vert}_p \geq {\Vertg\chi_K\Vert}_p$. Since $K = {\mathrm{supp\,}}([f^+]-g^+)$, we also have $${\Vert([f^+]-g^+)\chi_{\Omega\setminus K}\Vert}_p = {\Vert([f^-]-g^-)\chi_{\Omega\setminus K}\Vert}_p = {\Vert0\Vert}_p = 0,$$ from which we conclude that (2) indeed holds.
Suppose now that $[f]\in L^\infty(\Omega)$. If ${\Vertf\Vert}_\infty \not\in{{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, then inequality (2) holds. If ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, let $c_f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ satisfy $c_f(x) = {\Vertf\Vert}_\infty$ for all $x \in {\Omega_{{\mathbb{X}}}}$. Then $[c_f](x) = {\,\!^\circ{\Vertf\Vert}_\infty}$ for all $x \in \Omega$, so that $[c_f] \in L^\infty(\Omega)$. Since $c_f(x) \geq \max\{f^+(x), |f^-(x)|\}$ for all $x \in {\Omega_{{\mathbb{X}}}}$, then also $[c_f](x) \geq [f](x)$ for all $x \in {\Omega_{{\mathbb{X}}}}$. This is sufficient to conclude that inequality (2) holds.
If $[f] \in L^p({\Omega_{{\mathbb{X}}}})$ and ${\,\!^\circ{\Vertf\Vert}_{p}} > {\Vert[f]\Vert}_{p}$, then $f$ features some oscillations that are compensated by the linearity of $\Phi$. In this case, we can interpret $f$ as the representative of a weak or (weak-$\star$ when $p = \infty$) limit of a sequence of functions whose $L^p$ norm is uniformly bounded by ${\,\!^\circ{\Vertf\Vert}_{p}}$. In the next section, we will see how the behaviour of this weak-$\star$ limit can be described by a parametrized measure associated to $f$.
If ${\Vertf\Vert}_p \not\in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ but nevertheless $[f] \in L^p(\Omega)$, then $f$ also features concentrations that are compensated by the linearity of $\Phi$. An example is given by the function $f = {\mathbb{D}}\chi_0 = N\chi_{-\varepsilon}-N\chi_0$. The ${\,\!^\astL}^p$ norm of $f$ is ${\Vertf\Vert}_p = 2N^{p-1/p}$ for $p \not = \infty$ and $N$ for $p = \infty$; however, from Theorem \[teorema equivalenza derivate2\], we deduce that $[f]=D[\chi_0]=0$. In the next section, we will discuss how these concentrations affect the parametrized measure associated to $f$.
We will now address the coherence between the nonstandard extension of a $L^2$ function and its projection in the space of grid functions. These technical results will be used in Section \[solutions\].
Let ${P}: {\,\!^\astL^2({\widehat{\Omega}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ be the ${\,\!^\astL^2}$ projection over the closed subspace ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. Recall that ${P}(f)$ is the unique grid function satisfying $$\langle {P}(f), g \rangle = {\,\!^\ast\int}_{{\widehat{\Omega}}} f(x) {\widehat{g}}(x) dx$$ for all $g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$.
\[proiez s-c\] For all $f \in C^0(\Omega)$, ${P}({\,\!^\astf}) \in S^0({\Omega_{{\mathbb{X}}}})$ and ${\,\!^\astf}(x) {\approx}{P}({\,\!^\astf})(x)$ for all $x \in {\Omega_{{\mathbb{X}}}}$.
Let $f \in C^0(\Omega)$. Since for all $g \in{\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ we have the equality $$\langle {P}({\,\!^\astf}), g\rangle
=
{\,\!^\ast\int}_{{\widehat{\Omega}}} {\,\!^\astf}(x) {\widehat{g}}(x) dx,$$ by choosing $g = \varepsilon^{-k} \chi_{y}$ , we obtain $${P}({\,\!^\astf})(y)
=
\langle {P}({\,\!^\astf}), {\widehat{\varepsilon^{-k} \chi_{y}}}\rangle
=
\varepsilon^{-k}{\,\!^\ast\int}_{[y, y+\varepsilon]^k} {\,\!^\astf}(x) dx$$ for all $y \in{\Omega_{{\mathbb{X}}}}$. Since $$\min_{x \in [y, y+\varepsilon]^k} \{{\,\!^\astf}(x)\}
\leq
\varepsilon^{-k}{\,\!^\ast\int}_{[y, y+\varepsilon]^k} {\,\!^\astf}(x) dx
\leq
\max_{x \in [y, y+\varepsilon]^k} \{{\,\!^\astf}(x)\},$$ by S-continuty of ${\,\!^\astf}$, we deduce the thesis.
\[questo corollario\] For all $f \in L^2(\Omega)$, $[{P}({\,\!^\astf})] = f$.
For all $\varphi \in {\mathscr{D}}'(\Omega)$ we have $$\langle {P}({\,\!^\astf}), {\,\!^\ast\varphi}_{|{\mathbb{X}}} \rangle = {\,\!^\ast}\int_{{\widehat{\Omega}}} {\,\!^\astf} {\widehat{{\,\!^\ast\varphi}_{|{\mathbb{X}}}}} dx$$ and, by S-continuity of ${\,\!^\ast\varphi}$, $$ {\,\!^\ast\int}_{{\widehat{\Omega}}} {\,\!^\astf} {\widehat{{\,\!^\ast\varphi}_{|{\mathbb{X}}}}} dx
{\approx}{\,\!^\ast\int}_{{\,\!^\ast\Omega}} {\,\!^\astf} {\,\!^\ast\varphi} dx
=
\int_{\Omega} f\varphi dx.$$ This implies $[{P}({\,\!^\astf})] = f$.
The above Lemma can be sharpened under the hypothesis that $\Omega$ has finite measure.
\[norma r\] Let $\mu_L(\Omega)<+\infty$. For all $f \in L^2(\Omega)$, ${\Vert{\,\!^\astf}-{P}({\,\!^\astf})\Vert}_2 {\approx}0$.
Let $f \in L^2(\Omega)$, and let $r = {\,\!^\astf}-{P}({\,\!^\astf})$. By the properties of the ${\,\!^\astL^2}$ projection, we have $$\label{lpproj}
{\Vert{\,\!^\astf}\Vert}_2 = {\VertP({\,\!^\astf})\Vert}_2 + {\Vertr\Vert}_2.$$
By the nonstandard Lusin’s Theorem, there exists a ${\,\!^\ast}$compact set $K\subseteq {\,\!^\ast\Omega}$ that satisfies ${\,\!^\ast{\mu_L}}({\,\!^\ast\Omega} \setminus K) {\approx}0$ and ${\Vertr \chi_{K}\Vert}_2 {\approx}0$. Since ${\,\!^\ast{\mu_L}}({\,\!^\ast\Omega} \setminus K) {\approx}0$ and since $f \in L^2(\Omega)$, we have also ${\Vert{\,\!^\astf} \chi_{K}\Vert}_2 {\approx}{\Vert{\,\!^\astf}\Vert}_2$ and, as a consequence, $${\Vert{\,\!^\astf} \Vert}_2 {\approx}{\Vert{\,\!^\astf} \chi_{K}\Vert}_2 = {\Vert{P}({\,\!^\astf}) \chi_{K}\Vert}_2 + {\Vertr\chi_K\Vert} {\approx}{\Vert{P}({\,\!^\astf}) \chi_{K}\Vert}_2.$$ From the inequality chain $${\Vert{\,\!^\astf} \Vert}_2{\approx}{\Vert{P}({\,\!^\astf}) \chi_{K}\Vert}_2 \leq {\Vert{P}({\,\!^\astf})\Vert}_2 \leq {\Vert{\,\!^\astf}\Vert}_2$$ we deduce that ${\Vert{\,\!^\astf}\Vert}_2 {\approx}{\VertP({\,\!^\astf})\Vert}_2$ that, by equality \[lpproj\], implies ${\Vert{\,\!^\astf}-{P}({\,\!^\astf})\Vert}_2 {\approx}0$, as we wanted.
The previous Lemma suggests a definition of nearstandardness that will be useful in the sequel of the paper.
Let $\mu_L(\Omega)<+\infty$. We will say that $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ is nearstandard in $L^2(\Omega)$ iff there exists $g \in L^2(\Omega)$ such that ${\Vertf-P({\,\!^\astg})\Vert}_2 {\approx}0$.
Notice that, under the hypothesis that $\mu_L(\Omega)$ is finite, Corollary \[corollario lp\] and Lemma \[norma r\] entail that $f$ is nearstandard in $L^2(\Omega)$ if and only if $[f] \in L^2(\Omega)$ and ${\Vertf-P({\,\!^\ast[f]})\Vert}_2 {\approx}0$.
An extension of the Robinson-Bernstein embedding
------------------------------------------------
We conclude the study of the properties of grid functions as ${\,\!^\astL^p}$ functions by discussing the generalization of an embedding due to Robinson and Bernstein $$L^2(\Omega) \subset V \subset {\,\!^\astL^2}(\Omega),$$ where $V$ is a vector space of a hyperfinite dimension (for the details, we refer to [@invariant; @da]). In our case, by considering the embedding $l$ of the space of distributions to the space of grid functions defined in Theorem \[schwartz\] and by modifying the extension of $f$ to ${\widehat{f}}$, we will obtain the inclusions $$L^p(\Omega) \subset {\mathscr{D}}'(\Omega) \subset {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \subset {\,\!^\astL^p}(\Omega)$$ for all $1\leq p \leq \infty$.
\[proposition embedding\] Let $l$ be defined as in the proof of Theorem \[schwartz\]. There is an embedding $l': {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow \bigcap_{1\leq p \leq \infty} {\,\!^\astL^p}(\Omega)$ such that $$\label{embedding equality}
{\,\!^\ast\int}_{{{\,\!^\ast{\mathbb{R}}}}^k} (l'\circ l)(f) {\,\!^\ast\varphi} dx {\approx}\int_{{\mathbb{R}}^k} f \varphi dx$$ for all $1 \leq p \leq \infty$, for all $f \in L^p(\Omega)$ and for all $\varphi \in {\mathscr{D}}(\Omega)$. As a consequence, if we identify ${\mathscr{D}}'(\Omega)$ with $l({\mathscr{D}}'(\Omega)) \subseteq {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ and ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ with $l'({\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}) \subseteq {\,\!^\astL^p}(\Omega)$, we have the inclusions $$L^p(\Omega) \subset {\mathscr{D}}'(\Omega) \subset {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \subset {\,\!^\astL^p}(\Omega)$$ for all $1\leq p \leq \infty$.
Define $l'$ by $l'(f) = {\widehat{f}}\chi_{{\,\!^\ast\Omega}}$ for all $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. Since $l'(f)$ is an internal ${\,\!^\ast}$simple function, it belongs to ${\,\!^\astL^p}(\Omega)$ for all $1 \leq p \leq \infty$. We will now prove that, for this choice of $l'$, equality \[embedding equality\] holds.
Notice that for all $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, if $l'(f)(x) \not = {\widehat{f}}(x)$, then $x \in {\,\!^\ast\Omega}\setminus{\widehat{\Omega}}$ or $x \in {\widehat{\Omega}}\setminus{\,\!^\ast\Omega}$. By the definition of ${\widehat{\Omega}}$, this entails ${\,\!^\circx} \in \partial\Omega$. In particular, if $\varphi \in {\mathscr{D}}(\Omega)$, then ${\,\!^\circx} \not \in {\mathrm{supp\,}}\varphi$. As a consequence, for all $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ and for all $\varphi \in {\mathscr{D}}(\Omega)$, it holds $${\,\!^\ast\int}_{{{\,\!^\ast{\mathbb{R}}}}^k} l'(f)\ {\,\!^\ast\varphi} dx = {\,\!^\ast\int}_{{{\,\!^\ast{\mathbb{R}}}}^k} {\widehat{f}}\ {\,\!^\ast\varphi} dx.$$ By S-continuity of ${\,\!^\ast\varphi}$, we have also $${\,\!^\ast\int}_{{{\,\!^\ast{\mathbb{R}}}}^k} {\widehat{f}}\ {\,\!^\ast\varphi} dx {\approx}\langle f, {\,\!^\ast\varphi} \rangle.$$ If we let $f = l(g)$ for some $g \in L^p(\Omega)$, from Theorem \[schwartz\] we have $$\langle l(g),{\,\!^\ast\varphi} \rangle {\approx}{\langle}g, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= \int_{{\mathbb{R}}^k} g\varphi dx.$$ By putting together the previous equalities, we conclude that equation \[embedding equality\] holds.
We conjecture that for $p = 2$ and under the hypothesis that $\Omega$ has finite Lebesgue measure, we can choose the embedding $l$ in a way that the equality ${\Vert(l' \circ l)(f) - {\,\!^\astf}\Vert}_2 {\approx}0$ holds, as in the original embedding by Robinson and Bernstein.
Grid functions as parametrized measures
---------------------------------------
It is well known that weak limits of $L^p$ functions behave badly with respect to composition with a nonlinear function [@balder; @evans; @nonlinear; @sychev; @webbym]. Consider for instance a bounded sequence $\{ f_n \}_{n \in {\mathbb{N}}}$ of $L^\infty(\Omega)$ functions: by the Banach–Alaoglu theorem, there is a subsequence of $\{f_n\}_{n \in {\mathbb{N}}}$ that has a weak-$\star$ limit $f_\infty \in L^\infty(\Omega)$. Now let $\Psi \in {C^0_b}({\mathbb{R}})$: the sequence $\{ \Psi(f_n)\}_{n \in {\mathbb{N}}}$ is still bounded in $L^\infty(\Omega)$, so it has a weak-$\star$ limit $\Psi_\infty$. However, in general $\Psi_\infty \not = \Psi(f_\infty)$. To overcome this difficulty, the weak-$\star$ limit of the sequence $\{ f_n \}_{n \in {\mathbb{N}}}$ can be represented by a Young measure. In particular, the main theorem of Young measures states that for every bounded sequence $\{ f_n \}_{n \in {\mathbb{N}}}$ of $L^\infty(\Omega)$ functions there exists a measurable function $\nu : \Omega \rightarrow {{\mathbb{M}}^{\mathbb{P}}}({\mathbb{R}})$ that satisfies the following property: for all $\Psi \in {C^0_b}({\mathbb{R}})$, the weak-$\star$ limit of $\{ \Psi(f_n) \}_{n \in {\mathbb{N}}}$ is the function defined by $\overline{\Psi}(x) = \int_{{\mathbb{R}}} \Psi d\nu_x$, in the sense that the equality $$\label{young definition}
\lim_{n \rightarrow \infty} \int_{\Omega} \Psi(f_n(x)) \varphi(x) dx
=
\int_{\Omega} \left(\int_{{\mathbb{R}}} \Psi d\nu_x\right) \varphi(x) dx
=
\int_{\Omega} \overline{\Psi}(x) \varphi(x) dx$$ holds for all $\varphi \in L^1(\Omega)$.
The following example is discussed in detail in [@webbym]. Consider the Rademacher functions $f_n(x) = f_0(n^2x)$, with $f_0(x) = \chi_{[0,1/2)}(x) - \chi_{[1/2,1)}(x)$ extended periodically over ${\mathbb{R}}$. It can be calculated that the Young measure $\nu$ associated to the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ is constant and that $$\nu_x = \frac{1}{2} \delta_{1} + \frac{1}{2} \delta_{-1}$$ for almost every $x \in \Omega$, i.e. that for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in L^1(\Omega)$, $$\lim_{n \rightarrow \infty} \int_{{\mathbb{R}}} \Psi(f_n(x)) \varphi(x) dx
=
\left(\frac{1}{2}\Psi(1) + \frac{1}{2}\Psi(-1)\right) \int_{{\mathbb{R}}} \varphi(x) dx.$$
In the setting of grid functions, instead of bounded sequences of $L^\infty$ functions, we have grid functions with finite ${\,\!^\astL^\infty}$ norm. These functions can be used to represent weak-$\star$ limits of $L^\infty$ functions.
The function $f(n\varepsilon) = (-1)^n$ can be thought as a grid function representative for the weak-$\star$ limit of the Rademacher functions: in fact, for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c(\Omega)$, $${\,\!^\circ\langle {\,\!^\ast\Psi}(f), {\,\!^\ast\varphi} \rangle} = \left(\frac{1}{2}\Psi(1) + \frac{1}{2}\Psi(-1)\right) \int_{{\mathbb{R}}} \varphi(x) dx.$$ Since $C^0_c(\Omega)$ is dense in $L^1(\Omega)$, this is sufficient to conclude that the above formula holds for all $\varphi \in L^1(\Omega)$.
In [@cutland; @controls3], Cutland showed that every grid function that has finite ${\,\!^\astL}^\infty$ norm corresponds to a Young measure.
\[young\] For every $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ with ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, there exists a Young measure $\nu^f : \Omega \rightarrow {{\mathbb{M}}^{\mathbb{P}}}({\mathbb{R}})$ such that, for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c(\Omega)$, $$\label{young equivalence equation}
{\,\!^\circ\langle {\,\!^\ast\Psi}(f), {\,\!^\ast\varphi} \rangle}
=
\int_{\Omega} \left( \int_{{\mathbb{R}}} \Psi d \nu^f_x \right) \varphi(x) dx.$$
Since ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, there exists $n \in {\mathbb{R}}$ such that $|f(x)| < n$. We can identify $f$ with a function $\tilde{f} : {\widehat{\Omega}} \rightarrow {\,\!^\ast{{\mathbb{M}}^{\mathbb{P}}}({\,\!^\ast[-n,n]})}$ defined by $\tilde{f}(x) = \delta_{{\widehat{f}}(x)}$. Notice that for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c(\Omega)$ it holds $$\begin{aligned}
\notag
\langle {\,\!^\ast\Psi(f)}, {\,\!^\ast\varphi} \rangle
&{\approx}&
{\,\!^\ast\int_{{\widehat{\Omega}}}} {\,\!^\ast\Psi({\widehat{f}}(x))} {\,\!^\ast\varphi(x)} dx\\
&=&\label{quiqui}
{\,\!^\ast\int_{{\widehat{\Omega}}}} \left({\,\!^\ast\int}_{{\,\!^\ast[-n,n]}} {\,\!^\ast\Psi} d \tilde{f}(x) \right) {\,\!^\ast\varphi}(x)dx.
\end{aligned}$$
We define an internal measure $\mu$ over ${\,\!^\ast\Omega} \times {\,\!^\ast[-n,n]}$ by posing $$\mu(A\times B) = {\,\!^\ast\int}_A \tilde{f}_x(B) dx$$ for all Borel $A \subseteq \Omega$ and for all Borel $B \subseteq {\,\!^\ast[-n,n]}$. Let $L_\mu$ be the Loeb measure obtained from $\mu$ (for the properties of the Loeb measure, we refer for instance to [@nsa; @theory; @apps; @lo; @nsa; @working; @math]). We can define a standard measure $\mu_s$ over $\Omega \times [-n,n]$ by posing $$\mu_s(A \times B) = L_\mu(\{x \in {\,\!^\ast\Omega} \times {\,\!^\ast[-n,n]} : {\,\!^\circx} \in A \times B\}).$$ Since $\mu_s$ satisfies $\mu_s(A \times [-n,n]) = {\mu_L}(A)$ for all Borel $A \subseteq \Omega$, by Rohlin’s Disintegration Theorem the measure $\mu_s$ can be disintegrated as $$\mu_s(A \times B) = \int_A \nu^f_x(B) dx,$$ with $\nu^f : \Omega \rightarrow {{\mathbb{M}}^{\mathbb{P}}}([-n,n])$. By Lemma 2.6 of [@cutland; @controls3], $\nu^f$ satisfies $${\,\!^\circ\left({\,\!^\ast\int_{{\,\!^\ast\Omega}}} \left({\,\!^\ast\int}_{{\,\!^\ast[-n,n]}} {\,\!^\ast\Psi} d \tilde{f}(x) \right) {\,\!^\ast\varphi}(x)dx\right)}
=
\int_\Omega \left(\int_{[-n,n]} \Psi d \nu^f_x\right) \varphi(x) d x.$$ for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c(\Omega)$. Thanks to equality \[quiqui\], we deduce that $\nu^f$ satisfies \[young equivalence equation\]. We can extend $\nu^f_x$ to all of ${{\mathbb{M}}^{\mathbb{P}}}{({\mathbb{R}})}$ by defining $\nu^f_x(A) = \nu^f_x(A\cap[-n,n])$ for all Borel sets $A \subseteq {\mathbb{R}}$ and for all $x \in \Omega$, thus obtaining a Young measure that satisfies equation \[young equivalence equation\].
In [@ball; @webbym], it is shown that Young measures describe weak-$\star$ limits of bounded sequences of $L^\infty$ functions. We will now show that grid functions with finite $L^\infty$ norm can be similarly used to represent weak-$\star$ limits of $L^\infty$ functions in the setting of grid functions. This is a consequence of a more general property of the correspondence between grid functions and Young measures: if $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ satisfies ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ and $\nu^f$ is the Young measure associated to $f$ in the sense of Theorem \[young\], then $[f]$ corresponds to the barycentre of $\nu^f$.
\[lemma dirac\] Let $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ with ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, and let $\nu^f$ be the Young measure that satisfies equality \[young equivalence equation\]. Then $[f] \in L^\infty(\Omega)$ and the following equality holds for a.e. $x \in \Omega$: $$\label{uguaglianza inutile}
[f](x) = \int_{{\mathbb{R}}} \tau d\nu^f_x.$$ Moreover,
1. if $\{f_n\}_{n \in {\mathbb{N}}}$ is a sequence of $L^\infty$ functions that converges weakly-$\star$ to $\nu^f$ in the sense of equation \[young definition\], then $f_n \stackrel{\star}{{\rightharpoonup}} [f]$ in $L^\infty$;
2. if $\nu^f$ is Dirac, then $\nu^f_x$ is the Dirac measure centred at $[f](x)$ for a.e. $x \in \Omega$.
Define a function $g$ by posing $g(x) = \int_{{\mathbb{R}}} \tau d\nu^f_x$ for all $x \in \Omega$. Since $|g(x)| \leq {\,\!^\circ{\Vertf\Vert}_\infty}$ for a. e. $x \in \Omega$ and since ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, $g \in L^\infty(\Omega)$. By Theorem \[young\], for all $\varphi \in C^0_c(\Omega)$ we have the following equalities: $$\int_{\Omega} g(x) \varphi(x) dx = \int_{\Omega} \int_{{\mathbb{R}}} \tau d\nu^f_x \varphi(x) dx
=
{\,\!^\circ\langle f, {\,\!^\ast\varphi} \rangle}
=
\int_\Omega [f] \varphi dx.$$ Since $C^0_c(\Omega)$ is dense in $L^1(\Omega)$, we deduce that $g = [f]$ in $L^\infty(\Omega)$, as we wanted. We will now prove (1). By hypothesis, from equation \[young definition\] and from equation \[young equivalence equation\], it holds $$\lim_{n \rightarrow \infty} \int_{\Omega} \Psi(f_n(x)) \varphi(x) dx
=
\int_{\Omega} \left(\int_{{\mathbb{R}}} \Psi d\nu^f_x\right) \varphi(x) dx
=
{\,\!^\circ\langle {\,\!^\ast\Psi}(f), {\,\!^\ast\varphi} \rangle}$$ for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c(\Omega)$. As a consequence, by considering a function $\Psi \in {C^0_b}({\mathbb{R}})$ with $\Psi(x) = 1$ for all $x$ satisfying $|x| \leq {\,\!^\circ{\Vertf\Vert}_\infty}$, we obtain that the weak-$\star$ limit of the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ is equal to $[f]$.
Assertion (2) is a consequence of equality \[uguaglianza inutile\].
If the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ is not bounded in $L^\infty$, then it can be proved that there exists a parametrized measure $\nu : \Omega \rightarrow {\mathbb{M}}({\mathbb{R}})$ such that for all $\Psi \in {C^0_b}({\mathbb{R}})$ the weak-$\star$ limit of the sequence $\{ \Psi(f_n) \}_{n \in {\mathbb{N}}}$ is the function defined a.e. by $\overline{\Psi}(x) = \int_{{\mathbb{R}}} \Psi d\nu_x$ (for an in-depth discussion of this result, we refer to [@ball]). Notice that $\nu$ takes values in ${\mathbb{M}}({\mathbb{R}})$ instead of ${{\mathbb{M}}^{\mathbb{P}}}({\mathbb{R}})$, since the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ could diverge in a subset of $\Omega$ with positive measure.
The grid function counterpart of this result is that for any $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ there exists a function $\nu^f : \Omega \rightarrow {\mathbb{M}}({\mathbb{R}})$ that satisfies equation \[young equivalence equation\], even if ${\Vertf\Vert}_\infty \not \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$. If ${\Vertf\Vert}_\infty \not \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, $\nu_x^f$ might not be a probability measure, but it still satisfies the inequalities $0 \leq \nu_x^f({\mathbb{R}}) \leq 1$ for all $x \in \Omega$. In particular, the difference between $\nu_x^f({\mathbb{R}})$ and $1$ is due to $f$ being unlimited at some non-negligible fraction of $\mu(x)\cap{\mathbb{X}}^k$.
\[parametrized measures\] For every $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, there exists a parametrized measure $\nu^f : \Omega \rightarrow {\mathbb{M}}({\mathbb{R}})$ such that, for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c(\Omega)$, equality \[young equivalence equation\] holds. Moreover, for all $x \in \Omega$ and for all Borel $A \subseteq {\mathbb{R}}$, $0\leq \nu^f_x(A) \leq 1$.
Let $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, and for all $n \in {\mathbb{N}}$ define $$f_n(x) = \left\{\begin{array}{ll}
f(x)& \text{if } |f(x)| \leq n,\\
n & \text{if } f(x) > n,\\
-n & \text{if } f(x) < -n.
\end{array}\right.$$ Since for all $n \in {\mathbb{N}}$ it holds ${\Vertf_n\Vert}_\infty \leq n \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, by Theorem \[young\] there exists a Young measure $\nu^{n}$ that satisfies $$\label{you1}
{\,\!^\circ\langle {\,\!^\ast\Psi}(f_n), {\,\!^\ast\varphi} \rangle}
=
\int_{\Omega} \left( \int_{{\mathbb{R}}} \Psi d \nu^{n}_x \right) \varphi(x) dx.$$ for all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c(\Omega)$.
Recall that a sequence of parametrized measures $\{\mu^n\}_{n \in {\mathbb{N}}}$ converges weakly-$\star$ to a parametrized measure $\mu$ if for all $\Psi \in {C^0_b}({\mathbb{R}})$, the sequence $\{\Psi_n\}_{n \in {\mathbb{N}}}$ of $L^\infty$ functions defined by $$\Psi_n(x) = \int_{{\mathbb{R}}} \Psi d \mu^n_x$$ converges weakly-$\star$ to a function $\Psi_{\infty} \in L^\infty(\Omega)$ defined by $$\Psi_{\infty}(x) = \int_{{\mathbb{R}}} \Psi d \mu_x.$$ Define $\nu^f$ as the parametrized measure satisfying $\nu^{n} \stackrel{\star}{{\rightharpoonup}} \nu^f$ for some subsequence (not relabelled) of $\{\nu^{n}\}_{n \in {\mathbb{N}}}$. The existence of such a weak-$\star$ limit can be obtained as a consequence of the Banach-Alaouglu theorem (for further details about the weak-$\star$ limit of measures, we refer to to [@evans; @nonlinear]). We claim that $\nu^f$ satisfies equality \[young equivalence equation\] and that for all $x \in \Omega$, $0\leq \nu^f_x({\mathbb{R}}) \leq 1$.
Let $\Psi \in{C^0_b}({\mathbb{R}})$. Since $\lim_{|x| \rightarrow \infty} \Psi(x) = 0$, there is an increasing sequence of natural numbers $\{n_i\}_{i \in {\mathbb{N}}}$ such that if $|x| \geq n_i$, then $|\Psi(x)| \leq 1/i$. As a consequence of this inequality, for all $i \in {\mathbb{N}}$ and for all $\varphi \in C^0_c(\Omega)$ it holds $$\left| \langle {\,\!^\ast\Psi}(f_{n_i}), {\,\!^\ast\varphi} \rangle - \langle {\,\!^\ast\Psi}(f), {\,\!^\ast\varphi} \rangle \right| \leq 2/i {\Vert{\,\!^\ast\varphi}\Vert}_1.$$ Taking into account equation \[you1\], from the previous inequality we obtain $$\left| \int_{\Omega} \left( \int_{{\mathbb{R}}} \Psi d \nu^{n_i}_x \right) \varphi(x) dx - {\,\!^\circ\langle {\,\!^\ast\Psi}(f), {\,\!^\ast\varphi} \rangle} \right| \leq 2/i {\Vert\varphi\Vert}_1.$$ As a consequence, we deduce that $$\lim_{i \rightarrow \infty} \int_{\Omega} \left( \int_{{\mathbb{R}}} \Psi d \nu^{n_i}_x \right) \varphi(x) dx
=
{\,\!^\circ\langle {\,\!^\ast\Psi}(f), {\,\!^\ast\varphi} \rangle}.$$ This is sufficient to entail that $\nu^{n} \stackrel{\star}{{\rightharpoonup}} \nu^f$ and that $\nu^f$ satisfies equality \[young equivalence equation\].
The inequality $0 \leq \nu^f_x(A) \leq 1$ for all Borel $A \subseteq {\mathbb{R}}$ is a consequence of the lower semicontinuity of the weak-$\star$ limit of measures (see for instance theorem 3 of [@evans; @nonlinear]).
As a consequence of Theorem \[parametrized measures\], we deduce that the hypothesis ${\Vertf\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ in Theorem \[young\] can be relaxed. In particular, if $g$ is a grid function that differs from $f$ at some null set, then $f$ and $g$ induce the same parametrized measure, even if $f \not \equiv g$.
\[corollario importante young\] Let $L_{N}$ be the Loeb measure obtained from the measure $\mu_N(A) = |A|/N^k$ for all internal $A \subseteq {\mathbb{X}}^k$. If for $f, g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ it holds $L_{N}(\{x \in {\Omega_{{\mathbb{X}}}}: f(x) \not {\approx}g(x) \}) = 0$, then $\nu^f = \nu^g$. If ${\Vertf-g\Vert}_p {\approx}0$, then $\nu^f = \nu^g$.
If $L_{N}(\{x \in {\Omega_{{\mathbb{X}}}}: f(x) \not {\approx}g(x) \}) = 0$, then also $$L_{N}(\{x \in {\Omega_{{\mathbb{X}}}}: {\,\!^\ast\Psi}(f(x)) \not {\approx}{\,\!^\ast\Psi}(g(x)) \}) = 0$$ for all $\Psi \in {C^0_b}({\mathbb{R}})$. This is and the hypothesis $\Psi\in{C^0_b}({\mathbb{R}})$ are sufficient to deduce $\langle {\,\!^\ast\Psi}(f), {\,\!^\ast\varphi} \rangle {\approx}\langle {\,\!^\ast\Psi}(g), {\,\!^\ast\varphi} \rangle$ for all $\varphi \in C^0_c(\Omega)$ that, thanks to equation \[young equivalence equation\], is equivalent to the equality $\nu^f = \nu^g$.
The hypothesis ${\Vertf-g\Vert}_p {\approx}0$ implies $L_{N}(\{x \in {\Omega_{{\mathbb{X}}}}: f(x) \not {\approx}g(x) \}) = 0$, so the equality between $\nu^f$ and $\nu^g$ is a consequence of the previous part of the proof.
The above corollary can be seen as the grid function counterpart of Corollary 3.14 of [@webbym], that shows how Young measure ignore concentration phenomena. We find it useful to discuss this behaviour with an example, that also highlights how a grid function can describe simultaneously very different properties of a sequence of $L^p$ functions.
The following example is discussed from the standard viewpoint in [@webbym]. Consider the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ defined by $f_n(x) = n \chi_{[1-1/n,1]}$. Notice that ${\Vertf_n\Vert}_\infty = n$, so that the sequence is not bounded in $L^\infty({\mathbb{R}})$. For all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c({\mathbb{R}})$, it holds $$\lim_{n \rightarrow \infty} \int_{{\mathbb{R}}} \Psi(f_n) \varphi dx = \Psi(0) \int_{{\mathbb{R}}} \varphi dx$$ so that the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ converges weakly-$\star$ to the constant Young measure $\nu_x = \delta_0$ for all $x \in {\mathbb{R}}$.
The sequence $\{f_n\}_{n \in {\mathbb{N}}}$ satisfies the $L^1$ uniform bound ${\Vertf_n\Vert}_1 = 1$ for all $n \in {\mathbb{N}}$. Since for all $\varphi \in {\mathscr{D}}({\mathbb{R}})$ it holds $$\lim_{n \rightarrow \infty} \int_{{\mathbb{R}}} f_n \varphi dx = \lim_{n \rightarrow \infty} n \int_{[1-1/n,1]} \varphi dx = \varphi(1)$$ the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ converges in the sense of distributions to $\delta_1$, the Dirac distribution centred at $1$. Indeed, it can be proved that the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ converges weakly-$\star$ to $\delta_1$ in the space ${\mathbb{M}}({\mathbb{R}})$ of Radon measures. In the setting of grid functions, a representative for the limit of the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ is given by $f_N = N \chi_{1}$. For all $\Psi \in {C^0_b}({\mathbb{R}})$ and for all $\varphi \in C^0_c({\mathbb{R}})$, it holds $$\langle {\,\!^\ast\Psi}(f_N) , {\,\!^\ast\varphi} \rangle = \varepsilon \sum_{x \in {\mathbb{X}}, \, x \not = 1} \Psi(0) {\,\!^\ast\varphi}(x) + \varepsilon {\,\!^\ast\Psi}(N) \varphi(1).$$ Since $\Psi \in {C^0_b}({\mathbb{R}})$, ${\,\!^\ast\Psi}(N) {\approx}0$ and, by Lemma \[equivalenza integrali\], we deduce $${\,\!^\circ\langle {\,\!^\ast\Psi}(f_N) , {\,\!^\ast\varphi} \rangle} = \Psi(0) \int_{{\mathbb{R}}} \varphi(x)dx.$$ From the above equality and from equation \[young equivalence equation\], we deduce that the Young measure associated to $f_N$ is the constant Young measure $\nu_x = \delta_0$ for all $x \in {\mathbb{R}}$. Notice that the same result could have been deduced from Corollary \[corollario importante young\] by noticing that, since $L_{N}(\{x \in {\Omega_{{\mathbb{X}}}}: f_N(x) \not {\approx}0 \}) = 0$, the Young measure associated to $f_N$ is the same as the Young measure associated to the constant function $c(x) = 0$ for all $x \in {{\,\!^\ast{\mathbb{R}}}}$.
As for the distribution $[f_N]$, since for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\mathbb{X}})$ it holds $
\langle N\chi_1, \varphi \rangle = \varphi(1),
$ we deduce that $[f_N] = \delta_1$. In particular, the grid function $f_N$ coherently describes the behaviour of the limit of the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ both in the sense of Young measures and in the sense of distributions.
In the previous example we have considered a grid function $f$ with ${\Vertf\Vert}_1 \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, and we verified that the parametrized measure associated to $f$ was indeed a Young measure. This result holds under the more general hypothesis that ${\Vertf\Vert}_p \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$.
\[proposition young lp\] If ${\Vertf\Vert}_p \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, then $\nu^f_x$ is a probability measure for a.e. $x \in \Omega$.
If for some $x \in \Omega$ it holds $\nu^f_x({\mathbb{R}}) < 1$, then there exists $y \in {\Omega_{{\mathbb{X}}}}$, $y {\approx}x$ such that $f(y) \not \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$. The hypothesis ${\Vertf\Vert}_p \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$ implies $L_{N}(\{y \in {\Omega_{{\mathbb{X}}}}: f(y) \not \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}\}) = 0$: this is sufficient to conclude that ${\mu_L}(\{x \in \Omega : \nu^f_x({\mathbb{R}}) < 1 \}) = 0$, as desired.
We will conclude the discussion of the relations between grid functions and parametrized measures by determining the parametrized measure associated to a periodic grid function with an infinitesimal period. This is the grid function counterpart of the formula for the Young measure associated to the limit of a sequence of periodic functions (see Example 3.5 of [@balder]). We will prove this result for $k = 1$, as the generalization to an arbitrary dimension is mostly a matter of notation.
\[homogeneous\] If $f \in {\mathbb{G}({{\mathbb{X}}})}$ is periodic of period $M\varepsilon {\approx}0$, then the parametrized measure $\nu$ associated to $f$ is constant, and $$\int_{{\mathbb{R}}} \Psi d\nu_x = {\,\!^\circ\left( \frac{1}{M} \sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))\right)}$$ for all $x \in \Omega$ and for all $\Psi \in {C^0_b}({\mathbb{R}})$.
Without loss of generality, let $M \in {\,\!^\ast{\mathbb{N}}}$ and let $f$ be periodic over $[0,(M-1)\varepsilon]\cap{\mathbb{X}}$, with $M\varepsilon{\approx}0$.
Let $\Psi \in {C^0_b}({\mathbb{R}})$. At first, we will prove that $\frac{1}{M} \sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))$ is finite: in fact, $$\label{finiteness}
\inf_{x\in{{\,\!^\ast{\mathbb{R}}}}}{\,\!^\ast\Psi}(x) \leq
\frac{1}{M} \sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))
\leq \sup_{x \in {{\,\!^\ast{\mathbb{R}}}}} {\,\!^\ast\Psi}(x)$$ and by the boundedness of $\Psi$, we deduce that $\frac{1}{M} \sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))$ is finite.
Let now $\varphi \in S^0({\mathbb{X}})$ with ${\mathrm{supp\,}}\varphi \subset [a,b]$, $a, b \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$. Then there exists $h,k \in {\,\!^\ast{\mathbb{N}}}$ satisfying $a {\approx}Mh\varepsilon$ and $b {\approx}Mk\varepsilon$. We have the equalities $$\begin{aligned}
\notag
\langle {\,\!^\ast\Psi}(f), \varphi \rangle
&{\approx}&
\varepsilon \sum_{x\in{[Mh\varepsilon,Mk\varepsilon]_{{\mathbb{X}}}}} \Psi(f(x))\varphi(x)\\\notag
&=&
\varepsilon \sum_{j = h}^{k} \left(\sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon)) \varphi(jM\varepsilon + i\varepsilon) \right)\\\notag
&=&
\varepsilon \sum_{j = h}^{k} \left( \left(\sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))\right) (\varphi(jM\varepsilon) + e(j)) \right)\\\label{equiv misura 3}
&=&
\left(\frac{1}{M} \sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))\right) \left(M\varepsilon \sum_{j = h}^{k} ( \varphi(jM\varepsilon) + e(j))\right).
\end{aligned}$$ Let $$e = \max_{0 \leq i \leq M,\ k\leq j \leq h}\{|\varphi(jM\varepsilon)-\varphi(jM\varepsilon+i\varepsilon)|\}.$$ Since $\varphi \in S^0({\mathbb{X}})$ and ${\mathrm{supp\,}}\varphi \subset {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, $e {\approx}0$ and, as a consequence, $|e(j)| \leq e {\approx}0$. We deduce $$\left|M\varepsilon \sum_{j = k}^{h} e(j)\right| \leq M\varepsilon (k-h) e {\approx}(b-a)e {\approx}0$$ and, by equation \[finiteness\], $$\label{equiv misura 4}
\left( \frac{1}{M} \sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))\right) \left( M\varepsilon \sum_{j = h}^{k} e(j)\right) {\approx}0.$$ Since $M\varepsilon {\approx}0$, $$\label{equiv misura 5}
M\varepsilon \sum_{j = h}^{k} (\varphi(jM\varepsilon)
{\approx}\int_{{\,\!^\circa}}^{{\,\!^\circb}} {\,\!^\circ\varphi(x)} dx.$$ Putting together equalities \[equiv misura 3\], \[equiv misura 4\] and \[equiv misura 5\], we conclude $${\,\!^\circ\langle {\,\!^\ast\Psi}(f), \varphi(x) \rangle}
=
{\,\!^\circ\left(\frac{1}{M} \sum_{i = 0}^{M-1} {\,\!^\ast\Psi}(f(i\varepsilon))\right)} \int_{{\,\!^\circa}}^{{\,\!^\circb}} {\,\!^\circ\varphi(x)} dx$$ as we wanted.
The grid function formulation of partial differential equations {#solutions}
===============================================================
We have seen that the space of grid functions extends coherently both the space of distributions and the space of Young measures. For this reason, we believe they can successfully applied to the study of partial differential equations. In this section, we will give some results that allow to give a coherent grid function formulation of stationary and time-dependent PDEs. A solution to the grid function formulation can then be used to define a standard solution to the original problem. It turns out that, for some nonlinear problems, this process will give rise to measure-valued solutions.
The grid function formulation of linear PDEs {#linear pdes}
--------------------------------------------
A linear PDE can be written in the most general form as $$\label{lift2}
L(u) = T,$$ with $T \in {\mathscr{D}}'(\Omega)$, where $L : {\mathscr{D}}'(\Omega) \rightarrow {\mathscr{D}}'(\Omega)$ is linear, and where the equality is meant in the sense of distributions, i.e.$${\langle}L(u), \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= {\langle}T, \varphi {\rangle_{{\mathscr{D}}(\Omega)}}$$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}(\Omega)$. We would like to turn problem \[lift2\] in a problem in the sense of grid functions, i.e. $$\label{lift1}
{{L_{{\mathbb{X}}}}}(u) = {T_{{\mathbb{X}}}},$$ with ${T_{{\mathbb{X}}}}\in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, where ${{L_{{\mathbb{X}}}}}: {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}\rightarrow {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ is ${{\,\!^\ast{\mathbb{R}}}}$-linear, and where the equality is pointwise equality. We would like to determine sufficient conditions that ensure equivalence between problem \[lift1\] and problem \[lift2\], in the sense that \[lift2\] has a solution if and only if \[lift1\] has a solution.
Such a coherent formulation of linear PDEs relies upon the existence of ${{\,\!^\ast{\mathbb{R}}}}$-linear extensions of linear functionals over the space of distributions. Recall that every linear functional $L : {\mathscr{D}}'(\Omega) \rightarrow {\mathscr{D}}'(\Omega)$ induces an adjoint ${L^{\dagger}}: {\mathscr{D}_{\mathbb{X}}}(\Omega) \rightarrow {\mathscr{D}_{\mathbb{X}}}(\Omega)$ that satisfies $${\langle}L(T), \varphi {\rangle_{{\mathscr{D}}(\Omega)}}= {\langle}T, {L^{\dagger}}(\varphi) {\rangle_{{\mathscr{D}}(\Omega)}}$$ for all $T \in {\mathscr{D}}'(\Omega)$ and for all $\varphi \in {\mathscr{D}}(\Omega)$. If we find a ${{\,\!^\ast{\mathbb{R}}}}$-linear extension of ${L^{\dagger}}$ in the sense of grid functions, by taking the adjoint we are able to define a ${{\,\!^\ast{\mathbb{R}}}}$-linear extension of $L$.
\[lemma lift duale\] For every linear $L: {\mathscr{D}}(\Omega) \rightarrow {\mathscr{D}}(\Omega)$ there is a ${{\,\!^\ast{\mathbb{R}}}}$-linear ${{L_{{\mathbb{X}}}}}:{\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ such that ${{L_{{\mathbb{X}}}}}({\,\!^\ast\varphi}) = {\,\!^\ast(}L(\varphi))_{|{\Omega_{{\mathbb{X}}}}}$ for all $\varphi \in {\mathscr{D}}(\Omega)$.
For $\varphi \in {\mathscr{D}}'(\Omega)$ define $$U(\varphi) = \{ {{L_{{\mathbb{X}}}}}: {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \text{ such that } {{L_{{\mathbb{X}}}}}\text{ is } {{\,\!^\ast{\mathbb{R}}}}\text{-linear and } {{L_{{\mathbb{X}}}}}({\,\!^\ast\varphi}) = {\,\!^\astL(\varphi)}_{|{\Omega_{{\mathbb{X}}}}} \}$$ and let $U = \{ U(\varphi) : \varphi \in {\mathscr{D}}(\Omega)\}$. If we prove that $U$ has the finite intersection property, then, by saturation, $\bigcap U \not = \emptyset$, and any ${{L_{{\mathbb{X}}}}}\in \bigcap U$ is a ${{\,\!^\ast{\mathbb{R}}}}$-linear function that satisfies ${{L_{{\mathbb{X}}}}}({\,\!^\ast\varphi}) = {\,\!^\ast(}L(\varphi))_{{\mathbb{X}}}$ for all $\varphi \in {\mathscr{D}}(\Omega)$.
We will prove that, if $\varphi_1, \ldots, \varphi_n \in {\mathscr{D}}$, then $\bigcap_{i = 1}^n U(\varphi_i) \not = \emptyset$ by induction over $n$. If $n = 1$, we need to show that $U(\varphi) \not = \emptyset$ for all $\varphi \in {\mathscr{D}}'(\Omega)$. If $\varphi = 0$, then the constant function ${{L_{{\mathbb{X}}}}}(f) = 0$ for all $f \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ belongs to $U(\varphi)$. If $\varphi \not = 0$, let $f = {\,\!^\ast\varphi}_{|{\Omega_{{\mathbb{X}}}}}$, $g = {\,\!^\ast(L(\varphi)})_{|{\Omega_{{\mathbb{X}}}}}$, and let $\{f, b_2, \ldots, b_M\}$ be a ${\,\!^\ast}$basis of ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. Define also $${{L_{{\mathbb{X}}}}}\left(a_1f + \sum_{i = 2}^M a_i b_i \right) = a_1g$$ for all $a_1, \ldots a_M \in {{\,\!^\ast{\mathbb{R}}}}$. By definition, ${{L_{{\mathbb{X}}}}}$ is ${{\,\!^\ast{\mathbb{R}}}}$-linear and ${{L_{{\mathbb{X}}}}}\in U(\varphi)$.
We will now show that if $\bigcap_{i = 1}^{n-1} U(\varphi_i) \not = \emptyset$ for any choice of $\varphi_1, \ldots, \varphi_{n-1} \in {\mathscr{D}}(\Omega)$, then also $\bigcap_{i = 1}^{n} U(\varphi_i) \not = \emptyset$ for any choice of $\varphi_1, \ldots, \varphi_{n} \in {\mathscr{D}}(\Omega)$. If $\{\varphi_1, \ldots, \varphi_n\}$ are linearly dependent, thanks to linearity of $L$, any ${{L_{{\mathbb{X}}}}}\in \bigcap_{i = 1}^{n-1} U(\varphi_i)$ satisfies ${{L_{{\mathbb{X}}}}}\in \bigcap_{i = 1}^n U(\varphi_i)$. If $\{\varphi_1, \ldots, \varphi_n\}$ are linearly independent, let $f_n = ({\,\!^\ast\varphi_n})_{|{\Omega_{{\mathbb{X}}}}}$, let $g_n = {\,\!^\ast(L(\varphi_n)})_{|{\Omega_{{\mathbb{X}}}}}$ and let $\{f_n, b_{2}, \ldots, b_M\}$ be a ${\,\!^\ast}$basis of ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. For any ${{L_{{\mathbb{X}}}}}\in \bigcap_{i = 1}^{n-1} U(\varphi_i)$, define $\overline{{{L_{{\mathbb{X}}}}}} : {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ by $$\overline{{{L_{{\mathbb{X}}}}}}\left(a_1f_n + \sum_{i = 2}^M a_i b_i \right) =
a_1 g_n + {{L_{{\mathbb{X}}}}}\left(\sum_{i = 2}^M a_i b_i\right)$$ for all $a_1, \ldots a_M \in {{\,\!^\ast{\mathbb{R}}}}$. Then $\overline{{{L_{{\mathbb{X}}}}}}\in \bigcap_{i = 1}^n U(\varphi_i)$. This concludes the proof.
\[teorema lift lineare\] For every linear $L: {\mathscr{D}}'(\Omega) \rightarrow {\mathscr{D}}'(\Omega)$ there is a ${{\,\!^\ast{\mathbb{R}}}}$-linear ${{L_{{\mathbb{X}}}}}:{\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ such that ${\,\!^\circ\langle {{L_{{\mathbb{X}}}}}(f), {\,\!^\ast\varphi} \rangle} = {\langle}L [f], \varphi{\rangle_{{\mathscr{D}}(\Omega)}}$ for all $f \in {\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}})$ and for all $\varphi \in {\mathscr{D}}(\Omega)$. As a consequence, the following diagram commutes: $$\label{lift lineare}
\begin{array}{ccc}
{{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}& \stackrel{{{L_{{\mathbb{X}}}}}}{\longrightarrow} & {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}\\
\Phi \circ \pi \downarrow & & \downarrow \Phi \circ \pi\\
{\mathscr{D}}'(\Omega) & \stackrel{L}{\longrightarrow} & {\mathscr{D}}'(\Omega).
\end{array}$$
Let ${L^{\dagger}}$ be the adjoint of $L$, and let ${L^{\dagger}}_{\mathbb{X}}$ be the ${{\,\!^\ast{\mathbb{R}}}}$-linear operator coherent with ${L^{\dagger}}$ in the sense of Lemma \[lemma lift duale\]. Define $\langle {{L_{{\mathbb{X}}}}}(f), \varphi \rangle = \langle f, {L^{\dagger}}_{\mathbb{X}}(\varphi)\rangle$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$. From this definition, ${{\,\!^\ast{\mathbb{R}}}}$-linearity of ${{L_{{\mathbb{X}}}}}$ can be deduced from the ${{\,\!^\ast{\mathbb{R}}}}$-linearity of ${L^{\dagger}}_{\mathbb{X}}$.
We will now prove that ${{L_{{\mathbb{X}}}}}$ satisfies ${\,\!^\circ\langle {{L_{{\mathbb{X}}}}}(f), {\,\!^\ast\varphi} \rangle} = {\langle}L [f], \varphi{\rangle_{{\mathscr{D}}(\Omega)}}$ for all $f \in {\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}})$ and for all $\varphi \in {\mathscr{D}}(\Omega)$. Let $f \in {\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}})$: for any $\varphi \in {\mathscr{D}}(\Omega)$, thanks to Lemma \[lemma lift duale\] we have the equalities $$\langle {{L_{{\mathbb{X}}}}}(f), {\,\!^\ast\varphi} \rangle
=
\langle f, {L^{\dagger}}_{\mathbb{X}}({\,\!^\ast\varphi}_{|{\Omega_{{\mathbb{X}}}}})\rangle
=
\langle f, {\,\!^\ast{L^{\dagger}}(\varphi)}\rangle
{\approx}{\langle}[f], {L^{\dagger}}(\varphi){\rangle_{{\mathscr{D}}(\Omega)}}=
{\langle}L[f], \varphi{\rangle_{{\mathscr{D}}(\Omega)}},$$ as we wanted.
By Lemma \[character\] and thanks to the previous equality, if $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, then ${{L_{{\mathbb{X}}}}}(f) \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$. This concludes the proof of the commutativity of diagram \[lift lineare\].
From the previous Theorem, we obtain some sufficient conditions that ensure the equivalence between the linear problem \[lift2\] in the sense of distributions and the linear problem \[lift1\] in the sense of grid functions.
\[theorem lift lineare\] Let $L : {\mathscr{D}}'(\Omega) \rightarrow {\mathscr{D}}'(\Omega)$ be linear, and let ${{L_{{\mathbb{X}}}}}: {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ any function such that diagram \[lift lineare\] commutes. Let also $T \in {\mathscr{D}}'(\Omega)$. Then problem \[lift2\] has a solution if and only if problem \[lift1\] has a solution $u \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$ for some ${T_{{\mathbb{X}}}}$ satisfying $[{T_{{\mathbb{X}}}}] = T$.
By Theorem \[teorema lift lineare\], if problem \[lift1\] has a solution $u$, then $[u]$ satisfies problem \[lift2\].
The other implication is a consequence of Theorem \[teorema lift lineare\] and of surjectivity of $\Phi$: suppose that \[lift2\] has a solution $v$. The commutativity of diagram \[lift lineare\] ensures that for any $u \in \Phi^{-1}(v)$ it holds $[{{L_{{\mathbb{X}}}}}(u)] = T$, hence for ${T_{{\mathbb{X}}}}= {{L_{{\mathbb{X}}}}}(u)$ problem \[lift1\] has a solution.
Thanks to this equivalence result, any linear PDE can be studied in the setting of grid functions with the techniques from linear algebra.
As an example of the grid function formulation of a linear PDE, we find it useful to discuss the Dirichlet problem.
\[def dirichlet\] Let $\Omega \subset {\mathbb{R}}^k$ be open and bounded, $h \in {\mathbb{N}}$, $a_{\alpha, \beta} \in C^\infty(\Omega)$, and let $$L(v) = \sum_{0\leq|\alpha|,|\beta|\leq h} (-1)^{|\alpha|}D^{\alpha}(a_{\alpha, \beta}D^\beta v).$$ The Dirichlet problem is the problem of finding $v$ satisfying $$\label{dirichlet}
\left\{
\begin{array}{l}
L(v) = f \text{ in } \Omega\\
D^{\alpha}u = 0 \text{ for } |\alpha|\leq h-1 \text{ in } \partial\Omega.
\end{array}\right.$$ If $f \in C_b(\Omega)$, then $v$ is a classical solution of the Dirichlet problem if $$\label{classi dirichlet}
v \in C^{2h}_b(\Omega) \cap C^{2h-1}_b(\overline{\Omega}) \text{ and } L(v) = f.$$ If $f \in L^2(\Omega)$, then $v$ is a strong solution of the Dirichlet problem if $$v \in H^{2h}(\Omega) \cap H^{h}_0(\overline{\Omega}) \text{ and } L(v) = f \text{ a.e.}$$ If $f \in H^{-h}(\Omega)$, then $v$ is a weak solution of the Dirichlet problem if $$\label{weak dirichlet}
v \in H^{h}_0(\Omega) \text{ and }
\sum_{0\leq|\alpha|,|\beta|\leq h} \int a_{\alpha, \beta}D^\beta v D^\alpha w = f(w)
\text{ for all } w \in H^h_0(\Omega).$$
A grid function formulation of the Dirichlet problem \[dirichlet\] is the following: let $${{L_{{\mathbb{X}}}}}(u) = \sum_{0\leq|\alpha|,|\beta|\leq h} (-1)^{|\alpha|}{\mathbb{D}}^{\alpha}({\,\!^\asta}_{\alpha, \beta}{\mathbb{D}}^\beta u).$$ The Dirichlet problem is the problem of finding $u \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ satisfying $$\label{grid dirichlet}
\left\{
\begin{array}{l}
{{L_{{\mathbb{X}}}}}(u) = P({\,\!^\astf}) \text{ in } {\Omega_{{\mathbb{X}}}}\\
{\mathbb{D}}^{\alpha}u = 0 \text{ in } \partial_{\mathbb{X}}^\alpha{\Omega_{{\mathbb{X}}}}\text{ for } |\alpha|\leq s-1.
\end{array}\right.$$
Notice that equation \[grid dirichlet\] is satisfied in the sense of grid functions, i.e. pointwise, while equation \[dirichlet\] assumes the different meanings shown in Definition \[def dirichlet\].
A priori, a solution $u$ of problem \[lift1\] induces a solution $[u]$ of problem \[lift2\] in the sense of distributions. However, if $[u]$ is more regular, it is a solution to \[lift2\] in a stronger sense.
\[teorema dirichlet\] Let $u$ be a solution of problem \[grid dirichlet\]. Then
1. if $f \in C_b(\Omega)$ and $[u] \in C^{2h}_b(\Omega) \cap C^{2h-1}_b(\overline{\Omega})$, then $[u]$ is a classical solution of the Dirichlet problem;
2. if $f \in L^2(\Omega)$ and $[u] \in H^{2h}(\Omega) \cap H^{h}_0(\Omega)$, then $[u]$ is a strong solution of the Dirichlet problem;
3. if $f \in H^{-h}(\Omega)$ and $[u] \in H^{h}_0(\Omega)$, then $[u]$ is a weak solution of the Dirichlet problem, i.e. $[u]$ satisfies \[weak dirichlet\].
A solution $u$ of problem \[grid dirichlet\] satisfies the equality $$\begin{aligned}
\langle P({\,\!^\astf}), \varphi \rangle
&=& \sum_{0\leq|\alpha|,|\beta|\leq h} (-1)^{|\alpha|} \langle {\mathbb{D}}^{\alpha}({\,\!^\asta}_{\alpha, \beta}{\mathbb{D}}^\beta u), \varphi \rangle \\
&=& \sum_{0\leq|\alpha|,|\beta|\leq h} \langle {\,\!^\asta_{\alpha, \beta}} {\mathbb{D}}^\beta u, {\mathbb{D}}^{\alpha}\varphi \rangle.
\end{aligned}$$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}'({\Omega_{{\mathbb{X}}}})$.
We will now prove (1). If $f \in C_b(\Omega)$, then by Lemma \[proiez s-c\], $[P({\,\!^\astf})]=f$. By Theorem \[teorema equivalenza derivate2\], $[{\mathbb{D}}^{\beta}u] = D^{\beta}[u]$, and $[{\,\!^\asta}_{\alpha, \beta}{\mathbb{D}}^\beta u] = a_{\alpha, \beta} D^{\beta} [u]$, so that $$[(-1)^{|\alpha|}{\mathbb{D}}^{\alpha}({\,\!^\asta}_{\alpha, \beta}{\mathbb{D}}^\beta u)]
=
(-1)^{|\alpha|}D^{\alpha}(a_{\alpha, \beta}D^\beta [u]).$$ We deduce that $[u]$ satisfies equation \[classi dirichlet\] in the classical sense, as desired.
The proof of parts (2) and (3) is similar to that of part (1). The only difference is that it relies on Lemma \[corollario lp\] instead of Lemma \[proiez s-c\].
While Theorem \[teorema lift lineare\] and Theorem \[theorem lift lineare\] do not explicitly determine an extension ${{L_{{\mathbb{X}}}}}$ for a given linear PDE, they determine a sufficient condition for problem \[lift1\] to be a coherent representation of problem \[lift2\] in the sense of grid function. In the practice, an explicit extension ${{L_{{\mathbb{X}}}}}$ of a linear $L : {\mathscr{D}}'(\Omega) \rightarrow {\mathscr{D}}'(\Omega)$ can be determined from $L$ by taking into account that
- thanks to Theorem \[teorema equivalenza derivate2\], derivatives can be replaced by finite difference operators;
- shifts can be represented in accord to Corollary \[corollario shift\];
- if $a \in C^\infty(\Omega)$, then $[{\,\!^\asta}f]=a[f]$ for all $f \in {{\mathscr{D}_{\mathbb{X}}}'({{\Omega_{{\mathbb{X}}}}})}$, since for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, ${\,\!^\asta}\varphi \in {\mathscr{D}_{\mathbb{X}}}({\Omega_{{\mathbb{X}}}})$, and we have the equalities $${\,\!^\circ\langle {\,\!^\asta} f, \varphi \rangle} = {\,\!^\circ\langle f, {\,\!^\asta}\varphi \rangle} = {\langle}[f], a\varphi{\rangle_{{\mathscr{D}}(\Omega)}}= {\langle}a[f], \varphi{\rangle_{{\mathscr{D}}(\Omega)}}.$$
Similarly, we have not established a canonical representative ${T_{{\mathbb{X}}}}$ for $T$. However, observe that for all $g \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ and for all $x \in {\Omega_{{\mathbb{X}}}}$ it holds $$g(x) = \sum_{y \in {\Omega_{{\mathbb{X}}}}} g(y) N^k \chi_{y}(x)$$ Moreover, $\chi_{y}(x) = \chi_0(x-y)$, so that once a solution $u_0$ for the problem ${{L_{{\mathbb{X}}}}}u = N^k\chi_0$ is determined, a solution for ${{L_{{\mathbb{X}}}}}(u) = g$ can be determined from the above equality by posing $$\label{convo}
u_{g}(x) = \sum_{y \in {\Omega_{{\mathbb{X}}}}} g(y) u_0(x-y).$$ In fact, by linearity of ${{L_{{\mathbb{X}}}}}$ we have that, for all $x \in {\Omega_{{\mathbb{X}}}}$, $$\begin{aligned}
{{L_{{\mathbb{X}}}}}(u_g(x)) &=& \displaystyle {{L_{{\mathbb{X}}}}}\left( \sum_{y \in {\Omega_{{\mathbb{X}}}}} g(y) u_0(x-y) \right)\\ \\
& = & \displaystyle \sum_{y \in {\Omega_{{\mathbb{X}}}}} g(y) {{L_{{\mathbb{X}}}}}(u_0(x-y))\\
& = & \displaystyle\sum_{y \in {\Omega_{{\mathbb{X}}}}} g(y) N^k \chi_{0}(x-y)\\
& = & g(x).
\end{aligned}$$ In particular, $u_0$ plays the role of a fundamental solution for problem \[lift1\], while equality \[convo\] can be interpreted as the discrete convolution between $g$ and $u_0$. As a consequence, the study of a linear problem \[lift1\] can be carried out by determining the solutions to the problem ${{L_{{\mathbb{X}}}}}(u) = N^k\chi_0$.
The grid function formulation of nonlinear PDEs {#section nonli}
-----------------------------------------------
A nonlinear PDE can be written in the most general form as $$F(u) = f,$$usually with $u \in V \subseteq L^2(\Omega)$ and $F: V \rightarrow W \subseteq L^2(\Omega)$. As in the linear case, the grid function formulation of nonlinear problems is based upon the possibility to coherently extend every continuous $F : L^2(\Omega) \rightarrow L^2(\Omega)$ to all of ${\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$. Since the proofs of the following theorems are based upon Lemma \[norma r\], we will impose the additional hypothesis that the Lebesgue measure of $\Omega$ is finite. Notice that, in contrast to what happened for Theorem \[teorema lift lineare\], in the proof of Theorem \[teorema lifting nonlineari\], we will be able to explicitly determine a particular extension ${F_{{\mathbb{X}}}}$ for a given continuous $F : L^2(\Omega) \rightarrow L^2(\Omega)$.
\[teorema lifting nonlineari\] Let ${\mu_L}(\Omega) < +\infty$ and let $F : L^2(\Omega) \rightarrow L^2(\Omega)$ be continuous. Then there is a function ${F_{{\mathbb{X}}}}: {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ that satisfies
1. whenever $u, v \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ are nearstandard in $L^2(\Omega)$, ${\Vertu-v\Vert}_2 {\approx}0$ implies ${\Vert{F_{{\mathbb{X}}}}(u)-{F_{{\mathbb{X}}}}(v)\Vert}_2 {\approx}0$;
2. for all $f \in L^2(\Omega)$, $[{F_{{\mathbb{X}}}}({P}({\,\!^\astf}))] = F(f)$.
We will show that the function defined by ${F_{{\mathbb{X}}}}(u) = {P}({\,\!^\astF({\widehat{u}})})$ for all $u \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ satisfies the thesis. By continuity of $F$, whenever $u$ and $v$ are nearstandard in $L^2(\Omega)$ we have $${\Vertu-v\Vert}_2 {\approx}0 \text{ implies } {\Vert{\,\!^\astF(u)}-{\,\!^\astF(v)}\Vert}_2 {\approx}0,$$ and, by Lemma \[norma r\], $${\Vert{\,\!^\astF(u)}-{\,\!^\astF(v)}\Vert}_2 {\approx}0 \text{ implies } {\Vert{F_{{\mathbb{X}}}}(u)-{F_{{\mathbb{X}}}}(v)\Vert}_2 {\approx}0,$$ hence (1) is proved. We will now prove that $[{F_{{\mathbb{X}}}}({P}({\,\!^\astf}))] = F(f)$. By Lemma \[norma r\], we have ${\Vert{\,\!^\astf}-{P}({\,\!^\astf})\Vert}_2 {\approx}0$ and, by continuity of ${\,\!^\astF}$, ${\Vert{\,\!^\astF}({\,\!^\astf})-{\,\!^\astF}({P}({\,\!^\astf}))\Vert}_2 {\approx}0$. From Lemma \[questo corollario\] we have $[{F_{{\mathbb{X}}}}({\,\!^\astf})] = [{P}({\,\!^\astF({\,\!^\astf})})] = F(f)$, as desired.
\[remark 1\] In the same spirit, if $F: V \rightarrow W$ is continuous and the space of grid functions can be continuously embedded in ${\,\!^\astV}$ and ${\,\!^\astW}$, then one can prove similar theorems by varying condition (1) in order to properly represent the topologies on the domain and the range of $F$. For instance, if $F: H^1(\Omega) \rightarrow L^2(\Omega)$, then (1) would be replaced by $${\Vertu-v\Vert}_{H^1} {\approx}0 \text{ implies } {\Vert{F_{{\mathbb{X}}}}(u)-{F_{{\mathbb{X}}}}(v)\Vert}_2 {\approx}0,$$ where ${\Vertu-v\Vert}_{H^1}$ is defined in the expected way as $${\Vertu-v\Vert}_{H^1} = {\Vertu-v\Vert}_{2}+{\Vert{\nabla_{\mathbb{X}}}(u-v)\Vert}_2.$$
Condition (1) of Theorem \[teorema lifting nonlineari\] is a continuity requirement for ${F_{{\mathbb{X}}}}$, and condition (2) implies coherence of ${F_{{\mathbb{X}}}}$ with the original function $F$, so that theorem \[teorema lifting nonlineari\] ensures that for all continuous $F : L^2 \rightarrow L^2$ there is a function ${F_{{\mathbb{X}}}}:{\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}\rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ which is continuous and coherent with $F$. This result allows to formulate nonlinear PDEs in the setting of grid functions.
\[nonlinear pdes\] Let ${\mu_L}(\Omega) < +\infty$, let $F:L^2(\Omega)\rightarrow L^2(\Omega)$ and let ${F_{{\mathbb{X}}}}: {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})} \rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ satisfy conditions (1) and (2) of Theorem \[teorema lifting nonlineari\]. Let also $f \in L^2(\Omega)$. Then the problem of finding $v \in L^2(\Omega)$ satisfying $$\label{eq nonli}
F(v) = f$$ has a solution if and only if there exists a solution $u \in {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, $u$ nearstandard in $L^2(\Omega)$, that satisfy $$\label{grid nonli}
{F_{{\mathbb{X}}}}(u) = f_{\mathbb{X}}$$ for some $f_{\mathbb{X}}\in{\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$ with $[f_{\mathbb{X}}]=f$, and in particular for $f_{\mathbb{X}}= P({\,\!^\astf})$.
Suppose that \[grid nonli\] with $f_{\mathbb{X}}= {P}({\,\!^\astf})$ has a solution $u$. Since $[{P}({\,\!^\astf})] = f$ by Corollary \[questo corollario\], $u$ satisfies the equality $[{F_{{\mathbb{X}}}}(u)] = f$ in the sense of distributions. At this point, if $u$ is nearstandard in $L^2(\Omega)$, by Lemma \[norma r\] we have ${\Vert{\,\!^\ast[u]} - u\Vert}_2 {\approx}0$, so that $[u] \in L^2(\Omega)$, and condition (2) of Theorem \[teorema lifting nonlineari\] ensures that $[{F_{{\mathbb{X}}}}(u)] = F([u])$, so that $[u]$ is a solution of \[eq nonli\].
For the other implication, suppose that $v$ is a solution to \[eq nonli\]. Then, by condition (2) of Theorem \[teorema lifting nonlineari\], $[{F_{{\mathbb{X}}}}({P}({\,\!^\astv}))] = F(v)=f$, so that problem \[grid nonli\] has a solution.
If $u$ is a solution to \[grid nonli\] but it is not nearstandard in $L^2(\Omega)$, i.e. if ${\Vert{\,\!^\ast[u]} - u\Vert}_2 \not {\approx}0$, $[{F_{{\mathbb{X}}}}(u)]$ needs not be equal to $F([u])$. In fact, if $[u] \in L^2(\Omega)$ and ${\Vert{\,\!^\ast[u]} - u\Vert}_2 \not {\approx}0$, we have argued in Section \[sub lp\] that we expect $u$ to feature either strong oscillations or concentrations. Due to these irregularities, we have no reasons to expect that $[{F_{{\mathbb{X}}}}(u)](x)$, that represents the mean of the values assumed by ${F_{{\mathbb{X}}}}(u)$ at points infinitely close to $x$, is related to $F([u])(x)$, that represents the function $F$ applied to the mean of the values assumed by $u$ at points infinitely close to $x$. However, as we have seen in Section \[young\], if ${\Vertu\Vert}_\infty \in {{{\,\!^\ast{\mathbb{R}}}}_{fin}}$, then $u$ can be interpreted as a Young measure $\nu^u$. If the composition $F(\nu^u)$ is defined in the sense of equation \[young equivalence equation\], then $\nu^u$ satisfies $$\int_{\Omega} \int_{{\mathbb{R}}} F(\tau) d\nu^u(x) \varphi(x) dx = {\,\!^\circ \langle {F_{{\mathbb{X}}}}(u), \varphi \rangle}
= {\,\!^\circ\langle {P}{({\,\!^\astf})}, \varphi\rangle}
=
\int_{\Omega} f\varphi dx$$ for all $\phi \in {\mathscr{D}}'(\Omega)$, and can be regarded as a Young measure solution to equation \[eq nonli\]. In particular, since Young measures describe weak-$\star$ limits of sequences of $L^\infty$ functions, the relation between $F(\nu^u)$ and problem \[eq nonli\] is the following: there exists a family of regularized problems $$F_\eta(u) = f_\eta$$ and a family $\{u_\eta\}_{\eta > 0}$ of $L^2(\Omega)\cap L^\infty(\Omega)$ solutions of these problems such that $\nu^u$ represents the weak-$\star$ limit of a subsequence of $\{u_\eta\}_{\eta > 0}$, and $F(\nu^u)$ is the corresponding weak limit of the sequence $\{F(u_\eta)\}_{\eta > 0}$.
In the case that ${\Vertu\Vert}_\infty$ is infinite or that $[u] \not \in L^2(\Omega)$, we consider $u$ as a generalized solution of problem \[eq nonli\] in the sense of grid functions. Moreover, we expect $u$ to capture both the oscillations and the concentrations we would expect from a sequence of solutions of some family of regularized problems of \[eq nonli\]. A more in-depth example of this behaviour is discussed in the grid function formulation of a class of ill-posed PDEs in [@illposed].
\[remark 2\] Notice that if ${F_{{\mathbb{X}}}}$ satisfies the stronger continuity hypothesis $$\label{weak continuity}
u \equiv v \text{ implies } {F_{{\mathbb{X}}}}(u) \equiv {F_{{\mathbb{X}}}}(v),$$ then ${F_{{\mathbb{X}}}}$ has a standard part $\tilde{F}$ defined by $$\tilde{F}(g) = [{F_{{\mathbb{X}}}}(P({\,\!^\astg}))]$$ for any $g \in L^2(\Omega)$. Moreover, from Lemma \[norma r\] and from Theorem \[teorema lifting nonlineari\], we deduce that $\tilde{F} = F$. As a consequence, any grid function $u$ that satisfies ${F_{{\mathbb{X}}}}(u) = P({\,\!^\astf})$ induces a solution to problem \[eq nonli\].
However, the continuity condition \[weak continuity\] holds only for very regular functions, and it fails for many of the functions that still satisfy the hypotheses of Theorem \[teorema lifting nonlineari\].
If the function $F$ appearing in equation \[eq nonli\] can be expressed as $$F = L \circ G,$$ where $G$ is nonlinear and $L$ is linear, the equivalence between the standard notions of solutions for the PDE \[eq nonli\] and one of its formulations in the sense of grid functions can be obtained by a suitable combination of the results of Theorem \[theorem lift lineare\] and of Theorem \[nonlinear pdes\].
Time dependent PDEs
-------------------
Time dependent PDEs have been studied in the setting of nonstandard analysis by a variety of means. A possibility is to give a nonstandard representation of a given time dependent PDE by discretizing in time as well as in space, and by defining a standard solution to the original problem by the technique of stroboscopy. In [@imme1], van den Berg showed how the stroboscopy technique can be extended to the study of a class of partial differential equations of the first and the second order by imposing additional regularity hypotheses on the time-step of the discretization. For an in-depth discussion on the stroboscopy technique and its applications to partial differential equations, we remand to [@imme1; @petite; @sari].
A delicate point in the time discretization of PDEs is that the discrete time step cannot be chosen arbitrarily. In fact, it is often the case that the time-step of the discretization must be chosen in accord to some bounds that depend upon the specific problem. As an example, consider the nonstandard model for the heat equation discussed in [@watt], where the time-step is dependent upon the diameter of the grid and upon the diffusion coefficients. In general, if the discrete timeline $\mathbb{T}$ is a deformation of the grid ${\mathbb{X}}$, then the finite difference in time does not generalize faithfully the partial difference in time, and Theorem \[teorema equivalenza derivate\] fails. However, it is possible to determine sufficient conditions over $\mathbb{T}$ that imply the existence of $k \in {\mathbb{N}}$ such that Theorem \[teorema equivalenza derivate\] holds for derivatives up to order $k$. This study has been carried out in depth by van den Berg in [@imme2].
Another possible approach to time-dependent PDEs is to follow the idea of Capińsky and Cutland in [@capicutland1; @capicutland; @statistic] and subsequent works: the authors did not discretize in time, but instead worked with functions defined on ${{\,\!^\ast{\mathbb{R}}}}\times {\mathbb{X}}^k$, where the first variable represents time, and the other $k$ variables represent space. Following this idea, we formulate the problem $$\label{time dependent standard}
u_t - Fu = f$$ with $u : {\mathbb{R}}\rightarrow V \subseteq L^2(\Omega)$, $F : V \rightarrow W \subseteq L^2(\Omega)$ by the grid function problem $$\label{time dependent grid}
u_t - {F_{{\mathbb{X}}}}u = f_{\mathbb{X}}$$ with $u : {{\,\!^\ast{\mathbb{R}}}}\rightarrow {\mathbb{G}({{\Omega_{{\mathbb{X}}}}})}$, with $[f_{\mathbb{X}}] = f$, and where ${F_{{\mathbb{X}}}}$ is a suitable extension of $F$ in the sense of Theorems \[teorema lift lineare\] and \[teorema lifting nonlineari\]. Notice that, by Theorem \[teorema equivalenza derivate2\] and by Theorem \[teorema lift lineare\], the grid function formulation of a time dependent PDE is formally a hyperfinite system of ordinary differential equations, and it can be solved by exploiting the standard theory of dynamical systems.
Once we have a grid function formulation for a time dependent PDE, we would like to study the relation between its solutions and the solutions to the original problem. If ${\Vertu(t)\Vert}_\infty$ is finite and uniformly bounded in $t$, by the same argument of Theorem \[young\] $u$ corresponds to a Young measure $\nu^u : [0,T] \times \Omega \rightarrow {{\mathbb{M}}^{\mathbb{P}}}({\mathbb{R}})$. If the composition $F(\nu^u)$ is defined in the sense of equation \[young equivalence equation\], then $\nu^u$ satisfies the equality $$\begin{aligned}
\int_{[0,T]\times\Omega} \int_{{\mathbb{R}}} \tau d\nu^u(t,x) \varphi_t + \int_{{\mathbb{R}}}F(\tau)d\nu^u(t,x) \varphi d(t,x) + \\
+ \int_{\Omega} \int_{{\mathbb{R}}} \tau d\nu^u(0,x)\varphi(0,x) dx
&=& \int_{[0,T]\times\Omega} [f_{\mathbb{X}}]\varphi d(t,x)\end{aligned}$$ for all $\varphi \in C^1([0,T],{\mathscr{D}}(\Omega))$ with $\varphi(T,x) = 0$. If $u$ is more regular, the above equality can be exploited to prove that $[u]$ is a weak, strong or classical solution to the original problem, in the same spirit as in Theorem \[teorema dirichlet\]. See also the discussion in Remark \[remark estensioni\].
If ${\Vertu(t)\Vert}_\infty$ is not finite, the sense in which $[u]$ is a solution to problem \[time dependent standard\] has to be addressed on a case-by-case basis. In [@illposed], we will discuss an example where ${\Vertu(t)\Vert}_1$ is finite and uniformly bounded in time, and $[u]$ can be interpreted as a Radon measure solution to problem \[time dependent standard\].
Selected applications {#selected applications}
=====================
We believe that the grid functions are a very general theory that provide a unifying approach to a variety of problems from different areas of functional analysis. To support our claim, we will use the theory of grid functions to study two classic problems from functional analysis that are usually studied with very different techniques: the first problem concerns the nonlinear theory of distributions, and the second is a minimization problem from the calculus of variations. For an in-depth discussion of a grid function formulation of a class of ill-posed PDEs, we refer to [@illposed].
The product $HH'$ {#section hh}
-----------------
The following example is discussed in the setting of Colombeau algebras in [@colombeau; @advances], and it can also be formalized in the framework of algebras of asymptotic functions [@oberbuggenberg].
Let $H$ be the Heaviside function $$H(x) = \left\{
\begin{array}{ll}
0 & \text{if } x \leq 0\\
1 & \text{if } x > 0
\end{array}
\right.$$ and let $H'$ be the derivative of the Heaviside function in the sense of distributions, i.e. the Dirac distribution centered at $0$. It is well-known that the product $HH'$ is not well-defined in the sense of distributions. However, this product arises quite naturally in the description of some physical phenomena. For instance, in the study of shock waves discussed in [@colombeau; @advances], it is convenient to treat $H$ and $H'$ as a smooth functions and performing calculations such as $$\label{silly}
\int_{{\mathbb{R}}} (H^m-H^n)H'dx
=
\left[ \frac{H^{m+1}}{m+1} \right]^{+\infty}_{-\infty} - \left[ \frac{H^{n+1}}{n+1} \right]^{+\infty}_{-\infty}
=
\frac{1}{m+1}-\frac{1}{n+1}.$$ This calculation is not justified in the theory of distributions: on the one hand, $H^m=H^n$ for all $m, n\in{\mathbb{N}}$, so that we intuitively expect that the integral should equal $0$; on the other hand, since the products $H^mH'$ and $H^nH'$ are not defined, the integrand is not well-defined.
We will now show how in the setting of grid functions one can rigorously formulate the integral \[silly\] and compute the product $HH'$. Let $M \in {\,\!^\ast{\mathbb{N}}}\setminus{\mathbb{N}}$ satisfy $M\varepsilon {\approx}0$, and consider the grid function $h\in{\mathscr{D}_{\mathbb{X}}}'({\mathbb{X}})$ defined by $$h(x) = \left\{
\begin{array}{ll}
0 & \text{if } x \leq 0\\
x/(M\varepsilon) & \text{if } 0 < x < M\varepsilon\\
1 & \text{if } x \geq M\varepsilon
\end{array}
\right.$$ The function ${\mathbb{D}}h$ is given by $${\mathbb{D}}h(x) = \left\{
\begin{array}{ll}
0 & \text{if } x \leq 0 \text{ and } x \geq M\varepsilon\\
1/(M\varepsilon) & \text{if } 0 < x < M\varepsilon
\end{array}
\right.$$ In the next Lemma, we will prove that $h$ is a representative of the Heaviside function for which the calculation \[silly\] makes sense.
\[lemma prodotto\] The function $h$ has the following properties:
1. $[h^m]=H$ and $[{\mathbb{D}}h^m]=\delta_0$ for all $m \in {\,\!^\ast{\mathbb{N}}}$;
2. $h^m \not = h^n$ whenever $m \not = n$;
3. $\langle h^m-h^n, {\mathbb{D}}h \rangle {\approx}\frac{1}{m+1}-\frac{1}{n+1}$.
(1). Let $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\mathbb{X}})$ and, without loss of generality, suppose that $\varphi(x) \geq 0$ for all $x \in {\mathbb{X}}$. Then for all $m \in {\,\!^\ast{\mathbb{N}}}$ we have the inequalities $$\varepsilon \sum_{x \geq M\varepsilon} \varphi(x) \leq \langle h^m, \varphi \rangle \leq \varepsilon \sum_{x \geq 0} \varphi(x),$$ and, by taking the standard part of all the sides of the inequalities, we deduce $$\int_{0}^{+\infty} {\,\!^\circ\varphi(x)} dx \leq {\,\!^\circ\langle h^m, \varphi \rangle} \leq \int_{0}^{+\infty} {\,\!^\circ\varphi(x)} dx.$$ This is sufficient to conclude that $[h^m]=H$ for all $m \in {\,\!^\ast{\mathbb{N}}}$. By Theorem \[teorema equivalenza derivate2\], $[{\mathbb{D}}h^m] = H' = \delta_0$.
(2). Let $m \not = n$. Then, $$(h^m-h^n)(x) = \left\{
\begin{array}{ll}
0 & \text{if } x \leq 0 \text{ and } x \geq M\varepsilon\\
(x/(M\varepsilon))^m-(x/(M\varepsilon))^n & \text{if } 0 < x < M\varepsilon.
\end{array}
\right.$$ In particular, $h^m - h^n \not = 0$, even if $[h^m] - [h^n] = 0$.
(3). By the previous point, $$\langle h^m-h^n, {\mathbb{D}}h \rangle
=
\frac{1}{M} \sum_{j = 1}^{M} (j/M)^m-(j/M)^n.$$ Since $M$ is infinite, $$\frac{1}{M} \sum_{j = 1}^{M} (j/M)^m-(j/M)^n
{\approx}\int_0^1 x^m-x^n dx
=
\frac{1}{m+1}-\frac{1}{n+1}.$$
Thanks to the lemma above, we can compute the equivalence class in ${\mathscr{D}}'({\mathbb{R}})$ of the product $hh'$.
\[corollario prodotto\] $[h {\mathbb{D}}h] = \frac{1}{2} H'$.
For any $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\mathbb{X}})$, we have $$\langle h {\mathbb{D}}h, \varphi \rangle
=
\frac{1}{M^2} \sum_{j = 1}^M j \psi(j\varepsilon).$$ Let $
\underline{m} = \min_{1 \leq j \leq M}\{\varphi(j\varepsilon)\}$ and $\overline{m} = \max_{1 \leq j \leq M}\{\varphi(j\varepsilon)\}$. We have the following inequalities: $$\begin{aligned}
\frac{\underline{m}}{M} \sum_{j = 1}^M j/M
\leq
\frac{1}{M^2} \sum_{j = 1}^M j \varphi(j\varepsilon)
\leq
\frac{\overline{m}}{M} \sum_{j = 1}^M j/M.
\end{aligned}$$ Since $M$ is infinite, $$\frac{1}{M} \sum_{j = 1}^M j/M {\approx}\int_0^1 x dx=\frac{1}{2},$$ so that $${\,\!^\circ\left(\frac{\underline{m}}{2}\right)} \leq {\,\!^\circ\langle h {\mathbb{D}}h, \varphi \rangle} \leq {\,\!^\circ\left(\frac{\overline{m}}{2}\right)}.$$ By S-continuity of $\varphi$, $\underline{m} {\approx}\overline{m} {\approx}\varphi(0)$, so that ${\,\!^\circ\langle h {\mathbb{D}}h, \varphi \rangle } = \frac{1}{2} \,{\,\!^\circ\varphi(0)}$ for all $\varphi \in {\mathscr{D}_{\mathbb{X}}}({\mathbb{X}})$, which is equivalent to $[h {\mathbb{D}}h]=\frac{1}{2} H'$.
Notice that $h$ is not the only function satisfying Lemma \[lemma prodotto\] and Corollary \[corollario prodotto\]. In fact, we conjecture that Lemma \[lemma prodotto\] and Corollary \[corollario prodotto\] hold for a class of grid functions that satisfy some regularity conditions yet to be determined.
A variational problem without a minimum
---------------------------------------
We will now discuss a grid function formulation of a classic example of a variational problem without a minimum. For an in-depth analysis of the Young measure solutions to this problem we refer to [@sychev], and for a discussion of a similar problem in the setting of ultrafunctions, we refer to [@ultraapps]. The grid function formulation consists in a hyperfinite discretization, as in Cutland [@cutland; @controls2].
Consider the problem of minimizing the functional $$\label{variational}
J(u) = \int_0^1 \left( \int_0^x f(t)dt\right)^2 + (f(x)^2-1)^2 dx$$ with $f \in L^2([0,1])$. Intuitively, a minimizer for $J$ should have a small mean, but nevertheless it should assume values in the set $\{-1, +1\}$. Let us make precise this idea: define $$f_0 = \chi_{[k,k+1/2)}-\chi_{[k+1/2,k+1)},\ k \in {\mathbb{Z}}$$ and let $f_n : [0,1] \rightarrow {\mathbb{R}}$ be defined by $f_n(x) = f_0(nx)$. It can be verified that $\{f_n\}_{n \in {\mathbb{N}}}$ is a minimizing sequence for $J$, but $J$ has no minimum. However, the sequence $\{f_n\}_{n \in {\mathbb{N}}}$ is uniformly bounded in $L^\infty([0,1])$, hence it admits a weak\* limit in the sense of Young measures. The limit is given by the constant Young measure $$\nu_x = \frac{1}{2} (\delta_1 + \delta_{-1}).$$ We can evaluate $J(\nu)$: for the first term of the integral \[variational\], we have $$\int_0^x \nu(t) dt = \int_0^x \left(\int_{\mathbb{R}}\tau d\nu_x\right) dt = 0,$$ meaning that the barycentre of $\nu$ is $0$. Since the support of $\nu$ is the set $\{-1, +1\}$, the second term of the integral becomes $$(\nu(x)^2-1)^2 = \int_{\mathbb{R}}(\tau^2-1)^2 d\nu_x = 0.$$ As a consequence, $J(\nu) = 0$, and $\nu$ can be interpreted as a minimum of $J$ in the sense of Young measures.
In the setting of grid functions, the functional \[variational\] can be represented by $${J_{{\mathbb{X}}}}(u) = \varepsilon \sum_{n=0}^N \left[\left(\varepsilon \sum_{i = 0}^n f(i\varepsilon)\right)^2 + (f(n\varepsilon)^2-1)^2\right].$$ Observe that this representation is coherent with the informal description of $J$, and that the only difference between $J$ and ${J_{{\mathbb{X}}}}$ is the replacement of the integrals with the hyperfinite sums. Let us now minimize ${J_{{\mathbb{X}}}}$ in the sense of grid functions. The minimizing sequence found in the classical case suggests us that a minimizer of ${J_{{\mathbb{X}}}}$ should assume values $\pm 1$, and that it should be piecewise constant in an interval of an infinitesimal length. For $M \in {\,\!^\ast{\mathbb{N}}}$, let $f_M = {\,\!^\astu_0(Mx)}$. If $M < M' \leq N/2$, then $$\varepsilon \sum_{i = 0}^n f_M(i\varepsilon) > \varepsilon \sum_{i = 0}^n f_{M'}(i\varepsilon).$$ We deduce that a minimizer for ${J_{{\mathbb{X}}}}$ is the grid function $f_{N/2}$, that is explicitly defined by $f_{N/2}(n\varepsilon) = (-1)^n$.
We will now show that this solution is coherent with the one obtained with the classic approach, i.e. that the Young measure associated to $f_{N/2}$ corresponds to $\frac{1}{2} (\delta_1 + \delta_{-1})$. Since ${\Vertf_{N/2}\Vert}_\infty = 1$, Theorem \[young\] guarantees the existence of a Young measure $\nu$ that corresponds to $f_{N/2}$. Moreover, by Proposition \[homogeneous\], $\nu$ is constant, and $$\int_{{\mathbb{R}}} \Psi d\nu_x = \frac{1}{2} \sum_{i = 0}^1 \Psi(f_{N/2}(i\varepsilon)) = \frac{1}{2} (\Psi(1)+\Psi(-1))$$ for all $\Psi \in C^0_b({\mathbb{R}})$. We deduce that the Young measure associated to $f_{N/2}$ is constant and equal to $\frac{1}{2} (\delta_1 + \delta_{-1})$, the minimizer of $J$ in the sense of Young measures.
[1]{}
L. O. Arkeryd, N. J. Cutland and C. W. Henson (editors), *Nonstandard analysis - theory and applications*, Kluwer Academic Publications (1997).
E. J. Balder, *Lectures on Young measures*, Cahiers de mathématiques de la décision 9514, CEREMADE, Université Paris-Dauphine (1995).
J. M. Ball, *A version of the fundamental theorem for Young measures*, PDEs and Continuum Models of Phase Transitions: Proceedings of an NSF-CNRS Joint Seminar Held in Nice, France, January 18–22, (1988), doi:10.1007/BFb0024945.
V. Benci *Ultrafunctions and generalized solutions*, in: Advanced Nonlinear Studies, 13 (2013), 461–486, arXiv:1206.2257.
V. Benci, L. Luperi Baglini, *Generalized functions beyond distributions*, Arabian Journal of Mathematics vol 4 (2015), pp.231–253.
V. Benci, L. Luperi Baglini, *A model problem for ultrafunctions*, Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (2012). Electronic Journal of Differential Equations, Conference 21 (2014). ISSN: 1072-6691.
V. Benci, L. Luperi Baglini, *A non-archimedean algebra and the Schwartz impossibility theorem*, Monatshefte f[ü]{}r Mathematik vol. 176 no.4 (2015), pp. 503–520.
V. Benci, L. Luperi Baglini, *Ultrafunctions and applications*, Discrete & Continuous Dynamical Systems - Series S Vol. 7 Issue 4 (2014), p593-616. 24p.
I. P. van den Berg, *Discretisations of higher order and the theorems of Fa‘a di Bruno and DeMoivre-Laplace*, Journal of Logic & Analysis 5:6 (2013) 1–35 ISSN 17599008
I. P. van den Berg, *On the relation between elementary partial difference equations and partial differential equations*, Annals of Pure and Applied Logic 92 (1998), pp. 235–265.
A. Bernstein, A. Robinson, *Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos*, Pacific Journal of Mathematics 16:3 (1966), 421-431.
E. Bottazzi, *Two nonstandard models for an ill-posed parabolic equation*, in preparation.
M. Capiński, N. J. Cutland, *A Simple Proof of Existence of Weak and Statistical Solutions of Navier-Stokes Equations*, Proceedings: Mathematical and Physical Sciences, vol. 436, no. 1896 (1992), pp. 1–11., [www.jstor.org/stable/52016](www.jstor.org/stable/52016).
M. Capiński, N. J. Cutland, *Attractors for threedimensional NavierStokes equations*, The Royal Society, vol. 453, no. 1966 (1997), pp. 2413–2426, doi:[10.1098/rspa.1997.0129](10.1098/rspa.1997.0129).
M. Capiński, N. J. Cutland, *Navier-Stokes equations with multiplicative noise*, Nonlinearity, vol. 6, no. 1 (1993), <http://stacks.iop.org/0951-7715/6/i=1/a=005>.
M. Capiński, N. J. Cutland, *Statistical solutions of PDEs by nonstandard densities*, Monatshefte f[ü]{}r Mathematik, vol. 111, no. 2 (1991), pp. 99–117, doi:<http://dx.doi.org/10.1007/BF01332349>.
J. F. Colombeau, *A general multiplication of distributions*, Comptes Rendus Acad. Sci. Paris 296 (1983) 357–360, and subsequent notes presented by L. Schwartz.
J. F. Colombeau, *Nonlinear Generalized Functions: their origin, some developments and recent advances*, The São Paulo Journal of Mathematical Sciences, \[S.l.\], v. 7, n. 2, p. 201-239 (2013). ISSN 2316-9028.
J. G. van der Corput, *Introduction to the neutrix calculus*, Journal d’Analyse Mathematique vol. 7 (1960), pp. 281–399.
N. J. Cutland, *Infinitesimal Methods in Control Theory: Deterministic and Stochastic*, Acta Applicandae Mathematicae 5 (1986), pp. 105–135.
N. J. Cutland, *Internal controls and relaxed controls*, J. London Math. Soc. 15 (1983), pp. 529–589.
N. J. Cutland, B. Enright, *Stochastic nonhomogeneous incompressible Navier–Stokes equations*, Journal of Differential Equations, Volume 228, Issue 1, 2006, Pages 140-170, ISSN 0022-0396, doi:<http://dx.doi.org/10.1016/j.jde.2006.04.009>.
N. J. Cutland, H. J. Keiser, *Global Attractors for 3-Dimensional Stochastic Navier–Stokes Equations*, Journal of Dynamics and Differential Equations, vol. 16, no. 1 (2004), pp. 205–266, doi:[10.1023/B:JODY.0000041286.51881.39](10.1023/B:JODY.0000041286.51881.39).
N. J. Cutland, T. Lindstrom, *Random relaxed controls and partially observed stochastic systems*, Acta Applicandae Mathematicae 32(2) (1993), pp. 157–182.
M. Davis, *Applied Nonstandard Analysis*, Dover Publications, Inc., Mi- neola, New York, 2005.
H. Deguchi, M. Oberguggenberger, *Propagation of Singularities for Generalized Solutions to Wave Equations with Discontinuous Coefficients*, SIAM J. Math. Anal., 48(1) (2016), 397–442, doi:10.1137/15M1032661.
S. Demoulini, *Young measure solutions for a nonlinear parabolic equation of forward-backward type*, SIAM J. Math. Anal., 27 (1996), pp. 376–403.
P. A. M. Dirac, *On the Theory of Quantum Mechanics*, Proc. R. Soc. Lond. (1926), doi: 10.1098/rspa.1926.0133.
L. C. Evans, *Weak convergence methods for nonlinear partial differential equations*, CBMS Regional Conference Series in Mathematics Volume 74 (1990), ISBN: 978-0-8218-0724-8.
R. Goldblatt, *Lectures on the Hyperreals – An Introduction to Nonstandard Analysis*, Graduate Texts in Mathematics, vol. 188 (1998), Springer, doi: <http://dx.doi.org/10.1007/978-1-4612-0615-6>.
F. Hanqiao, D. F. St. Mary and F. Wattenberg, *Applications of nonstandard analysis to partial differential equations-I. The diffusion equation*, Mathematical Modelling, Vol. 7 (1986), pp. 507–523.
R.F. Hoskins, J.S. Pinto, *Hyperfinite representation of distributions*, Proc. Indian Acad. Sci. (Math. Sci.) (2000) 110: 363. doi:10.1007/BF02829532
H. J. Keiser, *Elementary calculus – An infinitesimal approach*, On-line Edition (2000).
M. Kinoshita, *Non-standard representations of distributions I*, Osaka J. Math. 25 (1988), pp. 805–824.
C. K. Li, *A review on the product of distributions*, Mathematical Methods in Engineering (2007), pp. 71–96, DOI: 10.1007/978-1-4020-5678-9\_5.
P.A. Loeb, *Conversion from nonstandard to standard measure spaces and applications in probability theory*, Transactions of the American Mathematical Society, vol. 211 (1975), 113–122, doi: <http://dx.doi.org/10.1090/S0002-9947-1975-0390154-8>.
P. A. Loeb, M. Wolff editors, *Nonstandard analysis for the working mathematician*, Kluwer Academic Publishers (2000).
C. Mascia, A. Terracina and A. Tesei, *Evolution of Stable Phases in Forward–Backward Parabolic Equations*, Advanced Studies in Pure Mathematics (2005).
C. Mascia, A. Terracina and A. Tesei, *Two-phase Entropy Solutions of a Forward–Backward Parabolic Equation*, Arch. Rational Mech. Anal. 194 (2009), pp. 887–925.
M. Nedeljkov, M. Oberguggenberger, *Ordinary differential equations with delta function terms*, Publ. Inst. Math. (Beograd) (N.S.) 91(105) (2012), 125 - 135.
V. Neves, *Nonstandard Calculus of Variations*, Journal of Mathematical Sciences (2004) 120: 940. doi:10.1023/B:JOTH.0000013557.48018.2a
M. Oberguggenberger, T. Todorov, *An embedding of Schwartz distributions in the algebra of asymptotic functions*, Int J Math Math Sci 21:417–428 (1998).
P. I. Plotnikov, *Passing to the limit with respect to viscosity in an equation with variable parabolicity direction*, Differential Equations 30 (1994), pp. 614–622.
A. Robinson, *Non-standard analysis* Amsterdam: North-Holland (1966).
T. Sari, *Petite histoire de la stroboscopie*, Colloque Trajectorien à la Mémoire de Georges Reeb et Jean-Louis Callot (Strasbourg-Obernai, 1995), 5–15, Prépubl. Inst. Rech. Math. Av., 1995/13, Univ. Louis Pasteur, Strasbourg, 1995.
T. Sari, *Stroboscopy and averaging*, A. Fruchard and A. Troesch (eds.), Colloque Trajectorien à la Mémoire de Georges Reeb et Jean-Louis Callot (Strasbourg-Obernai, 1995), pp. 95–124.
L. Schwartz, *Sur l’impossibilité de la multiplication des distributions*, C. R. Acad. Sci. Paris, 29 (1954), pp. 847–848.
M. Slemrod, *Dynamics of measured valued solutions to a backward-forward heat equation*, Journal of Dynamics and Differential Equations (1991).
F. Smarrazzo, *On a class of equations with variable parabolicity direction*, Discrete and continuous dynamical systems 22 (2008), pp. 729–758.
R. Strichartz, *A guide to distribution theory and Fourier Transforms*, CRC Press (1994).
M. A. Sychev, *Young measures as measurable functions and their applications to variational problems*, Journal of Mathematical Sciences, doi 10.1007/s10958-005-0503-6
L. Tartar, *An introduction to Sobolev spaces and interpolation spaces*, Springer-Verlag Berlin Heidelberg (2007).
T. D. Todorov, *An axiomatic approach to the nonlinear theory of generalized functions and consistency of Laplace transforms*, Integral Transforms and Special Functions, 22:9, 695-708 (2011), DOI: 10.1080/10652469.2011.604960.
T. D. Todorov, *Pointwise values and fundamental theorem in the algebra of asymptotic functions*, Grosser M, Hörmann G, Kunzinger M, Oberguggenberger M (eds) Non-linear theory of generalized functions. CRC Research Notes in Mathematics, vol 401. Chapman & Hall, Boca Raton, pp 369–383 (1999).
T. D. Todorov, H. Vernaeve, *Full algebra of generalized functions and non-standard asymptotic analysis*, J Log Anal 1: 205 (2008). doi:10.1007/s11813-008-0008-y
C. Tuckey, *Nonstandard Methods in the Calculus of Variations*, SIAM Review 37 (1995), pp. 474–475, 10.1137/1037114.
M. Webb, *Classical Young Measures in the Calculus of Variations*, <http://www.damtp.cam.ac.uk/user/mdw42/webbyoungmeasures.pdf> (2013).
L. C. Young, *Generalized curves and the existence of an attained absolute minimum in the calculus of variations*, C. R. Sci. Lettres Varsovie, C III 30 (1937), 212-234.
|
---
abstract: 'This letter presents a theoretical derivation of an estimate for a radio source jet kinetic luminosity. The expression yields jet powers that are quantitatively similar to a more sophisticated empirical relation published by the Willott, Blundell and Rawlings at Oxford. The formula allows one to estimate the jet kinetic luminosity from the measurement of the optically thin radio lobe emission in quasars and radio galaxies. Motivated by recent X-ray observation, the derivation assumes that most of the energy in the lobes is in plasma thermal energy with a negligible contribution from magnetic energy (not equipartition). The close agreement of the two independent expressions makes the veracity of these estimates seem very plausible.'
author:
- Brian Punsly
title: An Independent Derivation of the Oxford Jet Kinetic Luminosity Formula
---
Introduction
============
The purpose of this letter is to discuss estimates of the power transported by the radio jets in quasars and radio galaxies. An accurate estimate of the jet power is of fundamental physical interest, since it can be used to quantify the power emerging from the central engine of the radio source. In actuality, the radio luminosity is merely an indirect measure of the energy transported through the jets from the central engine that is not readily interpretable. Most of the energy flux is in mechanical form (kinetic luminosity) - the particles and fields necessary to produce the synchrotron luminosity that is detected in the radio lobes. The radiation losses, manifested as radio emission from the jet, are merely the waste energy of this kinematic flow.
Surprisingly, the most difficult methods of estimating jet power rely on observations of the jets themselves. Due to significant Doppler enhancement in relativistic jets, the synchrotron radio emission is a poor indicator of intrinsic jet power. For example, Cygnus A has extremely powerful radio lobes and faint radio jets. Most of the energy in the jets is not radiated away, but is transported to the lobes in the classical FRII double lobe morphology. Even the inclusion of observations of high energy emission such as optical or X-ray (inverse Compton) in one’s analysis of jet energetics does not tightly constrain the bulk jet flow. If the resolution is poor at high frequency (as is often the case) then one can not necessarily associate the plasma emitting the high frequency photons with the radio emitting plasma. If one has high resolution images then the high frequency emission can be detected in enhancement regions or knots in the jets. One can use this information to get an estimate of the plasma conditions within the dissipative knot, but this do not necessarily constrain the plasma state in the bulk of the jet. Furthermore, there are still ambiguities with the Doppler factor that affect the estimates quite dramatically.
The Doppler enhancement of relativistic flows in jets is a crucial parameter since the total luminosity of an unresolved jet scales as the Doppler factor to the fourth power and to the third power for a resolved cylindrical jet [@lin85]. This is the reason why the implementation of 5 GHz flux densities, as is common in studies of radio loudness of large quasar samples, is a poor indicator of the true intrinsic kinetic luminosity of the jets. More specifically, the majority of core dominated blazar-like quasars have incredibly strong 5 GHz flux densities from emission on the subkiloparsec scale, yet they have weak or moderate radio lobe emission @pun95. This is interpreted as the jet being of modest kinetic luminosity (at most) because there is not a large amount of hot plasma and gas that has been transported through the jets to the radio lobes. The 5 GHz flux only represents the dissipation in the jet itself and it has been extremely Doppler boosted. An estimate of kinetic luminosity based on 5 GHz flux density can be off by four or more orders of magnitude for a core dominated blazar.
A far better way to estimate the kinetic luminosity from a jet is to study the isotropic properties of the material ejected from the ends of the jets in the radio lobes. The radiation from the lobe material is generally considered to be of low enough bulk velocity so that Doppler enhancement is not much of an issue. The basic idea is that lobe expansion is dictated by the internal dynamics of the lobes and the physical state of the enveloping extragalactic gas. X-ray observations can indicate a bremsstrahlung spectrum of the surrounding gas that can be used to find the pressure of the extragalactic medium. X-rays also provide information on the working surfaces at the end of the lobes, “the hot spots.” One can associate the X-ray emission as inverse Compton radiation from the hot spots and the radio luminosity is the synchrotron emission from the hot spots. This constrains the plasma state within the luminous hot spots. However, most of the energy stored in the lobes is in the large diffuse regions of radio emission that constitutes the majority of the large volume of the radio lobes. It is the enormous volume of synchrotron emitting plasma within the lobes ($\sim 10^{4} -10^{5}\, \mathrm{kpc}^{3}$) that is the most direct indicator that the jets must be supplying huge quantities of hot plasma and magnetic field energy to the lobes. One can also use the curvature of the lobe synchrotron radio spectra to estimate parameters in the diffuse lobe gas - this is known as spectral ageing. Of course, all of these plasma state estimations are most accurate when applied to situations in which one has deep X-ray and radio data of a relaxed classical double lobe structure. This only occurs in a few instances, so such detailed analysis are not compatible with large sample studies. Motivated by these limitations, this letter presents two techniques for estimating jet energy based on partial information on the lobe parameters. The two methods involve different assumptions and have different ambiguities.
The most sophisticated calculation of the jet kinetic luminosity incorporates deviations from the minimum energy estimates in a multiplicative factor $f$ that represents the small departures from minimum energy, geometric effects, filling factors, protonic contributions and low frequency cutoff (see @wil99 for details). The quantity, $f$, is argued to constrained to be between 1 and 20. In @blu00, it was further determined that $f$ is most likely in the range of 10 to 20. Therefore we choose $f=15$ in order to convert 151 MHz flux densities, $F_{151}$, to estimates of kinetic luminosity, $Q_{151}$, using equation (12) and figure 7 of @wil99, $$\begin{aligned}
&& Q_{151}\approx 1.1\times
10^{45}\left[(1+z)^{1+\alpha}Z^{2}F_{151}\right]^{\frac{6}{7}}\mathrm{ergs/sec}\;,\\
&& Z \approx 3.31-3.65\times \nonumber\\
&&
\left(\left[(1+z)^{4}-0.203(1+z)^{3}+0.749(1+z)^{2}+0.444(1+z)+0.205\right]^{-0.125}\right)\;.\end{aligned}$$ The quantity $F_{151}$ is the optically thin flux density from the lobes (i.e., no contribution from Doppler boosted jets or radio cores) measured at 151 MHz in Jy. The flux density spectral index is defined as $F_{\nu}\sim\nu^{-\alpha}$. We have assumed a cosmology with $H_{0}$=70 km/s/Mpc, $\Omega_{\Lambda}=0.7$ and $\Omega_{m}=0.3$. The expression for Z is from @pen99.
In the following, a new estimate of the jet kinetic luminosity is derived that is motivated by the wealth of X-ray data on radio lobes that has been published since @wil99 [@blu00]. Both the current manuscript and the @wil99 derivations rest on the basic premise that $Q = U/T$ + radiation losses, where $U$ is the energy stored in the lobes and $T$ is the elapsed time. In @wil99, $U$ is found by assuming the lobes are near equipartition and there is uncertainty in the energy from protonic components and low frequency cutoffs which are incorporated in the empirical factor $f$ discussed above. Conversely, motivated by the new X-ray data described below, $U$ is computed theoretically in the limit that the lobes are very far from equipartition. In @wil99, $T$ is determined by an empirical estimator for the age of the radio source based on their lengths and head advance speeds. Conversely, in this treatment $T$ is computed from spectral ageing. In spite of the fact that $U$ and $T$ are determined from completely different methods and assumptions, the expressions for $Q$ that are found in (1.1) and (3.9) yield similar values (to within a factor of 2) for the jet luminosity. This close agreement lends credence to the claim that these formulae are robust estimators of jet kinetic luminosity.
Motivation: X-ray Observations
==============================
The minimum energy condition in the lobes seems to be in conflict with the X-ray data on the surrounding extragalactic gas. This was first noted for Cygnus A (see @pun01 and references therein) and later for 3C 388 in @lea01 as well as for a large sample of FRII radio sources in @har00 based on ROSAT data. These studies concluded that typically the pressure in the external gas greatly exceeds (by at least an order of magnitude) the lobe pressure associated with the minimum energy assumption. The general picture that seems to be emerging from the X-ray data of ROSAT, ASCA, XMM and Chandra is that the hot spot energies seem to slightly exceed the minimum energy requirement based on inverse Compton calculations of X-rays from the hot spots (see for example @wil02 [@brn02]), but the lobes themselves are far from equipartition. In @lea01, it was deduced that the departure from equipartion was most likely the consequence of a low energy population of positrons and electrons that is not an extension of the power law distribution responsible for the radio emission, but a low energy excess. The other possibility is protonic matter which would also drastically increase the energy content of the lobes over the equipartition estimates. In either case, the field energy is only a few percent of the particle energy in the lobes. For example, high resolution Chandra data was used to model the X-rays as inverse Compton emission in the lobes in the radio galaxy 3C 219 [@bru02]. It was concluded that the particle energy exceeded the magnetic energy by a factor of 60 in the radio lobes. Similarly, the FRII radio galaxy 3C 452 was studied with Chandra in @iso02 who estimate that the energy density in the particles is 27 times that of the magnetic field in the lobes. For most FR II sources, typical magnetic field strengths in the lobes that are estimated from X-ray data and pressure balance are only a third to a fifth of the minimum energy value [@har00; @lea01].
These X-ray observations yield valuable information on the energy content of the lobes, but are disjoint from the spectral ageing estimates of the lobe advance speeds. In the following estimation of jet kinetic energy, it is assumed based on the X-ray data presented above that the energy content of the lobes is purely in particle form to first order (accurate to a few percent). Yet, the notion that spectral ageing provides an estimate of lobe age is retained. By choosing a subequipartion field strength in the lobes, spectral ageing estimates are found to be longer than the corresponding minimum energy estimates [@ale96]. Similarly, by setting $t_{sep}=t_{syn}$, the subequipartion fields yield lower lobe separation velocities if this spectral ageing argument is viable. Thus, the problem of the large lobe advanced speeds in the minimum energy assumption is remedied by this modification [@ale96]. This method of computing jet kinetic luminosity is the lowest order improvement to the minimum energy estimate and was implemented in @pun01 to study Cygnus A.
Particle Energy Dominated Lobes
===============================
Motivated by the X-ray observations, we proceed to compute the jet power based on the limit that all of the lobe energy is in the hot particles. We also assume that the time to convert a jet energy flux to the stored lobe energy is the time that it has taken the lobes to propagate from the central engine to their current separation, $t_{sep}$.
Spectral Ageing
---------------
Spectral ageing within the radio lobes is often used to determine the lobe plasma age. The results are predicated on the assumption that the lobe plasma is primarily back flowing plasma in the sense that jet plasma is deflected backward at the working surfaces in the hot spots to form the lobe plasma. By studying the curvature of the radio spectra at different points within the lobes, one can in principle (if there is no reheating or re-injection of the plasma) determine the gradient in the high energy cutoff of the electron distribution due to synchrotron cooling and hence the plasma age. The age of the lobe plasma closest to the central engine should be the oldest plasma. Thus, one has an estimate of lobe age and therefore the lobe advance speed. Defining the spectral break frequency as $\nu_{_{b}}$, the synchrotron lifetime is expressed in cgs units as [@liu92] $$\begin{aligned}
&& t_{syn}\approx 1.58\times 10^{12}
B^{-\frac{3}{2}}\nu_{_{b}}^{-\frac{1}{2}}\;.\end{aligned}$$
The Energy Contained Within the Synchrotron Emitting Plasma
-----------------------------------------------------------
Consider a power law distribution of energetic particles (probably electrons and positrons) expressed in terms of the thermal Lorentz factor, $\gamma$, for a uniform source in a volume, $V$. The total number of particles contributing to the synchrotron radiation in the frequency interval $\nu_{_1}\leq\nu
\leq \nu_{_2}$ is: $$\begin{aligned}
&&
N_{r}=N_{0}V\int_{\gamma_{_1}}^{\gamma_{_2}}\gamma^{-n}\,\mathrm{d}\gamma
\; .\end{aligned}$$ The minimum and maximum Lorentz factors in the expression above are related to lower and upper cutoff frequencies in the synchrotron spectrum $\nu_{_{1}}$ and $\nu_{_{2}}$, respectively by [@gin79] $$\begin{aligned}
&&
\gamma_{_1}=\left[\frac{2\nu_{_1}y_{_1}(n)}{3\nu_{_B}}\right]^{\frac{1}{2}}\;,\quad
\gamma_{_2}=\left[\frac{2\nu_{_2}y_{_2}(n)}{3\nu_{_B}}\right]^{\frac{1}{2}},\end{aligned}$$ where $\nu_{_{B}}=(eB)/(2\pi{m_{e}c})$ is the cyclotron frequency and note that $$\begin{aligned}
&& y_{_1}(n) = 2.2,\, y_{_2}(n)=0.10 ,\mathrm{if}\;
n=2.5:\mathrm{and}\, y_{_1}(n) = 2.7 ,\,
y_{_2}(n)=0.18,\mathrm{if}\; n=3.0\;.\end{aligned}$$ The synchrotron spectral luminosity of the plasma, $L(\nu)$, is a function of both the particle distribution in momentum space and the magnetic field strength. Integrating the synchrotron power formula over the particle distribution yields [@gin79] $$\begin{aligned}
&& U_{e}\approx\frac{2\times 10^{11}B^{-\frac{3}{2}}}
{a(n)(n-2)}L(\nu_{_1})\nu_{_1}^{\frac{1}{2}}
[y_{_1}(n)]^{\frac{n-1}{2}}\nonumber \\
&& \hspace{0.3in} \times
\left[1-\left(\frac{y_{_2}(n)\nu_{_1}}{y_{_1}(n)\nu_{_2}}\right)^{\frac{n-1}{2}}\right
]\;,\end{aligned}$$ where $$\begin{aligned}
&& a(n)=\frac{\left(2^{\frac{n-1}{2}}\sqrt{3}\right)
\Gamma\left(\frac{3n-1}{12}\right)\Gamma\left(\frac{3n+19}{12}\right)
\Gamma\left(\frac{n+5}{4}\right)}
{8\sqrt\pi(n+1)\Gamma\left(\frac{n+7}{4}\right)} \; .\end{aligned}$$
Estimating the Jet Power
------------------------
Set $t_{sep}$ equal to the synchrotron ageing timescale $t_{syn}$, associated with the spectral break in the flux density, $F_{\nu}$, of the lobe plasma closest to the quasar (the emission just above the spectral break is from the lowest energy electrons that have synchrotron radiated away their energy and hence the oldest subpopulation of charges that have experienced synchrotron decay in the lobes). By combining $t_{syn}$ from (3.1) with the expression for the plasma energy, (3.5), one obtains an estimate for the energy stored in the lobes as a function of spectral luminosity, $L(\nu)$, in the limit of particle energy dominance, $$\begin{aligned}
&& U_{e}\approx
\frac{L(\nu_{1})(\nu_{1})^{1/2}(\nu_{b})^{1/2}}{7.9(n-2)a(n)}\left[y_{1}(n)\right]^{\frac{n-1}{2}}t_{syn}\;.\end{aligned}$$
Evaluation of the formula above requires numerous characteristic frequencies that need to be determined. Since expressions that are applicable to sparse data are desired for evaluating large samples, we choose a common set of “typical” parameters for an FRII radio source. First of all, determining $\nu_{b}$ requires high resolution maps of the lobes at a variety of frequencies. This data has been obtained for only a limited number of bright sources. The largest sample of these detailed observations is from @liu92. The average rest frame break frequency from the sample of @liu92 is $\nu_{b}=8.9\pm7.0$ GHz. Secondly, in order to estimate the minimum synchrotron frequency we note that from @bra69 (even though the measurements are likely to be extremely inaccurate) it is clear that many FRII sources are very strong emitters down to frequencies at least as low as 12.6 MHz. Thus, we pick $\nu_{1}=10\,\mathrm{MHz}$ in the quasar rest frame. Finally, in order to approximate the total radio luminosity, $L\equiv\int
L(\nu)\,d\nu$ (including the significant contribution at frequencies above the spectral break), with a single spectral index, a value of $\nu_{2}=100\,\mathrm{GHz}$ is chosen. Inserting these “typical” frequency values into (3.7), one obtains a simple estimator of lobe power in the limit of particle dominance, and noting that at the spectral break frequency, $t_{syn}\approx
t_{sep}$ $$\begin{aligned}
&& Q\approx\frac{U_{e}}{t_{sep}}+L\approx
\frac{\left[y_{1}(n)\right]^{\frac{n-1}{2}}(15.1)^{\alpha}}{(n-2)a(n)}10^{42}(1+z)^{1+\alpha}Z^{2}F_{151}\,\mathrm{ergs/sec}+L\;,\end{aligned}$$ where the spectral index $\alpha=(n-1)/2$ has been introduced ($L(\nu)\sim\nu^{-\alpha}$). It should be noted that the estimates above are very conservative. The existence of a substantial proton component to the lobe gas or an extension of the low frequency portion of the electron spectrum would increase the energy flux estimates significantly. Observations suggest that $\alpha\approx
1$ is a good fiducial value for the expression (3.8)[@kel69], $$\begin{aligned}
&&Q_{par}\approx
5.7\times10^{44}(1+z)^{1+\alpha}Z^{2}F_{151}\,\mathrm{ergs/sec}\;,\quad\alpha\approx
1\;.\end{aligned}$$
Conclusion
==========
An independent formula for the jet kinetic luminosity estimator in (1.1) is derived in (3.9) that was motivated by different physical assumptions. The two estimates agree to within a factor of 2. This lends credence to the idea that (1.1) and (3.9) are robust estimators of jet kinetic luminosity when the optically thin extended emission is measured in a deep radio map. The main result of this paper is that (1.1), although very ambitious in its intent, is likely to be correct to within a factor of a few even if some of the assumptions in its derivation are inaccurate.
Alexander, P., Pooley, G. 1996 in *Cygnus A - Study of a Radio Galaxy*, p. 149 eds C.L. Carilli and D.E. Harris Antonucci, R.J. 1993, Annu. Rev. Astron. Astrophys. **31** 473 Blundell, K., Rawlings, S. 2000, AJ **119** 1111 Braude, S.Ya. et al 1969, MNRAS **143**, 295 Brunetti, G. et al 2002, Proceedings of the Symposium ’New Visions of the X-ray Universe in the XMM-Newton and Chandra Era’, ESTEC (astro-ph/0202373) Brunetti, G., Bondi, M., Comastri, A., Setti, G. 2002, Astron. and Astrophys. **381** 795 Ginzburg, V. 1979, *Theoretical Physics and Astrophysics* Translated by D. Ter Haar (pergamon, New York) Hardcastle, M., Worrall,D. 2000, MNRAS **319** 562 Isobe et al 2002, ApJL **580** 111 Kellerman, K.I., Pauliny-Toth, K.I.I., Williams, P.J.S. 1969, ApJ **157** 1 Leahy, J.P., Gizani, N. 2001, ApJ **555** 709 Lind, K., Blandford, R. 1985, ApJ **295** 358 Liu, R., Pooley, G., Riley, J. 1992 MNRAS **257** 545 Pen, U.-L. 1999, ApJS **120** 49 Punsly, B. 1995, AJ **109** 1555 Punsly, B. 2001, *Black Hole Gravitohydromagnetics* (Springer-Verlag, New York) Tucker, W. 1975, *Radiation Processes in Astrophysics* (MIT Press, Cambridge). Willott, C., Rawlings, S., Blundell, K., Lacy, M. 1999, MNRAS **309** 1017 Wilson, A., Young, A., Shopbell, P. 2001, Apj **547** 740
|
---
abstract: 'We analysed the optical and radio properties of lobe-dominated giant-sized ($>$ 0.72 Mpc) radio quasars and compared the results with those derived for a sample of smaller radio sources to determine whether the large size of some extragalactic radio sources is related to the properties of their nuclei. We compiled the largest (to date) sample of giant radio quasars, including 24 new and 21 previously-known objects, and calculated a number of important parameters of their nuclei such as the black hole mass and the accretion rate. We conclude that giant radio quasars have properties similar to those of smaller size and that giant quasars do not have more powerful central engines than other radio quasars. The results obtained are consistent with evolutionary models of extragalactic radio sources which predict that giant radio quasars could be more evolved (aged) sources compared to smaller radio quasars. In addition we found out that the environment may play only a minor role in formation of large-scale radio structures.'
author:
- |
A. Kuźmicz$^{1}$[^1] and M. Jamrozy$^{1}$[^2]\
$^{1}$Astronomical Observatory, Jagiellonian University, ul. Orla 171 , 30-244 Kraków, Poland
date: 'Accepted 2011 Month 00. Received 2011 Month 00; in original form 2011 Month 00'
title: 'Optical and radio properties of giant radio quasars: Central black hole characteristics.'
---
\[firstpage\]
galaxies: active - quasars: emission lines - radio continuum: galaxies
Introduction
============
Giant radio sources (GRSs) are defined as powerful extragalactic radio sources, hosted by galaxies or quasars, for which the projected linear size of their radio structure is larger than 0.72 Mpc[[^3]]{} (assuming $H_0=71$ km s$^{-1}$Mpc$^{-1}$, $\Omega_M=0.27$, $\Omega_{\lambda}=0.73$; @b59). Looking through the new, “all-sky” radio surveys such as the Westerbork Northern Sky Survey (@b51), the NRAO VLA Sky Survey (NVSS; @b13), the Faint Images of the Radio Sky at Twenty centimeters (FIRST; @b3), the Sydney University Molonglo Sky Survey (@b10) and the Seventh Cambridge Survey (@b41) a large number of new giant sources were identified. Almost all of these GRSs are included in the samples of giants presented by [@b14], [@b34], [@b36], [@b37], [@b54], [@b56], as well as in the list of giants known before 2000 published by [@b25]. GRSs are very useful in studying a number of astrophysical problems, for example the evolution of radio sources, the properties of the intergalactic medium (IGM) at different redshifts, and the nature of the central active galactic nuclei (AGN). It is still unclear why such a small fraction of radio sources reach such a large size – it may be due to special external conditions, such as lower IGM density, or due to the internal properties of the “central engine”. Our knowledge about the nature of GRSs has improved somewhat following studies conducted in the last decade. However, these were focused almost exclusively on: the role of the properties of the IGM (@b1111 [@b32]), the advanced age of the radio structure (e.g. @b1112 [@b37b]), and recurrent radio activity (e.g. @b1114 [@b1130]) as responsible for gigantic size.
To date, there are about 230 GRSs known, and just a small fraction of them ($\sim 8$ per cent) are actually related to quasars. The lobe-dominated radio quasars usually have a classical FRII (@b19) morphology and most of their radio emission originates in the extended regions with steep radio spectra. The ratio of the flux density of the core to that of the lobes at 5 GHz is usually less than 1 (@b24). Optically, the lobe-dominated radio quasars are similar to the core-dominated quasars (@b2) and their lobe dominance is just an orientation effect.
In general, it is believed that strong jet activity in an AGN is related to the parsec or sub-parsec scale condition of its host galaxy and specifically to the properties of its central black hole (BH; @b6 [@b8; @b7]). Furthermore, if assuming that the power of the “central engine” is responsible for the linear size evolution of an radio source, we should expect the largest objects to be related to radio quasars, which host the most energetic AGNs, as opposed to radio galaxies. It is because there is observational evidence for a correlation between jet power and the expansion speed of a radio source’s lobes (e.g. @b1119 [@b1120]). The jets of high-power sources carry a larger momentum flux, which in turn implies a greater flux of kinetic energy. Therefore, radio quasars, which are on average more luminous than radio galaxies, should have higher lobe expansion speeds than radio galaxies and, assuming similar mean lifetimes of both these types of AGNs, radio quasars have the potential to reach larger size. A typical lobe expansion speed could be of the order of a few hundredths (or more) of the speed of light (e.g. @b1121), and this, together with the typical lifetime of an evolved radio source of the order of a few times $10^7$ years, gives a size of the order of a Mpc.
The radio loudness of quasars still remains a debated issue. Radio observations of optically-selected samples of quasars showed that only 10-40 per cent of the objects are powerful radio sources (for reference see e.g. @b1122 [@b1123]). Recently, thanks to FIRST – the large-area radio survey – the number of quasars with faint radio fluxes has grown enormously. Therefore, it is now possible to investigate the optical and radio properties of quasars based on statistically large samples of objects (e.g. @b69 [@b1128; @b1124; @b1122; @b1117; @b1123]), and to try to understand the connection between the optical emission (luminosity, BH mass and spin, accretion rate) and the radio (jet) activity. Evidence that the spin of the BH plays a significant role in radio activity has recently been found (e.g. @b1115 [@b1116; @b1117]). The relation between BH mass and radio loudness has also been intensively studied, but so far the results are equivocal. Many authors (e.g. @b35 [@b42; @b17; @b39; @b44]) have found that, on average, radio-louder AGNs possess larger BH masses. However, there are also many reports arguing against any dependence between these quantities (e.g., @b48 [@b23; @b71; @b1122; @b58]). Furthermore, the importance of the mechanical energy of jets and lobes released by BHs and the feedback on the surroundings has recently been realized (@b1125 [@b1127; @b1126]). AGNs deposit large amounts of energy into their galactic environment which may, for example, be responsible for halting star formation. There is broad observational evidence that mechanical heating by jets plays an important role in balancing radiative losses from the intra-cluster medium. Radio jets and lobes of quasars can modify the structure of the environment not only on kpc scales but also, by the giant-size sources, on Mpc scales.
The aim of this study is to investigate the radio and optical properties for a sample of lobe-dominated giant radio quasars (GRQs). We would like to answer the question whether the size of GRQs is related to the internal properties of their hosts. To investigate the role and importance of the central engine in generation of Mpc-scale structures we have compiled the largest sample of GRQs to date.
The paper is organized as follows: in Sect. 2 we describe our source samples and in Sect. 3 the possible biases of the samples. In Sect. 4 we present definitions of the parameters used in the analysis; in Sect. 5 we investigate the relations between the optical and radio properties of our sources. Sect. 6 presents our conclusions.\
Throughout the paper we assume the standard cosmology, with parameters as provided at the beginning of this Section.
The sample
==========
In our analysis we use 45 GRQs of which 21 are taken from the existing literature (for details see Table 1). The remaining 24, which were not previously identified as GRQs, we selected from catalogues of radio quasars compiled by [@b20], [@b4], [@b69], and [@b65]. The presented sample of giant-sized radio quasars is the largest to date. It contains sources even at large redshifts ($z\sim 2$).
As a comparison sample, we selected 31 smaller lobe-dominated radio quasars from a list of radio sources given by [@b47]. In order to obtain a number of objects comparable to the GRQ sample, 18 quasars selected from the catalogues cited were added to the comparison sample. The linear sizes of these objects are close to the limiting size of 0.72 Mpc, as we wanted to have a smooth transition in linear size between the smaller radio quasars and the GRQs. The sources from the comparison sample of lobe-dominated radio quasar meet the following criteria:
1. Have optical spectra in the Sloan Digital Sky Survey (SDSS; @b1).
2. Possess the MgII(2798Å) broad emission line in their spectra (as most of our GRQ spectra contain the MgII(2798Å) line). This condition limits the range of the redshift to $0.4\lesssim z \lesssim 2$; it was adopted in order to have similar properties of the optical spectra for all quasars and hence allow homogeneous measurements using the same methods for both samples.
3. Have a projected angular size of the radio structure larger than 02, to properly separate the components (lobes and core) of the source in the FIRST maps (which have 5$\times5$ angular resolution).
All our quasars possess a classical FRII radio morphology. These objects lie almost in the plane of the sky and therefore the influence of relativistic beaming is small. It is easy to determine the physical size of such sources based on radio maps, even at high redshift.
The final samples contain 45 GRQs and 49 smaller radio quasars, whose basic parameters are provided in Tables 1 and 2 respectively. The new, previously unrecognised, GRQs are marked in bold type in Table 1. Optical spectra from the SDSS as well as radio maps from the NVSS and FIRST surveys are available for almost all of these objects. In addition, the spectra of nine quasars published by [@b69], [@b4], [@b65] and [@b21] were provided to us in electronic FITS format by R. White (these are marked by the letter W in Tables 1 and 2). The columns of Table 1 and 2 contain respectively: (1) - J2000.0 IAU name; (2) and (3) J2000.0 right ascension and declination of the central position of the optical quasar; (4) redshift of the host object; (5) angular size in arcmin; (6) projected linear size in Mpc; (7) availability of the spectrum from the SDSS survey (S), or provided by White (W); availability of radio maps from NVSS or FIRST (N or F, respectively); (8) references to the identified object. Unfortunately, for two GRQs, J0631$-$5405 and J0810$-$6800, we had neither spectral nor radio data at hand, therefore we excluded them from further analysis.
---------------------- ------------------- ------------------- ------- -------- ------- -------- --------
IAU $\alpha$(J2000.0) $\delta$(J2000.0) z d D Avail. Ref.
name (h m s) ($^{o}$ ) arcmin Mpc Data
(1) (2) (3) (4) (5) (6) (7) (8)
[**J0204$-$0944**]{} 02 04 48.29 $-$09 44 09.5 1.004 6.035 2.914 S,N,F 1
[**J0210$+$0118**]{} 02 10 08.26 $+$01 18 42.3 0.870 2.618 1.214 W,N,F 1
J0313$-$0631 03 13 32.88 $-$06 31 58.0 0.389 3.090 0.973 S,N 2
J0439$-$2422 04 39 09.20 $-$24 22 08.0 0.840 1.960 0.899 N 3
J0631$-$5405 06 32 01.00 $-$54 04 58.7 0.204 5.200 1.04 - 4
J0750$+$6541 07 50 34.43 $+$65 41 25.4 0.747 3.271 1.439 N 5
[**J0754$+$3033**]{} 07 54 48.86 $+$30 33 55.0 0.796 3.842 1.730 S,N,F 6
[**J0754$+$4316**]{} 07 54 07.96 $+$43 16 10.6 0.347 8.061 2.360 S,N,F 7
[**J0801$+$4736**]{} 08 01 31.97 $+$47 36 16.0 0.157 5.438 0.876 S,N,F 7
J0809$+$2912 08 09 06.22 $+$29 12 35.6 1.481 2.184 1.118 S,N,F 6, 8
J0810$-$6800 08 10 55.10 $-$68 00 07.7 0.231 6.500 1.42 - 9
J0812$+$3031 08 12 40.08 $+$30 31 09.4 1.312 2.427 1.203 S,N,F 8
J0819$+$0549 08 19 41.12 $+$05 49 42.7 1.701 1.923 0.987 S,N,F 8
J0842$+$2147 08 42 39.96 $+$21 47 10.4 1.182 2.314 1.156 S,N,F 8
J0902$+$5707 09 02 07.20 $+$57 07 37.9 1.595 1.678 0.862 S,N,F 9, 8
[**J0918$+$2325**]{} 09 18 58.15 $+$23 25 55.4 0.688 2.079 0.885 S,N,F 10
[**J0925$+$4004**]{} 09 25 54.72 $+$40 04 14.2 0.471 4.379 1.546 S,N,F 10
[**J0937$+$2937**]{} 09 37 04.04 $+$29 37 04.8 0.451 2.640 0.909 S,N,F 6, 10
[**J0944$+$2331**]{} 09 44 18.80 $+$23 31 18.5 0.987 1.870 0.899 S,N,F 10
[**J0959$+$1216**]{} 09 59 34.49 $+$12 16 31.6 1.089 1.964 0.966 S,N,F 11
[**J1012$+$4229**]{} 10 12 44.29 $+$42 29 57.0 0.364 3.088 0.933 S,N,F 9
[**J1020$+$0447**]{} 10 20 26.87 $+$04 47 52.0 1.131 1.478 0.733 S,N,F 11
[**J1020$+$3958**]{} 10 20 41.15 $+$39 58 11.2 0.830 2.663 1.217 W,N,F 9
J1027$-$2312 10 27 54.91 $-$23 12 02.0 0.309 2.860 0.774 N 3
J1030$+$5310 10 30 50.91 $+$53 10 28.6 1.197 1.698 0.749 S,N,F 8
[**J1054$+$4152**]{} 10 54 03.27 $+$41 52 57.6 1.090 4.702 2.314 S,N,F 10
[**J1056$+$4100**]{} 10 56 36.26 $+$41 00 41.3 1.785 1.543 0.791 S,N,F 11
J1130$-$1320 11 30 19.90 $-$13 20 50.0 0.634 4.812 1.977 N 12
[**J1145$-$0033**]{} 11 45 53.67 $-$00 33 04.6 2.054 2.642 1.340 S,N,F 13
J1148$-$0403 11 48 55.89 $-$04 04 09.6 0.341 3.265 0.945 N,F 14
[**J1151$+$3355**]{} 11 51 39.68 $+$33 55 41.8 0.851 2.083 0.959 S,N,F 10
[**J1229$+$3555**]{} 12 29 25.56 $+$35 55 32.5 0.828 1.672 0.761 S,N,F 15
[**J1304$+$2454**]{} 13 04 51.42 $+$24 54 45.9 0.605 2.431 0.977 W,N,F 10
[**J1321$+$3741**]{} 13 21 06.42 $+$37 41 54.0 1.135 1.531 0.759 S,N,F 10
[**J1340$+$4232**]{} 13 40 34.70 $+$42 32 32.2 1.343 2.309 1.173 S,N,F 10
J1353$+$2631 13 53 35.92 $+$26 31 47.5 0.310 2.803 0.761 W,N,F 10, 16
[**J1408$+$3054**]{} 14 08 06.21 $+$30 54 48.5 0.837 3.618 1.658 S,N,F 10
[**J1410$+$2955**]{} 14 10 36.80 $+$29 55 50.9 0.570 2.483 0.970 W,N,F 6
J1427$+$2632 14 27 35.61 $+$26 32 14.5 0.363 3.822 1.158 S,N,F 16
J1432$+$1548 14 32 15.54 $+$15 48 22.4 1.005 2.824 1.364 S,N,F 14
J1504$+$6856 15 04 12.77 $+$68 56 12.8 0.318 3.140 0.867 N 5
J1723$+$3417 17 23 20.80 $+$34 17 58.0 0.206 3.787 0.760 W,N,F 17
J2042$+$7508 20 42 37.30 $+$75 08 02.5 0.104 10.052 1.138 N 18
J2234$-$0224 22 34 58.76 $-$02 24 18.9 0.550 3.236 1.241 N,F 1
[**J2344$-$0032**]{} 23 44 40.04 $-$00 32 31.7 0.503 2.658 0.973 W,N,F 1
---------------------- ------------------- ------------------- ------- -------- ------- -------- --------
\
References:(1) [@b4]; (2) [@b38]; (3) [@b25]; (4) [@b54]; (5) [@b34]; (6) [@b20]; (7) [@b55]; (8) [@b32]; (9) [@b65]; (10) [@b69]; (11) [@b31]; (12) [@b5]; (13) [@b33]; (14) [@b22]; (15) [@b57]; (16) [@b47]; (17) [@b26]; (18) [@b53].
-------------- ------------------- ------------------- ------- -------- ------- -------- ------
IAU $\alpha$(J2000.0) $\delta$(J2000.0) z d D Avail. Ref.
name (h m s) ($^{o}$ ) arcmin Mpc Data
(1) (2) (3) (4) (5) (6) (7) (8)
J0022$-$0145 00 22 44.29 $-$01 45 51.1 0.691 1.432 0.610 N,F 1
J0034$+$0118 00 34 19.18 $+$01 18 35.8 0.841 1.364 0.664 W,N,F 1
J0051$-$0902 00 51 15.12 $-$09 02 08.5 1.265 1.379 0.696 S,N,F 1
J0130$-$0135 01 30 43.00 $-$01 35 08.2 1.160 1.306 0.650 W,N,F 1
J0245$+$0108 02 45 34.07 $+$01 08 14.2 1.537 0.883 0.453 S,N,F 3
J0745$+$3142 07 45 41.66 $+$31 42 56.5 0.461 1.795 0.626 S,N,F 3
J0811$+$2845 08 11 36.90 $+$28 45 03.6 1.890 0.507 0.259 S,N,F 3
J0814$+$3237 08 14 09.23 $+$32 37 31.7 0.844 0.239 0.187 S,N,F 3
J0817$+$2237 08 17 35.07 $+$22 37 18.0 0.982 0.395 0.190 S,N,F 3
J0828$+$3935 08 28 06.85 $+$39 35 40.3 0.761 1.077 0.477 S,N,F 3
J0839$+$1921 08 39 06.95 $+$19 21 48.9 1.691 0.523 0.269 S,N,F 3
J0904$+$2819 09 04 29.63 $+$28 19 32.8 1.121 0.379 0.188 S,N,F 3
J0906$+$0832 09 06 49.81 $+$08 32 58.8 1.617 1.307 0.671 S,N,F 4
J0924$+$3547 09 24 25.03 $+$35 47 12.8 1.342 1.345 0.683 S,N,F 5
J0925$+$1444 09 25 07.26 $+$14 44 25.9 0.896 0.665 0.311 S,N,F 3
J0935$+$0204 09 35 18.51 $+$02 04 19.0 0.649 1.200 0.498 S,N,F 3
J0941$+$3853 09 41 04.17 $+$38 53 49.1 0.616 0.853 0.346 S,N,F 3
J0952$+$2352 09 52 06.36 $+$23 52 43.2 0.970 1.466 0.702 S,N,F 2
J1000$+$0005 10 00 17.65 $+$00 05 23.9 0.905 0.521 0.245 S,N,F 3
J1004$+$2225 10 04 45.75 $+$22 25 19.4 0.982 1.097 0.526 S,N,F 3
J1005$+$5019 10 05 07.10 $+$50 19 31.5 2.023 1.300 0.660 S,N,F 2
J1006$+$3236 10 06 07.58 $+$32 36 27.9 1.026 0.246 0.119 S,N,F 3
J1009$+$0529 10 09 43.56 $+$05 29 53.9 0.942 1.377 0.654 S,N,F 2
J1010$+$4132 10 10 27.50 $+$41 32 39.0 0.612 0.525 0.212 S,N,F 3
J1023$+$6357 10 23 14.61 $+$63 57 09.3 1.194 1.294 0.648 S,N,F 6
J1100$+$1046 11 00 47.81 $+$10 46 13.6 0.422 0.549 0.182 S,N,F 3
J1100$+$2314 11 00 01.14 $+$23 14 13.1 0.559 1.577 0.610 S,N,F 5
J1107$+$0547 11 07 09.51 $+$05 47 44.7 1.799 1.324 0.678 S,N,F 2
J1107$+$1628 11 07 15.04 $+$16 28 02.2 0.632 0.652 0.267 S,N,F 3
J1110$+$0321 11 10 23.84 $+$03 21 36.4 0.966 1.055 0.504 S,N,F 3
J1118$+$3828 11 18 58.53 $+$38 28 53.5 0.747 1.407 0.619 S,N,F 5
J1119$+$3858 11 19 03.20 $+$38 58 53.6 0.734 1.419 0.620 S,N,F 5
J1158$+$6254 11 58 39.76 $+$62 54 27.1 0.592 0.968 0.385 S,N,F 3
J1217$+$1019 12 17 01.28 $+$10 19 52.0 1.883 0.466 0.238 S,N,F 3
J1223$+$3707 12 23 11.23 $+$37 07 01.8 0.491 0.597 0.216 S,N,F 3
J1236$+$1034 12 36 04.52 $+$10 34 49.2 0.667 1.694 0.711 S,N,F 3
J1256$+$1008 12 56 07.66 $+$10 08 53.5 0.824 0.382 0.174 S,N,F 3
J1319$+$5148 13 19 46.25 $+$51 48 05.5 1.061 0.466 0.228 S,N,F 3
J1334$+$5501 13 34 11.71 $+$55 01 24.8 1.245 1.274 0.641 S,N,F 3
J1358$+$5752 13 58 17.60 $+$57 52 04.5 1.373 0.733 0.373 S,N,F 3
J1425$+$2404 14 25 50.65 $+$24 04 02.8 0.653 0.339 0.141 S,N,F 3
J1433$+$3209 14 33 34.26 $+$32 09 09.5 0.935 0.630 0.299 S,N,F 3
J1513$+$1011 15 13 29.30 $+$10 11 05.4 1.546 0.586 0.301 S,N,F 3
J1550$+$3652 15 50 02.01 $+$36 52 16.8 2.061 1.334 0.676 S,N,F 4
J1557$+$0253 15 57 52.83 $+$02 53 28.9 1.988 1.121 0.571 S,N,F 2
J1557$+$3304 15 57 29.94 $+$33 04 47.0 0.953 0.562 0.268 S,N,F 3
J1622$+$3531 16 22 29.90 $+$35 31 25.1 1.475 0.365 0.187 S,N,F 3
J1623$+$3419 16 23 36.45 $+$34 19 46.3 1.981 0.984 0.501 S,N,F 2
J2335$-$0927 23 35 34.68 $-$09 27 39.2 1.814 1.305 0.668 S,N,F 1
-------------- ------------------- ------------------- ------- -------- ------- -------- ------
References: (1) [@b4]; (2) [@b65]; (3) [@b47]; (4) [@b32]; (5) [@b69]; (6) [@b31].
Sample Biases
=============
Due to the method used to complete our sample, the results may in some cases be influenced by selection effects, for example related to the sensitivity of the radio surveys used for selecting extended sources. The sample of giant radio quasars was compiled in three stages and each of them may be affected by bias. First, compact radio objects were selected, then the optical counterparts were checked for spectra typical of quasars. The selection criteria for these steps were described in detail in the papers referenced in Sect. 2 and we will not focus on them here. In the third stage of selection, we inspected the radio maps of several hundred candidates, looking for targets which have extended radio lobes in addition to radio cores. The NVSS and FIRST surveys have a completeness of 96 and 89 per cent and a reliability of 99 and 94 per cent to the $5\sigma$ limits of 2.3 and 1.0 mJy respectively (@b1131). Since the resolution effect causes FIRST to become more incomplete for extended objects, we supplemented our search with the NVSS maps which have larger restoring beam size and hence larger surface brightness sensitivity. However, because of the limited baselines NVSS is insensitive to very extended coherent structures (larger than 15). Fortunately, we do not expect the existence of objects with such large angular size, at least at high redshifts. In addition, extended and aged radio sources could have weak double lobes not connected with a visible bridge of high frequency radio emission. Therefore, it may be hard to recognise such a source as just one homogeneous object, especially at high redshift where the inverse Compton losses against the cosmic microwave background are large. Detecting a steep-spectrum and low surface-brightness radio bridge connecting the radio core with hot spots for distant objects is therefore challenging and this may have caused us to overlook some objects.
It is worth noting that most of the recent works on quasars based on optical and radio data first select the candidates from optical catalogues of quasars and then correlate their coordinates with catalogues of radio sources. The authors usually concentrate on point-like radio sources, not extended objects (there are some exceptions, however, for example @b65). [@b1123] considered extended radio structures, but analysed only those objects whose lobe separation was smaller than 1. The authors stressed that the extended radio quasars represented a very small fraction of the SDSS quasars. They also wrote that quasars for which the radio structure diameter is greater then 1are even rarer. Thus one has to realize that objects of the class studied here are extremely rare.
As we pointed out in Sect. 2, the lobe-dominated radio quasars lie almost in the plane of the sky. Therefore, their measured radio luminosity is weakly influenced by relativistic beaming. In addition, it is easy to determine the proper physical size and volume occupied by the radio plasma for sources oriented in this manner. On the other hand, one should keep in mind that, besides lobe-dominated giant radio quasars as focused on here, there exist giant radio quasars located at a small angle to the line of sight which we have completely ignored because of the inability to determine their physical size.
Given all the drawbacks described above, we have nonetheless shown that giant radio quasars do not comprise just a few objects as previously thought, but constitute a larger group. In addition to the sample of newly identified giants, we also added the set of previously known giant quasars to increase the number of objects tested. Summing up, the sample we presented here is limited by the described selection criteria and is not fully homogeneous. Therefore, applying the conclusions obtained here to the whole population of radio-loud quasars should be done with caution.
Data analysis
=============
Radio data
----------
Using the Astronomical Image Processing System[^4] package for radio data reduction and analysis and maps from the NVSS and FIRST surveys, we measured the basic parameters of the selected radio quasars, which were further used to calculate their characteristics – defined in the following way:
1. The arm-length-ratio, $Q$, which is the ratio of distances ($d_1$ and $d_2$) between the core and the hot spots (peaks of radio emission), normalized in such a way that always $Q>1$ (for details see Fig. 1).
2. The bending angle, $B$, which is the complement of the angle between the lines connecting the lobes with the core.
3. The lobes’ flux-density ratio, $F=S_1/S_2$, where $S_1$ is the flux density of the lobe further from the core and $S_2$ is the flux density of the lobe closer to the core.
4. The source total luminosity at 1.4 GHz, $P\rm_{tot}$, which is calculated following the formula given by [@b12]:
$$\begin{aligned}
\lefteqn{log P_{tot}(WHz^{-1})=log S_{tot}(mJy)-(1+\alpha) \cdot log(1+z)}
\nonumber\\
& & {}+2log(D_L(Mpc))+17.08
\label{eq1}
\end{aligned}$$
where $\alpha$ is the spectral index (the convention we use here is $S_{\nu}\sim\nu^{\alpha}$) and $D_L$ is a luminosity distance. The total flux density, $S_{\rm tot}$, of individual sources is measured from NVSS maps and the average spectral index, in accordance with [@b68], is taken for all sources as $\alpha=-0.6$. The core luminosity at 1.4 GHz, $P\rm_{core}$, is calculated in a similar manner, but instead of $S_{\rm tot}$ in equation (\[eq1\]) we substitute the core flux density, $S_{\rm core}$, which is measured from FIRST maps and the average spectral index value, according to [@b72], is adopted as $\alpha = -0.3$.
5. The inclination angle, $i$, which is the angle between the jet axis and the line of sight (i.e. $i = 90$means that the object lies in the sky plane). The inclination angle was calculated, assuming that the Doppler boosting is the main factor underlying the asymmetries of a source, in the following way:
$$i=[acos(\frac{1}{\beta_j} \cdot \frac{(s-1)}{(s+1)})]
\label{eq2}$$
where $s=(S_j/S_{cj})^{1/2-\alpha}$, $S_j$ is the peak flux-density of the lobe closer to the core. $S_{cj}$ is the peak flux-density of the lobe further from the core and $\beta_j$ is the jet velocity. For all our objects, according to [@b68] and [@b1129], we assume $\beta_j=0.6$c.
The resulting values of the above parameters for our sources are listed in Table 3. For two objects, i.e. J0439$-$2422 and J1100$+$2314, we were not able to measure all the parameters, as for the source J0439$-$2422 the FIRST map was not available, and J1100$+$2314 has a too asymmetric radio structure.
{width="0.75\linewidth"}
Optical data
------------
### Spectra reduction
The quasar spectra were reduced through the standard procedures of the Image Reduction and Analysis Facility [^5] package including galactic extinction and redshift correction. Each spectrum was corrected for galactic extinction taking into account values of the colour excess $E(B-V)$ and the $B$-band extinction, $A_B$, taken from the NASA/IPAC Extragalactic Database. We calculated the extinction parameter $R=E(B-V)/A_B$ for each quasar in our samples. The extinction-corrected spectrum was then transformed to its rest frame using the redshift value given in the SDSS (or if the SDSS spectrum was unavailable, from other publications).
### Continuum subtraction and line parameters measurement
In order to obtain reliable measurements of emission lines, we need to subtract continuum emission, as optical and UV spectra of quasars are dominated by the power-law and Balmer continuum. Using the Image Reduction and Analysis Facility package, we subtracted the power-law continuum from our spectra. The continuum was fitted in several windows where we had not observed any emission lines (i.e. 1320–1350Å, 1430–1460Å, 1790–1830Å, 3030–3090Å, 3540–3600Å and 5600–5800Å). Particularly in the UV band, we also observe significant iron emission, which is often blended with the MgII(2798Å) line. The procedure of subtracting the iron emission was similar to that described by [@b11]. We used an Fe template in the UV band (1250–3090Å) as developed by [@b64], and in the optical band (3535–7530Å) given by [@b60]. First, we broadened the iron template by convolving it with Gaussian functions of various widths and multiplying by a scalar factor. Next, we chose the best fit of this modified template to each particular spectrum, and then subtracted it. After the subtraction of Fe line emission, we added the previously determined power-law continuum fit and refitted it once again (in a similar manner as suggested by @b64). An example of a “cleaned-up” spectrum is presented in Fig. 2.\
For the purpose of our analysis we needed to measure the parameters of broad emission lines like CIV(1549Å), MgII(2798Å) and H$_\beta$(4861Å). In some cases, performing this measurement was difficult due to asymmetries in the line profiles (particularly of highly ionized lines such as CIV), where it was hard to fit a Gaussian function. In order to overcome the problem, we used the method described in [@b50]. In Tables 4 and 5 (cols. 2–4) we provide the respective widths of broad emission lines for GRQs and smaller quasars, respectively. We were unable to measure the MgII emission line parameters in the spectrum of the GRQ J1408+3054, as it showed strong broad-absorption features which considerably affected the emission line profile.
### Black hole mass determination
The issue of determination of BH mass in AGNs has recently been often studied. The knowledge of the BH mass is of great importance in determining a number of physical parameters of AGNs and their evolution. In the first place, all of the commonly known techniques based on kinematic or dynamical studies (e.g. @b52) are only useful for inactive galaxies. Therefore, they cannot be applied directly for AGNs, which are very luminous and distant. The most promising method for AGNs is reverberation mapping of the broad emission lines from the broad-line region (@b49). This method works particularly well for type I AGNs (e.g. @b66), where the broad line region is not obscured by a dusty and gaseous torus. Assuming that the gas in the broad-line region is virialized in the gravitational field of a BH, we can calculate its mass as:
$$M_{\rm BH}=\frac{R_{\rm BLR} V^2_{\rm BLR}}{\rm G}
\label{eq3}$$
where G is the gravitational constant, $R_{\rm BLR}$ is the distance from broad-line region clouds to the central BH, $V_{\rm BLR}$ is the broad-line region virial velocity, which can be estimated from the FWHM (Full Width at Half Maximum) of a respective emission line as:
$$V_{\rm BLR}=f \cdot FWHM
\label{eq4}$$
where $f$ is the scaling factor, which depends on structure, kinematics, and orientation of the broad line region (for randomly distributed broad line region clouds $f=\sqrt{3}/2$). Basing on this method, [@b28; @b29] obtained an empirical relation between the broad line region size of an AGN and its optical continuum luminosity ($\lambda L_{\lambda}$) at 5100$\rm\AA$ (and later also at 1450$\rm\AA$, 1350$\rm\AA$ and in the 2–10 keV range):
$$R_{\rm BLR} \sim \lambda L_{\lambda}(5100\rm\AA)^{0.70 \pm 0.03}
\label{aga5}$$
This relation makes it possible to use an approximation to the reverberation mapping method, called the mass-scaling relation, which allows to determine BH mass using measurements of the FWHM of broad emission lines (e.g. CIV, MgII, H$\beta$) and the monochromatic continuum luminosity ($\lambda L_{\lambda}$) of a single-epoch spectrum only. In order to determine BH mass of our objects basing on the FWHM measurements of different emission lines, we applied the following equations:
$$\begin{aligned}
\lefteqn{M_{\rm BH}(CIV1549\rm\AA)= 4.57 \cdot 10^6 (\frac{\lambda
L_{\lambda}(1350\rm\AA)}{10^{44}erg s^{-1}})^{0.53\pm0.06} \cdot}
\nonumber\\
& & {}(\frac{FWHM(CIV1549\rm\AA)}{1000 km s^{-1}})^2 M_{\odot}
\label{eq6}\end{aligned}$$
$$\begin{aligned}
\lefteqn{M_{\rm BH}(MgII2798\rm\AA)=7.24 \cdot 10^6(\frac{\lambda
L_{\lambda}(3000\rm\AA)}{10^{44}erg s^{-1}})^{0.5} \cdot}
\nonumber\\
& & {}(\frac{FWHM(MgII2798\rm\AA)}{1000 km s^{-1}})^2 M_{\odot}
\label{eq7}\end{aligned}$$
$$\begin{aligned}
\lefteqn{M_{\rm BH}(H\beta4861\rm\AA)= 8.13 \cdot 10^6 (\frac{\lambda
L_{\lambda}(5100\rm\AA)}{10^{44}erg s^{-1}})^{0.50\pm0.06} \cdot}
\nonumber\\
& & {}(\frac{FWHM(H\beta4861\rm\AA)}{1000 km s^{-1}})^2 M_{\odot}
\label{eq8}\end{aligned}$$
Equations (\[eq6\]) and (\[eq8\]) were taken from [@b63], while equation (\[eq7\]) from [@b62]. The monochromatic continuum luminosities $\lambda L_{\lambda}$ can be computed as follows:
$$\lambda L_{\lambda} = 4 \pi D_{\rm Hubble}^2 \lambda f_{\lambda}
\label{eq9}$$
where $D_{\rm Hubble}$ is the comoving radial distance and $f_{\lambda}$ is the flux in the rest frame at wavelength $\lambda$ equal to 3000Å, 5100Å, or 1350Å. The resulting rest frame fluxes, monochromatic continuum luminosities and BH masses for GRQs and smaller quasars are given in Table 4 and 5 (cols. 5–7, 8–10 and 11–13) respectively.\
{width="0.99\linewidth"}
Results
=======
Radio properties
----------------
In our analysis we checked some general relations between radio parameters for our sample sources, similar to those shown for the sample of GRSs (mostly galaxies) described by [@b25]. On the optical- versus radio-luminosity plane our objects trace the regime of radio loudness (ratio of radio-to-optical luminosity) between 50 and 1000 and overlap with the FIRST-2dF sample of quasars of [@b1122].
In Fig. 3 we present the dependence between 1.4 GHz total luminosity and redshift for our quasars. It is important to note that our comparison sample of smaller radio quasars (sources marked as open circles in Fig. 3 and subsequent figures) contains only objects in the redshift range of 0.4$\lesssim$z$\lesssim$2 due to our selection criteria, i.e. the presence of the MgII(2798Å) emission line in the spectra (for details see Sect. 2). Such a cut-off in the redshift range of the quasars from the comparison sample should not, however, affect our main results, since the majority of GRQs have redshifts in a similar range. Therefore, the non-existence of smaller radio quasars in the upper-left part of Fig. 3 is artificial, whereas the absence of GRQs in the lower-right corner of this figure is the result of sensitivity limit of the radio surveys which we used for source recognition and measurements of source’s radio properties. It is known that in flux-limited samples we should expect a correlation between radio luminosity and redshift, since for larger distances we are able to detect only those sources which are luminous enough, and faint sources at higher redshifts are beyond the detection limit. For our quasar sample a dependence between redshift and total radio luminosity can be seen, but the correlation is not as strong as for the sample of GRSs from [@b25]. The Spearman rank correlation coefficient for the GRQs is 0.49, whereas for the GRSs from the paper cited above it is 0.90. This shows that the selection effects for our quasar sample are not as strong as for other radio galaxies and GRS samples of [@b25], though they may still have affected some of our results.
{width="0.99\linewidth"}
In Fig. 4 we present the luminosity, $P$, versus linear size, $D$, relation. The $P$–$D$ diagram is a helpful tool in investigating the evolution of radio sources and was frequently used to test evolutionary models (e.g. @b27 [@b9]). In order to draw this diagram we used the unprojected linear size of the sources, which was derived by taking into account the inclination angle, $i$, as $D^* = D/sin(i)$, where $D$ is the projected linear size (given in Tables 1 and 2 derived as the sum of $d_1$ and $d_2$ - for details see Fig. 1). The diagrams show that GRQs have, on average, lower core and total radio luminosities. The trend which we observe in our $P$–$D$ diagrams is consistent with the predictions of evolutionary models and can suggest that, under favourable conditions, the luminous smaller, and probably younger, radio quasars may evolve in time into the lower-luminosity aged GRQs. The non-existence of objects in the bottom-left part of Fig. 4 may be due to selection effects. Because of the surface-brightness limit we may overlook some extended objects with low total radio luminosities.
{width="0.99\linewidth"}\
{width="0.99\linewidth"}
In Fig. 5 we present the relation between the total and core radio luminosity. There is a strong correlation between those two quantities for radio quasars. We obtained a correlation coefficient of 0.76 and the slope of the linear fit equal to $0.82 \pm 0.08$, steeper than the slope of $0.59 \pm 0.05$ obtained by [@b25] for GRSs. The strong correlation between the core luminosity and the total luminosity in the population of giant-size radio galaxies was also mentioned by [@b37b]. On the one hand, this correlation can be attributed to the Doppler beaming of a parsec-scale jet and can reflect the different inclination angle of the nuclear jets, and thus inclination of the entire radio source’s axis to the observer’s line of sight. Relatively more luminous cores (in comparison to the total luminosity) should be observed in more strongly projected sources (i.e. quasars). Therefore, in GRQs one could expect to observe relatively stronger cores than in giant-sized radio galaxies. On the other hand, evolutionary effects (well visible in Fig. 4) can explain the clear difference in radio luminosity between GRQs and smaller quasars. Some authors (e.g. @b29) have suggested that giants should have more prominent cores, as stronger nuclear activity is necessary to produce the larger linear sizes of their radio structure. [@b25] attempted to verify this hypothesis for giant-sized radio galaxies by plotting a diagram of the core prominence, $f_c$, which is the ratio of core luminosity to the total luminosity of the radio source, but found no trend of this kind. For GRQs investigated in this paper we also plotted such a diagram (see Fig. 6) and came to a similar conclusion. We can reconcile this with the existence of the core luminosity – total luminosity correlation visible in Fig. 5 as a result of smaller quasars having more luminous cores but also larger total luminosities than GRQs. The resulting mean values of $f_c$ are 0.20 and 0.18 for GRQs and smaller quasars respectively. In Fig. 7 we plot the core prominence against linear size of the extended radio structure. The distribution of the core prominence is similar for GRQs and smaller radio quasars, which allows the claim that the strength of the central engine of GRQs is similar to that of smaller radio quasars.
{width="0.99\linewidth"}
{width="0.99\linewidth"}
{width="0.99\linewidth"}
We also investigated the asymmetries of radio structures in both our lobe-dominated radio quasar samples. It is well known that non-uniform environment (i.e. non-uniform density on both sides of the core) is one of the factors underlying radio structure asymmetries, which can be described by the arm-length ratio $Q$ (e.g. @bsch). The distribution of this parameter for the GRQs and the quasars from the comparison sample is presented in Fig. 8. We found that the GRQs seem to be more symmetric than the smaller radio quasars (there were no GRQs with $Q > 2.4$ in our sample). However, the obtained mean values of the $Q$ parameter for GRQs and for the comparison sample are $1.41 \pm 0.33$ and $1.65 \pm 0.61$, respectively, and therefore indistinguishable within the error limits. This suggests that the IGM in which the giants evolve is not more symmetrical than that around the smaller sources. Our results for the large radio sources are comparable to those by [@b25], which found that the mean value of the $Q$ parameter for the GRSs is 1.39, but for a comparison sample based on smaller 3CR sources they obtained a smaller $Q$ value equal to 1.19. We also compared the arm-length ratio value of GRQs and GRSs (see the bottom panel in Fig. 8). It can be seen clearly that the distributions for giant quasars and galaxies are similar.
The values of the bending angle $B$ and lobe flux-density ratio $F$ give similar result for both samples of quasars with mean values of $B=7.40 \pm 5.89$, $F=1.45 \pm 1.16$ and $B=8.50 \pm 7.31$, $F=2.28 \pm 5.53$ for GRQs and comparison sample, respectively.
In summary, there is no significant difference in the environmental properties of the IGM within which giant- and smaller-sized radio quasars evolve.
Furthermore, we checked distribution of the inclination angle, $i$ (see Fig. 9). For our sample of radio quasars, we obtained that most objects have inclinations between 60$^{ \rm o}$ and 90$^{ \rm o}$. This result is inconsistent with the models of AGN unification scheme, where – following [@b67] – the inclination angle for quasars has a value between 0$^{ \rm o}$–45$^{ \rm o}$. In the objects with the angle larger than 45$^{ \rm o}$, the broad-line region should be partially or totally obscured by a dusty torus and the broad emission lines should not be as prominent as we observe in the spectra from our quasar sample. A plausible explanation of the observed distribution of inclinations is that there is no dusty torus in some AGNs (@b18) or we are dealing with a clumpy, or receding torus (i.e. @b2222), thus broad emission lines could have been observed even in quasars with large inclinations. The quasar with the largest asymmetry of its radio structure is J1623+3419 with $i=13^{ \rm o}$. Such a small value of the inclination angle can suggest that it should rather be classified as a BL Lac object. Further observations are needed to confirm if its observed radio structure is actually related to a unique radio source.
{width="0.99\linewidth"} {width="0.99\linewidth"}
{width="0.99\linewidth"}
Black hole mass estimations
---------------------------
In order to obtain the central BH mass of quasars from our samples we used measurements of the CIV, MgII and H$\beta$ emission lines and the mass-scaling relations (equations (6), (7) and (8)). The mass values obtained are in the range of $1.6 \cdot 10^8 M_{\odot} < M_{\rm BH} < 12.3 \cdot 10^8 M_{\odot}$ when using the MgII emission line, and $1.5 \cdot 10^8 M_{\odot} < M_{\rm BH} < 29.2 \cdot 10^8 M_{\odot}$ when using the $H_{\beta}$ emission line. For some GRQs and quasars from the comparison sample it was possible to compare the results obtained on the basis of different emission-line measurements. In Fig. 10 we present the relation between the mass values calculated from MgII vs $H_{\beta}$ lines and those from CIV vs MgII lines, respectively. We found that the mass estimations based on the MgII line on average tend to be smaller than those obtained using the $H_{\beta}$ emission line (the linear fit to the data points is given by the relation: $M_{BH}H_{\beta}=2.87(\pm 0.98)\cdot M_{BH}MgII+5.00(\pm 9.25)$), and the mass estimations based on CIV line are larger than those obtained from the MgII line ($M_{BH}CIV=0.68(\pm 0.14) \cdot M_{BH}MgII+1.08(\pm 0.61)$). The above results are consistent with the earlier comparisons of BH masses estimated by other authors (e.g. @b63 [@b16; @b62]).
![Comparison of BH mass values estimated using measurement of different emission lines. [**top diagram:**]{} MgII versus $H_{\beta}$ BH masses; [**bottom diagram:**]{} MgII versus CIV BH mass. The linear fits to data points are described in the text.](Mbhmghb_1.eps "fig:"){width="0.99\linewidth"}\
![Comparison of BH mass values estimated using measurement of different emission lines. [**top diagram:**]{} MgII versus $H_{\beta}$ BH masses; [**bottom diagram:**]{} MgII versus CIV BH mass. The linear fits to data points are described in the text.](Mbhmgc4_1.eps "fig:"){width="0.99\linewidth"}
Black hole mass vs radio properties
-----------------------------------
In the paper by [@bkp], it is claimed that in the jet-formation models some dependence of jet power on BH mass should be expected. The assumption that the giants are formed due to a longer activity phase of the central AGN and/or more frequent duty cycles can imply that their BH masses should be larger because of longer accretion episodes. In Fig. 11 we present the relations between the total and core radio luminosity and the BH masse. It can be distinctly seen, however, that there is no correlation between the BH mass and either the core luminosity or the total luminosity for GRQs as well as smaller radio quasars.
![Relations between BH mass and radio luminosity at 1.4 GHz. [**top diagram:**]{} BH mass vs core luminosity; [**bottom diagram:**]{} BH mass vs total luminosity.](MbhPcore_1.eps "fig:"){width="0.99\linewidth"}\
![Relations between BH mass and radio luminosity at 1.4 GHz. [**top diagram:**]{} BH mass vs core luminosity; [**bottom diagram:**]{} BH mass vs total luminosity.](MbhPtot_1.eps "fig:"){width="0.99\linewidth"}
We also looked for a relation between BH masse and the unprojected radio linear size of a radio quasars. For the MgII BH mass estimations (Fig. 12), no obvious dependence has been found. Some interesting results can, however, be seen in Fig. 13. For the H$\beta$ and CIV BH mass estimations it can be clearly observed that the dependence between linear size of radio structures and their BH mass is quite significant. Surprisingly enough, the relation based on the H$\beta$ mass estimations for GRQs does not at all resemble that for quasars from the comparison sample. The slope of the linear fit for the sample of smaller quasars is steeper than that for the GRQs sample. This result suggests that the GRQs can be considered to represent another group of objects which differ physically from smaller quasars. We fitted linear functions independently to the data of GRQs and to the comparison sample (left panel of Fig. 13). The best fits obtained are as follows: $M_{\rm BH}H\beta = 10.995(\pm 7.023) \cdot D^*+1.629(\pm 10.268)$ and $M_{\rm BH}H\beta = 95.830(\pm 25.392) \cdot D^*-5.996(\pm 12.011)$ for the GRQs and for the comparison sample, with correlation coefficients of 0.48 and 0.74, respectively. We also plotted these lines on Fig. 12, taking into account the scaling factor between H$\beta$ and MgII BH mass estimations (equal to 2.87). It is obvious that the giants and the smaller radio quasars fulfil these relations quite well. Moreover, for the CIV mass estimation a weak correlation is also observed. The best fit is represented by a line $M_{\rm BH}CIV = 9.720(\pm 4.589) \cdot D^*+1.620(\pm 3.276)$ with a correlation coefficient of 0.51. The result obtained (particularly for the $H\beta$ mass estimations) can indicate that there may be some difference between GRQs and smaller radio quasars. It is hard to find a physical process to account for such a behaviour, especially as it is not found in the diagram for CIV BH masse. Some authors (e.g. @b30 [@b15]) suggested that the formation of different emission lines occurred in different regions of the broad line region, in the sense that the CIV emission should originate below the $H\beta$ emission. Therefore, GRQs and smaller radio quasars may differ with respect to the external structures of the broad line region, while their central parts would be similar. The question now is how to reconcile the fact that, according to the previously analysed relations for GRQs and smaller quasars, we did not see any clear distinction between these two types and here there is a clear difference. The possibility which comes to mind is that there is a difference in age between GRQs and smaller quasars and the composition of the broad-line region could be different for young and old quasars. However, the number of sources analysed using the CIV mass estimations is too small to allow for any definite conclusions, particularly relating to the smaller radio quasars. For example, this correlation deteriorates if we artificially shift the defining minimum GRQ size from 0.72 Mpc to a smaller value. Generally, apart from the above speculations on the composition of the broad line region, we can conclude that the apparent relationship between the linear size of the radio structure and the BH mass supports the evolutionary origin of GRQs: as time increases, the BH mass becomes larger and the size of radio structure grows.
{width="0.99\linewidth"}
![Dependence between the BH mass derived from the H$\beta$ - [**left panel**]{}, and the CIV - [**right panel**]{} - emission line, and the unprojected linear size of the radio structure.](panel_1a.eps){width="0.99\linewidth"}
Accretion rate
--------------
Using the obtained BH mass and the optical monochromatic continuum luminosity ($\lambda L_{\lambda}$) we calculated the accretion rate for our sample of quasars. The accretion rate is computed as m$(\lambda)$ = $L_{\rm bol}/L_{\rm Edd}$, where $L_{\rm bol}$ is the bolometric luminosity, assumed as:
$$L_{bol}=C_{\lambda} \lambda L_{\lambda}\label{eq10}$$
where $C_{\lambda}$ is equal: 9.0 for ${\lambda}=5100\rm\AA$ (according to @b28), 5.9 for ${\lambda}=3000\rm\AA$ (according to @b45) and 4.6 for ${\lambda}=1350\rm\AA$ (according to @b61). Following [@b16] the Eddington luminosity $L_{\rm Edd}$ is given by:
$$L_{\rm Edd}=1.45\cdot10^{38} M_{\rm BH}/M_{\odot} erg s^{-1}
\label{eq11}$$
The resulting values of $L_{\rm bol}$, $L_{\rm Edd}$ and m$(\lambda)$ for GRQs and smaller quasars are listed in Table 6 and 7, respectively. In Fig. 14 we present the BH mass as a function of accretion-rate values, which are calculated basing on the CIV, MgII and H$\beta$ emission lines as well as on the respective continuum luminosities, taking into account the scaling factor between H$\beta$, CIV and MgII mass estimations. As can be seen, the accretion rate is apparently higher for less massive BHs. A similar result was obtained by [@b16] for a sample of quasars and by [@b40] for narrow-line Seyfert galaxies. The result is consistent with the scenario of quasars increasing their BH mass during the accretion process solely. When there is no matter left, the accretion rate decreases, while a large amount of mass could have been accumulated in the central BH during the previous accretion episodes. In the scenario described by [@b40], the accretion rate is high in the early stages of AGN evolution and drops later on, so we could expect that at higher redshifts we should observe objects with larger accretion rates. However, Fig. 15 shows that, for our samples of quasars, no dependence between accretion rate and redshift is seen.\
The accretion rates for GRQs and for the comparison sample are consistent with typical values (0.01 $\div$ 1) for AGNs. Given the observed accretion rate we can constrain the lifetimes of the BHs in our samples. The obtained lower value for GRQs imply, that these sources are more evolved systems, for which the e-folding time to increase their BH mass (for a definition see e.g. @b2223) is longer than in the case of smaller-size quasars. The obtained mean values of accretion rate (m(3000Å)) are $0.07 \pm 0.03$ and $0.09 \pm 0.07$, respectively, for GRGs and smaller-size radio quasar. The dependence between accretion rate and unprojected linear size of radio structure is presented in Fig. 16.
{width="0.99\linewidth"}
{width="0.99\linewidth"}
In Fig. 17 we present the dependence of accretion rate m(3000Å) onto the core, as well as total, radio luminosity. There is a distinct trend for larger accretion rates to be observed in quasars with larger radio luminosity. The linear fits for m$(3000\rm\AA)$ are described by:\
m(3000Å)=0.114($\pm$0.044)log($P_{tot}$)-4.193($\pm$1.185),\
m(3000Å)=0.138($\pm$0.038)log$(P_{core})$-4.696($\pm$0.984)\
with correlation coefficients equal to 0.29 and 0.39 respectively.
{width="0.99\linewidth"}\
![Accretion rate as a function of total radio luminosity - [**top panel**]{} and core radio luminosity - [**bottom panel**]{}. ](m3000Ptot_1.eps "fig:"){width="0.99\linewidth"}\
![Accretion rate as a function of total radio luminosity - [**top panel**]{} and core radio luminosity - [**bottom panel**]{}. ](m3000Pcore_1.eps "fig:"){width="0.99\linewidth"}
Conclusions
===========
We have presented a comparison of radio and optical properties for a sample of GRQs and smaller radio quasars. It is important to mention that the measurements were obtained in a similar, homogeneous, manner for all sources from both the GRQ and comparison samples. Only the absolute values may be affected by some global calibration errors, if at all. The final conclusions are summarized below:\
1. Based on the $P$–$D$ diagram, we found that there is a continuous distribution of GRQs and smaller radio quasars. Therefore we can conclude that the GRQs could have evolved over time out of smaller radio quasars, which is consistent with the predictions of evolutionary models. We did not find that GRQs should have more prominent radio cores, which could suggest that the giants are similar to the smaller objects if we take into account their radio energetics.\
2. The arm-length-ratio and bending angle values for both GRQs and smaller radio quasars are similar, which indicates that there is no significant difference of the environmental properties of the IGM within which giant- and smaller radio quasars evolve.\
3. Statistically, the inclination angles obtained for our samples of quasars are inconsistent with traditional AGN unification scheme. Inclinations larger than 45$^{ \rm o}$ could, however, be explained based on recent results from studies of dusty torus properties.\
4. The values of BH masses estimated here are similar to those for the powerful AGNs. The BH masses estimated using the MgII emission line are in the range of $1.6 \cdot 10^8 M_{\odot} < M_{\rm BH} < 12.2 \cdot 10^8 M_{\odot}$ and $1.0 \cdot 10^8 M_{\odot} < M_{\rm BH} < 20.3 \cdot 10^8 M_{\odot}$ for GRQs and for the smaller radio quasars respectively. We did not find any constraints for more massive BHs to be located in GRQs.\
5. We did not find any significant correlation between the BH mass and the radio luminosity. However, using the $H_{\beta}$ and CIV line BH mass estimations a weak correlation between the linear size of the radio structure and the BH mass has been revealed. This might suggest that the linear size of giants could be related to their “central engines”. Surprisingly enough, the same relation, but based on the $H_{\beta}$ analysis results, is different for the GRQs and for the smaller radio quasars, which could suggest an inherent difference between these types of objects. However, this result should be taken with some caution as it was obtained only for a small number of quasars. The relation between the linear size of the radio structure and the BH mass supports the evolutionary origin of GRQs.\
6. The accretion rate for the more massive BHs is smaller than that for the less massive BHs. It is consistent with the scenario that quasars increase their BH mass during accretion process. The obtained mean value of accretion rate is equal to 0.07 for GRQs and 0.09 for smaller radio quasars. The lower value for GRQs suggests that GRQs are more evolved (aged) sources whose accretion process has slowed down or is almost over. The difference of m($\lambda$) and BH mass between the small-size radio quasars and large-size ones is, however, not significant, which could indicate similarities in their evolution. We found also a weak correlation between the accretion rates and the core radio luminosity, which confirms a connection between the accretion processes and the radio emission.\
7. The results obtained from the measurements based on the H$\beta$ and CIV emission lines seem to be more homogeneous than those based on MgII. The BH masses derived from the H$\beta$ and CIV mass-scaling relations have smaller uncertainties than those of the MgII line. The large uncertainties in the case of MgII measurements are due to the fact that this line is strongly affected by the Fe emission. Moreover, the large uncertainty of the mass-scaling relation slope for the MgII line is also due to the absence of reverberation data from systematic monitoring.\
In summary, taking into account the optical and radio properties, we can conclude that except for their size, the GRQs are similar to the smaller radio quasars. Their BH mass, accretion rate, prominence of radio core are comparable. The environment properties of GRQs and smaller radio quasars are also similar. Therefore, GRQs could be just evolved (aged) radio sources in which the accretion process has been diminished or is almost over and the large size is the consequence of their evolution. The sample of GRQs presented here, which is the largest one known to date, can be used for other astrophysical studies, such as on the evolution of radio sources.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Richard White for providing us with a number of quasar spectra and Marianne Vestergaard for the template of the Fe emission. We thank J. Machalski, S. Zo[ł]{}a and D. Kozie[ł]{}-Wierzbowska for their detailed and very helpful comments on the manuscript. We thank also the anonymous referee for her/his very valuable comments. This project was supported in part by the Polish National Center of Science under decision DEC-2011/01/N/ST9/00726.
-------------- ---------------- ----------------- ---------- ------ ------ ---------- -------------- ---------------- ----------------- ---------- ------ ------- ----------
IAU log(P$_{tot})$ log(P$_{core}$) B Q F i IAU log(P$_{tot})$ log(P$_{core}$) B Q F i
name W/Hz W/Hz \[$^o$\] \[$^o$\] name W/Hz W/Hz \[$^o$\] \[$^o$\]
(1) (2) (3) (4) (5) (6) (7) (1) (2) (3) (4) (5) (6) (7)
J0204$-$0944 25.76 24.92 0.0 2.06 0.59 81 J0022$-$0145 26.62 25.13 6.7 1.07 2.08 71
J0210$+$0118 25.99 25.31 25.6 1.38 0.30 63 J0034$+$0118 27.20 24.51 7.9 1.90 0.26 60
J0313$-$0631 25.89 24.45 5.6 1.11 0.97 87 J0051$-$0902 26.61 24.60 9.1 1.39 5.35 55
J0439$-$2422 27.09 $-$ 4.5 1.67 0.55 79 J0130$-$0135 26.18 24.68 5.9 1.90 0.15 62
J0750$+$6541 26.39 25.73 5.5 1.05 0.33 65 J0245$+$0108 27.55 25.85 12.0 2.28 0.68 85
J0754$+$3033 26.10 25.97 18.8 1.77 1.20 87 J0745$+$3142 26.96 26.59 5.3 1.10 0.62 88
J0754$+$4316 25.63 24.68 0.2 1.07 0.36 85 J0811$+$2845 27.23 26.71 6.1 2.35 1.26 80
J0801$+$4736 25.00 24.58 8.7 1.05 2.21 37 J0814$+$3237 26.84 26.49 12.8 2.40 0.37 72
J0809$+$2912 27.47 26.21 1.5 1.25 0.04 28 J0817$+$2237 27.69 26.42 5.7 1.06 2.38 71
J0812$+$3031 26.07 25.10 2.4 1.37 2.91 71 J0828$+$3935 26.26 24.94 0.4 1.15 0.23 79
J0819$+$0549 26.58 25.19 0.0 1.24 1.54 81 J0839$+$1921 27.78 26.97 11.4 1.36 0.11 41
J0842$+$2147 26.45 25.46 0.9 2.28 0.92 85 J0904$+$2819 26.83 26.22 2.3 1.07 3.15 45
J0902$+$5707 26.53 25.80 9.3 1.38 2.74 79 J0906$+$0832 26.81 26.05 0.5 1.29 0.88 86
J0918$+$2325 26.17 25.59 9.4 1.48 0.99 81 J0924$+$3547 26.49 25.90 1.9 1.05 0.68 84
J0925$+$4004 25.73 24.76 5.7 1.13 1.18 80 J0925$+$1444 27.36 26.04 5.6 1.26 1.25 84
J0937$+$2937 25.27 24.13 4.3 1.56 0.69 81 J0935$+$0204 27.06 26.47 4.3 1.62 37.20 -
J0944$+$2331 26.95 25.57 7.1 1.67 0.44 83 J0941$+$3853 26.95 26.03 0.3 1.44 0.72 85
J0959$+$1216 25.96 25.21 11.0 1.17 1.80 73 J0952$+$2352 26.23 26.02 17.7 2.56 0.97 89
J1012$+$4229 25.52 25.46 15.6 1.39 3.71 59 J1000$+$0005 27.46 26.38 18.4 1.35 1.17 85
J1020$+$0447 26.00 27.85 6.7 1.08 3.71 61 J1004$+$2225 27.40 26.05 4.7 1.25 1.03 85
J1020$+$3958 25.39 24.67 1.9 1.12 3.62 58 J1005$+$5019 27.18 26.96 16.2 2.73 1.54 76
J1027$-$2312 26.21 25.38 8.2 1.15 0.82 88 J1006$+$3236 27.31 26.97 0.9 2.89 1.06 80
J1030$+$5310 26.54 25.70 9.4 1.58 3.22 77 J1009$+$0529 26.79 25.89 8.7 1.01 1.75 78
J1054$+$4152 25.82 24.48 18.8 1.18 4.23 75 J1010$+$4132 27.35 26.57 5.5 1.69 12.73 42
J1056$+$4100 26.39 25.63 7.8 2.10 0.89 87 J1023$+$6357 27.01 26.00 0.2 1.37 0.44 70
J1130$-$1320 27.20 25.40 0.7 1.13 0.29 67 J1100$+$1046 26.61 26.19 1.8 1.20 0.61 78
J1145$-$0033 26.47 25.72 10.2 1.29 0.59 82 J1100$+$2314 26.38 25.14 $-$ $-$ 3.64 68
J1148$-$0403 26.32 25.76 12.8 1.06 1.20 88 J1107$+$0547 26.42 25.52 7.2 1.46 0.63 80
J1151$+$3355 26.26 25.16 11.3 1.98 0.33 32 J1107$+$1628 27.09 26.53 3.4 1.05 0.67 87
J1229$+$3555 26.20 24.65 14.9 1.33 0.39 57 J1110$+$0321 27.35 25.50 10.7 2.38 0.55 85
J1304$+$2454 25.76 25.41 1.6 1.46 1.82 77 J1118$+$3828 26.10 24.89 3.5 1.25 3.25 55
J1321$+$3741 26.53 25.55 17.0 1.06 0.72 79 J1119$+$3858 26.43 25.28 8.4 1.14 2.66 64
J1340$+$4232 26.24 25.48 3.5 1.92 0.59 89 J1158$+$6254 27.03 25.32 4.3 1.53 0.46 75
J1353$+$2631 25.83 24.78 13.4 1.21 2.98 34 J1217$+$1019 27.44 26.13 26.8 1.43 0.73 87
J1408$+$3054 25.93 24.82 4.9 1.41 2.72 80 J1223$+$3707 26.57 25.48 3.9 1.75 0.72 83
J1410$+$2955 25.26 24.56 10.4 1.30 1.00 81 J1236$+$1034 26.55 25.18 0.4 1.42 0.39 63
J1427$+$2632 26.17 25.23 9.1 1.70 0.40 45 J1256$+$1008 26.98 26.51 20.3 1.14 0.30 69
J1432$+$1548 26.83 25.71 2.9 1.39 0.99 87 J1319$+$5148 27.70 27.13 27.9 1.81 0.54 61
J1504$+$6856 26.13 25.52 4.0 1.85 1.66 81 J1334$+$5501 27.46 25.69 0.4 1.13 0.79 89
J1723$+$3417 26.26 25.67 1.1 1.05 2.11 51 J1358$+$5752 27.66 25.86 1.3 1.20 1.24 85
J2042$+$7508 25.67 24.72 7.1 1.03 2.69 61 J1425$+$2404 27.37 26.62 18.9 1.41 1.60 83
J2234$-$0224 25.89 24.71 1.9 1.49 0.18 87 J1433$+$3209 26.89 25.51 13.8 1.14 0.96 88
J2344$-$0032 25.46 25.12 0.8 1.54 0.76 79 J1513$+$1011 27.38 26.36 20.9 1.46 1.07 83
J1550$+$3652 26.99 25.39 1.0 1.78 0.28 64
J1557$+$0253 26.78 26.40 7.4 3.62 8.05 61
J1557$+$3304 27.45 27.26 20.2 1.54 1.50 89
J1622$+$3531 27.56 26.36 17.2 2.67 0.59 82
J1623$+$3419 26.31 25.77 7.9 2.19 0.64 13
J2335$-$0927 26.72 26.11 8.0 2.74 1.66 83
-------------- ---------------- ----------------- ---------- ------ ------ ---------- -------------- ---------------- ----------------- ---------- ------ ------- ----------
-------------- ------- ------- ----------- ------- ------------- -------- ------- -------------------------- ------- ------------------- ------------------- -------------------
IAU FWHM f$_\lambda$ Log$\lambda L_{\lambda}$ $M_{BH}$
name CIV MgII H$_\beta$ 1350Å 3000Å 5100Å 1350Å 3000Å 5100Å CIV MgII H$\beta$
J0204$-$0944 $-$ 34.19 $-$ $-$ 10.13 $-$ $-$ 44.61 $-$ $-$ 1.93$^{\pm0.25}$ $-$
J0210$+$0118 $-$ 49.61 $-$ $-$ 46.57 $-$ $-$ 45.17 $-$ $-$ 7.80$^{\pm0.22}$ $-$
J0754$+$3033 $-$ 48.59 139.90 $-$ 53.77 10.65 $-$ 45.17 44.70 $-$ 7.49$^{\pm0.25}$ 13.56$^{\pm2.57}$
J0754$+$4316 $-$ $-$ 226.37 $-$ $-$ 30.2 $-$ $-$ 44.53 $-$ $-$ 29.17$^{\pm1.32}$
J0801$+$4736 $-$ $-$ 125.23 $-$ $-$ 3.69 $-$ $-$ 42.97 $-$ $-$ 1.48$^{\pm0.76}$
J0809$+$2912 $-$ 46.91 $-$ $-$ 44.45 $-$ $-$ 45.48 $-$ $-$ 9.98$^{\pm0.51}$ $-$
J0812$+$3031 $-$ 30.91 $-$ $-$ 11.14 $-$ $-$ 44.72 $-$ $-$ 1.80$^{\pm0.15}$ $-$
J0819$+$0549 75.57 58.94 $-$ $-$ 1.50 $-$ $-$ 44.09 $-$ $-$ 3.16$^{\pm2.77}$ $-$
J0842$+$2147 $-$ 33.16 $-$ $-$ 10.96 $-$ $-$ 44.74 $-$ $-$ 2.12$^{\pm0.48}$ $-$
J0902$+$5707 31.09 47.90 $-$ $-$ 10.40 $-$ $-$ 44.89 $-$ $-$ 5.28$^{\pm1.28}$ $-$
J0918$+$2325 $-$ 55.91 173.43 $-$ 54.28 8.92 $-$ 45.08 44.52 $-$ 8.86$^{\pm0.46}$ 16.95$^{\pm3.37}$
J0925$+$4004 $-$ 62.30 196.95 $-$ 109.20 20.40 $-$ 45.10 44.60 $-$ 11.28$^{\pm4.24}$ 23.90$^{\pm2.01}$
J0937$+$2937 $-$ 41.23 98.86 $-$ 78.80 14.70 $-$ 44.92 44.42 $-$ 4.03$^{\pm0.51}$ 4.91$^{\pm0.24}$
J0944$+$2331 $-$ 43.36 $-$ $-$ 34.92 $-$ $-$ 45.13 $-$ $-$ 5.68$^{\pm0.54}$ $-$
J0959$+$1216 $-$ 46.95 $-$ $-$ 14.32 $-$ $-$ 44.81 $-$ $-$ 4.59$^{\pm3.26}$ $-$
J1020$+$0447 $-$ 57.04 $-$ $-$ 6.56 $-$ $-$ 44.49 $-$ $-$ 4.71$^{\pm1.45}$ $-$
J1020$+$3958 $-$ 66.70 $-$ $-$ 33.48 $-$ $-$ 45.00 $-$ $-$ 11.52$^{\pm5.16}$ $-$
J1030$+$5310 $-$ 36.46 $-$ $-$ 26.04 $-$ $-$ 45.13 $-$ $-$ 3.99$^{\pm0.27}$ $-$
J1054$+$4152 $-$ 49.00 $-$ $-$ 23.03 $-$ $-$ 45.01 $-$ $-$ 6.34$^{\pm3.40}$ $-$
J1056$+$4100 34.41 51.61 $-$ $-$ 2.841 $-$ $-$ 44.39 $-$ $-$ 3.43$^{\pm1.13}$ $-$
J1145$-$0033 61.12 $-$ $-$ 16.07 5.38 $-$ 44.87 44.74 $-$ 18.42$^{\pm2.43}$ $-$ $-$
J1151$+$3355 $-$ 51.11 $-$ $-$ 29.15 $-$ $-$ 44.95 $-$ $-$ 6.44$^{\pm2.48}$ $-$
J1229$+$3555 $-$ 31.93 $-$ $-$ 13.52 $-$ $-$ 44.60 $-$ $-$ 1.67$^{\pm0.29}$ $-$
J1304$+$2454 $-$ 47.98 189.27 $-$ 79.83 $-$ $-$ 45.15 $-$ $-$ 7.10$^{\pm0.43}$ $-$
J1321$+$3741 $-$ 73.96 $-$ $-$ 15.56 $-$ $-$ 44.87 $-$ $-$ 12.23$^{\pm3.98}$ $-$
J1340$+$4232 $-$ 52.94 $-$ $-$ 8.22 $-$ $-$ 44.69 $-$ $-$ 5.12$^{\pm2.67}$ $-$
J1353$+$2631 $-$ 41.75 206.40 $-$ 136.10 35.48 $-$ 44.86 44.51 $-$ 3.87$^{\pm2.63}$ 23.69$^{\pm2.59}$
J1410$+$2955 $-$ 69.94 $-$ $-$ 47.28 $-$ $-$ 44.88 $-$ $-$ 11.04$^{\pm6.67}$ $-$
J1427$+$2632 $-$ $-$ 195.90 $-$ $-$ 42.12 $-$ $-$ 44.72 $-$ $-$ 27.08$^{\pm4.74}$
J1432$+$1548 $-$ 52.83 $-$ $-$ 18.82 $-$ $-$ 44.88 $-$ $-$ 6.34$^{\pm1.02}$ $-$
J1723$+$3417 $-$ $-$ 64.88 $-$ 168.40 139.60 $-$ 44.62 44.77 $-$ $-$ 3.16$^{\pm0.407}$
J2344$-$0032 $-$ 41.18 $-$ $-$ 70.72 $-$ $-$ 44.96 $-$ $-$ 4.20$^{\pm0.31}$ $-$
-------------- ------- ------- ----------- ------- ------------- -------- ------- -------------------------- ------- ------------------- ------------------- -------------------
-------------- -------- ------- ----------- ------- ------------- -------- ------- -------------------------- ------- ------------------ ------------------- --------------------
IAU FWHM f$_\lambda$ Log$\lambda L_{\lambda}$ $M_{BH}$
name CIV MgII H$_\beta$ 1350Å 3000Å 5100Å 1350Å 3000Å 5100Å CIV MgII H$\beta$
J0034$+$0118 $-$ 56.34 $-$ $-$ 11.80 $-$ $-$ 44.58 $-$ $-$ 5.06$^{\pm0.11}$ $-$
J0051$-$0902 $-$ 64.84 $-$ $-$ 11.51 $-$ $-$ 44.81 $-$ $-$ 8.73$^{\pm3.13}$ $-$
J0130$-$0135 $-$ 61.47 $-$ $-$ 23.58 $-$ $-$ 45.06 $-$ $-$ 10.56$^{\pm0.48}$ $-$
J0245$+$0108 35.54 61.03 $-$ $-$ 12.67 $-$ $-$ 44.96 $-$ $-$ 9.23$^{\pm3.07}$ $-$
J0745$+$3142 186.40 53.84 173.80 $-$ 499.50 107.50 $-$ 45.74 45.30 $-$ 17.67$^{\pm0.76}$ 41.92$^{\pm4.52}$
J0811$+$2845 27.10 54.05 $-$ 59.34 12.92 $-$ 45.40 45.08 $-$ 6.88$^{\pm0.88}$ 8.30$^{\pm1.16}$ $-$
J0814$+$3237 $-$ 32.17 $-$ $-$ 17.93 $-$ $-$ 44.74 $-$ $-$ 1.99$^{\pm0.14}$ $-$
J0817$+$2237 $-$ 45.24 $-$ $-$ 41.95 $-$ $-$ 45.21 $-$ $-$ 6.72$^{\pm0.95}$ $-$
J0828$+$3935 $-$ 41.57 $-$ $-$ 15.63 $-$ $-$ 44.61 $-$ $-$ 2.85$^{\pm0.54}$ $-$
J0839$+$1921 21.60 36.12 $-$ $-$ 23.96 $-$ $-$ 45.29 $-$ $-$ 4.72$^{\pm0.10}$ $-$
J0904$+$2819 $-$ 36.98 $-$ $-$ 58.68 $-$ $-$ 45.44 $-$ $-$ 5.88$^{\pm0.26}$ $-$
J0906$+$0832 38.41 48.46 $-$ $-$ 8.01 $-$ $-$ 44.79 $-$ $-$ 4.78$^{\pm2.13}$ $-$
J0924$+$3547 $-$ 46.75 $-$ $-$ 19.14 $-$ $-$ 45.06 $-$ $-$ 6.09$^{\pm0.73}$ $-$
J0925$+$1444 $-$ 39.36 $-$ $-$ 32.33 $-$ $-$ 45.03 $-$ $-$ 4.18$^{\pm0.41}$ $-$
J0935$+$0204 $-$ 61.33 142.40 $-$ 92.04 17.1 $-$ 45.26 44.76 $-$ 13.22$^{\pm0.75}$ 15.07$^{\pm1.44}$
J0941$+$3853 $-$ 59.18 234.00 $-$ 42.59 9.19 $-$ 44.89 44.45 $-$ 8.01$^{\pm1.03}$ 28.53$^{\pm2.11}$
J0952$+$2352 $-$ 36.19 $-$ $-$ 53.89 $-$ $-$ 45.31 $-$ $-$ 4.85$^{\pm0.49}$ $-$
J1000$+$0005 $-$ 30.90 $-$ $-$ 12.99 $-$ $-$ 44.65 $-$ $-$ 1.65$^{\pm0.20}$ $-$
J1004$+$2225 $-$ 44.52 $-$ $-$ 18.23 $-$ $-$ 44.84 $-$ $-$ 4.29$^{\pm0.29}$ $-$
J1005$+$5019 18.18 46.60 $-$ 78.57 9.60 $-$ $-$ 44.98 $-$ 3.74$^{\pm0.46}$ 5.53$^{\pm2.99}$ $-$
J1006$+$3236 $-$ 34.69 $-$ $-$ 8.954 $-$ $-$ 44.57 $-$ $-$ 1.90$^{\pm0.20}$ $-$
J1009$+$0529 $-$ 68.01 $-$ $-$ 71.05 $-$ $-$ 45.41 $-$ $-$ 19.25$^{\pm1.07}$ $-$
J1010$+$4132 $-$ 28.88 65.89 $-$ 233.60 35.86 $-$ 45.62 45.04 $-$ 4.45$^{\pm0.34}$ 4.45$^{\pm0.76}$
J1023$+$6357 $-$ 48.48 $-$ $-$ 53.00 $-$ $-$ 45.43 $-$ $-$ 10.05$^{\pm0.97}$ $-$
J1100$+$1046 $-$ 49.35 21.30 $-$ 60.94 11.07 $-$ 44.76 44.25 $-$ 4.79$^{\pm2.17}$ 0.19$^{\pm0.10}$
J1100$+$2314 $-$ 66.57 296.03 $-$ 100.50 31.02 $-$ 45.19 44.91 $-$ 14.34$^{\pm4.02}$ 77.29$^{\pm5.26}$
J1107$+$0547 33.64 40.26 $-$ $-$ 6.66 $-$ $-$ 44.76 $-$ $-$ 3.21$^{\pm1.52}$ $-$
J1107$+$1628 $-$ 36.98 88.07 $-$ 180.00 33.76 $-$ 45.53 45.04 $-$ 6.57$^{\pm0.39}$ 7.92$^{\pm0.86}$
J1110$+$0321 $-$ 29.41 $-$ $-$ 16.59 $-$ $-$ 44.79 $-$ $-$ 1.77$^{\pm0.74}$ $-$
J1118$+$3828 $-$ 39.03 431.77 $-$ 29.22 3.46 $-$ 44.87 44.17 $-$ 3.39$^{\pm1.22}$ 69.97$^{\pm29.70}$
J1119$+$3858 $-$ 64.70 345.40 $-$ 31.08 6.27 $-$ 44.88 44.41 $-$ 9.47$^{\pm3.54}$ 59.42$^{\pm20.60}$
J1158$+$6254 $-$ 66.27 284.30 $-$ 185.80 44.74 $-$ 45.50 45.11 $-$ 20.31$^{\pm2.35}$ 89.96$^{\pm19.39}$
J1217$+$1019 20.56 38.45 $-$ 59.73 5.42 $-$ 45.39 44.70 $-$ 3.96$^{\pm0.53}$ 2.71$^{\pm1.00}$ $-$
J1223$+$3707 $-$ 50.07 264.00 $-$ 34.43 9.27 $-$ 44.63 44.29 $-$ 4.24$^{\pm0.95}$ 30.00$^{\pm5.01}$
J1236$+$1034 $-$ 38.83 272.90 $-$ 53.47 11.31 $-$ 45.05 44.60 $-$ 4.13$^{\pm1.94}$ 46.03$^{\pm9.20}$
J1256$+$1008 $-$ 33.46 33.74 $-$ 15.93 $-$ $-$ 44.67 $-$ $-$ 1.99$^{\pm0.18}$ $-$
J1319$+$5148 $-$ 39.67 $-$ $-$ 76.29 $-$ $-$ 45.52 $-$ $-$ 7.42$^{\pm0.80}$ $-$
J1334$+$5501 $-$ 55.70 $-$ $-$ 21.12 $-$ $-$ 45.06 $-$ $-$ 8.63$^{\pm0.92}$ $-$
J1358$+$5752 $-$ 45.54 $-$ $-$ 67.78 $-$ $-$ 45.62 $-$ $-$ 11.05$^{\pm1.92}$ $-$
J1425$+$2404 $-$ 56.62 131.60 $-$ 89.40 19.66 $-$ 45.25 44.83 $-$ 11.16$^{\pm1.62}$ 13.87$^{\pm1.64}$
J1433$+$3209 $-$ 45.42 $-$ $-$ 2.924 $-$ $-$ 44.02 $-$ $-$ 1.73$^{\pm1.85}$ $-$
J1513$+$1011 23.23 42.70 $-$ $-$ 36.67 $-$ $-$ 45.42 $-$ $-$ 7.72$^{\pm0.57}$ $-$
J1550$+$3652 21.99 47.58 $-$ 41.78 4.80 $-$ 45.28 44.69 $-$ 3.97$^{\pm0.52}$ 4.12$^{\pm1.81}$ $-$
J1557$+$0253 10.42 24.83 $-$ 34.97 4.69 $-$ 45.19 44.66 $-$ 0.79$^{\pm0.16}$ 1.09$^{\pm1.28}$ $-$
J1557$+$3304 $-$ 52.94 $-$ $-$ 25.46 $-$ $-$ 44.97 $-$ $-$ 7.05$^{\pm1.15}$ $-$
J1622$+$3531 $-$ 43.02 $-$ $-$ 10.56 $-$ $-$ 44.86 $-$ $-$ 4.08$^{\pm0.83}$ $-$
J1623$+$3419 25.17 48.20 $-$ 14.37 2.59 $-$ 44.80 44.40 $-$ 2.88$^{\pm0.59}$ 3.04$^{\pm1.78}$ $-$
J2335$-$0927 14.18 35.59 $-$ 60.11 1.14 $-$ 45.38 44.00 $-$ 1.85$^{\pm0.26}$ 1.04$^{\pm0.99}$ $-$
-------------- -------- ------- ----------- ------- ------------- -------- ------- -------------------------- ------- ------------------ ------------------- --------------------
-------------- ------- ---------------- ------- ------- ---------------- ---------- ------- ------- -------
IAU log(L$_{bol})$ log(L$_{Edd}$) m1350 m3000 m5100
name 1350Å 3000Å 5100Å CIV MgII $H\beta$
J0204$-$0944 $-$ 45.38 $-$ $-$ 46.45 $-$ $-$ 0.09 $-$
J0210$+$0118 $-$ 45.94 $-$ $-$ 47.05 $-$ $-$ 0.08 $-$
J0754$+$3033 $-$ 45.94 45.66 $-$ 47.04 47.29 $-$ 0.08 0.02
J0754$+$4316 $-$ $-$ 45.48 $-$ $-$ 47.63 $-$ $-$ 0.01
J0801$+$4736 $-$ $-$ 43.92 $-$ $-$ 46.33 $-$ $-$ 0.004
J0809$+$2912 $-$ 46.25 $-$ $-$ 47.16 $-$ $-$ 0.12 $-$
J0812$+$3031 $-$ 45.49 $-$ $-$ 46.42 $-$ $-$ 0.12 $-$
J0819$+$0549 45.29 44.86 $-$ 47.49 46.66 $-$ $-$ 0.02 $-$
J0842$+$2147 $-$ 45.51 $-$ $-$ 46.49 $-$ $-$ 0.11 $-$
J0902$+$5707 45.97 45.67 $-$ 47.08 46.88 $-$ $-$ 0.06 $-$
J0918$+$2325 $-$ 45.85 45.48 $-$ 47.11 47.39 $-$ 0.06 0.01
J0925$+$4004 $-$ 45.87 45.55 $-$ 47.21 47.54 $-$ 0.05 0.01
J0937$+$2937 $-$ 45.69 45.38 $-$ 46.77 46.85 $-$ 0.08 0.03
J0944$+$2331 $-$ 45.90 $-$ $-$ 46.92 $-$ $-$ 0.10 $-$
J0959$+$1216 $-$ 45.58 $-$ $-$ 46.82 $-$ $-$ 0.06 $-$
J1020$+$0447 $-$ 45.26 $-$ $-$ 46.83 $-$ $-$ 0.03 $-$
J1020$+$3958 $-$ 45.77 $-$ $-$ 47.22 $-$ $-$ 0.04 $-$
J1030$+$5310 $-$ 45.90 $-$ $-$ 46.76 $-$ $-$ 0.14 $-$
J1054$+$4152 $-$ 45.78 $-$ $-$ 46.96 $-$ $-$ 0.07 $-$
J1056$+$4100 45.34 45.16 $-$ 46.83 46.70 $-$ $-$ 0.03 $-$
J1145$-$0033 45.53 45.51 $-$ 47.43 $-$ $-$ 0.01 $-$ $-$
J1151$+$3355 $-$ 45.73 $-$ $-$ 46.97 $-$ $-$ 0.06 $-$
J1229$+$3555 $-$ 45.37 $-$ $-$ 46.39 $-$ $-$ 0.10 $-$
J1304$+$2454 $-$ 45.92 $-$ $-$ 47.01 47.57 $-$ 0.08 $-$
J1321$+$3741 $-$ 45.64 $-$ $-$ 47.25 $-$ $-$ 0.03 $-$
J1340$+$4232 $-$ 45.47 $-$ $-$ 46.87 $-$ $-$ 0.04 $-$
J1353$+$2631 $-$ 45.63 45.46 $-$ 46.75 47.54 $-$ 0.08 0.01
J1410$+$2955 $-$ 45.65 $-$ $-$ 47.20 $-$ $-$ 0.03 $-$
J1427$+$2632 $-$ $-$ 45.67 $-$ $-$ 47.59 $-$ $-$ 0.01
J1432$+$1548 $-$ 45.65 $-$ $-$ 46.96 $-$ $-$ 0.05 $-$
J1723$+$3417 $-$ 45.39 45.73 $-$ $-$ 46.66 $-$ $-$ 0.12
J2344$-$0032 $-$ 45.723 $-$ $-$ 46.78 $-$ $-$ 0.09 $-$
-------------- ------- ---------------- ------- ------- ---------------- ---------- ------- ------- -------
\
-------------- ------- ---------------- ------- ------- ---------------- ---------- ------- ------- -------
IAU log(L$_{bol})$ log(L$_{Edd}$) m1350 m3000 m5100
name 1350Å 3000Å 5100Å CIV MgII $H\beta$
J0034$+$0118 $-$ 45.35 $-$ $-$ 46.87 $-$ $-$ 0.03 $-$
J0051$-$0902 $-$ 45.58 $-$ $-$ 47.10 $-$ $-$ 0.03 $-$
J0130$-$0135 $-$ 45.83 $-$ $-$ 47.19 $-$ $-$ 0.05 $-$
J0245$+$0108 46.05 45.73 $-$ 47.23 47.13 $-$ 0.07 0.04 $-$
J0745$+$3142 $-$ 46.51 46.26 $-$ 47.41 47.78 $-$ 0.13 0.03
J0811$+$2845 46.06 45.85 $-$ 47.00 47.08 $-$ 0.11 0.06 $-$
J0814$+$3237 $-$ 45.51 $-$ $-$ 46.46 45.89 $-$ 0.11 $-$
J0817$+$2237 $-$ 45.98 $-$ $-$ 46.99 $-$ $-$ 0.10 $-$
J0828$+$3935 $-$ 45.38 $-$ $-$ 46.62 $-$ $-$ 0.06 $-$
J0839$+$1921 46.33 46.06 $-$ 46.95 46.84 $-$ 0.24 0.17 $-$
J0904$+$2819 $-$ 46.21 $-$ $-$ 46.93 $-$ $-$ 0.19 $-$
J0906$+$0832 45.82 45.56 $-$ 47.18 46.84 $-$ 0.04 0.05 $-$
J0924$+$3547 $-$ 45.83 $-$ $-$ 46.95 $-$ $-$ 0.08 $-$
J0925$+$1444 $-$ 45.81 $-$ $-$ 46.78 $-$ $-$ 0.11 $-$
J0935$+$0204 $-$ 46.03 45.72 $-$ 47.28 47.34 $-$ 0.06 0.02
J0941$+$3853 $-$ 45.66 45.41 $-$ 47.07 47.62 $-$ 0.04 0.01
J0952$+$2352 $-$ 46.080 $-$ $-$ 46.85 $-$ $-$ 0.17 $-$
J1000$+$0005 $-$ 45.42 $-$ $-$ 46.38 $-$ $-$ 0.11 $-$
J1004$+$2225 $-$ 45.61 $-$ $-$ 46.79 $-$ $-$ 0.07 $-$
J1005$+$5019 46.21 45.75 $-$ 46.74 46.90 $-$ 0.30 0.07 $-$
J1006$+$3236 $-$ 45.34 $-$ $-$ 46.44 $-$ $-$ 0.08 $-$
J1009$+$0529 $-$ 46.18 $-$ $-$ 47.45 $-$ $-$ 0.05 $-$
J1010$+$4132 $-$ 46.40 46.00 $-$ 46.81 46.81 $-$ 0.39 0.15
J1023$+$6357 $-$ 46.20 $-$ $-$ 47.16 $-$ $-$ 0.11 $-$
J1100$+$1046 $-$ 45.53 45.20 $-$ 46.84 45.43 $-$ 0.05 0.59
J1100$+$2314 $-$ 45.96 45.87 $-$ 47.32 48.05 $-$ 0.04 0.01
J1107$+$0547 45.87 45.54 $-$ 47.09 46.67 $-$ 0.06 0.07 $-$
J1107$+$1628 $-$ 46.31 45.99 $-$ 46.98 47.06 $-$ 0.21 0.09
J1110$+$0321 $-$ 45.57 $-$ $-$ 46.41 $-$ $-$ 0.14 $-$
J1118$+$3828 $-$ 45.64 45.12 $-$ 46.69 48.01 $-$ 0.09 0.001
J1119$+$3858 $-$ 45.65 45.37 $-$ 47.14 47.94 $-$ 0.03 0.003
J1158$+$6254 $-$ 46.27 46.07 $-$ 47.47 48.12 $-$ 0.06 0.01
J1217$+$1019 46.06 45.47 $-$ 46.76 46.60 $-$ 0.20 0.08 $-$
J1223$+$3707 $-$ 45.40 45.24 $-$ 46.79 47.64 $-$ 0.04 0.004
J1236$+$1034 $-$ 45.82 45.56 $-$ 46.78 47.82 $-$ 0.11 0.01
J1256$+$1008 $-$ 45.44 $-$ $-$ 46.46 45.70 $-$ 0.10 $-$
J1319$+$5148 $-$ 46.29 $-$ $-$ 47.03 $-$ $-$ 0.18 $-$
J1334$+$5501 $-$ 45.83 $-$ $-$ 47.10 $-$ $-$ 0.05 $-$
J1358$+$5752 $-$ 46.39 $-$ $-$ 47.20 $-$ $-$ 0.16 $-$
J1425$+$2404 $-$ 46.03 45.78 $-$ 47.21 47.30 $-$ 0.07 0.03
J1433$+$3209 $-$ 44.79 $-$ $-$ 46.40 $-$ $-$ 0.03 $-$
J1513$+$1011 46.59 46.20 $-$ 47.15 47.05 $-$ 0.27 0.14 $-$
J1550$+$3652 45.95 45.46 $-$ 46.76 46.78 $-$ 0.15 0.05 $-$
J1557$+$0253 45.85 45.43 $-$ 46.06 46.20 $-$ 0.61 0.17 $-$
J1557$+$3304 $-$ 45.74 $-$ $-$ 47.01 $-$ $-$ 0.05 $-$
J1622$+$3531 $-$ 45.63 $-$ $-$ 46.77 $-$ $-$ 0.07 $-$
J1623$+$3419 45.46 45.17 $-$ 46.62 46.64 $-$ 0.07 0.03 $-$
J2335$-$0927 46.04 44.77 $-$ 46.43 46.18 $-$ 0.41 0.04 $-$
-------------- ------- ---------------- ------- ------- ---------------- ---------- ------- ------- -------
\
[151]{} Adelman-McCarthy J.K., et al., 2008, ApJS, 175, 297 Alexander P., Leahy J.P., 1987, MNRAS, 225, 1 Antonucci, R.R.J., Barvainis, R., 1988, ApJ, 325L, 21 Arshakian T.G., Longair M.S., 2004, MNRAS, 351, 727 Becker R.H., White R.L., Helfand D.J., 1995, ApJ, 450, 559 Becker R.H., White R.L., Gregg M.D., Laurent-Muehleisen S.A., et al., 2001, ApJS, 135, 227 Bhatnagar S., Krishna G., Wisotzki L., 1998, MNRAS, 299, 25 Blandford R.D.,Payne D.G., 1982, MNRAS, 199, 883 Blandford R.D., Rees M.J., 1974, MNRAS, 169, 395 Blandford R.D., Znajeck R., 1977, MNRAS, 179, 433 Blundell K.M., Rawlings S., Willott C.J., 1999, AJ, 117, 677 Bock D.C.-J., Large M.I., Sadler E.M., 1999, AJ, 117, 1578 Boroson T.A., Green R.F., 1992, ApJS, 80, 109 Brown M.J.I., Webster R.L., Boyle B.J., 2001, AJ, 121, 2381 Cattaneo A., Best P.N., 2009, MNRAS, 395, 518 Ciliegi P., et al., 1999, MNRAS, 302, 222 Cirasuolo M., Magliocchetti M., Celotti A., Danese L., 2003, MNRAS, 341, 993 Condon J.J., Cotton W.D., Greisen E.W., Yin Q.F., Perley R.A., Taylor G.B., Broderick J.J., 1998, AJ, 115, 1693 Cotter G., Rawlings S., Saunders R., 1996, MNRAS, 281, 1081 Decarli, R., Labita, M., Treves, A., Falomo, R., 2008, MNRAS, 387, 1237 Dietrich M., Mathur S., Grupe D., Komossa S., 2009, ApJ, 696, 1998 Dunlop J.S., McLure R.J., Kukula M.J., Baum S.A. O’Dea C.P. Huges D.H., 2003, MNRAS, 340, 1095 Elitzur M., 2008, NewAR, 52, 274 Fanaroff B.L., Riley J.M., 1974, MNRAS, 167, 31 Ghisellini G., Tavecchio F., 2008, MNRAS, 387, 1669 Gopal-Krishna, Wiita P.J., Saripalli L., 1989, MNRAS, 239, 173 Gregg M.D., Becker R.H., White R.L., Helfand D.J., McMahon R.G., Hook I.M., 1996, AJ, 112, 407 Hewett P.C., Foltz C.B., Chaffee F.H., 2001, AJ, 122, 518 Hintzen P., Ulvestad J., Owen F., 1983, AJ, 88, 709 Ho L.C., 2002, ApJ, 564, 120 Hough D.H., Vermeulen R.C., Readhead A.C.S., Cross L.L., Barth E.L., Yu L.H., Beyer P.J., Phifer E.M., 2002, AJ, 123, 1258 Ishwara-Chandra C.H., Saikia D.J.,1999, MNRAS, 309, 100 Ivezic, M.J., et al., 2002, AJ, 124, 2364 Jägers W.J., Miley G.K., van Breugel W.J.M., Schilizzi R.T., Conway R.G., 1982, A&A, 105, 278 Jiang L., Fan X., Ivezic Z., Richards G.T., Schneider D.P., Strauss M.A., Kelly B.C., 2007, ApJ, 656, 680 Kaiser C.R., Dennett-Thorpe J., Alexander P., 1997, MNRAS, 292, 723 Kaspi S., Smith P.S., Netzer H., Maoz D., Jannuzi B.T., Giveon U., 2000, ApJ, 533, 631 Kaspi S., Maoz D., Netzer H., Peterson B.M., Vestergaard M., Jannuzi B.T., 2005, ApJ, 629, 61 Kaspi, S., Brandt, W.N., Maoz, D. Netzer, H., Schneider, D.P., Shemmer, O., 2007, ApJ, 659, 997 Komberg B.V., Pashchenko, I.N., 2009, ARep, 53, 1086 Kuligowska E., 2007, Master Thesies “Search for giant size radio galaxies at large redshift”, Jagiellonian University Kuligowska E., Jamrozy M., Kozieł-Wierzbowska D., Machalski J., 2009, AcA, 59, 431 Kuźmicz, A., Kuligowska, E., Jamrozy, M., 2011, AcA, 61, 71 Lara L., Marquez I., Cotton W.D., Feretti L., Giovannini G., Marcaide J.M., Venturi T., 2001, A&A, 378, 826 Leahy J.P., Muxlow T.W.B., Stephens P.W., 1989, MNRAS, 239, 401 Liu R., Pooley G., Riley J. M., 1992, MNRAS, 257, 545 Laor A., 2000, ApJ, 543, L111 Machalski J., Jamrozy M., Zoła S., 2001, A&A, 371, 445 Machalski J., Jamrozy M., 2006, A&A, 454, 95 Machalski J., Jamrozy M., Zoła S., Kozieł D., 2006, A&A, 454, 85 Machalski J., Kozieł-Wierzbowska D., Jamrozy M., 2007, AcA, 57, 227 Machalski J., Jamrozy M., Saikia D.J., 2009, MNRAS, 395, 812 Machalski J., Jamrozy, M., Stawarz Ł., Kozieł-Wierzbowska, D., 2011, ApJ, 740, 58 Marziani P., Zamanov R.K., Sulentic J.W., Calvani C., 2003, MNRAS, 345, 1133 Mathur S., 2000, MNRAS, 314, 17 McGilchrist M.M., Baldwin J.E., Riley J.M., Titterington D.J., Waldram E.M., Warner P.J., 1990, MNRAS, 246, 110 McLure R.J., Dunlop J.S., 2002, MNRAS, 331, 795 McLure R.J., Jarvis M.J., 2004, MNRAS, 353, L45 Merloni A., Heinz S., 2008, MNRAS, 388, 1011 Metcalf R. B., Magliocchetti M., 2006, MNRAS, 365, 101 Nenkova M., Sirocky M.M., Nikutta R., Ivezic, Z., Elitzur, M., 2008, ApJ, 685, 160 Nilsson K., 1998, A&AS, 132, 31 Oshlack A., Webster R., Whiting M., 2002, ApJ, 576, 81 Peterson B.M., 1993, PASP, 105, 207 Peterson B.M., et al., 2004, AJ, 613, 682 Rengelink R.B., Tang Y., de Bruyn A.G., Miley G.K., Bremer M.N., Röttgering H.J.A., Bremer M.A.R., 1997, A&AS, 124, 259 Richstone D., et al., 1998, Natur, 395, 14 Riley J.M., Warner P.J.,1990, MNRAS, 246, 1 Saripalli L., Hunstead R.W., Subrahmanyan R., Boyce E., 2005, AJ, 130, 896 Scheuer, P.A.G., 1995, MNRAS, 277, 331 Schoenmakers A.P., 1999, PhD-thesies “A population of Giant Radio Galaxies”, Utreht University Schoenmakers A. P., de Bruyn A. G., Röttgering H. J. A., van der Laan H., Mack K.-H., Kaiser C. R., 2000, in van Haarlem M. P., ed., Proc. Perspectives on Radio Astronomy: Science with Large Antenna Arrays. ASTRON, Dwingeloo, p. 165 Schoenmakers A.P., de Bruyn A.G., Röttgering H.J.A., van der Laan H., 2001, A&A, 374, 861 Shankar F., Salucci P., Granato G.L., De Zotti G., Danese L., 2004, MNRAS, 354, 1020 Shankar F., Cavaliere A., Cirasuolo M., Maraschi L., 2008, ApJ, 676, 131 Shankar F., Sivakoff G.R., Vestergaard M., Dai X., 2010, MNRAS, 401, 1869 Shen Y., Greene J.E., Strauss M.A., Richards G.T., Schneider D.P., 2008, ApJ, 680, 169 Sikora M., Stawarz Ł., Lasota J.-P., 2007, ApJ, 658, 815 Snellen I.A.G., Lehnert M.D., Bremer M.N., Schilizzi R.T., 2003, MNRAS, 342, 889 Spergel D.N., et al., 2003, ApJS, 148, 175 Subrahmanyan R., Saripalli L., Safouris V., Hunstead R. W., 2008, ApJ, 677, 63 Veron-Cetty M.-P., Joly M., Veron P., 2004, A&A, 417, 515 Vestergaard M., 2004, ApJ, 601, 676 Vestergaard M., Osmer P., 2009, ApJ, 699, 800 Vestergaard M., Peterson B.M., 2006, ApJ, 641, 689 Vestergaard M., Wilkes B.J., 2001, ApJS, 134, 1 de Vries W.H., Becker R.H., White R.L., 2006, AJ, 131, 666 Urry C., 2004, in Gordon T. R., Patrick B. H., eds, ASP Conf. Ser. Vol. 311, AGN Physics with the Sloan Digital Sky Survey. Astron. Soc. Pac., San Francisco, p. 49 Urry C.M., Padovani P., 1995, PASP, 107, 803 Wardle J.F.C., Aaron S.E., 1997, MNRAS, 286, 425 White R.L., et al., 2000, ApJS, 126, 133 Woo J-H., Urry M., 2002, ApJ, 581, L5 Zhang J.S., Fan J.H., 2003, ChJAA, 3, 415
\[lastpage\]
[^1]: E-mail: cygnus@oa.uj.edu.pl
[^2]: E-mail: jamrozy@oa.uj.edu.pl
[^3]: Many authors, assuming $H_0=50$ km s$^{-1}$Mpc$^{-1}$, have used 1 Mpc as the defining size for GRSs. For the currently accepted cosmological parameters as given above, this size decreases to $\sim$0.72 Mpc.
[^4]: http://www.aips.nrao.edu/
[^5]: http://iraf.noao.edu/
|
---
abstract: |
For very heavy quarks $Q$, relations derived from heavy-quark symmetry predict the existence of novel narrow doubly heavy tetraquark states of the form $Q_iQ_j \bar q_k \bar q_l$ (subscripts label flavors), where $q$ designates a light quark. By evaluating finite-mass corrections, we predict that double-beauty states composed of $bb\bar u \bar d$, $bb\bar u \bar s$, and $bb\bar d \bar s$ will be stable against strong decays, whereas the double-charm states $cc \bar q_k \bar q_l$, mixed beauty+charm states $bc \bar q_k \bar q_l$, and heavier $bb \bar q_k \bar q_l$ states will dissociate into pairs of heavy-light mesons. Observation of a new double-beauty state through its weak decays would establish the existence of tetraquarks and illuminate the role of heavy color-antitriplet diquarks as hadron constituents.
[DOI: 10.1103/PhysRevLett.119.202002](http://dx.doi.org/10.1103/PhysRevLett.119.202002)
author:
- 'Estia J. Eichten'
- Chris Quigg
bibliography:
- 'DHTQ.bib'
title: 'Heavy-Quark Symmetry Implies Stable Heavy Tetraquark Mesons $Q_iQ_j \bar q_k \bar q_l$'
---
[[^1]]{}
[[^2]]{}
Following the discovery of the charmonium-associated state $X(3872)$ by the BELLE collaboration [@Choi:2003ue], experiments have led a renaissance in hadron spectroscopy .
Many of the newly observed states invite identification with compositions less spare than the traditional quark–antiquark meson and three-quark baryon schemes . Tetraquark states composed of a heavy quark and antiquark plus a light quark and antiquark have attracted much attention. The observed candidates all fit the form $c \bar c q_k \bar q_l$, where the light quarks $q$ may be $u, d, \hbox{or } s$. No such states are observed significantly below threshold for strong decays into two heavy-light meson states $\bar c q_k + c \bar q_l$; all have strong decays to $c \bar c$ charmonium + light mesons.
In this Letter we examine the possibility of tetraquark configurations for which all strong decays are kinematically forbidden. We show that, in the heavy-quark limit, stable—hence exceedingly narrow—$Q_iQ_j \bar q_k \bar q_l$ mesons must exist. To apply this insight, we take into account corrections for finite heavy-quark masses to deduce which tetraquark states containing $b$ or $c$ quarks should be stable. The most promising example is a $J^P=1^+$ isoscalar double-$b$ meson, $\mathcal{T}^{\{bb\}-}_{[\bar u \bar d]}$.
In the heavy-quark limit, the lowest-lying tetraquark configurations resemble the helium atom, a factorized system with separate dynamics for the compact heavy color-$\mathbf{\bar 3}$ $Q_iQ_j$ “nucleus” and for the light quarks bound to the stationary color charge. (We recall that the one-gluon-exchange interaction is attractive for two quarks forming a color antitriplet, with half the strength of the attraction between a quark and antiquark bound in a color singlet.) At large $Q_i\hbox{ -- }Q_j$ separations, which become increasingly important as the heavy-quark masses decrease, the light $\bar q_k \bar q_l$ cloud screens the $Q_iQ_j$ interaction, so that the $Q_iQ_j \bar q_k \bar q_l$ complex may rearrange into a pair of heavy-light mesons . For heavy quarks $Q_iQ_j$ bound in a color $\mathbf{\bar 3}$ by an effective potential of the “Cornell” Coulomb$\,+\,$linear form at half strength for both components , the rms core radii are $\langle r^2\rangle^{1/2} = 0.28{\ensuremath{\hbox{ fm}}}\, (cc); 0.24{\ensuremath{\hbox{ fm}}}\, (bc); 0.19{\ensuremath{\hbox{ fm}}}\, (bb)$, all considerably smaller than the size of the associated tetraquark states. Hence the core-plus-light (anti)quarks idealization should be a reliable guide to the masses of ground-state tetraquarks containing charms and bottoms.
The ground state of the attractive $\mathbf{\bar 3}$ $Q_iQ_j$ configuration may have total spin $S_{Q_iQ_j} = 1$ for identical quarks ($i=j$) or for quarks of different flavors ($i \ne j$) in a symmetric flavor configuration $\{Q_iQ_j\}$ or total spin $S_{Q_iQ_j} = 0$ for quarks of different flavors ($i \ne j$) in an antisymmetric flavor configuration $[Q_iQ_j]$. To construct a color-singlet $Q_iQ_j \bar q_k \bar q_l$ state, the light $\bar q_k \bar q_l$ must be in a color-$\mathbf{3}$. For the tetraquark ground state, both the heavy $Q_iQ_j$ and light $\bar q_k \bar q_l$ pairs must be in ($\ell = 0$) $s$-waves. To satisfy the Pauli principle, the flavor-symmetric $\{\bar q_k \bar q_l\}$ state must have total (light-quark) spin $j_\ell = 1$, whereas the flavor-antisymmetric $[\bar q_k \bar q_l]$ must have $j_\ell = 0$.
*Stability in the heavy-quark limit.* For very heavy quarks, a hadron mass receives negligible contributions from the motion of the heavy quarks and spin interactions. Accordingly, the following relations hold among the masses of heavy-light and doubly-heavy-light mesons and baryons :
$$\begin{aligned}
m(\{Q_iQ_j\} \{\bar q_k \bar q_l\}) - m(\{Q_iQ_j\} q_y) &=& m(Q_x \{q_k q_l\}) - m(Q_x \bar q_y) \nonumber \\
m(\{Q_iQ_j\} [\bar q_k \bar q_l]) - m(\{Q_iQ_j\} q_y) &=& m(Q_x [q_k q_l]) - m(Q_x \bar q_y) \label{eq:hqs}\\
m([Q_iQ_j] \{\bar q_k \bar q_l\}) - m([Q_iQ_j] q_y) &=& m(Q_x \{q_k q_l\}) - m(Q_x \bar q_y) \nonumber \\
m([Q_iQ_j] [\bar q_k \bar q_l]) - m([Q_iQ_j] q_y) &=& m(Q_x [q_k q_l]) - m(Q_x \bar q_y) \;. \nonumber
\end{aligned}$$
(In the limit, a heavy core is a heavy core.)
It is easy to see that the dissociation of $Q_iQ_j \bar q_k \bar q_l$ into two heavy-light mesons is kinematically forbidden, for sufficiently heavy quarks. The $\mathcal{Q}$ value for the decay is $$\begin{array}{r}
\mathcal{Q} \equiv m(Q_i Q_j \bar q_k \bar q_l) - [m(Q_i \bar q_k) + m(Q_j \bar q_l)] = \qquad \\[3pt]
\Delta(q_k, q_l) - {{{{\textstyle \frac{1}{2}}}}}\!\left({{\textstyle \frac{2}{3}}}{\ensuremath{\alpha_{\mathrm{s}}}}\right)^2\![1 + O(v^2)]\overline M + O(1/\overline M)\;,
\end{array}
\label{eq:twomesons}$$ where $\Delta(q_k, q_l)$, the contribution due to light dynamics, becomes independent of the heavy-quark masses, $\overline M \equiv (1/{m_Q}_i + 1/{m_Q}_j)^{-1}$ is the reduced mass of $Q_i$ and $Q_j$, and [$\alpha_{\mathrm{s}}$]{} is the strong coupling. The velocity-dependent hyperfine corrections, here negligible, are calculable in the nonrelativistic QCD formalism [@Caswell:1985ui]. For large enough values of $\overline M$, the middle term dominates, so the tetraquark is stable against decay into two heavy-light mesons.
The other possible decay channel is to a doubly heavy baryon and a light antibaryon, $$(Q_iQ_j \bar q_k \bar q_l) \to (Q_iQ_j q_m) + (\bar q_k \bar q_l\bar q_m) \;.
\label{eq:baryons}$$ By [Eq. \[eq:hqs\]]{}, we have $$m(Q_iQ_j \bar q_k \bar q_l) - m(Q_iQ_j q_m) = m(Q_x q_k q_l) - m(Q_x \bar q_m) \;.
\label{eq:QvalB}$$ In the heavy-quark regime, the flavored-baryon–flavored-meson mass difference on the right-hand side of [Eq. \[eq:QvalB\]]{} has the generic form $\Delta_0 + \Delta_1/{M_Q}_x$. Using the observed mass differences, $m(\Lambda_c) - m(D) = 416.87{\ensuremath{\hbox{ MeV}}}$ and $m(\Lambda_b) - m(B) = 340.26{\ensuremath{\hbox{ MeV}}}$, and choosing effective quark masses $m_c \equiv m({\ensuremath{J\!/\!\psi}})/2 = 1.55{\ensuremath{\hbox{ GeV}}}$, $m_b \equiv m(\Upsilon)/2 = 4.73{\ensuremath{\hbox{ GeV}}}$, we find $\Delta_1 = 176.6{\ensuremath{\hbox{ MeV}}}^2$ and $\Delta_0 =303{\ensuremath{\hbox{ MeV}}}$, hence the mass difference in the heavy-quark limit is $303{\ensuremath{\hbox{ MeV}}}$. All of these mass differences are smaller than the mass of the lightest antibaryon, $m(\bar p) = 938.27{\ensuremath{\hbox{ MeV}}}$, so we conclude that no decay to a doubly heavy baryon and a light antibaryon is kinematically allowed. *This completes the demonstration that, in the heavy-quark limit, stable $Q_iQ_j \bar q_k \bar q_l$ mesons must exist.*
*Beyond the heavy-quark limit.* To ascertain whether stable tetraquark mesons might be observed, we must estimate masses of the candidate configurations. Numerous model calculations exist in the literature , but it is informative to make estimates in the spirit of heavy-quark symmetry.
The leading-order corrections for finite heavy-quark mass correspond to hyperfine spin-dependent terms and a kinetic energy shift that depends only on the light degrees of freedom, $$\delta m = \mathcal{S}\frac{\vec{S}\cdot \vec{j_\ell}}{2 {\mathcal{M}}} + \frac{\mathcal{K}}{2{\mathcal{M}}} \;,
\label{eq:dm}$$ where $\mathcal{M} = {m_Q}_i\mathrm{~or~}{m_Q}_i + {m_Q}_j$ denotes the mass of the heavy-quark core for hadrons containing one or two heavy quarks and the coefficients $\mathcal{S}$ and $\mathcal{K}$ are to be determined from experimental data summarized in Table \[tab:expmasses\]. The spin splittings lead directly to the coefficients $\mathcal{S}$ tabulated in the last column.
-------------------------------------- --------------- ------------------------------------------------ -------------------------------------------------- ------------------------- --------------------- ------------------------------
State[^3] $j_\ell$ Mass $(j_\ell+{{{{\textstyle \frac{1}{2}}}}})$ Mass $(j_\ell - {{{{\textstyle \frac{1}{2}}}}})$ Centroid Spin Splitting $\mathcal{S}\hbox{ [GeV}^2]$
\[2pt\] $D^{(*)}$ $(c\bar d)$ $\frac{1}{2}$ $2010.26$ $1869.59$ $1975.09$ $140.7$ 0.436
\[0.2mm\] $D_s^{(*)}$ $(c\bar s)$ $\frac{1}{2}$ $2112.1$ $1968.28$ $2076.15$ $143.8$ 0.446
$\Lambda_c$ $(cud)_\mathbf{\bar{3}}$ $0$ $2286.46$ $\cdots$ $\cdots$ $\cdots$
$\Sigma_c$ $(cud)_\mathbf{6}$ $1$ $2518.41$ $2453.97$ $2496.93$ $64.44$ 0.132
$\Xi_c$ $(cus)_\mathbf{\bar{3}}$ $0$ $2467.87$ $\cdots$ $\cdots$ $\cdots$
$\Xi_c^\prime$ $(cus)_\mathbf{6}$ $1$ $2645.53$ $2577.4$ $2622.82$ $68.13$ 0.141
$\Omega_c$ $(css)_\mathbf{6}$ $1$ $2765.9$ $2695.2$ $2742.33$ $70.7$ 0.146
$\Xi_{cc}$ $(ccu)_\mathbf{\bar{3}}$ $0$ $3621.40$[^4] $\cdots$ $\cdots$
\[1mm\] $B^{(*)}$ $(b\bar d)$ $\frac{1}{2}$ $5324.65$ $5279.32$ $5313.32$ $45.33$ 0.427
\[0.2mm\] $B_s^{(*)}$ $(b\bar s)$ $\frac{1}{2}$ $5415.4$ $5366.89$ $5403.3$ $48.5$ 0.459
$\Lambda_b$ $(bud)_\mathbf{\bar{3}}$ $0$ $5619.58$ $\cdots$ $\cdots$
$\Sigma_b$ $(bud)_\mathbf{6}$ $1$ $5832.1$ $5811.3$ $5825.2$ $20.8$ 0.131
$\Xi_b$ $(bds)_\mathbf{\bar{3}}$ $0$ $5794.5$ $\cdots$ $\cdots$
$\Xi_b^\prime$ $(bds)_\mathbf{6}$ $1$ $5955.33$ $5935.02$ $5948.56$ $20.31$ 0.128
$\Omega_b$ $(bss)_\mathbf{6}$ $1$ $6046.1$
\[1mm\] $B_c$ $(b\bar c)$ $\frac{1}{2}$ [6329]{}[^5] $6274.9$ [6315.4]{}$^\mathrm{c}$ [54]{}$^\mathrm{c}$ [0.340]{}$^\mathrm{c}$
\[1mm\]
-------------------------------------- --------------- ------------------------------------------------ -------------------------------------------------- ------------------------- --------------------- ------------------------------
The pattern of the spin coefficients is entirely consistent with the expectations of heavy-quark symmetry.
The kinetic energy shift due to light quarks will be different in $Q\bar{q}$ mesons and $Qqq$ baryons. By comparing the centroid (or center-of-gravity, c.g.) masses for the charm and bottom systems we can extract the difference of the kinetic-energy coefficients $\mathcal{K}$ for states that contain one or two light quarks, viz. $\delta \mathcal{K} \equiv \mathcal{K}_{(ud)} - \mathcal{K}_d$. For example, $$\begin{array}{r}
\{m[(cud)_\mathbf{\bar{3}}] - m(c\bar d)\} - \{m[(bud)_\mathbf{\bar{3}}] - m(b\bar d)\} \qquad \\[3pt]
= \delta \mathcal{K}\left(\displaystyle\frac{1}{2m_c} - \frac{1}{2m_b}\right) = 5.11{\ensuremath{\hbox{ MeV}}}\;,
\end{array}
\label{eq:finda}$$ from which we extract $\delta \mathcal{K} = 0.0235{\ensuremath{\hbox{ GeV}}}^2$. The resulting mass shifts are $$\begin{aligned}
\label{eq:KEshifts}
m[\{cc\} (\bar u\bar d)] - m(\{cc\}d)\!: & & \frac{\delta \mathcal{K}}{4m_c} = 2.80{\ensuremath{\hbox{ MeV}}}\\
m[(bc) (\bar u\bar d)] - m(\{bc\}d)\!: & & \frac{\delta \mathcal{K}}{2(m_c+m_b)} = 1.87{\ensuremath{\hbox{ MeV}}}\nonumber \\
m[\{bb\} (\bar u\bar d)] - m(\{bb\}d)\!: & \phantom{x} & \frac{\delta \mathcal{K}}{4m_b} = 1.24{\ensuremath{\hbox{ MeV}}}\nonumber\end{aligned}$$ These values are small—only slightly larger than the isospin breaking effects that we neglect as too small to affect the question of stability .
Combining the heavy-quark-symmetry relations of [Eq. \[eq:hqs\]]{} with the leading-order corrections we obtain the masses of ground-state $Q_iQ_j \bar q_k \bar q_l$ tetraquarks summarized in Table \[tab:2Q2q\] [^6]. As inputs for the doubly heavy baryons not yet experimentally measured, we use the model calculations of Karliner and Rosner .
[@ lccccccc @]{} State & $J^P$ & $j_\ell$ & $m(Q_iQ_j q_m)$ (c.g.) & HQS relation & $m(Q_iQ_j \bar q_k \bar q_l)$ & Decay Channel & $\mathcal{Q}$ \[MeV\]\
$\{cc\}[\bar u \bar d]$ & $1^+$ & $0$ & $3663$[^7] &$m(\{cc\}u) + 315$ & $3978$ & $D^+{D}^{*0}$ 3876 & $102$\
$\{cc\}[\bar q_k \bar s]$ & $1^+$ & $0$ & $3764$[^8]&$m(\{cc\}s) + 392$ & $4156$ & $D^+{D}^{*-}_s$ $3977$ & $179$\
$\{cc\}\{\bar q_k \bar q_l\}$ & $0^+,1^+,2^+$ & $1$ & $3663$ &$m(\{cc\}u) + 526$ & $4146,4167,4210$ & $D^+{D^0}, D^+{D}^{*0}$ $3734, 3876$ & $412, 292, 476$\
$[bc][\bar u \bar d]$ & $0^+$ & $0$ & $6914$ & $m([bc]u) + 315$ & $7229$ & $B^-D^+/B^0D^0$ $7146$& $83$\
$[bc][\bar q_k\bar s]$ & $0^+$ & 0 & $7010$[^9] & $m([bc]s) + 392$ & 7406 & $B_s D$ $7236$ & 170\
$[bc]\{\bar q_k \bar q_l\}$ & $1^+$ & $1$ & $6914$ & $m([bc]u) + 526$ & $7439$ & $B^*D/BD^*$ $7190/7290$ & $249$\
$\{bc\}[\bar u \bar d]$ & $1^+$ & $0$ & $6957$ & $m(\{bc\}u) + 315$ & $7272$ & $B^*D/BD^*$ $7190/7290$& $82$\
$\{bc\}[\bar q_k \bar s]$ & $1^+$ & 0 & $7053^{d}$ & $ m(\{bc\}s) + 392$ & 7445 & $ DB_s^*$ 7282 & 163\
$\{bc\}\{\bar q_k \bar q_l\}$ & $0^+,1^+,2^+$ & $1$ & $6957$ & $m(\{bc\}u) + 526$ & $7461,7472,7493$ & $BD/B^*D$ $7146/7190$ & $317,282,349$\
$\{bb\}[\bar u \bar d]$ & $1^+$ & $0$ & $10\,176$ & $m(\{bb\}u) + 306$ & $10\,482$ & $B^-\bar{B}^{*0}$ $10\,603$&\
$\{bb\}[\bar q_k \bar s]$ & $1^+$ & $0$ & $10\,252^{\mathrm{c}}$ & $m(\{bb\}s) + 391$ & $10\,643$ & $\bar{B}\bar{B}_s^*/\bar{B}_s\bar{B}^*$ $10\,695/10\,691$ &\
$\{bb\}\{\bar q_k \bar q_l\}$ & $0^+,1^+,2^+$ & $1$ & $10\,176$ & $m(\{bb\}u) + 512$ & $10\,674,10\,681,10\,695$ & $B^-{B^0},B^-{B}^{*0}$ $10\,559, 10\,603$ & $115,78, 136$\
*Narrow Tetraquark States.* As we explained in the discussion surrounding [Eq. \[eq:QvalB\]]{}, strong decays of $Q_iQ_j \bar q_k \bar q_l$ tetraquarks to a doubly heavy baryon and a light antibaryon are kinematically forbidden for all the ground states. Strong decay to a pair of heavy-light mesons will occur if the tetraquark state lies above threshold. For $J^P = 0^+\hbox{ or }2^+$, a $Q_iQ_j \bar q_k \bar q_l$ meson might decay to a pair of heavy-light pseudoscalar mesons while for $J^P=1^+$ the allowed decay channel would be a pseudoscalar plus a vector meson. According to our mass estimates, the only tetraquark mesons below threshold are the axial vector $\{bb\}[\bar u \bar d]$ meson, $\mathcal{T}^{\{bb\}-}_{[\bar u \bar d]}$, that is bound by $121{\ensuremath{\hbox{ MeV}}}$ and the axial vector $\{bb\}[\bar u \bar s]$ and $\{bb\}[\bar d \bar s]$ mesons bound by $48{\ensuremath{\hbox{ MeV}}}$. We expect all the other $Q_iQ_j \bar q_k \bar q_l$ tetraquarks to lie at least $78{\ensuremath{\hbox{ MeV}}}$ above the corresponding thresholds for strong decay . Promising final states include $\mathcal{T}^{\{bb\}-}_{[\bar u \bar d]}\! \to \Xi^0_{bc}\bar{p}$, $B^-D^+\pi^-$, and $B^-D^+\ell^-\bar{\nu}$ (which establishes a weak decay), $\mathcal{T}^{\{bb\}-}_{[\bar u \bar s]}\! \to \Xi^0_{bc}\bar{\Sigma}^-$, $\mathcal{T}^{\{bb\}0}_{[\bar d \bar s]}\! \to \Xi^0_{bc}(\bar{\Lambda},\bar{\Sigma}^0)$, and so on.
As others have noted [@Esposito:2013fma; @Luo:2017eub], unstable doubly heavy tetraquarks might be reconstructed as resonances in the “wrong-sign” combinations of $DD, DB,$ and $BB$. The doubly charged $\mathcal{T}^{\{cc\}++}_{[\bar d \bar s]} \!\to D^+ D_s^+, \hbox{ etc.}$ would stand out as *prima facie* evidence for a non-$q\bar{q}$ level.
While the production of $Q_iQ_j \bar q_k \bar q_l$ mesons is undoubtedly a rare event, we draw some encouragement for near-term searches from the large yield of $B_c$ mesons recorded in the LHC$b$ experiment and the not inconsiderable rate of Double-$\Upsilon$ production observed in 8-TeV $pp$ collisions by the CMS experiment, $\sigma(pp \to \Upsilon\Upsilon+\hbox{ anything}) = 68 \pm 15{\ensuremath{\hbox{ pb}}}$ [@Khachatryan:2016ydm]. The ultimate search instrument might be a future electron–positron Tera-$Z$ factory, for which the branching fractions [@Olive:2016xmw] $Z \to b\bar{b} =15.12 \pm 0.05\%$ and $Z \to b\bar{b}b\bar{b} = (3.6 \pm 1.3) \times 10^{-4}$ offer hope of many events containing multiple heavy quarks.
*Concluding remarks.* We have shown that, in the heavy-quark limit, stable $Q_iQ_j \bar q_k \bar q_l$ tetraquarks must exist. Our estimates of tetraquark masses lead us to expect that strong decays of the $J^P = 1^+$ $\{bb\}[\bar u \bar d]$, $\{bb\}[\bar u \bar s]$, and $\{bb\}[\bar d \bar s]$ states are kinematically forbidden, so that these states should be exceedingly narrow, decaying only through the charged-current weak interaction. Observation of any of these states would signal the existence of a new form of stable matter, in which the doubly heavy color-$\mathbf{\bar 3}$ $Q_iQ_j$ diquark is a basic building block. The unstable $Q_iQ_j \bar q_k \bar q_l$ tetraquarks—particularly those with small $\mathcal{Q}$ values—may be observable as resonances decaying into pairs of heavy-light mesons, if they are not too broad to stand out above backgrounds.
This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.
*Note added.*—We recently learned of interesting calculations of tetraquark masses that also highlight the likelihood of a stable doubly heavy tetraquark .
[^1]: [ORCID: 0000-0003-0532-2300](http://orcid.org/0000-0003-0532-2300)
[^2]: [ORCID: 0000-0002-2728-2445](http://orcid.org/0000-0002-2728-2445)
[^3]: Subscripts denote flavor-SU(3) representations for heavy baryons.
[^4]: From the LHC$b$ observation, Ref. [@Aaij:2017ueg].
[^5]: Inferred from the lattice QCD calculation of Ref. [@Dowdall:2012ab].
[^6]: Communication with decay channels tends to push the bound-state levels deeper. Open channels would induce mixing between the color-$\mathbf{\bar 3}$-core–$\mathbf{3}$-light quark configuration and meson–meson configurations.
[^7]: Based on the mass of the LHC$b$ $\Xi_{cc}^{++}$ candidate, $3621.40{\ensuremath{\hbox{ MeV}}}$, Ref. [@Aaij:2017ueg].
[^8]: Using the $s/d$ mass differences of the corresponding heavy-light mesons.
[^9]: Evaluated as ${{{{\textstyle \frac{1}{2}}}}}[m(c\bar s) - m(c\bar d) + m(b\bar s) -m(b\bar d)] + m(bcd)$.
|
---
abstract: |
For a given Lagrangian $L:[0,T]\times M\times M^\ast\rightarrow \mathbb{R}_+$ and probability measures $\mu\in\mathcal{P}(M^\ast)$, $\nu\in \mathcal{P}(M)$, we introduce the stochastic ballistic transportation problems $$\begin{aligned}
\tag{$\star$}
\underline{B}(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t,X,\beta(t,X))\,dt\right]\middle\rvert V\sim\mu,X_T\sim \nu\right\}\\\tag{$\star\star$}
\overline{B}(\nu,\mu):=\sup\left\{\mathbb{E}\left[\langle V,X_T\rangle -\int_0^T L(t,X,\beta(t,X))\,dt\right]\middle\rvert V\sim\mu,X_0\sim \nu\right\}\end{aligned}$$ where $X$ is a diffusion process with drift $\beta$. This cost is based on the stochastic optimal transport problem presented by Mikami and the deterministic ballistic transport introduced by Ghoussoub. We obtain a Kantorovich-style duality result that reformulates this problem in terms of solutions to the Hamilton-Jacobi-Bellman equation $$\frac{\partial\phi}{\partial t}+\frac{1}{2}\Delta \phi+H(t,x,\nabla\phi)=0,$$ and show how optimal processes may be thereby attained.
address: |
Mathematics Department\
University of British Columbia\
Vancouver, BC
author:
- Alistair Barton
- Nassif Ghoussoub
title: On Optimal Stochastic Ballistic Transports
---
Introduction & Related works
============================
In its modern incarnation, the problem of optimal transportation conceived by Monge [@Monge] is formulated as follows: let $X,Y$ be measure spaces with a cost function $c:X\times Y\rightarrow \mathbb{R}$. For two probability measures $\mu\in\mathcal{P}(X)$, $\nu\in\mathcal{P}(Y)$ the minimum cost of transportation between them is then $$\label{eq:OTclassic}
C(\mu,\nu)=\inf\left\{\int_{X\times Y} c(x,y)\,d\pi(x,y)\middle\rvert \pi\in\mathcal{T}(\mu,\nu)\right\},$$ where $\mathcal{T}(\mu,\nu)$ is the set of admissable *transport plans* for $(\mu,\nu)$. A transport plan $\pi$ is admissable if it is a probability measure on $X\times Y$ with first marginal $({\ensuremath{\text{proj}_{1}}})_\#\pi=\mu$ and second marginal $({\ensuremath{\text{proj}_{2}}})_\#\pi=\nu$.
A well known cost on $\mathbb{R}^n\times \mathbb{R}^n$ is the Wasserstein cost $c(x,y):=\langle x,y\rangle$ studied by Brenier [@Brenier], whose associated transportation costs are $$\begin{aligned}
\tag{$\underline{W}$}
\underline{W}(\mu,\nu):=&\inf\left\{\int_{\mathbb{R}^n\times \mathbb{R}^n} \langle x,y\rangle \,d\pi(x,y)\middle\rvert \pi\in\mathcal{T}(\mu,\nu)\right\},\\
\tag{$\overline{W}$}
\overline{W}(\mu,\nu):=&\sup\left\{\int_{\mathbb{R}^n\times \mathbb{R}^n} \langle x,y\rangle \,d\pi(x,y)\middle\rvert \pi\in\mathcal{T}(\mu,\nu)\right\}.\end{aligned}$$ These problems are clearly related by noting that $\underline{W}(\mu,\nu)=-\overline{W}({\hat{\mu}},\nu)$ where ${\hat{\mu}}$ is the reflection of $\mu$ (i.e., ${\hat{\mu}}(A):=\mu(-A)$). Probabilistically, we can rewrite these as: $$\begin{aligned}
\underline{W}(\mu,\nu):=&\inf_{\pi}\left\{{\ensuremath{\mathbb{E}_{}\left[ \langle X,Y\rangle \right]}}\middle\rvert X\sim \mu,Y\sim\nu\right\},\\
\overline{W}(\mu,\nu):=&\sup_\pi\left\{{\ensuremath{\mathbb{E}_{}\left[ \langle X,Y\rangle \right]}}\middle\rvert \rvert X\sim \mu,Y\sim\nu\right\},\end{aligned}$$ respectively. Note that, in general, $(X,Y)$ are not independent. Indeed Brenier showed that in the case where $\mu,\nu$ are absolutely continuous with respect to the Lebesgue measure then, in the optimal case for $\underline{W}$, $Y$ is completely determined by $X$ (in particular, there exists a convex function $\phi$ such that $Y=\nabla\phi(X)$) [@Brenier].
The *dual* formulation, discovered by Kantorovich [@kantorovich], allows us to equate the *primal* problem considered in \[eq:OTclassic\] to a problem on the space of functions: $$\label{eq:gendual}
C(\mu,\nu)=\sup\left\{\int_X f(x)\,d\mu(x)+\int_Y g(y)\,d\nu(y)\middle\rvert (f,g)\in\mathcal{K}(c)\right\}.$$ Here the set of *Kantorovich potentials* $\mathcal{K}(c)$ is the set of functions $(f,g)\in L^1(d\mu)\times L^1(d\nu)$ such that $f(x)+g(y)\le c(x,y)$. It is clear that the Kantorovich potentials may be assumed to satisfy $$\begin{aligned}
f(x)=&\inf_y\{c(x,y)-g(y)\}&
g(y)=&\inf_x\{c(x,y)-f(x)\}.\end{aligned}$$ For Wasserstein costs, this amounts to $f$ and $g$ being Legendre duals of each other: $g(y)=\tilde{f}(y)$ and $f(x)=\tilde{g}(x)$ for $\underline{W}$, and $g(y)=f^{\ast}(y)$ and $f(x)=g^\ast(x)$ for $\overline{W}$. Here $\tilde{f}$ (tesp., $f^*$) denote the concave (resp., convex) Legendre transform of $f$.
In this paper, we will investigate the ballistic stochastic dynamic transportation problem. This is a variant of the transportation problem developed by combining the ballistic and stochastic dynamic problems that we will briefly summarize for contextual purposes.
Bernard and Buffoni [@BernardBuffoni] introduced the idea of a dynamic cost function derived from the idea of minimizing the total action associated with a Lagrangian $L:[0,T]\times TM\rightarrow\mathbb{R}_+$ in the transportation between two measures $\nu_0,\nu_T$ on a manifold $M$ (here $TM$ is the tangent bundle on the manifold $M$). This was motivated to connect Mather theory with optimal transport. Here the cost $c_s^t(x,y)$ associated with the transportation problem denoted $C_s^t(\nu_s,\nu_t)$ is given by $$\label{eq:dyncost}
c_s^t(x,y):=\inf\left\{\int_s^t L(t,\gamma(t),\dot{\gamma}(t))\,dt\middle\rvert \gamma\in C^1([s,t],M), \gamma(s)=x, \gamma(t)=y\right\},$$ where $s<t$ and $(s,t)\in [0,T]^2$. They show [@BernardBuffoni Proposition 17] that this problem has a dual formulation akin to \[eq:gendual\], where the Kantorovich potentials may be restricted to the set $\{(u(x,T),-u(x,0))\rvert u\in HJ\}$ where $HJ$ is the set of continuous viscosity solutions of the Hamilton-Jacobi equation $$\label{eq:HJ}\tag{HJ}
{\ensuremath{\frac{\partial{u}}{\partial{t}}}}+H(t,x,\nabla u)= 0,$$ and $H(t,x,p):=\sup_v\{\langle p,v\rangle -L(t,x,v)\}$ is the hamiltonian associated with $L$. They further showed [@BernardBuffoni Theorem B] that, in the case where the measures are absolutely continuous, the optimal transportation measure can be described as the Hamiltonian flow of an initial momentum measure.
The ballistic cost was recently introduced by Ghoussoub [@Ghoussoub]. This is a cost on $M^\ast\times M$ (where $M=\mathbb{R}^n$) taking the form: $$ b_s^t(v,x):=\inf\left\{\langle v, \gamma (s)\rangle +\int_s^t L(t,\gamma(t),\dot{\gamma}(t))\,dt\middle\rvert \gamma\in C^1([s,t],M), \gamma(t)=x\right\},$$ This was used to lift the Hopf-Lax formula to Wasserstein space. Ghoussoub also showed that this cost corresponds to the dual problem $$\underline{B}_c(\mu_0,\nu_T)=\sup\left\{\int_M V(T,x)\,d\nu_T+\int_{M^*}\tilde{V}_0(v)\,d\mu_0\middle\rvert V\in HJ_c\right\}$$ where $HJ_c$ is the set of solutions to the Hamilton-Jacobi equation (\[eq:HJ\]) with initial condition $u(x,0)$ concave. Here $\tilde{V}_0(v):=\inf_{x\in M}\{\langle v,x\rangle +V_0(x)\}$ is the concave legendre transform. This result was obtained through the following interpolation result $$\underline{B}_c(\mu_0,\nu_T)=\inf\left\{\underline{W}(\mu_0,\nu_0)+C_c(\nu_0,\nu_T)\middle\rvert \nu_0\in\mathcal{P}(M)\right\}.$$ This is akin to the Hopf-Lax formula on Wasserstein space for the functional $\mathcal{U}^{\mu_0}:\rho\mapsto \underline{W}(\mu_0,\rho)$ under the Lagrangian $$\mathcal{L}(t,\rho,v)=\int_M L(t,x,v(x))\,d\rho(x)$$ where $v$ is a vector field related to $\rho$ via the continuity equation $\frac{d\rho}{dt}+{\rm div} (\rho v)=0$. (See Villani [@villani Chapter 8]).
The following stochastic variant of the action based transportation cost was considered by Mikami and Thieullen [@mikami]. $$\label{eq:stoc}
C_S^\epsilon(\nu_0,\nu_T):=\inf\left\{{\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X,\beta_X(t,X))\,dt\right]}}\middle\rvert X\in \mathcal{A}_{\nu_0}^{\nu_T}(\epsilon),X_T\sim\nu_T\right\}.$$ The set of *transportation processes* $\mathcal{A}_{\nu_0}^{\nu_T}(\epsilon)$ may be understood as the set of stochastic processes solving the stochastic differential equation $dX=\beta_X(t,X)\,dt+\epsilon dW_t$ for some $\beta_X(t,X)$, where $W_t$ is $\sigma(X_s:0\le s\le t)$-Brownian motion, initially distributed according to the measure $\nu_0$ and distributed at time $T$ according to $\nu_T$. An earlier result from Mikami [@Mikamilsc] shows that the deterministic case considered by Brenier [@Brenier], i.e., when $L(x, v)=\frac{1}{2}|v|^2$, is recovered from the corresponding stochastic transport as one takes $\epsilon\rightarrow 0$. This problem cannot be formulated in the same way as deterministic transportation problems (i.e., \[eq:OTclassic\])—indeed, the corresponding cost function could be $c(x,y)=C(\delta_x,\delta_y)=\infty$ everywhere even under reasonable assumptions on the Lagrangian. Therefore, classical Monge-Kantorovich duality need not apply here. Nevertheless, Mikami and Thieullen [@mikami Theorem 2.1] derived the following dual formulation through Legendre transforms considerations: $$C(\nu_0,\nu_T):=\sup\left\{\int_M \phi(T,x)\,d\nu_T(x)-\int_M \phi(0,x)\,d\nu_0(x)\middle\rvert \phi\in HJB\right\}.$$ Here, $HJB$ is the set of classical solutions to the following Hamilton-Jacobi-Bellman equation $$\tag{HJB}
{\ensuremath{\frac{\partial{\phi}}{\partial{t}}}}+\frac{\epsilon}{2}\Delta \phi+H(t,x,\nabla \phi)=0,$$ with final condition smooth and bounded. Furthermore for any optimal process $X$ there is a sequence of solutions $\phi_n$ to (\[eq:HJB\]) such that the SDE $$dX=\lim_{n\rightarrow\infty}\nabla_p H(t,X,\nabla\phi_n(t,x))dt+\epsilon dW_t$$is satisfied—this is the counterpart to the Hamiltonian flow for the stochastic cost.
In this paper we aim to combine the results of Ghoussoub, and Mikami and Thieullen, by considering the stochastic ballistic transportation problem: $$\underline{B}(\mu_0,\nu_T):=\inf\left\{{\ensuremath{\mathbb{E}_{}\left[\langle V,X(0)\rangle +\int_0^T L(t,X,\beta_X(t,X))\,dt\right]}}\middle\rvert V\sim\mu_0, X(\cdot)\in \mathcal{A}^{\nu_T}\right\}.$$ This may be considered in either the framework of stochastic control theory, or a stochastic version of the Hopf-Lax equation in measure space.
Assumptions & Main Results
==========================
Assumptions
-----------
We will operate on the following assumptions on the Lagrangian, that reduce to those considered by Mikami and Thieullen in [@mikami]:
1. $(t,x,v)\mapsto L(t,x,v)$ is $\mathcal{C}^3$, always positive, and has $\nabla_v^2L(t,x,v)>0$.
2. There exists $\delta>1$ such that $$\liminf_{{\ensuremath{\left\lvertu\right\rvert}}\rightarrow\infty}\frac{\inf_{t,x} L(t,x,u)}{{\ensuremath{\left\lvertu\right\rvert}}^\delta}>0.$$ (In \[prop:A1convex\] we show that this is equivalent $L(t,x,v)$ being bound below by a convex function $\ell(v)\in\Omega({\ensuremath{\left\lvertv\right\rvert}}^\delta)$).\
3. $$\tag{A2}\hspace{1mm}
\Delta L(\epsilon_1,\epsilon_2):=\sup\left\{\frac{1+L(t,x,u)}{1+L(s,y,u)}-1\middle\rvert{\ensuremath{\left\lvertt-s\right\rvert}}<\epsilon_1,{\ensuremath{\left\lvertx-y\right\rvert}}<\epsilon_2\right\}\overset{\epsilon_1,\epsilon_2\rightarrow0}{\longrightarrow} 0.$$
4. 1. $\sup_{t,x} L(t,x,0)<\infty$.
2. ${\ensuremath{\left\lvert\nabla_x L(t,x,v)\right\rvert}}/(1+L(t,x,v))$ is bounded.
3. $\sup\left\{{\ensuremath{\left\lvert\nabla_v L(t,x,u)\right\rvert}}: {\ensuremath{\left\lvertu\right\rvert}}\le R\right\}<\infty$ for all $R$.
5. \(i) $\Delta L(0,\infty)<\infty$ or (ii) $\delta =2$ in (A1).
We give a brief summary of the role of the above assumptions:\
(A1) is a coercivity result that is necessary for sequential compactness of minimizing transportation processes (\[prop:Ccoerc\]).\
(A2) is used to show that the expected action of a transportation process is lower semi-continuous [@Mikamilsc eqs. (3.17),(3.38)-(3.41)].\
(A3) is used with (A0) to derive the Hamilton-Jacobi-Bellman equation in \[prop:HJB\] [@HJB p. 210, Remark 11.2].\
(A4,i) allows us to uniformly bound the ratio $\frac{1+L(t,x,u)}{1+L(t,y,u)}$, while (A4,ii) ensures the minimizing $\beta_X$ satisfies $\int_0^T {\ensuremath{\left\lvert\beta_X(t,X)\right\rvert}}^2\,dt<\infty$ a.s., permitting us to assume the process $X(\cdot)$ is absolutely continuous with respect to $W(\cdot)$ [@LipsterStatofRP Theorem 7.16]. Either of these results with (A0)-(A3) is sufficient to show convexity of $(\nu_0,\nu_T)\mapsto C(\nu_0,\nu_T)$ [@mikami Lemma 3.2].
\[prop:A1convex\] (A1) is equivalent to $L(t,x,v)$ being bound below by a convex function $\ell(v)$ such that $$\liminf_{{\ensuremath{\left\lvertv\right\rvert}}\rightarrow\infty} \frac{\ell(v)}{{\ensuremath{\left\lvertv\right\rvert}}^\delta}>0.$$ Indeed, if $L$ is bound by such an $\ell$, it is simple to see (A1) holds. We prove the reverse direction by construction. By (A1), we may assume that there exists a $U\in\mathbb{R}$ such that for all ${\ensuremath{\left\lvertu\right\rvert}}>U$ $$\frac{\inf_{t,x} L(t,x,u)}{{\ensuremath{\left\lvertu\right\rvert}}^\delta}>\alpha,\qquad
(\alpha>0,\delta>1),$$ defining $\alpha,\delta$. Then the convex function $\ell:M^\ast\rightarrow \mathbb{R}$ defined by $$\ell(v):=\begin{cases}0& {\ensuremath{\left\lvertv\right\rvert}}<U\\
\alpha{\ensuremath{\left\lvert{\ensuremath{\left\lvertv\right\rvert}}-U\right\rvert}}^\delta& {\ensuremath{\left\lvertv\right\rvert}}\ge U
\end{cases}$$ is a lower bound on $L(t,x,v)$ and is asymptotically bounded below by ${\ensuremath{\left\lvertv\right\rvert}}^\delta$.
Notation
--------
The space of probability measures on a space $X$ will be indicated by $\mathcal{P}(X)$, while the subset of measures with finite barycenter will be denoted $\mathcal{P}_1(X):=\{\mu\in\mathcal{P}(X): \int {\ensuremath{\left\lvertx\right\rvert}}\,d\mu<\infty\}$. We will work on the space $M:=\mathbb{R}^d$ however we will still refer to the dual space $M^\ast$ explicitly, for clarity. Measures in $\mathcal{P}(M)$ will be designated by $\nu$, while $\mu$ will designate measures in $\mathcal{P}(M^\ast)$.
Given a Lagrangian $L:[0,T]\times M\times M^\ast\rightarrow\mathbb{R}$ satisfying properties (A), we define the (stochastic) dynamic and ballistic variants of the optimal transportation problem mentioned in the introduction: $$\begin{aligned}
C(\nu_0,\nu_T):=&\inf\left\{ {\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X(t),\beta_X(t,X(t)))\,dt\right]}}\middle\rvert X(\cdot)\in \mathcal{A}_{\nu_0}^{\nu_T}\right\},\\
{\underline{B}}(\mu_0,\nu_T):=&\inf\left\{{\ensuremath{\mathbb{E}_{}\left[\langle Y,X(0)\rangle+ \int_0^T L(t,X(t),\beta_X(t,X(t)))\,dt\right]}}\middle\rvert Y\sim \mu_0,X(\cdot)\in \mathcal{A}^{\nu_T}\right\}.\end{aligned}$$ Here $\mathcal{A}_{\nu_0}^{\nu_T}$ refers to the set of $\mathbb{R}^d$-valued continuous semimartingales $X(\cdot)$—initially distributed according to $\nu_0$ and finally distributed according to $\nu_T$—such that there exists a measurable drift $\beta_X:[0,T]\times C([0,T])\rightarrow M^\ast$ where
1. $\omega\mapsto\beta_X(t,\omega)$ is $\mathcal{B}(C([0,t]))_+$-measurable for all $t$.
2. $W(t):=X(t)-X(0)-\int_0^t \beta_X(s,X)\,ds$ is a $\sigma[X(s):s\in[0,t]]$-Brownian motion.
$\mathcal{A}^{\nu_T}$ refers to the same processes, except with initial distribution unconstrained, likewise $\mathcal{A}_{\nu_0}$ with the final distribution unconstrained and $\mathcal{A}$ with initial and final distribution unconstrained.
For convenience, we define the expected action of stochastic processes $X(\cdot)\in\mathcal{A}$: $$\mathscr{A}(X):={\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X(t),\beta_X(t,X(t)))\,dt\right]}}.$$ We will also consider another, analogous, ballistic stochastic cost defined by: $$\overline{B}(\nu_0,\mu_T):=\sup\left\{{\ensuremath{\mathbb{E}_{}\left[\langle V,X(T)\rangle- \int_0^T L(t,X(t),\beta_X(t,X(t)))\,dt\right]}}\middle\rvert V\sim\mu_T,X(\cdot)\in \mathcal{A}_{\nu_0}\right\},$$ which we will refer to as the *maximizing ballistic cost*—in comparison, $\underline{B}$ may be referred to as the *minimizing ballistic cost*.
We recall the convex and concave Legendre transforms, which we will denote $$\begin{aligned}
f^\ast(v):=&\sup_{x\in M}\{\langle v,x\rangle -f(x)\},\\
\tilde{f}(v):=&\inf_{x\in M}\{\langle v,x\rangle-f(x)\}=-(-f)^\ast(-v),\end{aligned}$$ respectively, where $v$ is in the dual space $M^\ast$. This notation will also be used for the legendre duals of functions of measures, such as $C_{\nu_0}:\nu\mapsto C(\nu_0,\nu)$ where the dual space is a linear subspace of continuous functions.
A set of functions that is of particular importance in this regard is the set of smooth functions with bound derivatives—notably, this may be realized as the set of mollified Lipschitz functions—which we will denote ${C^\infty_\text{db}}$, and the subset of convex such functions ${c}{C^\infty_\text{db}}$.
Main Results
------------
We present an interpolation result that allows us to interpret the ballistic cost in terms of the Wasserstein cost and the stochastic action cost.
If $L$ satisfies the assumptions (A), then our stochastic ballistic cost may be written as: $${\underline{B}}(\mu_0,\nu_T)=\inf_\nu\{\underline{W}(\mu_0,\nu)+C(\nu,\nu_T)\}.$$ This minimum is attained in the case where $\mu_0\in\mathcal{P}_1(M^\ast)$ and $\nu_T\in\mathcal{P}_1(M)$.
This leads to our main result, the dual of the stochastic ballistic cost:
If $\nu_T\in\mathcal{P}_1(M)$ and $\mu_0$ with compact support are such that ${\underline{B}}(\mu_0,\nu_T)<\infty$, and the Lagrangian satisfies (A0)-(A4) then we have $${\underline{B}}(\mu_0,\nu_T)=\sup_{-f\in {c}{\text{Lip}(M)}}\left\{\int_M f(x)\, d\nu_T(x)+\int_{M^*} \widetilde{\phi^f}(0,v)\,d\mu_0(v)\right\},$$ where $\widetilde{g}$ denotes the concave legendre transform of $g$ and $\phi^{f}$ is the solution to the Hamilton-Jacobi-Bellman equation $$\begin{aligned}
\label{eq:HJB}\tag{HJB}
{\ensuremath{\frac{\partial{\phi}}{\partial{t}}}}+\frac{1}{2}\Delta\phi(t,x)+H(t,x,\nabla\phi)=&0, & \phi(T,x)=&f(x).\end{aligned}$$
This formulation of the problem allows us to write optimal processes in terms of a maximizing sequence of $\phi$:
Under the assumptions of Theorem 2, with $d\mu_0\ll d\lambda$, we have that $(V,X(t))$ is a minimizing process for $\underline{B}(\mu_0,\nu_T)$ if and only if it is a solution to the SDE $$\begin{aligned}
dX =& \nabla_p\lim_{n\rightarrow\infty} H(t,X,\nabla\phi_n(t,X))\, dt + dW_t\\
X(0) =& \nabla\lim_{n\rightarrow\infty}{\ensuremath{\left(\widetilde{\phi_n^0}\right)}}(V),\end{aligned}$$ where $\phi_n$ is a sequence of solutions to (\[eq:HJB\]) that is maximizing for the dual problem. (Here we use the notation $\phi^t_n(x):=\phi_n(t,x)$).
### Maximizing Cost
We obtain parallel results for the maximizing cost $\overline{B}$:
If $L$ matches assumptions (A), then $$\overline{B}(\nu_0,\mu_T)=\sup\{\overline{W}(\nu,\mu_T)-C(\nu_0,\nu)\},$$ where, again, the maximum is attained when $\nu_0\in\mathcal{P}_1(M)$ and $\mu_T\in\mathcal{P}_1(M^\ast)$.
Assume the Lagrangian satisfies the assumptions (A). If $\nu_0\in\mathcal{P}_1(M)$, $\mu_T$ has compact support, and $\overline{B}(\nu_0,\mu_T)<\infty$, then $$\overline{B}(\nu_0,\mu_T)=\inf_{g\in{c}{C^\infty_\text{db}}}\left\{\int_{M^*} g^\ast\,d\mu_T+\int_M\phi^{g}\,d\nu_0\right\},$$ where $\phi^g$ solves the Hamilton-Jacobi-Bellman equation $$\begin{aligned}
\tag{HJB2}
{\ensuremath{\frac{\partial{\phi}}{\partial{t}}}}+\frac{1}{2}\Delta \phi -H(t,x,\nabla\phi)=&0&\phi(x,T)=g(x)\end{aligned}$$
Once more we may identify optimal processes by this formulation:
If the assumptions of \[thm:Boverdual\] are satisfied with $d\mu_T\ll d\lambda$, then $(V,X(t))$ is a maximizing process for $\overline{B}(\nu_0,\mu_T)$ iff it is a solution to the SDE $$\begin{aligned}
dX=&\lim_{n\rightarrow\infty}\nabla_p H(t,x,\nabla\phi_n(t,X))\,dt+dW_t\\
V=&\lim_{n\rightarrow\infty}\nabla\phi_n^T(X(T)), \end{aligned}$$ for a sequence of solutions $\phi_n$ to (\[eq:HJB2\]) that is minimizing for the dual problem.
Proofs
======
We first show a coercivity result that allows us to explore the structure of the problem and some of the consequences of our assumptions. This proposition is useful for showing existence of minimizers to our interpolation formula in \[thm:interpol\].
\[prop:Ccoerc\] For any fixed $\nu_T\in\mathcal{P}_1(M)$, $N\in\mathbb{R}$, the set of measures $\nu\in \mathcal{P}_1(M)$ satisfying $$\label{eq:tight}
C(\nu,\nu_T)\le N\int {\ensuremath{\left\lvertx\right\rvert}}\,d\nu(x)$$is tight. In other words, for every $\epsilon>0$ there exists an $R$ such that if $\nu\in \mathcal{P}_1(M)$ satsifies \[eq:tight\]), then $\nu({\ensuremath{B(R,0)}}^c)\leq \epsilon$.
First we define the set of measures $\mathcal{T}_{\epsilon,R}:=\{\nu\in\mathcal{P}_1(M):\nu({\ensuremath{B(R,0)}}^c)>\epsilon\}$. We will assume $\nu\in\mathcal{T}_{\epsilon,R}$ for what follows.
We begin by defining $R_1$ to be such that $\nu_T({\ensuremath{B(R_1,0)}}^c)<\epsilon$. Let $X\in\mathcal{A}$ be a stochastic process with $X(0)\sim \nu$ and $X(T)\sim \nu_T$. We leave $R$ to be defined later, but note that if we define the set $\Omega_R:=\{ {\ensuremath{\left\lvertX(0)\right\rvert}}>R\}$, then our assumption on $\nu$ yields $\mathbb{P}(\Omega_R)>\epsilon$. By positivity of $L$, this allows us to say that $\mathscr{A}(X)\ge \mathscr{A}(1_{\Omega_R}X)$ (henceforth we define the process $Y(t):=1_{\Omega_R}(\omega)X(t)$). Furthermore we can assume the probability of a path originating at a distance beyond $R$ from the origin and terminating within distance $R_1$ from the origin is $$\label{eq:crossbound}\begin{split}
{\ensuremath{\mathbb{P}(\{{\ensuremath{\left\lvertX(0)\right\rvert}}>R\}\cap\{{\ensuremath{\left\lvertX(T)\right\rvert}}\le R_1\})}}\ge &1-{\ensuremath{\mathbb{P}({\ensuremath{\left\lvertX(0)\right\rvert}}\le R)}}-{\ensuremath{\mathbb{P}({\ensuremath{\left\lvertX(T)\right\rvert}}> R_1)}}\\
=& {\ensuremath{\mathbb{P}({\ensuremath{\left\lvertX(0)\right\rvert}}>R)}}-{\ensuremath{\mathbb{P}({\ensuremath{\left\lvertX(T)\right\rvert}}> R_1)}}\\
=&\nu({\ensuremath{B(R,0)}}^c)-\nu_T({\ensuremath{B(R_1,0)}}^c)> \epsilon/2.
\end{split}$$
Applying (A1) via our remark in \[prop:A1convex\], we assume that there is a convex function $\ell:M^\ast\rightarrow\mathbb{R}$ such that for all ${\ensuremath{\left\lvertu\right\rvert}}>U$ $$\label{eq:A1coerc}
\frac{\ell(u)}{{\ensuremath{\left\lvertu\right\rvert}}^\delta}>\alpha,\qquad
(\alpha>0,\delta>1),$$ defining $\alpha,\delta$. Recall that $\ell({\ensuremath{\left\lvertv\right\rvert}})$ is a lower bound on $L(t,x,v)$.
Examining the process $Y$, we can take advantage of the convexity of $\ell$ to apply Jensen’s inequality: $$\begin{split}\label{eq:Jensencoerc}
{\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,Y,\beta_Y(t,Y))\,dt\right]}}\ge\,& {\ensuremath{\mathbb{E}_{}\left[\int_0^T \ell({\ensuremath{\left\lvert\beta_Y(t,Y)\right\rvert}})\,dt\right]}}\overset{(\text{J})}{\ge}{\ensuremath{\mathbb{E}_{}\left[\ell({\ensuremath{\left\lvertV\right\rvert}})T\right]}}\\
\overset{(\ref{eq:A1coerc})}{>}&\alpha T{\ensuremath{\mathbb{E}_{}\left[1_{{\ensuremath{\left\lvertV\right\rvert}}>U}{\ensuremath{\left\lvertV\right\rvert}}^\delta\right]}}\ge\alpha T{\ensuremath{\left({\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left\lvertV\right\rvert}}^\delta\right]}}-U^\delta\right)}},
\end{split}$$ where $\beta_Y:=1_{\Omega_R}\beta_X$ is the drift associated with the process $Y$ and $V:=(Y(T)-Y(0))/T$ is its time-average.
Putting this all together, we get $$\label{eq:Coerctogether}\begin{split}
\mathscr{A}(X)\ge& \mathscr{A}(Y)\overset{(\ref{eq:Jensencoerc})}{>} \alpha T{\ensuremath{\left({\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left\lvertV\right\rvert}}^\delta\right]}}-U^\delta\right)}}=\alpha T{\ensuremath{\left({\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left\lvert\frac{Y(T)-Y(0)}{T}\right\rvert}}^\delta \right]}}-U^\delta\right)}}\\
\overset{(\triangle)}{>}&\frac{\alpha}{T^{\delta-1}}{\ensuremath{\left({\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left\lvert{\ensuremath{\left\lvertY(0)\right\rvert}}-{\ensuremath{\left\lvertY(T)\right\rvert}}\right\rvert}}^\delta\right]}}-{\ensuremath{\left(UT\right)}}^\delta\right)}}
\end{split}$$ Minimizing the expectation over $(Y(0),Y(T))$ is a type of well known optimal transport problem, with the optimal plan given by the monotone Hoeffding-Frechet mapping $x\mapsto G_{{\ensuremath{\left\lvertY(0)\right\rvert}}}(G_{{\ensuremath{\left\lvertY(T)\right\rvert}}}^{-1}(x))$, where $G_Z(t):=\inf\{x\in\mathbb{R}:t\ge{\ensuremath{\mathbb{P}(Z\le x)}}\}$ is the quantile function associated with the random variable $Z$ [@beiglbock Theorem 1.1]. Thus the optimal plan maps each quantile in one measure to the corresponding quantile in the other.
Focussing on the expectation in \[eq:Coerctogether\], we get $$\begin{split}\label{eq:finalcoerce}
{\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left\lvert{\ensuremath{\left\lvertY(0)\right\rvert}}-{\ensuremath{\left\lvertY(T)\right\rvert}}\right\rvert}}^\delta\right]}}\ge &\int{\ensuremath{\left\lvertx-y\right\rvert}}^\delta\,d{\ensuremath{\left({\ensuremath{\left(G_{{\ensuremath{\left\lvertY(0)\right\rvert}}}\times G_{{\ensuremath{\left\lvertY(T)\right\rvert}}} \right)}}_\#\lambda_{[0,1]}\right)}}(x,y)\\
\overset{(\text{J})}{\ge}& {\ensuremath{\left\lvert\int x\,d{\ensuremath{\left(G_{{\ensuremath{\left\lvertY(0)\right\rvert}}}\right)}}_\#\lambda_{[0,1]}(x)-b\right\rvert}}^\delta\\
=& {\ensuremath{\left(\int_{{\ensuremath{B(R,0)}}^c}{\ensuremath{\left\lvertx\right\rvert}}\,d\nu(x)-b\right)}}^\delta,
\end{split}$$ where $b:=\int{\ensuremath{\left\lvertx\right\rvert}}\,d\nu_T/\epsilon>{\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left\lvertY(T)\right\rvert}}\right]}}$ and we require that $R>b/\epsilon$. We thus want to find $R$ such that $$\frac{\alpha}{T^{\delta-1}} {\ensuremath{\left({\ensuremath{\left(\int_{{\ensuremath{B(R,0)}}^c}{\ensuremath{\left\lvertx\right\rvert}}\,d\nu(x)-b\right)}}^\delta-{\ensuremath{\left(UT\right)}}^\delta\right)}}>N{\ensuremath{\left(\int_{{\ensuremath{B(R,0)}}^c}{\ensuremath{\left\lvertx\right\rvert}}\,d\nu(x)+R(1-\epsilon)\right)}}\ge N\int{\ensuremath{\left\lvertx\right\rvert}}\,d\nu(x)$$ Letting $I_\nu(R):=\int_{{\ensuremath{B(R,0)}}}{\ensuremath{\left\lvertx\right\rvert}}\,d\nu(x)$ (which is at least $R\epsilon$ for all $\nu\in\mathcal{T}_{\epsilon,R}$), we find the condition $$\frac{\alpha}{T^{\delta-1}} {\ensuremath{\left({\ensuremath{\left(I_\nu(R)^{1-\frac{1}{\delta}}-bI_\nu(R)^{-\frac{1}{\delta}}\right)}}^\delta-\frac{{\ensuremath{\left(UT\right)}}^\delta}{I_\nu(R)}\right)}}>N{\ensuremath{\left(1+\frac{R(1-\epsilon)}{I_\nu(R)}\right)}},$$ which by using the mentioned bound on $I_\nu(R)$ is satisfied if $R$ satisfies $$\frac{\alpha}{T^{\delta-1}} {\ensuremath{\left({\ensuremath{\left((R\epsilon)^{1-\frac{1}{\delta}}-b(R\epsilon)^{-\frac{1}{\delta}}\right)}}^\delta-\frac{{\ensuremath{\left(UT\right)}}^\delta}{R\epsilon}\right)}}>\frac{N}{\epsilon}.$$ Taking $R\rightarrow\infty$ satisfies this inequality, implying this set is non-empty and intersects with our earlier constraints that $R>R_1+UT$ and $R>b/\epsilon$.
The same argument holds if we add any fixed constant to the right side of \[eq:tight\].
Furthermore, \[eq:finalcoerce\] can be used to show that if $\nu_1\in\mathcal{P}_1(M)$ and $\nu_0\in\mathcal{P}(M)\setminus\mathcal{P}_1(M)$, then $C(\nu_0,\nu_1)=C(\nu_1,\nu_0)=\infty$. This leads to our focus on distributions with finite first moment in the following.
We now state some properties of $C$ shown by Mikami:
\[prop:Cdualsc\]([@mikami Theorem 2.1]) If the Lagrangian satisfies (A1)-(A4), then $$C(\nu_0,\nu_T)=\sup\left\{\int_M f(x)\,d\nu_T-\int_M \phi^f(0,x)\,d\nu_0\middle\rvert f\in\mathcal{C}_b^\infty\right\},$$ where $\phi_f$ solves (\[eq:HJB\]). Furthermore, $(\mu,\nu)\mapsto C(\mu,\nu)$ is convex and lsc under the weak topology.
Minimizing Ballistic Cost
-------------------------
Now that we have demonstrated the coercivity and lower semicontinuity of $C$, we can move on to investigating the minimizing ballistic cost: $\underline{B}$. These previous results are sufficient to show the existence of a minimizer to the interpolation:
\[thm:interpol\] If $L$ satisfies the assumptions (A), then our stochastic ballistic cost may be written $$\label{eq:interpol}
{\underline{B}}(\mu_0,\nu_T)=\inf_\nu\{\underline{W}(\mu_0,\nu)+C(\nu,\nu_T)\}.$$ Furthermore, this minimum is attained in the case where $\mu_0\in\mathcal{P}_1(M^\ast)$ and $\nu_T\in\mathcal{P}_1(M)$.
First, to show that $\underline{B}(\mu_0,\nu_T)$ is greater or equal to its interpolations, consider $(Y_n,X_n(\cdot))$ such that $${\ensuremath{\mathbb{E}_{}\left[\langle Y_n,X_n(0)\rangle +\int_0^T L(t,X_n(t),\beta_{X_n}(X_n,t))\,dt\right]}}< {\underline{B}}(\mu_0,\nu_T)+\tfrac{1}{n}.$$ Let $\nu_n:=PX_n(0)^{-1}$, then $$\begin{aligned}
\underline{W}(\mu_0,\nu_n)\le&{\ensuremath{\mathbb{E}_{}\left[\langle Y_n,X_n(0)\rangle\right]}}, \\
{\ensuremath{C}}(\nu_n,\nu_T)\le&{\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X_n(t),\beta_{X_n}(X_n,t))\,dt\right]}},\end{aligned}$$ so $$\inf_\nu\left\{\underline{W}(\mu_0,\nu)+{\ensuremath{C}}(\nu,\nu_T)\right\}\le\underline{W}(\mu_0,\nu_n)+{\ensuremath{C}}(\nu_n,\nu_T)< {\underline{B}}(\mu_0,\nu_T)+\tfrac{1}{n}.$$ Taking $n\rightarrow\infty$ gives the inequality.
To get the reverse inequality, take $\nu_n$ to be a sequence such that $$\underline{W}(\mu_0,\nu_n)+{\ensuremath{C}}(\nu_n,\nu_T)<\inf_\nu\left\{\underline{W}(\mu_0,\nu)+{\ensuremath{C}}(\nu,\nu_T)\right\}+\tfrac{1}{n}.$$ We may then define an admissible stochastic process $Z_n(\cdot)\in\mathcal{A}_{\nu_n}^{\nu_T}$ such that $${\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,Z_n(t),\beta_{Z_n}(Z_n,t))\,dt\right]}}< {\ensuremath{C}}(\nu_n,\nu_T)+\tfrac{1}{n}.$$ Then, let $W_n(\omega)\sim \gamma_{Z_n(0)(\omega)}^n$, where $d\gamma_x^n(y)\otimes d\mu_0(x)=d\gamma_n(x,y)$ is the disintegration of a measure $\gamma_n$ such that $$\int \langle y,x\rangle \,d\gamma_n(x,y)<\underline{W}(\mu_0,\nu_n)+\tfrac{1}{n}.$$ Thus $$\begin{split}
{\underline{B}}(\mu_0,\nu_T)\le&\langle W_n,Z_n\rangle+\int_0^T L(t,Z_n(t),\beta_{Z_n}(Z_n,t))\,dt<\underline{W}(\mu_0,\nu_n)+{\ensuremath{C}}(\nu_n,\nu_T)+\tfrac{2}{n}\\
<&\inf_\nu\left\{\underline{W}(\mu_0,\nu)+{\ensuremath{C}}(\nu,\nu_T)\right\}+\tfrac{3}{n}.
\end{split}$$ Lastly, to show this minimum is achieved in the case where $\mu_0$ and $\nu_T$ have finite mean, we must show that the minimizing sequence $\nu_n$ is sequentially compact in the weak topology. To do this, we note that the set of measures $\nu$ such that $$C(\nu,\nu_T)<N\int{\ensuremath{\left\lverty\right\rvert}}\,d\nu(y)+{\underline{B}}(\mu_0,\nu_T)+1$$ is tight by \[prop:Ccoerc\]. If we let $N:=\int {\ensuremath{\left\lvertx\right\rvert}}\,d\mu_0(x)$, then the collection of measures such that $$\begin{split}
{\underline{B}}(\mu_0,\nu_T)+1>&C(\nu,\nu_T)+\underline{W}(\mu_0,\nu)\\
>&C(\nu,\nu_T)-\int{\ensuremath{\left\lvertx\right\rvert}}{\ensuremath{\left\lverty\right\rvert}}\,d\mu_0(x)\,d\nu(y)\\
\overset{(\text{F})}{=}&C(\nu,\nu_T)-N\int{\ensuremath{\left\lverty\right\rvert}}\,d\nu(y),
\end{split}$$ is tight, where (F) is application of Fubini’s theorem. Thus, by Prokhorov’s theorem our minimizing sequence of interpolating measures necessarily weakly converges to a minimizing measure.
The attainment of a minimizing $\nu_0$ of \[eq:interpol\] is sufficient to show that of a minimizer of $\underline{B}$ when it is finite, since the minimum is achieved for both $\underline{W}$ and $C$ [@mikami Proposition 2.1].
\[cor:Blsc\] If the Lagrangian satisfies (A), then:
1. $\nu\mapsto{\underline{B}}(\mu_0,\nu)$ is lower semi-continuous on $\mathcal{P}_1(M)$ for all $\mu_0\in\mathcal{P}_1(M^\ast)$.
2. $(\mu_0,\nu_T)\mapsto {\underline{B}}(\mu_0,\nu_T)$ is jointly convex.
Consider a sequence $\nu_{T,n}\in\mathcal{P}_1(M)$ that converges weakly to $\nu_T\in\mathcal{P}_1(M)$. We have $${\underline{B}}(\mu_0,\nu_{T,n})=\underline{W}(\mu_0,\nu_n)+C(\nu_n,\nu_{T,n}).$$ for a sequence of measures $\nu_n\in\mathcal{P}_1(M)$ by \[thm:interpol\]. Since $C$ is lsc (\[prop:Cdualsc\]), we get $$\liminf_{n\rightarrow\infty} \underline{W}(\mu_0,\nu_n)+C(\nu_n,\nu_{T,n})\ge \liminf_{n\rightarrow\infty} \underline{W}(\mu_0,\nu_n)+C(\nu_n,\nu_{T})\ge {\underline{B}}(\mu_0,\nu_T).$$
\[lem:convexlemma\] If $(x,y)\mapsto h(x,y)$ jointly convex, then $\bar{h}(x):=\inf_y h(x,y)$ is convex.
Consider $x_1,x_2\in X$. Then there exist sequences $y^n_1,y^n_2\in Y$ such that $$\lim_n h(x_i,y_i^n)=\bar{h}(x_i).$$ Now take $x_\lambda:=\lambda x_1+(1-\lambda)x_2$, and $y^n_\lambda:=\lambda y_1^n+(1-\lambda)y_2^n$, then $$\bar{h}(x_\lambda)\le h(x_\lambda,y_\lambda^n)\le \lambda h(x_1,y_1^n)+ (1-\lambda) h(x_2,y_2^n)\overset{n\rightarrow\infty}{\longrightarrow} \lambda \bar{h}(x_1)+ (1-\lambda) \bar{h}(x_2)$$
Define the function $$\label{eq:convexinterpol}
h((\mu_0,\nu_T),\nu):=\underline{W}(\mu_0,\nu)+{\ensuremath{C}}(\nu,\nu_T).$$ Then by \[thm:interpol\] $$\label{eq:convexlemma}
{\underline{B}}(\mu_0,\nu_T)=\inf_\nu \left\{h((\mu_0,\nu_T),\nu)\right\}.$$ But $h((\mu_0,\nu_T),\nu)$ is jointly convex—as $(\mu_0,\nu)\mapsto\underline{W}(\mu_0,\nu)$ and $(\nu,\nu_T)\mapsto {\ensuremath{C}}(\nu,\nu_T)$ are both jointly convex (\[prop:Cdualsc\]). Hence applying \[lem:convexlemma\] gives the joint convexity of \[eq:convexlemma\].
Now that we have shown that $\nu_T\mapsto \underline{B}(\mu_0,\nu_T)$ is convex and lsc, we are able apply convex duality to get our dual formulation.
\[thm:stocbdual\] If $\nu_T\in\mathcal{P}_1(M^\ast)$ and $\mu_0$ with compact support are such that ${\underline{B}}(\mu_0,\nu_T)<\infty$, and the Lagrangian satisfies (A0)-(A4), then we have $$\label{eq:mindual}
{\underline{B}}(\mu_0,\nu_T)=\sup_{-f\in {c}{\text{Lip}(M)}}\left\{\int f(x)\, d\nu_T(x)+\int \widetilde{\phi^f}(0,v)\,d\mu_0(v)\right\},$$ where $\widetilde{g}$ is the concave legendre transform of $g$ and $\phi^{f}$ is the solution to the Hamilton-Jacobi-Bellman equation $$\begin{aligned}
\label{eq:HJB}\tag{HJB}
{\ensuremath{\frac{\partial{\phi}}{\partial{t}}}}+\frac{1}{2}\Delta\phi(t,x)+H(t,x,\nabla\phi)=&0, & \phi(T,x)=&f(x).\end{aligned}$$
\[lem:Wlegendre\] For $\mu\in\mathcal{P}(M)$ with compact support, define $\underline{W}_\mu:\mathcal{M}_1(M^\ast)\rightarrow \mathbb{R}$, (where $\mathcal{M}_1(M^\ast)$ is the linear space of measures $\nu$ on $M^\ast$ such that $\int_{M^*} {\ensuremath{\left(1+{\ensuremath{\left\lvertx\right\rvert}}\right)}}\,d\mu<\infty$) to be $$\underline{W}_{\mu}(\nu):=\begin{cases} \underline{W}(\mu,\nu)&\nu\in\mathcal{P}_1(M)\\
\infty&\text{otherwise}.
\end{cases}$$Then, for $f\in {\text{Lip}(M)}$, the convex Legendre transform of $\underline{W}_\mu$ is given by, $$\underline{W}_{\mu}^\ast(f)=-\int \widetilde{f}\,d\mu.$$
The Lipschitz condition is a consequence of the dual space of $\mathcal{M}_1(M)$ being the space of Lipschitz functions ${\text{Lip}(M)}$ [@lipschalg].
It is well known that $\underline{W}_\mu$ is convex and lsc (e.g. via the dual formulation, \[eq:gendual\]). Thus, by convex analysis, $\underline{W}_\mu=\underline{W}_\mu^{\ast\ast}$, giving us: $$\underline{W}_\mu(\nu)=\sup_{f\in{\text{Lip}(M)}}\left\{\int f\,d\nu -\underline{W}_\mu^\ast(f)\right\}.$$ But Kantorovich duality (when $\mu$ has compact support) also gives us $$\underline{W}_\mu(\nu)=\sup_{f\in{\text{Lip}(M)}}\left\{\int f\,d\nu +\int \widetilde{f}\,d\mu\right\}=F_\mu^\ast(\nu),$$ where $F_\mu:f\mapsto -\int \widetilde{f} \,d\mu=\int (-f)^\ast \,d\hat{\mu}(v)$ (recall $d\hat{\mu}(v)=d\mu(-v)$). To show that $\underline{W}_\mu^\ast(f)=F_\mu(f)$ it suffices to show that $f\mapsto F_\mu(f)$ is convex and lower semicontinuous, in which case $\underline{W}_\mu^\ast(\nu)=F_\mu^{\ast\ast}(f)=F_\mu(f)$. We will do this by showing $g\mapsto g^\ast(v)$ is a convex mapping for all $v\in M^\ast$: Fix $\lambda\in[0,1]$ and $v\in M^\ast$ arbitrary. $$\begin{split}
(\lambda g_1+(1-\lambda)g_2)^\ast (v)=&\sup_x \{\langle x,v\rangle -\lambda g_1(x)-(1-\lambda)g_2(x)\}\\
\overset{(1)}{\le} &\sup_x \{\langle x,v\rangle -\lambda {\ensuremath{\left(\langle x,v\rangle -g^\ast_1(v)\right)}}-(1-\lambda){\ensuremath{\left(\langle x,v\rangle -g^\ast_2(v)\right)}}\}\\
=& \lambda g_1^\ast(v)+(1-\lambda)g_2^\ast(v),
\end{split}$$ where we use $g(x)\ge \langle x,v'\rangle-g^\ast (v')$ for all $v'\in M^\ast$ in (1)—taking the liberty to let $v'=v$.
To show $F_\mu$ is weakly lsc, consider $f_n$ that converge pointwise to $f$ (a weaker condition than weak convergence). Then $$\begin{split}
\liminf_{n\rightarrow\infty} F_\mu(f_n)=&\liminf_{n\rightarrow\infty} \int (-f_n)^\ast\,d\hat{\mu}\overset{\text{(F)}}{\ge}\int \liminf_{n\rightarrow\infty} \sup_x \{\langle v,x\rangle +f_n(x)\}\,d\hat{\mu}\\
\ge & \int \sup_x \{\langle v,x\rangle +\liminf_{n\rightarrow\infty} f_n(x)\}\,d\hat{\mu}=\int (-f)^\ast\,d\hat{\mu}= F_\mu(f),
\end{split}$$ where we use $\{(-f_n)^\ast\}$ being bounded below by $\inf_n\{f_n(0)\}>-\infty$ to apply Fatou’s lemma (F) and use $\sup g\ge g$ to get to the second line.
Note that we can limit Kantorovich duality to Lipschitz functions only because $\mu$ is compact, hence the optimal $f$ has bounded gradient. We could extend this result to non-compactly supported $\mu$ if we consider $\alpha$-Hölder continuous functions for any $\alpha>1$ rather than Lipschitz functions. This case is precluded from us here because of the constraint to functions with bounded derivatives in the following proposition.
Since the double dual of $\underline{W}_\mu$ is equivalent to the dual formulation, Brenier’s theorem applies. And the maximizing $f$ provides the optimal transport $\nabla f$ of the primal problem.
We now state a similar result regarding the dynamic cost:
\[prop:HJB\] Assume (A) and let $f\in {C^\infty_\text{db}}$. Then, the unique solution to \[eq:HJB\] is given by: $$\tag{HJB'}\label{eq:dynprog}
\phi^f(t,x)=\sup_{X\in\mathcal{A}}\left\{{\ensuremath{\mathbb{E}\left[f(X(T))-\int_t^T L(s,X(s),\beta_X(s,X))\,ds\middle\rvert X(t)=x\right]}}\right\}.$$ Moreover, there exists an optimal process $X$ with drift $\beta_X(t,X)=\operatorname*{arg\,min}_v\{v\cdot \nabla \phi(t,x)+L(t,x,v)\}$.
This proposition may be shown by removing the boundedness condition in [@HJB Remark IV.11.2]. In fact it suffices to only have the first three derivatives bounded. Note that integrating \[eq:dynprog\] over $d\nu_0$ yields the legendre transform of $\nu_T\mapsto C(\nu_0,\nu_T)$ for $f\in{C^\infty_\text{db}}$. A key step in proving \[thm:stocbdual\] is showing that it is not necessary to consider functions outside of this set.
That this proposition does not necessarily hold for functions with unbounded derivatives confines us to $\mu$ with bounded support as we saw in \[lem:Wlegendre\].
For $\mu_0\in \mathcal{P}_1(M^\ast)$, define the function $\underline{B}_{\mu_0}:\mathcal{M}_1(M)\rightarrow\mathbb{R}\cup\{\infty\}$ to be $$\underline{B}_{\mu_0}(\nu):=\begin{cases}\underline{B}(\mu_0,\nu)&\nu\in\mathcal{P}_1(M)\\
\infty &\text{otherwise}.\end{cases}$$ From \[cor:Blsc\], we get that $\underline{B}_{\mu_0}$ is convex and lsc. Thus, by convex analysis, we have $$\label{eq:Bddual}
\underline{B}_{\mu_0}(\nu)=\underline{B}_{\mu_0}^{\ast\ast}(\nu)=\sup_{f\in {\text{Lip}(M)}}\left\{\int f\, d\nu-\underline{B}_{\mu_0}^\ast(f)\right\}.$$ We break this into two steps. First we show that when $f\in {C^\infty_\text{db}}$ the dual is appropriate: $$\begin{aligned}
\notag
\underline{B}_{\mu_0}^\ast(f)&\,:=\sup_{\nu_T\in \mathcal{P}_1(M)}\left\{\int f\,d\nu_T-\underline{B}(\mu_0,\nu_T)\right\}\\\label{eq:dualexp}
&\,\overset{(\ref{eq:interpol})}{=}\sup_{\substack{\nu_T\in\mathcal{P}_1(M)\\\nu\in\mathcal{P}_1(M)}}\left\{\int f\,d\nu_T-C(\nu,\nu_T)-\underline{W}(\mu_0,\nu)\right\}\\\notag
&\overset{(\ref{eq:dynprog})}{=}\sup_{\nu\in\mathcal{P}_1(M)}\left\{\int \phi^f(0,x)\,d\nu-\underline{W}(\mu_0,\nu)\right\}\\\notag
&\hspace{2mm}= \underline{W}_{\mu_0}^\ast(\phi^f)=-\int \widetilde{\phi^f}\,d\mu_0.\end{aligned}$$ Thus, plugging this into our dual formula (\[eq:Bddual\]) and restricting our supremum to ${C^\infty_\text{db}}$ gives $$\underline{B}_{\mu_0}(\nu)=\underline{B}_{\mu_0}^{\ast\ast}(\nu)\ge\sup_{f\in {C^\infty_\text{db}}}\left\{\int f\, d\nu+\int \widetilde{\phi^f}\,d\mu_0\right\}.$$ To show the reverse inequality we will adapt the mollification argument set out in [@mikami Proof of Theorem 2.1]. We assume our mollifier $\eta_\epsilon(x)$ is such that $\eta_1(x)$ is a smooth function on $[-1,1]^d$ that satisfies $\int \eta_1(x)\,dx=1$ and $\int x\eta_1(x)\,dx=0$, then define $\eta_\epsilon(x)=\epsilon^{-d}\eta_1(x/\epsilon)$. Then for Lipschitz $f$, $f_\epsilon:=f\ast \eta_\epsilon$ is smooth with bounded derivatives. We can derive a bound on $\underline{B}_{\mu\ast\eta_\epsilon}^\ast(f)$ by removing the supremum in \[eq:dualexp\] and fixing a process $X\in\mathcal{A}$. This gives us: $$\begin{gathered}
{\ensuremath{\mathbb{E}_{}\left[f_\epsilon(X(T))-\int_0^T L(s,X(s),\beta_X(s,X))\,ds-\langle X(0),Y\rangle\right]}}\overset{(\text{A2})}{\le}\\
{\ensuremath{\mathbb{E}_{}\left[f(X(T)+H_\epsilon)-\int_0^T \frac{L(s,X(s)+H_\epsilon,\beta_X(s,X))-\Delta L(0,\epsilon)}{1+\Delta L(0,\epsilon)}\,ds-\langle X(0)+H_\epsilon,Y+H_\epsilon\rangle+{\ensuremath{\left\lvertH_\epsilon\right\rvert}}^2\right]}}\le \\
\frac{\underline{B}_{\mu_0\ast\eta_\epsilon}^\ast(f{\ensuremath{\left(1+\Delta L(0,\epsilon)\right)}})}{1+\Delta L(0,\epsilon)}+T\frac{\Delta L(0,\epsilon)}{1+\Delta L(0,\epsilon)}+d\epsilon^2, \end{gathered}$$ where $H_\epsilon$ is a random variable with distribution $\eta_\epsilon\,dx$ that is independent of $X(\cdot),Y$ (we use this to get the second line), thus $X(T)+H_\epsilon\sim d(\eta_\epsilon\ast \nu_T)$. The third line arises by maximizing over processes $(X(\cdot)+H_\epsilon,Y+H_\epsilon)$.
Taking the supremum over $X\in\mathcal{A}_{\mu_0}$ of the left side above, we can retrieve a bound on $\underline{B}_{\mu_0}^\ast(f_\epsilon)$. This bound allows us to say $$\begin{split}
\int f_\epsilon\,d\nu-\underline{B}_{\mu}^\ast(f_\epsilon)\ge \int f\,d\nu_\epsilon-\frac{\underline{B}_{\mu_\epsilon}^\ast(f{\ensuremath{\left(1+\Delta L(0,\epsilon)\right)}})}{1+\Delta L(0,\epsilon)}+T\frac{\Delta L(0,\epsilon)}{1+\Delta L(0,\epsilon)}+d\epsilon^2,
\end{split}$$ where we use $\epsilon$-subscript to indicate convolution of a measure with $\eta_\epsilon$. Taking the supremum over $f\in{\text{Lip}(M)}$, we get the reverse inequality: $$\begin{split}
\sup_{f\in {C^\infty_\text{db}}}\left\{\int f\,d\nu-\underline{B}_{\mu}^\ast(f)\right\}\ge \frac{\underline{B}(\mu_\epsilon,\nu_\epsilon)}{1+\Delta L(0,\epsilon)}+T\frac{\Delta L(0,\epsilon)}{1+\Delta L(0,\epsilon)}+d\epsilon^2\overset{\epsilon\searrow 0}{\ge}\underline{B}(\mu_0,\nu_T),
\end{split}$$
In some sense $\nabla\phi$ is more fundamental than $\phi$, since our dual is invariant under $\phi\mapsto \phi+c$. Thus when discussing the convergence of a sequence of $\phi$, we refer to the convergence of their gradients. Notably the optimal gradient may not be bounded or smooth, hence may not be achieved within the set ${C^\infty_\text{db}}$. In the subsequent corollary, we denote $\mathcal{P}_X$ the measure on $M\times [0,T]$ associated with the process $X$ and refer to solutions to (\[eq:HJB\]) by $\phi_n^t(x):=\phi_n(t,x)$ without reference to their final condtion for convenience.
\[cor:minoptX\] Suppose the assumptions on \[thm:stocbdual\] are satisfied and $d\mu_0\ll d\lambda$. Then $(V,X(t))$ minimizes $\underline{B}(\mu_0,\nu_T)$ iff it is a solution to the SDE $$\begin{aligned}
\label{eq:optX}
dX =& \nabla_p H(t,X, \nabla\phi(t,X))\, dt + dW_t\\\label{eq:optV}
V =& \nabla\bar{\phi}(X(0)),\end{aligned}$$ where $\nabla\phi_n(t,x)\rightarrow \nabla\phi(t,x)$ $\mathcal{P}_X$-a.s. and $\nabla\phi_n(0,x)\rightarrow \nabla\bar{\phi}(x)$ $\nu_0$-a.s. for some sequence of $\phi_n$ that approach the supremum in \[eq:mindual\]. Furthermore $\bar{\phi}$ is concave.
First note that there exists optimal $(V,X)$, by \[thm:interpol\].
We begin with the forward direction. By \[thm:stocbdual\] $(V,X)$ is optimal iff there exists a sequence of solutions $\phi_n$ to \[eq:HJB\] that is maximizing in \[eq:maxdual\] such that $$\label{eq:converge}
{\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X,\beta_X(t,X))\,dt+\langle X(0),V\rangle\right]}}=\lim_{n\rightarrow\infty}{\ensuremath{\mathbb{E}_{}\left[\phi_n^T(X(T))+\widetilde{\phi}_n^0(V)\right]}}$$ We add $0$ to the right hand side to get $$\label{eq:ineqs}
\lim_{n\rightarrow\infty}{\ensuremath{\mathbb{E}_{}\left[\underbrace{\phi_n^T(X(T))-\phi_n^0(X(0))}_{\text{(a)}}+\underbrace{\phi_n^0(X(0))-\widetilde{\widetilde{\phi}_n^0}(X(0))}_{\text{(b)}}+\underbrace{\widetilde{\widetilde{\phi}_n^0}(X(0))+\widetilde{\phi}_n^0(V)}_{\text{(c)}}\right]}}.$$ where $\widetilde{\widetilde{f}}$ is the concave double dual of $f$, or its concave hull. Applying Ito’s formula to the first two terms, with the knowledge that they satisfy (\[eq:HJB\]) we get $${\ensuremath{\mathbb{E}_{}\left[\phi_n^T(X(T))-\phi_n^0(X(0))\right]}}={\ensuremath{\mathbb{E}_{}\left[\int_0^T\langle \beta_X,\nabla\phi_n^t(X(t))\rangle-H(t,X,\nabla\phi_n^t(X(t)))\,dt\right]}}$$ However, by the definition of the Hamiltonian, we have $\langle v,b\rangle - H(t,x,v)\le L(t,x,b)$, this along with results from convex analysis provides us with three inequalities on \[eq:ineqs\] $$\begin{aligned}
\tag{a} \langle \beta_X,\nabla\phi_n^t(X(t))\rangle-H(t,X,\nabla\phi_n^t(X(t)))\le& L(t,X,\beta_X(t,X))\\
\tag{b} \phi_n^0(X(0))-\widetilde{\widetilde{\phi}_n^0}(X(0))\le& 0\\
\tag{c} \widetilde{\widetilde{\phi}_n^0}(X(0))+\widetilde{\phi}_n^0(V)\le& \langle V,X(0)\rangle.\end{aligned}$$ These inequalities demonstrate that \[eq:ineqs\] breaks our problem into a stochastic and a Wasserstein transport problem (in the flavour of \[thm:interpol\]), along with a correction term to account for $\phi^0_n$ not being concave. Adding \[eq:converge\] to the mix, allows us to obtain $L^1$ convergence in the (a,b,c) inequalities, hence a.s. convergence of a subsequence $\phi_{n_k}$.
By properties on convex analysis, convergence in (b,c) is equivalent to $\phi_n^0$ converging $\nu_0$-a.s. to a concave function $\overline{\phi}$ such that $x\mapsto \nabla\overline{\phi}$ is the optimal transport plan for $\underline{W}(\nu_0,\mu_0)$ [@Brenier].
To obtain the optimal control for the stochastic process, one needs the uniqueness of the point $p$ achieving equality in (a). This is a consequence of the strict convexity and coercivity of $b\mapsto L(t,x,b)$ for all $t,x$. The differentiability of $L$ further ensures this value is achieved by $p=\nabla_v L(t,x,b)$. Hence (a) holds iff $$\nabla\phi_n^t(X_t)\longrightarrow \nabla_v L(t,X,\beta_X(t,X))\qquad\mathcal{P}_X\text{-a.s.}$$ Since $\phi_n^t$ are deterministic functions, this demonstrates that $X_t$ is a Markov process with drift $\beta_X$ determined by the inverse transform: $\beta_X(t,X)=\nabla_p H(t,X,\nabla\phi(t,X))$, i.e., \[eq:optX\].
It is not possible to conclude from the above work that $\bar{\phi}(x)=\phi(0,x)$ since $\bar{\phi}$ is defined on a $\mathcal{P}_X$-null set. To do so, we would require a regularity result on the optimal $\phi$ in time.
Maximizing Ballistic Cost {#sec:Bupper}
-------------------------
We now turn our focus to the related stochastic cost: $$\label{eq:Bover}
\overline{B}(\nu_0,\mu_T):=\sup\left\{{\ensuremath{\mathbb{E}_{}\left[\langle V,X(T)\rangle\right]}}-{\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X,\beta_X(t,X))\,dt\right]}}\middle\rvert X\in \mathcal{A}_{\nu_0}, V\sim\mu_T\right\},$$ which we term the *maximizing ballistic cost*.
The case where $\mu_T$ is given by a dirac measure $\delta_u$ proves suggestive. Here the maximizing ballistic cost may be interpreted literally in terms of the HJB equation by \[prop:HJB\]—with $f:x\mapsto \langle u,x\rangle$. Thus, in this particular case we can recover $$\overline{B}(\nu_0,\delta_u)=\int \phi^f(0,x)\,d\nu_0(x),$$ without the aid of any duality theorem. Notably, $$f^\ast(v)=\sup_z\{\langle v,z\rangle - \langle u,z\rangle\}=\begin{cases}
0& v=u\\
\infty & \text{otherwise}.
\end{cases}$$ further gives us the minimizing process, described by the SDE $$X(t)=X(0)+\int_0^t \nabla_p H(s,X,\nabla \phi(s,X))\,ds+W_t.$$ In the case where $L(t,x,v)=\frac{{\ensuremath{\left\lvertv\right\rvert}}^2}{2}+\ell(x,t)$, the maximizing ballistic cost may be considered as finding the optimal stochastic process with the final drift constrained a.s. to be $\beta_X(T,X)=u$. In this section we will show that these results generalize to all measures in $\mathcal{P}_1(M^\ast)$.
We begin by obtaining a similar interpolation result:
\[thm:Bointerpol\] If the Lagrangian matches assumptions (A), then $$\label{eq:maxinterpol}
\overline{B}(\nu_0,\mu_T)=\sup\{\overline{W}(\nu,\mu_T)-C_{L}(\nu_0,\nu)\},$$ where $C_{L}$ is the action corresponding to the Lagrangian $L$. Furthermore, if $\nu_0\in\mathcal{P}_1(M)$, and $\mu_T\in\mathcal{P}_1(M^\ast)$ there exist an optimal interpolant $\nu_T$.
We will proceed as in our proof of the first inequality. First, consider a sequence of rvs $(X_n(\cdot),V_n)$ approximating $\overline{B}(\nu_0,\mu_T)$. Then $$\begin{aligned}
{\ensuremath{\mathbb{E}_{}\left[\langle V_n,X_n(T)\rangle\right]}}-{\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X_n,\beta_{X_n}(t,X_n))\,dt\right]}}<\overline{B}(\nu_0,\mu_T)+\frac{1}{n}\end{aligned}$$ but if $X_n(T)\sim\mu_n$, $$\begin{aligned}
{\ensuremath{\mathbb{E}_{}\left[\langle V_n,X_n(T)\rangle\right]}}\le&\overline{W}(\mu_n,\mu_T)\\
{\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X_n,\beta_{X_n}(t,X_n))\,dt\right]}}\ge& C_{L}(\nu_0,\mu_n),\end{aligned}$$ showing that $\overline{B}(\nu_0,\mu_T)$ is less than its interpolations. To get the reverse inequality, take $\mu_n$ to be a sequence approaching the supremum. Then there exists a random variable $X_n(\cdot)\in\mathcal{A}_{\nu_0}^{\mu_n}$ such that $${\ensuremath{\mathbb{E}_{}\left[\int_0^T L(t,X_n,\beta_{X_n}(t,X_n))\,dt\right]}}<C(\nu_0,\mu_n)+\tfrac{1}{n}.$$ Since $X_n(T)\sim\mu_n$, we may also define $V_n(\omega)\sim \gamma^n_{X_n(0)(\omega)}$ to be distributed according to the disintegration of $d\gamma_x^n(y)\otimes d\mu_T(x)=d\gamma_n$ such that $$\iint\langle y,x\rangle \, d\gamma_n(x,y)>\overline{W}(\mu_n,\mu_T)-\tfrac{1}{n}.$$ We may further assume the minimizing sequence $\mu_n$ is tight in the case where $\nu_0\in \mathcal{P}_1(M^\ast)$ and $\mu_T\in \mathcal{P}_1(M)$ since by \[prop:Ccoerc\], $$\overline{B}(\nu_0,\mu_T)-1<\overline{W}(\mu_n,\mu_T)-C_{L}(\nu_0,\mu_n)<M\int {\ensuremath{\left\lvertv\right\rvert}}\,d\mu_n-C_{L}(\nu_0,\mu_n).$$Again, Prokhorov’s theorem concludes the proof.
\[thm:Boverdual\] Assume the Lagrangian satisfies the assumptions (A). If $\nu_0\in\mathcal{P}_1(M)$, $\mu_T$ has compact support, and $\overline{B}(\nu_0,\mu_T)<\infty$, then $$\label{eq:maxdual}
\overline{B}(\nu_0,\mu_T)=\inf_{g\in{c}{C^\infty_\text{db}}}\left\{\int_{M^*} g^\ast\,d\mu_T+\int_M\phi^{g}\,d\nu_0 \right\},$$ where $\phi$ solves the Hamilton-Jacobi-Bellman equation $$\begin{aligned}
\label{eq:HJB2}\tag{HJB2}
{\ensuremath{\frac{\partial{\phi}}{\partial{t}}}}+\frac{1}{2}\Delta \phi -H(t,x,\nabla\phi)=&0&\phi(x,T)=g(x)\end{aligned}$$
Let $\mu\in\mathcal{P}_1(M^\ast)$ have compact support. If we define $\overline{W}_\mu(\nu):\nu\mapsto \overline{W}(\mu,\nu)$, then for all $f\in {\text{Lip}(M^\ast)}$ $$\sup_{\nu\in\mathcal{P}_1(M)}\left\{\int_M f\,d\nu+\overline{W}(\mu,\nu)\right\}=\int (-f)^\ast\,d\mu.$$
This lemma follows from applying \[lem:Wlegendre\] to $\underline{W}({\hat{\mu}},\nu)$ (recall ${\hat{\mu}}(A)=\mu(-A)$), and noting that $$-\int \widetilde{f}(v)\,d{\hat{\mu}}(v)=-\int \widetilde{f}(-v)\,d\mu(v)=\int(-f)^\ast(v)\,d\mu(v).$$
Define $\overline{B}_{\nu_0}:\mu_T\mapsto\overline{B}(\nu_0,\mu_T)$, and note that this is a concave function (which can be seen by applying \[lem:convexlemma\] to the negative of \[eq:maxinterpol\]). Furthermore it is upper semi-continuous on $\mathcal{P}_1(M^\ast)$ by the same reasoning as \[cor:Blsc\]a). Thus we have $$\label{eq:maxddual}
\overline{B}_{\nu_0}(\mu_T)=-(-\overline{B}_{\nu_0})^{\ast\ast}(\mu_T)=\inf_{f\in {\text{Lip}(M^\ast)}}\left\{-\int_{M^*} f\,d\mu_T+(-\overline{B}_{\nu_0})^\ast(f)\right\}.$$ Investigating the dual, we find $$\begin{aligned}
\notag
(-\overline{B}_{\nu_0})^\ast(f)=&\sup_{\mu_T\in\mathcal{P}_1(M^\ast)} \left\{\int_{M^*} f\,d\mu_T +\overline{B}_{\nu_0}(\mu_T)\right\}\\\notag
=&\sup_{\substack{\mu_T\in\mathcal{P}_1(M^\ast)\\\nu_T\in\mathcal{P}_1(M^\ast)}} \left\{\int_{M^*} f\,d\mu_T +\overline{W}(\nu_T,\mu_T)-C_{L}(\nu_0,\nu_T)\right\}\\ \label{eq:lipmaxdual}
=&\sup_{\nu_T\in\mathcal{P}_1(M^\ast)} \left\{\int_{M^*} (-f)^\ast\,d\nu_T -C_{L}(\nu_0,\nu_T)\right\}.\end{aligned}$$ Note that in the case where $(-f)^\ast\in{C^\infty_\text{db}}$, this is simply $\int_M\phi^{(-f)^\ast}\,d\nu_0$, giving us $$\overline{B}_{\nu_0}(\mu_T)\le \inf_{(-f)^\ast\in {C^\infty_\text{db}}}\left\{-\int_{M^*} f\,d\mu_T+\int_M \phi^{(-f)^\ast}\,d\nu_0\right\}.$$ In either case, we can restrict our $f$ to be concave by noting that if we fix $g=(-f)^\ast$, then the set of corresponding $\{-f\rvert (-f)^\ast=g\}$ is minimized by the convex function $g^\ast=(-f)^{\ast\ast}\le -f$ [@ConvexEkeland Proposition 4.1]. Thus it suffices to consider $f$ convex.
We now show that it is sufficient to consider this infimum over $g\in {c}{C^\infty_\text{db}}$ by a similar mollification argument to \[thm:stocbdual\] (note that the mollifying preserves convexity). Maintaining the same assumptions and notation as in our earlier argument, we first note a useful application of Jensen’s inequality to the legendre dual of a mollified function: $$\begin{split}
g_\epsilon^\ast(v)=&\sup_x\left\{\langle v,x\rangle -{\ensuremath{\mathbb{E}_{}\left[g(x+H_\epsilon)\right]}}\right\}
\overset{(\text{J})}{\le}\sup_x\left\{\langle v,x\rangle -g(x)\right\}= g^\ast(v).
\end{split}$$ Mikami [@mikami Proof of Theorem 2.1] further shows that $$(\ref{eq:lipmaxdual})=C^\ast_{\nu_0}(g_\epsilon)\le \frac{C^\ast_{\nu_0\ast\eta_\epsilon}((1+\Delta L(0,\epsilon))g)}{1+\Delta L(0,\epsilon)}+T\frac{\Delta L(0,\epsilon)}{1+\Delta L(0,\epsilon)}.$$ Putting these together we get $$\int g^\ast_\epsilon\,d\mu_T+(-\overline{B}_{\nu_0})^\ast(g_\epsilon^\ast)\,d\nu_0\le \int g^\ast\,d\mu_T+\frac{C^\ast_{\nu_0\ast\eta_\epsilon}((1+\Delta L(0,\epsilon))g)}{1+\Delta L(0,\epsilon)}+T\frac{\Delta L(0,\epsilon)}{1+\Delta L(0,\epsilon)}.$$ And once we take the infimum over $g\in{c}{\text{Lip}(M)}$, we get $$\inf_{g\in c{C^\infty_\text{db}}}\left\{\int g^\ast\,d\mu_T+{\ensuremath{\left(-\overline{B}_{\nu}\right)}}^\ast(-g^\ast)\right\}\le \frac{-(-\overline{B})^{\ast\ast}_{\nu_0\ast\eta_\epsilon}(\mu_{L,\epsilon})}{1+\Delta L(0,\epsilon)}+T\frac{\Delta L(0,\epsilon)}{1+\Delta L(0,\epsilon)},$$ where $d\mu_{L,\epsilon}(v):=d\mu_T({\ensuremath{\left(1+\Delta L(0,\epsilon)\right)}}v)$. Taking $\epsilon\searrow 0$ dominates the right side by $\overline{B}(\nu_0,\mu_T)$ (where we exploit the upper semi-continuity of $\overline{B}$), completing the reverse inequality.
\[cor:maxoptX\] Suppose the assumptions on \[thm:Boverdual\] are satisfied, with $d\mu_0\ll d\lambda$. Then, $(V,X(t))$ is an optimal process iff it is a solution to the SDE $$\begin{aligned}
\label{eq:maxoptX}
dX =& \nabla_p H(t,x,\nabla\phi(t,X))\, dt + dW_t\\\label{eq:maxoptV}
X(T) =& \nabla\bar{\phi}(V,T)\end{aligned}$$ where $\lim_{n\rightarrow \infty}\phi_n(T,x)\rightarrow \bar{\phi}(x)$ $\nu_T$-a.s. and $\lim_{n\rightarrow\infty}\phi_n(t,x)=\phi(t,x)$ $\mathcal{P}_X$-a.s. for some sequence of $\phi_n$ that is minimizing the dual problem.
If $(V,X)$ is optimal, then \[thm:Boverdual\] means there exists a sequence of solutions $\phi_n$ to \[eq:HJB\] with convex final condition such that $$\label{eq:maxconverge}
{\ensuremath{\mathbb{E}_{}\left[\langle X(T),V\rangle-\int_0^T L(t,X,\beta_X(t,X))\,dt\right]}}=\lim_{n\rightarrow\infty}{\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left(\phi_n^T\right)}}^\ast(V)+\phi_n^0(X(0))\right]}}$$ We add $0$ to the right hand side to get $$\lim_{n\rightarrow\infty}{\ensuremath{\mathbb{E}_{}\left[{\ensuremath{\left(\phi_n^T\right)}}^\ast(V)+\phi_n^T(X(T))-\phi_n^T(X(T))+\phi_n^0(X(0))\right]}}.$$ Applying Ito’s formula to the last two terms, with the knowledge that $\phi_n$ satisfies (\[eq:HJB\]) we get $${\ensuremath{\mathbb{E}_{}\left[-\phi_n^T(X(T))+\phi_n^0(X(0))\right]}}={\ensuremath{\mathbb{E}_{}\left[\int_0^T-\langle \beta_X,\nabla\phi_n^t(X(t))\rangle+H(t,X,\nabla\phi_n^t(X(t)))\,dt\right]}}$$ However, by the definition of the Hamiltonian, we have $-\langle v,b\rangle + H(t,x,v)\ge -L(t,x,b)$, similarly $\phi^\ast(v)+\phi(x)\ge\langle v,x\rangle$. These inequalities allow us to separate the limit in \[eq:maxconverge\] into two requirements: (a) $\langle \beta_X,\nabla\phi_n^t(X(t))\rangle-H(t,X,\nabla\phi_n^t(X(t)))$ must converge to $L(t,X,\beta_X(t,X))$ and (b) $\phi_n^T(X(T))+{\ensuremath{\left(\phi_n^T\right)}}^\ast(V)$ must converge to $\langle X(T),V\rangle$ in $L^1$ hence a subsequence $\phi_{n_k}$ exists such that this convergence is a.e.
The journey from (a) to \[eq:maxoptX\] is as in \[cor:minoptX\]. The only difference from the earlier corollary is that we know that $\phi_n$ must converge to a convex function, so (b) implies $V=\nabla\lim_{n\rightarrow \infty}\phi_n(X(T))$.
Like for the minimizing cost, it is impossible to conclude $\phi(T,x)=\bar{\phi}(x)$ without a regularity result. **If** one could make this connection, then one could formulate \[eq:maxoptV\] as $V=\nabla \phi(T,X(T))$, allowing us to interpret the measure $\mu_T$ as a condition on our final momentum distribution—i.e., our final drift $\beta(T)$ would be constrained to be distributed according to the measure $\rho$ where $\mu_T={\ensuremath{\left(\nabla L(t,x,\cdot)\right)}}_\#\rho$. In the particular case where $L(t,x,v)=\frac{{\ensuremath{\left\lvertv\right\rvert}}^2}{2}+\ell(x,t)$, this would impose a Neumann type boundary condition on our stochastic process: $\beta(T)\sim\mu_T$.
[13]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
M. [Beiglb[ö]{}ck]{} and N. [Juillet]{}. . *ArXiv e-prints*, August 2012.
P. Bernard and B. Buffoni. . *Journal of the European Mathematical Society*, 90 (1):0 85–121, 2007. Y. Brenier. Polar factorization and monotone rearrangement of vector-valued functions. *Communications on Pure and Applied Mathematics*, 440 (4):0 375–417, 1991. I. Ekeland and R. T[é]{}man. *Convex Analysis and Variational Problems*, Convex Functions. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1999. W.H. Fleming and H.M. Soner. *Controlled Markov Processes and Viscosity Solutions*. Applications of Mathematics. Springer-Verlag, 1993. N. Ghoussoub. . *ArXiv e-prints*, May 2017.
L. Kantorovitch. On the translocation of masses. *Management Science*, 50 (1):0 1–4, 1958.
R.S. Liptser and A.N. Shiryaev. *Statistics of Random Processes: I. General Theory*. Applications of mathematics : stochastic modelling and applied Probability. Springer, 2001. T. Mikami. Optimal control for absolutely continuous stochastic processes and the mass transportation problem. *Electron. Commun. Probab.*, 7:0 199–213, 2002. T. Mikami and M. Thieullen. . *Stochastic Processes and their Applications*, 2006. G. Monge. *M[é]{}moire sur la th[é]{}orie des d[é]{}blais et des remblais*. De l’Imprimerie Royale, 1781.
C. Villani. *[Topics in Optimal Transportation]{}*. Graduate studies in mathematics. American Mathematical Society, 2003. N. Weaver. *Lipschitz Algebras*. World Scientific, 1999.
|
---
abstract: |
In this paper, we explore the quantum spacetimes that are potentially connected with the generalized uncertainty principles. By analyzing the gravity-induced quantum interference pattern and the Gedanken for weighting photon, we find that the generalized uncertainty principles inspire the effective Newton constant as same as our previous proposal. A characteristic momentum associated with the tidal effect is suggested, which incorporates the quantum effect with the geometric nature of gravity. When the simplest generalized uncertainty principle is considered, the minimal model of the regular black holes is reproduced by the effective Newton constant. The black hole’s tunneling probability, accurate to the second order correction, is carefully analyzed. We find that the tunneling probability is regularized by the size of the black hole remnant. Moreover, the black hole remnant is the final state of a tunneling process that the probability is minimized. A theory of modified gravity is suggested, by substituting the effective Newton constant into the Hilbert-Einstein action.
: generalized uncertainty principle, effective Newton constant, characteristic momentum, regular black hole, quantum tunneling, WKB approximation.
author:
- |
Xiang Li[^1]\
[*Department of Physics*]{}\
[*Jimei University, 361021, Xiamen, Fujian province, P. R. China*]{}\
Yi Ling[^2]\
[*Institute of High Energy Physics*]{}\
[*Chinese Academy of Sciences, 100049, Beijing, P. R. China*]{}\
You-Gen Shen[^3]\
[*Shanghai Astronomical Observatory*]{}\
[*Chinese Academy of Sciences, 200030, Shanghai, P. R. China*]{}\
Cheng-Zhou Liu[^4]\
[*Department of Physics*]{}\
[*Shaoxing University, 312000, Shaoxing, Zhejiang province, P. R. China*]{}\
Hong-Sheng He[^5]\
[*Department of Physics*]{}\
[*Jimei University, 361021, Xiamen, Fujian province, P. R. China*]{}\
Lan-Fang Xu[^6]\
[*Department of Physics*]{}\
[*Jimei University, 361021, Xiamen, Fujian province, P. R. China*]{}\
title: 'Generalized uncertainty principles, effective Newton constant and regular black holes'
---
-1cm
Introduction
============
On the ground of dimensional analysis[@mtw], the Planck length is defined as $\ell_\textsc{p}\equiv\sqrt{\hbar G/c^3}$, where $c$ is the speed of light, $G$ Newton constant, and $\hbar$ Planck constant. This unit of length should appear in any theory reconciling general relativity and quantum theory. It is generally believed that $\ell_\textsc{p}$ is the shortest measurable length, and quantum gravity effects( or quantum fluctuations in spacetimes) become crucial to understanding the physics on this length scale. As a classical theory, general relativity involves only $c$ and $G$, and the minimal length cannot be predicted naturally by the theory itself. There are some problems are somewhat related to the defect that the Planck length is absent in the classical spacetimes.
One of problems is the spacetime singularity[@ph]. Following from Penrose and Hawking’s theorems, the spacetime singularity is inevitable in the framework of classical general relativity. In a certain extent, the singularity is characterized by the divergence of Kretschmann scalars ($K^2=R_{\rho\lambda\mu\nu}R^{\rho\lambda\mu\nu}$). For a Schwarzschild black hole, $K^2\sim M^2/r^6$ become divergent, as $r\rightarrow 0$.
Another problem is the fate of black hole evaporation[@hawk]. In the case of Hawking’s temperature expression( $T_H\sim M^{-1}$), the negative capacity( $C\sim -M^{-2}$) makes the black hole evaporation faster and faster. Both the temperature and the mass loss rate( $\dot{M}\sim -M^{-2}$) become divergent, if the black hole vanishes.
The third problem is related to the tunneling picture of the black hole radiation[@parik; @parikh2; @vagenas4; @vagenas1; @zjy2; @zjy1]. The tunneling probability accurate to the first order correction becomes explosive[@vagenas1; @zjy1], if the final size of the black hole is allowed to be arbitrary small. When the second order correction to the tunneling probability is considered, the situation becomes worse[@zjy1]. We are confronted with an unacceptable picture that a black hole of any mass could vanish in an instant. This difficulty is associated with the absence of the minimal length, and then it is not necessarily overcome by improving the WKB method.
These problems are expected to be solvable at the presence of quantum gravity effects. The generalized uncertainty principle(GUP[@garay1]), as one of methods of quantum gravity phenomenology, has been applied to the black hole thermodynamics in some heuristic manners[@pschen1; @vagenas2; @lx1]. GUP imposes a lower bound on the size of the black hole, and modifies the black hole’s thermodynamics. However, it is hard to understand that the temperature approaches a value of order of Planck temperature, while the heat capacity of the minimal black hole vanishes. In other words, GUP predicts the existence of black hole remnant, but the temperature puzzle has not been solved completely.
This dilemma may be associated with such a working hypothesis in the literature that the matter is dominated by GUP, while the spacetimes are classical. This hypothesis can be named as the GUP-revised semiclassical theory. However, the corrections to classical spacetimes should be considered seriously, especially on the Planck scale. After all, a spacetime dominated by quantum gravity effects may be essentially different from the classical and smooth background. The quantum spacetime should reflect the existence of the minimal length, when the GUP is considered in an appropriate manner.
Some regular black holes with finite Kretschmann scalars have been suggested in the literature[@bardeen; @reuter; @nicolini; @hayward; @myung; @lx2; @culetu; @vagenas3; @ghosh], and they give rise to the zero temperature remnants. As a tentative attempt, we suggest a regular black hole which is connected with the GUP by an effective Newton constant[@lx2]. This suggestion is based on an observation upon the role that the GUP plays in the relation between the gravitational acceleration and Newton potential, in the context of operators. The effective Newton constant is motivated by substituting the GUP for Heisenberg commutator. However, this direct substitution of commutators is in the shortage of a clear physical picture. Moreover, it not consistent with the GUP-modified Hamiton equation[@chang1], and its reliability should be checked by other methods. We expect that the effective Newton constant can be inspired by the GUP in some concrete physical processes. Another limitation of our previous work is that the simplest GUP(as presented in the next section) doesn’t give rise to a regular black hole, although it means the existence of the minimal length. This shortage is related to such a characteristic momentum as $\Delta p\sim\hbar/r$. Although this momentum scale is motivated by Heisenberg’s principle, it is irrelevant to the quantum fluctuations in spacetime, because it doesn’t involve the Newton constant and the mass of source of gravitational field.
The aim of this paper is to gain a better understanding of the GUP-inspired effective Newton constant and quantum spacetimes, at the level of quantum gravity phenomenology. Firstly, the effective Newton constant inspired by the GUP will be reexamined in two Gedankens involving gravitation and quantum theory, i.e. COW phase shift and Einstein-Bohr’s box. Secondly, the momentum scale will be reconsidered. In our opinion, an appropriate scale should reflect the amplitudes of the quantum fluctuations in the curved spacetime, and may be related to the geometric character of gravity. It would be different from that suggested in Ref.[@lx2]. Thirdly, we will consider a regular black hole, which is inspired by the simplest GUP, i.e. the most popular version. Finally, quantum radiation from this black hole will be discussed seriously in this work. For a mini black hole, the temperature may lose its usual meaning in the thermodynamics. It is more reasonable to consider the quantum tunneling from this black hole. We are interested in the role that the minimal length plays in the tunneling process, and in the question of whether the explosion of tunneling probability occurs in the quantum spacetime.
Effective Newton constant inspired by the generalized uncertainty principles
============================================================================
Based on some theoretical considerations and gedanken experiments for incorporating gravitation with quantum theory, Heisenberg’s uncertainty principle is likely to suffer a modification as follows[@garay1] $$\begin{aligned}
\label{gup0}
\Delta x\sim\frac{\hbar}{\Delta
p}+\frac{\alpha\ell_\textsc{p}^2}{\hbar}\Delta p,\end{aligned}$$ Where $\alpha$ is a dimensionless number of order of unity. Since GUP means the minimal length of order of $\ell_\textsc{p}$, it should be crucial for the Planck scale physics. Corresponding to (\[gup0\]), Heisenberg’s commutator is extended to[@kempf; @ahlu] $$\begin{aligned}
\label{commut0}
[\hat{x},\hat{p}]=i\hbar\left(1+\frac{\alpha\ell_\textsc{p}^2}{\hbar^2}\hat{p}^2\right),\end{aligned}$$ which will be considered seriously in this paper. However, there are some other types of the generalized uncertainty principles[@rama][@hh], and the commutation relations are not necessarily the same as (\[commut0\]). So we begin with a more general commutator as follows[@lx2] $$\begin{aligned}
\label{commut1}
[\hat{x},\hat{p}]=i\hbar z,\end{aligned}$$ where $z=z(\hat{p})$ is a function of momentum. The average value of $z$, should ensure a lower bound on the measurable distance, $$\begin{aligned}
\label{uncertainty2}
\Delta x\geq\frac{z\hbar}{\Delta p}\geq\ell_\textsc{p},\end{aligned}$$ which suggests the discreteness of the spacetime[@vagenas5]. Considering the relation between Heisenberg’s principle and the wave-particle duality, let us derive the modified de-Broglie formula from the generalized commutator (\[commut1\]).
Modified wave-particle duality
------------------------------
It is well known that the proposal of uncertainty principle is closely related to the wave particle duality. Uncertainty relation can be derived from de Broglie formula, by analyzing the Heisenberg’s microscope gedanken experiment or the single slit diffraction of light. However, once the framework of quantum mechanics is established and Heisenberg’s uncertainty relation is regarded as a fundamental principle, de Broglie formula becomes a deduction[@cdl]. Concretely speaking, the momentum eigenstate $\psi_p=\exp(ipx/\hbar)$ can be derived from the canonical commutation relation $[\hat{x},\hat{p}]=i\hbar$, and de Broglie formula is obtained by comparing the momentum eigenstate with a plane wave function $\exp(2\pi ix/\lambda)$. It is expectable that de Broglie formula should suffer a modification, when the Heisenberg’s commutator is changed. Corresponding to the generalized commutator (\[commut0\]), the modified de Broglie relation is given by[@kempf; @ahlu] $$\begin{aligned}
\label{mdb1}
\lambda=\frac{2\pi
\ell_\textsc{p}\sqrt{\alpha}}{\arctan(\ell_\textsc{p}\sqrt{\alpha}p/\hbar)}.\end{aligned}$$ It is easy to check that Eq.(\[mdb1\]) satisfies the following relation $$\begin{aligned}
\label{mdbb}
\frac{d}{dp}\left(\frac{2\pi}{\lambda}\right)=\hbar^{-1}\left(1+\frac{\alpha\ell_\textsc{p}^2}{\hbar^2}{p}^2\right)^{-1}.\end{aligned}$$ As argued in the following, a more general formula associated with the commutator (\[commut1\]) is given by $$\begin{aligned}
\label{mdb2}
\frac{d}{dp}\left(\frac{2\pi}{\lambda}\right)=\hbar^{-1}z^{-1}.\end{aligned}$$ In order to explain this formula, we first construct a commutator as follows $$\begin{aligned}
\label{commut2}
[\hat{x}, \hat{k}]=i,\end{aligned}$$ where $\hat{k}=k(\hat{p})$ is a function of momentum operator. Following from the law of operator algebra, we obtain $$\begin{aligned}
\label{commut3}
[\hat{x},\hat{p}]=[\hat{x},k]\frac{d\hat{p}}{dk}=i\frac{d\hat{p}}{dk}.\end{aligned}$$ Comparing (\[commut1\]) with (\[commut3\]), we have $$\begin{aligned}
\frac{dk}{d\hat{p}}=\hbar^{-1}z^{-1},\nonumber\end{aligned}$$ and then $$\begin{aligned}
k(\hat{p})=\hbar^{-1}\int z^{-1}d\hat{p}.\end{aligned}$$ Obviously, $[\hat{p},k(\hat{p})]=0$, this means that there is a common eigenstate $\psi_p$ of eigenvalue $p$, which satisfies $$\begin{aligned}
\label{kstate}
\hat{p}\psi_p&=&p\psi_p,\nonumber\\
k(\hat{p})\psi_p&=&k(p)\psi_p,\end{aligned}$$ where $$\begin{aligned}
\label{mdb3}
k(p)=\hbar^{-1}\int z^{-1}(p)dp,\end{aligned}$$ is the eigenvalue of the operator $k(\hat{p})$. Since $[\hat{x},\hat{p}]\neq i\hbar$, the momentum operator is no longer represented by $\hat{p}=-i\hbar\nabla$. However, comparing (\[commut2\]) with Heisenberg’s commutator, we obtain $\hat{k}=-i\nabla$. For one dimensional case, the second equation of (\[kstate\]) becomes $$\begin{aligned}
-i\frac{d\psi_p}{dx}=k(p)\psi_p.\end{aligned}$$ So the momentum eigenstate is given by $$\begin{aligned}
\psi_p=\exp(ikx),\end{aligned}$$ which describes a plane wave of wavelength $\lambda=2\pi/k$. Thus $k(\hat{p})$ introduced in (\[commut2\]) can be viewed as the wave-vector operator, and (\[mdb3\]) is just the wave-number. The modified wave-particle duality is characterized by (\[mdb2\]) or (\[mdb3\]), which is the basis for the following discussions.
COW phase shift
---------------
In 1975, Colella, Overhauser, and Werner(COW) observed the gravity-induced quantum interference pattern of two neutron beams[@cow1]. When the plane of two beams is vertical to the horizontal plane, the phase shift is given by[@cow2; @cow3] $$\begin{aligned}
\label{phase1}
\Delta\varphi=\frac{mgA}{\hbar v},\end{aligned}$$ where $g$ denotes the earth’s gravitational acceleration, $v$ the average speed of neutrons, and $A$ the area enclosed by two interfering neutron beams that propagate on two paths on a plane.
This famous experiment may be regarded as a test of the property of gravity in the microscopic world[@saha]. It is naturally expected to shed light on the quantum structure of spacetime, by attaching the GUP’s significance to the gravity-induced phase shift. In the following discussions, COW experiment will be revisited in a heuristic manner[@cow3]. Let us consider two interfering neutron beams. For simplicity, the plane of two beams is set to be vertical to the horizontal plane. The first (upper) beam propagates on a horizontal path and a vertical downward path in sequence. The second (lower) beam propagates on a vertical downward path and a horizontal path in sequence. The area enclosed by two beams is $A=yl$, where $l$ is the length of each horizontal path, and $y$ is the height of the upper horizontal path with respect to the lower horizontal one. It is shown by simple analysis that the change in the phase of one vertical beam cancel out that of another vertical beam, and then the phase shift is completely attributed to the gravity-induced difference in the wavelength of two horizontal beams, $$\begin{aligned}
\label{phase2}
\Delta\varphi^{\prime}&=&2\pi \left(\frac{l}{\lambda_2}-\frac{l}{\lambda_1}\right)\nonumber\\
&=&l(k_2-k_1)\nonumber\\
&=&l\Delta k=l\frac{\Delta k}{\Delta p}\Delta p,\end{aligned}$$ where $\Delta p=p_2-p_1$ is the difference in momentum of two horizontal beams. Since $\Delta p$ is a small quantity, Eq.(\[phase2\]) can be expressed as $$\begin{aligned}
\label{phase3}
\Delta\varphi^{\prime}&\approx&l\frac{dk}{dp}\Delta p\nonumber\\
&=&\hbar^{-1}z^{-1}l\Delta p,\end{aligned}$$ where Eq.(\[mdb2\]) has been considered. The neutron beams propagate in the earth’s gravitational field, and obey energy conservation law, so we have $$\begin{aligned}
\label{energy1}
mgy&=&\frac{p_2^2-p_1^2}{2m}\nonumber\\
&=&v\Delta p,\end{aligned}$$ where the earth’s rotation is neglected, and $v=(p_1+p_2)/2m$. Substituting (\[energy1\]) into (\[phase3\]), we obtain $$\begin{aligned}
\label{phase4}
\Delta\varphi^{\prime}=\frac{mgyl}{z\hbar
v}=\frac{mg^{\prime}A}{\hbar v},\end{aligned}$$ where $g^{\prime}=g/z, A=yl$. When $z=1$, Eq.(\[phase4\]) returns to (\[phase1\]), which is just the earlier result predicted by usual quantum theory.
As shown by Eqs.(\[phase1\]) and (\[phase4\]), the expression for the corrected phase shift is almost the same as the usual result, except a momentum-dependent factor $z$. The latter can be obtained from the former by replacing $g$ with $g/z$. The GUP’s significance to COW experiment is equivalent to the situation that two neutron beams propagate in a modified gravitational field characterized by the effective field strength $g^{\prime}=g/z$.
Weighting photon
----------------
In 1930, Einstein devised a subtle gedanken experiment for weighting photon[@photon-box1; @photon-box2], and tried to demonstrate the inconsistency of quantum mechanics. Einstein considered a box that contains photon gas and hangs from a spring scale. An ideal clock mechanism in the box can open a shutter. Einstein assumed that the ideal clock could determine the emission time exactly (i.e. $\Delta
t\rightarrow 0$), when a photon was emitted from the box. On the other hand, the energy of the emitted photon can be obtained by measuring the difference in the box’s mass. This seemed to result in $\Delta E\Delta t\rightarrow 0$, and violate the uncertainty relation for energy and time.
However, as pointed out by Bohr[@photon-box1; @photon-box2], Einstein neglected the time-dilation effect, and then his deduction was incorrect. Following from general relativity, the time-dilation is attributed to the difference in the gravitational potential. Two clocks tick at different rates if they are at different heights. For the clock in the box, the time uncertainty due to the vertical position uncertainty $\Delta x$ is given by[@photon-box1; @photon-box2] $$\begin{aligned}
\label{dilation1}
\Delta t=\frac{g\Delta x}{c^2}t,\end{aligned}$$ where $t$ denotes a period of weighting the photon. Bohr argue that the accuracy of the energy of the photon is restricted as $$\begin{aligned}
\label{accuracy0}
\Delta E\geq\frac{c^2\hbar}{gt\Delta x}.
\end{aligned}$$ When Eq. (\[dilation1\]) is considered, the inequality (\[accuracy0\]) gives rise to the uncertainty relation $\Delta E\Delta
t\geq\hbar$, and the consistency of quantum theory is still maintained. Obviously, gravity plays an important role in the Bohr’s argument.
In the following, GUP will be considered along the line of Bohr’s argument, and its significance to gravitation will be analyzed in the gedaken experiment for weighting the photon. We first read the original position of the pointer on the box before the shutter opens. After the photon is released, the pointer moves higher than its original position. In order to lower the pointer to its original position, we hang some little weights on the box. The pointer returns to its original position after a period $t$. The photon’s weight $g\Delta m$ equals the total weight that hangs on the box. Obviously, the accuracy of weighting the photon is determined by the minimum of the added weight. The measurement becomes meaningless, if the added weight is too small to be observable. The weight $g\Delta m$ should be restricted by quantum theory. Let $\Delta x$ denote the accuracy of measuring the position of the pointer(or of the clock), the minimum of the momentum uncertainty is given by $$\begin{aligned}
\Delta p_{min}=\frac{z\hbar}{\Delta x},\end{aligned}$$ where (\[uncertainty2\]) has been considered. Over a period $t$, the smallest weight is $\Delta p_{min}/t$, which is the quantum limit of weighting the photon. Thus we obtain $$\begin{aligned}
\frac{z\hbar}{t\Delta x}=\frac{\Delta p_{min}}{t}\leq g\Delta
m,\nonumber\end{aligned}$$ or $$\begin{aligned}
\label{zhbar1}
z\hbar=\Delta x\Delta p_{min}\leq \Delta m(g\Delta x)t.\end{aligned}$$ Substituting (\[dilation1\]) into (\[zhbar1\]), the latter becomes $$\begin{aligned}
\label{uncertainty3}
z\hbar\leq c^2\Delta m\Delta t=\Delta E\Delta t.\end{aligned}$$ Such a modified uncertainty relation means the shortest interval of the time. It can be explained as follows. Since GUP is required to ensure the shortest observable length, we have (\[uncertainty2\]), and then $z\hbar\geq\ell_\textsc{p}/\Delta p$. The time uncertainty is restricted by $\Delta t\geq z\hbar/\Delta
E\geq\ell_\textsc{p}(\Delta p/\Delta E)\approx
\ell_\textsc{p}(dp/dE)$. $dE/dp$ is the speed of the box, so we obtain $\Delta
t\geq\ell_\textsc{p}/v\geq\ell_\textsc{p}/c=t_\textsc{p}=\sqrt{G\hbar/c^5}$.
Now we turn our attention to the inequality (\[zhbar1\]), from which the accuracy of the energy of the emitted photon is restricted as $$\begin{aligned}
\label{accuracy}
\Delta E\geq\frac{z c^2\hbar}{gt\Delta
x}=\frac{c^2\hbar}{g^{\prime}t\Delta x},\end{aligned}$$ Comparing it with (\[accuracy0\]), the difference is only that $g$ is replaced by $g^{\prime}=g/z$. Let us define $p^{\prime}=\hbar k$, which denotes the canonical momentum and satisfies Heisenberg’s commutation relation as follows $$\begin{aligned}
[\hat{x},\hat{p}^{\prime}]=i\hbar.\end{aligned}$$ The corresponding uncertainty $\Delta p^{\prime}$ can be expressed as $$\begin{aligned}
\label{uncertainty4}
\Delta p^{\prime}=\hbar\Delta k\approx\hbar\frac{dk}{dp}\Delta
p=\Delta p/z,\end{aligned}$$ where (\[mdb3\]) has been considered. On the other hand, the inequality (\[zhbar1\]) can be rewritten as $$\begin{aligned}
\label{zhbar3}
\hbar=\Delta x\Delta p_{min}/z\leq \Delta m(g\Delta x)t/z.\end{aligned}$$ Considering (\[uncertainty4\]) and (\[zhbar1\]), we have $$\begin{aligned}
\label{zhbar4}
\hbar=\Delta x\Delta p^{\prime}_{min}&=&\Delta x\Delta
p_{min}/z\nonumber\\
&\leq& \Delta m(g\Delta x)t/z=\Delta
m(g^{\prime}\Delta x)t.\end{aligned}$$ We find that $g^{\prime}$ is reproduced in the inequality (\[zhbar4\]), accompany with the return of Heisenberg principle. The means that $g^{\prime}$ should be understood in the context of usual quantum theory. According to the formula (\[dilation1\]), when $g\rightarrow g^{\prime}$, the time uncertainty becomes $$\begin{aligned}
\label{dilation2}
\Delta t^{\prime}=\frac{g^{\prime}\Delta x}{c^2}t=\Delta t/z.\end{aligned}$$ Substituting it into (\[zhbar4\]), we obtain $$\begin{aligned}
\label{zhbar5}
\hbar=\Delta x\Delta p^{\prime}_{min}\leq \Delta E\Delta t^{\prime},\end{aligned}$$ as required by usual quantum theory. The consistency of theory is maintained by the transformation: $g\rightarrow g^{\prime}$, when the GUP’s significance is explained in the context of usual quantum mechanics.
In summary, we suggest two pictures for understanding the COW phase shift and the gedanken experiment of weighting the photon. One picture is that GUP is considered directly in a classical gravitational field. In another picture, the usual quantum theory is retained by introducing the effective gravitational field strength $g^{\prime}$. Two pictures are equivalent, since an observer cannot distinguish the effect of GUP from the effective field strength. This equivalence inspires an effective Newton constant $G^{\prime}$, since the effective field strength can be expressed as $$\begin{aligned}
g^{\prime}=g/z=\frac{GM}{zR^2}=\frac{G^{\prime}M}{R^2}, \nonumber\end{aligned}$$ where $G^{\prime}=G/z$ is just the same as the suggestion in Ref.[@lx2].
In view of the above analysis, we introduce two working hypotheses: (i) the matters obey Heisenberg’s uncertainty principle; (ii) quantum spacetimes are characterized by the GUP-inspired effective Newton constant. They are the basis for the following discussions. When $G$ is replaced by $G^{\prime}$, we obtain a modified Schwarzschild metric as follows $$\begin{aligned}
\label{metric1}
ds^2&=&-\left(1-\frac{2G^{\prime}M}{c^2r}\right)c^2dt^2+\left(1-\frac{2G^{\prime}M}{c^2r}\right)^{-1}dr^2+r^2d\Omega,\\
G^{\prime}&=&G/z.\nonumber
\end{aligned}$$ This metric describes a family of spacetimes that depend on different scales of momentum. In the following section, we will propose a characteristic momentum to incorporate quantum effect with geometric character of gravity.
Gravitational tidal force and the characteristic momentum
=========================================================
Now we consider a black hole described by (\[metric1\]), with $z=1+\alpha \ell_\textsc{p}^2 p^2/\hbar^2$. Let $T$ denote the black hole temperature, and the characteristic momentum is identified with $k_BT/c$[@ling1; @ling2], the metric (\[metric1\]) becomes $$\begin{aligned}
\label{metric2}
ds^2&=&-\left(1-\frac{2G\widetilde{M}}{c^2r}\right)c^2dt^2+\left(1-\frac{2G\widetilde{M}}{c^2r}\right)^{-1}dr^2+r^2d\Omega,\\
\widetilde{M}&=&\frac{M}{1+\alpha\ell_\textsc{p}^2k_B^2 T^2/c^2\hbar^2}.\nonumber
\end{aligned}$$ The horizon is located by $r_{T}=2G\widetilde{M}/c^2$. The temperature, proportional to the surface gravity, is determined by $$\begin{aligned}
T=\frac{m_\textsc{p}^2c^2}{8\pi k_B
\widetilde{M}}=\frac{m_\textsc{p}^2c^2}{8\pi k_B
M}\left(1+\frac{\alpha \ell_\textsc{p}^2k_B^2
T^2}{c^2\hbar^2}\right),\end{aligned}$$ where $m_\textsc{p}=\sqrt{\hbar c/G}$ is the Planck mass. So we obtain $$\begin{aligned}
\label{temperature1}
T=\frac{4\pi M-\sqrt{(4\pi M)^2-\alpha m_\textsc{p}^2}}{\alpha
k_B/c^2},\end{aligned}$$ which returns to the usual formula $T_H=m_\textsc{p}^2c^2/(8\pi k_B
M)$, as $\alpha\rightarrow 0$. Except an inessential factor, the modified temperature (\[temperature1\]) is consistent with the previous work in the literarure[@pschen1; @vagenas2; @lx1; @ling1; @ling2]. Certainly, those old problems have yet not been solved. The expression (\[temperature1\]) gives rise to the maximum temperature when the mass approaches the minimal value $\sqrt{\alpha}m_\textsc{p}/{4\pi}$. Furthermore, the metric (\[metric2\]) indicates that there is still a singularity at $r=0$. However, the minimal mass means a lower bound on the size of black hole, $r_T\geq
\sqrt{\alpha}\ell_\textsc{p}/{4\pi}$. It is of order of the shortest observable distance derived from GUP. This make us believe that quantum spacetime is still characterized by the effective Newton constant $G^{\prime}=G/z$, if an appropriate characteristic scale is taken into account. The shortage of the metric (\[metric2\]) may be attributed to the fact that the black hole temperature is not an universal scale of meaning, since it can’t describe an ordinary star. Moreover, the black hole temperature is position independent, and doesn’t reflect the difference between strong gravitational field and weak field. It is necessary to reinvest the momentum scale with new physical meaning, if we expect for something beyond the previous efforts. As an observable quantity, $p^2\geq\Delta p^2$. In view of the limitation of the black hole temperature, we require the quantum fluctuation $\Delta p$ to satisfy some reasonable expectations.
Firstly, $\Delta p$ should be associated with the gravity, and should increase with the strength of the gravitational field, i.e. $\Delta p\sim r^{-s}$, $s>0$.
Secondly, $\Delta p$ should play a crucial role that improves the spacetime singularity, and make the black hole regular. As argued in Refs.[@reuter; @hayward; @lx2], the asymptotic behavior of the regular potential must satisfy $\phi\rightarrow r^{2+\delta}$, as $r\rightarrow 0$. This demands $\Delta p\sim r^{-{3/2-\delta}}$, and $\delta\geq 0$.
Thirdly, the minimal value of $\Delta p$ is intrinsic, and reflect the universal property of the gravitational fields of black holes and ordinary stars. According to general relativity, the gravitational field is regarded as the curved spacetime characterized by Riemann tensor $R_{\rho\lambda\mu\nu}$. This suggests $\Delta p$ be associated with those quantities constructed by Riemann tensor. Such a characteristic momentum may be estimated by combining the gravitational tidal force with quantum theory, since the tidal force is associated with the curvature of spacetime[@mtw; @liang].
Let us consider a pair of virtual particles with energy $\Delta E$. When the virtual particles are separated by a distance $\Delta x$, according to the geodesic deviation equation, the tidal force reads[@mtw; @liang; @nalikar] $$\begin{aligned}
F=\frac{2GM}{r^3}\left(\frac{\Delta E}{c^2}\right)\Delta x.\end{aligned}$$ Let $\Delta t$ denote the life-time of the virtual particles, the momentum uncertainty due to the tidal force is given by $$\begin{aligned}
\label{tidal1}
\Delta p=F\Delta t=\frac{2GM}{r^3}\left(\frac{\Delta
E}{c^2}\right)\Delta t\Delta x.\end{aligned}$$ The observability requires $\Delta p\Delta x\geq\hbar,~\Delta
E\Delta t\geq\hbar$, when the virtual particles are subject to the tidal force and become real. Following from (\[tidal1\]), we obtain $$\begin{aligned}
\label{tidal2}
(\Delta p)^2&\geq&\frac{\hbar\Delta p}{\Delta x}=\frac{\hbar F}{\Delta x}\Delta t\nonumber\\
&=&\frac{2\hbar}{c^2}\left(\frac{GM}{r^3}\right)\Delta E\Delta
t\geq\frac{2\hbar^2}{c^2}\left(\frac{GM}{r^3}\right).\end{aligned}$$ The right hand side of the inequality suggests a characteristic momentum, $\Delta p_{m}\sim\sqrt{GM\hbar^2/c^2r^3}$. We find that $\Delta p_{m}\rightarrow 0$ as $G\rightarrow 0$, which is associated with a free particle traveling in the flat spacetime. This characteristic scale can be understood as the minimal momentum of those particles produced from the quantum fields in the curved spacetime.
We can also analyze a real particle which is detected by a photon with energy $\Delta E$. Let $\Delta x$ denote the uncertainty in the position of the particle, the momentum uncertainty of the particle is given by $$\begin{aligned}
\label{tidal3}
\Delta \tilde{p}\geq\frac{\hbar}{\Delta x}+\frac{2GMm}{r^3}\Delta
x\Delta t,\end{aligned}$$ where $m$ is the mass of the particle, and $\Delta
t\geq\hbar/\Delta E$ is the characteristic time in the process of the photon-particle collision. On the right hand side of the inequality (\[tidal3\]), the first and the second terms belong different stories respectively. The second term is attributed to the tidal effect of gravity, and it vanishes in a flat spacetime. Following from (\[tidal3\]), we obtain $$\begin{aligned}
\label{tidal4}
\Delta \tilde{p}\geq 2\sqrt{\frac{2G M\hbar m\Delta t}{r^3}}\geq
2\sqrt{\frac{2GM\hbar^2}{r^3}\left(\frac{m}{\Delta E}\right)},\end{aligned}$$ where the time-energy uncertainty relation is considered. In order to avoid the production of new particles, the energy should be restricted as $\Delta E< mc^2$, otherwise it becomes meaningless to measure the position of the particle[@lifshitz]. So we obtain $$\begin{aligned}
\label{tidal5}
\Delta \tilde{p}> 2\sqrt{\frac{2GM\hbar^2}{c^2r^3}}.\end{aligned}$$ The difference between (\[tidal2\]) and (\[tidal5\]) is only a constant coefficient. This inessential difference is caused by a rough estimate of the amplitude of $\Delta E$. It will vanish, if we take a different estimate, such as $\Delta E<mc^2/4$.
In the above discussion, we don’t discriminate the black hole from an ordinary star, so the characteristic momentum appeared in (\[tidal2\]) is suitable for both of the two spacetimes. A similar scale has also been suggested by a different way[@reuter], but its physical meaning is different from our understanding. As one of two scales suggested in Ref.[@reuter], it is identified with the inverse of the proper time of an observer falling into the Schwarzschild black hole. Obviously, it is different from that scale of an ordinary star.
Identifying the characteristic scale with the right hand side of (\[tidal2\]), we obtain an effective Newton constant as follows $$\begin{aligned}
\label{rung2}
G^{\prime}=\frac{G}{1+2\alpha\ell_\textsc{p}^2\Delta
p_m^2/\hbar^2}=\frac{G}{1+2\alpha\ell_\textsc{p}^2GM/r^3c^2}.\end{aligned}$$ Different from Ref.[@reuter], the scale $\Delta p\sim 1/r$ is no longer considered in the following discussions. This is because it is motivated by Heisenberg principle, while Heisenberg principle has been incorporated with the tidal effect in the inequalities (\[tidal2\]) and (\[tidal3\]). We will explore a black hole characterized by the effective Newton constant as presented by (\[rung2\]).
Quantum tunneling from the regular black hole
=============================================
In this section, we take the Planck units, $G=\hbar=c=k_B=1$. Substituting (\[rung2\]) into (\[metric1\]), we obtain a modified Schwarzschild black hole as follows $$\begin{aligned}
\label{metric3}
ds^2&=&-\left(1+2\Phi\right)dt^2+\left(1+2\Phi\right)^{-1}dr^2+r^2d\Omega,\\
\Phi&=&-\frac{Mr^2}{r^3+2\alpha M}.\nonumber\end{aligned}$$ It is just the minimal model of the regular black hole suggested in Ref.[@hayward]. Let $\rho$ denote the radius of this black hole and satisfy $g^{11}(\rho)=0$, we have $$\begin{aligned}
\label{horizon1}
1-\frac{2M\rho^2}{\rho^3+2\alpha M}=0.\end{aligned}$$ The horizon is located by $$\begin{aligned}
\rho=\frac{2M}{3}+\frac{4M}{3}\cos\left[\frac{1}{3}\arccos\left(1-\frac{27\alpha}{8M^2}\right)\right],\end{aligned}$$ provided $M\geq M_c=27\alpha/16$. When $M\leq M_c$, the metric (\[metric3\]) doesn’t describe a black hole, since the equation (\[horizon1\]) has no positive solution and then the horizon is absent in this spacetime[@hayward]. The critical mass is the lower bond on the mass of an object that forms a black hole. Corresponding to this critical mass, there is a minimal radius of the black hole, $\rho_{min}=4M_c/3=\sqrt{3\alpha}$. In the following, we will investigate the quantum tunneling of this regular black hole, and focus on the question of whether the tunneling probability is regularized by the minimal length.
In the tunneling picture of black hole radiation[@parik; @parikh2], the tunneling probability is determined by the imaginary part of the action for a particle which tunnels through the horizon along a classically forbidden trajectory. At the zeroth order WKB approximation, the tunneling probability is suppressed by the change in the Bekenstein-Hawking entropy. This is consistent with the unitarity of quantum theory. When the second order correction is considered, the tunneling probability is given by[@zjy1] $$\begin{aligned}
\label{tunnel1}
\Gamma\sim\frac{\rho_i^2}{\rho_f^2}\exp[-2\textbf{Im}(S_0-S_2)],\end{aligned}$$ where $\rho_i$ denotes the initial radius of black hole in the tunneling process, and $\rho_f$ the final radius. $S_0-S_2$ is the action for a particle crossing the horizon from $\rho_i$ to $\rho_f$. Concretely speaking, $S_0$ is associated with the zeroth order term of WKB wave function, and $S_2$ is related to the second order correction. The first order term $S_1$ doesn’t appear in the imaginary part of the action, since it is real. In order to evaluate the emission rate of the regular black hole, we first introduce the Painleve type coordinate[@parik; @vagenas4] $$\begin{aligned}
\tilde{t}=t+\int\frac{\sqrt{-2\Phi}}{1+2\Phi}dr.\nonumber\end{aligned}$$ The metric (\[metric3\]) is therefore rewritten as $$\begin{aligned}
ds^2=-(1+2\Phi)d\tilde{t}^2+2\sqrt{-2\Phi}d\tilde{t}dr+dr^2+r^2d\Omega.\end{aligned}$$ It is appropriate for describing the particle which tunnels through the horizon, since the coordinate singularity has been removed. Setting $ds^2=0=d\Omega$, we obtain the equation of the radial null geodesics as follows $$\begin{aligned}
\label{geodesics1}
\dot{r}=\frac{dr}{d\tilde{t}}=1-\sqrt{-2\Phi},\end{aligned}$$ where the ingoing geodesics is neglected. When a particle is emitted from the black hole and the energy conservation is considered, a shell with energy $\omega^{\prime}$ travals in a spacetime of mass $M^{\prime}=M-\omega^{\prime}$. So we have $$\begin{aligned}
\label{tildephi}
1+2\Phi&=&\frac{r^3-2M^{\prime}r^2+2\alpha M^{\prime}}{r^3+2\alpha M^{\prime}}\nonumber\\
&=&-\frac{r^2-\alpha}{r^3+2\alpha
M^{\prime}}\left(2M^{\prime}-\frac{r^3}{r^2-\alpha}\right).\end{aligned}$$ The zero order action is given by $$\begin{aligned}
\label{action0}
S_0=\int_{\rho_i}^{\rho_f}{S_0^{\prime}}dr=\int_{\rho_i}^{\rho_f}p_rdr,\end{aligned}$$ where $$\begin{aligned}
\label{pr1}
S_0^{\prime}=p_r=\int\frac{dM^{\prime}}{\dot{r}}=\int_{M}^{M-\omega}\frac{dM^{\prime}}{1-\sqrt{-2\Phi}}.\end{aligned}$$ Substituting (\[tildephi\]) into (\[pr1\]), we obtain $$\begin{aligned}
p_r&=&\int_{M}^{M-\omega}\frac{1+\sqrt{-2\Phi}}{1+2\Phi}dM^{\prime}\nonumber\\
&=&-\int_{M}^{M-\omega}\frac{(1+\sqrt{-2\Phi})(r^3+2\alpha
M^{\prime})}{(r^2-\alpha)[2M^{\prime}-r^3/(r^2-\alpha)]}dM^{\prime}.\end{aligned}$$ There exists a singularity at $2M^{\prime}=r^3/(r^2-\alpha)$. In order for the positive frequency modes to decay with time[@parik], we deform the contour into the lower half $\omega^{\prime}$ plane, or into the upper half $M^{\prime}$ plane. By residue theorem, we obtain $$\begin{aligned}
p_r&=&\left(\frac{-i\pi}{2}\right)~\frac{2\times[r^3+\alpha r^3/(r^2-\alpha)]}{r^2-\alpha}\nonumber\\
&=&(-i\pi)\frac{r^5}{(r^2-\alpha)^2}.\end{aligned}$$ Substituting it into (\[action0\]), we get the imaginary part of the action as follows $$\begin{aligned}
\label{ims0}
\textbf{Im}S_0=-\frac{\pi}{2}\left[r^2+2\alpha\ln(r^2-\alpha)-\frac{\alpha^2}{r^2-\alpha}\right]\Bigg{|}_{\rho_i}^{\rho_f}.\end{aligned}$$ Following the procedure of WKB method applied in Ref.[@zjy1], we can also evaluate the higher order terms of the action. The first order term is determined by the following equation $$\begin{aligned}
S_1^{\prime}=-\frac{S_0^{\prime\prime}}{2S_0^{\prime}}=-\frac{r^2-5\alpha}{2r(r^2-\alpha)}.\nonumber\end{aligned}$$ So we have $$\begin{aligned}
S_1^{\prime\prime}=\frac{r^4-14\alpha
r^2+5\alpha^2}{2r^2(r^2-\alpha)^2},\nonumber\end{aligned}$$ and then $$\begin{aligned}
S_2^{\prime}=-\frac{S_1^{\prime
2}+S_1^{\prime\prime}}{2S_0^{\prime}}=(-i)\times\frac{3r^4-38\alpha
r^2+35\alpha^2}{8\pi r^7}.\end{aligned}$$ The imaginary part of the second order term is given by $$\begin{aligned}
\label{ims2}
\textbf{Im}S_2&=&\textbf{Im}\int_{\rho_i}^{\rho_f}S_2^{\prime}dr\nonumber\\
&=&\frac{1}{96\pi}\left(\frac{18}{r^2}-\frac{114\alpha}{r^4}+\frac{70\alpha^2}{r^6}\right)\Bigg{|}_{\rho_i}^{\rho_f}.\end{aligned}$$ Substituting (\[ims0\]) and (\[ims2\]) into (\[tunnel1\]), and considering $\rho_i^2/\rho_f^2=\exp\ln (\rho_i^2/\rho_f^2)$, we obtain the tunneling probability accurate to the second order correction, $\Gamma\sim e^{\Delta S}$, where $$\begin{aligned}
\label{deltas}
\Delta S=\left[\pi r^2-\ln r^2
+\frac{3}{8\pi
r^2}+2\alpha\pi\ln(r^2-\alpha)-\frac{\alpha^2\pi}{r^2-\alpha}-\frac{19\alpha}{8\pi
r^4}+\frac{35\alpha^2}{24\pi r^6}\right]\Bigg{|}_{\rho_i}^{\rho_f}.\end{aligned}$$ In the consideration of the unitarity of quantum theory, $\Delta S$ should be understood as the change in the entropy of the regular black hole. The entropy, including the first and the second order corrections, reads $$\begin{aligned}
\label{regulars}
S=\pi \rho^2-\ln \rho^2
+\frac{3}{8\pi
\rho^2}+2\alpha\pi\ln(\rho^2-\alpha)-\frac{\alpha^2\pi}{\rho^2-\alpha}-\frac{19\alpha}{8\pi
\rho^4}+\frac{35\alpha^2}{24\pi \rho^6},\end{aligned}$$ where $\rho$ is the radius of the black hole. The first three terms are similar to the expression for the Schwarzschild black hole[@zjy1], while the last four terms are new. New corrections are relevant to the parameter $\alpha$, and denote the difference between the regular black hole and the Schwarzschild black hole.
Let us make some remarks on the expressions (\[deltas\]) and (\[regulars\]). Let us consider the thermodynamical entropy of the regular black hole. As the inverse period of the imaginary time of the regular spacetime (\[metric3\]), the black hole temperature is given by $$\begin{aligned}
T&=&\frac{1}{2\pi}\left(\frac{d\Phi}{dr}\right)_{r=\rho}\nonumber\\
&=&\frac{1}{8\pi
M}-\frac{\alpha}{2\pi\rho^3}=\frac{\rho^2-3\alpha}{4\pi\rho^3},\end{aligned}$$ where (\[horizon1\]) has been considered. The thermodynamical entropy is defined as $$\begin{aligned}
S^{(0)}&=&\int\frac{dM}{T}=\int T^{-1}\left(\frac{dM}{d\rho}\right)d\rho\nonumber\\
&=&2\pi\int\frac{\rho^5d\rho}{(\rho^2-\alpha)^2}\nonumber\\
&=&\pi\left[\rho^2+2\alpha\ln(\rho^2-\alpha)-\frac{\alpha^2}{\rho^2-\alpha}\right],\end{aligned}$$ which is different from (\[deltas\]). However, it is consistent with the entropy derived from the zero order action of WKB method, as shown by (\[ims0\]). This is similar to the Schwarzschild black hole: the zero order action for the tunneling particle is related to the change in the Bekenstein-Hawking entropy[@parik; @zjy1].
For the Schwarzschild black hole, the emission rate accurate to the second order approximation, is determined by the first three terms in (\[deltas\]). Since classical general relativity doesn’t restrict the size of black hole, $\Delta S$ and $\Gamma$ become divergent as $\rho_f\rightarrow 0$. However, the tunneling probability of the regular black hole is finite, because it is regularized by the minimal radius of the horizon. This conclusion is nontrivial, in view of the subtle relation between the entropy expression (\[regulars\]) and the minimal radius $\rho_{min}$. If $\rho_{min}$ is allowed to be less than $\sqrt{\alpha}$, the entropy would be ill defined for the black hole of radius $\rho=\sqrt{\alpha}$, because of the divergence of the fourth and the fifth terms in (\[regulars\]). It is gratifying that $\rho_{min}=\sqrt{3\alpha}>\sqrt{\alpha}$, and those dangerous terms such as $(\rho^2-3\alpha)^{-1}$ don’t appear in the entropy expression.
According to the third law of thermodynamics, the entropy vanishes when a system of matter is in the ground state and its temperature approaches zero. For a given excited state, the probability of the transition to the ground state should be minimal, because it is greatly suppressed by the change in the entropy. The regular black hole has similar property, if the parameter $\alpha$ is not too small. This is because the entropy expression (\[regulars\]) is a monotonic increasing function of the horizon area.[^7] For an initial black hole with radius $\rho_i$, the minimal value of the tunneling probability is given by $$\begin{aligned}
\label{mgamma}
\Gamma\sim \exp\left[-\pi \rho_i^2+\ln \rho_i^2
-\frac{3}{8\pi
\rho_i^2}-2\alpha\pi\ln(\rho_i^2-\alpha)+\frac{\alpha^2\pi}{\rho_i^2-\alpha}+\frac{19\alpha}{8\pi
\rho_i^4}-\frac{35\alpha^2}{24\pi \rho_i^6}\right],\end{aligned}$$ which points to the black hole remnant with final radius $\rho_f=\sqrt{3\alpha}$.
In the expression (\[regulars\]), the fourth and the fifth terms is a part of the thermodynamical entropy of the regular black hole. It is interesting and confusing that they tend to cancel out the similar corrections to the entropy of the Schwarzschild black hole, such as the second and the third terms in (\[regulars\]). This fact might indicate a subtle correlation between quantum spacetimes and the quantum matters, but we don’t know how to explain it. We also notice that the regular black hole gives rise to the higher order corrections to the entropy, such as the last two terms in (\[regulars\]). We predict that the similar and opposite corrections might appears in the tunneling probability of a Schwarzschild black hole, when the fourth order WKB approximation is considered.
Summary and outlook
===================
This work involves two parts. The first part is devoted to the question of what is the significance of the GUP for the quantum spacetime. The answer may point to the a scale-dependent Newton constant, which is motivated by analyzing the role that the GUP plays in the COW phase shift and Einstein-Bohr’s Gedanken for weighting photon. It is consistent with our previous suggestion in Ref.[@lx2]. The minimal model of the regular black hole can be reproduced by considering the simplest GUP and a momentum scale associated with the tidal force. The second part is to calculate the tunneling probability accurate to the second order WKB approximation. The tunneling probability is regular, because the black hole has a nonzero minimal radius. Not only this, the tunneling probability of an initial black hole is minimized by the black hole remnant, if the parameter $\alpha$ is of order of the unity. In other words, the tunneling probability is minimal, if the final state of the black hole is a remnant.
Let us consider the matter source of the regular black hole. In this paper, the quantum spacetime is understood by connecting the GUP with the running of Newton constant. It reflects the quantum gravitational effects on the classical spacetime. According to the general theory of relativity, the effective stress-energy tensor can be derived from Einstein’s field equation, when the regular black hole is regarded as an input. The quantum gravitational effects are simulated by a matter fluid described by the effective stress-energy tensor[@reuter; @hayward]. We hope that this matter fluid can be reproduced from the GUP dominated vacuum fluctuations. This problem will be investigated in the future.
Besides constructing the above regular black hole, we also explore a theory of modified gravity. This alternative theory is based on a generalization of the effective Newton constant, and it may be characterized by a modified Hilbert-Einstein action as follows $$\begin{aligned}
\label{actiong1}
I^{\prime}=\int\frac{R-2\Lambda}{16\pi G^{\prime}}\sqrt{-g}d^4{x},\end{aligned}$$ where $G=c=1$, $G^{\prime}=z^{-1}(p)$, and $\Lambda$ is the cosmological constant. In order for the Lagrangian to be an invariant, the momentum scale $p$ is restricted to be a scalar. For a Schwarzshild spacetime, $p^2\sim M/r^3$, as argued in the section 3. This suggests that the characteristic momentum should be identified as $p\sim\sqrt{K}$, and then $z=z(K)$, where $K$ is the square root of the Kretschmann scalar. Considering the simplest GUP\[as given by (\[commut0\])\], the gravitational action can be expressed as $$\begin{aligned}
\label{actiong3}
I=\frac{1}{16\pi G}\int(1+\gamma K)(R-2\Lambda)\sqrt{-g}d^4{x},\end{aligned}$$ where $\gamma$ is a parameter, which is not necessarily the same as that in the metric (\[metric3\]). The action (\[actiong3\]) belongs to a class of more general theories of modified gravity[@carr; @clifton; @gcj], and then the field equation is given by $$\begin{aligned}
\label{field1}
(1+\gamma K)(G_{\mu\nu}+\Lambda g_{\mu\nu})+\gamma H_{\mu\nu}=8\pi
T_{\mu\nu},\end{aligned}$$ where $G_{\mu\nu}$ is the Einstein tensor, and $$\begin{aligned}
H_{\mu\nu}=\frac{R-2\Lambda}{K}R_{\rho\lambda\sigma\mu}R^{\rho\lambda\sigma}_{~~~\nu}
+(g_{\mu\nu}\nabla_{\sigma}\nabla^{\sigma}-\nabla_{\mu}\nabla_{\nu})K-2\nabla_\rho\nabla_{\sigma}\left[\frac{R-2\Lambda}{K}R^{\rho~~\sigma}_{({\mu\nu})~}\right].\end{aligned}$$ Direct calculation shows that the metric (\[metric3\]) is not a solution for the field equation (\[field1\]). It is not strange, since the metric (\[metric3\]) and the equation (\[field1\]) are suggested along different lines of argument, even though they are motivated by the effective Newton constant. However, the regular spacetime (\[metric3\]) has a de Sitter core near $r=0$, which satisfies the field equation (\[field1\]). This implies that the field equation (\[field1\]) permits the existence of the regular black holes. We also take notice of those terms associated with the cosmological constant $\Lambda$ in (\[field1\]), i.e. $(1+\gamma K)\Lambda g_{\mu\nu}$. Usually, the first term $\Lambda g_{\mu\nu}$ is utilized to cancel out the huge contribution from the vacuum energy on the right hand side of the field equation, where the bare $\Lambda$ must be of order of unity. Thus the second term of $\gamma K\Lambda$ play the role of the effective cosmological constant. It is interesting that this term is of order of the observed value. The modified gravity with the square root of Kretschmann scalar seems to be ignored in the literature. The field equation (\[field1\]) and relevant problems will be discussed in detail elsewhere. We hope that the spacetime singularities and the cosmological constant problem can be improved in this alternative theory.
Acknowledgments {#acknowledgments .unnumbered}
===============
X. Li thanks X. J. Yang and Q. J. Cao for their helps and useful discussions. This work is supported by Natural Science Foundation of Science Foundation of China(grants No.11373020, No.11575195), and Natural Science Foundation of Zhejiang Province of China(grant No.LY14A030001).
[99]{} C.W.Misner, K.S.Thorne, J. A. Wheeler, [*Gravitation*]{}, W.H.FREEMAN and Company, 1970. S.W.Hawking and G.F.R. Ellis, [*The Large Scale Structure of Spacetime*]{}, Combridge Universe Press, 1973. S. W. Hawking, Commun. Math. Phys. 43(1975), 199. M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85(2000), 5042. M. K. Parikh, Gen.Rel.Grav.36(2004), 2419. arXiv:hep-th/0405160. E. C. Vagenas, Phys.Lett. B559(2003), 65. arXiv:hep-th/0209185. M. Arzano, A. J. M. Medved and E. C. Vagenas, JHEP, 0509(2005), 037. J. Y. Zhang and Z. Zhao, Phys.Lett. B638 (2006), 110. arXiv:gr-qc/0512153. Jingyi Zhang, Phys. Lett. B668(2008), 353. L. J. Garay, Int. J. Mod. Phys. A10(1995), 145, and references therein. R. J. Adler, P. Chen and D. I. Santiago, Gen. Rel. Grav. 33 (2001), 2101. A. J. M. Medved and E. C. Vagenas, Phys. Rev. D70(2004), 124021. Li Xiang and X. Q. Wen, JHEP, 10(2009),046. J. Bardeen, Proc. GR5, Tbilisi, USSR(1968). A. Bonanno and M. Reuter, Phys. Rev. D62(2000),043008. P. Nicolini, A. Smailagic and E. Spallucci, Phys. Letts. B632(2006), 547. S. A. Hayward, Phys. Rev. Lett. 96(2006), 031103. Y. S. Myung, Y-W. Kim and Y-J. Kim, Phys. Letts. B656(2007), 221. Li Xiang, Yi Ling and You-Gen Shen, Int. J. Mod. D22(2013), 1342016. H. Culetu, Int. J. Theor. Phys. 54(2015), 2855. arXiv:1408.3334\[gr-qc\]. L. Balart, E. C. Vagenas, Phys. Lett. B730(2014), 14. arXiv:1401.2136\[gr-qc\]. M. Amir, S. G. Ghosh, Phys. Rev. D 94(2016), 024054. arXiv:1603.06382\[gr-qc\]. S. Benczik, L. N. Chang, D. Minic and et al, Phys.Rev.D66(2002), 026003. S. K. Rama, Phys. Letts. B519(2001), 103. U. Harbach and S. Hossenfelder, Phys.Lett. B632(2006), 379. A. Kempf, G. Mangano and R.B. Mann, Phys. Rev. D52(1995), 1108. D.V. Ahluwalia, Phys. Letts. A275(2000), 31. S. Deb, S. Das and E. C. Vagenas, Phys. Lett. B755(2016), 17. arXiv:1601.07893\[gr-qc\]. C. Cohen-Tannoudji, B. Diu and F. Laoe, [*Quantum Mechanics*]{}, Hermann, 1977. Colella, A. W. Overhauser and S. A. Werner, Phys. Rev. Lett. 34(1975), 1472. D. M. Greenberg and A. W. Overhauser, Rev. Mod. Phys.51(1979), 43. Guang Hong, [*Elementary Concepts in Quantum Mechanics*]{}(in chinese), High Education Press, 1990. A. Saha, Phys. Rev. D. 89, 025010 (2014). arXiv:1306.4202\[hep-th\]. . J. Kalckar Ed.Vol.7. Elsevier, Amsterdam, 1996. Y. Aharonov and D. Rohrlich, [*Quantum Paradoxes, quantum theory for the perplexed*]{}, WILEY-VCH Verlag GmbH $\&$ CO. KGaA, 2005. Y. Ling, B. Hu and X. Li, Phys. Rev. D73(2006), 087702. Y. Ling, X. Li, and H. B. Zhang, Mod. Phys. Letts. A22(2007), 2749. C. B. Liang and B. Zhou, [*Elementary Differential Geometry and General Relativity*]{}(in chinese), Scinence Press, 2006. J. V. Narlikar and T. Padmanabhan, [*Gravity, Gauge Theories and Quantum Cosmology*]{}, Reidel, Dordrecht, 1986. E. M. Lifshitz, L.P. Pitaevskii and V. B. Berestetskii, [*Quantum Electrodynamics*]{}, Reed Educational and Professional Publishing, 1982. S. M. Carroll, et al., Phys. Rev. D71(2005), 063513. axXiv:astro-ph/0410031. T. Clifton and J. D. Barrow, Phys. Rev. D72(2005), 123003. arXiv:gr-qc/0511076. C.J. Gao, Phys. Rev. D86(2012), 103512. arXiv:1208.2790\[gr-qc\].
[^1]: xiang.lee@163.com
[^2]: lingy@ihep.ac.cn
[^3]: ygshen@shao.ac.cn
[^4]: czlbj20@163.com
[^5]: hsh@jmu.edu.cn
[^6]: lance\_xu01@163.com
[^7]: It can be shown by numerical method that $dS/d{\rho}
>0$, when the parameter satisfies $\alpha>0.024.$
|
---
author:
- |
Jason Krone\
Amazon AI\
Palo Alto, CA\
[kronej@amazon.com]{}\
Yi Zhang\
Amazon AI\
Seattle, WA\
[yizhngn@amazon.com]{}\
Mona Diab\
The George Washington University\
Washington, DC\
[mtdiab@gwu.edu]{}\
bibliography:
- 'anthology.bib'
- 'acl2020.bib'
title: Learning to Classify Intents and Slot Labels Given a Handful of Examples
---
|
---
abstract: |
We study large-scale winds driven from uniformly bright self-gravitating disks radiating near the Eddington limit. We show that the ratio of the radiation pressure force to the gravitational force increases with height to a maximum of twice its value at the disk surface. Thus, uniformly bright self-gravitating disks radiating at the Eddington limit are fundamentally unstable to driving large-scale winds. These results contrast with the spherically symmetric case, where super-Eddington luminosities are required for wind formation. We apply this theory to galactic winds from starburst galaxies that approach the Eddington limit for dust. For hydrodynamically coupled gas and dust, we find that the asymptotic velocity of the wind is $v_{\infty}\simeq3\langle v_{\rm rot}\rangle$ and that $v_{\infty}\propto {\rm SFR}^{0.36}$, where $\langle v_{\rm
rot}\rangle$ is the mean disk rotation velocity and SFR is the star formation rate, both of which are in agreement with observations. However, these results of the model neglect the gravitational potential of the surrounding dark matter halo and a (potentially massive) old stellar bulge, which both act to decrease $v_\infty$. A more realistic treatment shows that the flow can either be unbound, or bound, forming a “fountain flow” with a typical turning timescale of $t_{\rm turn}\sim0.1-1$ Gyr, depending on the ratio of the mass and radius of the starburst disk relative to the total mass and break (or scale) radius of the dark matter halo or bulge. We provide quantitative criteria and scaling relations for assessing whether or not a starburst of given properties can drive unbound flows via the mechanism described in this paper. Importantly, we note that because $t_{\rm turn}$ is longer than the star formation timescale (gas mass/star formation rate) in the starbursts and ultra-luminous infrared galaxies for which our theory is most applicable, if starbursts are selected as such, they may be observed to have strong outflows along the line of sight with a maximum velocity $v_{\rm max}$ comparable to $\sim3\langle v_{\rm rot}\rangle$, even though their winds are in fact bound on large scales.
author:
- 'Dong Zhang & Todd A. Thompson'
title: 'Radiation Pressure Driven Galactic Winds from Self-Gravitating Disks'
---
Introduction
============
Galactic-scale winds are ubiquitous in starburst galaxies in both the local and high-redshift universe (@Heckman90; @Heckman00; @Pettini01 [@Pettini02]; @Shapley03; @rupke05; @Sawi08). They are important for determining the chemical evolution of galaxies and the mass-metallicity relation (@DS86; @Tremonti04; @Erb06; @finlator; @Peeples11), and as a primary source of metals in the intergalactic medium (IGM; e.g., @aguirre01). Moreover, galactic winds are perhaps the most extreme manifestation of the feedback between star formation in a galaxy and its interstellar medium (ISM). This feedback mechanism is crucial for understanding galaxy formation and evolution over cosmic time (@SH03; @OD06; @OD08; @Oppen10).
The most well-developed model for galactic winds from starbursts is the supernova-driven model of [@CC85], which assumes that the energy from multiple stellar winds and core-collapse supernovae in the starburst is efficiently thermalized. The resulting hot flow drives gas out of the host, sweeping up the cool ISM (@Heckman93; @SS00; @strickland02; @SH09; @Fujita09). Although this model is successful in explaining the X-ray properties of starbursts, the recent observational results that galaxies with higher star formation rates (SFRs) accelerate the absorbing cold gas clouds to higher velocities ($v\propto$SFR$^{0.35}$) and that the wind velocity is correlated with the galaxy escape velocity may challenge the traditional hypothesis that the cool gas is accelerated by the ram pressure of the hot supernova-heated wind, whose X-ray emission temperature varies little with SFR, circular velocity, and host galaxy mass, indicating a critical galaxy mass below which most of the hot wind escapes. These new observations may instead favor momentum-driven or radiation pressure-driven models for the wind physics (e.g., @Martin99; @Martin05; @Weiner09; Murray, Quataert & Thompson 2005 \[hereafter MQT05\]).
The model that galactic winds may be driven by momentum deposition provided by radiation pressure from the continuum absorption and scattering of starlight on dust grains was developed by MQT05. However, the conclusions of MQT05 are based on an assumed isothermal potential and spherical geometry, and are thus most appropriate for bright elliptical/spheroidal galaxies in formation. On the other hand, the theory of radiation-driven winds from accretion disks from the stellar to galactic scales has also been studied (e.g., @TF96 [@TF98]; @Proga98 [@Proga99]; @Proga00 [@Proga03]), but none of these works considered radiation from self-gravitating disks.
In this paper we answer the question of whether or not large-scale winds can be driven by radiation pressure from self-gravitating disks radiating near the Eddington limit. These considerations are motivated by the work of Thompson, Quataert & Murray (2005; TQM05), who argued that radiation pressure on dust is the dominant feedback mechanism in starbursts, and that in these systems star formation is Eddington-limited. For simplicity, throughout this paper we assume that the disk is of uniform brightness and surface density. In §2, we show that such disks are fundamentally unstable to wind formation because the radiation pressure force dominates gravity in the vertical direction above the disk surface. This result is qualitatively different from the well-known case in spherical symmetry. In §2, we also discuss the applicability of this model to starbursts and then calculate the terminal velocity of the wind along the disk pole, and its dependence on both the SFR and galaxy escape velocity. In §3, we assess the importance of a spherical stellar bulge and dark matter halo potential. In §4, we discuss the 3-dimensional wind structure and estimate the total wind mass loss rate. We discuss our findings and conclude in §5.
Radiation-Driven Winds & The Terminal Velocity
==============================================
We consider a disk with uniform brightness and total surface density: $I(r\leq r_{\rm rad})=I$ and $\Sigma(r\leq r_D)=\Sigma$, where $r_{\rm rad}$ and $r_D$ define the outer radius of the luminous and gravitating portion of the disk, respectively. The flux-mean opacity to absorption and scattering of photons is $\kappa$. The gravitational force along the polar axis above the disk is $$\begin{aligned}
f_{\rm grav}(z)&=&-2\pi G\Sigma\int_{0}^{r_{D}}\frac{zrdr}{(r^{2}+z^{2})^{3/2}}\nonumber\\
&=&-2\pi G\Sigma\left(1-\frac{z}{\sqrt{z^{2}+r_{D}^{2}}}\right),\end{aligned}$$ and the vertical radiation force along the pole is (e.g., @Proga98; @TF98) $$f_{\rm rad}(z)=\frac{2\pi \kappa I}{c}\int_{0}^{r_{\rm rad}}\frac{z^{2}rdr}{(r^{2}+z^{2})^{2}}
=\frac{\pi\kappa I}{c}\frac{r_{\rm rad}^{2}}{z^{2}+r_{\rm rad}^{2}}.$$ The extra factor $z/\sqrt{r^{2}+z^{2}}$ in the radiation force integral compared with the gravitational force makes the Eddington ratio along the pole $\Gamma(z)=|f_{\rm rad}(z)/f_{\rm grav}(z)|$ a function of height $z$ $$\Gamma(z)=\Gamma_{0}\left(\frac{r_{\rm rad}}{r_{D}}\right)^{2}
\left(\frac{z^{2}+r_{D}^{2}}{z^{2}+r_{\rm rad}^{2}}+\frac{z\sqrt{z^{2}+r_{D}^{2}}}{z^{2}+r_{\rm rad}^{2}}\right)\label{ratio},$$ where $\Gamma_{0}=\Gamma(z=0)=\kappa I/(2cG\Sigma)$ is the Eddington ratio at the disk center. A disk at the Eddington limit ($\Gamma_{0}=1$) requires $I_{\rm Edd}=2cG\Sigma/\kappa$, or flux $F_{\rm Edd}=2\pi cG\Sigma/\kappa$. If $r_{\rm
rad}/r_{D}>1/\sqrt{2}\simeq0.7$, then $\Gamma(z)$ increases along the $z$-axis above the disk. In particular, for $r_{\rm rad}\simeq
r_{D}$, the radiation force becomes twice the gravitational force as $z\rightarrow\infty$: $$\Gamma_{\infty}=\Gamma(z\rightarrow\infty)=2.$$ Because $\Gamma(z)$ increases monotonically with $z$, an infinitesimal displacement of a test particle in the vertical direction yields a net vertical acceleration, and the disk is thus unstable to wind formation. This result for disks is qualitatively different from the spherical case with a central point source where $\Gamma=f_{\rm rad}/f_{\rm grav}$ is constant with radius.
If we consider the motion of a test particle in the outflow, the velocity along the $z$-axis can be written as $$\begin{aligned}
\frac{v(z)^{2}-v_{0}^{2}}{4\pi G\Sigma r_{D}}=\hat{r}\Gamma_{0}\arctan\left(\frac{\hat{z}}{\hat{r}}\right)
-\left(1+\hat{z}-\sqrt{1+\hat{z}^{2}}\right),\label{orbit01}\end{aligned}$$ where $\hat{r}=r_{\rm rad}/r_{D}$, $\hat{z}=z/r_{D}$, and $v_{0}$ is initial vertical velocity. The first term on the right side of equation (\[orbit01\]) is the “radiation potential” along the pole, while the second term is the gravitational potential. The right side is always positive if $\Gamma_{0}\geq1$ and $\hat{r}\Gamma_{0}>2/\pi\approx0.64$, and thus the gas can be accelerated to infinity. On the other hand, an unbound outflow is still possible in the sub-Eddington case $\Gamma_{0}<1$ if the initial velocity $v_{0}$ is sufficiently large to escape the gravitational potential above the disk until reaching the critical height $z^{*}$ where $\Gamma(z^{*})=1$, because the gas is decelerated from its initial $v_{0}$ until it reaches $z^{*}$, it will be accelerated to infinity if $v(z^{*})\geq0$. Using this constraint we calculate the minimum $v_{0}$ required to drive an unbound wind in the sub-Eddington case, and the critical height $z^{*}$ where the wind profile acceleration changes sign. Figure \[fig\_subEdd\] shows the results. We introduce a characteristic velocity $v_{c}=\sqrt{4\pi G\Sigma r_{D}}$. The calculation is applied for $v_{\infty}=v(z\rightarrow\infty)\geq v_{0}$, or $\hat{r}\Gamma_{0}>2/\pi\simeq0.64$. In this case the wind can be finally accelerated at infinity. The important point here is that for a disk with sub-Eddington brightness, and non-zero vertical velocity $v_0$, the gas can be first decelerated and then accelerated to infinity. For the typical random initial velocities of gas in galaxies $\rho
v_{0}^{2}\sim\pi G \Sigma^{2}$, we have $v_{0}/v_{c}\sim(h/2r_{D})^{1/2}\sim0.22(r/r_{D})^{1/2}$, where the galactic thickness scale $h=0.1r$ has been assumed for the second equality (@1998ApJ...507..615D). For this reason we expect that even somewhat sub-Eddington disks can drive outflows, as shown in Figure \[fig\_subEdd\]. In contrast with the spherical case for which $v_{z}^{2}-v_{0}^{2}\propto \Gamma_{0}-1$, uniformly bright self-gravitating disks allow the gas above the disk to be accelerated for the case with $\Gamma_{0}=1$ or even for some sub-Eddington cases. In §4, we show the 3-dimensional trajectories of wind particles launched from both Eddington and sub-Eddington disks.
![Minimum initial velocity $v_{0}/v_{c}$ to drive unbound winds (thin lines), and the critical height $z^{*}$ turning the gas from deceleration to accelerations (thick lines) as a function of Eddington ratio $\Gamma_{0}$ in the sub-Eddington case $2/(\pi\hat{r})<\Gamma_{0}<1$, where $\hat{r}=r_{\rm rad}/r_{D}$=1, 0.9 and 0.8.[]{data-label="fig_subEdd"}](fig1.EPS){width="9cm"}
We wish to apply this simple theory to starburst galaxies, which may reach the Eddington limit for dust (TQM05). However, this application depends on the extent to which a disk-like collection of point sources of radiation (stars) may be treated as a uniformly bright disk. In the limit that the dusty gas of starbursts is optically-thick to the re-radiated FIR emission ($\Sigma\gtrsim0.1-1$g cm$^{-2}$; $\kappa_{\rm
FIR}\sim1-10$cm$^2$ g$^{-1}$; see TQM05), as is reached in ULIRGs, the self-gravitating disks around bright AGN, and some starbursts, the approximation of a uniform brightness disk is likely valid. However, for $\Sigma\lesssim10^{-3}$g cm$^{-2}$ ($\kappa_{\rm
UV}\sim10^3$cm$^2$ g$^{-1}$) the disk is optically thin to the UV radiation of massive stars and the approximations of this paper are invalidated. In the intermediate regime $10^{-3}\lesssim\Sigma\lesssim0.1-1$g cm$^{-2}$,[^1] the disk is optically-thick to UV radiation, but optically-thin to the re-radiated FIR (the so-called “single-scattering” limit), the application of the simple uniformly bright disk model is tentative, as it depends on the distribution of the sources relative to the dusty gas, the inter-star spacing relative to the vertical scale of the dusty gas, and the grain size distribution, which affects the overall scattering albedo. We save a detailed investigation to a future work, but here note two important points: (1) even for low-$\Sigma$ galaxies, the fraction of diffuse (potentially scattered) UV light may be substantial (@Thilker05), and (2) if a finite thickness disk radiates at the single-scattering Eddington limit (i.e., all photons are scattered/absorbed once in the wind; e.g., MQT05), and if the disk is uniformly emissive and absorptive, then, for example, $\kappa_{\rm UV}F_{\rm UV}\sim\kappa_{\rm UV}F_{\rm tot}/\tau_{\rm
UV}\sim F_{\rm tot}/\Sigma>\kappa_{\rm FIR}F_{\rm FIR}$, where $\tau_{\rm UV}\sim \kappa_{\rm UV}\Sigma$ and $F_{\rm tot}$, $F_{\rm UV}$, and $F_{\rm FIR}$ are the total, UV, and FIR fluxes, respectivly. The radiation pressure is thus dominated by UV emission from the “skin” of the disk (the $\tau_{\rm UV}\sim1$ surface), and $F_{\rm
UV}\sim F_{\rm Edd,\,UV}\simeq2\pi cG\Sigma/\kappa_{\rm UV}$. A similar argument can be made for all other wavebands that are “single-scattering”.
To proceed with our application to starbursts, the characteristic velocity $v_{c}$ is written as $$v_{c}=\sqrt{4\pi G\Sigma r_{D}}=500\,\textrm{km}\,\textrm{s}^{-1}\Sigma_{0}^{1/2}r_{D,1kpc}^{1/2},$$ where we take $\Sigma_{0}=\Sigma/1$ g cm$^{-2}=\Sigma/(4800\,{\rm
M}_{\odot}$ pc$^{-2}$), and $r_{D,1kpc}=r_{D}/1$ kpc. Momentarily neglecting the importance of the surrounding dark matter halo and the potentially massive old stellar bulge (both of which we evaluate in §3), from equation (\[orbit01\]) with $\Gamma_0=1$ and $r_{\rm rad}\simeq r_{D}$, the asymptotic terminal velocity along the pole is $$v_{\infty}=v_{c}\sqrt{\pi/2-1}\simeq380\,\textrm{km}\,\textrm{s}^{-1}\Sigma_{0}^{1/2}r_{D,1kpc}^{1/2}.\label{terminal01}$$ This expression for $v_\infty$ can be related to the star formation rate (SFR) using the Schmidt law, which relates the star formation surface density and gas surface density in galactic disks: $\Sigma_{\rm SFR}\propto\Sigma_{\rm gas}^{1.4}$ (@Ken98). We approximate $\Sigma_{\rm gas}=0.5f_{g,0.5}\Sigma$, where $f_{g}$ is the gas fraction. Since SFR$\sim\Sigma_{\rm SFR}\pi r_{D}^{2}$, we have $v_{\infty}\propto\Sigma^{1/2}r_{D}^{1/2}\propto\Sigma_{\rm
gas}^{1/2}r_{D}^{1/2}\propto\textrm{SFR}^{0.36}r_{D}^{-0.21}$ or $$v_{\infty}\sim400\,\textrm{km}\,\textrm{s}^{-1}f_{g,0.5}^{-0.5}\left(\frac{\rm SFR}{50M_{\odot}\textrm{yr}^{-1}}\right)^{0.36}
r_{D,1kpc}^{-0.21},\label{terminal02}$$ which is consistent with the observation $v_{\infty}\propto$ SFR$^{0.35\pm0.06}$ in low-redshift ULIRGs (@Martin05) and $v_{\infty}\propto$ SFR$^{0.3}$ for high-stellar-mass and high-SFR galaxies at redshift $z\sim1$ (@Weiner09; but, see Fig. 17 from Chen et al. 2010). The observed scatter at a given SFR may be caused by different $r_{D}$, $f_g$, $r_{\rm rad}/r_{D}$, $v_{0}$, bulge and dark matter halo mass and the time dependence of wind fountain with the evolution of the stellar population (see §3). Since we only use a simplified uniform disk model to derive equations (\[terminal01\]) and (\[terminal02\]), a more definitive comparison with the data should await a model with realistic distributions of surface density, opacity, and brightness.
If we assume the galactic disk is in radial centrifugal balance with a Keplerian velocity $v_{\rm rot}\sim\sqrt{G\pi\Sigma r}$, and we take the typical terminal velocity from equation (\[terminal01\]) to compare to the mean rotation velocity as half of that at the radius $r_{D}$, we have $$v_{\infty}\simeq4\sqrt{\pi/2-1}\langle v_{\rm rot}\rangle
\simeq3\langle v_{\rm rot}\rangle\label{velocity_1}.$$ This estimate is again made in the absence of galactic bulge and dark matter halo. That the terminal velocity of the wind increases linearly with the galactic rotation velocity is also consistent with the observational results in Martin (2005, Fig 7). Moreover, in their cosmological simulations of structure formation and IGM enrichment by galactic winds, [@OD06] assumed $v_{\infty}=3\sigma\sqrt{\Gamma_0-1}\sim3\sigma$, where $\sigma$ is the galactic velocity dispersion (see also @OD08; @Oppen10) with the factor of 3 as an assumption essentially putting in by hand. Our equation (\[velocity\_1\]) shows that it can be derived consistently if the wind is driven by radiation pressure from a self-gravitating disk radiating at or near the Eddington limit. However, as we will show in §3, the gravitational potential of the dark matter halo or a spherical old stellar bulge decreases the factor “3” in equation (\[velocity\_1\]) to $\sim1-3$, and can even lead to bound fountain flows. Thus, the factor of 3 in equation (\[velocity\_1\]) cannot be strictly obtained by the mechanism explored in this paper since all starburst galaxies have dark matter halos. Nevertheless, equations (\[terminal01\]) - (\[velocity\_1\]) are essential for developing an understanding of the maximum envelope of values for $v_\infty$, and its dependence on the observed properties of starbursts. More physics of $v_\infty$ and its $z$-dependence is presented in §3.
Finally, we note that the characteristic timescale for the wind to reach its asymptotic velocity is $$t_{c}\sim\sqrt{r_D/(4G\Sigma)}\sim3.5\times10^{6}\Sigma_{0}^{-1/2}r_{D,1kpc}^{1/2}\,\,\textrm{yr},
\label{tc}$$ which can be compared with the timescale of a bright star-forming disk: $t_{\star}\sim M_{\rm gas}/$SFR. Again employing the Schmidt Law, we find that $$t_\star\sim2\times10^{8}f_{g,0.5}^{-0.4}\Sigma_{0}^{-0.4}\,\,\,{\rm yr}
\label{tstar}$$ or $t_c/t_\star\sim0.02
r_{D,1kpc}^{1/2}f_{g,0.5}^{0.4}\Sigma_0^{-0.1}$. Therefore, the wind can be accelerated to $\sim v_\infty$ before the gas supply is depleted by star formation.[^2]
Bulge and dark matter halo
==========================
{width="18cm"}
The galactic bulge and dark matter halo around the galactic disk are important to the wind dynamics. If we do not consider the luminosity from the galactic bulge,[^3] both the bulge and halo only act to decrease the asymptotic wind velocity, and may even cause the wind to fall back to the disk as a “fountain flow”. For the galactic bulge we employ a truncated constant density sphere, and we adopt the NFW potential (@NFW96) to describe the dark matter halo distribution $\rho_{\rm DM}(r)\propto R^{-1}(R+r_{s})^{-3}$, where $r_{s}$ is the scale radius and $R=\sqrt{r^{2}+z^{2}}$ is the distance to the halo center. For simplicity, in this section we take $r_{\rm rad}\simeq r_{D}$. The Eddington limit $\Gamma_{0}=1$ including the dark matter halo becomes $$\frac{\pi\kappa I}{c}=2\pi G\Sigma+\frac{1}{2}\frac{GM_{\rm halo}}{r_{s}^{2}f(c_{\rm vir})}=2\pi G\Sigma\left(1+\frac{f_{h}}{2c'}\right),$$ where $c_{\rm vir}=r_{\rm vir}/r_{s}$, $r_{\rm vir}$ is the virial radius, $f(c_{\rm vir})=\ln(1+c_{\rm vir})-c_{\rm vir}/(1+c_{\rm vir})$, and $$f_{h}=\frac{M_{\rm halo}}{2\pi r_{s}r_{D}\Sigma f(c_{\rm vir})}
\sim\frac{M_{\rm halo}}{2M_{\rm disk}}\left(\frac{r_{D}}{r_{s}}\right)\frac{1}{f(c_{\rm vir})}.
\label{fh}$$ We introduce the parameters $c'=r_{s}/r_{D}$ and $$f_{b}=\frac{1}{2}\left(\frac{M_{\rm bulge}}{M_{\rm disk}}\right)\left(\frac{r_{D}}{r_{\rm bulge}}\right).
\label{fb}$$ The parameters $f_h$ and $f_b$ measure the importance of the halo and bulge, respectively, in determining the dynamics of the flow. The asymptotic velocity in terms of these parameters is (see eq. \[orbit01\]) $$\frac{v_{\infty}^{2}-v_{0}^{2}}{v_{c}^{2}}=\Gamma_{0}\left(1+\frac{f_{h}}{2c'}\right)\frac{\pi}{2}-(1+f_{b}+f_{h}).\label{orbit02}$$ Note that for $f_{b},f_{h}\rightarrow0$, equations (\[orbit02\]) and (\[terminal01\]) are equivalent.
We combine the effects of the galactic bulge and dark matter halo using the parameter $f_{0}=f_{h}+f_{b}$. Assuming $\Gamma_{0}=1$, the condition for matter to be unbound is $$f_{0}<\left[\left(\frac{v_{0}}{v_{c}}\right)^{2}+\left(\frac{\pi}{2}-1\right)\right]\left(1-\frac{\pi}{4c'}\right)^{-1}\label{f0limt}$$ For 1 kpc-sized disk and a typical dark matter halo $c'=r_{s}/r_{D}\sim10-50$ kpc/1 kpc $\sim10-50$ (@Merritt06), and an unbound wind for such a disk with $\Gamma_{0}=1$ requires $f_{0}\leq0.6+(v_{0}/v_{c})^{2}$.
What is a typical value for $f_0$? If we assume a galaxy with negligible bulge ($f_{b}\ll f_{h}$), then $$f_0\sim0.25\left(\frac{M_{\rm halo}/M_{\rm disk}}{30}\right)
\left(\frac{r_D/r_s}{1/30}\right)\left(\frac{2}{f(c_{\rm vir})}\right),
\label{scalef0}$$ where we have taken representative values of $M_{\rm halo}/M_{\rm disk}\sim10-100$ (e.g., @Leauthaud11), $c^\prime=r_s/r_D\sim10-100$, and $f(c_{\rm vir})\sim2$ (for $c_{\rm vir}\sim15$; @Maccio08). We find that for a realistic range of paramaters, $f_{0}$ is less than $\sim0.6$ (eq. \[f0limt\]), which shows that matter accelerated by radiation pressure may be unbound in many cases. For ultra-luminous infrared galaxies like Arp 220, with physical sizes of order $\sim100$pc (e.g., Downes & Solomon 1998), the factor $c'=r_s/r_D$ may be as large as $\sim10^3$. However, the ratio of the star-forming disk mass to the total halo mass may be $M_{\rm
disk}/M_{\rm halo}\sim10^{-3}$, so that $f_h$ is again $\sim0.3$, potentially implying that in very bright systems outflows can be driven by radiation pressure via the mechanism described here. As equation (\[fh\]) shows, larger $f_{0}$ corresponds to larger $r_{D}$, or smaller $M_{\rm disk}$, for a fixed dark matter halo.
More generally, taking $f_{0}$ as a parameter, the terminal velocity from equation (\[orbit02\]) is less than the value from equation (\[terminal01\]) by a factor $${\cal D}=\left[1-\left(\frac{\pi}{2}-1\right)^{-1}\left(1-\frac{\pi}{4c'}\right)f_{0}\right]^{1/2}.\label{decreas01}$$ As a result, both equations (\[terminal02\]) and (\[velocity\_1\]) in §2 need to be multiplied by ${\cal D}$ when the halo or bulge potential is included. For the purposes of an estimate, taking $c'\sim10$, the relation between $v_{\infty}$ and $\langle v_{\rm
rot}\rangle$ becomes $$v_{\infty}\simeq3(1-1.6f_{0})^{1/2}\langle v_{\rm rot}\rangle,\label{decreas02}$$ which shows that the dark matter halo potential well is too deep for particles to escape for $f_{0}\gtrsim0.6$. For $f_{0}\simeq0.25$ in equation (\[scalef0\]), instead of the naive estimate without the dark matter halo in equation (\[velocity\_1\]), which yields $v_\infty\simeq3\langle v_{\rm rot}\rangle$, we obtain $v_\infty\simeq2\langle v_{\rm rot}\rangle$. Even so, equation (\[decreas02\]) makes it clear that the factor of 3 in the relation $v_\infty\simeq3\langle v_{\rm rot}\rangle$ assumed in the calculation by [@OD06] (see discussion after eq. \[velocity\_1\]) cannot be stricly achieved for Eddington-limited starbursts by the mechanism discussed in this paper since all systems have dark matter halos, and thus non-zero $f_0$. To the extent that the “3” is required by [@OD06], additional physics such as an effectively super-Eddington galaxy luminosity, supernova-driven hot flows, or the mechanism discussed in Murray, Menard, & Thompson (2011) would be necessary to add to the models here.
![Outflow velocity $v_{z}/v_{c}$ along the pole with $c'=10$ and $f_{0}$ from top down $f_{0}=0,0.2,0.4,0.6,0.8,1,5$ (thick lines), $c'=4$ (dashed line) and $c'=100$ (dotted line) with $f_{0}=1$. The horizon line labeled as $v_{\infty}/v_{c}$ is the analytical solution as in equation (\[terminal01\]).[]{data-label="fig_polevelocity"}](fig3.EPS){width="9cm"}
The quantitative scaling relations for bound and unbound outflows, encapsulated in the factors $f_0$ and $c'$, are illustrated in Figure \[fig\_halo\]. The left panel shows contours of the asymptotic kinetic energy $(v_{\infty}/v_{c})^{2}$ as a function of $f_{0}$ and $c'=r_s/r_D$ assuming $v_{0}=0$ for unbound outflows.[^4] The right panel of Figure \[fig\_halo\] shows contours of $\log_{10}[z_{\rm turn}/r_D]$, the turning point of the wind scaled to the disk radius for bound “outflows.” All else being equal, a disk with a more massive dark matter halo has larger $f_{0}$, smaller $z_{\rm turn}$, and the flow has a shorter timescale for reaching the turning point $t_{\rm turn}\sim z_{\rm turn}/v_{c}\sim t_{c}(z_{\rm
turn}/r_{D}$) (see eq. \[tc\]). The right panel of Figure \[fig\_halo\] shows that $z_{\rm turn}/r_D$ can be larger than $10^3$, implying that the flow reaches many tens of kpc in height above the disk before falling back towards the host on a timescale of $\sim$Gyr.
Importantly, even if $f_0$ is large enough that the flow is bound, one may still observe an outgoing wind from a starburst galaxy for two reasons. First, there is a region of parameter space where the time to reach the turning point $t_{\rm turn}$ is larger than the time for the starburst to deplete its gas supply $t_\star=M_g/{\rm SFR}$ (see eq. \[tstar\]). Since $t_{\rm turn}/t_\star\sim(z_{\rm turn}/r_D)
t_c/t_\star$ (see eq. \[tc\]), $z_{\rm turn}/r_D\gtrsim60r_{D,\,1kpc}^{-1/2}
f_{g,0.5}^{-0.4}\Sigma_0^{0.1}$ is required for $t_{\rm turn}/t_\star\gtrsim1$. Second, an observable flow along the line of sight to a non-edge-on disk can have a maximum velocity $v_{\rm max}$ that is comparable to $v_{\infty}$ in equation (\[terminal01\]), even though the flow is bound on large scales ($z_{\rm turn}$) by the dark matter halo. Figure \[fig\_polevelocity\] shows the one-dimensional outflow velocity $v_{z}/v_{c}$ along the pole with various $f_{0}$ and $c'$. In the bound case, the maximum outflow velocities $v_{\rm max}$ peak around $1-10r_{D}$ before decreasing to much lower values. The maximum $v_{\rm max}$ is still comparable to $v_{\infty}$ in the absence of a halo or bulge, except for very large $f_{0}\gtrsim5$ (see eq. \[scalef0\]). The velocity of the flow before changing sign and falling back to the host galaxy is time-dependent and approximately reaches its maximum at a time $\sim t_{c}$ (eq. \[tc\]).
The fact that $t_{\rm turn}>t_\star$ and that $v_{\rm max}$ approaches a few $\times\langle v_{\rm rot}\rangle$ together imply that if galaxies are selected as bright starbursts, they may appear to have unbound outflows even though the gas is in fact bound on large scales by the dark matter halo potential. Indeed, Figure \[fig\_polevelocity\] implies that starbursts with bound flows may exhibit a rough correlation of the form $v_{\rm max}\sim1-3 \langle
v_{\rm rot}\rangle$.
{width="18cm"}
{width="18cm"}
{width="18cm"}
3-Dimensional Winds and Mass Loss Rate
======================================
So far we have focused on the forces along the polar $z$-axis. Figure \[fig\_orbit\_1\] shows 2-dimensional (2D) projections of the 3D orbits of test particles accelerated by radiation pressure and gravity, as well as their constant time surfaces above a uniform disk, starting from an initial height $0.1r_{D}$, both with and without an NFW potential, and with no galactic bulge. In the absence of a halo, the upper middle panel shows that winds can be driven even when the disk is sub-Eddington ($\Gamma_{0}=0.88$). The lower panels show that the character of the flow changes significantly as $f_{0}$ increases from 0.2, to 0.6, to 1.0 (left to right; compare with eq. \[scalef0\]). For the case $f_{0}=0.2$, the wind is clearly unbound. For $f_{0}=0.6$, particles emerging near the $z$-axis are accelerated to very large vertical distances, whereas particles emerging from the outer disk region fall back to the disk rapidly. A more massive halo with $f_{0}=1.0$ produces only a “fountain flow” in which particles fall back to the disk with a timescale of $\sim0.1-1$ Gyr. Moreover, Figure \[fig\_velocity\_1\] gives the 3D orbit vertical component $v_{z}$ evolution of the particles driven from different disk regions. For the face-on disk, the velocity $v_{z}$ of the wind is just the velocity along the line of sight. The maximum velocities are reached roughly at $t\sim t_{c}$. Also, the maximum and terminal velocities of particles from the outer disk region are smaller than those from the inner disk region. As in Figure \[fig\_orbit\_1\], matter from the outer region of the disk can be bound by the halo’s gravitational potential even though matter from the inner region is unbound.
To estimate the rate of mass ejection from Eddington-limited disks, we first note that the flux and luminosity are $$\begin{aligned}
F_{\rm Edd}=2\pi cG\Sigma/\kappa\sim3\times10^{12}\Sigma_{0}\kappa_{1}^{-1}\;L_{\odot}\;\textrm{kpc}^{-2},
\label{fedd}\end{aligned}$$ and $$L_{\rm Edd}=\pi r_{\rm rad}^{2}F_{\rm Edd}\sim10^{13}\left(\frac{r_{\rm rad}}{r_{D}}\right)^{2}\Sigma_{0}\kappa_{1}^{-1}r_{D,1kpc}^{2}\;L_{\odot},$$ respectively, where $\kappa_{1}=(\kappa/10)$ cm$^{2}$ g$^{-1}$ is the flux-mean dust opacity. The total mass ejection rate from the disk surface $\dot{M}_{\rm ej}$ is a local quantity measured on a disk scale height, where the gravity of the surrounding bulge and halo is much weaker than the disk’s gravitational and radiation pressure forces (as long as $f_0$ is not too large). Therefore $\dot{M}_{\rm ej}$ is only determined by the disk, and it can be estimated in the absence of bulge and halo from the integrated momentum equation. In the single-scattering limit, an estimate of $\dot{M}_{\rm ej}$ is $$\dot{M}_{\rm ej}v_{\infty,f_{0}=0}\sim L_{\rm Edd}/c,\label{massloss01}$$ where $v_{\infty,f_{0}=0}$ is the terminal velocity of the flow without a bulge or dark matter halo (eq. \[terminal01\]). Since $v_{\infty,f_{0}=0}$ is an upper limit to the velocity of the flow (valid as $f_0\rightarrow0$; see eq. \[decreas02\]), equation (\[massloss01\]) is only approximate. Nevertheless, to the extent that $v_{\rm max}$ is of order $v_{\infty,f_{0}=0}$ (see Fig. \[fig\_polevelocity\]), equation (\[massloss01\]) should yield an order-of-magnitude estimate of the mass loss rate from the disk itself. Combining equations (\[terminal01\]) and (\[massloss01\]) we have $$\dot{M}_{\rm ej}\sim3\times10^{2}\left(\frac{r_{\rm rad}}{r_{D}}\right)^{2}\Sigma_{0}^{1/2}\kappa_{1}^{-1}r_{D,1kpc}^{3/2}
\;M_{\odot}\;\textrm{yr}^{-1}\label{massloss02},$$ which is similar to observational results (e.g., Martin 2005, 2006). Keep in mind that $\kappa_1$ is between the FIR limit $\kappa_{\rm
FIR}\sim1-10$ cm$^{2}$ g$^{-1}$ and UV limit $\kappa_{\rm
UV}\sim10^{3}$ cm$^{2}$ g$^{-1}$. The estimate for $\dot{M}_{\rm
ej}$ thus varies by $2-3$ orders of magnitude from FIR-thick to UV-thin disks. Using $L=\epsilon c^2\, {\rm
SFR}$, where $\epsilon_{-3}=\epsilon/10^{-3}$ is an IMF-dependent constant, to calculate the luminosity of a starburst (e.g., Kennicutt 1998), the single-scattering estimate for the ratio of the mass ejection rate to the SFR is $$\frac{\dot{M}_{\rm ej}}{\rm SFR}\sim\frac{\epsilon c}{v_{\infty,f_0=0}}
\sim0.8\,\epsilon_{-3}\Sigma_0^{-1/2}r_{D,\,1kpc}^{-1/2},\label{massloss03}$$ which implies that radiation pressure drives more matter from disks in low-mass galaxies (see MQT05): for example, $\dot{M}_{\rm ej}/{\rm
SFR}\gtrsim10$ for $\Sigma r_D\lesssim3.1\times10^4$M$_\odot$ pc$^{-1}$. However, as low mass galaxies are usually the most dark matter halo dominated (@Persic96), with larger $f_0$ (see eqs. \[f0limt\] & \[scalef0\]), the matter lost from low mass disks can hardly escape the halo potential, and will fall back on the timescale $t_{\rm turn}$ (see the right panel of Fig. \[fig\_halo\]). Otherwise, some additional physical mechanism beyond that described in this paper (e.g., a supernova-heated wind) is needed to unbind it from the halo.
The quantity $\dot{M}_{\rm ej}$ in the above expressions is the rate at which matter is ejected from the disk in either a bound or unbound flow. In the former case, as discussed in §3, the outflow reaches a maximum outward velocity $v_{\rm max}$ before returning to the disk on a timescale $t_{\rm turn}$, which in many cases is $>t_\star$. In the case of an unbound outflow, $\dot{M}_{\rm ej}$ is a rough estimate for the mass loss rate from the halo as a whole on large scales. As emphasized in §3, the critical parameter determining whether or not the flow is bound or unbound is $f_0$ (eqs. \[f0limt\] & \[scalef0\]).
Conclusions & Discussion
========================
In this paper we study the large-scale winds from uniformly bright self-gravitating disks radiating near the Eddington limit. Different from the spherical case, where the Eddington ratio $\Gamma=f_{\rm
rad}/f_{\rm grav}$ is a constant with distance from the source, for disks $\Gamma$ increases to a maximum of twice its value at the disk surface (§2). As a result, such disks radiating at (or even somewhat below; see Fig. \[fig\_subEdd\]) the Eddington limit are unstable to driving large-scale winds by radiation pressure.
We quantify the characteristics of the resulting outflow in the context of Eddington-limited starbursts, motivated by the work of TQM05 who argued that radiation pressure on dust is the dominant feedback process in starburst galaxies. We find that the asymptotic terminal velocity along the polar direction from disks without a stellar bulge or dark matter halo is $v_{\infty}\sim\sqrt{4\pi G\Sigma
r_{D}}\sim3\langle v_{\rm rot}\rangle$, where $r_{D}$ is the disk radius, $\Sigma$ is the surface density, $v_{\rm rot}$ is the disk rotation velocity (see eqs. \[terminal01\] and \[velocity\_1\]), and may range from $\sim50-1000$km s$^{-1}$ for starbursts, depending on the system considered. Furthermore, by employing the observed Schmidt law, we find that $v_{\infty}\propto$ SFR$^{0.36}$r$_{D}^{-0.21}$ (see eq. \[terminal02\]). These results, in the absence of dark matter halo and bulge, are in agreement with recent observations (e.g., Martin 2005, 2006; Weiner et al. 2009; see also Chen et al. 2010). The typical mass loss rate from an Eddington-limited disk in the single-scattering limit is given by equation (\[massloss02\]) and suggests these outflows may efficiently remove mass from the disk (eq. \[massloss03\]).
However, both wind velocities and outflow rates can be significantly decreased by the presence of a spherical old stellar bulge or dark matter halo potential (§3). Deeper or more extended spherical potentials cause the flow to be bound on large scales, and to produce only “fountain flows” where particles fall back to the disk on a typical timescale of $\sim0.1-1$ Gyr, depending on the parameters of the system considered (see Figs. \[fig\_halo\]-\[fig\_velocity\_1\]). The criterion for the flow to become bound is given in equation (\[f0limt\]). For typical values of the parameter $f_0$ (see eq. \[scalef0\]), we find that the winds from starbursts can be either bound or unbound (see Fig. \[fig\_halo\]). For $f_{0}\simeq0.25$ in equation (\[scalef0\]), the asymptotic velocity of the wind is decreased by a factor of $\simeq0.7$ from the case neglecting the dark matter halo completely (compare eqs. \[velocity\_1\] and \[decreas02\]), from $\simeq3\langle v_{\rm rot}\rangle$ to $\simeq2.3\langle v_{\rm rot}\rangle$. An increase in $f_0$ by a factor of 2 would decrease the asymptotic velocity further to $\simeq1.3\langle v_{\rm rot}\rangle$, whereas an increase of $f_0$ by a factor of $\sim3$ to $\sim 0.75$ would cause the flow to become bound.
Importantly, even in the limit of bound fountain flows, if the timescale $t_{\rm turn}$ for reaching the turning point $z_{\rm turn}$ is longer than the lifetime of the starburst $t_{\star}$ (eq. \[tstar\]), one may still observe an outward going wind while the starburst is active and bright. The maximum positive velocity of the flow $v_{\rm max}$ along the line of sight for non-edge-on disks is still correlated with $v_{\rm rot}$ of the disk (see §3). These facts may complicate the inference from observations of winds that they are unbound from the surrounding large-scale dark matter halo.
Clearly, more detailed work is required to fully assess radiation pressure on dust as the mechanism for launching cool gas from starburst disks. The main limitations of the work presented here are that (1) we consider only disks with constant brightness and surface density, (2) we do not compute the hydrodynamics of the flow, but instead treat the outflow in the test-particle limit, and (3) we ignore other physical effects such as the momentum and energy input from supernovae (Chevalier & Clegg 1985; Strickland & Heckman 2009), which may act in concert with radiation pressure (MQT05; Murray et al. 2010). The general case, with hydrodynamics, realistic disk brightness, surface density, and dust opacity profiles will modify the picture presented here. Such an effort is underway (Zhang & Thompson, in preparation). However, the basic conclusion that uniformly bright self-gravitating disks radiating near the Eddington limit are able to drive large-scale winds — particularly in the high-$\Sigma$ limit in starbursts (see §2)— should not be fundamentally changed by more elaborate considerations. Indeed, although we have specialized the discussion to starbursts and dust opacity, the instability derived in §2 is of general applicability.
We thank Norman Murray, Crystal Martin, Romeel Davé, the anonymous referee, and especially Mark Krumholz and Eliot Quataert for many stimulating discussions and for a critical reading of the text. This work is supported by NASA grant \# NNX10AD01G.
Aguirre, A., et al., 2001, ApJ, 561, 521 Chen, Y.-M., et al. 2010, AJ, 140, 445 Chevalier, R. A., & Clegg, A. W. 1985, Nature, 317, 44 Davé, R., Oppenheimer, B. D., & Sivanandam, S. 2008, MNRAS, 391, 110 Dekel, A., & Silk, J. 1986, ApJ, 303, 39 Downes, D., & Solomon, P. M. 1998, , 507, 615 Erb, D. K., et al. 2006, ApJ, 644, 813 Finlator, K., & Dav[é]{}, R. 2008, , 385, 2181 Fujita, A., et al. 2009, ApJ, 698, 693 Heckman, T. M., Armus, L., & Miley, G. K. 1990, ApJS, 74, 833 Heckman, T. M., Lehnert, M. D., & Armus, L. in Shull, J. M., Thronson, H. A. Jr., eds, The Environment and Evolution of Galaxies P. 455 (Kluwer, Dordrecht) Heckman, T. M., Lehnert, M. D., Strickland, D. K., & Armus, L., 2000, ApJ, 129, 493 Kennicutt, Jr. R. C. 1998, ApJ, 498, 541 Leauthaud, A. 2011, arXiv: 1104.0928 Macciò, A. V., et al. 2008, MNRAS, 391, 1940 Martin C. L. 1999, ApJ, 513, 156 Martin C. L. 2005, ApJ, 621, 227 Martin C. L. 2006, ApJ, 647, 222 Merritt, D., Graham, A. W., Moore, B., Diemand, J., & Terzić, B. 2006, AJ, 132, 2685 Murray, N., Quataert, E., & Thompson, T. A. 2005, ApJ, 618, 569 (MQT05) Murray, N., M[é]{}nard, B., & Thompson, T. A. 2010, arXiv:1005.4419 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563 Oppenheimer, B. D., & Davé, R. 2006, MNRAS, 373, 1265 Oppenheimer, B. D., & Davé, R. 2008, MNRAS, 387, 577 Oppenheimer, B. D., et al. 2010, MNRAS, 406, 2325 Peeples, M. S., & Shankar, F. arXiv: 1007.3743 Pettini, M., et al. 2001, ApJ, 554, 981 Pettini, M., et al. 2002, ApJ, 569, 96 Persic, M., Salucci, P., & Stel, F. 1996, MNRAS, 281, 27 Proga, D. 2000, ApJ, 538, 684 Proga, D. 2003, ApJ, 585, 406 Proga, D., Stone, J. M., & Drew, J. E. 1998, MNRAS, 295, 595 Proga, D., Stone, J. M., & Drew, J. E. 1999, MNRAS, 310, 476 Rupke, D. S., Veilleux, S., & Sanders, D. B. 2005, , 160, 115 Sawicki, M., et al. 2008, ApJ, 687, 884 Shapley, A. E., Steidel, C. C., Pettini, M., & Adelberger, K. L. 2003, ApJ, 588, 65 Springel, V., & Hernquist, L. 2003, MNRAS, 339, 312 Strickland, D. K., & Stevens, I. R. MNRAS, 2000, 314, 511 Strickland, D. K., et al. 2002, , 568, 689 Strickland, D. K., & Heckman, T. M. ApJ, 2009, 697, 2030 Tajima, Y., & Fukue, J. 1996, PASJ, 48, 529 Tajima, Y., & Fukue, J. 1998, PASJ, 50, 483 Thompson, T. A., Quataert, E., & Murray, N. 2005, , 630, 167 (TQM05) Thilker, D. A., et al. 2005, , 619, L67 Tremonti, C. A. et al. 2004, ApJ, 613, 898 Weiner, B. J., et al. 2009, ApJ, 692, 187
[^1]: These bounds on $\Sigma$ depend linearly on the gas-to-dust ratio.
[^2]: For an instantaneous burst of star formation, the bolometric luminosity decreases on a timescale $t_{\rm MS}\sim4\times10^6$yr, the main sequence lifetime of the most massive stars. Comparing with equation (\[tc\]) we see that $t_c\lesssim t_{\rm MS}$ only when $\Sigma/r\gtrsim3\times10^{9}$M$_\odot$ kpc$^{-3}$.
[^3]: The case of a bright spherical bulge was considered in MQT05.
[^4]: Increasing $v_{0}$ to $0.2v_{c}$ does not change the position of the contours appreciably.
|
---
abstract: 'We described a decentralized distributed deterministic asynchronous Dykstra’s algorithm that allows for time-varying graphs in an earlier paper. In this paper, we show how to incorporate subdifferentiable functions into the framework using a step similar to the bundle method. We point out that our algorithm also allows for partial data communications. We discuss a standard step for treating the composition of a convex and linear function.'
author:
- 'C.H. Jeffrey Pang'
bibliography:
- '../refs.bib'
title: 'Subdifferentiable functions and partial data communication in a distributed deterministic asynchronous Dykstra’s algorithm'
---
[^1]
Introduction
============
Consider a connected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ where a closed convex function $f_{i}:\mathbb{R}^{m}\to\mathbb{R}\cup\{\infty\}$ is defined on each vertex $i\in\mathcal{V}$. A problem of interest that occurs in problems with data too large to be stored in a single location is to minimize the sum $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\underset{i\in\mathcal{V}}{\sum}f_{i}(x)\end{array}\label{eq:distrib-primal}$$ in a distributed manner so that the communications of data occur only along the edges of the graph. In our earlier paper [@Pang_Dist_Dyk], we consider the regularized problem $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\underset{i\in\mathcal{V}}{\sum}[f_{i}(x)+\frac{1}{2}\|x-[\bar{\mathbf{{x}}}]_{i}\|^{2}]\end{array}\label{eq:distrib-dyk-primal}$$ instead, where $\bar{\mathbf{{x}}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$.
\[subsec:Distrib-algs\]Distributed optimization algorithms
----------------------------------------------------------
Since this paper builds on [@Pang_Dist_Dyk], we shall give a brief introduction. Our algorithm is for the case when the edges are undirected. But we remark that notable papers on the directed case. A notable paper based on the directed case using the subgradient algorithm is [@EXTRA_Shi_Ling_Wu_Yin], and surveys are [@Nedich_survey] and [@Nedich_talk_2017]. The papers [@Nedich_Olshevsky] and [@Nedich_Olshevsky_Shi] further touch on the case of time-varying graphs. The algorithm in [@Notarstefano_gang_Newton_2017] uses a Newton-Raphson method to design a distributed algorithm for directed graphs. Naturally, the communication requirements for directed graphs need to be more stringent that the requirements for undirected graphs.
From here on, we discuss only algorithms for undirected graphs. A product space formulation on the ADMM leads to a distributed algorithm [@Boyd_Eckstein_ADMM_review Chapter 7]. Such an algorithm is decentralized and distributed, but is not asynchronous and so can get slowed down by slow vertices. An approach based on [@Eckstein_Combettes_MAPR] allows for asynchronous operation, but is not decentralized.
Moving beyond deterministic algorithms, distributed decentralized asynchronous algorithms were proposed, but many of them involve some sort of randomization. For example, the work [@Iutzeler_Bianchi_Ciblat_Hachem_1st_paper_dist; @Bianchi_Hachem_Iutzeler_2nd_paper_dist] and the generalization [@AROCK_Peng_Xu_Yan_Yin] are based on monotone operator theory (see for example the textbook [@BauschkeCombettes11]), and require the computations in the nodes to follow specific probability distributions.
We now look at asynchronous distributed algorithm with deterministic convergence (rather than probabilistic convergence). Other than subgradient methods, we mention that the paper [@Aytekin_F_Johansson_2016] is an algorithm for strongly convex problems that is primal in nature, so can’t handle constraint sets as is. The method in [@Aybat_Hamedani_2016] may arguably be considered to have these properties.
\[subsec:Dyk-method\]Dykstra’s algorithm and the corresponding distributed algorithm
------------------------------------------------------------------------------------
Again, we shall be brief with the introduction, and defer to [@Pang_Dist_Dyk] for a more detailed introduction. Dykstra’s algorithm was first studied in [@Dykstra83] for projecting a point onto the intersection of a number of closed convex sets. The convergence proof without the existence of dual solutions was established in [@BD86] and rewritten in terms of duality in [@Gaffke_Mathar], and is sometimes called the Boyle-Dykstra theorem. Dykstra’s algorithm was independently noted in [@Han88] to be block coordinate minimization on the dual problem, but their proof depends on the existence of a dual solution. (For an example of a problem without dual solutions, look at [@Han88 page 9] where two circles in $\mathbb{R}^{2}$ intersect at only one point.) We pointed out in [@Pang_Dyk_spl] that the Boyle-Dykstra theorem can be extended to the case of minimization problems of the form $\min_{x}\frac{1}{2}\|x-\bar{x}\|^{2}+\sum_{i=1}^{k}f_{i}(x)$. For more on the background on Dykstra’s algorithm, we refer to [@BauschkeCombettes11; @BB96_survey; @Deustch01; @EsRa11].
Dykstra’s algorithm was extended to a distributed algorithm in [@Borkar_distrib_dyk], and they highlight the works [@Aybat_Hamedani_2016; @LeeNedich2013; @Nedic_Ram_Veeravalli_2010; @Ozdaglar_Nedich_Parrilo] on distributed optimization. The work in [@Borkar_distrib_dyk] is vastly different from how Dykstra’s algorithm is studied in [@BD86] and [@Gaffke_Mathar].
It turns out that [@Notars_asyn_distrib_2015] also makes use of the same Dykstra’s algorithm setting, but they solve with a randomized dual proximal gradient method. The differences between their setup and ours is detailed in [@Pang_Dist_Dyk].
In [@Pang_Dist_Dyk], we rewrote in a form similar to (see Remark \[rem:partial-comms-change\] for an explanation of the differences) and applied an extended Dykstra’s algorithm. We list down the features of the distributed Dyksyra’s algorithm:
1. distributed (with communications occurring only between adjacent agents $i$ and $j$ connected by an edge),
2. decentralized (i.e., there is no central node coordinating calculations),
3. asynchronous (contrast this to synchronous algorithms, where the faster agents would wait for slower agents before they can perform their next calculations),
4. able to allow for time-varying graphs in the sense of [@Nedich_Olshevsky; @Nedich_Olshevsky_Shi] (to be robust against failures of communication between two agents),
5. deterministic (i.e., not using any probabilistic methods, like stochastic gradient methods),
6. able to allow for constrained optimization, where the feasible region is the intersection of several sets (this largely rules out primal-only methods),
7. able to incorporate proximable functions naturally.
Since Dykstra’s algorithm is also dual block coordinate ascent, the following property is obtained:
1. choosing large number of dual variables to be maximized over gives a greedier increase of the dual objective value.
Also, the distributed Dykstra’s algorithm does not require the existence of a dual minimizer provided that the functions $f_{i}(\cdot)$ are proximable. Moreover, if some of the $f_{i}(\cdot)$ were defined to be the indicator functions of closed convex sets, then a greedy step for dual ascent [@Pang_DBAP] is possible. For the rest of this paper, we shall just refer to the algorithm in [@Pang_Dist_Dyk] as the distributed Dykstra’s algorithm.
Main contribution of this paper
-------------------------------
This paper builds on [@Pang_Dist_Dyk]. We now describe the main contribution of this paper without assuming any prior knowledge of [@Pang_Dist_Dyk].
For each node $i\in\mathcal{V}$, recall the function $f_{i}:\mathbb{R}^{m}\to\mathbb{R}$ in . Let $\mathbf{f}_{i}:[\mathbb{R}^{m}]^{|\mathcal{V}|}\to\bar{\mathbb{R}}$ be defined by $\mathbf{f}_{i}(\mathbf{{x}})=f_{i}(x_{i})$ (i.e., $\mathbf{f}_{i}$ depends only on $i$-th variable, where $i\in\mathcal{V}$). Recall the graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. Let the set $\bar{\mathcal{E}}$ be defined to be $$\bar{\mathcal{E}}:=\mathcal{E}\times\{1,\dots,m\}.$$ For $\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$, the component $\mathbf{{x}}_{i}\in\mathbb{R}^{m}$ is straightforward. We let $[\mathbf{{x}}_{i}]_{\bar{k}}\in\mathbb{R}$ be the $\bar{k}$-th component of $\mathbf{{x}}_{i}\in\mathbb{R}^{m}$. For each $((i,j),\bar{k})\in\bar{\mathcal{E}}$, the hyperplane $H_{((i,j),\bar{k})}\subset[\mathbb{R}^{m}]^{|\mathcal{V}|}$ is defined to be $$H_{((i,j),\bar{k})}:=\{\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}:[\mathbf{{x}}_{i}]_{\bar{k}}=[\mathbf{{x}}_{j}]_{\bar{k}}\}.\label{eq:def-halfspaces}$$
We can see that the regularized problem is equivalent to $$\min_{\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}}\frac{1}{2}\|\mathbf{{x}}-\bar{\mathbf{{x}}}\|^{2}+\sum_{((i,j),\bar{k})\in\bar{\mathcal{E}}}\underbrace{\delta_{H_{((i,j),\bar{k})}}(\mathbf{{x}})}_{\mathbf{f}_{((i,j),\bar{k})}(\mathbf{{x}})}+\sum_{i\in\mathcal{V}}\mathbf{{f}}_{i}(\mathbf{{x}}).\label{eq:Dyk-primal}$$ We let the functions $\mathbf{f}_{\alpha}:[\mathbb{R}^{m}]^{|\mathcal{V}|}\to\bar{\mathbb{R}}$ be as defined in for all $\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}$. The (Fenchel) dual of is
$$\max_{\mathbf{{z}}_{\alpha}\in[\mathbb{R}^{m}]^{|\mathcal{V}|},\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}F(\{\mathbf{{z}}_{\alpha}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}),\label{eq:dual-fn}$$
where $$\begin{array}{c}
F(\{\mathbf{{z}}_{\alpha}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}):=-\frac{1}{2}\bigg\|\bar{\mathbf{{x}}}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\bigg\|^{2}+\frac{1}{2}\|\bar{\mathbf{{x}}}\|^{2}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha}^{*}(\mathbf{{z}}_{\alpha}).\end{array}\label{eq:Dykstra-dual-defn}$$ To give further insight on the problems -, we note that if $\mathbf{{f}}_{i}\equiv0$, then the problems reduces to the averaged consensus algorithm in [@Boyd_distrib_averaging; @Distrib_averaging_Dimakis_Kar_Moura_Rabbat_Scaglione], where the primal variable $\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ converges to the vector $\mathbf{{x}}^{*}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ (where $\mathbf{{x}}^{*}$ is defined so that each $\mathbf{{x}}_{i}^{*}\in\mathbb{R}^{m}$ is the average $\frac{1}{|\mathcal{V}|}\sum_{i\in\mathcal{V}}\bar{\mathbf{{x}}}_{i}$) at a linear rate dependent on the properties of the graph $(\mathcal{V},\mathcal{E})$.
In [@Pang_Dist_Dyk], we applied the techniques of [@Gaffke_Mathar; @Hundal-Deutsch-97] to prove that a block coordinate optimization applied to leads to the increase of the objective value $F(\cdot)$ in to the maximal value, which is also the objective value of since strong duality can also be proven. Further work in [@Pang_Dist_Dyk] shows that the algorithm has properties (1)-(5).
We now note that block coodinate optimization applied to can be easily carried out only if the functions $\mathbf{f}_{i}(\cdot)$ are proximable. For illustration, we suppose that we only minimize with respect to $\mathbf{{z}}_{i^{*}}$ for some $i^{*}\in\mathcal{V}$, but leave all other $\{\mathbf{{z}}_{\alpha}\}_{\alpha\in[\mathcal{V}\cup\mathcal{E}]\backslash\{i^{*}\}}$ fixed. We showed in [@Pang_Dist_Dyk] that $\mathbf{{z}}_{i^{*}}$ is sparse, with $[\mathbf{{z}}_{i^{*}}]_{j}=0$ whenever $i^{*}\neq j$ (see Proposition \[prop:sparsity\]), so employing techniques in [@Pang_Dist_Dyk] (see Proposition \[prop:subproblems\]) shows that the primal problem to be solved is $$\mathbf{{x}}_{i^{*}}=\min_{\mathbf{{x}}'_{i^{*}}\in\mathbb{R}^{m}}\frac{1}{2}\|\mathbf{{x}}'_{i^{*}}-[\bar{\mathbf{{x}}}-\sum_{\alpha\in[\mathcal{E}\cup\mathcal{V}]\backslash\{i^{*}\}}\mathbf{{z}}_{i}]_{i^{*}}\|^{2}+f_{i}(\mathbf{{x}}'_{i^{*}}),$$ and the $[z_{i^{*}}]_{i^{*}}$ is the corresponding dual variable. This means that $f_{i}(\cdot)$ has to be proximable.
As it stands, the algorithm in [@Pang_Dist_Dyk] does not handle the case when $f_{i}(\cdot)$ are smooth for all $i\in\mathcal{V}$. Given an affine minorant of $f_{i^{*}}(\cdot)$, say $\tilde{f}_{i^{*}}(\cdot)$, the conjugate $\tilde{f}_{i^{*}}^{*}(\cdot)$ satisfies $\tilde{f}_{i^{*}}^{*}(\cdot)\geq f_{i^{*}}^{*}(\cdot)$. The main contribution of this paper is to show that for the dual function , if the $f_{i}^{*}(\cdot)$ are replaced by $\tilde{f}_{i}^{*}(\cdot)$ whenever $f_{i}(\cdot)$ is a subdifferentiable function and $\tilde{f}_{i}(\cdot)$ is defined as an affine minorant of $f_{i}(\cdot)$, then the minorized dual functions would have the values ascending and converging to the optimal objective value of the dual problem . This extends the algorithm in [@Pang_Dist_Dyk] to give an algorithm with properties (1)-(8) and also incorporating subdifferentiable $f_{i}(\cdot)$ naturally. (A more traditional method of majorizing $f_{i^{*}}^{*}(\cdot)$ through $f_{i^{*}}^{*}([\mathbf{z}_{i^{*}}]_{i^{*}})+\langle\mathbf{x}_{i},z-[\mathbf{z}_{i^{*}}]_{i^{*}}\rangle+\frac{\sigma}{2}\|z-[\mathbf{z}_{i^{*}}]_{i^{*}}\|^{2}$ would be problematic because a strongly convex modulus $\sigma$ of $f_{i^{*}}^{*}(\cdot)$ may not even exist, which is the case when $f_{i^{*}}(\cdot)$ is affine.) As far as we are aware, distributed algorithms for subdifferentiable functions include methods based on the subgradient algorithm as mentioned earlier as well as [@Wang_Bertsekas_incremental_unpub_2013]. (Since the problems we treat in this paper are strongly convex, it would be unfair to bring out the fact that subgradient methods are slow for problems that are not strongly convex due to the need of using diminishing stepsizes. But still, our dual approach has other advantages compared to the subgradient algorithm since not all of properties (1)-(8) are satisfied by the subgradient algorithm.)
In Section \[sec:First-alg\], we first show that this procedure is sound for the sum of one subdifferentiable function and a regularizing quadratic with convergence rates compatible with standard first order methods. In Section \[sec:main-alg\], we integrate this algorithm into our distributed Dykstra’s algorithm.
Other contributions of this paper
---------------------------------
In [@Pang_Dist_Dyk], we had used the hyperplanes $H_{(i,j)}:=\{\mathbf{x}\in\mathbf{X}:[\mathbf{x}]_{i}=[\mathbf{x}]_{j}\}$ for all $(i,j)\in\mathcal{E}$ instead of . We point out that using $H_{((i,j),k)}$ instead of $H_{(i,j)}$ allows for part of the data to be communicated at one time step to achieve convergence to the optimal solution, which in turn means that computation will not be held back by communications between nodes. See Subsection \[subsec:Partial-comm-prelim\] and Example \[exa:partial-comms\] for more details.
Finally, in Subsection \[subsec:composition-lin-op\], we point out that a standard step allows us to reduce matrix operations whenever the function $f_{i}(\cdot)$ of the form $\tilde{f}_{i}\circ A_{i}$ for some closed convex function $\tilde{f}_{i}(\cdot)$ and linear map $A_{i}$, although such a step now introduces additional regularizing functions.
Notation
--------
For much of the paper, we will be looking at functions with domain either $\mathbb{R}^{m}$ or $[\mathbb{R}^{m}]^{|\mathcal{V}|}$. We reserve bold letters for functions with domain $[\mathbb{R}^{m}]^{|\mathcal{V}|}$ (for example, ), and we usually use non-bold letters for functions with domain $\mathbb{R}^{m}$ (for example, ). For a vector $\mathbf{{z}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$, $\mathbf{{z}}_{i}\in\mathbb{R}^{m}$ and $[\mathbf{{z}}]_{i}\in\mathbb{R}^{m}$ are both understood to be the $i$-th component of $\mathbf{{z}}$, where $i\in|\mathcal{V}|$. Furthermore, $[\mathbf{{z}}_{i}]_{\bar{k}}$ and $[[\mathbf{{z}}]_{i}]_{\bar{k}}\in\mathbb{R}$ are both understood to be the $\bar{k}$-th component of $[\mathbf{{z}}]_{i}$.
We say that $f(\cdot)$ is proximable if the problem $\arg\min_{x}f(x)+\frac{1}{2}\|x-\bar{x}\|^{2}$ is easy to solve for any $\bar{x}$. For a closed convex set $C$, the indicator function is denoted by $\delta_{C}(\cdot)$. All other notation are standard.
\[sec:First-alg\]The algorithm for one function
===============================================
In this section, we consider the problem $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}f(x)+\frac{1}{2}\|x-\bar{x}\|^{2},\end{array}\label{eq:small-pblm}$$ where $f:\mathbb{R}^{m}\to\mathbb{R}$ is a subdifferentiable convex function such that ${\mbox{\rm dom}}(f)=\mathbb{R}^{m}$. We define our first dual ascent algorithm to solve before we show how to integrate it into the distributed Dykstra’s algorithm for solving through the increasing the dual objective value in -. Consider Algorithm , which is somewhat like the bundle method.
\[alg:basic-dual-ascent\]In this algorithm, we want to solve
Let $h_{0}:\mathbb{R}^{m}\to\mathbb{R}$ be an affine function such that $h_{0}(\cdot)\leq f(\cdot)$ defined by the parameters $(\tilde{x}_{0},\tilde{f}_{0},\tilde{y}_{0})\in\mathbb{R}^{m}\times\mathbb{R}\times\mathbb{R}^{m}$ where for all $w\geq0$, $h_{w}:\mathbb{R}^{m}\to\mathbb{R}$ is defined through $(\tilde{x}_{w},\tilde{f}_{w},\tilde{y}_{w})\in\mathbb{R}^{m}\times\mathbb{R}\times\mathbb{R}^{m}$ by $$h_{w}(x)=\tilde{y}_{w}^{T}(x-\tilde{x}_{w})+\tilde{f}_{w}.\label{eq:def-q}$$
Without loss of generality, let $\tilde{x}_{0}$ be the minimizer to $\min_{x}h_{0}(x)+\frac{1}{2}\|x-\bar{x}\|^{2}$.
01 For $w=0,\dots$
02 $\quad$Recall $\tilde{x}_{w}$ is the minimizer to $\min_{x}h_{w}(x)+\frac{1}{2}\|x-\bar{x}\|^{2}$.
03 $\quad$Evaluate $f(\tilde{x}_{w})$ and find a subgradient $\tilde{s}_{w}\in\partial f(\tilde{x}_{w})$.
04 $\quad$Construct the affine function $\tilde{h}_{w}:\mathbb{R}^{m}\to\mathbb{R}$ to be $$\tilde{h}_{w}(x)=\tilde{s}_{w}^{T}(x-\tilde{x}_{w})+f(\tilde{x}_{w}).\label{eq:def-h-tilde-w}$$
05 $\quad$Consider
$$\begin{array}{c}
\underset{x}{\min}[\max\{\tilde{h}_{w},h_{w}\}(x)+\frac{1}{2}\|x-\bar{x}\|^{2}].\end{array}\label{eq:max-quad}$$
06 $\quad$Let $\tilde{x}_{w+1}$ be the minimizer of .
07 $\quad$Let $\tilde{f}_{w+1}=\max\{\tilde{h}_{w},h_{w}\}(\tilde{x}_{w+1})$.
08 $\quad$Let $\tilde{y}_{w+1}$ be $\bar{x}-\tilde{x}_{w+1}$.
09 $\quad$Define $h_{w+1}(\cdot)$ through $(\tilde{x}_{w+1},\tilde{f}_{w+1},\tilde{y}_{w+1})$ and .
10 End for
We shall prove that each function of the form is a lower approximation of $f(\cdot)$ in Lemma \[lem:h-w-leq-f\]. With a sequence of such lower approximations like in the bundle method, we can then solve . We prove some lemmas before proving the convergence of Algorithm \[alg:basic-dual-ascent\].
\[lem:h-w-leq-f\]In Algorithm \[alg:basic-dual-ascent\], the functions $h_{w}(\cdot)$ are such that $h_{w}(\cdot)\leq f(\cdot)$.
We prove our result by induction. Note that $h_{0}(\cdot)$ was defined so that $h_{0}(\cdot)\leq f(\cdot)$. It is also clear from the definition of $\tilde{h}_{w}(\cdot)$ in that $\tilde{h}_{w}(\cdot)\leq f(\cdot)$ for all $w\geq0$. Suppose that $h_{w}(\cdot)\leq f(\cdot)$. We now show that $h_{w+1}(\cdot)\leq f(\cdot)$. We have $$\max\{\tilde{h}_{w},h_{w}\}(\cdot)\leq f(\cdot).\label{eq:max-leq-f}$$ The functions $\tilde{h}_{w}(\cdot)$ and $h_{w}(\cdot)$ are convex, so $\max\{\tilde{h}_{w},h_{w}\}(\cdot)$ is convex. Since the minimum of $\max\{\tilde{h}_{w},h_{w}\}(\cdot)+\frac{1}{2}\|\cdot-\bar{x}\|^{2}$ is attained at $\tilde{x}_{w+1}$, it follows that $0\in\partial[\max\{\tilde{h}_{w},h_{w}\}](\tilde{x}_{w+1})+\tilde{x}_{w+1}-\bar{x}$, or that $\bar{x}-\tilde{x}_{w+1}\in\partial[\max\{\tilde{h}_{w},h_{w}\}](\tilde{x}_{w+1})$. The construction of $h_{w+1}(\cdot)$ implies that $h_{w+1}(\tilde{x}_{w+1})=\max\{\tilde{h}_{w},h_{w}\}(\tilde{x}_{w+1})$ and $h_{w+1}(\cdot)\leq\max\{\tilde{h}_{w},h_{w}\}(\cdot)$. Together with , this implies $h_{w+1}(\cdot)\leq f(\cdot)$, which completes the proof.
Let $$\bar{\alpha}_{w}:=f(\tilde{x}_{w})-h_{w}(\tilde{x}_{w}).\label{eq:bar-alpha-w}$$ Let the minimizer of be $x^{*}$. We have $$\begin{aligned}
\begin{array}{c}
h_{w}(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}\end{array} & \overset{\scriptsize{\mbox{Lem }\ref{lem:h-w-leq-f}\mbox{, Alg \ref{alg:basic-dual-ascent} line 2}}}{\leq} & \begin{array}{c}
f(x^{*})+\frac{1}{2}\|x^{*}-\bar{x}\|^{2}\end{array}\label{eq:value-chain}\\
& \overset{\scriptsize{x^{*}\mbox{ solves \eqref{eq:small-pblm}}}}{\leq} & \begin{array}{c}
f(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}.\end{array}\nonumber \end{aligned}$$ Let the real number $\alpha_{w}$ be $$\begin{array}{c}
\alpha_{w}:=\left[f(x^{*})+\frac{1}{2}\|x^{*}-\bar{x}\|^{2}\right]-\left[h_{w}(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}\right].\end{array}\label{eq:alpha-w}$$ It is clear to see that translates to $0\leq\alpha_{w}\leq\bar{\alpha}_{w}$.
\[lem:alpha-recurrs\]Recall the definitions of $\alpha_{w}$ and $\bar{\alpha}_{w}$ in and .
1. We have $\alpha_{w+1}\leq\alpha_{w}-\frac{1}{2}t^{2}$, where $\frac{1}{2}t^{2}+t\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|=\bar{\alpha}_{w}$.
2. Next, $$\begin{array}{c}
\frac{1}{2\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|^{2}}\alpha_{w+1}^{2}+\alpha_{w+1}\leq\alpha_{w}.\end{array}\label{eq:target-quad}$$
Since the function $x\mapsto\tilde{h}_{w}(x)+\frac{1}{2}\|x-\bar{x}\|^{2}$ is convex, it is bounded from below by its linearization at $\tilde{x}_{w}$ using $\tilde{s}_{w}\in\partial f(\tilde{x}_{w})$ via . In other words, for all $x\in\mathbb{R}^{m}$, we have $$\begin{array}{c}
\tilde{h}_{w}(x)+\frac{1}{2}\|x-\bar{x}\|^{2}\overset{\eqref{eq:def-h-tilde-w}}{\geq}\underbrace{(\tilde{s}_{w}+\tilde{x}_{w}-\bar{x})^{T}(x-\tilde{x}_{w})+f(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}}_{l_{(\tilde{x}_{w},\tilde{s}_{w})}(x)},\end{array}\label{eq:q-hat-lower-bdd}$$ where $l_{(\tilde{x}_{w},\tilde{s}_{w})}(\cdot)$ as defined above is the affine function derived from taking a subgradient of $\tilde{h}_{w}(\cdot)+\frac{1}{2}\|\cdot-\bar{x}\|^{2}$ at $\tilde{x}_{w}$. Consider the problem $$\begin{array}{c}
\underset{x}{\min}\underbrace{\max\left\{ h_{w}(x)+\frac{1}{2}\|x-\bar{x}\|^{2},l_{(\tilde{x}_{w},\tilde{s}_{w})}(x)\right\} }_{h_{w}^{+}(x)},\end{array}\label{eq:def-h-plus-w}$$ where $h_{w}^{+}(\cdot)$ is as defined above.
We first look at the case when $\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}=0$. We have $h_{w}(\tilde{x}_{w})\leq f(\tilde{x}_{w})$ by Lemma \[lem:h-w-leq-f\]. So $$\begin{aligned}
& f(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}\overset{\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}=0}{=}\min_{x}l_{(\tilde{x}_{w},\tilde{s}_{w})}(x)\overset{\eqref{eq:def-h-plus-w}}{\leq}\min_{x}h_{w}^{+}(x)\\
\leq & h^{+}(\tilde{x}_{w})\overset{\scriptsize{\text{Lemma \ref{lem:h-w-leq-f}}}}{\leq}f(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}.\end{aligned}$$ We then have $h_{w}(\tilde{x}_{w})=f(\tilde{x}_{w})$, or $\alpha_{w}=0$. Also, $$\begin{array}{c}
0=\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\in\partial f(\tilde{x}_{w})+\partial(\frac{1}{2}\|\cdot-\bar{x}\|^{2})(\tilde{x}_{w}),\end{array}$$ so $\tilde{x}_{w}=x^{*}$. The remaining techniques in this proof shows that $\alpha_{w'}=0$ for all $w'\geq w$, which implies the claims in this lemma. Thus, we assume $\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\neq0$.
Since $l_{(\tilde{x}_{w},\tilde{s}_{w})}(\cdot)$ is affine and $h_{w}(\cdot)+\frac{1}{2}\|\cdot-\bar{x}\|^{2}$ is a quadratic with minimizer $\tilde{x}_{w}$ and Hessian $I$, some elementary calculations will show that the minimizer of is of the form $\tilde{x}_{w}-t\frac{\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}}{\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|}$ for some $t\geq0$. Let $\tilde{d}_{w}:=\frac{\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}}{\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|}$, and let this minimizer be $\tilde{x}_{w}-t\tilde{d}_{w}$. We can see that $t>0$, because if $t=0$, then $\tilde{x}_{w}-t\tilde{d}_{w}=\tilde{x}_{w}$, and $\tilde{x}_{w}$ would once again be the minimizer of $f(\cdot)+\frac{1}{2}\|\cdot-\bar{x}\|^{2}$. With $t>0$, the function values in are equal, which gives $$\begin{array}{c}
h_{w}(\tilde{x}_{w}-t\tilde{d}_{w})+\frac{1}{2}\|(\tilde{x}_{w}-t\tilde{d}_{w})-\bar{x}\|^{2}\overset{\scriptsize{\mbox{Alg \ref{alg:basic-dual-ascent}, line 2}}}{=}h_{w}(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}+\frac{1}{2}t^{2},\end{array}\label{eq:t-square-term}$$ and $$\begin{array}{c}
l_{(\tilde{x}_{w},\tilde{s})}(\tilde{x}_{w}-t\tilde{d}_{w})\overset{\eqref{eq:bar-alpha-w},\eqref{eq:q-hat-lower-bdd}}{=}h_{w}(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}+\bar{\alpha}_{w}-t\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|.\end{array}$$ Equating the last two formulas gives $$\begin{aligned}
& \begin{array}{c}
\frac{1}{2}t^{2}+t\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|=\bar{\alpha}_{w}.\end{array}\label{eq:end-of-bar-alpha}\end{aligned}$$ Next, we have $$\begin{aligned}
\begin{array}{c}
h_{w+1}(\tilde{x}_{w+1})+\frac{1}{2}\|\tilde{x}_{w+1}-\bar{x}\|^{2}\end{array} & \overset{\scriptsize{\mbox{Alg \ref{alg:basic-dual-ascent} line 9}}}{=} & \begin{array}{c}
\max\{h_{w},\tilde{h}_{w}\}(\tilde{x}_{w+1})+\frac{1}{2}\|\tilde{x}_{w+1}-\bar{x}\|^{2}\end{array}\nonumber \\
& \overset{\eqref{eq:q-hat-lower-bdd},\eqref{eq:def-h-plus-w}}{\geq} & \begin{array}{c}
h_{w}^{+}(\tilde{x}_{w+1})\end{array}\label{eq:alpha-w-smaller}\\
& \geq & \begin{array}{c}
h_{w}^{+}(\tilde{x}_{w}-t\tilde{d}_{w})\end{array}\nonumber \\
& \overset{\eqref{eq:def-h-plus-w}}{=} & \begin{array}{c}
h_{w}(\tilde{x}_{w}-t\tilde{d}_{w})+\frac{1}{2}\|\tilde{x}_{w}-t\tilde{d}_{w}-\bar{x}\|^{2}\end{array}\nonumber \\
& \overset{\eqref{eq:t-square-term}}{=} & \begin{array}{c}
h_{w}(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}+\frac{1}{2}t^{2}.\end{array}\nonumber \end{aligned}$$ The formulas and imply the first part of our lemma. Next, let $t_{2}$ be the positive root of $$\begin{array}{c}
\frac{1}{2}t_{2}^{2}+t_{2}\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|=\alpha_{w}.\end{array}\label{eq:quad-form}$$ Since $\alpha_{w}\leq\bar{\alpha}_{w}$, we have $t_{2}\leq t$. Recalling the definition of $\alpha_{w}$ in , we have $$\begin{array}{c}
\alpha_{w+1}\overset{\eqref{eq:alpha-w},\eqref{eq:alpha-w-smaller}}{\leq}\alpha_{w}-\frac{1}{2}t^{2}\leq\alpha_{w}-\frac{1}{2}t_{2}^{2}\overset{\eqref{eq:quad-form}}{=}t_{2}\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|,\end{array}$$ or $\frac{\alpha_{w+1}}{\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|}\leq t_{2}$. Substituting this into gives as needed.
It is clear that $h_{0}(\cdot)$ in Algorithm \[alg:basic-dual-ascent\] can be defined as an affine function based on the evaluation of $f(x)$ and a subgradient in $\partial f(x)$ for some point $x$. In line 9, instead of $h_{w+1}(\cdot)$ defined there, one can use the maximum of a number of affine functions like in the bundle method. We shall only limit to the easy case of using one affine function to model $h_{w+1}(\cdot)$ for pedagogical reasons.
We need the following result proved in [@Beck_Tetruashvili_2013] and [@Beck_alt_min_SIOPT_2015].
\[lem:seq-conv-rate\](Sequence convergence rate) Let $\alpha>0$. Suppose the sequence of nonnegative numbers $\{a_{k}\}_{k=0}^{\infty}$ is such that $$a_{k}\geq a_{k+1}+\alpha a_{k+1}^{2}\mbox{ for all }k\in\{1,2,\dots\}.$$
1. [@Beck_Tetruashvili_2013 Lemma 6.2] If furthermore, $\begin{array}{c}
a_{1}\leq\frac{1.5}{\alpha}\mbox{ and }a_{2}\leq\frac{1.5}{2\alpha}\end{array}$, then $$\begin{array}{c}
a_{k}\leq\frac{1.5}{\alpha k}\mbox{ for all }k\in\{1,2,\dots\}.\end{array}$$
2. [@Beck_alt_min_SIOPT_2015 Lemma 3.8] For any $k\geq2$, $$\begin{array}{c}
a_{k}\leq\max\left\{ \left(\frac{1}{2}\right)^{(k-1)/2}a_{0},\frac{4}{\alpha(k-1)}\right\} .\end{array}$$ In addition, for any $\epsilon>0$, if $$\begin{array}{c}
\begin{array}{c}
k\geq\max\left\{ \frac{2}{\ln(2)}[\ln(a_{0})+\ln(1/\epsilon)],\frac{4}{\alpha\epsilon}\right\} +1,\end{array}\end{array}$$ then $a_{k}\leq\epsilon$.
Theorem \[thm:basic-conv-rate\] shows that Algorithm \[alg:basic-dual-ascent\] has convergence rates consistent with standard first order methods.
\[thm:basic-conv-rate\](Convergence rate) Suppose Algorithm \[alg:basic-dual-ascent\] is used to solve , and ${\mbox{\rm dom}}(f)=\mathbb{R}^{m}$.
1. There is a $O(1/w)$ convergence rate.
2. If in addition, $\nabla f(\cdot)$ is Lipschitz with constant $L_{1}$, then there is a linear rate of convergence.
Recall that $h_{w}(\cdot)\leq f(\cdot)$ from Lemma \[lem:h-w-leq-f\], so $$\begin{aligned}
\begin{array}{c}
f(x^{*})+\frac{1}{2}\|x^{*}-\bar{x}\|^{2}\end{array} & \geq & \begin{array}{c}
h_{w}(x^{*})+\frac{1}{2}\|x^{*}-\bar{x}\|^{2}\end{array}\\
& \overset{\scriptsize{\mbox{Alg \ref{alg:basic-dual-ascent} line 2}}}{\geq} & \begin{array}{c}
h_{w}(\tilde{x}_{w})+\frac{1}{2}\|\tilde{x}_{w}-\bar{x}\|^{2}+\frac{1}{2}\|x^{*}-\tilde{x}_{w}\|^{2}.\end{array}\\
\begin{array}{c}
\Rightarrow\alpha_{w}\end{array} & \overset{\eqref{eq:alpha-w}}{\geq} & \begin{array}{c}
\frac{1}{2}\|x^{*}-\tilde{x}_{w}\|^{2}.\end{array}\end{aligned}$$ Therefore, $\|\tilde{x}_{w}-x^{*}\|\leq\sqrt{2\alpha_{w}}$. We have $0\in\partial[f(\cdot)+\frac{1}{2}\|\cdot-\bar{x}\|^{2}](x^{*})$ and $\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\in\partial[f(\cdot)+\frac{1}{2}\|\cdot-\bar{x}\|^{2}](\tilde{x}_{w})$.
****
Since ${\mbox{\rm dom}}(f)=\mathbb{R}^{m}$ and $\tilde{x}_{w}$ lies in a compact set, there is some constant $L>0$ such that $\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|\leq L$. Hence $$\begin{array}{c}
\frac{1}{2L^{2}}\alpha_{w+1}^{2}+\alpha_{w+1}\leq\frac{1}{2\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|^{2}}\alpha_{w+1}^{2}+\alpha_{w+1}\overset{\eqref{eq:target-quad}}{\leq}\alpha_{w}.\end{array}$$
This recurrence together with Lemma \[lem:seq-conv-rate\] gives us the $O(1/w)$ convergence rate we need.
**** **** Let $L_{1}$ be the Lipschitz constant on the gradient of $f(\cdot)+\frac{1}{2}\|\cdot-\bar{x}\|^{2}$. We then have $\|\tilde{s}_{w}+\tilde{x}_{w}-\bar{x}\|\leq L_{1}\|\tilde{x}_{w}-x^{*}\|\leq L_{1}\sqrt{2\alpha_{w}}$. Using the formula gives us $\frac{1}{4L_{1}\alpha_{w}}\alpha_{w+1}^{2}+\alpha_{w+1}\leq\alpha_{w}$, or $$\begin{array}{c}
\frac{1}{4L_{1}}\left(\frac{\alpha_{w+1}}{\alpha_{w}}\right)^{2}+\frac{\alpha_{w+1}}{\alpha_{w}}\leq1.\end{array}$$ This gives us the linear convergence as needed.
\[rem:min-max-of-2-quads\](Minimizing ) The quadratic program can be solved easily by noting that the minimizer must be a minimizer of one of the problems $$\begin{aligned}
& & \begin{array}{c}
\underset{x}{\min}\,\tilde{h}_{w}(x)+\frac{1}{2}\|x-\bar{x}\|^{2},\text{ }\underset{x}{\min}\,h_{w}(x)+\frac{1}{2}\|x-\bar{x}\|^{2},\end{array}\\
& \text{ or} & \begin{array}{c}
\underset{x}{\min}\{h_{w}(x)+\frac{1}{2}\|x-\bar{x}\|^{2}:\tilde{h}_{w}(x)=h_{w}(x)\},\end{array}\end{aligned}$$ all of which are rather easy to solve.
We now state a proposition that will be useful for the proof of convergence.
\[prop:P-D-of-one-block\]Consider the problem $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\frac{1}{2}\|\bar{x}-x\|^{2}+f(x),\end{array}$$ and its corresponding (Fenchel) dual $$\begin{array}{c}
\underset{y\in\mathbb{R}^{m}}{\max}-\frac{1}{2}\|\bar{x}-y\|^{2}+\frac{1}{2}\|\bar{x}\|^{2}-f^{*}(y).\end{array}$$ The optimal solutions $x^{*}$ and $y^{*}$ are related by $x^{*}+y^{*}=\bar{x}$, and strong duality holds.
This result can be seen to be Moreau’s decomposition theorem.
In view of Proposition \[prop:P-D-of-one-block\], we now explain that Algorithm \[alg:basic-dual-ascent\] can be interpreted as a dual ascent algorithm. We can see that Algorithm \[alg:basic-dual-ascent\] finds $h_{w}(\cdot)$ for $w=0,1,\dots$ such that $h_{w}^{*}(\cdot)\geq f^{*}(\cdot)$ and dual iterates $\{\tilde{y}_{w}\}_{w=0}^{\infty}$ so that $\{-h_{w}^{*}(\tilde{y}_{w})+\frac{1}{2}\|\bar{x}\|^{2}-\frac{1}{2}\|\tilde{y}_{w}-\bar{x}\|^{2}\}_{w=0}^{\infty}$ is a monotonically nondecreasing sequence that converges to $-f^{*}(y^{*})+\frac{1}{2}\|\bar{x}\|^{2}-\frac{1}{2}\|y^{*}-\bar{x}\|^{2}$, where $y^{*}$ is the optimal dual variable for . This interpretation shall be exploited in our subdifferentiable distributed Dykstra’s algorithm.
\[sec:main-alg\]Deterministic Distributed Asynchronous Dykstra Algorithm
========================================================================
We now proceed to integrate the dual ascent algorithm in Section \[sec:First-alg\] into the distributed Dykstra’s algorithm for the problem . We partition the vertex set $\mathcal{V}$ as the disjoint union *$\mathcal{V}=\mathcal{V}_{1}\cup\mathcal{V}_{2}\cup\mathcal{V}_{3}\cup\mathcal{V}_{4}$* so that
- $f_{i}(\cdot)$ are proximable functions for all $i\in\mathcal{V}_{1}$.
- $f_{i}(\cdot)$ are indicator functions of closed convex sets for all $i\in\mathcal{V}_{2}$.
- $f_{i}(\cdot)$ are proximable functions such that ${\mbox{\rm dom}}(f_{i})=\mathbb{R}^{m}$ for all $i\in\mathcal{V}_{3}$.
- *$f_{i}(\cdot)$* are subdifferentiable functions (i.e., a subgradient is easy to obtain) such that ${\mbox{\rm dom}}(f_{i})=\mathbb{R}^{m}$ for all $i\in\mathcal{V}_{4}$.
We had the 3 sets $\mathcal{V}_{1}$, $\mathcal{V}_{2}$ and $\mathcal{V}_{3}$ in [@Pang_Dyk_spl]. In principle, the vertices in $\mathcal{V}_{2}$ and $\mathcal{V}_{3}$ can be placed into $\mathcal{V}_{1}$. As explained in [@Pang_Dyk_spl], the advantage of separating $\mathcal{V}_{3}$ is that more than one function can be minimized at a time for vertices in this set without affecting the proof of convergence (see Proposition \[prop:control-growth\]), and the advantage of separating $\mathcal{V}_{2}$ is that one can apply a greedy SHQP step in [@Pang_DBAP]. The set $\mathcal{V}_{4}$ contains subdifferentiable functions, which is the subject of this paper.
To simplify calculations, we let $\mathbf{{v}}_{A}$, $\mathbf{{v}}_{H}$ and $\mathbf{{x}}$ be denoted by
\[eq\_m:all\_acronyms\] $$\begin{aligned}
\mathbf{{v}}_{H} & = & \sum_{((i,j),\bar{k})\in\bar{\mathcal{E}}}\mathbf{{z}}_{((i,j),\bar{k})}\label{eq:v-H-def}\\
\mathbf{{v}}_{A} & = & \mathbf{{v}}_{H}+\sum_{i\in\mathcal{V}}\mathbf{{z}}_{i}\label{eq:from-10}\\
\mathbf{{x}} & = & \bar{\mathbf{{x}}}-\mathbf{{v}}_{A}.\label{eq:x-from-v-A}\end{aligned}$$
Intuitively, $\mathbf{{v}}_{H}$ describes the sum of the dual variables due to $H_{((i,j),\bar{k})}$ for all $((i,j),\bar{k})\in\bar{\mathcal{E}}$, $\mathbf{{v}}_{A}$ is the sum of all dual variables, and $\mathbf{{x}}$ is the estimate of the primal variable.
\[subsec:Partial-comm-prelim\]Partial communication of data
-----------------------------------------------------------
One insight that we point out in this paper is that Algorithm \[alg:Ext-Dyk\] supports the partial communication of data. We lay down the foundations of the parts of Algorithm \[alg:Ext-Dyk\] relevant for this insight.
Let $D\subset[\mathbb{R}^{m}]^{|\mathcal{V}|}$ be the diagonal set defined by $$D:=\{\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}:\mathbf{{x}}_{1}=\mathbf{{x}}_{2}=\cdots=\mathbf{{x}}_{|\mathcal{V}|}\}.\label{eq:diagonal-set}$$ With the definition of the hyperplanes $H_{((i,j),\bar{k})}$ in and $\mathcal{G}=(\mathcal{V},\mathcal{E})$ being a connected graph, we have $$\bigcap_{((i,j),\bar{k})\in\bar{\mathcal{E}}}H_{((i,j),\bar{k})}=D\mbox{ and }\sum_{((i,j),\bar{k})\in\bar{\mathcal{E}}}H_{((i,j),\bar{k})}^{\perp}=D^{\perp}=\left\{ \mathbf{{z}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}:\sum_{i\in\mathcal{V}}\mathbf{{z}}_{i}=0\right\} .\label{eq:D-and-D-perp}$$
\[prop:E-connects-V\]Suppose $\mathcal{G}=(\mathcal{V},\mathcal{E})$ is a connected graph. Let $H_{((i,j),\bar{k})}$ be the hyperplane . Let $\bar{\mathcal{E}}'$ be a subset of $\bar{\mathcal{E}}$. The following conditions are equivalent:
1. $\cap_{((i,j),\bar{k})\in\bar{\mathcal{E}}'}H_{((i,j),\bar{k})}=D$
2. $\sum_{((i,j),\bar{k})\in\mathcal{\bar{E}}'}H_{((i,j),\bar{k})}^{\perp}=D^{\perp}.$
3. For each $\bar{k}\in\{1,\dots,m\}$, the graph $\mathcal{G}'=(\mathcal{V},\mathcal{E}_{\bar{k}}')$ is connected, where $\mathcal{E}_{\bar{k}}':=\{(i,j)\in\mathcal{E}:((i,j),\bar{k})\in\bar{\mathcal{E}}'\}$.
The equivalence between (1) and (3) is easy, and the equivalence between (1) and (2) is simple linear algebra.
\[def:E-connects-V\]We say that *$\bar{\mathcal{E}}'\subset\bar{\mathcal{E}}$ connects $\mathcal{V}$* if any of the equivalent properties in Proposition \[prop:E-connects-V\] is satisfied.
\[rem:partial-comms-change\](Change from [@Pang_Dist_Dyk]) The change in this paper from [@Pang_Dist_Dyk] is that each hyperplane $H_{((i,j),\bar{k})}$ is now of codimension 1. In [@Pang_Dist_Dyk], we defined the hyperplanes $H_{(i,j)}:=\{\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}:\mathbf{{x}}_{i}=\mathbf{{x}}_{j}\}$ of codimension $m$ which are indexed by $(i,j)\in\mathcal{E}$ instead. The advantage of introducing the additional variables is that we can have a partial transfer of the data between two vertices rather than a full transfer. This will be elaborated in Example \[exa:partial-comms\].
\[lem:express-v-as-sum\](Expressing $v$ as a sum) Recall the definitions of $D$ and $H_{((i,j),\bar{k})}$ in and . There is a $C_{1}>0$ such that for all $\mathbf{{v}}\in D^{\perp}$ and $\bar{\mathcal{E}}'\subset\bar{\mathcal{E}}$ such that $\bar{\mathcal{E}}'$ connects $\mathcal{V}$, we can find $\mathbf{{z}}_{((i,j),\bar{k})}\in H_{((i,j),\bar{k})}^{\perp}$ for all $((i,j),\bar{k})\in\bar{\mathcal{E}}'$ such that $\sum_{((i,j),\bar{k})\in\bar{\mathcal{E}}'}\mathbf{{z}}_{((i,j),\bar{k})}=\mathbf{{v}}$ and $\|\mathbf{{z}}_{((i,j),\bar{k})}\|\leq C_{1}\|\mathbf{{v}}\|$ for all $((i,j),\bar{k})\in\bar{\mathcal{E}}'$.
This is elementary linear algebra. We refer to [@Pang_Dist_Dyk] for a proof of a similar result.
Algorithm description and preliminaries
---------------------------------------
In this subsection, we present Algorithm \[alg:Ext-Dyk\] below and recall some of the results that were presented in [@Pang_Dist_Dyk] that are necessary for further discussions.
Recall that in the one node case in Section \[sec:First-alg\], the subdifferentiable function $f_{i}(\cdot)$ is handled using lower approximates. In addition to , we need to consider the function $$\begin{array}{c}
F_{n,w}(\{\mathbf{{z}}_{\alpha}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}):=-\frac{1}{2}\bigg\|\bar{\mathbf{{x}}}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\bigg\|^{2}+\frac{1}{2}\|\bar{\mathbf{{x}}}\|^{2}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha,n,w}^{*}(\mathbf{{z}}_{\alpha}),\end{array}\label{eq:Dykstra-dual-defn-1}$$ where for all $n\geq1$ and $w\in\{1,\dots,\bar{w}\}$, $\mathbf{f}_{\alpha,n,w}:[\mathbb{R}^{m}]^{|\mathcal{V}|}\to\mathbb{R}$ satisfies
\[eq\_m:h-a-n-w\] $$\begin{aligned}
\mathbf{f}_{\alpha,n,w}(\cdot) & = & \mathbf{f}_{\alpha}(\cdot)\mbox{ for all }\alpha\in[\bar{\mathcal{E}}\cup\mathcal{V}]\backslash\mathcal{V}_{4}\label{eq:h-a-n-w-eq-h-a}\\
\mbox{ and }\mathbf{f}_{\alpha,n,w}(\cdot) & \leq & \mathbf{f}_{\alpha}(\cdot)\mbox{ for all }\alpha\in\mathcal{V}_{4}.\label{eq:h-a-n-w-lesser}\end{aligned}$$
So $F_{n,w}(\cdot)\leq F(\cdot)$. We present Algorithm .
\[alg:Ext-Dyk\](Distributed Dykstra’s algorithm) Consider the problem along with the associated dual problem .
Let $\bar{w}$ be a positive integer. Let $C_{1}>0$ satisfy Lemma \[lem:express-v-as-sum\]. For each $\alpha\in[\bar{\mathcal{E}}\cup\mathcal{V}]\backslash\mathcal{V}_{4}$, $n\geq1$ and $w\in\{1,\dots,\bar{w}\}$, let $\mathbf{f}_{\alpha,n,w}:[\mathbb{R}^{m}]^{|\mathcal{V}|}\to\mathbb{R}$ be as defined in . Our distributed Dykstra’s algorithm is as follows:
01$\quad$Let
- $\mathbf{{z}}_{i}^{1,0}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ be a starting dual vector for $\mathbf{f}_{i}(\cdot)$ for each $i\in\mathcal{V}$ so that $[\mathbf{{z}}_{i}^{1,0}]_{j}=0\in\mathbb{R}^{m}$ for all $j\in\mathcal{V}\backslash\{i\}$.
- $\mathbf{{v}}_{H}^{1,0}\in D^{\perp}$ be a starting dual vector for .
- Note: $\{\mathbf{{z}}_{((i,j),\bar{k})}^{n,0}\}_{((i,j),\bar{k})\in\bar{\mathcal{E}}}$ is defined through $\mathbf{{v}}_{H}^{n,0}$ in .
- Let $\mathbf{{x}}^{1,0}$ be $\mathbf{{x}}^{1,0}=\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{1,0}-\sum_{i\in\mathcal{V}}\mathbf{{z}}_{i}^{1,0}$.
02$\quad$For each $i\in\mathcal{V}_{4}$, let $\mathbf{f}_{i,1,0}:[\mathbb{R}^{m}]^{|\mathcal{V}|}\to\mathbb{R}$ be a function such that $\mathbf{f}_{i,1,0}(\cdot)\leq\mathbf{f}_{i}(\cdot)$
03$\quad$For $n=1,2,\dots$
04$\quad$$\quad$
05$\quad$$\quad$
$\quad$$\quad$
06$\quad$$\quad$For $w=1,2,\dots,\bar{w}$
07$\quad$$\quad$$\quad$Choose a set $S_{n,w}\subset\bar{\mathcal{E}}_{n}\cup\mathcal{V}$ such that $S_{n,w}\neq\emptyset$.
08$\quad$$\quad$$\quad$If $S_{n,w}\subset\mathcal{V}_{4}$, then
09$\quad$$\quad$$\quad$$\quad$Apply Algorithm \[alg:subdiff-subalg\].
10$\quad$$\quad$$\quad$else
11$\quad$$\quad$$\quad$$\quad$Set $\mathbf{f}_{i,n,w}(\cdot):=\mathbf{f}_{i,n,w-1}(\cdot)$ for all $i\in\mathcal{V}_{4}$.
12$\quad$$\quad$$\quad$$\quad$Define $\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in S_{n,w}}$ by $$\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in S_{n,w}}=\underset{z_{\alpha},\alpha\in S_{n,w}}{\arg\min}\frac{1}{2}\left\Vert \bar{\mathbf{{x}}}-\sum_{\alpha\notin S_{n,w}}\mathbf{{z}}_{\alpha}^{n,w-1}-\sum_{\alpha\in S_{n,w}}\mathbf{{z}}_{\alpha}\right\Vert ^{2}+\sum_{\alpha\in S_{n,w}}\mathbf{f}_{\alpha,n,w}^{*}(\mathbf{{z}}_{\alpha}).\label{eq:Dykstra-min-subpblm}$$
13$\quad$$\quad$$\quad$end if
14$\quad$$\quad$$\quad$Set $\mathbf{{z}}_{\alpha}^{n,w}:=\mathbf{{z}}_{\alpha}^{n,w-1}$ for all $\alpha\notin S_{n,w}$.
15$\quad$$\quad$End For
16$\quad$$\quad$Let $\mathbf{{z}}_{i}^{n+1,0}=\mathbf{{z}}_{i}^{n,\bar{w}}$ for all $i\in\mathcal{V}$ and $\mathbf{{v}}_{H}^{n+1,0}=\mathbf{{v}}_{H}^{n,\bar{w}}=\sum_{((i,j),\bar{k})\in\bar{\mathcal{E}}}\mathbf{{z}}_{((i,j),\bar{k})}^{n,\bar{w}}$.
17$\quad$$\quad$Let $\mathbf{f}_{i,n+1,0}(\cdot)=\mathbf{f}_{i,n,\bar{w}}(\cdot)$ for all $i\in\mathcal{V}_{4}$.
18$\quad$End For
Even though Algorithm \[alg:Ext-Dyk\] is described so that each node $i\in\mathcal{V}$ and $((i,j),\bar{k})\in\bar{\mathcal{E}}$ is associated with a dual variable $\mathbf{{z}}_{\alpha}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$, we point out that the size of the dual variable $z_{\alpha}$ that needs to be stored in each node and edge is small due to sparsity.
\[prop:sparsity\](Sparsity of $\mathbf{{z}}_{\alpha}$) We have $[\mathbf{{z}}_{i}^{n,w}]_{j}=0$ for all $j\in\mathcal{V}\backslash\{i\}$, $n\geq1$ and $w\in\{0,1,\dots,\bar{w}\}$. Similarly, for all $n\geq1$, $w\in\{0,1,\dots,\bar{w}\}$ and $(e,\bar{k})\in\bar{\mathcal{E}}$, the vector $\mathbf{{z}}_{(e,\bar{k})}^{n,w}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ satisfies $[[\mathbf{{z}}_{(e,\bar{k})}^{n,w}]_{i'}]_{k'}=0$ unless $\bar{k}=k'$ and $i$ is an endpoint of $e$.
The proof of this result is similar to the corresponding result in [@Pang_Dist_Dyk]. The claim for $z_{i}^{n,w}$ relies on the fact that $\mathbf{f}_{i,n,w}(\cdot)$ depends only on the $i$-th component, and the claim for $\mathbf{{z}}_{(e,\bar{k})}^{n,w}$ relies on the fact that $\mathbf{f}_{(e,\bar{k})}(\cdot)=\delta_{H_{(e,\bar{k})}^{\perp}}(\cdot)$, with $H_{(e,\bar{k})}^{\perp}$ containing vectors that are zero in all but 2 coordinates.
Dykstra’s algorithm is traditionally written in terms of solving for the primal variable $x$. For completeness, we show the equivalence between and the primal minimization problem. The proof is easily extended from [@Pang_Dyk_spl].
\[prop:subproblems\](On solving ) If a minimizer $\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in S_{n,w}}$ for exists, then the $x^{n,w}$ in satisfies $$\begin{array}{c}
\mathbf{{x}}^{n,w}=\underset{\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}}{\arg\min}\underset{\alpha\in S_{n,w}}{\sum}\mathbf{f}_{\alpha,n,w}(\mathbf{{x}})+\frac{1}{2}\left\Vert \mathbf{{x}}-\left(\bar{\mathbf{{x}}}-\underset{\alpha\notin S_{n,w}}{\sum}\mathbf{{z}}_{\alpha}^{n,w}\right)\right\Vert ^{2}.\end{array}\label{eq:primal-subpblm}$$ Conversely, if $\mathbf{{x}}^{n,w}$ solves with the dual variables $\{\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\}_{\alpha\in S_{n,w}}$ satisfying $$\begin{array}{c}
\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\in\partial\mathbf{f}_{\alpha,n,w}(\mathbf{{x}}^{n,w})\mbox{ and }\mathbf{{x}}^{n,w}-\bar{\mathbf{{x}}}+\underset{\alpha\notin S_{n,w}}{\sum}\mathbf{{z}}_{\alpha}^{n,w}+\underset{\alpha\in S_{n,w}}{\sum}\tilde{\mathbf{{z}}}_{\alpha}^{n,w}=0,\end{array}\label{eq:primal-optim-cond}$$ then $\{\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\}_{\alpha\in S_{n,w}}$ solves .
\[rem:Irrelevance-of-z\](Irrelevance of $\mathbf{{z}}_{((i,j),\bar{k})}^{n,w}$) In [@Pang_Dist_Dyk], we explained that each node $i\in\mathcal{V}$ needs to keep track of just $[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}\in\mathbb{R}^{m}$ and $[\mathbf{{z}}_{i}^{n,w}]_{i}\in\mathbb{R}^{m}$, and does not have to keep track of any part of the vectors $\mathbf{{z}}_{((i,j),\bar{k})}^{n,w}\in\mathbb{R}^{m}$ for $((i,j),\bar{k})\in\bar{\mathcal{E}}$. The same is true for Algorithms \[alg:Ext-Dyk\] and \[alg:subdiff-subalg\] here. The reason for introducing $\mathbf{{z}}_{((i,j),\bar{k})}^{n,w}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ is that the proof of the convergence result in Theorem \[thm:convergence\] needs , which in turn needs the variables $\mathbf{{z}}_{((i,j),\bar{k})}^{n,w}$.
\[exa:partial-comms\](Partial communication of data) Fix some $(i,j)\in\mathcal{E}$ and some set $\bar{K}\subset\{1,\dots,m\}$. Suppose the set $S_{n,w}$ is chosen to be $\{((i,j),\bar{k}):\bar{k}\in\bar{K}\}$. Then $\mathbf{{x}}^{n,w}$ is obtained from , which tells that $\mathbf{{x}}^{n,w}$ is the projection of $[\bar{\mathbf{{x}}}-\sum_{\alpha\notin S_{n,w}}\mathbf{{z}}_{\alpha}^{n,w-1}]$ onto $\cap_{((i,j),\bar{k}):\bar{k}\in\bar{K}}H_{((i,j),\bar{k})}$. Since $H_{((i,j),\bar{k})}$ are all affine spaces with normals $\mathbf{{z}}_{((i,j),\bar{k})}^{n,w-1}$, $\mathbf{{x}}^{n,w}$ is also the projection of $$\begin{array}{c}
\bar{\mathbf{{x}}}-\underset{\alpha\notin S_{n,w}}{\sum}\mathbf{{z}}_{\alpha}^{n,w-1}-\underset{\alpha\in S_{n,w}}{\sum}\mathbf{{z}}_{\alpha}^{n,w-1},\end{array}$$ or $\mathbf{{x}}^{n,w-1}$, onto $\cap_{((i,j),\bar{k}):\bar{k}\in\bar{K}}H_{((i,j),\bar{k})}$. This gives $$[[\mathbf{{x}}^{n,w}]_{i'}]_{k'}=\begin{cases}
\frac{1}{2}([[\mathbf{{x}}^{n,w-1}]_{i}]_{\bar{k}}+[[\mathbf{{x}}^{n,w-1}]_{j}]_{\bar{k}}) & \mbox{ if }i'\in\{i,j\}\mbox{ and }\bar{k}\in\bar{K}\\{}
[[\mathbf{{x}}^{n,w-1}]_{i'}]_{k'} & \mbox{ otherwise.}
\end{cases}$$ As mentioned in Remark \[rem:Irrelevance-of-z\], there is no need to keep track of the dual variables $\mathbf{{z}}_{((i,j),\bar{k})}^{n,w}$ to run Algorithm \[alg:Ext-Dyk\]. So the larger $\bar{K}$ is, the more variables are updated. Thus in Algorithm \[alg:Ext-Dyk\], computations can be performed continuously even when not all the data is communicated. In other words, communications will not be a bottleneck for Algorithm \[alg:Ext-Dyk\].
Subroutine for subdifferentiable functions
------------------------------------------
If $\mathcal{V}_{4}=\emptyset$, then Algorithm \[alg:Ext-Dyk\] corresponds mostly to the algorithm in [@Pang_Dist_Dyk] because there are no subdifferentiable functions. In this subsection, we present and derive Algorithm \[alg:subdiff-subalg\], which is a subroutine within Algorithm \[alg:Ext-Dyk\] to handle subdifferentiable functions.
We state some notation necessary for further discussions. For any $\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}$ and $n\in\{1,2,\dots\}$, let $p(n,\alpha)$ be $$p(n,\alpha):=\max\{w':w'\leq\bar{w},\alpha\in S_{n,w'}\}.$$ In other words, $p(n,\alpha)$ is the index $w'$ such that $\alpha\in S_{n,w'}$ but $\alpha\notin S_{n,k}$ for all $k\in\{w'+1,\dots,\bar{w}\}$. It follows from line 14 in Algorithm \[alg:Ext-Dyk\] that $$\mathbf{{z}}_{\alpha}^{n,p(n,\alpha)}=\mathbf{{z}}_{\alpha}^{n,p(n,\alpha)+1}=\cdots=\mathbf{{z}}_{\alpha}^{n,\bar{w}}\mbox{ for all }\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}.\label{eq:stagnant-indices}$$ Moreover, $((i,j),\bar{k})\notin\bar{\mathcal{E}}_{n}$ implies $((i,j),\bar{k})\notin S_{n,w}$ for all $w\in\{1,\dots,\bar{w}\}$, so $$0\overset{\scriptsize\eqref{eq:reset-z-i-j-1}}{=}\mathbf{{z}}_{((i,j),\bar{k})}^{n,0}=\mathbf{{z}}_{((i,j),\bar{k})}^{n,1}=\cdots=\mathbf{{z}}_{((i,j),\bar{k})}^{n,\bar{w}}\mbox{ for all }((i,j),\bar{k})\in\bar{\mathcal{E}}\backslash\bar{\mathcal{E}}_{n}.\label{eq:zero-indices}$$
We present Algorithm .
\[alg:subdiff-subalg\](Subalgorithm for subdifferentiable functions) This algorithm is run when line 9 of Algorithm \[alg:Ext-Dyk\] is reached. Suppose $S_{n,w}\subset\mathcal{V}_{4}$ and Assumption \[assu:to-start-subalg\] holds.
01 For each $i\in S_{n,w}$
02 $\quad$For $\tilde{f}_{i,n,w-1}(\cdot)$ defined in , consider $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\left[\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}-x\|^{2}+\max\{f_{i,n,w-1},\tilde{f}_{i,n,w-1}\}(x)\right],\end{array}\label{eq:alg-primal-subpblm}$$
03 $\quad$Let the primal and dual solutions of be $x_{i}^{+}$ and $z_{i}^{+}$
04 $\quad$Define $f_{i,n,w}:\mathbb{R}^{m}\to\mathbb{R}$ to be the affine function $$f_{i,n,w}(x):=f_{i,n,w-1}(x_{i}^{+})+\langle x-x_{i}^{+},[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}-x_{i}^{+}\rangle.\label{eq:def-f-i-n-w}$$
05 $\quad$In other words, $f_{i,n,w}(\cdot)$ is chosen such that the
$\qquad\qquad$primal and dual optimizers to coincide with that of $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\left[\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}-x\|^{2}+f_{i,n,w}(x)\right].\end{array}\label{eq:finw-design}$$
06 $\quad$Define the function $\mathbf{f}_{i,n,w}:[\mathbb{R}^{m}]^{|\mathcal{V}|}\to\mathbb{R}$ and
$\qquad\qquad$the dual vector $z_{i}^{n,w}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ to be $$\mathbf{f}_{i,n,w}(\mathbf{{x}}):=f_{i,n,w}(\mathbf{{x}}_{i})\mbox{ and }[\mathbf{{z}}_{i}^{n,w}]_{j}:=\begin{cases}
\mathbf{{z}}_{i}^{+} & \mbox{ if }j=i\\
0 & \mbox{ if }j\neq i.
\end{cases}\label{eq:def-z-i}$$
07 End for
08 For all $i\in\mathcal{V}_{4}\backslash S_{n,w}$, $\mathbf{f}_{i,n,w}(\cdot)=\mathbf{f}_{i,n,w-1}(\cdot)$.
We make three assumptions that will be needed for the proof of convergence of Theorem \[thm:convergence\].
\[assu:to-start-subalg\](Start of Algorithm \[alg:subdiff-subalg\]) Recall that at the start of Algorithm \[alg:subdiff-subalg\], $S_{n,w}\subset\mathcal{V}_{4}$. We make three assumptions.
1. Suppose $(n,w)$ is such that $w>1$ and $S_{n,w}\subset\mathcal{V}_{4}$ so that Algorithm \[alg:subdiff-subalg\] is invoked. Then for all $i\in S_{n,w}$, $[\mathbf{{z}}_{i}^{n,w-1}]_{i}\in\mathbb{R}^{m}$ is the optimizer to the problem $$\begin{array}{c}
\underset{z\in\mathbb{R}^{m}}{\min}\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}-z\|^{2}+f_{i,n,w-1}^{*}(z).\end{array}\label{eq:multi-node-start}$$ In other words, suppose $w_{i}\geq1$ is the largest $w'$ such that $i\in S_{n,w'}$ and $i\notin S_{n,\tilde{w}}$ for all $\tilde{w}\in\{w'+1,w'+2,\dots,w-1\}$. Then for all $\tilde{w}\in\{w_{i}+1,\dots,w-1\}$, $(e,\bar{k})\notin S_{n,\tilde{w}}$ if $i$ is an endpoint of $e$.
2. Suppose that for all $i\in\mathcal{V}_{4}$ and $\tilde{w}\in\{p(n,i)+1,\dots,\bar{w}\}$, $(e,\bar{k})\notin S_{n,\tilde{w}}$ if $i$ is an endpoint of $e$. (This implies $\mathbf{x}_{i}^{n,p(n,i)}=\mathbf{x}_{i}^{n,\bar{w}}$.)
3. Suppose that $S_{n,1}=\mathcal{V}_{4}$ for all $n>1$.
We need Assumption \[assu:to-start-subalg\](1) for Proposition \[prop:quad-dec-case-2\], which is in turn needed for the proof of Theorem \[thm:convergence\](i). We need Assumption \[assu:to-start-subalg\](2) so that the analogue of Lemma \[lem:alpha-recurrs\](1) holds, which in turn is used in the proof of Theorem \[thm:convergence\](iv). Also, Assumption \[assu:to-start-subalg\](1) is seen to be satisfied if $S_{n,w}\subset S_{n,w-1}$ if $S_{n,w}\subset\mathcal{V}_{4}$.
(On the problem ) Consider the case where $S_{n,w}=\{i\}$ first, where $i\in\mathcal{V}_{4}$. If $i$ were in $\mathcal{V}\backslash\mathcal{V}_{4}$ instead, $\mathbf{{z}}_{i}^{n,w}$ is the minimizer of $$\begin{array}{c}
\underset{\mathbf{{z}}_{i}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}}{\min}\frac{1}{2}\bigg\|\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\underset{j\in\mathcal{V}\backslash\{i\}}{\sum}\mathbf{{z}}_{j}^{n,w-1}-\mathbf{{z}}_{i}\bigg\|^{2}+\mathbf{f}_{i}^{*}(\mathbf{{z}}_{i}).\end{array}\label{eq:block-dual}$$ When $i\in\mathcal{V}_{4}$, we use $f_{i,n,w-1}(\cdot)$, where $f_{i,n,w-1}(\cdot)\leq f_{i}(\cdot)$, instead of $f_{i}(\cdot)$. This gives $f_{i,n,w}^{*}(\cdot)\geq f_{i}^{*}(\cdot)$. Instead of , we now have $$\begin{array}{c}
\underset{\mathbf{{z}}_{i}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}}{\min}\frac{1}{2}\bigg\|\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\underset{j\in\mathcal{V}\backslash\{i\}}{\sum}\mathbf{{z}}_{j}^{n,w-1}-\mathbf{{z}}_{i}\bigg\|^{2}+\mathbf{f}_{i,n,w-1}^{*}(\mathbf{{z}}_{i}).\end{array}\label{eq:dual-of-approx}$$ The dual of is (up to a constant independent of $\mathbf{{x}}$) $$\begin{array}{c}
\underset{\mathbf{{x}}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}}{\min}\frac{1}{2}\bigg\|\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\underset{j\in\mathcal{V}\backslash\{i\}}{\sum}\mathbf{{z}}_{j}^{n,w-1}-\mathbf{{x}}\bigg\|^{2}+\mathbf{f}_{i,n,w-1}(\mathbf{{x}}).\end{array}\label{eq:dual-of-dual-subpblm}$$ Since $\mathbf{{z}}_{i}^{n,w-1}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ and $\mathbf{{z}}_{i}^{n,w}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ are such that the components in $\mathcal{V}\backslash\{i\}$ are all zero by Proposition \[prop:sparsity\], the problem reduces to $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}-x\|^{2}+f_{i,n,w-1}(x).\end{array}\label{eq:unmod-alg-primal}$$ Suppose that the minimizer of is $\mathbf{{z}}_{i}^{n,w-1}$, which is the case when Assumption \[assu:to-start-subalg\](1) holds. Then the minimizer of is $[\mathbf{{x}}^{n,w-1}]_{i}$, which is also $[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\mathbf{{z}}_{i}^{n,w-1}]_{i}$ by . Construct $\tilde{f}_{i,n,w-1}:\mathbb{R}^{m}\to\mathbb{R}$ by $$\tilde{f}_{i,n,w-1}(x):=f_{i}([\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\mathbf{{z}}_{i}^{n,w-1}]_{i})+\langle s,x-[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\mathbf{{z}}_{i}^{n,w-1}]_{i}\rangle,\label{eq:linearize-f-i-n-w}$$ where $s\in\partial f_{i}([\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\mathbf{{z}}_{i}^{n,w-1}]_{i})$. The primal problem that we now consider is . $\hfill\Delta$
(On the condition $S_{n,1}=V_{4}$) Throughout this paper, we assumed $S_{n,1}=V_{4}$ in Assumption \[assu:to-start-subalg\]. Algorithm \[alg:Ext-Dyk\] with this condition would not be truly asynchronous, but it is relatively easy to enforce this condition. One way to enforce this condition is to use a global clock. Another way to enforce this condition is to use the sparsity of $\mathbf{z}_{\alpha}$ in Proposition \[prop:sparsity\]. Suppose that $\{S_{n,w}\}_{w=1}^{\bar{w}}$ is such that for all $i\in V_{4}$, $S_{n,w_{i}}=\{i\}$ for some $w_{i}\in\{1,\dots,\bar{w}\}$. Suppose also that for all $i,j\in V_{4}$ such that $w_{i}<w_{j}$:
- There are no $(e,k)\in\bar{E}$ such that $i$ and $j$ are the two endpoints of $e$ and $(e,k)\in S_{n,w'}$ for some $w'$ such that $w_{i}<w'<w_{j}$.
If condition ($\star$) holds for some $i,j\in V_{4}$, then the sparsity of $\mathbf{z}_{\alpha}^{n,w}$ implies that if we changed from $S_{n,w_{i}}=\{i\}$ and $S_{n,w_{j}}=\{j\}$ to $S_{n,w_{i}}=\{i,j\}$ and $S_{n,w_{j}}=\emptyset$, then the iterates $\{\mathbf{x}^{n,w}\}_{w}$ obtained will remain equivalent. It is possible to ensure ($\star$) for all $i,j\in V_{4}$ using a signal from a fixed node in $V$ propagated as computations in the algorithm are carried out.
As mentioned in Remark \[rem:min-max-of-2-quads\], the problem is still easy to solve if $f_{i,n,w-1}(\cdot)$ and $\tilde{f}_{i,n,w-1}(\cdot)$ are affine functions with the known parameters $[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}$ and $\mathbf{{z}}_{i}^{n,w-1}$.
Next, for the primal optimizer $x_{i}^{+}$ defined in line 3 of Algorithm \[alg:subdiff-subalg\], we can construct the affine function $f_{i,n,w}:\mathbb{R}^{m}\to\mathbb{R}$ to be such that $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}-x\|^{2}+f_{i,n,w}(x)\end{array}$$ has the same minimizer and objective value as . The function $f_{i,n,w}(\cdot)$ can be checked to be . It is clear to see that $f_{i,n,w}(\cdot)\leq\max\{f_{i,n,w-1}(\cdot),\tilde{f}_{i,n,w-1}(\cdot)\}$. Since both $f_{i,n,w-1}(\cdot)$ and $\tilde{f}_{i,n,w-1}(\cdot)$ are both by definition lower approximates of $f_{i}(\cdot)$, $f_{i,n,w}(\cdot)$ will also be a lower approximate of $f_{i}(\cdot)$. The function $\mathbf{f}_{i,n,w}:[\mathbb{R}^{m}]^{|\mathcal{V}|}\to\mathbb{R}$ is constructed to be $$\mathbf{f}_{i,n,w}(\mathbf{{x}})=f_{i,n,w}([\mathbf{{x}}]_{i}).$$ The $\mathbf{{z}}_{i}^{n,w}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ defined by would be the optimal solution of the dual problem $$\begin{array}{c}
\underset{\mathbf{{z}}_{i}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}}{\min}\frac{1}{2}\bigg\|\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}-\underset{j\in\mathcal{V}\backslash\{i\}}{\sum}\mathbf{{z}}_{j}^{n,w-1}-\mathbf{{z}}_{i}\bigg\|^{2}+\mathbf{f}_{i,n,w}^{*}(\mathbf{{z}}_{i}).\end{array}$$
(Similarities to the one node case) Note that the problem corresponds to , the function to , the problem to , and the function $h_{w+1}(\cdot)$ in line 9 of Algorithm \[alg:basic-dual-ascent\] to .
One way to understand Proposition \[prop:P-D-of-one-block\] is to see that any change in the primal objective value gives the same change in the dual objective value. We have the following result.
\[prop:quad-dec-case-2\]Suppose $(n,w)$ is such that $w>1$ and $S_{n,w}\subset\mathcal{V}_{4}$ so that Algorithm \[alg:subdiff-subalg\] is run, and Assumption \[assu:to-start-subalg\](1) holds. Then we have $$\begin{array}{c}
\frac{1}{2}\|\mathbf{{x}}^{n,w}-\mathbf{{x}}^{n,w-1}\|^{2}\leq F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})-F_{n,w-1}(\{\mathbf{{z}}_{\alpha}^{n,w-1}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}).\end{array}\label{eq:quad-dec-case-2}$$ Similarly, if Assumption \[assu:to-start-subalg\](2) and (3) hold, then $$\begin{array}{c}
\frac{1}{2}\|\mathbf{{x}}^{n,1}-\mathbf{{x}}^{n,0}\|^{2}\leq F_{n,1}(\{\mathbf{{z}}_{\alpha}^{n,1}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})-F_{n,0}(\{\mathbf{{z}}_{\alpha}^{n,0}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}).\end{array}\label{eq:quad-dec-case-3}$$
Recall Proposition \[prop:sparsity\] on the sparsity of the $\mathbf{{z}}_{i}^{n,w}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$. Recall that in line 3 of Algorithm \[alg:subdiff-subalg\] the primal and dual optimal solutions of are $x_{i}^{+}$ and $z_{i}^{+}$. We can see that $x_{i}^{+}=[\mathbf{{x}}^{n,w}]_{i}$ and $z_{i}^{+}=[\mathbf{{z}}_{i}^{n,w}]_{i}$. Let the dual and primal optimal solutions of be $z_{i}^{\circ}$ and $x_{i}^{\circ}$, which are $z_{i}^{\circ}=[\mathbf{{z}}_{i}^{n,w-1}]_{i}$ and $x_{i}^{\circ}=[\mathbf{{x}}^{n,w-1}]_{i}$ respectively. By Proposition \[prop:P-D-of-one-block\] and the forms of the problems and , we have $x_{i}^{+}+z_{i}^{+}=x_{i}^{\circ}+z_{i}^{\circ}$. Thus $z_{i}^{+}-z_{i}^{\circ}=-(x_{i}^{+}-x_{i}^{\circ})$. In other words, $$[\mathbf{{x}}^{n,w}-\mathbf{{x}}^{n,w-1}]_{i}=-[\mathbf{{z}}_{i}^{n,w}-\mathbf{{z}}_{i}^{n,w-1}]_{i}.\label{eq:for-beta-identity}$$ Note that since $S_{n,w}\cap\bar{\mathcal{E}}=\emptyset$, $\mathbf{{v}}_{H}^{n,w}=\mathbf{{v}}_{H}^{n,w-1}$. We have the following inequality chain, which we explain in a moment. $$\begin{aligned}
& & \begin{array}{c}
f_{i,n,w}([\mathbf{{x}}^{n,w}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-[\mathbf{{x}}^{n,w}]_{i}\|^{2}\end{array}\label{eq:for-beta-chain}\\
& = & \begin{array}{c}
f_{i,n,w-1}([\mathbf{{x}}^{n,w}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-[\mathbf{{x}}^{n,w}]_{i}\|^{2}\end{array}\nonumber \\
& \geq & \begin{array}{c}
f_{i,n,w-1}([\mathbf{{x}}^{n,w-1}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-[\mathbf{{x}}^{n,w-1}]_{i}\|^{2}+\frac{1}{2}\|[\mathbf{{x}}^{n,w-1}]_{i}-[\mathbf{{x}}^{n,w}]_{i}\|^{2}.\end{array}\nonumber \end{aligned}$$ The equation in comes from the fact that $[\mathbf{{x}}^{n,w}]_{i}$ being the minimizer of is such that $f_{i,n,w-1}([\mathbf{{x}}^{n,w}]_{i})=\tilde{f}_{i,n,w-1}([\mathbf{{x}}^{n,w}]_{i})$, and $f_{i,n,w}(\cdot)$ is designed through so that $f_{i,n,w}([\mathbf{{x}}^{n,w}]_{i})=f_{i,n,w-1}([\mathbf{{x}}^{n,w}]_{i})$. The inequality in follows from the design of $f_{i,n,w-1}(\cdot)$ through , which implies that $[\mathbf{{x}}^{n,w-1}]_{i}$ is the minimizer of $f_{i,n,w-1}(\cdot)+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-\cdot\|^{2}$.
Since $S_{n,w}\cap\bar{\mathcal{E}}=\emptyset$, we have $\mathbf{{v}}_{H}^{n,w-1}=\mathbf{{v}}_{H}^{n,w}$. Let $\beta_{i}$ be defined by $$\begin{aligned}
\beta_{i} & := & \begin{array}{c}
\left(f_{i,n,w}^{*}([\mathbf{{z}}_{i}^{n,w}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-[\mathbf{{z}}_{i}^{n,w}]_{i}\|^{2}\right)\end{array}\label{eq:beta-form}\\
& & \begin{array}{c}
-\big(f_{i,n,w-1}^{*}([\mathbf{{z}}_{i}^{n,w-1}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w-1}]_{i}-[\mathbf{{z}}_{i}^{n,w-1}]_{i}\|^{2}\big).\end{array}\nonumber \end{aligned}$$ Proposition \[prop:P-D-of-one-block\] implies that $$\begin{aligned}
& & \begin{array}{c}
f_{i,n,w}^{*}([\mathbf{{z}}_{i}^{n,w}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-[\mathbf{{z}}_{i}^{n,w}]_{i}\|^{2}\end{array}\label{eq:use-prop}\\
& = & \begin{array}{c}
-f_{i,n,w}([\mathbf{{x}}^{n,w}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}\|^{2}-\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-[\mathbf{{x}}^{n,w}]_{i}\|^{2}.\end{array}\nonumber \end{aligned}$$ An equation similar to involving $f_{i,n,w-1}(\cdot)$ plugged into and , and the fact that $\mathbf{{v}}_{H}^{n,w}=\mathbf{{v}}_{H}^{n,w-1}$ gives $\beta_{i}\geq\frac{1}{2}\|[\mathbf{{z}}_{i}^{n,w-1}-\mathbf{{z}}_{i}^{n,w}]_{i}\|^{2}$. One can easily check from the definitions that $$\begin{array}{c}
F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})-F_{n,w-1}(\{\mathbf{{z}}_{\alpha}^{n,w-1}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})=\underset{i\in S_{n,w}}{\sum}\beta_{i},\end{array}$$ which leads to our result. The proof of the second statement is exactly the same.
We remark on the design of Algorithm \[alg:Ext-Dyk\].
(On improving the affine models) In our design of Algorithm \[alg:Ext-Dyk\], we improve the affine model $f_{i,n,w}(\cdot)$ for $i\in\mathcal{V}_{4}$ only if $S_{n,w}\subset\mathcal{V}_{4}$. It is easy to see that we can apply the observation in Remark \[rem:min-max-of-2-quads\] to minimize the maximum of two quadratics analytically, but doing so without Assumption \[assu:to-start-subalg\] would affect the convergence proof.
Further new steps in convergence proof
--------------------------------------
Since the proof of convergence shares many similarities to the original proof in [@Pang_Dist_Dyk], we describe the new steps of the proof in this subsection that were not already covered and defer the rest of the proof to the appendix.
Recall the definition of $\mathbf{f}_{\alpha,n,w}(\cdot)$ in . We have the following easy claim.
\[claim:Fenchel-duality\]In Algorithm \[alg:Ext-Dyk\], for all $\alpha\in S_{n,w}$, we have
1. $-\mathbf{{x}}^{n,w}+\partial\mathbf{f}_{\alpha,n,w}^{*}(\mathbf{{z}}_{\alpha}^{n,w})\ni0$,
2. $-\mathbf{{z}}_{\alpha}^{n,w}+\partial\mathbf{f}_{\alpha,n,w}(\mathbf{{x}}^{n,w})\ni0$, and
3. $\mathbf{f}_{\alpha,n,w}(\mathbf{{x}}^{n,w})+\mathbf{f}_{\alpha,n,w}^{*}(\mathbf{{z}}_{\alpha}^{n,w})=\langle\mathbf{{x}}^{n,w},\mathbf{{z}}_{\alpha}^{n,w}\rangle$.
There are two cases. The first case is when is invoked. By taking the optimality conditions in with respect to $z_{\alpha}$ for $\alpha\in S_{n,w}$ and making use of to get $\mathbf{{x}}^{n,w}=\bar{\mathbf{{x}}}-\sum_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}\mathbf{{z}}_{\alpha}^{n,w}$, we deduce (a). The second case is when Algorithm \[alg:subdiff-subalg\] is invoked, and is similar. The equivalence of (a), (b) and (c) is standard.
For all valid $(n,w)$, since $\mathbf{f}_{\alpha,n,w}(\cdot)\leq\mathbf{f}_{\alpha}(\cdot)$ for all $\alpha\in\mathcal{V}_{4}$, we have $\mathbf{f}_{\alpha,n,w}^{*}(\cdot)\geq\mathbf{f}_{\alpha}^{*}(\cdot)$. Let $D_{\alpha,n}$ and $E_{\alpha,n}$ be defined to be
\[eq\_m:error-def\] $$\begin{aligned}
D_{\alpha,n} & := & \mathbf{f}_{\alpha,n,p(n,\alpha)}^{*}(\mathbf{{z}}_{\alpha}^{n,p(n,\alpha)})-\mathbf{f}_{\alpha}^{*}(\mathbf{{z}}_{\alpha}^{n,p(n,\alpha)})\geq0\label{eq:D-error-def}\\
\mbox{and }E_{\alpha,n} & := & \mathbf{f}_{\alpha}(\bar{x}-v_{A}^{n,p(n,\alpha)})-\mathbf{f}_{\alpha,n,p(n,\alpha)}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,p(n,\alpha)})\geq0.\label{eq:E-error-def}\end{aligned}$$
When $\alpha\in[\mathcal{\bar{E}\cup}\mathcal{V}]\backslash\mathcal{V}_{4}$, then $E_{\alpha,n}=D_{\alpha,n}=0$ for all $n$. Next, we have $$\begin{aligned}
& & \mathbf{f}_{\alpha}^{*}(\mathbf{{z}}_{\alpha}^{n,p(n,\alpha)})+\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,p(n,\alpha)})\label{eq:error-deriv}\\
& \overset{\eqref{eq_m:error-def}}{=} & \mathbf{f}_{\alpha,n,p(n,\alpha)}^{*}(\mathbf{{z}}_{\alpha}^{n,p(n,\alpha)})+\mathbf{f}_{\alpha,n,p(n,\alpha)}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,p(n,\alpha)})+E_{\alpha,n}-D_{\alpha,n}\nonumber \\
& \overset{\scriptsize{\alpha\in S_{n,p(n,\alpha)},\mbox{ Claim \ref{claim:Fenchel-duality}}}}{=} & \langle\mathbf{{z}}_{\alpha}^{n,p(n,\alpha)},\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,p(n,\alpha)}\rangle+E_{\alpha,n}-D_{\alpha,n}.\nonumber \end{aligned}$$
We now state the main convergence theorem of this paper.
\[thm:convergence\] (Convergence to primal minimizer) Consider Algorithm \[alg:Ext-Dyk\]. Assume that the problem is feasible, and for all $n\geq1$, $\bar{\mathcal{E}}_{n}=[\cup_{w=1}^{\bar{w}}S_{n,w}]\cap\bar{\mathcal{E}}$, and $[\cup_{w=1}^{\bar{w}}S_{n,w}]\supset\mathcal{V}$. Suppose that Assumption \[assu:to-start-subalg\] holds.
For the sequence $\{\mathbf{{z}}_{\alpha}^{n,w}\}_{{1\leq n<\infty\atop 0\leq w\leq\bar{w}}}\subset[\mathbb{R}^{m}]^{|\mathcal{V}|}$ for each $\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}$ generated by Algorithm \[alg:Ext-Dyk\] and the sequences $\{\mathbf{{v}}_{H}^{n,w}\}_{{1\leq n<\infty\atop 0\leq w\leq\bar{w}}}\subset[\mathbb{R}^{m}]^{|\mathcal{V}|}$ and $\{\mathbf{{v}}_{A}^{n,w}\}_{{1\leq n<\infty\atop 0\leq w\leq\bar{w}}}\subset[\mathbb{R}^{m}]^{|\mathcal{V}|}$ thus derived, we have:
1. The sum $\sum_{n=1}^{\infty}\sum_{w=1}^{\bar{w}}\|\mathbf{{v}}_{A}^{n,w}-\mathbf{{v}}_{A}^{n,w-1}\|^{2}$ is finite and $\{F_{n,\bar{w}}(\{\mathbf{{z}}_{\alpha}^{n,\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\}_{n=1}^{\infty}$ is nondecreasing.
2. There is a constant $C$ such that $\|\mathbf{{v}}_{A}^{n,w}\|^{2}\leq C$ for all $n\in\mathbb{N}$ and $w\in\{1,\dots,\bar{w}\}$.
3. For all $i\in\mathcal{V}_{3}\cup\mathcal{V}_{4}$, $n\geq1$ and $w\in\{1,\dots,\bar{w}\}$, the vectors $\mathbf{{z}}_{i}^{n,w}$ are bounded.
Suppose also that
1. There are constants $A$ and $B$ such that $$\sum_{\alpha\in\mathcal{\bar{E}}\cup\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{n,\bar{w}}\|\leq A\sqrt{n}+B\mbox{ for all }n\geq0.\label{eq:sqrt-growth-sum-z}$$
Then
1. For all $\alpha\in[\mathcal{\bar{E}}\cup\mathcal{V}]\backslash\mathcal{V}_{4}$, we have $E_{\alpha,n}=0$. Also, for all $i\in\mathcal{V}_{4}$, we have $\lim_{n\to\infty}E_{i,n}=0$.
2. There exists a subsequence $\{\mathbf{{v}}_{A}^{n_{k},\bar{w}}\}_{k=1}^{\infty}$ of $\{\mathbf{{v}}_{A}^{n,\bar{w}}\}_{n=1}^{\infty}$ which converges to some $\mathbf{{v}}_{A}^{*}\in[\mathbb{R}^{m}]^{|\mathcal{V}|}$ and that $$\lim_{k\to\infty}\langle\mathbf{{v}}_{A}^{n_{k},\bar{w}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)},\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\rangle=0\mbox{ for all }\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}.$$
3. Let $\mathbf{f}(\cdot)=\sum_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}\mathbf{f}_{\alpha}(\cdot)$. For the $\mathbf{{v}}_{A}^{*}$ in (v), $\mathbf{{x}}_{0}-\mathbf{{v}}_{A}^{*}$ is the minimizer of the primal problem and $$\lim_{k\to\infty}F_{n_{k},w}(\{\mathbf{{z}}_{\alpha}^{n_{k},w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})=\lim_{k\to\infty}F(\{\mathbf{{z}}_{\alpha}^{n_{k},w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})=\frac{1}{2}\|\mathbf{{v}}_{A}^{*}\|^{2}+\mathbf{f}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}).\label{eq:thm-iv-concl}$$
The properties (i) to (vi) in turn imply that $\lim_{n\to\infty}\mathbf{{x}}^{n,\bar{w}}$ exists and equals $\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}$, which is the primal minimizer of .
The proofs of parts (i), (ii), (v) and (vi) are similar to the proof in [@Pang_Dist_Dyk], and (iii) and (iv) are new. We shall prove (iii) and (iv) here and defer the rest of the proof to the appendix.
\[Proof of Theorem \[thm:convergence\](iii)\]In view of line 14 in Algorithm \[alg:Ext-Dyk\], it suffices to prove that $\mathbf{{z}}_{i}^{n,w}$ is bounded if $i\in S_{n,w}$. By the sparsity pattern in Proposition \[prop:sparsity\], for each $i\in\mathcal{V}_{3}\cup\mathcal{V}_{4}$, $\mathbf{{z}}_{i}^{n,w}$ is bounded if and only if $[\mathbf{{z}}_{i}^{n,w}]_{i}$ is bounded. Since $\{[\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,w}]_{i}\}_{{1\leq n<\infty\atop 0\leq w\leq\bar{w}}}$ is bounded by (ii), it is clear that $\{[\mathbf{{z}}_{i}^{n,w}]_{i}\}_{{1\leq n<\infty\atop 0\leq w\leq\bar{w}}}$ is bounded if and only if $\{[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}\}_{{1\leq n<\infty\atop 0\leq w\leq\bar{w}}}$ is bounded. Seeking a contradiction, suppose $\{[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}\}_{{1\leq n<\infty\atop 0\leq w\leq\bar{w}}}$ is unbounded. We look at the problem $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-x\|^{2}+f_{i}(x)\end{array}\label{eq:in-pf-primal-1}$$ and consider two possibilities. Let $\tilde{x}_{i}^{n,w}$ be the primal solution to . Note that if $i\in\mathcal{V}_{3}$, then $\tilde{x}_{i}^{n,w}$ is $[\mathbf{{x}}^{n,w}]_{i}$. If the $\{\tilde{x}_{i}^{n,w}\}_{n,w}$ are bounded, then the dual solution of is $[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-\tilde{x}_{i}^{n,w}$, which will be unbounded. A standard compactness argument shows that there is a point $\tilde{x}\in\mathbb{R}^{m}$ for which the set $\partial f_{i}(\tilde{x})$ is unbounded, which contradicts ${\mbox{\rm dom}}(f_{i})=\mathbb{R}^{m}$.
If the corresponding primal solutions $\tilde{x}_{i}^{n,w}$ are unbounded, consider $\{\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}$, where $$\tilde{\mathbf{{z}}}_{\alpha}^{n,w}=\mathbf{{z}}_{\alpha}^{n,w}\mbox{ if }\alpha\in[\bar{\mathcal{E}}\cup\mathcal{V}]\backslash\{i\}\mbox{ and }[\tilde{\mathbf{{z}}}_{i}^{n,w}]_{j}=\begin{cases}
[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,w}]_{i}-\tilde{x}_{i}^{n,w} & \mbox{ if }j=i\\
0 & \mbox{ otherwise. }
\end{cases}$$ Let $\tilde{F}_{n,w}(\cdot)$ be defined to be $$\begin{array}{c}
\tilde{F}_{n,w}(\{\mathbf{{z}}_{\alpha}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}):=-\frac{1}{2}\left\Vert \bar{\mathbf{{x}}}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\right\Vert ^{2}+\frac{1}{2}\|\bar{\mathbf{{x}}}\|^{2}-\underset{\alpha\in[\bar{\mathcal{E}}\cup\mathcal{V}]\backslash\{i\}}{\sum}\mathbf{f}_{\alpha,n,w}(\mathbf{{z}}_{\alpha})-\mathbf{f}_{i}(\mathbf{{z}}_{i})\end{array}\label{eq:Dykstra-dual-defn-2}$$ Then $F_{n,w}(\cdot)\leq\tilde{F}_{n,w}(\cdot)\leq F(\cdot)$. Also, Proposition \[prop:P-D-of-one-block\] shows that $[\tilde{\mathbf{{z}}}_{i}^{n,w}]_{i}$ is the dual solution to . So $$F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\leq\tilde{F}_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\leq\tilde{F}_{n,w}(\{\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}\})\leq F(\{\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}\}).\label{eq:big-F-chain}$$ Next, suppose $\mathbf{{x}}^{*}$ is a solution of . Then $$\begin{aligned}
& & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}^{*}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha}(\mathbf{{x}}^{*})-F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\end{array}\\
& \overset{\eqref{eq:big-F-chain}}{\geq} & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}^{*}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha}(\mathbf{{x}}^{*})-F(\{\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\end{array}\\
& \overset{\eqref{eq:From-8}}{\geq} & \begin{array}{c}
\frac{1}{2}\left\Vert \bar{\mathbf{{x}}}-\mathbf{{x}}^{*}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\tilde{\mathbf{{z}}}_{\alpha}^{n,w}\right\Vert ^{2}\end{array}\\
& \overset{\scriptsize{\mbox{Take }i\mbox{-th component only}}}{\geq} & \begin{array}{c}
\frac{1}{2}\left\Vert \tilde{x}_{i}^{n,w}-[\mathbf{{x}}^{*}]_{i}\right\Vert ^{2}.\end{array}\end{aligned}$$ The above inequality and the unboundedness of $\tilde{x}_{i}^{n,w}$ implies that the duality gap would go to infinity, which contradicts part (i). Thus we are done.
\[Proof of Theorem \[thm:convergence\](iv)\]The first sentence of this claim is immediate from . We now prove the second sentence. Seeking a contradiction, suppose that $\limsup_{n\to\infty}E_{i,n}>0$. In Algorithm \[alg:subdiff-subalg\], in view of Assumption \[assu:to-start-subalg\](2), $[\mathbf{{z}}_{i}^{n,\bar{w}}]_{i}$ is the minimizer of the problem $$\begin{array}{c}
\underset{z\in\mathbb{R}^{m}}{\min}\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,\bar{w}}]_{i}-z\|^{2}+f_{i,n,\bar{w}}^{*}(z).\end{array}\label{eq:in-pf-dual}$$ The associated primal problem is, up to a constant independent of $x$, $$\begin{array}{c}
\underset{x\in\mathbb{R}^{m}}{\min}\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,\bar{w}}]_{i}-x\|^{2}+f_{i,n,\bar{w}}(x).\end{array}\label{eq:in-pf-primal}$$ The primal solution is $[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,\bar{w}}]_{i}-[\mathbf{{z}}_{i}^{n,\bar{w}}]_{i}\overset{\eqref{eq_m:all_acronyms}}{=}[\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,\bar{w}}]_{i}$. The dual solution is $[\mathbf{{z}}_{i}^{n,\bar{w}}]_{i}$. So $$[\mathbf{{x}}_{i}^{n,\bar{w}}]_{i}\in\partial f_{i,n,\bar{w}}([\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,\bar{w}}]_{i}).\label{eq:subgrad-birth}$$ Recall Assumption \[assu:to-start-subalg\](3) and $\mathbf{{v}}_{H}^{n,\bar{w}}=\mathbf{{v}}_{H}^{n+1,0}$ by line 16 in Algorithm \[alg:Ext-Dyk\]. We now analyze the increase in the dual objective value of each separate problem: $$\begin{aligned}
\Delta_{i,n} & := & \begin{array}{c}
\big[f_{i,n+1,0}^{*}([\mathbf{{z}}_{i}^{n+1,0}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,0}]_{i}-[\mathbf{{z}}_{i}^{n+1,0}]_{i}\|^{2}\big]\end{array}\\
& & \begin{array}{c}
-\big[f_{i,n+1,1}^{*}([\mathbf{{z}}_{i}^{n+1,1}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,1}]_{i}-[\mathbf{{z}}_{i}^{n+1,1}]_{i}\|^{2}\big].\end{array}\end{aligned}$$ Recall that $E_{i,n}$ is also $f_{i}([\mathbf{{x}}^{n,\bar{w}}]_{i})-f_{i,n,\bar{w}}([\mathbf{{x}}^{n,\bar{w}}]_{i})=f_{i}([\mathbf{{x}}^{n+1,0}]_{i})-f_{i,n+1,0}([\mathbf{{x}}^{n+1,0}]_{i}).$ Proposition \[prop:P-D-of-one-block\] and Assumption \[assu:to-start-subalg\](2) tell us that $$\begin{aligned}
& & \begin{array}{c}
f_{i,n+1,0}^{*}([\mathbf{{z}}_{i}^{n+1,0}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,0}]_{i}-[\mathbf{{z}}_{i}^{n+1,0}]_{i}\|^{2}\big]\end{array}\\
& = & \begin{array}{c}
-f_{i,n+1,0}([\mathbf{{x}}^{n+1,0}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,0}]_{i}\|^{2}-\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,0}]_{i}-[\mathbf{{x}}^{n+1,0}]_{i}\|^{2}\big].\end{array}\end{aligned}$$ A similar result holds for the problem involving $f_{i,n+1,1}(\cdot)$. Since $S_{n+1,1}\cap\bar{\mathcal{E}}=\emptyset$ and $\mathbf{{v}}_{H}^{n+1,0}=\mathbf{{v}}_{H}^{n+1,1}$, we have $$\begin{aligned}
\Delta_{i,n} & = & \begin{array}{c}
\big[f_{i,n+1,1}([\mathbf{{x}}^{n+1,1}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,1}]_{i}-[\mathbf{{x}}^{n+1,1}]_{i}\|^{2}\big]\end{array}\\
& & \begin{array}{c}
-\big[f_{i,n+1,0}([\mathbf{{x}}^{n+1,0}]_{i})+\frac{1}{2}\|[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,0}]_{i}-[\mathbf{{x}}^{n+1,0}]_{i}\|^{2}\big].\end{array}\end{aligned}$$ The analogue of Lemma \[lem:alpha-recurrs\](1) tells us that $\Delta_{i,n}\geq\frac{1}{2}t_{i,n}^{2}$, where $t_{i,n}$ is the positive root satisfying $$\begin{array}{c}
\frac{1}{2}t_{i,n}^{2}+\|s_{i,n,\bar{w}}+[\mathbf{{x}}^{n+1,0}]_{i}-[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n+1,0}]_{i}\|t_{i,n}=E_{i,n},\end{array}$$ where $s_{i,n,\bar{w}}\in\partial f_{i}([\mathbf{{x}}^{n,\bar{w}}]_{i})$ is the subgradient used to form the linearization of $f(\cdot)$ at $[\mathbf{{x}}^{n+1,0}]_{i}$. Note that $[\mathbf{{x}}^{n,\bar{w}}]_{i}-[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,\bar{w}}]_{i}=-[\mathbf{{z}}_{i}^{n,\bar{w}}]_{i}$, so the term $\|s_{i,n,\bar{w}}+[\mathbf{{x}}^{n,\bar{w}}]_{i}-[\bar{\mathbf{{x}}}-\mathbf{{v}}_{H}^{n,\bar{w}}]_{i}\|$ becomes $\|s_{i,n,\bar{w}}-[\mathbf{{z}}_{i}^{n,\bar{w}}]_{i}\|$. Since both $s_{i,n,\bar{w}}$ and $[\mathbf{{z}}_{i}^{n,\bar{w}}]_{i}$ are bounded, $\limsup_{n\to\infty}t_{i,n}>0$, and so $\limsup_{n\to\infty}\Delta_{i,n}>0$. We can check from the definitions that $$\begin{array}{c}
F_{n+1,1}(\{\mathbf{{z}}_{\alpha}^{n+1,1}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})-F_{n+1,0}(\{\mathbf{{z}}_{\alpha}^{n+1,0}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})=\underset{i\in S_{n,\bar{w}}}{\sum}\Delta_{i,n}.\end{array}$$ This means that the dual objective value can increase indefinitely, which then implies that the problem is infeasible, which is a contradiction.
Proposition \[prop:control-growth\] below shows some reasonable conditions to guarantee . The ideas of its proof were already present in [@Pang_Dyk_spl; @Pang_Dist_Dyk], so we defer its proof to the appendix.
\[prop:control-growth\](Growth of $\sum_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{n,w}\|$) In Algorithm \[alg:Ext-Dyk\], suppose:
1. There are only finitely many $S_{n,w}$ for which $S_{n,w}\cap[\mathcal{V}_{1}\cup\mathcal{V}_{2}]$ contains more than one element.
2. There are constants $M_{1}>0$ and $M_{2}>0$ such that the size of the set $$\big\{(n',w):n'\leq n,\,w\in\{1,\dots,\bar{w}\},\,|S_{n',w}\cap\mathcal{V}|>1\big\}$$ is bounded by $M_{1}\sqrt{n}+M_{2}$ for all $n$.
Then condition (1) in Theorem \[thm:convergence\] holds.
\[subsec:composition-lin-op\]Composition with a linear operator
----------------------------------------------------------------
Suppose some $f_{i}(\cdot)$ were defined as $f_{i,1}\circ A_{i}(\cdot)$, where $f:Y\to\mathbb{R}$ is a closed convex function, $Y$ is another finite dimensional Hilbert space and $A_{i}:\mathbb{R}^{m}\to Y$ is a linear map. One may either still try to take the proximal mapping of $f_{i}(\cdot)$, but it may involve some expensive operations on $A_{i}$. Alternatively, we can write or we can write $f_{i,1}\circ A_{i}(x_{i})$ as $$f_{i}(y_{i})+\delta_{\{(x,y):A_{i}x=y\}}(x_{i},y_{i}),$$ which splits into the sum of two functions. Note however that since we require the problem to be strongly convex, creating the new variable $y$ adds new regularizing terms to the objective function.
Conclusion
==========
The main contribution in this paper is to show that the distributed Dykstra’s algorithm can be extended to incorporate subdifferentiable functions in a natural manner so that the algorithm converges to the primal minimizer, even if there is no dual minimizer. A next question is to find convergence rates of the algorithm. The derivation of such rates uses rather different techniques from that of this paper, and requires additional conditions to ensure the existence of a dual minimizer. We defer this to [@Pang_rate_D_Dyk], where we also perform numerical experiments that show that the distributed Dykstra’s algorithm is sound.
Further proofs
==============
In this appendix, we completing the parts of the proofs of Theorem \[thm:convergence\] and \[prop:control-growth\] that we consider to be too similar to the ones in [@Pang_Dist_Dyk].
The following inequality describes the duality gap between and . $$\begin{aligned}
& & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha}(\mathbf{{x}})-F(\{\mathbf{{z}}_{\alpha}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\end{array}\label{eq:From-8}\\
& \overset{\eqref{eq:Dykstra-dual-defn}}{=} & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}[\mathbf{f}_{\alpha}(\mathbf{{x}})+\mathbf{f}_{\alpha}^{*}(\mathbf{{z}}_{\alpha})]-\left\langle \bar{\mathbf{{x}}},\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\right\rangle +\frac{1}{2}\left\Vert \underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\right\Vert ^{2}\end{array}\nonumber \\
& \overset{\scriptsize\mbox{Fenchel duality}}{\geq} & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}\|^{2}+\left\langle \mathbf{{x}},\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\right\rangle -\left\langle \bar{\mathbf{{x}}},\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\right\rangle +\frac{1}{2}\left\Vert \underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\right\Vert ^{2}\end{array}\nonumber \\
& = & \begin{array}{c}
\frac{1}{2}\left\Vert \bar{\mathbf{{x}}}-\mathbf{{x}}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{{z}}_{\alpha}\right\Vert ^{2}\geq0.\end{array}\nonumber \end{aligned}$$
We continue with proving the rest of Theorem \[thm:convergence\].
\[Proof of rest of Theorem \[thm:convergence\]\] We first show that (i) to (vi) implies the final assertion. For all $n\in\mathbb{N}$ we have, from weak duality, $$\begin{array}{c}
F(\{\mathbf{{z}}_{\alpha}^{n,\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\leq\frac{1}{2}\|\bar{\mathbf{{x}}}-(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}).\end{array}\label{eq:weak-duality}$$ Since the values $\{F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\}_{n=1}^{\infty}$ are nondecreasing in $n$, we make use of (v) to get $$\begin{array}{rcl}
\underset{n\to\infty}{\lim}F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}) & = & \frac{1}{2}\|\bar{\mathbf{{x}}}-(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}).\end{array}$$ Since $F_{n,w}(\cdot)\leq F(\cdot)\leq\frac{1}{2}\|\bar{\mathbf{{x}}}-(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})\|^{2}+\sum_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})$, we have . Hence $\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}=\arg\min_{\mathbf{{x}}}\mathbf{f}(\mathbf{{x}})+\frac{1}{2}\|\mathbf{{x}}-\bar{\mathbf{{x}}}\|^{2}$, and (substituting $\mathbf{{x}}=\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}$ in ) $$\begin{aligned}
& & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})\|^{2}+\mathbf{f}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})-F(\{\mathbf{{z}}_{\alpha}^{n,\bar{w}}\}_{\alpha\in\mathcal{E}\cup\mathcal{V}})\end{array}\\
& \overset{\eqref{eq:From-8},\eqref{eq:v-H-def},\eqref{eq:from-10}}{\geq} & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})-\mathbf{{v}}_{A}^{n,\bar{w}}\|^{2}\end{array}\\
& \overset{\eqref{eq:x-from-v-A}}{=} & \begin{array}{c}
\frac{1}{2}\|\mathbf{{x}}^{n,\bar{w}}-(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})\|^{2}.\end{array}\end{aligned}$$ Hence $\lim_{n\to\infty}\mathbf{{x}}^{n,\bar{w}}$ is the minimizer in (P).
It remains to prove assertions (i) to (vi).
**Proof of (i):** We separate into two cases.
We first consider the case when $S_{n,w}\not\subset\mathcal{V}_{4}$. From the fact that $\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in S_{n,w}}$ minimizes (which includes the quadratic regularizer) we have $$F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\overset{\eqref{eq:Dykstra-min-subpblm}}{\geq}\begin{array}{c}
F_{n,w-1}(\{\mathbf{{z}}_{\alpha}^{n,w-1}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})+\frac{1}{2}\|\mathbf{{v}}_{A}^{n,w}-\mathbf{{v}}_{A}^{n,w-1}\|^{2}.\end{array}\label{eq:SHQP-decrease}$$ (The last term in arises from the quadratic term in .) By line 16 of Algorithm \[alg:Ext-Dyk\], $\mathbf{{z}}_{i}^{n+1,0}=\mathbf{{z}}_{i}^{n,\bar{w}}$ for all $i\in\mathcal{V}$ and $\mathbf{{v}}_{H}^{n+1,0}=\mathbf{{v}}_{H}^{n,\bar{w}}$ (even though the decompositions of $\mathbf{{v}}_{H}^{n+1,0}$ and $\mathbf{{v}}_{H}^{n,\bar{w}}$ may be different).
In the second case when $S_{n,w}\subset\mathcal{V}_{4}$, Proposition \[prop:quad-dec-case-2\] and show that the inequality holds.
Combining over all $n'\in\{1,\dots,n\}$ and $w\in\{1,\dots,\bar{w}\}$, we have $$\begin{array}{c}
F_{1,0}(\{\mathbf{{z}}_{\alpha}^{1,0}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})+\underset{n'=1}{\overset{n}{\sum}}\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|^{2}\overset{\eqref{eq:SHQP-decrease}}{\leq}F_{n,\bar{w}}(\{\mathbf{{z}}_{\alpha}^{n,\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}).\end{array}$$ Next, $F_{n,\bar{w}}(\{\mathbf{{z}}_{\alpha}^{n,\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})$ is bounded from above by weak duality. The proof of the claim is complete.
**Proof of (ii):** From part (i) and the fact that $F_{n,w}(\cdot)\leq F(\cdot)$, we have $$-F_{1,0}(\{\mathbf{{z}}_{\alpha}^{1,0}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\overset{\scriptsize\mbox{part (i)}}{\geq}-F_{n,w}(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\geq-F(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}).\label{eq:three-Fs}$$ Substituting $\mathbf{{x}}$ in to be the primal minimizer $\mathbf{{x}}^{*}$ and $\{\mathbf{{z}}_{\alpha}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}$ to be $\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}$, we have $$\begin{aligned}
& & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}^{*}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\mathbf{{x}}^{*})-F_{1,0}(\{\mathbf{{z}}_{\alpha}^{1,0}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\end{array}\\
& \overset{\eqref{eq:three-Fs}}{\geq} & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}^{*}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\mathbf{{x}}^{*})-F(\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\end{array}\\
& \overset{\eqref{eq:From-8}}{\geq} & \begin{array}{c}
\frac{1}{2}\left\Vert \bar{\mathbf{{x}}}-\mathbf{{x}}^{*}-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{{z}}_{\alpha}^{n,w}\right\Vert ^{2}\overset{\eqref{eq:from-10}}{=}\frac{1}{2}\|\bar{\mathbf{{x}}}-\mathbf{{x}}^{*}-\mathbf{{v}}_{A}^{n,w}\|^{2}.\end{array}\end{aligned}$$ The conclusion is immediate.
**Proof of (v):** We first make use of the technique in [@BauschkeCombettes11 Lemma 29.1] (which is in turn largely attributed to [@BD86]) to show that $$\begin{array}{c}
\underset{n\to\infty}{\liminf}\left[\left(\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{A}^{n,w}-\mathbf{{v}}_{A}^{n,w-1}\|\right)\sqrt{n}\right]=0.\end{array}\label{eq:root-n-dec}$$ Seeking a contradiction, suppose instead that there is an $\epsilon>0$ and $\bar{n}>0$ such that if $n>\bar{n}$, then $\left(\sum_{w=1}^{\bar{w}}\|\mathbf{{v}}_{A}^{n,w}-\mathbf{{v}}_{A}^{n,w-1}\|\right)\sqrt{n}>\epsilon$. By the Cauchy Schwarz inequality, we have $\begin{array}{c}
\frac{\epsilon^{2}}{n}<\left(\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{A}^{n,w}-\mathbf{{v}}_{A}^{n,w-1}\|\right)^{2}\leq\bar{w}\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{A}^{n,w}-\mathbf{{v}}_{A}^{n,w-1}\|^{2}.\end{array}$ This contradicts the earlier claim in (i) that $\sum_{n=1}^{\infty}\sum_{w=1}^{\bar{w}}\|\mathbf{{v}}_{A}^{n,w}-\mathbf{{v}}_{A}^{n,w-1}\|^{2}$ is finite.
Through , we find a sequence $\{n_{k}\}_{k=1}^{\infty}$ such that $$\begin{array}{c}
\lim_{k\to\infty}\left[\left(\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{A}^{n_{k},w}-\mathbf{{v}}_{A}^{n_{k},w-1}\|\right)\sqrt{n_{k}}\right]=0.\end{array}\label{eq:subseq-sqrt-limit}$$ Recalling the assumption , we get $$\begin{array}{c}
\underset{k\to\infty}{\lim}\left[\left(\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{A}^{n_{k},w}-\mathbf{{v}}_{A}^{n_{k},w-1}\|\right)\|\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\|\right]\overset{\eqref{eq:sqrt-growth-sum-z},\eqref{eq:subseq-sqrt-limit}}{=}0\mbox{ for all }\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}.\end{array}\label{eq:lim-sum-norm-z}$$ Moreover, $$\begin{aligned}
|\langle\mathbf{{v}}_{A}^{n_{k},\bar{w}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)},\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\rangle| & \leq & \begin{array}{c}
\|\mathbf{{v}}_{A}^{n_{k},\bar{w}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)}\|\|\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\|\end{array}\label{eq:inn-pdt-sum-norm}\\
& \leq & \begin{array}{c}
\left(\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{A}^{n_{k},w}-\mathbf{{v}}_{A}^{n_{k},w-1}\|\right)\|\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\|.\end{array}\nonumber \end{aligned}$$ By (ii), there exists a further subsequence of $\{\mathbf{{v}}_{A}^{n_{k},\bar{w}}\}_{k=1}^{\infty}$ which converges to some $\mathbf{{v}}_{A}^{*}\in\mathbb{R}^{m}$. Combining and gives (v).
**Proof of (vi):** From earlier results, we obtain $$\begin{aligned}
& & \begin{array}{c}
-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\bar{\mathbf{{z}}}-\mathbf{{v}}_{A}^{*})\end{array}\label{eq:biggest-formula}\\
& \overset{\eqref{eq:From-8}}{\leq} & \begin{array}{c}
\frac{1}{2}\|\bar{\mathbf{{x}}}-(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})\|^{2}-F(\{\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})\end{array}\nonumber \\
& \overset{\eqref{eq:Dykstra-dual-defn},\eqref{eq:stagnant-indices}}{=} & \begin{array}{c}
\frac{1}{2}\|\mathbf{{v}}_{A}^{*}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}^{*}(\mathbf{{z}}_{\alpha}^{n_{k},p(n_{k},\alpha)})\end{array}\nonumber \\
& & \begin{array}{c}
+\underset{((i,j),\bar{k})\notin\bar{\mathcal{E}}_{n_{k}}}{\overset{}{\sum}}\mathbf{f}_{((i,j),\bar{k})}^{*}(\mathbf{{z}}_{((i,j),\bar{k})}^{n_{k},\bar{w}})-\langle\bar{\mathbf{{x}}},\mathbf{{v}}_{A}^{n_{k},\bar{w}}\rangle+\frac{1}{2}\|\mathbf{{v}}_{A}^{n_{k},\bar{w}}\|^{2}\end{array}\nonumber \\
& \overset{\scriptsize\eqref{eq:error-deriv},\eqref{eq:zero-indices}}{=} & \begin{array}{c}
\frac{1}{2}\|\mathbf{{v}}_{A}^{*}\|^{2}+\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\langle\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)},\mathbf{{z}}_{\alpha}^{n_{k},p(n_{k},\alpha)}\rangle+\underset{i\in\mathcal{V}_{4}}{\overset{}{\sum}}E_{i,n_{k}}-\underset{i\in\mathcal{V}_{4}}{\overset{}{\sum}}D_{i,n_{k}}\end{array}\nonumber \\
& & \begin{array}{c}
-\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)})-\langle\bar{\mathbf{{x}}},\mathbf{{v}}_{A}^{n_{k},\bar{w}}\rangle+\frac{1}{2}\|\mathbf{{v}}_{A}^{n_{k},\bar{w}}\|^{2}\end{array}\nonumber \\
& \overset{\eqref{eq:stagnant-indices}}{=} & \begin{array}{c}
\frac{1}{2}\|\mathbf{{v}}_{A}^{*}\|^{2}-\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\langle\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)}-\mathbf{{v}}_{A}^{n_{k},\bar{w}},\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\rangle\end{array}\nonumber \\
& & \begin{array}{c}
-\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)})-\langle\bar{\mathbf{{x}}},\mathbf{{v}}_{A}^{n_{k},\bar{w}}\rangle+\underset{i\in\mathcal{V}_{4}}{\overset{}{\sum}}E_{i,n_{k}}-\underset{i\in\mathcal{V}_{4}}{\overset{}{\sum}}D_{i,n_{k}}\end{array}\nonumber \\
& & \begin{array}{c}
+\left\langle \bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},\bar{w}},\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{{z}}_{\alpha}^{n_{k},p(n_{k},\alpha)}\right\rangle +\frac{1}{2}\|\mathbf{{v}}_{A}^{n_{k},\bar{w}}\|^{2}\end{array}\nonumber \\
& \overset{\eqref{eq:from-10},\eqref{eq:zero-indices}}{=} & \begin{array}{c}
\frac{1}{2}\|\mathbf{{v}}_{A}^{*}\|^{2}-\frac{1}{2}\|\mathbf{{v}}_{A}^{n_{k},\bar{w}}\|^{2}-\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\langle\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)}-\mathbf{{v}}_{A}^{n_{k},\bar{w}},\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\rangle\end{array}\nonumber \\
& & \begin{array}{c}
-\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\overset{}{\sum}}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)})+\underset{i\in\mathcal{V}_{4}}{\overset{}{\sum}}E_{i,n_{k}}-\underset{i\in\mathcal{V}_{4}}{\overset{}{\sum}}D_{i,n_{k}}.\end{array}\nonumber \end{aligned}$$
Since $\lim_{k\to\infty}\mathbf{{v}}_{A}^{n_{k},\bar{w}}=\mathbf{{v}}_{A}^{*}$, we have $\lim_{k\to\infty}\frac{1}{2}\|\mathbf{{v}}_{A}^{*}\|^{2}-\frac{1}{2}\|\mathbf{{v}}_{A}^{n_{k},\bar{w}}\|^{2}=0$. The third term in the last group of formulas (i.e., the sum involving the inner products) converges to 0 by (v). The term $\lim_{k\to\infty}\sum_{i\in\mathcal{V}_{4}}E_{i,n_{k}}$ equals to 0 by (iii).
Next, recall that if $((i,j),\bar{k})\in\bar{\mathcal{E}}_{n_{k}}$, by , we have $\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},((i,j),\bar{k}))}\in H_{((i,j),\bar{k})}$. Note that from Claim \[claim:Fenchel-duality\](b), we have $\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n,p(n,((i,j),\bar{k}))}\in H_{((i,j),\bar{k})}$ for all $((i,j),\bar{k})\in\bar{\mathcal{E}}_{n}$. There is a constant $\kappa_{\bar{\mathcal{E}}_{n_{k}}}>0$ such that $$\begin{aligned}
& & d(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},\bar{w}},\cap_{((i,j),\bar{k})\in\bar{\mathcal{E}}}H_{((i,j),\bar{k})})\label{eq:reg-argument}\\
& \overset{\scriptsize{\bar{\mathcal{E}}_{n_{k}}\mbox{ connects }\mathcal{V},\mbox{ Prop \ref{prop:E-connects-V}(1)}}}{=} & d(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},\bar{w}},\cap_{((i,j),\bar{k})\in\bar{\mathcal{E}}_{n_{k}}}H_{((i,j),\bar{k})})\nonumber \\
& \leq & \kappa_{\bar{\mathcal{E}}_{n_{k}}}\max_{((i,j),\bar{k})\in\bar{\mathcal{E}}_{n_{k}}}d(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},\bar{w}},H_{((i,j),\bar{k})})\nonumber \\
& \overset{\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},((i,j),\bar{k}))}\in H_{((i,j),\bar{k})}}{\leq} & \kappa_{\bar{\mathcal{E}}_{n_{k}}}\max_{((i,j),\bar{k})\in\bar{\mathcal{E}}_{n_{k}}}\|\mathbf{{v}}_{A}^{n_{k},\bar{w}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},((i,j),\bar{k}))}\|.\nonumber \end{aligned}$$ Let $\kappa:=\max\{\kappa_{\mathcal{\bar{E}}'}:\bar{\mathcal{E}}'\mbox{ connects }\mathcal{V}\}$. We have $\kappa_{\bar{\mathcal{E}}_{n_{k}}}\leq\kappa$. Taking limits of , the RHS converges to zero by (i), so $d(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*},\cap_{((i,j),\bar{k})\in\bar{\mathcal{E}}}H_{((i,j),\bar{k})})=0$, or $\bar{\mathbf{{x}}}-\mathbf{{\mathbf{v}}}_{A}^{*}\in\cap_{((i,j),\bar{k})\in\bar{\mathcal{E}}}H_{((i,j),\bar{k})}$. So $\sum_{((i,j),\bar{k})\in\bar{\mathcal{E}}}\mathbf{f}_{((i,j),\bar{k})}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})=0$. Together with the fact that $\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},((i,j),\bar{k}))}\in H_{((i,j),\bar{k})}$, we have $$\sum_{((i,j),\bar{k})\in\bar{\mathcal{E}}_{n_{k}}}\mathbf{f}_{((i,j),\bar{k})}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},((i,j),\bar{k}))})=0=\underset{((i,j),\bar{k})\in\bar{\mathcal{E}}}{\overset{}{\sum}}\mathbf{f}_{((i,j),\bar{k})}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}).\label{eq:all-indicator-edges-zero}$$
Lastly, by the lower semicontinuity of $\mathbf{f}_{i}(\cdot)$, we have $$\begin{array}{c}
-\underset{k\to\infty}{\lim}\underset{i\in\mathcal{V}}{\sum}\mathbf{f}_{i}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},i)})\leq-\underset{i\in\mathcal{V}}{\overset{\phantom{\mathcal{V}}}{\sum}}\mathbf{f}_{i}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}).\end{array}\label{eq:lsc-argument}$$ As mentioned after , taking the limits as $k\to\infty$ would result in the first three terms and the 5th term of the last formula in to be zero. Hence $$\begin{aligned}
& & \begin{array}{c}
-\underset{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*})\end{array}\\
& \overset{\eqref{eq:biggest-formula}}{\leq} & \begin{array}{c}
\underset{k\to\infty}{\lim}-\underset{\alpha\in\bar{\mathcal{E}}_{n_{k}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{n_{k},p(n_{k},\alpha)})-\underset{k\to\infty}{\lim}\underset{i\in\mathcal{V}_{4}}{\sum}D_{i,n_{k}}\end{array}\\
& \overset{\eqref{eq:all-indicator-edges-zero},\eqref{eq:lsc-argument},\eqref{eq:D-error-def}}{\leq} & \begin{array}{c}
-\underset{\alpha\in\mathcal{\bar{E}}\cup\mathcal{V}}{\sum}\mathbf{f}_{\alpha}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}).\end{array}\end{aligned}$$ So becomes an equation in the limit, and $\lim_{n_{k}\to\infty}D_{i,n_{k}}=0$ for all $i\in\mathcal{V}_{4}$. The first two lines of then gives $$\begin{array}{c}
\underset{k\to\infty}{\lim}F(\{\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})=\frac{1}{2}\|\mathbf{{v}}_{A}^{*}\|^{2}+\underset{i\in\mathcal{V}}{\sum}\mathbf{f}_{i}(\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}),\end{array}$$ which shows that $\bar{\mathbf{{x}}}-\mathbf{{v}}_{A}^{*}$ is the primal minimizer. Recall the definitions of $F_{n,w}(\cdot)$, $F(\cdot)$ and $D_{i,n}$ in , and . We recall . Also, from line 11 of Algorithm \[alg:Ext-Dyk\], we have $\mathbf{f}_{\alpha,n,w}(\cdot)=\mathbf{f}_{\alpha,n,p(n,\alpha)}(\cdot)$. This gives $F_{n_{k},\bar{w}}(\{\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})+\sum_{i\in\mathcal{V}_{4}}D_{i,n_{k}}=F(\{\mathbf{{z}}_{\alpha}^{n_{k},\bar{w}}\}_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}})$, from which we deduce the equation on the left of as well.
\[Proof of Proposition \[prop:control-growth\]\]Since this result is used only in the proof of Theorem \[thm:convergence\](v), we can make use of Theorem \[thm:convergence\](i) and (iii) in its proof. To address condition (1), we can assume that $S_{n,w}\cap[\mathcal{V}_{1}\cup\mathcal{V}_{2}]$ always contains at most one element. Define the sets $\bar{S}_{n,1}$ and $\bar{S}_{n,2}\subset\{1,2,\dots\}\times\{1,\dots,\bar{w}\}$ as $$\begin{aligned}
\bar{S}_{n,1} & := & \{(n',w):n'\leq n,\,w\in\{1,\dots,\bar{w}\},\,|S_{n',w}\cap\mathcal{V}|\leq1\}\\
\bar{S}_{n,2} & := & \{(n',w):n'\leq n,\,w\in\{1,\dots,\bar{w}\},\,|S_{n',w}\cap\mathcal{V}|>1\}.\end{aligned}$$ Either $S_{n',w}\cap[\mathcal{V}_{1}\cup\mathcal{V}_{2}]=\emptyset$ or $|S_{n',w}\cap[\mathcal{V}_{1}\cup\mathcal{V}_{2}]|=1$. In the second case, let $i^{*}$ be the index such that $i^{*}\in S_{n',w}\cap[\mathcal{V}_{1}\cup\mathcal{V}_{2}]$. Otherwise, in the first case, we let $i^{*}$ be any index in $[\mathcal{V}_{1}\cup\mathcal{V}_{2}]$. We prove claims based on whether $(n',w)$ lies in $\bar{S}_{n,1}$ or $\bar{S}_{n,2}$.
Without loss of generality, we can assume that $S_{n',w}\cap\bar{\mathcal{E}}$ are linearly independent constraints. This also means that for a $\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}$, each $\mathbf{{z}}_{((i,j),\bar{k})}^{n',w}-\mathbf{{z}}_{((i,j),\bar{k})}^{n',w-1}$ can be determined uniquely with a linear map from the relation $$\sum_{\alpha\in\bar{\mathcal{E}}}[\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}]\overset{\eqref{eq:v-H-def}}{=}\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}.$$ Therefore for all $\alpha\in S_{n',w}\cap\bar{\mathcal{E}}$, there is a constant $\kappa_{\alpha,S_{n',w}\cap\bar{\mathcal{E}}}>0$ such that $$\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|\leq\kappa_{\alpha,S_{n',w}\cap\bar{\mathcal{E}}}\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|.\label{eq:basic-bdd-z-i-j}$$ Thus there is a constant $\kappa>0$ such that $$\sum_{\alpha\in\bar{\mathcal{E}}}\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|\overset{\scriptsize{\mbox{Alg \ref{alg:Ext-Dyk} line 14}}}{=}\sum_{\alpha\in S_{n,w}\cap\bar{\mathcal{E}}}\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|\overset{\eqref{eq:basic-bdd-z-i-j}}{\leq}\kappa\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|.\label{eq:bdd-z-i-j}$$
**** $$\begin{array}{c}
\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|+\underset{\alpha\in\bar{\mathcal{E}}}{\sum}\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|+\underset{i\in\mathcal{V}}{\overset{\phantom{i\in\mathcal{V}}}{\sum}}\|\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w-1}\|\leq C_{2}\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|.\end{array}\label{eq:all-3-bdd}$$
We have $$\begin{aligned}
\begin{array}{c}
\underset{i\in\mathcal{V}}{\sum}[\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}]_{i}\end{array} & \overset{\scriptsize{\eqref{eq_m:all_acronyms}}}{=} & \begin{array}{c}
\underset{i\in\mathcal{V}}{\sum}\underset{\alpha\in S_{n',w}}{\sum}[\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}]_{i}\end{array}\nonumber \\
& \overset{\mathbf{{z}}_{((i,j),\bar{k})}\in D^{\perp},\eqref{eq:D-and-D-perp}}{=} & \begin{array}{c}
\underset{i\in\mathcal{V}}{\sum}[\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}]_{i}\end{array}\nonumber \\
& \overset{\scriptsize{\mbox{Prop \ref{prop:sparsity}}}}{=} & \begin{array}{c}
[\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}]_{i^{*}}.\end{array}\label{eq:for-norm-rate}\end{aligned}$$ Recall that the norm $\|\cdot\|$ always refers to the $2$-norm unless stated otherwise. By the equivalence of norms in finite dimensions, we can find a constant $c_{1}$ such that $$\begin{aligned}
\begin{array}{c}
\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|\end{array} & \geq & \begin{array}{c}
c_{1}\underset{i\in\mathcal{V}}{\sum}\|[\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}]_{i}\|\end{array}\label{eq:bdd-z-i}\\
& \geq & \begin{array}{c}
c_{1}\Big\|\underset{i\in\mathcal{V}}{\sum}[\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}]_{i}\Big\|\end{array}\nonumber \\
& \overset{\eqref{eq:for-norm-rate}}{=} & \begin{array}{c}
c_{1}\|\mathbf{{z}}_{i^{*}}^{n,w}-\mathbf{{z}}_{i^{*}}^{n,w-1}\|\end{array}\nonumber \\
& \overset{\scriptsize{\mbox{Alg \ref{alg:Ext-Dyk} line 14}}}{=} & \begin{array}{c}
c_{1}\underset{i\in\mathcal{V}}{\sum}\|\mathbf{{z}}_{i}^{n,w}-\mathbf{{z}}_{i}^{n,w-1}\|.\end{array}\nonumber \end{aligned}$$ Next, $\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\overset{\eqref{eq:from-10}}{=}\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}-(\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1})$, so $$\begin{aligned}
\begin{array}{c}
\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|\end{array} & \leq & \begin{array}{c}
\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|+\|\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}\|\end{array}\label{eq:bdd-v-H}\\
& \overset{\eqref{eq:bdd-z-i}}{\leq} & \begin{array}{c}
\left(1+\frac{1}{c_{1}}\right)\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|.\end{array}\nonumber \end{aligned}$$ We can choose $\{\mathbf{{z}}_{\alpha}^{n,w}\}_{\alpha\in\bar{\mathcal{E}}}$ such that $$\sum_{\alpha\in S_{n',w}\cap\bar{\mathcal{E}}}[\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}]\overset{\scriptsize{\mbox{Alg \ref{alg:Ext-Dyk} line 14}}}{=}\sum_{\alpha\in\bar{\mathcal{E}}}[\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}]\overset{\eqref{eq:v-H-def}}{=}\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}.\label{eq:decomp-v-H}$$ Combining , and together shows that there is a constant $C_{2}>1$ such that holds.$\hfill\triangle$
**** $$\begin{array}{c}
\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|+\underset{\alpha\in\bar{\mathcal{E}}}{\sum}\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|+\underset{i\in\mathcal{V}}{\overset{\phantom{i\in\mathcal{V}}}{\sum}}\|\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w-1}\|\leq C_{5}.\end{array}\label{eq:all-3-bdd-2}$$ We have
$$\begin{aligned}
& & \begin{array}{c}
\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|\end{array}\label{eq:to-bdd-i-star-terms}\\
& \geq & \begin{array}{c}
c_{1}\Big\|\underset{i\in\mathcal{V}}{\sum}[\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}]_{i}\Big\|\end{array}\nonumber \\
& \overset{\eqref{eq_m:all_acronyms},\mathbf{{z}}_{((i,j),\bar{k})}\in D^{\perp},\eqref{eq:D-and-D-perp}}{=} & \begin{array}{c}
c_{1}\Big\|\underset{i\in\mathcal{V}}{\sum}\underset{j\in\mathcal{V}}{\sum}[\mathbf{{z}}_{j}^{n',w}-\mathbf{{z}}_{j}^{n',w-1}]_{i}\Big\|\end{array}\nonumber \\
& \overset{\scriptsize{\mbox{Prop. }\ref{prop:sparsity}}}{=} & \begin{array}{c}
c_{1}\Big\|\underset{i\in\mathcal{V}}{\sum}[\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w-1}]_{i}\Big\|\end{array}\nonumber \\
& = & \begin{array}{c}
c_{1}\Big\|[\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}]_{i^{*}}+\underset{i\in\mathcal{V}_{3}\cup\mathcal{V}_{4}}{\sum}[\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w-1}]_{i}\Big\|.\end{array}\nonumber \end{aligned}$$
From Theorem \[thm:convergence\](i), there is a constant $C_{3}>0$ such that $\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|\leq C_{3}$. By Theorem \[thm:convergence\](iii), there is a constant $C_{4}>0$ such that $$\|\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w}\|\leq C_{4}\mbox{ for all }i\in\mathcal{V}_{3}\cup\mathcal{V}_{4}.\label{eq:final-i-bdd}$$ So $\|\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}\|=\|[\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}]_{i^{*}}\|\overset{\eqref{eq:to-bdd-i-star-terms},\eqref{eq:final-i-bdd}}{\leq}(|\mathcal{V}_{3}|+|\mathcal{V}_{4}|)C_{4}+\frac{1}{c_{1}}C_{3}$, and $$\begin{array}{c}
\underset{i\in\mathcal{V}}{\sum}\|\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w-1}\|\overset{\eqref{eq:to-bdd-i-star-terms},\eqref{eq:final-i-bdd}}{\leq}2(|\mathcal{V}_{3}|+|\mathcal{V}_{4}|)C_{4}+\frac{1}{c_{1}}C_{3}.\end{array}\label{eq:final-star-bdd}$$ Next, from , we have $$\begin{aligned}
\begin{array}{c}
\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\end{array} & = & \begin{array}{c}
\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}+[\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}]+\underset{i\in\mathcal{V}_{3}\cup\mathcal{V}_{4}}{\sum}[\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w-1}]\end{array}\nonumber \\
\begin{array}{c}
\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|\end{array} & \leq & \begin{array}{c}
\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|+\|\mathbf{{z}}_{i^{*}}^{n',w}-\mathbf{{z}}_{i^{*}}^{n',w-1}\|+\underset{i\in\mathcal{V}_{3}\cup\mathcal{V}_{4}}{\sum}\|\mathbf{{z}}_{i}^{n',w}-\mathbf{{z}}_{i}^{n',w-1}\|\end{array}\nonumber \\
& \leq & \begin{array}{c}
C_{3}+2(|\mathcal{V}_{3}|+|\mathcal{V}_{4}|)C_{4}+\frac{1}{c_{1}}C_{3}.\end{array}\label{eq:final-H-bdd}\end{aligned}$$ Combining , and , we can show that Claim 2 holds. $\hfill\triangle$
Since $\{\mathbf{{z}}_{\alpha}^{n,0}\}_{\alpha\in\bar{\mathcal{E}}}$ was chosen to satisfy , there is some $M>1$ such that $$\begin{array}{c}
\underset{\alpha\in\bar{\mathcal{E}}}{\sum}\|\mathbf{{z}}_{\alpha}^{n,0}\|\overset{\eqref{eq:reset-z-i-j-3}}{\leq}M\|\mathbf{{v}}_{H}^{n,0}\|\overset{\eqref{eq:reset-z-i-j-4}}{\leq}M\left(\|\mathbf{{v}}_{H}^{1,0}\|+\underset{n'=1}{\overset{n-1}{\sum}}\underset{w=1}{\overset{\bar{w}}{\sum}}\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|\right)\end{array}\label{eq:z-bdd-for-E}$$ Now for any $n\geq1$, we have $$\begin{aligned}
\sum_{\alpha\in\bar{\mathcal{E}}\cup\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{n,\bar{w}}\| & \leq & \sum_{\alpha\in\bar{\mathcal{E}}}\|\mathbf{{z}}_{\alpha}^{n,0}\|+\sum_{w=1}^{\bar{w}}\sum_{\alpha\in\bar{\mathcal{E}}}\|\mathbf{{z}}_{\alpha}^{n,w}-\mathbf{{z}}_{\alpha}^{n,w-1}\|\label{eq:2nd-big-ineq}\\
& & +\sum_{n'=1}^{n}\sum_{w=1}^{\bar{w}}\sum_{\alpha\in\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|+\sum_{\alpha\in\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{1,0}\|\nonumber \\
& \overset{\eqref{eq:z-bdd-for-E}}{\leq} & M\|\mathbf{{v}}_{H}^{1,0}\|+\sum_{\alpha\in\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{1,0}\|+\sum_{w=1}^{\bar{w}}\left(\sum_{\alpha\in\bar{\mathcal{E}}}\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|\right)\nonumber \\
& & +\sum_{n'=1}^{n-1}\sum_{w=1}^{\bar{w}}\left(M\|\mathbf{{v}}_{H}^{n',w}-\mathbf{{v}}_{H}^{n',w-1}\|+\sum_{\alpha\in\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{n',w}-\mathbf{{z}}_{\alpha}^{n',w-1}\|\right)\nonumber \\
& \overset{\eqref{eq:all-3-bdd},\eqref{eq:all-3-bdd-2}}{\leq} & M\|\mathbf{{v}}_{H}^{1,0}\|+\sum_{\alpha\in\mathcal{V}}\|\mathbf{{z}}_{\alpha}^{1,0}\|+MC_{2}\sum_{m=1}^{n}\sum_{w=1}^{\bar{w}}\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|\nonumber \\
& & +MC_{5}\left(M_{1}\sqrt{n}+M_{2}\right).\nonumber \end{aligned}$$ By the Cauchy Schwarz inequality, we have $$\sum_{n'=1}^{n}\sum_{w=1}^{\bar{w}}\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|\leq\sqrt{n\bar{w}}\sqrt{\sum_{n'=1}^{n}\sum_{w=1}^{\bar{w}}\|\mathbf{{v}}_{A}^{n',w}-\mathbf{{v}}_{A}^{n',w-1}\|^{2}}.\label{eq:sum-bdd-by-sqrt-n}$$ Since the second square root of the right hand side of is bounded by Theorem \[thm:convergence\](i), we make use of to obtain the conclusion as needed.
[^1]: C.H.J. Pang acknowledges grant R-146-000-214-112 from the Faculty of Science, National University of Singapore.
|
---
abstract: 'A hitting set for a collection of sets is a set that has a non-empty intersection with each set in the collection; the hitting set problem is to find a hitting set of minimum cardinality. Motivated by instances of the hitting set problem where the number of sets to be hit is large, we introduce the notion of *implicit hitting set problems*. In an implicit hitting set problem the collection of sets to be hit is typically too large to list explicitly; instead, an oracle is provided which, given a set $H$, either determines that $H$ is a hitting set or returns a set that $H$ does not hit. We show a number of examples of classic implicit hitting set problems, and give a generic algorithm for solving such problems optimally. The main contribution of this paper is to show that this framework is valuable in developing approximation algorithms. We illustrate this methodology by presenting a simple on-line algorithm for the minimum feedback vertex set problem on random graphs. In particular our algorithm gives a feedback vertex set of size $n-(1/p)\log{np}(1-o(1))$ with probability at least $3/4$ for the random graph $G_{n,p}$ (the smallest feedback vertex set is of size $n-(2/p)\log{np}(1+o(1))$). We also consider a planted model for the feedback vertex set in directed random graphs. Here we show that a hitting set for a polynomial-sized subset of cycles is a hitting set for the planted random graph and this allows us to exactly recover the planted feedback vertex set.'
author:
- 'Karthekeyan Chandrasekaran [^1]\'
- 'Richard Karp [^2]\'
- 'Erick Moreno-Centeno [^3]\'
- Santosh Vempala
bibliography:
- 'references.bib'
title: Algorithms for Implicit Hitting Set Problems
---
=1
Introduction
============
In the classic Hitting Set problem, we are given a universe $U$ of elements and a collection $\T$ of subsets $S_1, \ldots, S_m$ of $U$; the objective is to find a subset $H\subseteq U$ of minimum cardinality so that every subset $S_i$ in $\T$ contains at least one element from $H$. The problem is NP-hard [@karp-np-complete], approximable to within $\log_2 |U|$ using a greedy algorithm, and has been studied for many interesting special cases.
There are instances of the hitting set problem where the number of subsets $|\T|$ to hit is exponential in the size of the universe. Consequently, obtaining a hitting set with approximation factor $\log_2|U|$ using the greedy algorithm which examines all subsets is unreasonable for practical applications. Our motivation is the possibility of algorithms that run in time polynomial in the size of the universe. In this paper, we introduce a framework that could be useful in developing efficient approximation algorithms for instances of the hitting set problem with exponentially many subsets to hit.
We observe that in many combinatorial problems, $\T$ has a succinct representation that allows efficient verification of whether a candidate set hits every subset in $\T$. Formally, in an implicit hitting set problem, the input is a universe $U$ and a polynomial-time [*oracle*]{} that, given a set $H$, either determines that $H$ is a hitting set or returns a subset that is not hit by $H$. Thus, the collection $\T$ of subsets to hit is not specified explicitly. The objective is to find a small hitting set by making at most polynomial($|U|$) queries to the oracle. In Section 1.1, we show several well-known problems that can be formulated as implicit hitting set problems.
We present a generic algorithm to obtain the optimal solution of implicit hitting set problems in Section 2. As this algorithm solves optimally the NP-hard (classic) hitting set problem as a subroutine, its worst-case running time is exponential as a function of $|U|$. The main purpose of stating the generic algorithm is to develop an intuition towards using the oracle. It suggests a natural way to use the oracle: first (1) propose a candidate hitting set $H$, then (2) use the oracle to check if the candidate set hits all the subsets, and if not obtain a subset $S$ that has not been hit, and finally (3) refine $H$ based on $S$ and repeat until a hitting set is found. The generic algorithm for the implicit hitting set problem is in fact a generalization of online algorithms for hitting set problems. Here, the ground set is specified in advance as before and the subsets to be hit arrive online. On obtaining a subset, the algorithm has to decide which new element to include in the hitting set and commit to the element. Thus, the online algorithm is restricted in that the refinement procedure can only add elements. Moreover, only those subsets that have not been hit by the candidate set are revealed online thereby saving the algorithm from having to examine all subsets in $\T$. This is similar to the mistake bound learning model [@littlestone88].
We apply the implicit hitting set framework and specialize the generic algorithm to the *Minimum Feedback Vertex Set* (FVS) problem: given a graph $G(V,E)$, find a subset $S\subseteq V$ of smallest cardinality so that every cycle in the graph contains at least one vertex from $S$. Although the number of cycles could be exponential in the size of the graph, one can efficiently check whether a proposed set $H$ hits all cycles ([*i.e.,*]{} is a feedback vertex set) or find a cycle that is not hit by $H$ using a breadth-first search procedure after removing the subset of vertices $H$ from the graph. The existence of a polynomial time oracle shows that it is an instance of the implicit hitting set problem.
The main focus of this paper is to develop algorithms that find nearly optimal hitting sets in random graphs or graphs with planted feedback vertex sets, by examining only a polynomial number of cycles. For this to be possible, we need the oracle to pick cycles that have not yet been hit in a natural yet helpful manner. If the oracle is adversarial, this could force the algorithm to examine almost all cycles. We consider two natural oracles: one that picks cycles in breath-first search (BFS) order and another that picks cycles according to their size.
We prove that if cycles in the random graph $G_{n,p}$ are obtained in a breadth-first search ordering, there is an efficient algorithm that examines a polynomial collection $\T'$ of cycles to build a nearly optimal feedback vertex set for the graph. The algorithm builds a solution iteratively by (1) proposing a candidate for a feedback vertex set in each iteration, (2) finding the next cycle that is not hit in a breadth-first ordering of all cycles, (3) augmenting the proposed set and repeating. A similar result for directed random graphs using the same algorithm follows by ignoring the orientation of the edges. Our algorithm is an online algorithm i.e., it commits to only adding and not deleting vertices from the candidate feedback vertex set.
It is evident from our results that the size of the feedback vertex set in both directed and undirected random graphs is close to $n$, for sufficiently large $p$. This motivates us to ask if a smaller planted feedback vertex set in random graphs can be recovered by using the implicit hitting set framework. This question is similar in flavor to the well-studied planted clique problem [@jerrumClique92; @sudakov98; @frieze08], but posed in the framework of implicit hitting set problems. We consider a natural planted model for the feedback vertex set problem in directed graphs. In this model, a subset of $\delta n$ vertices, for some constant $0<\delta\leq 1$, is chosen to be the feedback vertex set. The subgraph induced on the complement is a random directed acyclic graph (DAG) and all the other arcs are chosen with probability $p$ independently. The objective is to recover the planted feedback vertex set. We prove that the optimal hitting set for cycles of bounded size is the planted feedback vertex set. Consequently, ordering the cycles according to their sizes and finding an approximately optimal hitting set for the small cycles is sufficient to recover the planted feedback vertex set. This also leads to an online algorithm when cycles are revealed in increasing order of their size with ties broken arbitrarily. We conclude this section with some well-known examples of implicit hitting set problems.
Implicit Hitting Set Problems
-----------------------------
An [*implicit hitting set problem*]{} is one in which, for each instance, the set of subsets is not listed explicitly but instead is specified implicitly by an [*oracle*]{}: a polynomial-time algorithm which, given a set $H \subset U$, either certifies that $H$ is a hitting set or returns a subset that is not hit by $H$.
Each of the following is an implicit hitting set problem:
- [**Feedback Vertex Set in a Graph or Digraph**]{}\
Ground Set: Set of vertices of graph or digraph $G$.\
Subsets: Vertex sets of simple cycles in $G$.
- [**Feedback Edge Set in a Digraph**]{}\
Ground Set: Set of edges of digraph $G$.\
Subsets: Edge sets of simple cycles in $G$.
- [**Max Cut**]{}\
Ground Set: Set of edges of graph $G$.\
Subsets: Edge sets of simple odd cycles in $G$.
- [**k-Matroid Intersection**]{}\
Ground Set: Common ground set of $k$ matroids.\
Subsets: Subsets in the $k$ matroids.
- [**Maximum Feasible Set of Linear Inequalities**]{}\
Ground Set: A finite set of linear inequalities.\
Subsets: Minimal infeasible subsets of the set of linear inequalities.
- [**Maximum Feasible Set of Equations of the Form $x_i - x_j = c_{ij}\ (\!\!\!\mod q)$**]{}\
This example is motivated by the Unique Games Conjecture.
- [**Synchronization in an Acyclic Digraph**]{}\
Ground Set: A collection $U$ of pairs of vertices drawn from the vertex set of an acyclic digraph $G$.\
Subsets: Minimal collection $C$ of pairs from $U$ with the property that, if each pair in $C$ is contracted to a single vertex, then the resulting digraph contains a cycle.
[**Organization.**]{} In Section 2, we present a generic algorithm for the optimal solution of implicit hitting set problems. Then, we focus on specializing this algorithm to obtain small feedback vertex sets in directed and undirected random graphs. We analyze the performance of this algorithm in Section 3. We then consider a planted model for the feedback vertex set problem in directed random graphs. In Section 4, we give an algorithm to recover the planted feedback vertex set by finding an approximate hitting set for a polynomial-sized subset of cycles. We prove a lower bound for the size of the feedback vertex set in random graphs in Section 5. We state our results more precisely in the next section.
Results for Feedback Vertex Set Problems
----------------------------------------
We consider the feedback vertex set problem for the random graph $G_{n,p}$, a graph on $n$ vertices in which each edge is chosen independently with probability $p$. Our main result here is that a simple augmenting approach based on ordering cycles according to a breadth-first search (Algorithm Augment-BFS described in the next section) has a strong performance guarantee.
\[theorem:undirected-MFVS\] For $G_{n,p}$, such that $p=o(1)$, there exists a polynomial time algorithm that produces a feedback vertex set of size at most $n-(1/p)\log{(np)}(1-o(1))$ with probability at least $3/4$.
Throughout, $o(1)$ is with respect to $n$. We complement our upper bound with a lower bound on the feedback vertex set for $G_{n,p}$ obtained using simple union bound arguments.
\[theorem:undirected-FVSlowerbound\] Let $r=\frac{2}{p}\log{(np)}(1+o(1))+1$. If $p<1/2$, then every subgraph induced by any subset of $r$ vertices in $G_{n,p}$ contains a cycle with high probability.
This gives an upper bound of $r-1$ on the maximum induced acyclic subgraph of $G_{n,p}$. So, the size of the minimum feedback vertex set for $G_{n,p}$ is at least $n-r+1=n-(2/p)\log{np}$. A result of Fernandez de la Vega [@delavega96] shows that $G_{n,p}$ has an induced tree of size at least $(2/p)\log{np}(1-o(1))$, when $p=o(1)$. This gives the best possible existential result: there exists a feedback vertex set of size at most $n-(2/p)\log{np}(1-o(1))$ with high probability in $G_{n,p}$, when $p=o(1)$. We note that this result is not algorithmic; Fernandez de la Vega gives a greedy algorithm to obtain the largest induced tree of size $(1/p)\log{np}(1-o(1))$ in [@delavega86]. This algorithm is based on growing the induced forest from the highest labeled vertex and does not fall in the implicit hitting set framework (when the graph is revealed as a set of cycles). In contrast, our main contribution to the FVS problem in random graphs is showing that a simple breadth-first ordering of the cycles is sufficient to find a nearly optimal feedback vertex set. We also note that our algorithm is an online algorithm with good performance guarantee when the cycles are revealed according to a breadth-first ordering. Improving on the size of the FVS returned by our algorithm appears to require making progress on the long-standing open problem of finding an independent set of size $((1+\eps)/p)\log {np}$ in $G_{n,p}$. Assuming an optimal algorithm for this problem leads to an asymptotically optimal guarantee matching Fernandez de la Vega’s existential bound.
Next, we turn our attention to the directed random graph $D_{n,p}$ on $n$ vertices. The directed random graph $D_{n,p}$ is obtained as follows: choose a set of undirected edges joining distinct elements of $V$ independently with probability $2p$. For each chosen undirected edge $\{u,v\}$, orient it in one of the two directions $\{u\rightarrow v, v\rightarrow u\}$ in $D_{n,p}$ with equal probability.
The undirected graph $G_D$ obtained by ignoring the orientation of the edges in $D_{n,p}$ is the random graph $G(n,2p)$. Moreover, a feedback vertex set in $G_D$ is also a feedback vertex set for $D_{n,p}$. Therefore, by ignoring the orientation of the arcs, the Augment-BFS algorithm as applied to undirected graphs can be used to obtain a feedback vertex set of size at most $n-(1/2p)\log{(2np)}$ with probability at least $3/4$. A theorem of Spencer and Subramanian [@spencer-subramanian] gives a nearly matching lower bound on the size of the feedback vertex set in $D_{n,p}$.
[@spencer-subramanian]\[theorem:directed-FVSlowerbound\] Consider the random graph $D_{n,p}$, where $np\geq W$, for some fixed constant $W$. Let $r=(2/\log{(1-p)^{-1}})(\log{(np)}+3e)$. Every subgraph induced by any subset of $r$ vertices in $G$ contains a cycle with high probability.
It is evident from the results above that the feedback vertex set in a random graph contains most of its vertices when $p=o(1)$. This motivates us to ask if a significantly smaller “planted” feedback vertex set in a random graph can be recovered with the implicit hitting set framework. In order to address this question, we present the following planted model.
The planted directed random graph ${D}_{n,\delta,p}$ on $n$ vertices for $0<\delta\leq 1$ is obtained as follows: Choose $\delta n$ vertices arbitrarily to be the planted subset $P$. Each pair $(u,v)$ where $u\in P, v\in V$, is adjacent independently with probability $2p$ and the corresponding edge is oriented in one of the two directions $\{u\rightarrow v, v\rightarrow u\}$ in $D_{n,\delta,p}$ with equal probability. The arcs between vertices in $V\setminus P$ are obtained in the following manner to ensure that the subgraph induced on $V\setminus P$ is a DAG: Pick an arbitrary permutation of the vertices in $V\setminus P$. With the vertices ordered according to this permutation, each forward arc is present with probability $p$ independently; no backward arcs occur according to this ordering.
\[fig:planted-model\]
We prove that for graphs $D_{n,\delta,p}$, for large enough $p$, it is sufficient to hit cycles of small size to recover the planted feedback vertex set. For example, if $p\geq C_0/n^{1/3}$ for some absolute constant $C_0$, then it is sufficient to find the best hitting set for triangles in $D_{n,\delta,p}$. This would be the planted feedback vertex set. We state the theorem for cycles of length $k$.
\[theorem:planted-directedFVS\] Let $D$ be a planted directed random graph $D_{n,\delta,p}$ with planted feedback vertex set $P$, where $p\geq C/n^{1-2/k}$ for some constant $C$ and $0<\delta\leq 9/19$. Then, with high probability, the smallest hitting set for the set of cycles of size $k$ in $D$ is the planted feedback vertex set $P$.
Thus, in order to recover the planted feedback vertex set, it is sufficient to obtain cycles in increasing order of their sizes and find the best hitting set for the subset of all cycles of size $k$. Moreover, the expected number of cycles of length $k$ is at most $(nkp)^{k}=\text{poly}(n)$ for the mentioned range of $p$ and constant $k$. Thus, we have a polynomial-sized collection $\T'$ of cycles, such that the optimal hitting set for $\T'$ is also the optimal hitting set for all cycles in $D_{n,\delta,p}$.
However, finding the smallest hitting set is NP-hard even for triangles. We give an efficient algorithm to recover the planted feedback vertex set using an *approximate* hitting set for the small cycles.
\[theorem:algorithm-planted-directedFVS\] Let $D$ be a planted directed random graph $D_{n,\delta,p}$ with planted feedback vertex set $P$, where $p\geq C/n^{1-2/k}$ for some constant $C$ and $k\geq 3$, $0<\delta\leq 1/2k$. Then, there exists an algorithm to recover the planted feedback vertex set $P$ with high probability; this algorithm has an expected running time of $(nkp)^{O(k)}$.
Algorithms
==========
In this section, we mention a generic algorithm for implicit hitting set problems. We then focus on specializing this algorithm to the feedback vertex set problems in directed and undirected graphs.
A Generic Algorithm
-------------------
We mention a generic algorithm for solving instances of the implicit hitting set problem optimally with the aid of an oracle and a subroutine for the exact solution of (explicit) hitting set problems. The guiding principle is to build up a short list of important subsets that dictate the solution, while limiting the number of times the subroutine is invoked, since its computational cost is high.
A set $H \subset U$ is called [*feasible*]{} if it is a hitting set for the implicit hitting set problem, and [*optimal*]{} if it is feasible and of minimum cardinality among all feasible hitting sets. Whenever the oracle reveals that a set $H$ is not feasible, it returns $c(H)$, a subset that H does not hit. Each generated subset $c(H)$ is added to a growing list $\Gamma$ of subsets. A set $H$ is called $\Gamma$-feasible if it hits every subset in $\Gamma$ and $\Gamma$-optimal if it is $\Gamma$-feasible and of minimum cardinality among all $\Gamma$-feasible subsets. If a $\Gamma$-optimal set $K$ is feasible then it is necessarily optimal since $K$ is a valid hitting set for the implicit hitting set problem which contains subsets in $\Gamma$, and $K$ is the minimum hitting set for subsets in $\Gamma$. Thus the goal of the algorithm is to construct a feasible $\Gamma$-optimal set. [**Generic Algorithm**]{}\
Initialize $\Gamma \leftarrow \emptyset$.
1. Repeat:
1. $H\leftarrow U$.
2. Repeat while there exists a $\Gamma$-feasible set $H' = (H \cup X) - Y$ such that $X,Y\subseteq U$, $|X|<|Y|$:
1. If $H'$ is feasible then $H\leftarrow H'$; else $\Gamma \leftarrow \Gamma \cup \{c(H')\}$.
3. Construct a $\Gamma$-optimal set $K$.
4. If $|H|=|K|$ then return $H$ and halt ($H$ is optimal); if $K$ is feasible then return $K$ and halt ($K$ is optimal); else $\Gamma \leftarrow \Gamma \cup \{c(K)\}$.
[**Remark 1**]{}. Since the generic algorithm solves optimally an NP-hard problem as a subroutine, its worst-case execution time is exponential in $|U|$. Its effectiveness in practice depends on the choice of the missed subset that the oracle returns.
A companion paper [@km] describes successful computational experience with an algorithm that formulates a multi-genome alignment problem as an implicit hitting set problem, and solves it using a specially tailored variant of the generic algorithm.
Algorithm Augment-BFS
---------------------
In this section, we give an algorithm to find the feedback vertex set in both undirected and directed graphs. Here, we use an oracle that returns cycles according to a breadth-first search ordering. Instead of the exact algorithm for the (explicit) hitting set problem, as suggested in the generic algorithm, we use a simpler strategy of picking a vertex from each missed cycle. Essentially, the algorithm considers cycles according to a breadth-first search ordering and maintains an induced tree on a set of vertices denoted as surviving vertices. The vertices deleted in the process will constitute a feedback vertex set. Having built an induced tree on surviving vertices up to a certain depth $i$, the algorithm is presented with cycles obtained by a one-step BFS exploration of the surviving vertices at depth $i$. For each such cycle, the algorithm picks a vertex at depth $i+1$ to delete. The vertices at depth $i+1$ that are not deleted are added to the set of surviving vertices, thereby leading to an induced tree on surviving vertices up to depth $i+1$.
[**Remark 2**]{}. Although a very similar algorithm can be used for other variants of the feedback set problem, we note that these problems in random graphs turn out to be easy. For example, the feedback edge set problem is equivalent to the maximum spanning tree problem, while the feedback arc set problem has tight bounds for random graphs using very simple algorithms.
Feedback Vertex Set in Random Graphs
====================================
In this section, we show that Augment-BFS can be used to find a nearly optimal feedback vertex set in the undirected random graph $G_{n,p}$. Our main contribution is a rigorous analysis of the heuristic of simple cycle elimination in BFS order. We say that a vertex $v$ is a [***unique***]{} neighbor of a subset of vertices $L$ if and only if $v$ is adjacent to exactly one vertex in $L$.
In Algorithm Augment-BFS, we obtain induced cycles in BFS order having deleted the vertices from the current candidate FVS $S$. We refine the candidate FVS $S$ precisely as follows to obtain an induced BFS tree with unit increase in height: Consider the set $c(S)$ of cycles obtained by one-step BFS exploration from the set of vertices at current depth. Let $K$ denote the set of unexplored vertices in the cycles in $c(S)$ ($K$ is a subset of the vertices obtained by one-step BFS exploration from the set of vertices at current depth). Among the vertices in $K$ include all non-unique neighbors of the set of vertices at current depth into $S$. Find a large independent set in the subgraph induced by the unique neighbors $R\subseteq K$ of the set of vertices at current depth. Include all vertices in $R$ that are not in the independent set into $S$. This iterative refinement process is a natural adaptation of the idea behind the generic algorithm to the feedback vertex set problem where one collects a subset of cycles to find a hitting set $H$ for these cycles and proposes $H$ as the candidate set to obtain more cycles that have not been hit.
\[fig:bfs-exploration\]
Essentially, the algorithm maintains an induced BFS tree by deleting vertices to remove cycles. The set of deleted vertices form a FVS. Consequently at each level of the BFS exploration, one would prefer to add as many vertices from the next level $K$ as possible maintaining the acyclic property. One way to do this is as follows: Delete all the non-unique neighbors of the current level from $K$ thus hitting all cycles across the current and next level. There could still be cycles using an edge through the unique neighbors. To hit these, add a large independent set from the subgraph induced by the unique neighbors and delete the rest. Observe that this induced subgraph is a random graph on a smaller number of vertices. However, even for random graphs, it is open to find the largest independent set efficiently and only a factor $2$ approximation is known.
In our analysis, instead of using the two approximate algorithm for the independent set problem, we use the simple heuristic of deleting a vertex for each edge that is present in the subgraph to find an independent set at each level. In order to lower bound the size of the induced tree, it suffices to consider growing the BFS-tree up to a certain height $T$ using this heuristic and then using the $2$-approximate algorithm for independent set at height $T$ to terminate the algorithm. The size of the induced tree obtained using Algorithm Augment-BFS is at least as large as the one produced by the process just described. To simplify our analysis, it will be useful to restate the algorithm as Algorithm Grow-induced-BFS.
We remark that improving the approximation factor of the largest independent set problem in $G_{n,p}$ would also improve the size of the FVS produced. Our analysis shows that most of the vertices in the induced BFS tree get added at depth $T$ as an independent set. Moreover, the size of this independent set is close to $(2/p)\log{np}(1-o(1))$. Consequently, any improvement on the approximation factor of the largest independent set problem in $G_{n,p}$ would also lead to improving the size of the independent set found at depth $T$. This would increase the number of vertices in the induced BFS tree and thereby reduce the number of vertices in the feedback vertex set.
Observe that Algorithm Grow-induced-BFS can be used for the directed random graph $D_{n,p}$ by ignoring the orientation of the edges to obtain a nearly optimal feedback vertex set. Such a graph obtained by ignoring the orientation of the edges is the random graph $G(n,2p)$. Further, a FVS in such a graph is also a FVS in the directed graph. Consequently, we have the following theorem.
\[directed-FVS\] For $D_{n,p}$, there exists a polynomial time algorithm that produces a FVS of size at most $n-(1/2p)(\log{(np)}-o(1))$ with probability at least $3/4$.
By Theorem \[theorem:directed-FVSlowerbound\], we see that the algorithm is nearly optimal for directed random graphs.
Next, we analyze Algorithm Grow-induced-BFS to find the size of the FVS that it returns. For $i=0,\cdots,T$, let $L_i$ be the set of surviving vertices at level $i$ with $l_i:=|L_i|$, $R_{i+1}$ be the set of *unique* neighbors of $L_i$ with $r_{i+1}:=|R_{i+1}|$, and $U_{i}$ be the set of unexposed vertices of the graph after $i$ levels of BFS exploration with $u_{i}:=|U_{i}|$. Observe that $U_i:=V\setminus (L_0\cup_{j=1}^{i} K_i)$. We will need the following theorem due to Frieze [@frieze-ind-set], about the size of the independent set.
[@frieze-ind-set] \[theorem:large-ind-set\] Let $d=np$ and $\eps>0$ be fixed. Suppose $d_{\eps}\leq d=o(n)$ for some sufficiently large fixed constant $d_{\eps}$. Then, almost surely, the size of the independent set in $G_{n,p}$ is at least $$\left(\frac{2}{p}\right)(\log{np}-\log{\log{np}}-\log{2}+1-0.5\eps).$$
Large Set of Unique Neighbors
-----------------------------
The following lemma gives a concentration of the number of surviving vertices, unexposed vertices and unique neighbors to survivors at a particular level. It shows that upon exploring $t$ levels according to the algorithm, the number of surviving vertices at the $t$-th level, $l_t$, is not too small while the number of unexposed vertices, $u_t$, is large. It also shows a lower bound on the number of unique neighbors $r_{t+1}$ to a level of survivors. This fact will be used in proving Theorem \[theorem:undirected-MFVS\].
\[lemma:largesurvivors\] Let $c:=np$ and $T$ be the largest integer that satisfies $16Tp(c+20\sqrt{c})^{T-1}\leq 1/2$. Then, with probability at least $3/4$, $\forall t \in \{0,1,\cdots,T-1\}$,
1. $$\begin{aligned}
u_t &\leq \left(n-\frac{1}{4}\sum_{i=0}^t(c-20\sqrt{c})^i \right)\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right) \\
u_t &\geq (n-\sum_{i=0}^t(c+20\sqrt{c})^i)\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\end{aligned}$$
2. $$\begin{aligned}
l_t &\leq \left(c+20\sqrt{c} \right)^t\\
l_t &\geq \left(c-20\sqrt{c}\right)^t(1-16Tp(c+20\sqrt{c})^t)\\
& \ \ \ \times \left(1-\frac{\sum_{i=0}^t(c+20\sqrt{c})^i}{n}\right)\end{aligned}$$
3. $$\begin{aligned}
r_t &\leq (c+20\sqrt{c})^{t+1}\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
r_t &\geq \frac{(c-20\sqrt{c})^{t+1}}{4} \left(1-\frac{\sum_{i=0}^{t+1}(c+20\sqrt{c})^i}{n}\right)\\
& \ \ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\end{aligned}$$
Now, we are ready to prove Theorem \[theorem:undirected-MFVS\].
Proof of main theorem
---------------------
\[Proof of Theorem \[theorem:undirected-MFVS\]\] Our objective is to use the fact that the size of the surviving set of vertices is large when the algorithm has explored $T-1$ levels. Moreover, the number of unexposed vertices is also large. Thus, there is a large independent set among the *unique* neighbors of the surviving vertices. This set along with the surviving vertices up to level $T-1$ will form a large induced tree. We will now prove that the size of the independent set among the *unique* neighbors of $L_{T-1}$ is large.
By Theorem \[theorem:large-ind-set\], if $r_{T}p>d_{\eps}$ for some constant $d_{\eps}$ and $r_{T}p=o(r_{T})$, then there exists an independent set of size $(2/p)\log{(r_{T}p)}(1-o(1))$. It suffices to prove that $r_{T}$ is large and is such that $r_Tp>d_{\eps}$.
Note that the choice of $T=\left \lceil \frac{\ln{(1/16p)}-\ln{\ln{(1/16p)}}}{\ln{(c+20\sqrt{c})}}\right \rceil$ used in the algorithm satisfies the hypothesis of Lemma \[lemma:largesurvivors\]. Therefore, using Lemma \[lemma:largesurvivors\], with probability at least $3/4$, we have $$\begin{aligned}
r_{T}&\geq \frac{(c-20\sqrt{c})^{T}}{4} \left(1-\frac{\sum_{i=0}^{T}(c+20\sqrt{c})^i}{n}\right)\\
& \ \ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\geq \frac{(c-20\sqrt{c})}{64p}\left(1-\frac{\sum_{i=0}^{T}(c+20\sqrt{c})^i}{n}\right)\\
& \ \ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\geq \frac{c-20\sqrt{c}}{2^8p} \geq \frac{d_{\eps}}{p}\end{aligned}$$ for sufficiently large $c$ since $$\left(1-\frac{\sum_{i=0}^{T}(c+20\sqrt{c})^i}{n}\right)\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\geq \frac{15}{16}\cdot\frac{1}{2}\\.$$
Consequently, by Theorem \[theorem:large-ind-set\], there exists an independent set of size at least $\left({2}/{p}\right)\log {(r_{T}p)}(1-o(1))$. Moreover, step 3 of the algorithm finds a $2$-approximate independent set (see [@grimmett-diarmid-1975; @diarmid-1984]). Therefore, the size of the independent set found in step 3 is at least $(1/p)\log{r_Tp}(1-o(1))$, which is greater than $$\left(\frac{1}{p}\right)\log {(c)}(1-o(1)) = \left(\frac{1}{p}\right)\log {(np)}(1-o(1)).$$ Note that this set gets added to the tree obtained by the algorithm which increases the number of vertices in the tree while maintaining the acyclic property of the induced subgraph. Hence, with probability at least $3/4$, the induced subgraph has $\sum_{i=0}^{T-1}l_{i} + (1/p)\log{np}(1-o(1))$ vertices. Consequently, the FVS obtained has size at most $n-(1/p)\log{np}(1-o(1))$ with probability at least $3/4$.
Planted Feedback Vertex Set Problem
===================================
We prove Theorems \[theorem:planted-directedFVS\] and \[theorem:algorithm-planted-directedFVS\] in this section.
The proof of Theorems \[theorem:planted-directedFVS\] and \[theorem:algorithm-planted-directedFVS\] are based on the following fact formalized in Lemma \[lemma:cycle-through-every-vertex\]: if $S\subseteq V\setminus P$ is a subset of vertices of size at least $(1-\delta)n/10$, then with high probability, every vertex $u\in P$ induces a $k$-cycle with vertices in $S$. Consequently, a small hitting set $H$ for the $k$-cycles should contain either all vertices in $P$ or most vertices from $V\setminus P$. If some vertex $u\in P$ is not present in $H$, then the size of $H$ will be large since it should contain most vertices from $V\setminus P$. This contradicts the fact that $H$ is a small hitting set. Thus $H$ should contain the planted feedback vertex set $P$. This fact is stated in a general form based on the size of $H$ in Lemma \[lemma:small-hitting-set-contains-P\].
For Theorem \[theorem:planted-directedFVS\], $H$ is the smallest hitting set. By the previous argument $H\supseteq P$, and we are done since no additional vertex $v\in V\setminus P$ will be present in $H$ (in fact, $P$ is a hitting set for all cycles since it is a feedback vertex set). We formalize these arguments in this section.
\[lemma:cycle-through-every-vertex\] Let $D_{n,\delta,p}$ be a planted directed random graph where $p\geq C/n^{1-2/k}$ for some constants $C,k,\delta$. Then, with high probability, for every vertex $v\in P$, there exists a cycle of size $k$ through $v$ in the subgraph induced by $S\cup \{v\}$ in $D_{n,\delta,p}$ if $S$ is a subset of $V\setminus P$ of size at least $|V\setminus P|/10=(1-\delta)n/10$.
We give a proof of this lemma by the second moment method later. It leads to the following important consequence which will be used to prove Theorems \[theorem:planted-directedFVS\] and \[theorem:algorithm-planted-directedFVS\]. It states that every sufficiently small hitting set for the $k$-cycles in $D_{n,\delta,p}$ should contain every vertex from the planted feedback vertex set.
\[lemma:small-hitting-set-contains-P\] Let $H$ be a hitting set for the $k$-cycles in $D_{n,\delta,p}$ where $p\geq C/n^{1-2/k}$ for some constants $C,k,\delta$. If $|H|\leq t\delta n$ where $t\leq 9(1-\delta)/10\delta$, then $H\supseteq P$.
Suppose $u\in P$ and $u\not \in H$. Then $H$ should contain at least $|V\setminus P|-|V\setminus P|/10$ vertices from $V\setminus P$, else by Lemma \[lemma:cycle-through-every-vertex\], there exists a $k$-cycle involving $u$ and some $k-1$ vertices among the $|V\setminus P|/10$ vertices that $H$ does not contain contradicting the fact that $H$ hits all cycles of length $k$. Therefore, $|H|>|V\setminus P|-|V\setminus P|/10=(1-\delta)9n/10 \geq t\delta n$ by the choice of $t$. Thus, the size of $H$ is greater than $t\delta n$, a contradiction.
We will first show that the smallest hitting set for the $k$-cycles in $D_{n,\delta,p}$ is of size exactly $|P|=\delta n$. By Lemma \[lemma:cycle-through-every-vertex\] there exists a $k$-cycle through every vertex $v\in P$ and some $\{u_1,\cdots,u_{k-1}\}\subset S$ if $S\subset V\setminus P$ and $|S|\geq (1-\delta)n/10$.
\[lemma:lower-bound-HS\] If a subset $H\subseteq V$ hits all cycles of length $k$ in $D_{n,\delta,p}$, then $|H|\geq |P|$.
If $H$ contains all vertices in $P$, then we are done. Suppose not. Let $u\in P$ and $u\not \in H$. Then $H$ should contain at least $|V\setminus P|-|V\setminus P|/10$ vertices from $V\setminus P$, else by Lemma \[lemma:cycle-through-every-vertex\], there exists a $k$-cycle involving $u$ and some $k-1$ vertices among the $|V\setminus P|/10$ vertices that $H$ does not contain. This would contradict the fact that $H$ hits all cycles of length $k$. Therefore, $|H|>|V\setminus P|-|V\setminus P|/10=(1-\delta)9n/10\geq \delta n=|P|$ since $\delta\leq 9/19$.
Therefore, every hitting set for the subset of $k$-cycles should be of size at least $|P|=\delta n$. Also, we know that $P$ is a hitting set for the $k$-cycles since $P$ is a feedback vertex set in $D_{n,\delta,p}$. Thus, the optimum hitting set for the $k$-cycles is of size exactly $|P|$.
Let $H$ be the smallest hitting set for the $k$-cycles. Then $|H|=\delta n$. It is easily verified that $t=1$ satisfies the conditions of Lemma \[lemma:small-hitting-set-contains-P\] if $\delta\leq 9/19$. Therefore, $H\supseteq P$. Along with the fact that $H=\delta n =|P|$, we conclude that $H=P$.
Algorithm to Recover Planted Feedback Vertex Set
------------------------------------------------
In this section, we give an algorithm to recover the planted feedback vertex set in $D_{n,\delta,p}$ thereby proving Theorem \[theorem:algorithm-planted-directedFVS\]. Theorem \[theorem:planted-directedFVS\] suggests an algorithm where one would obtain all cycles of length $k$ and find the best hitting set for these set of cycles. Even though the number of $k$-cycles is polynomial, we do not have a procedure to find the best hitting set for $k$-cycles. However, by repeatedly taking all vertices of a cycle into the hitting set and removing them from the graph, we do have a simple greedy strategy that finds a $k$-approximate hitting set. We will use this strategy to give an algorithm that recovers the planted feedback vertex set.
[**Algorithm Recover-Planted-FVS($D_{n,\delta,p}=D(V,E)$)**]{}
1. Obtain cycles in increasing order of size until all cycles of length $k$ are obtained. Let $\T'$ be the subset of cycles. Let $S$ be the empty set.
2. While there exists a cycle $T\in \T'$ such that $S$ does not hit $T$,
1. Add all vertices in $T$ to $S$.
3. Return $H$, where $H=\{u\in S:\exists$ $k$-cycle through $v$ in the subgraph induced by $V\setminus S\cup\{u\}\}$.
The idea behind the algorithm is the following: The set $S$ obtained at the end of step $2$ in the above algorithm is a $k$-approximate hitting set and hence is of size at most $k\delta n$. Using Lemma \[lemma:small-hitting-set-contains-P\], it is clear that $S$ contains $P$ - indeed, if $S$ does not contain all vertices in $P$, then $S$ should contain most of the vertices in $V\setminus P$ contradicting the fact that the size of $S$ is at most $k\delta n$. Further, owing to the choice of $\delta$, it can be shown that $S$ does not contain at least $|V\setminus P|/10$ vertices from $V\setminus P$. Therefore, by Lemma \[lemma:cycle-through-every-vertex\], every vertex $v\in P$ induces a $k$-cycle with some subset of vertices from $V\setminus S$. Also, since $V\setminus P$ is a DAG no vertex $v\in V\setminus P$ induces cycles with any subset of vertices from $V\setminus S\subseteq V\setminus P$. Consequently, a vertex $v$ induces a $k$-cycle with vertices in $V\setminus S$ if and only if $v\in P$. Thus, the vertices in $P$ are identified exactly.
We use Algorithm Recover-Planted-FVS to recover the planted feedback vertex set from the given graph $D=D_{n,\delta,p}$. Since we are using the greedy strategy to obtain a hitting set $S$ for $\T'$, it is clear that $S$ is a $k$-approximate hitting set. Therefore $|S|\leq k\delta n$. It is easily verified that $t=k$ satisfies the conditions of Lemma \[lemma:small-hitting-set-contains-P\] if $\delta\leq 1/2k$. Thus, all vertices from the planted feedback vertex set $P$ are present in the subset $S$ obtained at the end of step 2 in the algorithm.
By the choice of $\delta\leq 1/2k$, it is true that $|S|\leq k\delta n\leq 9(1-\delta)n/10=9|V\setminus P|/10$. Hence, $|V\setminus S|\geq |V\setminus P|/10$.
Since $S\supseteq P$, the subset of vertices $V\setminus S$ does not contain any vertices from the planted set. Also, the number of vertices in $V\setminus S$ is at least $|V\setminus P|/10$. Consequently, by Lemma \[lemma:cycle-through-every-vertex\], each vertex $v\in P$ induces at least one $k$-cycle with vertices in $V\setminus S$. Since $V\setminus P$ is a DAG, none of the vertices $u\in V\setminus P$ induce cycles with vertices in $V\setminus S$. Therefore, a vertex $v\in S$ induces a $k$-cycle with vertices in $V\setminus S$ if and only if $v\in P$. Hence, the subset $H$ output by Algorithm Recover-Planted-FVS is exactly the planted feedback vertex set $P$.
Next we prove that the algorithm runs in polynomial time in expectation. The following lemma shows an upper bound on the expected number of cycles of length $k$. It is proved later by a simple counting argument.
\[lemma:expected-no-of-cycles\] The expected number of cycles of length $k$ in $D_{n,\delta,p}$ is at most $(nkp)^{k}$.
Since the expected number of cycles obtained by the algorithm is $(nkp)^{k}$ by Lemma \[lemma:expected-no-of-cycles\], the algorithm uses $(nkp)^k$-sized storage memory. Finally, since the size of $\T'$ is $(nkp)^k$, steps 2 and 3 of the algorithm can be implemented to run in expected $(nkp)^{O(k)}$ time.
Proofs
======
Lower Bound for FVS in Random Graphs
------------------------------------
In this section, we prove the lower bound for the Feedback Vertex Set in random graphs. We consider the dual problem - namely the maximum induced acyclic subgraph.
We will need the following bound on the number of ways to partition a positive integer $n$ into $k$ positive integers.
\[theorem:partition-function\][@wladimir-partition-function] Let $p_k(n)$ denote the number of ways to partition $n$ into exactly $k$ parts. Then there exists an absolute constant $A<1$ such that $$p_k(n)<A\frac{e^{c\sqrt{n-k}}}{(n-k)^{3/4}}e^{\frac{-2\sqrt{n-k}}{c}}L_2(e^{-\frac{c(k+1/2)}{2\sqrt{n-k}}})$$ where $c=\pi\sqrt{2/3}$ and $L_2(x)=\sum_{m=1}^{\infty}\frac{x^m}{m^2}$ for $|x|\leq 1$.
[Remark 3.]{} Since we will not need such a tight bound, we will use $p_k(n)< C_1e^{C_2(n-k)}$ for some constants $C_1,C_2>0$.
We prove Theorem \[theorem:undirected-FVSlowerbound\] now based on simple counting arguments. We observe that the proof of Theorem \[theorem:directed-FVSlowerbound\] given by Spencer and Subramanian is also based on similar counting arguments while observing that if a directed graph is acyclic, then there exists an ordering of the vertices such that each arc is in the forward direction.
First note that every induced subgraph on $r$ vertices is a graph from the family $G(r,p)$. We bound the probability that a graph $H=G(r,p)$ is a forest.\
$$\begin{aligned}
{{\sf Pr}\left(H\ \text{is a forest}\right)}&\leq \sum_{k=1}^r \sum_{n_1+\cdots+n_k=r, n_i>0} \text{No. of forests with spanning}\\
&\ \ \ \ \ \text{trees on $n_1,\cdots,n_k$ vertices}\\
&\ \ \ \ \ \ \ \times {{\sf Pr}\left(\text{Forest with $k$ components}\right)}\\
&= \sum_{k=1}^r \sum_{n_1+\cdots+n_k=r, n_i>0} \left(\frac{r!}{\prod_{i=1}^k n_i!}\right) \left(\prod_{i=1}^k n_i^{n_i-2} \right)\\
&\ \ \ \ \ \ \ \times p^{r-k}(1-p)^{\binom{r}{2}-r+k}\\
&\leq r!(1-p)^{\binom{r}{2}} \sum_{k=1}^r \sum_{n_1+\cdots+n_k=r, n_i>0} \left(\frac{p}{1-p}\right)^{r-k}\\
&\leq r!(1-p)^{\binom{r}{2}} \sum_{k=1}^r \sum_{n_1+\cdots+n_k=r, n_i>0} \left(2p\right)^{r-k} \\
& \ \ \ \quad \quad \text{(since $p<1/2$)}\\
&\leq r!(1-p)^{\binom{r}{2}} \sum_{k=1}^r \left(2p\right)^{r-k} \sum_{n_1+\cdots+n_k=r, n_i>0} 1\\
&= r!(1-p)^{\binom{r}{2}} \sum_{k=1}^r \left(2p\right)^{r-k} p_k(r)\\
&\leq r!(1-p)^{\binom{r}{2}} \sum_{k=1}^r \left(2p\right)^{r-k} C_1 e^{C_2(r-k)} \\
& \ \ \ \quad \quad \text{(by Remark 3)}\\
&\leq C_1r^r(1-p)^{\binom{r}{2}} \sum_{k=1}^r (2e^{C_2}p)^{r-k}\end{aligned}$$ $$\begin{aligned}
&\leq C_1(1-p)^{\frac{r^2}{2}}n^r \sum_{k=1}^r(2e^{C_2}p)^{r-k} \\
& \ \ \ \quad \quad \text{(since $r\leq n$)}\\
&\leq C_1(1-p)^{\frac{r^2}{2}}r(2e^{C_2}np)^r \\
& \leq e^{-r\left(\frac{pr}{2}-\log{(2e^{C_2}np)}-\frac{\log{(C_1r)}}{r}\right)}\\\end{aligned}$$ which tends to zero when $r>\frac{2}{p}(\log{np})(1+o(1))$.
Feedback Vertex Set in Random Graphs
------------------------------------
We will use the following Chernoff bound for the concentration of the binomial distribution.
\[lemma:chernoff\] Let $X=\sum_{i=1}^n X_i$ where $X_i$ are i.i.d. Bernoulli random variables with ${{\sf Pr}\left(X_i=1\right)}=p$. Then $${{\sf Pr}\left(|X-np|\geq a\sqrt{np}\right)}\leq 2e^{-a^2/2}.$$
We prove the lemma by induction on $t$. We will prove the stronger induction hypothesis that every $l_i$, $u_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds with probability at least $$a_t:= 1-\frac{t}{16T} -\frac{1}{16}\sum_{i=1}^{t}1/i^2.$$ We will prove the concentration of $r_{i+1}$ as a consequence of $l_i$ and $u_i$ satisfying their respective concentration bounds. We will in fact show that the failure probability of $r_{i+1}$ satisfying its concentration bound conditioned on $l_i$ and $u_i$ satisfying their respective concentration bounds will be at most $1/(32(i+1)^2)$. It immediately follows that with failure probability at most $(t/16T)+(3/32)\sum_{i=1}^{t}(1/i^2)+(1/32(t+1)^2)\leq 1/4$, every $r_{i+1}$, $u_i$ and $l_i$, for $i\in\{0,1,\cdots,t\}$ satisfies its respective concentration bound leading to the conclusion of the lemma.
For the base case, consider $t=0$. It is clear that $u_0=n-1$ and $l_0=1$ satisfy the concentration bounds with probability $1$. For the induction step, the induction hypothesis is the following: With probability at least $a_t$, the concentration bounds are satisfied for $u_i$ and $l_i$ for every $i\in \{0,1,\cdots,t\}$. We will bound the probability that $u_{t+1}$ or $l_{t+1}$ fails to satisfy its corresponding concentration bound conditioned on the event that $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds.
1\. To prove the concentration bound for $u_{t+1}$, observe that $u_{t+1}$ is a binomial distribution with $u_t$ trials and success probability $(1-p)^{l_t}$. Indeed, $u_{t+1}$ is the number of vertices among $U_t$ which are not neighbors of vertices in $L_t$. For each vertex $x\in U_t$, ${{\sf Pr}\left(\text{$x$ has no neighbor in $L_t$}\right)}=(1-p)^{l_t}$.
Therefore, by Lemma \[lemma:chernoff\], we have that ${{\sf Pr}\left(|u_{t+1}-u_t(1-p)^{l_t}|>\gamma_{t+1}\sqrt{u_t(1-p)^{l_t}}\right)}$ $$\leq 2e^{-\gamma_{t+1}^2/2}=\frac{1}{32(t+1)^2}$$ with $\gamma_{t+1}=\sqrt{4\ln{8(t+1)}}$. Hence, with probability at least $1-(1/32(t+1)^2)$, $$\begin{aligned}
u_{t+1} &\leq u_t(1-p)^{l_t}\left(1+\sqrt{\frac{4\ln{8(t+1)}}{u_t(1-p)^{l_t}}}\right),\\
u_{t+1} &\geq u_t(1-p)^{l_t}\left(1-\sqrt{\frac{4\ln{8(t+1)}}{u_t(1-p)^{l_t}}}\right).
$$
Now, using the bounds on $u_t$ and $l_t$, $$\begin{aligned}
\frac{4\ln{8(t+1)}}{u_t(1-p)^{l_t}} & \leq \frac{10\ln{\ln{n}}}{n}
$$ since $t+1\leq T\leq \ln{n}$, $$\begin{aligned}
(n-\sum_{i=0}^t (c+20\sqrt{c})^i) &\geq \frac{15n}{16},\\
(1-p(c+20\sqrt{c})^t) &\geq \frac{15}{16} \quad \text{and}\\
\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right) &\geq \frac{1}{2}.\end{aligned}$$
Hence, $$\begin{aligned}
u_{t+1} &\leq u_t(1-p)^{l_t}\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right) \label{ineq:u-upperbound}\\
u_{t+1} &\geq u_t(1-p)^{l_t}\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right) \label{ineq:u-lowerbound}.
$$
Therefore, $$\begin{aligned}
u_{t+1} &\geq u_t(1-p)^{l_t}\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\ \ \ \quad \quad \text{(Using inequality \ref{ineq:u-lowerbound})}\\
&\geq u_t(1-l_tp)\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\geq (n-\sum_{i=0}^t (c+20\sqrt{c})^i)\left(1-\frac{c(c+20\sqrt{c})^t}{n}\right)\\
&\ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\quad \quad \text{(Using the bounds on $u_t$ and $l_t$)}\\
&\geq (n-\sum_{i=0}^t (c+20\sqrt{c})^i)\left(1-\frac{(c+20\sqrt{c})^{t+1}}{n}\right)\\
&\ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&= \left(n-\sum_{i=0}^t (c+20\sqrt{c})^i - (c+20\sqrt{c})^{t+1}\right. \\
&\ \ + \left.\frac{(c+20\sqrt{c})^{t+1}}{n}\sum_{i=0}^t (c+20\sqrt{c})^i\right)\\
&\ \ \ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\end{aligned}$$ $$\begin{aligned}
&\geq \left(n- \sum_{i=0}^{t+1} (c+20\sqrt{c})^i\right)\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\end{aligned}$$ which proves the lower bound. The upper bound is obtained by proceeding similarly: $$\begin{aligned}
u_{t+1} &\leq u_t(1-p)^{l_t}\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right) \\
& \ \ \quad \quad \text{(Using inequality \ref{ineq:u-upperbound})}\\
&\leq u_t\left(1-\frac{l_tp}{2}\right)\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\leq u_t\left(1-\frac{c(c-20\sqrt{c})^t}{n}(1-16Tp(c+20\sqrt{c})^t)\right.\\
&\ \ \ \left.\left(1-\frac{\sum_{i=0}^t(c+20\sqrt{c})^i}{n}\right)\right)\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\quad \quad \text{(Using the bound on $l_t$)}\\
&\leq u_t\left(1-\frac{c(c-20\sqrt{c})^t}{4n}\right)\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right) \\
& \quad \quad \left(\text{Since $(1-16Tp(c+20\sqrt{c})^t)\geq \frac{1}{2}$,}\right.\\
&\ \ \quad \quad \left.\text{$\left(1-\frac{\sum_{i=0}^t(c+20\sqrt{c})^i}{n}\right)\geq \frac{15}{16}$}\right)\\
&\leq \left(n-\frac{\sum_{i=0}^t(c-20\sqrt{c})^i}{4n}\right)\left(1-\frac{c(c-20\sqrt{c})^t}{4n}\right)\\
&\ \ \ \times \left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\leq \left(n-\frac{\sum_{i=0}^t(c-20\sqrt{c})^i}{4n}\right)\left(1-\frac{(c-20\sqrt{c})^{t+1}}{4n}\right)\\
&\ \ \ \times \left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\leq \left(n-\frac{\sum_{i=0}^{t+1}(c-20\sqrt{c})^i}{4n}\right)\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right).\end{aligned}$$
Thus, $u_{t+1}$ satisfies the concentration bound with failure probability at most $1/(32(t+1)^2)$ conditioned on the event that $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds.
2\. Next we address the failure probability of $r_{t+1}$ not satisfying its concentration bound conditioned on the event that $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds. Lemma \[lemma:lowerbound-R\] proves that the number of [***unique***]{} neighbors $r_{t+1}$ is concentrated around its expectation.
\[lemma:lowerbound-R\] Let $q_t:=pl_t(1-p)^{l_t-1}$. With probability at least $1-(1/32(t+1)^2)$ $$\begin{aligned}
q_tu_t\left(1+\frac{20}{\sqrt{c}}\right) \geq r_{t+1} \geq q_tu_t\left(1-\frac{20}{\sqrt{c}}\right)\\\end{aligned}$$ when $t+1\leq T$.
Observe that $r_{t+1}$ is a binomially distributed random variable with $u_t$ trials and success probability $q_t$. Indeed, $r_{t+1}$ is the number of vertices among $U_t$ which are adjacent to exactly one vertex in $L_t$. For each $u\in U_t$, ${{\sf Pr}\left(\text{$u$ is adjacent to exactly one vertex in $L_t$} \right)}=pl_t(1-p)^{l_t-1}=q_t$.
Using $\beta_{t+1}=\sqrt{4\ln{8(t+1)}}$, by Lemma \[lemma:chernoff\], we have that ${{\sf Pr}\left(|r_{t+1}-q_tu_t|>\beta_{t+1}\sqrt{q_tu_t}\right)}$ $$\begin{aligned}
&\leq 2e^{-\beta_{t+1}^2/2}=\frac{1}{32(t+1)^2}.\end{aligned}$$ Hence, with probability at least $1-(1/32(t+1)^2)$, $$\begin{aligned}
r_{t+1} &\leq q_tu_t\left(1+\sqrt{\frac{4\ln{8(t+1)}}{q_tu_t}}\right) \label{ineq:r-lowerbound}\\
r_{t+1} &\geq \geq q_tu_t\left(1-\sqrt{\frac{4\ln{8(t+1)}}{q_tu_t}}\right). \label{ineq:r-upperbound}
$$
Lemma \[lemma:lowerbound-qu\] proves the concentration of the expected number of unique neighbors of $L_t$ conditioned on the event that $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds. This in turn helps in proving that $r_{t+1}$ is concentrated.
\[lemma:lowerbound-qu\] For $t+1\leq T$, if $u_t$ and $l_t$ satisfy their respective concentration bounds, then
1. $q_tu_t \leq c(c+20\sqrt{c})^{t}\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)$,
2. $q_tu_t$\
$\geq \frac{c(c-20\sqrt{c})^{t}}{4} \left(1-\frac{\sum_{i=0}^{t+1}(c+20\sqrt{c})^i}{n}\right)\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)$.
Recall that $q_t=pl_t(1-p)^{l_t-1}$. Hence, $$\begin{aligned}
q_tu_t &\geq p(n-\sum_{i=0}^t(c+20\sqrt{c})^i)l_t(1-p)^{l_t-1}\\
&\ \ \times\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&= pn\left(1-\frac{\sum_{i=0}^t(c+20\sqrt{c})^i}{n}\right)l_t(1-p)^{l_t-1}\\
&\ \ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\geq c\left(1-\frac{\sum_{i=0}^t(c+20\sqrt{c})^i}{n}\right)l_t(1-l_tp)\\
&\ \ \ \times \left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\geq c(c-20\sqrt{c})^{t}\left(1-\frac{\sum_{i=0}^t(c+20\sqrt{c})^i}{n}\right)^2\\
&\ \ \ \times(1-16Tp(c+20\sqrt{c})^t)(1-p(c+20\sqrt{c})^t)\\
&\ \ \ \times\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\quad \quad \text{(By the bound on $l_t$)}\\
&\geq \frac{c(c-20\sqrt{c})^{t}}{4} \left(1-\frac{\sum_{i=0}^{t+1}(c+20\sqrt{c})^i}{n}\right)\\
&\ \ \ \times\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\end{aligned}$$ using Lemma \[lemma:handler1\] and $$\begin{aligned}
(1-16Tp(c+20\sqrt{c})^t) &\geq \frac{1}{2},\\
(1-p(c+20\sqrt{c})^t) &\geq \frac{1}{2} \quad \quad \text{when $t+1\leq T$.}\end{aligned}$$ For the upper bound: $$\begin{aligned}
q_tu_t &=pl_t(1-p)^{l_t-1}u_t\\
&\leq pl_tu_t\\
&\leq pl_t\left(n-\frac{\sum_{i=0}^t(c-20\sqrt{c})^i}{4}\right)\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\ \ \ \ \quad \quad \text{(Using the bound on $u_t$)}\\
&\leq cl_t\left(1-\frac{\sum_{i=0}^t(c-20\sqrt{c})^i}{4n}\right)\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\\end{aligned}$$ $$\begin{aligned}
&\leq c(c+20\sqrt{c})^t\left(1-\frac{\sum_{i=0}^t(c-20\sqrt{c})^i}{4n}\right)\\
&\ \ \ \times \left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\\
&\ \ \ \ \quad \quad \text{(Using the bound on $l_t$)}\\
&\leq c(c+20\sqrt{c})^t\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right) \\
&\ \quad \quad \left(\text{Since $\left(1-\frac{\sum_{i=0}^t(c-20\sqrt{c})^i}{4n}\right)\leq 1$}\right).\end{aligned}$$
Consequently, using Lemma \[lemma:lowerbound-qu\], $$\begin{aligned}
\frac{4\ln{8(t+1)}}{q_tu_t} &\leq \frac{400}{c}
$$ since, when $t+1\leq T$, $$\begin{aligned}
\left(1-\frac{\sum_{i=0}^{t+1}(c+20\sqrt{c})^i}{n}\right) &\geq \left(\frac{15}{16}\right)^2,\\
\left(1-\sqrt{\frac{\ln{\ln{n}}}{n}}\right) &\geq \frac{1}{2} \quad \text{and}\\
\frac{1}{2} &\geq \frac{4\ln{8(t+1)}}{(c-20\sqrt{c})^{t}}.\end{aligned}$$
Hence, by inequalities \[ineq:r-lowerbound\] and \[ineq:r-upperbound\], with probability at least $1-(1/32(t+1)^2)$, $$\begin{aligned}
q_tu_t\left(1+\frac{20}{\sqrt{c}}\right) \geq r_{t+1} \geq q_tu_t\left(1-\frac{20}{\sqrt{c}}\right)\end{aligned}$$ when $t+1\leq T$. This concludes the proof of Lemma \[lemma:lowerbound-R\]
Lemmas \[lemma:lowerbound-R\] and \[lemma:lowerbound-qu\] together show that $r_{t+1}$ satisfies the concentration bounds with failure probability at most $(1/32(t+1)^2)$ conditioned on the event that $u_t$ and $l_t$ satisfy their respective concentration bounds.
3\. Finally we address the failure probability of $l_{t+1}$ satisfying its concentration bound conditioned on the event that $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds. By Step 2(e) of the algorithm, the number of surviving vertices in level $t+1$ is $l_{t+1}:=r_{t+1} - m_{t+1}$, where $m_{t+1}$ denotes the number of edges among the vertices in $R_{t+1}$. In Lemma \[lemma:lowerbound-R\], we showed that the number of [***unique***]{} neighbors $r_{t+1}$ is concentrated around its expectation. Lemma \[lemma:upperbound-M\] proves a concentration which bounds the number of edges among the vertices in $R_t$. These two bounds will immediately lead to the induction step on $l_{t+1}$. Thus, the probability that $l_{t+1}$ does not satisfy its concentration bound will at most be the probability that either $m_{t+1}$ or $r_{t+1}$ does not satisfy its respective concentration bound.
\[lemma:upperbound-M\] $m_{t+1}\leq 8Tr_{t+1}^2p$ with probability at least $1-(1/16T)$.
Recall that $m_{t+1}$ denotes the number of edges among the vertices in $R_{t+1}$. Since the algorithm has not explored the edges among the vertices in $R_{t+1}$, $m_{t+1}$ is a random variable following the Binomial distribution with $\binom{r_{t+1}}{2}$ trials and success probability $p$. By Markov’s inequality, we have that for $t+1\leq T$, $${{\sf Pr}\left(m_{t+1}\geq 8Tr_{t+1}^2p\right)}\leq \frac{1}{16T}.$$ Hence, $m_{t+1}\leq 8Tr_{t+1}^2p$ with probability at least $1-(1/16T)$, .
Recollect that $l_{t+1}=r_{t+1}-m_{t+1}$. The upper bound of the induction step follows using Lemma \[lemma:lowerbound-qu\]: $$\begin{aligned}
l_{t+1} &\leq r_{t+1}\\
&\leq q_tu_t\left(1+\frac{20}{\sqrt{c}}\right)\\
&\leq c(c+20\sqrt{c})^{t}\left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right)\left(1+\frac{20}{\sqrt{c}}\right)\\
&\leq (c+20\sqrt{c})^{t+1} \left(1+\sqrt{\frac{\ln{\ln{n}}}{n}}\right).\end{aligned}$$
For the lower bound, we use Lemmas \[lemma:lowerbound-R\] and \[lemma:upperbound-M\] conditioned on the event that $l_t$ and $u_t$ satisfy their respective concentration bounds. With failure probability at most $$\frac{1}{32(t+1)^2}+\frac{1}{16T},$$ we have that $l_{t+1}$ $$\begin{aligned}
&=r_{t+1}-m_{t+1}\\
&\geq r_{t+1} - 8Tr_{t+1}^2p\\
&= r_{t+1}(1-8Tr_{t+1}p)\\
&\geq q_tu_t\left(1-\frac{20}{\sqrt{c}}\right)\left(1-8Tq_tu_tp\left(1+\frac{20}{\sqrt{c}}\right)\right)\\
&\quad \quad \text{(Using Lemma \ref{lemma:lowerbound-R})}\\
&= l_tp(1-p)^{l_t-1}u_t\left(1-8Tl_tp^2(1-p)^{l_t-1}u_t\right.\\
&\ \ \left.\times\left(1+\frac{20}{\sqrt{c}}\right)\right)\left(1-\frac{20}{\sqrt{c}}\right)\\
& \quad \quad \text{(Substituting for $q_t=pl_t(1-p)^{l_t-1}$)}\\
& \geq l_tp\left(1-\frac{20}{\sqrt{c}}\right)(1-l_tp)(1-12Tl_tp^2(1-p)^{l_t-1}u_t)\\\end{aligned}$$ $$\begin{aligned}
&\geq l_tp\left(1-\frac{20}{\sqrt{c}}\right)(1-l_tp)(1-12Tnp^2l_t(1-p)^{l_t-1})\\
& \quad \quad \text{(Since $u_t\leq n$)}\\
&\geq l_tp\left(1-\frac{20}{\sqrt{c}}\right)(1-l_tp)(1-12Tcpl_t(1-p)^{l_t-1})\\
&\geq l_tp\left(1-\frac{20}{\sqrt{c}}\right)(1-l_tp-12Tcpl_t(1-p)^{l_t}(1-l_tp))\\
&\geq l_tp\left(1-\frac{20}{\sqrt{c}}\right)(1-l_tp(1+12Tc))\\
&\geq l_tpu_t(1-l_tp(1+12Tc))\left(1-\frac{20}{\sqrt{c}}\right)\\
&\geq l_tp(n-\sum_{i=0}^t {(c+20\sqrt{c})^i})(1-l_tp(1+12Tc))\\
&\ \ \ \times \left(1-\frac{20}{\sqrt{c}}\right) \quad \quad \text{(Using the bound on $u_t$)}\\
&\geq l_tnp\left(1-\frac{\sum_{i=0}^t {(c+20\sqrt{c})^i}}{n}\right)(1-l_tp(1+12Tc))\\
&\ \ \ \times \left(1-\frac{20}{\sqrt{c}}\right)\\&= l_tc\left(1-\frac{\sum_{i=0}^t {(c+20\sqrt{c})^i}}{n}\right)(1-l_tp(1+12Tc))\\
&\ \ \ \times\left(1-\frac{20}{\sqrt{c}}\right)\\&\geq c(c-20\sqrt{c})^{t}\left(1-\frac{\sum_{i=0}^t {(c+20\sqrt{c})^i}}{n}\right)^2\\
&\ \ \ \times(1-16Tp(c+20\sqrt{c})^t)\\
&\quad \quad \times (1-(c+20\sqrt{c})^tp(1+12Tc))\left(1-\frac{20}{\sqrt{c}}\right)\\&\ \ \ \quad \text{(using the bound on $l_t$)}\\
&\geq (c-20\sqrt{c})^{t+1}\left(1-\frac{\sum_{i=0}^{t+1} {(c+20\sqrt{c})^i}}{n}\right)\\
&\ \ \ \times(1-16Tp(c+20\sqrt{c})^{t+1}) \quad \quad \text{(Using Lemma \ref{lemma:handler1})}\end{aligned}$$ proving the induction step of the lower bound for $l_{t+1}$.
Thus, $l_{t+1}$ satisfies the concentration bounds with failure probability at most $(1/32(t+1)^2)+(1/16T)$ conditioned on the event that $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds.
Finally, by the union bound, with probability at most $\frac{1}{32(t+1)^2}+\frac{1}{32(t+1)^2}+\frac{1}{16T}$, either $u_{t+1}$ or $l_{t+1}$ does not satisfy its respective concentration bounds conditioned on the event that $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ satisfy their respective concentration bounds. By induction hypothesis, the failure probability of some $u_i,l_i$ for $i\in\{0,1,\cdots,t\}$ not satisfying their respective concentration bound is at most $1-a_t$. Hence, the probability that $u_i,l_i$ satisfy their respective concentration bound for every $i\in \{0,1,\cdots,t+1\}$ is at least $a_t(1-(1/16(t+1)^2)-(1/16T))\geq a_{t+1}$. Therefore, with probability at least $a_{t+1}$, every $u_i,l_i$ for $i\in\{0,1,\cdots,t+1\}$ satisfy their respective concentration bounds. This proves the stronger induction hypothesis.
To complete the proof of Lemma \[lemma:largesurvivors\], recollect that we showed that the failure probability of $r_{i+1}$ satisfying its concentration bound conditioned on $l_i$ and $u_i$ satisfying their respective concentration bounds is at most $1/(32(i+1)^2)$. By the union bound argument, it immediately follows that with failure probability at most $(t/16T)+(3/32)\sum_{i=1}^{t}(1/i^2)+(1/32(t+1)^2)\leq 1/4$, every $r_{i+1}$, $u_i$ and $l_i$, for $i\in\{0,1,\cdots,t\}$ satisfies its respective concentration bound. This concludes the proof of Lemma \[lemma:largesurvivors\].
\[lemma:handler1\] For $t+1\leq T$,
1. $$\left(1-\frac{\sum_{i=0}^t (c+20\sqrt{c})^i}{n}\right)^{2} \geq 1-\frac{\sum_{i=0}^{t+1} (c+20\sqrt{c})^i}{n}$$
2. $\left(1-16Tp(c+20\sqrt{c})^t\right)(1-(c+20\sqrt{c})^tp(1+12Tc))$ $$\begin{aligned}
&\geq \left(1-16Tp(c+20\sqrt{c})^{t+1}\right)\end{aligned}$$
\[Proof of Lemma \[lemma:handler1\]\] We prove the first part of the Lemma by induction. For the base case, we need to prove that $$\begin{aligned}
1+\frac{1}{n^2}-\frac{2}{n}&\geq 1-\frac{c+20\sqrt{c}}{n}-\frac{1}{n}\\
\text{i.e., to prove that}\quad n-1&\leq (c+20\sqrt{c})n\end{aligned}$$ which is true. For the induction step, we need to prove that $$\left(1-\frac{\sum_{i=0}^t(c+20\sqrt{c})^i}{n}-\frac{(c+20\sqrt{c})^{t+1}}{n}\right)^2$$ $$\geq 1-\frac{\sum_{i=0}^{t+2}(c+20\sqrt{c})^i}{n}$$ Now, LHS $$\begin{aligned}
&= \left(1-\frac{\sum_{i=0}^{t}(c+20\sqrt{c})^i}{n}\right)^2 + \frac{(c+20\sqrt{c})^{2t+2}}{n^2}\\
&\ \ \ -\frac{2(c+20\sqrt{c})^{t+1}}{n}\left(1-\frac{\sum_{i=0}^{t}(c+20\sqrt{c})^i}{n}\right)\\
&\geq 1-\frac{\sum_{i=0}^{t+1} (c+20\sqrt{c})^i}{n} + \frac{(c+20\sqrt{c})^{2t+2}}{n^2}\\
&\ \ \ -\frac{2(c+20\sqrt{c})^{t+1}}{n}+\frac{2(c+20\sqrt{c})^{t+1}\sum_{i=0}^{t}(c+20\sqrt{c})^i}{n^2}.\end{aligned}$$ Hence, it is sufficient to prove that $$\begin{aligned}
-\frac{(c+20\sqrt{c})^{t+2}}{n} &\leq \frac{(c+20\sqrt{c})^{2t+2}}{n^2}-\frac{2(c+20\sqrt{c})^{t+1}}{n}\\
&\ \ \ +\frac{2(c+20\sqrt{c})^{t+1}\sum_{i=0}^{t}(c+20\sqrt{c})^i}{n^2} \\
(c+20\sqrt{c}) &\geq 2 - \frac{(c+20\sqrt{c})^{t+1}}{n}\\
&\ \ \ - 2\frac{\sum_{i=0}^{t}(c+20\sqrt{c})^i}{n},
$$ which is true for large enough $c$ when $t+1\leq T$.
For the second part of the Lemma, we need to prove that $\left(1-16Tp(c+20\sqrt{c})^t\right)(1-(c+20\sqrt{c})^tp(1+12Tc))$ $$\begin{aligned}
&\geq \left(1-16Tp(c+20\sqrt{c})^{t+1}\right)\end{aligned}$$ i.e., $1-16Tp(c+20\sqrt{c})^t-(c+20\sqrt{c})^tp(1+12Tc)+18Tp^2(c+20\sqrt{c})^{2t}(1+12Tc)$ $$\begin{aligned}
&\geq 1-16Tp(c+20\sqrt{c})^{t+1}\end{aligned}$$ i.e., $$(1-16Tp(c+20\sqrt{c})^t)(1+12Tc)\leq 16T(c+20\sqrt{c}-1)$$ which is true since $1+12Tc\leq 16T(c+20\sqrt{c}-1)$ for large $c$ and the rest of the terms are less than $1$ when $t+1\leq T$.
Planted Feedback Vertex Set
---------------------------
We prove Lemma \[lemma:cycle-through-every-vertex\] by the second moment method.
Let $S\subset V\setminus P$, $|S|\geq (1-\delta)n/10$, $v\in P$. Let $X_v$ denote the number of cycles of size $k$ through $v$ in the subgraph induced by $S\cup\{v\}$. Then, $\E(X_v)=\binom{(1-\delta)n/10}{k-1}p^k$. Using Chebyshev’s inequality, we can derive that $${{\sf Pr}\left(X_v=0\right)}\leq \frac{{{\sf Var}\left(X_v\right)}}{\E(X_v)^2}.$$ To compute the variance of $X_v$, we write $X_v=\sum_{A\subseteq S:|A|=k-1} X_{A}$, where the random variable $X_A$ is $1$ when the vertices in $A$ induce a cycle of length $k$ with $v$ and $0$ otherwise. $$\begin{aligned}
{{\sf Var}\left(X_v\right)}&\leq \E(X_v)\\
&\ \ +\sum_{A,B\subseteq S:|A|=|B|=k-1,A\neq B} {{\sf Cov}\left(X_A,X_B\right)}\end{aligned}$$
Now, for any fixed subsets $A,B\subseteq S$, $|A|=|B|=k-1$ and $|A\cap B|=r$, ${{\sf Cov}\left(X_A,X_B\right)}\leq p^{2k-r}$ and the number of such subsets is at most $\binom{|S|}{2k-2-r}\binom{k}{r}\leq \binom{n}{2k-2-r}\binom{k}{r}$. Therefore, $$\sum_{r=0}^{k-2} \sum_{A,B\subseteq S:|A|=|B|=k-1,|A\cap B|=r} \frac{{{\sf Cov}\left(X_A,X_B\right)}}{\E(X_v)^2}$$ $$\begin{aligned}
& \leq \sum_{r=0}^{k-2} \frac{\binom{k}{r}\binom{n}{2k-2-r}p^{2k-r}}{\binom{(1-\delta)n/10}{2k-2}p^{2k}}\\
& \leq \sum_{r=0}^{k-2}\frac{C_r}{(np)^{r}} \\
&\ \ \quad \quad (\text{for some constants $C_r$ dependent on $r,\delta$})\\
& \rightarrow 0\end{aligned}$$ as $n\rightarrow \infty$ if $p\geq C/n^{1-2/k}$ for some sufficiently large constant $C$ since each term in the summation tends to $0$ and the summation is over a finite number of terms. Thus $${{\sf Pr}\left(X_v=0\right)}\leq \frac{1}{\binom{(1-\delta)n/10}{k-1}p^k} \leq \frac{1}{((1-\delta)n/10)^{k-1}p^k} .$$ Therefore, $${{\sf Pr}\left(X_v\geq 1\right)}\geq 1-\frac{1}{((1-\delta)n/10)^{k-1}p^k}$$ and hence $$\begin{aligned}
{{\sf Pr}\left(X_v\geq 1 \forall v\in P\right)} &\geq \left(1-\frac{1}{((1-\delta)n/10)^{k-1}p^k}\right)^{|P|}\\
&= \left(1-\frac{1}{((1-\delta)n/10)^{k-1}p^k}\right)^{\delta n}\\
&\geq e^{-\frac{10^{k-1}\delta}{2(1-\delta)^{k-1}n^{k-2}p^k}} \rightarrow 1\end{aligned}$$ as $n\rightarrow \infty$ if $p\geq \frac{C}{n^{1-2/k}}$ for some large constant $C$.
Finally, we prove Lemma \[lemma:expected-no-of-cycles\] by computing the expectation.
$\E(\text{Number of cycles of length $k$})$ $$\begin{aligned}
&\leq \sum_{i=1}^k \binom{|P|}{i}\binom{|R|}{k-i}k!p^k\\
&=\sum_{i=1}^k \binom{\delta n}{i}\binom{(1-\delta)n}{k-i}k!p^k\\
&\leq \sum_{i=1}^k (\delta n)^i((1-\delta)n)^{k-i}(kp)^k\end{aligned}$$ $$\begin{aligned}
&= ((1-\delta)nkp)^k\sum_{i=1}^k \left(\frac{\delta}{1-\delta}\right)^i\\
&= ((1-\delta)nkp)^k(1-\delta) \leq (nkp)^k.\end{aligned}$$
Conclusion
==========
Several well-known combinatorial problems can be reformulated as hitting set problems with an exponential number of subsets to hit. However, there exist efficient procedures to verify whether a candidate set is a hitting set and if not, output a subset that is not hit. We introduced the implicit hitting set as a framework to encompass such problems. The motivation behind introducing this framework is in obtaining efficient algorithms where efficiency is determined by the running time as a function of the size of the ground set. We initiated the study towards developing such algorithms by showing an algorithm for a combinatorial problem that falls in this framework – the feedback vertex set problem on random graphs. It would be interesting to extend our results to other implicit hitting set problems mentioned in Section 1.1.
[^1]: Georgia Institute of Technology. Supported in part by NSF awards AF-0915903 and AF-0910584. Email: [karthe@gatech.edu,vempala@cc.gatech.edu]{}.
[^2]: University of California, Berkeley. Email: [karp@icsi.berkeley.edu]{}
[^3]: Texas A&M University. Email: [e.moreno@tamu.edu]{}
|
---
author:
- |
Mariko Arisawa\
GSIS, Tohoku University\
Aramaki 09, Aoba-ku, Sendai 980-8579, JAPAN\
E-mail: arisawa@math.is.tohoku.ac.jp
title: |
Corrigendum for the comparison theorems in\
“A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations”.
---
In this note, we shall present the correction of the proofs of the comparison results in the paper \[1\]. In order to show clearly the correct way of the demonstration, we shall simplify the problem to the following.\
$$\hbox{(Problem (I))}:\quad
F(x,u,\nabla u,\nabla^2 u) -\int_{{\bf R^N}}
u(x+z)-u(x)\qquad \qquad\qquad\qquad$$ $$\label{problem}
\qquad -{\bf 1}_{|z|\leq 1}\la z,\n u(x) \ra q(dz)=0 \qquad \hbox{in} \quad \Omega,$$
$$\hbox{(Problem (II))}:\quad
F(x,u,\nabla u,\nabla^2 u) -\int_{\{z\in {\bf R^N}|x+z\in \overline{\Omega}\}}
u(x+z)-u(x)\qquad \qquad$$ $$\label{problem2}
\qquad -{\bf 1}_{|z|\leq 1}\la z,\n u(x) \ra q(dz)=0 \qquad \hbox{in} \quad \Omega,$$ where $\Omega\subset {\bf R^N}$ is open, and $q(dz)$ is a positive Radon measure such that $\int_{|z|\leq 1} |z|^2 q(dz)+\int_{|z| > 1} 1 q(dz) <\infty$. Although in \[1\] only (II) was studied, in order to avoid the non-essential technical complexity, here, let us give the explanation mainly for (I). For (I), we consider the Dirichlet B.C.: $$\label{dirichletI}
u(x)=g(x) \quad \forall x\in \Omega^c,$$ where $g$ is a given continuous function in $\Omega^c$. For (II), we assume that $\Omega$ is a precompact convex open subset in ${\bf R^N}$ with $C^1$ boundary satisfying the uniform exterior sphere condition, and consider either the Dirichlet B.C.: $$\label{dirichletII}
u(x)=h(x) \quad \forall x\in \p\Omega,$$ where $h$ is a given continuous function on $\p\Omega$, or the Neumann B.C.: $$\label{neumann}
\la \n u(x), {\bf n}(x)\ra=0 \quad \forall x\in \p\Omega,$$ where ${\bf n}(x)\in {\bf R^N}$ the outward unit normal vector field defined on $\p\Omega$. The above problems are studied in the framework of the viscosity solutions introduced in \[1\]. Under all the assumptions in \[1\], for (I) the following comparison result holds, and for (II), although the proofs therein are incomplete, the comparison results stated in \[1\] hold, and we shall show in a future article.\
[**Theorem 1.1(Problem I with Dirichlet B.C.)**]{} Assume that $\Omega$ is bounded, and the conditions for $F$ in \[1\] hold. Let $u\in USC({\bf R^N})$ and $v\in LSC(\bf R^N)$ be respectively a viscosity subsolution and a supersolution of (\[problem\]) in $\Omega$, which satisfy $u\leq v$ on $\Omega^c$. Then, $u\leq v$ in $\Omega$.
To prove Theorem 1.1, we approximate the solutions $u$ and $v$ by the supconvolution: $u^r(x)=\sup_{y\in {\bf R^N}}\{
u(y)-\frac{1}{2r^2}|x-y|^2\}$ and the infconvolution: $v_r(x)=\inf_{y\in {\bf R^N}}\{
v(y)+\frac{1}{2r^2}|x-y|^2
\}$ $(x\in {\bf R^N})$, where $r>0$.\
[**Lemma 1.2(Approximation for Problem (I))**]{} Let $u$ and $v$ be respectively a viscosity subsolution and a supersolution of (\[problem\]). For any $\nu>0$ there exists $r>0$ such that $u^r$ and $v_r$ are respectively a subsolution and a supersolution of the following problems. $$\label{subproblem}
F(x,u,\nabla u,\nabla^2 u) -\int_{{\bf R^N}}
u(x+z)-u(x)-{\bf 1}_{|z|\leq 1}\la z,\n u(x) \ra q(dz)\}\leq \nu,\quad$$ $$\label{superproblem}
F(x,v,\nabla v,\nabla^2 v) -\int_{{\bf R^N}}
v(x+z)-v(x)-{\bf 1}_{|z|\leq 1}\la z,\n v(x) \ra q(dz)\}\geq -\nu,$$ in $\Omega_r=\{x\in {\Omega}|\quad dist(x,\p\Omega)> \sqrt{2M}r\}$, where $M=\max\{\sup_{\overline{\Omega}}|u|,\sup_{\overline{\Omega}}|v|\}$.
Remark that $u^r$ is semiconvex, $v_r$ is semiconcave, and both are Lipschitz continuous in ${\bf R^N}$. We then deduce from the Jensen’s maximum principle and the Alexandrov’s theorem (deep results in the convex analysis, see \[2\] and \[3\]), the following lemma, the last claim of which is quite important in the limit procedure in the nonlocal term.
[**Lemma 1.3**]{} Let $U$ be semiconvex and $V$ be semiconcave in $\Omega$. For $\phi(x,y)=\a |x-y|^2$ ($\a>0$) consider $\Phi(x,y)=U(x)-V(y)-\phi(x,y)$, and assume that $(\ox,\oy)$ is an interior maximum of $\Phi$ in $\overline{\Omega}\times \overline{\Omega}$. Assume also that there is an open precompact subset $O$ of $\Omega\times \Omega$ containing $(\ox,\oy)$, and that $\mu$ $=\sup_{O} \Phi(x,y) -\sup_{\p O} \Phi(x,y) >0$. Then, the following holds.\
(i) There exists a sequence of points $(x_m,y_m)\in O$ ($m\in {\bf N}$) such that $\lim_{m\to \infty} (x_m,y_m)=(\ox,\oy)$, and $(p_m,X_m)\in J^{2,+}_{\Omega}U(x_m)$, $(p_m',Y_m)\in J^{2,-}_{\Omega}V(y_m)$ such that $\lim_{m\to \infty}p_m$$=\lim_{m\to \infty}p'_m$$=2\a(x_m-y_m)=p$, and $X_m\leq Y_m\quad \forall m$.\
(ii) For $P_m=(p_m-p,-({p'}_m-p))$, $\Phi_m(x,y)=\Phi(x,y)-\la P_m,(x,y)\ra$ takes a maximum at $(x_m,y_m)$ in O.\
(iii) The following holds for any $z\in {\bf R^N}$ such that $(x_m+z,y_m+z)\in O$. $$\label{important}
U(x_m+z)-U(x_m)-\la p_m,z\ra
\leq V(y_m+z)-V(y_m)-\la p'_m,z\ra.$$
By admitting these lemmas here, let us show how Theorem 1.1 is proved.\
$Proof\quad of\quad Theorem\quad 1.1.$ We use the argument by contradiction, and assume that $\max_{\overline{\Omega}}(u-v)$$=(u-v)(x_0)=M_0>0$ for $x_0\in \Omega$. Then, we approximate $u$ by $u^r$ (supconvolution) and $v$ by $v_r$ (infconvolution), which are a subsolution and a supersolution of (\[subproblem\]) and (\[superproblem\]), respectively. Clearly, $\max_{\overline{\Omega}}(u^r-v_r)$$\geq M_0>0$. Let $\ox\in \Omega$ be the maximizer of $u^r-v_r$. In the following, we abbreviate the index and write $u=u^r$, $v=v_r$ without any confusion. As in the PDE theory, consider $\Phi(x,y)=u(x)-v(y)-\a|x-y|^2$, and let $(\hx,\hy)$ be the maximizer of $\Phi$. Then, from Lemma 1.3 there exists $(x_m,y_m)\in \Omega$ ($m\in {\bf N}$) such that $\lim_{m\to \infty} (x_m,y_m)=(\hx,\hy)$, and we can take $(\e_m,\d_m)$ a pair of positive numbers such that $u(x_m+z)\leq u(x_m)+\la p_m,z\ra+\frac{1}{2}\la X_m z,z\ra+\d_m|z|^2$, $v(y_m+z)\geq v(y_m)+\la p'_m,z\ra+\frac{1}{2}\la Y_m z,z\ra-\d_m|z|^2$, for $\forall |z|\leq \e_m$. From the definition of the viscosity solutions, we have $$F(x_m,u(x_m),p_m,X_m) -\int_{|z|\leq \e_m}
\frac{1}{2}\la (X_m+2\d_m I)z,z \ra dq(z)$$ $$-\int_{|z|\geq \e_m}
u(x_m+z)-u(x_m)
-{\bf 1}_{|z|\leq 1}\la z,p_m \ra q(dz)\leq \nu,$$ $$F(y_m,v(y_m),p'_m,Y_m) -\int_{|z|\leq \e_m}
\frac{1}{2}\la (Y_m-2\d_m I)z,z \ra dq(z)$$ $$-\int_{|z|\geq \e_m}
v(y_m+z)-v(y_m)
-{\bf 1}_{|z|\leq 1}\la z,p'_m \ra q(dz)\geq -\nu.$$ By taking the difference of the above two inequalities, by using (\[important\]), and by passing $m\to \infty$ (thanking to (\[important\]), it is now available), we can obtain the desired contradiction. The claim $u\leq v$ is proved.\
[**Remark 1.1.**]{} To prove the comparison results for (II) (in \[1\]), we do the approximation by the supconvolution: $u^r(x)=\sup_{y\in {\overline{\Omega}}}\{ {u}(y)-\frac{1}{2r^2}|x-y|^2\}$, and the infconvolution: $v_r(x)=\inf_{y\in \overline{\Omega}}\{ {v}(y)+\frac{1}{2r^2}|x-y|^2\}$ as in Lemma 1.2. Because of the restriction of the domain of the integral of the nonlocal term and the Neumann B.C., a slight technical complexity is added. The approximating problem for (\[problem2\])-(\[neumann\]) in $\overline{\Omega}$ is as follows. $$\min[F(x,u(x),\nabla u(x),\nabla^2 u(x)) +
\min_{y\in \overline{\Omega},|x-y|\leq \sqrt{2M}r }\{
-\int_{\{z\in \bf R^N|y+z\in \overline{\Omega}\}}
u(x+z)-u(x)$$ $$-{\bf 1}_{|z|\leq 1}\la z,\n u(x) \ra q(dz),\quad \min_{y\in \p{\Omega},|x-y|\leq \sqrt{2M}r}\{ \la {\bf n}(y),\n u(x) \ra+\rho\} ]
\leq \nu,$$ $$\max[F(x,v(x),\nabla v(x),\nabla^2 v(x)) +
\max_{y\in \overline{\Omega},|x-y|\leq \sqrt{2M}r}\{
-\int_{\{z\in \bf R^N|y+z\in \overline{\Omega}\}}
v(x+z)-v(x)$$ $$-{\bf 1}_{|z|\leq 1}\la z,\n v(x) \ra q(dz),\quad \max_{y\in \p{\Omega},|x-y|\leq \sqrt{2M}r}\{ \la {\bf n}(y),\n v(x) \ra-\rho\}]
\geq -\nu.$$ We deduce the comparison result from this approximation and Lemma 1.3, by using the similar argument as in the proof of Theorem 1.1.\
[31]{} M. Arisawa, A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations. M.G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the AMS, vol.27, no. 1 1992. W.H. Fleming and H.M. Soner, Controlled Markov processes and Viscosity solutions, Springrt-Verlag 1992.
|
---
abstract: 'We present quantum oscillations observed in the heavy fermion compound YbPtBi in magnetic fields far beyond its field-tuned, quantum critical point. Quantum oscillations are observed in magnetic fields as low as 60kOe at 60 mK and up to temperatures as high as 3K, which confirms the very high quality of the samples as well as the small effective mass of conduction carriers far from the quantum critical point. Although the electronic specific heat coefficient of YbPtBi reaches $\sim$ 7.4 J/mol K$^{2}$ in zero field, which is one of the highest effective mass value among heavy fermion systems, it is suppressed quickly by applied magnetic field. The quantum oscillations were used to extract the quasiparticle effective masses of the order of the bare electron mass, which is consistent with the behavior observed in specific heat measurements. Such a small effective masses at high fields can be understood by considering the suppression of Kondo screening.'
author:
- 'E. Mun$^{1, 2}$, S. L. Bud’ko$^{1}$, Y. Lee$^{1}$, C. Martin$^{1, *}$, M. A. Tanatar$^{1}$, R. Prozorov$^{1}$, P. C. Canfield$^{1}$'
title: Quantum oscillations in heavy fermion compound YbPtBi
---
Introduction
============
Heavy fermion (HF) compounds have provided some of the clearest evidence for quantum phase transitions [@Lohneysen2007]. A large effective mass introduces a characteristic low energy scale that can easily be tuned by non-thermal control parameters, making HF systems desirable for studying quantum criticality. An antiferromagnetic (AFM) quantum critical point (QCP) has been explored in many HF systems [@Stewart2001; @Stewart2006]. However, a magnetic field-tuned QCP in Yb-based materials has been limited to only few cases, among stoichiometric compounds, in particular tetragonal YbRh$_{2}$Si$_{2}$ [@Gegenwart2002] and hexagonal YbAgGe [@Budko2004]. Recently, the HF compound YbPtBi attracted attention because of a quantum phase transition driven by magnetic field [@Mun2013], in particular providing relatively easy access to the field-tuned QCP as well as the appearance of a new phase between the ordered (AFM) state and paramagnetic Fermi liquid state.
YbPtBi has face-centered cubic (F$\bar{4}$3m) structure and manifests AFM ordering below $T_{N}$ = 0.4 K with an electronic specific heat coefficient of $\gamma$ $\sim$ 7.4 J/mol K$^{2}$ at zero field [@Mun2013; @Fisk1991]. Recent neutron scattering experiments characterized the averaged ordered moment of $\sim$ 0.8 $\mu_{B}$/Yb with the AFM propagation vector $\tau$ = (1/2, 1/2, 1/2) [@Ueland2014]. A field-tuned QCP is readily accessible by suppressing AFM ordering below $T_{N}$ = 0.4 K by application of external magnetic field of $H_{c}$ $\sim$ 4 kOe [@Mun2013]. The temperature dependence of the resistivity indicates a recovery of the Fermi liquid (FL) state for $H^{*}$ $\gtrsim$ 7.8 kOe. In addition, of particular interest in YbPtBi is the fact that a new phase appears in the proximity of the QCP. In this new phase, strange metallic metallic behavior (non-Fermi liquid) manifests and separates the AFM state and FL regime. A crossover scale has been found in thermodynamic and transport measurements [@Mun2013], where the crossover line shows a tendency of converging toward to $H^{*}$ $\sim$ 7.8 kOe in the zero temperature limit. A similar phase near QCP and the crossover line have been observed in YbAgGe [@Budko2005; @Mun2010] and Ge-doped YbRh$_{2}$Si$_{2}$ [@Custers2010].
Although many examples of QCP have been reported in HF materials [@Stewart2001; @Stewart2006], theoretical classifications have not been firmly established yet. Unlike the traditional picture (e. g. for a spin density wave) of approaching a quantum criticality [@Hertz1976; @Millis1993; @Moriya1995], where the heavy quasiparticles survive near the QCP, the unconventional way (local quantum criticality) to explain quantum phase transition suggests a destruction of Kondo screening of the $f$-electrons [@Si2001; @Si2003; @Coleman2007], involving Fermi surface reconstruction, where the heavy quasiparticles decompose into the conduction sea and local magnetic moments. If the Fermi surfaces have different shapes or are completely reconstructed across the QCP, then a Lifshitz transition, associated with a reconstruction of the Fermi surface, must separate the two phases between ordered and disordered state. In general, because most of the HF compounds have multiple Fermi surfaces, it is difficult to probe such a Lifshitz transition at extremely low temperatures. Transport measurements can address this issue clearly for a system with a single band, however this may not be true for a multiband systems. In metallic systems, the interpretation of physical quantities requires careful consideration of the Fermi surface, especially at the QCP.
In this paper, we report the determination of the high-field stabilized Fermi surfaces for YbPtBi inferred from quantum oscillations and band structure calculations. Until now, there was no direct experimental information about the size or shape of the Fermi surface in YbPtBi. By producing high quality single crystals, we are able to infer Fermi surfaces by detecting quantum oscillations down to 60 mK, but for only above $\sim$ 60 kOe, in resistivity measurements.
In principle, the Fermi surface can be studied by comparing the quantum oscillation measurements with electronic structure calculations. However, quantum oscillation data in most HF compounds is likely to be incomplete mainly due to the experimental difficulties. This is a standing problem in HF physics: in order to detect the heavier effective masses, higher magnetic fields and extremely lower temperatures (order of $\sim$ mK) are needed, however the mass enhancement can be suppressed due to the application of these larger magnetic fields. Further, when there is a 4$f$ electron contribution to the Fermi volume, band structure calculations can’t accurately predict the Fermi surface topology in detail and an estimate of the density of states. Therefore, in this study, we focused data analysis on the high field-paramagnetic regime and compared it to the band structure calculations in zero field for simple, trivalent Yb. We confirm the multiband nature of this system, but are unable to identify the heavy Fermi surface. In addition, we compare our results to the family of $R$PtBi, where quantum oscillations have been observed for $R$ = Y, La, Ce, and Nd [@Butch2011; @Goll2002; @Wosnitza2006; @Morelli1996].
Experimantal
============
Single crystals of YbPtBi were grown out of a ternary melt with excess of Bi as has been described in earlier reports [@Canfield1991; @Canfield1992]. Four-probe ac resistivity ($f$ = 16 Hz) measurements were performed in a Oxford dilution refrigerator, where the samples were attached to the cold state using GE-varnish. Pt wires were used as leads and attached to the sample with Epotek H20E silver epoxy. The magnetic field (**H**) was applied along the \[100\] and \[111\] direction, and the electric current (**I**) across the sample was applied perpendicular to **H**: **H** $\perp$ **I**. Details of experimental conditions are given in Ref. [@Mun2013].
In order to compare the experimental observations of Shubnikov-de Haas frequencies to the topology of the Fermi surfaces, we calculated the zero field band structure of paramagnetic, trivalent, YbPtBi. For the Fermi surface calculation, we have used a full-potential Linear Augmented Plane Wave (fp-LAPW) [@Blaha2001] method with a local density functional [@Perdew1992]. The structure data was taken from reported experimental results [@Robinson1994]. To obtain the self consistent charge density we chose 1204 $k$-points in the irreducible Brillouin zone and set $R_{MT}\cdot K_{max}$ to 9.0, where $R_{MT}$ is the smallest muffin-tin radius and $K_{max}$ is the plane-wave cutoff. We used muffin-tin radii 2.5 for all Yb, Bi, and Pt atoms. The calculation was iterated with 0.0001 electrons of charge and 0.01 mRy of total energy convergence criteria. Although there was a discussion about 4$f$ electron pinning at the Fermi energy [@Oppeneer1997] and we were aware that Fermi surface is quite different under 4$f$ electrons influence [@McMullan1992] we treated 4$f$ electrons as core-electrons since we were interested in the high magnetic field, paramagnetic state. To obtain SdH frequencies we calculated 2-dimensional Fermi surfaces and integrated the Fermi surface area. We chose planes which were perpendicular to $k_{z}$-axis and had 0.01 (2$\pi/a$) interval. Each plane (-1$\leq$$k_{x}$,$k_{y}$$\leq$1) was divided with 100$\times$100 mesh. For a 3-dimensional Fermi surface, we used 2300 $k$-points in the irreducible Brillounin zone and a XcrysDen graphic program [@XcrysDen].
Result
======
![Magnetoresistance (MR), plotted as $[\rho(H)-\rho(0)]/\rho(0)$ vs. $H$, of YbPtBi at $T$ = 0.1K along **H**$\parallel$\[100\] and **H**$\parallel$\[111\]. At high magnetic fields quantum oscillations are discernible for both curves.[]{data-label="YbPtBiSdH1"}](Fig1.pdf){width="1\linewidth"}
Shubnikov-de Haas (SdH) quantum oscillations have been observed throughout the magnetoresistance (MR) measurements at low temperatures and high magnetic fields. Figure \[YbPtBiSdH1\] shows the MR at $T$ = 0.1K for magnetic field applied along \[100\] and \[111\] directions. At high magnetic fields a broad local extrema in MR is observed for both magnetic field directions. This behavior may be due to the change of scattering processes with CEF levels, or it may be the oscillatory component corresponding to extremely small Fermi surface area in which the extremely small frequency has been observed for NdPtBi [@Morelli1996] in the paramagnetic state. For YbPtBi though this is not likely to be the case because the frequency is so small that it would have an amplitude that would make it hard to observe in SdH measurements. What is unambiguous are the clear quantum oscillations at high magnetic fields.
{width="0.5\linewidth"}{width="0.5\linewidth"}
In Figs. \[YbPtBiSdH2\] (a) and (c) typical SdH data sets for YbPtBi, after subtracting the background contributions, are displayed as a function of 1/$H$ at selected temperatures. The amplitude of the oscillations decreases as temperature increases. Since the signals are comprised of a superposition of several oscillatory components, the data are most easily visualized by using a fast Fourier transform (FFT) as shown in Figs. \[YbPtBiSdH2\] (b) for **H** $\parallel$ \[100\] and \[YbPtBiSdH2\] (d) for **H** $\parallel$ \[111\]. The FFT spectra at $T$ = 0.06K show several frequencies, including, in the case of **H** $\parallel$ \[100\], second harmonics with very small amplitudes. The observed frequencies are summarized in Table \[YbPtBiSdHtable\].
$f$ (MOe) $m^{*}/m_{e}$
------------------------- ----------- --------------------- -------------------------
**H**$\parallel$\[100\]
$\alpha$ 7.27 1.41
$\beta$ 7.83 1.59
$\delta$ 13.06 0.83
$\gamma$ 13.99 0.80
$\eta$ 14.37 0.97
2$\beta$ 15.67
2$\delta$ 26.13
2$\gamma$ 27.99
2$\eta$ 28.74
**H**$\parallel$\[111\]
$\xi$ 8.95 1.22
$\xi_{1}$ 17.90 0.49
: Frequencies $f$ and effective masses $m^{*}$ obtained from the SdH oscillations. $m_{e}$ is the bare electron mass.[]{data-label="YbPtBiSdHtable"}
Quantum oscillations are observed in magnetic fields as low as 60kOe at the lowest temperature measured and at temperatures as high as 3K, which further confirms the very high quality of the samples as well as the relatively small effective masses of the associated conduction electrons. The frequencies in FFT spectra do not shift with temperature and most of the first harmonics of the frequencies are clearly observed as high as 2K.
The oscillation amplitudes and the fit curves using the temperature reduction factor are plotted in Figs. \[YbPtBiSdH4\] (a) and (b). The cyclotron effective masses, $m^{*}$, of the carriers from the various orbits were determined by fitting the temperature-dependent amplitude of the oscillations to the Lifshitz-Kosevich (L-K) formula [@Shoenberg1984] for each frequency. The calculated effective masses range from $m^{*}(\alpha)\,\sim$ 1.41$m_{e}$ to $m^{*}(\xi_{1})\,\sim$ 0.49$m_{e}$, where $m_{e}$ is the bare electron mass. The inferred effective masses are summarized in Table \[YbPtBiSdHtable\]. We were not able to estimate the effective masses, associated with the second harmonic frequencies due to the small amplitude of the signals. Although the frequency $\xi_{1}$ is integer-multiple of $\xi$, $\xi_{1}$$\simeq$2$\xi$, it does not appear to be a higher harmonic of $\xi$ because of the inconsistent effective masses. In addition, if these frequencies are originating from the same extremal orbit, the phase difference between two frequencies can not be explained; the oscillation curves are generated by L-K formula with the phase term, $A_{1} \sin(2\pi \xi/H + \pi/1.95)$ + $A_{2} \sin(2\pi \xi_{1}/H - \pi/7.7)$, as shown in Fig. \[YbPtBiSdH4\] (c). Therefore, $\xi_{1}$ really appears to be an independent frequency, coming from different extremal orbit. The frequency of the orbit $\eta$ is almost twice of the frequency of the $\alpha$, however these orbits are also apparently coming from different Fermi surfaces. If the orbit $\eta$ is the second harmonics of the $\alpha$, the oscillation amplitude of the $\eta$ should be smaller than that of $\alpha$, but the amplitudes of these frequencies are almost the same. Therefore, the orbit $\eta$ is not a second harmonic of $\alpha$. Note that the frequency, observed near 14MOe at 2K along **H**$\parallel$\[100\], seems to split from one component into two component of $\gamma$ and $\eta$ with decreasing temperature, as indicated by up arrow in Fig. \[YbPtBiSdH2\] (b). At present it is not clear whether two frequencies of $\gamma$ and $\eta$ are originating from the same extremal orbit, thus it needs to be clarified by further detailed measurements.
![(a) and (b) Temperature dependence of the SdH amplitudes. Solid lines represent the fit curves to the Lifshitz-Kosevich (L-K) formula. All data and fit curves are normalized to 1, indicated by horizontal arrows, and shifted for clarity. (c) Resistivity along **H**$\parallel$\[111\] at $T$ = 0.06 and 1K, plotted as a function of 1/$H$ after subtracting the background MR, where the solid lines represent the fit curves based on the L-K formula with the frequency $\xi$ and $\xi_{1}$.[]{data-label="YbPtBiSdH4"}](Fig3.pdf){width="1\linewidth"}
Discussion
==========
The low carrier density for YbPtBi, determined from Hall coefficient measurements [@Mun2013], implies a Fermi surface occupying a small portion of the Brillouin zone, which is consistent with the results of our analysis of the quantum oscillations. The frequency of the quantum oscillations is proportional to the extremal cross-section, $A_{FS}$, of the Fermi surface; $f\,=\,(\hbar/2\pi e)A_{FS}$ [@Shoenberg1984]. In the high-field, paramagnetic region direct evidence for small Fermi surfaces comes from SdH measurements, where several small extremal orbits, implying a small portion of occupation of the Brillouin zone, are observed. Quantum oscillations have also been observed for LaPtBi and CePtBi [@Goll2002; @Wosnitza2006] from the electrical resistivty measurements in pulsed magnetic fields up to 50Tesla. The oscillation frequencies for LaPtBi are approximately 10 times smaller, 0.65MOe for **H**$\parallel$\[100\] increasing to 0.95MOe for **H**$\parallel$\[110\], than for YbPtBi. In this family, SdH for YPtBi are observed up to 10 K with a frequency of 0.46 MOe [@Butch2011]. Note that a single SdH oscillation with $f$ = 0.74 T was inferred from magnetoresistiance measurements in NdPtBi [@Morelli1996], where the observed frequency is two order of magnitude smaller than the frequencies observed for other compounds in this family. It is not clear whether the origin of the extraordinarily small frequency for NdPtBi is related to the broad local extrema observed in YbPtBi (Fig. \[YbPtBiSdH1\]). For CePtBi the anomalous temperature dependence of SdH frequency, $f$ = 0.6MOe, was observed along **H**$\parallel$\[100\] and a very low SdH frequency of $\sim$0.2MOe, which is independent of temperature, was found along **H**$\parallel$\[111\]. In addition to the unusual temperature dependence of the SdH frequency for CePtBi, the disappearance of the oscillations was observed above about 25 T at which the magnetic field-induced band structure change was proposed [@Wosnitza2006].
Since the SdH frequencies for YbPtBi are not changed by temperature or magnetic field, within the temperature and magnetic field range of our measurements, such a band structure modification is not expected. The band calculations for LaPtBi [@Oguchi2001] and CePtBi [@Goll2002], assuming localized 4$f$ states, were found to be semimetals. In these calculations, two hole-like Fermi surface sheets are found around zone center, which are similar to the measured angular dependence of the Fermi surface cross-section area of LaPtBi. A number of small electron-like pockets are also predicted in the band calculations. The effective masses for both LaPtBi and CePtBi have been estimated to be $\sim$ 0.3$m_{e}$ [@Wosnitza2006], which is somewhat smaller than for YbPtBi. The observed trend of SdH frequencies suggested larger Fermi surface sheets for YbPtBi than LaPtBi, and these are consistent with earlier resistivity results of $R$PtBi [@Canfield1991], where the resistivity varied from metallic (semimetallic) to small gap semiconductor when rare-earth changes from Lu to Nd [@Canfield1991]; $\rho(T)$ of LuPtBi decreases and $\rho(T)$ of NdPtBi increases as temperature decreases. The carrier density for LuPtBi is approximately two order of magnitude bigger than that for LaPtBi [@Mun2013].
![(a) Band structure of nonmagnetic YbPtBi, calculated for localized 4$f$ states. (b) Calculated Fermi surface in the fcc Brillouin zone. Two three dimensional pockets (bands 55 and 56), located in the zone center, are surrounded by sixteen cigar shaped pockets (bands 57 and 58). (c) Enlarged Fermi surface for band 55, 56, 57, and 58.[]{data-label="YbPtBiBand"}](Fig4.pdf){width="1\linewidth"}
The results of band structure calculations are shown in Fig. \[YbPtBiBand\] (a). The overall features are very similar to the results of LaPtBi, however with a larger Fermi surface occupation of the Brillouin zone. Around zone center two hole-like Fermi surface (band 55 and 56) are surrounded by sixteen small electron-like pockets (band 57 and 58). These calculated Fermi surfaces of YbPtBi are plotted in Fig. \[YbPtBiBand\] (b) and (c). The SdH frequencies of these four bands are calculated to be 2.4, 3.5, 0.79, and 0.65MOe from the maximum area perpendicular to the $k_{z}$, where there are orbits very close to 2.4 and 3.5MOe due to the 3-dimensional shape of the Fermi surfaces at the zone center. These values are much larger than the predicted value for LaPtBi and CePtBi [@Goll2002; @Wosnitza2006], however four times smaller than the frequencies determined from our experimental results.
The Fermi surfaces of YbPtBi are highly sensitive to the 4$f$ electron contributions as predicted in Ref. [@McMullan1992]. When the 4$f$ electrons are included in the band calculations, the six hole-like pockets are located zone center in which the predicted frequencies range from $\sim$ 27 to $\sim$ 164MOe [@McMullan1992], which is much higher than the experimental observations. So treating 4$f$ electrons as included in core levels appears to be reasonable. If the Fermi level is shifted to lower energy, the experimentally observed frequencies can be matched to the hole-like pockets at the zone center, whereas the electron-like pocket surrounding the zone center will not be detected. As a conjecture, the effective masses and frequencies of the orbit $\alpha$ and $\beta$ along \[100\] direction, linked to the $\xi$ along \[111\] direction, are almost the same, expected that these two orbits are came from the band 55. Similarly the orbits $\delta$, $\gamma$, and $\eta$, connected to the $\xi_{1}$, all came from the 3-dimensional shape of the band 56. Without an angular dependence of the SdH measurements, the Fermi surface topology can not be determined unambiguously and further theoretical work is needed to unravel the discrepancies in the precise extremal orbit sizes.
Since we have observed only small effective masses for YbPtBi, it is expected that the hybridization between 4$f$ and conduction electrons has been suppressed for these high magnetic fields. This is consistent with the specific heat results; the enormous value of $\gamma$ $\sim$ 7.4 J/mol$\cdot$K$^{2}$ for $H$ = 0 is suppressed to $\gamma$ $\sim$ 0.15 J/mol$\cdot$K$^{2}$ for $H$ = 50kOe and would extrapolate to $\sim$ 0.030 J/mol$\cdot$K$^{2}$ at 140 kOe (using date from [@Mun2013]). Note that if there are still significant 4$f$ electron contributions in this magnetic field range, up to 140kOe, it would require much lower temperatures to observe heavy electrons in SdH measurements. This is a standing problem in HF physics, in order to detect the heavier effective masses, higher magnetic fields and very low temperatures are needed, however the mass enhancement can be suppressed due to the application of these larger magnetic fields. Thus, lower measurement temperatures and crystals with extremely low scattering in terms of Dingle temperature are necessary to detect heavier effective mass of carriers. Measuring de Haas-van Alphen (dHvA) oscillations as a complementary to SdH oscillations may be another experimental approach since oscillation amplitudes have different dependence of $m^{*}$ in dHvA and SdH. However, this task can be challenging due to high paramagnetic background signal.
summary
=======
In summary, we have presented Shubnikov-de Haas quantum oscillations detected in YbPtBi, which is the first report on quantum oscillations since the discovery of its heavy fermion behavior in 1991. The band structure calculations for the Fermi surface are also presented, where treating 4$f$ electrons as included in core levels appears to be important in high field regime far beyond the quantum critical point. Comparison is made between the high field oscillations and zero field band structure calculations, allowed us to infer Fermi surfaces only in the paramagnetic state. The current study clearly shows that i) small oscillation frequencies suggest a low carrier density, consistent with Hall effect measurements, ii) multiple oscillation frequencies imply a multiband nature, and iii) small effective masses at high fields indicate a suppression of Kondo screening, consistent with specific heat results. Although it was not possible to infer heavy fermion state, the current study may, in future, give useful information about the nature of the quantum criticality when the Fermi volume at low fields are available.
This work was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering. The research was performed at the Ames Laboratory. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. The work at Simon Fraser University was supported by Natural Sciences and Engineering Research Council of Canada.
$^{*}$Current address; Ramapo College of New Jersey, Mahwah, New Jersey 07430, USA.
[99]{}
Hilbert v. Löhneysen, Achim Rosch, Matthias Vojta, and Peter Wölfle, Rev. Mod. Phys. [**79**]{}, 1015 (2007).
G. R. Stewart, Rev. Mod. Phys. [**73**]{}, 797 (2001). G. R. Stewart, Rev. Mod. Phys. [**78**]{}, 743 (2006).
P. Gegenwart, J. Custers, C. Geibel, K. Neumaier, T. Tayama, K. Tenya, O. Trovarelli, and F. Steglich, Phys. Rev. Lett. [**89**]{}, 056402 (2002).
S. L. Bud’ko, E. Morosan, and P. C. Canfield, Phys. Rev. B [**69**]{}, 014415 (2004).
E. D. Mun, S. L. Bud’ko, C. Martin, H. Kim, M. A. Tanatar, J.-H. Park, T. Murphy, G. M. Schmiedeshoff, N. Dilley, R. Prozorov, and P. C. Canfield, Phys. Rev. B [**87**]{}, 075120 (2013).
Z. Fisk, P. C. Canfield, W. P. Beyermann, J. D. Thompson, M. F. Hundley, H. R. Ott, E. Felder, M. B. Maple, M. A. Lopez de la Torre, P. Visani, and C. L. Seaman, Phys. Rev. Lett. [**67**]{}, 3310 (1991).
B. G. Ueland, A. Kreyssig, K. Prokeš, J. W. Lynn, L. W. Harriger, D. K. Pratt, D. K. Singh, T. W. Heitmann, S. Sauerbrei, S. M. Saunders, E. D. Mun, S. L. Bud’ko, R. J. McQueeney, P. C. Canfield, and A. I. Goldman, Phys. Rev. B [**89**]{}, 180403(R) (2014).
S. L. Bud’ko, V. Zapf, E. Morosan, and P. C. Canfield, Phys. Rev. B [**72**]{}, 172413 (2005).
E. Mun, S. L. Bud’ko, and Paul C. Canfield, Phys. Rev. B [**82**]{}, 174403 (2010).
J. Custers, P. Gegenwart, C. Geibel, F. Steglich, P. Coleman, and S. Paschen, Phys. Rev. Lett. [**104**]{}, 186402 (2010).
J. A. Hertz, Phys. Rev. B [**14**]{}, 1165 (1976).
A. J. Millis, Phys. Rev. B [**48**]{}, 7183 (1993).
T. Moriya and T. Takimoto, J. Phys. Soc. Jpn. 64, 960 (1995).
Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature (London) [**413**]{}, 804 (2001).
Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Phys. Rev. B [**68**]{}, 115103 (2003).
P. Coleman, Handbook of Magnetism and Advanced Magnetic Materials (John Wiley & Sons, Inc., Vol. 1, Heavy Fermions: electrons at the edge of magnetism, 2007).
N. P. Butch, P. Syers, K. Kirshenbaum, A. P. Hope, and J. Paglione, Phys. Rev. B [**84**]{}, 220504(R) (2011).
G. Goll, J. Hagel, H. v. Lohneysen, T. Pietrus, S. Wanka, J. Wosnitza, G. Zwicknagl, T. Yoshino, T. Takabatake, and A. G. M. Jansen, Europhys. Lett. [**57**]{}, 233 (2002).
J. Wosnitza, G. Goll, A. D. Bianchi, B. Bergk, N. Kozlova, I. Opahle, S. Elgazzar, Manuel. Richter, O. Stockert, H. v. Löhneysen, T. Yoshino, and T. Takabatake, New J. Phys. [**8**]{}, 174 (2006).
D. T. Morelli, P. C. Canfield, and P. Drymiotis, Phys. Rev. B [**53**]{}, 12896 (1996).
P. C. Canfield, J. D. Thompson, W. P. Beyermann, A. Lacerda, M. F. Hundley, E. Peterson, Z. Fisk, and H. R. Ott, J. Appl. Phys. [**70**]{}, 5800 (1991).
P. C. Canfield and Z. Fisk, Phil. Mag. B [**65**]{}, 1117 (1992).
P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnick, and J. Luitz, WIEN2k, An augmented Plane wave + Local Orbitals Program for Calculation Crystal Properties (K. Schwarz, TU wien, Austria, 2001).
J. P. Perdew and Y. Wang, Phys. Rev. B [**45**]{}, 13244 (1992).
R. A. Robinson, A. Purwanto, M. Kohgi, P. C. Canfield, T. Kamiyama, T.Ishigaki, J. W. Lynn, R. Erwin, E. Peterson, R. Movshovich, Phys. Rev. B [**50**]{}, 9595 (1994).
P. M. Oppeneer, V. N. Antonov, A. N. Yaresko, A. Ya. Perlov, and H. Eschrig, Phys. Rev. Lett. [**78**]{}, 4079 (1997).
G. J. McMullan and M. P. Ray, J. Phys: Condens. Matter [**4**]{} , 7095 (1992).
http://www.xcrysden.org
D. Shoenberg, Magnetic Oscillations in Metals (Cambridge University Press, Cambridge, England, 1984).
T. Oguchi, Phys. Rev. B [**63**]{}, 125115 (2001).
|
---
abstract: 'The genesis of spurious solutions in finite basis approximations to operators which possess a continuum and a point spectrum is discussed and a simple solution for identifying these solutions is suggested.'
author:
- 'R. C. Andrew and H. G. Miller [^1]'
title: A short note on the presence of spurious states in finite basis approximations
---
Recently Ackad and Horbatsch[@AH05] have presented a nice numerical method for the solution of the Dirac equation for the hydrogenic Coulomb problem using the Rayleigh-Ritz method[@KH02]. Using a mapped Fourier grid method, a matrix representation of the Dirac Hamiltonian is constructed in a Fourier sine basis, which upon diagonalization yields reasonably numerically accurate eigenvalues for a mesh size which is not exceptionally large. Relativistic sum rules[@GD82] provide a simple means of checking whether or not the number of basis states is adequate. As with any attempt to construct a matrix representation of an operator which contains continuum states, spurious states can occur and must be eliminated. Ackad and Horbatsch[@AH05] have pointed out that in certain cases they can be identified by looking at the numerical structure of the large and small components of the corresponding eigenvector. Similar phenomenon occur in the mapped Fourier grid representation of the non-relativistic Schr[ö]{}dinger problem[@WDM04] in which non-physical roots are observed at random locations. Again the potentials considered support both bound as well as continuum states. The wave functions of these spurious states are characterized by their unphysical oscillations and non-vanishing amplitude in the classically forbidden regions. The authors point out that they have found no satisfactory mathematical explanation for the occurrence of these spurious levels.
In this note we wish to point out that the genesis of these spurious states can easily be understood and that there is a simple way to identify them. Consider an operator, $\hat{H}$, which possesses a continuum (or continua) as well as a point spectrum. The subspace spanned by its bound state eigenfunctions, $\mathcal{H_B}$, is by itself certainly not complete. As the composition of this space is generally not known beforehand, a set of basis states which is complete and spans a space, $\mathcal{F}$, is chosen to construct a matrix representation of the operator, $\hat{H}$, to be diagonalized. Mathematically this corresponds to projecting the operator $\hat{H}$ onto the space $\mathcal{F}$. Clearly the eigenpairs obtained from diagonalizing the projected operator, $\hat{H}_P$, need not all be the same as those of the operator $\hat{H}$. However, because the set of basis states is complete, any state contained in $\mathcal{H_B}$ can be expanded in terms of this set of basis states. Hence $\mathcal{H_B}$ may also be regarded as a subspace of $\mathcal{F}$ and the complete diagonalization of $\hat{H}_P$ will yield not only the exact eigenstates of $\hat{H}$ but additional spurious eigensolutions. Note these spurious eigenfunctions are eigenfunctions of $\hat{H}_P$ but not of $\hat{H}$. Furthermore in this case the Rayleigh-Ritz bounds discussed in the paper by Krauthauser and Hill[@KH02] apply now to the eigenstates of $\hat{H}_P$.
It is interesting to note that the same problem occurs in the Lanczos algorithm[@L50] when it is applied to operators which possess a bound state spectrum as well as a continuum[@AM03]. This is not surprising as the Lanczos algorithm can also be considered as an application of the Rayleigh-Ritz method[@P80]. In this case an orthonormalized set of Krylov basis vectors is used to construct iteratively a matrix represention of the operator which is then diagonalized. Again spurious states can occur for precisely the same reasons given above. In this case we have proposed identifying the exact bound states in the following manner[@AM03]. After each iteration, for each of the converging eigenpairs ($e_{l \lambda}$,$|e_{l
\lambda}\rangle$), $\Delta_{l \lambda}=|e_{l \lambda}^2-<e_{l
\lambda}|\hat{H}^2|e_{l \lambda}>|$ (where $l$ is the iteration number) is calculated and a determination is made as to whether $\Delta$ is converging toward zero or not. For the exact bound states of $\hat{H}$, $\Delta$ must be identically zero while the other eigenstates states of the projected operator should converge to some non-zero positive value. This method has been successfully implemented to identify spurious states in non-relativistic[@AM03] as well as relativistic[@AMY07] eigenvalue problems. A similar procedure can be implemented in any Rayleigh-Ritz application. One simply must check to see whether the eigensolutions from the diagonalization of $\hat{H}_P$ are also eigensolutions of $\hat{H}^2$.
[8]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ** (, , ).
, , , ** (), <http://www.citebase.org/abstract?id=oai:arXiv.org:0706.2236>.
[^1]: E-Mail:hmiller@maple.up.ac.za
|
---
abstract: 'In the framework of relativistic quark model the behaviour of electromagnetic form factors of diquarks with $ J^{P}=0^{+} $ at small and intermediate momentum transfer are determined. The charge radii of nonstrange and strange scalar diquarks are calculated.'
author:
- |
S.M. Gerasyuta[^1] , D.V. Ivanov\
[*Department of Theoretical Physics,*]{}\
[*St. Petersburg State University*]{}\
[*198904, St.Petersburg, Russia* ]{}
title: Relativistic quark model and scalar diquarks charge radii
---
=15.4cm
Introduction {#intro}
============
Diquarks have become now an efficient tool for studying various processes in hadron physics. Diquark model arises naturally if we assume, that the strong two-quark correlations determine the properties of baryons. As a result, baryons can be considered as the connected quark-diquark states \[1-3\].
As a first step towards describing baryon form factors, one calculate the on-shell electromagnetic form-factors of the constituent diquarks. The diquark form factors are important ingredient in the baryon form factor and contain information about the sizes of the correlated diquark states. Evidence for correlated diquark states in baryons is found in deep-inelastic lepton scattering \[4\] and in hyperon weak decays \[5\].
In the framework of the dispersion N/D-method with help of the iteration bootstrap procedure the scattering amplitudes of dressed quarks were constructed \[6\]. The mass values of the lowest mesons ($ J^{PC}=
0^{-+}, 1^{--}, 0^{++} $) and their quark content are obtained. The $qq$-amplitudes in the colour state $ {\bar{3}}_{c} $ have the diquark levels with $ J^{P}= 0^{+}$ and the masses $ M_{ud}=$0.72 GeV, $M_{us}=M_{ds}=$ 0.86 GeV.
The present paper is devoted to the calculation of electric form factors and charge radii of nonstrange and strange scalar diquarks. The dispersion relation technique allows us to consider the relativistic effects in the composite systems. The double dispersion relations over the masses of the composite particles are used for the consideration of the electric scalar diquark form factors in the infinite momentum frame (section 2). In the Conclusion the calculation results for the electric form factors, charge radii of scalar diquarks and the status of the considered model are discussed.
Electric diquark form factors in the infinite momentum frame {#sect:1}
============================================================
We consider electromagnetic form factor of two-quark system. Let the masses of quarks composing the system be equal $ m_{1} $ and $ m_{2} $ respectively. The Feynman amplitude for process of scattering of virtual photon on the diquark with $ J^{P} =0^{+} $ is described by the standard triangle diagram (Fig.1) on which the photon interacts with each of the two quarks. This amplitude is equal:
$$A(q^{2})=
\int \frac{d^{4} k_{2}} {i{(2\pi )}^{4}}
\frac{Tr[{\gamma}_{5}(\hat{ k_{1}}+m_{1}){\gamma }
_{\mu}({\hat{k'_{1}}} +m_{1}){\gamma }_{5}
( -\hat{P}+\hat{k_{1}}+m_{2})]}{(m^{2}_{1}-k^{2}_{1})
(m^{2}_{1}-k'^{2}_{1})(m^{2}_{2}-{(P-k_{1})}^{2})}\times$$ $$G({(k_{1}-k_{2})}^{2})
G({(k'_{1} -k_{2})}^2) {\varepsilon}_{\mu} e_1 f_1
(q^2) + [1\longleftrightarrow 2 ],$$
where the latter contribution correspons to the diagram with particles 1 and 2 rearranged among themselves. $G$ is the diquark vertex function. Here $ e_{1,2} $ and $ f_{1,2}(q^2) $ are charges and the form factors of quarks $ m_1 $ and $ m_2 $ respectively. Then we obtain:
$$A ( q^{2} )=\frac{1}{2} \int \frac{d^{4} k_{2}} {i{(2\pi )}^{4}}
\left[(P'^{2}-{(m_{1}-m_{2})}^{2}) k_{1\mu }+ \right.$$ $$\left. (P^{2}-{(m_{1}-m_{2})}^{2}) k'_{1\mu }+q^{2} k_{2\mu }\right] \times$$ $${\left[(m^{2}_{1}-k^{2}_{1})(m^{2}_{1}-k'^{2}_{1})
(m^{2}_{2}-{(P-k_{1})}^{2})\right]}^{-1}\times$$ $$G({(k_{1}-k_{2})}^{2}) G({(k'_{1}-k_{2})}^{2})
{\varepsilon}_{\mu} e_1 f_1 (q^2) + [1\longleftrightarrow 2 ]$$
Using the expression \[7\]:
$$k_{1\mu } + k'_{1\mu }=\alpha (P_{\mu }+P'_{\mu })+\beta q_{\mu}+
{(k_{1\mu }+k'_{1\mu })}_{\bot },$$ $$\alpha=\frac { P'^{2} + P^{2} + 2 {m}_
{1}^{2}-2 {m}_{2}^{2}-q^{2}} {2 ( P^{2} + P'^{2})-q^{2}-{(P'^{2}-P^{2})}^{2}
/ q^{2}},$$ $$\beta=-\frac{\alpha (P'^{2}-P^{2})}{q^{2}},$$ we calculate the amplitude $ A(q^2) $:
$$A(q^{2})={\varepsilon }_{\mu} (P_{\mu } + P'_{\mu })
e_{D} G_{D}^{E} (q^{2}) ,$$
where the diquark form factor is obtained:
$$G_{D}^{E}(q^{2})=
\int \frac{ d^{4} k_{2}}{i{(2\pi )}^{4}} \frac{((1-\alpha){q}^{2} +
\alpha (P^{2} + P'^{2}-2{(m_{2}-m_{1})}^{2}))}{4(m_{1}^{2}-k_{1}^{2})
(m_{2}^{2}-k_{2}^{2})(m_{1}^{2}-{k'}_{1}^{2})} \times$$ $$G({(k_{1}-k_{2})}^{2}) G({(k_{1}-k'_{1})}^{2})
\frac{e_{1}}{e_{D}} f_{1} (q^{2}) +
[1\longleftrightarrow 2 ],$$ $ {\varepsilon}_{\mu} q_{\mu} =0 $, $ e_{D}$ is the diquark charge.
We pass to the infinite momentum frame and use the dispersion integration over the masses of composite particles \[7\]. The momentum of the composite particle (diquark) along the $z$-axis is large, $ P_{z}\to \infty $. Hereafter we introduce the notation $ P=k_{1} + k_{2}, P^{\prime } =P + q $ for the initial and final state momenta ($ P^{2}=s, P^{\prime 2}=s^{\prime } $). $ s $ and $ s^{\prime } $ are the initial and final energy of the composite system. The double discontinuity defines the form factor of the two-quark system (diquark):
$$G_{D}^{E}(q^{2})=\int\limits_{{ ( m_{1} + m_{2} )}^{2}}^
{{ \Lambda }_{s}} \frac{ds}{ \pi}\int\limits_{{ ( m_{1} + m_{2
} ) }^{2}}^{{ \Lambda }_{s}} \frac { d {s}^{ \prime }} {
\pi}\hskip 2 pt \frac{{disc }_{s} { disc }_{ s^{\prime }} G_{D}^{E}
( s, s^{ \prime }, q^{2} ) } { ( s-M_{D}^{2} ) ( s^{ \prime } -
M_{D}^{2} ) },$$
$${ disc }_{s}{ disc }_{s'} G_{D}^{E} (s, s', q^{2} )=$$ $$\frac{1}{4} G G' \left[e_{1} f_{1} (q^{2})D_{1}(s, s', q^{2})
{\Delta }_{1}(s, s', q^{2}) + \right.$$ $$\left. e_{2} f_{2} (q^{2}) D_{2} (s, s', q^{2})
{\Delta}_{2}(s, s', q^{2})\right]/e_{D},$$
Further we calculate the following terms:
$$D_{1}(s_{1}, {s'}_{1}, q^{2} )=\frac{1}{4}[(1-{\alpha }_{1}) q^{2} +
{\alpha }_{1} ({s'}_{1} + s_{1}-2 {(m_{1}-m_{2})}^{2})]$$ $${\alpha }_{1}=\frac{b_{1} + q^{2} {a}_{1} / s_{1}}
{2(1-q^{2} {a}_{1}^{2} / s_{1})},
{\Delta }_{1}(s_{1}, {s'}_{1}, q^{2})=\frac { b_{1} {a}_{1} + 1}{b_{1} +
c_{1} {a}_{1}}$$
$$b_{1}=1 + \frac{m_{1\bot }^{2}-m_{ 2\bot }^{2}}{s_{1}},
c_{1}=b_{1}^{2}-\frac{4{k}_{\bot}^{2} {\cos}^{2}\phi }{s_1}$$
$${a}_{1}=\frac{-b_{1} + \sqrt{(b_{1}^{2}-c_{1})
(1-s_{1} c_{1} / q^{2})}}{c_{1}},$$
$$s_{1}=\frac { m_{ 1\bot }^{2} + x(m_{2\bot }^{2}-m_{1\bot }^{2})}
{x(1-x)}, {s'}_{1} =s_{1} + q^{2}(1 + 2 {a}_{1})$$ $$m_{i\bot}^{2} = m_{i}^{2} + k_{\bot}^{2} , i=1,2$$ and $$D_{2}=D_{1} ( 1\leftrightarrow 2 ), { \Delta }_{2}=
{\Delta }_{1} ( 1\leftrightarrow 2 )$$
Finally we obtain:
$$G_{D}^{E} (q^{2})=\frac{1}{{(4\pi )}^{3}}\int_{0}^{{\Lambda }_{k_{\bot }}}
d k^{2}_{ \bot}\int_{0}^{ 2\pi } d\phi
\int_{0}^{1} dx \frac{1}{x(1-x)} \times$$ $$\sum_{i=1,2 } \frac{ G G'}{(s_{i} -M_{D}^{2})({s'}_{i}- M_{D}^{2})}
\frac{e_{i}}{e_{D}} f_{i}(q^{2}) \times$$ $$D_{i}(s_{i}, {s'}_{i}, q^{2}){\Delta }_{i}
(s_{i}, {s'}_{i}, q^{2})$$
The eq.(10) was used in the calculation of the diquark form factors provided the normalization $ G_{D}^{E}(0)=1 $.
Conclusion {#sect:2}
==========
In the present paper in the framework of dispersion integration technique we investigate the behaviour of electric diquark form factors with $ J^{P}=
0^{+}$ at small and intermediate momentum transfer $ Q^{2} \leq
0.5~{\rm GeV}^{2}$. The charge radii values of nonstrange and strange scalar diquarks are calculated. The scalar diquark masses were calculated \[6\]: $ M(ud)=0.72$ GeV, $M(us)=M(ds)=0.86$ GeV. The quark masses are equal: $ m_{u}=m_{d}=0.385$ GeV, $m_{s}=0.510$ GeV. Analogously \[6\] we use the dimensionless pair energy cut-off parameter: $ \lambda =12.2 $, that allows us to define the momentum cut-offs: $ {\Lambda }_{ k_{\bot}}(qq)= 0.3~{\rm GeV}^{2}$, ${\Lambda }_{k_{\bot }}(qs)
=0.41~{\rm GeV}^{2} $, where $q=u,d$. We consider the interaction of constituent quark with electromagnetic field and take into account the nonstrange and strange quark form factors: $ f_{q}(q^2)= exp({\gamma}_q q^2)$, ${\gamma}_q =0.33~{\rm GeV}^{-2} $ and $ f_{s}(q^2)= exp({\gamma}_s q^2)$, ${\gamma}_s =0.2~{\rm GeV}^{-2} $ \[6\]. The behaviour of the scalar diquark electric form factors are shown in Fig.2. The calculated charge radii are equal:
$ {<{r}_{ud}^{2}>}^{\frac{1}{2}} = 0.55~fm$, ${<{r}_{us}^{2}>}^{\frac{1}{2}}
= 0.65~fm$, ${<{r}_{ds}^{2}>}^{\frac{1}{2}} = 0.5~fm $.
In the present paper electromagnetic properties of diquarks are investigated in the framework of the relativistic description. These values of scalar diquark charge radii are compatible with other results for the diquark effective radii \[8-11\] and experimental data \[1\]. In the papers \[8,9\] assuming soft symmetry breaking in the diquark sector, the bosonisation of a quasi-Goldstone ud-diquark is performed. In the chiral limit the ud-diquark mass and diquark charge radius are defined by the gluon condensate $ M_{ud}=300$ MeV, ${<{r}_{ud}^{2}>}^{\frac{1}{2}} \simeq 0.5~fm $. This model allows to explain the relatively low mass of the scalar diquark.
A approach is based on a local effective quark model, a Nambu-Jona-Lasinio model with a colour-octet current-current interaction \[10,11\]. One calculated the electromagnetic form factors of scalar and axial vector diquark bound states using the gauge-invariant proper-time regularization. In the paper \[11\] the scalar diquark masses $ M_{ud} $ and scalar diquark charge radii $ {<{r}_{ud}^{2}>}^{\frac{1}{2}} $ for different values of the effective diquark coupling constants are calculated. The scalar diquark charge radius $ {<{r}_{ud}^{2}>}^{\frac{1}{2}} $ is equal $(0.5-0.55)~fm $.
But the non-relativistic, QCD-based, potential quark model for the proton and neutron inevitably predicts a spin-0 diquark structure with a charge radius of the $ 0.35~fm $ or smaller \[12,13\]. Such conflict between model and experimental data might possibly as the influence of relativistic effects.
[Acknowledgements]{}
The authors would like to thank A.A. Andrianov, V.A. Franke, Yu.V. Novozhilov for useful discussions.
[15]{}
M. Anselmino et al., Rev. Mod. Phys. [**65**]{}, 1199 (1993)
S. Fredriksson, M. Jändel , T. Larsson, Z. Phys. C[**14**]{}, 35 (1982)
H.G. Dosch, M. Jamin, B. Stech, Z. Phys. C[**42**]{}, 167 (1989)
A. Donnachie, P.V. Landshoff, Phys. Lett. B[**95**]{}, 437 (1980)
B. Stech, Phys. Rev. D[**36**]{}, 975 (1987)
V.V. Anisovich, S.M. Gerasyuta, A.V. Sarantsev, Int. J. Mod. Phys. A[**6**]{}, 625 (1991)
V.V. Anisovich, A.V. Sarantsev, Sov. J. Part. Nucl. [**45**]{}, 1479 (1987)
Yu. Novozhilov, A. Pronko, D. Vassilevich, Phys. Lett. B[**321**]{}, 425 (1994)
Yu. Novozhilov, A. Pronko, D. Vassilevich, Phys. Lett. B[**343**]{}, 358 (1995)
A. Buck, R. Alkofer, H. Reinhardt, Phys. Lett. B[**286**]{}, 29 (1992)
C. Weiss et al., Phys. Lett. B[**312**]{}, 6 (1993)
I.M. Narodetski, Yu.A. Simonov, V.P. Yurov, Z. Phys. C[**55**]{}, 695 (1992)
S. Fredriksson, J. Hansson, S. Ekelin, Z. Phys. C[**75**]{}, 107 (1997)
0.6mm
(135.00,76.67) (25.00,30.00)[(1,0)[110.00]{}]{} (25.00,33.00)[(1,0)[25.00]{}]{} (50.00,33.00)[(1,1)[30.00]{}]{} (80.00,63.00)[(1,-1)[30.00]{}]{} (110.00,33.00)[(1,0)[25.00]{}]{} (48.33,30.00)
------------------------------------------------------------------------
(110.33,30.00)
------------------------------------------------------------------------
(110.00,30.00)[(-1,0)[30.00]{}]{} (50.00,33.00)[(1,1)[16.00]{}]{} (80.00,63.00)[(1,-1)[16.00]{}]{} (80.00,63.00)(75.67,67.00)(80.33,70.33) (80.00,70.33)(83.33,73.00)(80.33,76.67) (83.67,75.67)[(0,0)\[lc\][q]{}]{} (35.00,27.33)[(0,0)\[ct\][$P$]{}]{} (125.67,27.33)[(0,0)\[ct\][$P'$]{}]{} (49.33,27.00)[(0,0)\[ct\][${\gamma}_5 $]{}]{} (111.67,27.00)[(0,0)\[ct\][${\gamma}_5 $]{}]{} (76.00,63.00)[(0,0)\[rc\][${\gamma}_{\mu}$]{}]{} (61.67,48.00)[(0,0)\[rc\][$k_1$ ]{}]{} (99.00,48.00)[(0,0)\[lc\][${k'}_1$]{}]{} (80.00,26.67)[(0,0)\[ct\][$-k_2$]{}]{}
units <0.6mm,0.6mm> from 0 0 to 0 100 from 100 0 to 100 100 from 0 0 to 100 0 from 0 100 to 100 100
\[r\] at 0 0 \*10 0 10 / \[r\] at 0 0 \*2 0 50 / \[t\] at 0 0 \*10 10 0 / \[t\] at 0 0 \*2 50 0 /
\[t\] <0mm,-3mm> at 0 0 \[t\] <0mm,-3mm> at 50 0 \[t\] <0mm,-3mm> at 100 0 \[r\] <-3mm,0mm> at 0 0 \[r\] <-3mm,0mm> at 0 50 \[r\] <-3mm,0mm> at 0 100
\[lt\] at 0 112 \[lt\] at 108 0
\[lb\] <1mm,-0.5mm> at 60 50.6 \[lt\] <1mm,0.6mm> at 60 47 \[l\] <1mm,0mm> at 60 32.5
x from 0 to 100, y from 0 to 100 0 100 5 98.5 10 95 15 89.646 20 83.1 25 77.625 30 72.5 35 67.835 40 63.5 45 59.65 50 55.9 55 53.156 60 50.6 / 0 100 5 98.5 10 94.5 15 88.666 20 82.5 25 76.854 30 71.3 35 66.443 40 61.7 45 57.394 50 53.7 55 50.098 60 47 / 0 100 5 96.5 10 89.7 15 80.809 20 71.3 25 64.288 30 57.4 35 51.898 40 47.1 45 42.675 50 38.8 55 35.505 60 32.5 /
[^1]: Present address: Department of Physics, LTA, Institutski Per.5, St.Petersburg 194021, Russia
|
---
abstract: 'We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb R^d$. The result is based on a probabilistic version of the Falconer-Sloan condition $C(s)$ introduced in [@FS]. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.'
address:
- 'Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland'
- 'Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland'
- 'Department of Mathematics, South China University of Technology, Guangzhou, 510641, P.R. China'
- 'Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland'
- 'Department of Mathematics, Uppsala University, P.O. Box 480, 75106 Uppsala, Sweden'
author:
- Esa Järvenpää
- Maarit Järvenpää
- Bing Li
- Örjan Stenflo
title: 'Random affine code tree fractals and Falconer-Sloan condition'
---
[^1]
Introduction {#intro}
============
In the investigation of dimensional properties of self-similar and self-conformal sets an important tool is the thermodynamic formalism. There is a natural way to attach a pressure function to a self-similar or self-conformal iterated function system and, for example, the Hausdorff dimension and multifractal spectrum can be calculated using the pressure. Since the pressure is defined by an additive potential function, there are many tools available for the purpose of analysing it.
In his famous theorem from 1988 Falconer [@F88] proved that the dimension of any typical self-affine set is equal to the unique zero of the pressure function under the assumption that the norms of the linear parts are less than 1/3. Later Solomyak [@S] verified that 1/3 can be replaced by 1/2 which is the best possible bound, see [@PU]. The potential is defined by means of the singular value functions of the iterates of the linear parts, and contrary to the self-conformal setting, the potential $\phi$ is not additive. In the self-affine case $\phi$ is subadditive guaranteeing the existence of the pressure and its unique zero. However, $\phi$ is not superadditive – not even in the weak sense that $\phi(n+m)\ge\phi(n)+\phi(m)-C$ for some constant $C$. In many cases this causes severe problems, see for example [@BF], [@FM07], [@FS], [@Fe], [@FeSh], [@JJKKSS] and [@KS].
There are various ways to introduce randomness to the self-affine setting. In [@JPS] Jordan, Pollicott and Simon considered a fixed affine iterated function system with a small random perturbation in translations at each step of the construction. When investigating random subsets of self-affine attractors, Falconer and Miao [@FM09] selected at each step of the construction a random subfamily of the original function system independently. Both in [@JPS] and [@FM09] there is total independence both in space, that is, between different nodes at a fixed construction level, and in scale, meaning that once a node is chosen its descendants are chosen independently of the previous history. Such systems are called statistically self-affine, since the law controlling the construction is the same at every node. However, typical realisations are not self-affine. Inspired by the random $V$-variable fractals introduced by Barnsley, Hutchinson and Stenflo in [@BHS2005], a new class of random self-affine code tree fractals was proposed in [@JJKKSS]. In this class typical realisations mimic the self-affinity of deterministic iterated function systems. Moreover, the probability distributions have certain independence only in scale, and therefore, typical realisations are locally random but globally nearly homogeneous. In particular, the attractor is a finite union of self-affine copies of sets with arbitrarily small diameter. Thus typical realisations are close to deterministic self-affine sets. In a code tree fractal the linear parts of the iterated function system may depend on the construction step. For example, attractors of graph directed Markov systems generated by affine maps [@F], or more generally sub-self-affine sets [@F95], are code tree fractals.
In this paper we generalise the dimension results in [@JJKKSS] concerning random affine code tree fractals. In [@JJKKSS] the existence of the pressure was proven under quite general conditions (see Theorem \[pexists\]). However, when verifying the relation between the dimension and the zero of the pressure several additional assumptions were needed – the most restrictive one being that $d=2$. The main cause for the extra assumptions was the non-superadditivity of the potential defining the pressure. In the self-affine setting various approaches have been introduced to overcome the problems caused by the non-superadditivity of the potential. These include the cone condition [@BF], [@FM07], [@FeSh], [@KS], irreducibility [@Fe] and non-existence of parallelly mapped vectors [@JJKKSS]. In this paper we focus on a general condition (see Definition \[FSdef\]) introduced recently by Falconer and Sloan [@FS]. Under the Falconer-Sloan condition (for brevity, F-S condition) higher dimensional spaces can also be considered, see Theorem \[maintheorem\]. The only additional assumption compared to Theorem \[pexists\] is that some iterates of the system satisfy the F-S condition with positive probability.
The F-S condition is related to a family of linear maps on $\mathbb R^d$. The condition is open in the sense that the set of families of linear maps satisfying it is open in any natural topology. In this paper we also address a problem proposed by Falconer concerning the genericity of the F-S condition. In $\mathbb R^2$ the F-S condition is easy to check but in higher dimensional spaces the question is more delicate. It turns out that a family of linear maps $\{S_i\}_{i=1}^k$ on $\mathbb R^d$ does not satisfy the F-S condition unless $k$ is sufficiently large (see Remark \[FSremark\].(b)) – the minimal value of $k$ being much larger than $d$. However, in Corollary \[typical\] we prove that there exists a natural number $n$ depending only on $d$ such that for any generic family $\{S_i\}_{i=1}^k$ the family $\{S_{i_1}\circ\dots\circ S_{i_l}\mid i_j\in\{1,\dots,k\}\text{ for }
j=1,\dots,l\text{ and } 1\le l\le n\}$ satisfies the F-S condition. The set is generic both in the topological sense, that is, it is open and dense, and in the measure theoretic sense meaning that it has full Lebesgue measure. Theorem \[CsCm\] provides an explicit criterion guaranteeing that a family $\{S_i\}_{i=1}^k$ belongs to the generic set. In Remark \[suffcon\] we explain why the complement of this generic set is non-empty, that is, why Corollary \[typical\] is not valid for all families.
In many problems related to self-affine iterated function systems it is sufficient to study iterates of the maps. This is also the case in Theorem \[maintheorem\]. The applicability of the F-S condition is based on the fact that the upper bound $n$ for the number of iterates needed in order that the family $\{S_{i_1}\circ\dots\circ S_{i_l}\mid i_j\in\{1,\dots,k\}\text{ for }
j=1,\dots,l\text{ and } 1\le l\le n\}$ satisfies the F-S condition is a constant depending only on the dimension of the ambient space. In particular, Corollary \[typical\] implies that typical systems satisfy the assumptions of Theorem \[maintheorem\].
The paper is organised as follows. In Section \[FS\] we recall the Falconer-Sloan setting and prove that the F-S condition is valid for a family of iterates of a generic family (Corollary \[typical\]). Moreover, we give an explicit criterion implying that a family belongs to this generic set (Theorem \[CsCm\]). In Section \[codetree\] we recall the notation from [@JJKKSS] concerning random affine code tree fractals and prove that the dimension of a typical affine code tree fractal is given by the zero of the pressure (Theorem \[maintheorem\]).
Falconer-Sloan condition $C(s)$ {#FS}
===============================
In this section we consider the genericity of the F-S condition introduced in [@FS] for the purpose of overcoming problems caused by the fact that in the self-affine setting the natural potential defining the pressure (for definition see ) is not supermultiplicative. Intuitively, the reason behind the applicability of the F-S condition is as follows: Letting $A$ and $B$ be $d\times d$-matrices, the norm $\Vert AB\Vert$ may be much smaller than $\Vert A\Vert\cdot\Vert B\Vert$. This happens if the vector $v$ which determines the norm of $B$ is mapped by $B$ onto an eigenspace of $A$ which corresponds to some small eigenvalue of $A$. In the expression of the pressure (for $s=1$) there is a sum of terms of the form $\Vert A B\Vert$. The F-S condition guarantees that $\Vert A B\Vert$ is not much less than $\Vert A\Vert\cdot\Vert B\Vert$ simultaneously for all pairs $(A,B)$.
We begin by recalling the notion from [@FS]. For all $m\in\mathbb N$ with $0\le m\le d$ we denote by $\Lambda^m$ the $m$-th exterior power of $\mathbb R^d$ with the convention $\Lambda^0=\mathbb R$. An $m$-vector ${{\bf v}}\in\Lambda^m$ is [*decomposable*]{} if it can be written as ${{\bf v}}=v_1\wedge\dots\wedge v_m$ for some $v_1,\dots,v_m\in\mathbb R^d$. Let $\Lambda_0^m$ be the set of decomposable $m$-vectors. If $\{e_1,\dots,e_d\}$ is a basis of $\mathbb R^d$, then $\{e_{i_1}\wedge\dots\wedge e_{i_m}\mid 1\le i_1<\dots<i_m\le d\}$ is a basis of $\Lambda^m$. Supposing that $\{e_1,\dots,e_d\}$ is an orthonormal basis of $\mathbb R^d$, [*the Hodge star operator*]{} $*:\Lambda^m\to\Lambda^{d-m}$ is defined as the linear map satisfying $$*(e_{i_1}\wedge\dots\wedge e_{i_m})=e_{j_1}\wedge\dots\wedge e_{j_{d-m}}$$ for all $1\le i_1<\dots<i_m\le d$, where $1\le j_1<\dots<j_{d-m}\le d$ satisfy $\{i_1,\dots,i_m\}\cup\{j_1,\dots,j_{d-m}\}=\{1,\dots,d\}$. Let $\omega=e_1\wedge\dots\wedge e_d$ be the normalised volume form on $\mathbb R^d$. Recall that $\Lambda^d$ is one dimensional. We define the inner product $\langle\cdot\mid\cdot\rangle$ on $\Lambda^m$ by the (implicit) formula $$\langle{{\bf v}}\mid{{\bf w}}\rangle\omega={{\bf v}}\wedge*{{\bf w}}.$$ Then the inner product is independent of the choice of the orthonormal basis $\{e_1,\dots,e_d\}$, and moreover, $\{e_{i_1}\wedge\dots\wedge e_{i_m}\mid 1\le i_1<\dots<i_m\le d\}$ becomes an orthonormal basis of $\Lambda^m$. Any linear map $S:\mathbb R^d\to\mathbb R^d$ induces a linear map $S:\Lambda^m\to\Lambda^m$ such that $S(v_1\wedge\dots\wedge v_m)=Sv_1\wedge\dots\wedge Sv_m$ for all $v_1\wedge\dots\wedge v_m\in\Lambda_0^m$.
Now we are ready to recall the definition of the condition $C(s)$ from [@FS] – first for integer parameters and after that for non-integral parameters $s$.
\[FSdef\] Consider a family $\{S_i:\mathbb R^d\to\mathbb R^d\}_{i\in I}$ consisting of linear maps. Let $m\in\mathbb N$ with $0\le m\le d$. The family $\{S_i\}_{i\in I}$ satisfies [*condition $C(m)$*]{} if for all $0\ne{{\bf v}},{{\bf w}}\in\Lambda_0^m$ there is $i\in I$ such that $\langle S_i{{\bf v}}\mid{{\bf w}}\rangle\ne 0$. Let $0<s<d$ be non-integral and let $m$ be the integer part of $s$. The family $\{S_i\}_{i\in I}$ satisfies [*condition $C(s)$*]{} if for all $0\ne{{\bf v}},{{\bf w}}\in\Lambda_0^m$ and $0\ne{{\bf v}}\wedge v,{{\bf w}}\wedge w\in\Lambda_0^{m+1}$ there is $i\in I$ such that $\langle S_i{{\bf v}}\mid{{\bf w}}\rangle\ne 0$ and $\langle S_i({{\bf v}}\wedge v)\mid{{\bf w}}\wedge w\rangle\ne 0$.
\[FSremark\] (a) The family $\{S_i\}_{i\in I}$ satisfies condition $C(m)$ if and only if for all $0\ne{{\bf v}}\in\Lambda_0^m$ the set $\{S_i{{\bf v}}\mid i\in I\}$ spans $\Lambda^m$. Here the if-part is clear whereas the only if -part involves a slight subtilty. Indeed, Definition \[FSdef\] deals with decomposable vectors and $\Lambda_0^m$ is not a vector space when $m\notin\{0,1,d-1,d\}$. For the only if -part, assume that there exists $0\ne{{\bf v}}\in\Lambda_0^m$ such that the set $\{S_i{{\bf v}}\mid i\in I\}$ does not span $\Lambda^m$. Letting $k$ be the maximal number of linearly independent vectors in $\{S_i{{\bf v}}\mid i\in I\}$, we have $k<\binom dm=\dim\Lambda^m$. Denote these vectors by ${{\bf w}}^1,\dots,{{\bf w}}^k$ and consider $i=1,\dots,k$. Now ${{\bf u}}=u_1\wedge\dots\wedge u_m\in\Lambda_0^m$ is perpendicular to ${{\bf w}}^i=w_1^i\wedge\dots\wedge w_m^i$, if and only if the vectors $P_iu_1,\dots,P_iu_m$ are linearly dependent. Here $P_i$ is the orthogonal projection onto the $m$-dimensional linear subspace spanned by $w_1^i,\dots,w_m^i$. Using the notation $B$ for the $m\times m$-matrix whose columns are the vectors $P_iu_1,\dots,P_iu_m$ expressed in the basis $\{w_1^i,\dots,w_m^i\}$, we observe that the vectors $P_iu_1,\dots,P_iu_m$ are linearly dependent, if and only if the determinant of $B$ is zero. This implies the existence of a polynomial map $Q:\mathbb R^{d^m}\to\mathbb R$ such that $\langle{{\bf u}}\mid{{\bf w}}^i\rangle=0$ if and only if $Q(u_1,\dots,u_m)=0$. This, in turn, gives that for all $i=1,\dots,k$ the set $$M_i=\{(u_1,\dots,u_m)\in\mathbb R^{d^m}\mid\langle{{\bf u}}\mid{{\bf w}}^i\rangle=0\}$$ has codimension 1, and clearly, $0\in M_i$. Note that ${{\bf u}}=u_1\wedge\dots\wedge u_m=0$ if and only if the vectors $u_1,\dots,u_m$ are linearly dependent, that is, all the $m\times m$-minors are zero for the $d\times m$-matrix whose columns are the vectors $u_1,\dots,u_m$. Since there are $\binom dm$ such minors and $k<\binom dm$, there exists $\overline{{\bf u}}=(\overline u_1,\dots,\overline u_m)\in\cap_{i=1}^kM_i$ such that $\overline{{\bf u}}\ne 0$. In particular, $\langle\overline{{\bf u}}\mid{{\bf w}}^i\rangle=0$ for all $i=1,\dots,k$. Therefore condition $C(m)$ is not satisfied.
\(b) From (a) we see that there must be at least $\binom dm$ maps in the family $\{S_i\}_{i\in I}$ for condition $C(m)$ to be satisfied. Note that when $d$ is large and $1<m<d-1$ the number $\binom dm$ is much larger than $d$.
\(c) If $m<s<m+1$ and $\{S_i\}_{i\in I}$ satisfies condition $C(s)$ then it satisfies condition $C(t)$ for all $m\le t\le m+1$. In [@FS Lemma 2.6] it is shown that the irreducibility condition used by Feng in [@Fe] is (essentially) equivalent to the condition $C(1)$.
We proceed by introducing the notation needed for studying the validity of the F-S condition. Let $F,G:\mathbb R^d\to\mathbb R^d$ be linear mappings with $d$ different real eigenvalues $\{\lambda_1,\dots,\lambda_d\}$ and $\{t_1,\dots,t_d\}$, respectively. Let $\{\hat e_1,\dots,\hat e_d\}$ and $\{\tilde e_1,\dots,\tilde e_d\}$ be the corresponding normalised eigenvectors. We assume that for all $k=1,\dots,d$ $$\label{eigenspacecondition}
\begin{split}
&\lambda_{i_1}\cdots\lambda_{i_k}\ne\lambda_{j_1}\cdots\lambda_{j_k}\text{ and }
t_{i_1}\cdots t_{i_k}\neq t_{j_1}\cdots t_{j_k}\text{ for all pairs }\\
& (i_1,\dots,i_k)\neq (j_1,\dots,j_k).
\end{split}$$ Let $A=A(F,G):\mathbb R^d\to\mathbb R^d$ be the linear map satisfying $\tilde e_i=A^{-1}e_i$, that is, $e_i=A\tilde e_i$ for all $1\le i\le d$. Let $\mathcal S_k=\mathcal S_k(F,G)$ be the family of compositions of $F$ and $G$ up to level $k$, that is, $$\label{Skdef}
\mathcal S_k=\{T_1\circ\cdots\circ T_j\mid 1\le j\le k\text{ and }
T_i\in\{F,G\} \text{ for all } 1\le i\le j\}.$$ Using the eigenbasis $\{\hat e_1,\dots,\hat e_d\}$ of $F$ as the basis of $A$, we view $A$ as a $d\times d$-matrix. Denote by $\mathcal M_d$ the class of $d\times d$-matrices whose minors are all non-zero.
With the above notation we prove two lemmas.
\[takingall\] Let $1\le m\le d$ and $A\in\mathcal M_d$ be as above. For all $1\le i_1<\cdots<i_m\le d$ write $$\label{representation}
\hat e_{i_1}\wedge\cdots\wedge\hat e_{i_m}=\sum_{1\le j_1<\cdots<j_m\le d}
c_{i_1\cdots i_m}^{j_1\cdots j_m}\,\tilde e_{j_1}\wedge\cdots\wedge\tilde e_{j_m}.$$ Then $c_{i_1\cdots i_m}^{j_1\cdots j_m}\neq 0$ for all $(i_1,\dots, i_m)$ and $(j_1,\dots, j_m)$.
We denote the set of all permutations of $(j_1,\dots,j_m)$ by $\operatorname{Per}(j_1,\dots,j_m)$ and write $\operatorname{sgn}(\sigma)$ for the sign of a permutation $\sigma\in\operatorname{Per}(j_1,\dots,j_m)$. Since $\hat e_{i_l}=\sum_{j=1}^dA_{j i_l}\tilde e_j$ for all $1\le l\le m$ and the wedge product is antisymmetric and multilinear, we have $$\begin{aligned}
\hat e_{i_1}\wedge\cdots\wedge\hat e_{i_m}&=\sum_{j_1=1}^d\dots\sum_{j_m=1}^d
A_{j_1i_1}\cdots A_{j_mi_m}\tilde e_{j_1}\wedge\cdots\wedge\tilde e_{j_m}\\
&=\sum_{1\le j_1<\cdots<j_m\le d}\bigl(\sum_{\sigma\in\operatorname{Per}(j_1,\dots,j_m)}
\operatorname{sgn}(\sigma)A_{\sigma_1i_1}\cdots A_{\sigma_mi_m}\bigr)\tilde e_{j_1}\wedge\cdots
\wedge\tilde e_{j_m}\\
&=c_{i_1\cdots i_m}^{j_1\cdots j_m}\,\tilde e_{j_1}\wedge\cdots\wedge\tilde e_{j_m}.\end{aligned}$$ Thus the coefficient $c_{i_1\cdots i_m}^{j_1\cdots j_m}$ is the minor of $A$ determined by the columns $i_1,\dots,i_m$ and rows $j_1,\dots,j_m$, and by the definition of $\mathcal M_d$, we have $c_{i_1\cdots i_m}^{j_1\cdots j_m}\ne 0$.
For all $1\le m\le d$, define $$B_1=\{\hat e_{i_1}\wedge\cdots\wedge\hat e_{i_m}\mid 1\le i_1<\dots<i_m\le d\}$$ and $$B_2=\{\tilde e_{j_1}\wedge\cdots\wedge\tilde e_{j_m}\mid 1\le j_1<\dots<j_m
\le d\}.$$ Then $B_1$ and $B_2$ are bases of $\Lambda^m$. Furthermore, the elements of $B_1$ and $B_2$ are the eigenvectors of $F:\Lambda^m\to\Lambda^m$ and $G:\Lambda^m\to\Lambda^m$ with eigenvalues $\lambda_{i_1}\cdots\lambda_{i_m}$ and $t_{j_1}\cdots t_{j_m}$, respectively.
\[Vandermonde\] Let $a_1,\dots,a_d\in\mathbb R\setminus\{0\}$ with $a_i\ne a_j$ for $i\ne j$ and let $v=(v_1,\dots,v_d)\in\mathbb R^d$ with $v_i\ne 0$ for all $i=1,\dots,d$. Denoting by $(a_j)^i$ the $i$-th power of $a_j$, it follows from the Vandermonde determinant formula that the vectors $\{((a_1)^iv_1,\dots,(a_d)^iv_d)\mid i=k,\dots,k+d-1\}$ span $\mathbb R^d$ for all $k\in\mathbb N$. By induction it is easy to see that the vectors $$\{v^{i_j}=((a_1)^{i_j}v_1,\dots,(a_d)^{i_j}v_d)\mid j=1,\dots,d\text{ and }
i_1<\dots<i_d\}$$ span $\mathbb R^d$. Indeed, the case $d=1$ is obvious. Assuming that the claim is true for $d$, we show that the vectors $\{v^{i_1},\dots,v^{i_{d+1}}\}$ span $\mathbb R^{d+1}$. Suppose to the contrary that this is not the case, that is, there is $j$ such that $v^{i_j}=\sum_{k\ne j}\alpha_kv^{i_k}$. For all $k=1,\dots,d+1$ we denote by $\Pi_k:\mathbb R^{d+1}\to\mathbb R^d$ the projection which omits the $k^{\text{th}}$ coordinate. Fix $l\ne j$. Now the induction hypothesis implies that $\Pi_jv^{i_j}=\sum_{k\ne j}b_k\Pi_jv^{i_k}$ and $\Pi_lv^{i_j}=\sum_{k\ne l}c_k\Pi_lv^{i_k}$ where the coefficients $b_k$ and $c_k$ are unique. Since $a_l\ne a_j$, we have $b_k\ne c_k$ for some $k\ne j$. On the other hand, $\Pi_jv^{i_j}=\sum_{k\ne j}\alpha_k\Pi_jv^{i_k}$ and $\Pi_lv^{i_j}=\sum_{k\ne l}\alpha_k\Pi_lv^{i_k}$, and therefore, $\alpha_k=b_k=c_k$ for all $k\ne j$ which is a contradiction.
\[nonzerocoordinates\] Let $0\ne{{\bf v}}\in\Lambda^m$ and $n=\binom dm$. Then there are at most $n(n-1)$ numbers $i\in\mathbb N$ with the property that at least one coordinate of $F^i{{\bf v}}$ with respect to the basis $B_2$ is equal to zero.
Let ${{\bf v}}=(v_1,\dots,v_n)$ be the coordinates of ${{\bf v}}$ with respect to the basis $B_1$ and let $k$ be the number of non-zero coordinates. We denote by $V_{{{\bf v}}}$ the $k$-dimensional plane spanned by those basis vectors in $B_1$ that correspond to the non-zero coordinates of ${{\bf v}}$. Let $\gamma_1,\dots,\gamma_n$ be the eigenvalues of $F:\Lambda^m\to\Lambda^m$. Observe that for the $i$-th iterate $F^i$ of $F$ we have $F^i{{\bf v}}=(\gamma_1^iv_1,\dots,\gamma_n^iv_n)$. Combining with Remark \[Vandermonde\], implies that the set $\{F^{i_1}{{\bf v}},\dots,F^{i_k}{{\bf v}}\}$ spans $V_{{{\bf v}}}$ for all natural numbers $i_1<i_2<\cdots<i_k$. For all $j=1,\dots,n$, let $$W_j=\{{{\bf w}}\in\Lambda^m\mid{{\bf w}}=(w_1,\dots,w_n)\text{ with respect to }B_2
\text{ and }w_j=0\}.$$ Applying Lemma \[takingall\] gives for all $j=1,\dots,n$ and $1\le i_1<\dots<i_m\le d$ that $\hat e_{i_1}\wedge\cdots\wedge\hat e_{i_m}\notin W_j$. Thus the dimension of $V_{{\bf v}}\cap W_j$ is strictly less than $k$. We conclude that for all $j=1,\dots,n$, there are at most $k-1$ indices $i$ such that $F^i{{\bf v}}\in W_j$, and therefore, there are at most $n(k-1)$ indices $i$ such that $F^i{{\bf v}}\in W_j$ for some $j=1,\dots,n$. Since this is true for all $1\le k\le n$, the claim follows.
Now we are ready to prove our main theorem in this section. For this purpose, set $n_0=\max\limits_{0\le m\le d}\binom dm$. After proving Corollary \[typical\], we discuss the criterion which is based on the following theorem and gives a sufficient condition for the validity of the F-S condition (see Remark \[suffcon\]).
\[CsCm\] Let $F$ and $G$ be as in and assume that $A=A(F,G)\in\mathcal M_d$. Then the family $\mathcal S_{2n_0^2}$ defined in satisfies the condition $C(s)$ for all $0\le s\le d$.
By Remark \[FSremark\] (c) it is enough to prove that the family $\mathcal S_{2n_0^2}$ satisfies the condition $C(s)$ for non-integral $s$. Letting $m$ be the integer part of $s$, set $n_1=\binom dm$ and $n_2=\binom d{m+1}$ and define $M=n_1(n_1-1)+n_2(n_2-1)+1$ and $N=n_1+n_2-1$. Let $0\neq{{\bf v}},{{\bf w}}\in\Lambda^m$ and $0\neq{{\bf u}},{{\bf z}}\in\Lambda^{m+1}$. By applying Lemma \[nonzerocoordinates\] to the iterates $F^i{{\bf v}}$ and $F^i{{\bf u}}$, where $1\le i\le M$, we deduce that there exists $1\le i_0\le M$ such that all coordinates of the iterates $F^{i_0}{{\bf v}}$ and $F^{i_0}{{\bf u}}$ with respect to the basis $B_2$ are non-zero. Furthermore, from Remark \[Vandermonde\] we see that for all $j_1<\dots<j_{n_1}$ the vectors $G^{j_1}(F^{i_0}{{\bf v}}),\dots,G^{j_{n_1}}(F^{i_0}{{\bf v}})$ span $\Lambda^m$. Hence, there are at least $N-n_1+1$ indices $j=1,\dots,N$ such that the points $G^j(F^{i_0}{{\bf v}})$ do not belong to the orthogonal complement ${{\bf w}}^\perp$ of ${{\bf w}}$. A similar argument implies that among these $N-n_1+1$ indices there exists $j_0$ such that $G^{j_0}(F^{i_0}{{\bf u}})\notin{{\bf z}}^\perp$, and therefore, $$\langle G^{j_0}F^{i_0}{{\bf v}}\mid{{\bf w}}\rangle\ne 0\text{ and }
\langle G^{j_0}F^{i_0}{{\bf u}}\mid{{\bf z}}\rangle\ne 0$$ implying that $\mathcal S_{M+N}$ satisfies $C(s)$. Since $M+N\le 2n_0^2$ this completes the proof of the claim.
Let $k\in\mathbb N$. We identify the space of families $\mathcal F=\{S_i:\mathbb R^d\to\mathbb R^d\}_{i=1}^k$ of linear maps with $\mathbb R^{d^2k}$. For $\mathcal F\in\mathbb R^{d^2k}$ define $$\mathcal S_l(\mathcal F)=\{S_{i_1}\circ\cdots\circ S_{i_j}\mid 1\le j\le l
\text{ and }S_{i_m}\in\mathcal F\text{ for all } 1\le m\le j\}.$$ With this notation we have the following consequence of Theorem \[CsCm\].
\[typical\] Letting $k\ge 2$ be a natural number, the set $$\mathcal C=
\{\mathcal F\in\mathbb R^{d^2k}\mid\mathcal S_{2n_0^2}(\mathcal F)
\text{ satisfies }C(s)\text{ for all }0\le s\le d\}$$ is open, dense and has full Lebesgue measure. More precisely, $\mathbb R^{d^2k}\setminus\mathcal C$ is contained in a finite union of $(d^2k-1)$-dimensional algebraic varieties.
We start with an easy observation: assuming that $\mathcal F\subset\mathcal G$ are families of linear maps on $\mathbb R^d$ and $\mathcal F$ satisfies condition $C(s)$, then $\mathcal G$ satisfies it too. Thus it is enough to prove the claim in the case $k=2$. The set of $d\times d$-matrices with a fixed non-zero minor is a $(d^2-1)$-dimensional algebraic variety. Since the number of minors is finite, the set $\mathbb R^{d^2}\setminus\mathcal M_d$ can be represented as a finite union of $(d^2-1)$-dimensional algebraic varieties, implying that $\mathcal M_d\subset\mathbb R^{d^2}$ is open, dense and has full Lebesgue measure. Moreover, note that the set of pairs $(F,G)$ of linear maps having $d$ real eigenvalues and not satisfying is a finite union of $(2d^2-1)$-dimensional algebraic varieties. Thus the set of pairs $(F,G)$ satisfying the assumptions of Theorem \[CsCm\] is open and has positive Lebesgue measure. For the purpose of verifying that $\mathcal C$ is dense and has full Lebesgue measure, we need to extend our argument to the case where $F$ and $G$ are allowed to have complex eigenvalues satisfying .
Recall that if $\lambda=re^{i\theta}$ is a complex eigenvalue of $F$, also $\overline\lambda=re^{-i\theta}$ is an eigenvalue of $F$, and there is a two dimensional invariant subspace $V\subset\mathbb R^d$ where $F$ acts as the rotation by angle $\theta$ composed with scaling by $r$. Let $e_1, e_2\in\mathbb R^d$ be such that $V$ is spanned by $e_1$ and $e_2$ and let $e_3$ be an eigenvector of $F$ corresponding to a real eigenvalue $t$. Then $e_3\wedge e_1$ and $e_3\wedge e_2$ span an eigenspace of $F$ on $\Lambda^2$ corresponding to the eigenvalue $t\lambda$. If $\rho$ is another complex eigenvalue of $F$ and $e_4$ and $e_5$ span the corresponding eigenspace, then $e_1\wedge e_2$ and $e_4\wedge e_5$ are eigenvectors of $F$ on $\Lambda^2$ with eigenvalues $\lambda\overline\lambda$ and $\rho\overline\rho$, respectively. The 4-dimensional subspace spanned by $\{e_1\wedge e_4,e_1\wedge e_5,e_2\wedge e_4,e_2\wedge e_5\}$ is divided into two invariant 2-dimensional subspaces corresponding to the complex eigenvalues $\lambda\rho$ and $\lambda\overline\rho$. By , the numbers $\lambda\overline\lambda,\rho\overline\rho,\lambda\rho$ and $\lambda\overline\rho$ are different. In this way we find a basis of $\Lambda^m$ consisting of eigenvectors of $F$. Since the Vandermonde determinant formula applies also for complex entries, Theorem \[CsCm\] is valid for an open dense set of pairs of linear maps $(F,G)$ having full Lebesgue measure. This completes the proof.
\[suffcon\] (a) Let $\mathcal F=\{T_i:\mathbb R^d\to\mathbb R^d\}_{i=1}^m$ be an iterated function system consisting of affine mappings $T_i(x)=S_i(x)+a_i$. When considering the validity of the F-S condition, the translation parts $a_i$ play no role. From Theorem \[CsCm\] and Corollary \[typical\] we conclude that if there are $i\ne j$ such that the eigenvalues of $S_i$ and $S_j$ satisfy and the eigenvectors of $S_i$ are mapped to those of $S_j$ by some $A\in\mathcal M_d$ then $\mathcal S_{2n_0^2}(\mathcal F)$ satisfies the condition $C(s)$ for all $0\le s\le d$.
\(b) Let $\mathcal F$ be as in remark (a). If $\mathcal F$ is not irreducible, that is, if there exists a non-trivial proper subspace $V\subset\mathbb R^d$ satisfying $S_i(V)\subset V$ for all $i=1,\dots,m$, then by Remark \[FSremark\].(a) the family $\mathcal S_N(\mathcal F)$ does not satisfy the condition $C(s)$ for any $0<s<d$ and for any $N\in\mathbb N$.
Random affine code tree fractals {#codetree}
================================
In this section we consider the Falconer-Sloan setting for a class of random affine code tree fractals introduced in [@JJKKSS] which are locally random but globally nearly homogeneous. It turns out that the earlier results in [@JJKKSS] can be improved under a probabilistic version of the condition $C(s)$. We begin by recalling the notation from [@JJKKSS].
Let $\mathcal F=\{F^\lambda=\{f_1^\lambda,\dots,f_{M_\lambda}^\lambda\}
\mid\lambda\in\Lambda\}$ be a family of iterated function systems on $\mathbb R^d$. Here the index set $\Lambda$ is a topological space. Assume that for all $i=1,\dots,M_\lambda$ the maps $f_i^\lambda\colon\mathbb R^d\to\mathbb R^d$ are affine, that is, $f_i^\lambda(x)= T_i^\lambda(x)+a_i^\lambda$, where $T_i^\lambda$ is a non-singular linear mapping and $a_i^\lambda\in\mathbb R^d$. We consider the case where the norms and the numbers of the maps are uniformly bounded meaning that $$\sup_{\lambda\in\Lambda,i=1,\dots,M_\lambda}\Vert T_i^\lambda\Vert<1,
\sup_{\lambda\in\Lambda,i=1,\dots,M_\lambda}|a_i^\lambda|<\infty\text{ and }
M=\sup_{\lambda\in\Lambda}M_\lambda<\infty.$$ Identifying $F^\lambda$ with an element of $\mathbb R^{(d^2+d)M_\lambda}$, gives $\mathcal F\subset\bigcup_{i=1}^M\mathbb R^{(d^2+d)i}$, where the union is disjoint. We equip $\bigcup_{i=1}^M\mathbb R^{(d^2+d)i}$ with the natural topology and assume that $\lambda\mapsto F^\lambda$ is a Borel map. Similarly, the linear parts $T_i^\lambda$ are embedded in $\mathbb R^{d^2M_\lambda}$.
We continue by introducing the concept of a code tree which is a modification of the standard tree construction of the attractor of an iterated function system. Indeed, instead of using the same family of maps at each construction step, different families with different numbers of maps are allowed in a code tree. Setting $I=\{1,\dots,M\}$, the length of a word $\tau\in I^k$ is $|\tau|=k$. Consider a function $\omega\colon\bigcup_{k=0}^\infty I^k\to\Lambda$, where $I^0=\{\emptyset\}$. We associate to $\omega$ a natural tree rooted at $\emptyset$ as follows: Let $\Sigma^{\omega}_{*}\subset\bigcup_{k=0}^\infty I^k$ be the unique set satisfying the following conditions:
- $\emptyset\in\Sigma^\omega_*$,
- if $i_1\cdots i_k\in\Sigma^\omega_*$ and $\omega(i_1\cdots i_k)=\lambda$, then $i_1\cdots i_kl\in\Sigma^\omega_*$ if and only if $l\leq M_\lambda$,
- if $i_1\cdots i_k\notin\Sigma^\omega_*$, then for all $l$ we have $i_1\cdots i_kl\notin\Sigma^\omega_*$.
The function $\omega$ restricted to $\Sigma^\omega_*$ is called an [*$\mathcal F$-valued code tree*]{} and the set of all $\mathcal F$-valued code trees is denoted by $\Omega$. Note that in a code tree the vertex $i_1\cdots i_k$ may be identified with the function system $F^{\omega(i_1\cdots i_k)}$, and moreover, the edge connecting $i_1\cdots i_k$ to $i_1\cdots i_kl$ may be identified with the map $f_l^{\omega(i_1\cdots i_k)}$. [*A sub code tree*]{} of a code tree $\omega$ is the restriction of $\omega$ to a subset $B\subset\Sigma_*^\omega$, where $B$ is rooted at some vertex $i_1\cdots i_k\in\Sigma_*^\omega$ and $B$ contains all descendants of $i_1\cdots i_k$ which belong to $\Sigma_*^\omega$. We endow $\Omega$ with the topology generated by the sets $$\{\omega\in\Omega\mid\Sigma_*^\omega\cap\bigcup_{j=0}^k I^j=J\text{ and }
\omega({{\bf i}})\in U_{{\bf i}}\text{ for all }{{\bf i}}\in J\},$$ where $k\in\mathbb N$, $U_{{\bf i}}\subset\Lambda$ is open for all ${{\bf i}}\in J$ and $J\subset\bigcup_{j=0}^k I^j$ is a tree rooted at $\emptyset$ and having all leaves in $I^k$. With this topology functions $\omega_1$ and $\omega_2$ are “close” to each other if their supports $\Sigma_*^{\omega_1}$ and $\Sigma_*^{\omega_2}$ agree up to the level $k$ and the values $\omega_1({{\bf i}})$ and $\omega_2({{\bf i}})$ are “close” to each other for all words ${{\bf i}}$ with $|{{\bf i}}|\le k$.
We equip $I^{\mathbb N}$ with the product topology. For each code tree $\omega\in\Omega$, define $$\Sigma^\omega=\{{{\bf i}}=i_1i_2\cdots\in I^{\mathbb N}\mid
i_1\cdots i_n\in\Sigma^\omega_*\text{ for all }n\in\mathbb N\}.$$ Then $\Sigma^\omega$ is compact. For all $k\in\mathbb N$ and ${{\bf i}}\in\Sigma^\omega\cup\bigcup_{j=k}^\infty I^j$, let ${{\bf i}}_k=i_1\cdots i_k$ be the initial word of ${{\bf i}}$ with length $k$. We use the following type of natural abbreviations for compositions: $$f^\omega_{{{\bf i}}_k}=f_{i_1}^{\omega(\emptyset)}\circ f_{i_2}^{\omega(i_1)}
\circ\dotsb\circ f_{i_k}^{\omega(i_1\cdots i_{k-1})}\text{ and }
T_{{{\bf i}}_k}^\omega=T_{i_1}^{\omega(\emptyset)}T_{i_2}^{\omega(i_1)}\cdots
T_{i_k}^{\omega(i_1\cdots i_{k-1})}.$$ Observe that, by the definition of the topology on $\Omega$, the maps $\omega\mapsto f^\omega_{{{\bf i}}_k}$ and $\omega\mapsto T_{{{\bf i}}_k}^\omega$ are Borel measurable. The code tree fractal corresponding to $\omega\in\Omega$ is $A^\omega =\{ Z^\omega({{\bf i}})\mid {{\bf i}}\in\Sigma^\omega\}$, where $Z^{\omega}({{\bf i}})=\lim_{k\to\infty}f^\omega_{{{\bf i}}_k}(0)$. Note that the attractor $A^\omega$ is well-defined since the maps $f_i^\lambda$ are uniformly contracting and the translation vectors $a_i^\lambda$ belong to a bounded set. For $k\in\mathbb N$, $\omega\in\Omega$ and ${{\bf i}}\in\Sigma^\omega$, the [*cylinder of length $k$ determined by ${{\bf i}}$*]{} is $$[{{\bf i}}_k]=\{{{\bf j}}\in\Sigma^\omega\mid j_l=i_l\text{ for all }l=1,\dots,k\}.$$
Next we introduce the concept of a neck level which is an essential feature of our model. The existence of neck levels guarantees that in our setting the attractor is globally nearly homogeneous. In fact, if $N_m\in\mathbb N$ is a neck level of $\omega$, then all the sub code trees of $\omega$ rooted at vertices ${{\bf i}}\in\Sigma_*^\omega$ with $|{{\bf i}}|=N_m$ are identical. In particular, the attractor $A^\omega$ is a finite union of affine copies of the attractor of the common sub code tree. Neck levels play an important role in the study of $V$-variable fractals, see for example [@BHS2005], [@BHS2008] and [@BHS12].
[*A neck list*]{} $N=(N_m)_{m\in\mathbb N}$ is an increasing sequence of natural numbers. Let $\widetilde\Omega$ be the set of $(\omega,N)\in\Omega\times\mathbb N^{\mathbb N}$ satisfying
- $N_m<N_{m+1}$ for all $m\in\mathbb N$ and
- if ${{\bf i}}_{N_m}{{\bf j}}_l,{{\bf i}}'_{N_m}\in\Sigma_*^\omega$, then ${{\bf i}}'_{N_m}{{\bf j}}_l\in\Sigma_*^\omega$ and $\omega({{\bf i}}_{N_m}{{\bf j}}_l)=\omega({{\bf i}}'_{N_m}{{\bf j}}_l)$.
The first condition means that $N$ is a neck list and the second condition guarantees that the sub code trees rooted at a certain neck level are identical. [*A shift* ]{} $\Xi\colon\widetilde\Omega\to\widetilde\Omega$ is defined by means of neck levels, that is, $\Xi(\omega,N)=(\hat\omega,\hat N)$, where $\hat N_m=N_{m+1}-N_1$ and $\hat\omega({{\bf j}}_l)=\omega({{\bf i}}_{N_1}{{\bf j}}_l)$ for all $m,l\in\mathbb N$. We denote the elements of $\widetilde\Omega$ by $\tilde\omega$, and for all $i\in\mathbb N$ we write $N_i(\tilde\omega)=N_i$ for the projection of $\tilde\omega=(\omega,N)$ onto the $i$-th coordinate of $N$. Moreover, on $\widetilde\Omega$ we use the topology generated by the [*cylinders*]{} $$\begin{aligned}
[(\omega,N)_m]=\{(\hat\omega,\hat N)\in\widetilde\Omega &\mid
\hat N_i=N_i\text{ for all }i\le m\text{ and }\hat\omega(\tau)=\omega(\tau)\\
&\text{ for all }\tau\text{ with } |\tau|<N_m\}.\end{aligned}$$ For any function $\phi$ of $\omega$ we use the notation $\phi(\tilde\omega)$ to view $\phi$ as a function of $\tilde\omega$. Finally for all $n<m\in\mathbb N\cup\{0\}$, let $$\Sigma_*^{\tilde\omega}(n,m)=\{i_{N_n+1}\cdots i_{N_m}\mid{{\bf i}}_{N_n}i_{N_n+1}\cdots
i_{N_m}\in\Sigma_*^{\tilde\omega}\},$$ where $N_0=0$.
For the purpose of defining the pressure, we proceed by recalling the notation from [@F88]. Let $T:\mathbb R^d\to\mathbb R^d$ be a non-singular linear mapping and let $$0<\sigma_d\leq\sigma_{d-1}\leq\dots\leq\sigma_2\leq\sigma_1=\Vert T\Vert$$ be the singular values of $T$, that is, the lengths of the semi-axes of the ellipsoid $T(B(0,1))$, where $B(x,\rho)\subset\mathbb R^d$ is the closed ball with radius $\rho>0$ centred at $x\in\mathbb R^d$. We define the [*singular value function*]{} by $$\Phi^s(T)
=\begin{cases}\sigma_1\sigma_2\cdots\sigma_{m-1}\sigma_m^{s-m+1},&
\text{if } 0\le s\le d,\\
\sigma_1\sigma_2\cdots\sigma_{d-1}\sigma_d^{s-d+1},&\text{if }s>d,
\end{cases}$$ where $m$ is the integer such that $m-1\le s<m$. The singular value function is submultiplicative, that is, $$\Phi^s(TU)\le\Phi^s(T)\Phi^s(U)$$ for all linear maps $T,U:\mathbb R^d\to\mathbb R^d$. For further properties of the singular value function see for example [@F88]. We assume that there exist $\underline\sigma,\overline\sigma\in (0,1)$ such that $$0<\underline\sigma\le\sigma_d(T_i^\lambda)\le\sigma_1(T_i^\lambda)
\le\overline\sigma<1$$ for all $\lambda\in\Lambda$ and for all $i=1,\dots,M_\lambda$. Note that, whilst the condition $\overline\sigma<1$ follows from the uniform contractivity assumption, the existence of $\underline\sigma>0$ is an additional assumption.
For all $k\in\mathbb N$ and $s\ge 0$, let $$S^{\tilde\omega}(k,s)=\sum_{{{\bf i}}_k\in\Sigma_*^{\tilde\omega}}
\Phi^s(T_{{{\bf i}}_k}^{\tilde\omega}).$$ The [*pressure*]{} is defined as follows $$\label{pressure}
p^{\tilde\omega}(s)=\lim_{k\to\infty}\frac{\log S^{\tilde\omega}(k,s)}k$$ provided that the limit exists. Since $T\mapsto\Phi^s(T)$ is a continuous function, the map $\tilde\omega\mapsto p^{\tilde\omega}(s)$ is Borel measurable.
According to the following theorem, the pressure exists and has a unique zero for typical random affine code tree fractals.
\[pexists\] Assume that $P$ is an ergodic $\Xi$-invariant Borel probability measure on $\widetilde\Omega$ such that $\int_{\widetilde\Omega}N_1(\tilde\omega)\,dP(\tilde\omega)<\infty$. Then for $P$-almost all $\tilde\omega\in\widetilde\Omega$ the pressure $p^{\tilde\omega}(s)$ exists for all $s\in[0,\infty[$. Furthermore, $p^{\tilde\omega}$ is strictly decreasing and there exists a unique $s_0$ such that $p^{\tilde\omega}(s_0)=0$ for $P$-almost all $\tilde\omega\in\widetilde\Omega$.
See [@JJKKSS Theorem 4.3].
In [@JJKKSS Remark 2.1] it was shown that any compact subset of the attractor of an iterated function system is a code tree fractal and, in particular, any sub-self-affine set is a code tree fractal. While verifying this, one ends up studying subsystems of the original iterated function system. For example, suppose that $F^1=\{f_1,f_2,f_3\}$ and let $F^2=\{f_1,f_2\}$ and $F^3=\{f_2,f_3\}$. When changing the translation vector of the second map in $F^2$, one needs to modify also the translation vector of the first map in $F^3$ since these maps are the same. Therefore, it is useful to allow identifications of translation vectors between different families. For this purpose, we equip the set $\widehat\Lambda=\{(\lambda,i)\mid\lambda\in\Lambda, i=1,\dots,M_\lambda\}$ with an equivalence relation $\sim$ satisfying the following assumptions
- the cardinality $\mathcal A$ of the set of equivalence classes ${{\bf a}}:=\widehat\Lambda/\sim$ is finite,
- for every $\lambda\in\Lambda$ we have $(\lambda,i)\sim (\lambda,j)$ if and only if $i=j$ and
- the equivalence classes, regarded as subsets of $\Lambda$, are Borel sets.
The notation ${{\bf a}}$ for the set of equivalence classes refers to the fact that some translation vectors of the maps $f_i^\lambda$ are identified even though the maps are not. The second condition means that different translation vectors inside a system $F^\lambda$ are never identified. The first condition allows us to view the set of equivalence classes ${{\bf a}}$ as an element of $\mathbb R^{d\mathcal A}$. From now on we will write $A_{{\bf a}}^{\tilde\omega}$ for the attractor of a code tree $\tilde\omega$ to emphasise that it depends on the set of equivalence classes of translation vectors ${{\bf a}}$.
Now we are ready to state our main theorem in this section. Generalising the earlier results in [@JJKKSS], we prove that, under the assumptions of Theorem \[pexists\], for random affine code tree fractals the Hausdorff, packing and box counting dimensions, denoted by $\operatorname{dim_H}$, $\operatorname{dim_p}$ and $\operatorname{dim_B}$, respectively, are almost surely equal to the unique zero of the pressure provided that a probabilistic version of the F-S condition is satisfied. We denote by $s_0$ the unique zero of the pressure given by Theorem \[pexists\].
\[maintheorem\] Assume that $0<\underline\sigma\le\overline\sigma<\frac 12$. Let $P$ be an ergodic $\Xi$-invariant Borel probability measure on $\widetilde\Omega$ such that $\int_{\widetilde\Omega}N_1(\tilde\omega)\,dP(\tilde\omega)<\infty$. Suppose that for all $0<s<d$ $$\label{probcondcs}
P\{\tilde{\omega}\in\widetilde\Omega\mid\{T_{{\bf j}}^{\tilde{\omega}}\mid{{\bf j}}={{\bf i}}_l, 1\le l\le N_1\text{ and }{{\bf i}}_{N_1}\in\Sigma_*^{\tilde\omega}(0,1)\}
\text{ satisfies condition }C(s)\}>0.$$ Then for $P$-almost all $\tilde\omega\in\widetilde\Omega$, $$\operatorname{dim_H}(A_{{\bf a}}^{\tilde\omega})=\operatorname{dim_p}(A_{{\bf a}}^{\tilde\omega})=\operatorname{dim_B}(A_{{\bf a}}^{\tilde\omega})
=\min\{s_0,d\}$$ for $\mathcal{L}^{d\mathcal{A}}$-almost all ${{\bf a}}\in\mathbb R^{d\mathcal A}$.
\[oldthm\] a) In [@JJKKSS Theorem 5.1] a special case of Theorem \[maintheorem\] was proven under substantially stronger assumptions. First of all, [@JJKKSS Theorem 5.1] deals only with the planar case $d=2$. Moreover, instead of the following non-existence of parallelly mapped vectors is assumed $$\label{parallel}
\begin{split}
P\{\tilde{\omega}\in\widetilde\Omega\mid&\text{ there exists }v\in\mathbb R^2
\setminus\{0\}\text{ such that }T_{{{\bf i}}_{N_1}}^{\tilde{\omega}}(v)
\text{ are parallel}\\
&\text{ for all }{{\bf i}}_{N_1}\in\Sigma_*^{\tilde\omega}(0,1)\}<1.
\end{split}$$ Observe that in the case $d=2$ the condition $C(s)$ is equivalent to the condition $C(1)$ for all $0<s<2$. Furthermore, for a family $\{S_i\}_{i=1}^k$ condition $C(1)$ means that for all vectors $v,w\in\mathbb R^2\setminus\{0\}$ there exists $i$ such that $\langle S_iv\mid w\rangle\ne 0$. Therefore, condition implies condition in the case $d=2$. Condition is weaker than condition , since in the former one all iterates up to level $N_1$ are considered whilst in the second one only iterates at level $N_1$ play a role. In [@JJKKSS Theorem 5.1] there are also technical conditions concerning the measure $P$ which are not needed here. As explained in [@JJKKSS] the upper bound $\frac 12$ for $\overline\sigma$ is optimal in Theorem \[maintheorem\].
b\) The map $N_1(\tilde\omega)$ is Borel measurable as a projection. Since $\tilde\omega\mapsto T_{{\bf j}}^{\tilde\omega}$ is a Borel map for all finite words ${{\bf j}}$ and the set of families of linear maps satisfying condition $C(s)$ is open, the set in is a Borel set.
Before the proof of Theorem \[maintheorem\] we present an example which demonstrates how certain random $V$-variable and random graph directed systems fit in our framework.
\[graphexample\] Let $\Lambda$ be a finite set of directed labeled multigraphs $\lambda=(W, E^\lambda,\mathcal F^\lambda)$ where $W=\{1,2,...,V\}$ is the common finite set of vertices for all $\lambda\in\Lambda$, $E^\lambda$ is a finite set of directed edges and, for each directed edge $e\in E^\lambda$, there is an associated map $\phi_e^\lambda\in\mathcal F^\lambda $ which is a contraction on $\mathbb R^d$. For all edges $e$, we denote by $i(e)$ and $t(e)$ the initial and terminal vertices of $e$, respectively.
Recall that in the general setting of graph directed systems (see for example [@MU03]), for each vertex $v\in W$, there is an associated metric space $X_v$, and for each edge $e\in E^\lambda$, the associated map is $\phi_e^\lambda:X_{t(e)}\to X_{i(e)}$. Here we make the simplifying assumption that $X_v=\mathbb R^d$ for all $v\in V$. Let $$M=\max_{\substack{v\in W\\\lambda\in\Lambda}}\#\{e\in E^\lambda\mid i(e)=v\}$$ be the maximum number of maps within any fixed graph $\lambda\in\Lambda$ with the same range. Recall that in a deterministic graph directed system there is only one graph $\lambda$ and the composition $\phi_{e_1}\circ\phi_{e_2}$ is allowed provided that $t(e_1)=i(e_2)$. In some random graph directed models (see for example [@RU]) the graph $\lambda$ is fixed and the maps $\phi_e$ are random whereas in our model the graphs are allowed to be random as well.
Fix a probability measure $\mu$ on $\Lambda$ and set $\mathcal G=\Lambda^{\{0\}\cup\mathbb N}$. Let $\mu^\infty=\mu^{\{0\}\cup\mathbb N}$ be the product measure on $\mathcal G$ and let $\sigma:\mathcal G\to\mathcal G$, $$\sigma(g_0g_1\cdots)=g_1g_2\cdots\text{ for all }{{\bf g}}=g_0g_1\cdots\in\mathcal G,$$ be the left shift. To all ${{\bf g}}\in\mathcal G$, we associate a $V$-tuple of code trees $\omega=(\omega_1,...,\omega_V) $ as follows: For all $\lambda\in\Lambda$ and $v\in \{1,2,...,V\}$, let $\mathcal F_v^\lambda=\{\phi_e^\lambda\mid e\in E^\lambda\text{ and }i(e)=v\}$ be the iterated function system consisting of those maps in $\lambda$ whose ranges correspond to the vertex $v$. We write $I=\{1,\dots,M\}$ and rename the edges with $i(e)=v$ as $e_1,\dots,e_m$. Observe that $m$ may depend on $v\in W$ and $\lambda\in\Lambda$. The definition of $M$ implies that $m\le M$. For all $v\in W$, set $\omega_v(\emptyset)=\mathcal F_v^{g_0}$. Now we proceed inductively. Assuming that $\omega_v(i_1\cdots i_n)=\mathcal F_w^{g_n}
=\{\phi_{e_1}^{g_n},\ldots,\phi_{e_m}^{g_n}\}$ for some $w\in W$, define $\omega_v(i_1\cdots i_ni_{n+1})=\mathcal F_{t(e_{i_{n+1}})}^{g_{n+1}}$ for $i_{n+1}=1,\dots,m$. Observe that every ${{\bf g}}\in\mathcal G$ defines a sequence of graphs, which, in turn, determines a sequence of ordered walks starting from $v$. The code tree fractal corresponding to $\omega_v$ is the set of the limit points of the set of maps associated to all infinite paths starting from $v$. This code tree fractal is the $v$-th component in the graph directed set corresponding to the infinite sequence ${{\bf g}}$.
A $V$-tuple $\omega$ of code trees defines a $V$-tuple of code tree fractals $\bar A^\omega=(A_1^\omega,\dots,A_V^\omega)$ componentwise as described at the beginning of this section. Note that for fixed ${{\bf g}}\in\mathcal G$, any sub code tree rooted at level $n$ is determined by the code of its top node. Since this code is an element of the set $\{\mathcal F_k^{g_n}\}_{k=1}^V$, there are at most $V$ distinct code trees at a fixed level. By definition, this means that $\omega=(\omega_1,\dots,\omega_V)$ and the corresponding code tree fractals, $\{A_v^\omega\mid v\in W\}$, are $V$-variable.
In order to apply Theorem \[maintheorem\] to the above system, we need some further assumptions. Suppose that $\phi_e^\lambda(x)=T^\lambda_e(x)+a_e^\lambda$ is a non-singular affine map on $\mathbb R^d$ with singular values uniformly bounded from below by $\underline\sigma>0$ and from above by $\overline\sigma <\frac 12$ for all $\lambda\in\Lambda$ and $e\in E^\lambda$. We equip the set $\widehat\Lambda=\{(\lambda,e)\mid\lambda\in\Lambda\text{ and }e\in E^\lambda\}$ with the trivial equivalence relation, that is, $(\lambda,e)\sim(\lambda',e')$ if $(\lambda,e)=(\lambda',e')$. Then the set of equivalence classes ${{\bf a}}=\widehat\Lambda/\sim$ may be identified with the collection of all translation vectors. Since $\Lambda$ is finite and the number of edges is bounded, the number $\mathcal A$ of equivalence classes in ${{\bf a}}$ is finite, and therefore, ${{\bf a}}\in\mathbb R^{d\mathcal A}$. To ensure that the $V$-tuple of code trees corresponding to ${{\bf g}}$ has no “dying” branches and, in particular, defines a $V$-tuple of non-empty code tree fractals, we assume that in $\mu$-almost all graphs $\lambda\in\Lambda$ every vertex is an initial vertex of some edge, that is, $$\mu\{\lambda\in\Lambda\mid\text{ for all }v\in W\text{ there exists }
e\in E^\lambda\text{ with }i(e)=v\}=1.$$
In addition to the above assumptions, the existence of neck levels needs to be guaranteed. Recall that at a neck level all the sub code trees are identical. Such levels exist provided that there is a vertex $v_0\in W$ such that the $\mu$-measure of the set of graphs $\lambda\in\Lambda$ whose all edges have terminal vertex equal to $v_0$ is positive. Hence, we assume that there exists a vertex $v_0\in W$ such that $\mu(\Lambda_{\text{neck}})>0$ where $$\Lambda_{\text{neck}}=\{\lambda\in\Lambda\mid t(e)=v_0\text{ for all }
e\in E^\lambda\}.$$ We emphasise that this is a natural assumption for a collection of random graphs. For example, it is satisfied if the random graphs are constructed as follows: First choose for each $v\in W$ the number of edges with initial vertex equal to $v$. Then for each edge choose the terminal vertex independently according to a probability vector $(p_1,\dots,p_V)$ with $p_{v_0}>0$. We first define auxiliary neck levels inductively as follows: Set $$\tilde N_1({{\bf g}})=\min\{n\ge 0\mid t(e)=v_0\text{ for all }e\in E^{g_n}\}+1$$ and define $$\tilde N_{k+1}({{\bf g}})=\min\{n\ge \tilde N_k({{\bf g}})\mid t(e)=v_0\text{ for all }e\in
E^{g_n}\}+1.$$ This sequence is well defined for $\mu^\infty$-almost all ${{\bf g}}\in\mathcal G$ since the distances $\tilde N_{k+1}-\tilde N_k$ form a sequence of independent geometrically distributed random variables, and therefore, for the expectation we have $$\label{Ebounded}
\int\tilde N_k({{\bf g}})\,d\mu^\infty({{\bf g}})
=k\int\tilde N_1({{\bf g}})\,d\mu^\infty({{\bf g}})<\infty$$ for all $k\in\mathbb N$. The neck list is defined by $N_k=\tilde N_{2n_0^2k}$ for all $k\in\mathbb N$, where $n_0$ is as in Theorem \[CsCm\].
Observe that the existence of neck levels implies that $A_{v_0}^\omega$ is a finite union of affine copies of the attractor determined by the common sub code tree at the first neck level $N_1$. Since all the sub code trees at this level are identical, all the components of the $V$-tuble attractor $\bar A^\omega$ are finite unions of affine images of the same fixed set. Thus the dimensions of the components of $\bar A^\omega$ are equal to that of $A_{v_0}^\omega$. For the purpose of calculating the almost sure dimension value of $A_{v_0}^\omega$, we apply Theorem \[maintheorem\].
We proceed by verifying that the assumptions of Theorem \[maintheorem\] are satisfied. Since we attached to almost every code tree $\omega_{v_0}$ a unique neck list, we may identify $\widetilde\Omega$ with the space of all code trees $\omega_{v_0}$. Moreover, the product measure $\mu^\infty$ determines a mixing, thereby ergodic, $\Xi$-invariant measure $P$ on $\widetilde\Omega$. Now and the definition of $N_1$ imply that $\int N_1(\tilde\omega_{v_0})\,d P(\tilde\omega_{v_0})<\infty$.
Finally, we have to ensure that the F-S condition is valid. Intuitively, this is achieved if we assume that there are many allowed sequences of edges with initial and terminal vertices equal to $v_0$ such that the associated maps satisfy the assumptions of Theorem \[CsCm\]. More precisely, we suppose that there exists $l\in\mathbb N$ such that $$\label{nicemaps}
\begin{split}
\mu^l\{&(\lambda_1,\dots,\lambda_l)\in\Lambda^l\mid\lambda_j\not\in
\Lambda_{\text{preneck}}\text{ for }j=1,\dots,l-1,\,\lambda_l\in
\Lambda_{\text{preneck}},\text{ there exist }\\
&e_{i_1}^{\lambda_1}\cdots e_{i_l}^{\lambda_l}\text{ and }e_{j_1}^{\lambda_1}\cdots
e_{j_l}^{\lambda_l}\text{ with }i(e_{i_1}^{\lambda_1})=i(e_{j_1}^{\lambda_1})
=t(e_{i_l}^{\lambda_l})=t(e_{j_l}^{\lambda_l})=v_0\text{ and}\\
&F:=T_{e_{i_1}}^{\lambda_1}\cdots T_{e_{i_l}}^{\lambda_l}\text{ and }
G:=T_{e_{j_1}}^{\lambda_1}\cdots T_{e_{j_l}}^{\lambda_l}
\text{ satisfy the assumptions of Theorem }\ref{CsCm}\}\\
&>0.
\end{split}$$ Since we use the product measure $\mu^\infty$ on $\mathcal G$, there is positive probability that the same pair of maps $(F,G)$ appears successively $2n_0^2$ times. Therefore, from Theorem \[CsCm\] we see that the condition is satisfied. Observe that the condition is satisfied with $l=1$ if there are maps $\phi_e^\lambda$ and $\phi_{e'}^\lambda$ as in Theorem \[CsCm\] with $i(e)=i(e')=t(e)=t(e')=v_0$ and $\lambda\in\Lambda_{\text{preneck}}$ is chosen with positive probability. This, in turn, is true for typical families by Corollary \[typical\].
For the proof of Theorem \[maintheorem\] we need the following notation and auxiliary results.
\[full\] Let $c>0$ and $0<s<d$. We say that a family of non-singular linear mappings $\{S_j:\mathbb R^d\to\mathbb R^d\}_{j=1}^k$ is [*$(c,s)$-full*]{} if $$\sum_{j=1}^k\Phi^s(US_jV)\ge c\Phi^s(U)\Phi^s(V)$$ for all non-singular linear mappings $U,V:\mathbb R^d\to\mathbb R^d$.
In Lemmas \[fullok\] and \[numbersatcondition\] we explore consequences of the probabilistic version of the F-S condition .
\[fullok\] Assuming that the condition is satisfied, there exists $c>0$ such that $$\varrho=P\{\tilde\omega\in\widetilde\Omega|\{T_{{{\bf i}}_{N_1}}^{\tilde\omega}\}_
{{{\bf i}}_{N_1}\in\Sigma_*^{\tilde\omega}(0,1)}\text{ is }(c,s)\text{-full }\}>0.$$
Since the set of $(c,s)$-full families is a Borel set, the set in the definition of $\varrho$ is a Borel set. Let $U,V:\mathbb R^d\to\mathbb R^d$ be non-singular linear maps. Suppose that $$\mathcal F=\{T_{{\bf j}}^{\tilde{\omega}}\mid{{\bf j}}={{\bf i}}_l,
1\le l\le N_1\text{ and }{{\bf i}}_{N_1}\in\Sigma_*^{\tilde\omega}(0,1)\}$$ satisfies the condition $C(s)$. By the proof of [@FS Proposition 2.1] (see also [@FS Corollary 2.2]), there exists ${{\bf j}}$ such that $$\label{basicineq}
\Phi^s(UT_{{\bf j}}^{\tilde\omega}V)\ge C(\mathcal F)\Phi^s(U)
\Phi^s(V),$$ where the constant $C(\mathcal F)$ is independent of $U$ and $V$. Observe that $C(\mathcal F)$ depends on $s$ but it is an interpolation of the constants obtained by replacing $s$ by $m$ and $m+1$, where $m$ is the integer part of $s$ (recall Remark \[FSremark\]). Let $\overline{{\bf i}}_{N_1}\in\Sigma_*^\omega(0,1)$ be such that ${{\bf j}}=\overline{{\bf i}}_{|{{\bf j}}|}$. Writing $T_{\overline{{\bf i}}_{N_1}}^{\tilde\omega}
=T_{{{\bf j}}}^{\tilde\omega}T_{i_{|{{\bf j}}|+1}}^{\tilde\omega(i_{\vert{{\bf j}}\vert})}
\cdots T_{i_{N_1}}^{\tilde\omega(i_{N_1-1})}$ and applying , gives $$\Phi^s(UT_{\overline{{\bf i}}_{N_1}}^{\tilde\omega}V)
\ge\underline\sigma^{N_1-\vert{{\bf j}}\vert}\Phi^s(UT_{{{\bf j}}}V)
\ge C(\mathcal F) \underline\sigma^{N_1}\Phi^s(U)\Phi^s(V).$$ This implies that $$\label{FSCor}
\sum_{{{\bf i}}_{N_1}\in\Sigma_*^{\tilde\omega}(0,1)}\Phi^s(UT_{{{\bf i}}_{N_1}}^{\tilde\omega}V)
\ge C(\mathcal F)\underline\sigma^{N_1}\Phi^s(U)
\Phi^s(V)$$ for all linear mappings $U,V:\mathbb R^d\to\mathbb R^d$. From we conclude that there exists $c>0$ such that $$P\{\tilde{\omega}\in\widetilde\Omega\mid C(\mathcal F)
\underline\sigma^{N_1}>c\}>0,$$ giving the claim.
In the following lemma we denote by $\# A$ the number of elements in a set $A$.
\[numbersatcondition\] Assume that the condition is satisfied and let $\varrho$ and $c$ be as in Lemma \[fullok\]. Define for all $n,m\in\mathbb{N}$ $$E^{\tilde\omega}(n,n+m)=\#\{n<j\le n+m\mid\{T_{{{\bf i}}_{N_1}}^{\Xi^{j-1}
(\tilde\omega)}\}\text{ is }(c,s)\text{-full }\}$$ and suppose that $P$ is $\Xi$-invariant and ergodic. Then for $P$-almost all $\tilde\omega\in\widetilde\Omega$ the following is true: for all $\varepsilon>0$ there exists $n_1(\tilde\omega,\varepsilon)>0$ such that for all $n>n_1(\tilde\omega,\varepsilon)$ we have $$E^{\tilde\omega}(n,n+\lceil\varepsilon n\rceil)\ge 1,$$ where $\lceil x\rceil$ is the smallest integer $m$ with $x\le m$.
Let $\chi$ be the characteristic function of the set $\{\tilde\omega\in\widetilde\Omega\mid\{T_{{{\bf i}}_{N_1}}^{\tilde\omega}\}\text{ is }
(c,s)\text{-full}\,\}$. Since $$E^{\tilde\omega}(0,n)=\sum_{j=0}^{n-1}\chi(\Xi^j(\tilde{\omega})),$$ we obtain from the Birkhoff ergodic theorem that for $P$-almost all $\tilde\omega\in\widetilde\Omega$ $$\label{expectedfull}
\lim_{n\to\infty}\frac{E^{\tilde\omega}(0,n)}{n}=\int_{\tilde\Omega}
\chi(\tilde\omega)dP(\tilde\omega)=\varrho.$$
Fix $\tilde\omega\in\widetilde\Omega$ satisfying and let $\varepsilon>0$. Defining $0<\tilde{\varepsilon}=\frac{\varrho\varepsilon n-1}{(\varepsilon+2)n}<\varrho$ for sufficiently large $n$, there exists $n_1(\tilde{\omega},\varepsilon)>0$ such that for all $n>n_1(\tilde{\omega},\varepsilon)$ and for all $m\ge 0$ we have $$(\varrho-\tilde\varepsilon)(n+m)<E^{\tilde\omega}(0,n+m)<(\varrho
+\tilde\varepsilon)(n+m),$$ and therefore, $$E^{\tilde\omega}(n,n+m)=E^{\tilde\omega}(0,n+m)-E^{\tilde\omega}(0,n)
>(\varrho-\tilde\varepsilon)m-2\tilde\varepsilon n.$$ Finally, taking $m\ge \varepsilon n$, gives $(\varrho-\tilde\varepsilon)m-2\tilde\varepsilon n\ge 1$, which implies that $E^{\tilde\omega}(n,n+m)\ge 1$. In particular, $E^{\tilde\omega}(n,n+\lceil\varepsilon n\rceil)\ge 1$.
\[numbernecks\] Under the assumptions of Theorem \[pexists\], we have for $P$-almost all $\tilde\omega\in\widetilde\Omega$ that $$\lim_{n\to\infty}\frac{N_{n+\lceil\varepsilon n\rceil}(\tilde\omega)
-N_{n-1}(\tilde\omega)}{N_n(\tilde\omega)}=\varepsilon$$ for all $\varepsilon>0$.
Since $N_n({\tilde\omega})=\sum_{j=0}^{n-1}N_1(\Xi^j(\tilde\omega))$, the Birkhoff ergodic theorem implies that for $P$-almost all $\tilde\omega\in\widetilde\Omega$ $$\lim_{n\to\infty}\frac{N_n({\tilde\omega})}{n}=\int_{\tilde\Omega}
N_1(\tilde\omega)dP(\tilde\omega)=b<\infty.$$ Now for any typical $\tilde\omega$ we have $$\lim_{n\to\infty}\frac{N_{n+\lceil\varepsilon n\rceil}(\tilde\omega)}
{N_n(\tilde\omega)}
=\lim_{n\to\infty}\frac{N_{n+\lceil\varepsilon n\rceil}(\tilde\omega)}
{n+\lceil\varepsilon n\rceil}\cdot\frac{n+\lceil\varepsilon n\rceil}{n}\cdot
\frac{n}{N_n(\tilde\omega)}=b(1+\varepsilon)\frac 1b=1+\varepsilon,$$ and similarly we see that $\lim_{n\to\infty}\frac{N_{n-1}(\tilde\omega)}{N_n(\tilde\omega)}=1$. Therefore, $$\lim_{n\to\infty}\frac{N_{n+\lceil\varepsilon n\rceil}(\tilde\omega)
-N_{n-1}(\tilde\omega)}{N_n(\tilde\omega)}=\varepsilon.$$
Now we are ready to prove Theorem \[maintheorem\].
In [@JJKKSS (5.20)] it is proven that under the assumptions of Theorem \[pexists\] we have $\operatorname{\overline{dim}_B}(A_{{\bf a}}^{\tilde\omega})\le\min\{s_0,d\}$ for $P$-almost all $\tilde\omega\in\widetilde\Omega$. Here $\operatorname{\overline{dim}_B}$ is the upper box counting dimension. Note that the assumption $d=2$ is not needed in the proof of [@JJKKSS (5.20)]. Since always $\operatorname{dim_H}\le\operatorname{dim_p}\le\operatorname{\overline{dim}_B}$ (see for example [@F (3.17) and (3.29)]), it is sufficient to verify that $$\label{almostgoal}
\operatorname{dim_H}(A_{{\bf a}}^{\tilde\omega})\ge\min\{s_0,d\}$$ for $P$-almost all $\tilde\omega\in\widetilde\Omega$. Let $s<\min\{s_0,d\}$. In the proof of [@JJKKSS Theorem 3.2] it is shown that follows provided that for $P$-almost all $\tilde\omega\in\widetilde\Omega$ there exists a probability measure $\mu^{\tilde\omega}$ on $\Sigma^{\tilde\omega}$ and a constant $D(\tilde{\omega})>0$ such that $$\label{eq66}
\mu^{\tilde\omega}([{{\bf i}}_l])\le D(\tilde\omega)\Phi^s(T_{{{\bf i}}_l}^{\tilde\omega})$$ for all ${{\bf i}}\in\Sigma^{\tilde\omega}$ and $l\in\mathbb N$.
For the purpose of verifying , we define for all $\tilde\omega\in\widetilde\Omega$ and $m\in\mathbb N$ $$\label{muomega}
\mu_m^{\tilde\omega} = \frac{\sum_{{{\bf i}}_{N_m}\in\Sigma_*^{\tilde\omega}(0,m)}
\Phi^s(T_{{{\bf i}}_{N_m}}^{\tilde\omega})\delta_{{{\bf i}}_{N_m}}}{\sum_{{{\bf i}}_{N_m}
\in\Sigma_*^{\tilde\omega}(0,m)}
\Phi^s(T_{{{\bf i}}_{N_m}}^{\tilde\omega})},$$ where $\delta_{{{\bf i}}_{N_m}}$ is the Dirac measure at some fixed point of the cylinder $[{{\bf i}}_{N_m}]$. The choice of the cylinder point plays no role in what follows. Since $\Sigma^{\tilde\omega}$ is compact, the sequence $(\mu_m^{\tilde\omega})_{m\in\mathbb N}$ has a weak\*-converging subsequence with a limit measure $\mu^{\tilde\omega}$. We proceed by showing that $\mu^{\tilde\omega}$ satisfies .
By Lemma \[numbernecks\] the following is true for $P$-almost all $\tilde\omega\in\widetilde\Omega$: for all $\varepsilon>0$ there exists $n_2(\tilde\omega,\varepsilon)>0$ such that for all $n>n_2(\tilde\omega,\varepsilon)$ $$\label{neckbound}
N_{n+\lceil\varepsilon n\rceil}(\tilde\omega)-N_{n-1}(\tilde{\omega})
<2\varepsilon N_n(\tilde\omega).$$ Furthermore, it follows from the definition of the pressure that for $P$-almost all $\tilde\omega\in\widetilde\Omega$ there exists for all $\varepsilon>0$ a number $n_3(\tilde\omega,\varepsilon)>0$ such that for all $n>n_3(\tilde\omega,\varepsilon)$ we have $$\label{pressurebound}
e^{(p^{\tilde\omega}(s)-\varepsilon)N_n(\tilde\omega)}<\sum_{{{\bf i}}_{N_n}
\in\Sigma_*^{\tilde\omega}(0,n)}\Phi^s(T_{{{\bf i}}_{N_n}}^{\tilde\omega})
< e^{(p^{\tilde\omega}(s)+\varepsilon)N_n(\tilde\omega)}.$$
Let $\varepsilon>0$. Consider $\tilde\omega\in\widetilde\Omega$ satisfying Lemma \[numbersatcondition\], and and set $n_0(\tilde\omega,\varepsilon)=\max\{n_1(\tilde\omega,\varepsilon),
n_2(\tilde\omega,\varepsilon),n_3(\tilde\omega,\varepsilon)\}$. For all ${{\bf i}}_l\in\Sigma_*^{\tilde\omega}$ with $l>N_{n_0(\tilde\omega,\varepsilon)}$, there exists $n>n_0(\tilde\omega,\varepsilon)$ such that $N_{n-1}<l\leq N_n$. Now Lemma \[numbersatcondition\] implies the existence of $1\le k\le\lceil\varepsilon n\rceil$ such that $\{T_{{{\bf j}}_{N_1}}^{\Xi^{n+k-1}(\tilde{\omega})}\}$ is $(c,s)$-full. Let $m$ be a natural number with $m>\varepsilon n$. In the remaining part of the proof we use the following abbreviations $\sum_{{\bf j}}=\sum_{{{\bf j}}:{{\bf i}}_l{{\bf j}}\in\Sigma_*^{\tilde\omega}(0,n+k-1)}$, $\sum_{N_1}=\sum_{{{\bf j}}_{N_1}\in\Sigma_*^{\tilde\omega}(n+k-1,n+k)}$, $\sum_{N_{m-k}}=\sum_{{{\bf k}}_{N_{m-k}}\in\Sigma_*^{\tilde\omega}(n+k,n+m)}$ and $\sum_{N_{n+k-1}}=\sum_{{{\bf i}}_{N_{n+k-1}}\in\Sigma_*^{\tilde\omega}(0,n+k-1)}$, and denote by $T_{({{\bf i}}_l){{\bf j}}}^{\tilde\omega}$ the last $|{{\bf j}}|$ maps of $T_{{{\bf i}}_l{{\bf j}}}^{\tilde\omega}$. Using the definition of $\mu_{n+m}^{\tilde\omega}$, applying the submultiplicativity of $\Phi^s$ in the numerator and utilising the $(c,s)$-fullness in the denominator, we obtain $$\begin{aligned}
\mu_{n+m}^{\tilde\omega}([{{\bf i}}_l])&=\frac{\sum_{{\bf j}}\sum_{N_1}\sum_{N_{m-k}}
\Phi^s(T_{{{\bf i}}_l{{\bf j}}}^{\tilde\omega}T_{{{\bf j}}_{N_1}}^{\Xi^{n+k-1}(\tilde\omega)}
T_{{{\bf k}}_{N_{m-k}}}^{\Xi^{n+k}(\tilde\omega)})}
{\sum_{N_{n+k-1}}\sum_{N_1}\sum_{N_{m-k}}\Phi^s(T_{{{\bf i}}_{N_{n+k-1}}}^{\tilde\omega}
T_{{{\bf j}}_{N_1}}^{\Xi^{n+k-1}(\tilde\omega)}
T_{{{\bf k}}_{N_{m-k}}}^{\Xi^{n+k}(\tilde\omega)})}\\
&\le\frac{\Phi^s(T_{{{\bf i}}_l}^{\tilde\omega})\sum_{{\bf j}}\sum_{N_1}\sum_{N_{m-k}}
\Phi^s(T_{({{\bf i}}_l){{\bf j}}}^{\tilde\omega})
\Phi^s(T_{{{\bf j}}_{N_1}}^{\Xi^{n+k-1}(\tilde\omega)})
\Phi^s(T_{{{\bf k}}_{N_{m-k}}}^{\Xi^{n+k}(\tilde\omega)})}
{c\sum_{N_{n+k-1}}\sum_{N_{m-k}}\Phi^s(T_{{{\bf i}}_{N_{n+k-1}}}^{\tilde\omega})
\Phi^s(T_{{{\bf k}}_{N_{m-k}}}^{\Xi^{n+k}(\tilde\omega)})}\\
&=\frac{\Phi^s(T_{{{\bf i}}_l}^{\tilde\omega})\sum_{{\bf j}}\sum_{N_1}
\Phi^s(T_{({{\bf i}}_l){{\bf j}}}^{\tilde\omega})
\Phi^s(T_{{{\bf j}}_{N_1}}^{\Xi^{n+k-1}(\tilde\omega)})}
{c\sum_{N_{n+k-1}}\Phi^s(T_{{{\bf i}}_{N_{n+k-1}}}^{\tilde\omega})}.\end{aligned}$$ Recall that in every family there are at most $M$ maps, $\Phi^s(T_j)\le 1$ for all $j$ and $k\le\lceil\varepsilon n\rceil$, and suppose that $\varepsilon<p^{\tilde\omega}(s)$. Applying in the numerator and in the denominator, we obtain for all $l>N_{n_0(\tilde\omega,\varepsilon)}$ that $$\mu_{n+m}^{\tilde\omega}([{{\bf i}}_l])\le\frac{\Phi^s(T_{{{\bf i}}_l}^{\tilde\omega})
M^{N_n(\tilde\omega)-N_{n-1}(\tilde\omega)+N_{n+\lceil\varepsilon n\rceil}(
\tilde\omega)-N_n(\tilde\omega)}}
{c e^{(p^{\tilde\omega}(s)-\varepsilon)N_{n+k-1}(\tilde\omega)}}
\le\frac{\Phi^s(T_{{{\bf i}}_l}^{\tilde\omega})M^{2\varepsilon N_n(\tilde\omega)}}
{c e^{(p^{\tilde\omega}(s)-\varepsilon)N_n(\tilde\omega)}}.$$ Taking $\varepsilon$ so small that $M^{2\varepsilon}<e^{p^{\tilde\omega}(s)-\varepsilon}$, we set $$D(\tilde\omega)=\max\Bigl\{c^{-1},\max_{l\le N_{n_0(\tilde\omega,\varepsilon)}}
\Bigl\{\frac{\mu^{\tilde\omega}[{{\bf i}}_l]}{\Phi^s(T_{{{\bf i}}_l}^{\tilde\omega})}
\Bigr\}\Bigr\}.$$ Then for all $l>0$ we have $$\mu_{n+m}^{\tilde\omega}([{{\bf i}}_l])\le D(\tilde\omega)
\Phi^s(T_{{{\bf i}}_l}^{\tilde\omega}).$$ Letting $m$ tend to infinity and recalling that cylinders are open, we obtain from the Portmanteau theorem [@K Theorem 17.20].
[50]{}
M. Barnsley, J. E. Hutchinson and Ö. Stenflo, *A fractal valued random iteration algorithm and fractal hierarchy*, Fractals **13** (2005), 111–146.
M. Barnsley, J. E. Hutchinson and Ö. Stenflo, *V-variable fractals: Fractals with partial self similarity*, Adv. Math. **218** (2008), 2051–2088.
M. Barnsley, J. E. Hutchinson and Ö. Stenflo, *V-variable fractals: Dimension results*, Forum Math. **24** (2012), 445–470.
J. Barral and D.-J. Feng, *Multifractal formalism for almost all self-affine measures*, Comm. Math. Phys. **318** (2013), 473–504.
K. J. Falconer, *The Hausdorff dimension of self-affine fractals*, Math. Proc. Cambridge Philos. Soc. **103** (1988), 339–350.
K. J. Falconer, *Sub-self-similar sets*, Trans. Amer. Math. Soc. **347** (1995), 3121–3129.
K. J. Falconer, *Fractal Geometry*, Second edition, John Wiley & Sons, Chichester, 2003.
K. J. Falconer and J. Miao, *Dimensions of self-affine fractals and multifractals generated by upper triangular matrices*, Fractals **15** (2007), 289–299.
K. J. Falconer and J. Miao, *Random subsets of self-affine fractals*, Mathematika **56** (2010), 61–76.
K. J. Falconer and A. Sloan, *Continuity of subadditive pressure for self-affine sets*, Real Anal. Exchange **34** (2009), 413–427.
D.-J. Feng, *Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices*, Israel J. Math. **170** (2009), 355–394.
D.-J. Feng and P. Shmerkin, *Non-conformal repellers and the continuity of pressure for matrix cocycles*, to appear in Geom. Funct. Anal.
E. Järvenpää, M. Järvenpää, A. Käenmäki, H. Koivusalo, Ö. Stenflo and V. Suomala, *Dimensions of random affine code tree fractals*, Ergodic Theory Dynam. Systems **34** (2014), 854–875.
T. Jordan, M. Pollicott and K. Simon, *Hausdorff dimension for randomly perturbed self affine attractors*, Comm. Math. Phys. **270** (2007), 519–544.
A. Käenmäki and P. Shmerkin, *Overlapping self-affine sets of Kakeya type*, Ergodic Theory Dynam. Systems **29** (2009), 941–965.
A. Kechris, *Classical Descriptive Set Theory*, Springer-Verlag, New York, 1995.
R. D. Mauldin and M. Urbański, *Graph Directed Markov Systems–Geometry and Dynamics of Limit Sets*, Cambridge Tracts in Mathematics **148**, Cambridge University Press, Cambridge, 2003.
F. Przytycki and M. Urbański, *On the Hausdorff dimension of some fractal sets*, Studia Math. **93** (1989), 155–186.
M. Roy and M. Urbański, *Random graph directed Markov systems*, Discrete Contin. Dyn. Syst. **30** (2011), 261–298.
B. Solomyak, *Measure and dimensions for some fractal families*, Math. Proc. Cambridge Philos. Soc. **124** (1998), 531–546.
[^1]: We thank the referee for useful comments and we acknowledge the support of Academy of Finland, the Centre of Excellence in Analysis and Dynamics Research. BL is partially supported by a NSFC grant 11201155. ÖS thanks the Esseen foundation.
|
---
abstract: 'Recent theoretical and observational works indicate the presence of a correlation between the star formation rate (SFR) and the active galactic nuclei (AGN) luminosity (and, therefore, the black hole accretion rate, $\dot M_{\rm BH}$) of Seyfert galaxies. This suggests a physical connection between the gas forming stars on kpc scales and the gas on sub-pc scales that is feeding the black hole. We compiled the largest sample of Seyfert galaxies to date with high angular resolution ($\sim\,0.4-0.8$) mid-infrared (8–13) spectroscopy. The sample includes 29 Seyfert galaxies drawn from the AGN Revised Shapley-Ames catalogue. At a median distance of 33 Mpc, our data allow us to probe nuclear regions on scales of $\sim$65pc (median value). We found no general evidence of suppression of the 11.3 polycyclic aromatic hydrocarbon (PAH) emission in the vicinity of these AGN, and used this feature as a proxy for the SFR. We detected the 11.3 PAH feature in the nuclear spectra of 45% of our sample. The derived nuclear SFRs are, on average, five times lower than those measured in circumnuclear regions of $600\,$pc in size (median value). However, the projected nuclear SFR densities (median value of 22[$M_\odot$]{}yr$^{-1}\,{\rm kpc}^{-2}$) are a factor of 20 higher than those measured on circumnuclear scales. This indicates that the SF activity per unit area in the central $\sim$65pc of Seyfert galaxies is much higher than at larger distances from their nuclei. We studied the connection between the nuclear SFR and $\dot M_{\rm BH}$ and showed that numerical simulations reproduce fairly well our observed relation.'
author:
- 'Pilar Esquej, Almudena Alonso-Herrero, Omaira González-Martín, Sebastian F. Hönig, Antonio Hernán-Caballero, Patrick F. Roche, Cristina Ramos Almeida, Rachel E. Mason, Tanio Díaz-Santos, Nancy A. Levenson, Itziar Aretxaga, José Miguel Rodríguez Espinosa, Christopher Packham'
bibliography:
- 'bibliography.bib'
- 'coevalbhNotes.bib'
nocite:
- '[@Marinucci2012]'
- '[@Risaliti2005]'
- '[@Brightman2011]'
- '[@Dadina2007]'
- '[@Tueller2008]'
- '[@Woo2002]'
- '[@Hoenig2010]'
- '[@DiamondStanic2012]'
- '[@Wang2011]'
- '[@Beckmann2006]'
- '[@Guainazzi2000]'
- '[@Mueller2003]'
- '[@Guainazzi2004]'
- '[@Akylas2001]'
- '[@Asmus2011]'
- '[@Malizia2007]'
- '[@Beifiori2009]'
- '[@Piconcelli2007]'
- '[@Hoenig2008]'
- '[@Roche2006]'
- '[@Gonzalez2013]'
- '[@Hoenig2010]'
- '[@Young2007]'
- '[@Mason2006]'
- '[@AlonsoHerrero2012]'
- '[@Sales2013]'
- '[@Mason2009]'
- '[@Colling2011]'
- '[@Sales2011]'
- '[@Alonso2011]'
- '[@DiazSantos2010]'
- '[@Roche2007]'
title: |
Nuclear star formation activity and black hole accretion\
in nearby Seyfert galaxies
---
Introduction {#sec:sec1}
============
One of the most important challenges in modern cosmology is to disentangle the physics behind the processes underlying galaxy formation and evolution. Observations over the past decades have revealed that supermassive black holes (SMBHs) likely reside at the centers of all galaxies with a bulge and that the properties of these black holes and their host galaxies are tightly correlated [e.g. @Magorrian1998; @Ferrarese2000; @Gebhardt; @Kormendy2013]. The co-evolution of galaxies and their corresponding SMBHs depends on some physical mechanism, referred to as feedback, that links accretion and ejection of gas residing on a sub-pc scale in galactic nuclei to the rest of the galaxy [@Silk1998; @King2010; @Nayakshin2012]. The connection between star formation (SF) activity on different physical scales in a galaxy and the presence of an active galactic nucleus (AGN) has been a long discussed topic. However, there still are many uncertainties under consideration to disentangle the processes behind such a relation [see e.g. @Hopkins2010 and references therein].
In the standard unification model, the powering mechanism of AGN is gas accretion onto a central SMBH. However, the physics of angular momentum transfer to the vicinity of the black hole is still unclear [see @Alexander2012 for a recent review]. Given that the angular momentum of inflowing gas produced by galaxy mergers or other large scale structures (e.g., bars) cannot be removed instantaneously, many studies proposed that the inflowing gas could form a circumnuclear disk where SF can take place. @Kawakatu2008 [and references therein] put forward a model for such a circumnuclear disk, which might be coincident with the putative torus of the unification model of AGN [@Antonucci1993]. This model predicts that SF would mostly take place in the outer parts of a 100pc-size torus [@Wada2002]. @CidFernandes1995 proposed the presence of a starburst in the obscuring torus as a solution for the absence of conspicuous broad lines in Seyfert 2s. The starburst disk model of @Thompson2005 estimates that most of the gas is supplied from outside the inner 200pc, but this is better suited for ultra-luminous infrared galaxies due to the high star formation rates (SFRs) considered. @Ballantyne2008 presented an update of the @Thompson2005 model with typical maximum SFRs of $\sim\,1\,M_\odot\,{\rm yr}^{-1}$, that could also potentially obscure the AGN. These nuclear pc-sized starbursts will mostly be associated with low luminosity AGN (i.e. Seyferts and low-ionization nuclear emission-line regions – LINERs). From an observational point of view, nuclear starbursts have been detected in Seyfert 2 galaxies and LINERs using UV images obtained with the Hubble Space Telescope [@Heckman1995; @GonzalezDelgado1998; @Colina2002].
The numerical simulations of @Hopkins2010 predict a relation with some scatter between the SFR of the galaxy on different scales – going from 10kpc-scale to the central parsec – and the black hole accretion rate ($\dot M_{\rm BH}$). This correlation appeared to be more prominent on smaller physical scales. However, simulations also indicate dynamical delays between the peaks of the SF and the BH growth [@Hopkins2012], in agreement with results from observational works [e.g., @Davies2007; @Wild2010; @Ramos2013].
Mid-infrared (mid-IR) spectroscopy is a powerful tool to explore the nature of AGN and SF activity in galaxies. Among the most remarkable characteristics of the mid-IR spectra of galaxies is the presence of polycyclic aromatic hydrocarbons (PAH) emission, with the most prominent features being at 6.2, 7.7, 8.6, 11.3 and 17. They are due to the stretching and bending vibrations of aromatic hydrocarbon materials, where the shortest wavelength features are dominated by the smallest PAHs [e.g. @Tielens2010]. This type of emission mostly originates in photo-dissociation regions where aromatic molecules are heated by the radiation field produced by young massive stars [@Roche1985; @Roche1991]. Therefore, PAHs are often used as indicators of the *current* SFR of galaxies. Note that they can also be excited by UV emission from B stars and thus PAH emission probes SF over a few tens of million years [e.g., @Peeters2004; @DiazSantos2010].
PAH features are detected in AGN, although they generally appear weak when compared with those of star forming galaxies [@Roche1991]. It has been proposed that the PAH molecules might be destroyed in the vicinity of an active nucleus due to the presence of a hard radiation field [@Voit1992]. There is also evidence that different PAHs might behave differently. @DiamondStanic2010 showed that the 11.3 PAH feature emission is a reliable indicator of the SFR in AGN, at least for Seyfert-like AGN luminosities and kpc scales, while the 6.2, 7.7, and 8.6features appear suppressed. Sings of variations between the different features have been reported by many authors [e.g. @Peeters2004; @Galliano2008]. For instance, @Smith2007 found that the ratio of the PAH emission at 7.7 and 11.3 is relatively constant among pure starbursts, while it decreases by up to factor of 5 for galaxies hosting a weak AGN. They interpreted this as a selective destruction of the smallest PAH carriers by the hard radiation arising from the accretion disk, ruling out the explanation in terms of ionization of the molecules [see also @Siebenmorgen2004].
A number of works have studied the SF activity using PAH emission and its relation to the AGN activity. @Shi2007 demonstrated that the SF contribution increases from Palomar-Green QSO, to 2MASS QSO, and radio galaxies. Using measurements of the 7.7 and 11.3 PAH using [*Spitzer*]{}/IRS data, they found higher SFRs for more intense nuclear activity, which indicates that the AGN selection technique influences the level of SF activity detected in the corresponding host galaxies. @Watabe2008 investigated the nuclear vs. circumnuclear SF for a sample of Seyfert galaxies using ground-based observations of the 3.3 PAH feature. Assuming that the this PAH traces the SF activity, they found that both SF and AGN activity are correlated [see also, @Imanishi2003; @Imanishi2004]. Such a relation implied that SF in the inner region of the AGN (within a few hundred parsecs from the center) might have a greater influence on $\dot M_{\rm BH}$. On the other hand, @Mason2007 found weak or absent PAH emission in the central 20pc of the Seyfert 1 galaxy NGC1097, whilst in the circumnuclear region, strong 3.3 and 11.3 PAH bands were detected. In the case of NGC1097, the absence of PAH emission may be related to destruction/ionization of PAH molecules by hard photons from the nuclear star cluster.
@DiamondStanic2012 recently found a strong correlation between the kpc-scale SF derived using the 11.3PAH feature and 24observations for Seyfert galaxies. However, the limited angular resolution of their [*Spitzer*]{}data ($\sim\,4-5\,$) did not allow them to resolve nuclear ($\sim\,100\,$pc) scales, and it is unclear if the measured PAH feature is associated with the galaxy or to the nuclear environment.
In this work we compile a sample of 29 Seyfert galaxies from the revised Shapley-Ames (RSA) galaxy catalogue [@Sandage1987] with published ground-based mid-IR high angular resolution spectroscopy obtained on 8m-class telescopes. At a median distance of 33Mpc, this sample allows us to study the nuclear SF activity around AGN on scales of $\sim$65pc. We also use mid-IR spectra taken with the Infrared Spectrograph [IRS, @Houck2004] on board [*Spitzer*]{} for all objects in our sample to investigate the extended ($\sim\,600\,$pc) SF in the host galaxy. This enables us to study the relation SFR-$\dot M_{\rm BH}$ at different scales in the local Universe.
This paper is structured as follows: Section 2 describes the sample selection and data analysis. In Section 3 we study the nuclear 11.3 PAH feature emission. Section 4 compares the circumnuclear and nuclear SF activity and its relation with $\dot M_{\rm BH}$. Finally, our conclusions are summarized in Section 5. Throughout this work we assumed a $\Lambda$CDM cosmology with ($\Omega_{\rm M}$, $\Omega_{\Lambda}$) = (0.3, 0.7) and ${H}_{0}$ = 70 ${\rm km}~{\rm s}^{-1}~{\rm Mpc}^{-1}$.
[lrccrrrrrl]{} & & & & & & &\
& (Mpc) & & & ([erg s$^{-1}$]{}) & ([erg s$^{-1}$]{}) & ([$M_\odot$]{})\
Circinus & 4.2 & 0.4 & Sy2 & 42.6$^{\rm (*)}$ & 43.8 & 6.42 & (1)\
ESO 323–G077 & 65.0 & 0.7 & Sy1 & 42.7 & 43.9 & 7.40 & (2)\
IC 5063 & 49.0 & 0.7 & Sy2 & 42.8 & 44.0 & 7.74 & (1)\
Mrk 509 & 151.2 & 0.8 & Sy1 & 43.9 & 45.4 & 7.86 & (3,4,5)\
NGC 1068 & 16.3 & 0.9 & Sy2 & 43.0$^{\rm (*)}$ & 44.3 & 7.59 & (1)\
NGC 1365 & 23.5 & 0.5 & Sy1 & 42.1$^{\rm (d)}$ & 43.1 & 8.20 & (6)\
NGC 1386 & 12.4 & 0.4 & Sy2 & 41.6$^{\rm (*)}$ & 42.6 & 7.42 & (1)\
NGC 1808 & 14.3 & 0.6 & Sy2 & 40.4 & 41.2 & …& (7)\
NGC 2110 & 33.6 & 0.7 & Sy1 & 42.6 & 43.7 & 8.30 & (3,4,8)\
NGC 2992 & 33.2 & 0.3 & Sy1 & 43.1 & 44.4 & 7.72 & (8)\
NGC 3081 & 34.4 & 0.9 & Sy2 & 42.5 & 43.6 & 7.13 & (1)\
NGC 3227 & 16.6 & 0.7 & Sy1 & 42.4 & 43.5 & 7.62 & (9,4,10)\
NGC 3281 & 46.1 & 0.5 & Sy2 & 42.6 & 43.8 & 7.91 & (1)\
NGC 3783 & 42.0 & 0.9 & Sy1 & 43.2 & 44.5 & 7.48 & (3,4,10)\
NGC 4151 & 14.3 & 0.7 & Sy1 & 42.1 & 43.2 & 7.66 & (11,12)\
NGC 4388 & 36.3 & 0.2 & Sy2 & 42.9 & 44.1 & 7.23 & (1)\
NGC 4507 & 51.0 & 0.8 & Sy2 & 43.1 & 44.4 & 7.65 & (1)\
NGC 4945 & 3.6 & 0.2 & Sy2 & 42.3$^{\rm (*)}$ & 43.4 & 6.15 & (13,14)\
NGC 5128 & 3.7 & 0.8 & Sy2 & 41.9 & 42.9 & 7.84 & (4)\
NGC 5135 & 59.3 & 0.7 & Sy2 & 43.1$^{\rm (*)}$ & 44.4 & 7.29 & (1)\
NGC 5347 & 33.6 & 0.8 & Sy2 & 42.4$^{\rm (*)}$ & 43.5 & 6.97 & (1,15)\
NGC 5506 & 26.6 & 0.3 & Sy1 & 43.0 & 44.3 & 7.95 & (1)\
NGC 5643 & 17.2 & 0.9 & Sy2 & 41.4 & 42.3 & 7.40 & (16,15)\
NGC 7130 & 70.0 & 0.9 & Sy2 & 43.1$^{\rm (*)}$ & 44.4 & 7.59 & (1)\
NGC 7172 & 37.4 & 0.6 & Sy2 & 42.2 & 43.3 & 7.67 & (17)\
NGC 7213 & 25.1 & 0.9 & Sy1 & 42.1 & 43.1 & 7.74 & (5)\
NGC 7469 & 70.8 & 0.7 & Sy1 & 43.3 & 44.7 & 7.08 & (3,4,6)\
NGC 7479 & 34.2 & 0.7 & Sy1 & 42.0 & 43.0 & 7.68 & (7,10)\
NGC 7582 & 22.6 & 0.4 & Sy1 & 41.9$^{\rm (d)}$ & 42.9 & 7.13 & (1,18)\
\[1ex\]
\
[**Notes.**]{}– ${^{\rm (a)}}$Distances from NED. [${^{\rm (b)}}$Compton-thick sources according to @Marinucci2012. Hard X-ray luminosities are corrected by a factor 70.]{} [${^{\rm (c)}}$AGN bolometric luminosities calculated from X-ray luminosities after applying the bolometric corrections of @Marconi2004.]{} [${^{\rm (d)}}$Changing-look AGN [e.g. @Bianchi2005]. Data from an intermediate state.]{} [${^{\rm (*)}}$Compton-thick sources.]{}\
[**References.**]{} (1) Marinucci et al. (2012), (2) Malizia et al. (2007), (3) Dadina et al. (2007), (4) Tueller et al. (2008), (5) Asmus et al. (2011), (6) Risaliti et al. (2005), (7) Brightman et al. (2011), (8) Woo & Urry (2002), (9) Hönig et al. (2010), (10) Diamond-Stanic & Rieke (2012), (11) Wang et al. (2011), (12) Beckmann et al. (2006), (13) Guainazzi et al. (2000), (14) Müller et al. (2003), (15) Beifiori et al. (2009), (16) Guainazzi et al. (2004), (17) Akylas et al. (2001), (18) Piconcelli et al. (2007).
\[table:tab1\]
Sample selection and data analysis {#sec:sec2}
==================================
Sample {#sec:sample}
------
Our sample (see Table \[table:tab1\]) is drawn from the galaxy-magnitude-limited RSA Seyfert sample, which includes the 89 Seyfert galaxies brighter than $B_{T}$=13mag from @Maiolino1995 and @Ho1997. We selected galaxies with existing high angular resolution ($\sim\,0.4-0.8$) mid-IR spectra observed on 8m-class telescopes. The sample contains a total of 29 Seyfert galaxies, of which 16 (55%) are Type 2 and 13 (45%) are Type 1 AGN. We included in the Seyfert 1 category those galaxies classified as Seyfert 1.5, 1.8, and 1.9, as well as those with broad near-IR lines.
We used the hard 2$-$10keV X-ray luminosity (see Table \[table:tab1\] for references) as a proxy for the AGN bolometric luminosities after correcting for absorption and applying the bolometric corrections of @Marconi2004. The high column density in Compton-thick objects (defined as those having [$N_{\rm H}$]{}$>10^{24}\,{\rm cm}^{-2}$, see Table \[table:tab1\]) prevents us from measuring the intrinsic nuclear luminosity below 10 keV. Instead, one can only derive the reflection component from model fitting. Assuming that the \[O[iii]{}\] forbidden line is a tracer of the AGN intrinsic luminosity and comparing it with the observed hard X-ray emission of Compton-thick AGN, @Marinucci2012 derived a correction factor of 70, that we used to correct the observed $2-10\,$keV luminosities of these objects This large correction factor is also theoretically justified by the torus model proposed by @Ghisellini1994. This is the method commonly used for Compton-thick sources and applied in other several works, as in e.g. @Bassani1999 [@Panessa2006]. We expect the nuclear mid-IR and X-ray luminosities to be well correlated [e.g. @Levenson2009; @Asmus2011], which is fulfilled for sources in our sample [@Hoenig2010; @Gonzalez2013]. The only significant outlier, NGC1808, indicates that for this source the AGN does not dominate the continuum mid-IR emission [see figure 5 in @Gonzalez2013]. The uncertainties in $L_{\rm agn}$ are driven by the scatter on the relationship for the bolometric correction which, in general, is significantly larger than the error on the X-ray luminosities. Based on the $L_{\rm agn}$ determination, @Young2010 derived typical uncertainties of 0.4dex [see also @Marinucci2012].
Our sample spans AGN bolometric luminosities in the range $\log {L_{\rm agn}}=41.2-45.5\,{\rm erg \,s}^{-1}$, with a median value of 43.7[erg s$^{-1}$]{}. This is a fair representation of the full RSA Seyfert sample [see figure 1 of @DiamondStanic2012]$^{2}$. As can be seen from Figure \[fig:fig\_Lagn\_hist\], Type 1 and Type 2 sources have similar distributions of $L_{\rm agn}$, with median values (in logarithm scale) of 43.7 and 43.8 erg s$^{-1}$ for Sy1 and Sy2s, respectively. We also list in Table \[table:tab1\] the BH masses of the galaxies in our sample and corresponding references. There is no available black hole mass measurement for NGC 1808. The median value for our sample is 3.9$\times 10^7$[$M_\odot$]{}, which is similar to the 3.2$\times
10^7$[$M_\odot$]{} median value for the complete RSA Seyfert sample [@DiamondStanic2012]. In terms of the Eddington ratio, we sample values of ${L_{\rm agn}/L_{\rm edd}=10^{-4}-0.3}$.
![\[fig:fig\_Lagn\_hist\] Distribution of the AGN bolometric luminosities for Sy1 (black histogram, horizontal filling lines) and Sy2 (red histogram, vertical lines) galaxies in our sample.](Fig1_Hist_Lbol_type.ps)
---------------- ---------------- ------------- ----------
[Galaxy]{} [Instrument]{} [slit]{} [Refs]{}
($\arcsec$)
ESO 323$-$G077 VISIR 0.75 (4)
IC 5063 T-ReCS 0.67 (5,2)
VISIR 1.00 (4)
Mrk 509 VISIR 0.75 (4)
NGC 1068 Michelle 0.36 (6)
VISIR 0.40 (4)
NGC 1365 T-ReCS 0.35 (7,2)
NGC 1386 T-ReCS 0.31 (2)
NGC 1808 T-ReCS 0.35 (8,2)
NGC 2110 Michelle 0.36 (9)
VISIR 0.75 (4)
NGC 2992 Michelle 0.40 (10)
NGC 3081 T-ReCS 0.65 (2)
NGC 3281 T-ReCS 0.35 (2,11)
NGC 3227 VISIR 0.75 (4)
NGC 3783 VISIR 0.75 (4)
NGC 4151 Michelle 0.36 (12)
NGC 4388 Michelle 0.40 (10)
NGC 4507 VISIR 1.00 (4)
NGC 5135 T-ReCS 0.70 (13,2)
NGC 5347 Michelle 0.40 (10)
NGC 5506 T-ReCS 0.36 (14,2)
NGC 5643 T-ReCS 0.36 (2)
VISIR 0.75 (4)
NGC 7130 T-ReCS 0.70 (13,2)
NGC 7172 T-ReCS 0.36 (14,2)
NGC 7213 VISIR 0.75 (4)
NGC 7469 VISIR 0.75 (4)
NGC 7479 T-ReCS 0.35 (2)
NGC 7582 T-ReCS 0.70 (2)
VISIR 0.75 (3)
---------------- ---------------- ------------- ----------
: Ground-based high angular resolution mid-IR spectroscopy
\
[**References.**]{} (1) Roche et al. (2006), (2) González-Martín et al. (2013), (3) Hönig et al. (2008), (4) Hönig et al. (2010), (5) Young et al. (2007), (6) Mason et al. (2006), (7) Alonso Herrero et al. (2012), (8) Sales et al. (2013), (9) Mason et al. (2009), (10) Colling (2011), (11) Sales et al. (2011), (12) Alonso Herrero et al. (2011), (13) Díaz Santos et al. (2010), (14) Roche et al. (2007). \[table:tab2\]
Observations
------------
Ground-based mid-IR spectroscopic observations of the 29 Seyfert galaxies were taken with three different instruments. They operate on 8m-class telescopes and cover the $N$-band, $\sim 8-13\,\mu$m. Table \[table:tab2\] summarizes details of the mid-IR spectroscopic observations, along with references where the data were originally published. Observations taken with the Thermal-Region Camera Spectrograph [T-ReCS @Telesco1998] on the 8.1m Gemini-South Telescope used the low resolution mode, which provides a spectral resolution of $R=\Delta \lambda/\lambda \sim 100$, and slit widths between 0.31 and 0.70. Observations performed by Michelle [@Glasse1997] on the 8.1m Gemini-North telescope, which has a higher spectral resolution by a factor of two ($R\sim 200$), were obtained with slit widths of $\sim 0.4\arcsec$. Finally, observations with the VLT spectrometer and imager for the mid–infrared [VISIR, @Lagage2004] instrument mounted on the 8.2m VLT UT3 telescope at the ESO/Paranal observatory were obtained with the low spectral resolution mode ($R\sim 300$) and a slit width of 0.75 or 1 (and 0.4for NGC1068). For the typical distances of our sample the ground-based slit widths probe typical physical scales of $\sim$65pc. These range from $\sim$7–255pc for all objects except for Mrk509 (545 pc), which is by far the most distant galaxy in the sample.
Sixteen sources were observed with [*Gemini*]{}/T-ReCS [@Gonzalez2013 and references therein]. Thirteen sources have VLT/VISIR observations [see @Hoenig2010 for details], with 4 overlapping with the T-ReCS sample. Finally, 6 Seyfert galaxies were observed with Gemini/Michelle [@Mason2006; @Alonso2011; @Colling2011], of which two sources also have VISIR observations. We refer the reader to the original papers for details on the observations and the data reduction.
We retrieved mid-IR [*Spitzer*]{}/IRS spectra (for all sources except for NGC1068) from the Cornell Atlas of [*Spitzer*]{}/IRS Source [CASSIS v4, @Lebouteiller2011]. We used staring mode observations taken with the short-low (SL) module covering the spectral range $\sim 5-15$. The spectral resolution is $R \sim 60-120$. The CASSIS database provides spectra with optimal extraction regions to ensure the best signal-to-noise ratio and are fully reduced. We only needed to apply a small offset to stitch together the two short-wavelength modules SL1 and SL2 (but note that this does not affect the PAH measurement, see Section \[sec:sec\_measurePAH\]). NGC1068 is part of the GOALS programme [@Armus2009] and IRS SH data have been obtained from the NASA/IPAC Infrared Science Archive (IRSA). Assuming a typical spatial resolution of 3.7for the SL module of IRS given by the slit width, this corresponds to a physical scale of about $\sim$600pc for our sample, i.e. a factor of 10 less resolved than for the ground-based data. Figure \[fig:fig\_example\_spectrum\] shows a comparison of [*Spitzer*]{}against ground-based data of NGC 7130 for illustration. We present the spectra of the full sample in the Appendix (see Figure \[fig:all\_spectra\]).
[![[*Spitzer*]{}/IRS SL spectrum (thick line) from CASSIS compared with the ground-based nuclear T-ReCS spectrum (thin line) from @Gonzalez2013 of NGC 7130, one of the galaxies in our sample. We show the location of the 11.3PAH feature, with the blue shaded area indicating the spectral range used for obtaining the integrated flux. The red lines are the fitted local continua. The rms of the spectrum is shown in yellow. We note that the \[Ne[ii]{}\]12.81 emission line is contaminated by the 12.7PAH feature, which cannot be resolved. The complete sample is shown in Figure \[fig:all\_spectra\] of the Appendix.[]{data-label="fig:fig_example_spectrum"}](NGC7130_fit_example.eps "fig:")]{}\
Measuring the $11.3\,\mu$m PAH feature {#sec:sec_measurePAH}
--------------------------------------
A number of methods have been developed to provide accurate measurements of the PAH feature fluxes, specially for the relatively large spectral range covered by IRS. These include, among others, [PAHFIT]{} [@Smith2007], [DecompIR]{} [@Mullaney2011], and spline fit [e.g. @Uchida2000; @Peeters2002]. They are useful for decomposing IR spectra, especially when the AGN emission is contaminated by extra-nuclear emission. These techniques, however, might not be appropriate for the limited wavelength coverage of ground-based data and/or weak PAH features [see @Smith2007].
We measured the flux of the 11.3 PAH feature following the method described in @Hernan2011. We fitted a local continuum by linear interpolation of the average flux in two narrow bands on both sides of the PAH. We then subtracted this fitted local continuum and integrated the residual data in a spectral range centered on 11.3 ($\lambda_{\rm rest}$=11.05$-$11.55), to obtain the PAH flux. Figure \[fig:fig\_example\_spectrum\] illustrates the method. As can be seen from this figure, this procedure slightly underestimates the PAH feature flux due to losses at the wings of the line profiles and overlaps between adjacent PAH bands. We corrected for these effects by assuming that the line has a Lorentzian profile of known width and applying a multiplicative factor [see @Hernan2011 for details]. We measured the equivalent width (EW) of the 11.3 PAH feature by dividing the integrated PAH flux by the interpolated continuum at the center of the feature. We derived the uncertainties in the measurements by performing Monte Carlo simulations. This was done by calculating the dispersion around the measured fluxes and EWs in a hundred simulations of the original spectrum with random noise distributed as the rms.
Additionally, PAH fluxes measured using a local continuum tend to be smaller than those using a continuum fitted over a large spectral range. To scale up our flux values to the total emission in the PAH features, we used the multiplicative factor of 2. This value was derived by @Smith2007 for the 11.3PAH, after comparing results obtained by the spline fitting and the [PAHFIT]{} full decomposition. For consistency in all measurements, we used the same method described above for both the ground-based and IRS spectra. We detected the 11.3 PAH feature at the $2\sigma$ level or higher significance in all objects of our sample observed with IRS, except for NGC 3783 (see Figs. \[fig:fig\_example\_spectrum\] and \[fig:all\_spectra\]). Our measurements of the $11.3\,\mu$m PAH fluxes agree well with those of @DiamondStanic2012 using [PAHFIT]{}, even though they used their own spectral extraction from IRS data.
Nuclear 11.3 PAH feature emission {#sec:sec3}
=================================
------------------------ ---------- ---------- ------------------------------ ------------- -------------------------- ---------- ---------
(/pc) (10$^{40}$ [erg s$^{-1}$]{}) ($10^{-3}$) ([$M_\odot$]{}yr$^{-1}$)
Circinus${^{\rm (a)}}$ IRS 3.70/75 5.1$\pm$ 0.3 61$\pm$ 1 0.13 0.23 0.83
ESO323-G077 VISIR 0.75/235 $<$ 9.3 $<$ 9 $<$ 0.23 $<$ 0.07 $<$0.56
IC5063 T-ReCS 0.65/153 $<$ 8.4 $<$ 12 $<$ 0.21 $<$ 0.53 …
Mrk509 VISIR 0.75/545 44.5$\pm$ 17.6 10$\pm$ 4 1.11 0.18 …
NGC1068 Michelle 0.36/28 14.4$\pm$ 2.7 9$\pm$ 1 0.36 0.36 …
NGC1365 T-ReCS 0.35/40 $<$ 1.9 $<$ 18 $<$ 0.05 $<$ 0.06 $<$0.97
NGC1386 T-ReCS 0.31/19 $<$ 0.5 $<$ 31 $<$ 0.01 $<$ 0.26 $<$0.91
NGC1808 T-ReCS 0.35/24 8.2$\pm$ 0.5 365$\pm$ 22 0.21 0.17 0.74
NGC2110 VISIR 0.75/121 $<$ 1.3 $<$ 7 $<$ 0.03 $<$ 0.12 …
NGC2992 Michelle 0.40/64 $<$ 3.8 $<$ 30 $<$ 0.09 $<$ 0.14 $<$0.33
NGC3081 T-ReCS 0.65/107 $<$ 1.9 $<$ 18 $<$ 0.05 $<$ 0.25 $<$0.34
NGC3227 VISIR 0.75/60 2.9$\pm$ 0.5 63$\pm$ 11 0.07 0.18 0.58
NGC3281 T-ReCS 0.35/78 $<$ 6.2 $<$ 9 $<$ 0.16 $<$ 0.98 $<$0.40
NGC3783 VISIR 0.75/151 $<$ 3.3 $<$ 6 $<$ 0.08 $<$ 0.79 $<$0.22
NGC4151 Michelle 0.36/25 $<$ 2.4 $<$ 17 $<$ 0.06 $<$ 1.03 $<$0.36
NGC4388 Michelle 0.40/70 $<$ 7.7 $<$ 68 $<$ 0.19 $<$ 0.38 $<$0.18
NGC4507 VISIR 1.00/245 $<$ 4.5 $<$ 5 $<$ 0.11 $<$ 0.12 $<$0.10
NGC4945${^{\rm (b)}}$ IRS 3.70/64 0.4$\pm$ 0.1 358$\pm$ 17 0.01 0.13 0.13
NGC5128${^{\rm (a)}}$ IRS 3.70/66 0.6$\pm$ 0.1 65$\pm$ 2 0.01 0.18 0.43
NGC5135${^{\rm (c)}}$ T-ReCS 0.70/200 5.9$\pm$ 2.2 34$\pm$ 12 0.15 0.04 0.63
NGC5347 Michelle 0.40/65 $<$ 6.5 $<$ 56 $<$ 0.16 $<$ 0.76 $<$0.64
NGC5506 T-ReCS 0.35/45 6.4$\pm$ 2.8 15$\pm$ 6 0.16 0.32 0.40
NGC5643 VISIR 0.75/62 1.6$\pm$ 0.2 49$\pm$ 4 0.04 0.24 0.29
NGC7130${^{\rm (c)}}$ T-ReCS 0.70/236 47.1$\pm$ 3.3 166$\pm$ 11 1.18 0.22 1.11
NGC7172 T-ReCS 0.35/63 $<$ 2.8 $<$ 43 $<$ 0.07 $<$ 0.16 $<$0.27
NGC7213 VISIR 0.75/91 $<$ 0.7 $<$ 8 $<$ 0.02 $<$ 0.14 $<$0.06
NGC7469 VISIR 0.75/255 47.7$\pm$ 4.7 31$\pm$ 3 1.19 0.10 0.82
NGC7479 T-ReCS 0.35/58 $<$ 5.6 $<$ 37 $<$ 0.14 $<$ 0.82 …
NGC7582 T-ReCS 0.70/76 3.9$\pm$ 0.7 50$\pm$ 9 0.10 0.30 0.41
------------------------ ---------- ---------- ------------------------------ ------------- -------------------------- ---------- ---------
\
[**Notes.**]{}—\
The 11.3 PAH luminosities include a multiplicative factor of 2 for comparison with [PAHFIT]{} measurements (see Section \[sec:sec\_measurePAH\]).\
The values of the EW and $f_{11.3\mu{\rm m \,PAH}}/f_{\rm [NeII]}$ are derived from measurements using the fitted local continuum. Upper limits at a 2$\sigma$ significance are included for non detections with the $<$ symbol.\
${^{\rm (a,b)}}$Circumnuclear SFRs from [*ISO*]{} 11.3 PAH measurements from ${^{\rm (a)}}$@Siebenmorgen2004 and ${^{\rm (b)}}$@Galliano2008, for 24 and 20 apertures, respectively.\
${^{\rm (c)}}$Values for the [\[Ne[ii]{}\]]{}line from @DiazSantos2010.\
\[table:tab3\]
In this section we investigate the nuclear 11.3 PAH feature emission in our sample of galaxies. Hereinafter, nuclear scales generally refer to the physical regions observed with the T-ReCS/Michelle/VISIR instruments, whereas circumnuclear scales are those probed with the IRS spectroscopy. The only exceptions are the most nearby (distances of $\sim 4\,$Mpc) galaxies Circinus, NGC 4945, and NGC 5128. To explore similar physical regions in comparison to the rest of the sample we used the IRS observations as our nuclear spectra for the three galaxies. For these sources the circumnuclear data are from @Siebenmorgen2004 and @Galliano2008 (see Table \[table:tab3\] for more information).
The median value of the nuclear physical sizes probed with our data is 65pc (see Table \[table:tab3\]). If two nuclear spectra exist for the same galaxy, we used the one sampling a physical scale closest to the median value, for consistency with the rest of the galaxies. This information along with the physical scales probed with the nuclear spectra are given in Table \[table:tab3\]. Note that this is approximately a factor of 10 improvement in physical resolution with respect to the circumnuclear median value of 600pc. Hereinafter, we will use the term *size* as referring to the physical scale probed, which is determined by the slit widths of the observations.
Detection of the $11.3\,\mu$m PAH feature {#sec:detection_PAH}
-----------------------------------------
The 11.3PAH feature is weak or it might not even be present in a large fraction of galaxies in our sample, as can be seen from the nuclear spectra in Figs. \[fig:fig\_example\_spectrum\] and \[fig:all\_spectra\]. We deemed the feature as detected if the integrated flux is, at least, two times above the corresponding measured error. This is equivalent to having the PAH feature detected with a significance of $2{\rm \sigma}$ or higher. The non-detections are given as upper limits at a $2\sigma$ level, that is, with a 95% probability that the real flux is below the quoted value.
Table \[table:tab3\] gives the nuclear luminosities and EWs of the 11.3 PAH detections, as well as $2\sigma$ upper limits for the remaining objects. Note that the flux of the 11.3PAH feature is not corrected for extinction. Thus, its proper characterisation might be hampered in cases of high extinction, i.e. when the PAH molecules are embedded behind the silicate grains and the feature is buried within the silicate absorption at $\sim$9.7. This also depends on the location of the material causing the extinction relative to the PAH emitting region. Another additional complication is the presence of crystalline silicate absorption at around 11, which has been detected in local ultraluminous infrared galaxies [@Spoon2006] and in local Seyferts [@Roche2007]. In particular, Colling (2011) detected crystalline silicate absorption that could be blended with the 11.3PAH feature in some of the galaxies in our sample, namely, NGC 4388, NGC 5506, NGC 7172, and NGC 7479 (see also Section \[sec:sec\_stacking\]).
Using T-ReCS/VISIR/Michelle data we detected nuclear 11.3 PAH emission in 10galaxies. For the three most nearby Seyferts, the 11.3 PAH feature is detected in the IRS observations, while in the corresponding T-ReCS/VISIR spectra the feature is below the detection limits. Taking this into account, we detected nuclear 11.3 PAH feature emission in 13 out of 29 galaxies ($45\%$ of the sample). The detection rate is similar for Seyfert 1s and Seyfert 2s (40% and 50%, respectively). The observed EWs of the feature (using the fitted local continuum) are between $\sim 0.01-0.4$. These values are much lower than those typical of high metallicity star forming galaxies [see @Hernan2011]. This is expected given that we are probing smaller regions around the nucleus, and probably the continuum emission is mostly arising from dust heated by the AGN.
To study a possible extra-nuclear origin of the PAH feature we investigated the morphology of the galaxies in our sample. We compiled the [*b/a*]{} axial ratio (measurements from NED, RC3 $D_{25}/R_{25}$ isophotal $B$-band diameters) to determine the inclination of the host galaxy, where $b$ and $a$ are the minor and mayor axis, respectively. Axial ratios $b/a\,<$0.5 are considered as edge-on galaxies, whereas face-on galaxies have $b/a\,>$0.5 (see Table \[table:tab1\]). With this definition we find 11 edge-on and 18 face-on galaxies. Out of the 13 sources with detection of the nuclear 11.3PAH feature, we find 5 (44%) edge-on and 8 (45%) face-on galaxies. We do not find that a positive detection predominates in edge-on galaxies, where material of the host galaxy along our line of sight could be misinterpreted as nuclear SF. However, we cannot rule out a dominant contribution from extra-nuclear SF in the most edge-on galaxies in our sample. @Gonzalez2013 found that the host galaxies could significantly contribute to the nuclear component for sources with the deepest silicate absorption features.
The majority of the nuclear 11.3 PAH detections in our sample are galaxies with well-documented nuclear starbursts and/or [*recent*]{} SF activity based on UV and optical observations [@GonzalezDelgado1998 NGC 5135, NGC 7130], modelling of the optical spectra [@StorchiBergmann2000 NGC 5135, NGC 5643, NGC 7130, NGC 7582], and near-IR integral field spectroscopy [@Davies2007; @Tacconi2013 Circinus, NGC 1068, NGC 3227, NGC 3783, NGC 5128].
![\[fig:fig\_stacked\] Result from the stacking of nuclear spectra (from T-ReCS, Michelle, and VISIR) without 11.3 PAH detections. The stacked spectrum for sources with weak silicate features (six galaxies, thick line) shows a $2\sigma$ detection of the 11.3 PAH feature. In the stacked spectrum (thin line) of the seven galaxies with deep silicate features the PAH feature remains undetected. The flux density units are arbitrary. The individual spectra were normalized at 12. See Section \[sec:sec\_stacking\] for details.](stck_fit.eps)
Stacking nuclear spectra with undetected $11.3\,\mu$m PAH emission {#sec:sec_stacking}
------------------------------------------------------------------
In this section we further investigate those galaxies with weak or no detected 11.3 PAH feature emission. We stacked the individual spectra deemed to have undetected 11.3 PAH features according to our $2\sigma$ criterion (see Table \[table:tab3\] and Section \[sec:detection\_PAH\]). We divided them in two groups. The first includes galaxies with a weak silicate feature: ESO 323-G077, NGC 1365, NGC 3081, NGC 3783, NGC 4151, and NGC 4507. The second group contains galaxies with relatively deep silicate features: NGC 1386, NGC 2992, NGC 3281, NGC 4388, IC 5063, NGC 7172 and NGC 7479. We excluded from the stacking NGC 2110 and NGC 7213 because the silicate feature is strongly in emission, and NGC 5347 because the spectrum is very noisy. We normalized the spectra at 12 and then used the [IRAF]{} task [scombine]{} with the *average* option to combine the different observations.
Figure \[fig:fig\_stacked\] shows the stacked spectra for the two groups. We applied the same method as in Section \[sec:detection\_PAH\] to determine if the PAH feature is detected. We found that the 11.3 PAH appears detected in the stacked nuclear spectrum of the galaxies with weak silicate features at a $2\sigma$ level. The derived EW of the 11.3 PAH is $8\times 10^{-3}$. The feature remains undetected in the stacked nuclear spectrum of sources with deep silicate features. This could be explained in part as due to extinction effects, given that the silicate absorption in these galaxies likely comes from cold foreground material [@Goulding2012]. We also note that the minimum around 11 in the stacked spectrum of galaxies with deep silicate features could be from crystalline silicates. Indeed, @Colling2011 found that inclusion of crystalline silicates improved the fit of the silicate features in NGC 7172, NGC 7479, and NGC 4388. Therefore, it would be expected to also appear in the stacked spectrum.
Is the $11.3\,\mu$m PAH feature suppressed in the vicinity of AGN?
------------------------------------------------------------------
It has been known for more than 20 years now that PAH emission is weaker in local AGN than in high metallicity star forming galaxies, although some AGN do also show strong PAH features on circumnuclear scales [@Roche1991]. It is not clear, however, if the decreased detection of PAH emission and the smaller EWs of the PAH features in AGN are due to 1) an increased mid-IR continuum arising from the AGN, 2) destruction of the PAH carriers in the harsh environment near the AGN [@Roche1991; @Voit1992] or 3) decreased SF in the nuclear region [@Hoenig2010]. Additionally, there is a prediction that smaller PAH molecules would be destroyed more easily in strong radiation fields [see e.g. @Siebenmorgen2004], also indicating that different PAHs may behave differently.
The effects of an increasing continuum produced by the AGN is clearly seen in the 3.3 PAH map of the central region of NGC 5128. The ratio of the feature-to-continuum (i.e., the EW of the feature) decreases towards the AGN, whereas the feature peaks in the center [see @Tacconi2013 for more details]. This implies that the PAH molecules are not destroyed in the harsh environment around the AGN of this galaxy (see also Section \[sec:sec\_PAHshielded\]). Similarly, @DiazSantos2010 showed that at least the molecules responsible for the 11.3 PAH feature can survive within $<$100pc from the AGN.
Some recent observational works reached apparently opposing conclusions on the PAH emission of AGN on physical scales within a few kpc from the nucleus, but note that these are for much larger physical scales than those probed here. @DiamondStanic2010 demonstrated for the RSA Seyferts that the 11.3PAH feature is not suppressed, whereas other mid-IR PAH features are. @LaMassa2012, on the other hand, combined optical and mid-IR spectroscopy of a large sample of AGN and star forming galaxies and concluded that in AGN-dominated systems (higher luminosity AGN) the 11.3PAH feature does get suppressed.
To investigate this issue, we plotted in Figure \[fig:fig\_Lagn\] the observed nuclear EW of the 11.3 PAH feature against the AGN bolometric luminosity for our sample, with smaller symbols indicating regions closer to the AGN. This figure does not show any clear trend. If the decreased nuclear EW of the 11.3PAH feature were due to the AGN mid-IR continuum in more luminous AGN, we would expect a trend of decreasing EW for increasing AGN bolometric luminosities. Alternatively, we would expect the same trend if the PAH molecules responsible for the 11.3 feature were suppressed/destroyed more easily in luminous AGN. From this figure we can see that at a given AGN luminosity we sometimes detect nuclear 11.3 PAH emission, whereas in other cases we do not. In other words, we do not see clear evidence in our sample for the 11.3 PAH feature to be suppressed in more luminous AGN, at least for the AGN bolometric luminosities covered in our sample of Seyfert galaxies.
![\[fig:fig\_Lagn\] Nuclear EW of the 11.3 PAH feature versus the AGN bolometric luminosity. Filled star symbols are detections whereas open squares are upper limits. The sizes/colors of the symbols (see figure legend) indicate the different physical sizes probed, which are determined by the slit widths of the observations. Hereinafter, we have marked Compton-thick objects in all plots using a double-star or a double-square for detections or upper limits, respectively.](Fig4_EW_vs_Lagn.ps)
As can be seen from Figure \[fig:fig\_Lagn\], there is no clear influence of the probed physical region sizes on the observed EWs. Hence, we do not see a tendency for the EW of the PAH feature to decrease for smaller physical regions. This would be the case if we were to expect a higher AGN continuum contribution and/or PAH destruction as we get closer to the AGN. No trend is either present when plotting the observed EWs with respect to luminosity densities. We note, however, that for the three closest Seyferts (Circinus, NGC 4945, NGC 5128) the 11.3 PAH feature is not detected in ground-based high-resolution T-ReCS spectra, which probe scales of $\sim7-15\,$pc for these sources [@Roche2006; @Gonzalez2013].
 
Nuclear PAH molecules shielded by the dusty torus? {#sec:sec_PAHshielded}
--------------------------------------------------
As we have shown in the previous sections and as presented by others [e.g., @Miles1994; @Marco2003; @DiazSantos2010; @Hoenig2010; @Gonzalez2013; @Sales2013; @Tacconi2013] PAH emission is detected in the vicinity (from tens to a few hundreds parsecs) of the harsh environments of some AGN. Therefore, at least in some galaxies, the PAH molecules are not destroyed (or at least not completely) near the AGN. They must be shielded from the AGN by molecular material with sufficient X-ray absorbing column densities [@Voit1992; @Miles1994; @Watabe2008]. As pointed out by @Voit1992b, for PAH features to be absent due to destruction, they have to be fragmented more quickly than they can be rebuilt. In other words, PAHs will exist if the rate of reaccretion of carbon onto the PAHs is higher than the evaporation rate caused by the AGN. Using the parameters of the @Voit1991 model, @Miles1994 estimated the column density of hydrogen required to keep the evaporation rate of PAHs below the rate of reaccretion of carbon onto the PAHs. As derived in @Miles1994, the time scale for X-ray absorption in terms of the hydrogen column density of the intervening material $N_{\rm H}$(tot), the distance from the AGN fixed in our case by the slit width $D_{\rm agn}$, and the X-ray luminosity of the AGN, can be written as
$$\tau \approx 700\,{\rm yr} \left(\frac{N_{\rm H}{\rm (tot)}}{10^{22}\,{\rm cm}^{-2}}\right)^{1.5}
\left(\frac{D_{\rm agn}}{\rm
kpc}\right)^2 \left(\frac{10^{44}\,{\rm erg \,s}^{-1}}{L_{\rm X}}\right).
\label{eq:eq_PAHtimescale}$$
@Voit1992b estimated that the time scale needed for reaccretion of a carbon atom on to a fractured PAH should be at least 3000 years for the typical conditions of the interstellar medium.
The protecting material, which has to be located between the nuclear sites of SF and the AGN, is likely to be that in the dusty torus postulated by the unified model. In a number of works [@RamosAlmeida2009; @RamosAlmeida2011A; @Alonso2011; @AlonsoHerrero2012] we have demonstrated that the clumpy torus models of @Nenkova2008a [@Nenkova2008b] accurately reproduce the nuclear infrared emission of local Seyfert galaxies. These models are defined by six parameters describing the torus geometry and the properties of the dusty clouds. These are, the viewing angle ($i$) and radial extent ($Y$) of the torus, the angular ($\sigma$) and radial distributions ($q$) of the clouds along with its optical depth ($\tau_{\rm V}$), and the number of clouds along the equatorial direction ($N_{\rm 0}$). The optical extinction of the torus along the line of sight is computed from the model parameters as $A_{V}^{\rm LOS} = 1.086 N_{\rm 0} \tau_{\rm V} e^{(-(i-90)^2/\sigma^2)}$mag. According to @Bohlin1978, the absorbing hydrogen column density is then calculated following $N_{\rm H}^{\rm LOS}/A_{V}^{\rm LOS}$ = $1.9 \times 10^{21} \, {\rm cm}^{-2}\, {\rm mag}^{-1}$.
With the derived torus model parameters, we estimated the hydrogen column density of the torus material in our line of sight for a sample of Seyferts. This typically ranges from $N_{\rm H}^{\rm LOS}\simeq 10^{23}$ to a few times $10^{24}\,{\rm cm}^{-2}$ [see @RamosAlmeida2009; @RamosAlmeida2011A; @Alonso2011; @AlonsoHerrero2012 for further details]. Using Equation \[eq:eq\_PAHtimescale\] for the hard X-ray luminosities (Table \[table:tab1\]) and distances from the AGN probed by our spectroscopy (see Table \[table:tab3\]), and setting $\tau$=3000yr as proposed by @Voit1992, we require hydrogen column densities of at least a few $10^{23}\,{\rm cm}^{-2}$ to protect the PAHs from the AGN radiation. Evidence for such values for the [$N_{\rm H}$]{} are found for our sample, as we derived absorbing columns of the order of $10^{23}\,{\rm cm}^{-2}$ or even higher.
In @RamosAlmeida2011A, we also demonstrated that Seyfert 2s are more likely to have higher covering factors than Seyfert 1s. Assuming that the nuclear SF occurs inside the torus, the PAH molecules may be more shielded in the nuclear region of Seyfert 2s. However, even the [$N_{\rm H}$]{}values along our line of sight from the torus model fits of Seyfert 1s (which would be a lower limit to the total [$N_{\rm H}$]{} in the torus) with 11.3 PAH detections are sufficient to protect the PAH carriers. This is the case for four Seyfert 1s (NGC3227, NGC5506, NGC7469, and NGC7582), as can be seen from the modelling by @Alonso2011. Note that the column densities that we refer to are not only those absorbing the X-rays but also including material much farther away from the accreting BH.
Another interesting aspect to keep in mind from Equation \[eq:eq\_PAHtimescale\] is that the column densities needed to protect the PAH molecules from the AGN X-ray emission become higher for more luminous AGN as well as for distances closer to the AGN. However, we emphasize that for the AGN luminosities of the RSA Seyferts and distances from the AGN probed by the observations presented here, the PAH molecules are likely to be shielded from the AGN by material in the torus residing on smaller scales. Also, part of the obscuring material even on these nuclear scales can reside in the host galaxy as shown by @Gonzalez2013. Thus, another source of opacity that might prevent the PAHs from being destroyed are dust lanes in galaxies or dust in the nuclear regions of merger systems. This might be the case for five sources in our sample, namely NGC4945, NGC5128, NGC5506, NGC7130 and NGC7582.
Nuclear Star Formation Rates in Seyfert galaxies
================================================
Relation between the $11.3\,\mu$m PAH feature and the \[Ne[ii]{}\]$12.81\,\mu$m emission line on nuclear scales {#sec:sec4_1}
---------------------------------------------------------------------------------------------------------------
In star forming galaxies, the luminosity of the \[Ne[ii]{}\] 12.81 emission line is a good indicator of the SFR [@Roche1991; @Ho2007; @DiazSantos2010]. We note that these measurements are contaminated by the 12.7 PAH feature, which is not easily resolvable.
In Seyfert galaxies the situation is more complicated because this line can be excited by both SF and AGN activity. The AGN contribution to the \[Ne[ii]{}\] varies from galaxy to galaxy in local Seyfert galaxies and other AGN [see e.g. @Melendez2008; @PereiraSantaella2010]. For the RSA Seyfert sample, @DiamondStanic2010 used IRS spectroscopy to compare the circumnuclear SFRs computed with the \[Ne[ii]{}\] line and the 11.3 PAH feature as a function of EW of the PAH. They found that the ratio of the two circumnuclear SFRs (on a kpc scale) is on average unity, with some scatter for galaxies with large PAH EWs $>$ 0.3. The most discrepant measurements were for those galaxies with elevated \[O[iv]{}\]/\[Ne[ii]{}\] and low EW of the PAH, that is, AGN dominated galaxies.
In Figure \[fig:fig\_lineratio\_Lagn\] (left panel) we show the observed nuclear PAH/\[Ne[ii]{}\] ratio as a function of the AGN luminosity. To correct for the 12.7PAH contamination, we have used a median ratio of the 11.3 versus the 12.7PAH features of 1.8, derived in @Smith2007, and subtracted it from the [\[Ne[ii]{}\]]{}measurement. NGC 1365, NGC 1386 and NGC 5347 are not included in the plot because both lines are undetected and, therefore, the value in the Y axis is completely unconstrained. We derived the [\[Ne[ii]{}\]]{}fluxes using the same technique as explained for the PAH in Section \[sec:sec\_measurePAH\], integrating the line between 12.6 and 12.9. For seven galaxies, namely IC5063, Mrk509, NGC1068, NGC2110, NGC5135, NGC7130 and NGC7479, the [\[Ne[ii]{}\]]{}line falls outside the wavelength range covered by our observations. They are not included in the plot except for NGC5135 and NGC7130, whose values have been extracted from @DiazSantos2010. Most of the galaxies with detections of the nuclear 11.3 PAH feature and [\[Ne[ii]{}\]]{} show ratios similar to those of high metallicity star forming galaxies [$\sim 0.7-2$, see e.g., @Roche1991; @DiazSantos2010], even if the \[Ne[ii]{}\] fluxes are not corrected for AGN emission. On the other hand, most of the nuclear spectra with non detections show upper limits to the PAH/\[Ne[ii]{}\] ratio below 0.5.
To derive the nuclear \[Ne[ii]{}\] flux solely due to star formation, we can use the fractional SF contribution to the \[Ne[ii]{}\] line within the IRS aperture estimated by @Melendez2008. For sources with strong AGN contribution (higher than 50%), we estimated the \[Ne[ii]{}\] flux coming from the AGN, which can be subtracted from the observed nuclear \[Ne[ii]{}\] flux. This is shown in Figure \[fig:fig\_lineratio\_Lagn\] (right panel). We note that for NGC 2992, NGC 3227, NGC 4151, NGC 5506 and NGC 7172, the estimated AGN contribution to the total [\[Ne[ii]{}\]]{} is higher than the nuclear [\[Ne[ii]{}\]]{}value, indicating that the AGN \[NeII\] contributions were overestimated. For these galaxies, we did not apply any correction. Figure \[fig:fig\_lineratio\_Lagn\] shows that for those Seyferts with a nuclear 11.3 PAH detection the PAH/\[Ne[ii]{}\]$_{\rm SF}$ ratio does not decrease with the AGN bolometric luminosity. This would be expected if the PAH emission was to be suppressed. Therefore, given that the PAH/\[Ne[ii]{}\]$_{\rm SF}$ ratio does not show a dependence on $L_{\rm agn}$, we conclude that the 11.3 PAH feature emission can be used to estimate the nuclear SFRs (see next section).
Circumnuclear ($\sim 600\,$pc) vs Nuclear ($\sim 65\,$pc) scales {#sec:sec4}
----------------------------------------------------------------
In the vicinity of an AGN, we expect the chemistry and/or the heating as dominated by X-rays from the so-called X-ray dominated regions (XDRs). In principle, XDRs could also contribute to PAH heating through the photodissociation and photoionization by FUV photons produced via excitation of H and H$_2$ in collisions with secondary electrons. However, as we derived in Section \[sec:sec\_PAHshielded\], the torus appears to provide the appropriate environment to shield the PAH molecules from the AGN emission, on typical physical scales of a few parsecs up to a few tens of parsecs. We thus expect little or no contribution of UV AGN produced photons to the PAH heating in the nuclear scales of Seyfert galaxies. Hereinafter, we will assume that the aromatic molecules are heated by the radiation field produced by young massive stars and that the 11.3 PAH luminosity can be used to estimate the SFR.
We derived nuclear and circumnuclear SFRs using the PAH 11.3 feature luminosities and applying the relation derived in @DiamondStanic2012
$$\begin{aligned}
\label{eq:sfr_pah}
{\rm SFR} \left( M_\odot \, {\rm yr}^{-1} \right) = 9.6 \times 10^{-9}
L \left({\rm PAH}_{11.3\,\mu m}, L_{\odot} \right)\end{aligned}$$
using [PAHFIT]{} measurements of galaxies with IR (8–1000) luminosities $L_{\rm IR}<
10^{11}\,L_\odot$, using the @Rieke2009 templates and a Kroupa IMF. This is appropriate for our sample, as the median value of the IR luminosity of the individual galaxies is $5\times
10^{10}\,L_\odot$. The uncertainties in the derived SFRs using Equation \[eq:sfr\_pah\] are typically 0.28dex [see @DiamondStanic2012 for full details].
For the 13 galaxies with nuclear 11.3 PAH detections, the nuclear SFRs span two orders of magnitude between $\sim$0.01–1.2$M_\odot\,{\rm yr}^{-1}$ for regions of typically $\sim$65pc in size. The non-detections indicate that the nuclear SFR of RSA Seyfert galaxies from the high angular resolution spectroscopy can be $\sim0.01-0.2\,M_\odot\,{\rm
yr}^{-1}$, or even lower.
The projected nuclear SFR densities are between 2 and 93[$M_\odot$]{}yr$^{-1}\,{\rm kpc}^{-2}$ with a median value 22[$M_\odot$]{}yr$^{-1}\,{\rm kpc}^{-2}$. These are consistent with the simulations of @Hopkins2012b for similar physical scales. The two galaxies with the largest SFR densities are NGC1068 and NGC1808 with values of 414 and 329[$M_\odot$]{}yr$^{-1}\,{\rm kpc}^{-2}$, respectively. We notice that for those galaxies in common with @Davies2007 (namely Circinus, NGC 1068, NGC 3783, NGC 7469, NGC 3227) we find quite discrepant values for the SFR density, with ours lying below those in @Davies2007 except for NGC1068. It might be due to the use of different SF histories and SFR indicators. We note that with the 11.3PAH feature we cannot explore age effects [see @DiazSantos2008 Figure 8] as this feature can be excited by both O and B stars, and thus it integrates over ages of up to a few tens of millions of years [@Peeters2004], unlike the measurements in @Davies2007 that sample younger populations. In addition, we detect neither nuclear nor circumnuclear SF in NGC 3783 based on the PAH measurements. On the other hand, PAHs can also be found in the interstellar medium (ISM) as being excited in less-UV rich environments, such as reflection nebulae [e.g. @Li2002]. However, the decreased strength of the IR emission features in these objects seems to indicate the low efficiency of softer near-UV or optical photons in exciting PAHs in comparison to SF [@Tielens2008].
The circumnuclear SFRs in our sample are between 0.2 and 18.4[$M_\odot$]{}yr$^{-1}$ [see also @DiamondStanic2012], and the median circumnuclear SFR densities are 1.2[$M_\odot$]{}yr$^{-1}$ kpc$^{-2}$. These are similar to those of the CfA [@Huchra1992] and 12[@Rush1993] samples, derived using the 3.3 PAH feature [see @Imanishi2003; @Imanishi2004].
The comparison between the nuclear and circumnuclear SFRs for our sample clearly shows that, in absolute terms, the nuclear SFRs are much lower (see Table \[table:tab3\]). This is in good agreement with previous works based on smaller samples of local AGN [e.g. @Siebenmorgen2004; @Watabe2008; @Hoenig2010; @Gonzalez2013]. The median value of the ratio between the nuclear and circumnuclear SFRs for the detections of the 11.3feature is $\sim$0.18 (see also Table \[table:tab3\]), with no significant difference for type 1 and type 2 Seyferts ($\sim 0.18$ and $\sim 0.21$, respectively).
In Figure \[fig:fig\_sfr\] we plot the nuclear and circumnuclear SFRs probing typical physical scales of $\sim$65pc and $\sim$600pc, respectively. Again, non-detections are plotted as upper limits at the 2$\sigma$ level. Overall, for our detections, the fraction of the SFR accounted for by the central $\sim 65$pc region of our Seyferts ranges between $\sim$5–35% of that enclosed within the aperture corresponding to the circumnuclear data.
![Comparison between the SFR on different scales, where SFR$_{\rm circumnuclear}$ implies typical physical scales of $\sim$0.6kpc and SFR$_{\rm nuclear}$ is for $\sim$65pc scales. Non-detections are plotted at a $2\sigma$ level. Symbols as in Figure \[fig:fig\_Lagn\]. The dashed line shows the median value of the nuclear/circumnuclear SFR ratio for the detections of the nuclear PAH feature (see text and Table \[table:tab3\]). The dotted line indicates a nuclear/circumnuclear ratio of one.[]{data-label="fig:fig_sfr"}](Fig6_SFR_circ_vs_nuc.ps)
While the nuclear SFRs are lower than the circumnuclear SFRs, the median nuclear projected SFR densities are approximately a factor of 20 higher than the circumnuclear ones in our sample (median values of 22 and 1[$M_\odot$]{}yr$^{-1}$ kpc$^{-2}$, respectively). This shows that the SF is not uniformly distributed. Conversely, it is more highly concentrated in the nuclear regions of the RSA Seyferts studied here. This is in agreement with simulations of @Hopkins2012b. The molecular gas needed to maintain these nuclear SFR densities appears to have higher densities in Seyfert galaxies than those of quiescent (non Seyferts) galaxies [@Hicks2013].
![Observed nuclear SFR vs. $\dot M_{\rm BH}$ relation. Predictions from @Hopkins2010 are shown as dashed lines. We show the $\dot M_{\rm BH}$$\approx$0.1$\times$SFR relation, which is expected for r$<$100pc, and the 1:1 relation which, is expected for the smallest physical scales ($r<10\,$pc). The solid line represents the fit to our detections of the nuclear 11.3PAH feature (see text for details).[]{data-label="fig:fig_sfr_bhar"}](Fig7_SFR_vs_BHAR_linfit.ps "fig:")
Nuclear star formation rate vs. black hole accretion rate {#sec:sec4}
---------------------------------------------------------
@Hopkins2010 performed smoothed particle hydrodynamic simulations to study the inflow of gas from galactic scales ($\sim$10kpc) down to $\lesssim$0.1pc, where key ingredients are gas, stars, black holes (BHs), self-gravity, SF and stellar feedback. These numerical simulations indicate a relation (with significant scatter) between the SFR and $\dot M_{\rm BH}$ that holds for all scales, and that is more tightly coupled for the smaller physical scales. The model of @Kawakatu2008 predicts that the AGN luminosity should also be tightly correlated with the luminosity of the nuclear (100pc) SF in Seyferts and QSOs, and also that $L_{\rm nuclear,SB}$/$L_{\rm AGN}$ is larger for more luminous AGN.
According to @Alexander2012 [and references therein], $\dot M_{\rm BH}$ and AGN luminosities follow the relation
$$\label{eq:BHAR}
\dot M_{\rm BH} ({M}_\odot\,{\rm yr}^{-1})=0.15(0.1/\epsilon)(L_{\rm agn}/10^{45}\,{\rm erg\,s}^{-1})$$
where we used $\epsilon=0.1$ as the typical value for the mass-energy conversion efficiency in the local Universe [@Marconi2004]. We obtained $\dot M_{\rm BH}$ ranging between $5 \times 10^{-6}$ and $0.5\,M_\odot\,{\rm yr}^{-1}$ for our sample. Uncertainties in the $\dot M_{\rm BH}$ estimations are dominated by those in $L_{\rm agn}$, i.e. 0.4dex, as mentioned in Section \[sec:sample\].
Figure \[fig:fig\_sfr\_bhar\] shows the observed nuclear SFR against $\dot M_{\rm BH}$ for the Seyferts in our sample. The different sizes of the symbols indicate different physical sizes of the probed regions. The prominent outlier in this figure is NGC 1808. We also show in Figure \[fig:fig\_sfr\_bhar\] as dashed lines predictions from the @Hopkins2010 simulations for $r<100\,$pc ($\dot M_{\rm BH}$$\,\approx$0.1$\times$SFR) and $r<10\,$pc ($\dot M_{\rm BH}$$\,\approx$SFR). These radii encompass approximately the physical scales probed by our nuclear SFR. The prediction from the @Kawakatu2008 disk model for Seyfert luminosities and BH masses similar to those of our sample falls between the two dashed lines ($\sim$ 1:0.4 relation).
To derive a possible correlation between the nuclear SFR and $\dot M_{\rm BH}$, we applied a simple fit to the nuclear SFR detections (excluding NGC 1808) and obtained a nearly linear relation (slope of 1.01, and uncertainties of 0.4dex in both parameters, Equation \[eq:eq\_fit\]), which is close to the 1:0.1 relation (see Figure \[fig:fig\_sfr\_bhar\] and below).
$$\begin{aligned}
{\log} \,{\dot M}_{\rm BH} = 1.01 \times {\log \, \rm SFR}_{\rm nuclear} - 1.11
\label{eq:eq_fit}\end{aligned}$$
Also including the upper limits in the fit we obtained a very similar result (slope of 0.95). In contrast, @DiamondStanic2012 obtained a slightly superlinear relation ($\dot M_{\rm BH} \propto {\rm SFR}^{1.6}$), when the SFRs are measured in regions of 1kpc radius. This behaviour (i.e., the relations becoming linear on smaller scales) is nevertheless predicted by the @Hopkins2010 simulations.
As can be seen in Figure \[fig:fig\_sfr\_bhar\], the @Hopkins2010 predictions for $r<100\,$pc reproduce fairly well the observed relation for our sample. We do not find a tendency for galaxies with SFRs measured in regions closer to the AGN (slit sizes of less than 100pc) to have larger $\dot M_{\rm BH}$ to SFR ratios (i.e., to lie closer to the 1:1 relation) than the rest, as predicted by @Hopkins2010. It is worth noting that these authors caution that their work do not include the appropriate physics for low accretion rates ($<< 0.1\,M_\odot \,{\rm yr}^{-1}$). The scatter in the theoretical estimations and the limited size of our sample of Seyfert galaxies prevent us from further exploring this issue. Future planned observations with the mid-IR CanariCam [@Telesco2003] instrument on the 10.4m Gran Telescopio de Canarias (GTC) will allow a similar study for larger samples of Seyfert galaxies.
Summary and conclusions
=======================
We have presented the largest compilation to date of high angular resolution ($0.4-0.8$) mid-IR spectroscopy of nearby Seyfert galaxies obtained with the T-ReCS, VISIR, and Michelle instruments. We used the 11.3PAH feature to study the nuclear SF activity and its relation to the circumnuclear SF, as well as with $\dot M_{\rm BH}$. The sample includes 29 Seyfert galaxies (13 Seyfert 1 and 16 Seyfert 2 galaxies) belonging to the nearby RSA AGN sample [@Maiolino1995; @Ho1997]. It covers more than two orders of magnitude in AGN bolometric luminosity, with the galaxies located at a median distance of 33Mpc. Our data allow us to probe typical nuclear physical scales (given by the slit widths) of $\sim$65pc. We used the hard X-ray luminosity as a proxy for the AGN bolometric luminosity and $\dot M_{\rm BH}$. We used mid-infrared Spitzer/IRS spectroscopy to study the SF taking place in the circumnuclear regions (a factor of 5–10 larger scales).
The main results can be summarized as follows:
1. The detection rate of the nuclear 11.3 PAH feature in our sample of Seyferts is 45% (13 out of 29 sources), at a significance of 2$\sigma$ or higher. Additionally, the stacked spectra of six galaxies without a detection of the 11.3 PAH feature and weak silicate features resulted into a positive detection of the 11.3 PAH feature above 2$\sigma$.
2. There is no evidence of strong suppression of the nuclear 11.3 PAH feature in the vicinity of the AGN, at least for the Seyfert-like AGN luminosities and physical nuclear regions (65pc median value) sampled here. In particular, we do not see a tendency for the EW of the PAH to decrease for more luminous AGN. The hydrogen column densities predicted from clumpy torus model fitting (a few $10^{23}\,{\rm cm}^{-2}$ up to a few $10^{24}\,{\rm cm}^{-2}$) would be, in principle, sufficient to shield the PAH molecules from AGN X-ray photons in our Seyfert galaxies.
3. The nuclear SFRs in our sample derived from the 11.3 PAH feature luminosities are between 0.01 and 1.2$M_\odot\,{\rm yr}^{-1}$, where we assumed no XDR-contribution to the PAH heating. There is a significant reduction of the 11.3PAH flux from circumnuclear (median size of 600pc) to nuclear regions (median size of 65pc), with a typical ratio of $\sim$5. Although this indicates that the SFRs are lower near the AGN in absolute terms, the projected SFR rate density in the nuclear regions (median value of 22[$M_\odot$]{}yr$^{-1}\,{\rm kpc}^{-2}$) is approximately 20 higher than in the circumnuclear regions. This indicates that the SF activity is highly concentrated in the nuclear regions in our sample of Seyfert galaxies.
4. Predictions from numerical simulations for the appropriate physical regions are broadly consistent with the observed relation between the nuclear SFR and $\dot M_{\rm BH}$ in our sample (slope of 1.01$\pm 0.4$). Although limited by the relatively small number of sources in our sample, we do not find decreased nuclear SFR-to-$\dot M_{\rm BH}$ ratios for regions closer to the AGN, as predicted by the @Hopkins2010 simulations.
We thank the anonymous referee for comments that helped us to improve the paper, and E. Piconcelli for useful discussion. PE and AAH acknowledge support from the Spanish Plan Nacional de Astronomía y Astrofísica under grant AYA2009-05705-E. PE is partially funded by Spanish MINECO under grant AYA2012-39362-C02-01. AAH and AHC acknowledge support from the Augusto G. Linares Program through the Universidad de Cantabria. CRA ackowledges financial support from the Instituto de Astrofísica de Canarias and the Spanish Plan Nacional de Astronomía y Astrofísica under grant AYA2010-21887-C04.04 (Estallidos). OGM and JMRE acknowledge support from the Spanish MICINN through the grant AYA2012-39168-C03-01. SFH acknowledges support by Deutsche Forschungsgemeinschaft (DFG) in the framework of a research fellowship (Auslandsstipendium).
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). The Cornell Atlas of Spitzer/IRS Sources (CASSIS) is a product of the Infrared Science Center at Cornell University, supported by NASA and JPL.
[ccc]{}
[]{} []{} []{}\
\[-2ex\]
[]{} []{} []{}\
\[-2ex\]
[]{} []{} []{}\
\[-2ex\]
[]{} []{} []{}\
[]{} []{} []{}\
\[fig:all\_spectra\]
[cc]{}
[]{} []{} []{}\
[]{} []{} []{}\
[]{} []{} []{}\
[]{} []{} []{}\
[]{} []{}\
|
---
abstract: 'The problem of determining Herbrand equivalence of terms at each program point in a data flow framework is a central and well studied question in program analysis. Most of the well-known algorithms for the computation of Herbrand equivalence in data flow frameworks [@GULWANI2007; @RRPai2016; @Ruthing1999] proceed via iterative fix-point computation on some abstract lattice of short expressions *relevant* to the given flow graph. However the mathematical definition of Herbrand equivalence is based on a meet over all path characterization over the (infinite) set of all possible expressions (see [@Steffen90 p. 393]). The aim of this paper is to develop a lattice theoretic fix-point formulation of Herbrand equivalence on the (infinite) concrete lattice defined over the set of all terms constructible from variables, constants and operators of a program. The present characterization uses an axiomatic formulation of the notion of Herbrand congruence and defines the (infinite) concrete lattice of Herbrand congruences. Transfer functions and non-deterministic assignments are formulated as monotone functions over this concrete lattice. Herbrand equivalence is defined as the maximum fix point of a composite transfer function defined over an appropriate product lattice of the above concrete lattice. A re-formulation of the classical meet-over-all-paths definition of Herbrand equivalence ( [@Steffen90 p. 393]) in the above lattice theoretic framework is also presented and is proven to be equivalent to the new lattice theoretic fix-point characterization.'
author:
- Jasine Babu
- 'K. Murali Krishnan'
- Vineeth Paleri
title: 'A fix-point characterization of Herbrand equivalence of expressions in data flow frameworks'
---
Introduction
============
A data flow framework is an abstract representation of a program, used in program analysis and compiler optimizations [@aho2007compilers]. As detection of semantic equivalence of expressions at each point in a program is unsolvable [@Kam1977], all known algorithms try to detect a weaker, syntactic notion of expression equivalence over the set of all possible expressions, called *Herbrand equivalence*. Stated informally, Herbrand equivalence treats operators as uninterpreted functions, and two expressions are considered equivalent if they are obtained by applying the same operator on equivalent operands [@GULWANI2007; @Muller2005; @Ruthing1999; @Steffen90].
The pioneering work of Kildall [@Kildall1973], which essentially is an abstract interpretation [@Cousot1977] of terms, showed that at each program point, Herbrand equivalence of expressions that occur in a program could be computed using an iterative refinement algorithm. The algorithm models each iteration as the application of a monotone function over a meet semi-lattice, and terminates at a fix-point of the function [@Kam1976; @Kam1977]. Subsequently, several problems in program analysis have been shown to be solvable using iterative fix-point computation on lattice frameworks. (see [@aho2007compilers] for examples). Several algorithms for computation of Herbrand equivalence of program expressions also were proposed in the literature [@Alpern1988; @GULWANI2007; @RRPai2016; @Rosen1988; @Ruthing1999; @Steffen90].
Although algorithmic computation to detect Herbrand equivalence among *expressions that appear in a program* proceeds via iterative fix-point computation on an abstract lattice framework, the classical mathematical definition of Herbrand equivalence uses a meet over all path formulation over the (infinite) set of all possible expressions (see [@Steffen90 p. 393]). The main difficulty in constructing a fix-point based definition for Herbrand equivalence of expressions at each program point is that it requires consideration of all program paths and equivalence of all expressions - including expressions not appearing in the program. Consequently, such a characterization of Herbrand equivalence cannot be achieved without resorting to set theoretic machinery.
It may be noted that, while the algorithm presented by Steffen et. al. [@Steffen90] uses an iterative fix-point computation method, their definition of Herbrand equivalences was essentially a meet over all paths (MOP) formulation. Though the MOP based definition of Herbrand equivalences given by Steffen et. al. [@Steffen90] is known since 1990, proving the completeness of iterative fix-point based algorithms using this definition is non-trivial. For instance, the algorithm proposed by the same authors [@Ruthing1999] was proven to be incomplete [@GULWANI2007] after several years, though it was initially accepted to be complete. In comparison with an MOP based definition of Herbrand equivalences, a fix-point characterization will render completeness proofs of iterative fix-point algorithms for computing the equivalence of program expressions simpler. The completeness proofs would now essentially involve establishing an equivalence preserving continuous homomorphism from the infinite concrete lattice of all Herbrand congruences to the finite abstract lattice of congruences of expressions that are *relevant* to the program, and proceed via induction.
In this paper, we develop a lattice theoretic fix-point characterization of Herbrand equivalences at each program point in a data flow framework. We define the notion of a congruence relation on the set of all expressions, and show that the set of all congruences form a complete lattice. Given a data flow framework with $n$ program points, we show how to define a continuous composite transfer function over the $n$ fold product of the above lattice, such that the maximum fix-point of the function yields the set of Herbrand equivalence classes at various program points. This characterization is then shown to be equivalent to a meet over all paths formulation of expression equivalence over the same lattice framework.
Section \[sec:terminology\] introduces the basic notation. Sections \[sec:cong-1\] to \[sec:non-det-asg\] develop the basic theory of congruences and transfer functions, including non-deterministic assignment functions. Section \[sec:data-flow-frameworks\] and Section \[sec:Herbrand\] deal with the application of the formalism of congruences to derive a fix-point characterization of Herbrand equivalence at each program point. Section \[sec:MOP\] describes a meet-over-all-paths formulation for expression equivalence and establishes the equivalence between the fix-point characterization and the meet-over-all-paths formulation.
Terminology {#sec:terminology}
===========
Let $C$ be a countable set of constants and $X$ be a countable set of variables. For simplicity, we assume that the set of operators $Op=\{+\}$. (More operators can be added without any difficulty). The set of all terms over $C\cup X$, $\mathcal{T}=\mathcal{T}(X,C)$ is defined by $t::=c \mid x \mid (t+t)$, with $c\in C$ and $x\in X$. (Parenthesis is avoided when there is no confusion.) Let $\mathcal{P}$ be a partition of $\mathcal{T}$. Let $[t]_{\mathcal{P}}$ (or simply $[t]$ when there is no confusion) denote the equivalence class containing the term $t\in \mathcal{T}$. If $t'\in [t]_{\mathcal{P}}$, we write $t\cong_{\mathcal{P}}t'$ (or simply $t\cong t'$). Note that $\cong$ is reflexive, symmetric and transitive. For any $A \subseteq \mathcal{T}$, $A(x)$ denotes the set of all terms in $A$ in which the variable $x$ appears and $\overline{A}(x)$ denotes the set of all terms in $A$ in which $x$ does not appear. In particular, for any $x\in X$, $\mathcal{T}(x)$ is the set of all terms containing the variable $x$ and $\overline{\mathcal{T}}(x)$ denotes the set of terms in which $x$ does not appear.
\[def:substitution\] For $t,\alpha\in \mathcal{T}$, $x\in X$, substitution of $x$ with $\alpha$ in $t$, denoted by $t[x\leftarrow \alpha]$ is defined as follows:
1. If $t=x$, then $t[x\leftarrow \alpha]=\alpha$.
2. If $t\notin \mathcal{T}(x)$, $t[x\leftarrow \alpha]=t$
3. If $t=t_1+t_2$ then $t[x\leftarrow \alpha]=t_1[x\leftarrow \alpha]+t_2[x\leftarrow \alpha]$.
In the rest of the paper, complete proofs for statements that are left unproven in the main text are provided in the appendix. Proofs for some elementary properties of lattices, which are used in the main text, are also given in the appendix, to make the presentation self contained.
Congruences of Terms {#sec:cong-1}
====================
We define the notion of congruence (of terms). The notion of congruence will be useful later to model equivalence of terms at various program points in a data flow framework.
\[def:cong\] Let $\mathcal{P}$ be a partition of $\mathcal{T}$. $\mathcal{P}$ is a Congruence (of terms) if the following conditions hold:
1. For each $c,c'\in C$, if $c\neq c'$ then $c\ncong c'$. (No two distinct constants are congruent).
2. For $t,t',s,s'\in \mathcal{T}$, $t'\cong t$ and $s'\cong s$ if and only if $t'+s'\cong t+s$. (Congruences respect operators).
3. For any $c\in C$, $t\in \mathcal{T}$, if $t\cong c$ then either $t=c$ or $t\in X$. (The only non-constant terms that are allowed to be congruent to a constant are variables).
The motivation for the definition of congruence is the following. Given the representation of a program in a data flow framework (to be defined later), we will associate a congruence to each program point at each iteration. Each iteration refines the present congruence at each program point. We will see later that this process of refinement leads to a well defined “fix point congruence” at each program point. We will see that this fix point congruence captures Herbrand equivalence at that program point.
The set of all congruences over $\mathcal{T}$ is denoted by $\mathcal{G}(\mathcal{T})$. We first note the substitution property of congruences.
\[obs:substitution\] Let $\mathcal{P}$ be a congruence over $\mathcal{T}$. Then, for each $\alpha, \beta \in \mathcal{T}$, $\alpha\cong \beta$ if and only if for all $x\in X$ and $t\in \mathcal{T}$, $t[x\leftarrow \alpha]\cong t[x\leftarrow \beta]$.
One direction is easy. If for all $x\in X$ and $t\in \mathcal{T}$, $t[x\leftarrow \alpha]\cong t[x\leftarrow \beta]$, then setting $t=x$ we get $\alpha=t[x\leftarrow \alpha]\cong t[x\leftarrow \beta]=\beta$. Conversely, suppose $\alpha\cong \beta$. Let $x\in X$ and $t\in \mathcal{T}$ be chosen arbitrarily. To prove $t[x\leftarrow \alpha]\cong t[x\leftarrow \beta]$, we use induction. If $t\notin \mathcal{T}(x)$, then $t[x\leftarrow \alpha]=t\cong t=t[x\leftarrow \beta]$. If $t=x$, then $t[x\leftarrow \alpha]=\alpha\cong \beta =t[x\leftarrow \beta]$. Otherwise, if $t=t_1+t_2$, then $t[x\leftarrow \alpha]=t_1[x\leftarrow \alpha]+t_2[x\leftarrow \alpha]
\cong t_1[x\leftarrow \beta]+t_2[x\leftarrow \beta] \text{ (by induction hypothesis)}=t[x\leftarrow \beta]$.
The following observation is a direct consequence of condition (3) of the definition of congruence.
\[obs:const2\] Let $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$, $c\in C$ and let $t\in \mathcal{T}(y)$, with $t\neq y$. Then $c\ncong_{_{\mathcal{P}}} t[y\leftarrow c]$.
We define a binary [*confluence operation*]{} on the set of congruences, $\mathcal{G}(\mathcal{T})$. A confluence operation transforms a pair of congruences into a congruence.
\[def:confluence\] Let $\mathcal{P}_1=\{A_i\}_{i\in I}$ and $\mathcal{P}_2=\{B_j\}_{j\in J}$ be two congruences. For all $i\in I$ and $j\in J$, define $C_{i,j}=A_i\cap B_j$. The confluence of $\mathcal{P}_1$ and $\mathcal{P}_2$ is defined by:\
$\mathcal{P}_1\wedge \mathcal{P}_2=\{C_{i,j}: i\in I,j\in J, C_{i,j}\neq \emptyset\}$.
\[thm:confluence\] If $\mathcal{P}_1$ and $\mathcal{P}_2$ are congruences, then $\mathcal{P}_1\wedge \mathcal{P}_2$ is a congruence.
Structure of Congruences {#sec:cong-2}
========================
In this section we will define an ordering on the set ${\mathcal{G}(\mathcal{T})}$ and then extend it it to a complete lattice.
Let $\mathcal{P}$, $\mathcal{P'}$ be congruences over $\mathcal{T}$. We say $P\preceq P'$ (read $P$ is a refinement of $P'$) if for each equivalence class $A\in \mathcal{P}$, there exists an equivalence class $A'\in \mathcal{P'}$ such that $A\subseteq A'$.
\[def:bot\] The partition in which each term in $\mathcal{T}$ belongs to a different class is defined as: $\bot=\{ \{t\} : t\in \mathcal{T} \}$.
The following observation is a direct consequence of the definition of $\bot$.
\[obs:bot\] $\bot$ is a congruence. Moreover for any $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$, $\bot \wedge \mathcal{P}=\bot$.
\[def:semilattice\] A partially ordered set $(P,\leq)$ is a meet semi-lattice if every pair of elements $a,b\in P$ has a greatest lower bound (called the meet of $a$ and $b$).
\[lem:poset\] $(\mathcal{G}(\mathcal{T}),\preceq)$ is a meet semi-lattice with meet operation $\wedge$ and bottom element $\bot$.
The following lemma extends the meet operation to arbitrary non-empty collections of congruences. The proof relies on the axiom of choice.
\[lem:subsetInfimum\] Every non-empty subset of $(\mathcal{G}(\mathcal{T}),\preceq)$ has a greatest lower bound.
Next, we extend the meet semi-lattice $({\mathcal{G}(\mathcal{T})},\preceq)$ by artificially adding a top element $\top$, so that the greatest lower bound of the empty set is also well defined.
\[def:PBar\] The lattice $({\overline{\mathcal{G}(\mathcal{\mathcal{T}})}},\preceq, \bot, \top)$ is defined over the set ${\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}={\mathcal{G}(\mathcal{T})}\cup \{\top \}$ with $\mathcal{P}\preceq \top$ for each $\mathcal{P}\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$. In particular, $\top$ is the greatest lower bound of $\emptyset$ and $\top \wedge \mathcal{P}=\mathcal{P}$ for every $\mathcal{P}\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$.
Hereafter, we will be referring the element $\top$ as a congruence. It follows from Lemma \[lem:subsetInfimum\] and the above definition that every subset of ${\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ has a greatest lower bound. Since a meet semi-lattice in which every subset has a greatest lower bound is a complete lattice (Theorem \[Athm:complete\]), we have:
\[thm:complete\] $({\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}, \preceq, \bot, \top)$ is a complete lattice.
\[def:sup-inf\] Let $S=\{\mathcal{P}_i\}_{i\in I}$ be an arbitrary collection of congruences in ${\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ ($S$ may be empty or may contain $\top$). The infimum of the set $S$, denoted by $\bigwedge S$ or $\bigwedge_{i\in I}\mathcal{P}_i$, is defined as the greatest lower bound of the set $\{\mathcal{P}_i\}_{i\in I}$.
Transfer Functions {#sec:transfun-1}
==================
We now define a class of unary operators on ${\mathcal{G}(\mathcal{T})}$ called [*transfer functions*]{}. A transfer function specifies how the assignment of a term to a variable transforms the congruence before the assignment into a new one.
\[def:trans-function\] Let $y\in X$ and $\beta\in \overline{\mathcal{T}}(y)$. (Note that $y$ does not appear in $\beta$). Let $\mathcal{P}=\{A_i\}_{i\in I} $ be an arbitrary congruence. The transfer function ${f_{_{y=\beta}}(\mathcal{P})}: \mathcal{G}(\mathcal{T})\longrightarrow \mathcal{G}(\mathcal{T})$ transforms $\mathcal{P}$ to a congruence ${f_{_{y=\beta}}(\mathcal{P})}$ given by the following:
- For each $i\in I$, let $B_{i}=\{t \in \mathcal{T}$ : $t[y\leftarrow \beta]\in A_{i}\}$.
- ${f_{_{y=\beta}}(\mathcal{P})}=\{B_{i}$ : $i\in I$, $B_i\neq \emptyset\}$.
![Application of Transfer Functions[]{data-label="fig:trans"}](tran-function.pdf)
It follows from the above definition that $\overline{A_i}(y)=(A_i\setminus A_i(y))\subseteq B_i$. That is, $B_i$ will contain all terms in $A_i$ in which $y$ does not appear. See Figure \[fig:trans\] for an example.
In the following, we write $f(\mathcal{P})$ instead of ${f_{_{y=\beta}}(\mathcal{P})}$ to avoid cumbersome notation. The following is a direct consequence of Definition \[def:trans-function\].
\[obs:trans\] For any $t,t'\in \mathcal{T}$, $t\cong_{f(\mathcal{P})}t'$ if and only if $t[y\leftarrow \beta]\cong_{\mathcal{P}}t'[y\leftarrow \beta]$.
To make Definition \[def:trans-function\] well founded, we need to establish the following:
\[thm:trans-function\] If $\mathcal{P}$ is a congruence, then for any $y\in X$, $\beta \in \overline{\mathcal{T}}(y)$, ${f_{_{y=\beta}}(\mathcal{P})}$ is a congruence.
Next, we extend the definition of transfer functions to $({\overline{\mathcal{G}(\mathcal{\mathcal{T}})}},\preceq,\bot,\top)$.
\[def:ext-transfun\] Let $y\in X$ and $\beta\in \overline{ \mathcal{T}}(y)$. Let $\mathcal{P}\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$. The extended transfer function ${\overline{f}_{_{y=\beta}}(\mathcal{P}) }: {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}\longrightarrow {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ transforms $\mathcal{P}$ to ${\overline{f}_{_{y=\beta}}(\mathcal{P}) }\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ given by the following:
- If $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$, ${\overline{f}_{_{y=\beta}}(\mathcal{P}) }={f_{_{y=\beta}}(\mathcal{P})}$.
- $\overline{f}_{_{y=\beta}}(\top)=\top$.
To simplify the notation, we often write ${f_{_{y=\beta}}(\mathcal{P})}$ (or even simply $f(\mathcal{P})$) instead of ${\overline{f}_{_{y=\beta}}(\mathcal{P}) }$, and refer to extended transfer functions as simply transfer functions, when the underlying assignment operation is clear from the context.
Properties of Transfer Functions {#sec:transfun-2}
================================
In this section we show that transfer functions are continuous over the complete lattice $({\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}, \preceq, \bot, \top)$.
Consider the (extended) transfer function $f=\overline{f}_{_{y=\beta}}$, where $y\in X$, $\beta\in \overline{\mathcal{T}}(y)$. Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be congruences in ${\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$, not necessarily distinct.
\[lem:distrb\] ${f(\mathcal{P}_1)}\wedge {f(\mathcal{P}_2)}= {f(\mathcal{P}_1\wedge \mathcal{P}_2)}$.
Since distributive functions are monotone, we have:
\[cor:monotone\] If $\mathcal{P}_1\preceq \mathcal{P}_2$, then ${f(\mathcal{P}_1)}\preceq {f(\mathcal{P}_2)}$.
We next show that distributivity extends to arbitrary collections of congruences.
\[def:cts\] Let $(L,\leq,\bot,\top)$ and $(L'\leq',\bot',\top')$ be complete lattices. A function $f: L\rightarrow L'$ is continuous if for each $\emptyset \neq S\subseteq L$, $f(\bigwedge S)=\bigwedge' f(S)$, where $f(S)=\{f(s): s\in S\}$ and $\bigwedge, \bigwedge'$ denote the infimum operations in the lattices $L$ and $L'$ respectively.
The definition of continuity given above is more stringent than the standard definition in the literature, which requires continuity only for subsets that are chains. Moreover, the definition above exempts the continuity condition to hold for the empty set, because otherwise even constant maps will fail to be continuous.
The proof of the next theorem uses the axiom of choice. Let $f=f_{_{y=\beta}}$, where $y\in X$, $\beta\in \overline{ \mathcal{T}}(y)$. For arbitrary collections of congruences $S\subseteq {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$, The notation $f(S)$ denotes the set $\{f(s): s\in S\}$.
\[thm:continuous\] For any $\emptyset \neq S \subseteq {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$, $f(\bigwedge S)=\bigwedge f(S)$.
Non-deterministic assignment {#sec:non-det-asg}
============================
Next we define a special kind of transfer functions corresponding to input statements in the program. This kind of transfer functions are called non-deterministic assignments.
\[def:non-det-trans\] Let $y\in X$ and let $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$. The transfer function ${f_{_{y=*}}(\mathcal{P})}: \mathcal{G}(\mathcal{T})\mapsto \mathcal{G}(\mathcal{T})$ transforms $\mathcal{P}$ to a congruence $f(\mathcal{P})={f_{_{y=*}}(\mathcal{P})}$, given by: for every $t,t'\in \mathcal{T}$, $t{\cong_{_{f(\mathcal{P})}}}t'$ if and only if both the following conditions are satisfied:
1. $t{\cong_{_{\mathcal{P}}}}t'$
2. For every $\beta \in \mathcal{T}\setminus \mathcal{T}(y), t[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow \beta]$.
Since for every pair of terms $t,t'\in \mathcal{T}$ the above definition precisely decides whether $t{\cong_{_{f(\mathcal{P})}}}t'$ or not, ${f_{_{y=*}}(\mathcal{P})}$ is the unique relation containing exactly those pairs $t,t'\in \mathcal{T}$ satisfying both the conditions in Definition \[def:non-det-trans\]. The definition asserts that two terms that were equivalent before a non-deterministic assignment, will remain equivalent after the non-deterministic assignment to $y$ if and only if the equivalence between the two terms is preserved under all possible substitutions to $y$.
To make the above definition well founded, we need to prove that ${f_{_{y=*}}(\mathcal{P})}$ is a congruence.
\[thm:non-det-trans\] If $\mathcal{P}$ is a congruence, then for any $y\in X$, ${f_{_{y=*}}(\mathcal{P})}$ is a congruence.
We write $\bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P})}$ to denote the set $\bigwedge\{ {f_{_{y=\beta}}(\mathcal{P})}: \beta \in \overline{\mathcal{T}}(y)\}$. The next theorem shows that each non-deterministic assignment may be expressed as a confluence of (an infinite collection of) transfer function operations.
\[thm:ntrans-char\] If $\mathcal{P}$ is a congruence, then for any $y\in X$,\
${f_{_{y=*}}(\mathcal{P})}=\mathcal{P}\wedge \left( \bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P})}\right)$.
Next, we extend the definition of non-deterministic assignment transfer functions to the complete lattice $({\overline{\mathcal{G}(\mathcal{\mathcal{T}})}},\preceq,\bot,\top)$.
\[def:ext-ntransfun\] Let $y\in X$ and $\mathcal{P}\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$. The extended transfer function ${\overline{f}_{_{y=*}}(\mathcal{P}) }: {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}\mapsto {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ transforms $\mathcal{P}$ to ${\overline{f}_{_{y=*}}(\mathcal{P}) }\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ given by the following:
- If $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$, ${\overline{f}_{_{y=*}}(\mathcal{P}) }={f_{_{y=*}}(\mathcal{P})}$.
- $\overline{f}_{_{y=*}}(\top)=\top$.
The following theorem involves use of the axiom of choice. We will write ${f_{_{y=*}}(\mathcal{P})}$ instead of ${\overline{f}_{_{y=*}}(\mathcal{P}) }$ to simplify notation.
\[thm:ntrans-cts\] For any $\emptyset \neq S \subseteq {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$, ${f}_{_{y=*}}(\bigwedge S)=\bigwedge {f}_{_{y=*}}(S)$, where\
${f}_{_{y=*}}(S)=\{f_{_{y=*}}(s): s\in S\}$.
In the following, we derive a characterization for non-deterministic assignment that does not depend on the axiom of choice. Condition (3) of the definition of congruence (Definition \[def:cong\]) is necessary to derive this characterization. We first note a lemma which states that if the equivalence between two terms $t,t'$ is preserved under substitution of $y$ with any two distinct constants chosen arbitrarily, then the equivalence between the two terms will be preserved under substitution of $y$ with any other term $\beta$ in which $y$ does not appear.
\[lem:nondet-constant\] Let $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$. Let $t,t'\in \mathcal{T}$ and $c_1,c_2\in C$ with $c_1\neq c_2$. Then $t[y\leftarrow c_1]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_1]$ and $t[y\leftarrow c_2]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_2]$ if and only if $t[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow \beta]$ for all $\beta\in \overline{\mathcal{T}}(y)$.
The above observation leads to a characterization of non-deterministic assignment that does not involve the axiom of choice.
\[thm:nondet-const\] Let $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$ and let $c_1,c_2\in C$, $c_1\neq c_2$. Then, for any $y\in X$,\
${f_{_{y=*}}(\mathcal{P})}=\mathcal{P}\wedge f_{_{y\leftarrow c_1}}(\mathcal{P}) \wedge f_{_{y\leftarrow c_2}}(\mathcal{P})$
It follows from the above theorem that non-deterministic assignments can be characterized in terms of just three congruences (instead of dealing with infinitely many as in Theorem \[thm:ntrans-char\]).
Data Flow Analysis Frameworks {#sec:data-flow-frameworks}
=============================
We next formalize the notion of a data flow framework and apply the formalism developed above to characterize Herbrand equivalence at each point in a program.
\[def:control-flow-graph\] A control flow graph $G(V,E)$ is a directed graph over the vertex set $V=\{1,2,\ldots,n\}$ for some $n\geq 1$ satisfying the following properties:
- $1\in V$ is called the entry point and has no predecessors.
- Every vertex $i\in V$, $i\neq 1$ is reachable from $1$ and has at least one predecessor and at most two predecessors.
- Vertices with two predecessors are called confluence points.
- Vertices with a single predecessor are called (transfer) function points.
\[def:data-flow-framework\] A data flow framework over $\mathcal{T}$ is a pair $D=(G,F)$, where $G(V,E)$ is a control flow graph on the vertex set $V=\{1,2,\ldots, n\}$ and $F$ is a collection of transfer functions over ${\mathcal{G}(\mathcal{T})}$ such that for each function point $i\in V$, there is an associated transfer function $h_i\in F$, and $F=\{h_i: i\in V \text{ is a transfer function point}\}$.
Data flow frameworks can be used to represent programs. An example is given in Figure \[fig:programpoints\].
In the sections that follow, for any $h_i\in F$, we will simply write $h_i$ to actually denote the extended transfer function $\overline{h}_i$ (see Definition \[def:ext-transfun\] and Definition \[def:ext-ntransfun\]) without further explanation.
Herbrand Equivalence {#sec:Herbrand}
====================
Let $D=(G,F)$ be a data flow framework over $\mathcal{T}$. In the following, we will define the Herbrand Congruence function $H_D : V(G)\mapsto {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$. For each vertex $i\in V(G)$, the congruence $H_D(i)$ will be called the *Herbrand Congruence* associated with the vertex $i$ of the data flow framework $D$. The function $H_D$ will be defined as the maximum fix-point of a continuous function ${f_{_D}}:{\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}\mapsto {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}$. The function ${f_{_D}}$ will be called the *composite transfer function* associated with the data flow framework $D$.
\[def:Prod\_lattice\] Let $n$ a positive integer. The product lattice,\
$({\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n},\preceq_n,\bot_n,\top_n)$ is defined as follows: for $\overline{\mathcal{P}}=(\mathcal{P}_1, \mathcal{P}_2,\ldots, \mathcal{P}_n)$, $\overline{\mathcal{Q}}=(\mathcal{Q}_1, \mathcal{Q}_2,\ldots, \mathcal{Q}_n) \in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}$, $\overline{\mathcal{P}}\preceq_n \overline{\mathcal{Q}}$ if $\mathcal{P}_i\preceq \mathcal{Q}_i$ for each $1\leq i\leq n$, $\bot_n=(\bot, \bot, \ldots,\bot)$ and $\top_n=(\top,\top, \ldots, \top)$.
For $S\subset {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}$, the notation $\bigwedge_n S$ will be used to denote the least upper bound of $S$ in the product lattice.
By Theorem \[thm:complete\], Theorem \[Athm:prod\_complete\] and Corollary \[Acor:prod\_complete\], we have:
\[thm:prod\_complete\] The product lattice satisfies the following properties:
1. $({\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n},\preceq_n,\bot_n,\top_n)$ is a complete lattice.
2. If $\tilde{S}\subseteq {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}$ is non-empty, with $\tilde{S}=S_1\times S_2\times \cdots \times S_n$, where $S_i\subseteq {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ for $1\leq i\leq n$. Then $\bigwedge_n \tilde{S}=(\bigwedge S_1,\bigwedge S_2, \ldots, \bigwedge S_n)$.
As preparation for defining the composite transfer function, we introduce the following functions:
\[def:proj\_map\] Let $n$ be a positive integer. For each $i\in \{1,2,\ldots,n\}$,
- The projection map to the $i^{th}$ co-ordinate, $\pi_i:{\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}\mapsto {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ is defined by\
$\pi_i(\mathcal{P}_1,\mathcal{P}_2,\ldots,\mathcal{P}_n)=\mathcal{P}_i$ for any $(\mathcal{P}_1,\mathcal{P}_2,\ldots,\mathcal{P}_n)\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}$.
- The confluence map $\pi_{i,j}:{\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}\mapsto {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ is defined by\
$\pi_{i,j}(\mathcal{P}_1,\mathcal{P}_2,\ldots, \mathcal{P}_n)=\mathcal{P}_i\wedge \mathcal{P}_j$ for any $(\mathcal{P}_1,\mathcal{P}_2,\ldots, \mathcal{P}_n)\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}$.
In addition to the above functions, we will also use the constant map which maps each element in ${\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}$ to $\bot$. The following observation is a consequence of Theorem \[Athm:conf\_cts\].
\[obs:conf\_cts\] Constant maps, projection maps and confluence maps are continuous.
For each $k\in V(G)$, ${\operatorname{Pred}}(k)$ denotes the set of predecessors of the vertex $k$ in the control flow graph $G$.
\[def:comp-trans-fun\] Let $D=(G,F)$ be a data flow framework over $\mathcal{T}$. For each $k\in V(G)$, define the component map $f_k:{\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}^{n}\mapsto {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ as follows:
1. If $k=1$, the entry point, then $f_k=\bot$. ($f_1$ is the constant function that always returns the value $\bot$).
2. If $k$ is a function point with ${\operatorname{Pred}}(k)=\{j\}$, then $f_k=h_k\circ \pi_j$, where $h_k$ is the (extended) transfer function corresponding to the function point $k$, and $\pi_k$ the projection map to the $k^{th}$ coordinate as defined in Definition \[def:proj\_map\].
3. If $k$ is a confluence point with ${\operatorname{Pred}}(k)=\{i,j\}$, then $f_k=\pi_{i,j}$, where $\pi_{i,j}$ is the confluence map as defined in Definition \[def:proj\_map\].
The composite transfer function of $D$ is defined to be the unique function (Observation \[Aobs:component\]) ${f_{_D}}$ satisfying $\pi_k\circ {f_{_D}}=f_k$ for each $k\in V(G)$.
The purpose of defining ${f_{_D}}$ is the following. Suppose we have associated a congruence with each program point in a data flow framework. Then ${f_{_D}}$ specifies how a simultaneous and synchronous application of all the transfer functions/confluence operations at the respective program points modifies the congruences at each program point. The definition of ${f_{_D}}$ conservatively sets the confluence at the entry point to $\bot$, treating terms in ${\mathcal{G}(\mathcal{T})}$ to be inequivalent to each other at the entry point. See Figure \[fig:programpoints\] for an example. The following observation is a direct consequence of the above definition.
\[obs:comp-trans\] The composite transfer function ${f_{_D}}$ (Definition \[def:comp-trans-fun\]) satisfies the following properties:
1. If $k=1$, the entry point, then $\pi_k\circ {f_{_D}}=\bot$.
2. If $k$ is a function point with ${\operatorname{Pred}}(k)=\{j\}$, then $f_k=\pi_k\circ {f_{_D}}= h_k\circ \pi_j$, where $h_k$ is the (extended) transfer function corresponding to the function point $k$.
3. If $k$ is a confluence point with ${\operatorname{Pred}}(k)=\{i,j\}$, then $f_k=\pi_k \circ {f_{_D}}= \pi_{i,j}$.
The following lemma is a consequence of Observation \[obs:comp-trans\].
\[lem:fd-induction\] Let $D=(G,F)$ be a data flow framework over $\mathcal{T}$ and $k\in V(G)$. Let $S=\{{f_{_D}}(\top_n), {f_{_D}}^2(\top_n),\ldots \}$, where ${f_{_D}}$ is the composite transfer function of $D$.
1. If $k=1$, the entry point, then $\pi_k\circ {f_{_D}}^{l}(\top_n)=\bot$ for all $l\geq 1$, hence $\pi_k (\bigwedge_n S) =\top$.
2. If $k$ is a function point with ${\operatorname{Pred}}(k)=\{j\}$, then for all $l\geq 1$, $$\begin{aligned}
(\pi_k \circ {f_{_D}}^{l})(\top_n)&=& (\pi_k\circ {f_{_D}})({f_{_D}}^{l-1}(\top_n))\\
&=&(h_k\circ \pi_j \circ {f_{_D}}^{l-1})(\top_n)
\end{aligned}$$
3. If $k$ is a confluence point with ${\operatorname{Pred}}(k)=\{i,j\}$, then for all $l\geq 1$, $$\begin{aligned}
(\pi_k \circ {f_{_D}}^{l})(\top_n)&=&(\pi_k\circ {f_{_D}})({f_{_D}}^{l-1}(\top_n)) \\
&=& (\pi_{i, j})({f_{_D}}^{l-1}(\top_n))\\
&=& \left((\pi_i \circ {f_{_D}}^{l-1})(\top_n)\right) \wedge \left((\pi_j \circ {f_{_D}}^{l-1})(\top_n)\right) \end{aligned}$$
By Theorem \[Athm:prod\_cts\], Observation \[Aobs:monotone\] and Corollary \[Acor:MFP\], we have:
\[thm:prop-composite-trans\] The following properties hold for the composite transfer function ${f_{_D}}$ (Definition \[def:comp-trans-fun\]):
1. ${f_{_D}}$ is monotone, distributive and continuous.
2. The component maps $f_k=\pi_k\circ {f_{_D}}$ are continuous for all $k\in \{1,2,\ldots, n\}$.
3. ${f_{_D}}$ has a maximum fix-point.
4. If $S=\{\top, {f_{_D}}(\top_n), {f_{_D}}^2(\top_n),\ldots \}$, then $\bigwedge_n S$ is the maximum fix-point of ${f_{_D}}$.
![Component Maps of the Composite Transfer Function[]{data-label="fig:programpoints"}](programpoints.pdf)
The objective of defining Herbrand Congruence as the maximum fix point of the composite transfer function is possible now.
\[def:Herbrand\] Given a data flow framework $D=(G,F)$ over $\mathcal{T}$, the Herbrand Congruence function $H_D : V(G)\mapsto {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ is defined as the maximum fix point of the composite transfer function ${f_{_D}}$. For each $k\in V(G)$, the value $H_D(k)\in {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ is referred to as the Herbrand Congruence at program point $k$.
The following is a consequence of Theorem \[thm:prop-composite-trans\] and the definition of Herbrand Congruence.
\[obs:herbrand\] For each $k\in V(G)$, $H_D(k)=\bigwedge \{ (\pi_k \circ {{f_{_D}}}^{l})(\top_n) : l\geq 0 \}$.
$$\begin{aligned}
H_D(k)&=&\pi_k(\bigwedge\text{$_{_n}$} \{{{f_{_D}}}^{l}(\top_n) : l\geq 0\}) \text{ (by Theorem~\ref{thm:prop-composite-trans}) }\\
&=&\bigwedge \{ \pi_k({{f_{_D}}}^{l}(\top_n)) : l\geq 0 \} \text{ (by continuity of $\pi_k$) } \\
&=&\bigwedge \{ (\pi_k\circ {{f_{_D}}}^{l})(\top_n) : l\geq 0\}
\end{aligned}$$
The definition of Herbrand congruence must be shown to be consistent with the traditional meet-over-all-paths description of Herbrand equivalence of terms in a data flow framework. The next section addresses this issue.
MOP characterization {#sec:MOP}
====================
In this section, we give a meet over all paths characterization for the Herbrand Congruence at each program point. This is essentially a lattice theoretic reformulation of the characterization presented by Steffen et. al. [@Steffen90 p. 393]. In the following, assume that we are given a data flow framework $D=(G,F)$ over $\mathcal{T}$, with $V(G)=\{1,2,\ldots, n\}$.
\[def:prog-path\] For any non-negative integer $l$, a program path (or simply a path) of length $l$ to a vertex $k\in V(G)$ is a sequence $\alpha=(v_0,v_2,\ldots v_{l})$ satisfying $v_0=1$, $v_l=k$ and $(v_{i-1},v_{i})\in E(G)$ for each $i\in \{1,2,\ldots l\}$. For each $i\in\{0,1,\ldots, l\}$, $\alpha_i$ denotes the initial segment of $\alpha$ up to the $i^{th}$ vertex, given by $(v_0,v_1,\ldots, v_i)$.
Note that the vertices in a path need not be distinct under this definition.
The next definition associates a congruence in ${\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$ with each path in $D$. The path function captures the effect of application of transfer functions along the path on the initial congruence $\bot$, in the order in which the transfer functions appear along the path.
\[def:path-fun\] Let $\alpha=(v_0,v_1,\ldots, v_l)$ be a path of length $l$ to vertex $k\in V(G)$ for some $l\geq 0$. We define:
1. When $i=0$, $m_{\alpha_i}=\bot$.
2. If $i>0$ and $v_{i}=j$, where $j\in V(G)$ is a function point, then $m_{\alpha_i}=h_j(m_{\alpha_{i-1}})$, where $h_j\in F$ is the extended transfer function associated with the function point $j$.
3. If $i>0$ and $v_{i}$ is a confluence point, then $m_{\alpha_i}=m_{\alpha_{i-1}}$.
4. $m_\alpha=m_{\alpha_l}$.
The congruence $m_\alpha$ is defined as the path congruence associated with the path $\alpha$.
For $k\in V(G)$ and $l\geq 0$, let $\Phi_l(k)$ denote the set of all paths of length [*less than $l$*]{} from the entry point $1$ to the vertex $k$. In particular, $\Phi_0(k) =\emptyset$, for all $k \in V(G)$. The following observation is a consequence of the definition of $\Phi_l(k)$.
\[obs:obs-path\] If $k \in V(G)$ and $l\geq 1$,
1. If $k$ is the entry point, then $\Phi_{l}(k)=\{(1)\}$, the set containing only the path of length zero, starting and ending at vertex $1$.
2. If $k$ is a function point with ${\operatorname{Pred}}(k)=\{j\}$, then\
$\{\alpha_{l-1} : \alpha\in \Phi_l(k)\}=\{\alpha' : \alpha' \in \Phi_{l-1}(j)\}=\Phi_{l-1}(j)$.
3. If $k$ is a confluence point with ${\operatorname{Pred}}(k)=\{i,j\}$, then\
$\{\alpha_{l-1} : \alpha\in \Phi_l(k)\}=\{\alpha' : \alpha' \in \Phi_{l-1}(i)\} \cup \{\alpha' : \alpha' \in \Phi_{l-1}(j)\}=\Phi_{l-1}(i) \cup \Phi_{l-1}(j)$.
For $l\ge 0$, we define the congruence $M_l(k)$ to be the meet of all path congruences associated with paths of length less than $l$ from the entry point to vertex $k$ in $G$. Stated formally, $$M_l(k)=\bigwedge\{ m_\alpha: \alpha \in \Phi_{l}(k)\}, \text{for $l \ge 0$}$$
\[obs:base-cases\] If $l=0$, $\Phi_{l}(k)=\emptyset$ and hence $M_0(k)=\top$, for all $k \in V(G)$. Further, $M_1(1)=\bot$ and $M_1(k)=\top$, for $k\ne 1$. In general, $M_l(k)=\top$ if there are no paths of length less than $l$ from $1$ to $k$ in $G$.
We define $\Phi_k$ to be the set of all paths from vertex $1$ to vertex $k$ in $G$, i.e., $\Phi_k=\bigcup_{l\geq 1} \Phi_l(k)$ and $MOP(k)=\bigwedge\{m_\alpha: \alpha \in \Phi(k) \}=\bigwedge\{ M_l(k): l\geq 0\}$. (The second equality follows from Lemma \[Alem:meet-union\] and Observation \[obs:base-cases\].) The congruence $MOP(k)$ is the meet of all path congruences associated with paths in $\Phi_k$.
Our objective is to prove $MOP(k)=H_{D}(k)$ for each $k\in \{1,2,\ldots, n\}$ so that $H_D$ captures the meet over all paths information about equivalence of expressions in $\mathcal{T}$. We begin with the following observations.
\[lem:paths\] For each $k\in V(G)$ and $l\geq 1$
1. If $k=1$, the entry point, then $M_l(k)=\bot$.
2. If $k$ is a function point with ${\operatorname{Pred}}(k)=\{j\}$, then $M_l(k)=h_k(M_{l-1}(j))$, where $h_k$ is the (extended) transfer function corresponding to the function point $k$.
3. If $k$ is a confluence point with ${\operatorname{Pred}}(k)=\{i,j\}$, then $M_l(k)=M_{l-1}(i) \wedge M_{l-1}(j)$.
If $k=1$, $\Phi_l(k)=\{(1)\}$, by Observation \[obs:obs-path\]. Hence $M_l(k)=M_{1}(1)=\bot$, by Observation \[obs:base-cases\].\
If $k$ is a function point with ${\operatorname{Pred}}(k)=\{j\}$, then $$\begin{aligned}
M_l(k)&=&\bigwedge\{ m_\alpha: \alpha \in \Phi_l(k)\}\\
&=&\bigwedge\{ h_k (m_{\alpha_{l-1}}): \alpha \in \Phi_l(k)\}\text{ (by Definition~\ref{def:path-fun})} \\
&=&\bigwedge\{ h_k (m_{\alpha'}): \alpha' \in \Phi_{l-1}(j)\}\text{ (by Observation~\ref{obs:obs-path})}\\
&=&h_k\left(\bigwedge\{m_{\alpha'}: \alpha' \in \Phi_{l-1}(j)\}\right)\text{ (by continuity of $h_k$) }\\
&=&h_k(M_{l-1}(j))\text{ (by definition of $M_{l-1}(j)$)}
\end{aligned}$$ If $k$ is a confluence point with ${\operatorname{Pred}}(k)=\{i,j\}$, then $$\begin{aligned}
M_l(k)&=&\bigwedge\{ m_\alpha: \alpha \in \Phi_l(k)\}\\
&=&\bigwedge( \{m_{\alpha_{l-1}}: \alpha \in \Phi_l(k)\}) \text{ (by Definition~\ref{def:path-fun})}\\
&=&\bigwedge\left(\{m_{\alpha'}: \alpha' \in \Phi_{l-1}(i)\} \cup \{m_{\alpha'}: \alpha' \in \Phi_{l-1}(j) \}\right) \text{ (by Observation~\ref{obs:obs-path})} \\
&=&\left( \bigwedge \{m_{\alpha'}: \alpha' \in \Phi_{l-1}(i)\}\right) \wedge \left(\bigwedge\{m_{\alpha'}: \alpha' \in \Phi_{l-1}(j)\}\right)\text{ (by Lemma~\ref{Alem:meet-union})}\\
&=&M_{l-1}(i) \wedge M_{l-1}(j)\text{ (by definition of $M_{l-1}(i)$ and $M_{l-1}(j)$)}
\end{aligned}$$
\[lem:MOP-MFP\] For each $k\in V(G)$ and $l\geq 0$, $M_l(k)=(\pi_k \circ {{f_{_D}}}^{l})(\top_n)$.
Let $k \in V(G)$ be chosen arbitrarily. We prove the lemma by induction on $l$.\
When $l=0$, $M_l(k)=\top$, by Observation \[obs:base-cases\], as required. Otherwise,
1. If $k=1$, the entry point, then by Lemma \[lem:fd-induction\] and Lemma \[lem:paths\],\
$M_l(k)=\bot=(\pi_k \circ {{f_{_D}}}^{l})(\top_n)$.
2. If $k$ is a function point with ${\operatorname{Pred}}(k)=\{j\}$ and $h_k$ is the (extended) transfer function corresponding to the function point $k$, then $$\begin{aligned}
M_l(k)&=&h_k(M_{l-1}(j)) \text{ (By Lemma~\ref{lem:paths})} \\
&=&h_k((\pi_j \circ {{f_{_D}}}^{l-1})(\top_n)) \text{ (By induction hypothesis) } \\
&=&(h_k\circ \pi_j\circ {{f_{_D}}}^{l-1})(\top_n) \\
&=&(\pi_k \circ {f_{_D}}^{l})(\top_n) \text{ (By Lemma~\ref{lem:fd-induction}) }
\end{aligned}$$
3. If $k$ is a confluence point with ${\operatorname{Pred}}(k)=\{i,j\}$, then $$\begin{aligned}
M_l(k)&=& M_{l-1}(i) \wedge M_{l-1}(j) \text{ (By Lemma~\ref{lem:paths}) }\\
&=& \left((\pi_i \circ {{f_{_D}}}^{l-1})(\top_n)\right) \wedge \left((\pi_j \circ {{f_{_D}}}^{l-1})(\top_n)\right) \text{ (By induction hypothesis) } \\
&=& (\pi_k \circ {f_{_D}}^{l})(\top_n) \text{ (By Lemma~\ref{lem:fd-induction}) }
\end{aligned}$$
Finally, we show that the iterative fix-point characterization of Herbrand equivalence and the meet over all paths characterization coincide.
\[thm:MOP-MFP\] Let $D=(G,F)$ be a data flow framework. Then, for each $k\in V(G)$, $MOP(k)=H_D(k)$.
$$\begin{aligned}
MOP(k)&=&\bigwedge\{m_\alpha: \alpha \in \Phi(k) \} \\
&=&\bigwedge\{ M_l(k): l\geq 0\} \text{ ( by Lemma~\ref{Alem:meet-union} and Observation~\ref{obs:base-cases}) }\\
&=&\bigwedge\{ (\pi_k \circ {{f_{_D}}}^{l})(\top_n) : l\geq 0 \} \text{ (by Lemma~\ref{lem:MOP-MFP}) }\\
&=& H_D(k) \text{ (by Observation~\ref{obs:herbrand})}\end{aligned}$$
Conclusion
==========
We have shown that Herbrand equivalence of terms in a data flow framework admits a lattice theoretic fix-point characterization. Though not the concern addressed here, we note that this fix-point characterization naturally leads to algorithms that work on the restriction of congruences to terms that actually appear in given data flow framework and iteratively compute the maximum fix-point (see, for example, the algorithm presented in Appendix \[sec:appendixc\]). This allows the algorithmic detection of Herbrand equivalence among the expressions that actually appear in any given program. Moreover, as mentioned in the introduction, analysis of the completeness of iterative fix point based algorithms that detects Herbrand equivalence among program expressions would be potentially easier using the fix-point characterization of Herbrand equivalences presented here, when compared to the current practice of using a meet over all paths characterization as the reference point. Further, we have shown that the lattice theoretic fix-point characterization presented here is equivalent to a meet over all paths characterization.
[10]{}
Alfred V. Aho, Monica S. Lam, Ravi Sethi, and Jeffrey D. Ullman. , volume 2. Addison-wesley Reading, 2006.
B. Alpern, M. N. Wegman, and F. K. Zadeck. Detecting equality of variables in programs. In [*Proceedings of the 15th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages*]{}, POPL ’88, pages 1–11, New York, NY, USA, 1988. ACM. Available from: <http://doi.acm.org/10.1145/73560.73561>, [](http://dx.doi.org/10.1145/73560.73561).
Patrick Cousot and Radhia Cousot. Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In [*Proceedings of the 4th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages*]{}, POPL ’77, pages 238–252, New York, NY, USA, 1977. ACM. Available from: <http://doi.acm.org/10.1145/512950.512973>, [](http://dx.doi.org/10.1145/512950.512973).
Sumit Gulwani and George C. Necula. A polynomial-time algorithm for global value numbering. , 64(1):97–114, 2007. Special issue on the 11th Static Analysis Symposium - SAS 2004.
John B. Kam and Jeffrey D. Ullman. Global data flow analysis and iterative algorithms. , 23(1):158–171, Jan 1976. Available from: <http://doi.acm.org/10.1145/321921.321938>, [](http://dx.doi.org/10.1145/321921.321938).
John B. Kam and Jeffrey D. Ullman. Monotone data flow analysis frameworks. , 7(3):305–317, Sep 1977. Available from: <https://doi.org/10.1007/BF00290339>, [](http://dx.doi.org/10.1007/BF00290339).
Gary A. Kildall. A unified approach to global program optimization. In [*Proceedings of the 1st Annual ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages*]{}, POPL ’73, pages 194–206, New York, NY, USA, 1973. ACM. Available from: <http://doi.acm.org/10.1145/512927.512945>, [](http://dx.doi.org/10.1145/512927.512945).
Markus M[ü]{}ller-Olm, Oliver R[ü]{}thing, and Helmut Seidl. , pages 79–96. Springer Berlin Heidelberg, Berlin, Heidelberg, 2005. Available from: <https://doi.org/10.1007/978-3-540-30579-8_6>, [](http://dx.doi.org/10.1007/978-3-540-30579-8_6).
Rekha R. Pai. Detection of redundant expressions: A precise, efficient, and pragmatic algorithm in ssa. , 46:167–181, 2016.
B. K. Rosen, M. N. Wegman, and F. K. Zadeck. Global value numbers and redundant computations. In [*Proceedings of the 15th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages*]{}, POPL ’88, pages 12–27, New York, NY, USA, 1988. ACM. Available from: <http://doi.acm.org/10.1145/73560.73562>, [](http://dx.doi.org/10.1145/73560.73562).
Oliver R[ü]{}thing, Jens Knoop, and Bernhard Steffen. , pages 232–247. Springer Berlin Heidelberg, Berlin, Heidelberg, 1999. Available from: <https://doi.org/10.1007/3-540-48294-6_15>, [](http://dx.doi.org/10.1007/3-540-48294-6_15).
Bernhard Steffen, Jens Knoop, and Oliver R[ü]{}thing. The value flow graph: A program representation for optimal program transformations. In [*Proceedings of the European Symposium on Programming, pages 389–405. Springer-Verlag LNCS 432*]{}, pages 389–405. Springer-Verlag, 1990.
Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. , 5(2):285–309, 1955.
Proofs of Theorems
==================
**Theorem \[thm:confluence\].** If $\mathcal{P}_1$ and $\mathcal{P}_2$ are congruences, then $\mathcal{P}_1\wedge \mathcal{P}_2$ is a congruence.
Let $\mathcal{P}_1=\{A_i\}_{i\in I}$ and $\mathcal{P}_2=\{B_j\}_{j\in J}$. Let $C_{i,j}=A_i\cap B_j$ for all $i\in I$ and $j\in J$. Clearly $\mathcal{P}_1\wedge \mathcal{P}_2=\{C_{i,j}: i\in I,j\in J, C_{i,j}\neq \emptyset\}$ is a partition of $\mathcal{T}$. It suffices to prove that $\mathcal{P}_1\wedge \mathcal{P}_2$ satisfies properties (1) to (3) of Definition \[def:cong\].
1. For any $c,c'\in C$, $c{\cong_{_{\mathcal{P}_1\wedge \mathcal{P}_2}}}c' \iff c {\cong_{_{\mathcal{P}_1}}}c'$ and $c{\cong_{_{\mathcal{P}_2}}}c'\iff c=c'$.
2. For $t,t',s,s'\in \mathcal{T}$, $t+s {\cong_{_{\mathcal{P}_1\wedge \mathcal{P}_2}}}t'+s'$ if and only if $t+s {\cong_{_{\mathcal{P}_1}}}t'+s'$ and $t+s {\cong_{_{\mathcal{P}_2}}}t'+s'$\
$\iff t'{\cong_{_{\mathcal{P}_1}}}t$, $t'{\cong_{_{\mathcal{P}_2}}}t$, $s'{\cong_{_{\mathcal{P}_1}}}s$ and $s'{\cong_{_{\mathcal{P}_2}}}s$ $\iff t'{\cong_{_{\mathcal{P}_1\wedge \mathcal{P}_2}}}t$ and $s'{\cong_{_{\mathcal{P}_1\wedge \mathcal{P}_2}}}s$.
3. For any $c\in C$, $t\in \mathcal{T}$, $c{\cong_{_{\mathcal{P}_1\wedge \mathcal{P}_2}}}t$ if and only if $c{\cong_{_{\mathcal{P}_1}}}t$ and $c{\cong_{_{\mathcal{P}_2}}}t$, only if either $t=c$ or $t\in X$ (by condition (3) of the definition of congruence).
**Lemma \[lem:poset\].** $(\mathcal{G}(\mathcal{T}),\preceq)$ is a meet semi-lattice with meet operation $\wedge$ and bottom element $\bot$.
It is evident from the definition of $\preceq$ that $(\mathcal{G}(\mathcal{T}),\preceq)$ is a partial order. Let $\mathcal{P}_1=\{A_i\}_{i\in I}$ and $\mathcal{P}_2=\{B_j\}_{j\in J}$ be congruences in $\in \mathcal{G}(\mathcal{T})$. We next show that $\mathcal{P}_1\wedge \mathcal{P}_2$ is the meet of $\mathcal{P}_1$ and $\mathcal{P}_2$ with respect to $\preceq$. By Definition \[def:confluence\], the relations $\mathcal{P}_1\wedge \mathcal{P}_2\preceq \mathcal{P}_1$ and $\mathcal{P}_1\wedge \mathcal{P}_2\preceq \mathcal{P}_2$ must hold true. Suppose that a congruence $\mathcal{P}=\{C_k\}_{k\in K}$ satisfies $\mathcal{P}\preceq \mathcal{P}_1$ and $\mathcal{P}\preceq \mathcal{P}_2$. Then by definition of $\preceq$, for each $k\in K$, there must exist $i\in I$ and $j\in J$ such that $C_k\subseteq A_i$ and $C_k\subseteq B_j$. Consequently, $\emptyset \neq C_k\subseteq (A_i\cap B_j)$. By Definition \[def:confluence\] we have $A_i\cap B_j\in \mathcal{P}_1\wedge \mathcal{P}_2$ and hence $\mathcal{P}\preceq \mathcal{P}_1\wedge \mathcal{P}_2$. Thus, $\mathcal{P}_1\wedge \mathcal{P}_2$ is the meet of $\mathcal{P}_1$ and $\mathcal{P}_2$. Finally, $\bot$ is the bottom element of ${\mathcal{G}(\mathcal{T})}$ by Observation \[obs:bot\].
**Lemma \[lem:subsetInfimum\].** Every non-empty subset of $(\mathcal{G}(\mathcal{T}),\preceq)$ has a greatest lower bound.
We will show that any arbitrary non-empty family of congruences $\{\mathcal{P}_r\}_{r\in R}$ of $\mathcal{T}$ has a greatest lower bound. For each $r \in R$, let $\mathcal{P}_r =\{A _{r, i_r}\}_{i_r \in I_r}$, where $I_r\neq \emptyset$. Let $\mathcal{I}$ be the Cartesian product of the index sets of the congruences; i.e., $\mathcal{I}=\prod_{r \in R}{I_r}$. Note that, each element in $\mathcal{I}$ is a sequence $(i_r)_{r \in R}$. For each $(i_r)_{r\in R} \in \mathcal{I}$, define $B_{(i_r)}= \bigcap_{r\in R} A_{r,i_r}$. (Note that the index element $i_r \in I_r$ and $A_{r,i_r}$ is the set in partition $\{\mathcal{P}_r\}_{r\in R}$ having index $i_r$). Let $\mathcal{P}=\{B_{(i_r)} : (i_r) \in \mathcal{I}, B_{(i_r)} \ne \emptyset \}$.
We claim that $\mathcal{P}$ is a congruence and it is the greatest lower bound of $\{\mathcal{P}_r\}_{r\in R}$. To show that $\mathcal{P}$ is a partition, first assume $t\in \mathcal{T}$. Then, for each $r\in R$, there exists some $i_r\in I_r$ such that $t\in A_{r,i_r}$. Hence $t\in B_{(i_r)}$. Thus every element appears in at least one set $B_{(i_r)}$ in $\mathcal{P}$. Next, if $(i_r),(j_r)\in \mathcal{I}$ such that $(i_r)\neq (j_r)$, we show that $B_{(i_r)}\cap B_{(j_r)}=\emptyset$. To this end, assume that some $t\in \mathcal{T}$ satisfies $t\in B_{(i_r)}\cap B_{(j_r)}$. Since $(i_r)\neq (j_r)$, there exists some index $r_0$ in which the sequences $(i_r)$ and $(j_r)$ differ; i.e, $i_{r_0}\neq j_{r_0}$. As $t\in B_{(i_r)}\cap B_{(j_r)}$, it must be the case that $t\in A_{r_0, i_{r_0}}\cap A_{r_0, j_{r_0}}$, which contradicts our assumption that $\mathcal{P}_{r_0}$ is a congruence.
Next, we prove that $\mathcal{P}$ satisfies properties (1) to (3) of Definition \[def:cong\].
1. Let $c,c'\in C$ be constants such that $c\neq c'$. Suppose $c,c'\in B_{(i_r)}$ for some $(i_r)\in \mathcal{I}$. Then, for each $r\in R$, $c,c'\in A_{r,i_r}$. However, this is impossible as $\{A_{r,i_r}\}_{i_r\in I_r}$ is a congruence for each $r$. This proves (1).
2. Let $t,t',s,s'\in \mathcal{T}$. We have $t,t'\in B_{(i_r)}$ and $s,s'\in B_{(j_r)}$ for some $(i_r),(j_r)\in \mathcal{I}$ if and only if $t,t'\in A_{r,i_r}$ and $s,s'\in A_{r,j_r}$ for each $r\in R$ if and only if there exists $(k_r)\in \mathcal{I}$ such that $t+s,t'+s'\in A_{r,k_r}$ for each $r\in R$ (because each $\mathcal{P}_r$ is a congruence) if and only if $t+s,t'+s'\in B_{(k_r)}$ (by definition of $B_{(k_r)}$), proving (3).
3. For any $c\in C$ and $t\in \mathcal{T}$, suppose $c,t\in B_{(i_r)}$. Then, $c,t\in A_{r,i_r}$ for each $r\in R$. Hence either $t=c$ or $t\in X$ because $\mathcal{P}_r$ is a congruence for each $r\in R$.
Next we show that $\mathcal{P}\preceq \mathcal{P}_r$ for each $r\in R$. Let $B_{(i_r)}\in \mathcal{P}$. By definition, $B_{(i_r)}=\bigcap_{r\in R} A_{r,i_r}$. Thus $B_{(i_r)}\subseteq A_{r,i_r}$ for each $r\in R$. As $A_{r,i_r}\in \mathcal{P}_r$ for each $r\in R$, we have $\mathcal{P}\preceq \mathcal{P}_r$ for each $r\in R$.
To prove that $\mathcal{P}$ is the greatest lower bound of $\{\mathcal{P}_r\}_{r\in R}$, assume $\mathcal{Q}\preceq \mathcal{P}_r$ for each $r\in R$. Let $\mathcal{Q}=\{C_k\}_{k\in K}$. Then, for each $k\in K$ and each $r\in R$, there exists $i_r\in I_r$ such that $C_k\subseteq A_{r,i_r}$. Hence $C_k\subseteq B_{(i_r)}=\bigcap_{r\in R} A_{r,i_r}$. Since $B_{(i_r)}\in \mathcal{P}$, we have $\mathcal{Q}\preceq \mathcal{P}$.
The proof of the lemma is complete. Note that axiom of choice was used in assuming that the set $\mathcal{I}$ is non-empty.
**Theorem \[thm:trans-function\].** If $\mathcal{P}$ is a congruence, then for any $y\in X$, $\beta \in \overline{\mathcal{T}}(y)$, ${f_{_{y=\beta}}(\mathcal{P})}$ is a congruence.
Let $\mathcal{P}=\{A_{i}\}_{i\in I}$. Let $f(\mathcal{P})={f_{_{y=\beta}}(\mathcal{P})}$ and $\{B_i\}_{i\in I}$ be defined as in Definition \[def:trans-function\]. Since $\mathcal{P}$ is a congruence, for each $t\in \mathcal{T}$, there exists classes $A_i,A_j\in \mathcal{P}$ such that $t\in A_i$ and $t[y\leftarrow \beta]\in A_j$. (Note that $i=j$ is possible, but $A_i$ and $A_j$ are uniquely determined by the terms $t$ and $t[y\leftarrow \beta]$). First we show that $f(\mathcal{P})$ has $t$ in exactly one class.
- If $t\notin A_i(y)$ (i.e., $y$ does not appear in $t$), then clearly $t\in B_i$ and no other class of $f(\mathcal{P})$ by definition. In particular, when $t=c$ for any constant $c\in C$, $c\in B_i$ if and only if $c\in A_i$, establishing condition (1) of Definition \[def:cong\].
- If $t\in A_i(y)$, then by definition of $B_j$, $t\in B_j$ and no other class in $f(\mathcal{P})$ contains $t$.
We have shown that $f(\mathcal{P})$ is a partition that satisfies condition (1) of Definition \[def:cong\]. Next, we prove that $f(\mathcal{P})$ satisfies the remaining conditions of the definition of congruence.\
*Condition (2)*: we need to prove that for all $t,t',s,s'\in \mathcal{T}$, $t'{\cong_{_{f(\mathcal{P})}}}t$ and $s'{\cong_{_{f(\mathcal{P})}}}s$ if and only if $t'+s'{\cong_{_{f(\mathcal{P})}}}t+s$. We have: $$\begin{aligned}
&&t'+s'{\cong_{_{f(\mathcal{P})}}}t+s\\
&\iff& (t'+s')[y\leftarrow \beta] {\cong_{_{\mathcal{P}}}}(t+s)[y\leftarrow \beta] \text{ (by definition of ${f_{_{y=\beta}}(\mathcal{P})}$) }\\
&=& t'[y\leftarrow \beta]+s'[y\leftarrow \beta] {\cong_{_{\mathcal{P}}}}t[y\leftarrow \beta]+s[y\leftarrow \beta] \text{ (by Definition~\ref{def:substitution})}\\
&\iff& t'[y\leftarrow \beta] {\cong_{_{\mathcal{P}}}}t[y\leftarrow \beta] \text{ and } s'[y\leftarrow \beta] {\cong_{_{\mathcal{P}}}}s[y\leftarrow \beta] \text{( by definition of }{\cong_{_{\mathcal{P}}}}\text{)}\\
&\iff& t' {\cong_{_{f(\mathcal{P})}}}t \text{ and } s' {\cong_{_{f(\mathcal{P})}}}s \text{ (by definition of ${f_{_{y=\beta}}(\mathcal{P})}$) }\end{aligned}$$ *Condition (3)*: For any $c\in C$, let $t\in \mathcal{T}$ such that $t\neq c$. Suppose $c{\cong_{_{f(\mathcal{P})}}}t$, then by Observation \[obs:trans\] $t[y\leftarrow \beta]\in X\cup \{c\}$ (by condition (3) of the definition of congruence), which is possible only if one among the following cases are true: (1) $t=x$ for some $x\in X$, $x\neq y$ such that $x{\cong_{_{\mathcal{P}}}}c$ or (2) $t=y$ and $\beta=x$ for some $x\in X$, $x{\cong_{_{\mathcal{P}}}}c$ or (3) $t=y$ and $\beta=c$. In any case $t\in X$.
**Lemma \[lem:distrb\].** ${f(\mathcal{P}_1)}\wedge {f(\mathcal{P}_2)}= {f(\mathcal{P}_1\wedge \mathcal{P}_2)}$.
If $\mathcal{P}_1 =\top$, then ${f(\mathcal{P}_1)}\wedge {f(\mathcal{P}_2)}= \top \wedge {f(\mathcal{P}_2)}= {f(\mathcal{P}_2)}= f(\top \wedge \mathcal{P}_2) $ and the lemma holds true. Similarly, if $\mathcal{P}_2 =\top$, the lemma holds true. Otherwise, let $\mathcal{P}_1=\{A_i\}_{i\in I}$ and $\mathcal{P}_2=\{B_j\}_{j\in J}$. By Definition \[def:trans-function\], we have: $$\begin{aligned}
{f(\mathcal{P}_1)}&=&\{A'_i : i\in I, A'_i\neq \emptyset\} \text{, where } A'_i=\{t\in \mathcal{T} : t[y\leftarrow \beta]\in A_i\}\\
{f(\mathcal{P}_2)}&=&\{B'_j : j\in J, B'_j\neq \emptyset\} \text{, where } B'_j=\{t\in \mathcal{T} : t[y\leftarrow \beta]\in B_j\}
\end{aligned}$$ By the definition of meet, we have: $$\begin{aligned}
\label{eqn:eqdistr}
{f(\mathcal{P}_1)}\wedge {f(\mathcal{P}_2)}&=& \{A'_i\cap B'_j : i\in I, j\in J, A'_i\cap B'_j\neq \emptyset\}\\
\mathcal{P}_1\wedge \mathcal{P}_2 &=& \{A_i\cap B_j : i\in I, j\in J, A_i\cap B_j\neq \emptyset\}\end{aligned}$$ Moreover, $$\label{eqn:eqmeet}
{f(\mathcal{P}_1\wedge \mathcal{P}_2)}= \{D_{i,j} : i\in I, j\in J, D_{i,j}\neq \emptyset\}$$ where, $$\begin{aligned}
D_{i,j}&=&\{t\in \mathcal{T} : t[y\leftarrow \beta]\in A_i\cap B_j\}\\
&=&\{t\in \mathcal{T} : t[y\leftarrow \beta]\in A_i \text{ and } t[y\leftarrow \beta]\in B_j\}\\
&=&\{t\in \mathcal{T} : t\in A'_i \text{ and } t\in B'_j\}=A'_i\cap B'_j \end{aligned}$$ Thus, Equation (\[eqn:eqmeet\]) becomes, $$\label{eqn:eqfinalmeet}
{f(\mathcal{P}_1\wedge \mathcal{P}_2)}= \{A'_i\cap B'_j : i\in I, j\in J, A'_i\cap B'_j \neq \emptyset\}$$ The result follows by comparing equations (\[eqn:eqdistr\]) and (\[eqn:eqfinalmeet\]).
**Theorem \[thm:continuous\].** For any $\emptyset \neq S \subseteq {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$, $f(\bigwedge S)=\bigwedge f(S)$.
![Commutativity diagram for Theorem \[thm:continuous\][]{data-label="fig:commutativity"}](continuous.pdf)
If $S=\{\top\}$, $f(\bigwedge S)=\top=\bigwedge f(S)$. Otherwise, we may assume without loss of generality that $\top \notin S$ (else consider $S\setminus \{\top\}$).
Let $S=\{\mathcal{P}_r\}_{r\in R}$ be a non-empty collection of congruences, not containing $\top$. For each $r\in R$, let $\mathcal{P}_r=\{A_{r,i_r}\}_{i_r\in I_r}$ for some index set $I_r\neq \emptyset$. By Definition \[def:trans-function\] of transfer function, for each $r\in R$ we have: $$\begin{aligned}
\label{eqn:fmeetP}
f(\mathcal{P}_r)&=&\{A'_{r,i_r} : i_r\in I_r, A'_{r,i_r}\neq \emptyset\} \text{, where } A'_{r,i_r}=\{t\in \mathcal{T} : t[y\leftarrow \beta]\in A_{r,i_r}\}
\end{aligned}$$ Hence $f(S)=\{ f(\mathcal{P}_r) : r\in R\}$.
Let $\mathcal{I}$ be the Cartesian product of the index sets of the congruences; i.e., $\mathcal{I}=\prod_{r \in R}{I_r}$. Each element in $\mathcal{I}$ is a sequence $(i_r)_{r \in R}$. For each $(i_r)_{r\in R} \in \mathcal{I}$, define\
$B_{(i_r)}= \bigcap_{r\in R} A_{r,i_r}$ and $B'_{(i_r)}= \bigcap_{r\in R} A'_{r,i_r}$. As shown in the proof of Lemma \[lem:subsetInfimum\], we have: $$\label{eqn:meetS}
\bigwedge S = \{B_{(i_r)} : (i_r) \in \mathcal{I}, B_{(i_r)} \ne \emptyset \}$$ $$\label{eqn:meetfS}
\bigwedge f(S) = \{B'_{(i_r)} : (i_r) \in \mathcal{I}, B'_{(i_r)} \ne \emptyset \}$$ From Equation \[eqn:meetS\] and Definition \[def:trans-function\] of transfer function we write: $$\begin{aligned}
\label{eqn:fmeetS}
f(\bigwedge S)= \{D_{(i_r)} : (i_r) \in \mathcal{I}, D_{(i_r)} \ne \emptyset \} \text{ where, } D_{(i_r)} = \{t\in \mathcal{T}: t[y\leftarrow \beta]\in B_{(i_r)}\}\end{aligned}$$ Comparing Equation \[eqn:meetfS\] and Equation \[eqn:fmeetS\], we see that to complete the proof, it is enough to prove that $D_{(i_r)}= B'_{(i_r)}$ for each $(i_r)\in I$ (see Figure \[fig:commutativity\]). $$\begin{aligned}
D_{(i_r)}&=&\{t\in \mathcal{T}: t[y\leftarrow \beta]\in B_{(i_r)}\}\\
&=&\{t\in \mathcal{T}: t[y\leftarrow \beta]\in \cap_{r\in R} A_{r,i_r}\}\text{ (by definition of $B_{(i_r)}$) } \\
&=&\{t\in \mathcal{T}: t[y\leftarrow \beta]\in A_{r,i_r}\text{ for each }r\in R\} \\
&=&\{t\in \mathcal{T}: t\in A'_{r,i_r}\text{ for each }r\in R\} \text{ (by Equation~\ref{eqn:fmeetP}) }\\
&=&\{t\in \mathcal{T}: t\in \cap_{r\in R} A'_{r,i_r}\}\\
&=&\{t\in \mathcal{T}: t\in B'_{(i_r)}\} \text{ (by definition of $B'_{(i_r)}$) }\\
&=& B'_{(i_r)}\end{aligned}$$
The proof is now complete. Note that the axiom of choice was used in the definition of the set $\mathcal{I}$.
**Theorem \[thm:non-det-trans\].** If $\mathcal{P}$ is a congruence, then for any $y\in X$, ${f_{_{y=*}}(\mathcal{P})}$ is a congruence.
We write $f(\mathcal{P})$ instead of ${f_{_{y=*}}(\mathcal{P})}$ to avoid cumbersome notation. First we show that ${\cong_{_{f(\mathcal{P})}}}$ is an equivalence relation, thereby establishing that $f(\mathcal{P})$ is a partition of $\mathcal{T}$. Reflexivity, symmetry and transitivity of ${\cong_{_{f(\mathcal{P})}}}$ is clear from Definition \[def:non-det-trans\]. Moreover, each equivalence class in ${f_{_{y=*}}(\mathcal{P})}$ is a subset of some equivalence class in the congruence $\mathcal{P}$. Consequently, *Condition (1)* and *Condition (3)* in the definition of congruence (Definition \[def:cong\]) will be satisfied by $f(\mathcal{P})$.
It suffices to show that ${f_{_{y=*}}(\mathcal{P})}$ satisfies Condition (2) of Definition \[def:cong\].\
*Condition (2)*: we need to prove that for all $t,t',s,s'\in \mathcal{T}$, $t'{\cong_{_{f(\mathcal{P})}}}t$ and $s'{\cong_{_{f(\mathcal{P})}}}s$ if and only if $t'+s'{\cong_{_{f(\mathcal{P})}}}t+s$. We have: $$\begin{aligned}
t'+s'&{\cong_{_{f(\mathcal{P})}}}& t+s\\
&\iff& (t'+s'){\cong_{_{\mathcal{P}}}}(t+s) \text{ and, for every } \beta \in \mathcal{T}\setminus \mathcal{T}(y),\\
&&(t+s)[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}(t'+s')[y\leftarrow \beta]\text{ (by definition of ${f_{_{y=*}}(\mathcal{P})}$) }\\
&\iff& (t'+s'){\cong_{_{\mathcal{P}}}}(t+s) \text{ and, for every } \beta \in \mathcal{T}\setminus \mathcal{T}(y),\\
&&(t[y\leftarrow \beta]+s[y\leftarrow \beta]){\cong_{_{\mathcal{P}}}}(t'[y\leftarrow \beta]+s'[y\leftarrow \beta])\\
&\iff& (t' {\cong_{_{\mathcal{P}}}}t) \text{ and } (s' {\cong_{_{\mathcal{P}}}}s) \text{ and, for every } \beta \in \mathcal{T}\setminus \mathcal{T}(y),\\
&&(t[y\leftarrow \beta] {\cong_{_{\mathcal{P}}}}t'[y\leftarrow \beta]) \text{ and } s[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}s'[y\leftarrow \beta])\text{ (by Definition~\ref{def:cong}) }\\
&\iff& (t' {\cong_{_{f(\mathcal{P})}}}t) \text{ and } (s' {\cong_{_{f(\mathcal{P})}}}s) \text{ (by definition of ${f_{_{y=*}}(\mathcal{P})}$) }\\\end{aligned}$$ The proof of the theorem is complete.
**Theorem \[thm:ntrans-char\].** If $\mathcal{P}$ is a congruence, then for any $y\in X$,\
${f_{_{y=*}}(\mathcal{P})}=\mathcal{P}\wedge \left( \bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P})}\right)$.
Let $\mathcal{Q}=\mathcal{P}\wedge \left( \bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P})}\right)$. It follows from condition (1) of Definition \[def:non-det-trans\] and Theorem \[thm:non-det-trans\] that ${f_{_{y=*}}(\mathcal{P})}$ is a congruence, and a refinement of $\mathcal{P}$. That is, ${f_{_{y=*}}(\mathcal{P})}\preceq \mathcal{P}$. From condition (2) of Definition \[def:non-det-trans\], we have, $$\begin{aligned}
t{\cong_{_{f_{_{y=*}}(\mathcal{P})}}}t' &\implies& \text{ for every }\beta\in \mathcal{T}\setminus \mathcal{T}(y)\text{, }t[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow \beta] \\
&\iff& \text{ for every }\beta\in \mathcal{T}\setminus \mathcal{T}(y)\text{, }t \cong_{_{f_{_{y=\beta}}(\mathcal{P})}} t' \text{ (by Observation~\ref{obs:trans})} \\\end{aligned}$$ Hence for each $\beta\in \mathcal{T}\setminus \mathcal{T}(y)$, ${f_{_{y=*}}(\mathcal{P})}\preceq {f_{_{y=\beta}}(\mathcal{P})}$. It follows from the definition of $\bigwedge$ that ${f_{_{y=*}}(\mathcal{P})}\preceq \bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P})}$ . Combining this with the fact that ${f_{_{y=*}}(\mathcal{P})}\preceq \mathcal{P}$ and using the definition of $\wedge$, we get ${f_{_{y=*}}(\mathcal{P})}\preceq Q$.
Conversely let $t,t'\in \mathcal{T}$ such that $t{\cong_{_{\mathcal{Q}}}}t'$. Then by definition of $\mathcal{Q}$, we have $t{\cong_{_{\mathcal{P}}}}t'$ and, for all $\beta \in \mathcal{T}\setminus \mathcal{T}(y)$ $t \cong_{_{f_{_{y=\beta}}(\mathcal{P})}} t'$. By the definition of ${f_{_{y=*}}(\mathcal{P})}$ (Definition \[def:non-det-trans\]), we have $t{\cong_{_{f_{_{y=*}}(\mathcal{P})}}}t'$. Consequently $Q\preceq {f_{_{y=*}}(\mathcal{P})}$.
Hence, ${f_{_{y=*}}(\mathcal{P})}=\mathcal{Q}$. Note that the proof involves use of the axiom of choice in assuming existence of $\bigwedge$.
**Theorem \[thm:ntrans-cts\].** For any $\emptyset \neq S \subseteq {\overline{\mathcal{G}(\mathcal{\mathcal{T}})}}$, ${f}_{_{y=*}}(\bigwedge S)=\bigwedge {f}_{_{y=*}}(S)$, where\
${f}_{_{y=*}}(S)=\{f_{_{y=*}}(s): s\in S\}$.
To avoid cumbersome notation, we will write $f$ for $f_{_{y=*}}$. If $S=\{\top\}$, $f(\bigwedge S)=\top=\bigwedge f(S)$. Otherwise, we may assume without loss of generality that $\top \notin S$ (else consider $S\setminus \{\top\}$).
Let $S=\{\mathcal{P}_r\}_{r\in R}$ be a non-empty collection of congruences, not containing $\top$. We denote by $\bigwedge_{r\in R} \mathcal{P}_r$ the congruence $\bigwedge \{\mathcal{P}_r: r\in R\}$ and $\bigwedge_{r\in R}f(\mathcal{P}_r)$ for the congruence $\bigwedge \{ f(\mathcal{P}_r): r\in R\}$.
$$\begin{aligned}
f(\bigwedge S)&=& f(\bigwedge_{r\in R} \mathcal{P}_r)\\
&=& (\bigwedge_{r\in R} \mathcal{P}_r) \wedge ( \bigwedge_{\beta \in \overline{\mathcal{T}}(y)}f(\bigwedge_{r\in R} \mathcal{P}_r) ) \text{ (by Theorem~\ref{thm:ntrans-char}) } \\
&=& (\bigwedge_{r\in R} \mathcal{P}_r) \wedge ( \bigwedge_{\beta \in \overline{\mathcal{T}}(y)}\bigwedge_{r\in R}{f_{_{y=\beta}}(\mathcal{P}_r)}) \text{ (by continuity of $f_{y=\beta}$) } \\
&=& (\bigwedge_{r\in R} \mathcal{P}_r) \wedge ( \bigwedge_{r\in R} \bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P}_r)}) \text{ (by properties of $\bigwedge$) } \\
&=& \bigwedge_{r\in R} ( \mathcal{P}_r \wedge \bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P}_r)}) \text{ (by properties of $\bigwedge$) } \\
&=& \bigwedge_{r\in R} f(\mathcal{P}_r) \text{ (by Theorem~\ref{thm:ntrans-char}) } \\
&=& \bigwedge f(S)
\end{aligned}$$
Note that the axiom of choice was implicitly used in the proof.
**Lemma \[lem:nondet-constant\].** Let $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$. Let $t,t'\in \mathcal{T}$ and $c_1,c_2\in C$ with $c_1\neq c_2$. Then $t[y\leftarrow c_1]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_1]$ and $t[y\leftarrow c_2]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_2]$ if and only if for every $\beta\in \overline{\mathcal{T}}(y)$, $t[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow \beta]$.
One direction is trivial. For the other, assume that $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$ and $c_1,c_2 \in C$, $c_1\neq c_2$. If $t=t'$ or $t,t'\notin \mathcal{T}(y)$, the lemma holds true trivially. Hence we assume that $t\in \mathcal{T}(y)$ and $t'\neq t$. If $t'\notin \mathcal{T}(y)$, Then our assumption leads to $t[y\leftarrow c_1]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_1]=t'=t'[y\leftarrow c_2]{\cong_{_{\mathcal{P}}}}t[y\leftarrow c_2]$. This is impossible by Lemma \[lem:const1\]. Hence we may assume that $t,t'\in \mathcal{T}(y)$.
If $t=y$, then by Observation \[obs:const2\], $t[y\leftarrow c_1]=c_1\ncong_{_{\mathcal{P}}} t'[y\leftarrow c_1]$ for any $t'\in \mathcal{T}(y)$, $t'\neq y$. Hence we may assume that $t\neq y$. Similarly, we may assume that $t'\neq y$ as well. Hence the lemma holds true whenever at least one among $t,t'$ is in $X\cup C$. We now proceed by induction.
Suppose $t=t_1+t_2$ and $t'=t'_1+t'_2$. We have: $$\begin{aligned}
\label{eqn:const1}
t[y\leftarrow c_1]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_1] &\iff& t_1[y\leftarrow c_1]+t_2[y\leftarrow c_1] {\cong_{_{\mathcal{P}}}}t'_1[y\leftarrow c_1]+t'_2[y\leftarrow c_1] \\
&\iff& t_1[y\leftarrow c_1] {\cong_{_{\mathcal{P}}}}t'_1[y\leftarrow c_1] \text{ and } t_2[y\leftarrow c_1] {\cong_{_{\mathcal{P}}}}t'_2[y\leftarrow c_1]\\
&&\text{ (By condition (2) of Definition~\ref{def:cong})}\end{aligned}$$ Similarly, $$\begin{aligned}
\label{eqn:const2}
t[y\leftarrow c_2]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_2] &\iff&t_1[y\leftarrow c_2] {\cong_{_{\mathcal{P}}}}t'_1[y\leftarrow c_2] \text{ and } t_2[y\leftarrow c_2] {\cong_{_{\mathcal{P}}}}t'_2[y\leftarrow c_2]\end{aligned}$$ We may assume as induction hypothesis that: $$\begin{aligned}
t_1[y\leftarrow c_1] {\cong_{_{\mathcal{P}}}}t'_1[y\leftarrow c_1] &\text{ and }& t_1[y\leftarrow c_2] {\cong_{_{\mathcal{P}}}}t'_1[y\leftarrow c_2] \\
&\iff&t_1[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'_1[y\leftarrow \beta] \text{, for every }\beta\in \overline{\mathcal{T}}(y)\end{aligned}$$ $$\begin{aligned}
t_2[y\leftarrow c_1] {\cong_{_{\mathcal{P}}}}t'_2[y\leftarrow c_1] &\text{ and }& t_2[y\leftarrow c_2] {\cong_{_{\mathcal{P}}}}t'_2[y\leftarrow c_2] \\
&\iff&t_2[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'_2[y\leftarrow \beta] \text{, for every }\beta\in \overline{\mathcal{T}}(y)\end{aligned}$$ Since the left sides of the above equivalences hold by assumption, we have: $$\begin{aligned}
t_1[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'_1[y\leftarrow \beta] \text{ and }
t_2[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'_2[y\leftarrow \beta] \text{, for every }\beta\in \overline{\mathcal{T}}(y)\end{aligned}$$ Hence, by condition (3) of Definition \[def:cong\], $$t[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow \beta] \text{, for every }\beta\in \overline{\mathcal{T}}(y)$$
\[lem:const1\] Let $t\in \mathcal{T}(y)$ and $c_1,c_2\in C$, $c_1\neq c_2$. Let $\mathcal{P}$ be a congruence. Then, $t[y\leftarrow c_1]\ncong_{_{\mathcal{P}}} t[y\leftarrow c_2]$.
If $t=y$, the lemma holds by condition (1) of the definition of congruence (Definition \[def:cong\]). Otherwise, we proceed by induction. Let $t=t_1+t_2$. Since $t\in \mathcal{T}(y)$, either $t_1\in \mathcal{T}(y)$ or $t_2\in \mathcal{T}(y)$. Without loss of generality, we assume that $t_1\in \mathcal{T}(y)$. We have by induction hypothesis: $$t_1[y\leftarrow c_1] \ncong_{_{\mathcal{P}}} t_1[y\leftarrow c_2]$$ Hence we have: $$\begin{aligned}
t[y\leftarrow c_1]&=& t_1[y\leftarrow c_1]+t_2[y\leftarrow c_1]\\
&\ncong_{_{\mathcal{P}}}& t_1[y\leftarrow c_2]+t_2[y\leftarrow c_2] \text{ (by condition (2) of Definition~\ref{def:cong}) }\\
&=& t[y\leftarrow c_2]
\end{aligned}$$
**Theorem \[thm:nondet-const\].** Let $\mathcal{P}\in {\mathcal{G}(\mathcal{T})}$ and let $c_1,c_2\in C$, $c_1\neq c_2$. Then, for any $y\in X$,\
${f_{_{y=*}}(\mathcal{P})}=\mathcal{P}\wedge f_{_{y\leftarrow c_1}}(\mathcal{P}) \wedge f_{_{y\leftarrow c_2}}(\mathcal{P})$
Let $\mathcal{Q}=\mathcal{P}\wedge f_{_{y\leftarrow c_1}}(\mathcal{P}) \wedge f_{_{y\leftarrow c_2}}(\mathcal{P})$ and let $\mathcal{Q}'=\mathcal{P}\wedge \left( \bigwedge_{\beta \in \overline{\mathcal{T}}(y)} {f_{_{y=\beta}}(\mathcal{P})}\right)$ We have: $$\begin{aligned}
t{\cong_{_{\mathcal{Q}}}}t' &\iff& t{\cong_{_{\mathcal{P}}}}t' \text{ and }t {\cong_{_{f_{_{y=c_1}}(\mathcal{P})}}}t' \text{ and } t {\cong_{_{f_{_{y=c_2}}(\mathcal{P})}}}t' \text{ (by definition of confluence) }\\
&\iff& t{\cong_{_{\mathcal{P}}}}t'\text{ and } t[y\leftarrow c_1]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_1] \text{ and } t[y\leftarrow c_2]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow c_2]\\
&&\text{ (by Observation~\ref{obs:trans}) } \\
&\iff & t{\cong_{_{\mathcal{P}}}}t' \text{ and } \forall \beta\in \overline{\mathcal{T}}(y)\text{ } t[y\leftarrow \beta]{\cong_{_{\mathcal{P}}}}t'[y\leftarrow \beta] \text{ (by Lemma~\ref{lem:nondet-constant}) } \\
&\iff & t{\cong_{_{\mathcal{P}}}}t' \text{ and } \forall \beta\in \overline{\mathcal{T}}(y)\text{ } t \cong_{_{f_{_{y=\beta}}(\mathcal{P})}} t' \text{ (by Observation~\ref{obs:trans})}\\
&\iff & t\cong_{\mathcal{Q}'} t' \text{ (by definition of confluence) } \end{aligned}$$
Lattice Properties
==================
Certain standard facts from lattice theory used in the paper are proved here for easy reference.
In the following, $(L,\leq, \bot, \top)$ denotes a lattice with greatest element $\top$ and least element $\bot$. (We will refer to the lattice simply as $L$ when there is not scope for confusion). For $a,b\in L$, $a\wedge b$ (respectively $a\vee b$) denotes the meet (respectively join) of $a$ and $b$. For $S\subseteq L$, $\bigwedge S$ (respectively $\bigvee S$) denotes the greatest lower bound (respectively least upper bound) of the set $S$ whenever it exists. The lattice $L$ is meet complete if $\bigwedge S$ exists for every subset $S$ of $L$. In particular, $\bigwedge \emptyset = \top$ and $\bigwedge L=\bot$. A lattice $L$ is a complete lattice if $\bigwedge S$ and $\bigvee S$ exists for every $S \subseteq L$.
\[Athm:complete\] If $(L,\leq,\bot, \top)$ is meet complete, then it is a complete lattice.
It suffices to prove that for each $S\subseteq L$, $\bigvee S$ exists. Let $S$ be a subset of $L$. Let $T=\{t \in L: s\leq t \text{ for each } s \in S\}$. Since $L$ is meet complete there exists $t_0\in L$ such that $t_0=\bigwedge T$. We claim that $t_0$ is the least upper bound of $S$. By definitions of the set $T$, for each $t\in T$ and $s\in S$, $s\leq t$. Hence, by the definition of meet, $s\leq t_0$ for each $s\in S$. Consequently $t_0$ is an upper bound to $S$. Next, if any $t\in L$ satisfies $s\leq t$ for each $s\in S$, then by the definition of $T$, $t\in T$. Hence $t_0\leq t$ (by the definition of $t_0$). Thus $t_0$ is the least upper bound of $S$.
The following property of the meet operation will be frequently used.
\[Alem:meet-union\] If $(L,\leq,\bot,\top)$ is a complete lattice. Let $\{S_i\}_{i\in I}$ be subsets of $L$. Then, $\bigwedge (\bigcup_{i\in I} S_i)=\bigwedge \{ \bigwedge S_i : i\in I\}$.
For each $i\in I$, let $\alpha_i=\bigwedge S_i$ and $\alpha=\bigwedge \{\alpha_i : i\in I\}$. Let $S=\bigcup_{i\in I} S_i$. We have to prove that $\alpha = \bigwedge S$. Since $\alpha\leq \alpha_i$ for each $i\in I$, $\alpha \leq s$ for each $s\in S_i$ for each $i\in I$. Hence $\alpha$ is a lower bound to $S$. If $\beta \in L$ satisfies $\beta \leq s$ for each $s\in S$, then $\beta \leq s_i$ for each $s_i\in S_i$ for all $i\in I$. Hence $\beta \leq \alpha_i$ for each $i\in I$. Consequently, $\beta \leq \alpha$.
We next define the Cartesian product of lattices.
\[Adef:Prod\_lattice\] Let $(L,\leq,\bot, \top)$ be a lattice and $n$ a positive integer. The product lattice, $(L^{n},\leq_n,\bot_n,\top_n)$ is defined as follows: for $\overline{a}=(a_1, a_2,\ldots, a_n)$, $\overline{b}=(b_1, b_2,\ldots, b_n) \in L^{n}$, $\overline{a}\leq_n \overline{b}$ if $a_i\leq b_i$ for each $1\leq i\leq n$, $\bot_n=(\bot, \bot, \ldots,\bot)$ and $\top_n=(\top,\top, \ldots, \top)$.
It is easy to see that $L^{n}$ is a lattice with the meet of $\overline{a}$ and $\overline{b}$ given by $(a_1\wedge b_1, a_2\wedge b_2,\ldots, a_n\wedge b_n)$. The meet of $\overline{a}$ and $\overline{b}$ will be denoted by $\overline{a}\wedge_n \overline{b}$. Similarly, the join of $\overline{a}$ and $\overline{b}$ is $\overline{a}\vee_n \overline{b}=(a_1\vee b_1, a_2\vee b_2,\ldots, a_n\vee b_n)$. For $\tilde{S}\subseteq L^{n}$, the notation $\bigwedge_n \tilde{S}$ (respectively $\bigvee_n \tilde{S}$) denotes the greatest lower bound (respectively least upper bound) of $\tilde{S}$ in $L^{n}$ whenever it exists.
\[Athm:prod\_complete\] If $(L,\leq,\bot,\top)$ is a complete lattice, then $(L^{n},\leq_n,\bot_n,\top_n)$ is a complete lattice.
In view of Theorem \[Athm:complete\], it suffices to prove that for each $\tilde{S}\subseteq L^{n}$, $\bigwedge_n \tilde{S}$ exists in $L^{n}$. If $\tilde{S}=\emptyset$, the proof is trivial. Let $\tilde{S}=S_1\times S_2\times \cdots \times S_n$, where $S_i\subseteq L$ for $1\leq i\leq n$. We will show that $\bigwedge_n \tilde{S}=(\bigwedge S_1,\bigwedge S_2, \ldots, \bigwedge S_n)$. Let $\alpha_i=\bigwedge S_i$ for $1\leq i\leq n$. Let $\overline{\alpha}=(\alpha_1,\alpha_2,\ldots, \alpha_n)$. Since $\alpha_i\leq s_i$ for each $s_i\in S_i$, by the definition of $\leq_n$, we have $\overline{\alpha}\leq_n \overline{s}$ for each $\overline{s}\in \tilde{S}$. Suppose $\overline{\beta}=(\beta_1,\beta_2,\ldots,\beta_n)\in L^{n}$ satisfies $\overline{\beta}\leq_n \overline{s}$ for each $\overline{s}\in \tilde{S}$. Then, by the definition of $\leq_n$, we have $\beta_i\leq s_i$ for each $s_i\in S_i$, $1\leq i\leq n$. It follows from the definition of $\alpha_i$ that $\beta_i\leq \alpha_i$ for all $1\leq i\leq n$. Consequently, $\beta \leq_n \alpha$. The proof is complete.
The proof of Theorem \[Athm:prod\_complete\] yields the following corollary.
\[Acor:prod\_complete\] Let $\tilde{S}\subseteq L^{n}$ be non-empty with $\tilde{S}=S_1\times S_2\times \cdots \times S_n$, where $S_i\subseteq L$ for $1\leq i\leq n$. Then $\bigwedge_n \tilde{S}=(\bigwedge S_1,\bigwedge S_2, \ldots, \bigwedge S_n)$.
Next we define continuous maps between lattices and show that continuous maps are distributive and monotone. Let $(L,\leq,\bot,\top)$ and $(L'\leq',\bot',\top')$ be complete lattices. For arbitrary subsets $S\subseteq L$ and $S'\subseteq L'$, we use the notation $\bigwedge S$ and $\bigwedge' S'$ to denote the greatest lower bounds of $S$ and $S'$ in the respective lattices $L$ and $L'$.
\[Adef:cts\] Let $(L,\leq,\bot,\top)$ and $(L'\leq',\bot',\top')$ be complete lattices. A function $f: L\rightarrow L'$ is continuous if for each $\emptyset \neq S\subseteq L$, $f(\bigwedge S)=\bigwedge' f(S)$, where $f(S)=\{f(s): s\in S\}$.
\[Aobs:monotone\] Let $(L,\leq,\bot,\top)$ and $(L'\leq',\bot',\top')$ be complete lattices. A function $f: L\rightarrow L'$ be continuous. Let $s,s'\in L$ be chosen arbitrarily. Then $f$ satisfies:
1. Distributivity: $f(s\wedge s')=f(s)\wedge f(s')$.
2. Monotonicity: if $s\leq s'$ then $f(s)\leq f(s')$.
The first property is immediate from the definition of continuity. For the second, assume that $s\leq s'$. Then $s=s\wedge s'$ and we have $f(s)=f(s\wedge s')=f(s)\wedge f(s')\leq f(s')$.
We next show that two particular families of maps from $L^{n}$ to $L$ called projection maps and confluence maps are continuous.
\[Adef:proj\_map\] Let $(L,\leq,\bot,\top)$ be a lattice. For each $i\in \{1,2,\ldots,n\}$, the projection map to the $i^{th}$ co-ordinate, $\pi_i:L^{n}\mapsto L$ is defined by $\pi_i(s_1,s_2,\ldots,s_n)=s_i$ for any $(s_1,s_2,\ldots,s_n)\in L^{n}$.
\[Adef:conf\_map\] Let $(L,\leq,\bot,\top)$ be a lattice. For each $i,j\in \{1,2,\ldots,n\}$, the confluence map $\pi_{i,j}:L^{n}\mapsto L$ is defined by $\pi_{i,j}(s_1,s_2,\ldots, s_n)=s_i\wedge s_j$ for any $(s_1,s_2,\ldots, s_n)\in L^{n}$.
Note that $\pi_{i,j}(\overline{s})=\pi_i(\overline{s})\wedge \pi_j(\overline{s})$ and $\pi_{i,i}(\overline{s})=\pi_i(\overline{s})$ for all $\overline{s}\in L^{n}$ and $i,j\in \{1,2,\ldots, n\}$. Thus projection maps are special instances of confluence maps.
\[Athm:conf\_cts\] Projection maps and confluence maps over a complete lattice $(L,\leq,\bot,\top)$ are continuous.
Let $i,j\in\{1,2,\ldots,n\}$. Let $\emptyset \neq \tilde{S}\subseteq L^{n}$. We need to prove that $\pi_{i,j}(\bigwedge_n \tilde{S})=\bigwedge \pi_{i,j}(\tilde{S})$. Let $\tilde{S}=S_1\times S_2\times \cdots \times S_n$, $S_i\neq \emptyset$. Let $\alpha=\bigwedge S_i$ and $\beta =\bigwedge S_j$. By Corollary \[Acor:prod\_complete\] and the definition of $\pi_{i,j}$ we have: $$\begin{aligned}
\pi_{i,j}(\bigwedge_n \tilde{S})&=&\pi_{i,j}(\bigwedge S_1,\bigwedge S_2, \ldots, \bigwedge S_n)=\alpha \wedge \beta \\
\bigwedge \pi_{i,j}(\tilde{S})&=&\bigwedge \pi_{i,j}(S_1\times S_2\times \cdots \times S_n)=\bigwedge \{s_i\wedge s_j : s_i\in S_i, s_j\in S_j\}
\end{aligned}$$ The equality between the left sides of the two equations above follows from Lemma \[Alem:meet-lemma\].
\[Alem:meet-lemma\] Let $(L,\leq, \bot, \top)$ be a complete lattice. Let $S_1$ and $S_2$ be non empty subsets of $L$ with $\alpha =\bigwedge S_1$ and $\beta =\bigwedge S_2$. Let $S'=\{s_1\wedge s_2: s_1\in S_1, s_2\in S_2\}$. Then $\bigwedge S'=\alpha \wedge \beta$.
Since $\alpha$ and $\beta$ are lower bounds to $S_1$ and $S_2$, we have $\alpha \wedge \beta \leq \alpha \leq s_1$ and $\alpha\wedge \beta \leq \beta \leq s_2$, for each $s_1\in S_1$ and $s_2\in S_2$. Consequently, from the definition of meet, we have $\alpha\wedge \beta \leq s_1\wedge s_2$ for any $s_1\in S_1$, $s_2\in S_2$ and thus $\alpha\wedge \beta$ is a lower bound to the set $S'$. Now, if $\gamma \leq s_1\wedge s_2$ for all $s_1\in S_1$ and $s_2\in S_2$, then clearly $\gamma \leq s_1$ for all $s_1\in S_1$ and $\gamma \leq s_2$ for all $s_2\in S_2$. Since $\alpha=\bigwedge S_1$ and $\beta =\bigwedge S_2$, we have $\gamma \leq \alpha$ and $\gamma \leq \beta$. Consequently, by the definition of meet, $\gamma \leq \alpha\wedge \beta$. This shows that $\alpha\wedge \beta$ is the greatest lower bound of the set $S'$.
Next we show that continuous maps are closed under composition.
\[Athm:compos\] Let $(L,\leq,\bot,\top)$, $(L',\leq',\bot',\top')$ and $(L'',\leq'',\bot'',\top'')$ be complete lattices. Let $f:L\mapsto L'$ and $g:L'\mapsto L''$ be continuous maps. Then the composition map, $g\circ f: L\mapsto L''$, defined by $(g\circ f)(s)=g(f(s))$ for each $s\in L$, is continuous.
Let $\emptyset \neq S\subseteq L$. We need to prove that $(g\circ f)(\bigwedge S)=\bigwedge (g\circ f)(S)$. Let $f(S)=\{f(s): s\in S\}$ and $(g\circ f)(S)=g(f(S))=\{ (g\circ f)(s) : s\in S\}$. $$\begin{aligned}
(g\circ f)(\bigwedge S) &=& g(f(\bigwedge S))\\
&=& g(\bigwedge f(S)) \text{ (By the continuity of $f$) } \\
&=& \bigwedge g(f(S)) \text{ (By the continuity of $g$) } \\
&=& \bigwedge (g\circ f)(S).\end{aligned}$$
The next theorem shows that $f$ is continuous if and only if each of its component maps are continuous. We first note the following:
\[Aobs:component\] Let $(L,\leq,\bot,\top)$ be a complete lattice. For each $i\in \{1,2,\ldots, n\}$, let $f_i:L^{n}\mapsto L$ be arbitrary functions. Then, the function $f:L^{n}\mapsto L^{n}$ defined by $f(\overline{s})=(f_1(\overline{s}),f_2(\overline{s}),\ldots, f_n(\overline{s}))$ satisfies $\pi_i\circ f=f_i$ for each $1\leq i\leq n$. Conversely, for any function $f:L^{n}\mapsto L^{n}$, the $i^{th}$ component map $f_i:L^{n}\mapsto L$ defined by $f_i=\pi_i\circ f$ for each $i\in\{1,2,\ldots, n\}$ satisfies $f_i(\overline{s})=\pi_i(f(\overline{s}))$ for all $\overline{s}\in L^{n}$.
\[Athm:prod\_cts\] Let $(L,\leq,\bot,\top)$ be a complete lattice. The map $f:L^{n}\mapsto L^{n}$ is continuous if and only if $f_i=\pi_i\circ f$ is continuous for each $1\leq i\leq n$.
Suppose $f$ is continuous. Since $\pi_i$ is continuous (by Theorem \[Athm:conf\_cts\]) for each $1\leq i\leq n$, by Theorem \[Athm:compos\], $\pi_i\circ f$ is continuous. This establishes one direction of the theorem.
Conversely, suppose $f_i$ is continuous for each $i\in \{1,2,\ldots,n\}$. Let $\emptyset \neq \tilde{S}\subseteq L^{n}$ be chosen arbitrarily. We need to prove that $f(\bigwedge_n \tilde{S})=\bigwedge_n f(\tilde{S})$. Let $\tilde{S}=S_1\times S_2\times \cdots \times S_n$, where $S_i \subseteq L$ for each $1\leq i\leq n$. Let $\alpha_i=\bigwedge S_i$ for each $1\leq i\leq n$. Let $\overline{\alpha} =(\alpha_1,\alpha_2,\ldots, \alpha_n)$. By Corollary \[Acor:prod\_complete\] we have: $$\begin{aligned}
\label{Aeqn:alpha}
\bigwedge_n \tilde{S}=(\bigwedge S_1,\bigwedge S_2,\ldots,\bigwedge S_n)=(\alpha_1,\alpha_2,\ldots, \alpha_n)=\overline{\alpha} \end{aligned}$$ Hence, $$\begin{aligned}
f(\bigwedge_n \tilde{S})&=&f(\overline{\alpha})\end{aligned}$$ Since for each $\overline{s}\in \tilde{S}$, $f(\overline{s})=(f_1(\overline{s}),f_2(\overline{s}),\ldots, f_n(\overline{s}))$, the above equation becomes: $$\begin{aligned}
\label{Aeqn:LHS-II}
f(\bigwedge_n \tilde{S})&=&(f_1(\overline{\alpha}),f_2(\overline{\alpha}),\ldots, f_n(\overline{\alpha}))\end{aligned}$$ Now, $$\begin{aligned}
\bigwedge_n f(\tilde{S})&=& \bigwedge_n \{ (f_1(\overline{s}), f_2(\overline{s}),\ldots, f_n(\overline{s})) : \overline{s}\in \tilde{S} \}\\
&=& \bigwedge_n f_1(\tilde{S}) \times f_2(\tilde{S})\cdots \times f_n(\tilde{S}) \text{ (by definition of Cartesian product) }\\
&=& (\bigwedge f_1(\tilde{S}),\bigwedge f_2(\tilde{S}),\ldots, \bigwedge f_n(\tilde{S})) \text{ (by Corollary~\ref{Acor:prod_complete}) }\\
&=& (f_1(\bigwedge _n \tilde{S}),f_2(\bigwedge _n\tilde{S}),\ldots, f_n(\bigwedge _n \tilde{S})) \text{ (by continuity of $f_1,f_2,\ldots, f_n$) }\\
&=& (f_1(\overline{\alpha}),f_2(\overline{\alpha}),\ldots, f_n(\overline{\alpha})) \text{ (by Equation~\ref{Aeqn:alpha}) }\\\end{aligned}$$ The theorem follows by comparing the above equation with Equation \[Aeqn:LHS-II\].
Next we turn to the computation of the maximum fix-point of a continuous map on a complete lattice. The well known Knaster-Tarski Theorem [@tarski1955] asserts the existence of a maximum fix-point for every monotone function defined over a complete lattice. When the function is continuous, the maximum fix-point can be defined as the greatest lower bound of a specific iteratively defined subset of the lattice, as described below. In the following we use the notation $f^{2}=f\circ f$, $f^{3}=f\circ f^2$, etc.
Let $(L,\leq,\bot,\top)$ be a complete lattice. Let $f:L\mapsto L$ be any function. $s\in L$ is a maximum fix point of $f$ if:
- $s$ is a fix point of $f$. That is, $f(s)=s$.
- for any $s'\in L$, if $f(s')=s'$, then $s'\leq s$.
\[Athm:MFP\] Let $(L,\leq,\bot,\top)$ be a complete lattice. Let $f:L\mapsto L$ be continuous. Let $S=\{\top, f(\top), f^{2}(\top), f^{3}(\top),\ldots\}$. Then $\bigwedge S$ is the maximum fix point of $f$.
Let $s_0=\bigwedge S$. We need to prove that $s_0$ is the maximum fix point of $f$. If $f(\top)=\top$, then $S=\{\top\}$ and the theorem holds. Otherwise, as $f(\top)\leq \top$, it follows from the monotonicity of $f$ (Observation \[Aobs:monotone\]) that, $S$ is a descending chain with $f^{i+1}(\top)\leq f^{i}(\top)$ for each $i\geq 1$; and we have $f(S)=S\setminus \{\top \}\neq \emptyset$. By continuity of $f$, we get: $$f(s_0)=f(\bigwedge S)=\bigwedge f(S)=\bigwedge (S\setminus \{\top\})=\bigwedge S=s_0$$ This shows $s_0$ is a fix-point. Suppose $s'\in L$ satisfies $f(s')=s'$. Since $s'\leq \top$, $s'=f(s')\leq f(\top)$. Extending the argument, we see that $s'\leq f^{i}(\top)$ for each $i\geq 1$. Hence by the definition of $\bigwedge$, $s'\leq \bigwedge S=s_0$. This proves that $s_0$ is the maximum fix-point of $f$.
The following is a consequence of the above proof.
\[Acor:MFP\] Let $(L,\leq,\bot,\top)$ be a complete lattice. Let $f:L\mapsto L$ be continuous. Then the set $S'=\{\top, f(\top), f^{2}(\top), f^{3}(\top),\ldots\}$ is a decreasing sequence of lattice values with $f^{i+1}(\top)\leq f^{i}(\top)$ for each $i\geq 0$. Moreover, $f(\bigwedge S')$ is the maximum fix point of $f$.
An Algorithm for Computing Program Expression Equivalence {#sec:appendixc}
=========================================================
A program expression in a program $P$ is a term over $(X \cup C)$ that actually appear in the program $P$. Two program expressions $e$ and $e'$ are Herbrand equivalent at a program point if they belong to the same partition class of the Herbrand Congruence at that program point. The *Herbrand equivalence classes of program expressions* at a program point are the partition classes obtained by restricting the Herbrand Congruence of that point to only the set of program expressions. In this section we describe an algorithm that calculates the Herbrand equivalence classes of program expressions at each program point in any input program, represented as a data flow framework. We restrict our attention to only intra procedural analysis.
Description of Data Structures
------------------------------
An $ID$ (which stands for a value identifier) is a composite data type with four fields, namely $ftype$, $valueNum$, $idOperand1$ and $idOperand2$. If $ftype$ is $0$, we call it an atomic ID and it will have an associated value number called $valueNum$, which is a positive integer and the other two fields $idOperand1$, $idOperand2$ are set to $NIL$. For a non-atomic ID, its $ftype$ will be an operator and its $valueNum$ field will be set to $-1$. The field $idOperand1$ will point to the $ID$ of the first operand and $idOperand2$ will point to the $ID$ of the second operand. An $IdArray$ type represents an array of $ID$s that holds one index for each element $t$ in $(X \cup C) \cup [(X \cup C) \times (X \cup C) \times Op]$ in an order arbitrarily fixed in the beginning. Each array element indicates the value identifier of the corresponding element $t$.
Associated with each program point $p$, there is an $IdArray$ and this together forms an array $Partitions$ which has one index corresponding to each program point. This way, for each program point $i$, $Partitions[i]$ will be an array of value identifiers, one index corresponding to each program expression and two program expressions $t$ and $t'$ are considered equivalent at program point $i$ if and only if $Partitions[i][t]=Partitions[i][t']$. The notation $numClasses$ represents the cardinality of the set $(X \cup C) \cup ((X \cup C) \times (X \cup C) \times Op)$ and $numProgPoints$ represents the number of program points in the input program.
typedef struct {
> int ftype; int valueNum; struct ID\* idOperand1; struct ID\* idOperand2;
} ID; typedef $IdArray$ $ID[numClasses]$; $IdArray$ $Partitions[numProgPoints]$;
Description of the Algorithm
----------------------------
$StartCounter()$; $Partitions[1][t] \gets CreateAtomicId()$; $Partitions[1][t] \gets AssignCompoundId(\&Partitions[1][t_1],\&Partitions[1][t_2],+)$; $Partitions[k]=\top$; $ConvergeFlag \gets 0$; $ConvergeFlag \gets 1$; $OldPartition \gets Partitions[k]$; $Partitions[k] \gets AssignStmt(Partitions[j], y, \beta)$; $Partitions[k] \gets NonDetAssign(Partitions[j], y)$; $Partitions[k] \gets Confluence(Partitions[i],Partitions[j])$; $ConvergeFlag \gets 0$;
$IdArray$ $\mathcal{Q}$; Initialize $\mathcal{Q} = \mathcal{P}$ $\mathcal{Q}[y] \gets \mathcal{P}[\beta]$ $\mathcal{Q}[t] \gets AssignCompoundId(\&\mathcal{Q}[y], \&\mathcal{Q}[t'], +)$; $\mathcal{Q}[t] \gets AssignCompoundId(\&\mathcal{Q}[t'], \&\mathcal{Q}[y], +)$; **return** $\mathcal{Q}$;
$IdArray$ $\mathcal{Q}_1, \mathcal{Q}_2$, $\mathcal{Q}$; Initialize $\mathcal{Q}_1=\mathcal{P}$; $\mathcal{Q}_2=\mathcal{P}$; Let $c_1, c_2 \in C$. $\mathcal{Q}_1[y]=\mathcal{P}[c_1]$; $\mathcal{Q}_2[y]=\mathcal{P}[c_2]$; $\mathcal{Q}_1[t] =$ AssignCompoundId$(\&\mathcal{Q}_1[y], \&\mathcal{Q}_1[t'], +)$; $\mathcal{Q}_2[t] =$ AssignCompoundId$(\&\mathcal{Q}_2[y], \&\mathcal{Q}_2[t'], +)$; $\mathcal{Q}_1[t] =$ AssignCompoundId$(\&\mathcal{Q}_1[t'], \&\mathcal{Q}_1[y], +)$; $\mathcal{Q}_2[t] =$ AssignCompoundId$(\&\mathcal{Q}_2[t'], \&\mathcal{Q}_2[y], +)$; $\mathcal{Q}=Confluence(\mathcal{Q}_1, \mathcal{Q}_2)$; **return** $Confluence(\mathcal{Q}, \mathcal{P})$;
$IdArray$ $\mathcal{Q}$; $int$ $AccessFlag[numClasses]$; $AccessFlag[t]=0$; $AccessFlag[t]=1$; $\mathcal{Q}[t]=\mathcal{P}_1[t]$; $S_1 = getClass(t, \mathcal{P}_1)$; $S_2 = getClass(t, \mathcal{P}_2)$; $newId =CreateAtomicId()$; $AccessFlag[t']=1$; $\mathcal{Q}[t']=newId$; $\mathcal{Q}[t] =$ AssignCompoundId$(\&\mathcal{Q}[t_1], \&\mathcal{Q}[t_2], +)$; **return** $\mathcal{Q}$;
$Set$ $S_1, S_2$; $S_1 = GetClass(t, \mathcal{P}_1)$; $S_2 = GetClass(t, \mathcal{P}_2)$; **return** $0$; **return** $1$;
$Set$ $S = \emptyset$; $S = S \cup \{t'\}$; **return** $S$;
struct ID $idClass$; $idClass.valueNum \gets IncCounter()$; $idClass.ftype \gets 0$; $idClass.idOperand1 \gets NIL$; $idClass.idOperand2 \gets NIL$; **return** $idClass$;
struct ID $idClass$; $idClass.valueNum \gets-1$ $idClass.ftype \gets optype$; $idClass.idOperand1 \gets id1$; $idClass.idOperand2 \gets id2$; **return** $idClass$;
$counter \gets 0 $
$counter \gets counter +1 $ **return** $counter$
|
---
abstract: 'We discuss whether some perturbed Friedmann-Robertson-Walker (FRW) universes could be creatable, i. e., could have vanishing energy, linear momentum and angular momentum, as it could be expectable if the Universe arose as a quantum fluctuation. On account of previous results, the background is assumed to be either closed (with very small curvature) or flat. In the first case, fully arbitrary linear perturbations are considered; whereas in the flat case, it is assumed the existence of: (i) inflationary scalar perturbations, that is to say, Gaussian adiabatic scalar perturbations having an spectrum close to the Harrison-Zel’dovich one, and (ii) arbitrary tensor perturbations. We conclude that, any closed perturbed universe is creatable, and also that, irrespective of the spectrum and properties of the inflationary gravitational waves, perturbed flat FRW universes with standard inflation are not creatable. Some considerations on pre-inflationary scalar perturbations are also presented. The creatable character of perturbed FRW universes is studied, for the first time, in this paper.'
author:
- Ramon Lapiedra
- Diego Sáez
title: 'Probing the creatable character of perturbed Friedmann-Robertson-Walker universes'
---
Introduction {#intro}
============
In [@Ferrando] (see also the addenda in [@Ferrando-bis]), for a wide set of non asymptotic Minkowskian space-times, it was uniquely determined when one of these space-times can be said to have vanishing energy, vanishing linear 3-momentum and vanishing angular 4-momentum, that is, vanishing energy and momenta. These vanishing values could be expected in the case of a universe which rose from a quantum vacuum fluctuation [@Tryon][@Albrow]. In [@Ferrando], universes of this kind, with vanishing energy and momenta, were called ‘creatable universes’.
It is very well known that, whatever the energy-momentum complex may be, the definition of energy and momenta of the universe is strongly dependent on the coordinate system. One must then stress the uniqueness reached [@Ferrando] in the characterization of the family of creatable universes, within the above wide set. This uniqueness has been reached by using some physical criteria to select the appropriate coordinate systems. See [@Garecki] as an example of a more mathematical criterion, which leads the author to use conformally flat coordinates in the space-time, when it is possible.
In order to look for convenient physical criteria to select the appropriate coordinates, one should assume that the proper energy and momenta of any space-time representing the universe is conserved in time. Taking into account that the Universe is supposed to embrace everything, fluxes of energy and momenta going out of such an entity are not possible and, consequently, the above assumption about time conservation seems actually appropriate. Such an assumption only has a clear physical meaning if it is referred to a physical (proper) and universal (synchronous) time. In other words, the conserved energy and momenta of a universe should be calculated by using Gauss coordinates, which involve the required time coordinate. Energy and momenta would be then conserved in terms of Gauss time coordinates. Because of this conservation, no global Gauss coordinates are necessary in practise to calculate energy and momenta. Actually, Gauss coordinates defined in the elementary vicinity of a generic space-like 3-surface suffice.
Given a space-time, there are plenty of different Gauss coordinate systems. Thus, we will need to make sure that the characterization of any perturbed FRW universe as creatable or non creatable is independent on the Gauss coordinate used. We will consider various of these perturbed universes and, then, in order to see if they are creatable, we will apply different strategies, inspired by [@Ferrando], which are adapted to each particular case. Nevertheless, any of these strategies will obey the following protocol: (i) take any space-like 3-surface, $\Sigma_3$, and build the corresponding Gauss coordinates in its elementary vicinity, (ii) look for new Gauss coordinates leading to an ‘instantaneous’ 3-space metric, $dl_0^{2}\equiv {dl_0^2(t=t_0)}$, which explicitly exhibits its conformal character on the boundary, $\Sigma_2$, of $\Sigma_3$ (these coordinates always exist; see [@Ferrando]) and, (iii) calculate the energy and momenta of the universe in the resulting coordinate system. Despite the still remaining freedom in the election of these coordinates, it can be seen that the calculated values of energy and momenta are unique for the different cases we consider in the present paper. In all these cases, if these energy and momenta vanish, the universe is creatable, since it can be directly seen that they vanish irrespective of both the selected space-like 3-surface, $\Sigma_3$, and the chosen time, $t_{0}$ (energy and momenta conservation). This will be verified at the end of each particular case.
It has sometimes been argued (see, e. g., [@Garecki]) that, in a space-time which is not asymptotically flat, the global energy and momenta would have no physical meaning, since these quantities could never be measured; however, it is also claimed that the energy and momenta of any part of this space-time have physical meaning. This double claim is not fully consistent; in fact, if energy and momenta have physical meaning for any part of the space-time, the corresponding global quantities should be interpreted as the limit of the physically significant energy and momenta of the parts, as they grow to fill the entire space-time.
By using the Weinberg energy-momentum complex [@Weinberg] and the above protocol, it was proved that the closed and flat Friedmann-Robertson-Walker (FRW) universes ($K=0,+1$) are creatable, whereas the open version ($K=-1$) is not [@Ferrando]. This is in agreement with most of the papers on the subject, but not with all of them (see again [@Garecki]). Similar conclusions were obtained by using very different methods. For example, in papers [@Atkatz] and [@Vilenkin], it was proved that the tunneling amplitude for creation from ’nothing’ is finite in the case of closed cosmological backgrounds and for a flat De Sitter unperturbed universe. These two papers concluded that the tunneling amplitude vanishes in the open case. These conclusions were obtained in the framework of particular FRW models for tunneling. Realistic universes with perturbations were not considered at all. The main goal of the present paper is the study of perturbed FRW universes without any calculation of transition amplitudes. Instead of these calculations, we use the above protocol to look for creatable universes (those with vanishing energy and momenta) without modelling any quantum tunneling.
The observations seem to indicate that we live in a perturbed FRW universe and, consequently, if the above protocol makes sense, at least one of the perturbed FRW universes should be creatable. Our results confirm this expectation, since we have proved that any perturbed closed universe is creatable. The perturbed $K=-1$ universe does not need to be considered here because the corresponding background is not creatable. The perturbed flat case is studied in detail along the paper. It is not creatable, under very general conditions strongly supported by current observations.
This paper is organized as follows: In Sec. \[sec-2\], we consider absolutely arbitrary linear perturbations in the case $K=+1$. The case $K=0$ is studied in Sec. \[sec-3\]. The total energy due to the scalar perturbations arising in standard inflationary models is calculated in Sec. \[sec-3A\] (it is infinite). Sec. \[sec-3B\] contains the calculation of the same quantity in the case of fully arbitrary gravitational waves (it vanishes). Finally, in Sec. \[sec-4\], we summarize our main conclusions and present a general discussion, including comparisons between our results and methods and those of previous papers dealing with quantum creation from ’nothing’ (see references [@Atkatz]–[@Hartle]).
Some of these results have been briefly presented, with no calculations, in the meeting ERE-2007 [@Lapiedra]
Let us finish this section with some words about notation. Units are chosen in such a way that the speed of light is $c=1$. The gravitational constant, the scale factor, and the index of the 3-space curvature are denoted $G$, $a$, and $K$, respectively. Symbols $t$ and $\tau$ stand for the Gauss and the conformal times ($a d\tau= dt$) and, finally, the unit vector $\hat{\mathbf{k}}$ and the modulus $k$ define a generic vector $\mathbf{k}=k\hat{\mathbf{k}}$ in momentum space.
The case of a perturbed closed FRW universe {#sec-2}
===========================================
The line element of the closed FRW space-time can always be written in the form: $$ds^2 = -dt^2+dl^2, \, \,dl^2 =
\frac{a^2(t)}{\left(1+\frac{K}{4}r^2\right)^2} \delta_{ij} dx^i
dx^j \, , \quad r^2 \equiv \delta_{ij} x^i x^j \, \label{FRW
metric}$$ with $K=+1$. Here, $x^{i}$ and $t$ are global Gauss coordinates. Moreover, the 3-space metric exhibits a conformally flat form everywhere.
Now, let us consider the case of a perturbed closed FRW universe with scalar and tensor perturbations [@Bardeen]. Then, in the synchronous gauge (see the Appendix), conditions $g_{00}=-1$, $g_{i0}=0$ are satisfied. Hence, in this gauge, Gauss coordinates are used. In terms of them, the 3-space metric, $dl^2$, reads as follows: $$dl^2 = \frac{a^2(t)}{\left(1+\frac{1}{4}r^2\right)^2}
(\delta_{ij}+h_{ij}) dx^i dx^j, \label{perturbed FRW metric}$$ where the $h_{ij}(t,x^{i})$ functions are such that $h_{ij}<<1$.
Let it be a particular space-like 3-surface, $t=t_{0}$, and particularize the above 3-space metric on this 3-surface, that is, consider $dl(t=t_{0})^2 \equiv{dl_0^2}$. This instantaneous 3-space metric is a conformally flat 3-metric on the boundary 2-surface, $\Sigma_2$, of $t=t_{0}$ [@Ferrando]. Then, among the different Gauss coordinate systems, one always can select some ones such that $dl_0^2$ on the boundary $\Sigma_2$, say $dl_0^2|_{\Sigma_{2}}$, shows explicitly its conformally flat character. According to the protocol displayed in Sec. \[intro\], which can be used here to calculate the energy and momenta of perturbed closed universes, we can pick up any one of these last coordinate systems to compute the corresponding values of the energy and momenta of our perturbed universe. We will subsequently see that these values are unique.
Then, according to [@Weinberg], the energy, $P^{0}$, the 3-momentum, $P^{i}$, and the 4-angular momentum, $(J^{jk},J^{0i})$, of the universe are: $$\begin{aligned}
P^0 & = & \frac{1}{16 \pi G} \int(\partial_j g_{ij} - \partial_i
g)
d \Sigma_{2i}, \label{energy}\\[3mm]
P^i & = & \frac{1}{16 \pi G} \int(\dot{g} \delta_{ij} -
\dot{g}_{ij}) d \Sigma_{2j},
\label{three-momentum}\\[3mm]
J^{jk} & = & \frac{1}{16 \pi G} \int(x_k \dot{g}_{ij} - x_j
\dot{g}_{ki}) d \Sigma_{2i},\label{angular three-momentum}
\\[3mm]
J^{0i} & = & P^i t - \frac{1}{16 \pi G} \int[(\partial_k g_{kj} -
\partial_j g)x_i + g \delta_{ij} - g_{ij}] d \Sigma_{2j},
\label{angular time momentum}\end{aligned}$$ where $\dot{g_{ij}} \equiv
\partial_t {g_{ij}}$, $g\equiv \delta_{ij}g_{ij}$, and where $d\Sigma_{2i}$ stands for the integration element on $\Sigma_2$. In the present case, the 2-surface $\Sigma_2$ is $r=\infty$.
Now, it is obvious that the energy and momenta of our perturbed closed FRW universe must vanish; in fact, according to Eq. (\[perturbed FRW metric\]), all the integrands in the above 2-surface integrals go at least like $1/r^4$ as $r$ tends to $\infty$. (Notice that $h_{ij}$ cannot grow indefinitely with $r$, otherwise we would not have $h_{ij}<<1$ everywhere). Of course, this asymptotic behavior, in terms of $1/r$, is valid for all Gauss coordinates which preserve the perturbed character of the metric Eq. (\[perturbed FRW metric\]), and so, whatever be the time parameter $t_0$, it is valid for the particular Gauss coordinate system where the 3-metric, $dl_0^2$, shows explicitly its conformal flat character on ${\Sigma_{2}}$. Evidently, such a behavior of the integrands implies straightforward that all the integrals in Eqs. (\[energy\])–(\[angular time momentum\]) vanish irrespective of the above Gauss coordinate used. Consequently, we can state that, whatever the properties of the perturbations may be, perturbed closed FRW universes are creatable.
The case of a perturbed flat FRW universe {#sec-3}
=========================================
In the case of a flat background, we cannot invoke the strong decaying of the 3-space metric, as $r$ tends to $\infty$, to conclude that energy and momenta vanish; namely, arguments similar to those of the last section do not apply. Furthermore, one could erroneously think that, in the flat case, the vanishing of the energy would come straightforward from the cosmological principle; namely, from the assumed statistical homogeneity when averaging on large enough volumes. Actually, as we will see in detail along this section, the values of the energy and momenta of the universe depend on both the statistical character and the spectra of the perturbations.
Scalar perturbations {#sec-3A}
--------------------
Let us first consider the case of scalar perturbations [@Bardeen] in a flat FRW universe. In any synchronous gauge (Gauss coordinates), the perturbed 3-space metric is $g_{ij}=a^{2}(\tau)(\delta_{ij}+h_{ij})$ and, according to [@Ma], the metric perturbation, $h_{ij}$, can be expanded in scalar harmonics (plane waves) as follows: $$h_{ij}(\mathbf{x},\tau)=\int d^{3}k e^{i\mathbf{k}.\mathbf{x}} h_{ij}(\mathbf{k},\tau)
= \int d^{3}k
e^{i\mathbf{k}.\mathbf{x}}[{\hat{\mathbf{k}}}^{i}{\hat{\mathbf{k}}}^{j}h(\mathbf{k},\tau)
+({\hat{\mathbf{k}}}^{i}{\hat{\mathbf{k}}}^{j}-\frac{1}{3}\delta_{ij})6\eta(\mathbf{k},\tau)] \ ,
\label{original metric}$$ where two functions, $h(\mathbf{k},\tau)$ and $\eta(\mathbf{k},\tau)$, defined in momentum space, have been introduced. Notice that, in order to have a real value for $h_{ij}$, functions $h$ and $\eta$ must satisfy the conditions $h(-\mathbf{k})=-h^{\star}(\mathbf{k})$ and $\eta(-\mathbf{k})=-\eta^{\star}(\mathbf{k})$.
According to the above protocol to calculate energy and momenta, we must use a new Gauss coordinate system, in which, the transformed components of the instantaneous 3-space metric, $g'_{ij}(t=t_0)$, show explicitly its conformal flat character on the boundary $r=\infty$. In the present case, we can do more than this since our instantaneous 3-space metric is a conformally flat metric everywhere on the space-like 3-surface $t=t_{0}$ (not only on the 2-surface $r=\infty$). In order to prove this statement, let us work in the $\mathbf{k}$-space. Given a $\mathbf{k}$-value, let us consider the infinitesimal coordinate transformation $$x^{i}={x}'^i+{e}^{i}(\mathbf{x},\mathbf {k}), \label{new
coordinates}$$ which leads to the following equation: $${h}'_{ij}(\mathbf{k})e^{i\mathbf{k}.{x}}=h_{ij}(\mathbf{k})e^{i\mathbf{k}.{x}}+
\partial_{i}e_{j}+\partial_{j}e_{i} \ .
\label{new metric}$$ If we then choose $e^{i}(\mathbf{x},\mathbf{k})=f(\mathbf{k})e^{i\mathbf{k}.\mathbf{x}}\hat{\mathbf{k}}^{i}$, where $f(\mathbf{k})$ is defined by the relation $$(h+6\eta+2a^{2}k^{2}f)_{t=t_{0}}=0 \ ,$$ equation (\[new metric\]) leads to the following new Fourier component ${h}'_{ij}(\mathbf{k})$ for $t=t_{0}$ $${h}'_{ij}(\mathbf{k})_{t=t_{0}}\equiv{{h}'_{ij}(\mathbf{k})}_{0}=
-2\eta_{0}(\mathbf{k})\delta_{ij}, \label{new-h}$$ where $\eta_0(\mathbf{k})\equiv{\eta(\mathbf{k},t=t_0)}$. Equation (\[new-h\]) implies that the instantaneous metric $g_{ij}(t=t_{0})$ is conformally flat everywhere on the 3-space $t=t_{0}$, as we wanted to prove.
To calculate the energy, we must insert the new components of the 3-space metric ${g}'_{ij}(t=t_{0})=a^2_0[\delta_{ij}+{h}'_{ij}(t=t_{0})]$ in the integral of Eq. (\[energy\]), which must be performed on the 2-surface $r=\infty$. This integral is to be calculated in the new coordinates ${x}'^{i}$; however, in practice, the old coordinates, $x^{i}$, can be used in first order calculations; in fact, since the integrand in Eq. (\[energy\]) trivially vanishes for the background metric (a flat FRW universe), this integrand is a first order quantity in the perturbed universe and, consequently, the integral can be evaluated irrespective of the coordinate system (the differences between the integrands in the old and new coordinates are second order quantities to be neglected in any linear approach). In short, to first order, metric perturbations and its derivatives can be calculated in terms of the old coordinates, and the old 2-surface element, $d\Sigma_{2i}$, can be used instead of the new one. Thus, according to Eq. (\[new-h\]), the total energy $P^{0}$ can be written as follows: $$P^{0}=\lim_{r\rightarrow \infty}\frac{a_0^2 r^{2}}{8\pi G}\int
d\Omega n_{i}\partial{_i}\int dk^{3}\eta_{0}
e^{i\mathbf{k}.\mathbf{x}}, \label{special energy}$$ where $d\Omega$ is the solid angle element in spherical coordinates, i. e., $d\Omega=\sin\theta d\theta d\phi$, and where $n_{i}\equiv{x^i/r}$.
It is noticeable that, according to the above expression of $P^0$, the energy of the universe does not depend on the function $h$ in Eq. (\[original metric\]). It only depends on the function $\eta$, that is, on the traceless part of the perturbed metric (\[original metric\]).
We assume statistical isotropy, which characterizes cosmological processes as, e. g., standard inflation. Since there are no privileged directions, the power spectrum of $\eta(\mathbf{k})$, namely, the function $P_{\eta}(\mathbf{k})=\langle |\eta(\mathbf{k})|^{2} \rangle $ is independent on $\hat{\mathbf{k}}$. It only depends on $k$. Three $k$-intervals can be distinguished (see the Appendix). The interval ($0,k_{min}$) involving pre-inflationary perturbations with some unknown power spectrum $P_{\eta 1}(\mathbf{k})$, the ($k_{min},k_{max}$) interval with inflationary perturbations evolving outside the horizon, whose spectrum is $P_{\eta 2}(\mathbf{k})$, and the the interval ($k_{max},Â\infty$) with inflationary perturbations which have reentered the horizon; in this case, the spectrum is denoted $P_{\eta 3}(\mathbf{k})$.
Inflationary perturbations are all Gaussian in the evolution period under consideration (before entering in nonlinear processes, see the Appendix); hence, if pre-inflationary perturbations are also assumed to be Gaussian (the non Gaussian case is discussed below), functions $\eta $ and $h$, as well as the Fourier transform of any physical quantity (density contrasts, pressure, and so on) are complex functions with random phases [@Peebles] for $0 < k < \infty$ and, consequently, the $\eta $ values necessary to calculate $P^0$ –from Eq. (\[energy\])– can be written in the form $$\eta(\mathbf{k}) = \frac {1}{\sqrt{2}} \Big[P_{_{\eta}}(k)\Big]^{1/2}
\Big[y_1(\mathbf{k})+iy_2(\mathbf{k})\Big] \ , \label{gaussian eta}$$ where $y_1(\mathbf{k})+iy_2(\mathbf{k})$ are the phases, which must be assigned taking into account the relation $\eta(-\mathbf{k})=-\eta^*(\mathbf{k})$ and the relations: $$\langle y_1(\mathbf{k}) \rangle =\langle y_2(\mathbf{k}) \rangle =0 \ ,
\label{c1}$$ $$\langle y_1(\mathbf{k}) \, y_2({\mathbf{k}}')\rangle =0 \ ,
\label{c2}$$ $$\langle y_n(\mathbf{k}) \, y_n({\mathbf{k}}')\rangle
=\delta^3(\mathbf{k}-\mathbf{\mathbf{k}}') \ ,
\label{c3}$$ where index $n$ run from $1$ to $2$, vectors $\mathbf{k}$ and ${\mathbf{k}}'$ are arbitrary momenta, and the averages are to be performed in the set of the universe realizations. In the last equation, $\delta^3$ is the three dimensional Dirac $\delta$-distribution. Equations (\[c1\])–(\[c3\]) ensure (central limit theorem) that the distribution of $P^0$ values corresponding to all the possible realizations of the universe is Gaussian.
By using the relation $$\int d\Omega n_i e^{i\mathbf{k}.\mathbf{x}}=\frac{4\pi
i}{kr}\Big(\frac{\sin kr}{kr}-\cos kr\Big)\mathbf{\hat{k}}_i \ ,
\label{integral}$$ which can be easily obtained, plus Eqs. (\[special energy\]) and (\[gaussian eta\]), one easily finds: $$P^0=-(a_0^2/2G)\lim_{r\rightarrow \infty}r \sum_{j=1}^{3} \int_{j} d^3
k \Big[P_{\eta j}(k)\Big]^{1/2}
\Big(\frac{\sin kr}{kr}-\cos kr\Big) y_1(\mathbf{k}) \ ,
\label{xxx}$$ where the index $j$ number the three intervals defined above. The volume integrals in momentum space are extended to the regions $k<k_{min}$, $k_{min}<k<k_{max}$, and $k>k_{max}$, in cases j=1, j=2, and j=3, respectively. In order to derive the last equation, we have used the relation $\eta(-\mathbf{k})=-\eta^*(\mathbf{k})$ and, consequently, the energy $P^0$ appears to be real valued as it must be. It is also important that, as a result of the same relation, quantity $P^0$ only depends on the random variable $y_1(\mathbf{k})$; whereas $y_2(\mathbf{k})$ becomes fully irrelevant in our calculations.
Taking into account Eq. (\[xxx\]) and the fact that $\langle y_1(\mathbf{k}) \rangle$ vanishes, one easily finds the relation $\langle P^0 \rangle=0$. Of course, the average is performed for all the possible realizations of the universe, which correspond to distinct $y_{1}(\mathbf{k})$ values but to the same power spectrum. Since the phases have been chosen in such a way that the $P^0$ values are normally distributed, we must now calculate the variance, $\langle (P^{0})^{2} \rangle$, which fully characterize this normal (Gaussian) distribution with vanishing mean. From Eq. (\[xxx\]), one easily finds a formula for $(P^{0})^{2}$ which is the addition of six different terms. Three of them involve products of integrals corresponding to distinct $j$-values (hereafter crossed terms). Since two $\mathbf{k}$-vectors corresponding to distinct $j$ values cannot coincide, equation (\[c3\]) implies that, after averaging, the crossed terms vanish. From the remaining three terms plus Eq. (\[c3\]) one easily obtains the following variance: $$\langle (P^{0})^{2} \rangle=(a_0^4/4G^2)\lim_{r\rightarrow \infty} r^2 \sum_{j=1}^{3} \int_{j}
d^3k P_{\eta j}(k) \Big(\frac{\sin kr}{kr}-\cos kr \Big)^2 \ .
\label{second special energy}$$ Each of the three terms of this last formula is positive (it is the square of one of the three integrals $j=1,2,3$) and, consequently, if one of them diverges, the variance $\langle (P^{0})^{2} \rangle$ diverges.
In the case of inflationary adiabatic scalar perturbations which evolve outside the horizon in the radiation dominated era ($k_{min}<k<k_{max}$), the solution of the linearized Einstein field equations can be found in Ma and Bertschinger [@Ma] (see the Appendix for more details). In the synchronous gauge, function $\eta $ evolves as follows: $$\eta=C+\frac{3}{4}\frac{D}{k\tau}+\alpha Ck^{2}\tau^{2} \ ; \label{eta}$$ in this equation, $\alpha$ is a pure number (see the Appendix), whereas $C$ and $D$ are dimensionless integration constants. That is to say, $C$ and $D$ cannot depend on time, but they may depend on $\mathbf{k}$. Notice that here, as well as in [@Ma], the components of $\mathbf{x}$ and $\mathbf{k}$ are dimensionless, since they are comoving coordinates and their associated momenta. We see that $\eta$ is the addition of three different modes: one of them grows like $\tau^{2}$, another one is constant, and the third one decreases like $\tau^{-1}$.
Let us now study the behavior of the integrals involved in Eq. (\[second special energy\]) as $r$ tends to $\infty$. We begin with one of the contributions to the $j=2$ integral (inflationary Gaussian perturbations with super-horizon sizes). It is the contribution due to the constant mode $C(\mathbf{k})$ in Eq. (\[eta\]). This contribution is hereafter denoted $\langle (\tilde{P}^{0})^{2} \rangle$. The spectrum $P_{_{C}}(k)$ of the constant mode is defined by the relation $$º
\Big[P_{_{C}}(k)\Big]^{1/2}=Ak^n, \label{modulus C}$$ where $A$ is a normalization constant and the spectral index is $n=-(3/2)-\beta$. According to Eq. (\[ck2\]) in the Appendix, constant $\beta = (1-n_{s})/2$ is positive and small as compared to unity. From the spectrum (\[modulus C\]), one easily finds the following variance: $$\langle (\tilde{P}^{0})^{2} \rangle=(\pi a_0^4 A^2/G^2)\lim_{r\rightarrow \infty}
r^{2+2\beta} \int_{rk_{min}}^{rk_{max}} dy\Big(\frac{\sin y}{y}-\cos
y)^2\Big)/y^{1+2\beta},
\label{yyy}$$ where $y\equiv kr$. For any $k$, the new variable $y$ tends to infinity as $r$ does. On account of this fact, it is easily proved that the integral in Eq. (\[yyy\]) goes just like the power $r^{-2\beta}$ as $r$ tends to infinity. Hence, $\langle (\tilde{P}^{0})^{2} \rangle$ diverges as $r^{2}$.
Let us now consider the growing and decaying terms appearing in Eq. (\[eta\]). It is obvious that the contributions of these time dependent terms to the energy cannot compensate the infinite value of $\langle (\tilde{P}^{0})^{2} \rangle$ corresponding to the constant mode, which implies that the total contribution to $\langle (P^{0})^{2} \rangle$ due to the $j=2$ $k$-interval cannot become finite. More precisely, from Eq. (\[eta\]) we find six distinct terms contributing to the $j=2$ integral. The term used in the previous calculation does not depend on time, whereas the remaining ones are time dependent. By this reason, the variance $\langle (\tilde{P}^{0})^{2} \rangle$ is proportional to $a^{4}$ for the time independent mode, whereas it exhibits other time dependences in the remaining cases. In all, compensation of the resulting terms (with distinct time evolutions) to give a conserved finite total energy is not possible, which ensures that the entire contribution $j=2$ to $\langle P^{0})^{2} \rangle$ is infinite. Moreover, this positive contribution cannot be compensated by those of the $j=1$ and $j=3$ cases to give, finally, a finite global value for $\langle P^{0})^{2} \rangle$, the reason being that, as explained above, the $j=1$ and $j=3$ contributions are also positive. All in all, the following relation holds: $$\langle (P^{0})^{2} \rangle=\infty
\label{infif}$$ and, consequently, we have a Gaussian statistical distribution of $P^0$ values with zero mean and infinite variance. Since the Gaussian probability density is $$P(P^{0})= \frac {1} {\sqrt{2 \pi \langle (P^{0})^{2} \rangle}}
e^{-(P^{0}-\langle P^{0} \rangle)^{2}/2\langle (P^{0})^{2} \rangle} \ ,
\label{denpro}$$ it is evident that, in our case, the probability of any particular finite value of $P^0$ vanishes and, as a consequence, we can say that, in the flat perturbed FRW universe under consideration, the contribution of the inflationary scalar perturbations to the total energy is infinite. Nevertheless, in the standard inflationary paradigm, there are also tensor perturbations, whose contribution to the total energy of a flat perturbed universe is calculated in next section.
Let us now discuss the case of non-Gaussian pre-inflationary perturbations. These perturbations are assumed to be generated in some process fully independent on inflation and, as explained in the Appendix, they are assumed to be significant only in the interval $(0,k_{min})$; hence, Eqs. (\[c3\]) are satisfied except for pairs of vectors $\mathbf{k}$ and ${\mathbf{k}}'$ whose modulus are both inside the interval ($0,k_{min}$). Therefore, there are no crossed terms in the development of $\langle (P^{0})^{2} \rangle$ (see above). As in Eq. (\[second special energy\]), the variance $\langle (P^{0})^{2} \rangle$ is the addition of three positive terms. Terms $j=1$ and $j=2$ have the same form as in the Gaussian case, whereas the term $j=3$ would be different. In this situation, the term $j=2$ diverges (same discussion as above) and, consequently, Eq. (\[infif\]) holds (which strongly suggests a non creatable universe). However, the distribution of $P^{0} $ values is not Gaussian in this case, and its probability density should be calculated for each particular non Gaussian model. Since Eq. (\[denpro\]) does not apply, the meaning of an infinite variance is now less clear. By this reason, in order to properly prove that flat universes with standard inflation are not creatable, we prefer a general argument, which proves that, if $\langle (P^{0})^{2} \rangle$ diverges in the interval ($k_{min},\infty $), namely, if the total energy of the inflationary Gaussian scalar perturbations is infinite, the energy of the universe cannot vanish whatever the properties of the non-Gaussian pre-inflationary perturbations may be; namely, the universe is not creatable. This is trivially proved taking into account that, in any admissible universe, the total energy must be a conserved quantity (see Sec. \[intro\]) which vanishes in the creatable case. In fact, a vanishing energy after inflation is only possible if the pre-inflationary energy is infinite and it exactly compensates the infinite energy associate to the scalar inflationary perturbations. However, such a universe would have an infinite energy before inflation and a vanishing one after this process, which is not compatible with the required energy conservation.
Now, before ending this Section, we raise some comments about the consistence of the main result: the infinite value of the energy we have found.
First of all, although the infinite energy has been formally obtained for a given value of $t_0$ and a certain $\Sigma_3$, it is obvious, by following the implementation of the protocol we have used, that this $t_0$ value and the choice of $\Sigma_3$ are both arbitrary.
Finally, we could perform a conformal coordinate transformation and still retain the explicit conformal flat form of the 3-space metric on $\Sigma_3$ (remember that, according to Eq. (\[new-h\]), in the present case, the 3-space metric is a conformally flat one all over $\Sigma_3$). But, trivially, the above infinite value of the energy does not depend on the conformal transformations which are pertinent here: those belonging to the translation, rotation, and dilatation subgroups of the conformal group. For all these coordinate transformations the energy remains infinite.
Tensor perturbations {#sec-3B}
--------------------
In this section we are concerned with tensor perturbations evolving in a flat background. In such a case, the instantaneous 3-space line element is: $g_{ij}=a_0^2(\delta_{ij}+h^{^{T}}_{ij})$. In Fourier space, we can write $$h^{^{T}}_{ij}(\mathbf{k})=H(\mathbf{k},\tau_0)\epsilon_{ij}
(\mathbf{\hat{k}}),
\label{tensor perturbation}$$ where the quantities $\epsilon_{ij}$ satisfy the conditions given in the Appendix.
Let us calculate the energy, $P^0_{_{T}}$, of these tensor perturbations. According to our protocol, calculations must be performed in a new coordinate system, in which the instantaneous 3-dimensional metric explicitly exhibits its conformally flat character on the boundary 2-surface $r=\infty$. For each value of $\mathbf{k}$, let us consider a coordinate transformation of the form (\[new coordinates\]). Functions $e^{i}(\mathbf{x},\mathbf {k})$ must be chosen in such a way that, in the new coordinates ${x}'^i$, the line element on the surface $r=R$, where $R$ is an arbitrary constant (at the end of our calculations, this constant will tend to $\infty$), has the form $${h^{^{T}}}'_{ij}(\mathbf{x})\Big|_{r=R}=f(\mathbf{n},\mathbf{k})
e^{iR\mathbf{k}.\mathbf{n}}\delta_{ij} \ ,
\label{conformal metric}$$ where ${h^{^{T}}}'_{ij}(\mathbf{x}) = {h^{^{T}}}'_{ij}(\mathbf{k})
e^{i\mathbf{k}.\mathbf{x}} $ is the metric perturbation in the space-like hypersurface $\tau = \tau_{0} $ corresponding to the fixed mode $\mathbf{k}$, and where function $f(\mathbf{n},\mathbf{k})$ is the conformal factor of the metric on $r=R$. As it has been said above, there always exists a family of coordinate systems in which Eq. (\[conformal metric\]) is satisfied [@Ferrando]. The energy and momenta can be calculated in any of these coordinates. With the appropiate $e_i$ functions, from Eq. (\[new coordinates\]), one easily finds the relation $${h^{^{T}}}'_{ij}(\mathbf{x})= h^{^{T}}_{ij}(\mathbf{x})
+\partial_i e_j+\partial_j e_i \ , \label{h prime}$$ which is equivalent to Eq. (\[new metric\]).
The energy, $P^0_{_{T}}$, is calculated by using Eq. (\[energy\]) and the metric perturbation components ${h^{^{T}}}'_{ij}(\mathbf{x'})$; nevertheless, in the linear approach we are using (see Sec. \[sec-3A\] for details), the following approximations can be performed: (i) write these components in terms of the old coordinates $x^{i}$, (ii) perform the derivatives with respect to $x^{i}$, and (iii) use the old 2-surface element, $d\Sigma_2^{i}$, instead of $d{{\Sigma}'}_2^{i}$. Hence, we can write $$P^0_{_{T}}=(a_0^2/16\pi G)\int \partial_j\Big[{h^{^{T}}}'_{ij}(\mathbf{x})-\partial_i
{h^{^{T}}}'(\mathbf{x})\Big] d\Sigma_2^i \ .$$ Taking into account Eq. (\[h prime\]), this last equation can be rewritten as follows: $$P^0_{_{T}}=(P^0_{_{T}})_H+(a_0^2/16\pi G) \int
\Big[\partial_j(\partial_i e_j+\partial_j e_i)-2\partial_i
\partial_k e_k\Big] d\Sigma_2^i \ ,$$ where $(P^0_{_{T}})_H$ is the energy corresponding to the first term of the r.h.s. of Eq. (\[h prime\]), whose Fourier transform, $h^{^{T}}_{ij}(\mathbf{k})$, is given by Eq. (\[tensor perturbation\]). As a result of the conditions satisfied by the quantities $\epsilon_{ij}$ (see the Appendix), it is easily proved that the term $(P^0_{_{T}})_H$ vanishes. Hence, $$P^0_{_{T}}=(a_0^2/16\pi G)\int (\partial_j\partial_j
e_i-\partial_i\partial_j e_j) d\Sigma_2^i \ ,$$ and, finally, the Gauss theorem allow us to write the surface integral in the last equation as a vanishing volume integral whatever the functions $e_i$ may be and, in particular, for the functions leading to Eq. (\[conformal metric\]). Hence, we have proved that the energy associated to any distribution of gravitational waves (propagating in a flat universe) vanishes; namely, the equation $$P^0_{_{T}}=0.$$ is satisfied.
Therefore, according to our protocol, we conclude that the energy $P^{0}_{_{T}}$ due to tensor perturbations of a flat FRW vanishes. Notice that this conclusion does not depend either on the spectrum $P_{_{T}}(k)$ of the tensor perturbations (see the Appendix), or on the statistical character of the distribution of these perturbations.
Notice again that similarly to what has been explained to the end of Sec. \[sec-3A\], the resulting vanishing energy does neither depend on the chosen value of the $t_0$ parameter, nor on the choice of $\Sigma_3$.
Since the energies due to scalar and tensor perturbations add, realistic perturbed universes including inflationary scalar modes have in all an infinite energy and so are not creatable. The presence of arbitrary tensor perturbation (zero energy) is irrelevant. Coming back to the end of Sec. \[sec-3A\], we see now that the energy of the scalar perturbed closed FRW universes is infinite irrespectively of, not only the rotation, traslation and dilatation groups, but actually on any infinitesimal coordinate transformation.
Conclusions and discussion {#sec-4}
==========================
Our main conclusion is that perturbed flat FRW universes, including arbitrary tensor perturbations, and the adiabatic Gaussian scalar ones generated during standard inflation, have an infinite energy which is due to the scalar perturbations (see Secs. \[sec-3A\] and \[sec-3B\]). Since the total energy does not vanish, perturbed flat universes are not creatable, at least, in the framework of the standard inflationary paradigm, which appears to be compatible with most current observations. This conclusion implies that, among the perturbed FRW universes undergoing ordinary inflation, only the closed ones are creatable and, consequently, the slightly inhomogeneous universe where we live should be closed.
It is generally believed that, in classical terms, there is no any way to decide if our universe is flat or closed, at least, if the curvature is small enough. An exception can be found in [@Barrow], where it is claimed that the [*spiral geodesic effect*]{} could be used to decide, observationally, whether we live either in a flat or a closed perturbed FRW universe. Another different method to distinguish between flat and closed perturbed universes arises from this paper. It is not directly based on observations. In our case, the creatable character of closed, and flat models compatible with observations, is studied according to the protocol described in Sec. \[intro\]. The closed universes are creatable whatever the linear perturbations may be. This is a robust conclusion which privileges closed models against the flat ones. In the $K=0 $ case, we have studied the most accepted model based on standard inflation. It does not appear to be creatable; nevertheless, this conclusion is not valid whatever the perturbations may be. On the contrary, it is based on some assumptions and, consequently, it must be revised if some of such assumptions are modified in future. Particular attention deserve our hypothesis about: (i) the statistical isotropy of the universe and, (ii) the adiabatic and Gaussian character of any scalar perturbation in the post-inflationary era for $k_{min} < k < \infty$. Statistical isotropy has been recently questioned [@Jaffe] [@Ghosh], and isocurvarture and (or) non Gaussian perturbations (based on cosmic strings, pre-inflationary processes and, so on) are not completely forbidden for $k_{min} < k < \infty$. Perturbed flat universes violating condition (i), and (or) condition (ii), and (or) any other possible condition, require particular studies if one wants to probe its creatable character.
The subject of the creatable character of the (non perturbed) FRW universes has been considered in two papers in the first eighties [@Atkatz][@Vilenkin] (there is also related work in [@Coleman] [@Hartle]). In [@Atkatz][@Vilenkin], the authors discussed the possibility that the Universe could have arisen by quantum tunneling from ‘nothing’. In [@Vilenkin], a cosmological model is proposed in which the Universe is created by quantum tunneling from ‘nothing’ into a particular closed FRW model. Furthermore, in [@Atkatz], the authors find that within the context of FRW models, only the spatially closed and the flat de Sitter universes can originate in this manner, because they find that a finite tunneling amplitude exists only from initial spaces with finite three-volume (on the Euclidean section).
In the absence of perturbations, the results in the present paper essentially agree with those of [@Atkatz]–[@Hartle], since we find with many other authors (including paper [@Ferrando]) that the closed and flat FRW models have vanishing energy and momenta and so, according to our terminology, this kind of universes would be creatable. Nevertheless, the method and the scopes of our work are very different from those of papers [@Atkatz]–[@Hartle].
First of all, in these papers, the authors considered precise quantum mechanisms to originate our classical universe from a quantum one; however, we assume that the produced classical universes must have vanishing energy and momenta and, then, we apply a definite protocol to decide whether a given classical universe (in the present case, some perturbed FRW models) has or not vanishing energy and momenta. Since we expect that any reasonable quantum process could not produce a universe with non vanishing energy, we have called these universes with vanishing energy and momenta [*creatable*]{} universes. In the above quoted papers, the authors assume that energy and momenta can only be properly defined in asymptotic Minkowskian universes and, consequently, they could not follow our line of research. Nevertheless, in the present paper, we have been able to define the energy and momenta of some non asymptotic Minkowskian universes in a consistent and unambiguous way (the basic idea supporting our procedures). Asymptotically, perturbed and non perturbed FRW universes appear to be conformally flat, and this fact has allowed a definiton of energy and momenta which is a generalization of that used in the case of asymptotic Minkowskian space-times. In both cases, we are constrained to calculate the energy and momenta in appropriate cordinate systems making explicit a certain form of the spatial metric and, then, showing how cooordinate transformations preserving this form do not alter the energy and momenta.
Furthermore, whereas in papers [@Atkatz]–[@Hartle] only exact FRW universes were studied, we have considered realistic perturbed FRW universes (a related study considering departures from FRW geometries was proposed in [@Atkatz]). In this way, we have found that perturbed closed FRW universes are creatable, but perturbed flat FRW universes, in the framework of standard inflation, are not.
Notice that, attending the different criteria put forward to define a universe as creatable (to have a finite tunneling amplitude, in the case of the quoted authors, or to have vanishing energy and momenta, in our case), it is not obvious that both definitions must lead to the same conclusions. Comparison is possible for unperturbed FRW models (the quoted authors have not studied the perturbed ones). There is full agreement in the closed case, but for non perturbed flat models, there is some discrepancy, since the quoted authors find them creatable only in the de Sitter subcase, whereas we find all them creatable. Nevertheless, this discrepancy has perhaps a non significant meaning, since when we have considered a realistic perturbed flat FRW model, we have found that this model is not creatable. This result shows that the exact flat FRW universe is an unstable one to our concerning. In the case of perturbed FRW universes, the comparison with the results of these authors is not possible since, as we have said, they have not considered the case.
We would like to thank J. J. Ferrando and J. A. Morales for valuable discussions. This work has been supported by the Spanish Ministerio de Educación y Ciencia, MEC-FEDER project FITS2006-06062.
Scalar and tensor inflationary perturbations in a flat background
=================================================================
Our protocol begins with the use of Gauss coordinates, in which, the metric has the form: $$ds^{2}=-dt^{2}+dl^{2}, \,\,\,\,\,\, dl^{2}=g_{ij}dx^{i}dx^{j} \ ,$$ this means that, in Gauss coordinates, the conditions $$g_{00}=-1, \,\,\,\,\,\, g_{0i}=0
\label{gs}$$ are satisfied. In the case of a FRW universe with scalar perturbations, conditions (\[gs\]) define the so-called synchronous gauge. In this gauge, a detailed study about the evolution of scalar perturbations in a flat FRW universe can be found in reference [@Ma]. The authors of that paper (Ma and Bertschinger) studied scalar perturbations in a rather general FRW flat universe containing baryons, cold dark matter (CDM), neutrinos, and radiation. They also studied the evolution in the longitudinal gauge. In the synchronous gauge, Ma and Berstchinger expanded the most general scalar metric perturbation as it is done in Eq. (\[original metric\]). This expansion only involves two arbitrary functions: $\eta(\mathbf{k}, \tau)$ and $h(\mathbf{k}, \tau)$.
In the longitudinal gauge one can write: $$ds^{2}=a^{2}(\tau)[-(1+2\psi)d\tau^{2}+(1+2\phi)\delta_{ij}dx^{i}dx^{j}] \ ,
\label{lon}$$ where only two arbitrary functions are necessary to describe the most general scalar perturbation. Fourier expansions of $\psi $ and $\phi $ involve the coefficients $\psi (\mathbf{k}, \tau)$ and $\phi (\mathbf{k}, \tau)$.
There are other coefficients appearing in the expansions of physical quantities involved in the energy momentum tensor, e. g., those corresponding to the density contrasts of the different energy components: $\delta_{\gamma}(\mathbf{k}, \tau)$ for radiation, $\delta_{c}(\mathbf{k}, \tau)$ for CDM and so on. All these coefficients are coupled in a complicate system of equations (see [@Ma]).
Fortunately, in order to evaluate the integrals giving the total energy and momenta of the perturbed universe, only functions $\eta(\mathbf{k}, \tau)$ and $h(\mathbf{k}, \tau)$ (related to the metric) could be actually necessary; this fact facilitates our calculations. Moreover, only functions $\eta(\mathbf{k}, \tau_0)$ and $h(\mathbf{k}, \tau_0)$ are required, $\tau_{0} $ being an arbitrary time. We can say that our problem is identical to that solved in paper [@Ma], where initial conditions to solve the equations governing the evolution of the perturbations were calculated at a fixed time. This time was chosen to be in the time interval limited by electron positron annihilation and the time at which light massive neutrinos become non relativistic. The same choice is appropriate for us. Ma and Bertschinger solved the evolution equations in the mentioned time interval for linear adiabatic perturbations larger than the horizon ($k \tau <<<1$). In the synchronous gauge, these authors found: $$\eta=2C+\frac {5+4R_{\nu}}{6(15+4R_{\nu})}Ck^{2}\tau^{2}, \,\,\,\,\, h=Ck^{2}\tau^{2} \ ,
\label{ini}$$ where $C=C(\mathbf{k})$, $R_{\nu}=\bar{\rho}_{\nu}/(\bar{\rho}_{\gamma}+\bar{\rho}_{\nu})$, and $\bar{\rho}_{\nu}$ and $\bar{\rho}_{\gamma}$ are the background energy densities of neutrinos and photons, respectively. There is also a time decaying term whose explicit form is given in Eq. (\[eta\]).
The same study was also performed in the longitudinal gauge. Indeed, Ma and Bertschinger fixed their initial conditions by assuming that, in the longitudinal gauge, under the assumption of statistical isotropy, the power spectrum of the $\psi $ potential of Eq. (\[lon\]) is $$P(\psi) \propto k^{-3} \ ,
\label{spec}$$ (see below for comments about this choice). How can we obtain the corresponding initial conditions in the synchronous gauge? The answer is easily obtained from Eqs. (18) in [@Ma]. One of these equations reads as follows: $$\psi(\mathbf{k}, \tau) = \frac{1}{2k^{2}} \Big( \ddot{h}(\mathbf{k}, \tau)+6
\ddot{\eta}(\mathbf{k}, \tau)+ \tau^{-1} [\dot{h}(\mathbf{k}, \tau)+
6\dot{\eta}(\mathbf{k}, \tau)] \Big) \ ,$$ where each dot stands for a derivative with respect to the conformal time.
Taking into account Eqs. (\[ini\])–(\[spec\]) and this last equation, one easily gets $$C(k) \propto k^{-3/2} \ ,
\label{ck}$$ which is valid in the synchronous gauge. That could be our basic condition in order to compute the integrals giving the total energy and momenta of the universe. It is also the basic assumption leading to the initial conditions used by Ma and Berstchinger to solve the evolution equations in the synchronous gauge. The resulting numerical solution gave a very good description of both the power spectrum, $P(k)$, of the energy density perturbations and the angular power spectrum ($C_{\ell}$ coefficients) of the CMB.
Now, a question arises: why the $\psi$-spectrum defined in Eq. (\[spec\]) is appropriate? Let us try to answer this.
After evolution, large enough cosmological inhomogeneities reenter the horizon in the matter dominated era and, afterward, it is well known that the potential $\psi $ satisfies the equation (see [@Ma]) $$\Delta \psi \propto \delta \ .
\label{lapla}$$ Since $\delta $ is the total energy density contrast, this last equation indicates that function $\psi $ plays the role of the peculiar Newtonian gravitational potential. Finally, Eqs. (\[spec\]) and (\[lapla\]) lead to $$P(k) = \langle |\delta_{\mathbf{k}}|^{2} \rangle \propto k \ ,
\label{hz}$$ which means that the spectrum of the energy density perturbations is a Harrison-Zel’dovich (HZ) one. This result justifies the use of the $\psi $-spectrum defined in Eq. (\[spec\]). The above HZ spectrum is only valid at times close enough to horizon crossing, but afterward, as the inhomogeneities evolve inside the horizon, microphysics becomes important and this spectrum evolves toward a new one of the form $P(k)=k/T(k)$, where $T(k)$ is the so-called transfer function.
Inflationary predictions are compatible with a HZ spectrum $P(k) \propto k$, as well as with an spectrum of the form $P(k) \propto k^{n_{s}}$ having its spectral index $n_{s}$ close to unity. Accordingly, the analysis of the data obtained by the WMAP mission during three observation years leads to the inequality $0.942 < n_{s} < 0.974 $. Moreover, if other observational data (galaxy surveys, other CMB observations, and so on) are taken into account, the resulting inequality appears to be $0.932 < n_{s} < 0.962 $ (see [@hin07] and [@sper07]). From Eq. (\[lapla\]) one easily proves that the condition $C(k) \propto k^{(n_{s}-4)/2} $ leads to a final spectrum $P(k) = k^{n_{s}} $. On account of these considerations, our calculations of the total energy and momenta of the universe is based on the relation: $$C(k) \propto k^{(n_{s}-4)/2}, \,\,\,\, n_{s} < 1, \,\,\,\, n_{s} \simeq 1 \ ,
\label{ck2}$$ which coincides with Eq. (\[ck\]) for $n_{s} = 1$.
We must emphasize that Eqs. (\[ini\]) and (\[ck2\]) are only valid for adiabatic scalar perturbations evolving outside the horizon; hence, these relations only hold for $k < k_{max} $, where $k_{max} = 2 \pi / L_{0}$ and $L_{0}=H^{-1}(\tau_{0})$; here, $L_{0}$ is the horizon size at the conformal time $\tau_{0}$. Moreover, if the adiabatic perturbations are inflationary, another inequality, $k > k_{min} $, must be also satisfied, where $k_{min} = 2 \pi / L_{I0}$ and $L_{I0}$ is the size, at time $\tau_{0}$, of a region comparable to the effective horizon at the beginning of inflation; that it to say, $L_{I0}$ is the typical size of the huge inflationary bubbles. For $k > k_{max} $ the perturbations evolve inside the horizon (where microphysics is important) and, consequently, the spectra of super-horizon perturbations must be modified by means of transfer functions. If these perturbations are inflationary, they are initially Gaussian and afterward, during the radiation dominated era (in particular, in the period considered in paper [@Ma] and also in Sec. \[sec-3A\]), they keep Gaussian because nonlinear processes leading to deviations from Gaussianity had not developed yet. Moreover, in the interval $k_{min} < k < \infty $, we assume that the perturbations produced during inflation are absolutely dominant against possible residual pre-inflationary fluctuations. The main reason is that, at the end of inflation, inflationary supercooling had made the pre-inflationary radiation density negligible against the total energy of the inflationary field and, consequently, after reheating, the mean radiation energy is fully dominated by the energy coming from the mean inflationary field and, evidently, the resulting adiabatic perturbations are associated to the fluctuations of this dominant field, with negligible contributions from pre-inflationary supercooled sources. Finally, for $k < k_{min} $, the perturbations are so long that they will have a pre-inflationary origin without any inflationary contribution. Thus, though these pre-inflationary perturbations can be expected to be small, their contribution to the integral in Eq. (\[special energy\]) could be significant as $k$ tends to zero and $r$ tends to infinity. Thus, as a precaution, this interval has been also considered along the paper. Of course, it has been taken into account that these pre-inflationary perturbations could be non Gaussian. All these ideas are carefully taken into account in Secs. \[sec-3A\] and \[sec-4\].
In general, inflation produces both scalar and tensor perturbations of the background universe. Some general considerations about tensor perturbations are now worthwhile. The tensor metric perturbations of a flat universe can be written in the form: $$h^{^{T}}_{ij}(\mathbf{x}, \tau) = \int d^{3} k e^{i \mathbf{k} \cdot \mathbf{x}}
h^{^{T}}_{ij}(\mathbf{k}, \tau) =
\int d^{3} k \,e^{i \mathbf{k} \cdot \mathbf{x}}
H_{_{T}}(\mathbf{k}, \tau) \epsilon_{ij}(\hat{\mathbf{k}})
\ ,$$ where functions $\epsilon_{ij} $ satisfy the following equations: $$\epsilon_{ij} = \epsilon_{ji}, \,\,\,\, \epsilon_{ii}=0, \,\,\,\, \epsilon_{ij}k_{i} =0 \ ,$$ which ensure that quantities $h^{^{T}}_{ij}(\mathbf{x},\tau)$ are symmetric, traceless, and divergenceless, as it must be in the case of metric perturbations describing gravitational waves. It is noticeable that functions $\epsilon_{ij}$ only depend on the unit vector $\hat{\mathbf{k}}$ and, consequently, any dependence on the scale (on $k$) of the tensor metric perturbation is involved in the coefficient $H_{_{T}}(\mathbf{k}, \tau)$. This scale dependence is usually fixed by defining a new power spectrum ([@dur01]) $$P_{_{T}} (k) = k^{3} \langle |H_{_{T}}(k)|^{2} \rangle \propto k^{n_{_{T}}} \ ,$$ where $n_{_{T}} $ is the tensor spectral index. With this spectrum, quantities $h^{^{T}}_{ij}(\mathbf{k})$ are proportional to $k^{(n_{_{T}}-3)/2}$. Taking into account previous formulas and assumptions, we could calculate the total energy and momenta of the universe for different $n_{_{T}} $ values. The explicit form of functions $\epsilon_{ij} $ is not necessary. Nevertheless, an explicit representation of these quantities can be easily found from the definitions given in [@hu97]. What can we say about the spectral index $n_{_{T}} $?
The spectrum generated by most inflationary potentials has an spectral index $n_{_{T}} \simeq 1-n_{s} $ (see reference [@cri93]); hence, from previous comments about $n_{s} $-values and WMAP data, it follows that the power spectrum of these inflationary backgrounds of gravitational waves is very flat (small but non vanishing $n_{_{T}}$ value). In the presence of these gravitational waves, the spectral index $n_{s}$ deviates from unity; by this reason and with the essential aim of allowing the existence of an inflationary background of gravitational waves (which is studied in Sec. \[sec-3B\]), the $n_{s}$ values used in Sec. \[sec-3A\] have been assumed to be slightly smaller than unity.
J. J. Ferrando, R. Lapiedra, J. A. Morales, Phys. Rev. D [**75**]{}, 124003 (2007).
J. J. Ferrando, R. Lapiedra, J. A. Morales, arXiv:0705.1049v2 \[gr-qc\], (2007).
E. P. Tryon, Nature [**246**]{}, 396 (1973).
M. G. Albrow, Nature (London), Phys. Sc. [**241**]{}, 56 (1973).
J. Garecki, arXiv:0708.2783v3 \[gr-qc\], (2007).
D. Atkatz, H. Pagels, Phys. Rev. D [**25**]{}, 2065 (1982)
A. Vilenkin, Phys. Rev. D, [**27**]{}, 2848 (1983)
S. Coleman and F. De Luccia, Phys. Rev. D [**21**]{}, 3305 (1980)
J. B. Hartle and S. W. Hawking, Phys. Rev. D [**28**]{} 2960 (1983)
S. Weinberg, [*Gravitation and Cosmology*]{} (Wiley, New York, 1972).
R. Lapiedra and D. Sáez, in [*Proceeding of the XXX Spanish Relativity Meeting ERE-2007 (to be published)*]{}.
J. M. Bardeen, Phys. Rev. D, [**22**]{}, 1882 (1980)
C. Ma and E. Bertschinger, ApJ [**455**]{}, 7 (1995)
P.J.E. Peebles, [*The Large-Scale Structure of the Universe*]{} (Princeton University Press, Princeton, 1980).
J. D. Barrow, R. Juszkiewicz and D. H. Sonoda, Mon. Not. R. Astr. Soc. [**213**]{}, 917 (1985)
G. Hinshaw et al., ApJS, [**170**]{}, 288 (2007)
C. B. Collins and S. W. Hawking, Mon. Not. R. Astr. Soc. [**162**]{}, 307 (1973)
H. Stephani, D. Kramer, M. Maccallum, C. Hoenselaers and E. Herlt, [*Exact solutions to Einstein Field Equations*]{} (Cambridge Univ. Press., Second edition, 2003)
T. R. Jaffe, A. J. Banday, H. K. Eriksen, K. M. Gòrski and F. K. Hansen, ApJ [**629**]{}, L1 (2005)
T. Ghosh, A. Hajian, and T. Souradeep, Phys. Rev. D, [**75**]{}, 083007 (2007)
D.N. Spergel et al., ApJS, [**170**]{}, 377 (2007)
R. Durrer, J. Phys. Stud., [**5**]{}, 177, (2001)
W. Hu, M. White, Phys. Rev. D [**56**]{}, 596 (1997)
R. Crittenden, J.R. Bond, R.L. Davis, G. Efstathiou, P.J. Steinhardt, Phys. Rev. Lett., [**71**]{}, 324 (1993)
|
---
abstract: 'All coboundary Lie bialgebras and their corresponding Poisson–Lie structures are constructed for the oscillator algebra generated by $\{\aa,\ap,\am,\bb\}$. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal $R$–matrices.'
---
å[N]{}
\#1[[**\#1**]{}]{}
\#1\#2[ ]{}
=1ex = 0.5cm = 0.5cm =1.5em =23.0cm =-1.0cm
Angel Ballesteros and Francisco J. Herranz
[*[ Departamento de Fsica, Universidad de Burgos]{}\
Pza. Misael Bañuelos,\
E-09001, Burgos, Spain*]{}
Introduction
============
Deformed Heisenberg and oscillator algebras have recently focused many investigations coming from different directions. Among them, we would like to quote the construction of deformed statistics [@Greenberg], the use of $q$–Heisenberg algebras to describe composite particles [@Avancini], the description of certain classes of exactly solvable potentials in terms of a $q$–Heisenberg dynamical symmetry [@Spiridonov], the link between deformed oscillator algebras and superintegrable systems [@Bonatsosa; @Bonatsosb] and the relations between these deformed algebras and $q$–orthogonal polynomials [@Floreanini].
Quantum universal enveloping algebras (QUEA) are much more selective deformations than general modifications of the commutation rules of a given algebra. In particular, the interest of finding Hopf algebra deformations of the oscillator algebra is twofold: firstly, because of the relevant role played by Hopf algebras to build up second quantization, as it has been recently discussed in [@Celeghini]. On the other hand, a quasitriangular quantum oscillator algebra has been related to Yang–Baxter systems and link invariants in [@Sierra].
The aim of this paper is to provide a systematic study of the quantum universal enveloping oscillator algebras underlying possible further generalizations of these results. A brief summary of the oscillator algebra and group is given in section 2. Since every QUEA defines uniquely a Lie bialgebra structure on the undeformed algebra, in section 3 we obtain and classify all coboundary Lie bialgebra structures for the harmonic oscillator algebra, as well as their corresponding Poisson–Lie brackets. In section 4 we make use of the Lyakhovsky and Mudrov formalism [@Lyak] in order to build up the deformed coproducts linked to all these Lie harmonic oscillator bialgebras. A complete quantization (including universal $R$–matrices) of two particular classes of non-standard (triangular) bialgebras is provided: the former is the natural “extension" of the non-standard deformation of the 1+1 Poincaré algebra discussed in [@Tmatrix] and the latter is a new three parameter quantization.
To our knowledge, the literature on Hopf algebra deformations of the oscillator algebra includes only the deformation given in [@Sierra; @Celeghinidos] and some new results that have been recently given in [@Vero] by computing the dual of an arbitrary quantum oscillator group obtained by following an $R$–matrix approach in a particular matrix representation (see [@FRT; @BCGST; @HLR]). Among these known deformations, the former can be easily included within our clasification at the Lie bialgebra level, and can thus be obtained without making use of contraction procedures. On the other hand, our method gives explicit (and universal) expressions for the oscillator QUE algebras linked to the quantizations of [@Vero] which are coboundaries. The procedure here outlined precludes cumbersome duality computations and leads to rather simple candidates for universal $R$–matrices.
Classical oscillator algebra and group
======================================
The oscillator Lie algebra $\osc$ is generated by $\{\aa,\ap,\am,\bb\}$ with Lie brackets =,=-,=, \[,\]=0 . \[aa\] Besides the central generator $\bb$ there exists another Casimir invariant: C=2å- -. \[ab\] A $3\times 3$ real matrix representation $D$ of (\[aa\]) is given by: &&D(å)=(
[ccc]{} 0 &0 & 0\
0 & 1 & 0\
0 & 0 & 0
), D()=(
[ccc]{} 0 &0 & 0\
0 & 0 & 1\
0 & 0 & 0
), &&D()=(
[ccc]{} 0 &1 & 0\
0 & 0 & 0\
0 & 0 & 0
), D()=(
[ccc]{} 0 &0 & 1\
0 & 0 & 0\
0 & 0 & 0
). \[ac\] The expression for a generic element of the oscillator group $\Osc$ coming from this representation is: &&T\^D={D()}{D()} {D()}{D(å)}&&=(
[ccc]{} 1 &e\^& +\
0 & e\^&\
0 & 0 & 1
) . \[ad\] The group law for the coordinates $\bbb$, $\aam$, $\aap$ and $\aaa$ is obtained by means of matrix multiplication ${T^D}''={T^D}'\cdot {T^D}$: && =+,=+ -e\^[-]{},&& =+e\^, =+e\^[-]{} . \[ae\] Left and right invariant vector fields are also deduced from (\[ad\]) and read: X\^L\_å=\_,X\^L\_=e\^\_, X\^L\_=e\^[-]{}\_- e\^[-]{}\_, X\^L\_=\_; \[af\] X\^R\_å=\_+ \_+ \_, X\^R\_= \_- \_, X\^R\_= \_, X\^R\_=\_. \[ag\]
The Heisenberg algebra can be seen as the subalgebra $\langle
\ap,\am,\bb\rangle$ of $\osc$ and the Heisenberg group $\langle
\aap,\aam,\bbb\rangle$ is recovered by taking the coordinate $\aaa\equiv
0$ in $\Osc$. Moreover, $\osc$ can be seen as a centrally extended (1+1) Poincaré algebra (by $\bb$). This fact will be useful in the quantization process.
Coboundary oscillator Lie bialgebras
====================================
Let $g$ be a Lie algebra and let $r$ be an element of $g\wedge g$. The cocomutator $\delta:g\rightarrow g\wedge g$ given by (X)=\[1X + X 1, r\], Xg, \[bc\] defines a coboundary Lie bialgebra $(g,\delta (r))$ if and only if $r$ fulfills the modified classical Yang–Baxter equation (YBE) =0, Xg, \[mod\] where $[[r,r]]$ is the Schouten bracket defined by :=\[r\_[12]{},r\_[13]{}\] + \[r\_[12]{},r\_[23]{}\] + \[r\_[13]{},r\_[23]{}\], \[bb\] and, if $r=r^{i j} X_i\otimes X_j$, we have denoted $r_{12}=r^{i j} X_i\otimes X_j\otimes 1$, $r_{13}=r^{i j} X_i\otimes 1\otimes X_j$ and $r_{23}=r^{i j} 1\otimes X_i\otimes X_j$.
When the $r$–matrix is such that $[[r,r]]=0$ (classical YBE), we shall say that $(g,\delta (r))$ is a [*non-standard*]{} (or triangular) Lie bialgebra. On the contrary, a solution $r$ of (\[mod\]) with non vanishing Schouten bracket will give rise to a so called [*standard*]{} Lie bialgebra.
We recall that, if $g=Lie(G)$, the (unique) Poisson–Lie structure on $C^\infty(G)$ linked to a fixed bialgebra $(g,\delta (r))$ is given by the Sklyanin bracket {,}=r\^(X\_\^LX\_\^L-X\_\^RX\_\^R)\[bd\] , ,C\^(G), where $X_\alpha^L$ and $X_\bt^R$ are the left and right invariant vector fields of $G$, respectively.
In particular, for $\osc$ we shall consider an arbitrary element $r$, which can be written in terms of six (real) coefficients: r= å+ å+ å+ + + . \[ba\] It is a matter of computation to prove that the corresponding Schouten bracket for $r$ (\[ba\]) is && = (+) å+ (-) å\
&& -2 å+ ( + - \^2). \[bab\] From this expression follows that the modified classical YBE (\[mod\]) is fulfilled provided that: && =0,&& (+)=0,\[da\]\
&& ( -)=0 .The solutions of this system are splitted into three classes: $\xp\ne 0$, $\xm\ne 0$ and $\xp=\xm=0$. For each of them we shall distinguish between non-standard ($[[r,r]]=0$) and standard Lie bialgebras as follows.
[**Type I$_+$.**]{} If $\xp\ne 0$ we have $\xm=0$ and $\yy=-\xx$. The Schouten bracket reduces to: =(-\^2) (). \[db\] Therefore if $\xp\yp-\xx^2\ne 0$ we have standard solutions and when $\yp = \xx^2/\xp$ we are considering non-standard ones.
[**Type I$_-$.**]{} If $\xm\ne 0$ equations (\[da\]) imply $\xp=0$ and $\yy=\xx$. The Schouten bracket is now: =( -\^2) (). \[dbb\] Standard solutions are obtained when $\xm\ym-\xx^2\ne 0$, while non-standard ones correspond to $\ym = \xx^2/\xm$.
[**Type II.**]{} Finally, we consider the case with $\xp=0$; if $\xm\ne 0$ we are again in type I$_-$, so we must take also $\xm=0$ in order to have three disjoint sets of solutions. In this case equations (\[da\]) are automatically satisfied and the Schouten bracket is: =-\^2 . \[dbbb\] Then the condition $\yy\ne 0$ gives rise to standard solutions and $\yy=0$ to non-standard ones.
All the information concerning this classification of coboundary oscillator Lie bialgebras is summarized in table I. Poisson–Lie structures for the oscillator group are deduced via the Sklyanin bracket (\[bd\]) and presented in table II.
Note that this classification is based in the use of skew-symmetric $r$–matrices. This implies no loss of generality: given an arbitrary element of $g\otimes g$, the map $\delta$ generated by (\[bc\]) has to be skew-symmetric to give rise to a Lie bialgebra. This amounts to impose $Ad^{\otimes 2}$–invariance on the symmetric part of $r$ and, therefore, $r$ will generate the same Lie bialgebra than its skew-symmetric part [@Tjin]. In particular, it can be easily checked that the more general element $\eta$ of $\osc\otimes\osc$ such that =0,X{å,,,} ; is given by: =\_1 (å+ å- - ) + \_2 , \[eta\] i.e., a linear combination of two terms directly related to the two Casimirs of $\osc$.
Quantization
============
In this section we first show how the Lyakhovsky and Mudrov (LM) formalism [@Lyak] allows all the cocommutators of the oscillator bialgebras previously found to generate coassociative coproducts in a straightforward way. Afterwards, we shall construct commutation rules and universal quantum $R$–matrices for some of these bialgebra quantizations.
The Lyakhovsky–Mudrov formalism
-------------------------------
Let us start with a short resume of the LM formalism which applies to an associative algebra $E$ over $\bf C$ with unit and generated by $n$ commuting elements $H_i$ and $m$ additional elements $X_j$. For any $m\times m$ numerical matrix $\mu$, by $\mu\,H$ we understand the matrix $\mu$ with all its entries multiplied by $H$. If $P$ is an $m\times m$ matrix with entries $p_{kl}\in E$, the $k$–th component of $P
\dot\otimes \vec X$ is defined as (PX)\_k=\_[l=1]{}\^m p\_[kl]{}X\_l. \[gf\] The main LM statement [@Lyak] is that $E$ can be endowed with a coalgebra structure as follows (where we have denoted by $\sigma$ the permutation map $\sigma (a\otimes b)=b\otimes a$):
[**Proposition 1.**]{} [*Let $\{1, H_1,\dots,H_n,X_1,\dots,X_m\}$ a basis of an associative algebra $E$ over $\bf C$ verifying the conditions \[[H\_i]{},[H\_j]{}\]=0,i,j=1,…,n. \[ga\] Let $\mm_i$, $\nn_j$ $(i,j=1,\dots,n)$ be a set of $m\times m$ complex matrices such that =\[[\_i]{},[\_j]{}\]=\[[\_i]{},[\_j]{}\]=0 , i,j=1,…,n. \[gb\] Let $\vec X$ be a column vector with components $X_l$ $(l=1,\dots,m)$. The coproduct and the counit &&(1) =1 1, (H\_i) =1 H\_i + H\_i1,\[gc\]&&(X) = (\_[i=1]{}\^n[\_i H\_i]{}) X + ( (\_[i=1]{}\^n[\_i H\_i]{}) X ), \[gd\]\
&&(1)=1,(H\_i)=(X\_l)=0,i=1,…,n;l=1,…m; \[ge\] endow $(E,\Delta,\epsilon)$ with a coalgebra structure.*]{}
The resulting coalgebra can be seen as a multiparametric deformation where the deformation parameters are the entries of the matrices $\mm_i$ and $\nn_j$. If we are able to find a compatible multiplication with the coproduct (\[gd\]) we will have finally obtained a quantum algebra.
It is worth remarking that this formalism encodes in the set of matrices $\mu_i$ and $\nu_j$ the whole coalgebra structure. In fact, the role of these matrices is, essentially, to reflect the Lie bialgebra underlying a given quantum deformation. This can be clearly appreciated by taking the first order (in all the parameters) of (\[gd\]): \_[(1)]{} (X) = (\_[i=1]{}\^n[\_i H\_i]{}) X + ( (\_[i=1]{}\^n[\_i H\_i]{}) X ) \[gg\] and recalling that the cocommutator $\delta$ corresponds to the co-antisymmetric part of (\[gg\]). It can be written in “matrix" form as: (X)=\_[(1)]{} (X)-\_[(1)]{} (X). \[gh\]
We would like to emphasize the following points:
$\bullet$ The commuting elements $H_i$ are the primitive generators.
$\bullet$ The cocommutator $\delta(X_i)$ does not contain terms of the form $H_i\wedge H_j$.
$\bullet$ The same cocommutator (\[gh\]) can be obtained from different choices of the matrices $\mu_i$ and $\nu_j$. This means that different sets of matrices might lead to right quantizations, all of them having the same first order terms in the deformation parameters. Moreover, we can choose $\mu_i=0$ as a representative of all these quantizations and we shall obtain (X) = - (\_[i=1]{}\^n[\_i H\_i]{}) X =- (\_[i=1]{}\^n[\_i H\_i]{}) X + (\_[i=1]{}\^n[\_i H\_i]{}) X). \[gi\]
Now let us reverse somehow the LM formalism trying to find in which way the oscillator Lie bialgebras given in table I can be recovered by a suitable choice of the matrices $\mu_i$ and $\nu_j$. Of course, the benefit of such a situation is to be able to “exponentiate" directly the bialgebra (\[gi\]) to a full coalgebra (\[gc\]–\[ge\]).
Let us start with non-standard type I$_+$ oscillator bialgebras. By denoting $H_1\equiv \ap$, $H_2\equiv \bb$, $X_1\equiv \aa$, $X_2\equiv
\am$, we see that $[H_1,H_2]=0$ and there exists a term of the type $H_1\wedge H_2\equiv\ap\wedge\bb$ within the cocommutator $\delta(\aa)$; however, this obstruction can be circumvented by defining a new generator in the form: å’=å- (/). \[ggi\] Hence, the cocommutators for the non-primitive generators $\aa'$ and $\am$ can be written as (
[c]{} å’\
)= (
[cc]{} -&0\
0 & -
)(
[c]{} å’\
)+ (
[cc]{} 0 &(\^2/)\
-& -2
)(
[c]{} å’\
) \[gj\] In view of this expression, the matrices $\mm_i$ and $\nn_j$ can be chosen as: \_1=\_2=(
[cc]{} 0 &0\
0 & 0
),\_1=(
[cc]{} &0\
0 &
),\_2=(
[cc]{} 0 &-\^2/\
& 2
). \[gk\] Now, the set of conditions of Proposition 1 are fulfilled, and we can use this result to get the coproducts: &&(
[c]{} å’\
)= {(
[cc]{} 0 &0\
0 & 0
)} (
[c]{} å’\
) &&+ ( {(
[cc]{} &- (\^2/)\
& +2
)}(
[c]{} å’\
))&& =(
[c]{} 1å’ + å’(1-) e\^[+]{} -(\^2/)e\^[+]{}\
1+ (1+) e\^[+]{} +å’e\^[+]{}
) \[gm\] We can finally return to the initial basis elements, thus obtaining a three-parameter QUEA (denoted by $U_{\xp,\xx,\ym}^{(\Ipp\, n)}(\osc)$) such that: && (å)=1å+ å(1-) e\^[+]{} -(\^2/)e\^[+]{} &&+(/)(1-(1-)e\^[+]{}), \[ggm\]\
&& ()= 1+ (1+) e\^[+]{}&&+(å- )e\^[+]{}.
This quantization procedure can be applied to the remaining types of bialgebras in the same way. For the standard type I$_+$ bialgebras we also use (\[ggi\]), while for the bialgebras of type I$_-$ we introduce the new generator å’=å- (/). \[ggii\] On the contrary, no such a kind of transformation is necessary to get the coproducts for the Lie bialgebras of type II.
The coproducts for the corresponding QUEA of the coboundary oscillator Lie bialgebras of table I are written down in table III; we denote each multiparametric quantum coalgebra by $U_{\a_i}^{(t\, m)}(\osc)$ where $t$ is the type, $m=s$ or $m=n$ according either to the standard or non-standard oscillator deformations and with $\a_i$ being the deformation parameters. The explicit expressions for the coproducts of $U_{\xp,\xx,\ym,\yp}^{(\Ipp\, s)}(\osc)$ and $U_{\xm,\xx,\ym,\yp}^{(\Imm\, s)}(\osc)$ are rather complicated so we keep their matrix forms written in terms of the generator $\aa'$ defined by either (\[ggi\]) or by (\[ggii\]), respectively.
The final step in the quantization process of a fixed bialgebra is to find the commutation relations compatible with its deformed coproduct (counit and antipode can be obtained in the form explained in [@Lyak]). In the following, we solve completely this problem and construct the deformed Hopf algebras $U_{\a_i}^{(t\,m)}(\osc)$ for some representative cases among the ones included in Table III.
Non-standard type I$_+$: $U_{z}^{(n)}(\osc)$
--------------------------------------------
It is remarkable that the oscillator algebra with basis $\{\aa,\ap,\am,\bb\}$ can be interpreted as an extended (1+1) Poincaré algebra where $\aa$ is the boost generator, $\ap$ and $\am$ generate the translations along the light-cone and $\bb$ is the central generator. This fact rises the question about whether it is possible to implement in this extended case the universal (non-standard) quantum deformation of the Poincaré algebra studied in [@Tmatrix] from a $T$–matrix approach.
Let us consider the non-standard oscillator bialgebras of type I$_+$ with $\xx=\ym=0$ and $\xp\equiv z$. According to table I the Lie bialgebra is characterized by commutation relations (\[aa\]), classical $r$–matrix: r=z å; \[ea\] and cocommutators: &&()=0,()=0, (å)=z å,&& ()=z (+å) . \[eb\] Poisson–Lie brackets are easily deduced from table II: &&{,}=z (e\^-1),{,}=0,{,}=z ,&&{,}=z ,{,}=z, {,}=-z a\_-\^2 . \[ec\] A quantum deformation for this Lie bialgebra is given by the following statement:
[**Proposition 2.**]{} [*The coproduct $\Delta$, counit $\epsilon$, antipode $\gamma$ &&()=1+1, ()=1+1,&&(å)=1å+åe\^[z]{}, ()=1+e\^[z]{}+zåe\^[z]{} ;&& \[ed\] (X)=0,X{å,,,}; \[ee\] &&()=-,()=-, &&(å)=-åe\^[-z]{}, ()=-e\^[-z]{}+ z åe\^[-z]{}; \[ef\] and the commutation relations \[å,\]=,=-,=e\^[z]{}, \[,\]=0 , \[eg\] determine a Hopf algebra (denoted by $U_z^{(n)}(\osc)$) which quantizes the non-standard bialgebra generated by the classical $r$–matrix (\[ea\]).*]{}
The coproduct (\[ed\]) is obtained from table III. Note that $\bb$ remains as a central generator. There is another element belonging to the center of $U_z^{(n)}\osc$ whose classical limit is (\[ab\]), namely C\_z=2å+ + . \[eh\]
An important feature of the quantum algebra $U_z^{(n)}(\osc)$ is that the generators $\aa$ and $\ap$ form a Hopf subalgebra which coincides exactly with the corresponding to the quantum Poincaré algebra of [@Tmatrix]. We recall that for this Hopf subalgebra there is a universal $R$–matrix given by: R={-zå}{zå} . \[ei\] Obviously, (\[ei\]) satisfies the quantum YBE for $U_z^{(n)}\osc$, but moreover it verifies (X)=R(X)R\^[-1]{}, X{å,,,} . \[ej\] This assertion must be proved only for $\bb$ and $\am$; the proof for the former is trivial since it is a central generator, and for the latter we have &&{zå}(){-zå} =1+ 1=\_0() , && {-zå} \_0() {zå} =().
The fulfillment of relation (\[ej\]) allows to use the FRT approach in order to get a quantum deformation of $Fun(\Osc)$ by taking into account that in the matrix representation (\[ac\]) the universal $R$–matrix (\[ei\]) collapses into: D(R)=II + z(D(å)D() - D()D(å)), \[rr\] where $I$ is the $3\times 3$ identity matrix. Therefore, the Hopf structure of the associated oscillator quantum group is given by:
[**Proposition 3.**]{} [*The coproduct, counit, antipode &&( )=1+ 1,&&( )=e\^[ ]{}+ 1,&&( )=e\^[- ]{}+ 1,&&( )=1+ 1 -e\^[- ]{} ; \[ek\] (X)=0,X{,,,}; \[el\] &&()=-,()=-e\^[- ]{},\
&&()=-e\^, ()=-- (e\^[-]{}e\^); \[em\] together with the commutation relations &&\[ , \]=z (e\^[ ]{} -1),=0,=z ,&&\[ , \]=z ,=z , \[ , \]=-z[ ]{}\^2 , \[en\] constitute a Hopf algebra denoted by $Fun_z^{(n)}(\Osc)$.*]{}
The coproduct (\[ek\]), counit (\[el\]) and antipode (\[em\]) are obtained from the relations $\Delta(T)=T\dot\otimes T$, $\epsilon(T)=I$ and $\gamma(T)=T^{-1}$, where $T\equiv T^D$ is the generic element of the oscillator group $\Osc$ (\[ad\]). The commutation rules are deduced from $RT_1T_2=T_2T_1R$, where $T_1=T\otimes I$, $T_2=I\otimes T$ and $R$ given by (\[rr\]).
The commutation relations (\[en\]) can be seen as a Weyl quantization $\{\, ,\, \}\to z^{-1}[\, ,\,]$ of the fundamental Poisson brackets (\[ec\]). It is also clear that the coalgebra structure of $Fun_z^{(n)}(\Osc)$ determined by the coproduct (\[ek\]) and counit (\[el\]) is valid for any quantum group which deforms $Fun(\Osc)$.
Some features of this new quantum oscillator algebra can be emphasized:
$\bullet$ When the central extension $\bb$ and its corresponding quantum coordinate $\hbbb$ vanish all results concerning the quantum Poincaré algebra and group given in [@Tmatrix] are recovered. In this sense, the quantum coordinates $\haaa$, $\haap$ and $\haam$ close a quantum Hopf subalgebra which coincides exactly with the quantum Poincaré group just mentioned.
$\bullet$ The primitive generator involved in the deformation is now $\ap$. This fact will be relevant at a representation theory level and, consequently, from the point of view of the physical properties of this deformed oscillator.
$\bullet$ The deformed Heisenberg subalgebra generated by $\ap,\am$ and $\bb$ is not a Hopf subalgebra due to the appearence of $\aa$ in $\Delta(\am)$. However, the Hopf subalgebra structure can be recovered by working on a representation where the central generator $\bb$ is expressed as a multiple of the identity. In this situation, $\aa$ can be defined in terms of $\ap$ and $\am$ by using the Casimir (\[eh\]). In general, this type of non-standard deformed bosons can be expected to build up $q$–boson realizations of the already known non-standard quantum algebras [@Ohn; @Bey].
Non-standard type II: $U_{\xx,\ym,\yp}^{(\II\,n)}(\osc)$
--------------------------------------------------------
The classical $r$–matrix r= å+ + , \[ma\] originates a non-standard three-parametric oscillator bialgebra of type II whose cocommutators and associated Poisson–Lie brackets appear in tables I and II, respectively. A quantum deformation of this coboundary Lie bialgebra is given by:
[**Proposition 4.**]{} [*The Hopf algebra denoted by $U_{\xx,\ym,\yp}^{(\II\,n)}(\osc)$ which quantizes the oscillator bialgebra generated by (\[ma\]) has coproduct given in table III, counit (\[ee\]), antipode &&()=-,()=-e\^, ()=-e\^[-]{},&&(å)=-å- (/) (1-e\^) - (/) (1-e\^[-]{}) ; \[mb\] and commutation relations && \[å,\]=- (-), \[å,\]=-- (),&&\[,\]=,=0 , \[mc\] where (x):=1[x\^2]{}(e\^[x]{}-1-x) . \[md\]* ]{}
Note that $\lim_{x\to 0}\vv(x)= \bb^2/2$. The quantum analogue of (\[ab\]): C\_[,,]{}=2å- - +2 (-) - 2 (), \[me\] belongs to the center of $U_{\xx,\ym,\yp}^{(\II\,n)}(\osc)$.
It is worth remarking that this quantum oscillator algebra can be related with the results of [@Vero]: $U_{\xx,\ym,\yp}^{(\II\,n)}(\osc)$ can be seen as a Type II case with $p\equiv \xx$, $q\equiv -\xx$, $b\equiv \yp$ and $c\equiv -\ym$. Moreover,
[**Proposition 5.**]{} [*The element && R={r} ={ å+ + } && ={-( å+ + )} { ( å+ + ) } && \[mf\] satisfies both the quantum YBE and relation (\[ej\]), so it is a universal $R$–matrix for $U_{\xx,\ym,\yp}^{(\II\,n)}(\osc)$.*]{}
Since $\bb$ is a central generator, it is clear that (\[mf\]) is a solution of the quantum YBE. The proof for property (\[ej\]) is sketched in Appendix A. In the matrix representation (\[ac\]) we get: D(R)=II + D(å)D() + D()D() + D()D(). \[mg\] The FRT prescription leads now to another multiparametric quantum deformation of the algebra of the smooth functions on the oscillator group $Fun_{\xx,\ym,\yp}^{(\II\,n)}(\Osc)$, given by coproduct (\[ek\]), counit (\[el\]), antipode (\[em\]) and the non vanishing commutation rules \[,\]=- + (e\^-1) ,= + (e\^[-]{} -1). \[mh\] The classical limit (in the three parameters) is $Fun(\Osc)$ and, once more, commutators (\[mh\]) are a Weyl quantization of the Poisson–Lie brackets written in table II.
Standard type II: $U_{z}^{(s)}(\osc)$
-------------------------------------
The classical $r$–matrix which solves the classical YBE and underlies the quantum oscillator algebra obtained in [@Sierra; @Celeghinidos] by a contraction method can be expressed in our notation as: r= - z(å+ å) + 2z . \[pa\] Its symmetric ($r_+$) and skew-symmetric ($r_-$) parts are: &&r\_+=(r+r)/2= z (+) - z(å+ å), \[pb\]\
&&r\_-=(r-r)/2= z . \[pc\] The symmetric part $r_+$ corresponds to the element $\eta$ (\[eta\]) with the parameters $\bt_1=-z$ and $\bt_2=0$. On the other hand, $r_-$ can be identified with a standard classical $r$–matrix of type II with parameters $\xx=\ym=\yp=0$ and $\yy\equiv -z$ (see table I). Both the standard $r$–matrix (which coincides with $r_-$ (\[pc\])) and the non antisymmetric one (\[pa\]) give rise to the same oscillator bialgebra with cocommutators: (å)=()=0,()=z ,()=z . \[pd\] The associated non vanishing Poisson–Lie brackets (see table II) are: {,}=z ,{,}=z . \[pe\]
The quantum deformation of this coboundary oscillator bialgebra is given by:
[**Proposition 6.**]{} [*The quantum algebra which quantizes the standard bialgebra generated by (\[pa\]) has a Hopf structure denoted by $U_{z}^{(s)}(\osc)$ and characterized by the coproduct, counit, antipode &&(å)=1å+å1, (A\_+’)=e\^[-z]{}A\_+’ +A\_+’ 1, && ()=1+1 , ()=1+e\^[z]{} ; \[pf\] (X)=0,X{å,A\_+’,,}; \[pg\] (å)=-å,()=-, (A\_+’)=-A\_+’ e\^[z]{}, ()=-e\^[-z]{} ; \[ph\] together with the commutation relations \[å,A\_+’\]=A\_+’,=-,=, \[,\]=0 . \[pi\]* ]{}
The quantum Casimir is: C\_z=2å - A\_+’-A\_+’ . \[pj\]
The coproducts (\[pf\]) are just those given in table III but written in terms of a new generator $A_+'=e^{\yy\bb}\ap$ where $\yy =- z$. In this case the universal $R$–matrix adopts a much simpler form than the one already known from [@Sierra; @Celeghinidos]. Namely, && R ={-z(å+ å)}{2z A\_+’}&&= {-z å}{-z å}{2z A\_+’}. \[pk\] It is worth remarking that all the quantum $R$–matrices given in this section are obtained via a straightforward exponentiation process from their classical counterparts (compare, for instance, (\[pk\]) to (\[pa\])).
The FRT prescription can be applied leading to the commutation rules of the quantum group $Fun_z^{(s)}(\Osc)$ by taking into account that (\[pk\]) in the matrix representation (\[ac\]) is just D(R)=II + 2z D()D(A\_+’) - z(D(å)D() + D()D(å)) ; \[pl\] (note that $D(A_+')\equiv D(\aa)$). In this way, the non vanishing commutators of $Fun_z^{(s)}(\Osc)$ read: =z ,=z , \[pm\] and correspond to a Weyl quantization of the Poisson–Lie brackets (\[pe\]).
Concluding remarks
==================
We have presented a systematic procedure in order to study the coboundary Lie bialgebras of the oscillator algebra. The first order deformations given by the corresponding cocomutators have been used to construct, by a sort of “exponentiation" process, multiparametric quantum deformations of the oscillator algebra. We point out that we have not treated the question of the equivalence of the coboundary oscillator bialgebras we have obtained, indeed this is actually a problem by itself. For instance, from an algebraic point of view, bialgebras of types I$_+$ and I$_-$ can be related by interchanging generators $\ap$ and $\am$, although this result is not so straightforward if we look at their corresponding Poisson–Lie groups.
It is worth stressing that, in the case here analysed, the complete (and rich) classification of the classical $r$–matrices (and, therefore, of the corresponding Poisson structures on the oscillator group) is easily obtained. This seems to indicate that, at least for Lie algebras with a low enough dimension, the complete solution of the modified classical YBE for an arbitrary skew element of $g\otimes g$ can be explicitly deduced giving rise to a great amount of new results.
This kind of procedure is complementary (and dual) to that developed in [@Vero], since it allows us to focus on the deformation at the quantum algebra level and looking for universal quantum $R$–matrices. In fact, given a skew solution $r$ of the modified classical YBE and a matrix representation $D$ of the quantum algebra, the element $D(R)=1+z\,D(r)$ will lead us to the corresponding $R$–matrix method.
This approach can be seen as a part of a research program that, in order to construct and study quantum algebras, tries to extract as much information as possible from the associated Lie bialgebras (as far as contraction methods are concerned, see for instance [@LBC]). It would be interesting to apply it to other physically interesting algebras whose coboundary bialgebra structures are not well known, among them, we would like to mention the Schrödinger, optical and Galilean algebras, also with the aim of obtaining some (universal) quantum deformations.
The authors acknowledge M. Santander and M.A. del Olmo a careful reading of the manuscript and many helpful suggestions as well as the referees for some pertinent comments. This work has been partially supported by DGICYT (Projects PB92–0255 and PB94–1115) from the Ministerio de Educación y Ciencia de España.
The main steps necessary to prove that the $R$–matrix (\[mf\]) verifies the property (\[ej\]) for the generators $\ap$, $\am$ and $\aa$ (for $\bb$ the proof is trivial) are as follows. We perform the computations by writing the $R$–matrix in terms of two exponentials $R=\exp\{-\bb \otimes W\}\exp\{W\otimes \bb\}$, where $W\equiv \xx \aa
+ \ym \ap + \yp \am$. We note that &&{W}(){- W}&&=1+ 1 -(-)(1-e\^[-]{}) + (/) (1-e\^[-]{})&&= \_0() + (/\^2) (1-e\^[-]{}) (1-e\^[-]{}).\[xa\] Since the second term of (\[xa\]) is central, we compute: &&{- W }\_0(){W }&&=e\^[-]{}+ 1 +(1-e\^[-]{}) (-) - (/) (1-e\^[-]{})&&= () - (/\^2) (1-e\^[-]{}) (1-e\^[-]{}).\[xb\] From these expressions $R\,\Delta(\ap)\,R^{-1}=
\sigma \circ\Delta(\ap)$ is easily derived. The proof for $\am$ is rather similar, and for the generator $\aa$ we shall have: &&{W}(å){- W}&&=1å+ å1 -(/\^2) { ()(1-e\^) +(1-e\^[-]{})}&& +(/\^2) { (-)(1-e\^) -(1-e\^[-]{})} &&=\_0(å) +(/\^3) {(1-e\^) (1-e\^) -(1-e\^[-]{}) (1-e\^[-]{})} .&& \[xc\] Now we compute &&{- W }\_0(å){W }&&=\_0(å)+(/) (1-e\^[-]{})+(/) (1-e\^)&&+(/) {(1-e\^) () - (1-e\^[-]{}) (-)}&&+(/\^2) (2-e\^-e\^[-]{})&&=(å) -(/\^3) {(1-e\^) (1-e\^) -(1-e\^[-]{}) (1-e\^[-]{})}, && \[xd\] to obtain again $R\,\Delta(\aa)\,R^{-1}=
\sigma \circ\Delta(\aa)$.
[[**Table I.**]{} Coboundary oscillator Lie bialgebras.]{}
--------------- --------------------------------------------- --------------------------------------------------------------
$r$
$\delta(\aa)$ $\xp\, \aa\wedge\ap - $\xp\, \aa\wedge\ap -
\yp\, \am\wedge \bb +\ym\, \ap\wedge\bb$ (\xx^2/\xp)\, \am\wedge \bb +\ym\, \ap\wedge\bb$
$\delta(\ap)$ $\quad 0$ $\quad 0$
$\delta(\am)$ $\xp\, ( \am\wedge \ap +\aa\wedge\bb) $\xp\, ( \am\wedge \ap +\aa\wedge\bb ) +2\xx\, \am\wedge\bb$
+2\xx\, \am\wedge\bb$
$\delta(\bb)$ $\quad 0$ $\quad 0$
$r$
$\delta(\aa)$ $-\xm\, \aa\wedge\am + \ym\, \ap\wedge\bb - $-\xm\, \aa\wedge\am + (\xx^2/\xm)\, \ap\wedge\bb -
\yp\, \am\wedge \bb$ \yp\, \am\wedge \bb$
$\delta(\ap)$ $-\xm\, ( \ap\wedge \am + \aa\wedge\bb) $-\xm\, (\ap\wedge \am +\aa\wedge\bb )
-2\xx\, \ap\wedge\bb$ -2\xx\, \ap\wedge\bb$
$\delta(\am)$ $\quad 0$ $\quad 0$
$\delta(\bb)$ $\quad 0$ $\quad 0$
$r$
$\delta(\aa)$ $\ym\, \ap\wedge\bb -\yp\, \am\wedge\bb$ $\ym\, \ap\wedge\bb -\yp\, \am\wedge\bb$
$\delta(\ap)$ $-(\xx+\yy)\, \ap\wedge\bb$ $-\xx\, \ap\wedge\bb$
$\delta(\am)$ $(\xx-\yy)\, \am\wedge\bb$ $\xx\,
\am\wedge\bb$
$\delta(\bb)$ $\quad 0$ $\quad 0$
--------------- --------------------------------------------- --------------------------------------------------------------
[[**Table II.**]{} Poisson–Lie brackets on the oscillator group.]{}
----------------- ------------------------------------------ -------------------------------------------
$\{\aaa,\aap\}$ $\xp\, (e^\aaa -1)$ $\xp\, (e^\aaa -1)$
$\{\aaa,\aam\}$ $\quad 0$ $\quad 0$
$\{\aam,\aap\}$ $\xp\, \aam$ $\xp\, \aam$
$\{\aaa,\bbb\}$ $\xp\, \aam$ $\xp\, \aam$
$\{\aap,\bbb\}$ $\xp\,\aam\aap +\ym\, (e^\aaa -1)$ $\xp\,\aam\aap +\ym\, (e^\aaa -1)$
$\{\aam,\bbb\}$ $-\xp\, a_-^2 +2\xx\, \aam + $-\xp\, a_-^2 +2\xx\, \aam +
\yp\, (e^{-\aaa} -1)$ (\xx^2/\xp)\, (e^{-\aaa} -1)$
$\{\aaa,\aap\}$ $\quad 0$ $\quad 0$
$\{\aaa,\aam\}$ $\xm\, (e^{-\aaa} -1)$ $\xm\, (e^{-\aaa} -1)$
$\{\aam,\aap\}$ $\xm\, \aap$ $\xm\, \aap$
$\{\aaa,\bbb\}$ $-\xm\, \aap e^{-\aaa}$ $-\xm\, \aap e^{-\aaa}$
$\{\aap,\bbb\}$ $-2\xx\, \aap +\ym\, (e^\aaa -1)$ $-2\xx\, \aap +(\xx^2/\xm)\, (e^\aaa -1)$
$\{\aam,\bbb\}$ $\yp\, (e^{-\aaa} -1)$ $\yp\, (e^{-\aaa} -1)$
$\{\aaa,\aap\}$ $\quad 0$ $\quad 0$
$\{\aaa,\aam\}$ $\quad 0$ $\quad 0$
$\{\aam,\aap\}$ $\quad 0$ $\quad 0$
$\{\aaa,\bbb\}$ $\quad 0$ $\quad 0$
$\{\aap,\bbb\}$ $-(\xx+\yy)\, \aap +\ym\, (e^\aaa -1)$ $-\xx\, \aap +\ym\, (e^\aaa -1)$
$\{\aam,\bbb\}$ $(\xx-\yy)\, \aam +\yp\, (e^{-\aaa} -1)$ $
\xx\, \aam +\yp\, (e^{-\aaa} -1)$
----------------- ------------------------------------------ -------------------------------------------
[[**Table III.**]{} Coproducts for QUEA of the oscillator algebra.]{}
[|ll|]{}\
: $U_{\xp,\xx,\ym,\yp}^{(\Ipp\, s)}(\osc)$ &($\xp\ne0$ and $ \xp\yp-\xx^2\ne 0$)\
\
\
\
: $U_{\xp,\xx,\ym}^{(\Ipp\,n)}(\osc)$&($\xp\ne0$)\
\
\
\
\
\
\
: $U_{\xm,\xx,\ym,\yp}^{(\Imm\,s)}(\osc)$ &($\xm\ne0$ and $ \xm\ym-\xx^2\ne 0$)\
\
\
\
: $U_{\xm,\xx,\yp}^{(\Imm\,n)}(\osc)$&($\xm\ne0$)\
\
\
\
\
\
\
: $U_{\xx,\yy,\ym,\yp}^{(\II\, s)}(\osc)$ &($\yy\ne0$)\
\
\
\
\
\
: $U_{\xx,\ym,\yp}^{(II\,n)}(\osc)$ &\
\
\
\
\
[40]{}
Greenberg O W 1991 [*Phys. Rev. D*]{} [**43**]{} 4111
Avancini S S and Krein G 1995 [*J. Phys. A: Math. Gen.*]{} [**28**]{} 685
Spiridonov V 1992 [*Phys. Rev. Lett.*]{} [**69**]{} 398
Bonatsos D, Daskaloyannis and Kokkotas K 1993 [*Phys. Rev. A*]{} [**48**]{} R3407
Bonatsos D, Daskaloyannis and Kokkotas K 1994 [*Phys. Rev. A*]{} [**50**]{} 3700
Floreanini R, Le Tourneaux J and Vinet L 1995 [*J. Phys. A: Math. Gen.*]{} [**28**]{} L287
Celeghini E, Rasetti M and Vitiello G 1995 [*J. Phys. A: Math. Gen.*]{} [**28**]{} L239
Gómez C and Sierra G 1993 [*J. Math. Phys.*]{} [**34**]{} 2119
Lyakhovsky V and Mudrov A 1992 [*J. Phys. A: Math. Gen.*]{} [**25**]{} L1139
Ballesteros A, Herranz F J, Pereña C M, del Olmo M A and Santander M 1995 [*J. Phys. A: Math. Gen.*]{} [**28**]{} 7113
Celeghini E, Giachetti R, Sorace E and Tarlini M 1991 [*J. Math. Phys.*]{} [**32**]{} 1155
Hussin V, Lauzon A and Rideau G [*Oscillator quantum groups from $R$–matrix method*]{} [*J. Phys. A: Math. Gen.*]{} to appear
Reshetikhin N Y, Takhtadzhyan L A and Faddeev L D 1990 [*Leningrad Math. J.*]{} [**1**]{} 193
Ballesteros A, Celeghini E, Giachetti R, Sorace E and Tarlini M 1993 [*J. Phys. A: Math. Gen.*]{} [**26**]{} 7495
Hussin V, Lauzon A and Rideau G 1994 [*Lett. Math. Phys.*]{} [**31**]{} 159
Tjin T 1992 [*Int. J. Mod. Phys. A*]{} [**7**]{} 6175
Ohn C 1992 [*Lett. Math. Phys.*]{} [**25**]{} 85
Ballesteros A, Herranz F J, del Olmo M A, and Santander M 1995 [*J. Phys. A: Math. Gen.*]{} [**28**]{} 941
Ballesteros A, Gromov N A, Herranz F J, del Olmo M A, and Santander M 1995 [*J. Math. Phys.*]{} [**36**]{} 5916
|
---
abstract: |
------------------------------------------------------------------------
We analyze the deuteron bound state through the One Pion Exchange Potential. We pay attention to the short distance peculiar singularity structure of the bound state wave functions in coordinate space and the elimination of short distance ambiguities by selecting the regular solution at the origin. We determine the so far elusive amplitude of the converging exponential solutions at the origin. All bound state deuteron properties can then be uniquely deduced from the deuteron binding energy, the pion-nucleon coupling constant and pion mass. This generates correlations among deuteron properties. Scattering phase shifts and low energy parameters in the $^3S_1-{}^3D_1 $ channel are constructed by requiring orthogonality of the positive energy states to the deuteron bound state, yielding an energy independent combination of boundary conditions. We also analyze from the viewpoint of short distance boundary conditions the weak binding regime on the light of long distance perturbation theory and discuss the approach to the chiral limit.
author:
- 'M. Pavón Valderrama'
- 'E. Ruiz Arriola'
title: Renormalization of the Deuteron with One Pion Exchange
---
Introduction
============
Pion dynamics plays a dominant role in the low energy structure of the Nucleon-Nucleon interaction, and in particular in the description of light nuclei like the deuteron [@Ericson:1988gk]. The long distance part of the interaction is given by one, two and higher pion exchanges and the fact that the deuteron is a weakly bound state suggests that many of its properties can indeed be explained in terms of these dynamical degrees of freedom in a model independent way and regardless on the less known short distance interaction. Glendenning and Kramer [@Glendenning62] in the early sixties recognized clear correlations between several deuteron observables generated by truncating the One Pion Exchange (OPE) potential at a distance $R=0.4915 {\rm fm}$ and assuming a hard core inside. Tight constraints on deuteron observables were established by Klarsfeld, Martorell and Sprung [@Klarsfeld:1980; @Klarsfeld:1984es] by integrating the deuteron wave function from infinity down to a cut-off radius using the OPE potential and rigorous inequalities. An accurate determination of the D/S asymptotic ratio was made by Ericson and Rosa-Clot [@Ericson:1981tn; @Ericson:1982ei] based on assuming the OPE correlation between the S and D wave functions and taking realistic potential models to describe the S wave function. (for a review on these developments see e.g Ref. [@Ericson:1985]). Friar [*et al*]{}. use a multipole form factor [@Friar84] whereas Ballot [*et al.*]{} used separate monopole forms factor for the central and tensor part of the OPE potential mimicking the finite size of nucleons [@Ballot:1989jk; @Ballot:1992sc]. Along a similar line of investigation Sprung [*et al.*]{} used a square well potential [@Martorell94] for the central component and a vanishing potential for the tensor component.
Within the effective field theory (EFT) approach to nuclear physics proposed by Weinberg [@Weinberg:rz] the situation was revisited from a somewhat different perspective since the OPE potential appears as the lowest order of a perturbative hierarchy based on chiral symmetry [@Ordonez:1995rz] (for a review see e.g. Ref. [@Bedaque:2002mn]), and short distance ambiguities could be eliminated by the renormalization program if the auxiliary regulator is removed from the theory at the end of the calculation. This cut-off independence should occur at [*any level*]{} of approximation, no matter how many pions are exchanged. At long distances, renormalization group methods suggest that one is close to an infrared fixed point [@Birse:1998dk]. The renormalization procedure can be explicitly and analytically carried out within perturbation theory [@Kaplan:1998sz]. However, these nice features become a non trivial numerical problem beyond perturbation theory motivating the use of truncation cut-off schemes. The work of Ref. [@Park:1998cu] uses a gaussian cut-off in coordinate space to regulate the contact delta interaction, Ref. [@Frederico:1999ps] proposes the use of a subtraction method in momentum space regulating the central part, Ref. [@Epelbaum:1999dj] uses a sharp momentum cut-off and in Ref. [@Phillips:1999am] it was proposed to use a finite short distance cut-off, whereas Ref. [@Entem:2001cg] puts exponentially suppressed regulators in momentum space. It should be mentioned that in all cases the corresponding coordinate/momentum space cut-off parameter $a$/$\Lambda$ is uncomfortably large/small from the viewpoint of renormalization theory. Typically, one has $a \sim 1.4
{\rm fm}$ (see e.g. Ref. [@Phillips:1999am]) and $\Lambda=600 {\rm
MeV}$ [@Park:1998cu] respectively. So, it is not obvious that according to the basic principles of EFT the short distance ambiguities are, as one might expect, indeed under control. Moreover, the existence of a well behaved finite renormalized limit is never guaranteed [*a priori*]{} and one relies mainly on numerics. Actually, the fact that the results on deuteron observables look rather similar, regardless on the particular way how the potential is modeled at short distances, proves that the long distance pion dynamics dominates the physics confirming the findings of Glendenning and Kramer [@Glendenning62] more than 40 years ago, but does not resolve the mathematical problem whether the OPE potential can make unambiguous predictions regardless of any short distance physical scale.
The OPE potential is local in coordinate space where the problem is naturally formulated by the standard Schrödinger framework. Moreover, it is singular at the origin and giving boundary conditions at that point is not a well defined procedure for uniquely determining the solution [@Case:1950] (for a comprehensive review in the one channel case see e.g. Ref. [@Frank:1971] ). There is the added difficulty that we have two coupled second order differential equations. In the deuteron channel one has four independent solutions, which according to their singularity structure correspond to either two regular and two irregular solutions at infinity or three regular and one irregular solution at the origin. The normalizability condition of the deuteron wave functions eliminates all constants for a given deuteron binding energy, which instead of being predicted has to be treated as an independent parameter. One of the advantages of the coordinate space treatment of renormalization is that it can directly be extended to other singular cases such as the Two Pion Exchange (TPE) potential [@Pavon_TPE] which is also finite everywhere except the origin. In contrast, momentum space treatments require an extra regularization of the potential besides the standard cut-off regularization of the Lippmann-Schwinger equation.
The authors of Ref. [@Martorell94] found a discrete sequence of equivalent short distance cut-off radii having almost the same deuteron properties. In their analysis of the problem one regular solution at the origin with a converging exponential behavior, $\exp
(-4(2 R/r)^{\frac12})$ with $ R \sim 1 {\rm fm}$, was discarded on numerical grounds. The same result was also implicitly used in Ref. [@Beane:2001bc] and large $N_c$ arguments in favor of it were raised in Ref. [@Beane:2002ab]. This extra condition would actually [*predict* ]{} the deuteron binding energy from the OPE potential. As we will show in this paper, the converging exponential is non-vanishing although rather elusive because its contribution to the deuteron wave function only becomes sizeable at relatively large distances and accurate numerical work must be done to pin down its value with a certain degree of confidence.
In the present work we show that there is no need to truncate the OPE potential on a physical scale to produce unique and cut-off independent predictions for deuteron properties and scattering observables in terms of the OPE potential parameters and the deuteron binding energy. These might then legitimately be called OPE model independent predictions and paves the way for a systematic investigation on the case where more pions are exchanged and other effects are taken into account [@Pavon_TPE]. After presenting the basic notation in Sect. \[sec:ope\], we discuss in Sect. \[sec:short0\] the regular solutions at the origin and establish that the limit when the regulator is removed is finite. For numerical purposes it is useful to define [*some*]{} short distance regulator as an auxiliary tool. In Sect. \[sec:cut-off\] we use six different regulators based on boundary conditions of the wave function and check for stability to high precision for all regulators. This procedure generates correlations among deuteron observables if the deuteron binding energy is varied as a free and independent parameter as we do in Sect. \[sec:OPE-corr\]. A particularly interesting situation is provided by the weak binding limit, which can be taken with fixed OPE potential parameters. In such a case the long distance behavior should dominate and one might expect perturbative methods to apply and be compared to the exact OPE calculation. The details of the perturbative calculation are postponed to Appendix \[sec:pert\] where we present a coordinate space version of the method, in consonance with the exact treatment. A detailed comparison shows that the perturbative argument is too naive and would only hold in the weak coupling regime as well, due to the appearance of non-analytical contributions in the $\pi NN $ coupling constant. Mathematically, we show that it is not possible to go beyond first order since the coefficients of the expansion diverge. Numerically, the disagreement at first order is typically on the $30 \%$ level for physical values of the OPE parameters at zero binding. After Ref. [@Bulgac:1997ji] the chiral limit in nuclear physics has attracted considerable attention in recent works [@Beane:2002xf; @Beane:2002vs; @Epelbaum:2002gb] and also the limit of heavy pions in connection with lattice QCD calculations, where the pion mass is still far from its physical value. We study in Sect. \[sec:m\_dependence\] the correlations among those observables if the pion mass is varied away from its physical value by studying a suitable extension of the Feynman-Helmann theorem. Another remarkable property of the OPE potential which we deal with in Sect. \[sec:scattering\] is that low energy parameters as well as the scattering phase-shifts can be uniquely determined from the OPE potential parameters and the deuteron binding, due to orthogonality constraints of the bound state and scattering states. In Sect. \[sec:short\] the determination of the non-vanishing coefficient of the converging exponential at the origin is carried out by a short distance expansion to eighth order of the OPE deuteron wave functions. In Sect. \[sec:concl\] we come to the conclusions.
One of the surprising results in the OPE description of the deuteron has to do with the small asymptotic ratio between the D and S waves, $
w(\infty) /u(\infty) = \eta = 0.0256$ coming from a large ratio at short distances of order unity, $ w(0) / u(0) = 1/\sqrt{2} = 0.707
$. Although this feature is specific to the OPE potential it is somewhat a bit outside the main topic of this work. So we relegate this issue to Appendix \[sec:local\_rot\] where we show how this can be easily understood if a local rotation of the deuteron wave functions diagonalizing the coupled channel potential is carried out. Obviously, such a transformation cannot simultaneously diagonalize the kinetic terms, but the residual mixing is related to the derivative of a local mixing angle which numerically turns out to be a slowly varying function. Using this as a starting approximation we can determine in a perturbative fashion the asymptotic D/S ratio yielding the exact OPE value with a $1\%$ accuracy.
Bound state equations and their solutions
=========================================
The OPE deuteron Equations {#sec:ope}
--------------------------
The Deuteron coupled channel $^3S_1 - {}^3D_1 $ set of equations read $$\begin{aligned}
-u '' (r) + U_s (r) u (r) + U_{sd} (r) w (r) &=& -\gamma^2 u
(r) \, ,\nonumber \\ \\ -w '' (r) + U_{sd} (r) u (r) + \left[U_{d} (r) +
\frac{6}{r^2} \right] w (r) &=& -\gamma^2 w (r) \, , \nonumber \\
\label{eq:sch_coupled} \end{aligned}$$ together with the asymptotic conditions at infinity $$\begin{aligned}
u (r) &\to & A_S e^{-\gamma r} \, , \nonumber \\ w (r) & \to & A_D
e^{-\gamma r} \left( 1 + \frac{3}{\gamma r} + \frac{3}{(\gamma r)^2}
\right) \, ,
\label{eq:bcinfty_coupled} \end{aligned}$$ where $ \gamma = \sqrt{M B} $ is the deuteron wave number, $A_S$ is the normalization factor and the asymptotic D/S ratio parameter is defined by $\eta=A_D/A_S$. The $^3S_1-{}^3D_1 $ coupled channel potential is given by $$\begin{aligned}
U_s = U_c \, , \qquad U_{sd} = 2 \sqrt{2} U_T \, , \qquad U_d = U_C -
2 U_T \, ,\end{aligned}$$ where the OPE reduced potential ($U=2 \mu V $) is given for $r > 0 $ by $$\begin{aligned}
U_C &=& -\frac{m M g_A^2 }{16 \pi f_\pi^2 } \frac{e^{-m r
}}{r} \, \\ U_T &=& -\frac{m^2 M g_A^2 }{16 \pi f_\pi^2 }
\frac{e^{-m r }}{r} \left( 1 + \frac3{m r}+ \frac3{(m
r)^2} \right) \, , \end{aligned}$$ where $m$ is the pion mass, $M=2 \mu_{np} = 2 M_n M_p /(M_n+M_p) $ twice the reduced proton-neutron mass, $g_A$ the axial nucleon coupling constant and $f_\pi$ the pion weak decay constant. Note that we assume this potential to be valid for any strictly positive distance, $ r\neq 0$, so the limit $r \to 0^+ $ will be carefully taken, [*without*]{} subtracting any contribution in the potential.
It is convenient to define the length scale $$\begin{aligned}
R = \frac{3 g_A^2 M }{32 \pi f_\pi^2}
\label{eq:R_def} \end{aligned}$$ which value is around 1 fm. For numerical calculations we take $f_\pi
=92.4 {\rm MeV}$, $M = 938.918 {\rm MeV}$, $ g_A =1.29 $ and hence $R
= 1.07764 {\rm fm}$. Using the Goldberger-Treiman relation, $g_{\pi
NN}= g_A M /f_\pi $, the corresponding pion nucleon coupling constant is $ g_{\pi NN}=13.1083$ according to a phase shift analysis of NN scattering [@deSwart:1997ep]. Nevertheless, after the latest determinations from the GMO sum rule [@Ericson:2000md] we will also take the value $ g_{\pi NN} =13.3158$. As we will see this variation at the $5 \% $ level dominates the uncertainties in the OPE calculations.
The short distance regular solutions {#sec:short0}
------------------------------------
We look for normalized functions of the Eqs. (\[eq:sch\_coupled\]), $$\begin{aligned}
1= \int_0^\infty ( u(r)^2 + w(r)^2 ) dr \, , \end{aligned}$$ from which $A_S$ can be determined. The normalization at infinity is guaranteed due to the asymptotic conditions, Eq. (\[eq:bcinfty\_coupled\]). However, the coupled channel potential becomes singular at short distances, since $ U_T \to - 2R /r^3
$. Keeping only this term in Eqs. (\[eq:sch\_coupled\]) one can decouple the equations through the unitary transformation [@Martorell94] $$\begin{aligned}
u_A (r) &=& \sqrt{\frac23} u (r) + \frac{1}{\sqrt{3}}w (r) \, , \nonumber \\
u_R (r) &=& -\frac1{\sqrt{3}}u (r) + \sqrt{\frac23} w (r) \, ,
\label{eq:eigenvectors}\end{aligned}$$ yielding an attractive singular potential $U_A \to -4 R/r^3 $ for $u_A$ and $U_R \to 8 R/r^3 $ for $u_R$. Any solution obtained by integrating from infinity with the Eq. (\[eq:bcinfty\_coupled\]) down to the origin has the asymptotic short distance behavior [^1], $$\begin{aligned}
u_R (r) &\to & \left(\frac{r}{R}\right)^{3/4} \left[ C_{1R} e^{+ 4
\sqrt{2} \sqrt{\frac{ R}{r}}} + C_{2R} e^{- 4 \sqrt{2} \sqrt{\frac{
R}{r}}} \right] \, , \nonumber \\ \\ u_A (r) &\to &
\left(\frac{r}{R}\right)^{3/4} \left[ C_{1A} e^{- 4 i \sqrt{\frac{
R}{r}}} + C_{2A} e^{ 4 i\sqrt{\frac{ R}{r}}} \right] \, . \nonumber
\label{eq:short_bc}\end{aligned}$$ The constants $C_{1R}$, $C_{2R}$, $C_{1A}$ and $C_{2A}$ depend on both $\gamma $ and $\eta$ and the OPE potential parameters, $g_{\pi
NN} $ and $m$. Note that the leading short distance $r$ dependence does not involve the pion mass and the deuteron wave number. Higher order corrections to these solutions can be computed systematically to high orders and are presented below in Sect. \[sec:short\].
The regular solution at infinity contains the normalization constant $A_S$, which is customarily set to one for computational purposes, the deuteron wave number $ \gamma $ and the asymptotic D/S ratio parameter $\eta$. The normalizability of the wave function at the origin requires $$\begin{aligned}
C_{1R} ( \gamma , \eta ) =0 \, ,
\label{eq:c1r}\end{aligned}$$ which is a relation between $\eta$ and $\gamma$. The other remaining constants are then completely fixed. This means that for the OPE potential, the deuteron binding energy can be used as an independent parameter. Thus, one has three independent variables, $ \gamma$, the coupling constant with length scale dimension $R $ (or equivalently $ g_{\pi NN }$) and the pion mass $ m$. Obviously, this suggests integrating in from infinity and determining $\eta$ from the regularity condition at the origin (\[eq:c1r\]).
To analyze whether some additional condition arises let us check the selfadjointness of the coupled channel Hamiltonian. The flux at a point $r$ is given by $$\begin{aligned}
i J (r) &=& u^* (r)' u(r) - u^* (r) u'(r) \nonumber \\
&+& w^* (r)' w(r) - w^* (r) w'(r) \, , \end{aligned}$$ so that current probability conservation at the origin implies $$\begin{aligned}
| C_{1A}|^2-|C_{2A}|^2 = 2\sqrt{2} i \left(
C_{1R}^* C_{2R} - C_{2R}^* C_{1R} \right) \, . \end{aligned}$$ Thus, if we set $ C_{1R}=0$ there is no condition on $ C_{2R}$ and one has $ C_{1A} = C_A e^{i \varphi} $ and $ C_{2A} = C_A e^{-i \varphi} $ with $C_A $ and $ \varphi$ real. So, we have three constants, $ C_{2R}
(\gamma) $, $C_A (\gamma) $ and $\varphi(\gamma)$, characterizing the normalizable solutions at short distances for a given value of the deuteron wave number $\gamma$, $$\begin{aligned}
u_R (r) &\to & C_{R} (\gamma) \left(\frac{r}{R}\right)^{3/4} e^{- 4
\sqrt{2} \sqrt{\frac{ R}{r}}} \, , \nonumber \\
u_A (r) &\to & C_{A}(\gamma) \left(\frac{r}{R}\right)^{3/4} \sin \left[ 4
\sqrt{\frac{ R}{r}} + \varphi(\gamma) \right] \, .
\label{eq:short_bc_reg}\end{aligned}$$ Actually, if we have any other state, say a scattering state with positive energy, unitarity (i.e. orthogonality) requires that the constant $\varphi(k) $ coincides with the bound state phase $\varphi(\gamma)$. We will come back to this issue later when discussing low energy parameters and scattering solutions in Sect. \[sec:scattering\]. It is natural to expect that some combination of short distance constants is independent on the OPE potential parameters as they encode short distance physics. In Sect. \[sec:m\_dependence\] we establish, by demanding the standard Feynman-Hellmann theorem, that specifically the short distance phase $\varphi $ does not depend on the OPE potential parameters. In Sect. \[sec:short\] we determine the values of the three constants characterizing the three regular solutions by a detailed short distance analysis of the OPE deuteron wave functions.
Note that any additional condition would actually [*predict*]{} both $\gamma$ and $\eta$ from $m$ and $R$. This contradicts the claim of Ref. [@Martorell94] that $C_{2R} =0$, a conclusion implicitly used in Ref [@Beane:2001bc] and supported by the large $N_c$ argument of Ref. [@Beane:2002ab]. On the other hand, if one takes the experimental values of $\eta$ and $\gamma$ as done in Ref. [@Phillips:1999am] one obtains both non vanishing $C_{1R}$ and $C_{2R}$ i.e., the irregular non-normalizable solution, unless a short distance cut-off, $ R > 0.8 {\rm fm}$, is introduced as a physical scale and not as an auxiliary removable regulator.
Regularization with boundary conditions {#sec:cut-off}
---------------------------------------
Ideally, one would integrate in the large asymptotic solutions, Eq. (\[eq:bcinfty\_coupled\]), and match the short distance behavior of Eq. (\[eq:short\_bc\]) imposing the regularity condition (\[eq:c1r\]). In practice, however, the converging exponential at the origin is rather elusive since integrated-in solutions quickly run into the diverging exponentials due to round-off errors for $ r
\sim 0.05 {\rm fm}$ and dominate over the converging exponential. The reason has to do with the fact that the natural scale where both exponentials are comparable is rather large $r = 4 \sqrt{2} R \sim 6
{\rm fm}$, but in that region the lowest order short distance approximation does not hold.
Instead, we will also try putting several short distance boundary conditions corresponding to the choice of regular solutions at the origin, $$\begin{aligned}
u(a) &=& 0 \qquad ({\rm BC1}) \, , \nonumber \\
u'(a) &=& 0 \qquad ({\rm BC2}) \, , \nonumber \\
w(a) &=& 0 \qquad ({\rm BC3}) \, , \nonumber \\
w'(a) &=& 0 \qquad ({\rm BC4}) \, , \nonumber \\
u(a) - \sqrt{2} w(a) &=& 0 \qquad ({\rm BC5}) \, , \nonumber \\
u'(a) - \sqrt{2} w'(a) &=& 0 \qquad ({\rm BC6}) \, , \nonumber \\
\label{eq:bc_a}\end{aligned}$$ The advantage of using this kind of short distance cut-offs based on a boundary condition is that there is only a single scale in the problem as one naturally expects, and that one never needs to declare what is the wave function below the boundary radius. Putting a square well potential as a counter-term [@Beane:2001bc] with depth $U_0 $ appears natural from standard perturbative experience but needs specification of a further length scale, $1/\sqrt{U_0}$, and moreover, generates multi-valuation ambiguities [@PavonValderrama:2003np; @PavonValderrama:2004nb].
It is convenient to use the superposition principle of boundary conditions to write $$\begin{aligned}
u (r) &=& u_S (r) + \eta \, u_D (r) \nonumber \\ w (r) &=& w_S (r) + \eta\,
w_D (r) \, , \end{aligned}$$ where $(u_S,w_S)$ and $(u_D,w_D)$ correspond to the boundary conditions at infinity, Eq. (\[eq:bcinfty\_coupled\]) with $A_S=1$ and $A_D=0$ and with $A_S=0$ and $A_D=1$ respectively. Thus, at the boundary we can impose any of the conditions by just eliminating $\eta$ [^2]. The resulting $\eta $ value obtained by all these boundary conditions is presented in Fig. \[fig:eta\_a\]. Actually, we see that the boundary condition $
u'(a) - \sqrt{2} w'(a) $ is about the smoothest condition we can think of, since the $u_R $ combination goes to zero at small distances, its derivative, $u_R' $ also goes to zero, although a bit less faster since $u_R' / u_R \sim 1/r^{3/2}$. We see that all determinations of $\eta$ based on any of the proposed cut-offs yield the same value with great accuracy at cut-off radii below $0.2 {\rm fm} $. This is somewhat fortunate since arithmetic precision is outraged typically for $r< 0.06 {\rm fm}$. Obviously, any short distance cut-off generates finite cut-off effects in the wave functions for distances close to the cut-off radius. In Sect. \[sec:short\] we analyze this problem by matching solutions of the form of Eq. (\[eq:short\_bc\_reg\]) to the integrated in numerical solutions and find that for many practical purposes these finite cut-off effects are negligible. Thus, we will base most of our results on the “smoothest” condition BC6 of Eq. (\[eq:bc\_a\]).
The resulting deuteron wave functions $u$ and $w$ obtained by integrating in from infinity to the origin the OPE potential are plotted in Fig. \[fig:irregular+regular\] where the irregular solutions are obtained with the experimental $D/S$ ratio $\eta_d =
0.0256 $ and the regular ones with the OPE $D/S$ ratio $\eta_{\rm
OPE} = 0.026333$. For comparison we also plot the NijmII deuteron wave functions. We emphasize that the value of $\eta$ is a direct consequence of taking the OPE down to the origin seriously.
Deuteron observables
--------------------
Once the solutions are known we can determine several observables of interest. The matter radius reads, $$\begin{aligned}
r_m^2 = \frac14 \langle r^2 \rangle = \frac14 \int_0^\infty r^2 ( u(r)^2 +
w(r)^2 ) dr \end{aligned}$$ while potential contribution to the quadrupole moment (without meson exchange currents) $$\begin{aligned}
Q_d = \frac1{20} \int_0^\infty r^2 w(r) ( 2\sqrt{2} u(r)-w(r) ) dr \end{aligned}$$ An important observable is the deuteron inverse radius $$\begin{aligned}
\langle r^{-1} \rangle = \int_0^\infty dr \frac{u(r)^2 + w(r)^2}{r} \end{aligned}$$ which appears in low energy pion-deuteron scattering. Finally, the $D$-state probability is given by $$\begin{aligned}
P_D = \int_0^\infty w(r)^2 dr \end{aligned}$$ Both $P_D $ and $\langle r^{-1} \rangle $ are sensitive to the intermediate distance region around $2 {\rm fm}$ whereas $Q_d $ and $r_m$ get their contribution from larger distances $\sim 4 {\rm fm} $.
The results for the asymptotic S-wave normalization $A_S$, the matter radius $r_m$, the quadrupole moment, $Q_d$, and the D-state probability, $P_D$ are presented in Table \[tab:table\_pert\]. The errors in the numerical calculation have been assessed by varying the short distance cut-off in the range $a=0.1-0.2 {\rm fm}$ (in momentum space that would naively correspond to take $ \Lambda =1/a = 2-4 {\rm
GeV}$). As we see, the cut-off uncertainty is smaller than the one induced by variations at the $2\%$ level in the $g_{\pi NN}$ coupling constant in the range between the lowest value ($\sim$ 13.1) obtained by a fit to NN phase-shifts [@deSwart:1997ep] and the highest recent value ($\sim$ 13.3) determined from the GMO sum rule [@Ericson:2000md]. Equivalently, this uncertainty corresponds to take $R=1.0776 {\rm fm} $ and $R = 1.1108 {\rm fm}$ respectively. Our results are generally speaking in agreement with previous determinations where different sorts of cut-off methods have also been implemented.
Discussion
----------
At this point it may prove useful to ponder on the previous results from a wider perspective. Let us remind that the basic assumption of an EFT is that the study of long wave length phenomena such as low energy scattering or weakly bound systems do not require a detailed knowledge of short distance physics. This general and widely accepted principle requires some qualification because attractive and repulsive singular potentials behave quite differently in this respect. Singular attractive potentials, $\sim 1/r^n $, generate wave functions vanishing as a power law, $r^{n/4} \sin ( r^{-n/2+1} + \varphi) $, and which need a mixed boundary condition to specify the short distance phase $\varphi$. Thus, short distance details become less important, regardless on the value of $\varphi$. On the contrary, for singular repulsive potentials the wave functions behave as $ r^{n/4} e^{\pm r^{-n/2+1}}$ and only for the regular solution short distance details become irrelevant. In the OPE potential, it is precisely the repulsive short distance OPE component which requires a fine tuning of the solutions and eliminates one [*a priori*]{} independent parameter like, e.g., the asymptotic D/S ratio $\eta$. As we see, if $\eta $ is treated as an independent variable the short distance behavior of the deuteron wave functions precludes the definition of a normalizable state due to the onset of the [*irregular*]{} solution. This short distance insensitivity at low energies could only be implemented by keeping the experimental $\eta$ value and [*ignoring*]{} OPE physics below some scale. The lower limit established in Ref. [@Phillips:1999am] to obtain a normalizable state was $a\sim 1.3 {\rm fm}$ for the OPE potential. This obviously requires some extension of the wave function below that scale and the pretended model independence becomes a bit obscured. Our point is that the short distance insensitivity materializes automatically for the [*regular*]{} OPE deuteron wave functions since they vanish at the origin.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\gamma ({\rm fm}^{-1})$ $\eta$ $A_S ( {\rm fm}^{-1/2}) $ $r_m ({\rm fm})$ $Q_d ( {\rm fm}^2) $ $P_D $ $\langle r^{-1} $ \alpha_0 ({\rm fm}) $ $\alpha_{02} ({\rm fm}^3) $ $ $r_0 ({\rm fm} ) $
\rangle $ \alpha_2 ({\rm fm}^5) $
--------------- -------------------------- ----------- --------------------------- ------------------ ---------------------- ---------- ----------------- ------------------------- ----------------------------- ------------------------- --------------------
[Short]{} Input 0 0.6806 1.5265 0 0% $\infty $ 4.3177 0 0 0
[OPE]{}(pert) Input 0.051 0.7373 1.6429 0.4555 0% $\infty $ 4.6089 2.5365 0 0.4831
[OPE]{} Input 0.02633 0.8681(1) 1.9351(5) 0.2762(1) 7.88(1)% 0.476(3) 5.335(1) 1.673(1) 6.169(1) 1.638(1)
[OPE]{}$^*$ Input 0.02687 0.8718(2) 1.9429(6) 0.2826(2) 7.42(1)% 0.471(3) 5.353(1) 1.715(1) 6.4001(1) 1.663(1)
NijmII Input 0.02521 0.8845(8) 1.9675 0.2707 5.635% 0.4502 5.418 1.647 6.505 1.753
Reid93 Input 0.02514 0.8845(8) 1.9686 0.2703 5.699% 0.4515 5.422 1.645 6.453 1.755
Exp. 0.231605 0.0256(4) 0.8846(9) 1.9754(9) 0.2859(3) 5.67(4) 5.419(7) 1.753(8)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
OPE Correlations in deuteron observables {#sec:OPE-corr}
========================================
As we have said, in the OPE potential we can use the deuteron wave number as an input of the calculation on the same footing as $g_{\pi
NN} $ and the pion mass $m$. Then, other observables are predicted. We will study now the dependence of these observables on $\gamma$, $m $ and $R$.
Dependence on the Binding Energy
--------------------------------
\
In Fig. \[fig:eta\[gamma\]\] we show the dependence of the D/S ratio as a function of the deuteron wave number $\gamma$ keeping $m$ and $R$ fixed. In the weak binding limit $\gamma \ll m_\pi $, long distances dominate and the finiteness of the wave function at a point $ r \gg
1/m $ requires $\eta \sim \gamma^2 $. The radius of convergence of such an expansion for the observables is $|\gamma| < m /2 $, since the integrals involve the factor $e^{-(2 \gamma + m) r}$ at large distances, diverging for $ \gamma < -m/2 $. The experimental number is not far from $ \gamma = m/3 $, which is within the domain of analyticity but somewhat close to the convergence radius. So, one expects a slow convergence. As we see in the weak binding limit we have a quadratic behavior $ \eta_{\rm OPE} \sim \gamma^2 $ whereas for stronger binding a linear behavior sets in. It is remarkable that the experimental values in the intermediate regime. On the other hand, in the strong binding case $\gamma \gg m_\pi $, short distances dominate and we must have $ \eta \sim 1/\sqrt{2}$. Numerically we find for the deuteron observables, $$\begin{aligned}
\eta^{\rm OPE} &=& 0.9638 \gamma^2 - 3.46864 \gamma^3+ {\cal O}
(\gamma^4) \\ \frac{A_S^{\rm OPE}}{\sqrt{2 \gamma}} &=& 1 + 1.2455
\gamma -0.4705 \gamma^2 + {\cal O}(\gamma^3) \\
\sqrt{8} \, \gamma \,\, r_m^{\rm OPE}
&=& 1 + 1.2455 \gamma -0.4705 \gamma^2 + {\cal O}
(\gamma^3) \\
Q_d^{\rm OPE} &=& 0.6815 - 3.5437 \gamma + {\cal O} (\gamma^2)\end{aligned}$$ Note that we have the weak binding correlation $$\begin{aligned}
r_m = \frac{A_S}{4\,\gamma^{3/2}} + {\cal O} ( \gamma^3 ) \end{aligned}$$ which is compatible at the $2 \sigma $ confidence level with data; for the experimental value $A_S= 0.8845 (8) $ the value $r_m=1.984(2) $ to be compared with the experimental number $r_m=1.971(6) $. In the weak binding limit one also has the correlation $$\begin{aligned}
\frac{\sqrt{2} \gamma^2 Q_d}{\eta_d} = 1 + {\cal O} (\gamma) \end{aligned}$$ a dependence that one would expect on general grounds by just taking the asymptotic formulas and neglecting the $w(r)^2$ term in the expression for the quadrupole moment. Experimentally this relation is fulfilled with a $15\%$ accuracy. The non-perturbative OPE value is actually closer to potential models.
Comparison with perturbation theory
-----------------------------------
It is instructive to solve the coupled deuteron equations, Eq. (\[eq:sch\_coupled\]) in standard perturbation theory for the fixed energy bound state. One of the reasons is to check the correctness of our non-perturbative calculations in the weak binding regime. Another motivation is to establish contact with the perturbative calculations of Ref. [@Kaplan:1998sz] where dimensional regularization in the power divergence subtraction (PDS) scheme was implemented. Finally, there is the question of quantitatively assessing the validity of such an approximation. We relegate the calculation to Appendix \[sec:pert\]. At first order in perturbation theory one gets in the weak binding limit $$\begin{aligned}
\eta_{\rm pert} &=& 1.5497 \gamma^2 -4.15479 \gamma^3 + {\cal O} ( \gamma^4, R^2) \nonumber \\
\frac{A_{S,{\rm pert}}}{\sqrt{2 \gamma}} &=& 1 - 0.7184 \gamma -
2.7394 \gamma^2 + {\cal O} ( \gamma^3, R^2)\nonumber \\
r_{m,\rm pert} \sqrt{8} \gamma &=& 1 + 0.71843 \gamma - 2.7394
\gamma^2 + {\cal O} ( \gamma^3, R^2) \nonumber \\
Q_{\rm pert} &=& 1.09587 - 5.87576 \gamma + {\cal O} ( \gamma^2, R^2)
\nonumber \\\end{aligned}$$ The nominally ${\cal O} (R^2)$ second order contributions are in fact divergent because the leading order correction to the D-wave component $w(r) $ diverges at the origin (see Appendix \[sec:pert\]). In general terms we find that the exact OPE results are estimated within $ 30\%$ by the first order perturbative calculations of Appendix. \[sec:pert\].
dependence on the pion mass and chiral limit {#sec:m_dependence}
============================================
Recent works [@Bulgac:1997ji; @Beane:2002xf; @Beane:2002vs; @Epelbaum:2002gb] predict the change of the deuteron binding energy as a function of the pion mass by taking the experimental binding energy at the physical value of the pion mass and making the additional assumption short distance physics to be independent on the pion mass. While it is true that the leading short distance $r$ dependence of the deuteron wave functions are independent on the pion mass, the constants $C_A$, $ C_R $ and $ \varphi $ do in principle depend on the three independent parameters $ m$, $g_{\pi NN}
$ and $\gamma$. As we have noted $\gamma $ cannot be predicted for the OPE potential. So the approach pursued in Refs. [@Bulgac:1997ji; @Beane:2002xf; @Beane:2002vs; @Epelbaum:2002gb] is equivalent to integrate in with the physical pion mass and then integrate out fixing some combination of short distance constants with the unphysical pion mass and searching for the appropriate regular solution at infinity. If one makes the pion lighter long distance effects should dominate, and one could just use the OPE potential to estimate the chiral limit as a first approximation. It is thus interesting to analyze the pion mass dependence both explicitly (i.e. varying $m$ in the OPE potential) and implicitly (i.e. taking into account the dependence of the OPE coupling $R$ on the pions mass). We will determine the pertinent combination of short distance constants by demanding the Feynman-Hellmann theorem in the OPE potential.
Explicit pion mass dependence
-----------------------------
To proceed, let us assume that to an infinitesimal change in $m \to m
+ \Delta m $ there corresponds a change both in the deuteron wave number $\gamma \to \gamma + \Delta \gamma $ and in the coupled channel potential matrix $ U (r) \to U(r)+ \Delta U(r) $. We can write a Lagrange identity by varying the equation and its adjoint, yielding for a normalized state $$\begin{aligned}
- \frac{\partial \gamma^2}{\partial m} &=& \langle \Psi_m |
\frac{\partial U }{\partial m} | \Psi_m \rangle \nonumber \\ &+&
\left[ u' \frac{\partial u}{\partial m} - u \frac{\partial
u'}{\partial m } + w' \frac{\partial w}{\partial m} - w \frac{\partial
w'}{\partial m } \right] \Bigg|_0^\infty
\label{eq:chiral}\end{aligned}$$ This is an extended Feynman-Hellmann theorem where the second term in the l.h.s. corresponds to the short distance contribution (the term at infinity vanishes for a bound state). One of the advantages of the Feynman-Hellmann theorem is that one could in principle establish comparison theorems, provided the change in the coupled channel potential matrix, $\Delta U$, is a definite quadratic form. Note also that the derivative with respect to $m$ annihilates the centrifugal term, $6/r^2 $, and one can diagonalize the coupled channel potential by the unitary transformation Eq. (\[eq:eigenvectors\]) so that the result behaves additively in the attractive and repulsive eigenchannels. Using the leading short distance behavior, Eq. (\[eq:short\_bc\_reg\]), we therefore get $$\begin{aligned}
- \frac{\partial \gamma^2}{\partial m} &=& \int_0^\infty dr \left[
u_A(r)^2 \frac{\partial U_A}{\partial m} + u_R(r)^2 \frac{\partial
U_R}{\partial m} \right] \nonumber \\ &+& C_A^2 \frac{d \varphi}{d
m} \, .\end{aligned}$$ As we see, assuming as suggested in Ref. [@Bulgac:1997ji] that the short distance physics does not depend on the pion mass corresponds to demanding the standard Feynman-Hellmann theorem where only the OPE potential change contributes. For $C_A \neq 0 $ one obtains the condition $$\begin{aligned}
\frac{d }{dm} \varphi (\gamma , m) = \frac{\partial \varphi}{\partial
\gamma} \frac{d \gamma}{d m} + \frac{\partial \varphi}{\partial
m} =0 \, ,
\label{eq:varphi_m} \end{aligned}$$ whence a functional relation between the pion mass and the deuteron binding energy follows. However, note that even in this case the sign of the result is indefinite since $$\begin{aligned}
\frac{\partial \gamma^2}{\partial m} &=& \int_0^\infty dr \left[
u_A(r)^2 \frac{\partial U_A}{\partial m} + u_R(r)^2 \frac{\partial
U_R}{\partial m} \right] \, . \end{aligned}$$ So, we have to determine the sign numerically.
The relation (\[eq:varphi\_m\]) has an equivalent formulation in the boundary condition regularization. For instance, if we assume the same condition BC6 of Eq. (\[eq:bc6\]) for all values of the pion mass we get $$\begin{aligned}
\frac{d}{dm}\left[ \frac{w' (a) }{u (a)\sqrt{2} + w (a)} \right] =0 \, , \end{aligned}$$ where $ a$ is taken to be independent of $m$. In practice, one computes the ratio within the bracket for the physical pion mass and searches for $\gamma$ such that the ratio for the unphysical pion mass yields the same numerical value. If we take the chiral limit we get $
B_d ( 0 , g_{\pi NN} ) = 4.3539 {\rm MeV} $ This value is very close to the one found in Ref. [@Beane:2001bc] $ B_d ( 0 , g_{\pi NN} )
= 4.2 {\rm MeV}$ [^3]. Deuteron observables in the explicit $ m=0 $ limit are listed in Table \[tab:table0\].
Implicit pion mass dependence
-----------------------------
To take into account the implicit pion mass dependence we have to take into account the dependence of $ R= 3g_A^2 M / 32 \pi f_\pi^2 $ on the pion mass. In the chiral limit one gets a larger OPE coupling [@Epelbaum:2002gb]. The value is uncertain and as an educated guess we take $R_0 =1.06(2) R $. Using the same formulation as in the $m$ dependence, the change in the deuteron binding with respect to the $g_{\pi NN} $ coupling constant or equivalently the scale dimension $R$ we get (assuming as before the short distance angle $\varphi $ to be independent on m), $$\begin{aligned}
- R \frac{\partial \gamma^2}{\partial R } &=& \int_0^\infty dr \left[
u_A(r)^2 U_A + u_R(r)^2 U_R \right] \nonumber \\
\label{eq:dgamma/dR}\end{aligned}$$ Again, the result is indefinite since $U_A < 0 $ and $U_R >0 $ and it is not obvious, unlike naive expectations, that a stronger coupling provides stronger binding. The sign depends actually on the details of the wave functions and the particular values of the parameters. Numerically one finds $d \gamma / dR > 0 $, a trend that can be understood if the repulsive term in Eq. (\[eq:dgamma/dR\]) is neglected or on the basis of the inequality $|u_A| > |u_R| $ which is numerically fulfilled. In any case one has the differential inequality $ d \gamma / dR < \gamma / R $.
Numerically, we get $ \gamma_0 = 0.61(10) {\rm fm}^{-1}$ and hence $$\begin{aligned}
B_d^0 = 15 (5) {\rm MeV} \end{aligned}$$ a value compatible with the analysis of Ref. [@Epelbaum:2002gb] $B_d^0 =9.6 \pm 3 $ (perhaps with larger errors [@Beane:2002xf] [^4]) but in disagreement with Refs. [@Beane:2001bc; @Beane:2002vs] where the deuteron becomes unbound for $m < 90 {\rm MeV}$. In any case we confirm the trend of having a stronger binding of the deuteron in the chiral limit. The corresponding observables can be looked up in Table. \[tab:table0\].
$\gamma ({\rm fm}^{-1})$ $\eta$ $A_S ( {\rm fm}^{-1/2}) $ $r_d ({\rm fm})$ $Q_d ( {\rm fm}^2) $ $P_D $
------------------ -------------------------- --------- --------------------------- ------------------ ---------------------- ------------
$m=138.03$ MeV Input 0.02633 0.8681(1) 1.9351(5) 0.2762(1) 7.88(1)%
$m=0$ (explicit) 0.3240(1) 0.09452 0.8444(1) 1.550(1) 0.3006(3) 10.96(2) %
$m=0$ (implicit) 0.61(10) 0.15(2) 0.48(7) 0.98(10) 0.15(3) 15(1) %
Scattering properties in the $^3S_1-{}^3D_1$ channel {#sec:scattering}
====================================================
Orthogonality constraints and Phase Shifts {#sec:phase-shifts}
------------------------------------------
For the $\alpha$ and $\beta$ positive energy scattering states we choose the asymptotic normalization $$\begin{aligned}
u_{k,\alpha} (r) &\to & \frac{\cos \epsilon}{\sin \delta_1}\Big( \hat
j_0 (kr) \cos \delta_1 - \hat y_0 (kr) \sin \delta_1 \Big) \, , \nonumber \\ w_{k,\alpha}
(r) &\to & \frac{\sin \epsilon}{\sin \delta_1}\Big( \hat j_2 (kr) -
\hat y_2(kr) \sin \delta_1 \Big) \, , \nonumber \\ \\
u_{k,\beta} (r) & \to & -\frac1{\sin \delta_1}\Big( \hat j_0 (kr) \cos \delta_2 - y_0 (kr)
\sin \delta_2 \Big) \, , \nonumber \\
w_{k,\beta} (r) &\to & \frac{\tan \epsilon}{\sin \delta_1}\Big( \hat
j_2 (kr) \cos \delta_2 - \hat y_2(kr) \sin \delta_2 \Big) \, , \nonumber \\ \end{aligned}$$ where $ \hat j_l (x) = x j_l (x) $ and $ \hat y_l (x) = x y_l (x) $ are the reduced spherical Bessel functions and $\delta_1$ and $\delta_2$ are the eigen-phases in the $^3S_1$ and $^3D_1$ channels, and $\epsilon$ is the mixing angle $E_1$. Again, the general solution at short distances is given by the general Eq. (\[eq:short\_bc\_reg\]), where the constants $C_A $, $ C_R $ and $\varphi$ are now different since we have a zero energy state and depend whether we have an $\alpha$ or $\beta$ state, so we have the short distance constants, $
C_{A,\alpha} (k) , C_{R,\alpha} (k) , \varphi_\alpha (k) $ and $
C_{A,\beta} (k) , C_{R,\beta} (k) , \varphi_\beta (k) $ respectively. This implies certain correlations between $\delta_1$, $\delta_2$ and $\epsilon$.
For a regular self-adjoint potential the orthogonality of bound and scattering states comes out automatically. We look now for the consequences of [*demanding*]{} this property in the singular OPE potential. Using the standard manipulations to prove orthogonality between states of different energy we get the following relation between $\alpha$ and $\beta$ states and the bound deuteron state (which we denote by a subscript $\gamma$ in this section), $$\begin{aligned}
0 &=& (\gamma^2+k^2 ) \int_0^\infty dr \Big[ u_\gamma (r) u_k (r) +
w_\gamma (r) w_k (r) \Big] \nonumber \\ &=& \left[ u_\gamma' u_k -
u_\gamma u_k' + w_\gamma' w_k - w_\gamma w_k' \right] \Big|_0^\infty
\label{eq:orth}\end{aligned}$$ Using the short distance solution, Eq. (\[eq:short\_bc\_reg\]), we get $$\begin{aligned}
C_{A,i} (k) C_A (\gamma) \sin\left[ \varphi (\gamma) - \varphi_i (k)
\right] =0 \qquad \, , \, i = \alpha ,\beta \end{aligned}$$ Which yields $$\begin{aligned}
\varphi (\gamma) = \varphi_\alpha (k) = \varphi_\beta (k)
\label{eq:phi_const}\end{aligned}$$ Thus, the short distance phases $ \varphi_\alpha(k) $ and $
\varphi_\beta(k) $ of the $^3S_1-{}^3D_1$ channel wave functions in the OPE potential at short distances are all determined by deuteron properties. This means, in particular that the low energy parameters and scattering phase shifts are uniquely determined by the deuteron binding energy and the OPE potential parameters [^5].
The previous argument can also be implemented if we have a short distance cut-off at $r=a$, the orthogonality relation of Eq. (\[eq:orth\]) transforms into the condition $$\begin{aligned}
&& u_\gamma' (a) u_{k,i} (a) + w_\gamma' (a) w_{k,i} (a) \\
&=& u_\gamma (a)
u_{k,i}'(a) + w_\gamma (a) w_{k,i} '(a) \nonumber \\ && \quad
i=\alpha,\beta
\label{eq:orth_a}\end{aligned}$$ Thus, if we impose the same condition on both solutions, Eq. (\[eq:orth\_a\]) cannot be satisfied unless they are related at the boundary. For instance for the condition BC6 of Eqs. (\[eq:bc\_a\]), we get the two relations $$\begin{aligned}
u_{k,i}' (a) = \sqrt{2} w_{k,i}' (a) \, , \qquad i=\alpha,\beta
\label{eq:bc6} \end{aligned}$$ The orthogonality relation corresponding to boundary conditions of the form of Eq. (\[eq:bc6\]) implies then the orthogonality constraint $$\begin{aligned}
\frac{w_{k,i}'(a)}{u_{k,i} (a)\sqrt{2} + w_{k,i} (a)} = \frac{w_\gamma' (a)
}{u_\gamma (a)\sqrt{2} + w_\gamma (a)}\end{aligned}$$ Which is the analog finite cut-off condition of Eq. (\[eq:phi\_const\]). The remaining conditions in Eqs. (\[eq:bc\_a\]) generate analogous orthogonality constraints.
The results for the $^3S_1-{}^3D_1$ channel phase shifts using these conditions are presented in Fig. \[fig:phase-shifts\]. The description is rather satisfactory and it seems to work, as one might expect, up to the vicinity of the CM momentum which magnitude coincides with the two pion exchange left cut $ k={\rm i} \,m $.
Low energy parameters {#sec:low-energy}
---------------------
In the low energy limit one has $$\begin{aligned}
\delta_1 &\to& - \alpha_0 k \, , \nonumber \\
\delta_2 &\to & -\alpha_2 k^5 \, , \nonumber \\
\epsilon & \to & \frac{\alpha_{02}}{\alpha_0} k^2 \end{aligned}$$ so that the zero energy the wave functions behave asymptotically $$\begin{aligned}
u_{0,\alpha} (r) & \to & 1- \frac{r}{\alpha_0} \, , \nonumber \\ w_{0,\alpha} (r) &
\to & \frac{3 \alpha_{02}}{\alpha_0 r^2 } \, , \nonumber \\ u_{0,\beta} (r)
&\to & \frac{r}{\alpha_0} \, , \nonumber \\ w_{0,\beta} (r) &=& \frac{3
\alpha_{2}}{\alpha_{02} r^2 }- \frac{r^3}{15 \alpha_{02}} \, .
\label{eq:zero_energy}\end{aligned}$$ Using these zero energy solutions one can determine the effective range. The $^3S_1$ effective range parameter is given by $$\begin{aligned}
r_0 &=& 2 \int_0^\infty \left[ \left(1-\frac{r}{\alpha_0} \right)^2 -
u_\alpha (r)^2 - w_\alpha (r)^2 \right] dr \, . \nonumber \\
$$ In the zero energy case, the vanishing of the diverging exponentials at the origin imposes a condition on the $\alpha $ and $\beta$ states which generate a correlation between $\alpha_0$ , $\alpha_{02}$ and $\alpha_2$. Using the superposition principle of boundary conditions we may write the solutions in such a way that $$\begin{aligned}
u_{0,\alpha} (r) &=& u_1 (r) - \frac{1}{\alpha_0} u_2 (r) + \frac{3
\alpha_{02}}{\alpha_0} u_3 (r) \nonumber \\ w_{0,\alpha} (r) &=& w_1
(r) - \frac{1}{\alpha_0} w_2 (r) + \frac{3 \alpha_{02}}{\alpha_0} w_3
(r) \nonumber \\ u_\beta (r) &=& \frac{1}{\alpha_0} u_2 (r) + \frac{3
\alpha_{2}}{\alpha_{02}} u_3 (r) -\frac1{15 \alpha_{02}} u_4 (r)
\nonumber \\ w_\beta (r) &=& \frac{1}{\alpha_0} w_2 (r) + \frac{3
\alpha_{2}}{\alpha_{02}} w_3 (r) -\frac1{15\alpha_{02}} w_4 (r)
\nonumber\end{aligned}$$ where the functions $u_{1,2,3,4}$ and $w_{1,2,3,4}$ are independent on $\alpha_0$, $\alpha_{02}$ and $\alpha_2$ and fulfill suitable boundary conditions. As a consequence we get a linear correlation between $
1/\alpha_0$, $\alpha_{02}/\alpha_0$ and also a linear correlation between $\alpha_2 /\alpha_{02}$ and $1/\alpha_{02}$. This means in turn that according to the OPE potential both $\alpha_{02} $ and $\alpha_2$ depends linearly with $ \alpha_{0}$. Numerically we get the following correlations, $$\begin{aligned}
\alpha_{02} &=& 0.963571370240 \, \alpha_0 - 3.467616391389 \nonumber \\
\alpha_{2} &=& 3.467616391389 \, \frac{\alpha_{02}}{\alpha_0} + 5.080264230656
\label{eq:alpha_corr1}\end{aligned}$$ These relations are cut-off independent and unique consequences of the OPE potential. On the other hand, the orthogonality between the bound state and the scattering state yields $$\begin{aligned}
\alpha_0 &=& 1.037805911852 \,\alpha_{02} + 3.598712446758 \qquad \qquad (\alpha)
\nonumber \\
\alpha_{02} &=& 0.288382561043 \,\alpha_0\,\alpha_2 - 1.465059639612\,\alpha_0
\qquad (\beta)\nonumber\\
\label{eq:alpha_corr2}\end{aligned}$$ The provided high accuracy is indeed needed. The four equations, Eq. (\[eq:alpha\_corr1\]) and Eq. (\[eq:alpha\_corr2\]), overdetermine the values of the three scattering lengths and could be solved in triplets yielding four different solutions. Actually, there are only two independent solutions which differences are compatible within our numerical uncertainties. The scattering lengths and effective range are presented in Table \[tab:table\_pert\] and compared to their perturbative value ( see Appendix \[sec:pert\] and to the high quality Nijmegen potential models [@Stoks:1994wp] [^6]. As we see, the agreement with the high quality potentials is at the few percent level. Perturbation theory does not account for most of the contribution to the effective range since the orthogonality constraints preclude a short distance contribution to $r_0$ and also to the deuteron matter radius $r_m$. This means in practice that the counterterm named $C_2$ in Refs. [@Kaplan:1998sz; @Kaplan:1998we] must vanish (See Appendix \[sec:pert\] for a detailed discussion). The dependence of the scattering lengths $\alpha_0$, $\alpha_{02}$ and $\alpha_2$ on the deuteron wavenumber $\gamma$ can be seen in Fig. \[fig:a0\[Gamma\]\], where $\gamma$ dependent generalizations of the correlations, Eq. (\[eq:alpha\_corr1\]) and Eq. (\[eq:alpha\_corr2\]) hold.
The phase shifts look very similar to previous work [@Beane:2001bc] using an energy expansion of a square well potential as a counter-term and adjusting the depth of the two lowest orders to reproduce the $^3S_1$ scattering length $\alpha_0 $ and effective range $r_0$ as independent parameters or our variable phase approach with non-trivial initial conditions in Ref. [@PavonValderrama:2004nb] where the full coupled channel S-matrix was tailored to reproduce the effective range expansion to any order treating all parameters as independent. Roughly speaking, both approaches could be mapped to an energy dependent boundary condition with no [*a priori*]{} orthogonality constraints [^7]. The fact that the orthogonality constrained boundary condition generates the bulk of the low energy threshold parameters with only [*one parameter*]{} naturally explains the similarity between the present phase shifts and those in previous works [@Beane:2001bc] and suggests that there is perhaps no need to make the short distance boundary condition energy dependent if the short distance cut-off is removed.
Short distance solutions and determination of the coefficients {#sec:short}
==============================================================
In this section we determine the coefficients of the OPE deuteron wave functions appearing at short distances in Eq. (\[eq:short\_bc\]). In particular, we compute the energy independent and OPE potential parameters independent short distance phase $ \varphi $. Let us remind that any choice of $\varphi$ corresponds to a different choice of short distance physics; given $\varphi$ and the OPE potential all deuteron and scattering properties are uniquely determined. However, the leading asymptotic form cannot directly be used to match the numerical solution obtained by integrating in the large distance solution. On the one hand, if we use cut-off approaches to determine the regular solution, there are short distance cut-off effects when the distance gets close to the cut-off radius. On the other hand, the fact that the diverging exponential dominates over the converging one provides too weak a signal for the corresponding coefficient. To remedy the situation we improve on the short distance solution to provide a reliable approximation at larger distances ($\sim 1{\rm
fm}$) where the diverging exponential is less dominant, and look for plateaus in the matching radius. It turns out (see below) that one should go at eight order in this expansion for a robust determination of the short distance coefficients. Actually, one can then directly match the short distance improved wave functions to the numerical solution without no reference to cut-offs. We will try the two methods and see that they yield to compatible results for the short distance coefficients.
In the limit $r\to 0 $ the solutions to the coupled equations can be written in an expansion of the form [^8] $$\begin{aligned}
u(r) &=& u_0 \left(\frac{r}{R}\right)^{a_1} e^{a_0
\sqrt{\frac{R}{r}}} f(r) \nonumber \\
w(r) &=& w_0
\left(\frac{r}{R}\right)^{a_2} e^{a_0
\sqrt{\frac{R}{r}}} g(r)
\nonumber \\\end{aligned}$$ with $$\begin{aligned}
f(r) &=& \sum_{n=0}^\infty A_n
\left(\frac{r}{R}\right)^{n/2} \nonumber \\
g(r) &=& \sum_{n=0}^\infty B_n
\left(\frac{r}{R}\right)^{n/2} \nonumber \\\end{aligned}$$ At leading order we get the equations $$\begin{aligned}
u_0 a_0^2 + 16 \sqrt{2} w_0 &=&0 \nonumber \\
16 \sqrt{2} u_0 + (a_0^2 - 16) w_0 &=&0
\label{eq:short_LO}\end{aligned}$$ which have the four non trivial solutions, $$\begin{aligned}
(1A) \, , \quad a_0 &=& - 4 i \, , \qquad w_0 = \frac{u_0}{\sqrt{2}} \, , \\
(2A) \, , \quad a_0 &=& + 4 i \, , \qquad w_0 = \frac{u_0}{\sqrt{2}} \, , \\
(2R) \, , \quad a_0 &=& - 4 \sqrt{2} \, , \qquad w_0 = -\sqrt{2} u_0 \, , \\
(1R) \, , \quad a_0 &=& + 4 \sqrt{2} \, , \qquad w_0 = -\sqrt{2} u_0 \, . \end{aligned}$$ The next to leading order equation becomes compatible only if $$\begin{aligned}
a_1 = a_2 = 3/4 \end{aligned}$$ For any solution in Eq. (\[eq:short\_LO\]) we may then solve for the remaining coefficients. One peculiar feature of this expansion is that if one wants to determine the solution to a given order, one has to compute the coefficients at a higher order. The reason is that strictly speaking a truncation of the expansion involves also non-diagonal elements, and one has the freedom to choose between solving $u$ or $w$ to a given accuracy. The explicit result to eight order is presented in Appendix \[sec:short\_app\]. The general short distance solution is written as a linear combination of the four independent solutions, $$\begin{aligned}
u (r) &=& \frac1{\sqrt{3}}\left(\frac{r}{R}\right)^{3/4} \Big[ -C_{1R}
f_{1R} (r) e^{+ 4 \sqrt{2} \sqrt{\frac{ R}{r}}} \nonumber \\ &-&
C_{2R} f_{2R}(r) e^{- 4 \sqrt{2} \sqrt{\frac{ R}{r}}} + \sqrt{2}
C_{1A} f_{1A}(r) e^{- 4 i \sqrt{\frac{ R}{r}}} \nonumber \\ &+&
\sqrt{2} C_{2A} f_{2A}(r) e^{ 4 i\sqrt{\frac{ R}{r}}} \Big]
\nonumber \\ w (r) &=& \frac1{\sqrt{3}}
\left(\frac{r}{R}\right)^{3/4} \Big[ \sqrt{2} C_{1R} g_{1R} (r) e^{+
4 \sqrt{2} \sqrt{\frac{ R}{r}}} \nonumber \\ &+& \sqrt{2} C_{2R}
g_{2R}(r) e^{- 4 \sqrt{2} \sqrt{\frac{ R}{r}}} + C_{1A} g_{1A}(r)
e^{- 4 i \sqrt{\frac{ R}{r}}} \nonumber \\ &+& C_{2A} g_{2A}(r) e^{
4 i\sqrt{\frac{ R}{r}}} \Big] \nonumber \\\end{aligned}$$ This expansion converges rather fast for each solution up to distances of about $r \sim 0.6-0.9 {\rm fm}$. That is about what one needs, since that is sufficiently far above the cut-off radius $ a\sim 0.1
{\rm fm}$. Matching $u,w,u'$ and $w'$ at some point in this region we get a linear relation between $C_{1R},C_{2R},C_{1A},C_{2A}$ and $\eta$. Actually, we find that the signal of the converging exponential is about hundred to thousand times that of the diverging exponential in the range between 0.6 and 1 fm [^9]. Matching directly the integrated in solution to the short distance solution with a vanishing coefficient of the diverging exponential $C_{1R}=0$, we get at the scale $ 0.7 < r < 0.9 {\rm fm} $ $$\begin{aligned}
C_{2R}&=& -0.47(1) \, , \quad \eta = 0.0263333(1) \nonumber \\
\bar C_{1A} &=& 0.1327(3) \, \quad , \bar C_{2A} = 0.2277(5) \end{aligned}$$ We can instead determine $\eta=0.0263332 $ from the boundary condition BC6 in Eq. (\[eq:bc\_a\]) at $r=0.2 {\rm fm} $ and deduce the remaining constants yielding $$\begin{aligned}
|C_{1R}| &<& 10^{-7} \, , \quad C_{2R} = -0.47 (1) \nonumber \\
\bar C_{1A} &=& 0.1327(3)\, , \quad \bar C_{2A} = 0.2277 (5) \end{aligned}$$ The errors have been estimated by varying the matching point in the region $0. 7 {\rm fm } < r < 0.9 {\rm fm } $. Note that although the coefficient of the diverging exponential $C_{1R}$ is six orders of magnitude larger than the one of the converging exponential, the solution through the matching condition is about eight orders of magnitude smaller (may even change sign ). So that the result provides a sizeable signal for the converging exponential. With these values we show in Fig. \[fig:u+w\_short.epsi\] the short distance wave functions compared to the integrated in numerical ones when the matching is undertaken at $ r=0.8 {\rm fm}$. To improve on the short distance side we have taken $C_{1R}=0$. The error in the region $0. 7
{\rm fm } < r < 0.9 {\rm } $ never exceeds a $0.01\%$. We have checked that setting the constant $C_{2R}=0$ introduces a larger deviation from the numerical solution as compared to the computed value in the region above $1 {\rm fm}$. Finally, the corresponding short distance angle reads $$\begin{aligned}
\varphi = - \tan^{-1} \frac{\bar C_{2A}}{\bar C_{1A}}= -59.7 (1)^o \, . \end{aligned}$$ The discussion in this section explicitly shows that contrary to the findings in Ref. [@Martorell94] the coefficient of the converging exponential does not vanish.
Conclusions and Outlook {#sec:concl}
=======================
In this paper we have reanalyzed the OPE potential in the triplet $^3S_1-{}^3D_1$ channel both for bound and scattering states. Rather than modeling the interaction below some finite short distance we have adopted the viewpoint of taking the potential seriously down to the origin. This must be carefully done and in a way as to get rid of any short distance ambiguities. In addition, this procedure proves crucial to be able to disentangle the OPE contribution from other contributions, like TPE and higher, electromagnetic effects and relativistic corrections to deuteron and NN scattering observables. Our analysis is carried out entirely in coordinate space where these corrections generate a potentical which is finite everywhere except at the origin. Momentum space treatments require an additional regularization of the potential.
The OPE coupled channel potential is singular at short distances and additional conditions need to be specified on the wave functions at the origin. Actually, the singular eigen-potentials at short distances are attractive and repulsive and while in the attractive case a mixed boundary condition specifies the corresponding short distance eigenfunction, in the repulsive case one must impose a standard homogeneous boundary condition. This only leaves one free parameter, which we have chosen to be the deuteron binding energy and which cannot be determined from the OPE potential. All remaining deuteron observables come out for free. For the scattering states in the $^3S_1-{}^3D_1$ channel, we have demanded orthogonality constraints between all states of different energy. This condition is actually an additional requirement for singular potentials, since the orthogonality relation carries information on the peculiar short distance behavior of the wave functions, and is not necessarily satisfied. The most obvious example where orthogonality constraints are violated corresponds to energy dependent potentials and energy dependent boundary conditions in coordinate space. A less trivial but significant example is the case of dimensional regularization in the PDS scheme as a perturbative analysis in coordinate space of both bound and scattering states reveals. The power of the orthogonality constraints for singular potentials is that all scattering properties are then predicted from the OPE potential parameters and the deuteron binding energy.
In our analysis it turns out that the short distance form of all wave functions is characterized by some short distance constants. We have clarified the role played by the exponentially suppressed regular solution by determining its non-vanishing value numerically using short distance expansions to high order, to explore the region below $0.1 {\rm fm} $, not accessible to standard numerical integration methods. Another relevant constant is given by a short distance phase $\varphi$ which plays the role of a fundamental dimensionless constant in the OPE problem. It does not depend on the energy nor on the OPE parameters, but it is related to the form of the OPE potential in the chiral limit. The closeness of this phase to $\pi /3 $ is mysterious and suggestive and requires further investigation.
It is remarkable that indeed the bulk of the experimental results both for the bound state as well the scattering observables are accounted for at the $2-3 \%$ level by the OPE potential taken from zero to infinity. We interpret this success as a confirmation on the validity of our choice of regular solutions and the use of orthogonality constraints. The discrepancies can legitimately be attributed to other effects such as TPE, electromagnetic and relativistic corrections. Many of the methods and results obtained in this paper can be generalized in a straightforward manner to take these effects into account and to the study of higher partial waves without any substantial modifications. In particular, the number of independent constants in a given channel depends on the short distance behavior of the long range potential. The bonus of such a program would be the complete elimination of short distance ambiguities in the study of the NN interaction with known long distance forces as determined by chiral symmetry. In our view this an indispensable prerequisite to asses the relevance of chiral symmetry in nuclear physics in a model independent way. The systematic study of these effects will be reported elsewhere [@Pavon_TPE].
We thank J. Nieves for a critical reading of the manuscript. This research was supported by DGI and FEDER funds, under contract BFM2002-03218 and by the Junta de Andalucía.
Perturbative solutions {#sec:pert}
======================
Bound state
-----------
In this appendix we solve the coupled deuteron equations, Eq. (\[eq:sch\_coupled\]) in standard perturbation theory for the fixed negative energy bound state. A somewhat related approach looking for the equivalence with the PDS scheme of [@Kaplan:1998we] in the one-channel positive energy case can be looked up in Ref. [@Cohen:1998bv]. The problem of orthogonality was not discussed. The requirement of normalizability of the deuteron state requires the D wave component to vanish. Thus, at lowest order we have the normalizable solutions, $$\begin{aligned}
u_\gamma^{(0)} (r) &=& e^{-\gamma r} \nonumber \\
w _\gamma^{(0)}(r) &=&0\end{aligned}$$ At first order we have to solve the equations $$\begin{aligned}
-u_\gamma^{(1) \prime \prime } (r) + \gamma^2 u_\gamma^{(1)} (r) &=& - U_s (r)
e^{-\gamma r} \, ,\nonumber \\ -w_\gamma^{(1) \prime \prime } (r) +
\left[\frac{6}{r^2} + \gamma^2 \right] w_\gamma^{(1)} (r) &=&
-U_{sd}(r) e^{-\gamma r} \, , \nonumber \\
\label{eq:sch_coupled_pert} \end{aligned}$$ Using the regular and irregular solutions at the origin $$\begin{aligned}
u_{\rm reg}( r) &=& 2 \frac{\sinh ( \gamma r )}{\gamma r} \nonumber \\
w_{\rm reg}( r) &=& 2 \left( 1+ \frac{3}{(\gamma r)^2} \right) \sinh (
\gamma r ) - \frac{6}{\gamma r} \cosh( \gamma r ) \nonumber \\
u_{\rm irreg} (r) &= & e^{-\gamma r} \, . \\ w_{\rm irreg} (r) & = & \
e^{-\gamma r} \left( 1 + \frac{3}{\gamma r} + \frac{3}{(\gamma r)^2} \right)
\, .\end{aligned}$$ we get $$\begin{aligned}
u_\gamma^{(1)} (r) &=& \int_0^\infty G_s (r,r') U_s (r') e^{-\gamma
r'} dr' \\ w_\gamma^{(1)} (r) &=& \int_0^\infty G_d (r,r') U_{sd}
(r') e^{-\gamma r'} dr'\end{aligned}$$ where $G_s$ and $G_d$ are the corresponding Green functions. Explicit calculation yields
$$\begin{aligned}
u_\gamma^{(1)} (r) &=& e^{-\gamma r} \frac{m^2\,R\,
\Gamma(0,m\,r + 2\,r\,\gamma )-\Gamma(0,m\,r)}{3\,\gamma } - \frac{2\,m^2\,R\,
{\rm Ei}(- m\,r - 2\,r\,\gamma )\,
\sinh (r\,\gamma )}{3\,\gamma } \\
w_\gamma^{(1)} (r) &=& {e^{-r\,\gamma }} \left( 1 + \frac{3}{r^2\,{\gamma
}^2} + \frac{3}{r\,\gamma } \ \right) \times \nonumber \\ &&
\,\Big[\frac{m^2\,R\, \left( 3\,m^2 - 4\,{\gamma }^2 \right) \,
\Gamma(0,m\,r + 2\,r\,\gamma )
-\Gamma(0,m\,r )}{6\,{\sqrt{2}}\,{\gamma }^3} \nonumber \\
&+& \frac{R\, \left( -6\,m^3\,\gamma + 6\,m^2\,{\gamma }^2 +
4\,{\gamma }^4 + \left( 3\,m^4 - 4\,m^2\,{\gamma }^2 \right) \, \log
(1 + \frac{2\,\gamma }{m}) \right) }{6\,{\sqrt{2}}\, {\gamma }^3}
\nonumber \\ &+& \frac{e^{-m\,r - 2\,r\,\gamma }\,R\, \left( 6 +
6\,m\,r + m^2\,r^2 - m^3\,r^3 + 4\,r\,\gamma + 4\,m\,r^2\,\gamma +
2\,m^2\,r^3\,\gamma \right) }{2\, {\sqrt{2}}\,r^4\,{\gamma }^3}
\nonumber \\ &-& \frac{R\, \,e^{-m\,r} \left( 6 + 6\,m\,r + m^2\,r^2 -
m^3\,r^3 - 8\,r\,\gamma - 8\,m\,r^2\,\gamma + 4\,r^2\,{\gamma }^2 +
4\,m\,r^3\,{\gamma }^2 \right) }{2\,{\sqrt{2}}\,r^4\, {\gamma }^3}
\nonumber \Big] \nonumber \\ &+& \left( 2 \left( 1+ \frac{3}{\gamma^2
r^2} \right) \sinh ( \gamma r ) - \frac{6}{\gamma r} \cosh( \gamma r
)\right) \times \nonumber \\ && \Big[ \frac{e^{- m\,r - 2\,r\,\gamma
}\,R\, \left( 6 + 6\,m\,r + m^2\,r^2 - m^3\,r^3 + 4\,r\,\gamma +
4\,m\,r^2\,\gamma + 2\,m^2\,r^3\,\gamma \right) }{2\,{\sqrt{2}}\,
r^4\,{\gamma }^3} \nonumber \\ &-& \frac{m^2\,R\, \left( 3\,m^2 -
4\,{\gamma }^2 \right) \, {\rm Ei}(- m\,r - 2\,r\,\gamma
)}{6\, {\sqrt{2}}\,{\gamma }^3} \Big]\end{aligned}$$
where $\Gamma(0,z)$ and ${\rm Ei} (z) $ are the standard incomplete Gamma function and the Exponential integral function respectively $$\begin{aligned}
\Gamma(0,z) &=& \int_z^\infty dt \frac{e^{-t}}{t} \\
{\rm Ei} (z) &=& - P\int_{-z}^\infty dt \frac{e^{-t}}{t} \end{aligned}$$ At asymptotically large distances we have $$\begin{aligned}
u_\gamma^{(1)} (r) &\to & c_{\rm pert} e^{-\gamma r} \\ w_\gamma^{(1)} (r)
&\to & \eta_{\rm pert} e^{-\gamma r} \left( 1 + \frac{3}{\gamma r} +
\frac{3}{(\gamma r)^2} \right)\end{aligned}$$ where $$\begin{aligned}
c_{\rm pert} &=& \int_0^\infty U_{s} (r) u_{\rm reg} (r) e^{-\gamma r} dr \\
\eta_{\rm pert} &=& \int_0^\infty U_{sd} (r) w_{\rm reg} (r) e^{-\gamma r} dr \end{aligned}$$ Explicit calculation yields $$\begin{aligned}
c_{\rm pert} &=& \frac{ R m^2}{3 \gamma} \log \left( 1 +
\frac{2\gamma}m \right)\\ \eta_{\rm Pert} &=& \frac{R}{6\sqrt{2}m
\gamma^3} \Big[ 4 \gamma^4 + 6m^2 \gamma^2 - 6 m^3 \gamma \nonumber \\
&+& ( 3 m^4 - 4 m^2 \gamma^2 ) \log\left( 1 + \frac{2 \gamma}{m}
\right) \Big] \nonumber \\ &=& \frac{32 \sqrt{2} R}{45 m} \gamma^2 -
\frac{2 \sqrt{2} R\gamma^3}{3m^2} + \dots
\label{eq:eta-pert} \end{aligned}$$ The numerical value we get is $\eta_{\rm pert}=0.0510 $ almost twice the exact OPE result. Taking this perturbative value for $\eta $ we show in Fig. \[fig:u+w\_perturbative\] the perturbative deuteron wave functions as compared to the exact ones.
Unfortunately, if one wants to improve on this first order calculation going to second order perturbation theory there is a problem since the behavior of the perturbative wave functions at short distances is given by $$\begin{aligned}
u_\gamma^{(1)} (r) &=& -\frac{ R m^2}{3 \gamma} \log \left( 1 +
\frac{2\gamma}m \right) + \dots \\ w_\gamma^{(1)} (r) &=&
\frac{\sqrt{2}R}{r} - \frac23 \sqrt{2} R \gamma + \dots\end{aligned}$$ making the wave function non normalizable, unlike the exact regular wave function. This divergence at short distances actually precludes going to higher orders in perturbation theory.
The normalization at first order is given by $$\begin{aligned}
\frac1{A_S^2} = \int_0^\infty ( e^{-2 \gamma r} + 2 u_\gamma^{(1)} (r)
e^{-\gamma r} )\end{aligned}$$ and hence $$\begin{aligned}
\frac{A_S}{\sqrt{2 \gamma}} &=& 1 - \frac{ 2R m^2 }{3(m + 2 \gamma)}
+\frac{ R m^2}{3 \gamma} \log \left( 1 + \frac{2\gamma}m \right) \nonumber \\ &=& 1 - \frac{ 2R
\gamma}3 - \frac{16 R \gamma^2}{9m} + \dots \end{aligned}$$ The deuteron matter radius is given by $$\begin{aligned}
r_{m,\,\rm pert}^2 = \frac14 A_S^2 \int_0^\infty r^2 ( e^{-2 \gamma r} + 2
u_\gamma^{(1)} (r) e^{-\gamma r} )\end{aligned}$$ and hence to first order one has $$\begin{aligned}
r_{m,\,\rm pert}^2 = \frac{1}{8 \gamma^2} + \frac{m^2 R ( 3 m +10 \gamma)}{18\gamma (m + 2 \gamma)^3 } + \dots \end{aligned}$$ yielding in the weak binding regime $$\begin{aligned}
\sqrt{8} \gamma \,\, r_{m,\,\rm pert} &=& 1 + \frac{2 R \gamma}{3} - \frac{16
R \gamma^2}{9 m} + \dots \end{aligned}$$ Finally, the quadrupole moment at first order is given by $$\begin{aligned}
Q_{\rm pert}= \frac{\sqrt{2}}{10} \int_0^\infty r^2 w_\gamma^{(1)} (r)
e^{-\gamma r} dr\end{aligned}$$ The integral can be evaluated to give $$\begin{aligned}
Q_{\rm pert} &=& \frac{8 R ( 4 m^2 + 9 \gamma m + 6 \gamma^2 )}{45 (m
+ 2 \gamma )^3} \nonumber \\
&=& \frac{32 R}{45 m} - \frac{8 R \gamma}{3m^2} + \frac{128R \gamma^2}{15 m^3} + \dots \end{aligned}$$\
yielding $ Q_{\rm pert}= 0.4555 {\rm fm}^2 $.
Our perturbative expressions for $A_S$, $r_m$ and $Q$ coincide with those of Kaplan Savage and Wise [@Kaplan:1998sz] provided one takes in their expression for $r_m$ the renormalization scale in the PDS scheme to be $\mu = \gamma $, instead of taking $\mu = m$ as they do or else taking the $C_2$ counter-term identically equal to zero. Actually, the $C_2$ counter-term can be mapped into a short distance contribution to the effective range parameter $r_0$ in the $^3S_1$ channel. The value of $\eta$ was not given in that reference but can be deduced from the off-diagonal scattering amplitude in the $^3S_1-{}^3D_1 $ channel given in their previous work [@Kaplan:1998we] by evaluating the residue at the deuteron pole. The result also agrees with the calculation presented here.
Low energy parameters {#low-energy-parameters}
---------------------
To check the identification $C_2=0$ further let us compute the S-wave effective range $r_0$. For our purposes it is sufficient to analyze the zero energy scattering state. The lowest order solution is given by an $\alpha $ state $$\begin{aligned}
u_{0,\alpha}^{(0)} (r) &=& \left( 1 - \frac{r}{\alpha_0} \right)
\nonumber \\
w_{0,\alpha}^{(0)} (r) &=& 0 \end{aligned}$$ At zeroth order in the OPE coupling the orthogonality constraint yields $$\begin{aligned}
0 &=& \int_0^\infty u_{0,\alpha}^{(0)} (r) u_\gamma^{(0)} (r) dr \nonumber \\
&=& \frac{1}{\gamma^2 } \left[-\frac1{\alpha_0} + \gamma
\right]\end{aligned}$$ which yields the scattering length to lowest order, $$\begin{aligned}
\alpha_0^{(0)} = \frac1{\gamma} \end{aligned}$$ At first order we use the regular solution $u_{\rm reg} (r)= r $ and the irregular solution $u_{\rm irreg} (r) = (1-r/\alpha_0) $ and get similarly to the bound state case the first order correction to the $\alpha $ state, $$\begin{aligned}
u_{0,\alpha}^{(1)} (r) &=& \frac{2 R }{3\, \alpha_0} e^{-m\,r} \left(1
- \alpha_0 \,m \right) - \frac{2\,m^2\,r\,R}{3} {\rm Ei}(- m\,r )
\nonumber \\
w_{0,\alpha}^{(1)} (r) &=&
\frac{\,e^{-m\,r}\,R}{15\,{\sqrt{2}}\,\alpha_0 \,m^2\,r^2} \nonumber \\
&\times& \Big( 120
-64\,\alpha_0 \,m + 120\,m\,r - 34\,\alpha_0 \,m^2\,r \nonumber \\ &+& 40\,m^2\,r^2 -
2\,\alpha_0 \,m^3\,r^2 + \alpha_0 \,m^4\,r^3 - \alpha_0 \,m^5\,r^4
\Big) \nonumber \\ &+& \frac{R\,\left( -120 + 64\,\alpha_0 \,m +
\alpha_0 \,m^6\,r^5\,\Gamma(0,m\,r) \right) } {15\,{\sqrt{2}}\,\alpha_0
\,m^2\,r^2} \nonumber \\ \end{aligned}$$ Note that asymptotically the first order correction to the S-wave vanishes exponentially and hence cannot contribute to the scattering length. On the other hand, the orthogonality relation to first order reads $$\begin{aligned}
0 = \int_0^\infty dr \left[ u_{\gamma}^{(0)} u_{0,\alpha}^{(0)} +
u_{\gamma}^{(1)} u_{0,\alpha}^{(0)} + u_{\gamma}^{(1)}
u_{0,\alpha}^{(0)} \right] \end{aligned}$$ and after computing the integrals one gets $$\begin{aligned}
0= - \frac1{\alpha_0}+ \gamma &+& \frac{ R }{3 \alpha_0 \gamma} \Big[
m^2 ( 1 + \alpha_0 \gamma ) \log\left( 1 + \frac{2 \gamma}{m} \right)
\nonumber \\ &-& 2\gamma ( m - \gamma + \alpha_0 m \gamma ) \Big]\end{aligned}$$ Solving perturbatively for $\alpha_0$ we get at first order $$\begin{aligned}
\alpha_{0,{\rm pert}} = \frac{1}{\gamma} - \frac{ 2 m^2 R }{3
\gamma^2} \left[ \frac{\gamma ( \gamma- 2 m ) }{m^2} + \log\left( 1 +
\frac{2 \gamma}{m} \right) \right] + \dots \nonumber \\\end{aligned}$$ Numerically one gets $ \alpha_{0,{\rm pert}}= (4.3177 + 0.2912 +
\dots) {\rm fm} $ to be compared with the full OPE result $\alpha_0 =
5.34 $ and the experimental value $ \alpha_0 = 5.42 {\rm fm} $. The $E_1$ scattering length $\alpha_{02}$ can be read off from the D-wave, using the asymptotic condition in Eq. (\[eq:zero\_energy\]) $$\begin{aligned}
\alpha_{02} &=& \frac{4 \sqrt{2} R (15 - 8 \alpha_0 m ) }{45 m^2}
\nonumber \\ &=& \frac{4 \sqrt{2} R (15 \gamma - 8 m ) }{45 \gamma
m^2}\end{aligned}$$ in the second line we have substituted the perturbative relation $\alpha_0 = 1/ \gamma + {\cal O} (R) $. Note the linear correlation $\alpha_{02} = 1.5499 \alpha_0- 4.1530 $ to be compared with the exact OPE relation in Eq. (\[eq:alpha\_corr1\]). The numerical value one gets for the first and second lines taking the experimental values of $\alpha_0 = 5.42 $ and $\gamma $ are $\alpha_{02} = 4.24 {\rm fm^3}$ and $\alpha_{02} = 2.53 {\rm fm^3}$ respectively to be compared with the experimental $\alpha_{02}=1.64 {\rm fm^3}$. In the weak binding limit one obtains $$\begin{aligned}
\gamma \,\alpha_{0,{\rm pert}} = 1 + \frac{2R\gamma}3 - \frac{16 R
\gamma^2}{9 m} + \dots\end{aligned}$$ In this limit we have the perturbative linear correlation between the scattering length and the deuteron matter radius $$\begin{aligned}
r_m = \frac{\alpha_0}{2\sqrt{2}} + {\cal O} (\gamma^3 , R^2) \end{aligned}$$ which yields the value $r_m = 1.92 $ for the experimental scattering length $\alpha_0 = 5.42 {\rm fm} $. The linear correlation was established empirically with realistic potentials in Ref. [@Martorell:1986; @Martorell:1990].
To first order the effective range in the $^3S_1$ eigen-channel is given by $$\begin{aligned}
r_0 = - 4 \int_0^\infty dr \, u_{0,\alpha}^{(0)} (r) u_{0,\alpha}^{(1)} (r) \end{aligned}$$ yielding $$\begin{aligned}
r_{0,{\rm pert}} &=& \frac{4R(3m^2- 8 \gamma m + 6 \gamma^2 ) }{9m^2}
\nonumber \\ &=& 1.4369 - 5.4789 \gamma + 5.8758 \gamma^2 \nonumber \\
&=& 0.4831
{\rm fm} \end{aligned}$$ a result much smaller than the full OPE result ($ 1.64\,{\rm fm} $) and the experimental number ($1.75\,{\rm fm} $). Again, our result corresponds to a theory where the short distance contribution to the effective range vanishes, i.e. $C_2=0$. A non vanishing value of $C_2$ was needed to fit the experimental values of both the matter radius and the effective range. Our calculation shows that the scheme developed in Refs. [@Kaplan:1998sz] and Ref. [@Kaplan:1998we] does not fulfill perturbatively the orthogonality constraints.
Short distance expansion {#sec:short_app}
========================
For the $f(r)$ function we get (we use $ x=r/R$),
$$\begin{aligned}
f_{1A}&=& 1 - \frac{35\,\imag }{32}\,{\sqrt{x}} - \frac{1811\,x}{6144}
+ \frac{2441\,\imag }{65536}\,x^{\frac{3}{2}} -
\frac{34805\,x^2}{8388608} \nonumber \\ &+& x^3\,\left(
\frac{9873675}{17179869184} + \frac{m^2\,R^2}{36} -
\frac{m^3\,R^3}{32} - \frac{3\,R^2\,{\gamma }^2}{64} \right)
\nonumber \\ &+& x^{\frac{7}{2}}\,\left( \frac{193405905\,\imag
}{549755813888} + \frac{353\,\imag }{24192}\,m^2\,R^2 -
\frac{709\,\imag }{92160}\,m^3\,R^3 + \frac{\imag }{28}\,m^4\,R^4 -
\frac{709\,\imag }{61440}\,R^2\,{\gamma }^2 \right) \nonumber \\ &+&
x^{\frac{5}{2}}\,\left( \frac{-333725\,\imag }{268435456} -
\frac{\imag }{15}\,m^3\,R^3 - \frac{\imag }{10}\,R^2\,{\gamma }^2 \
\right) \\
f_{2A}&=& f_{1A}^* \\
f_{2R}&=&
1 + \frac{67\,{\sqrt{x}}}{32\,{\sqrt{2}}} + \frac{7763\,x}{12288} +
\left( \frac{8873}{131072\,{\sqrt{2}}} -
\frac{m^2\,R^2}{3\,{\sqrt{2}}} \right) \,x^{\frac{3}{2}} +
\left( -\frac{105845}{33554432} -
\frac{55\,m^2\,R^2}{192} \right) \,x^2 \nonumber \\ &+&
\left( \frac{881405}{1073741824\,{\sqrt{2}}} -
\frac{10807\,m^2\,R^2}{184320\,{\sqrt{2}}} +
\frac{m^3\,R^3}{15\,{\sqrt{2}}} +
\frac{R^2\,{\gamma }^2}{10\,{\sqrt{2}}} \right) \,x^{\frac{5}{2}} \nonumber \\ &+&
\left( - \frac{23360715}{137438953472} -
\frac{332899\,m^2\,R^2}{11796480} + \frac{47\,m^3\,R^3}{960} +
\frac{m^4\,R^4}{36} + \frac{47\,R^2\,{\gamma }^2}{640} \right) \,x^3 \nonumber \\ &+&
\left( \frac{419268465}{4398046511104\,{\sqrt{2}}} +
\frac{30559591\,m^2\,R^2}{31708938240\,{\sqrt{2}}} +
\frac{2141\,m^3\,R^3}{1290240\,{\sqrt{2}}} +
\frac{229\,m^4\,R^4}{8064\,{\sqrt{2}}} +
\frac{2141\,R^2\,{\gamma }^2}{860160\,{\sqrt{2}}} \right) \,
x^{\frac{7}{2}} \nonumber \\
f_{1R} &=& f_{2R} \qquad \left( x \to e^{2\pi i} x \right) \end{aligned}$$
and for the $g(r)$ function one has $$\begin{aligned}
g_{1A}&=& 1 - \frac{35\,\imag }{32}\,{\sqrt{x}} - \frac{4883\,x}{6144}
+ \frac{82075\,\imag }{196608}\,x^{\frac{3}{2}} +
\frac{1245195\,x^2}{8388608} \nonumber \\ &+& \left(
\frac{-5136285\,\imag }{268435456} - \frac{\imag }{15}\,m^3\,R^3 -
\frac{\imag }{10}\,R^2\,{\gamma }^2 \ \right) \,x^{\frac{5}{2}}
\nonumber \\ &+& \left( \frac{42237195}{17179869184} -
\frac{m^2\,R^2}{18} - \frac{m^3\,R^3}{32} - \frac{3\,R^2\,{\gamma
}^2}{64} \right) \,x^3 \nonumber \\ &+& \left(
\frac{494999505\,\imag }{549755813888} - \frac{65\,\imag
}{12096}\,m^2\,R^2 + \frac{2363\,\imag }{92160}\,m^3\,R^3 +
\frac{\imag }{28}\,m^4\,R^4 + \frac{2363\,\imag
}{61440}\,R^2\,{\gamma }^2 \right) \,x^{\frac{7}{2}} +{\cal
O}(x^{4}) \\ g_{2A} &=& g_{1A}^* \\ g_{2R}&=& 1 +
\frac{67\,{\sqrt{x}}}{32\,{\sqrt{2}}} + \frac{13907\,x}{12288} +
\left( \frac{307195}{393216\,{\sqrt{2}}} -
\frac{m^2\,R^2}{3\,{\sqrt{2}}} \right) \,x^{\frac{3}{2}} + \left(
\frac{5075595}{33554432} - \frac{55\,m^2\,R^2}{192} \right) \,x^2
\nonumber \\ &+& \left( \frac{19661565}{1073741824\,{\sqrt{2}}} -
\frac{41527\,m^2\,R^2}{184320\,{\sqrt{2}}} +
\frac{m^3\,R^3}{15\,{\sqrt{2}}} + \frac{R^2\,{\gamma
}^2}{10\,{\sqrt{2}}} \right) \,x^{\frac{5}{2}} \nonumber \\ &+&
\left( - \frac{143137995}{137438953472} -
\frac{128033\,m^2\,R^2}{3932160} + \frac{47\,m^3\,R^3}{960} +
\frac{m^4\,R^4}{36} + \frac{47\,R^2\,{\gamma }^2}{640} \right) \,x^3
\nonumber\\ &+ & \left( \frac{1476620145}{4398046511104\,{\sqrt{2}}}
- \frac{45736601\,m^2\,R^2}{31708938240\,{\sqrt{2}}} +
\frac{45149\,m^3\,R^3}{1290240\,{\sqrt{2}}} +
\frac{229\,m^4\,R^4}{8064\,{\sqrt{2}}} + \frac{45149\,R^2\,{\gamma
}^2}{860160\,{\sqrt{2}}} \right) \, x^{\frac{7}{2}} +{\cal O}(x^4)
\nonumber \\ g_{1R} &=& g_{2R} \qquad \left( x \to e^{2\pi i} x
\right)\end{aligned}$$
LOCAL Diagonalization and perturbative mixing {#sec:local_rot}
=============================================
One of the puzzles one encounters in the description of the deuteron with the OPE potential is that while the dimensionless D/S ratio parameter is rather small at long distances $w / u \to \eta =0.0256$, it actually comes from a strong mixing at short distances where $ w/u
\to 1/\sqrt{2} \sim 0.707 $. Actually, the analog question for scattering states is that there seems to be a natural hierarchy for the phase shifts in the $^3S_1-{}^3D_1$ channel, namely $ \delta_{3S1}
\gg \delta_{3D1} \gg \epsilon_1 $ even though the threshold behavior of the D-wave is more suppressed than that of the mixing angle. The question is whether one can think of an expansion in terms of the $\eta$ parameter. There are two obvious situations where the mixing does not occur. One is the absence of tensor force. In the OPE potential that would also eliminate the $D-wave$. Another situation is dropping the mixing terms in the OPE potential, which is questionable since they are actually larger than the diagonal terms. It is possible, however, to write the equations in a form that the mixing is manifestly small at [*all*]{} distances. To this end we make a local rotation of the deuteron wave functions $$\begin{aligned}
\begin{pmatrix}
u(r) \\ w(r)
\end{pmatrix} =
\begin{pmatrix}
\cos\theta(r) & \sin\theta(r) \\ -\sin\theta(r) & \cos \theta(r)
\end{pmatrix}
\begin{pmatrix}
u_A (r)\\ u_R (r)
\end{pmatrix} \end{aligned}$$ in such a way as to diagonalize the potential we have $$\begin{aligned}
\begin{pmatrix}
U_s & U_{sd} \\ U_{sd} & U_{d} + \frac6{r^2}
\end{pmatrix}
&&= \nonumber \\
\begin{pmatrix}
\cos\theta & \sin\theta \\ -\sin\theta & \cos \theta
\end{pmatrix} &&
\begin{pmatrix}
U_A & 0 \\ 0 & U_R
\end{pmatrix}
\begin{pmatrix}
\cos\theta & \sin\theta \\ -\sin\theta & \cos \theta
\end{pmatrix} \end{aligned}$$
The deuteron equations for the OPE potentials read after the local rotation
$$\begin{aligned}
-u_A '' (r) + \left[ U_A (r) + \theta' (r)^2 \right] u_A (r) +
\gamma^2 u_A (r) &=& \left[2 \theta'(r) u_R' (r) + \theta''(r) u_R (r)
\right] \, ,\nonumber \\
-u_R '' (r) + \left[ U_R (r) + \theta' (r)^2 \right] u_R (r) + \gamma^2 u_R (r) &=& -\left[2 \theta'(r) u_A' (r) + \theta''(r) u_A (r) \right]
\, ,\nonumber \\
\label{eq:sch_rotated} \end{aligned}$$
In the coupled channel space these equations can be visualized as a particle with spin in the presence of a gauge potential $\theta'(r)$.
At long distances we have the expansions $$\begin{aligned}
\theta &=& \frac{ 2 \sqrt{2} R}{9 r} e^{-m r } (m^2 r^2 + 3 m r +3 ) +
\dots \\ U_A &=& - \frac{2 m^2 R}{3 r} e^{-m r} + \dots \\ U_R &=&
\frac{6}{r^2} + \frac{2R }{3r^3} ( m^2 r^2 + 6 m r + 6 ) e^{-m r} +
\dots\end{aligned}$$ whereas at short distances we have the behavior $$\begin{aligned}
\theta &=& \cos^{-1} \sqrt{\frac23}- \frac{r}{3\sqrt{2} R} +
\frac{r^2}{18 \sqrt{2} R^2} \nonumber \\ && - \frac{ (18 m^2 R^2 -5) r^3 }{324
\sqrt{2}R^3 }+ \dots \\ U_A &=& -\frac{4 R}{r^3}+
\frac{2}{r^2} - \frac{2}{3 r R} + \dots \\ U_R &=& \frac{8 R}{r^3} +
\frac{4}{r^2} + \frac{2 - 6 m^2 R^2}{3 R r} + \dots
\label{eq:rot_short}\end{aligned}$$ Note that in the locally rotated basis the mixing is related to the derivative of the mixing angle, $\theta'$ which is small at all distances (See Fig. \[fig:theta\]). Actually, at asymptotically large distances we have $$\begin{aligned}
u_A (r) \to u(r) \qquad u_R(r) \to w(r)\end{aligned}$$ If we neglect the mixing term in Eq. (\[eq:sch\_rotated\]) the equations decouple and, actually, there is no non-trivial solution for the repulsive eigen-channel, since the energy is fixed arbitrarily. Hence in the absence of mixing we have $ u_R=0$. At this level of approximation we then get $$\begin{aligned}
u(r) &=& \cos \theta (r) u_A(r) \\
w(r) &=& \sin \theta (r) u_A(r) \end{aligned}$$ In Fig. \[fig:urotated\] we show the solutions of the decoupled equations compared to the exact ones. As we see, the difference in the wave functions and hence the $D/S$ mixing is indeed small. Note that this is [*not*]{} the same as to neglect the tensor force. The results for the deuteron observables are presented in Table \[tab:table1\]. As we see, the quality of the zeroth $\eta$ approximation is rather good.
$\gamma ({\rm fm}^{-1}$ $\eta$ $A_S ( {\rm fm}^{-1/2}) $ $r_m ({\rm fm})$ $Q_d ( {\rm fm}^2) $ $P_D $
--------------------- ------------------------- ----------- --------------------------- ------------------ ---------------------- ----------
[OPE]{}-$\eta$ (LO) Input 0 0.8752 1.9423 0.1321 6%
[OPE]{}-pert (NLO) Input 0.051 0.7373 1.6429 0.4555 0
[OPE]{}-exact Input 0.02633 0.8681(1) 1.9351(5) 0.2762(1) 7.88(1)%
NijmII Input 0.0253(2) 0.8845(8) 1.968(1) 0.271(1) 5.67(4)%
Exp. (non-rel.) 0.231605 0.0256(4) 0.8846(9) 1.971(6) 0.2859(3) 5.67(4)%
Actually, we can check [*a posteriori*]{} that the mixing is indeed small for the zeroth order solutions. The inhomogeneous term at short distances behaves as $$\begin{aligned}
2 \theta'(r) u_A' (r) \to - \frac{2}{3 \sqrt{2} R^2}
\left(\frac{r}{R}\right)^{-3/4} C_A \sin \left( 4 \sqrt{R/r} +
\alpha \right) \nonumber \\ \end{aligned}$$ which compared to the remaining terms in Eq. (\[eq:sch\_rotated\]) can indeed be considered small. Under these circumstances, the mixing can then be included perturbatively, yielding $$\begin{aligned}
u_R (r) = \int_0^\infty dr' G_R (r,r') \left[2 \theta'(r') u_A' (r') +
\theta''(r ') u_A(r') \right] \nonumber \\
\label{eq:u_R-pert}\end{aligned}$$ with $G_R (r,r')$ the Green function of the homogeneous equation in the repulsive eigen-channel, $$\begin{aligned}
G_R(r,r') &=& w_{\rm reg} (r) w_{\rm irreg} (r') \theta (r'-r) \nonumber \\
&+& w_{\rm reg} (r') w_{\rm irreg} (r) \theta( r-r') \end{aligned}$$ with Wronskian equal to unity and $w_{\rm reg} (r) $ and $w_{\rm irreg}
(r)$ the regular solution and irregular solutions at the origin respectively. Asymptotically one has, $$\begin{aligned}
w_{\rm reg} (r) &\to& C \left(\frac{r}{R}\right)^{3/4} e^{-4 \sqrt{2}
\sqrt{R/r}} \qquad \qquad ( r \to 0 ) \nonumber \\ w_{\rm reg} (r)
&\to& e^{+\gamma r} \left( 1 - \frac{3}{\gamma r} + \frac{3}{(\gamma
r)^2} \right) \qquad ( r \to \infty ) \nonumber \\ w_{\rm irreg} (r)
&\to& C \left(\frac{r}{R}\right)^{3/4} e^{+4 \sqrt{2} \sqrt{R/r}}
\qquad \qquad ( r \to 0 ) \nonumber \\ w_{\rm irreg} (r) &\to&
e^{-\gamma r} \left( 1 + \frac{3}{\gamma r} + \frac{3}{(\gamma r)^2}
\right) \qquad ( r \to \infty ) \nonumber \\
\label{eq:asym}\end{aligned}$$ To get in practice the coefficient $C$ we start with $C=1$ at short distances and build the ratio to the asymptotic form at a sufficiently large distance. With these conditions the solution $u_R (r) $ at large distances behaves as $$\begin{aligned}
u_R (r) &\to& \eta e^{-\gamma r} \left( 1 +
\frac{3}{\gamma r} + \frac{3}{(\gamma r)^2} \right) \qquad ( r \to
\infty ) \nonumber\\\end{aligned}$$ with $$\begin{aligned}
\eta = \int_0^\infty dr w_{\rm reg} (r) \left[2 \theta'(r) u_A' (r) +
\theta''(r) u_A (r) \right]\end{aligned}$$ At short distances we get, from Eq. (\[eq:u\_R-pert\]) and using the asymptotic forms of Eq. (\[eq:asym\]) and Eq. (\[eq:rot\_short\]), the result $$\begin{aligned}
u_R (r) &\to& C C_A \left(\frac{r}{R}\right)^{7/4} \cos\left( 4 \sqrt{R/r} +
\alpha \right)\end{aligned}$$ in agreement with the leading short distance behavior of the full solution for the combination $u (r) - \sqrt{2} w(r) $ (see Sect. \[sec:short\]). The perturbative value for the asymptotic $D/S$ ratio we get is $$\begin{aligned}
\eta_{\rm pert} = 0.0261\end{aligned}$$ quite close to the OPE exact one, $ \eta_{\rm OPE}= 0.0263$.
Long distance solutions
=======================
As a complement to the perturbative treatment of Appendix \[sec:pert\] we analyze the bound solutions at long distances. The asymptotic deuteron wave functions for the OPE potential can be written in the form $$\begin{aligned}
u (r) &=& e^{-\gamma r} \left[ \sum_{k} F_{k} (r) e^{- k m
r} \right] \\ w(r) &=& \eta e^{-\gamma r} \left( 1 +
\frac{3}{\gamma r} + \frac{3}{(\gamma r)^2} \right) \left[ \sum_{k}
G_{k} (r) e^{- k m r} \right] \nonumber \\ \end{aligned}$$ The first order solution can be evaluated analytically, yielding
$$\begin{aligned}
F_1(r) &=& \frac{R\,e^{- m\,r} \left( m^2\,r^2 - 2\,\left( 1 + r\,\gamma \right) -
2\,m\,r\,\left( 1 + r\,\gamma \right) \right) \,\eta }{{\sqrt{2}}\,\,r^3\,
{\gamma }^2}
\nonumber \\ &+&
\frac{m^2\,R\,\left( 3\,{\sqrt{2}}\,m^2\,\eta +
{\gamma }^2\,\left( 4 - 4\,{\sqrt{2}}\,\eta \right) \right) \,
{\rm Ei}(-m\,r )}{12\,{\gamma }^3} \nonumber \\ &+&
\frac{e^{2\,r\,\gamma }\,m^2\,R\,\left( -3\,{\sqrt{2}}\,m^2\,\eta +
4\,{\gamma }^2\,\left( -1 + {\sqrt{2}}\,\eta \right) \right) \,
{\rm Ei}(- m\,r - 2\,r\,\gamma )}{12\,{\gamma }} \\
G_1(r) &=& \frac{m^2\,R\,\left( -4\,{\gamma }^2\,\left(
{\sqrt{2}} - 2\,\eta \right) + 3\,m^2\,\left( {\sqrt{2}} - \eta
\right) \right) \, {\rm Ei}(- m\,r
)}{12\,{\gamma }^3\,\eta } \nonumber \\
&+& \frac{R\, e^{- m\,r} \left( 3\,m^3\,r\,\left( {\sqrt{2}} - \eta \right) +
m^2\,r\,\gamma \,\left( -3 + r\,\gamma \right) \,\left(
{\sqrt{2}} - \eta \right) + 2\,{\gamma }^2 (1 + m r) \,\left( 2\,\eta +
r\,\gamma \,\left( -{\sqrt{2}} + \eta \right) \right) \right) }{2\,\,r\,
{\gamma }^2\,\left( 3 + 3\,r\,\gamma + r^2\,{\gamma }^2 \right)
\,\eta } \nonumber \\
&+& \frac{e^{2\,r\,\gamma
}\,m^2\,R\,\left( 3 - 3\,r\,\gamma + r^2\,{\gamma }^2 \right)
\, \left( 4\,{\gamma }^2\,\left( {\sqrt{2}} - 2\,\eta \right) -
3\,m^2\,\left( {\sqrt{2}} - \eta \right) \right) \,
{\rm Ei}(- m\,r - 2\,r\,\gamma
)}{12\,{\gamma }^3\, \left( 3 + 3\,r\,\gamma + r^2\,{\gamma }^2
\right) \,\eta } \end{aligned}$$
The second order can also be evaluated but the expression is too long to be presented here. In Fig. \[fig:u+w\_long.epsi\] we present the first order solutions compared to the exact ones. The perturbative solutions of Appendix \[sec:pert\] are obtained from the requirement that the S-wave $u$, becomes normalizable when extended down to the origin. This can only happen in the $D/S$ asymptotic ratio, $\eta$ takes the value given by Eq. (\[eq:eta-pert\]). This illustrates the fact that perturbation theory can always be applied at long distances but fails at short distances.
Note that here one treats the coupling constant $R$ and the mixing parameter $\eta$ as independent variables.
[99]{}
T. E. O. Ericson and W. Weise, [*Pions And Nuclei*]{}. 1988. Oxford, UK. Clarendon (1988) 479 p. (The International Series of Monographs on Physics, 74). N. K. Glendenning and G. Kramer, Phys. Rev. [**126**]{}, 2159 (1962).
S. Klarsfeld, J. Martorell and D. W. L. Sprung, Nucl. Phys. A [**352**]{} (1980) 113. S. Klarsfeld, J. Martorell and D. W. L. Sprung, J. Phys. G [**10**]{} (1984) 165. T. E. O. Ericson and M. Rosa-Clot, Phys. Lett. B [**110**]{}, 193 (1982). T. E. O. Ericson and M. Rosa-Clot, Nucl. Phys. A [**405**]{}, 497 (1983). T. E. O. Ericson and M. Rosa-Clot, Ann. Rev. Nucl. Part. Sci. [**35**]{} 271 (1985).
J. L. Friar, B. F. Gibson and G. L. Payne, Phys. Rev. C [**30**]{}, 1084 (1984)
J. L. Ballot, A. M. Eiro and M. R. Robilotta, Phys. Rev. C [**40**]{}, 1459 (1989). J. L. Ballot and M. R. Robilotta, Phys. Rev. C [**45**]{}, 986 (1992). D. W. L. Sprung, W. van Dijk, E. Wang, D. C. Zheng, P. Sarriguren, and J. Martorell Phys. Rev. C [**49**]{}, 2942 (1994)
S. Weinberg, Phys. Lett. B [**251**]{} (1990) 288. S. Weinberg, Nucl. Phys. B [**363**]{} (1991) 3. C. Ordonez, L. Ray and U. van Kolck, Phys. Rev. C [**53**]{}, 2086 (1996) C. Ordonez, L. Ray and U. van Kolck, Phys. Rev. Lett. [**72**]{}, 1982 (1994). C. Ordonez and U. van Kolck, Phys. Lett. B [**291**]{}, 459 (1992). P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. [**52**]{}, 339 (2002) M. C. Birse, J. A. McGovern and K. G. Richardson, Phys. Lett. B [**464**]{}, 169 (1999) D. B. Kaplan, M. J. Savage and M. B. Wise, Phys. Rev. C [**59**]{}, 617 (1999) T. S. Park, K. Kubodera, D. P. Min and M. Rho, Nucl. Phys. A [**646**]{}, 83 (1999) T. Frederico, V. S. Timoteo and L. Tomio, Nucl. Phys. A [**653**]{}, 209 (1999) E. Epelbaum, W. Gloeckle and U. G. Meissner, Nucl. Phys. A [**671**]{}, 295 (2000) D. R. Phillips and T. D. Cohen, Nucl. Phys. A [**668**]{} (2000) 45 D. R. Entem and R. Machleidt, Phys. Lett. B [**524**]{}, 93 (2002) K. M Case, Phys. Rev. [**80**]{}, 797 (1950)
W. M. Frank, D. J. Land, and R .M. Spector, Rev. Mod. Phys. [**43**]{}, 36 (1971).
M. Pavon Valderrama and E. Ruiz Arriola (in preparation)
S. R. Beane, P. F. Bedaque, M. J. Savage and U. van Kolck, Nucl. Phys. A [**700**]{} (2002) 377 S. R. Beane, arXiv:hep-ph/0204107. A. Bulgac, G. A. Miller and M. Strikman, Phys. Rev. C [**56**]{} (1997) 3307 S. R. Beane and M. J. Savage, Nucl. Phys. A [**713**]{}, 148 (2003) E. Epelbaum, U. G. Meissner and W. Gloeckle, Nucl. Phys. A [**714**]{}, 535 (2003) S. R. Beane and M. J. Savage, Nucl. Phys. A [**717**]{}, 91 (2003) J. J. de Swart, M. C. M. Rentmeester and R. G. E. Timmermans, PiN Newslett. [**13**]{}, 96 (1997) T. E. O. Ericson, B. Loiseau and A. W. Thomas, Phys. Rev. C [**66**]{} (2002) 014005 V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen and J. J. de Swart, Phys. Rev. C [**49**]{}, 2950 (1994) M. Pavon Valderrama and E. Ruiz Arriola, Phys. Lett. B [**580**]{}, 149 (2004) M. Pavon Valderrama and E. Ruiz Arriola, Phys. Rev. C [**70**]{}, 044006 (2004)
D. B. Kaplan, M. J. Savage and M. B. Wise, Nucl. Phys. B [**534**]{}, 329 (1998) J. J. de Swart, C. P. F. Terheggen and V. G. J. Stoks, arXiv:nucl-th/9509032. V. G. Stoks, R. A. Kompl, M. C. Rentmeester and J. J. de Swart, Phys. Rev. C [**48**]{}, 792 (1993). M. Pavon Valderrama and E. Ruiz Arriola, arXiv:nucl-th/0407113.
T. D. Cohen and J. M. Hansen, Phys. Lett. B [**440**]{}, 233 (1998) S. Klarsfeld, J. Martorell, J. A. Oteo, M. Nishimura and D. W. L. Sprung, Nucl. Phys. A [**456**]{}, 373 (1986).
D. W. L. Sprung, Hua Wu and J. Martorell, Phys. Rev. C [**42**]{}, 863A (1990).
[^1]: The solutions for $u_A $ and $u_R$ are written in terms of spherical Bessel functions [@Martorell94]. We keep the leading short distance behavior only.
[^2]: Numerically we find at the cut-off boundary $r=0.2 {\rm fm}$ $$\begin{aligned}
u(0.2) &=& 1139.23 - 43263.2\,\eta \nonumber \\
w(0.2) &=& -1807.33 + 68632.5\,\eta \nonumber \\
u'(0.2) &=& -35529.8 + 1.34913 \cdot \,{10}^6\,\eta \nonumber \\
w'(0.2) &=& 55194.3 - 2.09606 \cdot \,{10}^6\,\eta \end{aligned}$$ These large numbers appear because the of the dominance of the diverging exponential at short distances.
[^3]: These authors look for poles of the $S-$matrix so constructed as to reproduce the physical scattering length $\alpha_0 = 5.42 {\rm fm} $ and effective range $r_0 = 1.75
{\rm fm}$ at the physical value of the pion mass in the $^3S_1$ eigen channel. By doing so the explicit dependence on $g_{\pi NN}$ becomes rather weak. Actually, they take $ g_{\pi NN}=12.73 $ and we would get instead $ B_d ( 0 , g_{\pi NN} ) = 0.98 {\rm MeV} $ instead. This apparent contradiction is resolved by noting that, as we will see below, for $g_{\pi NN} = 13.1083$ in the OPE we get an scattering length $\alpha_0 = 5.335 {\rm fm} $ an effective range of $r_0=1.63$ quite close to the experimental values.
[^4]: If we take $R_0 = 1.1 R $ we get $B_d^0= 33 {\rm MeV}$.
[^5]: This property does not hold for other triplet channels with higher partial waves, because there are no bound states in those channels. Nevertheless, it is also true that there is only one independent parameter. This means in practice that one can use one scattering length out of the three to predict the phase shifts also in other partial waves.
[^6]: The values of $\alpha_0$ and $r_0$ have been determined in Ref. [@deSwart:1995ui], whereas $\alpha_{02}$ and $\alpha_2$ have been determined by us in Ref. [@PavonValderrama:2004nb]. See also Ref. [@PavonValderrama:2004se] for a extensive determination in all partial waves.
[^7]: The main difference in this regard has to do with the multi-valuation problem of the potential counter-term in Ref. [@Beane:2001bc] typical of inverse scattering problems. The approach of Ref. [@PavonValderrama:2004nb] does not have this problem.
[^8]: This expansion looks similar to a coupled channel WKB expansion but it is free of some inconveniencies. The applicability condition of the coupled channel WKB method would be that the de Broglie local wavelength [*matrix*]{} should be a slowly varying function of distance, implying in turn three conditions on the corresponding local wavelength eigenvalues as well as the corresponding WKB mixing angle which need not be necessarily satisfied [*simultaneously*]{}, generating conversion mode problems.
[^9]: For instance at $ r= 0.8 {\rm fm}$ we get $$\begin{aligned}
u &=& 1.9683 - 59.4526\,\eta \nonumber \\ &=& 1.14054\, \bar
C_{1A} - 52.3866\,C_{1R} + 1.09959\, \bar C_{2A} -
0.00169556\,C_{2R}, \nonumber \\ w &=& -4.00531 + 159.28\,\eta
\nonumber \\ &=& -0.667631\, \bar C_{1A} - 551.52\, C_{1R} +
1.67549\, \bar C_{2A} + 0.0177228\, C_{2R} \nonumber \\ u' &=&
-6.84126 + 287.992\,\eta \nonumber \\ &=& -1.34949\, \bar C_{1a} +
273.671\, C_{1R} + 4.02872\, \bar C_{2A} - 0.00925419\, C_{2R}
\nonumber \\ w' &=& 16.2726 - 607.14\,\eta \nonumber \\ &=&
-0.667631\, \bar C_{1A} - 551.52\, C_{1R} + 1.67549\, \bar C_{2A} +
0.0177228\,C_{2R} \nonumber \end{aligned}$$ where the l.h.s. corresponds to the numerical solution and the r.h.s. to the short distance approximation, and the barred coefficients $\bar C_{1A} = (C_{1A}+ C_{2A})/2$ and $\bar C_{2A} =
(-C_{1A} + C_{2A})/2\,i$ have been introduced.
|
---
abstract: 'To date, no framework combining quantum field theory and general relativity and hence unifying all four fundamental interactions, exists. Violations of the equivalence principle (EEP), being the foundation of general relativity, may hold the key to a theory of . The universality of free fall (UFF), which is one of the three pillars of the EEP, has been extensively tested with classical bodies. Quantum tests of the UFF, e.g. by exploiting matter wave interferometry, allow for complementary sets of test masses, orders of magnitude larger test mass coherence lengths and investigation of spin-gravity coupling. We review our recent work towards highly sensitive matter wave tests of the UFF on ground. In this scope, the first quantum test of the UFF utilizing two different chemical elements, and , yielding an ratio $=\etav$ has been performed. We assess systematic effects currently limiting the measurement at a level of parts in $10^8$ and finally present our strategies to improve the current state-of-the-art with a test comparing the free fall of rubidium and ytterbium in a very long baseline atom interferometry setup. Here, a $\SI{10}{m}$ baseline combined with a precise control of systematic effects will enable a determination of the ratio at a level of parts in $10^{13}$ and beyond, thus reaching and overcoming the performance limit of the best classical tests.'
author:
- 'D. Schlippert'
- 'H. Albers'
- 'L. L. Richardson'
- 'D. Nath'
- 'H. Heine'
- 'C. Meiners'
-
- 'A. Billon'
- 'J. Hartwig'
- 'C. Schubert'
- 'N. Gaaloul'
- 'W. Ertmer'
- 'E. M. Rasel'
bibliography:
- 'RASEL.bib'
title: 'Ground Tests of Einstein’s Equivalence Principle: From Lab-based to 10-m Atomic Fountains'
---
Introduction
============
With the great success of the grand unification theory [@Georgi74PRL] the question arose whether the remaining fourth interaction, gravitation, could be unified with the other three yielding a . However, all approaches trying to merge quantum field theory and general relativity to a framework consistent over all energy scales have failed so far [@Laemmerzahl06APB]. Hence, in spite of both theories being confirmed at outstanding precision on their own, extensions of at least one of them, e.g. additional fields, are necessary in order to resolve their incompatibility.\
General relativity is fully based on the postulates constituting equivalence principle (EEP). Next to local position invariance and local invariance, the EEP comprises the universality of free fall (UFF), which states that in absence of other forces all bodies located at the same space-time point experience the same acceleration in a gravitational field independently of their composition when neglecting self-gravity. While scrutinizing the EEP, it moreover was identified that under certain circumstances the UFF can be treated as direct empirical foundation for EEP [@Will14]. Hence, tests of the UFF are a promising candidate in order to further investigate possible extensions of our understanding of gravity compatible with a theory of .\
A validity of the UFF implies the equality of inertial mass $m_{\text{in}}$ and gravitational mass $m_{\text{gr}}$ of any test body. In 1884, described the fact that gravity, unlike any other interaction, acts identically on all bodies independently of their gravitational charge as a ** [@Hertz99]. The so called ratio $$\label{eq:eotvos}
\eta_{\,\text{A,B}}\equiv 2\thickspace\frac{g_{\text{A}}-g_{\text{B}}}{g_{\text{A}}+g_{\text{B}}}
=2\thickspace\frac{\left(\frac{m_{\text{gr}}}{m_{\text{in}}}\right)_{\text{A}}
-\left(\frac{m_{\text{gr}}}{m_{\text{in}}}\right)_{\text{B}}}
{\left(\frac{m_{\text{gr}}}{m_{\text{in}}}\right)_{\text{A}}
+\left(\frac{m_{\text{gr}}}{m_{\text{in}}}\right)_{\text{B}}}\;,$$ where $g_{\text{i}}$ is the gravitational acceleration of test body $i=A,B$ is a comprehensive figure of merit when testing the UFF of test bodies A and B and is non-zero in case of a violation of the UFF.\
Tests of the UFF emerged from thought experiment in the 16^th^ century of comparing the free fall of different cannon balls dropped from the leaning tower of Pisa, commonly referred to as tests [@Stillman03]. A demonstration test of this kind was performed during the Apollo 15 mission in 1971 by dropping a hammer and a feather on the Moon [@Apollo15]. The most accurate measurements of the Eötvös ratio were performed by i) monitoring the distance between Earth and Moon in free fall around the Sun by means of laser ranging [@Williams04PRL; @Mueller12CQG], yielding $\eta_{\,\text{Earth,Moon}}=(-0.8\pm 1.3)\times 10^{-13}$ and ii) employing a torsion balance [@Eotvos89] with beryllium and titanium test masses [@Schlamminger08PRL] yielding $\eta_{\,\text{Be,Ti}}=(0.3\pm 1.8)\times 10^{-13}$. The best test used a laser interferometer to read out the differential free fall motion of copper and uranium test masses [@Niebauer87PRL] and found $\eta_{\,\text{Cu,U}}=(1.3\pm 5.0)\times 10^{-10}$.\
The aforementioned tests employ classical, macroscopic bodies as test masses. In a complementary approach, the UFF can also be tested with quantum objects by observing the interference of massive particles such as neutrons or atoms under the influence of gravity. As first demonstrated in 1973 by , , and [@Colella75PRL], the gravitationally induced phase shift imprinted on a particle’s wave function is either compared to a classical gravimeter or to a second quantum object.\
Quantum tests of the UFF differ from their classical counterparts in various aspects. Matter wave tests extend the set of test masses by allowing to employ any laser-coolable species. Furthermore, use of cold atoms add the spin as a degree of freedom and enables investigation of spin-gravity coupling [@Laemmerzahl06APB], and the accessible ultracold temperatures are inherently linked to macroscopic coherence lengths [@Goeklue08CQG] which stands in fundamental contrast to classical test masses.\
Quantum tests of the UFF that have been performed in the past can be classified in three categories: i) semi-classical tests, comparing an atom interferometer to a classical gravimeter [@Peters99Nature; @Merlet10Metrologia] and reaching accuracies on the ppb-level; ii) quantum tests at a level of parts in $10^7$ comparing the free fall of rubidium [@Fray04PRL; @Bonnin13PRA; @Zhou15arxiv] or strontium [@Tarallo14PRL] isotopes; iii) quantum tests comparing the free fall of two different chemical elements [@Schlippert14PRL].\
Analyzing a test mass pair in a given framework, e.g. a test theory [@Damour12CQG] or a parametrization [@Hohensee13PRL], allows to quantify the influence of a violation of the UFF ruled out with a given test mass pair. In general, a well-suited test mass pair fulfills $m_{\text{A}}\gg m_{\text{B}}$ or vice versa, making different chemical elements generally interesting test pairs. Accordingly, with their naturally low relative mass difference comparisons of heavier isotopes suffer from lower sensitivity to violations. On the other hand, however, they benefit from strong rejection of noise sources [@Bonnin13PRA] and systematic errors [@Aguilera14CQG; @Hogan08arXiv].\
\
This article is organized as follows: In \[sec:QTUFF\], we provide a brief overview on the underlying theory of dual species matter wave interferometry and summarize the first quantum test of the UFF using two different chemical elements, and . We furthermore discuss an assessment of the systematic biases influencing our measurement. focuses on our strategies aiming towards a state-of-the-art test of the UFF comparing the free fall of ytterbium and rubidium in a $\SI{10}{m}$ very long base line atom interferometry setup. This article closes with an outlook into the future of matter wave tests of the UFF and a conclusion in \[sec:conclusion\].
Quantum test of the universality of free fall of and {#sec:QTUFF}
======================================================
In order to observe the gravitational acceleration acting on and , we employ the -type matter wave interferometer geometry [@Kasevich91PRL] realized with stimulated transitions coupling the states $\ket{F_i=1,\,p}$ and $\ket{F_i=2,\,p\pm\hbar\,\keffi{i}}$ as displayed in \[fig:mzdual\]. In this configuration, we make use of an effective wavefront acceleration $\frac{\alpha}{\keff}$ caused by a linear frequency ramp $\alpha$ of the beam splitting light frequency difference with effective wave vector $\keff$. This acceleration enters the leading order phase shift as (throughout this Section, $i$ is Rb or K) $$\label{eq:phaseshift}
\Delta\phi_i=(g_i-\frac{\alpha_i}{\keffi{i}})\cdot\keffi{i}\cdot T^2\;.$$ An experimental cycle starts by collecting $\SI{8e8}{atoms}$ of and $\SI{3e7}{atoms}$ of from a transversely cooled atomic beam within $\SI{1}{s}$ in a three-dimensional magneto-optical trap. The ensembles are subsequently cooled down to sub- temperatures utilizing the techniques described in Refs. [@Landini11PRA; @Chu98; @Phillips98] yielding temperatures $T_{\text{Rb}}=\SI{27}{\micro K}$ and $T_{\text{K}}=\SI{32}{\micro K}$. Optical pumping is utilized to prepare the atoms in the $\ket{F_i=1}$ Zeeman manifold. By switching off all cooling light fields, the atoms are subsequently released into free fall.\
A sequence of three Raman light pulses separated by the time $T$ is employed to form a -type interferometer while applying a linear chirp $\alpha$ on the laser difference frequency causing an acceleration of the wavefronts of the beam splitters. Afterwards, the exit ports of the interferometer are selectively read out by optical pumping and detection of fluorescence driving the $\ket{F_i=2}\rightarrow\ket{F'_i=3}$ transition. A single experimental cycle takes $\approx\SI{1.6}{s}$.\
By varying the the effective wavefront acceleration, a global phase minimum appears independently of the free evolution time $T$ where $g-\frac{\alpha}{\keff}=0$ and thus allows to determine $g$. shows the determination of gravitational acceleration $a_i^{(\pm)}(g)$ of and for the upward and downward direction of momentum transfer. Here, observation of the phase shift for both directions allows to strongly suppress systematic phase shifts that do not invert their sign when changing directions of momentum transfer by computing the half difference signal [@McGuirk02PRA; @Louchet-Chauvet11NJP].
![Space-time diagram of a dual-species matter wave interferometer in a constant gravitational field for the downward (thick lines) and upward (thin lines) direction of momentum transfer. Stimulated transitions at times $0$, $T$, and $2\,T$ couple the states $\ket{F_i=1,\,p}$ and $\ket{F_i=2,\,p\pm\hbar\,\keffi{i}}$, where $i$ stands for Rb (blue lines) or K (red lines). The velocity change induced by the pulses is not to scale with respect to the gravitational acceleration.[]{data-label="fig:mzdual"}](standalone_img/mzdual){width=".4\linewidth"}
![Determination of the differential gravitational acceleration of rubidium and potassium. Typical fringe signals and sinusoidal fit functions are plotted in dependence of the effective wavefront acceleration for pulse separation times $T=\SI{8}{ms}$ (black squares and solid black line), $T=\SI{15}{ms}$ (red circles and dashed red line), and $T=\SI{20}{ms}$ (blue diamonds and dotted blue line) for upward $(+)$ and downward $(-)$ direction of momentum transfer. The central fringe positions $\mathbf{a_i^{(\pm)}(g)}$ (dashed vertical lines), where $i$ is Rb or K, are shifted symmetrically around $g_i=[a_i^{(+)}(g)-a_i^{(-)}(g)]/2$ (solid vertical line). The data sets are corrected for slow linear drifts and offsets. []{data-label="fig:qtufffringes"}](standalone_img/qtufffringes){width=".9\linewidth"}
Data analysis
-------------
For testing the universality of free fall, the global phase minimum positions $a_i^{(\pm)}(g)$ in \[fig:qtufffringes\] are monitored continuously over $\approx\SI{4}{h}$ by tuning the effective acceleration of the wavefronts $\alpha_i^{(\pm)}/\keffi{i}$ around $a_i^{(\pm)}(g)$ in 10 steps per direction of momentum transfer with pulse separation time $T=\SI{20}{ms}$. Accordingly, the acquisition of $g_i=[a_i^{(+)}(g)-a_i^{(-)}(g)]/2$ takes $\SI{32}{s}$ in total and yields one data point for the ratio (\[eq:eotvos\]). The statistical uncertainty of the ratio measurement after $\SI{4096}{s}$ of integration is $\sigma_\eta=\stat$, dominated by technical noise of the potassium interferometer.\
In Table \[tab:systematics\] we list systematic effects influencing our measurement with overall bias of $\Delta\eta_{\text{tot}}=-5.4\times 10^{-8}$ and an uncertainty $\delta\eta_{\text{tot}}=3.1\times 10^{-8}$. A third column $\delta\eta^{\text{adv}}$ shows expected improved uncertainties at an overall level of parts per billion when using a dual-species optical dipole trap [@Zaiser11PRA] as a common source which allows to precisely collocate the ensembles and to control their differential center of mass motion and expansion.
Summary
-------
Taking into account the statistical uncertainty $\sigma_\eta$ and the bias $\Delta\eta_{\text{tot}}$, the ratio can be determined to $=\etav$. At the current stage, the experiment is solely limited by technical noise dominating the short-term instability of the potassium interferometer. Hence, in the quadratic sum the statistical uncertainty fully overrules our systematic uncertainty. By reducing technical noise sources common mode noise rejection [@Barrett15arxiv] between the interferometers will allow to push the experiment towards its limit posed by systematic uncertainty.
Very long baseline atom interferometry {#sec:VLBAI}
======================================
Experimental setup
------------------
As shown in section \[sec:QTUFF\] for a -type geometry, the sensitivity to accelerations of an atom interferometer scales with the square of the pulse separation time $T^2$. A natural way to improve this sensitivity is to increase the free-fall time of the atoms enabling longer pulse separation times. This is the main driver for ground-based very long baseline devices and micro-gravity experiments. The latter feature free-fall times up to several seconds (droptower, parabolic flights), minutes (sounding rockets), or even days (space stations, satellites) in a small and thus well characterized volume. The practical and technological challenges combined with the high costs limit, however, the use of such platforms. In this section, we report about an on-going project of a ground-based very long baseline atom interferometer (VLBAI) device that will extend the baseline of the apparatus described in section \[sec:QTUFF\] from to more than , allowing atoms to experience free-fall times up to $2T\sim\SI{1.3}{\second}$ in drop mode or up to $2T\sim\SI{2.6}{\second}$ in fountain mode. Together with our choice of species described below, we expect to reach an inaccuracy of $7\cdot 10^{-13}$ in the ratio in the near future [@Hartwig15NJP].\
As a device targeting a quantum test of the UFF, the proposed apparatus is designed as a dual-species gravimeter using ultra-cold mixtures of rubidium and ytterbium. The relevance of this species choice is motivated by the constraints possible to put on UFF violating theories, such as the dilaton scenario [@Damour12CQG] and the standard model extension [@Hohensee13PRL] (SME). In particular, an analysis in the SME framework shows that the Rb-Yb test pair choice is complementary to the Rb-K pair which was chosen for the previously described project, the QUANTUS/MAIUS/PRIMUS micro-gravity experiments [@Rudolph15NJP; @Seidel13proceedings; @Herrmann12] and the STE-QUEST [@Schubert13arxiv; @Aguilera14CQG] M4 satellite proposal.\
The extended size of the apparatus triggers specific engineering challenges to reach the UFF test performance announced above. As already demonstrated in other precision atom interferometers, a rotation compensation [@Lan12PRL; @Sugarbaker13PRL; @Dickerson13PRL] of the inertial reference mirror at rates of $\sim\si{\micro\radian\per\second}$ is required in order to mitigate the systematic uncertainty linked to the effect. Moreover, the use of rubidium atoms with magnetic susceptibility [@Steck] of requires magnetic shielding of a factor at least along the entire interferometry region. In this case, it extends over more than . Finally, the reduced diameter of the vacuum tube (for efficient magnetic shielding) limits its conductance and makes its evacuation down to challenging.\
Atomic sources
--------------
In order to fully take advantage of the long baseline without severe systematics limiting the performance, the size of the clouds during their free-fall must be kept as small as possible. This can be achieved by delta-kick collimation (DKC) techniques [@Chu1986pro; @AmmanPRL1997] already demonstrated in the scope of micro-gravity experiments [@Muentinga13PRL] or very-long-baseline atom fountains [@KovachyPRL2015]. In the current design, we plan for a mixture of rubidium and ytterbium with $2\cdot 10^5$ and $1\cdot 10^5$ atoms, respectively. Preliminary estimations show that with a DKC pulsed at few tens of milliseconds, it is possible to keep the radius of the mixture at around $\SI{2}{mm}$ after $\SI{1.5}{s}$ of free evolution time. Within this regime, the leading systematics effects are not expected to deteriorate the uncertainty of the UFF test [@Hartwig15NJP].\
Furthermore, the preparation time of such an ultra-cold mixture should not exceed in order to enable sufficient repetition rates for reaching a statistical error of $7\cdot10^{-13}$ after one day of averaging. This cycling rates should be within reach in view of recent development in the production of high-flux sources of degenerate gases [@Rudolph15NJP].
Dual-species launch for precision tests
---------------------------------------
The initial collocation and differential velocity of the two atomic clouds need to be kept small and very well characterized. Indeed, gravity gradients couple to the initial spatial offset and differential velocity inducing detrimental phase shifts at the output ports of the dual-interferometer [@Hogan08arXiv]. More precisely, the desired accuracy for a UFF test implies a maximum offset between the two clouds of about and a maximum relative velocity of about derived in previous work [@Hartwig15NJP]. Beyond these limits, the characterization of the gravity gradients becomes extremely challenging. In the condensed regime, the interactions play a crucial role in defining the symmetry of the ground state of the mixture. For a large overlap between the two test species, the choice of isotopes has to be restricted to miscible pairs. In a previous study [@Hartwig15NJP], we showed that the isotopes and can be good candidates to mix with thanks to their scattering length properties. Their natural abundances of $3\%$ and $0.1\%$, respectively, increase, however, the challenge for a high-flux source of suitable cold ytterbium atoms. The collocation requirement implies the use of a common trap for both species. Since the ground state of bosonic ytterbium cannot be magnetically trapped, a mid-infrared dipole trap will be used for this purpose. In order to fully unfold the potential of the baseline in terms of achievable free fall time, a fountain launch is necessary. Due to the very small diffrerential velocity allowed here, molasses launch is not sufficiently accurate. In a recent proposal [@Chamakhi2015], it was shown that a single lattice cannot drive atoms with different masses to the same velocity after an acceleration ramp. The use of two lattices to control each species is not possible due to crosstalks between the atoms transitions and the two light frequencies. It was rather suggested in the latter proposal to utilize two lattices at *tune-out* or *zero-magic* frequencies of one atom each. For rubidium, light frequencies for which the contribution of the D$_1$ and D$_2$ lines to the dipole potential balance, were recently precisely measured [@HeroldPRL2012] to an uncertainty below $\SI{1}{pm}$. Concerning ytterbium, there are, to our knowledge, no experimental data available but only theoretical calculations [@ChengMitroyPRA2013] predicting tune-out wavelengths at $\SI{358.78}{nm}$ and $\SI{553.06}{nm}$ with a large uncertainty of a fraction of a nanometer. It is therefore highly interesting to experimentally determine these wavelength for fundamental as well as practical reasons. Once this is done, it is possible to engineer a selective lattice launch accelerating the two atomic species to equal velocities up to few nm/s as suggested [@Chamakhi2015] for rubidium and potassium.\
The baseline presented in this section, would in this case close the precision gap between classical and quantum UFF tests utilizing interferometers with free fall times of up to $\SI{2.8}{\second}$.
Outlook & Conclusion {#sec:conclusion}
====================
Matter wave interferometers are a new tool with fascinating prospects for future investigations of gravity, its relation to quantum mechanics and related open questions [@Hamilton15PRL; @Hamilton15arxiv]. We demonstrated a test of the UFF with the two different chemical elements Rb and K to a level of $5\cdot10^{-7}$. With the same apparatus we anticipate an improvement by two orders of magnitude with the implementation of an optical dipole trap. We are setting up a large scale experiment with increased free fall time, targetting a UFF test with Rb and Yb to the level of $7\cdot10^{-13}$ competitive with classical tests. Pursuing tests of the universality of free fall is a very promising strategy to find the missing piece for a self-consistent framework valid over all energy scales and complementary to tests in space [@Touboul12CQG]. Matter wave interferometry is not only enlarging the choice of test materials, but also allows to probe gravity with new states of matter such as entangled atoms or even Schrödinger cats.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to W. P. Schleich, A. Roura, and H. Ahlers for fruitful discussions.\
This work was funded by the German Research Foundation (DFG) via the Sonderforschungsbereich (SFB) 1128 Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q) and the Cluster of Excellence Centre for Quantum Engineering and Space-Time Research (QUEST). It is also supported by the German Space Agency (DLR) with funds provided by the Federal Ministry for Economic Affairs and Energy (BMWi) due to an enactment of the German Bundestag under grant numbers DLR 50WM1131-1137 (QUANTUS-III) and DLR 50WM1142 (PRIMUS-II).
References {#references .unnumbered}
==========
|
---
author:
- |
K. J. Eskola,\
Department of Physics, University of Jyväskylä and Helsinki Institute of Physics, Finland\
E-mail:
- |
C. A. Salgado\
Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, Spain\
E-mail:
title: 'EPS09 - Global NLO analysis of nuclear PDFs and their uncertainties'
---
Introduction
============
The global analyses of the free-nucleon parton distribution functions (PDFs) are based on three fundamental aspects of QCD: asymptotic freedom, collinear factorization and scale evolution of the PDFs. These properties allow to compute the inclusive hard process cross-sections, schematically $$\sigma_{AB\rightarrow h+X} = \sum_{i,j} f_{i}^A(Q^2) \otimes \hat{\sigma}_{ij\rightarrow h+X} \otimes f_{j}^B(Q^2) + \mathcal{O}\left( {Q^{2}}\right)^{-n},$$ where $f_{i}$s denote the universal, process-independent PDFs obeying the DGLAP [@DGLAP] evolution equations, and $\hat{\sigma}_{ij\rightarrow h+X}$ are perturbatively computable coefficients. This approach has proven to work extremely well with increasingly more different types of data included in the analyses. In the case of bound nucleons the validity of factorization is not as well established (see e.g. Ref. [@Armesto:2006ph]), but it has nevertheless turned out to provide a very good description of the world data from deep inelastic scattering (DIS) and Drell-Yan (DY) dilepton measurements involving nuclear targets [@Eskola:1998iy; @Eskola:1998df; @Eskola:2007my; @Eskola:2008ca; @Hirai:2007sx; @deFlorian:2003qf]: in the nuclear environment the shape of the PDFs, however, is different from the free-nucleon PDFs. Here, we give an overview of our recent NLO global analysis of the nuclear PDFs (nPDFs) and their uncertainties [@Eskola:2009uj].
Analysis method and framework
=============================
Our analysis follows the usual global DGLAP procedure:
- [**The PDFs are parametrized at a chosen initial scale $Q_0^2$ imposing the sum rules.**]{} In this work we do not parametrize the absolute nPDFs, but the nuclear modification factors $R_i^A(x,Q_0^2)$ encoding the relative difference to the free-proton PDFs through $$f_{i}^A(x,Q^2) \equiv R_{i}^A(x,Q^2) f_{i}^{\rm CTEQ6.1M}(x,Q^2).
\label{eq:partondefinition}$$ Above, $f_{i}^{\rm CTEQ6.1M}(x,Q^2)$ refers to the CTEQ6.1M set of free proton PDFs [@Stump:2003yu] in the $\overline{MS}$ scheme and zero-mass variable flavour-number scheme. We consider three different modification factors: $R_V^A(x,Q_0^2)$ for both $u$ and $d$ valence quarks, $R_S^A(x,Q_0^2)$ for all sea quarks, and $R_G^A(x,Q_0^2)$ for gluons.
- [**The absolute PDFs at the parametrization scale $Q_0^2$ are connected to other perturbative scales $Q^2 > Q_0^2$ through the DGLAP evolution.**]{} In this way, also the nuclear modification factors $R_i^A(x,Q^2)$ become scale-dependent and the initial flavour-independence may also disappear. An efficient numerical solver for the parton evolution is an indispensable tool for any global analysis, but in the case of nPDFs this is even more critical as we always need to perform the evolution separately for 13 different nuclei — an order of magnitude more computation time than what is needed in a free-proton analysis.
- [**The cross-sections are computed using the factorization theorem.**]{} In the present analysis the DIS and DY data constitute the main part of the experimental input, but we also employ the midrapdity $\pi^0$-production data measured in d+Au collisions at RHIC. Inclusion of the $\pi^0$-data provides important further constraints for the gluon modification that are partly complementary to the DIS and DY data.
- [**The computed cross-sections are compared with the experimental data, and the initial parametrization is varied until the best agreement with the data is found.**]{} Our definition for the “best agreement” is based on minimizing the following $\chi^2$-function $$\begin{aligned}
\chi^2(\{a\}) & \equiv & \sum _N w_N \, \chi^2_N(\{a\})
\label{eq:chi2mod_1}
\\
\chi^2_N(\{a\}) & \equiv & \left( \frac{1-f_N}{\sigma_N^{\rm norm}} \right)^2 + \sum_{i \in N}
\left[\frac{ f_N D_i - T_i(\{a\})}{\sigma_i}\right]^2.
\label{eq:chi2}\end{aligned}$$ Within each data set $N$, $D_i$ denotes the experimental data value with $\sigma_i$ point-to-point uncertainty, and $T_i$ is the theory prediction corresponding to a parameter set $\{a\}$. If an overall normalization uncertainty $\sigma_N^{\rm norm}$ is specified by the experiment, the normalization factor $f_N \in [1-\sigma_N^{\rm norm},1+\sigma_N^{\rm norm}]$ is introduced. Its value is determined by minimizing $\chi^2_N$ and the final $f_N$ is thus an output of the analysis. The weight factors $w_N$ are used to amplify the importance of those data sets whose content is physically relevant, but contribution to $\chi^2$ would otherwize be too small to have an effect on the automated $\chi^2$ minimization.
In addition to finding the parameter set $\{a^0\}$ that optimally fits the experimental data, quantifying the uncertainties stemming from the experimental errors has become an increasingly important topic in the context of global PDF analyses. The Hessian method [@Pumplin:2001ct] provides a practical way of treating this issue. It is based on a quadratic approximation for the $\chi^2$ around its minimum $\chi^2_0$, $$\chi^2 \approx \chi^2_0 + \sum_{ij} \frac{1}{2} \frac{\partial^2 \chi^2}{\partial a_i \partial a_j} (a_i-a_i^0)(a_j-a_j^0) \equiv \chi^2_0 + \sum_{ij} H_{ij}(a_i-a_i^0)(a_j-a_j^0),
\label{eq:chi_2approx}$$ which defines the Hessian matrix $H$. Non-zero off-diagonal elements in the Hessian matrix are a sign of correlations between the original fit parameters, invalidating the standard (diagonal) error propagation formula $$(\Delta X)^2 = \left( \frac{\partial X}{\partial a_1} \cdot \delta a_1 \right)^2 + \left( \frac{\partial X}{\partial a_2} \cdot \delta a_2 \right)^2 + \cdots$$ for a PDF-dependent quantity (cross section) $X$. Therefore, it is useful to diagonalize the Hessian matrix, such that $$\chi^2 \approx \chi^2_0 + \sum_{i} z_i^2,
\label{eq:chi_2diag}$$ where each $z_i$ is a certain linear combination of the original parameters around $\{a^0\}$. In these variables, the usual form of the error propagation $$(\Delta X)^2 = \left( \frac{\partial X}{\partial z_1} \cdot \delta z_1 \right)^2 + \left( \frac{\partial X}{\partial z_2} \cdot \delta z_2 \right)^2 + \cdots
\label{eq:error_better}$$ stands on a much more solid ground. How to determine the size of the deviations $\delta z_k$ is, however, a difficult issue where no universally agreed procedure exists. We consider 15 parameters in the expansion (\[eq:chi\_2diag\]) and the prescription which we adopt here is to choose each $\delta z_k$ such that $\chi^2$ grows by a certain fixed amount $\Delta \chi^2$. Requiring each data set to remain close to its 90%-confidence range, we end up with a choice $\Delta \chi^2=50$ (see Appendix A of Ref. [@Eskola:2009uj] for details).
Much of the practicality of the Hessian method resides in constructing PDF error sets, which we denote by $S_k^\pm$. Each $S_k^\pm$ is obtained by displacing the fit parameters to the positive/negative direction along $z_k$ such that $\chi^2$ grows by the chosen $\Delta \chi^2=50$. Approximating the derivatives in Eq. (\[eq:error\_better\]) by a finite difference, we may then re-write the error formula as $$(\Delta X)^2 = \frac{1}{4} \sum_k \left[ X(S^+_k)-X(S^-_k) \right]^2,
\label{eq:error_best}$$ where $X(S^\pm_k)$ denotes the value of the quantity $X$ computed by the set $S_k^\pm$. If the lower and upper uncertainties $\Delta X^\pm$ differ, they should be computed separately, using the prescription [@Nadolsky:2001yg] $$\begin{aligned}
(\Delta X^+)^2 & \approx & \sum_k \left[ \max\left\{ X(S^+_k)-X(S^0), X(S^-_k)-X(S^0),0 \right\} \right]^2 \label{eq:ASymmetricPDFerrors} \\
(\Delta X^-)^2 & \approx & \sum_k \left[ \max\left\{ X(S^0)-X(S^+_k), X(S^0)-X(S^-_k),0 \right\} \right]^2, \nonumber\end{aligned}$$ where $S^0$ denotes the best fit.
Along with the grids and interpolation routine for the best NLO and LO fits, our new release — a computer routine called EPS09 [@EPS09code] — contains also 30 error sets required for computation of the uncertainties to nuclear cross-section ratios similar to those we present in the following section. However, as can be understood from the definition (\[eq:partondefinition\]), the total uncertainty in the absolute nPDFs is a combination of uncertainties from the baseline set CTEQ6.1M and those from the nuclear modifications $R_i^A$. Therefore, if an absolute cross-section is computed, also the CTEQ6.1M error-sets (with the EPS09 central set) should be included when calculating its uncertainty.
Results
=======
![The nuclear modifications $R_V$, $R_S$, $R_G$ for Lead at our initial scale $Q^2_0=1.69 \, {\rm GeV}^2$ and at $Q^2=100 \, {\rm GeV}^2$. The thick black lines indicate the best-fit results, whereas the dotted green curves denote the individual error sets. The shaded bands are computed using Eq. (\[eq:ASymmetricPDFerrors\]).[]{data-label="Fig:PbPDFs"}](Pb_allsets.eps)
In this section we present a selection of results from the NLO analysis. First, in Fig. \[Fig:PbPDFs\] we plot the obtained modifications at two scales, at $Q^2_0=1.69 \, {\rm GeV}^2$ and at $Q^2=100 \, {\rm GeV}^2$, which is to demonstrate the scale-dependence of the modifications and their uncertainties. One prominent feature to be noticed is that even if there is a rather large uncertainty band for the small-$x$ gluon modification $R_G^A$ at $Q^2_0$, the scale evolution tends to bring even a very strong gluon shadowing close to no shadowing at $Q^2\sim 100 \, {\rm GeV}^2$ — a very clear prediction of the DGLAP approach.
The DIS data constitutes the bulk of the experimental data available for the global analysis of nPDFs. In Fig. \[Fig:RF2A1\] we show some of the measured nuclear modifications for ($l+A$) DIS structure functions with respect to Deuterium, $$R_{F_2}^{\rm A}(x,Q^2) \equiv \frac{F_2^A(x,Q^2)}{F_2^d(x,Q^2)},$$ and the comparison with our EPS09 NLO results.
![The calculated NLO $R_{F_2}^A(x,Q^2)$ (filled symbols and error bands) compared with the NMC 95 [@Arneodo:1995cs] and the reanalysed NMC 95 [@Amaudruz:1995tq] data (open symbols).[]{data-label="Fig:RF2A1"}](RF2AD.eps)
The shaded bands denote the uncertainty derived from the 30 EPS09NLO error sets and, as should be noticed, their width is comparable to the error bars in the experimental data. This *a posteriori* supports the validity of our procedure for determining the $\Delta \chi^2$. Similar conclusion can be drawn upon inspecting the nuclear effects in the Drell-Yan data $$R_{\rm DY}^{\rm A}(x_{1,2},M^2) \equiv \frac{\frac{1}{A}d\sigma^{\rm pA}_{\rm
DY}/dM^2dx_{1,2}}{\frac{1}{2}d\sigma^{\rm pd}_{\rm DY}/dM^2dx_{1,2}},$$ where $M^2$ is the invariant mass of the lepton pair and $x_{1,2} \equiv \sqrt{M^2/s}\,e^{\pm y}$ ($y$ is the pair rapidity). Comparison to the E772 and E866 data is shown in Fig. \[Fig:RDY\]. Let us mention that the E772 data at $x_2 > 0.1$ carry some residual sensitivity also to the gluons and sea quarks, which has not been noticed before.
![The computed NLO $R_{\rm DY}^{\rm A}(x,M^2)$ (filled squares and error bands) as a function of $x_1$ and $x_2$ compared with the E866 [@Vasilev:1999fa] and the E772 [@Alde:1990im] data (open squares).[]{data-label="Fig:RDY"}](RDYe866.eps "fig:") ![The computed NLO $R_{\rm DY}^{\rm A}(x,M^2)$ (filled squares and error bands) as a function of $x_1$ and $x_2$ compared with the E866 [@Vasilev:1999fa] and the E772 [@Alde:1990im] data (open squares).[]{data-label="Fig:RDY"}](RDY.eps "fig:")
![ for inclusive pion production compared with the PHENIX [@Adler:2006wg] data (open squares). The error bars are the statistical uncertainties, and the yellow band indicates the point-to-point systematic errors. The PHENIX data have been multiplied by the optimized normalization factor $f_N = 1.03$, which is an output of our analysis. The STAR data [@Adams:2006nd] (open circles), multiplied by $f_N = 0.90$, are also shown although they were not included in the EPS09 analysis.[]{data-label="Fig:PHENIX"}](Phenix2007pi0.eps){width="18pc"}
The nuclear modification for inclusive pion production in d+Au collisions relative to p+p is defined as $$R_{\rm dAu}^{\pi} \equiv \frac{1}{\langle N_{\rm coll}\rangle} \frac{d^2 N_{\pi}^{\rm dAu}/dp_T dy}{d^2 N_{\pi}^{\rm pp}/dp_T dy} \stackrel{\rm min. bias}{=} \frac{\frac{1}{2A} d^2\sigma_{\pi}^{\rm dAu}/dp_T dy}{d^2\sigma_{\pi}^{\rm pp}/dp_T dy}, \nonumber$$ where $p_T$, $y$ are the transverse momentum and rapidity of the pion, and $\langle N_{\rm coll}\rangle$ denotes the number of binary nucleon-nucleon collisions. A comparison with the PHENIX and STAR data is shown in Fig. \[Fig:PHENIX\], and evidently, the shape of the spectrum gets very well reproduced by our parametrization. The downward trend toward large $p_T$ provides direct evidence for an EMC-effect in the large-$x$ gluons, while the suppression toward small $p_T$ is in line with the gluon shadowing. No additional effects are needed to reproduce the observed spectra. It is also reassuring that this shape is practically independent of the fragmentation functions used in the calculation — modern sets like [@Kniehl:2000fe; @Albino:2008fy; @deFlorian:2007aj] all give equal results.
Special attention should be paid to the scale-breaking effects in the data and to their good description by the DGLAP evolution. These effects are clearly visible e.g. in the E886 Drell-Yan data in Fig. \[Fig:RDY\] where the trend of diminishing nuclear effects toward larger invariant mass $M^2$ is observed. Also, from the DIS data as a function of $Q^2$, shown in Fig. \[Fig:RF2\_slopes\], the general features can be filtered out: At small $x$ the $Q^2$-slopes tend to be positive, while toward larger $x$ the slopes gradually die out and become even slightly negative.
![The calculated NLO scale evolution (solid black lines and error bands) of the ratio $F_2^{\mathrm{Sn}}/F_2^{\mathrm{C}}$ and $F_2^{\mathrm{C}}/F_2^{\mathrm{D}}$, compared with the NMC data [@Arneodo:1996ru] for several fixed values of $x$.[]{data-label="Fig:RF2_slopes"}](RF2SnC.eps "fig:") ![The calculated NLO scale evolution (solid black lines and error bands) of the ratio $F_2^{\mathrm{Sn}}/F_2^{\mathrm{C}}$ and $F_2^{\mathrm{C}}/F_2^{\mathrm{D}}$, compared with the NMC data [@Arneodo:1996ru] for several fixed values of $x$.[]{data-label="Fig:RF2_slopes"}](RF2CD.eps "fig:")
To close this section we show, in Fig. \[Fig:NLOcomp\], a comparison of the nuclear modifications for Lead from all available global NLO analyses. The comparison is shown again at $Q^2_0=1.69 \, {\rm GeV}^2$ and at $Q^2=100 \, {\rm GeV}^2$. Most significant differences — that is, curves being outside our error bands — are found from the sea quarks and gluons. At low $x$, the differences shrink when the scale $Q^2$ is increased, but at the high-$x$ region notable discrepancies persist at all scales. Most of the differences are presumably explainable by the assumed behaviours of the fit functions, but also the choices for the considered data sets (e.g. HKN and nDS do not implement the pion data), and differences in the definition of $\chi^2$ carry some importance.
![Comparison of the nuclear modifications for Lead at $Q^2 = 1.69 \, {\rm GeV}^2$ and at $Q^2 = 100 \, {\rm GeV}^2$ from the NLO global DGLAP analyses HKN07 [@Hirai:2007sx], nDS [@deFlorian:2003qf] and this work, EPS09.[]{data-label="Fig:NLOcomp"}](allcomp_100.eps)
Hadrons in the forward direction in d+Au at RHIC?
=================================================
In our previous article [@Eskola:2008ca], we studied the case of a very strong gluon shadowing, motivated by the suppression in the nuclear modification $R_{\rm dAu}$ for the negatively-charged hadron yield in the forward rapidities ($\eta=2.2,3.2$) measured by the BRAHMS collaboration [@Arsene:2004ux] in d+Au collisions at RHIC. We now return to this issue by applying the EPS09NLO parametrization to this specific process. The EPS09 predictions (with fDSS fragmentation functions [@de; @Florian:2007hc]) compared with the BRAHMS data for the absolute $h^-$ spectra, $$\frac{d^3 N^{\rm pp}}{d^2p_T dy} \stackrel{\rm min. bias}{=} \frac{1}{\sigma_{NN}^{\rm inelastic}} \frac{d^3\sigma^{\rm pp}}{d^2p_T dy}
\quad ; \quad
\frac{d^3 N^{\rm dAu}}{d^2p_T dy} \stackrel{\rm min. bias}{=} \frac{{\langle N_{\rm coll}\rangle}}{\sigma_{NN}^{\rm inelastic}} \frac{\frac{1}{2A}d^3\sigma^{\rm dAu}}{d^2p_T dy},
\label{eq:BRAHMScrosssection}$$ are shown in Fig. \[Fig:BRAHMS1\]. In the $\eta=2.2$ bin, the measured p+p and d+Au spectra are both in good agreement with the NLO pQCD. However, in the most forward $\eta=3.2$ bin in the p+p case, there is a systematic and significant discrepancy between the measured and computed $p_T$ spectra. This observation casts doubts on any conclusion made from the nuclear modification $R_{\rm dAu}$ alone — clearly, one should first account for the absolute baseline spectrum in p+p collisions. This is why we have not included this BRAHMS data set in our global analysis, either.
![Inclusive $h^-$ yield in p+p and d+Au collisions. The experimental $p_T$-binned data [@Arsene:2004ux] are shown by open squares with statistical and systematical errors added in quadrature. The blue band indicates the 90% confidence range derived from the CTEQ6.1M and EPS09 uncertainties. The calculated cross-sections have been averaged over the $p_T$-bin width.[]{data-label="Fig:BRAHMS1"}](ppCrosssection.eps "fig:") ![Inclusive $h^-$ yield in p+p and d+Au collisions. The experimental $p_T$-binned data [@Arsene:2004ux] are shown by open squares with statistical and systematical errors added in quadrature. The blue band indicates the 90% confidence range derived from the CTEQ6.1M and EPS09 uncertainties. The calculated cross-sections have been averaged over the $p_T$-bin width.[]{data-label="Fig:BRAHMS1"}](dAuCrosssection.eps "fig:")
Summary
=======
We have here outlined our NLO analysis of nPDFs. The very good agreement with the experimental data $\chi^2/N \approx 0.79$ — especially the correct description of the scaling-violation effects — lends support to the validity of collinear factorization in nuclear collisions. In addition to the best fit, we have distilled the experimental errors into the 30 nPDF error-sets, which encode the relevant parameter-space neighbourhood of the $\chi^2$ minimum. All these sets will be available as a computer routine at [@EPS09code] for general use.
Although not discussed here, we have also performed the leading-order (LO) counterpart of the NLO analysis as we want to provide the uncertainty tools also for the widely-used LO framework. The best-fit quality is very similar both in LO and NLO, but the uncertainty bands become somewhat smaller when going to higher order. In the LO case, our error analysis indeed accommodates also the strong gluon shadowing suggested in our previous work EPS08 [@Eskola:2008ca].
In the near future, more RHIC data will become available and published, and factorization will be tested further. Also p+$A$ runs at the LHC would be welcome for this purpose. Even better possibilities for the nPDF studies would, however, be provided by the lepton-ion colliders like the planned eRHIC or LHeC.
During the recent years, the free-proton PDF analyses have gradually shifted to a better organized prescription for treating the heavy quarks than the zero-mass scheme employed in this analysis and in CTEQ6.1M. Such general-mass scheme should be especially important e.g. in the case of charged-current neutrino interactions. Interestingly, the CTEQ collaboration has lately looked also at this type of data among other neutrino-Iron measurements [@Schienbein:2007fs], and noticed that the best fit tends to point to somewhat different nuclear modifications than what have been obtained in the global nPDF analyses. Both the extension of the nPDF analysis to the general-mass scheme and a systematic investigation of the possible discrepancies between neutrino and other data, remains as future work.
[99]{}
Y. L. Dokshitzer, Sov. Phys. JETP [**46**]{} (1977) 641 \[Zh. Eksp. Teor. Fiz. [**73**]{} (1977) 1216\]; V. N. Gribov and L. N. Lipatov, Yad. Fiz. [**15**]{} (1972) 781 \[Sov. J. Nucl. Phys. [**15**]{} (1972) 438\]; V. N. Gribov and L. N. Lipatov, Yad. Fiz. [**15**]{} (1972) 1218 \[Sov. J. Nucl. Phys. [**15**]{} (1972) 675\]; G. Altarelli and G. Parisi, Nucl. Phys. B [**126**]{} (1977) 298. N. Armesto, J. Phys. G [**32**]{} (2006) R367 \[arXiv:hep-ph/0604108\].
K. J. Eskola, V. J. Kolhinen and P. V. Ruuskanen, Nucl. Phys. B [**535**]{} (1998) 351 \[arXiv:hep-ph/9802350\]. K. J. Eskola, V. J. Kolhinen and C. A. Salgado, Eur. Phys. J. C [**9**]{} (1999) 61 \[arXiv:hep-ph/9807297\]. K. J. Eskola, V. J. Kolhinen, H. Paukkunen and C. A. Salgado, JHEP [**0705**]{} (2007) 002 \[arXiv:hep-ph/0703104\]. K. J. Eskola, H. Paukkunen and C. A. Salgado, JHEP [**0807**]{} (2008) 102 \[arXiv:0802.0139 \[hep-ph\]\]. M. Hirai, S. Kumano and T. H. Nagai, arXiv:0709.3038 \[hep-ph\]. D. de Florian and R. Sassot, Phys. Rev. D [**69**]{} (2004) 074028 \[arXiv:hep-ph/0311227\]. K. J. Eskola, H. Paukkunen and C. A. Salgado, arXiv:0902.4154 \[hep-ph\], submitted to JHEP. D. Stump, J. Huston, J. Pumplin, W. K. Tung, H. L. Lai, S. Kuhlmann and J. F. Owens, JHEP [**0310**]{} (2003) 046 \[arXiv:hep-ph/0303013\].
J. Pumplin [*et al.*]{}, Phys. Rev. D [**65**]{} (2001) 014013 \[arXiv:hep-ph/0101032\]. P. M. Nadolsky and Z. Sullivan, in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)* ]{} ed. N. Graf, [*In the Proceedings of APS / DPF / DPB Summer Study on the Future of Particle Physics (Snowmass 2001), Snowmass, Colorado, 30 Jun - 21 Jul 2001, pp P510*]{} \[arXiv:hep-ph/0110378\]. https://www.jyu.fi/fysiikka/en/research/highenergy/urhic/nPDFs
M. Arneodo [*et al.*]{} \[New Muon Collaboration.\], Nucl. Phys. B [**441**]{} (1995) 12 \[arXiv:hep-ex/9504002\]. P. Amaudruz [*et al.*]{} \[New Muon Collaboration\], Nucl. Phys. B [**441**]{} (1995) 3 \[arXiv:hep-ph/9503291\]. D. M. Alde [*et al.*]{}, Phys. Rev. Lett. [**64**]{} (1990) 2479. M. A. Vasilev [*et al.*]{} \[FNAL E866 Collab.\], Phys. Rev. Lett. [**83**]{} (1999) 2304 \[arXiv:hep-ex/9906010\]. S. S. Adler [*et al.*]{} \[PHENIX Collab.\], Phys. Rev. Lett. [**98**]{} (2007) 172302 \[arXiv:nucl-ex/0610036\]. J. Adams [*et al.*]{} \[STAR Collaboration\], Phys. Lett. B [**637**]{} (2006) 161 \[arXiv:nucl-ex/0601033\]. B. A. Kniehl, G. Kramer and B. Potter, Nucl. Phys. B [**582**]{} (2000) 514 \[arXiv:hep-ph/0010289\]. S. Albino, B. A. Kniehl and G. Kramer, arXiv:0803.2768 \[hep-ph\]. D. de Florian, R. Sassot and M. Stratmann, Phys. Rev. D [**75**]{} (2007) 114010 \[arXiv:hep-ph/0703242\]. M. Arneodo [*et al.*]{} \[New Muon Collaboration\], Nucl. Phys. B [**481**]{} (1996) 23. I. Arsene [*et al.*]{} \[BRAHMS Collab.\], Phys. Rev. Lett. [**93**]{} (2004) 242303 \[arXiv:nucl-ex/0403005\]. D. de Florian, R. Sassot and M. Stratmann, Phys. Rev. D [**76**]{} (2007) 074033 \[arXiv:0707.1506 \[hep-ph\]\]. I. Schienbein, J. Y. Yu, C. Keppel, J. G. Morfin, F. Olness and J. F. Owens, arXiv:0710.4897 \[hep-ph\].
|
---
abstract: 'Bloch-like surface waves associated with a quasiperiodic structure are observed in a classic wave propagation experiment which consists of pulse propagation with a shallow fluid covering a quasiperiodically drilled bottom. We show that a transversal pulse propagates as a plane wave with quasiperiodic modulation, displaying the characteristic undulatory propagation in this quasiperiodic system and reinforcing the idea that analogous concepts to Bloch functions can be applied to quasicrystals under certain circumstances.'
author:
- 'M. Torres[^1]'
- 'J.L. Aragón[^2]'
- 'J. P. Adrados'
- 'P. Cobo'
- 'S. Tehuacanero'
title: 'Quasiperiodic Bloch-like states in a surface wave experiment'
---
The main difficulty towards the development of a systematic analytic approach to the transport properties of quasiperiodic systems has been the absence of an analogous Bloch theorem approach as used in the periodic case. In the first efforts to apply a modified version of the Bloch theorem, it was noticed that the dense spectrum of quasiperiodic systems is dominated by only a few special reciprocal lattice points that may be taken to construct a quasi-Brillouin zone [@Smith]. Thus, by considering only the dominant Fourier components, the atomic distribution can be expanded in terms of a discrete aperiodic lattice. Wave functions of the form $\Psi _\mathbf{k} = u_\mathbf{k}
(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}}$ will therefore solve the Schrodinger equation. In this case $u_\mathbf{k}$ is quasiperiodic and should formally be defined on a countable dense set of reciprocal lattice vectors. But, by the above considerations, this expansion is useful since the Fourier development of the modulation function $u_\mathbf{k}$ can be restricted to the few special reciprocal vectors that dominate the spectra. Thus, Bloch-like states could describe the plane wave propagation in so schematized quasicrystals and free-electron-like bands are expected. Recently this idea was experimentally tested showing that analogous concepts to Bloch functions can be applied to quasicrystals [@Rotenberg].
The classic wave propagation in quasicrystalline systems was addressed in a first seminal acoustic experiment of He and Maynard [@He] by the feature that acoustical waves are ideal tools to investigate formally similar quantum propagation effects [@Maynard]. On the other hand, appearance of the quasicrystalline symmetry in fluids dynamics was firstly predicted theoretically by Zaslavsky and co-workers [@Zaslavsky1] and a simulation similar to the conditions of the present experiments was reported in Ref. [@Beloshapkin]. Finally, compressible quasisymmetric flows were considered in Ref. [@Morgulis], whereas a general outlook on order and disorder in fluid motion can be found in the experiments of Gollub [@Gollub].
In this letter we shall see that a discrete restricted spectral scenario can be displayed by means of impulsive waves in hydrodynamic quasicrystals, where we observe Bloch-like surface waves. The waves are generated at the frequencies corresponding to the Fourier components of the quasiperiodic structure at the dominant diffraction spots. The observed Bloch-like waves are plane waves with quasiperiodic modulation generated when a pulse propagates transversally to the quasiperiodic structure. Liquid surface waves shape a quasiperiodic grid that obeys the so-called Octonacci sequence, previously studied [@Sire; @Borcherds] but never observed in any experiment.
The quasiperiodic structure involved in our experiment is the octagonal Ammann-Beenker tiling composed by squares and rhombuses [@Senechal]. Associated with this octagonal tiling is a quasiperiodic sequence, named the Octonacci sequence [@Sire], that can be generated starting from two steps $L$ and $S$, which are related according to the irrational ratio $L / S = 1 + \sqrt{2}$, by iteration of substitution rules: $L \rightarrow LSL$ and $S
\rightarrow L$.
The experiments are performed with surface waves generated on a shallow fluid that covers the quasiperiodically drilled bottom of a transparent vessel. Such experiments are similar to others realized in vessels with periodic bathymetry and described elsewhere [@Torres1; @Torres2] but here not only a continuous wave excitation driven by a vertical monofrequency vibration but also a new experiment of transversal pulse propagation is performed. The bottom dimples are located at $121$ vertices of the octagonal tiling. The edge length $l$ of the tiling is $8$ mm with an error lower than $0.4$%, the radius $r$ of the cylindrical bottom wells is $1.75$ mm with an error lower than $1$% and their depth $d$ is $2$ mm. The depth of the liquid layer over the cylindrical wells is given by $h_2 = h_1 + d$, where $h_1$ is the depth of the thin liquid layer covering the bottom of the vessel among holes.
Under conditions of continuous-wave excitation, an inertial hydrodynamical undulatory instability grows over the bottom wells when the system vibrates vertically at a frequency of $35$ Hz. Such an instability becomes remarkable (Fig. \[fig:fig1\]) due to the high density and the very low surface tension of the liquid [@Liquid]. Oscillating bulges over dimples are connected by surface waves with shorter wavelength that decorate the shallow liquid region among holes with $h_1$ being $0.4$ mm. This is a physical scenario similar to that of the Kronig-Penney model but adapted here to a $2$D quasiperiodic system. It should be remarked that Fig. \[fig:fig1\] is the first available experimental example of a quasiperiodic pattern of waves not arising from a non-linear instability such as the Faraday instability. Standing waves are coupled to only two rings of Fourier wave components of the quasiperiodic bottom structure, as shown in the inset of Fig. \[fig:fig1\]. Their wavenumber ratio is $1 +
\sqrt{2}$, as it can be measured in the diffraction pattern (see below). If the effective Fourier transform of a quasiperiodic structure is restricted to a discrete set of Fourier peaks as in this case, then Bloch-like modulation functions can be used to describe the wave propagation in such a simplified quasicrystal [@Janot]. The pattern of Fig. \[fig:fig1\] is due to the strong coupling between liquid surface waves and the bottom quasiperiodic topography and does not depend on the shape of the vessel boundary. Undistinguishable patterns are generated with octagonal or circular boundaries. Although the boundaries are reflecting vertical walls, the boundary symmetry matches the symmetry of this standing wave experiment allowing the mentioned boundary-independent strong wave coupling. Due to the accuracy on realizing the setup, localization phenomena do not appear in the described experiment. However, slight tiltings of the vessel generate wave domains [@Torres1; @Torres2]; furthermore, point and linear defects can be easily introduced in the system by dropping mercury on the bottom dimples to study new interesting wave localization phenomena [@Torres3].
The propagation of a plane wave through the octagonal quasiperiodic structure can be visualized by means of a experiment of wave pulse propagation. A coupling between the vertical waves of the vessel-liquid system and the surface waves generated by a transversal pulse is expected, modelling in this way the propagation of a plane wave through the quasiperiodic structure. This experiment was realized in a vessel with octagonal boundary. The octagon side $L$ is $4$ cm and it is perpendicular to the $\Gamma-X$ direction of the well structure. The surface ratio $f$ between bottom holes and the whole octagon is about $0.15$ with $h_1$ being $0.5$ mm. The system is excited near the octagonal boundary with a wave pulse parallelly to the liquid surface and perpendicularly to a side of the octagon. The signal is picked up by means of a Brüel & Kjaer 4344 accelerometer placed at the center of the vessel and it is processed by means of a digital acquisition system. The impulsive signal and the corresponding Fourier transform are shown in Fig. \[fig:fig2\] (inset and solid line, respectively). Three clear spectral peaks of the vibrational vessel-liquid system appear at about $20$, $30$ and $50$ Hz. As we shall see, such resonances indicate the existence of three narrow band gaps in the liquid surface wave propagation [@Torres2], *i.e*. standing liquid waves are generated at approximately the above generated frequencies.
At the start of each pulse the liquid *feels* the perturbation and a nice quasicrystalline surface wave pattern suddenly appears \[Fig. \[fig:fig3\](a)\]. A transitory weak turbulence arises in the system after scarcely $0.04$ s \[Fig. \[fig:fig3\](b)\], whereas robust standing waves drawing clear quasiperiodic grids can be observed between $0.08$ and $0.24$ s on the liquid surface \[Fig. \[fig:fig3\](c)\]. Finally, quasiperiodic grid patterns decay until the arrival of the next pulse. As wave phase velocities are about $11$ cm s$^{-1}$ and the wave group velocity is nearly null near the gaps, times for an echo at the boundaries to come back are much longer than observation time.
The robust quasiperiodically spaced standing waves shown in \[Fig. \[fig:fig3\](c)\] are generated by discrete Bragg resonances and thus can be considered quasiperiodic Bloch-like waves. To verify this, first note that the irrational ratio $LS / L = \sqrt{2}$ is apparent in our experiment \[Fig. \[fig:fig3\](c)\]. Now, using a crystallography-oriented computer program [@Hovmuller], the Fourier transform of the pattern of Fig. \[fig:fig3\](c) is calculated and shown in Fig. \[fig:fig4\] (top left). Such diffraction pattern matches with an adequate subset of the diffraction pattern of the direct product of both orthogonal Octonacci sequences calculated according to theoretical methods [@Sire; @Aragon] as shown in Fig. \[fig:fig4\] (top right). The absence of some diffraction peaks indicates the directional character of the impulsive action. The pulse runs along the $\Gamma-X$ direction from the upper left to the bottom right corner in both patterns at the top of Fig. \[fig:fig4\]. Along this direction, a intensity profile is taken in the experimental pattern and recovered the inverse Fourier transform of that unidimensional diffraction subset. The result is shown in Fig. \[fig:fig4\] (bottom) which displays an Octonacci sequence, and it matches with that generated theoretically starting from the above mentioned substitution rules. Finally, the above described intensity profile along the $\Gamma-X$ direction is scaled according to the wavenumber of the waves of the diffraction pattern shown in Fig. \[fig:fig3\](c). Such scale is then changed according to the approximate dispersion relationship given [@Torres2] by $$\omega ^2 = gk \left( 1 + \frac{T}{\rho g} k^2 \right) \tanh (k
h_0),$$ where $h_0 = h_1 (1-f) + h_2 f$, $\omega$ is the angular frequency, $k$ is the wave number, $g$ is the acceleration due to gravity, $T$ is the liquid surface tension and $\rho$ is the liquid density [@Liquid]. In Fig. \[fig:fig2\] (dashed lines) the gray scale intensity profile is plotted versus the frequency according to the above mentioned change of scale. The first maximum is scaled by the wavenumber $k = 11$ cm$^{-1}$, that corresponds to the main ubiquitous wave appearing in the experimental pattern of Fig. \[fig:fig3\](c). As it can be seen, diffraction peaks which represent narrow band gaps closely match in frequency with those independently measured also in Fig. \[fig:fig2\]. Thus, in this restricted scenario, where the resonances of the vibrational coupling generates a discrete spectrum, the wave pattern observed in Fig. \[fig:fig3\](c) corresponds to quasiperiodic Bloch-like states.
If experiments of pulse propagation are realized in vessels with periodic bathymetry [@Torres1; @Torres2] no signal of turbulence appears. Thus, as remarked in a different context [@Borcherds; @Zaslavsky2], the quasiperiodicity of the hydrodynamical system could be the origin of the weak chaos observed in the described experiments just at the start of pulses, when amplitudes are higher and hence the nonlinearity is stronger. Then, a rapidly increasing number of incommensurable Fourier harmonics can grow due to the finite frequency bandwidth of the pulse and the incommensurate nature of the system. This gives rise to the pre-turbulent state of the surface waves. When the multiscattering becomes weaker, the Fourier mode cascade decays and the propagative wave exhibits a clean quasiperiodic grid pattern.
Anyway, the quasiperiodic structure underlies in spite of the weak turbulence apparent in Fig. \[fig:fig3\](b), which must be looked at grazing incidence to recognize Octonacci quasiperiodic sequences. This is evident in Fig. \[fig:fig5\], which is the inverse Fourier transform of Fig. \[fig:fig3\](b). The direct product of two orthogonal Octonacci sequences is recovered there, showing a patch of the well known octagonal tiling filled with square and rhombic tiles [@Senechal].
In conclusion, we have shown Bloch-like surface waves associated with a quasiperiodic structure in a classic wave propagation experiment. These waves draw clear quasiperiodic grids that obey the Octonacci sequence. Our results along with earlier ones [@He] can be helpful to understand the characteristic undulatory propagation in quasiperiodic systems.
This work has been supported by MCYT (Project No.BFM20010202) and DGAPA-UNAM (Proyect 108199).
[10]{} A.P. Smith and N.W. Ashcroft, Phys. Rev. Lett. **59**, 1365 (1987); K. Niizeki and T. Akamatzu, J. Phys. Cond. Mat. **2**, 2759 (1990). E. Rotenberg, W. Theis, K. Horn and P. Gille, Nature (London) **406**, 602 (2000).
S. He and J.D. Maynard, Phys. Rev. Lett. **62**, 1888 (1989).
J.D. Maynard, Rev. Mod. Phys. **73**, 401 (2001), and references therein; S. He and J. D. Maynard, Phys. Rev. Lett. **57**, 3171 (1986); G. Bayer and T. Niederdränk, Phys Rev. Lett. **70**, 3884 (1993).
G.M. Zaslavsky, R.Z. Sagdeev and A.A. Chernikov, Sov. Phys. JETP **67**, 270 (1988).
V.V. Beloshapkin *et al.*, Nature (London) *337*, 133 (1989). See also P. Ball, *The self-made tapestry* (Oxford University Press, Oxford 1999).
A. Morgulis, V.I. Yudovich, and G.M. Zaslavsky, Commun. Pure Appl. Math. **48**, 571 (1995).
J.P. Gollub, Proc. Natl. Acad. Sci. USA **92**, 6705 (1995) C. Sire, R. Mosseri and J. F. Sadoc, J. Phys. France **50**, 3463 (1989). P. H. Borcherds and G. P. McCauley, J. Phys. A: Math. Gen. **24**, 3455 (1991). M. Senechal, *Quasicrystals and Geometry* (Cambridge University Press, Cambridge, 1995). M. Torres, J. P. Adrados and F. M. de Espinosa, Nature (London) **398**, 114 (1999). M. Torres *et al.*, Phys. Rev E **63**, 011204 (2001). Fluorinert FC75 of the 3M Company, a fully fluorinated organic liquid with a low kinematic viscosity = 0.8 10$^{\rm{}2}$ cm$^{\rm{}2}$/s, lower than that of the water, very low surface tension T = 15 dyn/cm and very high density = 1.77 g/cm$^{\rm{}3}$. C. Janot, *Quasicrystals: A Primer* (Oxford Univ. Press, Oxford, 1994).
M. Torres and J.L. Aragón, to be published.
CRISP: Crystallographic Image Processing Release 1.3a. See: S. Hovmoller, Ultramicroscopy **41**, 121 (1992). J.L. Aragón, G. Naumis and M. Torres, Acta Cryst. A **58**, 352 (2002). G. M. Zaslavsky, R.Z. Zagdeev, D.A. Usikov and A.A. Chernikov, *Weak Chaos and QuasiRegular Patterns* (Cambridge Univ. Press, Cambridge, 1991).
![Snapshot of the system vibrating vertically at a frequency of $35$ Hz, under conditions of continuous wave excitation. The inset shows its Fourier spectrum with a well defined and discrete set of relevant components.[]{data-label="fig:fig1"}](figura1.eps){width="7.0cm"}
![The solid line is the Fourier transform of the impulsive signal as picked up by an accelerometer at the center of the vessel. It shows clear resonances at approximately $20$, $30$ and $50$ Hz. Such resonances are specific of the system. The first peak at very low frequency, close to the origin, corresponds to the Fourier transform of the square pulse that excites the system. The signal in the time domain is presented in the inset. The dashed line is the gray scale of the subpattern along the $\Gamma-X$ direction as represented in Fig. \[fig:fig4\] versus frequency. The gray scale (between $0$ and $1$) was rescaled to match the peak of the Fourier transform at $30$ Hz. Three standing waves indicating narrow band gaps appear at approximately $20$, $32$ and $50$ Hz.[]{data-label="fig:fig2"}](figura2.ps){width="8.0cm"}
![Temporal sequence of patterns observed when the system is disturbed with a transverse pulse. (a) Quasicrystalline pattern observed at the start of each pulse. (b) A transitory weak turbulence is observed after $0.04$ s. (c) Standing waves draw clear quasiperiodic grids between $0.08$ and $0.24$ s. This figure should be looked diagonally at grazing incidence.[]{data-label="fig:fig3"}](figura3.ps){width="7.0cm"}
. At the bottom, the inverse Fourier transform of a subpattern along the $\Gamma-X$ is shown. The Octonacci sequence is clearly recovered.[]{data-label="fig:fig4"}](figura4.ps){width="7.5cm"}
. The structure of the octagonal tiling underlying on the well quasiperiodic arrangement of the vessel bottom is recovered.[]{data-label="fig:fig5"}](figura5.ps){width="7.5cm"}
[^1]: Electronic address: manolo@iec.csic.es
[^2]: Electronic address: aragon@fata.unam.mx
|
---
abstract: 'We show that topological transitions in electronic spin transport are feasible by a controlled manipulation of spin-guiding fields. The transitions are determined by the topology of the fields texture through an *effective* Berry phase (related to the winding parity of spin modes around poles in the Bloch sphere), irrespective of the actual complexity of the nonadiabatic spin dynamics. This manifests as a distinct dislocation of the interference pattern in the quantum conductance of mesoscopic loops. The phenomenon is robust against disorder, and can be experimentally exploited to determine the magnitude of inner spin-orbit fields.'
author:
- 'Henri Saarikoski,$^1$ J. Enrique Vázquez-Lozano,$^2$ José Pablo Baltanás,$^2$ Fumiya Nagasawa,$^3$ Junsaku Nitta,$^3$ and Diego Frustaglia$^2$'
title: Topological transitions in spin interferometers
---
In the early 1980s Berry showed that quantum states in a cyclic motion may acquire a phase component of geometric nature [@berry]. This opened a door to a class of topological quantum phenomena in optical and material systems [@BMKNZ03]. With the development of quantum electronics in semiconducting nanostructures, a possibility emerged to manipulate electronic quantum states via the control of spin geometric phases driven by magnetic field textures [@loss]. After several experimental attempts [@MHKWB98; @YPS02; @BKSN06; @KTHSHDSBBM06; @GLIERW07] indisputable signatures of spin geometric phases in conducting electrons were found in 2012 [@nagasawa1] in agreement with the theory [@frustaglia]. This paved the way for the development of a topological spin engineering [@nagasawa2].
An early proposal for the topological manipulation of electron spins by Lyanda-Geller involved the abrupt switching of Berry phases in spin interferometers [@lyanda-geller]. These are conducting rings of mesoscopic size subject to Rashba spin-orbit (SO) coupling, where a radial magnetic texture ${\bf B}_{\rm SO}$ steers the electronic spin (Fig. \[fig-1\]a). For relatively large field strengths (or, alternatively, slow orbital motion) the electronic spins follow the local field direction adiabatically during transport, acquiring a Berry phase factor $\pi$ of geometric origin (equal to half the solid angle subtended by the spins in a roundtrip) leading to destructive interference effects. By introducing an additional in-plane uniform field ${\bf B}$, it was assumed that the spin geometric phase undergoes a sharp transition at the critical point beyond which the corresponding solid angle vanishes together with the Berry phase, and interference turns constructive. The transition should manifest as a step-like characteristic in the ring’s conductance as a function of the coupling fields (so far unreported). However, this reasoning appears to be oversimplified: the adiabatic condition can not be satisfied in the vicinity of the transition point, since the local steering field vanishes and reverses direction abruptly at the rim of the ring. Moreover, typical experimental conditions correspond to moderate field strengths, resulting in nonadiabatic effects in analogy to the case of spin transport in helical magnetic fields [@betthausen]. Hence, a more sophisticated approach is required. This includes identifying the role played by nonadiabatic Aharonov-Anandan (AA) geometric phases [@aharonov]. Here, we report transport simulations showing that a topological phase transition is possible in loop-shaped spin interferometers away from the adiabatic limit. The transition is determined by the topology of the field texture through an effective Berry phase related to the winding parity of the spin eigenmodes around the poles in the Bloch sphere. This contrasts with the actual complexity of the emerging dynamic and AA geometric phases, which exhibit a correlated behavior close to the transition.
We consider a two-dimensional electron gas (2DEG) confined at the interface of a semiconducting heterostructure ($xy$ plane in Fig. \[fig-1\]a). The 2DEG is subject to SO interaction due to structure inversion asymmetry, which can be tuned by gate electrodes [@nitta]. The SO field ${\bf B}_{\rm SO}$ couples to conduction electron spin as [@bychkov] $$H_{\rm SO}=({\alpha}/{\hbar})({\bm \sigma} \times {\bf p})\cdot \hat z \equiv {\bf B}_{\rm SO} \cdot {\bm \sigma},
\label{HSO}$$ with ${\bf B}_{\rm SO} = B_{\rm SO} ( {{\hat k} \times {\hat z}})$, $\alpha$ the SO strength, $\bf p$ the electronic momentum, ${\bm \sigma}$ the vector of Pauli spin matrices, ${\hat k}$ the unit vector along the electron wave vector ${\bf k}$, and $\hat z$ the unit vector perpendicular to the 2DEG. This SO term gives rise to the Aharonov-Casher (AC) [@aharonovcasher] interference patterns in the conductance of ring ensembles [@nagasawa1; @nagasawa2]. Geometric and dynamical phases developed by electrons moving in circular orbits have been identified as distinct contributions to the AC phase in rings [@frustaglia]. Moreover, spin eigenstates subtend a regular cone in the Bloch sphere with solid angle $\Omega= - 2 \pi (1-1/\sqrt{Q^2+1})$ where $Q=2m^*\alpha r/\hbar^2$ is the adiabaticity parameter [@frustaglia], $m^*$ is the effective electron mass and $r$ the ring radius. This corresponds to a geometric AA phase $-\Omega/2$ acquired by the spins in a roundtrip [@frustaglia; @nagasawa1]. The spin states are radial only in the adiabatic limit $Q \gg 1$, giving a Berry phase $\pi$.
We add a homogeneous Zeeman field in the $xy$ plane $$H_{\rm Z}= {\bf B}\cdot {\bm \sigma}= B (\cos\gamma\, \sigma_x+\sin\gamma\, \sigma_y),
\label{HZ}$$ where $\gamma$ is the angle with respect to the axis of the wire. In geometries where the contact leads are symmetrically coupled to the rings, electron spins traveling along symmetric interference paths acquire equal Zeeman phases resulting in constructive interference for ${\bf B}_{\rm SO} =0$. Both constructive and destructive interference of Zeeman phases are possible in rings coupled tangentially to leads to form loops [@yang] due to interference of paths shown in Fig. \[fig-1\]a.
![a) The model system: a conducting wire of width $W$ is attached tangentially to a ring of radius $r$, forming a loop. The main interference paths are straight along the wire (A) and (counter)clockwise around the loop (B and C). The spin-orbit field ${\bf B}_{\rm SO}$ is radial and the homogeneous magnetic field ${\bf B}$ lies in the $xy$ plane. b) The Berry phases in the adiabatic limit. For $B_{\rm SO}\gg B$ the solid cone $\Omega=2\pi$ corresponding to the Berry phase $\pi$ (left). For $B \gg B_{\rm SO}$ the solid cone vanishes giving Berry phase 0 (right).[]{data-label="fig-1"}](FIG-1.png){width="\columnwidth"}
![ Conductance (in units of $e^2/h$) as a function of the SO and Zeeman couplings in a ballistic single-mode loop. Left: 2D simulations for a $r=1.2\;{\rm \mu m}$ loop in InGaAs at $E_{\rm F}=88\;{\rm meV}$. Right: 1D semiclassical model. The dashed lines show the wavefronts in an adiabatic treatment, Eq. (\[dynamic\]). The phase dislocation along $B_{\rm SO}=B$ is a signature of transition in field’s topology. The SO and Zeeman scales are in terms of $Q = 2 m^* \alpha r/\hbar^2$ and $2 m^* r B/(\hbar^2k)$, respectively, and $\gamma=\pi/2$. \[fig-2\]](FIG-2.png){width="0.95\columnwidth"}
![Upper panel: cosine of the total phase $\phi$ (left) and the dynamical phase component $\phi_{\rm d}$ (right) in the 1D model. Lower panel: a complementary complexity arises in the cosine of the AA geometric phase component $\phi_{\rm g}$ (left), evidenced by the spin-eigenmode textures calculated at the selected points (right). \[fig-3\]](FIG-3.pdf){width="\columnwidth"}
We adopt here the loop geometry to study the interplay between Zeeman and AC phases. In the presence of SO coupling the in-plane magnetic field manifests as a pure geometrical effect at the lowest order in $B$, without affecting the dynamical phase [@nagasawa2]. The perturbation approach fails as $B$ nears $B_{\rm SO}$. Instead, we use the following methods: i) one-dimensional (1D) calculations based on semiclassical methods, providing access to local spin dynamics and geometric phases in the ballistic regime, and ii) two-dimensional (2D) numerical simulations suitable for multi-mode systems with or without disorder. We assume that the leads are spin-compensated and that the largest energy scale is the Fermi energy $E_{\rm F}$, so that the SO and Zeeman energies can be considered small in comparison to the kinetic term. Minor anisotropies arise as a function of $\gamma$ but these are not crucial for our conclusions.
In the 1D semiclassical model we assume three possible and equally probable paths for transmitting spin carriers: a direct path along the wire and (counter)clockwise paths around the loop (Fig. \[fig-1\]a). The $2\times2$ transmission amplitude matrix for spins then reads $\Gamma \sim \mathds{I}+ \Gamma_+ + \Gamma_-$, where $\Gamma_\pm$ are the (counter)clockwise transmission amplitude matrices. These are calculated by approximating the circular loop as a regular polygon with a large number of vertices following the method used in [@BFG05], which is extended here to include in-plane magnetic fields. The conductance is obtained from the transmission probabilities (Landauer formula), given by the trace of $\Gamma \Gamma^\dagger$.
The 2D numerical calculations of electron transport are based on a tight-binding system of transport equations which was solved using the recursive Green’s function method (RGFM) [@wimmer] as well as the Kwant code [@groth]. Disorder in the system is introduced by a lattice disorder model [@ando]. We use the material parameters of InGaAs ($m^*=0.05m_0$ with $m_0$ the bare electron mass).
![Dashed line: cosine of half the solid angle subtended by the spin-guiding field along the dashed line in Fig. \[fig-3\] (lower panel, left) corresponding to the Berry phase in a hypothetic adiabatic evolution and subject to a topological transition at $\Delta=1$. Circles: cosine of the azimuthal component of $\ell \pi$ of the AA geometric phase $\phi_{\rm g}$ (winding parity) along this path acting as an effective Berry phase $\phi_{\rm B}$. Solid line (blue): 1D spin energy splitting between different spin species (normalized by the largest energy value in that window). Anomalies arise in $\ell \pi$ near the degeneracy points, typified by the dips. \[fig-4\]](Fig-4.pdf){width="\columnwidth"}
Figure \[fig-2\] shows the conductance in a single-mode ballistic loop calculated with both methods. It displays an interference pattern with two main characteristics: (i) radial wavefronts starting from the origin and (ii) a distinct phase dislocation along the critical line $\Delta \equiv B/B_{\rm SO} = 1$. The wavefronts correspond to Zeeman oscillations of period $2m^* r B/\hbar^2k=2.0$. In the adiabatic regime the dynamical spin phase $\phi_{\rm d}$ is proportional to the average field $\int_0^{2\pi} \sqrt{(B_{\rm SO}\sin \theta + B)^2 + (B_{\rm SO}\cos \theta )^2} \, d\theta$, giving $$\begin{aligned}
%\phi_{\rm d} \propto 2 B_+ \left [ E\left(\frac{\pi}{4}, \frac{4 B_{\rm SO} B}{B_+^2} \right ) + E\left(\frac{3\pi}{4}, \frac{4 B_{\rm SO} B}{B_+^2} \right ) \right ],
\phi_{\rm d} \propto 2 (B_{\rm SO} + B)\left [( E(\pi/4, {\cal B} ) + E(3\pi/4, {\cal B} ) \right ],
\label{dynamic}\end{aligned}$$ where $\theta$ is the angle in Fig. \[fig-1\], ${\cal B} = 4B_{\rm SO} B/(B_{\rm SO}+B)^2$, and $E(\varphi,m)$ are elliptic integrals of the 2nd kind. Lines of constant adiabatic $\phi_{\rm d}$ are plotted in Fig. \[fig-2\]. The fit with the calculated wavefronts is very good despite the fact that actual spin dynamics is nonadiabatic (some deviations are visible for $\Delta \ll 1$, where wavefronts are best described by geometric phase shifts [@nagasawa2; @joibari]). The critical line corresponds to the frontier where the field texture changes topology, which coincides with the spin-eigenstate texture only in the adiabatic regime. These results are intriguing, since the observed pattern presents properties recalling adiabatic dynamics in a nonadiabatic scenario. The 2D methods give results qualitatively similar to those obtained with the 1D model, indicating that the semiclassical approach captures the essential features.
The main contribution to the 1D results in Fig. \[fig-2\] is given by terms of the form $\Gamma_{\pm}+\Gamma_{\pm}^\dagger$. When diagonalized, these matrices have elements $\cos \phi_{\pm}^\sigma$ with $\sigma$ the spin-eigenmode label. The phases $\phi_{\pm}^\sigma$ ($\phi$ henceforth) consist of two parts: $\phi=\phi_{\rm d}+\phi_{\rm g}$, with a dynamical part $\phi_{\rm d}$ and a geometric AA one $\phi_{\rm g}$. A dimensionless conductance can then be conveniently simplified as $\mathcal{G} \equiv 1+\cos (\phi_{\rm d}+\phi_{\rm g})$. The dynamical spin phase can be obtained independently from the expectation value of the spin Hamiltonian $H_{\rm s}= H_{\rm SO}+H_{\rm Z}$ over the spin eigenmodes $|\chi(\theta)\rangle$ as $\phi_{\rm d} = -(m^*r/\hbar^2 k) \int_0^{2\pi} \langle \chi(\theta)|H_{\rm s}|\chi(\theta)\rangle {\rm d}\theta$. Spin phases $\phi$, $\phi_{\rm d}$ and $\phi_{\rm g}=\phi-\phi_{\rm d}$ together with some typical spin-eigenmode textures are shown in Fig. \[fig-3\]. The phase $\phi_{\rm g}$ behaves smoothly near the axes, approaching the adiabatic limit $\pi$ for a strong radial SO texture and vanishing for $B_{\rm SO}=0$. This is apparent from the simple dynamics of the spin eigenstates in those regions (textures A and C). In the vicinity of the critical line $\Delta = 1$, instead, $\phi_{\rm g}$ displays a complex pattern as a signature of a strongly nonadiabatic spin dynamics (texture B). This shows that an adiabatic treatment [@lyanda-geller] close to $\Delta=1$ is not suitable even in the limit of strong fields, and no signature of a topological transition is expected in $\phi_{\rm g}$. In contrast, such a transition is indeed present in the total phase $\phi$, visible as a characteristic dislocation in the interference pattern for conductance in Fig. \[fig-2\].
To understand the origin of the topological transition we generalize a treatment first introduced in Ref. for the study of spin (Berry) adiabatic phases to the case of nonadiabatic spin dynamics. In the absence of degeneracies, the AA geometric phase can be written as $\phi_{\rm g}= \frac{1}{2} \int_0^{2\pi} \frac{\partial\delta}{\partial\theta}(1+\sigma \cos\eta(\theta)) \text{d}\theta = \ell \pi + \frac{\sigma}{2} \int_0^{2\pi} \frac{\partial\delta}{\partial\theta} \cos\eta(\theta) \text{d}\theta$, where $\delta$ and $\eta$ are the azimuthal and polar angle coordinates on the Bloch sphere and $\ell$ is an integer accounting for the windings of the spin eigenmodes around its poles. The second term in $\phi_{\rm g}$ is responsible for the complex structure shown in Fig. \[fig-3\]. We find that this fluctuating term cancels out exactly with an identical component appearing in the dynamical phase such that the total phase reduces to $\phi= \phi_{\rm d}^0 + \ell \pi$, where $\phi_{\rm d}^0=\frac{\sigma}{2} \int_0^{2\pi} \frac{1}{\cos\eta(\theta)} \frac{\partial\delta}{\partial\theta} \text{d}\theta$ is a smooth component of $\phi_{\rm d}$. Our numerical results show that $\ell$ undergoes a parity transition near $\Delta=1$, with odd $\ell$ for $\Delta<1$ and even $\ell$ for $\Delta>1$ (Fig. \[fig-4\]). Hence, the simplified dimensionless conductance writes $\mathcal{G}=1+\cos (\phi_{\rm B})\cos (\phi^{0}_{\rm d})$, where we identify $\phi_{\rm B}=\ell\pi$ as an [*effective*]{} Berry phase causing the phase dislocation at $\Delta=1$ in Fig. \[fig-2\] as $\cos (\phi_{\rm B})$ jumps from $1$ to $-1$, while the smooth term $\phi^{0}_{\rm d}$ leads to wavefronts. This recalls a topological transition in the adiabatic limit [@lyanda-geller] (dashed line in Fig. \[fig-4\]) but involving an effective Berry phase.
The above picture fails near the degeneracy points [@resonances], where the analyticity of the geometric potentials is not guaranteed. The degeneracy points can be characterized as those for which the dynamical-phase difference between distinct spin species (which is equivalent to the spin energy splitting, Fig. \[fig-4\]) is equal to zero. When calculated numerically, $\phi_{\rm B}$ presents a series of anomalies roughly fitting these points. Still, these are compensated by corresponding anomalies arising in $\phi_{\rm d}^0$ such that the total phase $\phi$ is not affected. A full understanding of the role played by degeneracies deserve further efforts beyond the scope of this work. Despite that, our approach captures most of the physics relevant to the problem.
Experiments are often performed in ensembles of multi-mode rings where the interference signal is strengthened and nongeneric features from individual structures are averaged out [@richter]. Figure \[fig-5\] shows interference patterns in the conductance of multi-mode InGaAs loops in the presence of disorder calculated with the RGFM at low temperatures. Zeeman phases are susceptible to temperature and disorder since they are proportional to $1/k$, in contrast to the AC phase which is independent of $k$. Besides, the in-plane field leads to dephasing of the AC oscillations [@meijer]. However, the interference pattern persists in the whole diagram, due to the relevance of Zeeman phases in loops. The AC oscillation frequency doubles when the mean free path decreases as Altshuler-Aronov-Spivak (AAS) paths become relevant [@altshuler]. This effect is not seen for Zeeman phases. Since $B_{\rm SO}$ is proportional to the propagating velocity of a mode, multiple critical lines may arise. Even though, only the transition of the lowest transport mode is clearly visible since higher modes move at slower speed, being more prone to scattering and decoherence. Nevertheless, the triple-mode case in Fig. \[fig-5\]a fits remarkably well the single-mode results for the lowest transport mode (Fig. \[fig-2\]). These results show that the topological transition is robust, and could be detected in multichannel loops in the presence of moderate disorder.
![Simulated interference pattern in the conductance of multi-mode loops ($r=0.52\;{\rm \mu m}$) calculated with the 2D method. $E_{\rm F}=64.3\;{\rm meV}$ and $\gamma=0$. The wire width is $W=35.4\;{\rm nm}$ in (a) and (b), supporting 3 modes, and 53.9 nm in (c) and (d), supporting 5 modes. Electron mean free path $L_{\rm MF}=3.3\;\mu{\rm m}$ in (a) and (c) and $1.6\;\mu{\rm m}$ in (b) and (d). The dashed lines give wavefronts of $\phi_{\rm d}$, Eq. (\[dynamic\]). The solid red lines indicate the critical line $\Delta=1$ for the lowest transport mode. The topological transition is visible as a shift in the interference peak positions for $B_{\rm SO}>B$ (crosses). \[fig-5\]](FIG-5.png){width="\columnwidth"}
We have measured InGaAs samples with mean free paths of the order of a few micrometers [@nagasawa2]. Analysis of these samples indicates that it is possible to fabricate 0.5 to 1 micron radius loops where the gate voltage can change $Q$ by about 1.5 to 3 units. A strong 15 T magnetic field gives $Q$ above 10. These field ranges are high enough to reveal signatures of the topological transition. HgTe/HgCdTe is also a good candidate for experiments due to reports showing high mobility [@hgcdte1], strong $B_{\rm SO}$ [@hgcdte2], and high Zeeman coupling [@KTHSHDSBBM06].
Our findings open possible lines of future research. Alternative interferometer geometries could be studied with stronger wire-to-ring coupling in comparison to loop geometries allowing for higher signal strength in experiments, e.g., rings with asymmetric interference paths or symmetric rings with Aharonov-Bohm fluxes. Due to the robustness of the topological transition, a loop device could be used as a magnetometer measuring the in-situ intensity of the Rashba spin-orbit fields, while deviations from the critical line $\Delta=1$ may be used to estimate the strength of the Dresselhaus SO interaction [@dresselhaus]. Signatures of complex AA geometric phases may be revealed by studying transport of spin-polarized carriers [@oltscher]. Finally, we note that analogous topological transitions in geometric phases emerge also in classical physics [@bgoss]. We have studied magnetic moment dynamics [@landaulifshitz] under the combined action of rotating and homogeneous fields and found a topological transition that features a phase shift of $2\pi$ associated with SO(3) rotations.
[*Acknowledgments.—*]{} This work was supported by Grants-in-Aid for Scientific Research (C) No. 26390014 and (S) No. 22226001 from Japan Society for the Promotion of Science. DF and JPB acknowledge support from the Spanish Ministry of Science and Innovation’s project FIS2011-29400 and from the Junta de Andalucía’s Excellence Project No. P07-FQM-3037.
[99]{} M. V. Berry, Proc. R. Soc. London A [**392**]{}, 45 (1984). A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger, [*The Geometric Phase in Quantum Systems*]{} (Springer-Verlag, New York, 2003). D. Loss, P. Goldbart, and A. V. Balatsky, Phys. Rev. Lett. [**65**]{}, 1655 (1990). A. F. Morpurgo, J. P. Heida, T. M. Klapwijk, B. J. van Wees, and G. Borghs, Phys. Rev. Lett. [**80**]{}, 1050 (1998). J.-B. Yau, E. P. De Poortere, and M. Shayegan, Phys. Rev. Lett. [**88**]{}, 146801 (2002). T. Bergsten, T. Kobayashi, Y. Sekine, and J. Nitta, Phys. Rev. Lett. [**97**]{}, 196803 (2006). M. König, A. Tschetschetkin, E. M. Hankiewicz, J. Sinova, V. Hock, V. Daumer, M. Schäfer, C. R. Becker, H. Buhmann, and L. W. Molenkamp, Phys. Rev. Lett. [**96**]{}, 076804 (2006). B. Grbić, R. Leturcq, T. Ihn, K. Ensslin, D. Reuter, and A. D. Wieck, Phys. Rev. Lett. [**99**]{}, 176803 (2007). F. Nagasawa, J. Takagi, Y. Kunihashi, M. Kohda, and J. Nitta, Phys. Rev. Lett. [**108**]{}, 086801 (2012). D. Frustaglia and K. Richter, Phys. Rev. B [**69**]{}, 235310 (2004). F. Nagasawa, D. Frustaglia, H. Saarikoski, K. Richter, and J. Nitta, Nature Comm. [**4**]{}, 2526 (2013). Y. Lyanda-Geller, Phys. Rev. Lett. [**71**]{}, 657 (1993). C. Betthausen, T. Dollinger, H. Saarikoski, V. Kolkovsky, G. Karczewski, T. Wojtowicz, K. Richter, and D. Weiss, Science [**337**]{}, 324 (2012). Y. Aharonov and J. Anandan, Phys. Rev. Lett. [**58**]{}, 1593 (1987). J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. [**78**]{}, 1335 (1997). Y. A. Bychkov and E. I. Rashba, J. Phys. C [**17**]{}, 6039 (1984). Y. Aharonov and A. Casher, Phys. Rev. Lett. [**53**]{}, 319 (1984). M. J. Yang, C. H. Yang, and Y. B. Lyanda-Geller, Europhys. Lett. [**66**]{}, 826 (2004). D. Bercioux, D. Frustaglia, and M. Governale, Phys. Rev. B [**72**]{}, 113310 (2005). M. Wimmer and K. Richter, J. Comput. Phys. [**228**]{}, 8548 (2009). C. W. Groth, M. Wimmer, A. R. Akhmerov, X. Waintal, New J. Phys [**16**]{}, 063065 (2014). T. Ando, Phys. Rev. B [**44**]{}, 8017 (1991). F. K. Joibari, Y. M. Blanter, and G. E. W. Bauer, Phys. Rev. B [**88**]{}, 115410 (2013). Y. Aharonov, E. Ben-Reuven, S. Popescu, and D. Rohrlich, Phys. Rev. Lett. [**65**]{}, 3065 (1990); [*ibid.*]{}, Nucl. Phys. B [**350**]{}, 818 (1991). We notice that degeneracy points correspond to magnetic resonances due to the combined action of the magnetic and Rashba fields on the spin carriers, also accounting for generalized Bloch-Siegert shifts \[F. Bloch and A. Siegert, Phys. Rev. [**57**]{}, 522 (1940); J. H. Shirley, Phys. Rev. [**138**]{}, B979 (1965)\]. K. Richter, Physics [**5**]{}, 22 (2012). F. E. Meijer, A. F. Morpurgo, T. M. Klapwijk, T. Koga, and J. Nitta, Phys. Rev. B [**70**]{}, 201307 (2004). B. L. Altshuler, A. G. Aronov, and B. Z. Spivak, JETP Letters [**33**]{}, 94 (1981). G. Dresselhaus, Phys. Rev [**100**]{}, 580 (1955). M. Oltscher, M. Ciorga, M. Utz, D. Schuh, D. Bougeard, and D. Weiss, Phys. Rev. Lett. [**113**]{}, 236602 (2014). X. C. Zhang, A. Pfeuffer-Jeschke, K. Ortner, V. Hock, H. Buhmann, C. R. Becker, and G. Landwehr, Phys. Rev. B [**63**]{}, 245305 (2001). Y. S. Gui, C. R. Becker, N. Dai, J. Liu, Z. J. Qiu, E. G. Novik, M. Schäfer, X. Z. Shu, J. H. Chu, H. Buhmann, and L. W. Molenkamp, Phys. Rev. B [**70**]{}, 115328 (2004). B. Goss Levi, Phys. Today [**46**]{}, 17 (1993). L. D. Landau, E. M. Lifshitz, Phys. Z. Sowjetunion [**8**]{}, 153 (1935).
|
---
abstract: |
We show that a general solution of the Einstein equations that describes approach to an inhomogeneous and anisotropic sudden spacetime singularity does not experience geodesic incompleteness. This generalises the result established for isotropic and homogeneous universes. Further discussion of the weakness of the singularity is also included.
PACS number: 98.80.-k
author:
- |
[John D. Barrow$^{1}$ and S. Cotsakis$^{2}$ ]{}\
$^{1}$DAMTP, Centre for Mathematical Sciences,\
Cambridge University, Cambridge CB3 0WA, UK\
$^{2}$Research group of Geometry, Dynamical Systems and Cosmology,\
University of the Aegean, Karlovassi 83200, Samos, Greece.
title: '[Geodesics at Sudden Singularities]{}'
---
Introduction
============
There has been strong interest in the structure and ubiquity of finite-time singularities in general-relativistic cosmological models since they were first introduced by Barrow et al [@jb1], as a counter-example to the belief [@ellis] that closed Friedmann universes obeying the strong energy condition must collapse to a future singularity. They were characterised in detail as sudden singularities in refs. [@jb2; @jb3; @jb4] and are ‘weak’ singularities in the senses defined by Tipler [@tip] and Krolak [@kr]. A sudden future singularity at $t_{s}$ is defined informally in terms of the metric expansion scale factor, $a(t)$ with $t_{s}>0,$ by $0<a(t_{s})<\infty ,$ $0<\dot{a}(t_{s})<\infty ,\ddot{a}(t\rightarrow t_{s})\rightarrow -\infty $. These archetypal examples have finite values of the metric scale factor, its first time derivative and the density at a finite time but possess infinities in the second time derivative of the scale factor and in the pressure. Higher-order examples exist with infinities in the $(2+n)^{th}$ derivatives of the scale factor and the $n^{th}$ derivative of the matter pressure [@jb3; @jb4]. Other varieties of finite-time singularity have been found in which a different permutation of physical quantities take on finite and infinite values.[^1].
The general isotropic and homogeneous approach to a sudden finite-time singularity introduced in [@jb2] for the Friedmann universe has been used [@bct] to construct a quasi-isotropic, inhomogeneous series expansion around the finite-time singularity which contains nine independently arbitrary spatial functions, as required of a part of the general cosmological solution when the pressure and density are not related by an equation of state. The stability properties of a wide range of possible finite-time singularities were also studied in ref [@lip].
It has also been shown by Fernández-Jambrina and Lazkoz [@jambr; @jambr2; @jambr3] that, in the context of the Friedmann universe, the sudden singularity introduced in [@jb2] has the property that geodesics do not feel the sudden singularity and pass through it. In this note we will examine the evolution of geodesics in the general nine-function solution in the vicinity of an inhomogeneous and anisotropic sudden singularity to see if this result continues to hold. We will also formulate these earlier results more precisely.
We will use Latin indices for spacetime components, Greek indices for space components, and set $G=c=1$.
Geometric setup
===============
Let $\Sigma _{0}$ be the 3-space defined by the equations $x^{i}=\phi
^{i}(\xi ),\xi =(\xi _{1},\xi _{2},\xi _{3})$, located at $t=0$. We suppose that the sudden singularity is located at the time $t_{s}$ to the future, and denote by $\Sigma _{s}$ the 3-space $t=t_{s}$. We may attach geodesic normal (synchronous) coordinates at any point $B\in \Sigma _{s}$ as follows. Let $u^{i}(\xi )$ be a $\mathcal{C}^{0}$ vector field over $\Sigma _{0}$, and through any point on $\Sigma _{0}$ we draw causal geodesics tangent to $u^{i}(\xi )$ in both future and past directions parametrized by $t$. These geodesics have $dx^{i}/dt=u^{i}$ (and $t=0$ on $\Sigma _{0}$). Then the geodesic $x^{i}(t)$ that passes through $B$ cuts $\Sigma _{0}$ at the point $A$ with coordinates $(\xi _{1},\xi _{2},\xi _{3})$ where $t=0$ and $dx^{i}/dt=u^{i}$. The coordinates of $B$ are then $(t_{s},\xi )$, where $t_{s}$ is $t$ evaluated at $B$ and $\xi $ at $A$.
$\mathcal{C}^1$ quasi-isotropic metric
======================================
In [@bct] we found that near a sudden singularity the general form of the metric in geodesic normal coordinates is $$ds^{2}=dt^{2}-\gamma _{\alpha \beta }dx^{\alpha }dx^{\beta },\quad \gamma
_{\alpha \beta }=a_{_{\alpha \beta }}+b_{_{\alpha \beta }}t+c_{_{\alpha
\beta }}t^{n}+\cdots ,\quad n\in (1,2), \label{met}$$and the leading orders of the energy-momentum tensor components, defined by $$T_{j}^{i} =(\rho +p)u^{i}u_{j}-p\delta _{j}^{i}, \quad u_{a}u^{a} =1,$$are $$u_{\alpha }=-\frac{3(b_{\alpha ;\beta }^{\beta }-b_{;\alpha })}{2n(n-1)c}\;t^{2-n}\sim t^{2-n},\quad u^{\alpha }=\gamma ^{\alpha \beta }u_{\beta
}\sim t^{2},$$ $$16\pi \rho =\left( P+\frac{b^{2}-b^{\mu \nu }b_{\mu \nu }}{4}\right) -\frac{n}{2}(b^{\mu \nu }c_{\mu \nu }-bc)\;t^{n-1}+\cdots ,$$ $$16\pi p=-\frac{2n(n-1)c}{3}\;t^{n-2}-\frac{3b^{\mu \nu }b_{\mu \nu }+b^{2}+4P}{12}-\frac{n}{2}(b^{\mu \nu }c_{\mu \nu }+\frac{bc}{3})\;t^{n-1}+\cdots .$$ The Ricci scalar is $$R=R_{i}^{i}=-n(n-1)ct^{n-2}-\frac{b_{\mu \nu }b^{\mu \nu }+b^{2}+4P}{4}-\frac{n}{2}(b^{\mu \nu }c_{\mu \nu }+bc)\;t^{n-1}+\cdots ,$$ where $P$ is the trace of $P_{\alpha \beta }$, the spatial Ricci tensor associated with $a_{\alpha \beta }.$
This solution is only $\mathcal{C}^{1}$, meaning the the metric, its first derivatives as well as the Christoffel symbols will be continuous through the 3-slice $\Sigma _{s}$ containing the sudden singularity at $B$, but we expect discontinuities in the second and higher derivatives of the metric, and at least in the first derivatives of the Christoffel symbols.
Geodesic behaviour at $t_s$
===========================
The Christoffel symbols are $\mathcal{C}^{0}$, and so the geodesic equations, $$\ddot{x}^{i}+\Gamma _{jk}^{i}u^{j}u^{k}=0, \label{g}$$will have solutions, $x^{i}(t)$, with continuous derivatives up to and including ${d^{2}x^{i}}/{dt^{2}}$. Therefore, we can Taylor estimate these solutions as follows. For any $\delta >0$ and $t\in (t_{s}-\delta
,t_{s}+\delta )$, we have $$x^{i}(t)=x^{i}(t_{s})+(t-t_{s})u^{i}(t_{s})-\frac{1}{2}(t-t_{s})^{2}(\Gamma
_{\alpha \beta }^{i}u^{\alpha }u^{\beta })(t_{\ast }), \label{exp}$$with $t_{\ast }$ between $t$ and $t_{s}$. The last term is given in the Lagrange form for the remainder. Since the error term is quadratic in $t-t_{s}$, it vanishes asymptotically for both past and future sudden singularities. This means that the geodesic equations (\[g\]) have complete $\mathcal{C}^{2}$ solutions through the sudden singularity at $B$ to the future and the past given by this form. In higher-order lagrangian theories of gravity it is possible for sudden singularities to arise because there are infinities in the third, or higher, time derivatives of the metric scale factor. In these cases the effect of the singularity on the geodesics is weaker still and avoids a violation of the dominant energy condition [lake, jb4]{}.
A spacetime is Tipler(T)-strong [@tip] iff, as the affine parameter $\tau
\rightarrow t_{s}$, the integral $$T(u)\equiv \int_{0}^{\tau }d\tau ^{\prime }\int_{0}^{\tau ^{^{\prime
}}}R_{ij}u^{i}u^{j}d\tau ^{\prime \prime }\rightarrow \infty .$$The spacetime is Krolak(K)-strong [@kr] iff, as $\tau \rightarrow t_{s}$, the integral $$K(u)\equiv \int_{0}^{\tau }R_{ij}u^{i}u^{j}d\tau ^{\prime }\rightarrow
\infty .$$If these conditions do not hold the spacetime is T-weak or K-weak, respectively. It is possible for a singularity to be K-strong but T-weak, for example the so-called [@type] Type III singularities with $\rho
\rightarrow \infty ,\left\vert p\right\vert \rightarrow \infty $ as $a\rightarrow a_{s}$ have this property. In our case, the various components of the Ricci curvature have leading orders of the following forms: $R_{00}\sim t^{n-2},R_{0\alpha }\sim t^{0},R_{\alpha \gamma }\sim t^{2(n-1)},$ while $u^{0}\sim t^{0},u^{\alpha }\sim t^{2}$. Therefore $$R_{ij}u^{i}u^{j}\sim t^{n-2}+2t^{2}+t^{2n+2}.$$But since at the sudden singularity, $1<n<2$, we find that $$R_{ij}u^{i}u^{j}\sim t^{n-2},\quad \text{as}\quad t\rightarrow t_{s},$$and so after one integration we have, $$K(u)\sim \tau ^{n-1}\rightarrow t_{s}^{n-1},\quad \text{as}\quad \tau
\rightarrow t_{s},$$and after a second integration, $$T(u)\sim \tau ^{n}\rightarrow t_{s}^{n},\quad \text{as}\quad \tau
\rightarrow t_{s},$$and so the generic sudden singularity (\[met\]) is T-weak and K-weak[^2]. This weakness also suggests that we do not expect these singularity structures to be modified by quantum particle production effects. Some studies of the quantum cosmology of sudden singularities which confirm this have been made in refs [@quan] but quantum modifications can occur for particular regularisation procedures [@haro]. There are also interesting classical questions about the passage through a sudden singularity in certain examples where the background matter variables, $\rho $ and $p$, do not continue to be well defined. These problems can be avoided by a distributional redefinition of the cosmological quantities involved [@dist]. It is also interesting to note that extended objects like fundamental string loops can pass through weak singularities without their invariant sizes becoming infinite [@bal].
Conclusion
==========
This result generalizes the studies of Fernández-Jambrina and Lazkoz [@jambr; @jambr2; @jambr3] by showing that there is no geodesic incompleteness at a general inhomogeneous and anisotropic sudden singularity. The inclusion of anisotropy and inhomogeneity does not introduce geodesic incompleteness. We would expect that these results will also hold for sudden singularities in Loop Quantum Gravity cosmologies of the sort studied in ref. [@singh] and in higher-order lagrangian gravity theories [@jb3].
[99]{} J. D. Barrow, G. Galloway and F .J. Tipler, *Mon. Not. R. astron. Soc.* **223,** 835 (1986).
G. F. R. Ellis, In *General Relativity and Cosmology*, Proc. Fermi School of Physics 47, (Academic Press: New York, 1971), ed. R. K. Sachs, p. 139.
J. D. Barrow, *Class. Quantum Grav.* **21** L79 (2004).
J. D. Barrow, *Class. Quantum Grav.* **21** 5619 (2004).
J. D. Barrow and C. G. Tsagas, *Class. Quantum Grav*. **22** 1563 (2005).
F. J. Tipler, *Phys. Lett. A* **64**, 8 (1977).
A. Krolak *Class. Quantum Grav*. **3**, 267 (1986).
G. T. Fox, *Am. J. Phys.* **41**, 311 (1973), R. Stephenson, *Am. J. Phys.* **50,** 1150 (1982), J.D. Barrow, *Mathletics,* (W. Norton: New York, 2012), chap. 19.
J. D. Barrow, S. Cotsakis and A. Tsokaros, *Class. Quantum Grav.* **27,** 165017 (2010).
J. D. Barrow and S. Z. W. Lip, *Phys. Rev. D* **80,** 043518 (2009).
L. Fernández-Jambrina and R. Lazkoz, *Phys. Rev. D* **70,** 121503 (2004).
L. Fernández-Jambrina and R. Lazkoz, *J. Phys. Conf. Ser.* **66**, 012015 (2007); *AIP Conf. Proc.* **841**, 420 (2006).
L. Fernández-Jambrina and R. Lazkoz, *Phys. Rev. D* **74,** 064030 (2006).
K. Lake, *Class. Quantum Grav.* **21**, L129 (2004).
S. Nojiri, S. D. Odintsov and S.Tsujikawa, *Phys. Rev. D* **71**, 063004 (2005).
M. Dabrowski, *Phys. Lett. B* **702**, 320 (2011).
J. D. Barrow, A. B. Batista, J. C. Fabris, and M. J. S. Houndjo, *Phys. Rev. D* **78**, 123508 (2008); J. D. Barrow, A. B. Batista, G. Dito, J. C. Fabris and M. J. S. Houndjo, *Phys. Rev. D*** 84**, 123518 (2011).
J. de Haro, J. Amoros and E. Elizalde, *Phys. Rev. D* **85**, 123527 (2012).
Z. Keresztes, L. Á. Gergely and A. Yu. Kamenshchik, *Phys. Rev. D* **86**, 063522 (2012).
A. Balcerzak and M. Dabrowski. *Phys. Rev. D* **73,** 101301 (2006).
P. Singh, *Class. Quantum Grav.* **26**, 125005 (2009); P. Singh and F. Vidotto, *Phys. Rev. D* **83**, 064027 (2011).
[^1]: There is an interesting example in Newtonian mechanics of motion which formally begins from rest with infinite acceleration. It is motion at constant power. This means $v\dot{v}$ is constant, where $v=\dot{x}$ is the velocity in the $x$ direction and so $v\propto t^{1/2}$ and $x\propto t^{3/2}
$ if initially $v(0)=x(0)=0$. Thus we see that the acceleration formally has $\dot{v}\propto t^{-1/2}$ and diverges as $t\rightarrow 0$. This motion at constant power is an excellent model of drag-car racing. The singularity in the acceleration as $t\rightarrow 0$ is ameliorated in practice by the inclusion of frictional effects on the initial motion [@drag].
[^2]: If $0<n<1,$ and the metric contains a power of $(t-t_{s}\ )^{n}$, then it will only have a well-defined meaning as a real function when $t-t_{s}>0$. So, at any point $t_{s}$ (e.g., when $t_{s}=0$), it will be defined asymptotically only in the past direction and not to the future. Our argument also requires continuity of the Christoffel symbols, and so it will not be valid when the metric contains a power of $(t-t_{s}\ )^{n}$ with $0<n<1$, even if we restrict only to the past direction. For an isotropic solution with a sudden singularity at $t=0$, see [@dab]
|
---
abstract: 'The Grundy number of a graph $G$, denoted by $\Gamma(G)$, is the largest $k$ such that there exists a partition of $V(G)$, into $k$ independent sets $V_1,\ldots, V_k$ and every vertex of $V_i$ is adjacent to at least one vertex in $V_j$, for every $j<i$. The objects which are studied in this article are families of $r$-regular graphs such that $\Gamma(G) = r + 1$. Using the notion of independent module, a characterization of this family is given for $r=3$. Moreover, we determine classes of graphs in this family, in particular the class of $r$-regular graphs without induced $C_4$, for $r \le 4$. Furthermore, our propositions imply results on partial Grundy number.'
author:
- 'Nicolas Gastineau[^1]'
- Hamamache Kheddouci
- Olivier Togni
bibliography:
- 'bib.bib'
title: 'On the family of $r$-regular graphs with Grundy number $r+1$'
---
Introduction
============
We consider only undirected connected graphs in this paper. Given a graph $G=(V,E)$, a *proper $k$-coloring* of $G$ is a surjective mapping $c:V \rightarrow\{1,\ldots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E$; the *color class* $V_i$ is the set $\{u\in V| c(u)=i\}$ and a vertex $v$ has color $i$ if $v\in V_i$. A vertex $v$ of color $i$ is a *Grundy vertex* if $v$ is adjacent to at least one vertex colored $j$, for every $j<i$. A *Grundy $k$-coloring* is a proper $k$-coloring such that every vertex is a Grundy vertex. A *partial Grundy $k$-coloring* is a proper $k$-coloring such that every color class contains a Grundy vertex. The *Grundy number* (*partial Grundy number*, respectively) of $G$ denoted by $\Gamma(G)$ ($\partial\Gamma(G)$, respectively) is the largest $k$ such that $G$ admits a Grundy $k$-coloring (partial Grundy $k$-coloring, respectively).
Let $N(v)=\{u\in V(G)|uv\in E(G)\}$ be the neighborhood of $v$. A set $X$ of vertices is an *independent module* if $X$ is an independent set and all vertices in $X$ have the same neighborhood. The vertices in an independent module of size 2 are called *false twins*. Let $P_n$, $C_n$, $K_n$ and $I_n$ be respectively, the path, cycle complete and empty graph of order $n$. The concepts of Grundy $k$-coloring and domination are connected. In a Grundy coloring, $V_{1}$ is a dominating set. Given a graph $G$ and an ordering $\phi$ on $V(G)$ with $\phi=v_1,\ldots,v_n$, the greedy algorithm assigns to $v_i$ the minimum color that was not assigned in the set $\{v_1,\ldots,v_{i-1}\} \cap N(v_i)$. Let $\Gamma_{\phi}(G)$ be the number of colors used by the greedy algorithm with the ordering $\phi$ on $G$. We obtain the following result [@ER2003]: $\Gamma(G)=\max\limits_{\phi\in S_n}(\Gamma_{\phi}(G))$.
The Grundy coloring is a well studied problem. Zaker [@ZA2006] proved that determining the Grundy number of a given graph, even for complements of bipartite graphs, is an NP-complete problem. However, for a fixed $t$, determining if a given graph has Grundy number at least $t$ is decidable in polynomial time. This result follows from the existence of a finite list of graphs, called $t$-atoms, such that any graph with Grundy number at least $t$ contains a $t$-atom as an induced subgraph. It has been proven that there exists a Nordhaus-Gaddum type inequality for the Grundy number [@FU2008; @ZA2006], that there exist upper bounds for $d$-degenerate, planar and outerplanar graphs [@BA2008; @CH2012], and that there exist connections between the products of graphs and the Grundy number [@EF2007; @AS2010; @CA2012]. Recently, Havet and Sampaio [@HA2013] have proven that the problem of deciding if for a given graph $G$ we have $\Gamma(G)=\Delta(G)+1$, even if $G$ is bipartite, is NP-complete. Moreover, they have proven that the dual of Grundy $k$-coloring problem is in FPT by finding an algorithm in $O (2k^{2k}.|E|+2^{2k} k^{3k+5/2})$ time.Note that a Grundy $k$-coloring is a partial Grundy $k$-coloring, hence $\Gamma(G)\le\partial\Gamma(G)$. Given a graph $G$ and a positive integer $k$, the problem of determining if a partial Grundy $k$-coloring exists, even for chordal graphs, is NP-complete but there exists a polynomial algorithm for trees [@SH2005].
Another coloring parameter with domination constraints on the colors is the *$b$-chromatic number*, denoted by $\varphi(G)$, which is the largest $k$ such that there exists a proper $k$-coloring and for every color class $V_i$, there exists a vertex adjacent to at least one vertex colored $j$, for every $j$, with $j\neq i$. Note that a $b$-coloring is a partial Grundy $k$-coloring, hence $\varphi(G)\le\partial\Gamma(G)$. The $b$-chromatic number of regular graphs has been investigated in a series of papers ([@EL2009; @KL2010; @CA2011; @SH2012]). Our aim is to establish similar results for the Grundy coloring. We present two main results: A characterization of the Grundy number of every cubic graph and the following theorem: For $r\le 4$, every $r$-regular graphs without induced $C_4$ has Grundy number $r+1$. We conjecture that this assertion is also true for $r>4$.
For any integer $r\ge1$, every $r$-regular graph without induced $C_4$ has Grundy number $r+1$.
Section 2 gives characterizations of some classes of graphs with Grundy number at most $k$, $2\le k\le \Delta(G)$, using the notion of independent module. Section 3 contains the first main theorem: A description of the cubic graphs with Grundy number at most 3 that also allows us to prove that every cubic graph except $K_{3,3}$ has partial Grundy number $4$. This theorem implies the existence of a linear algorithm to determine the Grundy number of cubic graphs. In Section 4, we present examples of infinite families of regular graphs with Grundy number exactly or at most $k$, $3\le k\le r$. To determine these families we use recursive definitions. The last section contains the second main theorem of this article: 4-regular graphs without induced $C_4$ have Grundy number 5.
General results
===============
The reader has to be aware of the resemblance of name between the following notion and that of partial Grundy $k$-coloring.
Let $G$ be a graph. A Grundy partial $k$-coloring is a Grundy $k$-coloring of a subset $S$ of $V(G)$.
If $G$ admits a Grundy partial $k$-coloring, then $\Gamma(G)\ge k$.
This property has an important consequence: For a graph $G$, with $\Gamma(G)\ge t$ and any Grundy partial $t$-coloring, there exist smallest subgraphs $H$ of $G$ such that $\Gamma(H)=t$. The family of $t$-atoms corresponds to these subgraphs. This concept was introduced by Zaker [@ZA2006]. The family of $t$-atoms is finite and the presence of a $t$-atom can be determined in polynomial time for a fixed $t$. The following definition is slightly different from Zaker’s one, insisting more on the construction of every $t$-atom.
For any integer $t$, we define the family of $t$-atoms, denoted by $\mathcal{A}_t$, $t=1,\ldots$ by induction. Let the family $\mathcal{A}_1$ contain only $K_1$. A graph $G$ is in $\mathcal{A}_{t+1}$ if there exists a graph $G'$ in $\mathcal{A}_t$ and an integer $m$, $m\le|V(G')|$, such that $G$ is composed of $G'$ and an independent set $I_m$ of order $m$, adding edges between $G'$ and $I_m$ such that every vertex in $G'$ is connected to at least one vertex in $I_m$. Moreover a $t$-atom $A$ is minimal, if there is no $t$-atom included in $A$ other than itself.
For a given graph $G$, $\Gamma(G)\ge t$ if and only if $G$ contains an induced minimal $t$-atom.
We now present conditions related to the presence of modules that allows us to upper-bound the Grundy number.
Let $G$ be a graph and $X$ be an independent module. In every Grundy coloring of $G$, the vertices in $X$ must have the same color. \[ff\]
Let $G$ be an $r$-regular graph. A vertex $v$ is a $(0,\ell)$-twin-vertex if there exists an independent module of cardinality $r+2-\ell$ that contains $v$.
Let $G$ be an $r$-regular graph. The color of an $(0,\ell)$-twin-vertex is at most $\ell$ in every Grundy coloring of $G$. \[0twin\]
Let $v$ be a $(0,\ell)$-twin-vertex colored $\ell+1$ in $G$. By Definition, $v$ is in an independent module $X$ of cardinality $r+2-\ell$ and by Proposition \[ff\], every other vertex of $X$ should be colored $\ell+1$. Let $u$ be a neighbor of $v$. There are at most $\ell-2$ neighbors of $u$ in $V(G-X)$. Therefore, $u$ cannot be colored $\ell$.
A vertex $v$ of a graph $G$ is a $(1,\ell)$-twin-vertex if $N(v)$ can be partitioned into at least $\ell-1$ independent modules.
Let $G$ be a graph. The color of an $(1,\ell)$-twin-vertex is at most $\ell$ in every Grundy coloring of $G$.
By Proposition \[ff\], vertices of the neighborhood of $v$ can only have $\ell-1$ different colors. Therefore, the color of $v$ is at most $\ell$.
A vertex $v$ of a graph $G$ is a $(2,\ell)$-twin-vertex if $N(v)$ is independent and every vertex in $N(v)$ is a $(1,\ell)$-twin-vertex.
Let $G$ be a graph. The color of an $(2,\ell)$-twin-vertex is at most $\ell$ in every Grundy coloring of $G$.
Let $v$ be a $(2,\ell)$-twin-vertex in $G$. Every vertex in $N(v)$ is a $(1,\ell)$-twin-vertex. If a vertex in $N(v)$ is colored $\ell$, then $v$ could only have a color at most $\ell-1$. If the vertices in the neighborhood of $v$ have colors at most $\ell-1$, then in every Grundy coloring of $G$, $v$ has a color at most $\ell$.
Let $G$ be a graph. If every vertex is a $(1,\ell)$-twin-vertex or a $(2,\ell)$-twin-vertex, then $\Gamma(G)\le\ell$. \[2twin\]
Let $G$ be a regular graph. If every vertex is an $(i,\ell)$-twin-vertex, for some $i$, $0 \le i \le 2$, then $\Gamma(G)\le\ell$. \[itwin\]
Let $G$ be a graph. We have $\Gamma(G)\le 2$ if and only if $G=K_{n,m}$ for some integers $n>0$ and $m>0$. \[g2\]
Grundy numbers of cubic graphs
==============================
In the following sections, the figures describe Grundy partial $k$-colorings. By a dashed edge we denote a possible edge. The vertices not connected by edges in the figures cannot be adjacent as it would contradict the hypothesis.
Let $G$ be a connected 2-regular graph. $\partial\Gamma(G)=\Gamma(G)=2$ if and only if $G=C_4$. \[indc2\]
(0,0.5) – (1.2,1); (0,0.5) – (1.2,0); (1.2,0) – (2.4,1); (1.2,0) – (2.4,0); (1.2,1) – (2.4,1); (1.2,1) – (2.4,0);
(0+4,0.5) – (1.2+4,1); (0+4,0.5) – (1.2+4,0); (1.2+4,0) – (2.4+4,1); (1.2+4,0) – (2.4+4,0); (1.2+4,1) – (2.4+4,1); (1.2+4,1) – (2.4+4,0); (1.2+4,1) – (2.4+4,1); (2.4+4,1) – (3.6+4,0.5); (2.4+4,0) – (3.6+4,0.5); at (0,0.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.2,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.4,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.2,1) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.4,1) \[circle,draw=black,fill=black, scale=0.7\] ; at (0+4,0.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.2+4,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.4+4,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.2+4,1) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.4+4,1) \[circle,draw=black,fill=black, scale=0.7\] ; at (3.6+4,0.5) \[circle,draw=black,fill=black, scale=0.7\] ;
The following definition gives a construction of the cubic graphs in which every vertex is an $(i,3)$-twin-vertex, for some $i$, $0 \le i \le 2$. Figure \[figcuu\] gives the list of every graph of order at most 16 in this family.
Let $K_{2,3}$ and $K^{*}_{3,3}$ be the graphs from Figure \[figk\]. We define recursively the family of graphs $\mathcal{F}^{*}_3$ as follows:
1. $K_{2,3} \in \mathcal{F}^{*}_3$ and $K^{*}_{3,3}\in \mathcal{F}^{*}_3$;
2. the disjoint union of two elements of $\mathcal{F}^{*}_3$ is in $\mathcal{F}^{*}_3$;
3. if $G$ is a graph in $\mathcal{F}^{*}_3$, then the graph $H$ obtained from $G$ by adding an edge between two vertices of degree at most $2$ is also in $\mathcal{F}^{*}_3$;
4. if $G$ is a graph in $\mathcal{F}^{*}_3$, then the graph $H$ obtained from $G$ by adding a new vertex adjacent to three vertices of degree at most 2 is in $\mathcal{F}^{*}_3$.
The family $\mathcal{F}_3$ is the subfamily of cubic graphs in $\mathcal{F}^{*}_3$.
\[ggg2\] Let $G$ be a cubic graph. Every vertex of $V(G)$ is an $(i,3)$-twin vertex, for some $i$, $0\leq i\leq 2$, if and only if $G \in\mathcal{F}_3$.
(0,0) – (0.8,0); (0,0.8) – (0.8,0.8); (0,0) – (-0.7,0.4); (0,0.8) – (-0.7,0.4); (0.8,0) – (1.5,0.4); (0.8,0.8) – (1.5,0.4); (0.8,0) – (0,0.8); (0.8,0.8) – (0,0); (1.5,0.4) – (-0.7,0.4);
at (0,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.8,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (0,0.8) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.8,0.8) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.5,0.4) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.7,0.4) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.4,-0.6)[$K_{3,3}$]{};
(2.5,0.7) – (3,0); (3.5,0.7) – (3,0); (2.5,0.7) – (2.5,1.5); (3.5,0.7) – (3.5,1.5); (2.5,0.7) – (3.5,1.5); (3.5,0.7) – (2.5,1.5); (3.5,1.5) – (2.5,1.5); (4,0.7) – (4.5,0); (5,0.7) – (4.5,0); (4,0.7) – (4,1.5); (5,0.7) – (5,1.5); (4,0.7) – (5,1.5); (5,0.7) – (4,1.5); (5,1.5) – (4,1.5); (5,1.5) – (4,1.5); (3,0) – (4.5,0); at (3,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.5,0.7) \[circle,draw=black,fill=black, scale=0.7\] ; at (3.5,0.7) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.5,1.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (3.5,1.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.5,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (4,0.7) \[circle,draw=black,fill=black, scale=0.7\] ; at (5,0.7) \[circle,draw=black,fill=black, scale=0.7\] ; at (4,1.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (5,1.5) \[circle,draw=black,fill=black, scale=0.7\] ;
(6.7,0) – (9.1,0); (6.7,0.8) – (9.1,0.8); (6.7,0) – (6,0.4); (6.7,0.8) – (6,0.4); (9.1,0) – (8.3,0.8); (9.1,0.8) – (8.3,0); (7.5,0) – (6.7,0.8); (7.5,0.8) – (6.7,0); (9.8,0.4) – (9.1,0); (9.8,0.4) – (9.1,0.8); (9.8,0.4) – (6,0.4); at (6,0.4) \[circle,draw=black,fill=black, scale=0.7\] ; at (9.8,0.4) \[circle,draw=black,fill=black, scale=0.7\] ; at (6.7,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (7.5,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (8.3,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (9.1,0) \[circle,draw=black,fill=black, scale=0.7\] ; at (6.7,0.8) \[circle,draw=black,fill=black, scale=0.7\] ; at (7.5,0.8) \[circle,draw=black,fill=black, scale=0.7\] ; at (8.3,0.8) \[circle,draw=black,fill=black, scale=0.7\] ; at (9.1,0.8) \[circle,draw=black,fill=black, scale=0.7\] ;
(-0.3,0-3.3) – (-0.7,0.7-3.3); (-0.3,0-3.3) – (0.1,0.7-3.3); (-0.7,1.5-3.3) – (-0.7,0.7-3.3); (0.1,1.5-3.3) – (0.1,0.7-3.3); (-0.7,1.5-3.3) – (0.1,0.7-3.3); (0.1,1.5-3.3) – (-0.7,0.7-3.3); (-0.7,1.5-3.3) – (-0.3,2.2-3.3); (0.1,1.5-3.3) – (-0.3,2.2-3.3); (1,0-3.3) – (0.6,0.7-3.3); (1,0-3.3) – (1.4,0.7-3.3); (0.6,1.5-3.3) – (0.6,0.7-3.3); (1.4,1.5-3.3) – (1.4,0.7-3.3); (0.6,1.5-3.3) – (1.4,0.7-3.3); (1.4,1.5-3.3) – (0.6,0.7-3.3); (0.6,1.5-3.3) – (1,2.2-3.3); (1.4,1.5-3.3) – (1,2.2-3.3); (-0.3,0-3.3) – (1,0-3.3); (-0.3,2.2-3.3) – (1,2.2-3.3); at (-0.3,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.3,2.2-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (1,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (1,2.2-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.7,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.7,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.1,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.1,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.6,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.6,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.4,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.4,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ;
(-0.3+2.9,0-3.3) – (-0.7+2.9,0.7-3.3); (-0.3+2.9,0-3.3) – (0.1+2.9,0.7-3.3); (-0.7+2.9,1.5-3.3) – (-0.7+2.9,0.7-3.3); (0.1+2.9,1.5-3.3) – (0.1+2.9,0.7-3.3); (-0.7+2.9,1.5-3.3) – (0.1+2.9,0.7-3.3); (0.1+2.9,1.5-3.3) – (-0.7+2.9,0.7-3.3); (-0.7+2.9,1.5-3.3) – (-0.3+2.9,2.2-3.3); (0.1+2.9,1.5-3.3) – (-0.3+2.9,2.2-3.3); (-0.3+2.9,0-3.3) – (0.8+2.9,1.1-3.3); (-0.3+2.9,2.2-3.3) – (0.8+2.9,1.1-3.3); (0.8+2.9,1.1-3.3) – (0.8+0.5+2.9,1.1-3.3); (1.5+0.5+2.9,0.7-3.3) – (0.8+0.5+2.9,1.1-3.3); (1.5+0.5+2.9,1.5-3.3) – (0.8+0.5+2.9,1.1-3.3); (1.5+0.5+2.9,0.7-3.3) – (2.3+0.5+2.9,0.7-3.3); (1.5+0.5+2.9,1.5-3.3) – (2.3+0.5+2.9,1.5-3.3); (1.5+0.5+2.9,0.7-3.3) – (2.3+0.5+2.9,1.5-3.3); (1.5+0.5+2.9,1.5-3.3) – (2.3+0.5+2.9,0.7-3.3); (2.3+0.5+2.9,0.7-3.3) – (2.3+0.5+2.9,1.5-3.3);
at (-0.3+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.7+2.9,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.1+2.9,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.1+2.9,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.7+2.9,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.3+2.9,2.2-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.8+0.5+2.9,1.1-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.8+2.9,1.1-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.5+0.5+2.9,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.3+0.5+2.9,0.7-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.5+0.5+2.9,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.3+0.5+2.9,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ;
(3.3+2.9,0-3.3) – (3.3+2.9,0.8-3.3); (4.1+2.9,0-3.3) – (4.1+2.9,0.8-3.3); (3.3+2.9,0-3.3) – (4.1+2.9,0-3.3); (3.3+2.9,0-3.3) – (4.1+2.9,0.8-3.3); (3.3+2.9,0.8-3.3) – (4.1+2.9,0-3.3); (3.3+2.9,0.8-3.3) – (3.7+2.9,1.5-3.3); (4.1+2.9,0.8-3.3) – (3.7+2.9,1.5-3.3);
(4.6+2.9,0-3.3) – (4.6+2.9,0.8-3.3); (5.4+2.9,0-3.3) – (5.4+2.9,0.8-3.3); (4.6+2.9,0-3.3) – (5.4+2.9,0-3.3); (4.6+2.9,0-3.3) – (5.4+2.9,0.8-3.3); (4.6+2.9,0.8-3.3) – (5.4+2.9,0-3.3); (4.6+2.9,0.8-3.3) – (5+2.9,1.5-3.3); (5.4+2.9,0.8-3.3) – (5+2.9,1.5-3.3);
(5.9+2.9,0-3.3) – (5.9+2.9,0.8-3.3); (6.7+2.9,0-3.3) – (6.7+2.9,0.8-3.3); (5.9+2.9,0-3.3) – (6.7+2.9,0-3.3); (5.9+2.9,0-3.3) – (6.7+2.9,0.8-3.3); (5.9+2.9,0.8-3.3) – (6.7+2.9,0-3.3); (5.9+2.9,0.8-3.3) – (6.3+2.9,1.5-3.3); (6.7+2.9,0.8-3.3) – (6.3+2.9,1.5-3.3); (5+2.9,1.5-3.3) – (5+2.9,2-3.3); (6.3+2.9,1.5-3.3) – (5+2.9,2-3.3); (3.7+2.9,1.5-3.3) – (5+2.9,2-3.3);
at (3.3+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (3.3+2.9,0.8-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.1+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.1+2.9,0.8-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.6+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.6+2.9,0.8-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (5.4+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (5.4+2.9,0.8-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (5.9+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (5.9+2.9,0.8-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (6.7+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (6.7+2.9,0.8-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (5+2.9,2-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (5+2.9,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (6.3+2.9,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (3.7+2.9,1.5-3.3) \[circle,draw=black,fill=black, scale=0.7\] ;
(-0.7,0.7-5.5) – (-0.2,0-5.5); (0.3,0.7-5.5) – (-0.2,0-5.5); (-0.7,0.7-5.5) – (-0.7,1.5-5.5); (0.3,0.7-5.5) – (0.3,1.5-5.5); (-0.7,0.7-5.5) – (0.3,1.5-5.5); (0.3,0.7-5.5) – (-0.7,1.5-5.5); (0.3,1.5-5.5) – (-0.7,1.5-5.5); (-0.2,0-5.5) – (0.3,0-5.5); (0.3,0-5.5) – (1,0.4-5.5); (0.3,0-5.5) – (1,-0.4-5.5); (1,0.4-5.5) – (1.8,0.4-5.5); (1,-0.4-5.5) – (1.8,-0.4-5.5); (1,0.4-5.5) – (1.8,-0.4-5.5); (1,-0.4-5.5) – (1.8,0.4-5.5); (1.8,0.4-5.5) – (2.5,0-5.5); (1.8,-0.4-5.5) – (2.5,0-5.5); (3,0-5.5) – (2.5,0-5.5); (2.5,0.7-5.5) – (3,0-5.5); (3.5,0.7-5.5) – (3,0-5.5); (2.5,0.7-5.5) – (2.5,1.5-5.5); (3.5,0.7-5.5) – (3.5,1.5-5.5); (2.5,0.7-5.5) – (3.5,1.5-5.5); (3.5,0.7-5.5) – (2.5,1.5-5.5); (2.5,1.5-5.5) – (3.5,1.5-5.5); at (-0.2,0-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.7,0.7-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.3,0.7-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (-0.7,1.5-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.3,1.5-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.6+2.9,0-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.6+2.9,0.8-3.3) \[circle,draw=black,fill=black, scale=0.7\] ; at (0.3,0-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (1,0.4-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (1,-0.4-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.8,0.4-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (1.8,-0.4-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.5,0-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (3,0-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.5,0.7-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (2.5,1.5-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (3.5,0.7-5.5) \[circle,draw=black,fill=black, scale=0.7\] ; at (3.5,1.5-5.5) \[circle,draw=black,fill=black, scale=0.7\] ;
(6.7,0-6) – (9.1,0-6); (6.7,0.8-6) – (9.1,0.8-6); (6.7,0-6) – (6,0.4-6); (6.7,0.8-6) – (6,0.4-6); (9.1,0-6) – (8.3,0.8-6); (9.1,0.8-6) – (8.3,0-6); (7.5,0-6) – (6.7,0.8-6); (7.5,0.8-6) – (6.7,0-6); (9.8,0.4-6) – (9.1,0-6); (9.8,0.4-6) – (9.1,0.8-6); (9.8,0.4-6) .. controls (9.8,1.5-6) and (6,1.5-6) .. (5,0.4-6); (5,0.4-6) – (6,0.4-6); (5,0.4-6) – (5,1-6); (5,1-6) – (4.5,1.7-6); (5,1-6) – (5.5,1.7-6); (4.5,2.5-6) – (4.5,1.7-6); (5.5,2.5-6) – (5.5,1.7-6); (4.5,2.5-6) – (5.5,1.7-6); (5.5,2.5-6) – (4.5,1.7-6); (5.5,2.5-6) – (4.5,2.5-6); at (6,0.4-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (9.8,0.4-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (6.7,0-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (7.5,0-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (8.3,0-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (9,0-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (6.7,0.8-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (7.5,0.8-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (8.3,0.8-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (9,0.8-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (5,0.4-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (5,1-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.5,1.7-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (5.5,1.7-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (4.5,2.5-6) \[circle,draw=black,fill=black, scale=0.7\] ; at (5.5,2.5-6) \[circle,draw=black,fill=black, scale=0.7\] ;
Every graph $G$ in $\mathcal{F}_3$ has three kind of vertices: $(0,3)$-twin-vertices (called also false twins), vertices where an edge is added by Point 3 and vertices added by Point 4. Vertices where an edge is added by Point 3 are $(1,3)$-twin-vertex and vice versa. Vertices added by Point 4 are $(2,3)$-twin-vertices and vice versa.
Let $G$ be a cubic graph. $\Gamma(G)\le 3$ if and only if every vertex is an $(i,3)$-twin-vertex, for some $i$, $0 \le i \le 2$.
By Corollary \[itwin\], the “if” part is proven. Assume that $G$ contains a vertex $v$ which is not an $(i,3)$-twin-vertex, for some $i$, $0 \le i \le 2$ and $\Gamma(G)<4$. In every configuration we want to either find a Grundy partial 4-coloring, contradicting $\Gamma(G)<4$ or proving that $v$ is an $(i,3)$-twin-vertex, for some $i$, with $0 \le i \le 2$. We will refer to a given Grundy partial 4-coloring by its reference in Figure \[figc\]. We consider three cases: $v$ or a neighbor of $v$ is in a $C_3$, $v$ is in an induced $C_4$ and $v$ or a neighbor of $v$ are not in a $C_3$ and $v$ is not in an induced $C_4$. Let $C$ be an induced cycle of order 3 or 4 which contains $v$ or a neighbor of $v$ and let $D_1=\{ x\in V(G)|d(x,C)=1\}$, where $d(x,C)$ is the distance from $x$ to $C$ in the graph $G$. To simplify notation, $D_1$ will also denote the subgraph of $G$ induced by $D_1$.
Case 1:
: Assume that $v$ or a neighbor of $v$ is in $C$ and $C=C_3$. If $|D_1|=1$, then $G=K_4$ and $\Gamma(K_4)=4$. If $|D_1|=2$ and $D_1=P_2$, then $v$ is a $(0,3)$-twin-vertex or a $(1,3)$-twin-vertex. If $D_1=I_2$ then Figure 3.1.a yields a Grundy partial 4-coloring of $G$. If $|D_1|=3$, then we have four subcases: $D_1$ is $C_3$ or $P_3$ (Figure 3.1.b), $P_2\cup I_1$ (Figure 3.1.c) or $I_3$ (Figure 3.1.d). In every case $G$ admits a Grundy partial 4-coloring.
Case 2:
: Assume that $v$ is in $C$ and $C=C_4$. Note that for two non adjacent vertices of $C$ who have a common neighbor in $D_1$, the vertex $v$ is a $(0,3)$-twin-vertex or a $(1,3)$-twin-vertex. Hence, we will not consider these cases. If $|D_1|=2$, then $D_1=P_2$ or $D_1=I_2$ (Figure 3.2.a) and in both cases, $G$ admits a Grundy partial 4-coloring. If $|D_1|=3$, Figure 3.2.b yields a Grundy partial 4-coloring of $G$. In the case $|D_1|=4$, we first assume that two adjacent vertices of $C$ have their neighbors in $D_1$ adjacent (Figure 3.2.c). Afterwards, we suppose that the previous case does not happen and that two non adjacent vertices of $C$ have their neighbors in $D_1$ adjacent (Figure 3.2.d). In the case $D_1=I_4$, we first suppose that two vertices of $D_1$ which have two adjacent vertices of $C$ as neighbor, are not adjacent to two common vertices (Figure 3.2.e) and after consider they are (Figure 3.2.f).
Case 3:
: Assume that $v$ or a neighbor of $v$ is not in a $C_3$ and $v$ is not in an induced $C_4$. Firstly, suppose that a neighbor $u$ of $v$ is in an induced $C_4$. Using the coloring from the previous case, $G$ admits a Grundy partial 4-coloring in every cases except in the case where two neighbors of $v$ in the $C_4$ have a common neighbor outside the $C_4$. However, this case cannot happen for every neighbor of $v$, otherwise $v$ would be a $(2,3)$-twin-vertex. Assume that $u$ is the neighbor of $v$ not in the previous configuration. If $u$ is in an induced $C_4$, then using the coloring from the previous case, $G$ admits a Grundy partial 4-coloring. If $u$ is not in an induced $C_4$, then Figure 3.3.a yields a Grundy partial 4-coloring of $G$. In this figure, the color 2 is given to a neighbor of $u$ not adjacent to both $f_1$ and $f_2$. Secondly, suppose that $v$ is in an induced $C_5$. Figure 3.3.b yields a Grundy partial 4-coloring of $G$. Thirdly, if $v$ is not in an induced $C_5$, then Figure 3.3.c yields a Grundy partial 4-coloring of $G$.
Therefore, if $\Gamma(G)\le 3$, then every vertex is an $(i,3)$-twin-vertex, for some $i$, $0 \le i \le 2$.
Observe that if an edge is added between the two vertices of degree 2 in $K_{3,3}^{*}$, then we obtain $K_{3,3}$ which has Grundy number 2. By Proposition \[ggg2\], in all the remaining cases, the cubic graphs which have Grundy number at most 3 are different from complete bipartite graphs. Therefore, they have Grundy number 3.
(0,0) – (0.6,1); (1.2,0) – (0.6,1); (0,0) – (1.2,0); (0,0) – (-0.6,1); (-0.6,1) – (0.6,1); (1.8,-0.5) – (1.2,0); at (0,0) \[circle,draw=red!50,fill=red!20\] ; at (0,0) [2]{}; at (0.6,1) \[circle,draw=green!50,fill=green!20\] ; at (0.6,1) [3]{}; at (1.2,0)\[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2,0) [4]{}; at (-0.6,1) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.6,1) [1]{}; at (1.8,-0.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.8,-0.5) [1]{}; at (0.6,-1.2)[1.a]{};
(0+3.5,0) – (0.6+3.5,1); (1.2+3.5,0) – (0.6+3.5,1); (0+3.5,0) – (1.2+3.5,0); (0.6+3.5,1) – (0.6+3.5,1.5); (-0.6+3.5,-0.5) – (0+3.5,0); (1.8+3.5,-0.5) – (1.2+3.5,0); (0.6+3.5,1.5) – (-0.6+3.5,-0.5); (0.6+3.5,1.5) – (1.8+3.5,-0.5); (-0.6+3.5,-0.5) – (1.8+3.5,-0.5); at (-0.6+3.5,-0.5) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.6+3.5,-0.5) [1]{}; at (1.2+3.5,0) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+3.5,0) [1]{}; at (0+3.5,0) \[circle,draw=red!50,fill=red!20\] ; at (0+3.5,0) [2]{}; at (1.8+3.5,-0.5) \[circle,draw=red!50,fill=red!20\] ; at (1.8+3.5,-0.5) [2]{}; at (0.6+3.5,1) \[circle,draw=green!50,fill=green!20\] ; at (0.6+3.5,1) [3]{}; at (0.6+3.5,1.5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0.6+3.5,1.5) [4]{}; at (4.1,-1.2)[1.b]{};
(0+7,0) – (0.6+7,1); (1.2+7,0) – (0.6+7,1); (0+7,0) – (1.2+7,0); (0.6+7,1) – (0.6+7,1.5); (1.8+7,-0.5) – (1.2+7,0); (0.6+7,1.5) – (1.8+7,-0.5); (0.6+7,1.5) – (1.8+7,1.5); (1.8+7,1.5) – (1.8+7,-0.5); at (0.6+7,1) \[circle,draw=green!50,fill=green!20\] ; at (0.6+7,1) [3]{}; at (0.6+7,1.5)\[circle,draw=yellow!50,fill=yellow!20\] ; at (0.6+7,1.5) [4]{}; at (1.8+7,-0.5) \[circle,draw=red!50,fill=red!20\] ; at (1.8+7,-0.5) [2]{}; at (0+7,0) \[circle,draw=red!50,fill=red!20\] ; at (0+7,0) [2]{}; at (1.2+7,0) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+7,0) [1]{}; at (1.8+7,1.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.8+7,1.5) [1]{}; at (7.6,-1.2)[1.c]{};
(0,0-3.3) – (0.6,1-3.3); (1.2,0-3.3) – (0.6,1-3.3); (0,0-3.3) – (1.2,0-3.3); (0.6,1-3.3) – (0.6,1.5-3.3); (-0.6,-0.5-3.3) – (0,0-3.3); (1.8,-0.5-3.3) – (1.2,0-3.3); at (-0.6,-0.5-3.3) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.6,-0.5-3.3) [1]{}; at (1.8,-0.5-3.3) \[circle,draw=blue!50,fill=blue!20\] ; at (1.8,-0.5-3.3) [1]{}; at (0.6,1.5-3.3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.6,1.5-3.3) [1]{}; at (0.6,1-3.3) \[circle,draw=red!50,fill=red!20\] ; at (0.6,1-3.3) [2]{}; at (0,0-3.3) \[circle,draw=green!50,fill=green!20\] ; at (0,0-3.3) [3]{}; at (1.2,0-3.3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2,0-3.3)[4]{}; at (0.6,-4)[1.d]{};
(0+2.5,0.5-3) – (1.2+2.5,1-3); (0+2.5,0.5-3) – (1.2+2.5,0-3); (1.2+2.5,0-3) – (1.2+2.5,1-3); (1.2+2.5,0-3) – (2.4+2.5,0-3); (1.2+2.5,1-3) – (2.4+2.5,1-3); (2.4+2.5,0-3) – (2.4+2.5,1-3); (2.4+2.5,0-3) – (3.6+2.5,0.5-3); (2.4+2.5,1-3) – (3.6+2.5,0.5-3); (2.5,0.5-3) – (3.6+2.5,0.5-3); at (1.2+2.5,1-3) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+2.5,1-3) [1]{}; at (2.5,0.5-3) \[circle,draw=red!50,fill=red!20\] ; at (2.5,0.5-3) [2]{}; at (2.4+2.5,1-3) \[circle,draw=red!50,fill=red!20\] ; at (2.4+2.5,1-3) [2]{}; at (2.4+2.5,0-3) \[circle,draw=green!50,fill=green!20\] ; at (2.4+2.5,0-3) [3]{}; at (1.2+2.5,0-3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2+2.5,0-3)[4]{}; at (3.6+2.5,0.5-3) \[circle,draw=blue!50,fill=blue!20\] ; at (3.6+2.5,0.5-3) [1]{}; at (1.8+2.5,-4)[2.a]{};
(0+6.8,0.5-3) – (1.2+6.8,1-3); (0+6.8,0.5-3) – (1.2+6.8,0-3); (1.2+6.8,0-3) – (1.2+6.8,1-3); (1.2+6.8,0-3) – (2.4+6.8,0-3); (1.2+6.8,1-3) – (2.4+6.8,1-3); (2.4+6.8,0-3) – (2.4+6.8,1-3); (3.6+6.8,1-3) – (2.4+6.8,1-3); (3.6+6.8,0-3) – (2.4+6.8,0-3); (0+6.8,0.5-3) – (3.6+6.8,0-3); (0+6.8,0.5-3) – (3.6+6.8,1-3); (3.6+6.8,0-3) – (3.6+6.8,1-3);
at (3.6+6.8,0-3) \[circle,draw=blue!50,fill=blue!20\] ; at (3.6+6.8,0-3) [1]{}; at (1.2+6.8,1-3) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+6.8,1-3) [1]{}; at (0+6.8,0.5-3) \[circle,draw=red!50,fill=red!20\] ; at(0+6.8,0.5-3) [2]{}; at (2.4+6.8,1-3) \[circle,draw=red!50,fill=red!20\] ; at (2.4+6.8,1-3) [2]{}; at (2.4+6.8,0-3) \[circle,draw=green!50,fill=green!20\] ; at (2.4+6.8,0-3) [3]{}; at (1.2+6.8,0-3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2+6.8,0-3)[4]{}; at (3.6+6.8,1-3) \[circle,draw=black,fill=black\] ; at (1.8+6.8,-4)[2.b]{};
(1.2-0.3,0-5.6) – (1.2-0.3,1-5.6); (0-0.3,0-5.6) – (0-0.3,1-5.6); (0-0.3,0-5.6) – (3.6-0.3,0-5.6); (0-0.3,1-5.6) – (3.6-0.3,1-5.6); (2.4-0.3,0-5.6) – (2.4-0.3,1-5.6); at (2.4-0.3,0-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (2.4-0.3,0-5.6) [1]{};
at (0-0.3,1-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (0-0.3,1-5.6) [1]{}; at (0-0.3,0.-5.6) \[circle,draw=red!50,fill=red!20\] ; at (0-0.3,0-5.6) [2]{}; at (2.4-0.3,1-5.6) \[circle,draw=red!50,fill=red!20\] ; at (2.4-0.3,1-5.6) [2]{}; at (1.2-0.3,0-5.6) \[circle,draw=green!50,fill=green!20\] ; at (1.2-0.3,0-5.6) [3]{}; at (1.2-0.3,1-5.6) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2-0.3,1-5.6)[4]{}; at (3.6-0.3,0-5.6) \[circle,draw=black,fill=black\] ; at (3.6-0.3,1-5.6) \[circle,draw=black,fill=black\] ; at (1.8,-6.4)[2.c]{};
(0+4,0-5.6) – (0+4,1-5.6); (1.2+4,0-5.6) – (0+4,0-5.6); (1.2+4,1-5.6) – (0+4,1-5.6); (1.2+4,0-5.6) – (0.7+4,0.5-5.6); (1.2+4,1-5.6) – (0.7+4,0.5-5.6); (1.2+4,0-5.6) – (1.7+4,0.5-5.6); (1.2+4,1-5.6) – (1.7+4,0.5-5.6); (1.7+4,0.5-5.6) – (2.5+4,0.5-5.6); at (0+4,1-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (0+4,1-5.6) [1]{}; at (2.5+4,0.5-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (2.5+4,0.5-5.6) [1]{}; at (0.7+4,0.5-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (0.7+4,0.5-5.6) [1]{}; at (1.2+4,1-5.6) \[circle,draw=red!50,fill=red!20\] ; at (1.2+4,1-5.6) [2]{}; at (0+4,0-5.6) \[circle,draw=red!50,fill=red!20\] ; at (0+4,0-5.6) [2]{}; at (1.2+4,0-5.6) \[circle,draw=green!50,fill=green!20\] ; at (1.2+4,0-5.6) [3]{}; at (1.7+4,0.5-5.6) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1.7+4,0.5-5.6) [4]{}; at (5.2,-6.4)[2.d]{};
(1.2+7,0-5.6) – (1.2+7,1-5.6); (-0.7+7,0-5.6) – (3.6+7,0-5.6); (0+7,1-5.6) – (3.6+7,1-5.6); (2.4+7,0-5.6) – (2.4+7,1-5.6);
at (1.2+7,1-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+7,1-5.6) [1]{}; at (3.6+7,0-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (3.6+7,0-5.6) [1]{}; at (2.4+7,1-5.6) \[circle,draw=red!50,fill=red!20\] ; at (2.4+7,1-5.6) [2]{}; at (0+7,0-5.6) \[circle,draw=red!50,fill=red!20\] ; at (0+7,0-5.6) [2]{}; at (-0.7+7,0-5.6) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.7+7,0-5.6) [1]{}; at (1.2+7,0-5.6) \[circle,draw=green!50,fill=green!20\] ; at (1.2+7,0-5.6) [3]{}; at (2.4+7,0-5.6) \[circle,draw=yellow!50,fill=yellow!20\] ; at (2.4+7,0-5.6)[4]{}; at (0+7,1-5.6) \[circle,draw=black,fill=black\] ; at (3.6+7,1-5.6) \[circle,draw=black,fill=black\] ; at (8.8,-6.4)[2.e]{};
(1.2,0-8) – (1.2,1-8); (0,0-8) – (3.6,0-8); (0,1-8) – (4.8,1-8); (2.4,0-8) – (2.4,1-8); (0,0-8) – (1.8,0.35-8); (0,0-8) – (1.8,0.7-8); (3.6,0-8) – (1.8,0.35-8); (3.6,0-8) – (1.8,0.7-8);
at (1.2,1-8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2,1-8) [1]{}; at (3.6,0-8) \[circle,draw=blue!50,fill=blue!20\] ; at (3.6,0-8) [1]{}; at (4.8,1-8) \[circle,draw=blue!50,fill=blue!20\] ; at (4.8,1-8) [1]{}; at (1.2,0-8) \[circle,draw=red!50,fill=red!20\] ; at (1.2,0-8) [2]{}; at (3.6,1-8) \[circle,draw=red!50,fill=red!20\] ; at (3.6,1-8) [2]{}; at (2.4,1-8) \[circle,draw=green!50,fill=green!20\] ; at (2.4,1-8) [3]{}; at (2.4,0-8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (2.4,0-8)[4]{}; at (0,1-8) \[circle,draw=black,fill=black\] ; at (0,0-8) \[circle,draw=black,fill=black\] ; at (1.8,0.35-8) \[circle,draw=black,fill=black\] ; at (1.8,0.7-8) \[circle,draw=black,fill=black\] ; at (1.8,-8.8)[2.f]{};
(1.2+7.5,1-8) – (1.2+6,0.5-8); (1.2+7.5,1-8) – (1.2+7.5,0.5-8); (1.2+7.5,1-8) – (1.2+9,0.5-8); (1.2+6,0.5-8) – (1.2+5.5,0-8); (1.2+6,0.5-8) – (1.2+6.5,0-8); (1.2+7.5,0.5-8) – (1.2+7,0-8); (1.2+7.5,0.5-8) – (1.2+8,0-8); (1.2+9,0.5-8) – (1.2+8.5,0-8); (1.2+9,0.5-8) – (1.2+9.5,0-8); (1.2+5.5,0-8) – (1.2+5.5,-0.8-8); (1.2+7,0-8) – (1.2+7,-0.8-8); (1.2+8,0-8) – (1.2+8,-0.8-8); (1.2+8.5,0-8) – (1.2+8.5,-0.8-8); (1.2+9.5,0-8) – (1.2+9.5,-0.8-8); (1.2+7,0-8) – (1.2+8,-0.8-8); (1.2+8,0-8) – (1.2+7,-0.8-8); (1.2+8.5,0-8) – (1.2+9.5,-0.8-8); (1.2+9.5,0-8) – (1.2+8.5,-0.8-8); (1.2+6.5,0-8) – (1.2+7,-0.8-8); (1.2+6.5,0-8) – (1.2+8,-0.8-8); (1.2+5.5,0-8) .. controls (1.2+6.9,-9.6) .. (1.2+8.5,-0.8-8);
at (1.2+6,0.5-8) \[circle,draw=green!50,fill=green!20\] ; at (1.2+6,0.5-8) [3]{}; at (0.9+6,0.6-8) [$u$]{}; at (1.2+7.5,1-8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2+7.5,1-8) [4]{}; at (0.9+7.5,1.1-8) [$v$]{}; at (1.2+7.5,0.5-8) \[circle,draw=red!50,fill=red!20\] ; at (1.2+7.5,0.5-8) [2]{}; at (1.2+5.5,0-8) \[circle,draw=red!50,fill=red!20\] ; at (1.2+5.5,0-8) [2]{}; at (1.2+9,0.5-8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+9,0.5-8) [1]{}; at (1.2+7,0-8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+7,0-8) [1]{}; at (1.2+6.5,0-8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+6.5,0-8) [1]{}; at (1.2+5.5,-0.8-8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+5.5,-0.8-8) [1]{}; at (1.2+7,-0.8-8) \[circle,draw=black,fill=black\] ; at (0.8+7,-0.8-8) [$f_1$]{}; at (1.2+8,-0.8-8) \[circle,draw=black,fill=black\] ; at (0.8+8,-0.8-8) [$f_2$]{}; at (1.2+8,0-8) \[circle,draw=black,fill=black\] ; at (1.2+9.5,-0.8-8) \[circle,draw=black,fill=black\] ; at (1.2+9.5,0-8) \[circle,draw=black,fill=black\] ; at (1.2+8.5,-0.8-8) \[circle,draw=black,fill=black\] ; at (1.2+8.5,0-8) \[circle,draw=black,fill=black\] ;
at (1.2+7.5,-9.5)[3.a]{};
(1,1-10.3) – (2.5,1-10.3); (1,1-10.3) – (0.5,0.5-10.3); (1,1-10.3) – (1.5,0.5-10.3); (2.5,1-10.3) – (2,0.5-10.3); (2.5,1-10.3) – (3,0.5-10.3); (1.75,0-10.3) – (2,0.5-10.3); (1.75,0-10.3) – (1.5,0.5-10.3);
at (2.5,1-10.3) \[circle,draw=green!50,fill=green!20\] ; at (2.5,1-10.3) [3]{}; at (1,1-10.3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1,1-10.3) [4]{}; at (0.7,1.1-10.3) [$v$]{}; at (1.5,0.5-10.3) \[circle,draw=red!50,fill=red!20\] ; at (1.5,0.5-10.3) [2]{}; at (2,0.5-10.3) \[circle,draw=red!50,fill=red!20\] ; at (2,0.5-10.3) [2]{}; at (0.5,0.5-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.5,0.5-10.3) [1]{}; at (3,0.5-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (3,0.5-10.3) [1]{}; at (3,0.5-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (3,0.5-10.3) [1]{}; at (1.75,0-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (1.75,0-10.3) [1]{}; at (1.75,-0.7-10.3)[3.b]{};
(1+4,1-10.3) – (2.5+4,1-10.3); (1+4,1-10.3) – (0.5+4,0.5-10.3); (1+4,1-10.3) – (1.5+4,0.5-10.3); (2.5+4,1-10.3) – (2+4,0.5-10.3); (2.5+4,1-10.3) – (3+4,0.5-10.3); (2+4,0-10.3) – (2+4,0.5-10.3); (1.5+4,0-10.3) – (1.5+4,0.5-10.3); (1.5+4,0-10.3) .. controls (1.75+4,-0.5-10.3) .. (2+4,0-10.3); (1.5+4,0.5-10.3) – (1+4,0-10.3);
at (2.5+4,1-10.3) \[circle,draw=green!50,fill=green!20\] ; at (2.5+4,1-10.3) [3]{}; at (1+4,1-10.3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (1+4,1-10.3) [4]{}; at (0.7+4,1.1-10.3) [$v$]{}; at (1.5+4,0.5-10.3) \[circle,draw=red!50,fill=red!20\] ; at (1.5+4,0.5-10.3) [2]{}; at (2+4,0.5-10.3) \[circle,draw=red!50,fill=red!20\] ; at (2+4,0.5-10.3) [2]{}; at (0.5+4,0.5-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.5+4,0.5-10.3) [1]{}; at (3+4,0.5-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (3+4,0.5-10.3) [1]{}; at (3+4,0.5-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (3+4,0.5-10.3) [1]{}; at (1+4,0-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (1+4,0-10.3) [1]{}; at (2+4,0-10.3) \[circle,draw=blue!50,fill=blue!20\] ; at (2+4,0-10.3) [1]{}; at (1.5+4,0-10.3) \[circle,draw=black,fill=black\] ; at (1.75+4,-0.7-10.3)[3.c]{};
A cubic graph $G$ does not contain any induced minimal subcubic 4-atom if and only if every vertex is an $(i,3)$-twin-vertex, for some $i$, $0\le i \le 2$.
Let $G$ be a cubic graph. If $G$ is without induced $C_4$, then $\Gamma(G)=4$. \[indc3\]
As every graph $G$ with $\Gamma(G)<4$ is composed of copies of $K_{2,3}$ or $K^{*}_{3,3}$, the graph $G$ always contains a square if $\Gamma(G)<4$.
For a fixed integer $t$, the largest $(t+1)$-atom has order $2^{t}$. Thus, for a graph $G$ of maximum degree $t$, there exists an $O(n^{2^{t}})$-time algorithm to determine if $\Gamma(G)<t+1$ (which verifies if the graph contains an induced $(t+1)$-atom). For a cubic graph, we obtain an $O(n^{8})$-time algorithm, whereas our characterization yields a linear-time algorithm.
Let $G$ be a cubic graph of order $n$. There exists an $O(n)$-time algorithm[^2] to determine the Grundy number of $G$.
Suppose we have a cubic graph $G$ with its adjacency list. Verifying if $G$ is $K_{3,3}$ can be done in constant time. We suppose now that $G$ is not $K_{3,3}$. For each vertex $v$, the algorithm verifies that $v$ is an $(i,3)$-twin-vertex, for some $i$, $0\le i \le 2$. If the condition is true for all vertices, then $\Gamma(G)=3$, else $\Gamma(G)=4$. To determine if a vertex $v$ is a $(0,3)$-twin-vertex, it suffices to verify that there is a common vertex other than $v$ in the adjacency lists of the neighbors of $v$. To determine if a vertex $v$ is a $(1,3)$-twin-vertex, it suffices to verify that there are two neighbors of $v$ which have the same adjacency list. To determine if a vertex $v$ is a $(2,3)$-twin-vertex, it suffices to verify that the neighborhood of $v$ is independent and that every neighbor is a $(1,3)$-twin-vertex. Hence, checking if a vertex is an $(i,3)$-twin vertex can be done in constant time, so the algorithms runs in linear time.
If $G$ is a connected cubic graph and $G\neq K_{3,3}$, then $\partial \Gamma(G)=4$.
Let $G$ be a cubic connected graph. Note that if $\Gamma(G)=4$ then $\partial \Gamma(G)=4$. Every graph $G$ with $\Gamma(G)<4$ is composed of copies of $K_{2,3}$ or $K^{*}_{3,3}$. If $G$ contains more than two copies (so it is different from $K_{3,3}$), then a vertex can be colored 4 in the first copy and a vertex can be colored 3 in the second copy. Hence, $\partial \Gamma(G)=4$.
Only $K_{3,3}$ and three other cubic graphs have $b$-chromatic number at most 3 [@KL2010]. Thus, our result is coherent with the results on the $b$-chromatic number. Shi et al. [@SH2005] proved that there exists a smallest integer $N_r$ such that every $r$-regular graph $G$ with more than $N_r$ vertices has $\partial\Gamma(G)=r+1$. Observe that we have $N_2=4$ and $N_3=6$. It is an open question to determine $N_r$ for $r\ge 4$. However, using results on $b$-chromatic number [@CA2011], we have $N_r\le2 r^3-r^{2}+r$.
Properties on the Grundy number of $r$-regular graphs
=====================================================
Let $r\ge2$ be an integer. We define recursively the family of graphs $\mathcal{G}^{*}_r$ as follows:
1. $K_{r-k,k+2} \in \mathcal{G}^{*}_r$, for any $k$, $0 \le k \le (r-2)/2$;
2. the disjoint union of two elements of $\mathcal{G}^{*}_{r}$ is in $\mathcal{G}^{*}_r$;
3. if $G$ is a graph in $\mathcal{G}^{*}_r$, then the graph $H$ obtained from $G$ by adding an edge between two vertices of degree at most $r-1$ is also in $\mathcal{G}^{*}_r$;
4. if $G$ is a graph in $\mathcal{G}^{*}_r$, then the graph $H$ obtained from $G$ by adding a new vertex adjacent to $r$ vertices of degree at most $r-1$ is in $\mathcal{G}^{*}_r$.
The family $\mathcal{G}_r$ is the subfamily of $r$-regular graphs in $\mathcal{G}^{*}_r$.
Let $G$ be an $r$-regular graph. If $G \in \mathcal{G}_r$, then $\Gamma(G)<r+1$.
By $I_{r-k}$ and $I_{k+2}$, with $|I_{r-k}|=r-k$ and $|I_{k+2}|=k+2$, we denote the two sets of vertices in the bipartition of an induced subgraph $K_{r-k,k+2}$ in $G$. Firstly, suppose there exists a vertex $u$ in an induced subgraph $K_{r-k,k+2}$ colored $r+1$. Without loss of generality, suppose $u$ is in $I_{r-k}$. The $r$ neighbors of $u$ should have colors from 1 to $r$. Among the neighbors of $u$, $k+2$ neighbors are in $I_{k+2}$. Let $v$ be the neighbor of $u$ in $I_{k+2}$ with the largest color in $I_{k+2}$. The vertex $v$ has color at least $k+2$. Hence, there exists an integer $s\ge 0$ such that the color of $v$ is $k+2+s$. Note that there are $s$ vertices in $N(u)\setminus I_{k+2}$ which have colors at most $k+2+s$. The colors of the $s$ vertices are the only one possible remaining colors at most $k+2+s$ in $I_{r-k}$. Hence, as there are $k$ vertices in $N(v)\setminus I_{r-k}$, the neighbors of $v$ can only have at most $k+s$ different colors at most $k+2+s$. Therefore, we have a contradiction and $u$ cannot have color $r+1$. Secondly, suppose there exists a vertex $u$ added by Point 4 which has color $r+1$. As a neighbor of $u$ in an induced $K_{r-k,k+2}$ should be colored $r$, the argument is completely similar to the previous one.
Let $G$ be a 4-regular graph. If $G\in \mathcal{G}_4$, then $\Gamma(G)<5$.
The reader can believe that the family of 4-regular graphs with $\Gamma(G)< 5$ contains only the family $\mathcal{G}_4$. However, there exist graphs with Grundy number $r$ which are not inside this family. For example, the power graph (the graph where every pair of vertices at pairwise distance 2 become adjacent) of the 7-cycle $C_7^2$ satisfies $\Gamma(C_7^2)<5$ and is not in $\mathcal{G}_4$.
The next proposition shows that unlike the $b$-chromatic number, $r$-regular graphs of order arbitrarily large with Grundy number $k$ can be constructed for any $r$ and any $k$, $3\le k\le r+1$.
Let $r\ge4$ and $3\le k\le r+1$ be integers. There exists an infinite family $\mathcal{H}$ of connected $r$-regular graphs such that for all $G$ in $\mathcal{H}$, $\Gamma(G)=k$.
Let $i\ge 2$ be a positive integer and $r_1$, $\ldots$, $r_{k-1}$ be a sequence of positive integers such that $r=r_1+\ldots+r_{k-1}$. We construct a graph $G_{r,k,i}$ as follows: Take $2i$ copies of $K_{r_1,\ldots,r_{k-1}}$. Let $H_{j-1}$ be the copy number $j$ of $K_{r_1,\ldots,r_{k-1}}$ and $H_{j,r_l}$ be the independent $r_l$-set in $H_j$. If $j\equiv 1\pmod{2}$, do the graph join of $H_{j\pmod{2i},r_1}$ and $H_{j-1\pmod{2i},r_1}$ and for an integer $l$, $1<l<k$, do the graph join of $H_{j\pmod{2i},r_l}$ and $H_{j+1\pmod{2i},r_l}$. The $r$-regular graph obtained is the graph $G_{r,k,i}$. Figure \[figmr\] gives $G_{r,k,i}$, for $k=4$ and $i\ge2$. Note that $H_{j,r_i}$ is an independent module. Thus, every vertex is a $(0,k)$-twin-vertex. By Proposition \[0twin\], $\Gamma(G_{r,k,i})\le k$.For an integer $l$, $1<l<k$, color one vertex $l-1$ in $H_{1,r_l}$ and $H_{2,r_l}$. Afterwards, color one vertex $k-1$ in $H_{1,r_1}$ and one vertex $k$ in $H_{2,r_1}$. The given coloring is a Grundy partial $k$-coloring of $G_{r,k,i}$ for $i\ge2$. Therefore, $\Gamma(G_{r,k,i})=k$, for $i\ge2$.
at (0-3,0) (t1)\[circle,draw=black!50,fill=black!01\] [$I_{r_{1}}$]{}; at (1.3-3,0) (t2)\[circle,draw=black!50,fill=black!01\] [$I_{r_{2}}$]{}; at (0.65-3,1.3)(t3)\[circle,draw=black!50,fill=black!01\] [$I_{r_{3}}$]{}; at (0,0) (t4)\[circle,draw=black!50,fill=black!01\] [$I_{r_{2}}$]{}; at (1.3,0) (t5)\[circle,draw=black!50,fill=black!01\] [$I_{r_{1}}$]{}; at (0.65,1.3) (t6)\[circle,draw=black!50,fill=black!01\] [$I_{r_{3}}$]{}; at (0+3,0) (t7)\[circle,draw=black!50,fill=black!01\] [$I_{r_{1}}$]{}; at (1.3+3,0) (t8)\[circle,draw=black!50,fill=black!01\] [$I_{r_{2}}$]{}; at (0.65+3,1.3) (t9)\[circle,draw=black!50,fill=black!01\] [$I_{r_{3}}$]{}; at (0+6,0) (t10)\[circle,draw=black!50,fill=black!01\] [$I_{r_{2}}$]{}; at (1.3+6,0) (t11)\[circle,draw=black!50,fill=black!01\] [$I_{r_{1}}$]{}; at (0.65+6,1.3) (t12)\[circle,draw=black!50,fill=black!01\] [$I_{r_{3}}$]{}; (t1.east) – (t2.west); (t1.south) – (t2.south); (t1.north) – (t2.north); (t1.west) – (t3.west); (t1.north) – (t3.south); (t1.east) – (t3.east); (t2.west) – (t3.west); (t2.north) – (t3.south); (t2.east) – (t3.east); (t4.east) – (t5.west); (t4.south) – (t5.south); (t4.north) – (t5.north); (t4.west) – (t6.west); (t4.north) – (t6.south); (t4.east) – (t6.east); (t5.west) – (t6.west); (t5.north) – (t6.south); (t5.east) – (t6.east); (t7.east) – (t8.west); (t7.south) – (t8.south); (t7.north) – (t8.north); (t7.west) – (t9.west); (t7.north) – (t9.south); (t7.east) – (t9.east); (t8.west) – (t9.west); (t8.north) – (t9.south); (t8.east) – (t9.east); (t10.east) – (t11.west); (t10.south) – (t11.south); (t10.north) – (t11.north); (t10.west) – (t12.west); (t10.north) – (t12.south); (t10.east) – (t12.east); (t11.west) – (t12.west); (t11.north) – (t12.south); (t11.east) – (t12.east);
(t1.west) – (-4,0); (t2.east) – (t4.west); (t3.east) – (t6.west); (t5.east) – (t7.west); (t8.east) – (t10.west); (t9.east) – (t12.west); (t8.east) – (t10.west); (t11.east) – (8.3,0);
Grundy number of 4-regular graphs without induced $C_4$
=======================================================
The following lemmas will be useful to prove the second main theorem of this paper: The family of 4-regular graphs without induced $C_4$ contains only graphs with Grundy number 5.
Let $G$ be a 4-regular graph without induced $C_4$. If $G$ contains (an induced) $K_4$ then $\Gamma(G)=5$. \[k4\]
Note that if $G=K_5$, we have $\Gamma(G)=5$. If $G$ is not $K_5$ then every pair of neighbors of vertices of $K_4$ cannot be adjacent ($G$ would contain a $C_4$). Giving the color 1 to each neighbor of the vertices of $K_4$ and colors 2, 3, 4, 5 to the vertices of $K_4$, we obtain a Grundy partial 5-coloring of $G$.
Let $G$ be a 4-regular graph without induced $C_4$ and let $W$ be the graph from Figure \[figm31\]. If $G$ contains an induced $W$ then $\Gamma(G)=5$.
The names of the vertices of $W$ come from Figure \[figm31\]. Depending on the different cases that could happen, Grundy partial 5-colorings of $G$ will be given using their references on Figure \[figm31\]. Let $D_1$ be the set of vertices at distance 1 from vertices of $W$ in $G-W$. Suppose that two vertices of $W$ have a common neighbor in $D_1$. This two vertices could only be $u_4$ and $u_5$ or $u_3$ and $u_5$ (or $u_1$ and $u_4$, by symmetry). In the case that $u_4$ and $u_5$ have a common neighbor in $D_1$, colors will be given to neighbors of $u_3$ in $D_1$, depending if they are adjacent (Figure 5.1.a) or not (Figure 5.1.b). In the case that $u_3$ and $u_5$ have a common neighbor $w$ in $D_1$, $w$ can be adjacent with a neighbor of $u_3$ in $D_1$ (Figure 5.2.a) or not (Figure 5.2.b). Suppose now that no vertices in $W$ have a common neighbor in $D_1$. Let $w_1$ and $w_2$ be the neighbors of $u_3$ in $D_1$. We first consider that $w_1$ and $w_2$ are adjacent (Figure 5.3.a). Secondly, we consider that $w_1$ and $w_2$ are not adjacent and that $u_5$, $u_3$ and $w_1$ are in an induced $C_5$ (Figure 5.3.b). Finally, we consider that the previous configurations are impossible (Figure 5.3.c).
(0.8\*0.8,0) – (-0.8\*0.8,0); (-0.8\*0.8,0) – (-0.4\*0.8,0.8\*0.8); (0.8\*0.8,0) – (0.4\*0.8,0.8\*0.8); (0,0) – (0.4\*0.8,0.8\*0.8); (0,0) – (-0.4\*0.8,0.8\*0.8); (0.4\*0.8,0.8\*0.8) – (-0.4\*0.8,0.8\*0.8); at (0,-0.8)[The graph $W$.]{}; at (-0.8,-0.2)[$u_1$]{}; at (0,-0.2)[$u_2$]{}; at (0.8,-0.2)[$u_3$]{}; at (-0.4,0.8)[$u_4$]{}; at (0.4,0.8)[$u_5$]{};
(3.4\*0.8+1,0) – (1.2\*0.8+1,0); (1.2\*0.8+1,0) – (1.6\*0.8+1,0.8\*0.8); (2.8\*0.8+1,0) – (2.4\*0.8+1,0.8\*0.8); (2\*0.8+1,0) – (2.4\*0.8+1,0.8\*0.8); (2\*0.8+1,0) – (1.6\*0.8+1,0.8\*0.8); (2.4\*0.8+1,0.8\*0.8) – (1.6\*0.8+1,0.8\*0.8); (2.4\*0.8+1,0.8\*0.8) – (2\*0.8+1,1.6\*0.8); (1.6\*0.8+1,0.8\*0.8) – (2\*0.8+1,1.6\*0.8); (3.4\*0.8+1,0) – (3.1\*0.8+1,0.6\*0.8); (2.8\*0.8+1,0) – (3.1\*0.8+1,0.6\*0.8);
at (2.8\*0.8+1,0) \[circle,draw=green!50,fill=green!20\] ; at (2.8\*0.8+1,0) [3]{}; at (2.4\*0.8+1,0.8\*0.8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (2.4\*0.8+1,0.8\*0.8)[4]{}; at (2\*0.8+1,0) \[circle,draw=black!50,fill=black!20\] ; at (2\*0.8+1,0) [5]{}; at (1.2\*0.8+1,0) \[circle,draw=red!50,fill=red!20\] ; at (1.2\*0.8+1,0) [2]{}; at (1.6\*0.8+1,0.8\*0.8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6\*0.8+1,0.8\*0.8) [1]{}; at (2\*0.8+1,1.6\*0.8) \[circle,draw=red!50,fill=red!20\] ; at (2\*0.8+1,1.6\*0.8) [2]{}; at (3.1\*0.8+1,0.6\*0.8) \[circle,draw=blue!50,fill=blue!20\] ; at (3.1\*0.8+1,0.6\*0.8) [1]{}; at (3.4\*0.8+1,0) \[circle,draw=red!50,fill=red!20\] ; at (3.4\*0.8+1,0) [2]{}; at (2\*0.8+1,-0.8)[1.a]{};
(3.4\*0.8+3.7,0) – (1.2\*0.8+3.7,0); (1.2\*0.8+3.7,0) – (1.6\*0.8+3.7,0.8\*0.8); (2.8\*0.8+3.7,0) – (2.4\*0.8+3.7,0.8\*0.8); (2\*0.8+3.7,0) – (2.4\*0.8+3.7,0.8\*0.8); (2\*0.8+3.7,0) – (1.6\*0.8+3.7,0.8\*0.8); (2.4\*0.8+3.7,0.8\*0.8) – (1.6\*0.8+3.7,0.8\*0.8); (2.4\*0.8+3.7,0.8\*0.8) – (2\*0.8+3.7,1.6\*0.8); (1.6\*0.8+3.7,0.8\*0.8) – (2\*0.8+3.7,1.6\*0.8); (2.8\*0.8+3.7,0) – (3.1\*0.8+3.7,0.6\*0.8); (3.4\*0.8+3.7,1.2\*0.8)– (3.1\*0.8+3.7,0.6\*0.8);
at (2.8\*0.8+3.7,0) \[circle,draw=green!50,fill=green!20\] ; at (2.8\*0.8+3.7,0) [3]{}; at (2.4\*0.8+3.7,0.8\*0.8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (2.4\*0.8+3.7,0.8\*0.8)[4]{}; at (2\*0.8+3.7,0) \[circle,draw=black!50,fill=black!20\] ; at (2\*0.8+3.7,0) [5]{}; at (1.2\*0.8+3.7,0) \[circle,draw=red!50,fill=red!20\] ; at (1.2\*0.8+3.7,0) [2]{}; at (1.6\*0.8+3.7,0.8\*0.8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6\*0.8+3.7,0.8\*0.8) [1]{}; at (2\*0.8+3.7,1.6\*0.8) \[circle,draw=red!50,fill=red!20\] ; at (2\*0.8+3.7,1.6\*0.8) [2]{}; at (3.4\*0.8+3.7,0)\[circle,draw=blue!50,fill=blue!20\] ; at (3.4\*0.8+3.7,0) [1]{}; at (3.1\*0.8+3.7,0.6\*0.8) \[circle,draw=red!50,fill=red!20\] ; at (3.1\*0.8+3.7,0.6\*0.8) [2]{}; at (3.4\*0.8+3.7,1.2\*0.8) \[circle,draw=blue!50,fill=blue!20\] ; at (3.4\*0.8+3.7,1.2\*0.8) [1]{}; at (2\*0.8+3.7,-0.8)[1.b]{};
(3.6\*0.8+6.5,0) – (1.2\*0.8+6.5,0); (1.2\*0.8+6.5,0) – (1.6\*0.8+6.5,0.8\*0.8); (2.8\*0.8+6.5,0) – (2.4\*0.8+6.5,0.8\*0.8); (2\*0.8+6.5,0) – (2.4\*0.8+6.5,0.8\*0.8); (2\*0.8+6.5,0) – (1.6\*0.8+6.5,0.8\*0.8); (3.2\*0.8+6.5,0.8\*0.8) – (1.6\*0.8+6.5,0.8\*0.8); (3.6\*0.8+6.5,0) – (3.2\*0.8+6.5,0.8\*0.8); (2.8\*0.8+6.5,0) – (3.2\*0.8+6.5,0.8\*0.8);
at (2.8\*0.8+6.5,0) \[circle,draw=green!50,fill=green!20\] ; at (2.8\*0.8+6.5,0) [3]{}; at (2.4\*0.8+6.5,0.8\*0.8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (2.4\*0.8+6.5,0.8\*0.8)[4]{}; at (2\*0.8+6.5,0) \[circle,draw=black!50,fill=black!20\] ; at (2\*0.8+6.5,0) [5]{}; at (1.2\*0.8+6.5,0) \[circle,draw=red!50,fill=red!20\] ; at (1.2\*0.8+6.5,0) [2]{}; at (1.6\*0.8+6.5,0.8\*0.8) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6\*0.8+6.5,0.8\*0.8) [1]{}; at (3.2\*0.8+6.5,0.8\*0.8) \[circle,draw=red!50,fill=red!20\] ; at (3.2\*0.8+6.5,0.8\*0.8) [2]{}; at (3.6\*0.8+6.5,0) \[circle,draw=blue!50,fill=blue!20\] ; at (3.6\*0.8+6.5,0) [1]{}; at (8.5,-0.8)[2.a]{};
(3.4\*0.8-2,-2.2) – (1.2\*0.8-2,-2.2); (1.2\*0.8-2,-2.2) – (1.6\*0.8-2,0.8\*0.8-2.2); (2.8\*0.8-2,0-2.2) – (2.4\*0.8-2,0.8\*0.8-2.2); (2\*0.8-2,0-2.2) – (2.4\*0.8-2,0.8\*0.8-2.2); (2\*0.8-2,0-2.2) – (1.6\*0.8-2,0.8\*0.8-2.2); (4\*0.8-2,0.8\*0.8-2.2) – (1.6\*0.8-2,0.8\*0.8-2.2); (2.8\*0.8-2,-2.2) – (3.2\*0.8-2,0.8\*0.8-2.2); (1.2\*0.8-2,-2.2) .. controls (-0.8\*0.8-1,-0.8) and (1.2\*0.8,-0.8) .. (4\*0.8-2,0.8\*0.8-2.2);
at (2.8\*0.8-2,0-2.2) \[circle,draw=green!50,fill=green!20\] ; at (2.8\*0.8-2,0-2.2) [3]{}; at (2.4\*0.8-2,0.8\*0.8-2.2) \[circle,draw=yellow!50,fill=yellow!20\] ; at (2.4\*0.8-2,0.8\*0.8-2.2)[4]{}; at (2\*0.8-2,0-2.2) \[circle,draw=black!50,fill=black!20\] ; at (2\*0.8-2,0-2.2) [5]{}; at (1.2\*0.8-2,0-2.2) \[circle,draw=red!50,fill=red!20\] ; at (1.2\*0.8-2,0-2.2) [2]{}; at (1.6\*0.8-2,0.8\*0.8-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6\*0.8-2,0.8\*0.8-2.2) [1]{}; at (3.2\*0.8-2,0.8\*0.8-2.2) \[circle,draw=red!50,fill=red!20\] ; at (3.2\*0.8-2,0.8\*0.8-2.2) [2]{}; at (3.6\*0.8-2,0-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (3.6\*0.8-2,0-2.2) [1]{}; at (4\*0.8-2,0.8\*0.8-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (4\*0.8-2,0.8\*0.8-2.2) [1]{}; at (0,-2.8)[2.b]{};
(1.4\*0.8+2.5,-2.2) – (-0.8\*0.8+2.5,-2.2); (-0.8\*0.8+2.5,0-2.2) – (-0.4\*0.8+2.5,0.8\*0.8-2.2); (0.8\*0.8+2.5,0-2.2) – (0.4\*0.8+2.5,0.8\*0.8-2.2); (0+2.5,0-2.2) – (0.4\*0.8+2.5,0.8\*0.8-2.2); (0+2.5,0-2.2) – (-0.4\*0.8+2.5,0.8\*0.8-2.2); (0.4\*0.8+2.5,0.8\*0.8-2.2) – (-0.4\*0.8+2.5,0.8\*0.8-2.2); (0.4\*0.8+2.5,0.8\*0.8-2.2) – (1\*0.8+2.5,0.8\*0.8-1.9); (1.4\*0.8+2.5,-2.2) – (1.1\*0.8+2.5,0.8\*0.6-2.2); (0.8\*0.8+2.5,-2.2) – (1.1\*0.8+2.5,0.8\*0.6-2.2);
at (0+2.5,-2.2) \[circle,draw=green!50,fill=green!20\] ; at (0+2.5,-2.2) [3]{}; at (0.4\*0.8+2.5,0.8\*0.8-2.2) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0.4\*0.8+2.5,0.8\*0.8-2.2) [4]{}; at (0.8\*0.8+2.5,-2.2) \[circle,draw=black!50,fill=black!20\] ; at (0.8\*0.8+2.5,-2.2) [5]{}; at (1.4\*0.8+2.5,-2.2) \[circle,draw=red!50,fill=red!20\] ; at (1.4\*0.8+2.5,-2.2) [2]{}; at (-0.4\*0.8+2.5,0.8\*0.8-2.2) \[circle,draw=red!50,fill=red!20\] ; at (-0.4\*0.8+2.5,0.8\*0.8-2.2) [2]{}; at (-0.8\*0.8+2.5,-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8\*0.8+2.5,-2.2) [1]{}; at (1\*0.8+2.5,0.8\*0.8-1.9) \[circle,draw=blue!50,fill=blue!20\] ; at (1\*0.8+2.5,0.8\*0.8-1.9) [1]{}; at (1.1\*0.8+2.5,0.8\*0.6-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (1.1\*0.8+2.5,0.8\*0.6-2.2) [1]{}; at (0.2+2.5,-2.8)[3.a]{};
(1.6\*0.8+5.1,-2.2) – (-0.8\*0.8+5.1,-2.2); (-0.8\*0.8+5.1,0-2.2) – (-0.4\*0.8+5.1,0.8\*0.8-2.2); (0.8\*0.8+5.1,0-2.2) – (0.4\*0.8+5.1,0.8\*0.8-2.2); (0+5.1,0-2.2) – (0.4\*0.8+5.1,0.8\*0.8-2.2); (0+5.1,0-2.2) – (-0.4\*0.8+5.1,0.8\*0.8-2.2); (2\*0.8+5.1,0.8\*0.8-2.2) – (-0.4\*0.8+5.1,0.8\*0.8-2.2); (1.6\*0.8+5.1,-2.2) – (2\*0.8+5.1,0.8\*0.8-2.2); (0.8\*0.8+5.1,-2.2) – (1.4\*0.8+5.1,-2.6); (-0.8\*0.8+5.1,-2.2) .. controls (-0.8\*0.8+4.8,-0.8) and (1.2\*0.8+5.1,-0.8) .. (2\*0.8+5.1,0.8\*0.8-2.2);
at (0+5.1,-2.2) \[circle,draw=green!50,fill=green!20\] ; at (0+5.1,-2.2) [3]{}; at (0.4\*0.8+5.1,0.8\*0.8-2.2) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0.4\*0.8+5.1,0.8\*0.8-2.2) [4]{}; at (0.8\*0.8+5.1,-2.2) \[circle,draw=black!50,fill=black!20\] ; at (0.8\*0.8+5.1,-2.2) [5]{}; at (1.6\*0.8+5.1,-2.2) \[circle,draw=red!50,fill=red!20\] ; at (1.6\*0.8+5.1,-2.2) [2]{}; at (-0.8\*0.8+5.1,-2.2) \[circle,draw=red!50,fill=red!20\] ; at (-0.8\*0.8+5.1,-2.2) [2]{}; at (1.2\*0.8+5.1,0.8\*0.8-2.2) \[circle,draw=red!50,fill=red!20\] ; at (1.2\*0.8+5.1,0.8\*0.8-2.2) [2]{}; at (-0.4\*0.8+5.1,0.8\*0.8-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.4\*0.8+5.1,0.8\*0.8-2.2) [1]{}; at (2\*0.8+5.1,0.8\*0.8-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (2\*0.8+5.1,0.8\*0.8-2.2) [1]{}; at (1.4\*0.8+5.1,-2.6) \[circle,draw=blue!50,fill=blue!20\] ; at (1.4\*0.8+5.1,-2.6) [1]{}; at (0.4+5.1,-2.8)[3.b]{};
(2.4\*0.8+7.9,-2.2) – (-0.8\*0.8+7.9,-2.2); (-0.8\*0.8+7.9,0-2.2) – (-0.4\*0.8+7.9,0.8\*0.8-2.2); (0.8\*0.8+7.9,0-2.2) – (0.4\*0.8+7.9,0.8\*0.8-2.2); (0+7.9,0-2.2) – (0.4\*0.8+7.9,0.8\*0.8-2.2); (0+7.9,0-2.2) – (-0.4\*0.8+7.9,0.8\*0.8-2.2); (1.2\*0.8+7.9,0.8\*0.8-2.2) – (-0.4\*0.8+7.9,0.8\*0.8-2.2); (0.8\*0.8+7.9,-2.2) – (1.4\*0.8+7.9,-2.6); (1.6\*0.8+7.9,-2.2) – (2.2\*0.8+7.9,-1.8); (-0.8\*0.8+7.9,-2.2) .. controls (-0.8\*0.8+7.6,-0.8) and (1.2\*0.8+7.8,-0.8) .. (2.2\*0.8+7.9,-1.8);
at (0+7.9,-2.2) \[circle,draw=green!50,fill=green!20\] ; at (0+7.9,-2.2) [3]{}; at (0.4\*0.8+7.9,0.8\*0.8-2.2) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0.4\*0.8+7.9,0.8\*0.8-2.2) [4]{}; at (0.8\*0.8+7.9,-2.2) \[circle,draw=black!50,fill=black!20\] ; at (0.8\*0.8+7.9,-2.2) [5]{}; at (1.6\*0.8+7.9,-2.2) \[circle,draw=red!50,fill=red!20\] ; at (1.6\*0.8+7.9,-2.2) [2]{}; at (1.2\*0.8+7.9,0.8\*0.8-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2\*0.8+7.9,0.8\*0.8-2.2)[1]{}; at (-0.4\*0.8+7.9,0.8\*0.8-2.2) \[circle,draw=red!50,fill=red!20\] ; at (-0.4\*0.8+7.9,0.8\*0.8-2.2) [2]{}; at (-0.8\*0.8+7.9,-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8\*0.8+7.9,-2.2) [1]{}; at (2.4\*0.8+7.9,-2.2) \[circle,draw=blue!50,fill=blue!20\] ; at (2.4\*0.8+7.9,-2.2)[1]{}; at (1.4\*0.8+7.9,-2.6)\[circle,draw=blue!50,fill=blue!20\] ; at (1.4\*0.8+7.9,-2.6)[1]{}; at (2.2\*0.8+7.9,-1.8) \[circle,draw=black,fill=black\] ; at (0.8\*0.8+7.9,-2.8)[3.c]{};
Let $G$ be a 4-regular graph without induced $C_4$. If $G$ contains $C_3$ then $\Gamma(G)=5$. \[cycle3\]
(0-0.5,0) – (0-0.5,0.8); (-0.8-0.5,0.8) – (0-0.5,0); (-0.8-0.5,0) – (0-0.5,0.8); (-0.8-0.5,0) – (-0.8-0.5,0.8); (0.8-0.5,0) – (0-0.5,0); (0.8-0.5,0) – (0-0.5,0.8); (0.8-0.5,0.8) – (0-0.5,0); (0.8-0.5,0.8) – (0-0.5,0.8); (0.8-0.5,0.8) – (1.2-0.5,1.2); (0.8-0.5,0.8) – (1.2-0.5,0.4); (1.2-0.5,1.2) – (1.2-0.5,0.4); (1.2-0.5,1.2) .. controls (-0.7,1.6) .. (-0.8-0.5,0.8); (1.2-0.5,1.2) .. controls (-0.7,1.4) .. (-0.8-0.5,0); (1.2-0.5,0.4) .. controls (0.9-0.5,-0.5) .. (-0.8-0.5,0);
at (0-0.5,0) \[circle,draw=green!50,fill=green!20\] ; at (0-0.5,0) [3]{}; at (0-0.5,0.8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0-0.5,0.8) [4]{}; at (-0.4-0.5,0.4) \[circle,draw=black!50,fill=black!20\] ; at (-0.4-0.5,0.4) [5]{}; at (-0.8-0.5,0.8) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8-0.5,0.8) [1]{}; at (-0.8-0.5,0) \[circle,draw=red!50,fill=red!20\] ; at (-0.8-0.5,0) [2]{}; at (0.8-0.5,0) \[circle,draw=blue!50,fill=blue!20\] ; at (0.8-0.5,0) [1]{}; at (0.8-0.5,0.8) \[circle,draw=red!50,fill=red!20\] ; at (0.8-0.5,0.8) [2]{}; at (1.2-0.5,0.4) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2-0.5,0.4) [1]{}; at (1.2-0.5,1.2) \[circle,draw=black,fill=black\] ; at (0-0.5,-0.4) [$u_2$]{}; at (0-0.5,1.025) [$u_1$]{}; at (0-0.5,-0.9)[1.a]{};
(0+2.3,-0.4) – (0+2.3,1.2); (-0.8+2.3,0.8) – (0+2.3,0); (-0.8+2.3,0) – (0+2.3,0.8); (-0.8+2.3,0) – (-0.8+2.3,0.8); (0.4+2.3,0.4) – (0+2.3,0); (0.4+2.3,0.4) – (0+2.3,0.8); (0.4+2.3,0.4) – (0.8+2.3,0); (0.4+2.3,0.4) – (0.8+2.3,0.8); (0.8+2.3,0) – (0.8+2.3,0.8); (0.8+2.3,0) .. controls (2.3+0.2,1.8) .. (-0.8+2.3,0.8); (0.8+2.3,0) .. controls (2.3,-0.9) .. (-0.8+2.3,0); (0.8+2.3,0.8) .. controls (2.3,2) .. (-0.8+2.3,0.8); at (0+2.3,0) \[circle,draw=green!50,fill=green!20\] ; at (0+2.3,0) [3]{}; at (0+2.3,0.8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0+2.3,0.8) [4]{}; at (-0.4+2.3,0.4) \[circle,draw=black!50,fill=black!20\] ; at (-0.4+2.3,0.4) [5]{}; at (-0.8+2.3,0) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+2.3,0) [1]{}; at (-0.8+2.3,0.8) \[circle,draw=red!50,fill=red!20\] ; at (-0.8+2.3,0.8) [2]{}; at (0.4+2.3,0.4) \[circle,draw=red!50,fill=red!20\] ; at (0.4+2.3,0.4) [2]{}; at (0+2.3,-0.4) \[circle,draw=blue!50,fill=blue!20\] ; at (0+2.3,-0.4) [1]{}; at (0+2.3,1.2) \[circle,draw=blue!50,fill=blue!20\] ; at (0+2.3,1.2) [1]{}; at (0.8+2.3,0.8) \[circle,draw=blue!50,fill=blue!20\] ; at (0.8+2.3,0.8) [1]{}; at (0.8+2.3,0) \[circle,draw=black,fill=black\] ; at (0+2.3,-0.9)[1.b]{};
(0+4.1,0) – (0+4.1,0.8); (-0.4+4.1,0.4) – (0+4.1,0); (-0.4+4.1,0.4) – (0+4.1,0.8); (1.6+4.1,0) – (0+4.1,0); (0.8+4.1,0) – (0+4.1,0.8); (0.8+4.1,0.8) – (0+4.1,0); (1.6+4.1,0.8) – (0+4.1,0.8); (1.6+4.1,0.8) – (1.6+4.1,0); (0.8+4.1,0.8) – (0.8+4.1,1.4); at (0.8+4.1,0.8) \[circle,draw=green!50,fill=green!20\] ; at (0.8+4.1,0.8) [3]{}; at (0+4.1,0.8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0+4.1,0.8) [4]{}; at (0+4.1,0) \[circle,draw=black!50,fill=black!20\] ; at (0+4.1,0) [5]{}; at (-0.4+4.1,0.4) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.4+4.1,0.4) [1]{}; at (0.8+4.1,0) \[circle,draw=red!50,fill=red!20\] ; at (0.8+4.1,0) [2]{}; at (1.6+4.1,0) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6+4.1,0) [1]{}; at (1.6+4.1,0.8) \[circle,draw=red!50,fill=red!20\] ; at(1.6+4.1,0.8) [2]{}; at (0.8+4.1,1.4) \[circle,draw=blue!50,fill=blue!20\] ; at (0.8+4.1,1.4) [1]{}; at (0.8+4.1,-0.9)[2.a]{};
(0+6.7,0) – (0+6.7,0.8); (-0.4+6.7,0.4) – (0+6.7,0); (-0.4+6.7,0.4) – (0+6.7,0.8); (1.6+6.7,0) – (0+6.7,0); (0.8+6.7,0) – (0+6.7,0.8); (0.8+6.7,0.8) – (0+6.7,0); (1.6+6.7,0.8) – (0+6.7,0.8); (0.8+6.7,0.8) – (0.4+6.7,1.2); (1.6+6.7,0.8) – (1.2+6.7,0.4); (1.6+6.7,0.8) – (2+6.7,0.4); (1.2+6.7,0.4) – (2+6.7,0.4); (1.6+6.7,0) – (1.2+6.7,0.4); (1.6+6.7,0) – (2+6.7,0.4); (0.8+6.7,0) – (0.4+6.7,-0.4); (-0.4+6.7,0.4) – (1.2+6.7,0.4);
at (0.8+6.7,0.8) \[circle,draw=green!50,fill=green!20\] ; at (0.8+6.7,0.8) [3]{}; at (0+6.7,0.8) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0+6.7,0.8) [4]{}; at (0+6.7,0) \[circle,draw=black!50,fill=black!20\] ; at (0+6.7,0) [5]{}; at (-0.4+6.7,0.4) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.4+6.7,0.4) [1]{}; at (0.8+6.7,0) \[circle,draw=red!50,fill=red!20\] ; at (0.8+6.7,0) [2]{}; at (1.6+6.7,0) \[circle,draw=black,fill=black\] ; at (1.6+6.7,0.8) \[circle,draw=red!50,fill=red!20\] ; at (1.6+6.7,0.8) [2]{}; at (2+6.7,0.4) \[circle,draw=blue!50,fill=blue!20\] ; at (2+6.7,0.4) [1]{}; at (0.4+6.7,-0.4) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4+6.7,-0.4) [1]{}; at (0.4+6.7,1.2) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4+6.7,1.2) [1]{}; at (1.2+6.7,0.4) \[circle,draw=black,fill=black\] ; at (1.2+6.5,-0.9)[2.b]{};
(0-1.3,0-2.5) – (0-1.3,0.8-2.5); (-0.4-1.3,0.4-2.5) – (0-1.3,0-2.5); (-0.4-1.3,0.4-2.5) – (0-1.3,0.8-2.5); (1.6-1.3,0-2.5) – (0-1.3,0-2.5); (0.8-1.3,0-2.5) – (0+-1.3,0.8-2.5); (0.8-1.3,0.8-2.5) – (0-1.3,0-2.5); (1.6-1.3,0.8-2.5) – (0-1.3,0.8-2.5); (0.8-1.3,0.8-2.5) – (0.4-1.3,1.2-2.5); (-0.4-1.3,0.4-2.5) – (1.2-1.3,0.4-2.5); (1.6-1.3,0.8-2.5) – (1.2-1.3,0.4-2.5); (1.6-1.3,0.8-2.5) – (1.2-1.3,1.2-2.5);
at (0.8-1.3,0.8-2.5) \[circle,draw=green!50,fill=green!20\] ; at (0.8-1.3,0.8-2.5) [3]{}; at (0-1.3,0.8-2.5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0-1.3,0.8-2.5) [4]{}; at (0-1.3,0-2.5) \[circle,draw=black!50,fill=black!20\] ; at (0-1.3,0-2.5) [5]{}; at (0.8-1.3,0-2.5) \[circle,draw=red!50,fill=red!20\] ; at (0.8-1.3,0-2.5) [2]{}; at (1.6-1.3,0.8-2.5) \[circle,draw=red!50,fill=red!20\] ; at (1.6-1.3,0.8-2.5) [2]{}; at (-0.4-1.3,0.4-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.4-1.3,0.4-2.5) [1]{}; at (1.2-1.3,1.2-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2-1.3,1.2-2.5) [1]{}; at (0.4-1.3,1.2-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4-1.3,1.2-2.5) [1]{}; at (1.6-1.3,0-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6-1.3,0-2.5) [1]{}; at (1.2-1.3,0.4-2.5) \[circle,draw=black,fill=black\] ; at (0.4-1.3,-0.9-2.5)[2.c]{};
(0+1.9,0-2.5) – (0+1.9,0.8-2.5); (1.6+1.9,0-2.5) – (-0.8+1.9,0-2.5); (0.8+1.9,0-2.5) – (0+1.9,0.8-2.5); (0.8+1.9,0.8-2.5) – (0+1.9,0-2.5); (1.6+1.9,0.8-2.5) – (-0.8+1.9,0.8-2.5); (0.8+1.9,0.8-2.5) – (0.4+1.9,1.2-2.5); (1.6+1.9,0.8-2.5) – (1.6+1.9,0-2.5); at (0.8+1.9,0.8-2.5) \[circle,draw=green!50,fill=green!20\] ; at (0.8+1.9,0.8-2.5) [3]{}; at (0+1.9,0.8-2.5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0+1.9,0.8-2.5) [4]{}; at (0+1.9,0-2.5) \[circle,draw=black!50,fill=black!20\] ; at (0+1.9,0-2.5) [5]{}; at (0.8+1.9,0-2.5) \[circle,draw=red!50,fill=red!20\] ; at (0.8+1.9,0-2.5) [2]{}; at (-0.8+1.9,0-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+1.9,0-2.5) [1]{}; at (-0.8+1.9,0.8-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+1.9,0.8-2.5) [1]{}; at (1.6+1.9,0-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6+1.9,0-2.5) [1]{}; at (1.6+1.9,0.8-2.5) \[circle,draw=red!50,fill=red!20\] ; at (1.6+1.9,0.8-2.5) [2]{}; at (0.4+1.9,1.2-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4+1.9,1.2-2.5) [1]{}; at (0.4+1.9,-0.9-2.5)[3.a]{};
(0+4,-0.6-2.5) – (0+4,0.8-2.5); (0.8+4,0-2.5) – (0+4,0-2.5); (0.8+4,0-2.5) – (0+4,0.8-2.5); (0.8+4,0.8-2.5) – (0+4,0-2.5); (0.8+4,0.8-2.5) – (0+4,0.8-2.5); (0.8+4,0.8-2.5) – (1.6+4,0.4-2.5); (1.6+4,0.4-2.5) – (1.2+4,0.8-2.5); (1.6+4,0.4-2.5) – (2+4,0.8-2.5); (1.2+4,0.8-2.5) – (2+4,0.8-2.5); (1.6+4,1.2-2.5) – (1.2+4,0.8-2.5); (1.6+4,1.2-2.5) – (2+4,0.8-2.5); (0+4,0.8-2.5) – (1.6+4,1.2-2.5); (1.6+4,-0.4-2.5) – (1.6+4,-0.8-2.5); (1.6+4,-0.4-2.5) – (1.6+4,0-2.5); (1.6+4,-0.4-2.5) – (2+4,-0.4-2.5); (0.8+4,0.8-2.5) – (0.8+4,0.4-2.5); (0.8+4,0-2.5) – (1.6+4,-0.4-2.5); (0.8+4,0-2.5) – (0.5+4,-0.4-2.5); (0+4,-0.6-2.5) – (1.6+4,-0.8-2.5); (1.6+4,1.2-2.5) .. controls (2.5+4,1-2.5) .. (2+4,-0.4-2.5); (0.8+4,0.4-2.5) – (1.6+4,0-2.5);
at (0.8+4,0-2.5) \[circle,draw=green!50,fill=green!20\] ; at (0.8+4,0-2.5) [3]{}; at (0+4,0.8-2.5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0+4,0.8-2.5) [4]{}; at (0+4,0-2.5) \[circle,draw=black!50,fill=black!20\] ; at (0+4,0-2.5) [5]{}; at (0.8+4,0.8-2.5) \[circle,draw=red!50,fill=red!20\] ; at (0.8+4,0.8-2.5) [2]{}; at (1.6+4,0-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6+4,0-2.5) [1]{}; at (1.6+4,1.2-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6+4,1.2-2.5) [1]{}; at (1.6+4,0.4-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6+4,0.4-2.5) [1]{}; at (0+4,-0.6-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (0+4,-0.6-2.5) [1]{}; at (1.6+4,-0.4-2.5) \[circle,draw=red!50,fill=red!20\] ; at (1.6+4,-0.4-2.5) [2]{}; at (0.5+4,-0.4-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.5+4,-0.4-2.5) [1]{}; at (0.8+4,0.4-2.5) \[circle,draw=black,fill=black\] ; at (2+4,0.8-2.5) \[circle,draw=black,fill=black\] ; at (1.2+4,0.8-2.5) \[circle,draw=black,fill=black\] ; at (1.6+4,-0.8-2.5) \[circle,draw=black,fill=black\] ; at (2+4,-0.4-2.5) \[circle,draw=black,fill=black\] ; at (0.8+4,-0.9-2.5)[3.b]{};
(0+6.6,-0.6-2.5) – (0+6.6,1.4-2.5); (1.6+6.6,0-2.5) – (0+6.6,0-2.5); (0.8+6.6,0-2.5) – (0+6.6,0.8-2.5); (0.8+6.6,0.8-2.5) – (0+6.6,0-2.5); (1.6+6.6,0.8-2.5) – (0+6.6,0.8-2.5); (0.8+6.6,0.8-2.5) – (0.4+6.6,1.2-2.5); (1.6+6.6,0.8-2.5) – (1.2+6.6,0.4-2.5); (1.6+6.6,0.8-2.5) – (2+6.6,0.4-2.5); (1.6+6.6,0-2.5) – (1.2+6.6,0.4-2.5); (1.6+6.6,0-2.5) – (2+6.6,0.4-2.5); (1.2+6.6,0.4-2.5) – (2+6.6,0.4-2.5); (1.6+6.6,0.8-2.5) – (1.2+6.6,1.2-2.5); at (0.8+6.6,0.8-2.5) \[circle,draw=green!50,fill=green!20\] ; at (0.8+6.6,0.8-2.5) [3]{}; at (0+6.6,0.8-2.5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0+6.6,0.8-2.5) [4]{}; at (0+6.6,0-2.5) \[circle,draw=black!50,fill=black!20\] ; at (0+6.6,0-2.5) [5]{}; at (0.8+6.6,0-2.5) \[circle,draw=red!50,fill=red!20\] ; at (0.8+6.6,0-2.5) [2]{}; at (1.6+6.6,0.8-2.5) \[circle,draw=red!50,fill=red!20\] ; at (1.6+6.6,0.8-2.5) [2]{}; at (1.6+6.6,0-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6+6.6,0-2.5) [1]{}; at (0+6.6,-0.6-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (0+6.6,-0.6-2.5) [1]{}; at (0+6.6,1.4-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (0+6.6,1.4-2.5) [1]{}; at (0.4+6.6,1.2-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4+6.6,1.2-2.5) [1]{}; at (1.2+6.6,1.2-2.5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+6.6,1.2-2.5) [1]{}; at (1.2+6.6,0.4-2.5) \[circle,draw=black,fill=black\] ; at (2+6.6,0.4-2.5) \[circle,draw=black,fill=black\] ; at (0.8+6.6,-0.9-2.5)[3.c]{};
(0-1.7,-0.6-5) – (0-1.7,1.4-5); (1.6-1.7,0-5) – (0-1.7,0-5); (0.8-1.7,0-5) – (0-1.7,0.8-5); (0.8-1.7,0.8-5) – (0-1.7,0-5); (2.2-1.7,0.8-5) – (0-1.7,0.8-5); (0.8-1.7,0.8-5) – (0.4-1.7,1.2-5); (1.6-1.7,0.8-5) – (2.2-1.7,1.3-5); (1.6-1.7,0.8-5) – (2.2-1.7,0.3-5); at (0.8-1.7,0.8-5) \[circle,draw=green!50,fill=green!20\] ; at (0.8-1.7,0.8-5) [3]{}; at (0-1.7,0.8-5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0-1.7,0.8-5) [4]{}; at (0-1.7,0-5) \[circle,draw=black!50,fill=black!20\] ; at (0-1.7,0-5) [5]{}; at (0.8-1.7,0-5) \[circle,draw=red!50,fill=red!20\] ; at (0.8-1.7,0-5) [2]{}; at (1.6-1.7,0.8-5) \[circle,draw=red!50,fill=red!20\] ; at (1.6-1.7,0.8-5) [2]{}; at (1.6-1.7,0-5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.6-1.7,0-5) [1]{}; at (0-1.7,-0.6-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0-1.7,-0.6-5) [1]{}; at (0-1.7,1.4-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0-1.7,1.4-5) [1]{}; at (0.4-1.7,1.2-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4-1.7,1.2-5) [1]{}; at (2.2-1.7,0.8-5) \[circle,draw=blue!50,fill=blue!20\] ; at (2.2-1.7,0.8-5) [1]{}; at (2.2-1.7,1.3-5) \[circle,draw=black,fill=black\] ; at (2.2-1.7,0.3-5) \[circle,draw=black,fill=black\] ; at (1-2.3,-6.4)[3.d]{};
(-0.8+2,0-5) – (-0.4+2,0.8-5); (-0.8+2,0-5) – (-1+2,0-0.6-5); (-0.8+2,0-5) – (-0.6+2,0.-0.6-5); (-0.4+2,0.8-5) – (-0.8+2,1.1-5); (0+2,0-5) – (0+2,-0.6-5); (-0.8+2,0-5) – (0.8+2,0-5); (-0.4+2,0.8-5) – (0+2,0-5); (-0.4+2,0.8-5) – (1.2+2,0.8-5); (0.8+2,0-5) – (1.2+2,0.8-5); (-0.6+2,-0.6-5) .. controls (-0+2,-1-5) and (1.2+2,-1-5) .. (1.2+2,0.8-5); at (1.2+2,0.8-5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+2,0.8-5)[1]{}; at (-1+2,0.-0.6-5) \[circle,draw=blue!50,fill=blue!20\] ; at (-1+2,0.-0.6-5)[1]{}; at (-0.8+2,1.1-5) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+2,1.1-5)[1]{}; at (0+2,-0.6-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0+2,-0.6-5)[1]{}; at (-0.4+2,0.8-5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (-0.4+2,0.8-5) [4]{}; at (0+2,0-5) \[circle,draw=black!50,fill=black!20\] ; at (0+2,0-5) [5]{}; at (-0.6+2,-0.6-5) \[circle,draw=red!50,fill=red!20\] ; at (-0.6+2,-0.6-5) [2]{}; at (0.8+2,0-5) \[circle,draw=red!50,fill=red!20\] ; at (0.8+2,0-5) [2]{}; at (0.4+2,0.8-5) \[circle,draw=red!50,fill=red!20\] ; at (0.4+2,0.8-5) [2]{}; at (-0.8+2,0-5) \[circle,draw=green!50,fill=green!20\] ; at (-0.8+2,0-5) [3]{};
at (-0.4+2,1.1-5) [$u_1$]{}; at (0.2+2,0.3-5) [$u_2$]{}; at (-1.2+2,0-5) [$u_3$]{}; at (2.1,-5-1.4) [4.a]{};
(-0.8+4.6,0-5) – (-0.4+4.6,0.8-5); (-0.8+4.6,0-5) – (-0.8+4.6,0-1-5); (-0.8+4.6,-1-5) – (0.8+4.6,0-1-5); (-0.4+4.6,0.8-5) – (1.2+4.6,0.8-5); (0+4.6,0-5) – (0+4.6,-0.6-5); (-0.8+4.6,0-5) – (0.8+4.6,0-5); (-0.4+4.6,0.8-5) – (0+4.6,0-5); (0.8+4.6,0-5) – (1.2+4.6,0.8-5); (0.8+4.6,0-5) – (0.8+4.6,-1-5); (0.8+4.6,0-5) – (0.4+4.6,0.4-5); (0.8+4.6,-1-5) – (1.2+4.6,0.8-5); at (0+4.6,0-5) \[circle,draw=black!50,fill=black!20\] ; at (0+4.6,0-5) [5]{}; at (-0.8+4.6,0-5) \[circle,draw=green!50,fill=green!20\] ; at (-0.8+4.6,0-5) [3]{}; at (0.8+4.6,0-5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (0.8+4.6,0-5) [4]{}; at (0.8+4.6,-1-5) \[circle,draw=green!50,fill=green!20\] ; at (0.8+4.6,-1-5) [3]{}; at (-0.4+4.6,0.8-5) \[circle,draw=red!50,fill=red!20\] ; at (-0.4+4.6,0.8-5) [2]{}; at (1.2+4.6,0.8-5) \[circle,draw=red!50,fill=red!20\] ; at (1.2+4.6,0.8-5)[2]{}; at (0.4+4.6,0.8-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4+4.6,0.8-5) [1]{}; at (-0.8+4.6,0-1-5) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+4.6,0-1-5) [1]{}; at (0+4.6,-0.6-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0+4.6,-0.6-5) [1]{}; at (0.4+4.6,0.4-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.4+4.6,0.4-5) [1]{}; at (0.2+4.6,-5-1.4) [4.b]{};
(-0.8+7.2,0-5) – (-0.4+7.2,0.8-5); (-0.8+7.2,0-5) – (-0.8+7.2,0-1-5); (-0.8+7.2,-1-5) – (0.8+7.2,0-1-5); (-0.4+7.2,0.8-5) – (1.2+7.2,0.8-5); (0+7.2,0-5) – (0+7.2,-0.6-5); (-0.8+7.2,0-5) – (0.8+7.2,0-5); (-0.4+7.2,0.8-5) – (0+7.2,0-5); (0.8+7.2,0-5) – (1.2+7.2,0.8-5); (0.8+7.2,0-5) – (0.8+7.2,-1-5); (-0.8+7.2,0-5) – (-1.1+7.2,-0.6-5); (-0.4+7.2,0.8-5) – (-0.8+7.2,1.2-5);
at (-0.4+7.2,0.8-5) \[circle,draw=yellow!50,fill=yellow!20\] ; at (-0.4+7.2,0.8-5) [4]{}; at (0+7.2,0-5) \[circle,draw=black!50,fill=black!20\] ; at (0+7.2,0-5) [5]{}; at (0.8+7.2,0-5) \[circle,draw=red!50,fill=red!20\] ; at (0.8+7.2,0-5) [2]{}; at (0.4+7.2,0.8-5) \[circle,draw=red!50,fill=red!20\] ; at (0.4+7.2,0.8-5) [2]{}; at (-0.8+7.2,-1-5) \[circle,draw=red!50,fill=red!20\] ; at (-0.8+7.2,-1-5) [2]{}; at (-0.8+7.2,0-5) \[circle,draw=green!50,fill=green!20\] ; at (-0.8+7.2,0-5) [3]{}; at (1.2+7.2,0.8-5) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+7.2,0.8-5) [1]{}; at (0.8+7.2,-1-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0.8+7.2,-1-5) [1]{}; at (0+7.2,-0.6-5) \[circle,draw=blue!50,fill=blue!20\] ; at (0+7.2,-0.6-5) [1]{}; at (-1.1+7.2,-0.6-5) \[circle,draw=blue!50,fill=blue!20\] ; at (-1.1+7.2,-0.6-5) [1]{}; at (-0.8+7.2,1.2-5) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+7.2,1.2-5) [1]{}; at (0.2+7.2,-5-1.4) [4.c]{};
(-0.8+0.3,0-7.3) – (-0.4+0.3,0.8-7.3); (-0.8+0.3,0-7.3) – (-0.6+0.3,-0.5-7.3); (-0.4+0.3,0.8-7.3) – (1.2+0.3,0.8-7.3); (0+0.3,0-7.3) – (0+0.3,-0.6-7.3); (0.8+0.3,0-7.3) – (0.8+0.3,-0.6-7.3); (-0.8+0.3,0-7.3) – (0.8+0.3,0-7.3); (-0.4+0.3,0.8-7.3) – (0+0.3,0-7.3); (0.8+0.3,0-7.3) – (1.2+0.3,0.8-7.3); (-0.8+0.3,0-7.3) – (-1.1+0.3,-0.6-7.3); (-0.4+0.3,0.8-7.3) – (-0.8+0.3,1.2-7.3); (0.4+0.3,0.8-7.3) – (0+0.3,1.2-7.3); (0.4+0.3,0.8-7.3) – (0.8+0.3,1.2-7.3); (-1+0.3,-1-7.3) – (-0.6+0.3,-0.5-7.3); (-0.6+0.3,-1-7.3) – (-0.6+0.3,-0.5-7.3); (-0.2+0.3,-1-7.3) – (-0.6+0.3,-0.5-7.3); (-1+0.3,-1-7.3) .. controls (-1.5+0.3,-1-7.3) .. (-0.8+0.3,1.2-7.3); (-0.2+0.3,-1-7.3) .. controls (0.4+0.3,-1-7.3) .. (0+0.3,-0.6-7.3); (0.8+0.3,-0.6-7.3) .. controls (1.2+0.3,-0.4-7.3) .. (0.8+0.3,1.2-7.3);
at (-0.4+0.3,0.8-7.3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (-0.4+0.3,0.8-7.3) [4]{}; at (0+0.3,0-7.3) \[circle,draw=black!50,fill=black!20\] ; at (0+0.3,0-7.3) [5]{}; at (0.8+0.3,0-7.3) \[circle,draw=red!50,fill=red!20\] ; at (0.8+0.3,0-7.3) [2]{}; at (0.4+0.3,0.8-7.3) \[circle,draw=red!50,fill=red!20\] ; at (0.4+0.3,0.8-7.3) [2]{}; at (-0.8+0.3,0-7.3) \[circle,draw=green!50,fill=green!20\] ; at (-0.8+0.3,0-7.3) [3]{}; at (-0.6+0.3,-0.5-7.3) \[circle,draw=red!50,fill=red!20\] ; at (-0.6+0.3,-0.5-7.3) [2]{}; at (0+0.3,1.2-7.3) \[circle,draw=blue!50,fill=blue!20\] ; at (0+0.3,1.2-7.3) [1]{}; at (-0.8+0.3,1.2-7.3) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+0.3,1.2-7.3)[1]{}; at (-1.1+0.3,-0.6-7.3) \[circle,draw=blue!50,fill=blue!20\] ; at (-1.1+0.3,-0.6-7.3) [1]{}; at (0+0.3,-0.6-7.3) \[circle,draw=blue!50,fill=blue!20\] ; at (0+0.3,-0.6-7.3) [1]{}; at (0.8+0.3,-0.6-7.3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.8+0.3,-0.6-7.3) [1]{}; at (-0.6+0.3,-1-7.3) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.6+0.3,-1-7.3) [1]{}; at (1.2+0.3,0.8-7.3) \[circle,draw=black,fill=black\] ; at (-0.2+0.3,-1-7.3) \[circle,draw=black,fill=black\] ; at (-1+0.3,-1-7.3) \[circle,draw=black,fill=black\] ; at (0.8+0.3,1.2-7.3) \[circle,draw=black,fill=black\] ; at (0.2,-0.5-8.2) [4.d]{};
(-1.2+3.7,0-8.2) – (1.2+3.7,0-8.2); (0.4+3.7,0-8.2) – (0+3.7,0.8-8.2); (-0.4+3.7,0-8.2) – (0+3.7,0.8-8.2); (-0.4+3.7,0-8.2) – (-1+3.7,0.4-8.2); (-1+3.7,0.4-8.2) – (-1.8+3.7,0-8.2); (-1+3.7,0.4-8.2) – (-1.8+3.7,0.4-8.2); (-1+3.7,0.4-8.2) – (-1.8+3.7,0.8-8.2); (0+3.7,0.8-8.2) – (0+3.7,1.2-8.2); (0+3.7,1.2-8.2) – (-0.8+3.7,0.8-8.2); (0+3.7,1.2-8.2) – (-0.8+3.7,1.2-8.2); (0+3.7,1.2-8.2) – (-0.8+3.7,1.6-8.2); (0.4+3.7,0-8.2) – (1+3.7,0.4-8.2); (1+3.7,0.4-8.2) – (1.8+3.7,0-8.2); (1+3.7,0.4-8.2) – (1.8+3.7,0.4-8.2); (1+3.7,0.4-8.2) – (1.8+3.7,0.8-8.2); (0+3.7,0.8-8.2) – (0.6+3.7,1.2-8.2); (-1.8+3.7,0-8.2) .. controls (0+3.7,-0.4-8.2) .. (1.8+3.7,0-8.2); (-1.8+3.7,0.8-8.2) – (-0.8+3.7,0.8-8.2); (-0.8+3.7,1.6-8.2) .. controls (1.8+3.7,1.6-8.2) .. (1.8+3.7,0.8-8.2); at (-0.4+3.7,0-8.2) \[circle,draw=yellow!50,fill=yellow!20\] ; at (-0.4+3.7,0-8.2) [4]{}; at (0.4+3.7,0-8.2) \[circle,draw=black!50,fill=black!20\] ; at (0.4+3.7,0-8.2) [5]{}; at (0+3.7,0.8-8.2) \[circle,draw=green!50,fill=green!20\] ; at (0+3.7,0.8-8.2) [3]{}; at (0+3.7,1.2-8.2) \[circle,draw=red!50,fill=red!20\] ; at (0+3.7,1.2-8.2) [2]{}; at (-1+3.7,0.4-8.2)\[circle,draw=red!50,fill=red!20\] ; at (-1+3.7,0.4-8.2) [2]{}; at (1+3.7,0.4-8.2) \[circle,draw=red!50,fill=red!20\] ; at (1+3.7,0.4-8.2) [2]{}; at (0.6+3.7,1.2-8.2) \[circle,draw=blue!50,fill=blue!20\] ; at (0.6+3.7,1.2-8.2) [1]{}; at (1+3.7,0-8.2) \[circle,draw=blue!50,fill=blue!20\] ; at (1+3.7,0-8.2) [1]{}; at (-1+3.7,0-8.2) \[circle,draw=blue!50,fill=blue!20\] ; at (-1+3.7,0-8.2) [1]{}; at (-0.8+3.7,1.2-8.2) \[circle,draw=blue!50,fill=blue!20\] ; at (-0.8+3.7,1.2-8.2) [1]{}; at (1.8+3.7,0.4-8.2) \[circle,draw=blue!50,fill=blue!20\] ; at (1.8+3.7,0.4-8.2) [1]{}; at (-1.8+3.7,0.4-8.2) \[circle,draw=blue!50,fill=blue!20\] ; at (-1.8+3.7,0.4-8.2) [1]{}; at (-1.8+3.7,0.8-8.2) \[circle,draw=black,fill=black\] ; at (-1.8+3.7,0-8.2)\[circle,draw=black,fill=black\] ; at (1.8+3.7,0.8-8.2) \[circle,draw=black,fill=black\] ; at (1.8+3.7,0-8.2)\[circle,draw=black,fill=black\] ; at (-0.8+3.7,0.8-8.2) \[circle,draw=black,fill=black\] ; at (-0.8+3.7,1.6-8.2)\[circle,draw=black,fill=black\] ; at (0+3.7,-0.5-8.2) [4.e]{};
Depending on the different cases that could happen, a reference to the Grundy partial 5-coloring of $G$ in Figure \[figm32\] will be given. Let $M_i$, $i=2$ or 3, be the graph of order $2+i$ containing two adjacent vertices $u_1$ and $u_2$ which have exactly $i$ common neighbors, $\{v_1,\ldots,v_i \}$, that form an independent set. Let $D_1$ be the set of vertices at distance 1 from an induced $M_i$ in $G-M_i$, for $2\le i\le3$.
Case 1:
: Firstly, assume that $G$ contains an induced $M_3$ and a vertex of $M_3$ has its two neighbors in $D_1$ adjacent (Figure 6.1.a). Secondly, assume that $G$ contains an induced $M_2$ and a vertex of $M_2$ has its two neighbors in $D_1$ adjacent (Figure 6.1.b). Note that these Grundy partial 5-colorings use the fact that $G$ cannot contain a $K_4$ by Lemma \[k4\].
Case 2:
: Assume that $G$ contains an induced $M_3$ excluding the previous configuration. There are three cases: $u_1$, $v_2$ and $v_3$ are in an induced $C_5$ (Figure 6.2.a), $u_1$, $v_2$ and $v_3$ are in an induced $C_6$ and not in an induced $C_5$ (Figure 6.2.b) and $u_1$, $v_2$ and $v_3$ are neither in an induced $C_5$ nor $C_6$ (Figure 6.2.c).
Case 3:
: Suppose that $G$ contains an induced $M_2$ excluding the previous configurations. Firstly, we suppose that $u_1$, $v_1$ and $v_2$ are in an induced $C_5$ (Figure 6.3.a). Secondly, we suppose that $u_1$, $v_1$ are in an induced $C_5$ excluding the previous case (Figure 6.3.b). Thirdly, we suppose that $u_1$, $v_1$ and $v_2$ are in an induced $C_6$ and not in an induced $C_5$ (Figure 6.3.c) and finally neither in an induced $C_5$ nor $C_6$ (Figure 6.3.d).
Suppose that $G$ contains a 3-cycle $C$ and no induced $M_2$. Let $u_1$, $u_2$ and $u_3$ be the vertices of $C$. Let $w_1$ and $w_2$ be the neighbors of $u_1$ outside $C$, let $w'_1$ and $w'_2$ be the neighbors of $u_2$ outside $C$ and let $w''_1$ and $w''_2$ be the neighbors of $u_3$ outside $C$.
Case 4:
: Firstly, suppose that $u_1$, $u_2$, $w_1$ and $w'_1$ are in a 5-cycle and a neighbor of $u_1$, say $w_1$, has a common neighbor with $w'_1$ (Figure 6.4.a). Secondly, excluding the previous configuration, suppose that $u_1$, $u_2$, $w_1$ and $w'_1$ are in a 5-cycle; $w''_1$, $v_1$, $u_1$ and $w_1$ are in another 5-cycle and $w_1$ is in a triangle (Figure 6.4.b). We suppose that $w_1$ is not in a triangle (Figure 6.4.c). Thirdly, excluding the previous configurations, we obtain a Grundy partial 5-coloring if two vertices of $C$ are in a 5-cycle (Figure 6.4.d). Fourthly, we suppose that two vertices of $C$ cannot be in a 5-cycle (Figure 6.4.e).
In the following two lemmas, we consider a graph $G$ of girth $g=5$ and possibly containing an induced Petersen graph. Let $u_1$, $u_2$, $u_3$, $u_4$ and $u_5$ be the vertices in an induced $C_5$ (or in the the outer cycle of a Petersen graph, if any). Let $v_1$, $v'_1$, $v_2$, $v'_2$, $v_3$, $v'_3$, $v_4$, $v'_4$, $v_5$ and $v'_5$ be the remaining neighbors of respectively $u_1$, $u_2$, $u_3$, $u_4$ and $u_5$ (all different as $g=5$).
Let $G$ be a 4-regular graph with girth $g=5$. If $G$ contains the Petersen graph as induced subgraph then $\Gamma(G)=5$. \[lpet\]
Suppose that $v_1$, $v_2$, $v_3$, $v_4$ and $v_5$ form an induced $C_5$ (the inner cycle of the Petersen graph). Let $u'_2$ and $u'_5$ be the remaining neighbors of respectively $v_2$ and $v_5$. Observe that $v'_1$ can be adjacent with no more than three vertices among $v'_3$, $v'_4$, $u'_2$ and $u'_5$. Firstly, suppose that $v'_1$ is not adjacent with $v'_3$ (or $v'_4$, without loss of generality since the configuration is symmetric). The left part of Figure \[figm51\] illustrates a Grundy partial 5-coloring of the graph $G$. Secondly, assume that $v'_1$ is not adjacent with $u'_5$ (or $u'_2$, without loss of generality). The right part of Figure \[figm51\] illustrates a Grundy partial 5-coloring of the graph $G$.
(0,0) – (3\*2/3,0); (0,0) – (-1\*2/3,2\*2/3); (3\*2/3,0) – (4\*2/3,2\*2/3); (-1\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (4\*2/3, 2\*2/3) – (1.5\*2/3,4.4\*2/3); (1.5\*2/3,3.2\*2/3) – (1.5\*2/3,4.4\*2/3); (-1\*2/3,2\*2/3) – (0.2\*2/3, 2.2\*2/3); (4\*2/3, 2\*2/3) – (2.8\*2/3, 2.2\*2/3); (0,0) – (0.7\*2/3,0.7\*2/3); (3\*2/3,0) – (2.3\*2/3,0.7\*2/3); (0.7\*2/3,0.7\*2/3) – (2.8\*2/3,2.2\*2/3); (2.3\*2/3,0.7\*2/3) – (0.2\*2/3,2.2\*2/3); (0.7\*2/3,0.7\*2/3) – (1.5\*2/3,3.2\*2/3); (2.3\*2/3,0.7\*2/3) – (1.5\*2/3,3.2\*2/3); (1.5\*2/3,4.4\*2/3) – (1.5\*2/3,5.6\*2/3) ; (-1\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (0.2\*2/3,2.2\*2/3) – (2.8\*2/3,2.2\*2/3); (0,0) – (-1.2\*2/3,0); (4.2\*2/3,0) – (3\*2/3,0); (0.2\*2/3,2.2\*2/3) – (-1.5\*2/3,2.2\*2/3+0.9); (2.8\*2/3,2.2\*2/3) – (4.5\*2/3,2.2\*2/3+0.9); (1.5\*2/3,5.6\*2/3) – (-1.5\*2/3,2.2\*2/3+0.9); (1.5\*2/3,5.6\*2/3) – (4.5\*2/3,2.2\*2/3+0.9); (1.5\*2/3,5.6\*2/3) – (-1.2\*2/3,0); (4\*2/3,2\*2/3) – (4\*2/3+0.8,2\*2/3) ; at (0,0) \[circle,draw=red!50,fill=red!20\] ; at (0,0) [2]{}; at (-1\*2/3,2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (-1\*2/3,2\*2/3) [3]{}; at (3\*2/3,0) \[circle,draw=green!50,fill=green!20\] ; at (3\*2/3,0) [3]{}; at (4\*2/3,2\*2/3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (4\*2/3,2\*2/3)[4]{}; at (1.5\*2/3,4.4\*2/3) \[circle,draw=black!50,fill=black!20\] ; at (1.5\*2/3,4.4\*2/3) [5]{}; at (1.5\*2/3,3.2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (1.5\*2/3,3.2\*2/3) [2]{}; at (0.7\*2/3,0.7\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.7\*2/3,0.7\*2/3) [1]{}; at (2.8\*2/3,2.2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (2.8\*2/3,2.2\*2/3) [2]{}; at (0.2\*2/3,2.2\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.2\*2/3,2.2\*2/3) [1]{}; at (1.5\*2/3,5.6\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (1.5\*2/3,5.6\*2/3) [1]{}; at (4\*2/3+0.8,2\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (4\*2/3+0.8,2\*2/3) [1]{}; at (4.2\*2/3,0) \[circle,draw=blue!50,fill=blue!20\] ; at (4.2\*2/3,0) [1]{}; at (-1.5\*2/3,2.2\*2/3+0.9)\[ circle,draw=black,fill=black\] ; at (4.5\*2/3,2.2\*2/3+0.9)\[ circle,draw=black,fill=black\] ; at (-1.2\*2/3,0)\[ circle,draw=black,fill=black\] ; at (2.3\*2/3,0.7\*2/3)\[ circle,draw=black,fill=black\] ; at (-0.3,-0.3)[$u_{3}$]{}; at (-1\*2/3-0.4,2\*2/3)[$u_{2}$]{}; at (3\*2/3+0.3,-0.3)[$u_{4}$]{}; at (4\*2/3+0.4,2\*2/3-0.2)[$u_{5}$]{}; at (1.5\*2/3+0.4,4.4\*2/3+0.1)[$u_{1}$]{}; at (0.7\*2/3-0.4,0.7\*2/3+0.1)[$v_{3}$]{}; at (2.3\*2/3+0.4,0.7\*2/3+0.1)[$v_{4}$]{}; at (0.2\*2/3,2.2\*2/3+0.3)[$v_{2}$]{}; at (2.8\*2/3,2.2\*2/3+0.3)[$v_{5}$]{}; at (1.5\*2/3+0.4,3.2\*2/3)[$v_{1}$]{}; at (1.5\*2/3+0.4,5.6\*2/3)[$v'_{1}$]{}; at (-1.5\*2/3-0.5,2.2\*2/3+0.9)[$u'_{2}$]{}; at (4.5\*2/3+0.5,2.2\*2/3+0.9)[$u'_{5}$]{}; at (-1.2\*2/3-0.3,-0.3)[$v'_{3}$]{}; at (4.2\*2/3+0.3,-0.3)[$v'_{4}$]{}; at (4\*2/3+1.2,2\*2/3-0.2)[$v'_{5}$]{};
(0+6.3,0) – (3\*2/3+6.3,0); (0+6.3,0) – (-1\*2/3+6.3,2\*2/3); (3\*2/3+6.3,0) – (4\*2/3+6.3,2\*2/3); (-1\*2/3+6.3,2\*2/3) – (1.5\*2/3+6.3,4.4\*2/3); (4\*2/3+6.3,2\*2/3) – (1.5\*2/3+6.3,4.4\*2/3); (1.5\*2/3+6.3,3.2\*2/3) – (1.5\*2/3+6.3,4.4\*2/3); (-1\*2/3+6.3,2\*2/3) – (0.2\*2/3+6.3,2.2\*2/3); (4\*2/3+6.3,2\*2/3) – (2.8\*2/3+6.3,2.2\*2/3); (0+6.3,0) – (0.7\*2/3+6.3,0.7\*2/3); (3\*2/3+6.3,0) – (2.3\*2/3+6.3,0.7\*2/3); (0.7\*2/3+6.3,0.7\*2/3) – (2.8\*2/3+6.3,2.2\*2/3); (2.3\*2/3+6.3,0.7\*2/3) – (0.2\*2/3+6.3,2.2\*2/3); (0.7\*2/3+6.3,0.7\*2/3) – (1.5\*2/3+6.3,3.2\*2/3); (2.3\*2/3+6.3,0.7\*2/3) – (1.5\*2/3+6.3,3.2\*2/3); (1.5\*2/3+6.3,4.4\*2/3) – (1.5\*2/3+6.3,5.6\*2/3) ; (-1\*2/3+6.3,2\*2/3) – (1.5\*2/3+6.3,4.4\*2/3); (0.2\*2/3+6.3,2.2\*2/3) – (2.8\*2/3+6.3,2.2\*2/3); (0+6.3,0) – (-1.2\*2/3+6.3,0); (4.2\*2/3+6.3,0) – (3\*2/3+6.3,0); (0.2\*2/3+6.3,2.2\*2/3) – (-1.5\*2/3+6.3,2.2\*2/3+0.9); (2.8\*2/3+6.3,2.2\*2/3) – (4.5\*2/3+6.3,2.2\*2/3+0.9); (1.5\*2/3+6.3,5.6\*2/3) – (-1.5\*2/3+6.3,2.2\*2/3+0.9); (1.5\*2/3+6.3,5.6\*2/3) – (4.2\*2/3+6.3,0); (1.5\*2/3+6.3,5.6\*2/3) – (-1.2\*2/3+6.3,0); (4\*2/3+6.3,2\*2/3) – (4\*2/3+6.3+0.8,2\*2/3) ; at (0+6.3,0) \[circle,draw=blue!50,fill=blue!20\] ; at (0+6.3,0) [1]{}; at (-1\*2/3+6.3,2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (-1\*2/3+6.3,2\*2/3) [2]{}; at (3\*2/3+6.3,0) \[circle,draw=red!50,fill=red!20\] ; at (3\*2/3+6.3,0) [2]{}; at (4\*2/3+6.3,2\*2/3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (4\*2/3+6.3,2\*2/3)[4]{}; at (1.5\*2/3+6.3,4.4\*2/3) \[circle,draw=black!50,fill=black!20\] ; at (1.5\*2/3+6.3,4.4\*2/3) [5]{}; at (1.5\*2/3+6.3,3.2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (1.5\*2/3+6.3,3.2\*2/3) [3]{}; at (0.7\*2/3+6.3,0.7\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (0.7\*2/3+6.3,0.7\*2/3) [2]{}; at (2.8\*2/3+6.3,2.2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (2.8\*2/3+6.3,2.2\*2/3) [3]{}; at (2.3\*2/3+6.3,0.7\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.3\*2/3+6.3,0.7\*2/3) [1]{}; at (1.5\*2/3+6.3,5.6\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (1.5\*2/3+6.3,5.6\*2/3) [1]{}; at (4.5\*2/3+6.3,2.2\*2/3+0.9) \[circle,draw=blue!50,fill=blue!20\] ; at (4.5\*2/3+6.3,2.2\*2/3+0.9) [1]{}; at (4\*2/3+6.3+0.8,2\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (4\*2/3+6.3+0.8,2\*2/3) [1]{}; at (-1.5\*2/3+6.3,2.2\*2/3+0.9) \[circle,draw=black,fill=black\] ; at (4.2\*2/3+6.3,0) \[circle,draw=black,fill=black\] ; at (-1.2\*2/3+6.3,0) \[circle,draw=black,fill=black\] ; at (0.2\*2/3+6.3,2.2\*2/3) \[circle,draw=black,fill=black\] ; at (-0.3+6.3,-0.3)[$u_{3}$]{}; at (-1\*2/3-0.4+6.3,2\*2/3)[$u_{2}$]{}; at (3\*2/3+0.3+6.3,-0.3)[$u_{4}$]{}; at (4\*2/3+0.4+6.3,2\*2/3-0.2)[$u_{5}$]{}; at (1.5\*2/3+0.4+6.3,4.4\*2/3+0.1)[$u_{1}$]{}; at (0.7\*2/3-0.4+6.3,0.7\*2/3+0.1)[$v_{3}$]{}; at (2.3\*2/3+0.4+6.3,0.7\*2/3+0.1)[$v_{4}$]{}; at (0.2\*2/3+6.3,2.2\*2/3+0.3)[$v_{2}$]{}; at (2.8\*2/3+6.3,2.2\*2/3+0.3)[$v_{5}$]{}; at (1.5\*2/3+6.3+0.4,3.2\*2/3)[$v_{1}$]{}; at (1.5\*2/3+6.3+0.4,5.6\*2/3)[$v'_{1}$]{}; at (-1.5\*2/3+6.3-0.5,2.2\*2/3+0.9)[$u'_{2}$]{}; at (4.5\*2/3+6.3+0.5,2.2\*2/3+0.9)[$u'_{5}$]{}; at (-1.2\*2/3-0.3+6.3,-0.3)[$v'_{3}$]{}; at (4.2\*2/3+0.3+6.3,-0.3)[$v'_{4}$]{}; at (4\*2/3+1.2+6.3,2\*2/3-0.2)[$v'_{5}$]{};
In a graph $G$, let a *neighbor-connected* $C_n$ be an $n$-cycle $C$ such that the set of vertices of $G$ at distance 1 from $C$ is not independent.
Let $G$ be a 4-regular graph with girth $g=5$. If $G$ contains a neighbor-connected $C_5$ as induced subgraph, then $\Gamma(G)=5$.
Let $C$ be a neighbor-connected $C_5$ in $G$. By Lemma \[lpet\] we can suppose that the neighbors of the vertices of $C$ do not form an induced $C_5$ (otherwise a Petersen would be an induced subgraph). Hence, we can assume that the neighbors of the vertices of $C$ form a subgraph of a $C_{10}$. If there are two edges between the neighbors of the vertices of $C$, then Figure \[figm52\] illustrates Grundy partial 5-colorings of the graph $G$. Suppose that two neighbors are adjacent, say $v_1$ and $v'_3$ and the graph $G$ does not contain the previous configuration. Note that $v'_3$ can be adjacent with $v'_1$ and $v'_5$. Let $w_1$, $w_2$ and $w_3$ be the three neighbors of $v_2$ different from $u_2$. We suppose that $w_1$ can be possibly adjacent with $v'_3$ and $w_2$ can be possibly adjacent with $v'_1$. Figure \[figm53\] illustrates a Grundy partial 5-coloring of $G$ in this case. In this figure, the vertex $w_3$ can be possibly adjacent with $v'_5$ or $v_4$, but in this case we can switch the color 1 from $v'_5$ to $v_5$ or from $v'_4$ to $v_4$.
(0,0) – (3\*2/3,0); (0,0) – (-1\*2/3,2\*2/3); (3\*2/3,0) – (4\*2/3,2\*2/3); (-1\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (4\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (1.8\*2/3,3.2\*2/3) – (1.5\*2/3,4.4\*2/3); (1.2\*2/3,3.2\*2/3) – (1.5\*2/3,4.4\*2/3); (-1\*2/3,2\*2/3) – (0.2\*2/3,2.5\*2/3); (-1\*2/3,2\*2/3) – (0.2\*2/3,1.9\*2/3); (4\*2/3,2\*2/3) – (2.8\*2/3,2.5\*2/3); (4\*2/3,2\*2/3) – (2.8\*2/3,1.9\*2/3); (0,0) – (0.5\*2/3,0.9\*2/3); (0,0) – (0.9\*2/3,0.5\*2/3); (3\*2/3,0) – (2.5\*2/3,0.9\*2/3); (3\*2/3,0) – (2.1\*2/3,0.5\*2/3); (1.8\*2/3,3.2\*2/3) – (0.9\*2/3,0.5\*2/3); (2.8\*2/3,1.9\*2/3) – (0.9\*2/3,0.5\*2/3); (1.2\*2/3,3.2\*2/3) – (0.5\*2/3,0.9\*2/3); (2.8\*2/3,2.5\*2/3) – (0.5\*2/3,0.9\*2/3); (0.2\*2/3,2.5\*2/3) – (2.8\*2/3,2.5\*2/3); (0.2\*2/3,1.9\*2/3) – (2.8\*2/3,1.9\*2/3); (0.2\*2/3,2.5\*2/3) – (2.5\*2/3,0.9\*2/3); (0.2\*2/3,1.9\*2/3) – (2.1\*2/3,0.5\*2/3); (1.8\*2/3,3.2\*2/3) – (2.5\*2/3,0.9\*2/3); (1.2\*2/3,3.2\*2/3) – (2.1\*2/3,0.5\*2/3); at (0,0) \[circle,draw=red!50,fill=red!20\] ; at (0,0) [2]{}; at (-1\*2/3,2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (-1\*2/3,2\*2/3) [3]{}; at (3\*2/3,0) \[circle,draw=green!50,fill=green!20\] ; at (3\*2/3,0) [3]{}; at (4\*2/3,2\*2/3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (4\*2/3,2\*2/3)[4]{}; at (1.5\*2/3,4.4\*2/3) \[circle,draw=black!50,fill=black!20\] ; at (1.5\*2/3,4.4\*2/3) [5]{}; at (1.8\*2/3,3.2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (1.8\*2/3,3.2\*2/3)[2]{}; at (1.2\*2/3,3.2\*2/3)\[circle,draw=blue!50,fill=blue!20\] ; at (1.2\*2/3,3.2\*2/3)[1]{}; at (0.9\*2/3,0.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.9\*2/3,0.5\*2/3) [1]{}; at (2.8\*2/3,1.9\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (2.8\*2/3,1.9\*2/3) [2]{}; at (2.8\*2/3,2.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.8\*2/3,2.5\*2/3) [1]{}; at (0.2\*2/3,1.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.2\*2/3,1.9\*2/3) [1]{}; at (2.5\*2/3,0.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.5\*2/3,0.9\*2/3) [1]{}; at (0.5\*2/3,0.9\*2/3)\[ circle,draw=black,fill=black\] ; at (2.1\*2/3,0.5\*2/3)\[ circle,draw=black,fill=black\] ; at (0.2\*2/3,2.5\*2/3)\[ circle,draw=black,fill=black\] ; at (-0.3,-0.3)[$u_{3}$]{}; at (-1\*2/3-0.4,2\*2/3)[$u_{2}$]{}; at (3\*2/3+0.3,-0.3)[$u_{4}$]{}; at (4\*2/3+0.4,2\*2/3)[$u_{5}$]{}; at (1.5\*2/3+0.4,4.4\*2/3+0.1)[$u_{1}$]{}; at (0.7\*2/3-0.45,0.7\*2/3+0.15)[$v_{3}$]{}; at (0.7\*2/3+0.5,0.7\*2/3-0.15)[$v'_{3}$]{}; at (2.3\*2/3+0.45,0.7\*2/3+0.15)[$v_{4}$]{}; at (2.3\*2/3-0.3,0.7\*2/3-0.3)[$v'_{4}$]{}; at (0.2\*2/3+0.15,2.2\*2/3+0.5)[$v_{2}$]{}; at (0.2\*2/3,2.2\*2/3-0.5)[$v'_{2}$]{}; at (2.8\*2/3,2.2\*2/3+0.5)[$v_{5}$]{}; at (2.8\*2/3,2.2\*2/3-0.5)[$v'_{5}$]{}; at (1.5\*2/3+0.6,3.2\*2/3)[$v_{1}$]{}; at (0.75\*2/3,3.2\*2/3) [$v'_{1}$]{};
(0+5,0) – (3\*2/3+5,0); (0+5,0) – (-1\*2/3+5,2\*2/3); (3\*2/3+5,0) – (4\*2/3+5,2\*2/3); (-1\*2/3+5,2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (4\*2/3+5,2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (1.8\*2/3+5,3.2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (1.2\*2/3+5,3.2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (-1\*2/3+5,2\*2/3) – (0.2\*2/3+5,2.5\*2/3); (-1\*2/3+5,2\*2/3) – (0.2\*2/3+5,1.9\*2/3); (4\*2/3+5,2\*2/3) – (2.8\*2/3+5,2.5\*2/3); (4\*2/3+5,2\*2/3) – (2.8\*2/3+5,1.9\*2/3); (5,0) – (0.5\*2/3+5,0.9\*2/3); (5,0) – (0.9\*2/3+5,0.5\*2/3); (3\*2/3+5,0) – (2.5\*2/3+5,0.9\*2/3); (3\*2/3+5,0) – (2.1\*2/3+5,0.5\*2/3); (1.8\*2/3+5,3.2\*2/3) – (0.9\*2/3+5,0.5\*2/3); (2.8\*2/3+5,1.9\*2/3) – (0.9\*2/3+5,0.5\*2/3);
(1.2\*2/3+5,3.2\*2/3) – (0.5\*2/3+5,0.9\*2/3); (2.8\*2/3+5,2.5\*2/3) – (0.5\*2/3+5,0.9\*2/3); (0.2\*2/3+5,2.5\*2/3) – (2.8\*2/3+5,2.5\*2/3); (0.2\*2/3+5,1.9\*2/3) – (2.8\*2/3+5,1.9\*2/3); (0.2\*2/3+5,2.5\*2/3) – (2.5\*2/3+5,0.9\*2/3); (0.2\*2/3+5,1.9\*2/3) – (2.1\*2/3+5,0.5\*2/3); (1.8\*2/3+5,3.2\*2/3) – (2.5\*2/3+5,0.9\*2/3); (1.2\*2/3+5,3.2\*2/3) – (2.1\*2/3+5,0.5\*2/3); at (0+5,0) \[circle,draw=red!50,fill=red!20\] ; at (0+5,0) [2]{}; at (-1\*2/3+5,2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (-1\*2/3+5,2\*2/3) [3]{}; at (3\*2/3+5,0) \[circle,draw=green!50,fill=green!20\] ; at (3\*2/3+5,0) [3]{}; at (4\*2/3+5,2\*2/3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (4\*2/3+5,2\*2/3)[4]{}; at (1.5\*2/3+5,4.4\*2/3) \[circle,draw=black!50,fill=black!20\] ; at (1.5\*2/3+5,4.4\*2/3) [5]{}; at (1.8\*2/3+5,3.2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (1.8\*2/3+5,3.2\*2/3)[2]{}; at (0.9\*2/3+5,0.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.9\*2/3+5,0.5\*2/3) [1]{}; at (2.8\*2/3+5,1.9\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (2.8\*2/3+5,1.9\*2/3) [2]{}; at (2.8\*2/3+5,2.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.8\*2/3+5,2.5\*2/3) [1]{}; at (0.2\*2/3+5,1.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.2\*2/3+5,1.9\*2/3) [1]{}; at (2.5\*2/3+5,0.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.5\*2/3+5,0.9\*2/3) [1]{}; at (1.2\*2/3+5,3.2\*2/3)\[circle,draw=blue!50,fill=blue!20\] ; at (1.2\*2/3+5,3.2\*2/3)[1]{}; at (0.5\*2/3+5,0.9\*2/3)\[ circle,draw=black,fill=black\] ; at (2.1\*2/3+5,0.5\*2/3)\[ circle,draw=black,fill=black\] ; at (0.2\*2/3+5,2.5\*2/3)\[ circle,draw=black,fill=black\] ; at (-0.3+5,-0.3)[$u_{3}$]{}; at (-1\*2/3-0.4+5,2\*2/3)[$u_{2}$]{}; at (3\*2/3+0.3+5,-0.3)[$u_{4}$]{}; at (4\*2/3+0.4+5,2\*2/3)[$u_{5}$]{}; at (1.5\*2/3+0.4+5,4.4\*2/3+0.1)[$u_{1}$]{}; at (0.7\*2/3-0.45+5,0.7\*2/3+0.15)[$v_{3}$]{}; at (0.7\*2/3+0.5+5,0.7\*2/3-0.15)[$v'_{3}$]{}; at (2.3\*2/3+0.45+5,0.7\*2/3+0.15)[$v_{4}$]{}; at (2.3\*2/3-0.3+5,0.7\*2/3-0.3)[$v'_{4}$]{}; at (0.2\*2/3+5+0.15,2.2\*2/3+0.5)[$v_{2}$]{}; at (0.2\*2/3+5,2.2\*2/3-0.5)[$v'_{2}$]{}; at (2.8\*2/3+5,2.2\*2/3+0.5)[$v_{5}$]{}; at (2.8\*2/3+5,2.2\*2/3-0.5)[$v'_{5}$]{}; at (1.5\*2/3+0.6+5,3.2\*2/3)[$v_{1}$]{}; at (0.75\*2/3+5,3.2\*2/3) [$v'_{1}$]{};
(0,0) – (3\*2/3,0); (0,0) – (-1\*2/3,2\*2/3); (3\*2/3,0) – (4\*2/3,2\*2/3); (-1\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (4\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (1.8\*2/3,3.2\*2/3) – (1.5\*2/3,4.4\*2/3); (1.2\*2/3,3.2\*2/3) – (1.5\*2/3,4.4\*2/3); (-1\*2/3,2\*2/3) – (0.2\*2/3,2.5\*2/3); (-1\*2/3,2\*2/3) – (0.2\*2/3,1.9\*2/3); (4\*2/3,2\*2/3) – (2.8\*2/3,2.5\*2/3); (4\*2/3,2\*2/3) – (2.8\*2/3,1.9\*2/3); (0,0) – (0.5\*2/3,0.9\*2/3); (0,0) – (0.9\*2/3,0.5\*2/3); (3\*2/3,0) – (2.5\*2/3,0.9\*2/3); (3\*2/3,0) – (2.1\*2/3,0.5\*2/3); (1.8\*2/3,3.2\*2/3) – (0.9\*2/3,0.5\*2/3); (1.2\*2/3,3.2\*2/3) – (0.5\*2/3,0.9\*2/3); (2.8\*2/3,1.9\*2/3) – (0.5\*2/3,0.9\*2/3); (0.2\*2/3,2.5\*2/3) – (-1.2\*2/3,3.5\*2/3); (0.2\*2/3,2.5\*2/3) – (-1.2\*2/3,4.4\*2/3); (0.2\*2/3,2.5\*2/3) – (-1.2\*2/3,0.5\*2/3); (-1.2\*2/3,0.5\*2/3) – (0.9\*2/3,0.5\*2/3); (-1.2\*2/3,4.4\*2/3) – (1.2\*2/3,3.2\*2/3); at (3\*2/3,0) \[circle,draw=red!50,fill=red!20\] ; at (3\*2/3,0) [2]{}; at (4\*2/3,2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (4\*2/3,2\*2/3) [3]{}; at (0,0) \[circle,draw=green!50,fill=green!20\] ; at (0,0) [3]{}; at (-1\*2/3,2\*2/3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (-1\*2/3,2\*2/3) [4]{}; at (1.5\*2/3,4.4\*2/3) \[circle,draw=black!50,fill=black!20\] ; at (1.5\*2/3,4.4\*2/3) [5]{}; at (1.8\*2/3,3.2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (1.8\*2/3,3.2\*2/3)[2]{}; at (0.9\*2/3,0.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.9\*2/3,0.5\*2/3) [1]{}; at (0.2\*2/3,2.5\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (0.2\*2/3,2.5\*2/3) [2]{}; at (2.8\*2/3,2.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.8\*2/3,2.5\*2/3) [1]{}; at (0.2\*2/3,1.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.2\*2/3,1.9\*2/3) [1]{}; at (2.5\*2/3,0.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.5\*2/3,0.9\*2/3) [1]{}; at (1.2\*2/3,3.2\*2/3)\[circle,draw=blue!50,fill=blue!20\] ; at (1.2\*2/3,3.2\*2/3)[1]{}; at (-1.2\*2/3,3.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (-1.2\*2/3,3.5\*2/3) [1]{}; at (2.8\*2/3,1.9\*2/3) \[ circle,draw=black,fill=black\] ; at (0.5\*2/3,0.9\*2/3) \[ circle,draw=black,fill=black\] ; at (2.2\*2/3,0.5\*2/3) \[ circle,draw=black,fill=black\] ; at (-1.2\*2/3,0.5\*2/3) \[ circle,draw=black,fill=black\] ; at (-1.2\*2/3,4.4\*2/3) \[ circle,draw=black,fill=black\] ; at (-0.3,-0.3)[$u_{3}$]{}; at (-1\*2/3-0.4,2\*2/3)[$u_{2}$]{}; at (3\*2/3+0.3,-0.3)[$u_{4}$]{}; at (4\*2/3+0.4,2\*2/3)[$u_{5}$]{}; at (1.5\*2/3+0.4,4.4\*2/3+0.1)[$u_{1}$]{}; at (0.7\*2/3-0.45,0.7\*2/3+0.15)[$v_{3}$]{}; at (0.7\*2/3+0.5,0.7\*2/3-0.15)[$v'_{3}$]{}; at (2.3\*2/3+0.45,0.7\*2/3+0.15)[$v_{4}$]{}; at (2.3\*2/3-0.3,0.7\*2/3-0.3)[$v'_{4}$]{}; at (0.2\*2/3+0.15,2.2\*2/3+0.4)[$v_{2}$]{}; at (0.2\*2/3,2.2\*2/3-0.5)[$v'_{2}$]{}; at (2.8\*2/3,2.2\*2/3+0.5)[$v_{5}$]{}; at (2.8\*2/3,2.2\*2/3-0.5)[$v'_{5}$]{}; at (1.5\*2/3+0.6,3.2\*2/3)[$v_{1}$]{}; at (0.75\*2/3,3.2\*2/3) [$v'_{1}$]{}; at (-1.8\*2/3,4.4\*2/3) [$w_{2}$]{}; at (-1.8\*2/3,0.5\*2/3) [$w_{1}$]{}; at (-1.8\*2/3,3.5\*2/3) [$w_{3}$]{};
If $G$ is a 4-regular graph with girth $g=5$, then $\Gamma(G)=5$. \[cycle5\]
Let $C$ be a 5-cycle in $G$. Assume that two neighbors of consecutive vertices of $C$, for example $v_1$ and $v_5$, have a common neighbor $w_1$. The left part of Figure \[figm54\] illustrates a Grundy partial 5-coloring of the graph $G$. In this figure the vertex $w_1$ can be possibly adjacent with $v'_2$, $v'_3$ or $v_4$, but in this case we can switch the color 1 from $v'_2$ to $v_2$, from $v'_3$ to $v_3$ or from $v_4$ to $v'_4$. Hence, we can suppose that no neighbors of consecutive vertices of $C$ are adjacent. Among the neighbors of $v_1$, there exists one vertex $w_1$ not adjacent with both $v_4$ and $v'_4$ (otherwise $G$ would contain a $C_4$). Among the neighbor of $v'_5$, there exists one vertex, say $w_2$, not adjacent with $w_1$. The right part of Figure \[figm54\] illustrates a Grundy partial 5-coloring of the graph $G$. In this figure the vertex $w_1$ can be possibly adjacent with $v_4$ and the vertex $w_2$ can be possibly adjacent with $v'_2$ or $v_4$, but in these cases we can switch the color 1 from $v'_2$ to $v_2$ or from $v_4$ to $v'_4$.
(0,0) – (3\*2/3,0); (0,0) – (-1\*2/3,2\*2/3); (3\*2/3,0) – (4\*2/3,2\*2/3); (-1\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (4\*2/3,2\*2/3) – (1.5\*2/3,4.4\*2/3); (1.8\*2/3,3.2\*2/3) – (1.5\*2/3,4.4\*2/3); (1.2\*2/3,3.2\*2/3) – (1.5\*2/3,4.4\*2/3); (-1\*2/3,2\*2/3) – (0.2\*2/3,2.5\*2/3); (-1\*2/3,2\*2/3) – (0.2\*2/3,1.9\*2/3); (4\*2/3,2\*2/3) – (2.8\*2/3,2.5\*2/3); (4\*2/3,2\*2/3) – (2.8\*2/3,1.9\*2/3); (0,0) – (0.5\*2/3,0.9\*2/3); (0,0) – (0.9\*2/3,0.5\*2/3); (3\*2/3,0) – (2.5\*2/3,0.9\*2/3); (3\*2/3,0) – (2.1\*2/3,0.5\*2/3); (2.8\*2/3,2.5\*2/3) – (3.8\*2/3,4.4\*2/3); (1.8\*2/3,3.2\*2/3) – (3.8\*2/3,4.4\*2/3); at (0,0) \[circle,draw=red!50,fill=red!20\] ; at (0,0) [2]{}; at (-1\*2/3,2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (-1\*2/3,2\*2/3) [3]{}; at (3\*2/3,0) \[circle,draw=green!50,fill=green!20\] ; at (3\*2/3,0) [3]{}; at (4\*2/3,2\*2/3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (4\*2/3,2\*2/3)[4]{}; at (1.5\*2/3,4.4\*2/3) \[circle,draw=black!50,fill=black!20\] ; at (1.5\*2/3,4.4\*2/3) [5]{}; at (1.8\*2/3,3.2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (1.8\*2/3,3.2\*2/3)[2]{}; at (0.9\*2/3,0.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.9\*2/3,0.5\*2/3) [1]{}; at (2.8\*2/3,2.5\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (2.8\*2/3,2.5\*2/3) [2]{}; at (2.8\*2/3,1.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.8\*2/3,1.9\*2/3) [1]{}; at (0.2\*2/3,1.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.2\*2/3,1.9\*2/3) [1]{}; at (2.5\*2/3,0.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.5\*2/3,0.9\*2/3) [1]{}; at (1.2\*2/3,3.2\*2/3)\[circle,draw=blue!50,fill=blue!20\] ; at (1.2\*2/3,3.2\*2/3)[1]{}; at (3.8\*2/3,4.4\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (3.8\*2/3,4.4\*2/3)[1]{}; at (0.2\*2/3,2.5\*2/3) \[ circle,draw=black,fill=black\] ; at (0.5\*2/3,0.9\*2/3) \[ circle,draw=black,fill=black\] ; at (2.2\*2/3,0.5\*2/3) \[ circle,draw=black,fill=black\] ; at (-0.3,-0.3)[$u_{3}$]{}; at (-1\*2/3-0.4,2\*2/3)[$u_{2}$]{}; at (3\*2/3+0.3,-0.3)[$u_{4}$]{}; at (4\*2/3+0.4,2\*2/3)[$u_{5}$]{}; at (1.5\*2/3+0.4,4.4\*2/3+0.1)[$u_{1}$]{}; at (0.7\*2/3-0.45,0.7\*2/3+0.15)[$v_{3}$]{}; at (0.7\*2/3+0.5,0.7\*2/3-0.15)[$v'_{3}$]{}; at (2.3\*2/3+0.45,0.7\*2/3+0.15)[$v_{4}$]{}; at (2.3\*2/3-0.3,0.7\*2/3-0.3)[$v'_{4}$]{}; at (0.2\*2/3+0.1,2.2\*2/3+0.5)[$v_{2}$]{}; at (0.2\*2/3,2.2\*2/3-0.5)[$v'_{2}$]{}; at (2.8\*2/3,2.2\*2/3+0.5)[$v_{5}$]{}; at (2.8\*2/3,2.2\*2/3-0.5)[$v'_{5}$]{}; at (1.5\*2/3+0.6,3.2\*2/3)[$v_{1}$]{}; at (0.75\*2/3,3.2\*2/3) [$v'_{1}$]{}; at (3.8\*2/3+0.4,4.4\*2/3) [$w_{1}$]{};
(0+5,0) – (3\*2/3+5,0); (0+5,0) – (-1\*2/3+5,2\*2/3); (3\*2/3+5,0) – (4\*2/3+5,2\*2/3); (-1\*2/3+5,2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (4\*2/3+5,2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (1.8\*2/3+5,3.2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (1.2\*2/3+5,3.2\*2/3) – (1.5\*2/3+5,4.4\*2/3); (-1\*2/3+5,2\*2/3) – (0.2\*2/3+5,2.5\*2/3); (-1\*2/3+5,2\*2/3) – (0.2\*2/3+5,1.9\*2/3); (4\*2/3+5,2\*2/3) – (2.8\*2/3+5,2.5\*2/3); (4\*2/3+5,2\*2/3) – (2.8\*2/3+5,1.9\*2/3); (0+5,0) – (0.5\*2/3+5,0.9\*2/3); (0+5,0) – (0.9\*2/3+5,0.5\*2/3); (3\*2/3+5,0) – (2.5\*2/3+5,0.9\*2/3); (3\*2/3+5,0) – (2.1\*2/3+5,0.5\*2/3); (1.8\*2/3+5,3.2\*2/3) – (3.8\*2/3+4.2,4.4\*2/3); (1.8\*2/3+5,3.2\*2/3)– (1.8\*2/3+4.6,2\*2/3); (1.8\*2/3+5,3.2\*2/3)– (1.8\*2/3+5,2\*2/3); (2.8\*2/3+5,1.9\*2/3) – (3.8\*2/3+5,4.4\*2/3-0.1); (2.8\*2/3+5,1.9\*2/3) – (3.8\*2/3+5,4.4\*2/3-0.6); (2.8\*2/3+5,1.9\*2/3) – (3.8\*2/3+5,4.4\*2/3-1.1);
(3.8\*2/3+4.2,4.4\*2/3) – (3.8\*2/3+5,4.4\*2/3-0.1); (0.5\*2/3+5,0.9\*2/3) – (1.8\*2/3+4.6,2\*2/3); (0.9\*2/3+5,0.5\*2/3)– (1.8\*2/3+5,2\*2/3);
at (0+5,0) \[circle,draw=red!50,fill=red!20\] ; at (0+5,0) [2]{}; at (-1\*2/3+5,2\*2/3) \[circle,draw=green!50,fill=green!20\] ; at (-1\*2/3+5,2\*2/3) [3]{}; at (3\*2/3+5,0) \[circle,draw=green!50,fill=green!20\] ; at (3\*2/3+5,0) [3]{}; at (4\*2/3+5,2\*2/3) \[circle,draw=yellow!50,fill=yellow!20\] ; at (4\*2/3+5,2\*2/3)[4]{}; at (1.5\*2/3+5,4.4\*2/3) \[circle,draw=black!50,fill=black!20\] ; at (1.5\*2/3+5,4.4\*2/3) [5]{}; at (1.8\*2/3+5,3.2\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (1.8\*2/3+5,3.2\*2/3)[2]{}; at (0.9\*2/3+5,0.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.9\*2/3+5,0.5\*2/3) [1]{}; at (2.8\*2/3+5,1.9\*2/3) \[circle,draw=red!50,fill=red!20\] ; at (2.8\*2/3+5,1.9\*2/3) [2]{}; at (2.8\*2/3+5,2.5\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.8\*2/3+5,2.5\*2/3) [1]{}; at (0.2\*2/3+5,1.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (0.2\*2/3+5,1.9\*2/3) [1]{}; at (2.5\*2/3+5,0.9\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (2.5\*2/3+5,0.9\*2/3) [1]{}; at (1.2\*2/3+5,3.2\*2/3)\[circle,draw=blue!50,fill=blue!20\] ; at (1.2\*2/3+5,3.2\*2/3)[1]{}; at (3.8\*2/3+4.2,4.4\*2/3) \[circle,draw=blue!50,fill=blue!20\] ; at (3.8\*2/3+4.2,4.4\*2/3)[1]{}; at (3.8\*2/3+5,4.4\*2/3-1.1) \[circle,draw=blue!50,fill=blue!20\] ; at (3.8\*2/3+5,4.4\*2/3-1.1)[1]{}; at (0.2\*2/3+5,2.5\*2/3) \[ circle,draw=black,fill=black\] ; at (0.5\*2/3+5,0.9\*2/3) \[ circle,draw=black,fill=black\] ; at (2.2\*2/3+5,0.5\*2/3) \[ circle,draw=black,fill=black\] ; at (1.8\*2/3+4.6,2\*2/3) \[ circle,draw=black,fill=black\] ; at (1.8\*2/3+5,2\*2/3) \[ circle,draw=black,fill=black\] ; at (3.8\*2/3+5,4.4\*2/3-0.1) \[ circle,draw=black,fill=black\] ; at (3.8\*2/3+5,4.4\*2/3-0.6) \[ circle,draw=black,fill=black\] ; at (-0.3+5,-0.3)[$u_{3}$]{}; at (-1\*2/3-0.4+5,2\*2/3)[$u_{2}$]{}; at (3\*2/3+0.3+5,-0.3)[$u_{4}$]{}; at (4\*2/3+0.4+5,2\*2/3)[$u_{5}$]{}; at (1.5\*2/3+0.4+5,4.4\*2/3+0.1)[$u_{1}$]{}; at (0.7\*2/3-0.45+5,0.7\*2/3+0.15)[$v_{3}$]{}; at (0.7\*2/3+0.5+5,0.7\*2/3-0.15)[$v'_{3}$]{}; at (2.3\*2/3+0.45+5,0.7\*2/3+0.15)[$v_{4}$]{}; at (2.3\*2/3-0.3+5,0.7\*2/3-0.3)[$v'_{4}$]{}; at (0.2\*2/3+5+0.1,2.2\*2/3+0.5)[$v_{2}$]{}; at (0.2\*2/3+5,2.2\*2/3-0.5)[$v'_{2}$]{}; at (2.8\*2/3+4.7,2.2\*2/3+0.2)[$v_{5}$]{}; at (2.8\*2/3+5,2.2\*2/3-0.5)[$v'_{5}$]{}; at (1.5\*2/3+5+0.6,3.2\*2/3)[$v_{1}$]{}; at (0.75\*2/3+5,3.2\*2/3) [$v'_{1}$]{}; at (4\*2/3+0.4+5,4.4\*2/3-1.1)[$w_{2}$]{}; at (4\*2/3+0.2+4.2,4.4\*2/3-0.2)[$w_{1}$]{};
In the following lemma and proposition, we consider a graph $G$ of girth $g=6$. Let $u_1$, $u_2$, $u_3$, $u_4$, $u_5$ and $u_6$ be the vertices in an induced $C_6$. Let $v_1$, $v'_1$, $v_2$, $v'_2$, $v_3$, $v'_3$, $v_4$, $v'_4$, $v_5$, $v'_5$, $v_6$ and $v'_6$ be the remaining neighbors of respectively $u_1$, $u_2$, $u_3$, $u_4$, $u_5$ and $u_6$ (all different as $g=6$).
If $G$ is a 4-regular graph with girth $g=6$ which contains a neighbor-connected $C_6$ as induced subgraph, then $\Gamma(G)=5$. \[lnc6\]
Firstly, suppose that there are two edges which connect the neighbors in the same way than in the left part of Figure \[figm61\]. Let $w_1$ be a neighbor of $v'_1$ not adjacent with $v_4$. The graph $G$ admits a Grundy partial 5-coloring as the left part of Figure \[figm61\] illustrates it. Secondly, suppose that there is one edge (or more) which connect the neighbors without the configuration from the previous case. Let $w_1$ be a neighbor of $v_3$ not adjacent with $v_2$ and let $w_2$ be a neighbor of $v'_1$ not adjacent with $w_1$. The graph $G$ admits a Grundy partial 5-coloring as the right part of Figure \[figm61\] illustrates it.
(0,0) – (-1.2,1.2); (0,0) – (1.2,1.2); (1.2,1.2) – (1.2,2.6); (-1.2,1.2) – (-1.2,2.6); (1.2,1.2) – (1.2,2.6); (1.2,2.6) – (0,4); (-1.2,2.6) – (0,4); (0,0) – (0.2,0.7); (0,0) – (-0.2,0.7); (0,4) – (0.2,3.3); (0,4) – (-0.2,3.3); (1.2,1.2) – (0.5,1.2); (1.2,1.2) – (0.5,1.6); (-1.2,1.2) – (-0.5,1.2); (-1.2,1.2) – (-0.5,1.6); (1.2,2.6) – (0.5,2.6); (1.2,2.6) – (0.5,2.2); (-1.2,2.6) – (-0.5,2.6); (-1.2,2.6) – (-0.5,2.2);
(0.5,1.2) – (-0.5,2.2); (-0.5,1.2) – (0.5,2.2); (-0.5,1.6)–(0.5,2.6); (0.5,1.6) – (-0.5,2.6); (-0.2,0.7)–(-0.2,3.3); (0.2,0.7)–(0.2,3.3);
(0.2,3.3) – (0.7,4); (0.2,3.3) – (1.2,4); (-0.5,1.2)–(0.7,4); (0.5,1.2) .. controls (2.4,2.6) .. (1.2,4);
at (0,0)\[circle,draw=blue!50,fill=blue!20\] ; at (0,0) [1]{}; at (-1.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (-1.2,1.2) [2]{}; at (-1.2,2.6) \[circle,draw=green!50,fill=green!20\] ; at (-1.2,2.6) [3]{}; at (1.2,1.2) \[circle,draw=green!50,fill=green!20\] ; at (1.2,1.2) [3]{}; at (1.2,2.6)\[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2,2.6) [4]{}; at (0,4) \[circle,draw=black!50,fill=black!20\] ; at (0,4) [5]{};
at (0.5,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5,1.2) [2]{}; at (0.5,2.2)\[circle,draw=red!50,fill=red!20\] ; at(0.5,2.2) [2]{}; at (0.5,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5,1.2) [2]{}; at (0.5,2.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5,2.2) [2]{}; at (0.2,3.3)\[circle,draw=red!50,fill=red!20\] ; at (0.2,3.3) [2]{}; at (-0.5,2.2)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.5,2.2) [1]{}; at (-0.5,1.2)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.5,1.2) [1]{}; at (-0.2,3.3)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.2,3.3) [1]{}; at (0.5,2.6)\[circle,draw=blue!50,fill=blue!20\] ; at (0.5,2.6) [1]{}; at (1.2,4) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2,4) [1]{}; at (-0.5,1.6) \[ circle,draw=black,fill=black\] ; at (0.5,1.6) \[ circle,draw=black,fill=black\] ; at (-0.5,2.6) \[ circle,draw=black,fill=black\] ; at (0.2,0.7) \[ circle,draw=black,fill=black\] ; at (-0.2,0.7) \[ circle,draw=black,fill=black\] ; at (0.7,4) \[ circle,draw=black,fill=black\] ; at (0.4,0) [$u_6$]{}; at (-0.5,0.7) [$v_6$]{}; at (0.45,0.7) [$v'_6$]{}; at (-1.6,1.2) [$u_4$]{}; at (-0.5,0.9) [$v_4$]{}; at (-0.8,1.8) [$v'_4$]{}; at (-1.6,2.6) [$u_2$]{}; at (-0.8,2.2) [$v_2$]{}; at (-0.8,2.8) [$v'_2$]{};
at (1.6,1.2) [$u_5$]{}; at (0.5,0.9) [$v_5$]{}; at (0.8,1.8) [$v'_5$]{}; at (1.6,2.6) [$u_3$]{}; at (0.8,2.2) [$v_3$]{}; at (0.8,2.8) [$v'_3$]{}; at (0.4,4) [$u_1$]{}; at (-0.5,3.2) [$v_1$]{}; at (0.55,3.2) [$v'_1$]{}; at (1.6,4) [$w_1$]{};
(0+4.2,0) – (-1.2+4.2,1.2); (0+4.2,0) – (1.2+4.2,1.2); (1.2+4.2,1.2) – (1.2+4.2,2.6); (-1.2+4.2,1.2) – (-1.2+4.2,2.6); (1.2+4.2,1.2) – (1.2+4.2,2.6); (1.2+4.2,2.6) – (0+4.2,4); (-1.2+4.2,2.6) – (0+4.2,4);
(0+4.2,0) – (0.2+4.2,0.7); (0+4.2,0) – (-0.2+4.2,0.7); (0+4.2,4) – (0.2+4.2,3.3); (0+4.2,4) – (-0.2+4.2,3.3); (1.2+4.2,1.2) – (0.5+4.2,1.2); (1.2+4.2,1.2) – (0.5+4.2,1.6); (-1.2+4.2,1.2) – (-0.5+4.2,1.2); (-1.2+4.2,1.2) – (-0.5+4.2,1.6); (1.2+4.2,2.6) – (0.5+4.2,2.6); (1.2+4.2,2.6) – (0.5+4.2,2.2); (-1.2+4.2,2.6) – (-0.5+4.2,2.6); (-1.2+4.2,2.6) – (-0.5+4.2,2.2);
(0.5+4.2,1.2) – (-0.5+4.2,2.2); (0.5+4.2,1.6) – (-0.5+4.2,2.6);
(0.2+4.2,3.3) – (0.7+4.2,4); (0.2+4.2,3.3) – (1.2+4.2,4); (0.5+4.2,2.2) – (1.2+4.2,3.2); (0.5+4.2,2.2) – (0+4.2,2.7); (-0.5+4.2,2.2) – (0+4.2,2.7); (1.2+4.2,3.2) – (1.2+4.2,4); (0.5+4.2,1.2) .. controls (2.4+4.2,3) .. (0.7+4.2,4);
at (0+4.2,0)\[circle,draw=blue!50,fill=blue!20\] ; at (0+4.2,0) [1]{}; at (-1.2+4.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (-1.2+4.2,1.2) [2]{}; at (-1.2+4.2,2.6) \[circle,draw=green!50,fill=green!20\] ; at (-1.2+4.2,2.6) [3]{}; at (1.2+4.2,1.2) \[circle,draw=green!50,fill=green!20\] ; at (1.2+4.2,1.2) [3]{}; at (1.2+4.2,2.6)\[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2+4.2,2.6) [4]{}; at (0+4.2,4) \[circle,draw=black!50,fill=black!20\] ; at (0+4.2,4) [5]{}; at (0.5+4.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5+4.2,1.2) [2]{}; at (0.5+4.2,2.2)\[circle,draw=red!50,fill=red!20\] ; at(0.5+4.2,2.2) [2]{}; at (0.5+4.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5+4.2,1.2) [2]{}; at (0.5+4.2,2.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5+4.2,2.2) [2]{}; at (0.2+4.2,3.3)\[circle,draw=red!50,fill=red!20\] ; at (0.2+4.2,3.3) [2]{}; at (-0.5+4.2,2.2)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.5+4.2,2.2) [1]{}; at (-0.2+4.2,3.3)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.2+4.2,3.3) [1]{}; at (0.5+4.2,2.6)\[circle,draw=blue!50,fill=blue!20\] ; at (0.5+4.2,2.6) [1]{}; at (0.7+4.2,4) \[circle,draw=blue!50,fill=blue!20\] ; at (0.7+4.2,4) [1]{}; at (1.2+4.2,3.2) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+4.2,3.2) [1]{}; at (-0.5+4.2,1.6) \[ circle,draw=black,fill=black\] ; at (0.5+4.2,1.6) \[ circle,draw=black,fill=black\] ; at (-0.5+4.2,2.6) \[ circle,draw=black,fill=black\] ; at (-0.5+4.2,1.2) \[ circle,draw=black,fill=black\] ; at (0+4.2,2.7) \[ circle,draw=black,fill=black\] ; at (1.2+4.2,4) \[ circle,draw=black,fill=black\] ; at (0.2+4.2,0.7) \[ circle,draw=black,fill=black\] ; at (-0.2+4.2,0.7) \[ circle,draw=black,fill=black\] ; at (0.4+4.2,0) [$u_6$]{}; at (-0.5+4.2,0.7) [$v_6$]{}; at (0.45+4.2,0.7) [$v'_6$]{}; at (-1.6+4.2,1.2) [$u_4$]{}; at (-0.5+4.2,0.9) [$v_4$]{}; at (-0.8+4.2,1.8) [$v'_4$]{}; at (-1.6+4.2,2.6) [$u_2$]{}; at (-0.8+4.2,2.2) [$v_2$]{}; at (-0.8+4.2,2.8) [$v'_2$]{};
at (1.6+4.2,1.2) [$u_5$]{}; at (0.5+4.2,0.9) [$v_5$]{}; at (0.8+4.2,1.8) [$v'_5$]{}; at (1.6+4.2,2.6) [$u_3$]{}; at (0.8+4.2,2.2) [$v_3$]{}; at (0.8+4.2,2.8) [$v'_3$]{}; at (0.4+4.2,4) [$u_1$]{}; at (-0.5+4.2,3.2) [$v_1$]{}; at (0.55+4.2,3.2) [$v'_1$]{}; at (1.6+4.2,3.2) [$w_1$]{}; at (0.9+4.2,4.2) [$w_2$]{};
If $G$ is a 4-regular graph with girth $g=6$, then $\Gamma(G)=5$. \[cycle6\]
By Lemma \[lnc6\], assume that no neighbors of the vertices of the induced $C_6$ are adjacent. Firstly, suppose that there are two neighbors at distance 4 along the cycle $C_6$, for example $v'_1$ and $v_5$, which have a common neighbor $w_1$. Let $w_2$ be a neighbor of $v_3$ not adjacent with $w_1$. $G$ admits a Grundy partial 5-coloring as the left part of Figure \[figm62\] illustrates it. Secondly, suppose that there are no two neighbors at distance 4 along the cycle $C_6$ which have a common neighbor. Let $w_1$ be a neighbor of $v'_1$ not adjacent with a neighbor of $v_5$ or a neighbor of $v_3$, let $w_2$ be a neighbor of $v_3$ not adjacent with a neighbor of $v_5$, and let $w_3$ be a neighbor of $v_5$. The graph $G$ admits a Grundy partial 5-coloring as the right part of Figure \[figm62\] illustrates it.
(0,0) – (-1.2,1.2); (0,0) – (1.2,1.2); (1.2,1.2) – (1.2,2.6); (-1.2,1.2) – (-1.2,2.6); (1.2,1.2) – (1.2,2.6); (1.2,2.6) – (0,4); (-1.2,2.6) – (0,4);
(0,0) – (0.2,0.7); (0,0) – (-0.2,0.7); (0,4) – (0.2,3.3); (0,4) – (-0.2,3.3); (1.2,1.2) – (0.5,1.2); (1.2,1.2) – (0.5,1.6); (-1.2,1.2) – (-0.5,1.2); (-1.2,1.2) – (-0.5,1.6); (1.2,2.6) – (0.5,2.6); (1.2,2.6) – (0.5,2.2); (-1.2,2.6) – (-0.5,2.6); (-1.2,2.6) – (-0.5,2.2);
(0.2,3.3) – (1.2,4); (0.5,1.2) – (1.2,4); (0.5,2.2) – (1.2,3.35); (0.5,2.2) – (1.2,3); (1.2,4) – (1.2,3.35);
at (0,0)\[circle,draw=blue!50,fill=blue!20\] ; at (0,0) [1]{}; at (-1.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (-1.2,1.2) [2]{}; at (-1.2,2.6) \[circle,draw=green!50,fill=green!20\] ; at (-1.2,2.6) [3]{}; at (1.2,1.2) \[circle,draw=green!50,fill=green!20\] ; at (1.2,1.2) [3]{}; at (1.2,2.6)\[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2,2.6) [4]{}; at (0,4) \[circle,draw=black!50,fill=black!20\] ; at (0,4) [5]{};
at (0.5,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5,1.2) [2]{}; at (0.5,2.2)\[circle,draw=red!50,fill=red!20\] ; at(0.5,2.2) [2]{}; at (0.5,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5,1.2) [2]{}; at (0.5,2.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5,2.2) [2]{}; at (0.2,3.3)\[circle,draw=red!50,fill=red!20\] ; at (0.2,3.3) [2]{}; at (-0.5,2.2)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.5,2.2) [1]{}; at (-0.2,3.3)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.2,3.3) [1]{}; at (0.5,2.6)\[circle,draw=blue!50,fill=blue!20\] ; at (0.5,2.6) [1]{}; at (1.2,4) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2,4) [1]{}; at (1.2,3) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2,3) [1]{}; at (-0.5,1.6) \[ circle,draw=black,fill=black\] ; at (0.5,1.6) \[ circle,draw=black,fill=black\] ; at (-0.5,1.2) \[ circle,draw=black,fill=black\] ; at (-0.5,2.6) \[ circle,draw=black,fill=black\] ; at (0.2,0.7) \[ circle,draw=black,fill=black\] ; at (-0.2,0.7) \[ circle,draw=black,fill=black\] ; at (1.2,3.35) \[ circle,draw=black,fill=black\] ; at (0.4,0) [$u_6$]{}; at (-0.5,0.7) [$v_6$]{}; at (0.45,0.7) [$v'_6$]{}; at (-1.6,1.2) [$u_4$]{}; at (-0.5,0.9) [$v_4$]{}; at (-0.8,1.8) [$v'_4$]{}; at (-1.6,2.6) [$u_2$]{}; at (-0.8,2.2) [$v_2$]{}; at (-0.8,2.8) [$v'_2$]{};
at (1.6,1.2) [$u_5$]{}; at (0.8,0.9) [$v_5$]{}; at (0.8,1.8) [$v'_5$]{}; at (1.6,2.6) [$u_3$]{}; at (0.8,2.2) [$v_3$]{}; at (0.5,2.8) [$v'_3$]{}; at (0.4,4) [$u_1$]{}; at (-0.5,3.2) [$v_1$]{}; at (0.55,3.2) [$v'_1$]{}; at (1.6,4) [$w_1$]{}; at (1.6,3) [$w_2$]{};
(0+4.2,0) – (-1.2+4.2,1.2); (0+4.2,0) – (1.2+4.2,1.2); (1.2+4.2,1.2) – (1.2+4.2,2.6); (-1.2+4.2,1.2) – (-1.2+4.2,2.6); (1.2+4.2,1.2) – (1.2+4.2,2.6); (1.2+4.2,2.6) – (0+4.2,4); (-1.2+4.2,2.6) – (0+4.2,4);
(0+4.2,0) – (0.2+4.2,0.7); (0+4.2,0) – (-0.2+4.2,0.7); (0+4.2,4) – (0.2+4.2,3.3); (0+4.2,4) – (-0.2+4.2,3.3); (1.2+4.2,1.2) – (0.5+4.2,1.2); (1.2+4.2,1.2) – (0.5+4.2,1.6); (-1.2+4.2,1.2) – (-0.5+4.2,1.2); (-1.2+4.2,1.2) – (-0.5+4.2,1.6); (1.2+4.2,2.6) – (0.5+4.2,2.6); (1.2+4.2,2.6) – (0.5+4.2,2.2); (-1.2+4.2,2.6) – (-0.5+4.2,2.6); (-1.2+4.2,2.6) – (-0.5+4.2,2.2);
(0.2+4.2,3.3) – (-0.4+4.2,4); (0.2+4.2,3.3) – (-0.8+4.2,4); (0.2+4.2,3.3) – (-1.2+4.2,4); (0.5+4.2,2.2) – (1.2+4.2,3.6); (0.5+4.2,2.2) – (1.2+4.2,3.2); (0.5+4.2,1.2) – (1.2+4.2,0.6); (0.5+4.2,1.2) – (1.2+4.2,0.2); (-0.4+4.2,4) – (1.2+4.2,3.2); (-0.8+4.2,4) – (1.2+4.2,0.2); (1.2+4.2,3.6) .. controls (1.2+5.2,2.1).. (1.2+4.2,0.6);
at (0+4.2,0)\[circle,draw=blue!50,fill=blue!20\] ; at (0+4.2,0) [1]{}; at (-1.2+4.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (-1.2+4.2,1.2) [2]{}; at (-1.2+4.2,2.6) \[circle,draw=green!50,fill=green!20\] ; at (-1.2+4.2,2.6) [3]{}; at (1.2+4.2,1.2) \[circle,draw=green!50,fill=green!20\] ; at (1.2+4.2,1.2) [3]{}; at (1.2+4.2,2.6)\[circle,draw=yellow!50,fill=yellow!20\] ; at (1.2+4.2,2.6) [4]{}; at (0+4.2,4) \[circle,draw=black!50,fill=black!20\] ; at (0+4.2,4) [5]{};
at (0.5+4.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5+4.2,1.2) [2]{}; at (0.5+4.2,2.2)\[circle,draw=red!50,fill=red!20\] ; at(0.5+4.2,2.2) [2]{}; at (0.5+4.2,1.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5+4.2,1.2) [2]{}; at (0.5+4.2,2.2)\[circle,draw=red!50,fill=red!20\] ; at (0.5+4.2,2.2) [2]{}; at (0.2+4.2,3.3)\[circle,draw=red!50,fill=red!20\] ; at (0.2+4.2,3.3) [2]{}; at (-0.5+4.2,2.2)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.5+4.2,2.2) [1]{}; at (-0.2+4.2,3.3)\[circle,draw=blue!50,fill=blue!20\] ; at (-0.2+4.2,3.3) [1]{}; at (0.5+4.2,2.6)\[circle,draw=blue!50,fill=blue!20\] ; at (0.5+4.2,2.6) [1]{}; at (-1.2+4.2,4) \[circle,draw=blue!50,fill=blue!20\] ; at (-1.2+4.2,4) [1]{}; at (1.2+4.2,3.6) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+4.2,3.6) [1]{}; at (1.2+4.2,0.2) \[circle,draw=blue!50,fill=blue!20\] ; at (1.2+4.2,0.2)[1]{}; at (-0.5+4.2,1.6) \[ circle,draw=black,fill=black\] ; at (0.5+4.2,1.6) \[ circle,draw=black,fill=black\] ; at (-0.5+4.2,1.2) \[ circle,draw=black,fill=black\] ; at (-0.5+4.2,2.6) \[ circle,draw=black,fill=black\] ; at (0.2+4.2,0.7) \[ circle,draw=black,fill=black\] ; at (-0.2+4.2,0.7) \[ circle,draw=black,fill=black\] ; at (-0.4+4.2,4)\[ circle,draw=black,fill=black\] ; at (-0.8+4.2,4) \[ circle,draw=black,fill=black\] ; at (1.2+4.2,3.2) \[ circle,draw=black,fill=black\] ; at (1.2+4.2,0.6) \[ circle,draw=black,fill=black\] ; at (0.4+4.2,0) [$u_6$]{}; at (-0.5+4.2,0.7) [$v_6$]{}; at (0.45+4.2,0.7) [$v'_6$]{}; at (-1.6+4.2,1.2) [$u_4$]{}; at (-0.8+4.2,0.9) [$v_4$]{}; at (-0.8+4.2,1.8) [$v'_4$]{}; at (-1.6+4.2,2.6) [$u_2$]{}; at (-0.8+4.2,2.2) [$v_2$]{}; at (-0.8+4.2,2.8) [$v'_2$]{};
at (1.6+4.2,1.2) [$u_5$]{}; at (0.5+4.2,0.9) [$v_5$]{}; at (0.8+4.2,1.8) [$v'_5$]{}; at (1.6+4.2,2.6) [$u_3$]{}; at (0.8+4.2,2.2) [$v_3$]{}; at (0.5+4.2,2.8) [$v'_3$]{}; at (0.4+4.2,4) [$u_1$]{}; at (-0.5+4.2,3.2) [$v_1$]{}; at (0.55+4.2,3.2) [$v'_1$]{}; at (1.6+4.2,3.6) [$w_2$]{}; at (-1.6+4.2,4) [$w_1$]{}; at (1.6+4.2,0.2) [$w_3$]{};
If $G$ is a 4-regular graph with girth $g\ge 7$, then $\Gamma(G)=5$. \[propg\]
Suppose that $G$ contains a 7-cycle. We denote the 5-atom which is a tree by $T_5$ (the binomial tree with maximum degree 4). It can be easily verified that $G$ contains $T_5$ where two leaves are merged (which is a 5-atom). Moreover, if $G$ does not contain a 7-cycle, then it contains $T_5$ as induced subgraph.
Let $G$ be a 4-regular graph. If $G$ does not contain an induced $C_4$, then $\Gamma(G)=5$. \[indc4\]
Suppose that $G$ does not contain an induced $C_4$. Using Proposition \[propg\] for the case $g\ge 7$, Propositions \[cycle5\] and \[cycle6\] for the case $g=5,6$, and Proposition \[cycle3\] when $G$ contains a $C_3$ yields the desired result.
By Proposition \[indc2\], Corollary \[indc3\] and Theorem \[indc4\], any $r$-regular graph with $r\le4$ and without induced $C_4$ has Grundy number $r+1$. Therefore, it is natural to propose Conjecture 1.
[^1]: Author partially supported by the Burgundy Council
[^2]: Independently of our work, Yahiaoui et al. [@YA2012] have established a different algorithm to determine if the Grundy number of a cubic graph is 4.
|
---
author:
- |
David Glickenstein\
University of Arizona\
and\
Massachusetts Institute of Technology
title: Geometric triangulations and discrete Laplacians on manifolds
---
Introduction\[introductions\]
=============================
In this paper we shall explore Euclidean structures on manifolds which lead to Laplace operators. Euclidean structures can be introduced on a triangulation of a manifold by giving each simplex the geometric structure of a Euclidean simplex. This structure gives the manifold a length space structure in the same way a Riemannian metric gives a manifold a length structure: the length between two points is the infimum of the lengths of paths between the two points. The length of a path is determined by the fact that each simplex it passes through has the structure of Euclidean space.
The purpose of this paper is to be able to do analysis on the piecewise Euclidean space. The Laplace operator $\triangle$ is well defined on many geometric spaces, and is especially important as a natural operator on a Riemannian manifold and as a generator of Brownian motion. In this paper, we define a general Euclidean structure called a duality triangulation which not only allows one to measure length between points and volume of simplices, but also allows one to describe a geometric dual cell decomposition and the volume of dual cells. This allows one to define a Laplace operator in a natural way, which has been applied to fields such as image processing [@MDSB] [@Hir] and physics [@Mer].
The duality triangulation structure is very similar to other Euclidean structures used in both pure and applied math; specifically, we address the connection to weighted triangulations and Thurston triangulations. In addition, positivity of volumes of certain duals correspond to Delaunay or regular triangulations, which are used in a very wide range of applications from biology to physics to computer graphics.
This paper is organized as follows. We begin in Section \[euclidean structures\] with an introduction to Euclidean structures by recalling the definitions of weighted and Thurston triangulations, introducing dual triangulations, and relating the three types of triangulations. In Section \[regular triangulations\] we discuss regular triangulations and Delaunay triangulations and consider flip algorithms for constructing regular and Delaunay triangulations. In Section \[laplace\] we introduce the Laplace operator $\triangle$ associated to a given duality triangulation and derive some of its properties. Finally, in Section \[Riemannian\] we briefly discuss the status of piecewise linear Riemannian geometry.
The major new results in this paper are the result on the equivalence of weighted, Thurston, and duality triangulations in Section \[equivalence of triangulations\], the analysis of flip algorithms in Section \[regular triangulations\], the generalization of Rippa’s theorem to regular triangulations in Section \[rippa\], and the definiteness results in Section \[laplace and heat equations\].
Many of the results in this paper were motivated as generalizations of those described in [@BS].
Euclidean structures\[euclidean structures\]
============================================
Basic definitions\[Euclidean structures basic\]
-----------------------------------------------
In this section we shall introduce three types of Euclidean structures: weighted triangulations, Thurston triangulations, and duality triangulations. All structures begin with a topological triangulation $\mathcal{T=}\left\{
\mathcal{T}_{0},\mathcal{T}_{1},\ldots,\mathcal{T}_{n}\right\} $ of an $n$-dimensional manifold (we shall usually use $n$ to denote the dimension of the complex in this paper). The triangulation consists of lists of simplices $\sigma^{k}$, where the superscript denotes the dimension of the simplex, and $\mathcal{T}_{k}$ is a list of all $k$-dimensional simplices $\sigma
^{k}=\left\{ i_{0},\ldots,i_{k}\right\} $. We shall often refer to $0$-dimensional simplices as vertices, $1$-dimensional simplices as edges, $2$-dimensional simplices as faces or triangles, and $3$-dimensional simplices as tetrahedra. We shall often denote vertices as $j$ instead of $\left\{
j\right\} .$ Let $\mathcal{T}_{1}^{+}$ denote the directed edges, where we distinguish $\left( i,j\right) $ from $\left( j,i\right) $. When the order does not matter, we use $\left\{ i,j\right\} $ to denote an edge. A triangulation is said to be an $n$-dimensional manifold if a neighborhood of every vertex is homeomorphic to a ball in $\mathbb{R}^{n}.$ A two-dimensional manifold is often referred to as a surface. Throughout this paper we will be dealing exclusively with triangulations of manifolds or parts of manifolds.
In order to give the topological triangulation a geometric structure, each edge $\left\{ i,j\right\} $ is assigned a length $\ell_{ij}$ such that for each simplex in the triangulation there exists a Euclidean simplex with those edge lengths. We call such an assignment a *Euclidean triangulation* $\left( \mathcal{T},\mathcal{\ell}\right) $, where we think of $\ell$ as a function $$\ell:\mathcal{T}_{1}\rightarrow(0,\infty).$$ The conditions on $\ell$ include the triangle inequality, but there are further restrictions in higher dimensions which ensure that the simplices can be realized as (non-degenerate) Euclidean simplices. The restrictions can be expressed in terms of the square of volume, which can be expressed as a polynomial in the squares of the edge lengths by the Cayley-Menger determinant formula. Each pair of simplices $\sigma_{1}^{n}$ and $\sigma_{2}^{n}$ connected at a common boundary simplex $\sigma^{n-1}$ is called a *hinge*. In a Euclidean triangulation every hinge can be embedded isometrically in $\mathbb{R}^{n}$.
Euclidean triangulations have the structure of a distance space with an intrinsically defined distance. Given any curve $\gamma$ whose length can be computed on each Euclidean simplex, we can compute the total length of the curve $L\left( \gamma\right) $ as $L\left( \gamma\right) =\sum_{\sigma
}L_{\sigma}\left( \gamma\cap\sigma\right) $ where $L_{\sigma}\left(
\gamma\cap\sigma\right) $ is the length of the curve in the simplex $\sigma$ (if the curve intersects the simplex many times, we simply add the contributions of each piece of the intersection). In particular, we can consider curves which are differentiable when restricted to each simplex (these are called piecewise differentiable curves). The intrinsic distance is defined as $$d\left( P,Q\right) =\inf\left\{ L\left( \gamma\right) :\gamma\text{ is a
path from }P\text{ to }Q\right\} . \label{intrinsic distance}$$ The class of paths can be either taken to be piecewise differentiable or piecewise linear since length is minimized on piecewise linear paths, as explained in [@Sto Section 2]. A path which locally minimizes length is called a *geodesic* and one which globally minimizes is called a *minimizing geodesic*.
We are now ready to introduce more structures on Euclidean triangulations.
Weighted triangulations\[weighted triangulations\]
--------------------------------------------------
We begin with weighted triangulations.
A *weighted triangulation* is a Euclidean triangulation $\left(
\mathcal{T},\ell\right) $ together with weights $$w:\mathcal{T}_{0}\rightarrow\mathbb{R}.$$
We think of the weight $w_{i}$ as the square of the radius of a circle centered at the vertex $i.$ These weighted triangulations are used in the literature on regular triangulations such as [@ES] and [@AK]. Thinking of the weights in this way, in each $n$-dimensional simplex there exists an $\left( n-1\right) $-dimensional sphere which is orthogonal to each of the spheres centered at the vertices (this means they are perpendicular if they intersect, or else orthogonal in the sense described in [Ped]{}). In this way, each simplex $\sigma$ has a corresponding center $C\left( \sigma\right) ,$ which is the center of this sphere, and the center has a weight $w_{C\left( \sigma\right) }$ which is the square of the radius of this sphere. See Figures \[circletriangle\] and \[spheretetrahedron\].
\[ptb\]
\[ptb\]
An important particular case of weighted triangulations is that when $w_{i}=0$ for all vertices $i.$ This is the basis for Delaunay triangulations, but may not satisfy the Delaunay condition. We shall revisit this in Section \[regular triangulations\].
Thurston triangulations\[thurston triangulations\]
--------------------------------------------------
A *Thurston triangulation* is a collection $\left( \mathcal{T},w,c\right) ,$ where $$\begin{aligned}
w & :\mathcal{T}_{0}\rightarrow\mathbb{R},\\
c & :\mathcal{T}_{1}\rightarrow\mathbb{R},\end{aligned}$$ where $c_{ij}<w_{i}+w_{j}$ and such that the induced lengths $$\ell_{ij}=\sqrt{w_{i}+w_{j}-c_{ij}}$$ make $\left( \mathcal{T},\ell\right) $ into a Euclidean triangulation.
For a Thurston triangulation, one considers the weight $w_{i}$ to be the square of the radius $r_{i}$ of a sphere centered at vertex $i,$ just as for weighted triangulations, and one considers $c_{ij}=2r_{i}r_{j}\cos\left(
\pi-\theta_{ij}\right) $ where $\theta_{ij}$ is the angle between the spheres centered at vertices $i$ and $j.$ In this case, one derives the formula for $\ell_{ij}$ by the law of cosines. By considering $c_{ij}$ instead of $\theta_{ij},$ we have included some cases where the spheres do not intersect. These structures were studied by W. Thurston in the context of proving Andreev’s theorem (see [@Thurs] and [@MR]).
An important special case is that when $c_{ij}=-2r_{i}r_{j}$ (i.e. $\theta_{ij}=0$). This is the case of a sphere packing on each simplex, since it corresponds to the spheres being mutually tangent (as in [@CR] [@G1] [@G2]).
Duality triangulations\[duality triangulations\]
------------------------------------------------
A *duality triangulation* is a collection $\left( \mathcal{T},d\right)
,$ where$$d:\mathcal{T}_{1}^{+}\rightarrow\mathbb{R}$$ which satisfies $$d_{ij}^{2}+d_{jk}^{2}+d_{ki}^{2}=d_{ji}^{2}+d_{ik}^{2}+d_{kj}^{2}
\label{compatibility for local duality}$$ for each $\left\{ i,j,k\right\} \in\mathcal{T}_{2}$ and such that the induced lengths $$\ell_{ij}=d_{ij}+d_{ji}$$ make $\left( \mathcal{T},\ell\right) $ into a Euclidean triangulation.
We think of the weight $d_{ij}$ as representing the portion of the length $\ell_{ij}$ of edge $\left\{ i,j\right\} $ which has been assigned to vertex $i$ while $d_{ji}$ is the portion assigned to vertex $j.$ We thus call them *local lengths*. The total length of $\left\{ i,j\right\} $ is the sum of the contributions $d_{ij}$ from vertex $i$ and $d_{ji}$ from vertex $j.$ Hence each edge is assigned a center $C\left( \left\{ i,j\right\} \right)
$ which is distance $d_{ij}$ from vertex $i$ and distance $d_{ji}$ from vertex $j.$ The condition (\[compatibility for local duality\]) ensures that for each triangle $\left\{ i,j,k\right\} ,$ the perpendiculars to the three edges through the edge centers meet at one point, which can be called the center of the triangle, $C\left( \left\{ i,j,k\right\} \right) .$ We shall soon see that this condition on $2$-dimensional simplices allows us to define a center for every simplex in the triangulation.
There are two canonical examples which automatically satisfy the condition (\[compatibility for local duality\]). One is the case where $d_{ij}$ depends only on $i$ for all edges $\left( i,j\right) $ (that is, $d_{ij}=d_{ik},$ etc.). We call this a *circle* or *sphere packing* as in [@G1], and the dual comes from the inscripted circle, that is, the center $C\left( \left\{ i,j,k\right\} \right) $ is the center of the circle inscribed in $\left\{ i,j,k\right\} $ in 2D and the center $C\left(
\left\{ i,j,k,\ell\right\} \right) $ is the center of the sphere tangent to each of the edges of the tetrahedron $\left\{ i,j,k,\ell\right\} $ in 3D. Another important case is where $d_{ij}=d_{ji}.$ This corresponds to the center $C\left( \left\{ i,j,k\right\} \right) $ coming from the circle circumscribed about the triangle $\left\{ i,j,k\right\} $ and similar for all higher dimensions.
The structure is called a duality triangulation because the existence of a center $C\left( \sigma\right) $ for each $\sigma$ puts a piecewise-Euclidean length structure on the dual of the triangulation in such a way that dual simplices are orthogonal to ordinary simplices. For example, in two dimensions, if an edge $\left\{ i,j\right\} $ is part of the two simplices $\left\{ i,j,k\right\} $ and $\left\{ i,j,\ell\right\} ,$ then we can define the length of the dual edge $\bigstar\left\{ i,j\right\} $ to be equal to the distance from the center $C\left( \left\{ i,j,k\right\}
\right) $ of the triangle $\left\{ i,j,k\right\} $ to the center $C\left(
\left\{ i,j\right\} \right) $ of the edge $\left\{ i,j\right\} $ plus the distance from $C\left( \left\{ i,j,\ell\right\} \right) $ to $C\left(
\left\{ i,j\right\} \right) .$ When the hinge is isometrically embedded in $\mathbb{R}^{2},$ we see that $\bigstar\left\{ i,j\right\} $ is a straight line which is perpendicular to the edge $\left\{ i,j\right\} .$ We shall now show that this can be done in all dimensions, and no additional restrictions must be made besides (\[compatibility for local duality\]) for each triangle.
\[existence of duals\]A duality triangulation in any dimension has unique centers $C\left( \sigma^{m}\right) $ for each simplex $\sigma^{m}$ such that $C\left( \sigma^{m}\right) $ is at the intersection of the $\left(
m-1\right) $-dimensional hyperplanes through $C\left( \left\{ i,j\right\}
\right) $ and perpendicular to $\left\{ i,j\right\} $ for each $\left\{
i,j\right\} $ in $\sigma^{m}.$
We construct the centers $C\left( \sigma^{m}\right) $ inductively for $m$-dimensional simplices. Each pair of $m$-dimensional simplices meeting at an $\left( m-1\right) $-dimensional simplex (a hinge) can be embedded in $\mathbb{R}^{m}$ as two adjacent Euclidean simplices. To make the notation more readable, we shall not distinguish between the embedding of the hinge in $\mathbb{R}^{m}$ and the hinge as abstract simplices in the piecewise Euclidean manifold. A simplex $\sigma^{m}$ is assumed to be Euclidean with the assigned edge lengths given by $\ell_{ij}.$ We now inductively construct the centers of each simplex. First, $C\left( \left\{ i\right\} \right) =i$ and $C\left( \left\{
i,j\right\} \right) $ is the point on $\left\{ i,j\right\} $ which is a distance $d_{ij}$ to $\left\{ i\right\} $ and a distance $d_{ji}$ to $\left\{ j\right\} .$ Now, given centers $C\left( \sigma^{k}\right) $ for $k\leq m-1,$ we construct $C\left( \sigma^{m}\right) $ as follows. Label the vertices of $\sigma^{m}$ to be $\left\{ 0,1,\ldots,m\right\} .$
Let $\Pi_{\left\{ i,j\right\} }$ denote the plane in $\mathbb{R}^{m}$ through $C\left( \left\{ i,j\right\} \right) $ and perpendicular to $\left\{ i,j\right\} $ (this is a hyperplane in $\mathbb{R}^{m}$). First we construct the center of a simplex $\left\{ 0,1,2\right\} $ ($m=2$). One can embed the simplex in $\mathbb{R}^{2}$ as the three vertices $\left(
0,0\right) ,$ $\left( \ell_{01},0\right) ,$ and $\left( \ell_{02}\cos\gamma_{0},\ell_{02}\sin\gamma_{0}\right) ,$ where $\gamma_{0}$ is the angle at vertex $0.$ The centers of the three edges are realized as $C\left(
\left\{ 0,1\right\} \right) =\left( d_{01},0\right) ,$ $C\left( \left\{
0,2\right\} \right) =\left( d_{02}\cos\gamma_{0},d_{02}\sin\gamma
_{0}\right) ,$ and $C\left( \left\{ 1,2\right\} \right) =\left(
\ell_{01}-d_{12}\cos\gamma_{1},d_{12}\sin\gamma_{1}\right) .$ Hence $$\begin{aligned}
\Pi_{\left\{ 0,1\right\} } & =\left\{ \left( d_{01},t\right)
:t\in\mathbb{R}\right\} ,\\
\Pi_{\left\{ 0,2\right\} } & =\left\{ \left( d_{02}\cos\gamma_{0}+t\sin\gamma_{0},d_{02}\sin\gamma_{0}-t\cos\gamma_{0}\right) :t\in
\mathbb{R}\right\} ,\\
\Pi_{\left\{ 1,2\right\} } & =\left\{ \left( \ell_{01}-d_{12}\cos
\gamma_{1}+t\sin\gamma_{1},d_{12}\sin\gamma_{1}+t\cos\gamma_{1}\right)
:t\in\mathbb{R}\right\} .\end{aligned}$$ A quick calculation (using the law of cosines to compute $\cos\gamma_{i}$ and $\sin\gamma_{i}$ in terms of $d_{ij}$) shows that the three intersection points of these lines coincide if and only if (\[compatibility for local duality\]) holds.
We now construct $C\left( \sigma^{m}\right) $ given $C\left( \sigma
^{m-1}\right) $ for all $\left( m-1\right) $-dimensional simplices. Since $\sigma^{m}$ is a nondegenerate Euclidean simplex, the planes $\Pi_{\left\{
0,1\right\} },\ldots,\Pi_{\left\{ 0,m\right\} }$ intersect at one point, $c.$ We need only show that the planes $\Pi_{\left\{ i,j\right\} }$ also intersect $c.$ This is true because inside $\left\{ 0,i,j\right\} ,$ the planes $\Pi_{\left\{ 0,i\right\} }$ and $\Pi_{\left\{ 0,j\right\} }$ meet each other and the plane $\Pi_{\left\{ i,j\right\} }$ at $C\left( \left\{
0,i,j\right\} \right) .$ Furthermore, since these planes are all perpendicular to $\left\{ 0,i,j\right\} ,$ the intersection $\Pi_{\left\{
0,i\right\} }\cap\Pi_{\left\{ i,j\right\} }$ is equal to the intersection $\Pi_{\left\{ 0,i\right\} }\cap\Pi_{\left\{ 0,j\right\} }$ and hence contains $c.$ We call this point $C\left( \sigma^{m}\right) =c.$
Centers allow a geometric description of the Poincaré dual of the triangulation. Any triangulation of a manifold has a cell complex which is its Poincaré dual (see, for instance, [@Bre] or [@Hat]). As noted by Hirani [@Hir], the assignment of a center to each simplex allows one to assign a geometric Poincaré dual, or just dual for short. See Figures \[2ddualpic\] and \[3ddualpic\] for two-dimensional and three-dimensional simplices with dual cells included. Hirani restricted himself to well-centered triangulations, which means that the center of each simplex is inside the simplex. This is a very strong restriction, for even Delaunay triangulations may not be well-centered. Duality structures allow one to define geometric duals (a realization of the Poincaré dual), each of which has a volume. The structure may not be well-centered, and for this reason some volumes may be negative. The $k$-dimensional volume of a simplex $\sigma^{k}$ will be denoted $\left\vert
\sigma^{k}\right\vert $ (for instance $\left\vert \left\{ i,j\right\}
\right\vert =\ell_{ij}$) and the $\left( n-k\right) $-dimensional (signed) volume of the dual of a simplex $\bigstar\sigma^{k}$ will be denoted $\left\vert \bigstar\sigma^{k}\right\vert .$
\[ptb\]
\[ptbptb\]
It is helpful to consider an example before considering the general definitions. Given a triangulation of a three-dimensional manifold, one defines the duals as follows (compare with Figure \[3ddualpic\]):
1. The dual of a 3-simplex $\left\{ i,j,k,\ell\right\} $ is the center, $\bigstar\left\{ i,j,k,\ell\right\} =C\left( \left\{
i,j,k,\ell\right\} \right) ,$ and its volume is one.
2. The dual of a 2-simplex $\left\{ i,j,k\right\} $ contained in $\left\{ i,j,k,\ell\right\} $ and $\left\{ i,j,k,m\right\} $ is a 1-cell $\bigstar\left\{ i,j,k\right\} ,$ which is the union of the line from $C\left( \left\{ i,j,k,\ell\right\} \right) $ to $C\left( \left\{
i,j,k\right\} \right) $ and the line from $C\left( \left\{
i,j,k,m\right\} \right) $ to $C\left( \left\{ i,j,k\right\} \right) $. Its volume is slightly tricky. We define the volume as $$\begin{aligned}
\left\vert \bigstar\left\{ i,j,k\right\} \right\vert & =\pm d\left[
C\left( \left\{ i,j,k,\ell\right\} \right) ,C\left( \left\{
i,j,k\right\} \right) \right] \pm d\left[ C\left( \left\{
i,j,k,m\right\} \right) ,C\left( \left\{ i,j,k\right\} \right) \right]
\\
& =\pm d\left[ C\left( \left\{ i,j,k,\ell\right\} \right) ,C\left(
\left\{ i,j,k,m\right\} \right) \right]\end{aligned}$$ where $d$ is the Euclidean distance in $\mathbb{R}^{3}$ (these are well defined because we can embed the hinge in $\mathbb{R}^{3}$) and the signs are defined appropriately. In the first line, the sign is positive if $C\left(
\left\{ i,j,k,\ell\right\} \right) $ is on the same side of the plane containing the side $\left\{ i,j,k\right\} $ as the simplex $\left\{
i,j,k,\ell\right\} $ is, and negative if it is on the other side (similarly for $\left\{ i,j,k,m\right\} $). The sign on the second line is defined to be compatible with the previous definition. Note that it is possible for $\left\vert \bigstar\left\{ i,j,k\right\} \right\vert $ to be negative.
3. The dual of a 1-simplex $\left\{ i,j\right\} $ is the union of triangles. For each $k,\ell$ such that $\left\{ i,j,k,\ell\right\} $ is a simplex, the intersection of the simplex with the dual $\bigstar\left\{
i,j\right\} $ is the union of the right triangle with vertices $C\left(
\left\{ i,j,k,\ell\right\} \right) ,$ $C\left( \left\{ i,j,k\right\}
\right) ,$ $C\left( \left\{ i,j\right\} \right) $ and the right triangle with vertices $C\left( \left\{ i,j,k,\ell\right\} \right) ,$ $C\left(
\left\{ i,j,\ell\right\} \right) ,$ $C\left( \left\{ i,j\right\}
\right) .$ Each of these triangles has a signed area. The first is $$\pm\frac{1}{2}d\left[ C\left( \left\{ i,j,k,\ell\right\} \right)
,C\left( \left\{ i,j,k\right\} \right) \right] ~d\left[ C\left(
\left\{ i,j\right\} \right) ,C\left( \left\{ i,j,k\right\} \right)
\right]$$ and the second is defined similarly. The sign is defined as the product of the appropriate signs in each of the two distances.
4. The dual of a vertex $\left\{ i\right\} $ is a union of right tetrahedra. For each $j,k,\ell$ such that $\left\{ i,j,k,\ell\right\} $ is a simplex, the intersection of $\bigstar\left\{ i\right\} $ with $\left\{
i,j,k,\ell\right\} $ is the union of six tetrahedra:
1. the tetrahedron defined by the vertices $C\left( \left\{
i,j,k,\ell\right\} \right) ,$ $C\left( \left\{ i,j,k\right\} \right) ,$ $C\left( \left\{ i,j\right\} \right) ,$ and $i,$
2. the tetrahedron defined by $C\left( \left\{ i,j,k,\ell\right\}
\right) ,$ $C\left( \left\{ i,j,k\right\} \right) ,$ $C\left( \left\{
i,k\right\} \right) ,$ and $i,$
3. the tetrahedron defined by $C\left( \left\{ i,j,k,\ell\right\}
\right) ,$ $C\left( \left\{ i,j,\ell\right\} \right) ,$ $C\left(
\left\{ i,j\right\} \right) ,$ and $i,$
4. the tetrahedron defined by $C\left( \left\{ i,j,k,\ell\right\}
\right) ,$ $C\left( \left\{ i,j,\ell\right\} \right) ,$ $C\left(
\left\{ i,\ell\right\} \right) ,$ and $i,$
5. the tetrahedron defined by $C\left( \left\{ i,j,k,\ell\right\}
\right) ,$ $C\left( \left\{ i,k,\ell\right\} \right) ,$ $C\left(
\left\{ i,k\right\} \right) ,$ and $i,$
6. and the tetrahedron defined by $C\left( \left\{ i,j,k,\ell\right\}
\right) ,$ $C\left( \left\{ i,k,\ell\right\} \right) ,$ $C\left(
\left\{ i,\ell\right\} \right) ,$ and $i.$
The volume of $\bigstar\left\{ i\right\} $ is the sum of the volumes of these tetrahedra, namely$$\pm\frac{1}{6}d\left[ C\left( \left\{ i,j,k,\ell\right\} \right)
,C\left( \left\{ i,j,k\right\} \right) \right] ~d\left[ C\left(
\left\{ i,j\right\} \right) ,C\left( \left\{ i,j,k\right\} \right)
\right] ~d\left[ i,C\left( \left\{ i,j\right\} \right) \right]$$ for the first and similarly for the others, where the signs are defined appropriately.
We can define the geometric duals in a triangulation of an $n$-dimensional manifold inductively as follows.
Define the dual of $\left\{ 0,\ldots,n\right\} $ to be $\bigstar\left\{
0,\ldots,n\right\} =C\left( \left\{ 0,\ldots,n\right\} \right) ,$ and $\left\vert \bigstar\left\{ 0,\ldots,n\right\} \right\vert =1.$
The signed distance $$d_{\pm}\left[ C\left( \sigma^{n}\right) ,C\left( \sigma^{n-1}\right)
\right]$$ for $\sigma^{n-1}\subset\sigma^{n}$ is equal to the distance between $C\left(
\sigma^{n}\right) $ and $C\left( \sigma^{n-1}\right) $ in any isometric embedding $\sigma^{n}\subset\mathbb{R}^{n}$ with the sign positive if $C\left( \sigma^{n}\right) $ is on the same side of the hyperplane defined by $\sigma^{n-1}\subset\mathbb{R}^{n}$ as $\sigma^{n}$ is, and negative if $C\left( \sigma^{n}\right) $ is on the opposite side.
It will be useful to know the following formula for the distance between the center of a triangle and the center of a side. Consider a triangle $\left\{
i,j,k\right\} .$ Then some basic Euclidean geometry yields $$d_{\pm}\left[ C\left( \left\{ i,j,k\right\} \right) ,C\left( \left\{
i,j\right\} \right) \right] =\frac{d_{ik}-d_{ij}\cos\gamma_{i}}{\sin
\gamma_{i}} \label{center distance}$$ where $\gamma_{i}$ is the angle at vertex $i.$
\[volume with signed distance\]For any $k\geq1,$ the volume of a simplex $\sigma^{k}$ is$$\left\vert \sigma^{k}\right\vert =\frac{1}{k!}\sum_{\sigma^{0}\subset
\cdots\subset\sigma^{k}}\prod\limits_{j=0}^{k-1}d_{\pm}\left[ C\left(
\sigma^{j}\right) ,C\left( \sigma^{j+1}\right) \right]
\label{volume formula for a simplex}$$ where $\sigma^{k}$ is fixed and the sum is over all strings of simplices contained in $\sigma^{k}.$
The proof is by induction on $k.$ If $k=1,$ then $\left\vert \left\{
i,j\right\} \right\vert =d_{ij}+d_{ji}$. Assume (\[volume formula for a simplex\]) is true and consider $\sigma^{k+1}.$ Let the boundary of $\sigma^{k+1}$ be made up of $\sigma_{0}^{k},\ldots
,\sigma_{k+1}^{k}.$ The volume can be computed as $$\left\vert \sigma^{k+1}\right\vert =\frac{1}{k+1}\sum_{i=0}^{k+1}d_{\pm
}\left[ C\left( \sigma_{i}^{k}\right) ,C\left( \sigma^{k+1}\right)
\right] \left\vert \sigma_{i}^{k}\right\vert$$ where each term in the sum is the volume of the simplex consisting of the center $C\left( \sigma^{k+1}\right) $ union $\sigma_{i}^{k}$ and the signs for $d_{\pm}$ tell us whether to add the area or subtract the area. It follows from the inductive hypothesis that $$\left\vert \sigma^{k+1}\right\vert =\frac{1}{\left( k+1\right) !}\sum_{\sigma^{0}\subset\cdots\subset\sigma^{k+1}}\prod\limits_{j=0}^{k}d_{\pm
}\left[ C\left( \sigma^{j}\right) ,C\left( \sigma^{j+1}\right) \right]
.$$
Note that the above argument works for any choice of center $C\left(
\sigma^{k}\right) \in\mathbb{R}^{k}$ as long as $C\left( \sigma^{\ell
}\right) $ are the orthogonal projections onto the subspaces spanned by $\sigma^{\ell}$ for each subsimplex. The volume of a dual simplex is defined as follows.
\[definition of dual volume\]The volume of a dual simplex $\bigstar
\sigma^{k}$ is defined to be $$\left\vert \bigstar\sigma^{k}\right\vert =\frac{1}{\left( n-k\right) !}\sum_{\sigma^{k}\subset\cdots\subset\sigma^{n}}\prod\limits_{j=k}^{n-1}d_{\pm
}\left[ C\left( \sigma^{j}\right) ,C\left( \sigma^{j+1}\right) \right]
\label{dual volume fmla}$$ where $\sigma^{k}$ is fixed and the sum is over all strings of simplices containing $\sigma^{k}.$
Note that the volume is signed (it may be negative). We note that the total volume is expressible in terms of volumes of the dual simplices.
Given a duality triangulation $\mathcal{T}$ of dimension $n,$ the total volume is $$V=\sum_{\sigma^{n}\in\mathcal{T}_{n}}\left\vert \sigma^{n}\right\vert
=\sum_{i\in\mathcal{T}_{0}}\left\vert \bigstar\left\{ i\right\} \right\vert
. \label{volumes equal}$$
We know that $$\left\vert \bigstar\left\{ i\right\} \right\vert =\frac{1}{n!}\sum_{\left\{
i\right\} \subset\cdots\subset\sigma^{n}}\prod\limits_{j=0}^{n-1}d_{\pm
}\left[ C\left( \sigma^{j}\right) ,C\left( \sigma^{j+1}\right) \right]$$ by (\[dual volume fmla\]) and $$\left\vert \sigma^{n}\right\vert =\frac{1}{n!}\sum_{\sigma^{0}\subset
\cdots\subset\sigma^{n}}\prod\limits_{j=0}^{n-1}d_{\pm}\left[ C\left(
\sigma^{j}\right) ,C\left( \sigma^{j+1}\right) \right]$$ by (\[volume formula for a simplex\]). Hence it is sufficient to show that $$\sum_{i\in\mathcal{T}_{0}}\sum_{\left\{ i\right\} \subset\cdots\subset
\sigma^{n}}$$ is a reordering of $$\sum_{\sigma^{n}\in\mathcal{T}_{n}}\sum_{\sigma^{0}\subset\cdots\subset
\sigma^{n}}.$$ Here is one way to see this. Make a graph whose vertices are all simplices of all dimensions and whose edges connect two simplices if one simplex is in the boundary of the other. An easy way to draw the graph in the plane is to put vertices corresponding to $n$-dimensional simplices in a horizontal line on top, then $\left( n-1\right) $-dimensional simplices in a horizontal line below those, and so on until at the bottom is a horizontal line containing all of the vertices corresponding to $0$-dimensional simplices in the triangulation. Now draw the edges, which can only connect a vertex in a row to a vertex in the row above or below. Now we shall see that both sums are equal to the sum over all paths between the top and bottom of this graph. We can count this in two ways, first start at the bottom with each path starting at a $0$-dimensional simplex, or first start at the top with each path starting at an $n$-dimensional simplex. These are the two sums.
Equivalence of metric triangulations\[equivalence of triangulations\]
---------------------------------------------------------------------
We shall now show that weighted triangulations are equivalent to Thurston triangulations, and that, up to a universal scaling of the weights, both are almost equivalent to the set of duality triangulations. This is motivated by the geometric interpretations of the lengths, weights, angles, etc.
First we show the equivalence of weighted triangulations and Thurston triangulations.
There is a bijection between weighted triangulations and Thurston triangulations.
The definition of Thurston triangulation gives the map to weighted triangulations, keeping $w_{i}$ the same and assigning $$\ell_{ij}=\sqrt{w_{i}+w_{j}-c_{ij}}.$$ Since we assumed that $w_{i}+w_{j}-c_{ij}>0$, $\ell_{ij}$ must be positive. Similarly, we can map the other way as $$c_{ij}=w_{i}+w_{j}-\ell_{ij}^{2}.$$ Note that since $\ell_{ij}>0,$ we must have that $w_{i}+w_{j}-c_{ij}>0$.
Next we map weighted triangulations to duality triangulations. Notice that there is a one parameter family of deformations of a given weighted triangulation of a triangle $\left\{ i,j,k\right\} $ which fix the center $C\left( \left\{ i,j,k\right\} \right) $. These deformations are given by$$w_{i}\rightarrow w_{i}+t \label{deformation}$$ for varying $t.$ We call these *weight scaling deformations*, or just *weight scalings*.
\[equivalence of weighted and duality\]Weighted triangulations modulo weight scalings can be mapped injectively into the set of duality triangulations. It is a bijection if the set of duality triangulations are required to satisfy $$\sum_{k=0}^{r}\left( d_{i_{k}i_{k-1}}^{2}-d_{i_{k-1}i_{k}}^{2}\right) =0
\label{loop property}$$ for all loops $j=i_{0},i_{1},\ldots,i_{r}=j$, where $\left\{ i_{k},i_{k+1}\right\} \in\mathcal{T}_{1}.$
The key observation is that given spheres at the vertices of a simplex with given radii $\sqrt{w_{i}},$ one can always construct a sphere which is orthogonal to each of these spheres. The center of that sphere will be the center of the simplex, and for that reason is often called the *orthogonal center* [@ES]. By the arguments above, we need only construct the dual for triangles. One can do this very easily by embedding the circles in a vector space of signature $1,1,1,-1$ as in [@Ped 40.2]. Given a center, one can draw the lines perpendicular to the sides of the triangle through the center, and these determine $d_{ij}.$ A careful calculation yields$$d_{ij}=\frac{\ell_{ij}^{2}+w_{i}-w_{j}}{2\ell_{ij}}. \label{d_ij from (w,l)}$$ This is the map to duality triangulations. Note that the condition (\[compatibility for local duality\]) is automatically satisfied.
There appears to be more information in weighted triangulations, however, because the new circle centered at the orthogonal center has a radius, which can be calculated to be $$\begin{aligned}
r_{ijk}^{2} & =d_{ij}^{2}+\left( \frac{d_{ik}-d_{ij}\cos\gamma_{ijk}}{\sin\gamma_{ijk}}\right) ^{2}-w_{i}\label{r_ijk}\\
& =\frac{d_{ij}^{2}+d_{ik}^{2}-2d_{ij}d_{ik}\cos\gamma_{ijk}}{\sin^{2}\gamma_{ijk}}-w_{i},\nonumber\end{aligned}$$ where $\gamma_{ijk}$ is the angle at vertex $i$ in triangle $\left\{
i,j,k\right\} .$ Note that $r_{ijk}^{2}=w_{C\left( \left\{ i,j,k\right\}
\right) },$ the weight assigned to the center of $\left\{ i,j,k\right\} .$ The weight scalings allow, for any single triangle $\left\{ i,j,k\right\} ,$ one to specify the value of $r_{ijk}^{2}$ while fixing the center $C\left(
\left\{ i,j,k\right\} \right) .$ Fixing the center means that each would map to the same duality triangulation. It is easy to see that the formula (\[d\_ij from (w,l)\]) is unchanged by scaling deformations like (\[deformation\]). If one chooses $r_{ijk}$ then the map is unique. Once this scale is fixed in one triangle, however, the scale is determined on adjacent triangles, because weights on shared vertices have been fixed, and the deformation (\[deformation\]) must be done for all vertices $i$ in the triangle. Thus there is one free scaling parameter for the whole triangulation (if it is connected).
The inverse map from duality triangulations to weighted triangulations must take $d_{ij}+d_{ji}$ to $\ell_{ij}.$ In order to get the weights, we must first fix $w_{0}$ for a given vertex (this is a free parameter since we are considering the weighted triangulation modulo scaling). Then each neighboring weight can be calculated using (\[d\_ij from (w,l)\]):$$w_{j}=d_{ji}^{2}-d_{ij}^{2}+w_{i}. \label{w from d_ij}$$ We need only show that this is well defined. Suppose $\left\{ i,j,k\right\}
\in\mathcal{T}_{2}$ and consider a $w_{k}$ which can be defined from $w_{j}$ or $w_{i}.$ Then we need that $$d_{ki}^{2}-d_{ik}^{2}+w_{i}=d_{kj}^{2}-d_{jk}^{2}+w_{j}.$$ But since $w_{j}=d_{ji}^{2}-d_{ij}^{2}+w_{i},$ this follows from the fact that $d_{ki}^{2}-d_{ik}^{2}=d_{kj}^{2}-d_{jk}^{2}+d_{ji}^{2}-d_{ij}^{2}$ from (\[compatibility for local duality\]). It follows by a similar argument that any null-homotopic loop can be triangulated and property (\[loop property\]) holds automatically, showing that for any null-homotopic loop $j=i_{0},i_{1},\ldots,i_{L}=j$ of $L$ vertices with $\left\{ i_{k},i_{k+1}\right\}
\in\mathcal{T}_{1}$, $$w_{j}=\sum_{k=1}^{L}\left( d_{i_{k}i_{k-1}}^{2}-d_{i_{k-1}i_{k}}^{2}\right)
+w_{j}.$$ Thus, in general, we need to assume property (\[loop property\]) is satisfied for the weights to be well-defined. For example, the following triangulation of the torus does not satisfy (\[loop property\]) for all loops. Tile a torus with the two triangles $\left\{ 1,2,3\right\} ,\left\{
1,2,4\right\} $ where $d_{31}=d_{21}=d_{24}=1-\varepsilon,$ $d_{13}=d_{12}=d_{42}=\varepsilon,$ and $d_{32}=d_{23}=d_{14}=d_{41}=\frac{1}{2}$ for small $\varepsilon,$ see Figure \[torus\]. Note that $$\begin{aligned}
d_{12}^{2}+d_{23}^{2}+d_{31}^{2} & =\varepsilon^{2}+\frac{1}{4}+\left(
1-\varepsilon\right) ^{2}=d_{21}^{2}+d_{13}^{2}+d_{32}^{2}\\
d_{12}^{2}+d_{24}^{2}+d_{41}^{2} & =\varepsilon^{2}+\frac{1}{4}+\left(
1-\varepsilon\right) ^{2}=d_{21}^{2}+d_{14}^{2}+d_{42}^{2}$$ and so on. The homotopy-nontrivial loop containing $\left\{ 1,2\right\} $ will not satisfy property (\[loop property\]). However, if we started with a weighted triangulation, property (\[loop property\]) is automatically satisfied and thus the map from weighted triangulations to duality triangulations is injective.
\[ptb\]
For a triangulation of a simply connected manifold, there is a bijection between weighted triangulations up to scaling and duality triangulations.
Since the manifold is simply connected, any loop bounds a 2-dimensional disk, homeomorphic to $D^{2}=\left\{ x\in\mathbb{R}^{2}:\left\vert x\right\vert
^{2}\leq1\right\} $, which is triangulated. One can easily prove by induction on the number of triangles triangulating the disk that on the boundary of any such disk, (\[loop property\]) holds.
Regular triangulations\[regular triangulations\]
================================================
Introduction to regular triangulations
--------------------------------------
Recall the definition of a regular triangulation (see, for instance, [@ES] or [@AK]). Let $d\left( x,p\right) $ be the Euclidean distance between points $p$ and $x.$ Define the power distance $$\pi_{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$$ by$$\pi_{p}\left( x\right) =d\left( x,p\right) ^{2}-w_{p}
\label{power definition}$$ if $p$ is a point weighted with $w_{p}.$ The power is important as a function which is zero on the sphere centered at $p$ with radius $\sqrt{w_{p}}$, positive outside the sphere, and negative inside the sphere. Notice that if $p$ is a vertex of a simplex $\sigma$ and $c=C\left( \sigma\right) $ then $\pi_{c}\left( p\right) =w_{p}$ and $\pi_{p}\left( c\right) =w_{c},$ where the weight $w_{c}$ is defined as the square of the radius of the orthogonal sphere as described in Section \[weighted triangulations\].
Since we can embed any hinge in $\mathbb{R}^{n},$ the following local definition of regularity makes sense on a piecewise Euclidean manifold.
An $\left( n-1\right) $-dimensional simplex $\sigma^{n-1}$ incident on two $n$-dimensional simplices $\sigma_{1}^{n}=\sigma^{n-1}\cup\left\{
v_{1}\right\} $ and $\sigma_{2}^{n}=\sigma^{n-1}\cup\left\{ v_{2}\right\} $ is *locally regular* if $\pi_{c_{1}}\left( v_{2}\right) >w_{v_{2}}$ and $\pi_{c_{2}}\left( v_{1}\right) >w_{v_{1}},$ where $c_{i}=C\left(
\sigma_{i}^{n}\right) $ is the center of $\sigma_{i}^{n}$ for $i=1$ or $2.$ If the weights are all equal to zero, a locally regular simplex is said to be *locally Delaunay*.
Sometimes we will instead say that the *hinge* is locally regular. A hinge is locally Delaunay if and only if it satisfies the local empty circumsphere property: the sphere circumscribing $\sigma_{1}^{n}$ does not contain $v_{2}$. This is simply the interpretation of the definition when the weights are equal to zero. Note that the condition for being locally regular is unchanged by a weight scaling of the type (\[deformation\]) due to the formula (\[r\_ijk\]) for $w_{C\left( \left\{ i,j,k\right\} \right) }.$
There are actually global definitions of regular and Delaunay, since the definition of power (\[power definition\]) makes sense globally using the intrinsic distance (\[intrinsic distance\]) described in Section \[Euclidean structures basic\].
An $n$-dimensional weighted triangulation is *regular* if for every $\sigma^{n}\in\mathcal{T}_{n},$ we have $\pi_{C\left( \sigma^{n}\right)
}\left( v\right) >w_{v}$ for every vertex $v$ in the complement of $\sigma^{n}.$ In the case that the weights are all zero, we say the triangulation is *Delaunay*.
In the case of two-dimensional Delaunay, the condition on the power says that for every circle containing at least three vertices, there is no vertex inside that circle. It is a well known fact that for $n$-dimensional regular triangulations of points in $\mathbb{R}^{n}$ [@AK] and for 2-dimensional piecewise Euclidean surfaces with zero weights [@BS] [@Lei] that every hinge being locally regular is equivalent to the triangulation being regular. It is likely that the proof in [@Lei Chapter 3] can be generalized to regular triangulations of any dimension, but we do not do that here.
The argument in [@AK] uses the fact that a geodesic must be a straight line, and along a geodesic line the power increases in the manner listed below. To generalize that argument, one needs the following assumption:
\[monotonicity\]Suppose the hinge $\left\{ \sigma_{1}^{n},\sigma_{2}^{n},\sigma^{n-1}\right\} $ is locally regular. Consider a minimizing geodesic ray $\gamma$ starting at $X_{0}$ which intersects a hinge $\left\{
\sigma_{1}^{n},\sigma_{2}^{n},\sigma^{n-1}\right\} $ by first entering $\sigma_{1}^{n}$ and then $\sigma_{2}^{n}.$ The simplex $\sigma^{n-1}$ determines a plane which separates $\sigma_{1}^{n}$ and $\sigma_{2}^{n}$ and contains all points $x$ such that $\pi_{C\left( \sigma_{1}^{n}\right)
}\left( x\right) =\pi_{C\left( \sigma_{2}^{n}\right) }\left( x\right) .$ Then $\pi_{C\left( \sigma_{1}^{n}\right) }\left( X_{0}\right)
<\pi_{C\left( \sigma_{2}^{n}\right) }\left( X_{0}\right) .$
One might try to prove Criterion \[monotonicity\] by developing the geodesic in the plane in the following way (we consider two dimensions for simplicity). Start with a triangle and embed it in $\mathbb{R}^{2}.$ For each new triangle which the geodesic goes through, embed a copy in $\mathbb{R}^{2}$ adjacent to the previous triangle so that it looks like we are unfolding the manifold. The geodesic must be a straight line if it does not go through a vertex and so we may try to make comparisons on this development. Note also that by the following theorem of Gluck, every two points have a minimizing geodesic between them.
\[[[@Sto Prop. 2.1]]{}\]If a piecewise Euclidean manifold is complete with respect to the intrinsic distance, in particular if $M$ is a finite triangulation, then there is at least one minimizing geodesic between any two points of $M.$
The problem with this is that geodesics do go through vertices and even by varying the endpoints slightly, a minimizing geodesic may still go through the vertex (see [@MP Figure 14]). Hence it is not at all clear that Criterion \[monotonicity\] is always satisfied.
Note that Bobenko and Springborn [@BS] are able to prove that Delaunay is the same as all edges being locally Delaunay in general by developing the triangulation (not along a geodesic). Their argument appears to strongly use the fact that the edges are locally Delaunay (with all weights equal to zero), but does not use Criterion \[monotonicity\].
For completeness, we include the proof for regular triangulations of $n$-dimensional manifolds, assuming Criterion \[monotonicity\], which is proven using a similar method.
Under the assumptiong of Criterion \[monotonicity\], an $n$-dimensional weighted triangulation is regular if and only if all of its hinges are locally regular.
This proof is essentially the one seen in [@AK] for Delaunay triangulations. Clearly if the triangulation is regular, then all hinges are locally regular. Now suppose all of the hinges of a weighted triangulation are locally regular. Given a vertex $v$ and a simplex $\sigma^{n}$ such that $v$ is not in $\sigma^{n},$ we may consider the line $L$ from $v$ to a point in the simplex $\sigma^{n}.$ Possibly by adjusting the line slightly, it must intersect, in order, a sequence of $n$-dimensional simplices $\sigma_{1}^{n},\ldots\sigma_{k}^{n}=\sigma^{n}$ where $v$ is in a simplex bordering $\sigma_{1}^{n}.$ By Criterion \[monotonicity\] we know that $$\pi_{C\left( \sigma_{i}^{n}\right) }\left( v\right) <\pi_{C\left(
\sigma_{i+1}^{n}\right) }\left( v\right)$$ for $i=1,\ldots,k-1.$ Since the triangulation is locally regular, $$w_{v}<\pi_{C\left( \sigma_{1}^{n}\right) }\left( v\right) .$$ Stringing these together, we get that $$w_{v}<\pi_{C\left( \sigma^{n}\right) }\left( v\right) .$$
Although we have not proven that regular triangulations and locally regular triangulations are the same, we will often suppress the word local in the rest of this paper, always considering the local property.
Regular triangulations and duality structures\[regular triangulations and duality\]
-----------------------------------------------------------------------------------
In order to have a definition of locally regular in terms of duality structures, we first look at the two-dimensional case. A regular hinge $\left\{ \left\{ i,j,k\right\} ,\left\{ i,j,\ell\right\} \right\} $ must satisfy $$\begin{aligned}
\pi_{C\left( \left\{ i,j,k\right\} \right) }\left( \ell\right) &
=d\left( C\left( \left\{ i,j,k\right\} \right) ,\left\{ \ell\right\}
\right) ^{2}-r_{ijk}^{2}>w_{\ell}\\
\pi_{C\left( \left\{ i,j,\ell\right\} \right) }\left( k\right) &
=d\left( C\left( \left\{ i,j,\ell\right\} \right) ,\left\{ k\right\}
\right) ^{2}-r_{ij\ell}^{2}>w_{k}.\end{aligned}$$
The center $C\left( \left\{ i,j,k\right\} \right) $ and radius $r_{ijk}$ are uniquely determined by the three equations$$\begin{aligned}
d\left( C\left( \left\{ i,j,k\right\} \right) ,\left\{ i\right\}
\right) ^{2}-r_{ijk}^{2} & =w_{i}\\
d\left( C\left( \left\{ i,j,k\right\} \right) ,\left\{ j\right\}
\right) ^{2}-r_{ijk}^{2} & =w_{j}\\
d\left( C\left( \left\{ i,j,k\right\} \right) ,\left\{ k\right\}
\right) ^{2}-r_{ijk}^{2} & =w_{k}.\end{aligned}$$
Put the triangle in Euclidean space with vertices $v_{i}=\vec{0},v_{j},v_{k}.$ We know that $C\left( \left\{ i,j,k\right\} \right) =xv_{j}+yv_{k}$ for some $x$ and $y$ and let $z$ be the unknown radius. Now we can write the first two equations as $$\begin{aligned}
\left\vert xv_{j}+yv_{k}\right\vert ^{2}-z^{2} & =w_{i}\\
\left\vert \left( xv_{j}+yv_{k}\right) -v_{j}\right\vert ^{2}-z^{2} &
=w_{j}$$ so$$w_{i}-2v_{j}\cdot\left( xv_{j}+yv_{k}\right) +\ell_{ij}^{2}=w_{j}$$ which is linear in $x,y.$ Similarly, we have $$w_{i}-2v_{k}\cdot\left( xv_{j}+yv_{k}\right) +\ell_{ik}^{2}=w_{k}.$$ So the problem reduces to a linear system $$\begin{aligned}
w_{i}+\ell_{ij}^{2}-w_{j} & =2\ell_{ij}^{2}x+2\ell_{ij}\ell_{ik}\left(
\cos\gamma_{i}\right) y\\
w_{i}+\ell_{ik}^{2}-w_{k} & =2\ell_{ij}\ell_{ik}\left( \cos\gamma
_{i}\right) x+2\ell_{ik}^{2}y,\end{aligned}$$ where $\gamma_{i}$ is the angle at vertex $i,$ with solutions $$\begin{aligned}
x & =\frac{\left( w_{i}+\ell_{ij}^{2}-w_{j}\right) \ell_{ik}-\left(
w_{i}+\ell_{ik}^{2}-w_{k}\right) \ell_{ij}\cos\gamma_{i}}{2\left( \sin
^{2}\gamma_{i}\right) \ell_{ij}^{2}\ell_{ik}}\\
y & =\frac{\left( w_{i}+\ell_{ik}^{2}-w_{k}\right) \ell_{ij}-\left(
w_{i}+\ell_{ij}^{2}-w_{j}\right) \ell_{ik}\cos\gamma_{i}}{2\left( \sin
^{2}\gamma_{i}\right) \ell_{ij}\ell_{ik}^{2}}$$ $\allowbreak$and $$z^{2}=x^{2}\ell_{ij}^{2}+y^{2}\ell_{ik}^{2}+2xy\ell_{ij}\ell_{ik}\cos
\gamma_{i}-w_{i}.$$
\[boundary regular\]If an edge is on the boundary of regular, i.e. $$\pi_{C\left( \left\{ i,j,k\right\} \right) }\left( \ell\right) =d\left(
C\left( \left\{ i,j,k\right\} \right) ,\left\{ \ell\right\} \right)
^{2}-r_{ijk}^{2}=w_{\ell},$$ then $C\left( \left\{ i,j,k\right\} \right) =C\left( \left\{
i,j,\ell\right\} \right) $ and $r_{ijk}=r_{ij\ell}.$
If $d\left( C\left( \left\{ i,j,k\right\} \right) ,\ell\right)
^{2}-r_{ijk}^{2}=w_{\ell}$ then $\left( C\left( \left\{ i,j,k\right\}
\right) ,r_{ijk}\right) $ satisfy the same three equations as $\left(
C\left( \left\{ i,j,\ell\right\} \right) ,r_{ij\ell}\right) ,$ which determine these uniquely. Hence they must be equal.
\[edge regular duality\]An edge $\left\{ i,j\right\} $ is regular if and only if $\left\vert \bigstar\left\{ i,j\right\} \right\vert >0.$
Clearly $\left\vert \bigstar\left\{ i,j\right\} \right\vert =0$ on the boundary of regular as in Corollary \[boundary regular\] since the centers are the same. It is clear that $\left\vert \bigstar\left\{ i,j\right\}
\right\vert >0$ if the edge is regular.
One can now address the case of $n$ dimensions. The corresponding proofs go through essentially untouched, and one has the following characterization of regular triangulations.
An $\left( n-1\right) $-dimensional simplex $\sigma^{n-1}$ which forms a hinge with simplices $\sigma_{i}^{n}=\sigma^{n-1}\cup\left\{ i\right\} $ and $\sigma_{j}^{n}=\sigma^{n-1}\cup\left\{ j\right\} $ is regular if and only if $\left\vert \bigstar\sigma^{n-1}\right\vert >0.$
Note that $\bigstar\sigma^{n-1}$ is a one-dimensional simplex, so the property of being regular has to do with lengths dual to $\left( n-1\right)
$-simplices being positive. The previous discussion motivates the following definitions which, in light of Theorem \[equivalence of weighted and duality\], are slight generalizations of those for weighted triangulations.
An $n$-dimensional hinge at simplex $\sigma^{n-1}$ is said to be *locally regular* if $\left\vert \bigstar\sigma^{n-1}\right\vert >0.$ An $n$-dimensional duality triangulation $\mathcal{T}$ is said to be *locally regular* if $\left\vert \bigstar\sigma^{n-1}\right\vert >0$ for all $\sigma^{n-1}\in\mathcal{T}_{n-1}.$
The duality structure is called a Voronoi diagram in the case the triangulation is Delaunay. Voronoi diagrams can be described in a more direct way. A point $x$ is in the Voronoi cell $\bigstar\left\{ i\right\} $ if it is closer to $i$ than to any other vertex. The boundary of the Voronoi cells forms the $\left( n-1\right) $-dimensional complex called the Voronoi diagram. The analogue for regular triangulations is called a power diagram. A point $x$ is in the power cell $\bigstar\left\{ i\right\} $ if its power distance $\pi_{i}\left( x\right) $ is less than $\pi_{j}\left( x\right) $ for any $j\neq i$ (see [@AK] [@ES]). In the case of regular triangulations, the duality described in Section \[duality triangulations\] is the same as using power diagrams. However, our notion of duality is more general, making sense for weighted triangulations which are not regular.
An interesting question is how to find a regular triangulation of a given manifold with given weights. One method of construction is via so called flip algorithms.
Flips in 2D\[flips 2d\]
-----------------------
We first consider the case of two dimensions. One can imagine the following notion of a flip. Given a hinge consisting of two triangles $\left\{
i,j,k\right\} $ and $\left\{ i,j,\ell\right\} $ incident on one common edge $\left\{ i,j\right\} ,$ there exists a flip which exchanges this hinge with a new hinge, namely $\left\{ i,k,\ell\right\} $ and $\left\{ j,k,\ell
\right\} .$ Note that the flip fixes the boundary quadrilateral which consists cyclically of the vertices $i,k,j,\ell.$ This exchange is called a $2\rightarrow2$ bistellar flip, or Pachner move ([@Pac]). If the hinge is convex, then this can be done metrically. In fact, the flip can be made at the level of a duality structure. Given the hinge described above, to do the bistellar flip we need to construct $d_{k\ell}$ and $d_{\ell k}$ such that the condition (\[compatibility for local duality\]) is satisfied in each of the new triangles. This is done by solving the following system of equations for $d_{k\ell}$ and $d_{\ell k}$,$$\begin{aligned}
d_{ik}^{2}+d_{k\ell}^{2}+d_{\ell i}^{2} & =d_{ki}^{2}+d_{i\ell}^{2}+d_{\ell
k}^{2}\\
d_{k\ell}+d_{\ell k} & =d\left( k,\ell\right)\end{aligned}$$ where $d\left( k,\ell\right) $ is the distance between vertex $k$ and vertex $\ell.$ This distance is the Euclidean distance because the entire hinge can be embedded in $\mathbb{R}^{2}.$ Note that the first equation is equivalent to $$d_{jk}^{2}+d_{k\ell}^{2}+d_{\ell j}^{2}=d_{kj}^{2}+d_{j\ell}^{2}+d_{\ell
k}^{2}$$ using (\[compatibility for local duality\]) for triangles $\left\{
i,j,k\right\} $ and $\left\{ i,j,\ell\right\} .$ The system can actually be written in a form easier to solve:$$\begin{aligned}
d_{k\ell}-d_{\ell k} & =\frac{d_{ki}^{2}+d_{i\ell}^{2}-d_{\ell i}^{2}-d_{ik}^{2}}{d\left( k,\ell\right) }\label{flip system}\\
d_{k\ell}+d_{\ell k} & =d\left( k,\ell\right) \nonumber\end{aligned}$$ which is linear, although the dependence of $d\left( k,\ell\right) $ on the remaining $d$’s is not obvious (although easy to find using trigonometry). Hence the $2\rightarrow2$ bistellar flip is well defined on duality triangulations, and the triangle inequality follows automatically. The two hinges which are equivalent by bistellar flips are shown in Figure \[2dflip\].
\[ptb\]
The flip requires that the quadrilateral is convex, otherwise the flip would require that one part is folded back, which complicates matters. This motivates the following definition:
A hinge is *flippable* if the quadrilateral defined by the hinge when embedded in $\mathbb{R}^{2}$ is convex.
Now, given a convex quadrilateral, there exist two possible ways to make it into a hinge. The duals are uniquely determined by an assignment of centers to the edges on the quadrilateral. Let $L_{\left\{ i,j\right\} }$ be the line perpendicular to $\left\{ i,j\right\} $ and through $C\left( \left\{
i,j\right\} \right) .$ Then $L_{\left\{ i,k\right\} }$ and $L_{\left\{
j,k\right\} }$ meet at a point which is the center $C\left( \left\{
i,j,k\right\} \right) $ and similarly $L_{\left\{ i,\ell\right\} }$ and $L_{\left\{ j,\ell\right\} }$ meet at a point which is the center $C\left(
\left\{ i,j,\ell\right\} \right) .$ However, also $L_{\left\{ i,k\right\}
}$ and $L_{\left\{ i,\ell\right\} }$ meet at a point which becomes $C\left(
\left\{ i,k,\ell\right\} \right) $ after the flip, and similarly with $L_{\left\{ j,k\right\} }$ and $L_{\left\{ j,\ell\right\} }.$ Hence the centers in the hinge form another quadrilateral dual to the hinge (see the right side of Figure \[2dflip\]). One diagonal of the dual quadrilateral corresponds to $\bigstar\left\{ i,j\right\} $ and the other corresponds to $\bigstar\left\{ k,\ell\right\} .$ One must have positive length and the other negative length (or both are zero if all dual lines meet at a single point), so either the hinge is regular, or it will become regular by a flip. One can also think of the flip of the hinge corresponding to a flip of the dual hinge. To make this argument rigorous, one simply uses the fact that $\bigstar\left\{ i,j\right\} $ must be perpendicular to $\left\{
i,j\right\} ,$ and considers the possible cases for $\left\vert
\bigstar\left\{ i,j\right\} \right\vert $ being positive, negative, or zero. If it is negative, then it must look like the right side Figure \[2dflip\] and hence a flip makes $\left\vert \bigstar\left\{ k,\ell\right\}
\right\vert $ positive. If $\left\vert \bigstar\left\{ i,j\right\}
\right\vert $ is zero, then a flip maintains this.
Flip algorithms\[flip algorithms\]
----------------------------------
The most naive flip algorithm is to take a given weighted triangulation, look for a flippable edge which is not regular, and flip it. Continue until the triangulation is regular. This algorithm was first suggested by Lawson and shown to find Delaunay triangulations for points in $\mathbb{R}^{2}$ ([@Law1], see also exposition in [@Edel] and related result in [@Law2]). It was later shown to work for any 2D piecewise Euclidean triangulation (where the weights are all zero) independently in [@ILTC] and [@Riv]. This turns out not to work to find higher dimensional Delaunay triangulations or to find regular triangulations (if there are nonzero weights) even in dimension $2$. It was later found that points in $\mathbb{R}^{n}$ can be triangulated with regular triangulations (for any dimension) by incrementally adding one vertex at a time and doing all the flips before adding additional vertices. In this case one must pay close attention to the order of the flipping and the algorithm must either sort the hinges or dynamically decide which hinge to flip next [@Joe] [@ES]. Unfortunately, it is not yet clear how to extend these algorithms to piecewise Euclidean manifolds, since their proofs rely on the fact that the triangulations are in $\mathbb{R}^{n}.$ In this section we propose a subset of the space of all weighted triangulations for which the naive flip algorithm works, just as in the case of two-dimensional Delaunay triangulations.
Consider the following set.
A 2-dimensional duality triangulation is said to be *edge positive* if $d_{ij}>0$ for every directed edge $\left( i,j\right) $ of the triangulation and for any possible flip, i.e. any solution of (\[flip system\]).
Hence a triangulation is edge positive if the centers of each edge are inside the edge and if the center of the new edge after any flip is also inside that edge. This implies that any non-regular edge is flippable:
\[2D edge lemma\]Given a 2D edge positive duality triangulation, if an edge is not regular, then it is flippable.
We prove the contrapositive. Suppose a hinge consisting of $\left\{
i,j,k\right\} $ and $\left\{ i,j,\ell\right\} $ is not flippable, i.e. the quadrilateral is not convex. There can only be one interior angle larger than $\pi,$ and it must be at vertex $i$ or $j.$ Say it is at $i.$ Let $L_{k}$ be the line through vertex $i$ which is perpendicular to $\left\{ i,k\right\} $ and let $L_{\ell}$ be the line through vertex $i$ which is perpendicular to $\left\{ i,\ell\right\} $. Since $d_{ik}>0,$ the center $C\left( \left\{
i,j,k\right\} \right) $ must be on the side of $L_{k}$ on which $\left\{
i,k\right\} $ lies; call this open half-space $H_{k}.$ Similarly, $C\left(
\left\{ i,j,\ell\right\} \right) $ must lie on the side of $L_{\ell}$ on which $\left\{ i,\ell\right\} $ lies; call this half space $H_{\ell}.$ Let $H_{j}$ be the half-space containing $\left\{ i,j\right\} $ whose boundary is the line $L_{j}$ perpendicular to $\left\{ i,j\right\} $ through $i.$ Then $C\left( \left\{ i,j,k\right\} \right) $ must be in $H_{k}\cap H_{j}$ and $C\left( \left\{ i,j,\ell\right\} \right) $ must be in $H_{k}\cap
H_{\ell}.$ Since $L_{k},$ $L_{\ell},$ and $L_{j}$ intersect at $i$ and since the angle at $i$ is more than $\pi,$ $H_{k}\cap H_{j}$ and $H_{\ell}\cap
H_{j}$ are disjoint sectors in a half-space. Use Euclidean isometries to make put the hinge such that $i$ is at the origin, $\left\{ i,j\right\} $ is along the positive $x$-axis, and $k$ has positive $y$-value (and hence $\ell$ must have negative $y$-value). Any possible segment $\bigstar\left\{
i,j\right\} $ must be on a vertical line which intersects $\left\{
i,j\right\} .$ It is easy to see that any such line must intersect $H_{k}\cap
H_{j}$ with a larger $y$-value than it intersects $H_{\ell}\cap H_{j},$ implying that $\left\vert \bigstar\left\{ i,j\right\} \right\vert >0.$
\[2D edge flip algo\]The edge flip algorithm finds a regular triangulation given an edge positive duality triangulation.
Since every flip maintains the edge positive property and every nonregular edge is flippable, we can always do a flip if the triangulation is not regular. We now only need an monotone quantity which measures the progress of the algorithm to complete the proof in the same way as in [@AK], [@ES], [@ILTC], and [@Riv]. Since we are in two dimensions, we can use the Dirichlet energy for almost any function, since the energy increases if a flip makes the hinge regular (see Theorem \[generalized rippa theorem\]). Since this function increases every time we perform a flip and there are finitely many possible configurations, the algorithm must terminate.
Note that the edge flip algorithm to find Delaunay surfaces is a special case, since in that case, $d_{ij}=\ell_{ij}/2>0.$ In the next section, we suggest the analogue of this proof for higher dimensions. However, the analogue of edge positive is possibly less natural in this setting.
Higher dimensional flips\[higher dimensional flips\]
----------------------------------------------------
First let’s consider the analogue of the $2\rightarrow2$ bistellar move in higher dimensions. Recall that in any dimension, we can embed a hinge in $\mathbb{R}^{n},$ so the type of relevant flips must take place inside one or two simplices in $\mathbb{R}^{n}.$ The relevant flip is the $2\rightarrow n$ flip in $\mathbb{R}^{n}$ (see Figure \[flip3d\] for the 3D version). The flip takes two simplices $\sigma_{i}^{n}=\sigma_{0}^{n-1}\cup\left\{
i\right\} $ and $\sigma_{j}^{n}=\sigma_{0}^{n-1}\cup\left\{ j\right\} $ meeting at a common face $\sigma_{0}^{n-1}=\left\{ k_{1},\ldots
,k_{n}\right\} $ and replaces it with $n$ simplices $\sigma_{k_{p}}^{n}=\left\{ i,j,k_{1},\ldots,\hat{k}_{p},\ldots,k_{n}\right\} ,$ where $\hat{k}_{p}$ indicates that $k_{p}$ is not present. The same argument as above shows that $d_{ij}$ and $d_{ji}$ can be chosen so that the duality conditions (\[compatibility for local duality\]) hold for each face and the choice is consistent because of the duality conditions which already hold.
\[ptb\]
[fig7.eps]{}
Now the duality structure gives a hinge a dual hinge similarly to above. Look at the Figure \[flip3dwithdual\] to see the 3D case. The boundary of $\sigma_{i}^{n}$ consists of the faces $\sigma_{0}^{n}=\left\{ k_{1},\ldots,k_{n}\right\} $ and $\sigma_{ik_{p}}^{n-1}=\left\{ i,k_{1},\ldots,\hat{k}_{p},\ldots,k_{n}\right\} $ for $p=1,\ldots,n$ while the boundary of $\sigma_{j}^{n}$ is similarly decomposed. Let $L_{\sigma^{n-1}}$ be the line through $C\left( \sigma^{n-1}\right) $ and perpendicular to $\sigma^{n-1}$ for any $\left( n-1\right) $-dimensional simplex. We know that $L_{\sigma_{ik_{p}}}$ and $L_{\sigma_{ik_{q}}}$ intersect at the point $C\left( \sigma_{i}^{n}\right) $ for every $p,q=1,\ldots,n$ by Proposition \[existence of duals\]. We can also consider after the $2\rightarrow n$ flip. The boundary of $\sigma_{k_{p}}^{n}$ consists of $\sigma_{ik_{p}}^{n-1}$ and $\sigma_{jk_{p}}^{n-1}$ together with $\sigma_{k_{p}k_{q}}^{n-1}=\left\{
i,j,k_{1},\ldots,\hat{k}_{p},\ldots,\hat{k}_{q},\ldots,k_{n}\right\} $ for $q=1,\ldots,n$ and $q\neq p.$ Hence $L_{\sigma_{ik_{p}}}$ and $L_{\sigma
_{jk_{p}}}$ intersect at the point $C\left( \sigma_{k_{p}}^{n}\right) $ for each $p=1,\ldots,n.$ We find that there is a polytope with vertices $C\left(
\sigma_{i}^{n}\right) ,$ $C\left( \sigma_{j}^{n}\right) ,$ and $C\left(
\sigma_{k_{p}}^{n}\right) $ for $p=1,\ldots,n.$ This is the dual hinge. The centers $C\left( \sigma_{i}^{n}\right) $ and $C\left( \sigma_{j}^{n}\right) $ are connected via the edge $\bigstar\sigma_{0}^{n-1}.$ If $\left\vert \bigstar\sigma_{0}^{n-1}\right\vert <0$ then the flip on the hinge does a $n\rightarrow2$ flip on the dual hinge which results in removing $\bigstar\sigma_{0}^{n-1}$ and replaces it with $\bigstar\sigma_{k_{p}k_{q}}^{n-1},$ which are $\binom{n}{2}$ dual edges, each with positive length.
\[ptb\]
[fig8.eps]{}
We see that this sort of flipping is exactly what is needed to make regular triangulations via some sort of flip algorithm. However, the condition of flippability is harder to guarantee. We now examine flippability.
An $n$-dimensional triangulation is said to be $m$-central if $C\left(
\sigma^{k}\right) $ is inside $\sigma^{k}$ for all $k\leq m.$
So edge positive is the same as $1$-central. Furthermore, $n$-central is what is called well-centered in [@Hir]. We now show that $\left( n-1\right)
$-central assures that nonregular hinges are flippable.
Given an $\left( n-1\right) $-central triangulation of an $n$-dimensional manifold, if a hinge is not regular, then it is flippable.
The proof is essentially the same as the proof of Lemma \[2D edge flip algo\]. Consider a hinge consisting of the simplices $\left\{ i,k_{1},\ldots,k_{n}\right\} $ and $\left\{ j,k_{1},\ldots
,k_{n}\right\} $. The first claim is that if the hinge is unflippable, then at least one dihedral angle must be greater than $\pi.$ This is clear because if every dihedral angle is less than or equal to $\pi,$ then the hinge is the intersection of half-spaces defined by the $\left( n-1\right) $-simplices on the boundary and hence convex. Now consider the hyperplanes whose dihedral angle is greater than $\pi.$ By relabeling we may assume that the hyperplanes are determined by faces $\sigma_{ik_{n}}^{n-1}=\left\{ i,k_{1},\ldots
,k_{n-1}\right\} $ and $\sigma_{jk_{n}}^{n-1}=\left\{ j,k_{1},\ldots
,k_{n-1}\right\} $ and intersect at $\sigma_{0}^{n-2}=\left\{ k_{1},\ldots,k_{n-1}\right\} .$ Because $C\left( \sigma_{ik_{n}}^{n-1}\right)
\subset\sigma_{ik_{n}}^{n-1},$ the $C\left( \sigma_{i}^{n}\right) $ must be inside the half-space defined by the plane $\Pi_{ik_{n}}$, the plane through $\sigma_{0}^{n-2}$ and perpendicular to $\sigma_{ik_{n}}^{n-1},$ on the side containing $\sigma_{ik_{n}}^{n-1}.$ We have the same for $C\left( \sigma
_{j}^{n}\right) $ and since the angle is larger than $\pi$ we must have that $\left\vert \bigstar\sigma_{0}^{n-1}\right\vert >0$ by a similar argument to that in the proof of Lemma \[2D edge lemma\].
Regular triangulations of points in $\mathbb{R}^{n}$ are usually produced via some sort of incremental algorithm (see [@ES], [@Joe]). The key observation is that if a new point is inserted into a regular triangulation, then there is at least one non-regular hinge which is flippable (or there are no non-regular hinges and it is regular). The generalization to the manifold setting is the following. Let $Star\left( v\right) ,$ the star of a vertex $v$, be defined as all simplices containing $v.$
Suppose Criterion \[monotonicity\] is true. If every hinge in a triangulation is regular except for hinges intersecting $Star\left( v\right)
$ for some vertex $v,$ then some if some hinge is not regular, there exists a flippable nonregular hinge. Hence the triangulation can be made regular via a flipping algorithm.
\[Proof (sketch)\]The proof in [@ES] (also with exposition in [@Edel Section 12]) can be applied to this situation. We are able to prove this lemma in the generality of manifolds because we have supposed Criterion \[monotonicity\] in that generality.
Using this lemma on subsets of $\mathbb{R}^{n}$, one is able to construct regular triangulations by: insert one vertex, make the triangulation regular, and then insert the next vertex, make the triangulation regular, etc. Unfortunately, on a manifold, it is not clear what the intermediate triangulations are so the algorithm does not quite work. Also, if one starts with any triangulation, one may not have a regular triangulation which is reachable only by flips, as seen in the example [@ES Fig. 5.1].
Laplacians\[laplace\]
=====================
Laplace operators on graphs and on piecewise Euclidean manifolds have been studied in many different contexts, for instance [@BS], [@CL], [@Chun], [@G1], [@G2], [@He], [@Hipt], [@Hir], [@MDSB], [@PP]. The purpose of this section is to consider the comments from Bobenko and Springborn in [@BS], which suggests the use of Delaunay triangulations as a natural context in which to describe Laplace operators, and look at the generalization of these comments to regular triangulations.
Laplace operator defined\[laplace defined\]
-------------------------------------------
The suggested Laplace operator on two-dimensional surfaces in [@BS] (also seen in [@Hir], [@MDSB]) is the following operator on functions $f:\mathcal{T}_{0}\rightarrow\mathbb{R},$$$\left( \triangle f\right) _{i}=\sum_{j:\left\{ i,j\right\} \in
\mathcal{T}_{1}}w_{ij}\left( f_{j}-f_{i}\right) \label{delaunay laplacian}$$ where $w_{ij}$ is defined by $$w_{ij}=\frac{1}{2}\left( \cot\gamma_{kij}+\cot\gamma_{\ell ij}\right)$$ if $\gamma_{kij}$ is the angle at vertex $k$ in triangle $\left\{
i,j,k\right\} ,$ and the hinge containing $\left\{ i,j\right\} $ consists of the triangles $\left\{ i,j,k\right\} $ and $\left\{ i,j,\ell\right\} .$ Note that if $w_{ij}>0$ then this is a Laplacian with weights on the graph defined by the one-skeleton of the triangulation, and that $\triangle f_{i}>0$ if $f_{i}$ is the minimal value of $f$ and $\triangle f_{i}<0$ if $f_{i}$ is the maximal value of $f.$ Bobenko and Springborn note that if the triangulation is Delaunay, then $w_{ij}>0$ and the Laplacian is, in fact, a Laplacian on graphs in the classical sense (see [@Chun]).
A simple calculation shows that if we take the weights at all vertices to be zero, then the signed distance $$d_{\pm}\left[ C\left( \left\{ i,j,k\right\} \right) ,C\left( \left\{
i,j\right\} \right) \right] =r_{ijk}\cos\gamma_{kij}$$ where $r_{ijk}$ is the circumradius of triangle $\left\{ i,j,k\right\} $. Since the circumradius can be computed to be $$r_{ijk}=\frac{1}{2}\frac{\ell_{ij}}{\sin\gamma_{kij}}$$ we find that $$d_{\pm}\left[ C\left( \left\{ i,j,k\right\} \right) ,C\left( \left\{
i,j\right\} \right) \right] =\frac{1}{2}\ell_{ij}\cot\gamma_{kij}.$$ It immediately follows that$$w_{ij}=\frac{\left\vert \bigstar\left\{ i,j\right\} \right\vert }{\left\vert
\left\{ i,j\right\} \right\vert }.$$ We see that the Delaunay condition is equivalent to $w_{ij}>0,$ which is equivalent to $\left\vert \bigstar\left\{ i,j\right\} \right\vert >0.$
In general, Hirani [@Hir] suggests the following definition of Laplacian:$$\left( \triangle f\right) _{i}=\frac{1}{\left\vert \bigstar\left\{
i\right\} \right\vert }\sum_{j:\left\{ i,j\right\} \in\mathcal{T}_{1}}\frac{\left\vert \bigstar\left\{ i,j\right\} \right\vert }{\left\vert
\left\{ i,j\right\} \right\vert }\left( f_{j}-f_{i}\right) .
\label{laplacian}$$ This formula has roots in the following integration by parts formula for the smooth Laplacian:$$\int_{U}\triangle f~dV=\int_{\partial U}\nabla f\cdot n~dS
\label{integration by parts}$$ where $n$ is the unit normal to $\partial U.$ Taking $U=\bigstar\left\{
i\right\} $ and slightly rearranging terms, we get the corresponding formula on piecewise Euclidean manifolds$$\left( \triangle f\right) _{i}~\left\vert \bigstar\left\{ i\right\}
\right\vert =\sum_{j:\left\{ i,j\right\} \in\mathcal{T}_{1}}\frac
{f_{j}-f_{i}}{\left\vert \left\{ i,j\right\} \right\vert }\left\vert
\bigstar\left\{ i,j\right\} \right\vert$$ where $\frac{f_{j}-f_{i}}{\left\vert \left\{ i,j\right\} \right\vert }$ is the normal derivative and $\left\vert \bigstar\left\{ i,j\right\}
\right\vert $ is the surface area measure on the boundary of $\bigstar\left\{
i\right\} .$ This formula is well defined on any duality triangulation (which is the motivation for the definition) and coincides with (\[delaunay laplacian\]) in the case of Delaunay triangulations, except for the factor of $\left\vert \bigstar\left\{ i\right\} \right\vert $. One can think of the difference between considering the induced measure $\triangle
f~dV$ instead of the pointwise Laplacian $\triangle f.$ It is, in fact, natural to consider the measure instead since, if we consider the discrete Laplacian approximating a smooth one, the pointwise Laplacian is only accurate when considered on scales larger than the scale of the discretization.
We note that the Laplacian given by (\[laplacian\]) is also the same as the Laplacian considered by Chow-Luo [@CL] in two dimensions as observed by Z. He, where the duality is defined by Thurston triangulations as described above. It also appears in [@G1] [@G2] in three dimensions, where Thurston triangulations are considered such that $d_{ij}$ depend only on $i.$ Also, the Laplacian described in [@Luo] is actually the Laplacian described above in (\[delaunay laplacian\]) with the same weights $w_{ij}.$ The interest in these Laplacians is that they are not derived from means such as (\[integration by parts\]) but instead as the induced time derivative of curvature quantities under geometric evolutions.
The Laplacian defined in (\[laplacian\]) is a Laplacian with weights on graphs in the usual sense (see [@Chun]) if the coefficients $$\frac{\left\vert \bigstar\left\{ i,j\right\} \right\vert }{\left\vert
\bigstar\left\{ i\right\} \right\vert }$$ are each nonnegative. In two dimensions we see that this is implied by $d_{ij}>0$ and $\left\vert \bigstar\left\{ i,j\right\} \right\vert >0,$ which is the condition that the triangulation is regular.
Note that the Laplacian can be considered the gradient of a Dirichlet energy functional as described in [@BS], which is the analogue of the smooth functional $$E\left( f\right) =\int_{M}\left\vert \nabla f\right\vert ^{2}dV.$$ The Dirichlet energy functional induced by the duality triangulation is $$E\left( f\right) =\frac{1}{2}\sum_{\left\{ i,j\right\} \in\mathcal{T}_{1}}\frac{\left\vert \bigstar\left\{ i,j\right\} \right\vert }{\left\vert
\left\{ i,j\right\} \right\vert }\left( f_{j}-f_{i}\right) ^{2}.
\label{dirichlet energy}$$ This specializes in the case where the $w_{i}=0$ for all $i\in\mathcal{T}_{0}$ (or, equivalently, $d_{ij}=d_{ji}=\ell_{ij}/2$ for all $\left\{ i,j\right\}
\in\mathcal{T}_{1}$) to the Dirichlet energy in [@BS]. Note that this energy is positive if $\left\vert \bigstar\left\{ i,j\right\} \right\vert
>0.$
A generalization of Rippa’s theorem\[rippa\]
--------------------------------------------
Rippa [@Rip] showed that if one considers the Dirichlet energy (\[dirichlet energy\]) on a triangulation of points in $\mathbb{R}^{2}$ where the weights are zero (or equivalently, $d_{ij}=d_{ji}=\ell_{ij}/2$ for all edges $\left\{ i,j\right\} $), flipping to make an edge Delaunay increases the Dirichlet energy. Bobenko and Springborn [@BS] note that his proof extends trivially to piecewise Euclidean surfaces (2-dimensional manifolds). We shall express Rippa’s theorem in a way closer to the exposition on [@BS], which is in line with the notation in this paper.
\[[@Rip]\]Let $\left( \mathcal{T},\ell\right) $ be a piecewise Euclidean, triangulated surface with assigned edge lengths $\ell,$ which we think of as a weighted triangulation with all weights equal to zero. Let $\mathcal{T}_{0}$ be the vertices of the triangulation and let $f:\mathcal{T}_{0}\rightarrow
\mathbb{R}$ be a function. Suppose $\mathcal{T}^{\prime}$ is another triangulation which is gotten from $\mathcal{T}$ by a $2\rightarrow2$ bistellar flip on edge $e$ (in particular, $\mathcal{T}_{0}=\mathcal{T}_{0}^{\prime},$) such that the hinge is locally Delaunay after the flip. Then $$E_{\mathcal{T}^{\prime}}\left( f\right) \leq E_{\mathcal{T}}\left(
f\right) ,$$ where $E_{\mathcal{T}}$ and $E_{\mathcal{T}^{\prime}}$ are the Dirichlet energies corresponding to $\mathcal{T}$ and $\mathcal{T}^{\prime}.$ As a consequence, the minimum is attained when all edges are Delaunay (and hence the triangulation is a Delaunay triangulation).
Rippa’s proof involves calculating $E\left( f_{\mathcal{T}^{\prime}}\right)
-E\left( f_{\mathcal{T}}\right) $ and showing that it is negative. The key is a lemma which factors $E\left( f_{\mathcal{T}^{\prime}}\right) -E\left(
f_{\mathcal{T}}\right) $ and for which we shall give a direct proof later for the more general case of regular triangulations. The only thing missing is the proof of the final sentence, which requires that flipping edges eventually produces a Delaunay triangulation, which is proved in [@ILTC] and [@Riv]. We can generalize the first part of Rippa’s theorem to regular triangulations:
\[generalized rippa theorem\]Let $\left( \mathcal{T},d\right) $ be a duality triangulation of a surface with assigned local lengths $d$. Let $\mathcal{T}_{0}$ be the vertices of the triangulation and let $f:\mathcal{T}_{0}\rightarrow\mathbb{R}$ be a function. Suppose $\left( \mathcal{T}^{\prime},d^{\prime}\right) $ is another duality triangulation which is gotten from $\left( \mathcal{T},d\right) $ by a $2\rightarrow2$ bistellar flip on edge $e$ such that the hinge is locally regular after the flip. Then $$E_{\mathcal{T}^{\prime}}\left( f\right) \leq E_{\mathcal{T}}\left(
f\right) ,$$ where $E_{\mathcal{T}}$ and $E_{\mathcal{T}^{\prime}}$ are the Dirichlet energies corresponding to $\left( \mathcal{T},d\right) $ and $\left(
\mathcal{T}^{\prime},d^{\prime}\right) .$
The proof depends on the following important generalization of Rippa’s key lemma [@Rip Lemma 2.2] (see also [@Pow]).
Let $\mathcal{T=}\left\{ \left\{ 1,2,3\right\} ,\left\{ 1,2,4\right\}
\right\} $ and $\mathcal{T}^{\prime}=\left\{ \left\{ 1,3,4\right\}
,\left\{ 2,3,4\right\} \right\} $ be two hinges differing by a flip along $\left\{ 1,2\right\} $. Then $$E\left( f_{\mathcal{T}^{\prime}}\right) -E\left( f_{\mathcal{T}}\right)
=\left( f_{\mathcal{T}^{\prime}}\left( c\right) -f_{\mathcal{T}}\left(
c\right) \right) ^{2}A_{1234}^{2}\Phi$$ where $$\Phi=\frac{2\left( r_{3}r_{4}-r_{1}r_{2}\right) A_{1234}+w_{1}A_{234}+w_{2}A_{134}-w_{3}A_{124}-w_{4}A_{123}}{8A_{123}A_{134}A_{234}A_{124}},$$ $A_{ijk}$ is the area of $\left\{ i,j,k\right\} ,$ $A_{1234}=A_{123}+A_{124}=A_{134}+A_{234}$ is the area of the hinge, $c$ is the intersection of the diagonals, $r_{i}$ is the distance between $c$ and vertex $i,$ and $f_{\mathcal{T}^{\prime}}$ and $f_{\mathcal{T}}$ are the piecewise linear interpolations of $f$ with respect to the different triangulations. One can write$$\begin{aligned}
f_{\mathcal{T}}\left( c\right) & =\frac{r_{1}}{\ell_{12}}f_{2}+\frac
{r_{2}}{\ell_{12}}f_{1}\\
f_{\mathcal{T}^{\prime}}\left( c\right) & =\frac{r_{3}}{\ell_{34}}f_{4}+\frac{r_{4}}{\ell_{34}}f_{3}.\end{aligned}$$
The proof is somewhat involved although straightforward. We use a proof which is more direct than the ones given by Rippa [@Rip] and Powar [@Pow] for the case of Delaunay triangulations.
Because we are on a single hinge, it is equivalent to use weighted triangulations by Theorem \[equivalence of weighted and duality\]. Let $\left( \ell,w\right) $ be the corresponding lengths and weights. A simple calculation tells us that $$\frac{d_{\pm}\left( C\left( \left\{ i,j\right\} \right) ,C\left(
\left\{ i,j,k\right\} \right) \right) }{\ell_{ij}}=\frac{1}{2}\cot
\gamma_{kij}+\frac{w_{i}}{2\ell_{ij}^{2}}\cot\gamma_{jik}+\frac{w_{j}}{2\ell_{ij}^{2}}\cot\gamma_{ijk}-\frac{w_{k}}{4A_{ijk}},$$ where $\gamma_{ijk}$ is the angle at vertex $i$ in triangle $\left\{
i,j,k\right\} $ and $A_{ijk}=\left\vert \left\{ i,j,k\right\} \right\vert $ is the area. For simplicity, we shall use the notation $h_{ij,k}=d_{\pm
}\left( C\left( \left\{ i,j\right\} \right) ,C\left( \left\{
i,j,k\right\} \right) \right) ,$ which we think of as the height of the triangle $\left\{ i,j,C\left( \left\{ i,j,k\right\} \right) \right\} .$ Note that $\left\vert \bigstar\left\{ 1,2\right\} \right\vert =h_{12,3}+h_{12,4},$ for instance. For any function $f,$ we can compute$$E\left( f_{\mathcal{T}^{\prime}}\right) -E\left( f_{\mathcal{T}}\right)
=\frac{1}{2}\sum_{i,j=1}^{4}a_{ij}f_{i}f_{j},$$ where $$\begin{aligned}
a_{12} & =\frac{h_{12,3}}{\ell_{12}}+\frac{h_{12,4}}{\ell_{12}},\;\;\;a_{13}=\frac{h_{13,2}}{\ell_{13}}-\frac{h_{13,4}}{\ell_{13}},\\
a_{14} & =\frac{h_{14,2}}{\ell_{14}}-\frac{h_{14,3}}{\ell_{14}},\;\;\;a_{23}=\frac{h_{23,1}}{\ell_{23}}-\frac{h_{23,4}}{\ell_{23}},\\
a_{24} & =\frac{h_{24,1}}{\ell_{24}}-\frac{h_{24,3}}{\ell_{24}},\;\;\;a_{34}=-\frac{h_{34,1}}{\ell_{34}}-\frac{h_{34,2}}{\ell_{34}},\end{aligned}$$ and $a_{ii}=-\sum_{j\neq i}a_{ij}$ (where we have symmetrized $a_{ij}=a_{ji}$). We now wish to factor the coefficients.
We can easily figure out $r_{i}$ in terms of areas in the following way. For a realization of the hinge, with $v_{i}$ representing the coordinates of $\left\{ i\right\} ,$ we see that $c=v_{1}+\frac{r_{1}}{\ell_{12}}\left(
v_{2}-v_{1}\right) =v_{3}+\frac{r_{3}}{\ell_{13}}\left( v_{4}-v_{3}\right)
.$ By taking the cross product with $v_{2}-v_{1}$ or $v_{4}-v_{3}$ we find that $$r_{1}=\frac{\ell_{12}A_{134}}{A_{1234}}~~\text{and~~}r_{3}=\frac{\ell
_{34}A_{123}}{A_{1234}},$$ where $A_{1234}=A_{123}+A_{124}=A_{134}+A_{234}$ is the area of the entire hinge. Similarly, $$r_{2}=\frac{\ell_{12}A_{234}}{A_{1234}}~~\text{and~~}r_{4}=\frac{\ell
_{34}A_{124}}{A_{1234}}.$$ Thus $$\begin{aligned}
f_{\mathcal{T}^{\prime}}\left( c\right) -f_{\mathcal{T}}\left( c\right)
& =\frac{r_{3}}{\ell_{34}}f_{4}+\frac{r_{4}}{\ell_{34}}f_{3}-\frac{r_{1}}{\ell_{12}}f_{2}-\frac{r_{2}}{\ell_{12}}f_{1}\\
& =\frac{1}{A_{1234}}\left( A_{123}f_{4}+A_{124}f_{3}-A_{134}f_{2}-A_{234}f_{1}\right) .\end{aligned}$$ Also useful will be the calculation$$r_{3}r_{4}-r_{1}r_{2}=\frac{1}{A_{1234}^{2}}\left( \ell_{34}^{2}A_{123}A_{124}-\ell_{12}^{2}A_{234}A_{134}\right) .$$
There are essentially two different types of coefficients to consider. We need only consider $a_{12}$ and $a_{13}$ since the others are similar. Let $\gamma_{ijk}$ be the angle at vertex $i$ in triangle $\left\{ i,j,k\right\}
.$ Consider $a_{12}.$ $$\begin{aligned}
a_{12} & =\frac{h_{12,3}}{\ell_{12}}+\frac{h_{12,4}}{\ell_{12}}\\
& =\frac{1}{2}\cot\gamma_{312}+\frac{w_{1}}{2\ell_{12}^{2}}\cot\gamma
_{213}+\frac{w_{2}}{2\ell_{12}^{2}}\cot\gamma_{123}-\frac{w_{3}}{4A_{123}}\\
& \;\;\;+\frac{1}{2}\cot\gamma_{412}+\frac{w_{1}}{2\ell_{12}^{2}}\cot
\gamma_{214}+\frac{w_{2}}{2\ell_{12}^{2}}\cot\gamma_{124}-\frac{w_{4}}{4A_{124}}\\
& =\frac{1}{2}\left( \cot\gamma_{312}+\cot\gamma_{412}\right) +\frac{w_{1}}{2\ell_{12}^{2}}\left( \cot\gamma_{213}+\cot\gamma_{214}\right) \\
& \;\;\;+\frac{w_{2}}{2\ell_{12}^{2}}\left( \cot\gamma_{123}+\cot
\gamma_{124}\right) -\frac{w_{3}}{4A_{123}}-\frac{w_{4}}{4A_{124}}.\end{aligned}$$ Let $\theta$ be the angle at $c$ in the triangle $\left\{ 1,3,c\right\} .$ We shall use the fact that in any triangle $\left\{ i,j,k\right\} $ we have $\ell_{ij}=\ell_{ik}\cos\gamma_{ijk}+\ell_{jk}\cos\gamma_{jik}$ to compute the parts.$$\begin{aligned}
\cot\gamma_{312}+\cot\gamma_{412} & =\frac{\ell_{13}\ell_{23}\cos
\gamma_{312}}{2A_{123}}+\frac{\ell_{14}\ell_{24}\cos\gamma_{412}}{2A_{124}}\\
& =\frac{\ell_{13}^{2}-\ell_{12}\ell_{13}\cos\gamma_{123}}{2A_{123}}+\frac{\ell_{14}^{2}-\ell_{12}\ell_{14}\cos\gamma_{124}}{2A_{124}}\\
& =\frac{\ell_{13}^{2}}{2A_{123}}+\frac{\ell_{14}^{2}}{2A_{124}}-\left(
\left( \frac{\sin\gamma_{314}}{\sin\theta\sin\gamma_{123}}-\cot\theta\right)
+\left( \frac{\sin\gamma_{413}}{\sin\theta\sin\gamma_{124}}+\cot
\theta\right) \right) \\
& =\frac{\ell_{13}^{2}}{2A_{123}}+\frac{\ell_{14}^{2}}{2A_{124}}-\frac
{1}{\sin\theta}\left( \frac{\sin\gamma_{314}}{\sin\gamma_{123}}+\frac
{\sin\gamma_{413}}{\sin\gamma_{124}}\right) \\
& =\frac{\ell_{13}^{2}}{2A_{123}}+\frac{\ell_{14}^{2}}{2A_{124}}-\frac
{1}{\sin\theta}\frac{\ell_{12}A_{134}A_{1234}}{\ell_{34}A_{123}A_{124}}\\
& =\frac{\ell_{13}^{2}A_{124}+\ell_{14}^{2}A_{123}-\ell_{12}^{2}A_{134}}{2A_{123}A_{124}}\\
& =\frac{\ell_{13}^{2}+\ell_{14}^{2}}{2A_{1234}}+\frac{\ell_{13}^{2}A_{124}^{2}+\ell_{14}^{2}A_{123}^{2}-\ell_{12}^{2}A_{134}^{2}}{2A_{123}A_{124}A_{1234}}-\frac{\ell_{12}^{2}A_{134}A_{234}}{2A_{123}A_{124}A_{1234}}$$ since $$\sin\gamma_{314}=\cos\gamma_{123}\sin\theta+\sin\gamma_{123}\cos\theta$$ and $$\sin\gamma_{413}=\cos\gamma_{124}\sin\theta-\sin\gamma_{124}\cos\theta.$$ Furthermore,$$\begin{aligned}
\ell_{13}^{2}A_{124}^{2}+\ell_{14}^{2}A_{123}^{2}-\ell_{12}^{2}A_{134}^{2} &
=\frac{1}{4}\ell_{12}^{2}\ell_{13}^{2}\ell_{14}^{2}\left( \sin^{2}\gamma_{124}+\sin^{2}\gamma_{123}-\sin^{2}\left( \gamma_{123}+\gamma
_{124}\right) \right) \\
& =-\frac{1}{2}\ell_{12}^{2}\ell_{13}^{2}\ell_{14}^{2}\left( \sin
\gamma_{123}\sin\gamma_{124}\cos\gamma_{134}\right) \\
& =-2A_{123}A_{124}\ell_{13}\ell_{14}\cos\gamma_{134}$$ since $$\sin^{2}A+\sin^{2}B-\sin^{2}\left( A+B\right) =-2\sin A\sin B\cos\left(
A+B\right) .$$ Thus we have$$\begin{aligned}
\cot\gamma_{312}+\cot\gamma_{412} & =\frac{\left( \ell_{13}^{2}+\ell
_{14}^{2}-2\ell_{13}\ell_{14}\cos\gamma_{134}\right) }{2A_{1234}}-\frac
{\ell_{12}^{2}A_{134}A_{234}}{2A_{123}A_{124}A_{1234}}\\
& =\frac{\ell_{34}^{2}A_{123}A_{124}-\ell_{12}^{2}A_{134}A_{234}}{2A_{1234}A_{123}A_{124}}\\
& =\frac{A_{1234}}{A_{123}A_{124}}\left( r_{3}r_{4}-r_{1}r_{2}\right) .\end{aligned}$$ For the other parts,$$\begin{aligned}
\cot\gamma_{213}+\cot\gamma_{214} & =\frac{\cos\gamma_{213}}{\sin
\gamma_{213}}+\frac{\cos\gamma_{214}}{\sin\gamma_{214}}\\
& =\frac{\sin\gamma_{234}}{\sin\gamma_{213}\sin\gamma_{214}}\\
& =\frac{\ell_{12}^{2}A_{234}}{2A_{123}A_{124}}$$ and$$\cot\gamma_{123}+\cot\gamma_{124}=\frac{\ell_{12}^{2}A_{134}}{2A_{123}A_{124}}.$$ Thus$$\begin{aligned}
a_{12} & =\frac{1}{2}\left( \cot\gamma_{312}+\cot\gamma_{412}\right)
+\frac{w_{1}}{2\ell_{12}^{2}}\left( \cot\gamma_{213}+\cot\gamma_{214}\right)
\\
& +\frac{w_{2}}{2\ell_{12}^{2}}\left( \cot\gamma_{123}+\cot\gamma
_{124}\right) -\frac{w_{3}}{4A_{123}}-\frac{w_{4}}{4A_{124}}.\end{aligned}$$ implies that$$\begin{aligned}
a_{12} & =\frac{A_{234}A_{134}}{4A_{123}A_{134}A_{234}A_{124}}\left(
2A_{1234}\left( r_{3}r_{4}-r_{1}r_{2}\right) +w_{1}A_{234}+w_{2}A_{134}-w_{3}A_{124}-w_{4}A_{123}\right) \\
& =2A_{234}A_{134}\Phi.\end{aligned}$$ (Recall $$\Phi=\frac{2A_{1234}\left( r_{3}r_{4}-r_{1}r_{2}\right) +w_{1}A_{234}+w_{2}A_{134}-w_{3}A_{124}-w_{4}A_{123}}{8A_{123}A_{134}A_{234}A_{124}}$$ as in the statement of the lemma.)
Now consider $a_{13}.$ We can compute $$\begin{aligned}
a_{13} & =\frac{h_{13,2}}{\ell_{13}}-\frac{h_{13,4}}{\ell_{13}}\\
& =\frac{1}{2}\cot\gamma_{213}+\frac{w_{1}}{2\ell_{13}^{2}}\cot\gamma
_{312}+\frac{w_{3}}{2\ell_{13}^{2}}\cot\gamma_{123}-\frac{w_{2}}{4A_{123}}\\
& \;\;\;-\left( \frac{1}{2}\cot\gamma_{413}+\frac{w_{1}}{2\ell_{13}^{2}}\cot\gamma_{314}+\frac{w_{3}}{2\ell_{13}^{2}}\cot\gamma_{134}-\frac{w_{4}}{4A_{134}}\right) \\
& =\frac{1}{2}\left( \cot\gamma_{213}-\cot\gamma_{413}\right) +\frac{w_{1}}{2\ell_{13}^{2}}\left( \cot\gamma_{312}-\cot\gamma_{314}\right) \\
& \;\;\;+\frac{w_{3}}{2\ell_{13}^{2}}\left( \cot\gamma_{123}-\cot
\gamma_{134}\right) -\frac{w_{2}}{4A_{123}}+\frac{w_{4}}{4A_{134}}.\end{aligned}$$ We see that $$\begin{aligned}
\cot\gamma_{213}-\cot\gamma_{413} & =\frac{\sin\gamma_{324}}{\sin
\gamma_{213}\sin\theta}-\frac{\sin\gamma_{124}}{\sin\gamma_{413}\sin\theta}\\
& =\frac{\ell_{12}^{2}A_{134}A_{234}-\ell_{34}^{2}A_{123}A_{124}}{2A_{1234}A_{123}A_{134}}$$ since $\sin\gamma_{324}=-\cos\theta\sin\gamma_{213}+\sin\theta\cos\gamma
_{213}$ and similarly $\sin\gamma_{124}=-\cos\theta\sin\gamma_{413}+\sin
\theta\cos\gamma_{413}.$ We also get$$\begin{aligned}
\cot\gamma_{312}-\cot\gamma_{314} & =\frac{\cos\gamma_{312}\sin\gamma
_{314}-\cos\gamma_{314}\sin\gamma_{312}}{\sin\gamma_{312}\sin\gamma_{314}}\\
& =-\frac{\sin\gamma_{324}}{\sin\gamma_{312}\sin\gamma_{314}}\\
& =-\frac{\ell_{13}^{2}A_{234}}{2A_{123}A_{134}}$$ and$$\cot\gamma_{123}-\cot\gamma_{134}=\frac{\ell_{13}^{2}A_{124}}{2A_{123}A_{134}}.$$ And so $$\begin{aligned}
a_{13} & =\frac{-A_{234}A_{124}}{4A_{123}A_{134}A_{234}A_{124}}\left(
2A_{1234}\left( r_{3}r_{4}-r_{1}r_{2}\right) +w_{1}A_{234}-w_{3}A_{124}+w_{2}A_{134}-w_{4}A_{123}\right) \\
& =-2A_{234}A_{124}\Phi.\end{aligned}$$
A similar argument gives the other coefficients. Then we see, for instance, that $$\begin{aligned}
a_{11} & =-a_{12}-a_{13}-a_{14}\\
& =2\left( -A_{234}A_{134}+A_{234}A_{124}+A_{234}A_{123}\right) \Phi\\
& =2A_{234}^{2}\Phi\end{aligned}$$ with similar expressions for $a_{22},$ $a_{33},$ and $a_{44}.$ Finally, we get that $$E\left( f_{\mathcal{T}^{\prime}}\right) -E\left( f_{\mathcal{T}}\right)
=\left( A_{123}f_{4}+A_{124}f_{3}-A_{234}f_{1}-A_{134}f_{2}\right) ^{2}\Phi,$$ which is equivalent to the lemma.
Now we can prove the theorem.
\[Proof of Theorem \[generalized rippa theorem\]\]Note that since we are only concerned with a hinge, it is equivalent to consider weighted triangulations or duality triangulations. Since the coefficient $a_{12}=\frac{\left\vert
\bigstar\left\{ 1,2\right\} \right\vert }{\left\vert \left\{ 1,2\right\}
\right\vert }$ and $a_{34}=-\frac{\left\vert \bigstar\left\{ 3,4\right\}
\right\vert }{\left\vert \left\{ 3,4\right\} \right\vert },$ we see that $a_{12}<0$ and $a_{34}<0$ if and only if $\mathcal{T}^{\prime}$ is regular after the flip and not regular before the flip. Since all areas $A_{ijk}$ are positive, $a_{12}<0$ if and only if $\Phi<0$ and hence the result is proven.
Note that in the proof we have shown that $\Phi<0$ if and only if $\mathcal{T}$ is not regular and $\mathcal{T}^{\prime}$ is regular.
In order to get the global statement, one needs to know that a regular triangulation can be found using flips. This is not true in general (see [@ES]). However, we investigated some conditions when a flip algorithm does work in Section \[flip algorithms\].
As a corollary of Rippa’s theorem, we get an entropy quantity that increases under the action of flipping to make a hinge regular.
\[eigenvalue cor\]Consider the entropy defined by $$\Lambda=\inf\left\{ E\left( f\right) :\sum_{i\in\mathcal{T}_{0}}f_{i}^{2}=1\text{ and }\sum_{i\in\mathcal{T}_{0}}f_{i}=0\right\} .$$ Then $\Lambda$ decreases when an edge is flipped to make the hinge regular.
Let $\Lambda^{\prime}$ denote the entropy after the flip and let $f_{0}$ be the $f$ which realize $\Lambda$ (since $f$ is in a compact set, there must be an actual $f$ which minimizes $E\left( f\right) $). Then $$\Lambda^{\prime}=\inf_{f}E_{\mathcal{T}^{\prime}}\left( f\right) \leq
E_{\mathcal{T}^{\prime}}\left( f_{0}\right) \leq E_{\mathcal{T}}\left(
f_{0}\right) =\Lambda.$$
Note that $\Lambda$ can be considered an eigenvalue of a particular operator closely related to $\triangle.$ We remark that Corollary \[eigenvalue cor\] is similar in spirit to what is proven by G. Perelman at the beginning of his paper [@Per], where he shows that a slightly more complicated entropy, $$\inf\left\{ \int\left( Rf^{2}+4\left\vert \nabla f\right\vert ^{2}\right)
dV:\int f^{2}dV=1\right\} ,$$ where $R$ is the scalar curvature, increases under Ricci flow.
Note that in $n$ dimensions, the regularity condition corresponds to $\left\vert \bigstar\sigma^{n-1}\right\vert >0$ while good Dirichlet energy corresponds to $\left\vert \bigstar\sigma^{1}\right\vert >0.$ Hence the correspondence between regular triangulations and the Dirichlet energy only occurs in dimension $2$ because $1=2-1,$ which is why the theorem is only described for dimension $2.$ Although we do not pursue it here, this may indicate that the Laplacian should instead be defined on functions on vertices of the dual complex, $f:\bigstar\mathcal{T}_{n}\rightarrow\mathbb{R}$, in which case the Laplacian would be $$\left( \triangle f\right) _{\bigstar\sigma_{0}^{n}}=\frac{1}{\left\vert
\sigma_{0}^{n}\right\vert }\sum_{\sigma^{n}\in\mathcal{T}_{n}}\frac{\left\vert
\sigma^{n}\cap\sigma_{0}^{n}\right\vert }{\left\vert \bigstar\left(
\sigma^{n}\cap\sigma_{0}^{n}\right) \right\vert }\left( f_{\bigstar
\sigma^{n}}-f_{\bigstar\sigma_{0}^{n}}\right)$$ where the sum is over all $n$-simplices. In this case, positivity of the coefficients corresponds to being regular.
Laplace and heat equations\[laplace and heat equations\]
--------------------------------------------------------
Given a Laplace operator, we can now consider the standard elliptic and parabolic equations, namely the Laplace equation $$\triangle u=0 \label{laplace equation}$$ and the heat equation$$\frac{du}{dt}=\triangle u, \label{heat equation}$$ where the heat equation is an ordinary differential equation since $\triangle$ is a difference operator. A solution $u$ to the Laplace equation is called a *harmonic function*.
In order to study these equations, it will sometimes be easier to consider $\triangle u=0$ as a matrix equation. We think of $u:\mathcal{T}_{0}\rightarrow\mathbb{R}$ as a vector and $\triangle$ corresponds to a matrix $L$ whose off-diagonal pieces are $$L_{ij}=\frac{\left\vert \bigstar\left\{ i,j\right\} \right\vert }{\left\vert
\left\{ i,j\right\} \right\vert }$$ and whose diagonal pieces are$$L_{ii}=-\sum_{j:\left\{ i,j\right\} \in\mathcal{T}_{1}}\frac{\left\vert
\bigstar\left\{ i,j\right\} \right\vert }{\left\vert \left\{ i,j\right\}
\right\vert }.$$ Then one can write the Laplace equation as $$Lu=0.$$ Note that if we wish to consider Poisson’s equation $$\triangle u=f \label{poisson equation}$$ then this is equivalent to $$Lu=fV$$ where $\left( fV\right) _{i}=f_{i}\left\vert \bigstar\left\{ i\right\}
\right\vert .$ It is clear that $L$ has the constant functions $f_{i}=a$ (or the vector $\left( a,a,\ldots,a\right) $) in the nullspace. If $\left\vert
\bigstar\left\{ i,j\right\} \right\vert >0$ then we find the following.
\[laplace neg semidef 1\]If $\left\vert \bigstar\left\{ i,j\right\}
\right\vert >0$ for all edges $\left\{ i,j\right\} $ then $L$ is negative semidefinite with nullspace spanned by the constant vectors.
In this case we have an $N\times N$ matrix $L$ with diagonal entries negative and off-diagonal entries positive and with $\sum_{j=1}^{N}L_{ij}=0.$ We reiterate an argument from [@CR]. Let $\left( v_{1},\ldots,v_{N}\right)
$ be an eigenvector corresponding to $\lambda\geq0.$ We may assume that $v_{1}>0$ is the maximum of $v_{i}.$ We wish to show that $v_{i}=v_{j}$ for all $i,j.$ Observe$$\lambda v_{1}=\sum_{i=1}^{N}L_{1i}v_{i}\leq\sum_{i=1}^{N}L_{1i}v_{1}=0.$$ Equality holds if and only if $v_{i}=v_{1}$ for all $i.$
If $\left\vert \bigstar\left\{ i,j\right\} \right\vert >0$ for all edges $\left\{ i,j\right\} $ then Poisson’s equation has a solution for any $f$ such that $$\sum_{i\in\mathcal{T}_{0}}f_{i}V_{i}=0.$$
This is the analogue of the smooth result that $\triangle u=f$ has a solution if $\int_{M}fdV=0.$ One may also consider boundary conditions such as Dirichlet and Neumann conditions. These cases for Delaunay triangulations in two dimensions were studied by Bobenko and Springborn [@BS].
The condition $\left\vert \bigstar\left\{ i,j\right\} \right\vert >0$ is obviously very important for the proof of Theorem \[laplace neg semidef 1\]. In two dimensions, this condition is equivalent to being regular by Corollary \[edge regular duality\]. It is not always necessary to assume $\left\vert
\bigstar\left\{ i,j\right\} \right\vert >0$, as seen in the following special cases.
Recall that in two dimensions, if a duality triangulation is edge-positive, then the flip algorithm finds a regular triangulation (Theorem \[2D edge flip algo\]). For a similar set of two-dimensional triangulations, the Laplacian is negative semidefinite.
\[laplace neg semidef 2\]For any triangulation such that $d_{ij}>0$ for all $\left( i,j\right) \in\mathcal{T}_{1}^{+},$ the Laplacian matrix $L$ is negative semidefinite with nullspace spanned by the constant vectors.
We begin with a series of claims and an important lemma before beginning the proof. We shall prove this by a sequence of claims. For all of the claims it is assumed that the weights $d_{ij}$ are all positive. We shall use $h_{ij}=d_{\pm}\left[ C\left( \left\{ 1,2,3\right\} \right) ,C\left(
\left\{ i,j\right\} \right) \right] $ and $\gamma_{i}$ is the angle at vertex $i.$ Consider only the $3\times3$ matrix $M$ corresponding to $\left\{
1,2,3\right\} $ with entries $M_{ij}=h_{ij}/\ell_{ij}$ if $i\neq j$ and $M_{ii}=-\sum_{j\neq i}M_{ij}.$
If $h_{ij}<0$ then $\gamma_{i}<\frac{\pi}{2}$ and $\gamma_{j}<\frac{\pi}{2}.$
Let $k$ be the third vertex so that $\left\{ i,j,k\right\} =\left\{
1,2,3\right\} .$ We know that $$h_{ij}=\frac{d_{ik}-d_{ij}\cos\gamma_{i}}{\sin\gamma_{i}}$$ by formula \[center distance\]). If $h_{ij}<0$ then $0<d_{ik}<d_{ij}\cos\gamma_{i}.$ Hence $\cos\gamma_{i}>0$ and $\gamma_{i}<\pi/2.$ We can also express $h_{ij}$ as $$h_{ij}=\frac{d_{jk}-d_{ji}\cos\gamma_{j}}{\sin\gamma_{j}}$$ and follow the same logic.
Thus only one $M_{ij}$ may be negative. Suppose it is $M_{12}.$
$M_{12}+M_{13}=\frac{\ell_{23}\left( d_{12}\cos\gamma_{2}+d_{13}\cos
\gamma_{3}\right) }{2A_{123}}.$
We calculate$$\begin{aligned}
M_{12}+M_{13} & =\frac{d_{23}-d_{21}\cos\gamma_{2}}{\ell_{12}\sin\gamma_{2}}+\frac{d_{32}-d_{31}\cos\gamma_{3}}{\ell_{13}\sin\gamma_{3}}\\
& =\frac{\ell_{23}\left( \ell_{23}-d_{21}\cos\gamma_{2}-d_{31}\cos\gamma
_{3}\right) }{2A_{123}}$$ and finally we use that $\ell_{23}=\ell_{12}\cos\gamma_{2}+\ell_{13}\cos
\gamma_{2}.$
$d_{12}\cos\gamma_{2}+d_{13}\cos\gamma_{3}>0.$
If both $\gamma_{2}$ and $\gamma_{3}$ are less than or equal to $\pi/2$ then this is clear (since both may not be equal to $\pi/2$). Since $M_{12}<0,$ and hence $h_{12}<0,$ we can only have $\gamma_{3}>\pi/2.$ Since $h_{12}<0$ and $h_{13}>0$ we have that $$\frac{d_{13}}{d_{12}}<\cos\gamma_{1}<\frac{d_{12}}{d_{13}}$$ so $d_{12}>d_{13}.$ Furthermore, since $\gamma_{1}+\gamma_{2}<\pi$ we have that$$0<-\cos\gamma_{3}=\cos\left( \gamma_{1}+\gamma_{2}\right) <\cos\gamma_{2}$$ so $$-d_{13}\cos\gamma_{3}<d_{12}\cos\gamma_{2}.$$
\[diagonal entries negative lemma\]$M_{ii}<0.$
By the above argument, we know that $M_{11}=-M_{12}-M_{13}<0.$ Similar arguments hold for the other coefficients.
\[Proof of Theorem \[laplace neg semidef 2\]\]It is sufficient to prove that for any matrix $M_{ij},$ $1\leq i,j\leq3,$ is negative semidefinite. We know that the vector $\left( 1,1,1\right) $ is in the nullspace and we have already shown in Lemma \[diagonal entries negative lemma\] that the diagonal entries are negative. Hence it is sufficient to show that the determinant of the $2\times2$ submatrix $M_{ij},$ $1\leq i,j\leq2,$ is positive. We find that the $2\times2$ determinant is equal to $M_{12}M_{13}+M_{12}M_{23}+M_{13}M_{23}.$ We compute the determinant to be equal to $$\frac{\left( d_{13}h_{23}+d_{23}h_{13}\right) \sin\gamma_{2}}{\ell_{12}\ell_{13}}$$ (to do this calculation, begin by writing the terms in the determinant using formula (\[center distance\]) choosing all of the denominators to contain $\sin\gamma_{1}\sin\gamma_{2},$ then rearrange the terms using the facts that $\gamma_{1}+\gamma_{2}+\gamma_{3}=\pi,$ $d_{ij}+d_{ji}=\ell_{ij},$ and $\ell_{ij}=\ell_{ik}\cos\gamma_{i}+\ell_{jk}\cos\gamma_{k}$ several times and finally recollecting $h_{23}$ and $h_{13}$ again using formula (\[center distance\])). Note that the determinant is symmetric in all permutations in $1,2,3.$ We know by the claim above that two of the three $h_{ij}$ must be positive, so choosing the two that are positive, we must have that the determinant is positive. Hence the matrix is negative semidefinite.
We consider $d_{ij}$ to be the length of a vector located at $i$ and in the direction towards $j.$ Thus the condition $d_{ij}>0$ is like a positivity (or Riemannian) condition for a metric (which measures the length of vectors) and is thus a somewhat natural condition. The following is another result on definiteness of the Laplacian with different assumptions.
For a three-dimensional sphere packing triangulation, $L$ is negative semidefinite with nullspace spanned by the constant vectors.
It is proven in [@G2] (see also [@Riv2]) that the matrix $A_{\left\{
1,2,3,4\right\} }=\left( \frac{\partial\alpha_{i}}{\partial r_{j}}\right)
_{1\leq i,j\leq4}$ is negative semidefinite with nullspace spanned by the vector $\left( r_{1},\ldots,r_{4}\right) $. If we let $R_{\left\{
1,2,3,4\right\} }$ be the diagonal matrix with $r_{i},$ $i=1,\ldots,4$ on the diagonal, we see that $$L=\sum_{\sigma^{3}\in\mathcal{T}_{3}}\left( R_{\sigma^{3}}A_{\sigma^{3}}R_{\sigma^{3}}\right) _{E}.$$ where $\left( M_{\sigma^{3}}\right) _{E}$ is the matrix extended by zeroes to a $\left\vert \mathcal{T}_{0}\right\vert \times\left\vert \mathcal{T}_{0}\right\vert $ matrix so that the $\left( M_{\sigma^{3}}\right) _{E}$ acts on a vector $\left( v_{1},\ldots,v_{\left\vert \mathcal{T}_{0}\right\vert }\right) $ only on the coordinates corresponding to vertices in $\sigma^{3}.$ Since $r_{i}>0$ for all $i\in\mathcal{T}_{0},$ it follows that $L$ is negative semidefinite with nullspace spanned by $\left(
1,\ldots,1\right) .$
The importance of this result is it does not assume any positivity of the dual area, which appears to be stronger than the assumption that $L$ is negative definite. If $L$ is negative semi-definite with nullspace spanned by the constant vector $\left( 1,\ldots,1\right) $ then one can always solve the Poisson equation for $f$ such that $\sum f_{i}A_{i}=0.$
The heat equation is an time-dependent, linear ordinary differential equation$$\frac{du}{dt}=Lu$$ whose short time existence is guaranteed by the existence theorem for ordinary differential equations. One of the key properties of the heat equation is the maximum principle, which says that the maximum decreases and the minimum increases. This is true if $\left\vert \bigstar\left\{ i,j\right\}
\right\vert >0$.
If $\left\vert \bigstar\left\{ i,j\right\} \right\vert >0$ then for a solution $u_{i}\left( t\right) $ of the heat equation, $u_{\max}\left(
t\right) $ decreases and $u_{\min}\left( t\right) $ increases, where $u_{\max}=\max\left\{ u_{i}:i\in\mathcal{T}_{0}\right\} $ and $u_{\min}=\min\left\{ u_{i}:i\in\mathcal{T}_{0}\right\} .$
The proof is standard and is simply that for any operator $Eu$ defined by $$\left( Eu\right) _{i}=\sum_{j\neq i}e_{ij}\left( u_{j}-u_{i}\right)$$ for some weights $e_{ij}>0,$ then $\left( Eu\right) _{i}<0$ if $u_{i}=u_{\max}$ and $\left( Eu\right) _{i}>0$ if $u_{i}=u_{\min}.$
Note that the maximum principle is not equivalent to $L$ being negative semidefinite; it is a stronger condition and the proof uses that the coefficients off the diagonal are positive. However, for certain functions (geometric ones which are related to the coefficients of the Laplacian), it may be possible to show that the maximum decreases and the minimum increases. We call this a maximum principle for the function $f$ and we say that the operator is *parabolic-like* for the function $f.$ In [@G2] it is proven that the sphere-packing case is parabolic-like for a curvature function $K.$
Toward discrete Riemannian manifolds\[Riemannian\]
==================================================
Much of this work arose out of an attempt to describe Riemannian manifolds using piecewise Euclidean methods. In this final section, we try to describe some of the work already done toward this end. There are two different philosophies. One is to find analogues of the Riemannian setting. The idea is to set up a framework on which variational-type arguments may be made analogously to those in the smooth setting. The other is to actually approximate smooth Riemannian geometry with discrete geometric structures. We shall briefly consider both of these.
Analogues of Riemannian geometry
--------------------------------
In this paper we gave a discrete operator on duality triangulations which, it was argued, is an analogue of the Laplacian on a Riemannian manifold. This gives rise also to a discrete heat equation, which is an ordinary differential equation in this setting. It is not hard to imagine that similar arguments give rise to Laplace-Beltrami operators on forms with the proper definition of forms. A $k$-form can be defined to be an element of the dual space to the vector space spanned by the $k$-dimensional simplices. There are also dual $k$-forms which are elements of the dual space to the vector space spanned by the duals of the $\left( n-k\right) $-dimensional simplices. Hirani [@Hir] describes how to use duality information as we have described to define the Hodge star operation, and thereby the Laplace-Beltrami operator on these forms. One may then ask about an analogue of the Hodge theorem. This has been studied somewhat by R. Hiptmair [@Hipt]. Study of the Laplace-Beltrami operator on manifolds is also related to the study of the Laplacian and harmonic analysis on metrized graphs and electrical networks (see [@DS], [@BF], [@BR]).
Another important aspect of Riemannian geometry is the study of geodesics, which we recall are locally length-minimizing curves. In the setting of piecewise Euclidean manifolds, the geodesics are piecewise linear. One may then ask many questions about geodesics, such as the number of closed geodesics (see Pogorelov’s work on quasi-geodesics on convex surfaces [@Pog]) and the size of the cut locus to a basepoint, the locus of points with two or more geodesics connecting it to the basepoint (see Miller-Pak [@MP]). Many results on geodesics on piecewise Euclidean manifolds were found by D. Stone [@Sto], which lead him to some possible definitions of curvature. The discrete geodesic problem for polytopes in $\mathbb{R}^{3}$ was studied extensively in [@MMP].
Much of modern Riemannian geometry is concerned with different notions of curvature, such as sectional, Ricci, and scalar. In the piecewise Euclidean setting, there are a number of definitions of curvatures, although it is still somewhat an open question which ones are the proper ones for classification purposes. Since the literature in this area is vast, we simply indicate some of the principle works. D. Stone [@Sto] was successful in proving analogues of the Cartan-Hadamard theorem (that negatively curved manifolds have universal cover homeomorphic to $\mathbb{R}^{n}$) and Myer’s theorem (that positively curved manifolds are compact with a bound on the diameter) on piecewise Euclidean manifolds using a quantity which he calls bounds on sectional curvature. T. Regge introduced a notion of scalar curvature which is described at each $\left( n-2\right) $-dimensional simplex as $2\pi$ minus the sum of the dihedral angles at that simplex [@Reg]. This has been widely studied as the so-called Regge calculus (see, for instance, [@Fro], [@HW], [@Ham], [@ACM]). There are even some convergence results, which we mention in the next section. Another potential curvature quantity in three dimensions is described by Cooper and Rivin in [@CR]. They consider the curvature at a vertex to be $4\pi$ minus the sum of the solid (or trihedral) angles at the vertex. This curvature is certainly weaker than the curvature introduced by Regge, but may be related to scalar curvature. It is possible that the right curvature quantity will lead to a geometric flow which simplifies geometry in a way similar to the way Ricci or Yamabe flow do in the smooth category. This has been studies a bit in [@CL], [@Luo], [@G1], [@G2], and actually was the initial motivation for the definitions of Laplacian described in this paper. Other applications of discrete analogues of Riemannian geometry or geometric operators can be found in [@BS], [@ILTC], [@MDSB], [@Mer], [@PP], and [@WGCTY]. In addition, techniques applying to metric spaces with sectional curvature bounded in the sense of Alexandrov may apply (see [@BBI]).
Approximating Riemannian geometry
---------------------------------
Another goal is to approximate Riemannian geometry by a discrete geometry such as piecewise Euclidean triangulations. One would hope to be able to find elements of Riemannian geometry such as Laplacian, Levi-Civita connection, sectional curvature, scalar curvature, and so forth and not only have analogous structures, but be able to show that as the triangulation gets finer and finer, the discrete versions converge to the smooth versions. We mention here some of the results which have been successful in this direction.
One of the most influential works is by Cheeger, Müller, and Schrader, who were able to relate discrete curvatures to Lipschitz-Killing curvatures [@CMS]. The relevant discrete curvature is the sum certain angles and volumes of hinges. In particular, the scalar curvature measure ($RdV$) is concentrated on $\left( n-2\right) $-dimensional hinges in a triangulation, and under a condition that the triangulation does not degenerate, they find that the curvature quantity $2\pi$ minus the sum of the dihedral angles multiplied by the volume of the $\left( n-2\right) $-dimensional hinge converges to the scalar curvature measure. This version of scalar curvature is also the one suggested by Regge [@Reg] and used extensively in the Regge calculus. They prove convergence for each of the Lipschitz-Killing curvatures. In addition, Barrett and Parker [@BP] proved a pointwise convergence of piecewise-linear approximations of the Riemannian metric tensor and certain types of tensor fields.
In regards to the Laplacian, some experimental work has been done by G. Xu studying pointwise convergence of different discretized Laplace-Beltrami operators to the smooth ones [@Xu1] [@Xu2]. Some of the discretizations are the same or similar to those considered in this paper, while some are not. On graphs (one-dimensional manifolds and generalizations), it has been shown that the eigenvalues of the discrete Laplacians on metrized graphs converge to the eigenvalues of the smooth Laplacian on a metrized graph [@Fuj1] [@Fuj2] [@Fab].
It was W. Thurston’s idea to approximate the Riemann mapping between subsets of $\mathbb{C}$ by mappings of circle packings. Such a discretization has been shown to actually converge to the Riemann mapping [@RS].
I would like to thank Herbert Edelsbrunner, Feng Luo, and Igor Pak for some very helpful conversations related to parts of this paper.
[99]{}
J. Ambjørn, M. Carfora, and A. Marzuoli. *The geometry of dynamical triangulations*, Lecture Notes in Physics. New Series m: Monographs, vol. 50, Springer-Verlag, Berlin, 1997.
F. Aurenhammer and R. Klein. Voronoi diagrams. Handbook of computational geometry, 201–290, North-Holland, Amsterdam, 2000.
M. Baker and X. Faber. Metrized graphs, electrical networks, and Fourier analysis, preprint at arXiv:math.CO/0407428.
M. Baker and R. Rumely. Harmonic analysis on metrized graphs, preprint at arXiv:math.CO/0407427.
J. W. Barrett and P. E. Parker. Smooth limits of piecewise-linear approximations, J. Approx. Theory 76 (1994), no. 1, 107–122.
A. I. Bobenko and B. A. Springborn. A discrete Laplace-Beltrami operator for simplicial surfaces, preprint at arXiv:math.DG/0503219.
G. E. Bredon. *Topology and geometry*. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1993.
D. Burago, Y. Burago and S. Ivanov. *A course in metric geometry.* Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. *Corrections of typos and small errors to the book “A Course in Metric Geometry”*: http://www.pdmi.ras.ru/staff/burago.html\#English.
J. Cheeger, W. Müller, and R. Schrader. On the curvature of piecewise flat spaces, Comm. Math. Phys. 92, no. 3 (1984), 405–454.
B. Chow and F. Luo. Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003), 97–129.
F. R. K. Chung. *Spectral graph theory*. CBMS Regional Conference Series in Mathematics, 92. American Mathematical Society, Providence, RI, 1997.
D. Cooper and I. Rivin. Combinatorial scalar curvature and rigidity of ball packings, Math. Res. Lett. 3 (1996), no. 1, 51–60.
P. G. Doyle and J. L. Snell. *Random walks and electric networks*. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984.
H. Edelsbrunner. Triangulations and meshes in computational geometry, Acta Numerica (2000), 133-213.
H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangulations. Algorithmica 15 (1996), no. 3, 223–241.
X. W. C. Faber. Spectral convergence of the discrete Laplacian on models of a metrized graph, preprint at arXiv:math.CA/0502347.
J. Fröhlich. Regge calculus and discretized gravitational functional integrals, Nonperturbative quantum field theory: Mathematical aspects and applications, Selected papers, Advanced Series in Mathematical Physics, vol. 15, World Scientific Publishing Co. Inc., River Edge, NJ, 1992, 523–545.
K. Fujiwara. Convergence of the eigenvalues of Laplacians in a class of finite graphs, Geometry of the spectrum (Seattle, WA, 1993), 115–120, Contemp. Math., 173, Amer. Math. Soc., Providence, RI, 1994.
K. Fujiwara. Eigenvalues of Laplacians on a closed Riemannian manifold and its nets, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2585–2594.
D. Glickenstein. A combinatorial Yamabe flow in three dimensions, Topology 44 (2005), No. 4, 791-808.
D. Glickenstein. A maximum principle for combinatorial Yamabe flow, Topology 44 (2005), No. 4, 809-825.
H. W. Hamber. Simplicial quantum gravity, Phénomènes critiques, systèmes aléatoires, théories de jauge, Part I, II (Les Houches, 1984), North-Holland, Amsterdam, 1986, 375–439.
H. W. Hamber and R. M. Williams. Simplicial quantum gravity in three dimensions: analytical and numerical results, Phys. Rev. D (3) 47 (1993), no. 2, 510–532.
A. Hatcher. *Algebraic topology*. Cambridge University Press, Cambridge, 2002.
Z.-X. He. Rigidity of infinite disk patterns, Ann. of Math. (2) 149 (1999), no. 1, 1–33.
R. Hiptmair. Discrete Hodge operators, Numer. Math., 90(2):265–289, 2001a.
A. N. Hirani. Discrete exterior calculus, Ph.D. thesis, California Institute of Technology, Pasadena, CA, May 2003.
C. Indermitte, Th. M. Liebling, M. Troyanov, and H. Clémençon. Voronoi diagrams on piecewise flat surfaces and an application to biological growth. Theoret. Comput. Sci. 263 (2001), no. 1-2, 263–274.
B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations, Computer Aided Geometric Design, v.8 n.2 (May 1991), 123-142.
C. L. Lawson. Software for $C^{1}$ suface interpolation, in *Mathematical Software III*, Academic Press, New York, 1977, 161-194.
C. L. Lawson. Transforming triangulations, Discrete Math. 3 (1972), 365–372.
G. Leibon. Random Delaunay triangulations, the Thurston-Andreev theorem, and metric uniformization, Ph.D. thesis, University of California at San Diego, La Jolla, CA, 1999.
F. Luo. Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6 (2004), no. 5, 765–780.
A. Marden and B. Rodin. On Thurston’s formulation and proof of Andreev’s theorem, Computational methods and function theory (Valparaíso, 1989), Springer, Berlin, 1990, 103–115.
M. Meyer, M. Desbrun, P. Schröder, and A. H. Barr. Discrete differential geometry operators for triangulated 2-manifolds,Visualization and mathematics III, Math. Vis., Springer, Berlin, 2003, pp. 35–57.
C. Mercat. Discrete Riemannian surfaces and the Ising model, Commun. Math. Phys. 218 (2001), 177-216.
E. Miller and I. Pak. Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings, preprint at arXiv:math.MG/0312253.
J. S. B. Mitchell, D. M. Mount, and C. H. Papadimitriou. The discrete geodesic problem, SIAM J. Comput. 16 (1987), no. 4, 647–668.
U. Pachner. Über die bistellare Äquivalenz simplizialer Sphären und Polytope. (German) Math. Z. 176 (1981), no. 4, 565–576.
D. Pedoe. *Geometry, a comprehensive course*, second ed., Dover Publications Inc., New York, 1988.
G. Perelman. The entropy formula for the Ricci flow and its geometric applications, preprint at arXiv:math.DG/0211159.
U. Pinkall and K. Polthier. Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2 (1993), no. 1, 15–36.
A. V. Pogorelov. Quasi-geodesic lines on a convex surface. Amer. Math. Soc. Translation 1952, (1952). no. 74, 45 pp.
P. L. Powar. Minimal roughness property of the Delaunay triangulation: a shorter approach. Comput. Aided Geom. Design 9 (1992), no. 6, 491–494.
T. Regge. General relativity without coordinates, Nuovo Cimento (10) 19 (1961), 558–571.
S. Rippa. Minimal roughness property of the Delaunay triangulation, Computer Aided Geometric Design 7 (1990), 489–497.
I. Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. of Math. (2) 139 (1994), no. 3, 553–580.
I. Rivin. An extended correction to Combinatorial Scalar Curvature and Rigidity of Ball Packings, (by D. Cooper and I. Rivin), preprint at arXiv:math.MG/0302069.
B. Rodin and D. Sullivan. The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349-360.
D. A. Stone. Geodesics in piecewise linear manifolds, Trans. Amer. Math. Soc. 215 (1976), 1-44.
W. P. Thurston. *The geometry and topology of 3-manifolds*, Chapter 13, Princeton University Math. Dept. Notes, 1980, available at http://www.msri.org/publications/books/gt3m.
Y. Wang, X. Gu, T. F. Chan, P. M. Thompson, and S.-T. Yau. Intrinsic brain surface conformal mapping using a variational method, SPIE International Symposium on Medical Imaging, 2004.
G. Xu. Convergence of discrete Laplace-Beltrami operators over surfaces. Comput. Math. Appl. 48 (2004), no. 3-4, 347–360.
G. Xu. Discrete Laplace-Beltrami operators and their convergence. Comput. Aided Geom. Design 21 (2004), no. 8, 767–784.
|
---
abstract: 'We give a bijective proof of the fact that the number of $k$-prefixes of minimal factorisations of the $n$-cycle $(1\ldots n)$ as a product of $n-1$ transpositions is $n^{k-1}\binom{n}{k+1}$. Rather than a bijection, we construct a surjection with fibres of constant size. This surjection is inspired by a bijection exhibited by Stanley between minimal factorisations of an $n$-cycle and parking functions, and by a counting argument for parking functions due to Pollak.'
address: 'Thierry Lévy Université Pierre et Marie Curie - Laboratoire de Probabilités et Modèles Aléatoires Case courrier 188 - 4, place Jussieu - F-75252 Paris Cedex 05 '
author:
- Thierry Lévy
bibliography:
- 'geodesics.bib'
title: Prefixes of minimal factorisations of a cycle
---
Introduction
============
It is very well known that the $n$-cycle $(1\ldots n)$ cannot be written as a product of less than $n-1$ transpositions, and that there are $n^{n-2}$ ways of writing it as a product of exactly $n-1$ transpositions. Among other proofs, the one given by R. Stanley in [@Stanley] relies on a bijection between minimal factorisations of $(1\ldots n)$ and parking functions of length $n-1$. The bijection is straightforward in one direction, from factorisations to parking functions, and more complicated in the other, but parking functions are very easily counted, thanks to a cunning argument which Stanley attributes to Pollak.
In [@Biane], P. Biane generalised this result and proved that if $a_{1},\ldots,a_{k}$ are integers at least equal to $2$ such that $(a_{1}-1)+\ldots+(a_{k}-1)=n-1$, then there are $n^{k-1}$ ways of writing the $n$-cycle $(1\ldots n)$ as a product $c_{1}\ldots c_{k}$ where $c_{i}$ is an $a_{i}$-cycle for all $i\in \int{1}{k}$.
In this paper, we generalise the result in another direction by counting the initial segments of length $k\in \int{0}{n-1}$ of minimal factorisations of $(1\ldots n)$ by transpositions (see for a precise definition). The number of these prefixes appears in the computation of the repartition of the eigenvalues of a large random unitary matrix taken under the heat kernel measure (see [@LevyAIM]). Using the deep relations between the unitary groups and the symmetric groups, it is possible to make these number appear under their combinatorial definition in this computation, and it is then crucial to be able to determine their value. This was done in [@LevyAIM], where it was proved that the number of these segments is $n^{k-1}\binom{n}{k+1}$. However, the proof given there was rather obscure and the goal of the present paper is to give a bijective proof of this identity.
The present proof consists in constructing a surjective mapping from the set $\int{1}{n}^{k} \times \binom{\int{1}{n}}{k+1}$ to the set of $k$-prefixes of minimal factorisations, with the property that the fibres of this surjection are exactly the orbits of the shift modulo $n$.
The paper is organised as follows. In Section \[postintro\], we describe the set which we want to enumerate and recall some classical facts about the geometry of the Cayley graph of the symmetric group. As a guide and motivation, we also give an informal description of the surjection. In Section \[path properties\], we collect various elementary properties of the prefixes of minimal factorisations of an $n$-cycle, in particular those for which the sequence of the smallest terms displaced by each successive transposition is non-decreasing. In Section \[permutation\], we describe an action of the symmetric group of order $k$ on the set $k$-prefixes of minimal factorisations which plays a crucial role in the construction of the surjection. This construction is finally presented in Section \[surjection\], together with the study of the surjection and the proof of our counting result.
The Cayley graph of the symmetric group {#postintro}
=======================================
The beginning of this section is meant to set up the notation and describe the problem. To start with, given two integers $k$ and $l$ such that $k<l$, we denote by $\int{k}{l}$ the set of integers $\{k,k+1,\ldots,l\}$.
Let $n\geq 1$ be an integer. Let $\S_{n}$ be the symmetric group of order $n$. Let $\T_{n}\subset \S_{n}$ be the subset which consists of all transpositions. It is a conjugacy class of $\S_{n}$ and the Cayley graph of the pair $(\S_{n},\T_{n})$ is defined without ambiguity regarding the order of multiplications. In this note, we endow $\S_{n}$ with the graph distance of this Cayley graph.
This distance can be computed easily by counting the number of cycles of permutations. For all $\sigma\in \S_{n}$, we denote by $\ell(\sigma)$ the number of cycles of $\sigma$, including the trivial cycles. For example, $\sigma$ is a transposition if and only if $\ell(\sigma)=n-1$. The distance between two permutations $\sigma_{1}$ and $\sigma_{2}$ is simply $n-\ell(\sigma_{1}\sigma_{2}^{-1})$. We denote by $|\sigma|$ the number $n-\ell(\sigma)$. Note that for all permutation $\sigma$, one has $|\sigma^{-1}|=|\sigma|$.
The distance on $\S_{n}$ allows one to define a partial order on $\S_{n}$, by setting $\sigma_{1}\preccurlyeq \sigma_{2}$ if and only if $|\sigma_{2}|=|\sigma_{1}|+|\sigma_{1}^{-1}\sigma_{2}|$. We are interested in computing the number of elements of the following set, defined for all $k\geq 0$ : $$\label{def Sigma}
\Sigma_{n}(k)=\left\{(\tau_{1},\ldots,\tau_{k})\in (\T_{n})^{k} : |\tau_{1}\ldots \tau_{k}|=k,\; \tau_{1}\ldots \tau_{k}\preccurlyeq (1\ldots n)\right\}.$$ We will see the elements $\Sigma_{n}(k)$ as paths in the symmetric group, according to the following convention : if $\gamma=(\tau_{1},\ldots,\tau_{k})$ is an element of $\Sigma_{n}(k)$, we denote for all $l\in \int{0}{k}$ by $\gamma_{l}$ the permutation $\tau_{1}\ldots\tau_{l}$. In particular, $\gamma_{0}$ is the identity.
The condition $|\tau_{1}\ldots\tau_{k}|=k$ in the definition of $\Sigma_{n}(k)$ means that for each $l\in \int{1}{k}$, the multiplication on the right by $\tau_{l}$ reduces by $1$ the number of cycles of the permutation $\tau_{1}\ldots \tau_{l-1}$. This is equivalent to saying that the two elements of $\int{1}{n}$ which are exchanged by $\tau_{l}$ belong to distinct cycles of $\tau_{1}\ldots \tau_{l-1}$.
The condition $\tau_{1}\ldots\tau_{k}\preccurlyeq (1\ldots n)$ means, according to the definition of the partial order, that the chain $(\tau_{1},\ldots,\tau_{k})$ of transpositions can be completed into a minimal factorisation of $(1\ldots n)$, that is, a chain $(\tau_{1},\ldots,\tau_{n-1})\in (\T_{n})^{n-1}$ such that $\tau_{1}\ldots \tau_{n-1}=(1\ldots n)$, or yet in other words, a shortest path from the identity to $(1\ldots n)$.
From this description, it follows that $\Sigma_{n}(k)$ is
- empty if $k\geq n$,
- the set of minimal factorisations of $(1\ldots n)$ if $k=n-1$,
- the projection of $\Sigma_{n}(n-1)$ on the first $k$ coordinates of $(\T_{n})^{n-1}$ if $k\leq n-1$.
In particular, if $(\tau_{1},\ldots,\tau_{k})$ belongs to $\Sigma_{n}(k)$, then $\tau_{1}\ldots \tau_{l}\preccurlyeq (1\ldots n)$ for all $l\in \int{0}{k}$. The following classical lemma enables us to decide when a permutation $\sigma$ satisfies $\sigma \preccurlyeq (1\ldots n)$.
Let $\sigma\in \S_{n}$ be a permutation. The relation $\sigma\preccurlyeq (1\ldots n)$ holds if and only if the following two conditions hold:
1. Each cycle of $\sigma$ has the cyclic order induced by $(1\ldots n)$.
2. The partition of $\{1,\ldots,n\}$ by the cycles of $\sigma$ is non-crossing with respect to the cyclic order defined by $(1\ldots n)$.
The first condition is equivalent to the following: each cycle of $\sigma$ can be written $(i_{1}\ldots i_{r})$ with $i_{1}<\ldots <i_{r}$. The second condition means that there exist no subset $\{i,j,k,l\}$ of $\int{1}{n}$ with $i<j<k<l$ such that $i$ and $k$ belong to a cycle of $\sigma$ and $j$ and $l$ belong to another cycle of $\sigma$.
It is well known that $\Sigma_{n}(n-1)$ has $n^{n-2}$ elements. On the other extreme, $\Sigma_{n}(0)$ consists in the empty path and $\Sigma_{n}(1)=\T_{n}$ has $\binom{n}{2}$ elements. Our main goal is to give a bijective proof of the equality $$\label{main card}
|\Sigma_{n}(k)|=n^{k-1} \binom{n}{k+1} ,$$ by the means of a surjective mapping $$\int{1}{n}^{k} \times \binom{\int{1}{n}}{k+1} \to \Sigma_{n}(k),$$ such that the preimage of each element of $\Sigma_{n}(k)$ consists in $n$ elements. In order to construct and study this mapping, we will need to get fairly concretely into the structure of the elements of $\Sigma_{n}(k)$ and this is what we begin in the next section. Before this, let us describe informally the surjection.
Let us start with a sequence $(a_{1},\ldots,a_{k})\in \int{1}{n}^{k}$ and a subset $\{b_{1},\ldots,b_{k+1}\} \subset \int{1}{n}$. Let us reorder $(a_{1},\ldots,a_{k})$ into a non-decreasing sequence $(i_{1}\leq \ldots \leq i_{k})$. Consider a circular bike shed with $n$ spaces labelled from $1$ to $n$ counterclockwise in the natural order, and in which only the spaces labelled $\{b_{1},\ldots,b_{k+1}\}$ are open. A first cyclist enters the shed just after the space $i_{k}$, expores the shed counterclockwise, thus starting from space $i_{k}+1$, and parks into the first open and available space. We denote this space by $j_{k}$. Then, $k-1$ other cyclists park one after the other, starting respectively just after the spaces $i_{k-1},\ldots,i_{1}$. We record the spaces which they occupy as $j_{k-1},\ldots,j_{1}$. At the end of the process, there is exactly one space left vacant among the $k+1$ open ones. If this space is not labelled by $1$, we consider that the procedure has failed and we redo it from the beginning after applying to $(a_{1},\ldots,a_{k})$ and $\{b_{1},\ldots,b_{k+1}\}$ the unique shift modulo $n$ which ensures that the second attempt will not fail. We assume now that our initial data is such that the procedure does not fail.
Since the space $1$ has not been occupied, no cyclist has gone past it in the process and the inequalities $i_{1}<j_{1},\ldots,i_{k}<j_{k}$ hold. Moreover, we shall prove that $((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ belongs to $\Sigma_{n}(k)$ (see Lemma \[sig\]).
Now, let $\sigma\in \S_{k}$ be a permutation such that $(a_{1},\ldots,a_{k})=(i_{\sigma(1)},\ldots,i_{\sigma(k)})$. Let us emphasize that if the first attempt of our parking procedure failed, the sequences which we are considering here are those which we obtained after the shift. This permutation is not unique in general, but we shall prove that the result of the construction is independent of our choice (see Proposition \[action\]). We want to let $\sigma$ act on the path $((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$. For this, write $\sigma$ as a product of transpositions of the form $(l \, l+1)$ with $l\in \int{1}{k-1}$ and let these transpositions act successively on $((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ as follows: if $i_{l}=i_{l+1}$, do nothing, but if $i_{l}\neq i_{l+1}$, exchange $(i_{l}\, j_{l})$ and $(i_{l+1}\, j_{l+1})$ and conjugate the one with the smallest $i$ by the other.
Let us illustrate this on an example. Take $n=8$, $k=4$, consider the sequence $(1,3,7,1)$ and the subset $\{1,3,5,6,7\}$. The bikes enter the shed just after the spaces $7,3,1,1$ in this order and park respectively in the spaces $1,5,3,6$ (see Figure \[shed\] below).
![\[shed\]The paths of the bikes are labelled by their order of entrance in the shed. On the left-hand side, the original situation. On the right-hand side, the shifted one so that the space left vacant is $1$. Observe that the order of entrance has been modified by the shift.](shed.eps){width="13cm"}
The procedure fails : the empty space is labelled $7$. We must shift everything by $2$ modulo $8$ and redo the parking. The new sequence is $(3,5,1,3)$, the new subset $\{1,3,5,7,8\}$. The bikes enter the shed after the spaces $(5,3,3,1)$ and park at $(7,5,8,3)$. We obtain the chain $((1\, 3),(3\, 8),(3\, 5),(5\, 7))$. A permutation which transforms $(3,5,1,3)$ into $(1,3,3,5)$ is $(1\, 3)(2\, 4)=(2\, 3)(1\, 2)(3\, 4)(2\, 3)$. The transposition $(2\, 3)$ does not change the chain, then $(3\, 4)$ changes it to $((1\, 3),(3\, 8),(5\, 7),(3\, 7))$, then $(1\, 2)$ to $((3\, 8),(1\, 8),(5\, 7),(3\, 7))$ and finally $(2\, 3)$ to $((3\, 8),(5\, 7),(1\, 8),(3\, 7))$. This is the element of $\Sigma_{8}(4)$ which the surjection produces. It is indeed an element of $\Sigma_{8}(4)$, since $(3\, 8)(5\, 7)(1\, 8)(3\, 7)=(13578)\preccurlyeq (1\ldots 8)$ and $|(13578)|=4$.
Non-decreasing geodesic paths {#path properties}
=============================
Let us agree on the convention that every time we write a transposition under the form $(i\, j)$, we mean $i<j$.
For all permutation $\pi\in \S_{n}$ and all $x\in \int{1}{n}$, we denote by $C_{\pi}(x)$ the cycle of $\pi$ which contains $x$. We will sometimes forget the cyclic order on $C_{\pi}(x)$ and consider it merely as a subset of $\int{1}{n}$. The following result is largely inspired by the proof of Theorem 3.1 in the work [@Stanley] of R. Stanley.
\[prop mini\] Let $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\; j_{k}))$ be an element of $\Sigma_{n}(k)$. Choose $l\in \int{1}{k}$. The following properties hold.
1. $i_{l}<C_{\gamma_{l-1}}(j_{l})$ and $i_{l}$ is the largest element of $C_{\gamma_{l-1}}(i_{l})$ with this property.
2. $j_{l}=\max C_{\gamma_{l-1}}(j_{l})$.
3. $i_{l}<C_{\gamma_{l-1}}(i_{l}+1)$.
4. If $i_{l}+1\notin \{i_{1},i_{2},\ldots,i_{l-1}\}$, then $C_{\gamma_{l-1}}(i_{l}+1)=\{i_{l}+1\}$.
5. If $k=n-1$, $i_{l}=\max\{i_{1},i_{2},\ldots,i_{n-1}\}$ and $l=\max\{s\in \int{1}{k} : i_{s}=i_{l}\}$, then $j_{l}=i_{l}+1$.
Since $|\gamma_{l}|=|\gamma_{l-1}|+1$, $i_{l}$ and $j_{l}$ belong to distinct cycles of $\gamma_{l-1}$ and to the same cycle of $\gamma_{l}$. The cycle of $\gamma_{l}$ which contains $i_{l}$ and $j_{l}$ has the cyclic order induced by $(1\ldots n)$, so that it is of the form $(x_{1}<\ldots < x_{r} < i_{l} < y_{1}<\ldots < y_{s}< j_{l} < z_{1}<\ldots <z_{t})$. The cycles of $\gamma_{l-1}$ which contain $i_{l}$ and $j_{l}$ are thus respectively $(x_{1}\ldots x_{r} \; i_{l} \; z_{1} \ldots z_{t})$ and $(y_{1}\ldots y_{s}\; j_{l})$. This proves the first two assertions.
The second part of the first assertion implies that $i_{l}+1\notin C_{\gamma_{l-1}}(i_{l})$. If $i_{l}+1\in C_{\gamma_{l-1}}(j_{l})$, then third assertion follows from the first. Let us assume that $i_{l}+1\notin C_{\gamma_{l-1}}(j_{l})$. In this case, $C_{\gamma_{l-1}}(i_{l}+1)=C_{\gamma_{l}}(i_{l}+1)$. Suppose that there is an element $x$ in $C_{\gamma_{l-1}}(i_{l}+1)$ such that $x<i_{l}$. Then the quadruplet $x<i_{l}<i_{l}+1<y_{l}$ would violate the non-crossing condition on the cycles of $\gamma_{l}$ imposed by the condition $\gamma_{l}\preccurlyeq (1\ldots n)$. This concludes the proof of the third assertion.
Let us assume that $i_{l}+1\notin \{i_{1},\ldots,i_{l-1}\}$. Let $r$ be the smallest element of $\int{1}{k}$, if it exists, such that the cycle of $i_{l}+1$ in $\gamma_{r}$ is not reduced to the singleton $\{i_{l}+1\}$. We must have $i_{r}=i_{l}+1$ or $j_{r}=i_{l}+1$. If $i_{r}=i_{l}+1$, then our assumption implies $r\geq l$, so that $C_{\gamma_{l-1}}(i_{l}+1)=\{i_{l}+1\}$. If $j_{r}=i_{l}+1$, then $i_{r}\in C_{\gamma_{r}}(i_{l}+1)$. Since $i_{r}\leq i_{l}$ and thanks to the third assertion, this implies that $r\geq l$, so that in this case also we have $C_{\gamma_{l-1}}(i_{l}+1)=\{i_{l}+1\}$. This proves the fourth assertion.
Let us assume that $k=n-1$, $i_{l}=\max\{i_{1},i_{2},\ldots,i_{n-1}\}$ and $l=\max\{s \in \int{1}{k}: i_{s}=i_{l}\}$. We are thus looking, in a minimal factorisation of $(1\ldots n)$, at the last occurrence of the largest $i$. Let, as before, $r$ be the smallest element of $\int{1}{n-1}$ such that $C_{\gamma_{r}}(i_{l}+1)$ is not reduced to the singleton $\{i_{l}+1\}$. Since $(1\ldots n)$ has no fixed point, we know for sure that $r$ exists. By maximality of $i_{l}$, we have $i_{l}+1=j_{r}$. If $r>l$, then by maximality of $i_{l}$ and of $l$, we have $i_{r}<i_{l}$. Thus, the quadruplet $i_{r}<i_{l}<i_{l}+1<j_{l}$ violates the non-crossing condition on the cycles of the permutation $\gamma_{r}$. This proves the fifth assertion.
We now make an observation of monotonicity.
\[monotone\] Consider $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\; j_{k})) \in \Sigma_{n}(k)$ and $l,m\in \int{1}{k}$ with $l<m$.
1. If $i_{l}=i_{m}$, then $j_{l}>j_{m}$.
2. If $j_{l}=j_{m}$, then $i_{l}>i_{m}$.
Let us assume that $i_{l}=i_{m}$. We have $j_{m}\notin C_{\gamma_{m-1}}(i_{m})$ and $j_{l}\in C_{\gamma_{m-1}}(i_{l})=C_{\gamma_{m-1}}(i_{m})$. In particular, $j_{l}\neq j_{m}$. Both $i_{m}$ and $j_{l}$ belong to $C_{\gamma_{m-1}}(i_{m})$ but, according to the first assertion of Lemma \[prop mini\], $i_{m}$ is the largest element of $C_{\gamma_{m-1}}(i_{m})$ which is smaller than any element of $C_{\gamma_{m-1}}(j_{m})$. Hence, there exists $x\in C_{\gamma_{m-1}}(j_{m})$ such that $i_{m}=i_{l}<x<j_{l}$. The inequality $j_{l}<j_{m}$ cannot hold, for then the quadruplet $i_{l}<x<j_{l}<j_{m}$ would violate the non-crossing property of the cycles of $\gamma_{m-1}$.
The second assertion follows from the first and the existence of a simple involution of $\Sigma_{n}(k)$, which we describe in Lemma \[involution\] below.
\[involution\] Let $((i_{1}\, j_{1}),\ldots,(i_{k}\; j_{k}))$ be an element of $\Sigma_{n}(k)$. Then the chain of transpositions $((n+1-j_{k}\, n+1-i_{k}),\ldots,(n+1-j_{1}\; n+1-i_{1}))$ is also an element of $\Sigma_{n}(k)$.
Let $\phi\in \S_{n}$ be the involution which exchanges $i$ and $n+1-i$ for all $i\in \{1,\ldots,n\}$. The point is the identity $\phi (1\ldots n)^{-1}\phi^{-1}=(1\ldots n)$. Let $(\tau_{1},\ldots,\tau_{k})$ be an element of $\Sigma_{n}(k)$. Then on one hand $|\phi \tau_{k}\ldots \tau_{1} \phi^{-1}|=|\tau_{k}\ldots \tau_{1}|=|\tau_{1}\ldots \tau_{k}|=k$. On the other hand, we have the equality $|(1\ldots n)^{-1} \phi \tau_{k}\ldots \tau_{1} \phi^{-1}|=|\phi^{-1} (1\ldots n)^{-1} \phi \tau_{k}\ldots \tau_{1}|=|(1\ldots n) (\tau_{1}\ldots \tau_{k})^{-1}|=n-1-k$. Hence, $\phi \tau_{k}\ldots \tau_{1} \phi^{-1} \preccurlyeq (1\ldots n)$. Finally, $(\phi \tau_{k}\phi^{-1},\ldots,\phi\tau_{1}\phi^{-1})$ belongs to $\Sigma_{n}(k)$.
It turns out that the elements of $\Sigma_{n}(k)$ for which the sequence $(i_{1},\ldots,i_{n})$ is non-decreasing are easy to describe and to characterise. We call them [*non-decreasing paths*]{} and we denote by $\Sigma_{n}^{*}(k)$ the subset of $\Sigma_{n}(k)$ which they constitute.
\[sm\] Let $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\; j_{k}))$ be an element of $\Sigma_{n}^{*}(k)$. The following properties hold.
1. The sequence $(j_{1},\ldots,j_{k})$ has no repetitions.
2. For all $l\in \int{1}{k}$, $j_{l}$ is a fixed point of $\gamma_{l-1}$.
3. For all $l\in \int{1}{k}$, $\gamma_{l}$ is obtained from $\gamma_{l-1}$ by inserting $j_{l}$ into the cycle of $i_{l}$ immediately after $i_{l}$.
4. For all $m\in \int{1}{k}$, the support of $\gamma_{m}$ is $\bigcup_{l=1}^{m} \{i_{l}\}\cup \{j_{l}\}$.
The second assertion of Lemma \[monotone\] implies that each repetition in the sequence $(j_{1},\ldots,j_{k})$ corresponds to a descent in the sequence $(i_{1},\ldots,i_{k})$, hence the first assertion.
For all $l\in \int{1}{k}$, we have $j_{l}>i_{l}\geq \ldots \geq i_{1}$ and, by the first assertion, $j_{l} \notin\{j_{1},\ldots,j_{l-1}\}$, so that $j_{l}$ is a fixed point of $\gamma_{l-1}$. This is the second assertion.
For all $l\in \int{1}{k}$, the second assertion implies that $\gamma_{l}(i_{l})=j_{l}$, and we have $\gamma_{l}(j_{l})=\gamma_{l-1}(i_{l})$. This is exactly the third assertion.
The fourth assertion follows from the second and third assertions by induction on $k$.
\[carac\] Consider $((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))\in (\T_{n})^{k}$. Assume that $i_{1}\leq \ldots \leq i_{k}$. The following properties are equivalent.
1. $((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))\in \Sigma_{n}(k)$.
2. For all $l,m\in \{1,\ldots,n-1\}$ such that $l<m$, one either has $j_{l}\leq i_{m}$ or $j_{l}>j_{m}$.
Let us prove that the first property implies the second. For this, let us choose $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))\in \Sigma_{n}(k)$ and $l,m$ with $1\leq l<m\leq n-1$. It follows from the first assertion of Lemma \[sm\] that $j_{l}\neq j_{m}$. Let us assume by contradiction that $i_{m}<j_{l}< j_{m}$. Then, by Lemma \[monotone\], $i_{l}<i_{m}$. Hence, $i_{l}<i_{m}<j_{l}<j_{m}$. We know, by the second assertion of Lemma \[sm\], that $C_{\gamma_{m-1}}(j_{m})=\{j_{m}\}$. We claim that $i_{m}\notin C_{\gamma_{m-1}}(i_{l})$. Otherwise, since $j_{l}\in C_{\gamma_{m-1}}(i_{l})$, the element $j_{l}$ of $C_{\gamma_{m-1}}(i_{m})$ would satisfy both $j_{l}>i_{m}$ and $j_{l}<C_{\gamma_{m-1}}(j_{m})$, in contradiction with the first assertion of Lemma \[prop mini\].
It follows from this argument that neither $i_{m}$ nor $j_{m}$ belong to the common cycle of $i_{l}$ and $j_{l}$ in $\gamma_{m-1}$. Hence, the two cycles $C_{\gamma_{m}}(i_{l})=C_{\gamma_{m}}(j_{l})$ and $C_{\gamma_{m}}(i_{m})=C_{\gamma_{m}}(j_{m})$ are distinct. Since $i_{l}<i_{m}<j_{l}<j_{m}$, this contradicts the non-crossing property of the cycles of $\gamma_{m}$.\
Let us now prove that the second property implies the first. To start with, observe that the second property implies that $j_{1},\ldots,j_{k}$ are pairwise distinct and that the equality $i_{l}=i_{m}$ for $l<m$ implies $j_{l}>j_{m}$.
We now proceed by induction on $k$. If $k=1$, then the result is true because $\Sigma_{n}(1)=\T_{n}$. Let us assume that the result holds for paths of length up to $k-1$ and let us consider a path $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))\in (\T_{n})^{k}$ such that $i_{1}\leq \ldots \leq i_{k}$ and the second property holds. By induction, $\gamma_{k-1}$ is a product of $n-k+1$ cycles with the cyclic order induced by $(1\ldots n)$ and which form a non-crossing partition of $\{1,\ldots,n\}$.
By the third assertion of Lemma \[sm\], $\gamma_{k}$ a product of $n-k$ cycles.
Let us prove that the cyclic order of the new cycle is the order induced by $(1\ldots n)$. We certainly have $i_{k}<j_{k}$ and we claim that $i_{k}<j_{k}<\gamma_{k-1}(i_{k})$ in the cyclic order of $(1\ldots n)$, which means exactly that $\gamma_{k-1}(i_{k})\leq i_{k}$ or $\gamma_{k-1}(i_{k})>j_{k}$. But $\gamma_{k-1}(i_{k})$ is either $i_{l}$ for some $l\in \int{1}{k-1}$, in which case $\gamma_{k-1}(i_{k})\leq i_{k}$, or $\gamma_{k-1}(i_{k})$ is $j_{l}$ for some $l\in \int{1}{k-1}$, in which case $\gamma_{k-1}(i_{k})\leq i_{k}$ or $\gamma_{k-1}(i_{k})> j_{k}$, by the main assumption.
Let us finally prove that the cycles of $\gamma_{k}$ form a non-crossing partition. The only way this could not be true is if some cycle contained two elements $x$ and $y$ such that $i_{k} < x <j_{k}<y < \gamma_{k-1}(i_{k})$ in the cyclic order. But the any $x$ such that $i_{k}<x<j_{k}$ does neither belong to $\{i_{1},i_{2},\ldots,i_{k}\}$ nor to $\{j_{1},j_{2},\ldots,j_{k}\}$ and hence is a fixed point of $\gamma_{k}$.
This proposition allows us to prove that a non-decreasing path $\gamma\in \Sigma_{n}^{*}(k)$ is completely determined by the sequence $(i_{1},\ldots,i_{k})$ and the support of $\gamma_{k}$.
\[determine\] Let $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\; j_{k}))$ be an element of $\Sigma_{n}^{*}(k)$. For all $l\in \int{1}{k}$, $j_{l}$ is the minimum of the intersection of $\int{i_{l}+1}{n}$ with the support of $\gamma_{l}$.
Moreover, if $\tilde\gamma=((i_{1}\, \tilde j_{1}),\ldots,(i_{k}\; \tilde j_{k}))$ is another element of $\Sigma_{n}^{*}(k)$ such that $\tilde \gamma_k$ and $\gamma_{k}$ have the same support, then $\tilde \gamma=\gamma$.
The support of $\gamma_{l}$ is $\bigcup_{s=1}^{l} \{i_{s}\} \cup \{j_{1},\ldots,j_{l}\}$. For all $s<l$, we have $i_{s}\leq i_{l}$ and, by Proposition \[carac\], $j_{s}\leq i_{l}$ or $j_{s}>j_{l}$. The first assertion follows.
Let us prove the second assertion by induction on $k$. The result is true for $k=0$. Let us assume that is has been proved for paths of length up to $k-1$. By the first assertion, $\tilde j_{k}=j_{k}$. Hence, $\delta=((i_{1}\, j_{1}),\ldots,(i_{k-1}\; j_{k-1}))$ and $\tilde \delta=((i_{1}\, \tilde j_{1}),\ldots,(i_{k-1}\; \tilde j_{k-1}))$ are two elements of $\Sigma_{n}(k-1)$ such that $\tilde \delta_{k-1}$ and $\delta_{k-1}$ have the same support. By induction, they are equal.
Permutation of geodesic paths {#permutation}
=============================
In this section, we will describe an action of the group $\S_{k}$ on $\Sigma_{n}(k)$. More precisely, let us consider the projection $P:\Sigma_{n}(k)\to \int{1}{n-1}^{k}$ which sends the chain $((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ to the sequence $(i_{1},\ldots,i_{k})$. The group $\S_{k}$ acts naturally on $\int{1}{n-1}^{k}$ by the formula $\sigma\cdot (i_{1},\ldots,i_{k})=(i_{\sigma^{-1}(1)},\ldots,i_{\sigma^{-1}(k)})$ and we will endow $\Sigma_{n}(k)$ with an action of $\S_{k}$ such that $P$ is an equivariant mapping which preserves the stabilisers. This last condition is equivalent to the fact that the restriction of $P$ to each orbit of $\S_{k}$ in $\Sigma_{n}(k)$ is an injection.
In order to define the action of $\S_{k}$ on $\Sigma_{n}(k)$, we will use the classical action of the braid group $B_{k}$ on the product of $k$ copies of an arbitrary group $G$ (see for example [@Artin]). If $\beta_{1},\ldots,\beta_{k}$ are the usual generators of $B_{k}$, this action is given by the formula $$\beta_{l}\cdot (g_{1},\ldots,g_{k})=(g_{1},\ldots,g_{l+1},g_{l+1}^{-1}g_{l}g_{l+1},\ldots,g_{k}),$$ valid for all $(g_{1},\ldots,g_{k})\in G^{k}$ and all $l\in \int{1}{k-1}$. Observe that if $T\subset G$ is a conjugacy class, then $T^{k}$ is stable under this action. Moreover, the product map $(g_{1},\ldots,g_{n})\mapsto g_{1}\ldots g_{n}$ is invariant under this action.
Let us denote by $\sigma_{1}=(1\, 2), \ldots,\sigma_{k-1}=(k-1\, k)$ the Coxeter generators of $\S_{k}$, so that the natural mophism $B_{k}\to \S_{k}$ sends $\beta_{l}$ to $\sigma_{l}$ for all $l\in \int{1}{k-1}$. Consider $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ in $\Sigma_{n}(k)$ and $l\in \int{1}{k-1}$. Set $$\label{def action}
\sigma_{l}\cdot \gamma=\left\{\begin{array}{cc} \gamma & \mbox{if } i_{l}=i_{l+1}, \\
\beta_{l}\cdot \gamma & \mbox{if } i_{l}<i_{l+1}, \\
\beta_{l}^{-1}\cdot \gamma & \mbox{if } i_{l}>i_{l+1}.\end{array}\right.$$ Since the action of the braid group preserves the ordered product of the components, $\sigma_{l}\cdot \gamma$ belongs to $\Sigma_{n}(k)$.
Practically, $\sigma_{l}\cdot \gamma$ is obtained from $\gamma$ by doing nothing if $i_{l}=i_{l+1}$, and otherwise, by swapping the $l$-th and $(l+1)$-th elements of $\gamma$ and conjugating the one with the smallest $i$ by the other. In this way, the transposition with the largest $i$ is not modified, and only the $j$ of the other is affected. For example, if $k=2$, $$\begin{aligned}
\sigma_{1}\cdot((1\, 3),(1\, 2))&=((1\, 3),(1\, 2)),\\
\sigma_{1}\cdot((1\, 2),(2\, 3))&=((2\, 3),(1\, 3)),\\
\sigma_{1}\cdot((2\, 3),(1\, 3))&=((1\, 2),(2\, 3)).\end{aligned}$$ A straightforward inspection will convince the reader of the following fact.
\[equiv\] For all $\gamma \in \Sigma_{n}(k)$ and $l\in \int{1}{k-1}$, one has $P(\sigma_{l}\cdot \gamma)=\sigma_{l}\cdot P(\gamma)$.
Moreover, if $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ and $\sigma_{l}\cdot \gamma=((i_{\sigma_{l}^{-1}(1)}\, \tilde j_{1}),\ldots,(i_{\sigma_{l}^{-1}(k)}\, \tilde j_{k}))$, then the sets $\bigcup_{l=1}^{k}\{i_{l}\}\cup \{j_{l}\}$ and $\bigcup_{l=1}^{k}\{i_{l}\}\cup \{\tilde j_{l}\}$ are equal.
We will show at the end of this section that the set $\bigcup_{l=1}^{k}\{i_{l}\}\cup \{j_{l}\}$ is the support of $\gamma_{k}$. For the time being, let us prove that defines an action of $\S_{k}$ on $\Sigma_{n}(k)$.
\[action\] The action of the Coxeter generators of $\S_{k}$ on $\Sigma_{n}(k)$ defined by extends to an action of $\S_{k}$.
Moreover, the mapping $P:\Sigma_{n}(k)\to \int{1}{n-1}^{k}$ is equivariant and preserves the stabilisers : for all $\gamma\in \Sigma_{n}(k)$ and all $\pi\in \S_{k}$, one has $\pi\cdot P(\gamma)=P(\gamma)$ if and only if $\pi\cdot \gamma=\gamma$.
We must prove that the operations which we have defined satisfy the Coxeter relations $\sigma_{l}^{2}=\id$ for $l\in \int{1}{k-1}$, $(\sigma_{l}\sigma_{m})^{2}=\id$ for $l,m\in \int{1}{k-1}$ with $|l-m|\geq 2$, and $(\sigma_{l}\sigma_{l+1})^{3}=\id$ for $l\in \int{1}{n-2}$.
The first relation follows from Lemma \[equiv\]. Indeed, $\sigma_{l}\cdot (\sigma_{l}\cdot \gamma)$ is either $\gamma$ or $\beta_{l}\beta_{l}^{-1}\cdot \gamma$ or $\beta_{l}^{-1}\beta_{l}\cdot \gamma$, hence in any case $\gamma$. The second relation is equivalent to $\sigma_{l}\cdot(\sigma_{m}\cdot \gamma)=\sigma_{m}\cdot(\sigma_{l}\cdot \gamma)$ and it clearly holds for $|l-m|\geq 2$. In order to prove the third relation, there are six cases to consider, correponding to the possible relative positions of $i_{l}$, $i_{l+1}$ and $i_{l+2}$. In each case, the relation $\beta_{l}\beta_{l+1}\beta_{l}=\beta_{l+1}\beta_{l}\beta_{l+1}$ implies the relation $(\sigma_{l}\sigma_{l+1})^{3}=\id$.
We have thus an action of the symmetric group $\S_{k}$ on $\Sigma_{n}(k)$. By Lemma \[equiv\], the mapping $P$ is equivariant under this action and the natural action on $\int{1}{n-1}^{k}$. If $\gamma\in \Sigma_{n}(k)$ and $\pi\in \S_{k}$ satisfy $\pi\cdot \gamma=\gamma$, then $\pi\cdot P(\gamma)=P(\pi\cdot\gamma)=P(\gamma)$. Finally, let us prove that $\pi\cdot P(\gamma)=P(\gamma)$ implies $\pi\cdot \gamma=\gamma$. Let us choose $\gamma\in \Sigma_{n}(k)$. A permutation $\pi$ stabilises $P(\gamma)$ if and only if its cycles are contained in the level sets of the mapping $1\mapsto i_{1},\ldots,k\mapsto i_{k}$. Thus, the stabiliser of $P(\gamma)$ is generated by the transpositions which it contains, and we may restrict ourselves to the case where $\pi$ is a transposition $(l\, m)$ with $i_{l}=i_{m}$. We have $(l\, m)=\sigma_{l}\ldots \sigma_{m-2}\sigma_{m-1}\sigma_{m-2}\ldots \sigma_{l}$ and $\sigma_{m-2}\ldots \sigma_{l}=(m-1\ldots l)$. Since $P$ is equivariant, the transpositions which are at the positions $m-1$ and $m$ in the chain $\sigma_{m-2}\ldots \sigma_{l}\cdot \gamma$ have respectively $i_{l}$ and $i_{m}$ as their smallest element. Since $i_{l}=i_{m}$, we find $$(l\, m)\cdot \gamma=\sigma_{l}\ldots \sigma_{m-2}\sigma_{m-1}\sigma_{m-2}\ldots \sigma_{l}\cdot \gamma=\sigma_{l}\ldots \sigma_{m-2}\sigma_{m-2}\ldots \sigma_{l}\cdot \gamma=\gamma,$$ as expected.
\[support\] Let $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ be an element of $\Sigma_{n}(k)$. The support of $\gamma_{k}=(i_{1}\, j_{1})\ldots (i_{k}\, j_{k})$ is the set $\bigcup_{l=1}^{k}\{i_{l}\}\cup \{j_{l}\}$.
The action of $\S_{k}$ on $\Sigma_{n}(k)$ preserves both the support of $\gamma_{k}$ and the set to which we wish to show that it is equal. Since every orbit contains a non-decreasing chain, that is, a chain for which the sequence $(i_{1},\ldots,i_{n})$ is non-decreasing, we may assume that the element $\gamma$ which we are considering has this property, and apply the fourth assertion of Lemma \[sm\].
In the context of minimal factorisations of a cycle, the natural action of the braid group is called the Hurwitz action and it is known to be transitive (see for example [@Ripoll]). The action which we have defined here is germane to this action but different, as it is an action of the symmetric group. In [@Biane], P. Biane defined yet another similar action of the symmetric group on minimal factorisations of a cycle as a product of cycles. The proof of Lemma \[equiv\] is inspired by this work.
The main surjection {#surjection}
===================
We have now gathered the information necessary to define the surjection which is our main goal. Although we do not develop this point, our construction is inspired by the enumeration of parking functions by an argument due to Pollak, and the bijection constructed by Stanley between parking functions and minimal factorisations of an $n$-cycle (see [@Stanley]).
Let us start by formalising the parking process in a bike shed described in Section \[postintro\]. Given a sequence $E=(e_{1},\ldots,e_{k})\in \int{1}{n}^{k}$ of entry points and a set $O=\{o_{1},\ldots,o_{k+1}\}\subset \int{1}{n}$ of open spaces, we define a sequence of parking spaces $(p_{1},\ldots,p_{k})$ by backwards induction, by setting $$\label{def jk}
p_{k}=(1\ldots n)^{r} e_{k}, \mbox{ where } r=\min\left\{s\in \int{1}{n} : (1\ldots n)^{s} e_{k} \in \{o_{1},\ldots,o_{k+1}\}\right\}$$ and, assuming that $p_{k},\ldots,p_{l+1}$ have been defined, $$\label{def other j}
p_{l}=(1\ldots n)^{r} e_{l}, \mbox{ where } r=\min\{s\in \int{1}{n} : (1\ldots n)^{s} e_{l} \in \{o_{1},\ldots,o_{k+1}\}\setminus\{p_{l+1},\ldots,p_{k}\}\}.$$ We call this construction the parking process and write $\Pi(E,O)=(p_{1},\ldots,p_{k})$. The set $O\setminus \{p_{1},\ldots,p_{k}\}$ consists of a single element, which we call the residue and denote by $\rho(E,O)$.
Let us state the properties of the parking process which matter for our construction. In what follows, we call shift modulo $n$ the action of $\Z/n\Z$ on $\int{1}{n}^{k}$ and $\binom{\int{1}{n}}{k+1}$ determined componentwise and elementwise in the ovious way by the $n$-cycle $(1\ldots n)$.
\[prop park\] 1. The parking process is equivariant with respect to the shift modulo $n$, that is, $\Pi((1\ldots n)E,(1\ldots n)O)=(1\ldots n)\Pi(E,O)$ and $\rho((1\ldots n)E,(1\ldots n)O)=(1\ldots n)\rho(E,O)$\
2. If $E'$ differs from $E$ by a permutation, then $\Pi(E',O)$ differs from $\Pi(E,O)$ by a permutation. In particular, $\rho(E',O)=\rho(E,O)$.\
3. If $\rho(E,O)=1$, then for all $l\in \int{1}{k}$, one has $e_{l}<p_{l}$.
The shift modulo $n$ is an automorphism of the set $\int{1}{n}$ endowed with the cyclic order determined by $(1\ldots n)$. The definition of the parking process by the equations and uses only this structure of cyclic order. Hence, the first assertion holds.
In order to check the second assertion, it suffices to check that the set $\{p_{1},\ldots,p_{k}\}$ is not modified by the permutation of two neighbours in the sequence $E$. This is a simple verification which we leave to the reader.
Let us assume that $\rho(E,O)=1$. Then, for all $l\in \int{1}{k}$, the integer $1$ belongs to $\{o_{1},\ldots,o_{k+1}\}\setminus\{p_{l+1},\ldots,p_{k}\}$. Hence, the integer $r \in \int{1}{n}$ such that $p_{l}=(1\ldots n)^{r} e_{l}$ satisfies $r\leq n+1-e_{l}$, actually even $r<n+1-e_{l}$ because $p_{l}\neq 1$, so that $e_{l}<p_{l}\leq n$.
Let us now begin the construction of the surjection itself. Consider $A=(a_{1},\ldots,a_{k})\in \int{1}{n}^{k}$ and $B=\{b_{1},\ldots,b_{k+1}\} \subset \int{1}{n}$. Let us apply to $A$ and $B$ the shift modulo $n$ which ensures that the residue of the parking process applied to $A$ and $B$ is $1$. Thus, let us define $\tilde A=(1\ldots n)^{1-\rho(A,B)}A$ and $\tilde B=(1\ldots n)^{1-\rho(A,B)}B$.
Let $I=(i_{1}\leq \ldots \leq i_{k})$ be the non-decreasing reordering of $\tilde A$. Let $J=(j_{1},\ldots,j_{k})=\Pi(I,\tilde B)$ be the result of the parking process applied to $I$ and $\tilde B$.
\[i<j\] The inequalities $i_{1}<j_{1}, \ldots, i_{k}<j_{k}$ hold.
By the first assertion of Lemma \[prop park\], we have $\rho(\tilde A,\tilde B)=1$. Since $I$ differs from $\tilde A$ by a permutation, the second assertion of the same lemma implies that $\rho(I,\tilde B)=1$. The third assertion of the same lemma concludes the proof.
The main property of the construction so far is the following.
\[sig\] The chain $((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ belongs to $\Sigma_{n}^{*}(k)$.
Since the sequence $(i_{1},\ldots,i_{k})$ is non-decreasing, it suffices to check that the second property of Proposition \[carac\] is satisfied. Let us choose $l,m\in \int{1}{k}$ with $l<m$ and let us assume that $j_{l}>i_{m}$. We need to prove that $j_{l}>j_{m}$.
We have $j_{m}=\min\left(\int{i_{m}+1}{n}\cap (\{\tilde b_{1},\ldots,\tilde b_{k+1}\}\setminus \{j_{m+1},\ldots,j_{k}\})\right)$ and, since we are assuming that $j_{l}>i_{m}$, $j_{l}=\min\left(\int{i_{m}+1}{n}\cap (\{\tilde b_{1},\ldots,\tilde b_{k+1}\}\setminus \{j_{l+1},\ldots,j_{k}\})\right)$. Thus, $j_{l}$ is the minimum of a set which is contained in another set of which $j_{m}$ is the minimum.
In order to complete the construction, let us choose a permutation $\sigma\in \S_{k}$ such that $\sigma\cdot \tilde A=I$. There is in general more than one choice for $\sigma$, but two different choices belong to the same right coset of the stabilizer of $I$. Since the mapping $P$ preserves the stabilisers (see Proposition \[action\]), the element $$\Gamma_{n,k}(A,B)=\sigma^{-1}\cdot ((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$$ of $\Sigma_{n}(k)$ is well defined.
The mapping $\displaystyle \Gamma_{n,k}:\int{1}{n}^{k} \times \binom{\int{1}{n}}{k+1} \to \Sigma_{n}(k)$ is a surjection whose fibres are the orbits of the shift modulo $n$. In particular, the preimage of each element of $\Sigma_{n}(k)$ contains $n$ elements and $\displaystyle |\Sigma_{n}(k)|=n^{k-1}\binom{n}{k+1}$.
In order to prove that the mapping $\Gamma_{n,k}$ is surjective, let us construct a section of it.
Let $\gamma=((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$ be an element of $\Sigma_{n}(k)$. Let $\sigma\in \S_{k}$ be a permutation such that $\sigma\cdot \gamma=((a_{1}\, b_{1}),\ldots,(a_{k}\, b_{k}))$ satisfies $a_{1}\leq \ldots \leq a_{k}$. Set $b_{k+1}=1$. By Lemma \[sm\], the set $\{b_{1},\ldots,b_{k+1}\}$ contains $k+1$ elements. Proposition \[carac\] implies that for all $l\in \int{1}{k}$, the set $\{a_{l}+1,\ldots,b_{l}\}\cap\{b_{1},\ldots,b_{l-1}\}$ is empty. Thus, $$b_{l}=\min\left(\int{i_{l}+1}{n} \cap (\{b_{1},\ldots,b_{k+1}\}\setminus\{b_{l+1},\ldots,b_{k})\right),$$ so that $\Pi((a_{1},\ldots,a_{k}),\{b_{1},\ldots,b_{k+1}\})=(b_{1},\ldots,b_{k})$. Moreover, the residue of this parking process is $b_{k+1}=1$. It follows from the definition of $\Gamma_{n,k}$ that $\Gamma_{n,k}((i_{1},\ldots,i_{k}),\{b_{1},\ldots,b_{k+1}\})=\gamma$.
The definition of $\Gamma_{n,k}(A,B)$ shows that it is actually a function of $(1\ldots n)^{1-\rho(A,B)}A$ and $(1\ldots n)^{1-\rho(A,B)}B$. This observation and the first assertion of Lemma \[prop park\] imply that $\Gamma_{n,k}$ is invariant under the shift modulo $n$.
Let us finally prove that each fibre of $\Gamma_{n,k}$ consists in one single orbit of the shift. Let $(A,B)\in \int{1}{n}^{k} \times \binom{\int{1}{n}}{k+1}$ be such that $\rho(A,B)=1$. Let $\sigma$ be a permutation which reorders $P(\Gamma_{n,k}(A,B))$ into a non-decreasing sequence and write $\sigma\cdot \Gamma_{n,k}(A,B)=((i_{1}\, j_{1}),\ldots,(i_{k}\, j_{k}))$. Then $A=\tilde A=P(\Gamma_{n,k}(A,B))$ and $B=\tilde B=\{1,j_{1},\ldots,j_{k}\}$. Hence, each fibre of $\Gamma_{n,k}$ contains a unique pair $(A,B)$ such that $\rho(A,B)=1$.
To be complete, one should conclude by observing that the action of $\Z/n\Z$ on $\int{1}{n}^{k} \times \binom{\int{1}{n}}{k+1}$ is free.
[**Acknowledgement.**]{} It is a pleasure to thank Philippe Marchal who brought parking functions and their relation to the enumeration of minimal factorisations to my attention.
|
---
abstract: 'It is shown that a paradigm of classical statistical mechanics — the thermalization of a Brownian particle — has a low-dimensional, deterministic analogue: when a heavy, slow system is coupled to fast deterministic chaos, the resultant forces drive the slow degrees of freedom toward a state of statistical equilibrium with the fast degrees. This illustrates how concepts useful in statistical mechanics may apply in situations where low-dimensional chaos exists.'
address: |
Institute for Nuclear Theory, University of Washington,\
Seattle, WA 98195
author:
- 'C. Jarzynski'
title: 'Thermalization of a Brownian particle via coupling to low-dimensional chaos'
---
Since the study of chaotic dynamics has clarified fundamental issues in classical statistical mechanics [@rasetti], it is worthwhile to consider the converse: when does intuition from statistical mechanics carry over to [*low-dimensional*]{} chaos? We all know, for instance, that a heavy particle immersed in a heat bath —a Brownian particle — is subject to both an average frictional force, and stochastic fluctuations around this average, and that the balance between these two [*thermalizes*]{} the particle. Now suppose the “Brownian” particle is coupled to a fast, low-dimensional, chaotic trajectory, rather than to a true heat bath. It is known that the particle then feels a dissipative force[@ott; @wilk; @br]; does the particle also (in some sense yet to be defined) “thermalize” with the chaotic trajectory? That is, does the fast chaos behave as a kind of “miniature heat reservoir”, exchanging energy with the particle in a way that brings the two into statistical equilibrium? In this paper, we pursue this question by considering the reaction forces acting on a heavy, slow system (our Brownian particle) due to its coupling to a light, fast trajectory. When the fast motion is chaotic, the forces on the particle include a conservative force, and two velocity-dependent forces, one magnetic-like, the other dissipative[@br]. However (as in the case of coupling to a true thermal bath), there also exists a rapidly fluctuating, effectively [*stochastic*]{} force, which has not been studied in detail. We describe an approach which incorporates this force, with the others, into a unified framework. It is shown that the inclusion of this stochastic force — related to the frictional force by a fluctuation-dissipation relation[@br] — causes the slow Brownian particle and the fast chaotic trajectory to evolve toward statistical equilibrium.
This result provides some justification for applying statistical arguments (involving, e.g., relaxation toward equipartition of energy) to physical situations of only a few degrees of freedom. A discussion of examples — including one-body dissipation in nuclear dynamics[@wf], the Fermi mechanism of cosmic ray acceleration[@fermi], and the diffusive transport of comets[@comets] — where such “thermal” arguments may provide insight into the physics behind more explicit calculations, will be presented in Ref.[@long].
As a starting point for our discussion, we consider the framework of Ref.[@br], where the position [**R**]{} of the slow particle parametrizes the Hamiltonian $h$ governing the fast motion: $h=h({\bf z},{\bf R})$, where [**z**]{} denotes the fast phase space coordinates. (The nature of the fast system will remain unspecified, but we take it to have [*a few*]{}, $N\sim 2$, degrees of freedom.) This classical version of the Born-Oppenheimer framework has received considerable interest in recent years[@wilk; @br; @recent]. We assume that, if [**R**]{} were held [*fixed*]{}, then a fast trajectory evolving under $h$ would ergodically and chaotically explore its [*energy shell*]{} (surface of constant $h$) in the fast phase space. This sets a fast time scale, $\tau_{f}$, which we may take to be the Lyapunov time associated with the fast chaos. A slow time scale, $\tau_{s}$, is set by the motion of the slow particle: it is the time required for the Hamiltonian $h$ to change significantly. We assume $\tau_f\ll\tau_s$; thus, the fast trajectory ${\bf z}(t)$ evolves under a slowly time-dependent Hamiltonian $h$. The full Hamiltonian for the combined system of slow and fast degrees is given by $H({\bf R},{\bf P},{\bf z})=
P^2/2M+h({\bf z},{\bf R})$, where ${\bf P}$ is the momentum of the slow particle, and $M$ is its mass. $({\bf R},{\bf P},{\bf z})$ thus specifies a point in the full phase space of slow and fast variables. It is assumed that surfaces of constant $H$ are bounded in the full phase space.
Given this formulation, the force on the slow particle is ${\bf F}(t)=-\partial h/\partial{\bf R}$, evaluated along the trajectory ${\bf z}(t)$. From the point of view of the slow particle, this force fluctuates rapidly, so it is natural to separate ${\bf F}(t)$ into a slowly-changing [*average*]{} component, and rapid fluctuations $\tilde{\bf F}(t)$ around this average. In Ref.[@br], Berry and Robbins introduce an approximation scheme for obtaining the net [*average*]{} reaction force. At leading (zeroth) order of approximation, the [*ergodic adiabatic invariant*]{}[@ott] dictates the energy of the fast system as a function of the slow coordinates, and this energy in turn serves as a potential for the slow system, giving rise to a conservative “Born-Oppenheimer” force ${\bf F}_0$. At next order, the Berry-Robbins framework yields two velocity-dependent reaction forces: [*deterministic friction*]{} (${\bf F}_{df}$) and [*geometric magnetism*]{} (${\bf F}_{gm}$)[@memory]. Geometric magnetism is a gauge force related to the geometric phase; deterministic friction (see also Ref.[@wilk]) describes the irreversible flow of energy from the slow to the fast variables. Thus, while at leading order the fast degrees of freedom create a potential well for the slow degrees, at first order the fast motion effectively adds a magnetic field, and drains the slow system of its energy.
What about the effets of the rapidly fluctuating component, $\tilde{\bf F}(t)$? If the analogy with ordinary Brownian motion is correct and some sort of statistical equilibration occurs, then $\tilde{\bf F}(t)$ ought to play a central role in the process. We now describe a framework which incorporates the effects of $\tilde{\bf F}(t)$ into a description of the slow particle’s evolution.
In our framework we consider an [*ensemble*]{} of systems. Each member of the ensemble consists of a single slow particle coupled to a single fast trajectory, and represents one possible realization of the combined system of slow and fast variables. Representing this ensemble by a density $\phi$ in the full phase space, Liouville’s equation is: $$\label{eq:liou}
{\partial\phi\over\partial t}+
{{\bf P}\over M}\cdot{\partial\phi\over\partial{\bf R}}-
{\partial h\over\partial{\bf R}}\cdot
{\partial\phi\over\partial{\bf P}}+
\{\phi,h\}=0,$$ where $\{\cdot,\cdot\}$ denotes the Poisson bracket with respect to the fast variables, [**z**]{}. Henceforth, we will ignore all information about the fast trajectory except its energy, $E(t)\equiv h[{\bf z}(t),{\bf R}(t)]$ (which evolves on the slow time scale[@ott]). Thus, what we are really after is the evolution of $W({\bf R},{\bf P},E,t)$, the distribution of our ensemble in the reduced space where all fast variables other than $E$ have been projected out. In this reduced space, $\tilde{\bf F}(t)$ is [*stochastic*]{}, which in turn suggests that $W$ evolves [*diffusively*]{}.
The derivation of an evolution equation for $W$ is somewhat involved, and is sketched in the Appendix. Here we simply state the result: $$\label{eq:central}
{\partial W\over\partial t}=
-{{\bf P}\over M}\cdot{\partial W\over\partial{\bf R}}+
\hat{\bf D}\cdot({\bf u}W)+
{\epsilon\over 2}
\hat D_i\Biggl[\Sigma L_{ij}\hat D_j
\Biggl({W\over\Sigma}\Biggr)\Biggr].$$ (Summation over repeated indices is implied.) To explain notation, we first define $\Omega(E,{\bf R})=\int d{\bf z}\,
\theta[E-h({\bf z},{\bf R})]$. Then $\Sigma\equiv\partial\Omega/\partial E$, and ${\bf u}\equiv-(1/\Sigma)
(\partial\Omega/\partial{\bf R})$. Next, $$\hat{\bf D}\equiv
{\partial\over\partial{\bf P}}-
{{\bf P}\over M}{\partial\over\partial E}.$$ $L_{ij}(E,{\bf R})$ is an integrated correlation function defined by Eq.\[eq:lij\]. Finally, $\epsilon\sim\tau_{f}/\tau_{s}\ll 1$ is an ordering parameter; Eq.\[eq:central\] is valid to ${\it O}(\epsilon)$.
$\Omega$, $\Sigma$, and [**u**]{} have simple interpretations in terms of the [*energy shell*]{} $(E,{\bf R})$ \[the surface [*in [**z**]{}-space*]{} defined by $h({\bf z},{\bf R})=E$\]. $\Omega(E,{\bf R})$ is the volume of [**z**]{}-space enclosed by this shell. $\Sigma(E,{\bf R})=\int d{\bf z}\,\delta(E-h)$ represents the statistical weight of the shell — i.e. the amount of fast phase space occupied by this shell — and is useful for evaluating energy shell averages: $$\langle Q\rangle_{E,{\bf R}}=
{1\over\Sigma(E,{\bf R})}
\int d{\bf z}\,\delta(E-h)\,Q({\bf z}),$$ where $\langle Q\rangle_{E,{\bf R}}$ denotes the average of $Q({\bf z})$ over the energy shell $(E,{\bf R})$. Finally, ${\bf u}(E,{\bf R})=\langle
\partial h/\partial{\bf R}\rangle_{E,{\bf R}}$.
What does Eq.\[eq:central\] reveal about the reaction forces on the slow particle? Below, we outline calculations behind the following assertions regarding the content of Eq.\[eq:central\]: (1) it reproduces the average reaction forces ${\bf F}_0$, ${\bf F}_{df}$, and ${\bf F}_{gm}$; (2) it describes the effects of the rapidly fluctuating force $\tilde{\bf F}(t)$; and (3) it predicts that the Brownian particle does indeed thermalize with the fast trajectory. For a more detailed treatment of this problem, see Ref.[@long].
First, letting ${\cal E}=P^2/2M+E$ denote the total energy of the system, note that $\hat{\bf D}{\cal E}=0$. Thus, $\hat{\bf D}$ is a constrained derivative: $\hat{\bf D}=(\partial/\partial{\bf P})_{\cal E}$, where the notation indicates that ${\cal E}$, not $E$, is held fixed. This means that the evolution depicted by Eq.\[eq:central\] takes place along surfaces of constant ${\cal E}$ in $({\bf R},{\bf P},E)$-space; this is simply a statement of energy conservation.
Next, if we explicitly separate drift terms from diffusion terms, Eq.\[eq:central\] becomes $$\label{eq:driftdiff}
{\partial W\over\partial t}=
-{\partial\over\partial{\bf R}}\cdot\Biggl(
{{\bf P}\over M}W\Biggr)
-{\partial\over\partial{\bf P}}\cdot\Biggl(
{\bf f}W\Biggr)
+{\epsilon\over 2}
{\partial^2\over\partial P_i\partial P_j}
\Biggl(L_{ij}W\Biggr).$$ Here, the derivatives w.r.t. ${\bf P}$ are the constrained derivatives $(\partial/\partial{\bf P})_{\cal E}$, and $$\begin{aligned}
\label{eq:f}
{\bf f}=-{\bf u}(E,{\bf R})-\epsilon
K\cdot{{\bf P}\over M},\\
\label{eq:fluctdiss}
K_{ij}(E,{\bf R})={1\over 2\Sigma}{\partial\over\partial E}
\Biggl(\Sigma L_{ij}\Biggr).\end{aligned}$$ In Eq.\[eq:driftdiff\], ${\bf f}$ plays the role of a [*drift coefficient*]{} for the slow momentum, and thus represents the [*average force acting on the Brownian particle*]{}. A comparison with Ref.[@br], Sec. 2, reveals that the first term of [**f**]{} is the leading (Born-Oppenheimer) force ${\bf F}_0$; the second is a sum of the two velocity-dependent forces, ${\bf F}_{df}$ and ${\bf F}_{gm}$: if we express the matrix $K$ in Eq.\[eq:f\] as the sum of its symmetric and anti-symmetric components, then the former gives us ${\bf F}_{df}$, the latter ${\bf F}_{gm}$. Eq.\[eq:driftdiff\] thus reproduces the [*average*]{} forces ${\bf F}_0$, ${\bf F}_{df}$, and ${\bf F}_{gm}$ acting on the Brownian particle.
The last term in Eq.\[eq:driftdiff\] describes the [*diffusion of slow momenta*]{} due to the fluctuating force $\tilde{\bf F}(t)$. The diffusion coefficient is the matrix $L$, or more precisely its symmetric component $L^{sym}$. By Eq.\[eq:fluctdiss\], however, $L^{sym}$ is related to $K^{sym}$, which as mentioned is responsible for the [*dissipative*]{} force acting on the slow particle. Eq.\[eq:fluctdiss\] thus emerges as a fluctuation-dissipation relation. This relation was first noted by Berry and Robbins[@br].
Finally, do the forces acting on the slow Brownian particle cause it to thermalize with the fast trajectory? To answer, we must first define what we mean by “thermalization” in the context of the present problem (where temperature plays no role). In ordinary statistical mechanics, [*thermalization*]{} means, fundamentally, a [*statistical sharing of the total energy*]{} $X$: after a Brownian particle has long been in contact with a heat bath, the probability for finding it in some state of energy $x$ is simply proportional to the amount of phase space available for the bath to have the remaining energy, $X-x$. (This leads to the Boltzmann factor $P\propto\exp -x/k_BT$.) Similarly in the present context, we take the “thermalization” of slow and fast degrees to mean a statistical sharing of the total energy $\cal E$: the slow and fast variables have [*thermalized*]{}, if the probability for finding the former in a state $({\bf R},{\bf P})$ is simply proportional to the amount of phase space available for the latter to have energy $E={\cal E}-P^2/2M$, namely $\Sigma(E,{\bf R})$. (Thus, $\Sigma(E,{\bf R})$ plays the role of the Boltzmann constant here.) For our [*ensemble*]{}, this condition implies that an initial distribution $W({\bf R},{\bf P},E,t_0)$ evolves toward one that has the form $$\label{eq:asymp}
W({\bf R},{\bf P},E,t_\infty)=G({\cal E})\,\Sigma(E,{\bf R}).$$ \[$G({\cal E})$ is determined by the distribution of total energies, $\eta({\cal E})$, which remains constant.\]
We now make some formal arguments to show that Eq.\[eq:asymp\] indeed represents the ultimate fate of a distribution $W$ evolving under Eq.\[eq:central\]. Consider an [*entropy*]{} $S[W]\equiv-\int W\ln (W/\Sigma)$, where $\int\equiv\int d^3R\int d^3P\int dE$. Using the identity $\partial\Sigma/\partial{\bf R}=-
(\partial/\partial E)(\Sigma{\bf u})$, Eq.\[eq:central\] gives $$\label{eq:dsdt}
{dS\over dt}=
{\epsilon\over 2}\int
{\Sigma^2\over W}
L_{ij}\Gamma_i\Gamma_j\ge 0,$$ where ${\bf\Gamma}=\hat {\bf D}(W/\Sigma)$. (The inequality follows from the fact that the eigenvalues of $L^{sym}$ are non-negative. A proof of the latter is given in Ref.[@long]; less formally, recall that the eigenvalues of $L^{sym}$ are diffusion coefficients, and as such have no business being negative.) Now, the distribution of total energies, $\eta({\cal E})$, is conserved as $W$ evolves with time. However, within the set of all densities $W$ corresponding to a particular $\eta({\cal E})$, $S[W]$ is bounded from above[@long]. Thus as $W$ evolves with time, the value of $S[W]$ never exceeds a certain upper limit. Since $dS/dt\ge 0$, the entropy must eventually [*saturate*]{}, i.e. ${\bf\Gamma}\rightarrow 0$ as $t\rightarrow\infty$[@nofrict]. This in turn implies that $$\label{eq:thermal}
W({\bf R},{\bf P},E,t)\rightarrow
g({\cal E},{\bf R},t)\,\Sigma(E,{\bf R}).$$ However, Eq.\[eq:thermal\] is a solution of Eq.\[eq:central\] only if $g$ is independent of both [**R**]{} and $t$, so we finally conclude that $W\rightarrow G({\cal E})\,\Sigma(E,{\bf R})$ asymptotically with time. Thus, the ensemble [*thermalizes*]{}, in the sense defined in the previous paragraph; this is our central result.
This result may be restated as follows[@long]. If we start with a fast chaotic, ergodic system, which we then enlarge by coupling a few slow degrees of freedom to the fast ones, then the combined system is itself ergodic (over the surface of constant $H$) in the enlarged phase space. Thus the property of ergodicity is promoted, from the fast phase space, to the full phase space of slow and fast variables.
Note also that this “thermalization” proceeds on a time scale much longer than that characterizing the chaotic evolution ($\tau_{f}$). This is again similar to the case of ordinary Brownian motion — where such a separation of time scales is central[@kubo] — but stands in contrast to the more familiar examples of low-dimensional chaos (e.g. the $N=2$ Sinai billiard[@sinai]), where the [*mixing time*]{} and the Lyapunov time are comparable.
It is no new thing to say that a chaotic, ergodic trajectory offers a low-dimensional ($N\sim 2$) analogue for a truly thermal ($N\gg 1$) system. The novelty of the present work is that it extends this analogy to encompass the important paradigm of Brownian motion, where the thermal system or chaotic trajectory is coupled to a few degrees of freedom characterized by a much longer time scale. Then, in either case, the forces acting on the slow system drive it toward a state of genuine [*statistical*]{} equilibrium with its environment.
Finally, it would be interesting to study the quantal version of this problem. Srednicki[@sred] has recently argued that concepts from quantum chaos may provide a solid foundation for quantum statistical mechanics. The focus in Ref.[@sred] is on genuinely thermal systems ($N\gg 1$), and does not deal specifically with the case when a few degrees of freedom are slower than the rest. Nevertheless, Srednicki’s application of [*Berry’s conjecture*]{}[@bc] to the quantal evolution of a classically chaotic system might serve as a guide to a quantal analysis of the purely classical problem studied here. (To the best of my knowledge, no one has looked explicitly at the application of Berry’s conjecture to a system which classically exhibits two widely separated time scales.)
It is a pleasure to acknowledge that conversations with Greg Flynn, Allan Kaufman, Robert Littlejohn, Jim Morehead, and Władek ' Swiatecki were very useful in obtaining the results presented in this papers. This work has been supported by the Department of Energy under Grants No.DE-AC03-76SF00098 and DE-FG06-90ER40561, and by the National Science Foundation under Grant No. NSF-PYI-84-51276.
[**Appendix**]{}
Here we sketch the derivation of Eq.\[eq:central\] from Eq.\[eq:liou\], using what is essentially (though not explicitly) a multiple-time-scale analysis, and is similar to that of Ref.[@br].
To begin, we use our adiabaticity parameter $\epsilon\ll 1$ to formally incorporate into Eq.\[eq:liou\] the assumption that $({\bf R},{\bf P})$ is “slow and heavy”, whereas [**z**]{} is “fast and light”: $$\label{eq:lioutwo}
\epsilon{\partial\phi\over\partial t}+
\epsilon
{{\bf P}\over M}\cdot{\partial\phi\over\partial{\bf R}}-
\epsilon
{\partial h\over\partial{\bf R}}\cdot
{\partial\phi\over\partial{\bf P}}+
\{\phi,h\}=0.$$ With this modification, changes in $({\bf R},{\bf P})$ take place over times of order unity, whereas changes in [**z**]{} occur over times of order $\epsilon$.
Next, we insert the Ansatz $\phi=\phi_0+\epsilon\phi_1+\epsilon^2\phi_2+\cdots$ into Eq.\[eq:lioutwo\] and order by powers of $\epsilon$: $$\begin{aligned}
\label{eq:zero}
\Bigl\{h,\phi_0\Bigr\}&=&0\\
\label{eq:r}
\Bigl\{h,\phi_r\Bigr\}&=&
\Biggl({\partial\over\partial t}+
{{\bf P}\over M}\cdot{\partial\over\partial{\bf R}}-
{\partial h\over\partial{\bf R}}\cdot
{\partial\over\partial{\bf P}}\Biggr)
\phi_{r-1},\end{aligned}$$ $r\ge 1$. Since $h$ commutes only with functions of itself under the Poisson bracket (by assumption of ergodicity), the solution to Eq.\[eq:zero\] has the form $\phi_0({\bf R},{\bf P},{\bf z},t)
=A\Bigl({\bf R},{\bf P},h({\bf z},{\bf R}),t\Bigr)$. To solve for the dependence of $A$ on its arguments, we must examine Eq.\[eq:r\], with $r=1$: $$\label{eq:one}
\Bigl\{h,\phi_1\Bigr\}=
\Biggl({\partial\over\partial t}+
{{\bf P}\over M}\cdot{\partial\over\partial{\bf R}}-
{\partial h\over\partial{\bf R}}\cdot
{\partial\over\partial{\bf P}}\Biggr)
\phi_0.$$ Taking a phase space average of both sides over some energy shell $(E,{\bf R})$ in the fast phase space, we get $$\label{eq:solna}
0={\partial A\over\partial t}+
{{\bf P}\over M}\cdot{\partial A\over\partial{\bf R}}-
{\bf u}\cdot\hat{\bf D}A,$$ where the third argument of $A$ is now $E$, the energy of the shell over which the average is taken. With the identity $\hat{\bf D}\cdot(\Sigma{\bf u})=({\bf P}/M)
\cdot\partial\Sigma/\partial{\bf R}$, we rewrite Eq.\[eq:solna\] as $$\label{eq:secondsolna}
{\partial\over\partial t}\Bigl(\Sigma A\Bigr)
=-{{\bf P}\over M}\cdot{\partial\over\partial{\bf R}}
\Bigl(\Sigma A\Bigr)
+\hat{\bf D}\cdot
\Bigl({\bf u}\Sigma A\Bigr).$$
To solve for $\phi_1$, we first use Eq.\[eq:solna\] to rewrite Eq.\[eq:one\]: $$\label{eq:oneagain}
\Bigl\{h,\phi_1\Bigr\}=
-\Biggl({\partial h\over\partial{\bf R}}-{\bf u}
\Biggr)\cdot\hat{\bf D}A\equiv
-{\partial\tilde h\over\partial{\bf R}}
\cdot\hat{\bf D}A,$$ adopting the notation of Ref.[@br]. \[The left side of this equation is evaluated at $({\bf R},{\bf P},{\bf z},t)$; the value of the third argument of $A$ on the right side is $E=h({\bf z},{\bf R})$.\] This has the form $\{h,f\}=g$; the general solution[@br] consists of both a homogeneous term, $\phi_{1H}=B({\bf R},{\bf P},h,t)$, and an inhomogeneous term, $$\phi_{1I}({\bf R},{\bf P},{\bf z},t)=
\int_{-\infty}^0 ds\,
{\partial\tilde h\over\partial{\bf R}}
({\bf z}_s,{\bf R})\cdot\hat{\bf D}A,$$ where ${\bf z}_s({\bf z},{\bf R})$ is the point in phase space reached by evolving a trajectory from [**z**]{}, for a time $s$, under the Hamiltonian $h({\bf z},{\bf R})$. Note that $\langle\phi_{1I}\rangle_{E,{\bf R}}=0$ for any $(E,{\bf R})$.
We solve for $B$ much as we did for $A$: writing Eq.\[eq:r\], with $r=2$, we average each side over an energy shell $(E,{\bf R})$. After manipulation, this gives $$\label{eq:solnb}
0={\partial\over\partial t}\Bigl(\Sigma B\Bigr)+
{{\bf P}\over M}\cdot{\partial\over\partial{\bf R}}
\Bigl(\Sigma B\Bigr)-\hat{\bf D}\cdot
\Bigl({\bf u}\Sigma B\Bigr)-
{1\over 2}
\hat D_i\Bigl(\Sigma L_{ij}\hat D_j A\Bigr),$$ where $$\label{eq:lij}
L_{ij}=2\int_{-\infty}^0 ds
\Biggl\langle{\partial\tilde h\over\partial R_i}
\Biggl({\partial\tilde h\over\partial R_j}\Biggr)_s
\Biggr\rangle_{E,{\bf R}}.$$ The first factor inside angular brackets is evaluated at [**z**]{}, the second at ${\bf z}_s({\bf z},{\bf R})$; the average is over all points [**z**]{} on the energy shell $(E,{\bf R})$. It is assumed that the integral converges.
Finally, $W({\bf R},{\bf P},E,t)$ is given by a projection of $\phi$ from $({\bf R},{\bf P},{\bf z})$ to $({\bf R},{\bf P},E)$: $$\label{eq:w}
W=\int d{\bf z}\,\delta(E-h)\phi=
\Sigma(E,{\bf R})\,\langle\phi\rangle_{E,{\bf R}}.$$ Since $\langle\phi\rangle_{E,{\bf R}}=A+\epsilon B$ (to order $\epsilon$), we combine Eqs.\[eq:secondsolna\], \[eq:solnb\], and \[eq:w\] to obtain the desired result, Eq.\[eq:central\].
See e.g. M. Rasetti, [*Modern Methods in Equilibrium Statistical Mechanics*]{} (World Scientific, Philadelphia, 1986).
E. Ott, Phys.Rev.Lett. [**42**]{}, 1628 (1979).
M. Wilkinson, J.Phys. A[**23**]{}, 3603 (1990).
M.V. Berry and J.M. Robbins, Proc.Roy.Soc.Lond. A[**442**]{}, 659 (1993).
J.Błocki et al., Ann.Phys. [**113**]{}, 338 (1978).
E. Fermi, Phys.Rev.[**75**]{}, 1169 (1949).
R.Z. Sagdeev and G.M. Zaslavsky, Nuovo Cimento [**97**]{}, 119 (1987).
Manuscript in preparation.
E. Gozzi and W.D. Thacker, Phys.Rev. D [**35**]{}, 2398 (1987); Y. Aharonov et. al., Nucl.Phys. B [**350**]{}, 818 (1991); Y. Aharonov and A. Stern, Phys.Rev.Lett. [**25**]{}, 3593 (1992); M.V. Berry and J.M. Robbins, Proc.Roy.Soc.Lond. A [**442**]{}, 641 (1993).
In the Berry-Robbins model, the slow particle is coupled to a [*microcanonical ensemble*]{} of fast trajectories. This model suppresses the stochastic force (by design), and gives rise to a [*memory*]{} force which does not appear in the present formulation.
Except in the non-generic case when ${\rm det}(L^{sym})=0$.
R. Kubo, M. Toda, and N. Hashitsume, [*Statistical Physics II*]{} (Springer-Verlag, Berlin, 1985), Chapter 1.
Ya.G. Sinai, Russ.Math.Surv. [**25**]{}, 137 (1970).
M.Srednicki, Phys.Rev.E [**50**]{}, 888 (1994); UCSB-TH-94-17 (cond-mat/9406056), 1994; UCSB-TH-94-40 (cond-mat/9410046), 1994.
M.V.Berry, J.Phys.A [**10**]{}, 2083 (1977).
|
---
abstract: 'We investigate a lattice-fluid model defined on a two-dimensional triangular lattice, with the aim of reproducing qualitatively some anomalous properties of water. Model molecules are of the “Mercedes Benz” type, i.e., they possess a $D_3$ (equilateral triangle) symmetry, with three bonding arms. Bond formation depends both on orientation and local density. We work out phase diagrams, response functions, and stability limits for the liquid phase, making use of a generalized first order approximation on a triangle cluster, whose accuracy is verified, in some cases, by Monte Carlo simulations. The phase diagram displays one ordered (solid) phase which is less dense than the liquid one. At fixed pressure the liquid phase response functions show the typical anomalous behavior observed in liquid water, while, in the supercooled region, a reentrant spinodal is observed.'
author:
- 'C. Buzano, E. de Stefanis, A. Pelizzola, and M. Pretti'
title: 'Two-dimensional lattice-fluid model with water-like anomalies'
---
Introduction
============
Water is an anomalous fluid with respect to several thermodynamic properties [@EisenbergKauzmann1969; @Franks1982; @Stanley2003]. At ordinary pressures the solid phase (ice) is less dense than the corresponding liquid, the liquid phase has a temperature of maximum density, while both isothermal compressibility and isobaric heat capacity display a minimum as a function of temperature. Moreover, the heat capacity is unusually large. There is general agreement, among physicists, that an explanation of such anomalous properties is to be found in the peculiar features of hydrogen bonds, and the ability of water molecules to form such kind of bonds [@Stanley1998; @Poole1994]. It is also widely believed that the same physics should be responsible of the unusual properties of water as a solvent for apolar compounds [@FrankEvans1945; @Stillinger1980], that is of the hydrophobic effect, of high importance in biophysics [@Dill1990]. Nevertheless, a comprehensive theory which explains all of these phenomena has not been developed yet. A lot of work has been done in “realistic” simulations [@StillingerRahman1974; @Jorgensen1983; @MahoneyJorgensen2000; @Stanley2002], based on different interaction potentials, but they generally require a large computational effort, and it is not always easy to understand which detail of the model is important to determine certain properties. On the contrary, simplified models generally need easier numerical calculations and allow quite easily to trace connections between microscopic interactions and macroscopic properties [@BellLavis1970; @BenNaim1971; @Bell1972; @Lavis1973; @BellSalt1976; @LavisChristou1977; @LavisChristou1979; @MeijerKikuchiVanRoyen1982; @HuckabyHanna1987; @SastrySciortinoStanley1993jcp; @RobertsDebenedetti1996; @SilversteinHaymetDill1998]. A simplified mechanism which has been proposed to describe the relevant physics of hydrogen bonding is the following one (see for instance Refs. ). Hydrogen bond formation requires that the two involved molecules are in certain relative orientations and stay (on average) at a distance which is larger than the optimal distance for Van der Waals interaction. In other words there exists a competition between Van der Waals interaction (allowing [*higher density*]{} and [*higher orientational entropy*]{}, but resulting in a [*weaker bonding*]{}) and hydrogen bonding (requiring [*lower density*]{} and [*lower orientational entropy*]{}, but resulting in a [*stronger bonding*]{}). This simple mechanism has been implemented in different models, both on- [@SastrySciortinoStanley1993jcp; @RobertsDebenedetti1996; @PatrykiejewPizioSokolowski1999; @BruscoliniPelizzolaCasetti2002] and off-lattice [@SilversteinHaymetDill1998], in 3 [@SastrySciortinoStanley1993jcp; @RobertsDebenedetti1996] as well as 2 dimensions [@SilversteinHaymetDill1998; @PatrykiejewPizioSokolowski1999; @BruscoliniPelizzolaCasetti2002]. One of them is the 2-dimensional Mercedes Benz model, originally proposed by Ben-Naim [@BenNaim1971], in which model molecules possess three bonding arms arranged as in the Mercedes Benz logo. In recent papers by Dill and coworkers [@SilversteinHaymetDill1998; @SilversteinHaymetDill1999], a similar (off-lattice) model has been simulated at constant pressure by a Monte Carlo method, allowing to describe in a qualitatively correct way several anomalous properties of liquid water and also of hydrophobic solvation. Nevertheless, in view of investigations on the behavior of water in contact with other chemical species, as it happens for instance in several biological processes, it would be desirable to obtain an even simpler representation of the physics of hydrogen bonding.
In this paper we investigate a model of the Mercedes Benz type on the triangular lattice, with a twofold purpose. As mentioned above, we are first meant to explore the possibility of obtaining a simpler model with the same underlying physical mechanism, and with qualitatively the same macroscopic properties. Moreover, we are interested in extending the model analysis to the global phase diagram and in particular to the supercooled regime, in which water anomalies are thought to find an explanation. Such a detailed analysis is just made easier by increased simplicity. Working on a lattice, we have to resort to a trick to describe hydrogen bond weakening, when the two participating molecules are too close to each other. Such a trick is similar to the one proposed by Roberts and Debenedetti for their 3-dimensional model [@RobertsDebenedetti1996; @RobertsPanagiotopoulosDebenedetti1996]. The energy of any formed bond is increased (weakened bond) of some fraction by the presence of a third molecule on a site close to the bond (i.e., on the third site of the triangle). Due to the presence of only three bonding arms, it is not possible to distinguish between hydrogen bond donors and acceptors, but this seems to be of minor importance to the physics of hydrogen bonding [@SilversteinHaymetDill1998]. Let us notice that the model has the same bonding properties as the early model proposed by Bell and Lavis [@BellLavis1970], and the same weakening criterion as the model recently investigated by Patrykiejew and coworkers [@PatrykiejewPizioSokolowski1999; @BruscoliniPelizzolaCasetti2002], but here non-bonding orientations are added. Such a feature is essential to describe directional selectivity of hydrogen bonds.
The paper is organized as follows. In Sec. II we define the model in detail and analyze its ground state. In Sec III we introduce the first order approximation in a cluster variational formulation, which we employ for the analysis. Sec. IV describes the results and Sec. V is devoted to some concluding remarks.
Model formulation and ground state
==================================
The model is defined on a two dimensional triangular lattice. A lattice site can be empty or occupied by a molecule with three equivalent bonding arms separated by $2\pi/3$ angles. Two nearest-neighbor molecules interact with an attractive energy $-\epsilon$ ($\epsilon > 0$) representing Van der Waals forces. Moreover, if two arms are pointing to each other, an orientational term $-\eta$ ($\eta > 0$) is added to mimic the formation of a hydrogen (H) bond. Due to the lattice symmetry, a particle can form three bonds at most and there are only 2 bonding orientations, when the arms are aligned with the lattice, while we assume that $w$ non-bonding configurations exist ($w$ is another input parameter of the model). Finally, the H bond energy is weakened by a term $c\eta/2$ ($c \in [0,1]$) when a third molecule is on a site near a formed bond. In the two dimensional triangular lattice there are two such weakening sites per bond, so that a fully weakened H bond energy turns out to be $-(1 - c)\eta$. Let us notice that, in the above description, H bonding is a 3-body interaction. The hamiltonian of the system can be written as a sum over the triangles $${\mathcal{H}}=
\frac{1}{2} \sum_{\langle r,r',r'' \rangle}
{\mathcal{H}}_{i_r {i_r}_{'} {i_r}_{''}}
,
\label{eq:ham}$$ where ${\mathcal{H}}_{ijk}$ is a contribution which will be referred to as triangle hamiltonian, and $i_r,{i_r}_{'},{i_r}_{''}$ label site configurations for the 3 vertices $r,r',r''$, respectively. Possible configurations are empty site ($i=0$), site with a molecule in one of the 2 bonding orientations ($i=1,2$) or in one of the $w$ non-bonding ones ($i=3$) (see Tab. \[tab-site-conf\]). The triangle hamiltonian reads $$\begin{aligned}
{\mathcal{H}}_{ijk} & = & -\epsilon(n_i n_j + n_j n_k + n_k n_i)
\\ &&
-\eta[h_{ij}(1 - c n_k) + h_{jk}(1 - c n_i) + h_{ki}(1 - c n_j)]
\nonumber
,
\label{eq:triham}\end{aligned}$$ where $n_i$ is an occupation variable, defined as $n_i=0$ for $i=0$ (empty site) and $n_i=1$ otherwise (occupied site), while $h_{ij}=1$ if the pair configuration $(i,j)$ forms a H bond, and $h_{ij}=0$ otherwise. Let us notice that triangle vertices are set on three triangular sublattices, say $A,B,C$, and $i,j,k$ are assumed to denote configurations of sites placed on $A,B,C$ sublattices respectively. Assuming also that $A,B,C$ are ordered counterclockwise on up-pointing triangles (and then clockwise on down-pointing triangles), we can define $h_{ij}=1$ if $i=1$ and $j=2$ and $h_{ij}=0$ otherwise. Let us notice that both Van der Waals ($-\epsilon n_i n_j$) and H bond energies ($-\eta h_{ij}$), that are 2-body terms, are split between two triangles, whence the $1/2$ prefactor in Eq. . On the contrary the 3-body weakening terms ($\eta h_{ij} c n_k/2$) are associated each one to a given triangle, and the $1/2$ factor is absorbed in the prefactor. Let us denote the triangle configuration probability by $p_{ijk}$, and assume that the probability distribution is equal for every triangle (no distinction between up- or down-pointing triangles). Taking into account that there are 2 triangles per site, we can write the following expression for the internal energy per site of an infinite lattice $$u = \sum_{i=0}^3 \sum_{j=0}^3 \sum_{k=0}^3 w_i w_j w_k p_{ijk} {\mathcal{H}}_{ijk},
\label{eq:intenergy}$$ The multiplicity for the triangle configuration $(i,j,k)$ is given by $w_i w_j w_k$, where $w_i = w$ for $i=3$ (non-bonding configuration) and $w_i = 1$ otherwise (bonding configuration or vacancy).
[l|cccc]{} config. & empty &  &  &  $i$ & 0 & 1 & 2 & 3 $w_i$ & 1 & 1 & 1 & $w$
\[tab-site-conf\]
Let us now have a look at the ground state properties of the model. In order to do so, let us investigate the zero temperature grand-canonical free energy $\omega^\circ = u - \mu \rho$ ($\mu$ being the chemical potential and $\rho$ the density, i.e., the average site occupation probability), which can be formally written in the same way as the internal energy $u$ of Eq. , by replacing the triangle hamiltonian ${\mathcal{H}}_{ijk}$ by $$\tilde{{\mathcal{H}}}_{ijk} = {\mathcal{H}}_{ijk} - \mu \frac{n_i + n_j + n_k}{3}
\label{eq:trihamtilde}
.$$ We find an infinitely dilute “gas” phase (G) with zero density and zero free energy, and an ordered “open ice” phase ($\mathrm{I_o}$) with maximum number of H bonds per molecule. The latter configuration is realized through the formation of an open (honeycomb) H bond network with density $2/3$ and free energy $$\omega^\circ_\mathrm{I_o} = -\epsilon -\eta - 2\mu/3
.
\label{eq:icefreenergy}$$ Another possibility is the “closed ice” phase ($\mathrm{I_c}$), in which all interstitial sites are occupied and all hydrogen bonds are fully weakened. The resulting free energy is $$\omega^\circ_\mathrm{I_c} = -3\epsilon -\eta(1 - c) -\mu
.
\label{eq:fluidfreenergy}$$ Let us notice that it is never possible to form 3 bonds in a triangle, which means that we have frustration. It is easy to show that the G phase is stable ($\omega^\circ_\mathrm{I_o}>0$) for $\mu < \mu_\mathrm{G-I_o}$, where $$\mu_\mathrm{G-I_o} = -3(\epsilon + \eta)/2
,$$ the $\mathrm{I_o}$ phase is stable ($\omega^\circ_\mathrm{I_o}<0$ and $\omega^\circ_\mathrm{I_o}<\omega^\circ_\mathrm{I_c}$) for $\mu_\mathrm{G-I_o} < \mu < \mu_\mathrm{I_o-I_c}$, where $$\mu_\mathrm{I_o-I_c} = - 6\epsilon + 3c\eta
,$$ and the $\mathrm{I_c}$ phase is stable ($\omega^\circ_\mathrm{I_c}<0$ and $\omega^\circ_\mathrm{I_c}<\omega^\circ_\mathrm{I_o}$) for $\mu >
\mu_\mathrm{I_o-I_c}$. The $\mathrm{I_o}$ phase has actually a stability region, i.e., $\mu_\mathrm{G-I_o} <
\mu_\mathrm{I_o-I_c}$, provided $$\eta > \frac{3}{2c+1} \epsilon
,$$ which, in the worst case ($c=0$), reads $\eta > 3\epsilon$. We shall always work in the latter regime, which is the most significant one to describe real water properties. It is also possible to show that, at the transition point between the open and closed ice phases ($\mu = \mu_\mathrm{I_o-I_c}$), any configuration built up of a honeycomb H bond network with any number of occupied interstitial sites has the same free energy. Hence we expect that the $\mathrm{I_o-I_c}$ transition does not exist at finite temperature, and actually we shall observe a unique ice (I) phase, in which the interstitial site occupation probability gradually increases upon increasing the chemical potential.
Let us finally notice that another possible phase is a homogeneous and isotropic one in which the lattice is fully occupied and molecules can assume only bonding configurations ($i=1,2$). This “bonded liquid” phase, whose free energy coincides with that of the $\mathrm{I_c}$ phase in Eq. , is observed in the $w = 0$ case, studied by Patrykiejew and others [@PatrykiejewPizioSokolowski1999; @BruscoliniPelizzolaCasetti2002]. In this scenario, non-bonding configurations are absent and the bonded liquid ground state has, for $c \neq 1$, the same degeneracy as the Ising triangular antiferromagnet [@BruscoliniPelizzolaCasetti2002]. Nevertheless, in this work we shall deal with the case $w \gg 1$, which is relevant to describe H bond directionality. In this case the closed ice phase is entropically favored with respect to the bonded liquid phase, which cannot appear at finite temperature. In conclusion, because of the introduction of non-bonding configurations, the ground state degeneracy is removed at $T =
0^+$, where only an infinitely dilute (gas) phase and a symmetry-broken (ice) phase are present. Such a phase behavior is closer to the one of water than the one obtained for $w=0$.
First order approximation
=========================
We shall carry out the finite temperature analysis of the model mainly by means of a generalized first order approximation on a triangle cluster, which we introduce in the framework of the cluster variation method. The cluster variation method is an improved mean-field theory based on an approximate expression for the entropy. In Kikuchi’s original formulation [@Kikuchi1951] the entropy is obtained by an approximate counting of the number of microstates. In a modern formulation [@An1988] the approximate entropy can be viewed as a truncation of a cluster cumulant expansion. The truncation is justified by the expected rapid vanishing of the cumulants upon increasing the cluster size, namely when the cluster size becomes larger than the correlation length of the system (the method necessarily fails near critical points) [@Morita1972]. The approximation is completely defined by the maximum clusters left in the truncated expansion, usually denoted as basic clusters. One obtains a free energy functional in the cluster probability distributions, to be minimized, according to the variational principle of statistical mechanics.
For our model we choose up-pointing triangles as basic clusters (an analogous treatment works for down-pointing triangles). This approximation, which seems to be good in particular for frustrated models [@NagaharaFujikiKatsura1981; @Pretti2003], is easily shown to be equivalent to a first order approximation on a triangle cluster [@BellLavis1970]. Let us notice that the internal energy is treated exactly, because the range of interactions does not exceed the basic cluster size, unlike the ordinary mean-field approximation. The grand-canonical free energy per site $\omega = u - \mu \rho - Ts$ ($s$ being the entropy per site), can be written as a functional in the triangle probability distribution as $$\begin{aligned}
\beta \omega & = &
\sum_{i=0}^3 \sum_{j=0}^3 \sum_{k=0}^3 w_i w_j w_k p_{ijk}
\times \label{eq:func} \\ &&
\left[
\beta \tilde{{\mathcal{H}}}_{ijk} + \ln p_{ijk}
- \frac{2}{3} \ln \left( p^A_i p^B_j p^C_k \right)
\right]
\nonumber
,\end{aligned}$$ where $\beta \equiv 1/T$ (temperature is expressed in energy units, whence entropy in natural units) and $p^X_i$ is the probability of the $i$ configuration for a site on the $X$ sublattice ($X=A,B,C$). The latter can be obtained as a marginal of the triangle configuration probability $p_{ijk}$, namely $$\begin{aligned}
p^A_i & = & \sum_{j=0}^3 \sum_{k=0}^3 w_j w_k p_{ijk}
\nonumber \\
p^B_j & = & \sum_{i=0}^3 \sum_{k=0}^3 w_i w_k p_{ijk}
\label{eq:marginals} \\
p^C_k & = & \sum_{i=0}^3 \sum_{j=0}^3 w_i w_j p_{ijk}
\nonumber
.\end{aligned}$$ The above expressions show that the only variational parameter in $\omega$ is the triangle probability distribution, that is the 64 variables $\{p_{ijk}\}$.
The minimization of $\omega$ with respect to these variables, with the normalization constraint $$\sum_{i=0}^3 \sum_{j=0}^3 \sum_{k=0}^3 w_i w_j w_k p_{ijk} = 1
,
\label{eq:constraint}$$ can be performed by the Lagrange multiplier method, yielding the equations $$p_{ijk} = \xi^{-1} e^{-\beta \tilde{{\mathcal{H}}}_{ijk}}
{\left( p^A_i p^B_j p^C_k \right)}^{2/3},
\label{eq:cvmeq}$$ where $\xi$, related to the Lagrange multiplier, is obtained by imposing the constraint Eq. : $$\xi =
\sum_{i=0}^3 \sum_{j=0}^3 \sum_{k=0}^3 w_i w_j w_k
e^{-\beta \tilde{{\mathcal{H}}}_{ijk}}
{\left( p^A_i p^B_j p^C_k \right)}^{2/3}
.$$ Eq. is in a fixed point form, and can be solved numerically by simple iteration (natural iteration method [@Kikuchi1974]). In our case the numerical procedure can be proved to lower the free energy at each iteration [@Kikuchi1974; @Pretti2003], and therefore to converge to local minima. The solution of Eq. gives the equilibrium $\{p_{ijk}\}$ values, from which one can compute the thermal average of every observable. Inserting these values into Eqs. and gives respectively the equilibrium internal energy and free energy. The latter can be also easily expressed through the normalization constant as $$\beta \omega = -\ln \xi
,$$ whence $\xi$ can be viewed as the approximate (single site) grand-canonical partition function. It is also worth mentioning that Eq. preserves homogeneity ($p^X_i = p^Y_i$; $\forall i,X,Y$), due to the invariance of $\tilde{{\mathcal{H}}}_{ijk}$ under cycle permutation of the subscripts (see Eqs. and ). Let us finally notice that the free energy expression Eq. can be also derived by considering the model on a triangular Husimi tree (triangle cactus) [@Pretti2003] as a bulk free energy density, that is the free energy contribution far enough from the boundary, where an invariance condition for the configuration probability of the triangles is assumed to hold.
Results
=======
Phase diagrams
--------------
In order to provide a first insight into the model, let us report in Fig. \[phasediag\] the phase diagram in the chemical potential-temperature plane, for $\eta/\epsilon = 4$, $c = 0.5$, and $w = 50$. Three phases can be observed: An ice (I) phase, with broken symmetry among the three sublattices, a liquid (L) phase and a gas (G) phase. The latter two phases preserve the sublattice symmetry but the liquid phase has a higher density. The ice phase has a lower density than the liquid phase, and its structure reminds that of ground state ice, with interstitial sites occupied by molecules in non-bonding configurations. We can observe a triple point (TRP), in which the three phases coexist, and a gas-liquid critical point (CP). All displayed transition lines are first-order. The above phase diagram shares several properties with the one of real water. Other crystalline phases, such as a real close-packed ice, cannot be reproduced by the model.
Let us now investigate the role of model parameters, by analyzing phase diagrams obtained for different values. In Fig. \[manyphase\]a, $\eta/\epsilon$ and $c$ are left unchanged, while the number of non-bonding configurations $w$ is varied within the interval $[20,100]$. Upon increasing $w$, the liquid phase turns out to be more stable with respect to the ice phase, and the I-L transition temperature decreases. On the contrary, for lower $w$ values, the I phase is increasingly stabilized and the I-L transition temperature increases. For $w=20$ the whole L-G coexistence and also the critical point disappears. Such a behavior can be explained by the fact that the L phase is characterized by a higher number of non-bonding molecules than the I phase, in which bonding molecules tend to form an ordered structure. Therefore high $w$ values largely increase the liquid phase entropy.
In Fig. \[manyphase\]b, $w$ and $c$ are held fixed and the ratio $\eta/\epsilon$ is varied within the interval $[3,5]$. Let us notice that we have restricted the investigation to cases in which the orientational (H bond) interaction is stronger than the non-orientational one, which is the case for real water. It turns out that the ratio $\eta/\epsilon$ affects the stability of the I phase with respect to both the G and L phases. In fact higher values of $\eta$ means stronger H bond, which favors the I phase, that is the only extensively H-bonded phase. On the contrary the L and G phases are dominated by non-oriented interactions with coupling constants $\epsilon$, therefore both these two phases are unfavored by high $\eta/\epsilon$ values. Even in this case the L-G coexistence may become metastable.
The ice phase at high pressures has maximum density and number of weakening molecules per H bond. Raising $c$, the stability of this configuration is lowered with respect to the liquid phase with few H bonds. This is shown in Fig. \[manyphase\]c where $\eta/\epsilon$ and $w$ are fixed and the weakening parameter $c$ is varied in its interval of definition $[0,1]$. This trend is reversed for low $w$ values ($w = 0$ as well), because in the latter case the liquid has the maximum number of fully weakened bonds.
In the next part of this work we focus on a particular choice of parameters ($w = 20$, $\eta / \epsilon = 3$ and $c = 0.8$) which, from the above analysis, turn out to correspond to a water-like phase diagram. Fig. \[pT\_spin\_max\] shows the temperature-pressure phase diagram, and Fig. \[density\] the temperature-density phase diagram. Let us notice that pressure $P$ is simply given by $P = -\omega$ (the volume per site is assumed to be equal to 1, i.e., pressure is expressed in energy units), due to the fact that the free energy has been defined as a grand-canonical potential.
TMD locus and stability limits
------------------------------
One of the water anomalies that the present model is able to reproduce is the temperature of maximum density (TMD) along isobars for the liquid phase. Joining TMD at different pressures defines the so called TMD locus, which is a negatively sloped line in the $T$-$P$ phase diagram of real water. We determine the TMD locus numerically, by adjusting the chemical potential in order to fix the pressure and then imposing that the (isobaric) thermal expansion coefficient vanishes.
The limit of stability of the liquid phase (spinodal) is the locus in which the metastable liquid ceases to be a minimum of the free energy, and becomes a saddle point. The stability limit can be obtained by studying the eigenvalues of the hessian matrix of the free energy [@Debenedetti1996] $$\begin{aligned}
&&
\frac{\partial^2 (\beta \omega)}
{\partial p_{i{^{\,\!}}j{^{\,\!}}k{^{\,\!}}} \partial p_{i'j'k'}}
= w_{i}w_{j}w_{k} \biggr\{ \frac{\delta_{ii'}\delta_{jj'} \delta_{kk'}}{p_{ijk}}
\label{eq:hessian} \\ &&
- \frac{2}{3} \biggr[
\frac{\delta_{ii'} w_{j'} w_{k'}}{p^A_i} +
\frac{w_{i'} \delta_{jj'} w_{k'}}{p^B_j} +
\frac{w_{i'} w_{j'} \delta_{kk'}}{p^C_k}
\biggr]\biggr\}
\nonumber
.\end{aligned}$$ Let us notice that, when the liquid phase stability is lost (some eigenvalue of the above matrix vanishes), also the corresponding fixed point of the natural iteration equations becomes unstable. In order to determine the stability limit with respect to the symmetry-broken ice phase, it is sufficient to impose homogeneity during the iterative procedure, which is done by replacing Eqs. with $$p^A_i = p^B_i = p^C_i =
\sum_{j=0}^3 \sum_{k=0}^3 w_j w_k
\frac{p_{ijk} + p_{kij} + p_{jki}}{3}
.
\label{eq:homogeneity}$$ This trick cannot be applied when the liquid stability is lost with respect to a homogeneous phase, because the liquid fixed point of equations becomes definitely unstable, due to divergence of the density response functions. In the latter case the spinodal is determined by solving the eigenvalue problem for the hessian matrix rewritten by forcing the homogeneity condition .
The results are shown in Fig. \[pT\_spin\_max\]. The stability limit of the liquid with respect to the gas phase starts from the critical point and reaches a minimum in the negative pressure region. After this point the line becomes negatively sloped and joins continuously the stability limit with respect to the ordered ice phase. The TMD locus intersects the limit of stability in its minimum in the $T$-$P$ plane, according to the predictions of Speedy and Debenedetti [@Speedy1982I; @Speedy1982II; @Speedy1987; @DebenedettiDantonio1986I; @DebenedettiDantonio1986II; @DantonioDebenedetti1987; @DebenedettiDantonio1988], based on thermodynamic consistency arguments. In fact the TMD locus causes the liquid limit of stability line to retrace, giving rise to a tensile strength maximum and to a continuous boundary. Let us recall that, while at the stability limit with respect to the gas phase, the density response functions diverge, this is not the case at the stability limit with respect to the ordered phase. Nevertheless we can observe that the density response functions tend to diverge also upon decreasing temperature, as observed experimentally. The locus of divergence, terminating at some kind of critical point, can be defined, in the framework of a simplified variational free energy forced to describe a homogeneous system, as an additional stability limit with respect to a low density liquid phase. Such “phase” corresponds to a saddle point of the original (not symmetrized) free energy, unstable with respect to the solid phase. As the low pressure solid phase reminds the ground state “open ice” structure, which is three-fold degenerate, the triangle probability distribution of the low density liquid phase turns out to be essentially an arithmetic average over the three ice distributions. The unphysical nature of this solution is also reflected in its negative entropy. The divergence locus, together with the locus at which the liquid phase entropy vanishes (Kauzmann line), are shown for completeness in the inset of Fig. \[pT\_spin\_max\]. Upon increasing temperature the divergence locus meets the spinodal tangentially and they become the same curve ending in the “true” gas-liquid critical point.
Response functions
------------------
Let us now investigate the density response functions and the specific heat of the liquid at constant pressure $P/\epsilon = 1$ (pressure is kept fixed by numerically adjusting the chemical potential $\mu$). It turns out that these functions display an anomalous behavior similar to that of real liquid water. The first response function we consider is the thermal expansion coefficient $\alpha_P = (-\partial\ln\rho/\partial T)_P$, which is proportional to the entropy-specific volume cross-correlation. For a typical fluid $\alpha_P$ is always positive because if in a region of the system the specific volume is a little larger then the average, then the local entropy is also larger, i.e., the two quantities are positively correlated. On the contrary, for our model $\alpha_P$ (Fig. \[response\]a) displays an anomalous behavior. As temperature is lowered $\alpha_P$ vanishes (at the TMD), becomes negative, and finally tends to diverge. As previously mentioned, divergence can be observed only for pressure values less than some “critical” pressure. Anyway, before divergence is actually reached, the liquid loses stability with respect to the ice phase.
The trend of the isothermal compressibility $\kappa_T =
(\partial\ln\rho/\partial P)_T$ is also anomalous (Fig. \[response\]c). For a typical liquid $\kappa_T$ decreases as one lowers temperature, because it is proportional to density fluctuations, which decrease upon decreasing temperature. On the contrary, in Fig. \[response\]c we can observe that $\kappa_T$, once reached a minimum, begins to increase upon decreasing temperature. Such a behavior is observed in real liquid water. An analogous behavior characterizes the constant pressure specific heat $c_P = (-T
\partial^2 \mu/\partial T^2)_P$ (Fig. \[response\]b).
Numerical simulation
--------------------
We have studied the model in the first order approximation to obtain easily detailed information about phase diagrams and in particular the metastable region. In order to check this approximation and obtain an estimate of its quantitative accuracy, we have also performed some (grand-canonical) Monte Carlo simulations on a $60 \times 60$ triangular lattice with periodic boundary conditions. From the very beginning, we have chosen quite a low number of non-bonding configurations for our analysis ($w=20$), in order to increase the speed of simulation dynamics. In fact a lower $w$ value corresponds to a smaller configuration space. We report some results in the following.
In Fig. \[transition\] we show a first order transition between the gas and the liquid phases along a constant temperature path $T/\epsilon = 1.05$, quite less than the critical temperature. At the critical point the correlation length increases and the approximation may give worse predictions. Fig. \[transition\] suggests that the first order approximation well localizes the transition and that far enough from the critical point its predictions are nearly quantitative. Of course Monte Carlo simulations display smooth density variations, due to finite size effects, but the Binder cumulant (inset), displaying a minimum, gives evidence of a first order transition.
The reentrance of the liquid stability limit, which is one of the striking features of the (metastable) phase diagram of this model, is also confirmed by simulations. Performing simulations in the metastable region, the spinodal has been determined by an arbitrary criterion for the life time of the metastable phase (100 Monte Carlo steps), as it has been done in previous studies [@SastrySciortinoStanley1993jcp]. Such a criterion allows us to find the kinetically controlled limit of supercooling (homogeneous nucleation locus), shown in Fig. \[MCspin\], along with the corresponding first order approximation result. Both methods show a reentrant spinodal forming a continuous boundary. The simulations also confirm the distinction between liquid limit of stability with respect to the gas or to the ice phase, as in the first order approximation.
Discussion and conclusions
==========================
In this paper we have investigated a 2-dimensional lattice model in which model molecules possess three equivalent bonding arms, and bonding energy depends on the presence of neighbor molecules, giving rise to a 3-particle interaction. The observed behavior is qualitatively similar to that of water, exhibiting the correct anomalies. Upon supercooling, $\kappa_T$ and $c_P$ increase and $\alpha_P$ becomes negative and large in magnitude. Nevertheless, at ordinary pressures (less than the critical pressure) the density anomaly ($\alpha_P = 0$) is found in the metastable liquid region. We have also determined the spinodal limits to the liquid state, and pointed out the relationship between these limits and the TMD locus. The growth in the response functions upon decreasing temperature can be interpreted on the basis of a reentrant spinodal scenario. The liquid-gas spinodal meets the TMD locus at the reentrance point, as required by thermodynamic consistency. Actually the reentrant spinodal conjecture is one of the possible theoretical explanations of water anomalies, and some experimental results are consistent with this explanation [@ZhengDurbenWolfAngell1991]. Nevertheless, it is important to note that, for the specific case of water, alternative interpretations of the stability problem exist, based on the second critical point conjecture [@Stanley1998]. The latter, supported by molecular dynamics simulations [@Stanley2002], seems to be more consistent with the existence, in the negative pressure region, of a monotonic liquid-gas spinodal and a reentrant TMD locus. On the contrary, our model displays a metastable liquid state which is bounded by a spinodal both at positive as well as negative pressures, forming a continuous boundary. The lower temperature part of the boundary is the limit of stability with respect to the ordered ice phase, while the higher temperature part is the limit of stability with respect to the gas phase. While the response functions diverge at the liquid-gas spinodal, at the liquid-solid spinodal they do not, even if they tend to higher values. Anyway, in our framework, it is also possible to investigate the behavior of the unstable liquid (a saddle point of the variational free energy) and determine the locus of divergence. The latter always turns out to lie at a temperature less than the limit of stability, according to experiments [@SpeedyAngell1976]. It also turns out that the divergence locus terminates at some kind of critical point, meaning that response functions should not show divergent-like behavior for pressure values greater than some critical pressure.
Let us notice that a previous lattice model on the 3-dimensional body centered cubic lattice had pointed out a qualitatively similar behavior [@SastrySciortinoStanley1993jcp]. Nevertheless, in such a model, orientational degrees of freedom of water are not treated explicitly and two equivalent sublattices are artificially distinguished by the hamiltonian. This is necessary to favor an open structured phase. Moreover, the analytical treatment is based on the determination of a temperature dependent 2-particle interaction. On the contrary in our model there exists an explicit, though simplified, modelling of hydrogen bonding and no temperature dependent interaction is introduced. The open structured phase is favored in principle by the triangular lattice structure.
We have mentioned in the Introduction that the present model is actually an extension over an early model proposed by Bell and Lavis [@BellLavis1970] (corresponding to the case in which $w=0$ and $c=0$) and over a recent model investigated by Patrykiejew and coworkers [@PatrykiejewPizioSokolowski1999; @BruscoliniPelizzolaCasetti2002] (corresponding to $w=0$). The former model in the same approximation actually displays, for $\eta/\epsilon > 3$, a density anomaly (without singularities), but we have verified that the anomaly occurs in a negative entropy region. The latter model shows an unrealistic phase diagram, in which, for high enough pressure, the liquid phase extends its stability region down to zero temperature. In the present work we have shown that the addition of non-bonding configurations to such a simple class of 2-dimensional lattice models allows us to reproduce a qualitatively correct water-like behavior. Moreover, this result has been obtained in a computationally much simpler way than a conceptually similar model with continuous degrees of freedom, that is the Mercedes-Benz one. The latter model is highly appealing, because of its ability to explain most phenomena related to hydrophobicity [@SilversteinHaymetDill1999]. Therefore it would be interesting to analyze also the properties of the present model for a solution of an inert (apolar) solute, whose peculiar properties are thought to be strictly related to hydrogen bonding. This goes beyond the scope of the present paper and will be the subject of a forthcoming article.
|
---
abstract: 'The emerging Internet of Things (IoT) is facing significant scalability and security challenges. On the one hand, IoT devices are “weak” and need external assistance. Edge computing provides a promising direction addressing the deficiency of centralized cloud computing in scaling massive number of devices. On the other hand, IoT devices are also relatively “vulnerable” facing malicious hackers due to resource constraints. The emerging blockchain and smart contracts technologies bring a series of new security features for IoT and edge computing. In this paper, to address the challenges, we design and prototype an edge-IoT framework named “EdgeChain” based on blockchain and smart contracts. The core idea is to integrate a permissioned blockchain and the internal currency or “coin” system to link the edge cloud resource pool with each IoT device’ account and resource usage, and hence behavior of the IoT devices. EdgeChain uses a credit-based resource management system to control how much resource IoT devices can obtain from edge servers, based on pre-defined rules on priority, application types and past behaviors. Smart contracts are used to enforce the rules and policies to regulate the IoT device behavior in a non-deniable and automated manner. All the IoT activities and transactions are recorded into blockchain for secure data logging and auditing. We implement an EdgeChain prototype and conduct extensive experiments to evaluate the ideas. The results show that while gaining the security benefits of blockchain and smart contracts, the cost of integrating them into EdgeChain is within a reasonable and acceptable range.'
author:
- 'Jianli Pan, Jianyu Wang, Austin Hester, Ismail Alqerm, Yuanni Liu, Ying Zhao [^1] [^2] [^3]'
title: 'EdgeChain: An Edge-IoT Framework and Prototype Based on Blockchain and Smart Contracts'
---
Edge computing, fog computing, EdgeChain, Internet of Things, IoT, blockchain, smart contracts, scalability, security.
Introduction {#sec:intro}
============
It is predicted that the emerging Internet of Things (IoT) will connect more than 50 billion smart devices by the year 2025 [@ERIC20]. It will inevitably change the way we live and work with smart houses, workspaces, transport and even cities on the horizon. However, such trends create significant scalability and security challenges. First, the IoT devices are relatively [*“weak”*]{} and most of their data are sent to remote clouds to be processed. Examples include the majority of the smart phones applications and smart home devices such as Google Home and Amazon Echo. But the existing centralized cloud computing model is very difficult to scale with the projected massive number of devices due to the large amount of generated data and the relatively long distance between IoT devices and clouds. Second, the IoT devices are relatively [*“vulnerable”*]{} and could be relatively easily controlled by malicious hackers to form “botnet” for various attacks [@DDOS162; @BOTNET16]. This is aggravated by the fact that most of the cheap IoT devices are with very limited security capabilities, and very poor or even no technical upgrading or maintenance services, though recently Google’s Android Things 1.0 [@AND18] started pushing this.
Edge computing [^4] [@SHI16; @PAN18a; @BON12; @MEC16; @SAT09] is an emerging direction to provide solutions for the IoT scalability issue. It pushes more computing, networking, storage, and intelligence resources closer to the IoT devices, and provide various benefits such as faster response, handling big data, reducing backbone network traffic, and providing edge intelligence. Typical benefited IoT applications include emergency response, augmented reality, video surveillance, speech recognition, computer vision, and self-driving.
Many works have also been devoted to IoT security. Traditional general-purpose security solutions are not suitable to run on the IoT devices due to the capability constraints [@GAR18]. A typical compromise is to use lightweight IoT security protocols [@LEE14; @RFC7252; @RAZ12; @YAO15; @RFC7815; @LWIG]. Perimeter based security through firewall [@OPP97; @CHE05] does not require running additional software on IoT devices but cannot prevent internal attacks and has been proved ineffective in securing billions of weak devices. Compared with perimeter based trust, zero-trust approaches [@OSB16; @WAR14; @ZERO16] are proved to be more effective and seem promising. Direct or indirect system-level security approaches, which do not put intensive security-related loads on IoT devices and do not assume the IoT devices being well-maintained, and if enabled with a zero-trust or trustless capabilities, are much needed. Blockchain [@NAK08; @NAR16] combined with smart contracts [@SZA96; @CHR16] enable a [*trustless*]{} environment and are recently attracting more attention due to unique features such as data/transactions persistence, tampering resistance, validity, traceability, and distributed fault tolerance. Limited efforts have been made applying them into decentralized IoT and edge computing systems, and two typical work are *Xiong et al.* [@XIO18; @XIO17] using game theory and *Chatzopoulos et al.* [@CHA17] focused on computation offloading. In comparison, our research focus is not on consensus mechanism and mining. Instead, we use permissioned blockchain and smart contracts as carrying vehicle, and our major focus is to provision resources for various IoT applications and control and regulate IoT devices’ behavior.
In this paper, we seek a fundamentally different approach to tackle these key challenges collectively through a blockchain based and resource oriented edge-IoT framework named *EdgeChain*. The EdgeChain’s position in the multi-tier edge-IoT system is illustrated in the Fig. \[fig:position\]. As we can see that EdgeChain locates between the edge cloud platforms and the various IoT applications that are launched in the shared infrastructure. It means that EdgeChain can run on different edge cloud platforms such as HomeCloud [@ict16] or Cloudlet [@SAT09].
![EdgeChain position in the multi-tier edge-IoT system network topology.[]{data-label="fig:position"}](img/fig2.pdf){width="\linewidth"}
The core EdgeChain idea is to integrate a permissioned blockchain and the internal currency or “coin” system to link the edge cloud resource pool with each IoT device’ account and resource usage, and hence behavior of the IoT devices. EdgeChain uses a credit-based resource management system to control how much resource IoT devices can obtain from edge servers, based on pre-defined rules on priority, application types and past behavior. Smart contracts are used to enforce the rules and policies to regulate the IoT device behavior in a non-deniable and automated manner. All the IoT activities and transactions are recorded into blockchain for secure data logging and auditing. As a short summary, the major contributions of the EdgeChain framework include:
1. A new EdgeChain framework integrating permissioned blockchain and smart contracts capabilities.
2. An internal currency or coin system linking the edge cloud resource pool with IoT device accounts and resource usage behavior.
3. A credit-based resource management system to control how much resources IoT devices can obtain from edge servers.
4. A resource-oritented and smart contracts based policy enforcement method to regulate the IoT device behavior.
5. A prototype implementation and experimentation to validate and evaluate the EdgeChain ideas.
Our latest EdgeChain accomplishments have been included in two provisional patents we recently filed [@patent2018a; @patent2018b]. Note that EdgeChain is still an ongoing project and some of the work are still in progress. We will discuss the status accordingly in the following sections. The rest of this paper is organized as follows. In Section \[sec:design\], we discuss several key approaches and designs of EdgeChain. We present the EdgeChain framework and functional modules in Section \[sec:framework\]. Section \[sec:workflows\] is about the key processes and workflows. In Section \[sec:evaluation\], we discuss the prototype and evaluation. Section \[sec:relatedwork\] is the related work, while the conclusions and future work follow in Section \[sec:conclusion\].
EdgeChain Key Approaches and Designs {#sec:design}
====================================
In this section, we discuss some key EdgeChain design considerations. Fig. \[fig:vision\] shows the overall EdgeChain vision including the problem space and the solution space.
![EdgeChain vision: the problem space and solution space.[]{data-label="fig:vision"}](img/fig1.pdf){width="\linewidth"}
Permissioned Blockchain {#subsec:permissioned}
-----------------------
Blockchain networks can be generally categorized into permissionless or public blockchain, and permissioned or private blockchain [@CHR16]. Permissionless blockchain such as Bitcoin network is a peer-to-peer decentralized network. It is usually not controlled by any private organization and the whole network runs on broad consensus of all the members in the network. The trade-off is relatively lower transaction processing throughput and higher latency. Permissioned Blockchain, however, is not a pure peer-to-peer network. The stakeholders such as the application owners of this type of blockchain will have a more controlled and regulated environment, and higer transaction throughput. The consensus mechanisms used for permissionless and permissioned blockchain are also different.
The EdgeChain system uses a permissoned blockchain since the major goal is to support miscellaneous distributed IoT applications that generally have owners and customers. The system stakeholders need more control and higher throughput and performance. It is not necessary to run very resource-intensive proof-of-work algorithms for consensus and sybil attacks cannot happen. It also removes the necessity of economic incentive for mining, which is usually very resource-consuming in the Bitcoin network. More effective but less resource-intensive consensus protocols are available and a typical example is Practical Byzantine Fault Tolerance (PBFT) [@CAS99] for such an environment.
In EdgeChain, the mining work is only done by the edge servers that have more resources than the IoT devices. It is never done by the resource constrained IoT devices. The mining is much less resource intensive compared with permissionless blockchain network. In other words, the edge servers will be in charge of monitoring the transactions, creating and appending new blocks when new transactions happen. The IoT devices in EdgeChain are only blockchain and smart contracts clients. If they are EdgeChain-aware devices and installed with blockchain and smart contracts software, they are able to interact with the edge servers and get resources and assistance for their tasks through procedures such as cloud offloading [@offloading15]. If they are legacy devices and do not need resources from the edge servers, then they do not even need to install the blockchain and smart contracts software. The EdgeChain is totally transparent to them, but still can create blockchain accounts and manage these IoT devices from the back end.
Credit-based Resource Management
--------------------------------
EdgeChain uses an internal currency or coin system enabled by blockchain to link the edge resource pool with the IoT device accounts and resource usage behavior. EdgeChain consists of a novel credit-based resource management system where each IoT device is created a blockchain account and given an initial amount of credit coins. The credit coin balance determines the device’s ability to obtain resources from the edge servers. Generally speaking, the device with a larger balance is afforded quicker and faster access. The edge server records credits and debits and provides the necessary resources requested by the IoT device based on a set of rules that takes pre-defined priority, application types and past behavior into account. As an ongoing research effort, we are designing detailed intelligent resource provisioning mechanism at the edge clouds for the Quality of Experience (QoE) of multiple applications and heterogeneous devices.
In fact, we observe that this resource credit management mechanism not necessarily has to be implemented by the internal currency system. The edge server can maintain a traditional credit score system and decide how to grant resources to different devices. However, by utilizing the internal currency system, EdgeChain can gain a series of intrinsic security benefits coming with blockchain. For example, all the coin transactions are automatically logged into the secure and unmodifiable database on blockchain, and it is good for future auditing purposes. Also, it enables smart contracts that could facilitate non-deniable and automated execution of the scheduling rules and policy enforcement in the edge-IoT systems. All these new benefits are not possible without the blockchain and the internal currency system.
Resource-oriented, Smart Contracts Based Policy Enforcement and IoT Behavior Regulating
---------------------------------------------------------------------------------------
EdgeChain controls the IoT devices based on their behavior and resource use instead of their locations which results in better security control. This overcomes limitations in existing Edge-IoT solutions which are usually “perimeter” based security, i.e., deploying a firewall or a filtering system between the internal and external network and by default trusting the users and nodes “inside” the network. If internal IoT devices were hacked and turned to botnet, it is hard to control them.
EdgeChain uses a resource-oriented, smart contracts based, and indirect security scheme for IoT behavior regulating and auditing. EdgeChain adopts an indirect system-level security approach, which means that we do not require the IoT devices to run resource-intensive security software. Instead, EdgeChain monitors, controls and regulates the behavior of IoT devices based on their resource usage and activities. Based on the application types, priority, device’s past behavior, the pre-programmed smart contracts enforce the resource policy automatically. It means that if some IoT devices were compromised and controlled by hackers for malicious activities such as behaving erratically, making continuous resource requests that are out-of-line with its profile or application intent, or initiating Denial of Service (DoS) attacks, the smart contracts will execute automatically based on the pre-programmed policies. It will be very soon the device’s currency account will run out of balance, through which EdgeChain will be able to quickly identify, control, and contain malicious nodes or devices in the network without requiring them actually to be involved in specific security procedures. EdgeChain can easily take further measures such as putting the devices into the blacklist or blocking the specific devices for further actions. Since smart contracts are based on blockchain, all the activities are recorded into the blockchain. Thus, it is very difficult for any malicious nodes to cause sustained damage or run away with no traces. As an ongoing research effort, we are designing intelligent methods to learn the devices’ history behavior pattern based on the data logged in the blockchain to more accurately identify and recognize potential malicious behavior.
![A simple example of a standalone EdgeChain box deployment in smart home.[]{data-label="fig:standalone"}](img/fig3.pdf){width="\linewidth"}
Evolutionary and Backward Compatible Approach {#subsec:evolutionary}
---------------------------------------------
We realize the fact that there are a large number of cheap IoT devices that may have very limited security capabilities or are being very poorly maintained and barely upgraded. Though the Google’s Android Things 1.0 [@AND18] has just been released trying to work on this, it still has a long way to go. There are some extremely incapable IoT devices such as Narrowband IoT (NB-IoT) devices. It may be infeasible to run even the most lightweight blockchain client software. We classify these devices as legacy devices which are EdgeChain-unaware. The other type of devices are relatively capable enough to install with blockchain and smart contracts software and act as a blockchain client. We classify them as non-legacy devices. Non-legacy devices are able to interact with EdgeChain directly and request resources and assistance from the edge servers. Legacy nodes are unaware of the existence of and incapable of working with edge servers.
The EdgeChain framework adopts an evolutionary and backward compatible approach allowing legacy or extremely incapable IoT devices to work in the new paradigm without assuming them to install new blockchain software or to be updated regularly. The EdgeChain system level capability enables measuring, monitoring, and controlling resource usage of both current and previously installed IoT devices. This goal is achieved through a proxy that works between the legacy IoT devices and the blockchain and smart contract modules, through which the blockchain and smart contracts run transparently to the legacy devices. The proxy sniffs the activities of the legacy nodes and creates blockchain accounts for them just as for non-legacy nodes. In such case, EdgeChain only monitors the behavior and take necessary action if detecting malicious activities. It will not involve allocating edge server resource for the devices. Through the proxy, the legacy IoT devices are not required to know anything about blockchain and smart contracts but they can still be monitored, managed, and controlled by the new Edge-IoT framework. Even if they are compromised by hackers, their malicious behavior can be identified and damages can be contained.
Standalone Deployment vs. Distributed Deployment
------------------------------------------------
Another important advantage with EdgeChain is the ability to be tailored to the need of the intended application. This allows it to be deployed in both stand-alone modes such as in a smart home as well as distributed modes such as a smart campus or smart city scenario. Fig \[fig:standalone\] shows a simple example of a standalone EdgeChain box that is deployed in a smart home. In larger scale use cases and applications such as smart campus and smart cities, multiple such EdgeChain boxes could work in a fully distributed environment, in which cases the distributed boxes work together and share the blockchain and smart contracts data. The edge servers are also able to offload and handover workloads with each other in busy situations. The edge servers can also run appropriate incentive or gaming algorithms associated with their resource pool and blockchain coin accounts to optimize specific goals in revenue, cost, or service latency.
EdgeChain Framework and Functional Modules {#sec:framework}
==========================================
In this section, we discuss the overall EdgeChain framework and functional modules. The overall system framework is shown in Fig. \[fig:framework\]. We can see that the EdgeChain sits between the IoT devices and the edge servers listening to messages and performing corresponding tasks which include device registration and device requests processing. Along the message path, the key modules of EdgeChain include IoT Proxy, Smart Contracts Interface, Smart Contracts, Blockchain Server, and Application Interface. We discuss these modules in a bit more details.
![EdgeChain framework and functional modules.[]{data-label="fig:framework"}](img/fig4.pdf){width="\linewidth"}
IoT Proxy
---------
As we discussed in Subsection \[subsec:evolutionary\], the major function of the IoT Proxy module is to accommodate the legacy devices and facilitate their interactions with the blockchain and smart contracts modules. The proxy listens and sniffs the legacy nodes’ activities and creates blockchain accounts for them. Registration is done for them in the same way as non-legacy nodes so that the IoT behavior regulating and auditing functions work for them as well. Thus, all their activities are recorded in blockchain as non-legacy nodes. In contrast, the non-legacy devices can interact with smart contract directly and get can get accounts created themselves through the smart contracts interface. Implementing this proxy server function requires appropriate sniffing software and we are currently investigating the most effective open-source tools for the EdgeChain project purposes.
Smart Contracts Interface
-------------------------
When the IoT activities occur such as registration, communicating between IoT devices, requesting edge server resources, or sending data to outside servers on the Internet, pre-programmed and deployed smart contracts will be triggered to automatically perform corresponding operations and enforce the predefined management rules or policies. Smart Contracts Interface builds a bridge between the IoT applications and the smart contracts. In our implementation, we utilize the Javascript based APIs, named Web3 protocol, to create the smart contract instances for IoT devices. Smart contract instances can call the functions and perform the rules that were encoded in the contracts on behalf of the specific IoT devices.
Smart Contracts
---------------
The smart contracts, as the containers of all the rules and policies, consist of two main modules in the EdgeChain system. First, we build a digital currency system whose token are virtual coins to represent the trust levels of IoT devices or their quotas of edge resources they can get. Since every IoT device is bound with a blockchain account, it will be assigned with a certain amount of coins based on its history behavior and resource type. For example, if a device keep behaving well without any malicious actions, it will receive more coins to pay for more service resources. Otherwise, the device may be penalized by being charged more coins to receive the same services or never being rewarded. Second, a module of policy management maintains all the rules that were determined at the time of their creation. The policies can be divided into two types: (1) rules to analyze behavior of IoT devices and handle harmful ones; (2) resource allocation policies to dynamical assign resource to the requests and schedule tasks.
Blockchain Server
-----------------
In our implementation, the smart contracts are deployed and distributed on the blockchain. The blockchain server provides blockchain service where the IoT devices connect to it as clients. Two functions are performed on the blockchain server. First, the server executes the smart contracts by collecting the transactions among devices and generating the new blocks to run the code embedded in the contracts. Seconds, all the activities in our system are recorded on the blockchain by automatically logging device information, requests and other activities on blocks. This process is also called “mining” in the permissonless blockchain. However, as discussed in Subsection \[subsec:permissioned\], the EdgeChain mining process is a lot less resource intensive due to the possible usage of more effective consensus mechanism such as PBFT [@CAS99] and no need for proof-of-work mechanism.
Application Interface
---------------------
After the interaction with smart contracts and blockchain, there are two possible outcomes: the requests are either rejected due to limited balance in their accounts or malicious behavior identified, or the request are accepted and granted with permission to receive extra edge resource from the edge servers. If the requests are granted, then the IoT devices can interact with the edge server IoT applications directly, e.g., the resource-intensive work such as face recognition from the video stream can be offloaded to the edge servers for faster processing. In this case, Application Interface opens the channels between smart contracts and the edge cloud to trigger resource provision based on the execution results from smart contracts. We achieve this function using Node.js frameworks to listen to the events on the channels and build communications for IoT devices and edge cloud accordingly.
Note that in terms of delay and time cost, it is true that smart contracts and blockchain operations are not for free and it could take a certain amount of time to finish. The good news is that registration is usually a one-time operation for a specific device. For resource request with the edge servers, after the initial request is granted, the resource provisioning and interactions happen directly between IoT devices and edge servers which will not cause further delay. We conduct very detailed evaluation and experimentation in Section \[sec:evaluation\].
Edge Resource Provisioning
--------------------------
Once the IoT devices are granted with resources and their accounts are with enough balance for the requested resources, the edge cloud will provision resources in computation, memory, storage, networking, and intelligence accordingly. Since the application may have various requirements for computer capability, bandwidth, latency and privacy, individual virtual machines work as the basic units to meet the specific resource requests. For example, for the video stream based face recognition application example we mentioned, the edge servers could spawn and launch additional virtual machines to process the video stream and get the face recognized. If not sufficient resource available from this edge server, EdgeChain can coordinate with neighbor edge servers to get additional resource. Additional incentive mechanisms and dynamic pricing schemes using game theory or auction can be useful to optimize certain goals in revenue or cost. The IoT devices accounts will be charged accordingly based on the service amount and quality they receive.
EdgeChain Key Processes and Workflows {#sec:workflows}
=====================================
With all EdgeChain framework and modules, we will discuss the critical processes and workflows in this section.
Blockchain Deployment
---------------------
Blockchain implementation can be performed in a distributed way on the edge servers and user devices, and get synchronized across these nodes. We begin by installing blockchain software on the edge server, non-legacy devices, and the IoT Proxy. Our blockchain is built on the Ethereum platform[@ethereum14] which is initialized by default to sync with a live public network. However, our EdgeChain system is currently developed for the experimental purpose, so we configure it for use on a private network on campus.
Fig. \[fig:bcworkflow\] shows the workflow of blockchain deployment. The blockchain begins with creating a “genesis” block, which holds configuration information such as the hash value of blockhead, timestamp, and difficulty of block mining. It is worth noting that the amount of difficulty makes a significant influence on the mining speed and then on the global system performance since the mining process is realized by solving a Proof-of-Work (PoW) problem with a certain difficulty. Given that only the edge server is permitted to do the mining job, there is no need for a rigorous PoW mechanism to solve the consensus problem. Therefore, our EdgeChain system sets the difficulty to a reasonable low level to balance between over quick mining to avoid storage waste and efficiency of packing transactions. To further reduce the resource consumption of the edge server, we implement an auto-mining function only occurring when there exist unconfirmed transactions.
![Blockchain implementation workflow.[]{data-label="fig:bcworkflow"}](images/bcWorkflow.png)
To sync with one another, all devices must have the same genesis block. The initialization process will provide each node with same genesis configuration. Next, a primary account must be created for each node and public keys are assigned for unique identification. The account gives each node a blockchain address with which it can interact with other nodes and smart contracts. To isolate our system from other public or private blockchains, all nodes are set “no discovery” so they cannot connect to other peers without explicit addresses. Such isolation secures the devices from being hooked by external attackers. Thus, each node maintains a specific whitelist called “enode addresses” which contains the public keys, IP addresses and network ports of the edge server and some dependent IoT devices. Adding the enode addresses to each node’s configuration will allow syncing to occur. Upon completion of the above steps, each node is ready to launch. They will begin seeking friend nodes, syncing and shortly be prepared for use.
Development and Deployment of Smart Contracts
---------------------------------------------
The proper development of smart contracts guarantees the correct execution of management rules. In our EdgeChain system, the key functional operations including device registration and edge resource allocation are enforced by the corresponding contracts. We deploy smart contracts following the workflow in Fig. \[fig:scworkflow\]. When developing a smart contract on the blockchain, it is important to run thorough tests because once deployed, a contract can only be redeployed and lose any data associated with the previous version. Such a redeployment would migrate the contract to the new location and the users may be outdated with an unsupported contract. After deployment, smart contracts are assigned with addresses and treated as normal accounts on blockchain. In order to interact with them, a user must have a copy of the correct address to create an instance as an interface utilizing remote procedure calls (RPC) protocol. The edge server is the performer to execute the functions in the contracts when the IoT devices are the initiators to trigger them.
![Smart contracts implementation workflow.[]{data-label="fig:scworkflow"}](images/scWorkflow.png)
The smart contracts specify various permissions to different devices where the edge server owns the higher authority to access all the functions but the IoT devices are only limited to some basic functions. Such a setting reduces the impact even if some weak devices are hacked to perform malicious activities. To help engage the legacy nodes into the system, a proxy is deployed in order to fulfill their interaction requests. Other than the direct interaction launched by the nodes, smart contracts are also able to indirectly interface with the outside world by triggering “events” which are watched by application interfaces running on the edge server or other nodes on the network. Upon noticing an event of an application, some smart contract can be automatically triggered to execute the predefined tasks. For example, after the edge server finishes serving one user requests, the related service data like service time would be recorded on blockchain by executing a specific contract.
Device Registration on Blockchain
---------------------------------
Registration is the first step to engage the IoT devices to be managed and monitored in the EdgeChain system. As illustrated in Fig. \[fig:regworkflow\], the registration starts from determining the type of devices. If there are legacy devices lacking the capability to run blockchain, the proxy can create accounts for each device and register the device specifications stored in the registration smart contracts. If there are non-legacy devices, they can interact with contracts directly to save their attributes by sending transactions.
![Devices registration workflow.[]{data-label="fig:regworkflow"}](images/regWorkflow.png)
The registered information makes decisive effect on the request admission introduced in the next section. Specifically, the device specifications partially reference the Manufacturers Usage Description (MUD) [@mud18] files which list the activities and communications allowed for IoT devices. Such specifications contain input/output data type, requests of edge resources, MAC address, IP address, network port, communication protocol, and indication flags. Besides, each device registers a unique account address to join blockchain. Upon registration, the edge server will verify the above information and take control of the modification rights of registration data. More parameters will be appended such as priority, coin balance, credit, and requests timestamp to benefit device management. As a summary, Table \[tab: regTable\] represents the key device attributes we defined in the registration database which include all the devices key information, value units, and examples. Edge server and IoT devices have different authorities to modify the registry. The attributes marked with “\*” can only be updated by the edge server. The other basic attributes are filled up during the first registration process initialized by IoT devices.
**Device Specifications** **Value Unit** **Example**
--------------------------- ---------------- ------------------------------
account address string 0xc968efa8019d (hash value)
network port int 42024
input/output data string video,audio,text
bandwidth request double \[minValue, maxValue\]
CPU request double \[minValue, maxValue\]
memory request double \[minValue, maxValue\]
storage request double \[minValue, maxValue\]
MAC address string 00-14-22-01-23-45
priority\* int 1 / 2 / 3 / 4
coin balance\* double 200.00
credit\* int 100
isBlocked\* bool false
isRegistered\* bool false
last request id\* string 0xcf30613db6a84 (hash value)
: Registered device attributes.[]{data-label="tab: regTable"}
IoT Behavior Regulation and Activities Management
-------------------------------------------------
The IoT behavior regulation and activities management is the core function of our EdgeChain system for IoT scalability and security. In this subsection, we explain the critical designs in the following order: detailed workflows, edge resource allocation algorithms and behavior management scheme.
### IoT Behavior Regulation Workflow
EdgeChain not only regulates the activities among IoT devices but also provides the extra edge computing service to boost the resource-intensive applications. When the activities or the requests from IoT devices are recieved, they are treated differently based on the type of devices, either legacy or non-legacy devices. Legacy devices have no request for the support of edge cloud to handle the additional workload. Non-legacy devices could request to obtain edge resource and services under the enforced rules of smart contracts. The detailed workflow is shown in Fig. \[fig:regworkflow\] and discussed below.
{width="6.5in" height="4.5in"}
For a legacy device, the blockchain server monitors its data flow to other IoT devices or outside network through a sniffer deployed on the IoT gateway such as a WiFi router. During the work process, its activities or behaviors, such as network port and data destination, are logged on blockchain. Then the smart contracts start analyzing the behavior of the device by matching the above observation with the registered attributes. Based on the analyzing results, the blockchain server will choose to keep monitoring the normal behavior. Or it will trigger the smart contract to block any malicious legacy devices and update flags in their registration files. Their future activities will be detected and blocked automatically without performing behavior analysis again. Finally, the execution results of the related smart contracts will be stored on blockchain automatically.
For non-legacy devices, they may send service requests for additional resources for resource-intensive applications such as Virtual Reality (VR) gaming. Once received, the requests are recorded on blockchain in the form of transactions. Next, the resource allocation contracts are executed by the edge server to retrieve the attributes of the devices and analyze the resource requirements in the service requests. If the devices are found to attempt malicious behavior, they will be penalized by reducing their coin balance, lowering credit points and even blocking service for all future requests. If the devices behave normally, the edge cloud will first check the remaining available resource before further process the requests. If the resource pool is exhausted, the requests is rejected and logged. Otherwise, smart contracts perform the resource allocation strategy based on the device types, request details and payable coins. After obtaining the decisions, the edge server starts to schedule the service for the devices immediately. In the meantime, coins will be charged from the devices’ account when the edge service begins. Again, the decisions and coin exchanges are all recorded on blockchain.
### Resource Allocation Based on Pricing Mechanism
In this specific instance, our optimization goal of resource allocation is to maximize the acceptance rate of user requests. In this case, the currency system plays as the connector among edge server, IoT devices and blockchain by linking edge resource with coins. Our proposed currency system is built on a pricing mechanism to decide: (a) the ordering of the requests may be served; (b) the specific service fee.
The price of a resource request dynamically changes according to the following environmental parameters:
- Total amount of edge resources
- Current available edge resources
- Requested edge resource
- Application priority
Considering the QoE requirements, we categorize the priority of IoT applications into 4 levels, from highest to lowest: (1) Urgent monitoring: patient monitoring, people crowd sensing; (2) Latency sensitive tasks: virtual reality (VR), augmented reality (AR); (3) Reliable data transmission: bank transactions, privacy transferring; (4) Tolerant tasks: light control, sensors based passive monitoring.
**Symbol** **Definition**
------------ ------------------------------------------------------------ --
$N$ amount of requests at timeslot $t$
$M$ number of resource types
$R$ requested resource $R=\{r_{i\_1},r_{i\_2},...,r_{i\_M} \}$
$C$ current availabe resource $C=\{c_{1},c_{2},...,c_{M} \}$
$W$ total resources $W=\{w_{1},w_{2},...,w_{M} \}$
$L$ priority level
$K$ amount of accepted requested at timeslot
$\alpha$ constant basic price value
$\beta$ influence factor of priority
: Parameters of pricing mechanism.[]{data-label="tab: priceTable"}
Table \[tab: priceTable\] shows the symbol notations used to calculate the price. We first define the unit price of resource $ j$ for the request $i$: $${ P }_{ i\_ j }={ \alpha }^{ \frac { { r }_{ i\_ j } }{ { c }_{ j } } }*{ \beta }^{ { L }_{ i } }$$ Then the total price for request $i$ is defined as, where ${ c }_{ j }\in [0,{ w }_{ j }]$: $${ P }_{ i }=\sum _{ j }^{ M }{ { { r }_{ i\_ j }*[{ \alpha }^{ \frac { { r }_{ i\_ j } }{ c\_ j } }*{ \beta }^{ { L }_{ i } }] } } =\quad { \beta }^{ { L }_{ i } }\sum _{ j }^{ M }{ { { r }_{ i\_ j }{ *\alpha }^{ \frac { { r }_{ i\_ j } }{ c\_ j } } } }$$
Using the dynamic pricing, we propose a heuristic request admission algorithm as illustrated in Algorithm \[algPrice\]. The proposed algorithm proceeds as follows. At the beginning of timeslot $t$, the number of requests is $N$ and the number of resource types is $M$. For each request ${ r }_{ i\_ j }\in R $, judge if any kind of left edge resource ${ r }_{ i\_ j } $ is less $c_{j}$. If yes, the request is rejected without consideration in this timeslot. If there still have enough resources, calculate the total price of the requests. After all the requests are estimated, the one with the lowest price value is accepted and added to acceptance queue. Then the amounts of available resources $C$ are updated. The rest of requests are reestimated in the next iteration. The algorithm continuous until no request can be admitted. Assume the final acceptance number of request is $K$, we can conclude the time complexity is $O[(N*M+1+M)*K] = O(N*M*K)$, where $K < N$. Therefore, the algorithm can be solved in polynomial time.
$N$ requests $\{req_{1},req_{2},...,req_{N}\}$ at time $t$, request queue $Q(t)$, current available amount of resources $C=\{c_{1},c_{2},...,c_{M} \}$, requested resource $R=\{r_{i\_1},r_{i\_2},...,r_{i\_M} \}$, priority of $req_{i} L_{i}$ . accept or deny request $req_{i}$
deny request $req_{i}$; continue next iteration; calculate the total price ${ P }_{ i }$; accpet $req_{i}$ with minimal price ${ P }_{ i }$; remove the accepted request from $Q(t)$; update the avaible edge resources $C$; EXIT;
### Behavior Management Based on Credit System
Behavior management aims at detecting the potentially malicious activities or requests and taking action to avoid further damage to the system. We propose a credit system to perform the behavior management. Our credit system is distinguished from other similar schemes in the IoT environment because the credit affects resource allocation on the edge server instead of the coorperations between IoT devices. On the other hand, the credit is not directly related to price strategy for edge service but make up the incentive or punishment scheme to restrict the request activities. In this paper, we present the ongoing design and the primary model to show how the credit system works. We consider the following features:
- Price threshold: Assume each device only runs one kind of application and sends one kind of resource request, a specific threshold $P_{thres}$ is set for this device i. If $P_{total}$ exceeds $P_{thres}$, the request is regard as potential bad behavior so the deivce credit is reduced. Otherwise, the request is regard as good behavior and credit increases.
- Request frequency: If a device continuously send requests in an overhigh frequency, it tends to occupy resource than the common use. So we reduce its credit.
- Network port: A device should communicate with the edge server using the predefined network port in the MUD file. Otherwise, some abnormal behavior happens.
- Data traffic destination: A device usually has fixed communication targets, so the strange destination indicates the possibility the device is hacked or under control.
Each new registered devices owns same initial credits. With the changes of the real-time credit values, we propose two kinds of management actions: (1) If the credit of a device has already been reduced to 0, it is blocked for any future activities; (2) otherwise, the device will get various coin returned based on the credit changes. The equation is defined as follow: $${ Coin }s_{ return }={ Coins }_{ charged }+\Delta Credit*\eta$$ where $\Delta Credit$ is the change of credit value and $\eta$ is the influence factors of changes.
We can conclude that the ability to pay for edge service is under the control of the credit system. The better manner receives higher chance to obtain more resources.
Prototype and Evaluation {#sec:evaluation}
========================
In this section, we first introduce our experimental testbed built as the EdgeChain prototype. Then, we implement the key functions to verify if it is feasible with acceptable performance overhead. In the third part, two typical IoT applications in different service priorities are deployed on the EdgeChain system to show the compatibility between blockchain and applications. Finally, we test the performance of the pricing-based resource allocation system.
EdgeChain Prototype Environment Setup
-------------------------------------
The testbed includes the back-end edge cloud cluster and the front-end IoT devices, proxy, and access point. The edge cloud cluster is an OpenStack deployment including 4 high-performance Dell PowerEdge R630 rack servers, 1 high-performance Dell PowerEdge C730x rack server, and 1 high-performance Cisco 3850 switch. The front end consists of several Raspberry Pi 3 Model B single board computers, a Google AIY voice kit, a Google AIY vision kit, and a laptop. One desktop is configured as the proxy for legacy IoT devices, and a high-performance Cisco WiFi Access Point, as illustrated in Fig. \[fig:testbed\]
![EdgeChain testbed.[]{data-label="fig:testbed"}](images/testbed.png){width="\linewidth"}
The detailed hardware and software configurations are as follows. From the aspect of hardware, each OpenStack compute node rack server is equipped with 18 independent CPU cores and 256GB RAM. The mining environment is set up using one core and the rest of the processor cores are reserved for the edge computing service. The miner can boost up to 3.5GHz CPU, 8GB RAM, 1TB storage. As the IoT devices, a Raspberry Pi has 1.2GHz CPU, 1 GB RAM and 32GB storage with several accessory modules including cameras, sense hat, microphone and Google bonnet. The laptop has 2.2GHz CPU, 4GB RAM and 256 GB storage. As for the desktop proxy, 3.2 GHz CPU, 16GB RAM and 1TB storage are installed to manage the multiple blockchain accounts of IoT devices.
Regarding the software, the edge server has installed with *CentOS 7* as the operating system, *Go-ethereum* as the blockchain running framework, *Solidity* as the smart contract development language, *Truffle* as the contract deployment tool, and *Node.js* as the interface of interactions between IoT applications and blockchain. Except for the blockchain part, the edge computing resources are virtualized using OpenStack cloud platform which helps scale up or down the resource pool flexibly. The edge service is provided in the form of virtual machines to fit the variant specifications of user requests. The Raspberry Pis have been installed with Raspbian operating system and *Go-ethereum* to work in the light mode without block mining function. The laptop is with MacOS and the desktop installs Ubuntu 16.
In the testbed, the rack server works as the edge service provider and the block miner solving Proof-of-Work (PoW) puzzle. The Raspberry Pis and the laptop act as blockchain clients generating and sending transactions of resource requests to the edge server. The desktop interacts with the blockchain on behalf on the legacy devices as a proxy. Given the above installations, the edge server works as a “full” blockchain node which stores all the transactions, executes the predefined smart contracts and mines new blocks. The IoT devices work as “light” blockchain nodes which only store the transactions data. Fig. \[fig:packagesize\] shows the storage requirements for the prerequisites software modules, where *Ethash* is the PoW system used to mine blocks. We put most of the computation work occurring on the blockchain to the full node in order to reduce the overhead on the light nodes.
Overhead of Blockchain and Smart Contracts Operation
----------------------------------------------------
We evaluate the blockchain operation based on the two primary functions: IoT devices registration and edge server resource allocation to illustrate the extra overhead caused by the block mining and the interactions of smart contracts. The source of overhead can be divided into three aspects: computation, communication, and storage.
### Computation Cost of Mining Process on Edge Server
We first evaluate the overhead of device registration in which device specifications are loaded in the transactions signed by their generators. Then the transactions are broadcasted to all the other devices engaged in our system. Finally, these new transactions are packed in the blocks and verified by the miner. We observe the average usage of computation resource on the edge server during mining and no mining, as illustrated in Fig. \[mining\]. During the block mining, the edge server consumes much higher CPU and memory resource to commit and packs transactions into new blocks. In contrast, in the idle situation, it only listens to coming transactions such as mining new block caused by new transactions thus consume much less CPU and memory resource.
### Communication and Storage Cost for Blocks Synchronization
Given that blockchain is the fully distributed, each device is required to be synchronized with the mainstream chain. The synchronization mechanism relies on the automatic updates and leads to the communication and storage overhead to the system, where the former results from the data transmission and the later from the writing to the local disk. In our system, the edge server maintains the mainstream blockchain and other devices download the chain data from it. In order to evaluate the synchronization delay intuitively, we compare IoT devices to the edge server. Since the edge server as the miner has more computing and bandwidth resource than IoT devices, it completes the validation and transmission of the new blocks faster. As illustrated in Fig. \[miningDelay\], we find the average time to synchronize a new block is 4.09 ms for edge server and 35.9 ms for IoT devices. With higher delay, the IoT devices still meet the latency requirements even for the real-time applications that response time is less than 100ms.
The average size of a block is 128.78 KB and each block can store up to 208 device registrations. Fig. \[blocksize\] presents a sample of 50 blocks which have various sizes ranging from 108 KB to 223 KB. Thus, the system will generate around 1.8 MB blockchain data on average for 1,000 devices’ registration.
### Computation and Communication Cost of Smart Contract Transactions
In addition to block mining and synchronization, blockchain operation relies on the transactions triggered by the smart contracts. Taking the resource request transaction as an example, we evaluate the computation cost and the interaction delay with smart contracts. The CPU and memory usage are compared between the edge server and IoT devices, as illustrated in Fig. \[tranx\]. We observe that the regular transactions take a very low percentage of CPU resource while the memory usage is little higher since the blockchain client occupies 8 % even in idel time. We also evaluate the interaction delay of smart contracts which is significant to guarantee system efficiency. Fig. \[tranxDelay\] shows the completion of one transaction is less than 50 ms. Such delay should satisfy the latency requirement of the real-time applications.
Overhead Comparison of Two Typical IoT Applications
---------------------------------------------------
To evaluate the feasibility and compatibility of the proposed system, we compare blockchain overhead of two typical Edge-IoT applications. We evaluate the face recognition and the natural-language processing applications by testing the computation and communication cost. Face recognition is widely used in the security monitoring applications such as city surveillance, crowd control and door guarding which is latency-sensitive to achieve quick reaction. The typical application of the natural-language processing or voice recognition is the smart home assistant such as Google Home and Amazon echo.
For the face recognition, the Raspberry Pi captures video frames with camera module in 1080p resolution and 60Hz frequency, uploads them to the edge server for image processing and waits for the detection results in the form of location coordinates of detected faces. With regard to the natural-language processing, the Raspberry Pi records the human voice with a USB microphone, transfers it to the edge server, and then the translated text is returned.
We first evaluate the computation cost of blockchain comparing with the two applications. Fig. \[appCPU\] shows that the blockchain has the lowest CPU usage compared with the two applications. In addtion, Fig. \[appMem\] shows the blockchain has the highest memory usage but still in a low percentage when working with other applications in parallel. Thus, the IoT devices will not suffer from the overload problem. Second, we evaluate the difference of communication data rate among sending blockchain transactions, video and audio data on a Raspberry Pi, as reported in Table \[tab: appComm\]. We observe that the regular transactions of resource requests bring very low overhead to the I/O performance and overall network bandwidth.
In summary, we observe that the blockchain can support and collaborate with the IoT application in a distributed and secure way. The overhead is within a reasonable and acceptable range, and the system is feasible to satisfy the requirements to build a multi-application EdgeChain platform for future demands.
Applications Data Rate
----------------------------- -----------
Blockchain Transactions 0.54 KB/s
Face Recognition 1.64 MB/s
Natural-language Processing 8.12 KB/s
: Comparison of communication rate[]{data-label="tab: appComm"}
Resource Allocation Performance of the Pricing Scheme
-----------------------------------------------------
At last, we evaluate the resource allocation performance of the proposed pricing scheme. The goal of resource allocation is to improve the acceptance rate of use requests, which mainly depends on the proposed pricing mechanism.
We first evaluate the influence of $\alpha$ and $\beta$. $\alpha$ has no effect on the performance since it determines the range of ${ \alpha }^{ \frac { { r }_{ i\_ j } }{ c\_ j } } $ is located in $[1, \alpha] $. In contrast, $\beta$ adjusts the impact of application priority where the high-priority requests are more likely to be served. We do three random simulations and each one contains 2,000 iteration of random numbers of user requests with different resource requirements. The system parameters are set in Table \[tab: sysConfi\] and the request parameters are set in Table \[tab: reqConfi\]. Fig. \[beta\] shows the best range of Beta is in \[1.3, 1.4\] and the too large beta will lead to decrement of acceptance rate since the requests admission simply depends on the priority.
Second, we compare the acceptance rates among the pricing mechanism, First-Come-First-Serve (FCFS) and multi-level scheduling based on priority, where $\beta=1.35$. Fig. \[acRate1\] shows that our proposed algorithm performs best. Then, we evaluate the performance with the change of total edge resources, as illustrated in Fig. \[acRate2\]. Starting from the configuration in Table \[tab: sysConfi\], the amount of resources gradually decreases to lower percentages. Our pricing algorithm performs better.
$\alpha$ 100
-------------------- -----
CPU capacity 300
Memory capacity 250
Storage capacity 250
Bandwidth capacity 250
: System parameters[]{data-label="tab: sysConfi"}
**Priority** **CPU** **Memory** **Storage** **Bandwidth** **Lifetime**
-------------- ----------- ------------ ------------- --------------- --------------
Level 1 \[1,5\] \[1,5\] \[1,5\] \[1,5\] \[1,5\]
Level 2 \[10,15\] \[5,10\] \[5,10\] \[1,10\] \[1,5\]
Level 3 \[1,5\] \[1,5\] \[1,5\] \[1,5\] \[1,5\]
Level 4 \[1,3\] \[1,3\] \[1,3\] \[1,3\] \[1,3\]
: Requests parameters[]{data-label="tab: reqConfi"}
Related Work {#sec:relatedwork}
============
Due to the interdisciplinary essence of EdgeChain, related work comes from different aspects such as IoT, edge computing, blockchain, and smart contracts. A great amount of efforts have been focused on these individual topics, thus, limited by the space, we will not enumerate all the separate efforts. Instead, we will focus on those directly or closely related work. The most closely related work are *Xiong et al.* [@XIO18; @XIO17] that uses game theory and a pricing mechanism to optimize the profits of the miners at the edge servers. It focuses on the blockchain running costs. *Chatzopoulos et al.* [@CHA17] focuses on computation offloading between devices themselves by using some incentive and reputation schemes. *Sharma et al.* [@SHA17] proposes a conceptual software-defined edge nodes scheme using multi-layer blockchain. Different from these work, our research focus is not on blockchain itself. Instead, we use blockchain as carrying vehicle to provision resources for various IoT applications and control and regulate IoT devices’ behavior. More reviewing articles [@CHR16; @SUB17; @YEO17; @DOR16; @KSH17] present the overall future prospects in combining blockchain and IoT.
Blockchain and smart contracts are being used to secure many different areas and we will not enumerate them here, but a few example efforts include securing smart home [@DOR17], securing 5G fog network handover [@SHA18], securing virtual machine orchestration [@BOZ17], securing access control in IoT [@ZHA18], and secure data provenance management [@RAM17].
Another thrust of related work is about edge computing research. A large amount of existing work are either on specific applications such as video analytics, vehicular network, cognitive assistance, and emergency response, or very heavily focused on optimizing specific targets such as revenue, cost, delay, or energy consumption associated with operations such as mobile edge offloading, service migration, virtual machines chaining, placement, and orchestration. We will not list all of these works but two good start reading points are [@SHI16; @PAN18a].
Conclusions and Future Work {#sec:conclusion}
===========================
In this paper, we discussed the design and prototype of the EdgeChain framework which is a novel edge-IoT framework based on blockchain and smart contracts. EdgeChain integrates a permissioned blockchain to link the edge cloud resources with each IoT device’s account, resource usage and hence behavior of the IoT device. EdgeChain uses a credit-based resource management system to control the IoT deivces’ resource that can be obtained from the edge server. Smart contracts are used to regulate IoT devices’ behavior and enforce policies. We implemented an EdgeChain prototype and conducted extensive experiments which showed that the cost for EdgeChain to integrate blockchain and smart contracts are within reasonable range while gaining various intrinsic benefits from blockchain and smart contracts. EdgeChain is still an ongoing project and we are currently working on various issues within the framework such as IoT Proxy, intelligent resource provisioning for multiple heterogeneous applications, and better IoT device behavior regulations.
Acknowledgment {#acknowledgment .unnumbered}
==============
The work is supported in part by National Security Agency (NSA) under grants No.: H98230-17-1-0393 and H98230-17-1-0352, and by National Aeronautics and Space Administration (NASA) EPSCoR Missouri RID research grant under No.: NNX15AK38A.
[Jianli Pan]{} is currently an Assistant Professor in the Department of Mathematics and Computer Science at the University of Missouri, St. Louis. He obtained his Ph.D. degree from the Department of Computer Science and Engineering of Washington University in St. Louis. He also holds a M.S. degree in Computer Engineering from Washington University in Saint Louis and a M.S. degree in Information Engineering from Beijing University of Posts and Telecommunications (BUPT), China. He is currently an associate editor for both IEEE Communication Magazine and IEEE Access. His current research interests include edge clouds, Internet of Things (IoT), Cybersecurity, Network Function Virtualization (NFV), and smart energy.
[Jianyu Wang]{} is currently a Ph.D. student with the Department of Mathematics and Computer Science at the University of Missouri, St. Louis. He received an M.S. in Electrical and Computer Engineering from the Rutgers University, New Brunswick. His current research interests include edge cloud and mobile cloud computing.
[Austin Hester]{} is currently an undergraduate student with the Department of Mathematics and Computer Science at the University of Missouri, St. Louis. His current research interests include Internet of Things and Blockchain.
[Yuanni Liu]{} is an associate professor at the Institute of Future Network Technologies, Chong Qing University of Posts and Telecommunications. She received her Ph.D. from the Department of network technology Institute, Beijing University of Posts and Telecommunications, China, in 2011. Her research interests include mobile crowd sensing, IoT security, and data virtualization.
[^1]: First manuscript: June 1st, 2018.
[^2]: J. Pan, J. Wang, A. Hester, and I. Alqerm are with the Department of Mathematics and Computer Science in University of Missouri, St. Louis, MO 63121, USA. (Email: [pan, jwgxc, arh5w6, alqermi]{}@umsl.edu).
[^3]: Y. Liu is with the Institute of Future Network Technologies, Chong Qing University of Posts and Telecommunications, China. (Email: liuyn@cqupt.edu.cn).
[^4]: Edge computing is also often referred as “fog computing”, “Mobile Edge Computing”, or “Cloudlet” in different literature, despite slightly different definitions and scopes. We use edge computing or edge cloud in this paper.
|
---
abstract: 'Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator $\mathcal{A}$ given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially, $\mathcal{A}$ takes weighted averages over translates of the curve. In this paper a variant of this problem is discussed where averages over both translates and dilates of a fixed curve are considered. The sharp range of estimates for the resulting operator is obtained in all dimensions, except for an endpoint. The techniques used are redolent of those previously applied in the study of $\mathcal{A}$. In particular, the arguments are based upon the refinement method of Christ, although a significant adaptation of this method is required to fully understand the additional smoothing afforded by averaging over dilates.'
address: 'Jonathan Hickman, Room 5409, James Clerk Maxwell Building, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD.'
author:
- Jonathan Hickman
bibliography:
- 'Reference.bib'
title: 'Uniform $L_x^p - L^q_{x,r}$ Improving for Dilated Averages over Polynomial Curves'
---
Introduction and statement of results
=====================================
Let $\gamma: I \rightarrow {\mathbb{R}}^d$ denote a (parametrisation of a) smooth curve in ${\mathbb{R}}^d$ where $I \subseteq {\mathbb{R}}$ is an interval. Consider the operator $A$ defined, at least initially, on the space of for all test functions $f$ on ${\mathbb{R}}^d$ by $$\label{operator}
Af(x,r) := \int_I f(x-r \gamma (t)) \alpha(t)\,{\mathrm{d}}t \qquad \textrm{for all $(x,r) \in {\mathbb{R}}^d \times [1,2]$.}$$ Here $\alpha$ denotes some density which is assumed to be smooth and non-negative. Thus $A$ takes averages over translates of dilates of $\gamma$. A natural problem is to establish the range of $(p,q_1,q_2)$ for which there is an a priori mixed norm estimate either of the form $$\label{mixed1}
\|Af\|_{L^{q_2}_tL^{q_1}_x({\mathbb{R}}^d \times [1,2])} \leq C_{d, \gamma} \|f\|_{L^p({\mathbb{R}}^d)}$$ or $$\label{mixed2}
\|Af\|_{L^{q_1}_xL^{q_2}_t({\mathbb{R}}^d \times [1,2])} \leq C_{d, \gamma} \|f\|_{L^p({\mathbb{R}}^d)}.$$ This question subsumes the study of the $L^p$ mapping properties of both single averages and maximal functions associated to space curves. The archetypical case to consider is when $\gamma := h \colon [0,1] \to {\mathbb{R}}^d$ is the so-called *moment curve* given by $$h(t) := (t, t^2, \dots, t^d)$$ with constant density $\alpha \equiv 1$. With this choice of curve and density the following results are known:
- Taking $q_2 = \infty$ in reduces matters to determining the set of $(p,q)$ for which the single averages $$\label{single}
\mathcal{A}_rf(x) := \int_0^1 f(x - rh(t))\,{\mathrm{d}}t$$ are type $(p,q)$ uniformly for $r \in [1,2]$. The $L^p-L^q$ mapping properties of $\mathcal{A}:= \mathcal{A}_1$ were investigated in low dimensions in a number of papers [@Littman1973; @Oberlin1987; @Oberlin1997; @Greenleaf1999; @Oberlin1999] before being completely determined in all dimensions by Stovall [@Stovall2009] using powerful new methods developed by Christ [@Christb; @Christ1998].
- On the other hand, if one sets $q_2 = \infty$ in the situation is very different. In particular, one now wishes to understand the $L^p-L^q$ mapping properties of the maximal function $\mathcal{M}$ associated to $h$, defined by $$\mathcal{M}f(x) := \sup_{1 \leq r \leq 2} |\mathcal{A}_rf(x)|.$$ A celebrated theorem of Bourgain [@Bourgain1986] established $L^p-L^p$ mapping properties for $d=2$; this result was extended by Schlag [@Schlag1997a] who proved an almost-sharp range of $L^p-L^q$ estimates.[^1] However, the problem of determining even the $L^p-L^p$ range remains open in all other dimensions. Some partial results in this direction are given in [@Pramanik2007].[^2]
In this paper the special case of and where $q_1 = q_2$ is considered. In particular, Theorem \[momentthm\] below almost completely determines the set of $(p,q)$ for which $A$ is bounded from $L^p_x({\mathbb{R}}^d)$ to $L^q_{x,r}({\mathbb{R}}^d \times [1,2])$ when $\gamma(t) := h(t)$ is the moment curve. Testing the inequality on some simple examples (see Section \[necessity\]) shows such a bound is possible only if $(1/p, 1/q)$ lies in the trapezium $\mathcal{T}_d$ given by the closed, convex hull of the set $$\{(0, 0), (1,1), (1/p_1, 1/q_1), (1/p_2, 1/q_2)\}$$ where $$\left(\frac{1}{p_1}, \frac{1}{q_1} \right):= \left(\frac{1}{d}, \frac{d-1}{d(d+1)}\right) \quad \textrm{and} \quad \left(\frac{1}{p_2}, \frac{1}{q_2} \right):= \left( \frac{d^2 - d+2}{d(d+1)}, \frac{d-1}{d+1} \right).$$
This condition is shown to be sufficient, at least up to an endpoint.
\[momentthm\] For $d \geq 2$ the operator $Af(x,r) := \mathcal{A}_rf(x)$ is bounded from $L^p_x({\mathbb{R}}^d)$ to $L^q_{x,r}({\mathbb{R}}^d\times [1,2])$ for all $(1/p,1/q) \in \mathcal{T}_d\setminus \{(1/p_1, 1/q_1)\}$. If $(1/p,1/q) \notin \mathcal{T}_d$, then $A$ is not restricted weak-type $(p,q)$.
The proof of Theorem \[momentthm\] proceeds by establishing a restricted weak-type inequality at the endpoint $(p_1, q_1)$. Therefore, except for the question of whether this weak-type endpoint inequality can be strengthened the theorem completely determines the $L^p$ mapping of $A$ for the given choice of curve.
The $d=2$ case of Theorem \[momentthm\] (when the curve is also a hypersurface) is already known to hold with a strong-type inequality at the endpoint. This result essentially appears, for example, in the work of Strichartz [@Strichartz1977] and Schlag and Sogge [@Schlag1997]. Furthermore, in [@Strichartz1977; @Schlag1997] it is observed that the critical $L^2_x-L^6_{x,r}$ inequality for dilated averages over circles is equivalent to a Stein-Tomas Fourier restriction theorem for a conic surface and connections between this theory and estimates for certain evolution equations are also discussed. In addition, the $d=2$ case follows from more recent work of Gressman [@Gressman2006; @Gressman2013] utilising methods which are rather combinatorial in nature. The combinatorial techniques found in [@Gressman2006] are akin to the arguments found in the present article; both are based on earlier work of Christ [@Christ1998], discussed later in the introduction.
For $d \geq 3$ the results appear to be new and, indeed, no previous (non-trivial) partial results are known to the author. It is remarked that the connection between the theory of dilated averages, Fourier restriction and analysis of PDE appears to be confined to the hypersurface setting but nevertheless Theorem \[momentthm\] is arguably of interest in its own right.
Theorem \[momentthm\] is in fact a special case of a more general result, Theorem \[bigthm\], described below. Indeed, rather than restricting attention to $h$, this paper considers $A$ defined with respect to any polynomial curve. In this setting the statement of the results requires some preliminary motivation and definitions.
Given an arbitrary curve $\gamma$, when investigating the mapping properties of the key consideration is curvature. One would expect $A$ is non-degenerate (in the sense that the largest possible range of estimates hold for $A$) if and only if the $d-1$ curvature functions associated to $\gamma$ are non-vanishing in the support of the density $\alpha$. This kind of phenomenon is well-known in the context of single averages and maximal functions[^3] (the latter over curves in ${\mathbb{R}}^2$) and, indeed, many other operators whose definition depends on some submanifold (or family of submanifolds) of ${\mathbb{R}}^d$ (see, for instance, [@Christ1999; @Tao2003]). One method for quantifying the relationship between the curvature of $\gamma$ and the boundedness of $A$ is to introduce a specific choice of weight $\lambda_{\gamma}$ in the definition of the operator. In particular, $\lambda_{\gamma}$ is carefully chosen to vanish at the flat points of the curve so as to ameliorate the effect of the degeneracies. One can then hope to achieve $L^p_x - L^q_{x,r}$ boundedness for the full range of exponents corresponding to the non-degenerate case under mild hypotheses on the curve. This strategy follows the example of numerous authors (notably Drury [@Drury1990], Oberlin [@Oberlin2002], Dendrinos, Laghi and Wright [@Dendrinos2009], Stovall [@Stovall2014; @Stovall2010] and Dendrinos and Stovall [@Dendrinos]) who, in considering averages defined with respect to degenerate curves, have chosen the underlying measure in the definition of the operator to be the so-called *affine arc-length* measure, described below. This measure has the desired effect of dampening any degeneracies of the curve or surface and also makes the problem both affine and parametrisation invariant.
To make this discussion precise, define the torsion $L_{\gamma}$ of the curve to be the function $$L_{\gamma}(t) := \det (\gamma^{(1)}(t) \dots \gamma^{(d)}(t) )$$ where $\gamma^{(i)}$ denotes the $i$th derivative of $\gamma$, viewed as a column vector. This function vanishes precisely when any of the $d-1$ curvature functions associated to $\gamma$ vanish. The affine arc-length measure ${\mathrm{d}}\mu_{\gamma}$ on $\gamma$ is then defined by $$\int f \,{\mathrm{d}}\mu_{\gamma} := \int_{{\mathbb{R}}} f(\gamma(t)) \lambda_{\gamma}(t)\,{\mathrm{d}}t$$ whenever $f \in C_c({\mathbb{R}}^d)$, say, where $\lambda_{\gamma}(t) := |L_{\gamma}(t)|^{2/d(d+1)}$. In this paper the operator $A_{\gamma}$ given by convolution with dilates of this measure is studied. Explicitly, $$\label{Poperator}
A_{\gamma}f(x,r) := \int_{{\mathbb{R}}} f(x-r \gamma(t)) \lambda_{\gamma}(t)\,{\mathrm{d}}t$$ for all test functions $f$ on ${\mathbb{R}}^d$.
The choice of weight $\lambda_{\gamma}$ is further motivated by the fact that the resulting measure exhibits both parametrisation and affine invariance (for a detailed discussion see, for example, [@Christc]). Consequently, for all exponents $1 \leq p,q < \infty$ satisfying the relation $1/q = 1/p - 2/d(d+1)$ any $L^p_x - L^q_{x,r}$ inequality for $A_{\gamma}$ is affine invariant in the sense that if $\gamma$ is replaced with $X \circ \gamma$ for some invertible linear transformation $X$, then the estimate still holds with the same constant. One can therefore hope to achieve estimates for $A_{\gamma}$ which are uniform over all $\gamma$ belonging to a large class of curves. A counter-example due to Sjölin [@Sjolin1974] demonstrates such uniformity is impossible if the class includes curves which exhibit an arbitrarily large number of oscillations (see also [@Dendrinos]). It is therefore natural to postulate uniform estimates are possible over all polynomial curves of some fixed degree, since the degree controls the number of oscillations.
\[bigthm\] Let $d \geq 2$ and $P \colon {\mathbb{R}}\to {\mathbb{R}}^d$ be a polynomial curve whose components have maximum degree $n$. Then $$\|A_Pf\|_{L^q_{x,r}({\mathbb{R}}^d \times [1,2])} \leq C_{n,d} \|f\|_{L^p_x({\mathbb{R}}^d)}$$ whenever $(1/p, 1/q)$ lies in $\mathcal{T}_d \setminus\{(1/p_1, 1/q_1)\}$ and satisfies $1/q = 1/p - 2/d(d+1)$. Here the constant $C_{n,d}$ depends only on $p$, $q$, the dimension $d$ and the degree $n$ of the polynomial mapping.
Again, the theorem will follow by demonstrating a uniform restricted weak-type $(p_1, q_1)$ inequality holds. In addition, Theorem \[bigthm\] is almost-sharp in the sense that if $(1/p, 1/q)$ does not lie in the intersection of $\mathcal{T}_d$ with the line $1/q = 1/p - 2/d(d+1)$, then such estimates do not hold with a finite constant.
It is remarked that the result of Theorem \[bigthm\] is new for dimensions $d \geq 3$ whilst the $d=2$ case follows from a very general theorem due to Gressman [@Gressman2013]. Indeed, Gressman’s theorem, *inter alia*, establishes the hypersurface analogue of Theorem \[bigthm\] in all dimensions, up to and including all the relevant endpoints.
If ${\mathbb{R}}$ is replaced with a bounded interval $I$ in the definition of $A_P$, then the resulting operator (which is denoted by $A_P^{\mathrm{c}}$) is trivially seen to be bounded from $L^p_{x}$ to $L^p_{x,r}$ for all $1 \leq p \leq \infty$. Real interpolation therefore yields the following corollary.
The operator $A_P^{\mathrm{c}}$ is bounded from $L^p_{x,r}$ to $L^q_x$ for all $(1/p,1/q) \in \mathcal{T}_d\setminus \{(1/p_1, 1/q_1)\}$. If $(1/p,1/q) \notin \mathcal{T}_d$ and $L_P \not\equiv 0$, then $A_P$ is not restricted weak-type $(p,q)$.
Notice, when $P = \gamma$ is the moment curve defined above, the torsion function $L_{\gamma}$ is constant and thus Theorem \[momentthm\] is indeed a special case of Theorem \[bigthm\].
It is natural to ask whether the restricted weak-type $(p_1, q_1)$ endpoint can be strengthened to a strong-type estimate. This is certainly the case in dimension $d=2$ where the inequality is a consequence of the aforementioned theorem due to Gressman [@Gressman2013]. Furthermore, one may recover the strong-type bound for $d=2$ by combining the analysis contained within the present article with an extrapolation method due to Christ [@Christb] (see also [@Stovall2009]). It is possible that the argument can be adapted to the case where $d$ belongs to a certain congruence class modulo $3$ to (potentially) establish the strong-type bound in this situation. A more detailed discussion of the validity of the strong-type endpoint appears below in Remark \[strong type remark\].
Theorems \[momentthm\] and \[bigthm\] belong to a growing body of works which have applied variants of the geometric and combinatorial arguments due to Christ [@Christ1998] to the study of operators collectively known as generalised Radon transforms, of which $A$ is an example. Essentially these operators are defined for any point $y$ belonging to $\Sigma$ an $n$-dimensional manifold by integration over a $k$-dimensional manifold $M_y$ which depends on $y$, where $k < n$ is referred to here as the *dimension of the associated family*. The techniques of [@Christ1998] have fruitfully been applied and developed in, for instance, [@Christ; @Christa; @Christ2002; @Christ2008; @Dendrinos2009; @Gressman2004; @Gressman2009; @Stovall2014; @Stovall2009; @Stovall2010; @Tao2003] to study the Lebesgue mapping properties of one-dimensional generalised Radon transforms $R$ which are, roughly, operators $R$ for which $R$ and its adjoint $R^*$ are both generalised Radon transforms given by integration over some family of curves. The approach has been less successful when considering $R$ which are *unbalanced* in the sense that $R$ and $R^*$ are both generalised Radon transforms but the dimensions of the associated families are not equal, although it has still produced results in some specific cases, for example [@Erdogan2010; @Gressman2006; @Gressman2013]. The dilated averaging operator fits into this framework by setting $\Sigma := {\mathbb{R}}^d \times (1,2)$ and for each $(x,r) \in \Sigma$ defining $M_{(x,r)}$ to be the curve parametrised by $t \mapsto x - r \gamma(t)$. Observe that although $A$ is defined by integration over curves, the adjoint of $A$ is defined by integration over $2$-surfaces and hence the operator is unbalanced.
The structure of the paper is as follows. In the following section the necessary conditions on $(p,q)$ for $A$ to be restricted weak type $(p,q)$ are discussed. In Section \[overview of refinement method\] standard methods together with estimates for single averages are combined to reduce the proof of Theorem \[bigthm\] to proving a single restricted weak-type inequality. Christ’s method of refinements is also reviewed and used to establish the simple case of Theorem \[momentthm\] when $d=3$. The remaining sections develop this method to be applicable in the general situation.
A word of explanation concerning notation is in order: throughout the paper $C$ and $c$ will be used to denote various positive constants whose value may change from line to line but will always depend only on the dimension $d$ and degree $\deg P$ of some fixed polynomial. If $X, Y \geq 0$, then the notation $X \lesssim Y$ or $Y \gtrsim X$ signifies $X \leq C Y$ and this situation is also described by “$X$ is $O(Y)$”. In addition, $X \sim Y$ indicates $X \lesssim Y \lesssim X$. Finally, the cardinality of any finite set $B$ will be denoted by $\# B$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The author wishes acknowledge his PhD supervisor, Prof. Jim Wright, for all his kind and patient guidance relating to this work. He would also like to thank both Marco Vitturi and Betsy Stovall for elucidating discussions regarding some of the references.
Necessary conditions {#necessity}
====================
Suppose the operator $A$ from Theorem \[momentthm\] satisfies a restricted weak-type $(p,q)$ inequality for some $1 \leq p,q < \infty$. Here it is shown that the exponents $p,q$ must satisfy four conditions, each corresponding to an edge of the trapezium $\mathcal{T}_d$. The first three conditions also appear in the study of the averaging operator $\mathcal{A}$ defined in the introduction and are deduced by the same reasoning. The remaining condition does not appear in the theory of single averages and here the dilation parameter plays a non-trivial rôle, although the arguments are only marginally different from those used to examine $\mathcal{A}$.
(3.1,0) -| (0,3.1) ; at (1.5, 0) [$1/p$]{}; at (0, 1.5) [$1/q$]{}; (0,0) coordinate (es) – (3.0,3.0) coordinate (ee); (0,0) coordinate (es) – (1.0,0.5) coordinate (ee); (2.0,1.5) coordinate (es) – (3.0,3.0) coordinate (ee); (1.0,0.5) coordinate (es) – (2.0,1.5) coordinate (ee); (0,0) coordinate (es) – (1.5,1.0) coordinate (ee); (0,3) coordinate (es) – (3,3) coordinate (ee); (3,0) coordinate (es) – (3,3) coordinate (ee);
(1.0,0.5) node\[dot\] (int2) ; at (1.0,0.5) [$ \big(\frac{1}{p_1}, \frac{1}{q_1}\big)$]{}; (2.0,1.5) node\[dot\] (int2) ; at (2.0,1.5) [$ \big(\frac{1}{p_2}, \frac{1}{q_2}\big)$]{}; (1.5,1) node\[dot\] (int2) ; at (1.5,1) [$ \big(\frac{1}{q_2'}, \frac{1}{p_2'}\big)$]{}; (0,0) node\[dot,label=below:[$(0,0)$]{}\] (int2) ; (3,3) node\[dot,label=right:[$(1,1)$]{}\] (int2) ;
To begin, a slight modification of a general theorem of Hörmander [@Hormander1960] implies $p \leq q$.
For the second condition, let $R(\delta) : = \prod_{j=1}^d [-\delta^j,\delta^j]$ and note that $$A \chi_{R(\delta)}(x,r) = |\{ t \in [0,1] : x - r\gamma (t) \in R(\delta) \}|.$$ If $x \in (1/2) R(\delta)$, then whenever $t \in [0, \delta/4]$ it follows that $$|x_j -rt^j| \leq \delta^j/2 + 2 (\delta/4)^j \leq \delta^j \qquad \textrm{for $j = 1, \dots, d$}$$ and therefore $$A \chi_{R(\delta)}(x,r) \geq \frac{\delta}{4} \chi_{(1/2)R(\delta)}(x).$$ Consequently, applying the hypothesised restricted weak-type estimate, $$|R(\delta)| \lesssim \left|\left\{ (x,r) \in {\mathbb{R}}^d \times [1,2] : A \chi_{R(\delta)}(x,r) > \delta/8 \right\}\right| \lesssim \left( \frac{1}{\delta}|R(\delta)|^{1/p}\right)^{q}.$$ Observe $|R(\delta)| \sim_d \delta^{d(d+1)/2}$ and so the preceding inequality implies $$\delta^{d(d+1)/(2q)} \lesssim \delta^{d(d+1)/(2p) - 1} \qquad \textrm{for all $0 < \delta < 1$.}$$ The exponents $(p,q)$ must therefore satisfy the relation $$\frac{1}{q} \geq \frac{1}{p} - \frac{2}{d(d+1)}.$$ The third condition is established by testing $A$ on $\chi_{B(\delta)}$, the characteristic function of a ball $B(\delta) \subset {\mathbb{R}}^d$ of radius $0 < \delta < 1$, centred at the origin. It is easy to see $$\label{nec2}
A \chi_{B(\delta)}(x,r) \gtrsim \delta \chi_{\mathcal{N}_r(\delta)}(x)$$ where $\mathcal{N}_r(\delta)$ is a $\delta/3$-neighbourhood of the $r$-dilate of the moment curve; that is, the set of all points $x \in {\mathbb{R}}^d$ for which $|x- r\gamma(t_0)| < \delta/3$ for some $t_0 \in [0,1]$. The hypothesised restricted weak-type estimate together with imply $$\begin{aligned}
\left|\left\{(x,r) \in {\mathbb{R}}^d \times [1,2] : x \in \mathcal{N}_r(\delta)\right\}\right| &\leq& \left|\left\{ (x,r) \in {\mathbb{R}}^d \times [1,2] : A \chi_{B(\delta)}(x,r) > C \delta \right\}\right| \\
&\lesssim& \left( \frac{1}{\delta}|B(\delta)|^{1/p}\right)^{q}.\end{aligned}$$ Observe $|B(\delta)| \sim_d \delta^d$ whilst $|\mathcal{N}_r(\delta)| \gtrsim \delta^{d-1}$ for all $r \in [1,2]$ and so the preceding inequality implies $$\delta^{(d-1)/q} \lesssim \delta^{d/p - 1}\qquad \textrm{for all $0 < \delta < 1$.}$$ Thus the exponents must satisfy the relation $$\frac{1}{q} \geq \frac{d}{d-1} \frac{1}{p} - \frac{1}{d-1}.$$
The final condition on $(1/p, 1/q)$ is deduced by considering the adjoint $A^*$ of $A$. A simple computation yields $$A^*g(x) = \int_1^2 \int_0^1 g(x + r\gamma(t), r)\,{\mathrm{d}}t {\mathrm{d}}r$$ for suitable functions $g$ defined on ${\mathbb{R}}^d \times [1,2]$. The hypothesis on $(p,q)$ is equivalent to the assumption that $A^*$ is restricted weak-type $(q',p')$. For $B(\delta)$ as above, let $F(\delta)$ denote the set $B(\delta) \times [1, 1+c\delta]$ for some small constant $c$. Observe $$A^* \chi_{F(\delta)}(x) \gtrsim \delta^2 \chi_{\mathcal{N}_1(\delta)}(-x)$$ where $\mathcal{N}_1(\delta)$ is as defined above. Therefore, $$|\mathcal{N}_1(\delta)| \leq \left|\left\{ x \in {\mathbb{R}}^d : A^* \chi_{F(\delta)}(x) \gtrsim \delta^2 \right\}\right| \lesssim \left( \frac{1}{\delta^{2}}|F(\delta)|^{1/q'}\right)^{p'}.$$ Finally, $|F(\delta)| \sim_d \delta^{d+1}$ whilst $|\mathcal{N}_1(\delta)| \gtrsim \delta^{d-1}$ and so the preceding inequality implies $$\delta^{(d-1)/p'} \lesssim \delta^{(d+1)/q' - 2} \qquad \textrm{for all $0 < \delta < 1$.}$$ It follows that the exponents must satisfy the relation $(d+1)/q' - 2 \leq (d-1)/p'$ which can be rewritten as $$\frac{1}{q} \geq \frac{d-1}{d+1} \frac{1}{p}.$$
An overview of the refinement method {#overview of refinement method}
====================================
It remains to show the conditions on $(p,q)$ described in Theorem \[bigthm\] are sufficient to ensure $A_P$ satisfies a type $(p,q)$ inequality with the desired uniformity. Real interpolation immediately reduces matters to establishing a uniform restricted weak-type $(p_1, q_1)$ and strong type $(p_2, q_2)$ estimate for $A_P$. The latter is easily dealt with by appealing to the existing literature. Indeed, a theorem of Stovall [@Stovall2010] implies the estimate $$\label{stovallest}
\|A_Pf( \,\cdot\, , r) \|_{L^{q_2}_x({\mathbb{R}}^d)} \lesssim \|f\|_{L^{p_2}_x({\mathbb{R}}^d)}$$ holds for all $r \in [1,2]$. Taking $L^{q_2}_r([1,2])$-norms of both sides of yields the uniform type $(p_2, q_2)$ inequality for $A_P$ and the proof of Theorem \[bigthm\] is therefore reduced to establishing the following Proposition.
\[weakthm\] For $d \geq 2$ the inequality $$\label{weak}
\langle A_P\chi_E, \chi_F \rangle \lesssim |E|^{1/d}|F|^{(d^2+1)/d(d+1)}$$ is valid for all pairs of Borel sets $E \subset {\mathbb{R}}^d$ and $F \subset {\mathbb{R}}^d \times [1,2]$ of finite Lebesgue measure.
The proof of Proposition \[weakthm\] will utilise the geometric and combinatorial techniques introduced by Christ in [@Christ1998], which were briefly discussed in the introduction. Collectively these techniques are referred to as the method of refinements. In this section the rudiments of the method are reviewed. It is instructive to consider the proof of the analogue of Proposition \[weakthm\] in three dimensions ($d=3$) for the operator $A$ from the statement of Theorem \[momentthm\]. In this situation the arguments are extremely simple and only a crude version of the refinement procedure is required.
Let $E$ and $F$ denote fixed sets satisfying the hypotheses of Proposition \[weakthm\] for $d=3$. Assume, without loss of generality, that $\langle A\chi_E, \chi_F \rangle \neq 0$ where $A$ is the operator from Theorem \[momentthm\]. One wishes to establish the inequality $$\langle A\chi_E, \chi_F \rangle \lesssim |E|^{1/3}|F|^{5/6},$$ from which Theorem \[momentthm\] follows for the case $d=3$. Defining constants $\alpha$ and $\beta$ by the equation $\langle A\chi_E, \chi_F \rangle = \alpha|F| = \beta|E|$, one may rewrite the preceding inequality as a lower bound on the measure of $E$; explicitly, $$\label{momentequivweake}
|E| \gtrsim \alpha^{6} (\beta/\alpha).$$
The basic idea behind Christ’s method is to attempt to prove by using iterates of $A$ and $A^*$ to construct a natural parameter set $\Omega \subset {\mathbb{R}}^3$ and parametrising function $\Phi : \Omega \rightarrow E$ with a number of special properties. First of all, $\Phi$ must have bounded multiplicity so, by applying the change of variables formula, $$|E| \gtrsim \int_{\Omega} |J_{\Phi}(t)| \,{\mathrm{d}}t$$ where $J_{\Phi}$ denotes the Jacobian of $\Phi$. It then remains to bounded this integral from below by some expression in terms of $\alpha$ and $\beta$, which is possible provided that the parametrisation has been carefully constructed.
Following [@Christ1998], define $$\begin{aligned}
F_1 &:=& \big\{ (x,r) \in F : A \chi_E(x,r) > \alpha / 2 \big\}, \\
E_1 &:=& \big\{ y \in E : A^* \chi_{F_1}(y) > \beta / 4 \big\}.\end{aligned}$$ It is not difficult to see the assumptions on $E$ and $F$ imply $\langle A\chi_{E_1}, \chi_{F_1} \rangle \neq 0$ and therefore $E_1$ is non-empty. Fix $y_0 \in E_1$ and define a map $\Phi_1 \colon [1,2] \times [0,1] \to {\mathbb{R}}^3 \times [1,2]$ by $$\label{moment Phi1}
\Phi_1(r_1,t_1) := \left(\begin{array}{c}
y_0 + r_1h(t_1) \\
r_1
\end{array} \right).$$ Note that the set $$\Omega_1:= \left\{ (r_1, t_1) \in [1,2] \times [0,1] : \Phi_1(r_1,t_1) \in F_1 \right\}$$ satisfies $|\Omega_1| > \beta / 4$. Similarly, define a map $\Phi_2 \colon [1,2] \times [0,1]^2 \to {\mathbb{R}}^3$ by $$\label{moment Phi2}
\Phi_2(r_1, t_1, t_2) := y_0 + r_1h(t_1) - r_1h(t_2)$$ and observe for each $(r_1, t_1) \in \Omega_1$ the set $$\Omega_2(r_1, t_1):= \left\{ t_2 \in [0,1] : \Phi_2(r_1, t_1, t_2) \in E \right\}$$ satisfies $|\Omega_2(r_1, t_1)| > \alpha / 2$. Finally, define the structured set $$\Omega_2 := \big\{(r_1, t_1, t_2) \in [1,2] \times [0,1]^2 : (r_1, t_1) \in \Omega_1 \textrm{ and } t_2 \in \Omega_2(r_1, t_1) \big\}.$$ Now, $\Omega := \Omega_2 \subset {\mathbb{R}}^3$ is the parameter set alluded to above and $\Phi := \Phi_2 |_{\Omega} : \Omega \rightarrow E$ the parametrising function. Observe $\Phi$ is well-defined by the preceding observations and the polynomial nature of this map ensures it has almost everywhere bounded multiplicity.[^4] The absolute value of the Jacobian $J_{\Phi}(r_1, t_1, t_2)$ of $\Phi$ may be expressed as $$r_1^2 \left| \det \left(\begin{array}{ccc}
1 & 1 & t_2 - t_1 \\
2t_1 & 2t_2 & t_2^2 - t_1^2 \\
3t_1^2 & 3t_2^2 & t_2^3 - t_1^3 \end{array}\right) \right| = 6r_1^2 \bigg| \int_{t_1}^{t_2} V(t_1, t_2, x) \,{\mathrm{d}}x \bigg|$$ where $V(x_1, \dots, x_m) := \prod_{1\leq i<j \leq m} (x_j - x_i)$ denotes, and will always denote, the $m$-variable Vandermonde polynomial. The sign of $V(t_1, t_2, x)$ does not change as $x$ varies between $t_1$ and $t_2$ and so modulus signs can be placed inside the integral in the above expression. Thus, $$|J_{\Phi}(r_1, t_1, t_2)| \gtrsim |t_1 - t_2| \int_{t_1}^{t_2} |x-t_1||t_2-x| \,{\mathrm{d}}x \gtrsim |t_1 - t_2|^4,$$ where the last inequality may easily be deduced by removing a $|t_1-t_2|/8$-neighbourhood of the endpoints $\{t_1,t_2\}$ from the domain of integration. Consequently, by applying the change of variables formula, $$|E| \gtrsim \int_{\Omega} |J_{\Phi}(r_1, t_1, t_2)| \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1 \gtrsim \int_{\Omega_1}\int_{\Omega_2(r_1, t_1)} |t_1 - t_2|^4 \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1.$$ For each $(t_1,r_1) \in \Omega_1$ define $\widetilde{\Omega}_2(r_1, t_1) := \Omega_2(r_1, t_1)\setminus (t_1 + c\alpha, t_1 - c\alpha)$ for a suitably small constant $c$, chosen so that $|\widetilde{\Omega}_2(r_1, t_1)| \gtrsim \alpha$. Hence, $$|E| \gtrsim \int_{\Omega_1}\int_{\widetilde{\Omega}_2(r_1, t_1)} |t_1 - t_2|^4 \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1 \gtrsim \alpha^4 \int_{\Omega_1}\int_{\widetilde{\Omega}_2(r_1, t_1)} \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1 \gtrsim \alpha^5 \beta,$$ and this concludes the proof of and thereby establishes Theorem \[momentthm\] for $d=3$.
The remainder of the paper will develop this elementary argument in order to prove Proposition \[weakthm\] in any dimension $d$ and for any polynomial curve $P$.
The polynomial decomposition theorem of Dendrinos and Wright {#decomposition 1}
============================================================
The refinement method essentially reduces the problem of establishing the restricted weak-type inequality from Proposition \[weakthm\] to estimating a Jacobian determinant associated with a certain naturally arising change of variables. In the case of the moment curve this Jacobian takes a particularly simple form involving a Vandermonde polynomial $V(t)$. For a general polynomial curve $P \colon {\mathbb{R}}\to {\mathbb{R}}^d$ one is led to consider expressions of the form $$\label{JP}
J_P(t) := \det(P'(t_1) \dots P'(t_d))$$ for $t = (t_1, \dots, t_d) \in {\mathbb{R}}^d$.[^5] The multivariate polynomial $J_P$ can be effectively estimated by comparing it with the Vandermonde polynomial and a certain geometric quantity expressed in terms of the torsion function (whose definition is recalled below). This leads to what is referred to here (and in [@Dendrinos2010]) as a *geometric inequality* for $J_P$. It is often the case that such a comparison is not possible globally; however, an important theorem due to Dendrinos and Wright [@Dendrinos2010] demonstrates the existence of a decomposition of the real line into a bounded number of intervals, ${\mathbb{R}}= \bigcup_{m=1}^C \overline{I_m}$, such that such a geometric inequality holds on each constituent interval $I_m$. Furthermore, the torsion function has a particularly simple form when restricted to an $I_m$: it is comparable to a centred monomial. Restricting the analysis to an interval arising from the Dendrinos-Wright decomposition therefore significantly simplifies the situation and allows for an effective estimation of the Jacobian $J_P$.
In order to state the decomposition lemma, recall the torsion of the curve $P$ is defined to be the polynomial function $$L_P(t) := \det(P^{(1)}(t) \dots P^{(d)}(t))$$ where $P^{(i)}$ denotes the $i$th derivative of $P$.
\[Dendrinos Wright theorem\] Let $P \colon {\mathbb{R}}\to {\mathbb{R}}^d$ be a polynomial curve of degree $n$ such that $L_P \not\equiv 0$. There exists an integer $C = C_{d,n}$ and a decomposition ${\mathbb{R}}= \bigcup_{m=1}^{C} \overline{I_m}$ where the $I_m$ are pairwise disjoint open intervals with the following properties:
1) Whenever $\mathbf{t} = (t_1, \dots, t_d) \in I_m^d$ the geometric inequality $$|J_P(\mathbf{t})| \gtrsim \prod_{i=1}^d|L_P(t_i)|^{1/d} |V(\mathbf{t})|$$ holds.
2) For every $1 \leq m \leq C$ there exists a positive constant $D_{m}$, a non-negative integer $K_{m} \lesssim 1$ and a real number $b_{m} \in {\mathbb{R}}\setminus I_m$ such that $$|L_{P}(t)| \sim D_{m}|t - b_m|^{K_{m}} \qquad \textrm{for all $t \in I_m$.}$$
Theorem \[Dendrinos Wright theorem\] originally appeared in [@Dendrinos2010] where it was used to study Fourier restriction operators associated to polynomial curves (see [@Stovall] for further developments in this direction). Concurrently, Dendrinos, Laghi and Wright [@Dendrinos2009] applied the decomposition to establish uniform estimates for convolution with affine arc-length on polynomial curves in low dimensions; their results were subsequently extended to all dimensions by Stovall [@Stovall2010]. Many of the methods of this paper are based on those found in [@Dendrinos2009; @Stovall2010].
Fixing a polynomial $P$ for which $L_P \not\equiv 0$, to prove Proposition \[weakthm\] it suffices to establish the analogous uniform restricted weak-type inequalities for the local operators $$A^c_P(x,r) := \int_I f(x - r P(t)) \, \lambda_P(t){\mathrm{d}}t$$ where $I$ is any bounded interval. Furthermore, one may assume $I$ lies completely within one of the intervals $I_m$ produced by the decomposition (indeed, $A^c_P$ can always be expressed as a sum of a bounded number of operators of the same form for which this property holds). Observing the translation, reflection and scaling invariance of the problem, one may assume $D_{m} = 1$, $b_m = 0$ and $I \subset (0, \infty)$ with $|I| = 1$ without any loss of generality. Similar reductions were made in [@Stovall2010] where further details can be found. Notice under these hypotheses, $|L_P(t)| \sim t^K$ uniformly on $I$ for some non-negative integer $K \lesssim 1$.
Henceforth $A$ will denote the operator defined by $$\label{reduced operator}
Af(x, r) := \int_I f(x - r P(t)) \,{\mathrm{d}}\mu_P(t)$$ where $\mu_P$ is now the weighted measure ${\mathrm{d}}\mu_P(t) := \lambda_P(t){\mathrm{d}}t$; $\lambda_P$ is redefined as $\lambda_P(t):= t^{2K/d(d+1)}$ and the integer $K$ and interval $I$ satisfy the above properties. It remains to prove the analogue of the restricted weak-type inequality from Proposition \[weakthm\] for this operator.
To close this section it is remarked that Stovall [@Stovall2010] established an upper bound for certain derivatives of $J_P$ on the set $I^d$ in terms of $J_P$ itself. This estimate will be of use in the forthcoming analysis and is recorded presently for the reader’s convenience.
\[Stovall’s observation\] Let $S \subseteq \{1, \dots, d\}$ be a non-empty set of indices. Whenever $t = (t_1, \dots, t_d) \in I^d$, one has the estimate $$\bigg| \prod_{j \in S} \frac{\partial}{\partial t_j} J_P(t) \bigg| \lesssim \sum_{T \subseteq S} \sum_{u, \epsilon} \bigg(\prod_{{j} \in S\setminus T} t_{j}^{-1}\bigg)\bigg( \prod_{{j} \in T} t_{j}^{-\epsilon(j)}|t_{j} - t_{u({j})}|^{\epsilon(j) - 1}\bigg)|J_P(t)|$$ where the outer sum is over all subsets $T$ of $S$ and the inner sum is over all functions $u \colon T \to \{1, \dots, d\}$ with the property $u(j) \neq j$ for all $j \in T$ and all $\epsilon \colon T \to \{0,1\}$.
Parameter towers
================
Having made the reductions of the previous section, fix Borel sets $E \subseteq {\mathbb{R}}^d$ and $F \subseteq {\mathbb{R}}^d \times [1,2]$ of finite Lebesgue measure such that $\langle A \chi_E\,,\, \chi_F\rangle \neq 0$ where $A$ is of the special form described in . As in Section \[overview of refinement method\], the quantities $$\alpha := \frac{1}{|F|} \langle A\chi_E, \chi_F \rangle \quad \textrm{and} \quad \beta := \frac{1}{|E|} \langle \chi_E, A^*\chi_F \rangle$$ play a dominant rôle in the analysis. Indeed, by some simple algebra the inequality can be restated in terms of $\alpha$ and $\beta$ as either $$\label{equivweake}
|E| \gtrsim \alpha^{d(d+1)/2} (\beta/\alpha)^{(d-1)/2}$$ or $$\label{equivweakf}
|F| \gtrsim \alpha^{d(d+1)/2} (\beta/\alpha)^{(d+1)/2}.$$ The proof will proceed by attempting to establish either one of these estimates by applying a variant of the refinement procedure described earlier. In view of the $L^{p_2}_x - L^{q_2}_{x,r}$ estimate established in Section \[overview of refinement method\], henceforth it is assumed without loss of generality that $\alpha > \beta$. Indeed, the restricted weak-type $(p_2, q_2)$ inequality implies $$|E| \gtrsim \alpha^{d(d+1)/2} (\beta/\alpha)^{d}$$ from which follows in the case $\alpha \leq \beta$.
As in Section \[overview of refinement method\], either or will be established by constructing suitable parameter domain $\Omega$ and parametrising function $\Phi$ where $\Omega$ is some structured set. In this section the basic structure of such a domain $\Omega$ is described.
Consider a collection $\{\Omega_j\}_{j=1}^D$ of Borel measurable sets either of the form[^6] $$\label{floortower}
\Omega_j \subseteq [1,2]^{\floor{j/2}} \times I^j \qquad \textrm{for $j = 1, \dots, D$}$$ or $$\label{ceiltower}
\Omega_j \subseteq [1,2]^{\ceil{j/2}} \times I^j \qquad \textrm{for $j = 1, \dots, D$}.$$ In order to be concise it is useful to let $\bracket{x}$ ambiguously denote either $\ceil{x}$ or $\floor{x}$ for any $x \in {\mathbb{R}}$, where it is understood the notation is consistent within any given equation. Thus and are considered simultaneously by writing $$\Omega_j \subseteq [1,2]^{\bracket{j/2}} \times I^j \qquad \textrm{for $j = 1, \dots, D$.}$$ Assume each $\Omega_j$ has positive $(j + \bracket{j/2})$-dimensional measure. The following definitions, which borrow terminology from [@Christ; @Christa], are fundamental in what follows.
i) A collection $\{\Omega_j\}_{j=1}^D$ of the above form is a (parameter) tower of height $D \in {\mathbb{N}}$ if for any $1 < j \leq D$ and $r_1, \dots, r_{\bracket{j/2}} \in [1,2]$ and $t_1, \dots, t_j \in I$ the following holds: $$(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j \Rightarrow (\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$$ where $\mathbf{r}_k := (r_1, \dots, r_{\bracket{k/2}})$ and $\mathbf{t}_k := (t_1, \dots, t_k)$ for $k = j-1, j$.
ii) If a tower is described as “type 1” (respectively, “type 2”) this indicates the constituent sets are of the form described in (respectively, ). Thus, when considering type 1 (respectively, type 2) towers the symbol $\bracket{x}$ is interpreted as $\floor{x}$ (respectively, $\ceil{x}$) for any $x \in {\mathbb{R}}$.
iii) Given a type 1 (respectively, type 2) tower $\{\Omega_j\}_{j=1}^D$, fix $1< j \leq D$. For each $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$ define the associated fibre $\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$ to be the set $$\left\{\begin{array}{ll}
\big\{ t_j \in I : (\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j \big\}
& \textrm{if $j$ is odd (respectively even)}\\
&\\
\left\{ (r_{\bracket{j/2}}, t_j) \in [1,2] \times I : (\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j \right\} & \textrm{if $j$ is even (respectively odd).}
\end{array}\right.$$
A type $j$ tower is characterised by the property that the initial set $\Omega_1$ is $j$-dimensional, for $j=1,2$.
For example, the collection $\{\Omega_j\}_{j=1}^2$ defined in Section \[overview of refinement method\] constitutes a type 2 tower. In what follows, type 1 towers will be of primary interest. The elements of the various levels of a type 1 tower are typically denoted using the following notation: $$\mathbf{t}_1 = t_1 \in \Omega_1, \quad (\mathbf{r}_2, \mathbf{t}_2) = (r_1, t_1,t_2) \in \Omega_2, \quad (\mathbf{r}_3, \mathbf{t}_3) = (r_1, t_1, t_2, t_3) \in \Omega_3,$$ $$(\mathbf{r}_4, \mathbf{t}_4) = (r_1, r_2, t_1, t_2, t_3, t_4) \in \Omega_4,\quad (\mathbf{r}_5, \mathbf{t}_5) = (r_1, r_2, t_1, t_2, t_3, t_4, t_5) \in \Omega_5, \dots .$$
Recall that certain mappings $\Phi_1$ and $\Phi_2$, defined in and , were associated to the tower constructed in Section \[overview of refinement method\]. Presently the analogues of these mappings in the general situation are discussed. First of all one associates to every $(x_0, r_0) \in {\mathbb{R}}^d \times [1,2]$ and $y_0 \in {\mathbb{R}}^d$ a family of functions.
i) Given $(x_0, r_0) \in {\mathbb{R}}^d \times [1,2]$ define the functions $\Psi_j(x_0, r_0;\,\cdot\,) : [1,2]^{\floor{j/2}} \times I^j \to {\mathbb{R}}^d$ by $$\label{parf}
\Psi_j(x_0, r_0;\mathbf{r}_j, \mathbf{t}_j) = x_0 + \sum_{k = 1}^j (-1)^{k} r_{\floor{k/2}} P(t_k).$$ for all $\mathbf{r}_j = (r_1, \dots, r_{\floor{j/2}}) \in [1,2]^{\floor{j/2}}$ and $\mathbf{t}_j = (t_1, \dots, t_j) \in I^j$.
ii) Given $y_0 \in {\mathbb{R}}^d$ define the functions $\Psi_j(y_0;\,\cdot\,) : [1,2]^{\ceil{j/2}} \times I^j \to {\mathbb{R}}^d$ by $$\label{pare}
\Psi_j(y_0; \mathbf{r}_j, \mathbf{t}_j) = y_0 + \sum_{k = 1}^j (-1)^{k+1} r_{\ceil{k/2}} P(t_k)$$ for all $\mathbf{r}_j = (r_1, \dots, r_{\ceil{j/2}}) \in [1,2]^{\ceil{j/2}}$ and $\mathbf{t}_j = (t_1, \dots, t_j) \in I^j$.
To any tower one associates a family of mappings on the constituent sets, defined in terms of the $\Psi_j$ functions.
\[associated mappings\] Suppose $\{\Omega_j\}_{j=1}^D$ is a type 1 (respectively, type 2) tower and fix some $z_0 = (x_0, r_0) \in {\mathbb{R}}^d \times [1,2]$ (respectively, $z_0 = y_0 \in {\mathbb{R}}^d$). The family of mappings $\{\Phi_j\}_{j=1}^D$ associated to these objects is defined as follows:
i) For $1 \leq j \leq D$ odd (respectively, even) let $\Phi_j : \Omega_j \rightarrow {\mathbb{R}}^d$ denote the map $$\Phi_j(\mathbf{r}_j, \mathbf{t}_j) := \Psi_j(z_0; \mathbf{r}_j, \mathbf{t}_j).$$
ii) For $1 \leq j \leq D$ even (respectively, odd) let $\Phi_j : \Omega_j \rightarrow {\mathbb{R}}^d \times [1,2]$ denote the map $$\Phi_j(\mathbf{r}_j, \mathbf{t}_j) := \left( \begin{array}{c}
\Psi_k(z_0, \mathbf{r}_j, \mathbf{t}_j)\\
r_{\bracket{j/2}} \end{array}\right).$$
Referring back to the simple case discussed earlier, (appropriate restrictions of) the functions defined in and are easily seen to constitute the family associated to the point $y_0$ and tower $\{\Omega_j\}_{j=1}^2$ constructed in Section \[overview of refinement method\].
For notational convenience define the following quantity $$\label{kappa}
\kappa := \frac{d(d+1)}{2K + d(d+1)}.$$ Recalling the definition of $\mu_P$ from , it is also useful to let $\nu_P$ denote the measure given by the product of Lebesgue measure on $[1,2]$ with $\mu_P$. Hence, for any Borel set $R \subseteq [1,2] \times I$, $$\nu_P(R) = \int_1^2 \int_I \chi_R(r,t)\,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}r.$$ Initially the following lemma is used to construct a suitable parameter tower.
\[dendrinosstovall\] There exists a point $(x_0, r_0) \in F$ and a type 1 tower $\{\Omega_j\}_{j = 1}^{d+1}$ with the following properties:
1) Whenever $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$ it follows that $$\alpha^{\kappa} = \max\{\alpha, \beta\}^{\kappa} \lesssim t_1 < t_2 < \dots < t_j.$$
2) For $1 \leq j \leq d+1$ odd:
i) $\Phi_j(\Omega_j) \subseteq E$;
ii) $\mu_P(\Omega_1) \gtrsim \alpha$ and if $j > 1$, then $\mu_P\left(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\right) \gtrsim \alpha$ whenever $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$;
iii) If $j > 1$ and $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, then $$\int_{t_{j-1}}^{t_j} \,\lambda_P(t){\mathrm{d}}t \gtrsim \alpha.$$
3) For $1 < j \leq d+1$ even:
i) $\Phi_j(\Omega_j) \subseteq F$;
ii) $\nu_P \left(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\right) \gtrsim \beta$ whenever $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$;
iii) If $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, then $$\int_{t_{j-1}}^{t_j} \,\lambda_P(t){\mathrm{d}}t \gtrsim \beta.$$
Lemma \[dendrinosstovall\] is a slight modification of a recent result due to Dendrinos and Stovall [@Dendrinos], based on a fundamental construction due to Christ [@Christ1998]. Rather than present a proof of Lemma \[dendrinosstovall\] a stronger statement, Lemma \[towerlem\], is established below.
To conclude this section it is noted that a tower admitting all the properties described in the previous lemma automatically satisfies a certain separation condition. This observation was also used in [@Dendrinos].
\[separation\] Let $\{\Omega_j\}_{j=1}^{d+1}$ be a tower with all the properties described in Lemma \[dendrinosstovall\].
i) Suppose $1 < j \leq d+1$ is odd. Then, for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$ it follows that $$t_j - t_i \gtrsim \alpha t_i^{-2K/d(d+1)} \qquad \textrm{for $1 \leq i \leq j-1$.}$$
ii) Suppose $1 < j \leq d+1$ is even. Then, for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$ it follows that $$t_j - t_{j-1} \gtrsim \beta t_{j-1}^{-2K/d(d+1)}; \qquad
t_j - t_{i} \gtrsim \alpha t_i^{-2K/d(d+1)} \quad \textrm{for $1 \leq i \leq j-2$.}$$
Let $1 < j \leq n+1$ be either odd or even and $1 \leq i \leq j-1$. If $j$ is even, then further suppose $i\leq j-2$. For $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, properties 1) and 2) iii) of the construction ensure there exists some $t_{i} < s_{i} < t_{j}$ for which $$\int_{t_{i}}^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t \sim \alpha.$$ Consequently, $$s_{i}^{1/\kappa} \sim \int_0^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t = \int_0^{t_{i}} \lambda_P(t)\,{\mathrm{d}}t + \int_{t_{i}}^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t \sim t_{i}^{1/\kappa} + \alpha$$ and, since $\alpha \lesssim t_{i}^{1/\kappa}$ holds by property 1), one concludes that $s_{i} \sim t_{i}$. Whence, $$|t_j - t_i| \geq |s_{i} - t_{i}| \gtrsim \bigg(\int_{t_{i}}^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t\bigg)t_{i}^{-2K/d(d+1)} \gtrsim \alpha t_{i}^{-2K/d(d+1)}.$$ The remaining case when $j$ is even and $i = j-1$ can be dealt with in a similar fashion, applying property 3) iii).
Improved parameter towers
=========================
The properties detailed in Lemma \[dendrinosstovall\], though useful, are insufficient for the present purpose. Observe that although the even fibres of the tower constructed in Lemma \[dendrinosstovall\] are two-dimensional sets, consisting of points $(r_{j/2}, t_j) \in [1,2] \times I$, all the bounds are decidedly one-dimensional in the sense that they are in terms of the $t_j$ variables and there is little reference to the dilation parameters. An additional refinement is necessary to take advantage the higher dimensionality of the even fibres.
\[towerlem\] Fix $0 < \delta \ll 1$ a small parameter. There exists a point $(x_0, r_0) \in F$ and a tower $\{\Omega_k\}_{k = 1}^{d+1}$ satisfying all the properties of Lemma \[dendrinosstovall\] with the additional property that for each even $1 < j \leq d+1$ either $$|t_j - t_{j-1}| \geq \delta (\alpha\beta)^{1/2} t_{j-1}^{-2K/d(d+1)}$$ holds for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, or $$|t_j - t_{j-1}| < \delta (\alpha\beta)^{1/2} t_{j-1}^{-2K/d(d+1)} \quad \textrm{ and } \quad |r_{j/2} - r_{j/2-1}| \gtrsim (\beta/\alpha)^{1/2}$$ both hold for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$. If the former case the index $j$ is designated “pre-red”, whilst in the latter $j$ is designated “pre-blue”. The odd vertices are “pre-achromatic”; that is, they are not assigned a “pre-colour”.
The partition of the indices into the sets of odd, pre-red and pre-blue indices plays a very similar rôle to the construction of the band structure in [@Christ1998] and in particular the “slicing method” of [@Christ; @Christ1998] will be utilised.
As the prefix “pre-” suggests, later in the argument it will be convenient to relabel the indices. In particular, in the following section an updated labelling will be introduced which designates the indices either “red”, “blue” or “achromatic”.
The result is essentially established as follows. By applying the argument of Dendrinos and Stovall [@Dendrinos] one obtains an initial tower with the properties stated in Lemma \[dendrinosstovall\]. To ensure the additional property described in Lemma \[towerlem\] holds one further refines the tower, appealing to the following elementary (but notationally-involved) result.
\[refinecor\] Let $\{ \widetilde{\Omega}_j\}_{j=1}^D$ be a tower of height $D$ and for each even $1 < k \leq D$ let $$\big\{A_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) : (\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \in \widetilde{\Omega}_{k-1} \big\}$$ a collection of measurable subsets of $[1,2] \times I$. Then there exists a tower[^7] $\{\Omega_j\}_{j=1}^D$ satisfying:
a) $\Omega_1 \subseteq \widetilde{\Omega}_1$ and $\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \subseteq \widetilde{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$ for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$ and all $1 < j \leq D$;
b) The following estimates hold for the fibres:
i) $\mu_P(\Omega_1) \geq 2^{-\floor{D/2}} \mu_P(\widetilde{\Omega}_1)$.
ii) Whenever $1 < j \leq D$ is odd, $$\mu_P\big(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) \geq 2^{-\floor{D/2}} \mu_P\big(\widetilde{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big)$$ holds for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$.
iii) Whenever $1 < j \leq D$ is even, $$\nu_P\big(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) \geq 2^{-\floor{D/2}} \nu_P\big(\widetilde{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big)$$ holds for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$.
c) For each even $1 < k \leq D$ precisely one of the following holds:
i) $\Omega_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \subseteq A_k(\mathbf{r}_{k-1}, \mathbf{t}_{k-1})$ for all $(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \in \Omega_{k-1}$;
ii) $\Omega_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \cap A_k(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) = \emptyset$ for all $(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \in \Omega_{k-1}$.
\[measure remark\] Strictly speaking, the proof below will not address the issue of whether the $\Omega_j$ are measurable (as required in the definition of a tower), although the fibres certainly will be (and therefore b) i) and b) ii) make sense). In practice, when the lemma is applied below it will be clear from the choice of sets $A_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1})$ that the resulting $\Omega_j$ are measurable and so this omission is unimportant for the present purpose.
Proceed by induction on $D$, the case $D = 1$ being vacuous. Let $1 < D$ and fix a tower $\{\widetilde{\Omega}_j\}_{j=1}^D$. Apply the induction hypothesis to $\{\widetilde{\Omega}_j\}_{j=1}^{D-1}$ to obtain a tower $\{\widehat{\Omega}_j\}_{j=1}^{D-1}$ satisfying the properties a), b) and c) of the corollary, with $D$ replaced by $D-1$. For each $(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1}$ define $\widehat{\Omega}_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) := \widetilde{\Omega}_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) $. If $D$ is odd define $\widehat{\Omega}_{D}$ to be $$\big\{ (\mathbf{r}_{D}, \mathbf{t}_{D}) \in \widetilde{\Omega}_D : t_D \in \omega_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\textrm{ and } (\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1} \big\}$$ where $\mathbf{r}_{D} = \mathbf{r}_{D-1}$ and $\mathbf{t}_{D} = (\mathbf{t}_{D-1}, t_{D})$; throughout this article, similar notation will be used for elements belonging to levels of various parameter towers without further comment. Similarly, if $D$ is even, then define $\widehat{\Omega}_{D}$ to be $$\big\{ (\mathbf{r}_{D}, \mathbf{t}_{D}) \in \widetilde{\Omega}_D : (r_{D/2}, t_D) \in \omega_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\textrm{ and } (\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1} \big\}.$$
If $D$ is odd, then the proof is immediately completed by letting $\Omega_j := \widehat{\Omega}_j$ for $j=1,\dots, D$. It remains to consider the case when $D$ is even, which is more involved. Define a sequence of sets $\omega_{D-k} \subseteq \widehat{\Omega}_{D-k}$ for $1 \leq k \leq D-1$ recursively as follows. For all $(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1}$ let $$\omega_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) := \widehat{\Omega}_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \cap A(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})$$ and define $\omega_{D-1}$ to be the set $$\left\{ (\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1} : \nu_P\big(\omega_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\big) \geq \frac{1}{2}\nu_P\big(\widehat{\Omega}_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\big) \right\}.$$ Hence, $\omega_{D-1}$ is the set of points $(\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-1}$ with the property that most of the associated fibre $\widehat{\Omega}_D(\mathbf{r}, \mathbf{t})$ lies in $A(\mathbf{r}, \mathbf{t})$.
Now suppose $\omega_{D-k}$ has been defined for some $1 \leq k \leq D-2$ and let $(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1}) \in \widehat{\Omega}_{D-k-1}$. If $k$ is odd, then $D- k$ is also odd and $\omega_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1})$ is defined to be $$\big\{ t_{D-k} \in \widehat{\Omega}_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1}) : (\mathbf{r}_{D-k}, \mathbf{t}_{D-k}) \in \omega_{D-k}\big\}.$$ Let $$\omega_{D-k-1} := \left\{ (\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-k-1} : \mu_P\big(\omega_{D-k}(\mathbf{r}, \mathbf{t})\big) \geq \frac{1}{2}\mu_P\big(\widehat{\Omega}_{D-k}(\mathbf{r}, \mathbf{t})\big) \right\}$$ so that $\omega_{D - k -1}$ is the set of points $(\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-k -1}$ with the property that most of the associated fibre $\widehat{\Omega}_{D-k}(\mathbf{r}, \mathbf{t})$ lies in $\omega_{D-k}(\mathbf{r}, \mathbf{t})$.
Similarly, if $k$ is even, then $D - k$ is even and $\omega_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1})$ is defined to be $$\big\{ (r_{(D-k)/2}, t_{D-k}) \in \widehat{\Omega}_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1}) : (\mathbf{r}_{D-k}, \mathbf{t}_{D-k}) \in \omega_{D-k}\big\}$$ and one completes the recursive definition by letting $$\omega_{D-k-1} := \left\{ (\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-k-1} : \nu_P\big(\omega_{D-k}(\mathbf{r}, \mathbf{t})\big) \geq \frac{1}{2}\nu_P\big(\widehat{\Omega}_{D-k}(\mathbf{r}, \mathbf{t})\big) \right\}.$$ Having constructed the sequence $\omega_{D-k}$ for $1 \leq k \leq D-1$, suppose $\mu_P(\omega_1) \geq \tfrac{1}{2}\mu_P(\widehat{\Omega}_1)$. If one defines $\Omega_1 := \omega_1$ and $\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) := \omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$ for $1 < j \leq D$, then one may construct a tower inductively by setting $$\label{tower recursion 1}
\Omega_j := \{(\mathbf{r}_{j}, \mathbf{t}_{j}) \in [1,2]^{(j-1)/2} \times [0,1]^{j} : (\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1} \textrm{ and } t_j \in \Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \}$$ for $j > 1$ odd and $$\label{tower recursion 2}
\Omega_j := \{(\mathbf{r}_{j}, \mathbf{t}_{j}) \in [1,2]^{j/2} \times [0,1]^{j} : (\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1} \textrm{ and } (r_{j/2},t_j) \in \Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \}$$ for $j$ even. It immediately follows from the definitions that the resulting tower $\{\Omega_j\}_{j-1}^{D}$ satisfies the properties stated in the lemma with c) i) holding for $k=D$.
On the other hand, if $\mu_P(\omega_1) < \tfrac{1}{2}\mu_P(\widehat{\Omega}_1)$, then define $\Omega_1 := \widehat{\Omega}_1 \setminus \omega_1$ and let $$\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) := \widehat{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \setminus\omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$$ for $1 < j \leq D$ so that properties a) and b) i) and c) clearly hold for the resulting tower $\{\Omega_j\}_{j-1}^{D}$, which is again defined by and . To prove b) ii), suppose $1 < j \leq D$ is odd and $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$. Thus, $(r_{(j-1)/2}, t_{j-1}) \in \Omega_{j-1}(\mathbf{r}_{j-2}, \mathbf{t}_{j-2})$ and, by the definition of $\omega_{j-1}(\mathbf{r}_{j-2}, \mathbf{t}_{j-2})$, it follows that $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \widehat{\Omega}_{j-1} \setminus\omega_{j-1}$. Finally, the definition of $\omega_{j-1}$ ensures $$\begin{aligned}
\mu_P\big(\Omega_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) &=\mu_P\big(\widehat{\Omega}_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) - \mu_P\big(\omega_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) \\
&\geq 2^{-1}\mu_P\big(\widehat{\Omega}_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big)\\
&\geq 2^{-\floor{(D-1)/2}+1}\mu_P\big(\widetilde{\Omega}_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big),\end{aligned}$$ where the induction hypothesis has been applied in the last inequality. A similar argument shows b) iii) also holds, completing the proof.
Having stated this refinement result one may proceed to prove Lemma \[towerlem\].
Observe, defining $\Sigma := {\mathbb{R}}^d \times [1,2] \times I$ it follows that $$\langle A \chi_E, \chi_F \rangle = \int_{\Sigma} \chi_{U}(x,r,t)\,\lambda_P(t) {\mathrm{d}}x {\mathrm{d}}r{\mathrm{d}}t$$ where $\lambda_P(t):= t^{2K/d(d+1)}$ and $$U := \big\{ (x,r,t) \in \Sigma : x - r P(t) \in E \textrm{ and } (x,r) \in F \big\}.$$ Writing $I := (a,b)$ where $a \geq 0$ and $b - a =1$, a method due to Dendrinos and Stovall [@Dendrinos] may be applied to produce a sequence $\{U_k\}_{k=0}^{\infty}$ of decreasing subsets of $U$ of pairwise comparable measure such that for all $k \geq 1$ either $$\label{ds1}
\int_t^b \chi_{U_{k-1}}(x , r, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau \geq 4^{-(k+1)} \alpha$$ or $$\label{ds2}
\int_1^2 \int_t^b \chi_{U_{k-1}}(x - r P(t) +\rho P(\tau), \rho, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau {\mathrm{d}}\rho \geq 4^{-(k+1)} \beta$$ holds for all $(x,r,t) \in U_k$. Specifically, the $\{U_k\}_{k=0}^{\infty}$ can be chosen so that $$\int_{\Sigma} \chi_{U_0}(x,r,t)\,\lambda_P(t) {\mathrm{d}}x {\mathrm{d}}r{\mathrm{d}}t \geq \frac{1}{2}\langle A \chi_E, \chi_F \rangle$$ and for all $k \geq 1$:
i) $U_k \subseteq U_{k-1}$ and $$\int_{\Sigma} \chi_{U_{k}}(x,r,t) \,\lambda_P(t) {\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t \geq \frac{1}{4}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t;$$
ii) If $k \not\equiv d \mod 2$ (respectively, $k \equiv d \mod 2$), then (respectively ) holds for all $(x,r,t) \in U_k$.
iii) Furthermore, for each $k$, if $t \in I$ is such that $(x,r,t) \in U_k$ for some $(x,r) \in {\mathbb{R}}^d \times [1,2]$, then $t \geq (\alpha/2 \kappa)^{\kappa}$ where $\kappa$ is as in .
This construction is due to Dendrinos and Stovall [@Dendrinos], however the details are appended for completeness.
Fix $(x_0, r_0, t_0) \in U_{d+1}$. The next step is to use the sets $\{U_k\}_{k=0}^{d+1}$ to construct an initial tower $\{\widetilde{\Omega}_j\}_{j=1}^{d+1}$ satisfying the properties of Lemma \[dendrinosstovall\] and such that whenever $(\mathbf{r}_j, \mathbf{t}_j) \in \widetilde{\Omega}_j$ for some $1 \leq j \leq d+1$, it follows that $$\left\{\begin{array}{ll}
\left(\Psi_j(\mathbf{r}_j, \mathbf{t}_j), r_{j/2}, t_{j}\right) \in U_{d+1-j} & \textrm{if $0 \leq j \leq d+1$ is even}\\
&\\
\left(\Psi_{j-1}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}), r_{\floor{j/2}}, t_{j}\right) \in U_{d+1-j} & \textrm{if $1 \leq j \leq d+1$ is odd}
\end{array}\right.$$ where $\Psi_j(\mathbf{r}_j, \mathbf{t}_j) := \Psi_j(x_0, r_0;\mathbf{r}_j, \mathbf{t}_j)$ for $j \geq 1$ is as defined in and $\Psi_0(\mathbf{r}_0, \mathbf{t}_0) := x_0$. It is convenient to consider $(\mathbf{r}_0, \mathbf{t}_0)$ as some arbitrary object (say, $(\mathbf{r}_0, \mathbf{t}_0) := (r_0, t_0)$) and $\Psi_0$ as a function on the singleton set $\widetilde{\Omega}_0 := \{(\mathbf{r}_0, \mathbf{t}_0)\}$, taking the value $x_0$.
The tower $\{\widetilde{\Omega}_j\}_{j=1}^{d+1}$ is constructed recursively. To begin, define $\widetilde{\Omega}_0$ as above; suppose $\widetilde{\Omega}_{j}$ has been defined for some $0 \leq j \leq d$ and fix $(\mathbf{r}_{j}, \mathbf{t}_{j}) \in \widetilde{\Omega}_{j}$. The argument splits into two cases, depending on the parity of $j$.
Case 1: $j$ is even. {#case-1-j-is-even. .unnumbered}
--------------------
Since $(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}), r_{j/2}, t_{j}) \in U_{d+1-j}$ and $d+1-j \not\equiv d \mod 2$, one may apply to deduce $$\label{ds3}
\int_{t_{j}}^b \chi_{U_{d-j}}(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}), r_{j/2}, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau \geq 4^{-(d+2-j)} \alpha.$$ It is therefore possible to choose $t_j < s_j < b$ with the property $$\label{ds4}
\int_{t_j}^{s_j} \,\lambda_P(\tau){\mathrm{d}}\tau = 4^{-(d+5/2-j)} \alpha.$$ Define the set $$\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j) := \big\{ t_{j+1} \in (s_j, b] : (\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}), r_{j/2}, t_{j+1}) \in U_{d-j} \big\},$$ noting, by and , that this has measure $\mu_P(\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j)) \geq 4^{-(d+5/2-j)} \alpha$. To complete the recursive step in this case, if $j=0$ let $\widetilde{\Omega}_1 := \widetilde{\Omega}_1(\mathbf{r}_0, \mathbf{t}_0)$ whilst for $j > 0$ define $$\widetilde{\Omega}_{j+1} := \big\{ (\mathbf{r}_{j+1}, \mathbf{t}_{j+1} ) : (\mathbf{r}_{j}, \mathbf{t}_{j} ) \in \widetilde{\Omega}_j \textrm{ and } t_{j+1} \in \widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j) \big\}.$$
Case 2: $j$ is odd. {#case-2-j-is-odd. .unnumbered}
-------------------
Since $(\Psi_{j-1}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}), r_{\floor{j/2}}, t_{j}) \in U_{d+1-j}$ and $d+1-j \equiv d \mod 2$, one may apply to deduce $$\label{ds5}
\int_1^2 \int_{t_{j}}^b \chi_{U_{d-j}}(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}) +\rho P(\tau), \rho, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau {\mathrm{d}}\rho \geq 4^{-(d+2-j)} \beta.$$ Here the identity $$\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}) = \Psi_{j-1}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) - r_{\floor{j/2}}P(t_j)$$ has been applied, which is a consequence of the definition . It is therefore possible to choose $t_j < s_j < b$ and $t_j < \tilde{s}_j$ with the properties $$\label{ds6}
\int_{t_j}^{s_j} \,\lambda_P(\tau){\mathrm{d}}\tau = 4^{-(d+5/2-j)} \beta, \quad \int_{t_j}^{\tilde{s}_j} \,\lambda_P(\tau){\mathrm{d}}\tau = 2^{-(d+3-j)} (\alpha\beta)^{1/2}.$$ Define $\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j)$ to be the set $$\left\{ (r_{\ceil{j/2}}, t_{j+1}) \in D(\mathbf{r}_j, \mathbf{t}_j) : \left(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}) + r_{\ceil{j/2}} P(t_{j+1}), r_{\ceil{j/2}}, t_{j+1}\right) \in U_{d-j} \right\}$$ where $D(\mathbf{r}_j, \mathbf{t}_j) := ([1,2] \times (s_j, b])\setminus R(\mathbf{r}_j, \mathbf{t}_j)$ for the rectangle $$R(\mathbf{r}_j, \mathbf{t}_j):=\big\{ (r,t) \in {\mathbb{R}}^2 : |r - r_{\floor{j/2}}| \leq 2^{-(d+4-j)} (\beta/\alpha)^{1/2} \textrm{ and } t_j \leq t \leq \tilde{s}_j \big\}.$$ Observe, by it follows that $$\int_1^2 \int_I \chi_{R(\mathbf{r}_j, \mathbf{t}_j)}(r,t)\,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}r \leq 4^{-(d+3-j)}\beta.$$ Thus, by and , the set $\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j)$ has measure $\nu_P(\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j))\geq 4^{-(d+3-j)} \beta$.
Finally, to complete the recursive definition let $$\widetilde{\Omega}_{j+1} := \big\{ (\mathbf{r}_{j+1}, \mathbf{t}_{j+1} ) : (\mathbf{r}_{j}, \mathbf{t}_{j} ) \in \Omega_j \textrm{ and } (r_{\ceil{j/2}}, t_{j+1}) \in \Omega_{j+1}(\mathbf{r}_j, \mathbf{t}_j) \big\}.$$
It is easy to verify the collection $\{\widetilde{\Omega}_j\}_{j=1}^{d+1}$ forms a tower satisfying all the properties of Lemma \[dendrinosstovall\]. Finally, Lemma \[refinecor\] is applied to further refine this tower to ensure the additional property stated in Lemma \[towerlem\] holds. For each $1 < j \leq d+1$ even, define $$A(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) := \left\{ (r, t) \in [1,2] \times I : t - t_{j-1} > \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)} \right\}$$ for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \widetilde{\Omega}_{j-1}$. Letting $\{\Omega_j\}_{j=1}^{d+1}$ denote the refined tower, the existence of which is guaranteed by Lemma \[refinecor\], it is easy to see this has all the desired properties. In particular, for each $1 < j \leq d+1$ even, precisely one of the following holds:
a) For all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_{j}$ one has $|t_j - t_{j-1}| > \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)}$.
b) For all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_{j}$ one has $|t_j - t_{j-1}| \leq \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)}$. In this case, observe for $t_{j-1} \leq \tau \leq t_j$ it follows that $$|\tau - t_{j-1}| \leq \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)} \lesssim \delta \alpha (\alpha^{\kappa})^{-2K/d(d+1)} \lesssim \delta t_{j-1}$$ by condition iii) of the sets $U_k$ described above. Thus $\tau \lesssim t_{j-1}$ and consequently $$\int_{t_{j-1}}^{t_j}\,\lambda_P(\tau){\mathrm{d}}\tau \lesssim \delta (\alpha\beta)^{1/2}.$$ If $\delta$ is chosen from the outset to be sufficiently small depending only on $d$ and $K$, then it follows that $t_{j-1} < t_j < \tilde{s}_{j-1}$ by the preceding inequality and the definition of $\tilde{s}_{j-1}$ from . Since $(r_{j/2}, t_j) \notin R(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$, one concludes that $|r_{j/2} - r_{j/2-1}| > 2^{-(d+5-j)} (\beta / \alpha)^{1/2}$.
Definition of the parameter domain {#definitionparameter}
==================================
Henceforth fix a tower $\{\Omega_j\}_{j=1}^{d+1}$ satisfying the properties stated in Lemma \[towerlem\] with a suitably small choice of $0 < \delta \ll 1$ so as to satisfy all the forthcoming requirements of the proof. It is stressed that the subsequent argument will require $\delta$ to be chosen depending only on the admissible parameters $d$ and $\deg P$; a careful examination of what follows shows such a choice of $\delta$ is always possible.
Observe that the set $\Omega_{d+1}$ is of dimension $\floor{3(d+1)/2}$; one requires a domain of either dimension $d$ or $d+1$ to effectively parametrise either the set $E$ or $F$. Two methods can be applied to remedy this excess of variables. The first is to simply consider the tower defined only to a lower level; that is, only work with $\{\Omega_j\}_{j=1}^N$ for some $N \leq d+1$. The second method is to consider the whole tower $\{\Omega_j\}_{j=1}^{d+1}$ and *freeze* a number of the variables $t_j$. What is essentially meant by this is that for some set of indices $\mathcal{I} \subset \{1, \dots, d+1\}$ and choice of $(s_i)_{i\in \mathcal{I}} \subset {\mathbb{R}}^{\# \mathcal{I}}$ each set $\Omega_j$ is replaced with $$\big\{ (r, t) \in \Omega_j : t_i = s_i \textrm{ for all }i \in \mathcal{I} \cap \{1, \dots, j\} \big\}.$$ In order to optimise the subsequent Jacobian estimates, both methods are combined below and the variables to be frozen are chosen according to the “pre-colour” of their indices. In particular only $t_i$ for $i$ a pre-blue index will be frozen. In light of this discussion define the function $\zeta: \{1, \dots, d+1\} \rightarrow \{1,2\}$ as follows: $$\zeta(j) := \left\{\begin{array}{ll}
2 & \textrm{if $j$ is pre-red} \\
1 & \textrm{otherwise}
\end{array} \right. \qquad \textrm{for $j = 1, \dots, d+1$.}$$ One can think of $\zeta(j)$ as the number of variables contributed by the fibres of the $j$th floor of the tower after the pre-blue variables have been frozen. Note that there exists a least $1 < N \leq d+1$ such that $$\label{defn N}
Z(N) := \sum_{j=1}^{N} \zeta(j) \in \{d, d+1\};$$ in particular, the number of variables contributed by the first $N$ floors corresponds to the dimension of either $E$ or $F$. At this point the proof splits into a number of different cases depending on the parity of $N$ and the value of $Z(N)$.
If $N$ is odd, then $\mathrm{Case}(1, Z(N))$ holds. If $N$ is even, then $\mathrm{Case}(2, Z(N))$ holds.
Ostensibly there are four distinct cases; however, the minimality of $N$ precludes $\mathrm{Case}(1, d+1)$ and so it suffices to consider the remaining three cases.
The preceding observations lead to the definition of a suitable parameter domain $\Omega$ and mapping $\Phi$ which will form the focus of study for the remainder of the paper. By the nature of the definitions of $\Omega$ and $\Phi$, it will be convenient to introduce a relabelling of the relevant indices as “red,” “blue” or “achromatic” to replace the established “pre-red, pre-blue, pre-achromatic” system. All these definitions depend on which $\mathrm{Case}(i, j)$ happens to hold.
Case$\mathbf{(1, d)}$ and Case$\mathbf{(2, d+1)}$: {#casemathbf1-d-and-casemathbf2-d1 .unnumbered}
--------------------------------------------------
The simplest situation corresponds to when either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d+1)$ holds. In both instances one defines $\Omega := \Omega_{N}$ and $\Phi := \Phi_{N}$. In the former case $\Phi : \Omega \rightarrow E$ whilst in the latter $\Phi : \Omega \rightarrow F$ by the properties 2. i) and 3. i) of Lemma \[dendrinosstovall\], respectively. Here essentially no relabelling is required: each pre-red (respectively, pre-blue) index $1 \leq j \leq N$ is designated red (respectively, blue) whilst the odd indices are achromatic (that is, they have no colour assigned to them).
Case$\mathbf{(2, d)}$: {#casemathbf2-d .unnumbered}
----------------------
This situation is slightly more complicated. Fix $t_0 \in \Omega_1$ and consider the family of sets $\{\Omega_j^*\}_{j=1}^{N}$ defined by $$\Omega_j^* := \big\{ (t_1, \dots, t_{j}) : (t_0, t_1, \dots, t_{j}) \in \Omega_{j+1} \big\} \qquad \textrm{for $j = 1, \dots, N$.}$$ It is easy to see $\{\Omega_j^*\}_{j=1}^{N}$ constitutes a type 2 tower. Let $y_0 := x_0 - r_0P(t_0)$ and let $\{\Phi^*_j\}_{j=1}^N$ denote the family of mappings associated to $\{\Omega_j^*\}_{j=1}^{N}$ and the point $y_0$, as defined in Definition \[associated mappings\]. Define $\Omega := \Omega^*_N$ and $\Phi := \Phi^*_{N}$ and observe, by property 2. i) of Lemma \[dendrinosstovall\], that $\Phi : \Omega \rightarrow E$. In this case the colouring system of the indices is redefined. In particular,
i) The index 1 is designated red;
ii) The odd indices $1 < j \leq N$ are designated red (respectively, blue) if $j+1$ was pre-red (respectively, pre-blue) in the previous scheme;
iii) The even indices are designated achromatic.
Freezing variables and families of parametrisations {#freeze}
===================================================
Rather than use a single function to parametrise $E$ or $F$, here the slicing method of Christ [@Christ; @Christ1998] is used to construct a family of maps $G_{\sigma}$. To begin some notation is introduced.
Let $M$ denote the number of non-blue indices in $\{1, \dots, N\}$ and label these indices $l_1 < l_2 < \dots < l_M$. Similarly, let $m$ (respectively, $n$) denote the number of red (respectively, blue) indices so that $M = N - n$. For the reader’s convenience the following table indicates the relationship between these parameters in the various cases.
$$\label{counting variables}
\begin{tabular}{ | c | c | c |}
\hline
& N & d \\ \hline
$\mathrm{Case}(1,d)$ & 2m +2n + 1 & 3m + 2n + 1 \\ \hline
$\mathrm{Case}(2,d+1)$ & 2m +2n & 3m + 2n - 1 \\ \hline
$\mathrm{Case}(2,d)$ & 2m +2n & 3m + 2n \\
\hline
\end{tabular}$$
These computations follow immediately from the definition of $N$ in .
Now let $ 1 \leq \mu_1 < \dots < \mu_m \leq M$ be such that $\{ l_{\mu_i}\}_{i=1}^m$ is precisely the set of red indices. Rather than the blue indices themselves, it will be useful to enumerate those indices which lie directly before a blue index. Irrespective of which case happens to hold, any blue index is at least 2 and so there are precisely $n$ indices lying directly before a blue index. In particular, let $1 \leq \nu_1 < \dots < \nu_{n} \leq M$ be such that $\{l_{\nu_j}+1\}_{j=1}^{n}$ are precisely the blue indices.
Define functions $\tau$ and $\sigma$ on $\Omega$ by $$\begin{aligned}
\tau = (\tau_1, \dots, \tau_M) &:=& ( t_{l_1}, \dots, t_{l_M}) \\
s = (s_{\nu_1}, \dots, s_{\nu_{n}}) &:=& (t_{l_{\nu_j}+1} - t_{l_{\nu_j}})_{j=1}^{n} \\
\sigma = (\sigma_{\nu_1}, \dots, \sigma_{\nu_n}) &:=& (s_{\nu_j} \tau_{\nu_j}^{2K/d(d+1)})_{j=1}^{n}.\end{aligned}$$
Finally let $\rho = (\rho_{\mu_1}, \dots, \rho_{\mu_m}, \rho_{\nu_1}, \dots, \rho_{\nu_{n}})$ where $\rho_{\mu_i}$ (respectively, $\rho_{\nu_j}$) is the dilation variable arising from the fibres of floor $l_{\mu_i}$ (respectively, $l_{\nu_j} +1$) of the tower. More precisely, $\rho_{\mu_j} := r_{\ceil{l_{\mu_i}/2}}$ for $1 \leq i \leq m$ whilst $\rho_{\nu_j} := r_{\ceil{(l_{\nu_j}+1)/2}}$ for $1 \leq j \leq n$.
Observe that the map $$\label{phi change of variables}
\varphi: (r, t) \mapsto (\rho, \tau, \sigma)$$ is a valid change of variables with Jacobian determinant satisfying $$\left|\det \frac{\partial \varphi}{\partial (r, t)}\right| = \prod_{j=1}^{n} \tau_{\nu_j}^{2K/d(d+1)}.$$ For $\sigma \in {\mathbb{R}}^n$ define the parameter set $\omega(\sigma) := \big\{ (\rho, \tau) : \varphi^{-1}(\rho, \tau, \sigma) \in \Omega \big\}$ and let $$\label{setw}
W := \big\{ \sigma \in {\mathbb{R}}^{n} : \omega(\sigma) \neq \emptyset \} \subseteq \big[0, \delta(\alpha\beta)^{1/2}\big]^n$$ where the inclusion follows from properties of the blue indices. Finally, consider the mapping $G_{\sigma}$ on $\omega(\sigma)$ by $$G_{\sigma}(\rho, \tau):= \Phi \circ \varphi^{-1}(\rho, \tau, \sigma).$$ By , it follows that in $\mathrm{Case}(1, d)$ and $\mathrm{Case}(2, d)$ the maps $G_{\sigma}$ are functions of $d$ variables and take values in $E$ whilst in $\mathrm{Case}(2, d+1)$ the $G_{\sigma}$ are functions of $d+1$ variables and take values in $F$. Hence in each case the maps $G_{\sigma}$ have the desirable property that the domain and codomain are of equal dimension.
Furthermore, the polynomial nature of maps $G_{\sigma}$ imply each has bounded multiplicity. In particular, the following well-known multiplicity lemma applies to this situation.
\[multiplicity\] Let $Q: {\mathbb{R}}^d \rightarrow {\mathbb{R}}^d$ be a polynomial mapping; that is, $Q(t) = (Q_j(t))_{j=1}^d$ for all $t \in {\mathbb{R}}^d$ where each $Q_j : {\mathbb{R}}^d \rightarrow {\mathbb{R}}$ is a polynomial in $d$ variables. Suppose the Jacobian determinant $J_Q$ of $Q$ does not vanish everywhere. Then for almost every $x \in {\mathbb{R}}^d$ the set $Q^{-1}(\{x\})$ is finite. Moreover, for almost every $x \in {\mathbb{R}}^d$ the inequality $$\label{mult1}
\# Q^{-1}(\{x\}) \leq \prod_{j=1}^d \deg (Q_j)$$ holds, where $\deg(Q_j)$ denotes the degree of $Q_j$.
The simple proof of this lemma appears in [@Christ1998]; however, it is included at the end of this section for completeness.
As a consequence of the Multiplicity Lemma, if $J_{\sigma}$ denotes the Jacobian of $G_{\sigma}$, then in $\mathrm{Case}(1, d)$ and $\mathrm{Case}(2, d)$ one concludes that the estimate $$\label{Jacobian estimate}
|E| \gtrsim \int_{\omega(\sigma)} |J_{\sigma}(\rho, \tau)| \,{\mathrm{d}}\rho{\mathrm{d}}\tau$$ holds for all $\sigma \in W$. In $\mathrm{Case}(2, d+1)$ there is a similar estimate but with $|E|$ replaced with $|F|$ on the left-hand side of the above expression. Thus, in order to establish either or in the present cases it suffices to prove a suitable estimate for the Jacobian $|J_{\sigma}(\rho, \tau)|$ on the set $\omega(\sigma)$.
This section is concluded with the proof of the Multiplicity Lemma.
Since the zero locus $Z$ of $J_Q$ is a proper algebraic subset of ${\mathbb{R}}^d$ it has measure zero. Furthermore, as $Q$ is a polynomial (and therefore locally Lipschitz) mapping the image $Q(Z)$ of $Z$ under $Q$ has measure zero. It is claimed that holds for all $x \in {\mathbb{R}}^d \setminus Q(Z)$. Indeed, fixing such an $x$ notice that $Q^{-1}(\{x\}) = \bigcap_{j=1}^d V_j^x$ where $\{V_j^x\}_{j=1}^d$ are algebraic sets given by $$V_j^x := \{ t \in {\mathbb{R}}^d : Q_j(t) - x_j = 0 \}.$$ Bezout’s theorem implies that the cardinality of this intersection is either uncountable or at most $\prod_{j=1}^d \deg (Q_j)$.[^8] It therefore suffices to show that $Q^{-1}(\{x\})$ is not uncountable; this is achieved by proving each point of the set is isolated. By the choice of $x$, whenever $t_0 \in Q^{-1}(\{x\})$ the vectors $\{ \nabla Q_j(t_0)\}_{j=1}^d$ span ${\mathbb{R}}^d$. Thus the $V_j^x$ are smooth hypersurfaces in a neighbourhood of $t_0$ which, of course, intersect at $t_0$ and are transversal at this point of intersection. It follows that $t_0$ must be an isolated point of $Q^{-1}(\{x\})$, as required.
Reduction to Jacobian estimates
===============================
Recall, in order to prove the main theorem it suffices to obtain a lower bound on either $|E|$ or $|F|$ in terms of $\alpha$ and $\beta$, as discussed in and . In the previous sections, it was shown that when either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d)$ holds a family of useful mappings $G_{\sigma} \colon \omega(\sigma) \to E$ can be constructed. These mappings effectively parametrise the set $E$ and, in particular, one has the estimate . A similar construction is available in $\mathrm{Case}(2, d+1)$ (this time parametrising the set $F$) and so it remains to find effective bounds for integrals such as that appearing in the right-hand side of . This will be achieved by estimating pointwise the Jacobian determinant $J_{\sigma}$ of the map $G_{\sigma}$. In order to state the main result in this direction, it is convenient to introduce the notation $$\eta := \left\{ \begin{array}{ll}
0 & \textrm{if either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d+1)$ holds} \\
1 & \textrm{if $\mathrm{Case}(2, d)$ holds}
\end{array}\right. .$$
\[jaclem1\] Let $\sigma \in W$, where $W$ is as defined in . Then $$|J_{\sigma}(\rho, \tau)| \gtrsim \alpha^{d(d+1)/2 - M}(\beta / \alpha )^{(m+n - \eta)/2} \prod_{l=1}^M \tau_l^{2K/d(d+1)}$$ for all $(\rho, \tau) \in \omega(\sigma)$.
Theorem \[bigthm\] is a direct consequence of Lemma \[jaclem1\].
To prove Theorem \[bigthm\] it suffices to show the estimate holds in both $\mathrm{Case}(1, d)$ and $\mathrm{Case}(2,d)$ and holds in $\mathrm{Case}(2, d+1)$. Indeed, recall both and are equivalent to the desired endpoint restricted weak-type $(p_1, q_1)$ inequality for $A$. For notational convenience, let $$|X| := \left\{ \begin{array}{ll}
|E| & \textrm{if either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d)$ holds} \\
|F| & \textrm{if $\mathrm{Case}(2, d+1)$ holds}
\end{array}\right. .$$
Apply Lemma \[jaclem1\] to each $\sigma \in W$ together with the Multiplicity Lemma to deduce in all cases $$\begin{aligned}
|X| &\gtrsim& \int_{\omega(\sigma)} |J_{\sigma}(\rho, \tau)| \,{\mathrm{d}}\rho{\mathrm{d}}\tau \\
&\gtrsim& \alpha^{d(d+1)/2 - M}(\beta / \alpha )^{(m+n - \eta)/2}\int_{\omega(\sigma)} \prod_{k=1}^M \tau_k^{2K/d(d+1)} {\mathrm{d}}\rho{\mathrm{d}}\tau .\end{aligned}$$ Integrating both sides of the preceding inequality over $W$, it follows that $$\label{wkpf2}
(\alpha\beta)^{n/2} |X| \gtrsim \alpha^{d(d+1)/2 - M}(\beta / \alpha )^{(m+n - \eta)/2} \int_{\varphi(\Omega)} \prod_{k=1}^M \tau_k^{2K/d(d+1)} \,{\mathrm{d}}\rho {\mathrm{d}}\tau {\mathrm{d}}\sigma$$ where $\varphi$ is the map defined in . By a change of variables, the integral on the right-hand side of can be written as $$\begin{aligned}
\int_{\varphi(\Omega)} \prod_{k=1}^M \tau_k^{2K/d(d+1)} \,{\mathrm{d}}\rho {\mathrm{d}}\tau {\mathrm{d}}\sigma &=& \int_{\Omega} \prod_{k=1}^M t_{l_k}^{2K/d(d+1)}\left| \det \frac{\partial \varphi}{\partial (r, t)}(r,t) \right| \,{\mathrm{d}}r {\mathrm{d}}t \\
&=& \int_{\Omega} \prod_{k=1}^M t_{l_k}^{2K/d(d+1)}\prod_{j=1}^{n} t_{l_{\nu_j}}^{2K/d(d+1)}\,{\mathrm{d}}r {\mathrm{d}}t.\end{aligned}$$ Arguing as in the last step of the proof of Lemma \[towerlem\], one may deduce $t_{l_{\nu_j}} \sim t_{l_{\nu_j}+1}$ for $1 \leq j \leq n$ provided that the parameter $\delta$ from Lemma \[towerlem\] is chosen to be sufficiently small (depending only on the degree of $P$ and $d$). The previous expression is therefore bounded below by a constant multiple of $$\int_{\Omega} \prod_{k=1}^N t_{k}^{2K/d(d+1)}\,{\mathrm{d}}r {\mathrm{d}}t.$$ Applying Fubini’s theorem and the estimates for the $\mu_P$-measure of the fibres of the $\Omega_j$, one may easily deduce the above integral is at least a constant multiple of $\alpha^N(\beta/\alpha)^{\floor{N/2}}$. Whence, combining these observations and multiplying both sides of by $\alpha^{-n}(\beta/\alpha)^{-n/2}$, one arrives at the estimate $$|X| \gtrsim \alpha^{d(d+1)/2 - M + N-n}(\beta / \alpha )^{(m-\eta)/2 + \floor{N/2}}.$$ Recalling $M=N-n$ and , this is easily seen to be the desired estimate.
To complete the proof of Theorem \[weakthm\] it remains to prove Lemma \[jaclem1\].
The proof of the Jacobian estimates: $\mathrm{Case}(1,d)$
=========================================================
In the previous section the proof of the main theorem was reduced to establishing the pointwise estimates for the Jacobian function described in Lemma \[jaclem1\]. Here the proof of Lemma \[jaclem1\] in $\mathrm{Case}(1,d)$ is discussed in detail. The same arguments can be adapted to treat the remaining cases, as demonstrated in the following section.
The arguments here, which are based primarily on those of [@Christ1998; @Stovall2010], are somewhat lengthy; it is convenient, therefore, to present the proof as a series of steps.
Compute the Jacobian matrix. {#compute-the-jacobian-matrix. .unnumbered}
----------------------------
Recalling the definition of the mapping $\Phi := \Phi_N$, one may use the established index notation to express $\Phi \circ \varphi^{-1}$ as $$\begin{aligned}
\Phi\circ \varphi^{-1}(\rho, \tau, \sigma) &=& x_0 -r_0P(\tau_1) + \sum_{i=1}^m \rho_{\mu_i}\big( P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \big) \\
&& + \sum_{j=1}^n \rho_{\nu_j}\big( P(\tau_{\nu_j}+ s_{\nu_j}(\tau,\sigma)) - P(\tau_{\nu_j +1}) \big).\end{aligned}$$ One immediately deduces that $$\frac{\partial G_{\sigma}}{\partial \rho_{\mu_i}} (\rho, \tau) = P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \qquad \textrm{for $i = 1, \dots, m$}$$ whilst $$\frac{\partial G_{\sigma}}{\partial \rho_{\nu_j}} (\rho, \tau) = P(\tau_{\nu_j} + s_{\nu_j}(\tau,\sigma)) - P(\tau_{\nu_j +1}) \qquad \textrm{for $j = 1, \dots, n$}$$ which identifies $m+n$ of the columns of the Jacobian matrix. The remaining columns correspond to differentiation with respect to the $\tau$ variables and are readily computed by expressing $\Phi\circ \varphi^{-1}$ as $$\begin{aligned}
\Phi\circ \varphi^{-1}(\rho, \tau, \sigma) &=& x_0 + \sum_{i=1}^m \big( \rho_{\mu_i}P(\tau_{\mu_i}) - \rho^*_{\mu_i-1}P(\tau_{\mu_i - 1})\big) \\
&&+ \sum_{j=1}^n \big( \rho_{\nu_j}P(\tau_{\nu_j} + s_{\nu_j}(\tau,\sigma)) - \rho^*_{\nu_j-1}P(\tau_{\nu_j})\big) - \rho^*_MP(\tau_M)\end{aligned}$$ where the $\rho^*_{\mu_i-1}$, $\rho^*_{\nu_j-1}$ and $\rho^*_M$ are defined in the obvious manner; for instance, if $\rho_{\nu_j}$ corresponds to the parameter $r_k$ via the change of variables $\varphi$, then $\rho^*_{\nu_j-1}$ is understood to correspond to the parameter $r_{k-1}$. Thus for $i=1, \dots, m$ one has $$\frac{\partial G_{\sigma}}{\partial \tau_{\mu_i}} (\rho, \tau) = \rho_{\mu_i}P'(\tau_{\mu_i}) \quad \textrm{and} \quad \frac{\partial G_{\sigma}}{\partial \tau_{\mu_i - 1}} (\rho, \tau) = - \rho^*_{\mu_i-1}P'(\tau_{\mu_i - 1})$$ whilst $$\frac{\partial G_{\sigma}}{\partial \tau_{M}} (\rho, \tau) = - \rho^*_MP'(\tau_M),$$ accounting for a further $2m+1$ columns to the Jacobian matrix. To compute the remaining $n$ columns differentiate $G_{\sigma}$ with respect to the $\tau_{\nu_j}$ to give $$\begin{aligned}
\label{taunucolumn}
\frac{\partial G_{\sigma}}{\partial \tau_{\nu_j}} (\rho, \tau) &=& \rho_{\nu_j}P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma)) - \rho^*_{\nu_j-1}P'(\tau_{\nu_j}) \\
\nonumber
&& - \frac{2K}{d(d+1)} \frac{s_{\nu_j}(\tau, \sigma)}{\tau_{\nu_j}}\rho_{\nu_j}P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma))\end{aligned}$$ for $j=1, \dots, n$.
Compare $J_{\sigma}$ with $J_P$. {#compare-j_sigma-with-j_p. .unnumbered}
--------------------------------
The estimation of the Jacobian $J_{\sigma}$ will be achieved by comparing it to the more tractable expression $J_P$, introduced in . Once such a comparison is established, $J_{\sigma}$ can then be bounded by means of the geometric inequality of Dendrinos and Wright (that is, Theorem \[Dendrinos Wright theorem\]). This inequality is guaranteed to hold in the appropriate setting due to the reductions made earlier in the article.
To begin, express the Jacobian determinant in the form of an integral $$J_{\sigma}(\rho, \tau)= \pm\int_{R(\tau) \times B_{\sigma}(\tau)} \wp_{\sigma}(\rho, \tau, x) \,{\mathrm{d}}x$$ where $\wp_{\sigma}(\rho, \tau, x)$ is a multi-variate polynomial and $$R(\tau) := \prod_{i=1}^{m} (\tau_{\mu_i}, \tau_{\mu_i+1}) \quad \textrm{and} \quad
B_{\sigma}(\tau) := \prod_{i=1}^{n} (\tau_{\nu_j} + s_{\nu_j}, \tau_{\nu_j+1})$$ are rectangles.[^9] The polynomial $\wp_{\sigma}(\rho, \tau, x)$ is the product of $C(\rho) = \rho_M^* \prod_{i=1}^m \rho_{\mu_i - 1}^* \rho_{\mu_i}$ and the determinant of the matrix $A_{\sigma}(\rho, \tau, x)$ obtained from original Jacobian matrix by making the following changes:
- The column $P(\tau_{\mu_i}) - P(\tau_{\mu_i +1})$ is replaced with $P'(x_i)$ for $i=1, \dots, m$.
- The column $P(\tau_{\nu_j}+ s_{\nu_j}) - P(\tau_{\nu_j +1})$ is replaced with $P'(x_{m+j})$ for $j = 1, \dots, n$.
- The columns $\rho_{\mu_i}P'(\tau_{\mu_i})$ and $- \rho^*_{\mu_i-1}P'(\tau_{\mu_i - 1})$ are replaced with $P'(\tau_{\mu_i})$ and $P'(\tau_{\mu_i - 1})$, respectively, for all $i = 1, \dots, m$. In addition, $- \rho^*_MP'(\tau_M)$ is replaced with and $P'(\tau_M)$.
- The remaining columns $n$ are unaltered; in other words, they agree with the corresponding columns of the Jacobian matrix.
Notice the unaltered columns are those corresponding to differentiation by $\tau_{\nu_j}$ and are of the form given in . Each may be expressed as the sum of three terms $$\label{taylor1}
\frac{\partial G_{\sigma}}{\partial \tau_{\nu_j}} (\rho, \tau) = \sum_{i=1}^3 T^i_{\sigma, j}(\rho,\tau)$$ where, writing $c := - 2K/d(d+1)$, $$\begin{aligned}
T^1_{\sigma, j}(\rho,\tau) &:=& (\rho_{\nu_j} - \rho^*_{\nu_j-1})P'(\tau_{\nu_j}), \\
T^2_{\sigma, j}(\rho,\tau) &:=& c \frac{s_{\nu_j}(\tau, \sigma)}{\tau_{\nu_j}}\rho_{\nu_j}P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma)), \\
T^3_{\sigma, j}(\rho,\tau) &:=& \rho_{\nu_j}(P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma)) - P'(\tau_{\nu_j})). \end{aligned}$$ The multi-linearity of the determinant and are now applied to express $\det A_{\sigma}(\rho, \tau, x)$ as a sum of determinants of more elementary matrices. In order to present concisely the resulting expression it is useful to introduce some notation. In particular, for $S \subseteq \mathcal{N} := \{\nu_1, \dots, \nu_n\}$, let $\Delta_S$ denote the function of $\rho$ given by $$\Delta_S(\rho) := \prod_{\nu \in S} (\rho_{\nu} - \rho^*_{\nu - 1})$$ and $R_{\sigma, S}(\tau) \subset {\mathbb{R}}^{\#S}$ the rectangle $$R_{\sigma, S}(\tau) := \prod_{\nu \in S} (\tau_{\nu}, \tau_{\nu}+ s_{\nu}).$$
With this notation $\det A_{\sigma}(\rho, \tau, x)$ equals $$\label{integrand1}
\sum_{\mathcal{S}} \Delta_{S_1}(\rho) \bigg(\prod_{\nu \in S_2}\frac{c \rho_{\nu}s_{\nu}}{\tau_{\nu}}\bigg)\bigg( \prod_{\nu \in S_3} \rho_{\nu}\bigg) \int_{R_{\sigma, S_3}(\tau)} \bigg(\prod_{\nu \in S_3}\frac{\partial}{\partial y_{\nu}}\bigg) J_P(\xi_{\mathcal{S}}(y), x)\,{\mathrm{d}}y,$$ at least up to a sign, where the sum ranges over all partitions $\mathcal{S}:=(S_1, S_2, S_3)$ of $\mathcal{N}$ and for any such partition $\xi_{\mathcal{S}}(y) = (\xi_{\mathcal{S},l}(\tau,\sigma, y))_{l=1}^M$ is defined by $$\xi_{\mathcal{S},l}(\tau, \sigma,y) := \left\{\begin{array}{ll}
\tau_{l} + s_{l} & \textrm{if $l \in S_2$,}\\
y_{l} & \textrm{if $l \in S_3$,} \\
\tau_{l} & \textrm{otherwise.}
\end{array} \right.$$ If $S_3 = \emptyset$, then the integral appearing in is interpreted as $J_P(\xi_{\mathcal{S}}, x)$.
The term of the sum in corresponding to the unique partition for which $S_1 = \mathcal{N}$ is simply $\Delta(\rho)J_P(\tau, x)$ where $\Delta(\rho) := \Delta_{\mathcal{N}}(\rho)$; the sum of the remaining terms is denoted by $E_{\sigma}(\rho, \tau, x)$. Thus, equals $$\label{error}
\Delta(\rho)J_P(\tau, x) + E_{\sigma}(\rho, \tau, x).$$ In conclusion, the Jacobian $J_{\sigma}$ can be expressed in terms of (an integral of) the function $J_P$ together with some error term.
Control the error. {#control-the-error. .unnumbered}
------------------
It will be shown that provided that $\delta$ is chosen sufficiently small, depending only on $d$ and $\deg P$, the right-hand summand of is subordinate to the left-hand summand. Only a bounded number of terms of are non-zero and the error is therefore a sum of $O(1)$ terms which will be estimated individually.
By the properties of the parameter tower, $s_{\nu} \lesssim \delta (\beta/\alpha)\tau_{\nu}$ for all $\nu \in \mathcal{N}$. Hence, for any $S_2 \subseteq \mathcal{N}$ one has $$\label{error estimate 1}
\prod_{\nu \in S_2}\frac{c \rho_{\nu}s_{\nu}}{\tau_{\nu}} \lesssim \delta^{\#S_2}(\beta/\alpha)^{\#S_2} \lesssim \delta^{\#S_2} \Delta_{S_2}(\rho)$$ where the final inequality is due to the definition of the blue indices. A suitable error bound would follow from a similar estimate for each of the integrals appearing in . In particular, fixing some partition $\mathcal{S} = (S_1, S_2, S_3)$ of $\mathcal{N}$, it suffices to prove $$\label{error estimate 2}
\int_{R_{\sigma, S_3}(\tau)} \left|\left(\prod_{\nu \in S_3}\frac{\partial}{\partial y_{\nu}}\right) J_P(\xi_{\mathcal{S}}(y), x)\right|\,{\mathrm{d}}y \lesssim \delta^{\#S_3} \Delta_{S_3}(\rho)|J_P(\tau, x)|.$$ Indeed, once is established, the error bound $$\begin{aligned}
\label{errorest1}
|E_{\sigma}(\rho, \tau, x)| &\lesssim& \bigg(\sum_{\substack{\mathcal{S} = (S_1, S_2, S_3) \\ S_1 \neq \mathcal{N}}} \delta^{\#S_2 + \#S_3} \prod_{j=1}^3\Delta_{S_j}(\rho)\bigg) |J_P(\tau, x)| \\
\nonumber
&\lesssim& \delta \Delta(\rho)|J_P(\tau, x)|\end{aligned}$$ immediately follows, noting the factor $\prod_{\nu \in S_3} \rho_{\nu}$ from is $O(1)$ whenever it appears in a non-zero term of the sum.
If $S_3 = \emptyset$, then is trivial. Fix a partition $\mathcal{S}$ as above with $S_3$ non-empty and some $y \in R_{\sigma, S_3}(\tau)$ and consider the ratio $$\label{ratio1}
\left| \frac{\left(\prod_{\nu \in S_3}\frac{\partial}{\partial y_{\nu}}\right)J_P(\xi_{\mathcal{S}}(y), x)}{J_P(\xi_{\mathcal{S}}(y), x)} \right|.$$ For notational convenience write $\xi = (\xi_1, \dots, \xi_n) := \xi_{\mathcal{S}}(y)$. Using the derivative estimate from Proposition \[Stovall’s observation\] one may bound by a linear combination of $O(1)$ terms (with $O(1)$ coefficients) of the form $$\label{complicated thing}
\bigg(\prod_{{\nu} \in T_1} y_{\nu}^{-1}\bigg)\bigg( \prod_{{\nu} \in T_2} y_{\nu}^{-\epsilon(\nu)}|y_{\nu} - \xi_{u({\nu})}|^{\epsilon(\nu) - 1}\bigg)\bigg( \prod_{{\nu} \in T_3}y_{\nu}^{-\epsilon(\nu)}|y_{\nu} - x_{v({\nu})}|^{\epsilon(\nu) - 1}\bigg)$$ where:
- $(T_1, T_2, T_3)$ is a partition of $S_3$;
- $u \colon T_2 \to \{1, \dots, M\}$ is a function with the property $u(j) \neq j$ for all $j \in T_2$;
- $v \colon T_3 \to \{1, \dots, n+m\}$ (with no additional conditions) and
- $\epsilon \colon T_2 \cup T_3 \to \{0,1\}$.
To prove it therefore suffices to establish a suitable bound for the integral of the product of and $|J_P(\xi, x)|$ over the set $R_{\sigma, S_3}(\tau)$. The first step is to estimate by applying the following observations.
i) Given $y \in R_{\sigma, S_3}(\tau)$, by the definition of the parameter tower the estimates $$\begin{aligned}
y_{\nu} &\geq& \tau_{\nu} = \tau_{\nu}^{1/\kappa} \tau_{\nu}^{-2K/d(d+1)} \gtrsim \alpha\tau_{\nu}^{-2K/d(d+1)};\\
|y_{\nu} - \xi_{u({\nu})}| &\geq& |\tau_{\nu} - \tau_{u({\nu})}| - \delta \alpha (\tau_{\nu}^{-2K/d(d+1)} + \tau_{u({\nu})}^{-2K/d(d+1)}); \\
|y_{\nu} - x_{v({\nu})}| &\geq& |\tau_{\nu} - x_{v({\nu})}| - \delta \alpha \tau_{\nu}^{-2K/d(d+1)}\end{aligned}$$ hold for ${\nu} \in \mathcal{N}$.
ii) Since the indices $l_{\nu}$ for $\nu \in \mathcal{N}$ are those that directly precede a blue index (and so $l_{\nu}$ is odd), Corollary \[separation\] ensures $\tau_{\nu} - \tau_u \gtrsim \alpha \tau_u^{-2K/d(d+1)}$ for all $1 \leq u < \nu$. Moreover, the ordering of the variables then guarantees $$\tau_{\nu} - \tau_u \gtrsim \alpha \tau_{\nu}^{-2K/d(d+1)} \qquad \textrm{whenever $1 \leq u < \nu$}.$$
iii) On the other hand, since the labelling $l_k$ omits the blue indices, for any $\nu \in \mathcal{N}$ and $\nu < u \leq M$ one must have $l - l_{\nu} \geq 2$ where $l$ is the index such that $\tau_u = t_l$. Consequently, by applying Corollary \[separation\] in this case one concludes that $$\tau_u-\tau_{\nu} \gtrsim \alpha \tau_{\nu}^{-2K/d(d+1)} \qquad \textrm{whenever $\nu < u \leq M$}.$$
Combining these observations one immediately deduces that $$|y_{\nu} - \xi_{u(\nu)}| \gtrsim\alpha \tau_{\nu}^{-2K/d(d+1)}$$ for all $\nu \in \mathcal{N}$, provided that $\delta$ is chosen initially to be sufficiently small in the earlier application of Lemma \[towerlem\].
It would be useful to have a similar bound for the terms $|y_{\nu} - x_l|$. At present such an estimate is not possible due to the potential lack of separation between the $\tau_{\nu}$ and $x_l$ variables. To remedy this, temporarily assume the addition separation hypothesis $$\label{sephypo1}
|\tau_{\nu} - x_l| \gtrsim \alpha\tau_{\nu}^{-2K/d(d+1)}$$ for all $\nu \in \mathcal{N}$ and all $1 \leq l \leq m+n$. Presently it is shown that this separation hypothesis leads to desirable control over the error term $E_{\sigma}(\rho, \tau, x)$; the following step is then to modify the existing set-up so that indeed holds without the need of additional assumptions.
The preceding discussion, together with the identity $|R_{\sigma, S}(\tau)| = \prod_{\nu \in S} s_{\nu}$, implies is controlled by $$\begin{aligned}
\alpha^{-\# S_3} \prod_{\nu \in S_3} \tau_{\nu}^{2K/d(d+1)} &\lesssim& \delta^{\#S_3} (\beta / \alpha)^{\#S_3/2} |R_{\sigma, S_3}(\tau)|^{-1}\\
&\lesssim& \delta^{\#S_3} |\Delta_{S_3}(\rho)||R_{\sigma, S_3}(\tau)|^{-1}\end{aligned}$$ provided that $\delta$ is chosen to be sufficiently small. Observe, both of the above inequalities are simple consequences of the definition of the blue indices. Consequently, the left-hand side of may be bounded by $$\delta^{\#S_3} \Delta_{S_3}(\rho)\frac{1}{|R_{\sigma, S_3}(\tau)|}\int_{R_{\sigma, S_3}(\tau)}|J_P(\xi_{\mathcal{S}}, x)|\,{\mathrm{d}}y$$ and so , and thence , would follow if $$|J_P(\xi(y), x)| \sim |J_P(\tau, x)| \qquad \textrm{ for all $y \in R_{\sigma, S_3}(\tau)$.}$$ This approximation is readily deduced by combining Proposition \[Stovall’s observation\] with Grönwall’s inequality (for a proof of Grönwall’s inequality see, for instance, [@Tao2006 Chapter 1]).
Hence, the estimate is established under the assumption of the separation hypothesis .
Enforce separation. {#enforce-separation. .unnumbered}
-------------------
In the previous section it was shown if were to hold for each $\nu \in \mathcal{N}$ uniformly over all $x = (x_1, \dots, x_{m+n}) \in R_{\sigma, S_3}(\tau)$, then by choosing $0< \delta \ll 1$ sufficiently small one may control the integrand by the easily-understood function $|\Delta(\rho)| |J_P(\tau, x)|$. Clearly for fixed $i$ the estimate cannot hold for at least one value of $l$, since as $x$ varies over $R(\tau)\times B_{\sigma}(\tau)$ some $x_l$ can stray close to $\tau_{\nu_i}$ in the boundary regions. To remedy this problem one simply removes a suitable small portion of $R(\tau) \times B_{\sigma}(\tau)$ from the boundary, observing that this can be done without greatly diminishing the size of the integral to be estimated. Given $0< \epsilon < 1/2$, $ 1 \leq i \leq m$ and $1 \leq j \leq n$, define the $\epsilon$-truncate of $R_i(\tau) := (\tau_{\mu_i}, \tau_{\mu_i+1})$ and $B_{\sigma, j}(\tau) := (\tau_{\nu_j} + s_{\nu_j}, \tau_{\nu_j+1})$ by $$R^{\epsilon}_i(\tau) := (\tau_{\mu_i} +\epsilon |R_i(\tau)|, \tau_{\mu_i+1}- \epsilon |R_i(\tau)|)$$ and $$B^{\epsilon}_{\sigma, j}(\tau) := (\tau_{\nu_j} + s_{\nu_j} +\epsilon |B_{\sigma, j}(\tau)|, \tau_{\nu_j+1}- \epsilon |B_{\sigma, j}(\tau)|),$$ respectively. Moreover, define the $\epsilon$-truncates of the associated rectangles to be $R^{\epsilon}(\tau) := \prod_{i=1}^m R^{\epsilon}_i(\tau)$ and $B^{\epsilon}_{\sigma}(\tau) := \prod_{j=1}^n B^{\epsilon}_{\sigma, j}(\tau)$. Lemma \[polylem\] below establishes the existence of some constant $0< c_0 < 1/2$, depending only on $d$ and $\deg P$, such that $$\label{jacoint1}
|J_{\sigma}(\rho, \tau)| \geq \bigg|\int_{D(\tau)} \wp_{\sigma}(\rho, \tau, x) \,{\mathrm{d}}x\bigg| - \frac{1}{2} \int_{D(\tau)} |\wp_{\sigma}(\rho, \tau, x)| \,{\mathrm{d}}x.$$ where $D(\tau) := R^{c_0}(\tau) \times B^{c_0}_{\sigma}(\tau)$.
It is easy to show that for all $x \in D(\tau)$ the condition holds with a uniform constant. Observe $$|B_{\sigma, j}(\tau)| = \tau_{\nu_{j+1}} - (\tau_{\nu_j} + s_{\nu_j}) = t_{l_{(\nu_j +1)}} - t_{(l_{\nu_j} +1)},$$ where the brackets in the subscript are included to aid the clarity of exposition. Since $l_{\nu_j} +1$ is, by definition, a blue index it follows that $l_{(\nu_j +1)}$ is odd and, consequently, $$|B_{\sigma, j}(\tau)| \gtrsim \alpha t_{(l_{\nu_j} +1)}^{-2K/d(d+1)} = \alpha (\tau_{\nu_j} + s_{\nu_j})^{-2K/d(d+1)}$$ by Corollary \[separation\] part i). Futhermore, recalling $s_{\nu_j} \lesssim \delta (\beta/\alpha)\tau_{\nu_j} \lesssim \tau_{\nu_j}$, it follows that $$\label{B length}
|B_{\sigma, j}(\tau)| \gtrsim \alpha \tau_{\nu_j}^{-2K/d(d+1)}.$$ Now suppose $x_l \in B^{c_0}_{\sigma, j_0}(\tau)$ for some fixed $j_0 \in \{1, \dots, n\}$. It is clear from the definition of the parameter domain that if $j \neq j_0$, then holds for $\nu = \nu_j$. Similarly, if $x_l \in R^{c_0}_{i_0}(\tau)$ for some fixed $i_0 \in \{1, \dots, m\}$, then holds for all $\nu \in \mathcal{N}$. It remains to verify when $x_l \in B^{c_0}_{\sigma, j_0}(\tau)$ and $j = j_0$, but this is immediate from the definition of the truncation and the bound .
Consequently, for $x \in D(\tau)$ and $\delta$ sufficiently small holds and thus the estimate $$\label{polynomial estimate}
|\wp_{\sigma}(\rho, \tau, x)| \gtrsim |\Delta(\rho)||J_P(\tau, x)|$$ is valid on $D(\tau)$. Furthermore, it is claimed that as $x$ varies over $D(\tau)$ the sign of $\wp_{\sigma}(\rho, \tau, x)$ is unchanged. Once this observation is established the right-hand side of can be written as $$\frac{1}{2} \int_{D(\tau)} |\wp_{\sigma}(\rho, \tau, x)| \,{\mathrm{d}}x \gtrsim |\Delta(\rho) |\int_{D(\tau)}| J_P(\tau, x)|\,{\mathrm{d}}x$$ To prove the claim, note that the ordering of the components of the $(r,t) \in \Omega$ implies the sign of $V(\tau, x)$ is fixed as $x$ varies over $D(\tau)$; the geometric inequality guaranteed by Theorem \[Dendrinos Wright theorem\] therefore ensures that the sign of $J_P(\tau, x)$ is also fixed (and is non-zero). The estimate now implies the claim.
Bound $J_P$ and apply the properties of $\Omega$. {#bound-j_p-and-apply-the-properties-of-omega. .unnumbered}
-------------------------------------------------
Combining the estimate guaranteed by Theorem \[Dendrinos Wright theorem\] and the preceding observations one deduces that $$|J_{\sigma}(\rho, \tau)| \gtrsim |\Delta(\rho) | \prod_{l=1}^{M}|L_P(\tau_l)|^{1/d} \int_{D(\tau)}\prod_{k=1}^{m+n}|L_P(x_k)|^{1/d} | V(\tau, x)| \,{\mathrm{d}}x.$$ Over the domain of integration the estimate $$| V(\tau, x)| \gtrsim \alpha^{d(d-1)/2-M(M-1)/2} |V(\tau)| \prod_{l=1}^M \tau_l^{-K(d-M)/d(d+1)} \prod_{k=1}^{m+n} x_k^{-K(d-1)/d(d+1)}$$ is valid owing to both and the additional separation enforced by truncating the set $R(\tau)$. Furthermore, the construction of the (type 1) parameter tower ensures $$\label{Vandermonde estimate}
| V(\tau)| \gtrsim \alpha^{M(M-1)/2} (\beta/\alpha)^{m/2} \prod_{l=1}^M \tau_l^{-K(M-1)/d(d+1)}.$$ Since the properties of the blue intervals imply $|\Delta(\rho) |\gtrsim (\beta/\alpha)^{n/2}$, one may combine the preceding inequalities to deduce $$\label{Jacobian bound}
|J_{\sigma}(\rho, \tau)| \gtrsim \alpha^{d(d-1)/2}(\beta/\alpha)^{(m+n)/2} \bigg(\int_{D(\tau)}\prod_{k=1}^{d-M}x_k^{2K/d(d+1)} \,{\mathrm{d}}x \bigg)\prod_{l=1}^{M}\tau_l^{2K/d(d+1)} .$$ Here the approximation $L_P(t) \sim t^{K}$ has been applied, which was a consequence of the decomposition theorem.
Finally, the integral on the right-hand side of the above expression is easily seen to satisfy $$\int_{D(\tau)}\prod_{k=1}^{d-M}x_k^{2K/d(d+1)} \,{\mathrm{d}}x \gtrsim \alpha^{d - M},$$ concluding the proof.
It remains to state and prove the lemma which justifies the estimate . In general, for $0 < \epsilon < 1/2$ the $\epsilon$-truncation $I^{\epsilon}$ of a finite open interval $I = (a, b)$ is defined as $I^{\epsilon} := (a +\epsilon(b-a), b - \epsilon(b-a))$. If $I_1, \dots, I_K$ is a family of finite open intervals, the $\epsilon$-truncation $R^{\epsilon}$ of the associated rectangle $R := \prod_{j=1}^K I_j$ is defined simply by $R^{\epsilon} := \prod_{j=1}^K I_j^{\epsilon}$.
\[polylem\] Given any $M, K \in {\mathbb{N}}$ there exists a constant $0 < c_{M,K} < 1/2$ with the following property. For all $0< \epsilon < c_{M,K}$ there exists $C_{M,K}(\epsilon) >0$ such that for any collection $I_1, \dots, I_K$ of finite open intervals with associated rectangle $R$ one has $$\int_{R \setminus R^{\epsilon}} |p(x)| \,{\mathrm{d}}x \leq C_{M,K}(\epsilon) \int_{R^{\epsilon}} |p(x)| \,{\mathrm{d}}x$$ whenever $p$ is a polynomial of degree at most $M$ in $x =(x_1, \dots, x_K)$. Moreover, $\lim_{\epsilon \rightarrow 0} C_{M,K}(\epsilon) = 0$ for any fixed $M, K$.
Once the lemma is established, taking $n,m $ and $M$ to be as defined in the previous proof and $K := m + n$, the inequality (at least in $\mathrm{Case}(1,d)$) follows by choosing $c_0$ sufficiently small so that $0 < C_{M,K}(c_0) < 1/2$.
By homogeneity it suffices to consider the case $I_1 = \dots = I_K = (0,1)$ and a simple inductive procedure further reduces the problem to the case $K = 1$. Fixing $M$ and letting $I = (0,1)$, the proof is now a simple consequence of the equivalence of norms on finite-dimensional spaces: if $C_M < \infty$ is defined to be the supremum of the ratio $\|p\|_{L^{\infty}(I)} / \|p\|_{L^1(I)}$ over all polynomials of degree at most $M$, then $$\int_{I \setminus I^{\epsilon}} |p(x)| \,{\mathrm{d}}x \leq 2\epsilon C_M \bigg(\int_{I \setminus I^{\epsilon}} |p(x)| \,{\mathrm{d}}x + \int_{I^{\epsilon}} |p(x)| \,{\mathrm{d}}x \bigg).$$ Provided that $0 < \epsilon < C_M/2$ one may take $C_{M,1}(\epsilon) := 2\epsilon C_M / (1 - 2\epsilon C_M)$, completing the proof.
The proof of the Jacobian estimates: $\mathrm{Case}(2,d+1)$ and $\mathrm{Case}(2, d)$
=====================================================================================
The argument used to prove Lemma \[jaclem1\] in $\mathrm{Case}(1,d)$ can easily be adapted to establish the result in the remaining cases. The necessary modifications are sketched below; the precise details are left to the patient reader.
Adapting the arguments to $\mathrm{Case}(2,d+1)$. {#adapting-the-arguments-to-mathrmcase2d1. .unnumbered}
-------------------------------------------------
To prove the inequality in $\mathrm{Case}(2,d+1)$ only a minor modification of the preceding argument is needed. Notice by the minimality of the parameter $N$ defined in it follows that the index $N$ is red and so $\mu_m = M$. Here $\Phi \circ \varphi^{-1}$ maps into ${\mathbb{R}}^d \times [1,2]$ and is given by $$\Phi \circ \varphi^{-1}(\rho, \tau, \sigma) = \left( \begin{array}{c}
\Psi_N(x_0, r_0; \varphi^{-1}(\rho, \tau, \sigma))\\
\rho_{\mu_m} \end{array}\right)$$ where $$\begin{aligned}
\Psi_N(x_0, r_0; \varphi^{-1}(\rho, \tau, \sigma)) &=& x_0 -r_0P(\tau_1) + \sum_{i=1}^{m-1} \rho_{\mu_i}\big( P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \big) \\
&&+ \sum_{j=1}^n \rho_{\nu_j}\big( P(\tau_{\nu_j}+ s_{\nu_j}) - P(\tau_{\nu_j +1}) \big) + \rho_{\mu_m}P(\tau_{\mu_m}).\end{aligned}$$
The Jacobian matrix is now a $(d+1)\times (d+1)$ matrix. The columns given by differentiating $G_{\sigma}$ with respect to $\rho_{\mu_i}$ are $$\left(\begin{array}{c}
P(\tau_{\mu_i}) - P(\tau_{\mu_i+1}) \\
0
\end{array}\right) \ \ \textrm{for $j = 1, \dots, m-1$ and} \ \ \left(\begin{array}{c}
P(\tau_{\mu_m}) \\
1
\end{array}\right).$$ For remaining columns, the first $d$ components are precisely the components of the corresponding columns in the previous case and the $d+1$ component is 0. Expanding the determinant across row $(d+1)$, the methods used earlier in the proof can be applied to deduce $$J_{\sigma}(\rho, \tau) = \pm\int_{R(\tau)\times B_{\sigma}(\tau)} \wp_{\sigma}(\rho, \tau, x) \,{\mathrm{d}}x$$ where $\wp_{\sigma}(\rho, \tau, x) $ is the determinant of a $d \times d$ matrix and $$R(\tau) := \prod_{i=1}^{m-1} (\tau_{\mu_i}, \tau_{\mu_i+1}); \qquad B_{\sigma}(\tau) := \prod_{j=1}^n (\tau_{\nu_{j}} + s_{\nu_j}, \tau_{\nu_{j}+1}).$$ The key difference is now the integral is over a rectangle of dimension $m+n -1$ (rather than $m+n$). Define the truncated domain $D(\tau)$ in analogous manner to the previous case. Notice from it follows that $d-M = m+n -1$, which is precisely the dimension of the set $D(\tau)$ in the present situation. Arguing as before, the inequality also holds in this setting and from this one obtains the required estimate.
Adapting the argument to $\mathrm{Case}(2, d)$. {#adapting-the-argument-to-mathrmcase2-d. .unnumbered}
-----------------------------------------------
Here the map $\Phi \circ \varphi^{-1}$ is given by $$\begin{aligned}
\Phi\circ \varphi^{-1}(\rho, \tau, \sigma) &=& y_0 + \sum_{i=1}^m \rho_{\mu_i}\big( P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \big) \\
&& + \sum_{j=1}^n \rho_{\nu_j}\big( P(\tau_{\nu_j}+ s_{\nu_j}(\tau,\sigma)) - P(\tau_{\nu_j +1}) \big)\end{aligned}$$ and thus the columns of the Jacobian matrix essentially agree with those of $\mathrm{Case}(1,d)$, with the exception that now there is no column corresponding to $\partial G_{\sigma} / \partial \tau_{\mu_1 - 1}$.
The above arguments now carry through almost verbatim; the only substantial difference in this situation is that the Vandermonde estimate becomes $$|V(\tau)| \gtrsim \alpha^{M(M-1)/2} (\beta/\alpha)^{(m-1)/2}$$ due to the fact that the parameter tower in this situation is of type 2, as opposed to type 1 in both of the previous cases.
A final remark
==============
\[strong type remark\] In the introduction the possibility of strengthening the restricted weak-type $(p_1, q_1)$ estimate from Proposition \[weakthm\] to a strong-type estimate was discussed. It was remarked that the strong-type estimate in dimension $d=2$ follows from a result of Gressman [@Gressman2013], but can also be established by combining the analysis contained within the present article with an extrapolation method due to Christ [@Christb] (see also [@Stovall2009]). Here some further details are sketched. The key ingredients in Christ’s extrapolation technique are certain ‘trilinear’ variants of the estimates and . Recall, to prove the weak-type bound it sufficed to show *either* *or* holds since both these estimates are equivalent. This equivalence breaks down when one passes to the trilinear setting and to establish the strong-type inequality one must prove *both* the trilinear version of *and* the trilinear version of hold. This can be achieved in the $d=2$ case by introducing an “inflation” argument (see [@Christ] and also [@Gressman2006]). One may attempt to apply the same techniques in higher dimensions but now the Jacobian arising from the inflation is rather complicated. The question of whether or not this Jacobian can be effectively estimated remains unresolved. It is possible that the inflation argument is not required when $d$ belongs to a certain congruence class modulo $3$ and potentially the strong-type bound could be established more directly from existing arguments in this situation.
Appendix: The method of Dendrinos and Stovall {#appendix-the-method-of-dendrinos-and-stovall .unnumbered}
=============================================
This final section details the construction of the sequence of sets $\{U_k\}_{k=1}^{\infty}$ featured in Lemma \[towerlem\]. The argument here is due to Dendrinos and Stovall [@Dendrinos]. At this point some preliminary definitions and remarks are pertinent. Observe $$\langle A\chi_E\,,\, \chi_F \rangle = \int_{\Sigma} \chi_F(\pi_1(x,r,t)) \chi_E(\pi_2(x,r,t)) \lambda_P(t)\,{\mathrm{d}}x{\mathrm{d}}r {\mathrm{d}}t$$ where $\Sigma := {\mathbb{R}}^d \times [1,2] \times I$ and $\pi_1 \colon \Sigma \to {\mathbb{R}}^d \times [1,2]$ and $\pi_2 \colon \Sigma \to {\mathbb{R}}^d$ are the mappings $$\pi_1(x,r,t) := (x,r), \qquad \pi_2(x,r,t) := x - rP(t).$$ Define the $\pi_j$-fibres to be the sets $\pi_j^{-1}\circ\pi_j(x,r,t)$ for $(x,r,t) \in \Sigma$ and $j = 1,2$. Thus, the $\pi_1$-fibres form a partition of $\Sigma$ into a continuum of curves (which are simply parallel lines) whilst the $\pi_2$-fibres partition $\Sigma$ into a continuum of 2-surfaces. Writing $$U := \pi_1^{-1}(F) \cap \pi_2^{-1}(E) = \left\{ (x,r,t) \in \Sigma : \pi_1(x,r,t) \in F\textrm{ and } \pi_2(x,r,t) \in E \right\}$$ it follows that $$\langle A\chi_E\,,\, \chi_F \rangle = \int_{\Sigma} \chi_U(x,r,t)\lambda_P(t)\,{\mathrm{d}}x{\mathrm{d}}r {\mathrm{d}}t.$$ The sets $\{U_k\}_{k=0}^{\infty}$ are defined recursively. To construct the initial set $U_0$, let $$B_0 := \{(x,r,t) \in U : 0 < t < (\alpha/2\kappa)^{\kappa}\}.$$ Then, recalling the definition of $\lambda_P$ and applying Fubini’s theorem, it follows that $$\begin{aligned}
\int_{\Sigma} \chi_{B_0}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t &=& \int_{F} \int_0^{(\alpha/2\kappa)^{\kappa}} \chi_E(x - rP(t))\, \lambda_P(t){\mathrm{d}}t {\mathrm{d}}x {\mathrm{d}}r \\
&\leq& \frac{1}{2}\alpha|F| = \frac{1}{2} \int_{\Sigma} \chi_U(x,r,t)\,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.\end{aligned}$$ Define $U_0 := U\setminus B_0$ so that $$\int_{\Sigma} \chi_{U_0}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t \geq \frac{1}{2}\int_{\Sigma} \chi_{U}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.$$ Note that this definition will ensure property iii) holds for the sequence of refinements. Now suppose the set $U_{k-1}$ has been defined for some $k \geq 1$ and satisfies the conditions stipulated in the proof of Lemma \[towerlem\].
Case $k \equiv d \mod 2$. {#case-k-equiv-d-mod-2. .unnumbered}
-------------------------
In order to ensure the property holds in this case, the following refinement procedure is applied. Let $B_{k-1}$ denote the set $$\left\{ (x,r,t) \in U_{k-1} : \int_1^2\int_I \chi_{U_{k-1}}(x - rP(t) + \rho P(\tau), \rho, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau{\mathrm{d}}\rho \leq 4^{-(k+1/2)} \beta \right\}.$$ The map $(\rho, \tau) \mapsto (x - rP(t) + \rho P(\tau),\rho, \tau)$ parametrises the fibre $\pi_2^{-1}(\pi_2(x,r, t))$ and so $B_{k-1}$ is precisely the set of all points belonging to $\pi_2$-fibres which have a “small” intersection with $U_{k-1}$. Removing the parts of $U_{k-1}$ lying in these fibres should not significantly diminish the measure of the set and indeed, by Fubini’s theorem and a simple change of variables, $$\begin{aligned}
\nonumber
\int_{\Sigma} \chi_{B_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}x{\mathrm{d}}r &=& \int_{{\mathbb{R}}^n\times[1,2]}\int_I \chi_{B_{k-1}}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}x{\mathrm{d}}r \\
\nonumber
&\leq& \int_{\{ x \in E \,:\, T_{k-1}(x) \leq 4^{-(k+1/2)}\beta\}} T_{k-1}(x)\, {\mathrm{d}}x \\
\nonumber
&\leq& 4^{-(k+1/2)}\beta|E| \\
\label{Dendrinos and Stovall 1}
&\leq& \frac{1}{2}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t\end{aligned}$$ where $$T_{k-1}(x) := \int_1^2\int_I \chi_{U_{k-1}}(x+\rho P(\tau),\rho,\tau) \,\lambda_P(\tau){\mathrm{d}}\tau{\mathrm{d}}\rho.$$ Note that the inequality is due to property i) of the sets $U_j$ for $1 \leq j \leq k-1$, stated in Lemma \[towerlem\]. Thence, letting $U_{k-1}' := U_{k-1} \setminus B_{k-1}$ it follows that $$\label{U bound}
\int_{\Sigma} \chi_{U_{k-1}'}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t \geq \frac{1}{2}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.$$ Now, recalling $I = (a,b)$, define $B_{k-1}'$ to be the set $$\left\{ (x,r,t) \in U_{k-1}' : \int_1^2\int_t^b \chi_{U_{k-1}'}(x - rP(t) + \rho P(\tau),\rho, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau{\mathrm{d}}\rho \leq 4^{-(k+1)}\beta \right\}.$$ Given $x \in \pi_2(U_{k-1}')$, the fibre-wise nature of the definition of $U_{k-1}'$ implies for $(x+rP(t),r,t) \in U_{k-1}'$ if and only if $(x+rP(t),r,t) \in U_{k-1}$ and consequently $$\label{ref1}
\int_1^2\int_I \chi_{U_{k-1}'}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r \geq 4^{-(k+1/2)} \beta.$$ On the other hand, $$\label{ref2}
\int_1^2\int_I \chi_{B_{k-1}'}(x+rP(t),r,t) \, \lambda_P(t){\mathrm{d}}t{\mathrm{d}}r = 4^{-(k+1)}\beta.$$ Indeed, the left-hand side can be expressed as $$\nu_P\big(\big\{(r,t) \in K(x) : \nu_P\big(K(x) \cap ([1,2] \times (t,b))\big) \leq 4^{-(k+1)}\beta\big\}\big)$$ for $K(x) \subseteq [1,2] \times I$ a measurable subset. The identity is now a consequence of the fact that for any measure $\nu$ on ${\mathbb{R}}^2$ which is, say, absolutely continuous with respect to Lebesgue measure, $$\nu\big(\big\{ (r,t) \in K : \nu\big(K \cap ({\mathbb{R}}\times (t, \infty))\big) \leq u \big\}\big) = u$$ for all $0< u < \nu(K)$ and all $K \subseteq {\mathbb{R}}^2$ measurable.
Thence, combining and it follows that $$\int_1^2\int_I \chi_{B_{k-1}'}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r \leq \frac{1}{2} \int_1^2\int_I \chi_{U_{k-1}'}(x+rP(t),r,t) \lambda_P(t){\mathrm{d}}t{\mathrm{d}}r$$ whenever $x \in \pi_2(U_{k-1}')$. Defining $U_{k}:= U_{k-1}' \setminus B_{k-1}'$, one observes that $$\begin{aligned}
\int_{\Sigma} \chi_{U_{k}}(x,r,t) \,\lambda_P(t) {\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t &=& \int_{\pi_2(U_{k-1}')} \int_1^2\int_I \chi_{U_k}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r {\mathrm{d}}x \\
&\geq& \frac{1}{2}\int_{\pi_2(U_{k-1}')} \int_1^2\int_I \chi_{U_{k-1}'}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r {\mathrm{d}}x \\
&=& \frac{1}{4}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t.\end{aligned}$$ Moreover, the set $U_k$ is easily seen to satisfy .
Case $k \not \equiv d \mod 2$. {#case-k-not-equiv-d-mod-2. .unnumbered}
------------------------------
It remains to define the set $U_k$ under the assumption $k \not \equiv d \mod 2$, ensuring property is satisfied. Here one is concerned with the fibres of the map $\pi_1$. Define $$B_{k-1} := \left\{ (x,r,t) \in U_{k-1} : \int_I \chi_{U_{k-1}}(x,r, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau \leq 4^{-(k+1/2)} \alpha \right\}.$$ Notice that the map $\tau \mapsto (x,r, \tau)$ parametrises the fibre $\pi_1^{-1}(\pi_1(x,r,t))$ and so $B_{k-1}$ is the collection of all points $(x,r,t)$ in $U_{k-1}$ which belong to $\pi_1$-fibres which have a “small” intersection with $U_{k-1}$. Reasoning analogously to the previous case, if one defines $U_{k-1}' := U_{k-1} \setminus B_{k-1}$ it follows that holds in this case. Finally, let $$B_{k-1}' := \left\{ (x,r,t) \in U_{k-1}' : \int_t^b \chi_{U_{k-1}'}(x,r, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau \leq 4^{-(k+1)} \alpha \right\}$$ and $U_{k} := U_{k-1}'\setminus B_{k-1}'$. Again arguing as in the previous case, it follows that $$\int_{\Sigma} \chi_{U_{k}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t \geq \frac{1}{4}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.$$ This recursive procedure defines a sequence of sets with all the desired properties.
[^1]: More precisely, both Bourgain and Schlag studied the circular maximal function rather than the parabolic variant discussed here. However, in this context both objects can be understood via the same techniques.
[^2]: It is remarked that this brief survey is far from complete: there are many other results and, in particular, an extensive literature investigating these problems for more general classes of curves.
[^3]: Although in the case of maximal functions one must also consider the position of the curve in the plane, see [@Iosevich1994].
[^4]: That is, for almost every $x \in {\mathbb{R}}^d$ the cardinality of the pre-image $\Phi^{-1}(\{x\})$ is no greater than some fixed (finite) constant. For further details see Lemma \[multiplicity\] below.
[^5]: For the moment curve $h(t) := (t, t^2, \dots, t^d)$, one immediately observes that $J_h(t) = cV(t)$.
[^6]: Here $\ceil{x} := \min\{ n \in {\mathbb{N}}: n \geq x\}$ and $\floor{x} := \max\{ n \in {\mathbb{N}}: n \leq x\}$ for any $x \in {\mathbb{R}}$.
[^7]: Strictly speaking, $\{\Omega_j\}_{j=1}^D$ will only satisfy a weak definition of a tower: see Remark \[measure remark\] below.
[^8]: Recall, Bezout’s theorem states that for any collection $Q_1, \dots, Q_n$ of homogeneous polynomials on ${\mathbb{C}}\mathbb{P}^n$ the number of intersection points of the associated hypersurfaces $ \{z \in {\mathbb{C}}\mathbb{P}^n : Q_j(z) = 0 \}$ (counted with multiplicity) is either uncountable or precisely $\prod_{j=1}^n \deg (Q_j)$. The real version used here follows by homogenising the polynomials and taking the domain of the resulting functions to be $\mathbb{CP}^n$. One then applies Bezout’s theorem in complex projective space, de-homogenises and restricts to real-value intersection points. See, for example, [@Hassett2007] pp 223-224.
[^9]: For notational convenience the dependence of $s_{\nu}$ on $(\tau, \sigma)$ has been suppressed.
|
---
abstract: 'Degeneracy effects for bosons are more important for smaller particle mass, smaller temperature and higher number density. Bose condensation requires that particles be in the same lowest energy quantum state. We propose a cosmic background Bose condensation, present everywhere, whith its particles having the lowest quantum energy state, $\hbar c / \lambda$, with $\lambda$ about the size of the visible universe, and therefore unlocalized. This we identify with the quantum of the self gravitational potential energy of any particle, and with the bit of information of minimum energy. The entropy of the universe ($\sim 10^{122} \ bits$) has the highest number density ($ \sim 10^{36} \ bits / cm^3$) of particles inside the visible universe, the smallest mass, $\sim 10^{-66} g$, and the smallest temperature, $\sim 10^{-29} K$. Therefore it is the best candidate for a Cosmic Background Bose Condensation (CBBC), a completely calmed fluid, with no viscosity, in a superfluidity state, and possibly responsible for the expansion of the universe.'
author:
- 'Antonio Alfonso-Faus'
- Màrius Josep Fullana i Alfonso
title: 'Cosmic Background Bose Condensation (CBBC) '
---
Introduction
============
[@Wei] advanced a clue to suggest that large numbers are determined by both, microphysics and the influence of the whole universe. He constructed a mass using the physical constants $G, \hbar, c$ and the Hubble parameter $H$. This mass was not too different from the mass of a typical elementary particle (like a pion) and is given by
$$m \approx (\hbar^2 H /Gc)^{1/3}
\label{e1}$$
In our work here we consider a general elementary particle of mass $m$. This particle may include not only baryons but the possible quantum masses of dark matter and dark energy in the universe. Since the mass $m$ will disappear from the resultant relation, the conclusion is totally independent on the kind of elementary particle that we may consider. The self gravitational potential energy $E_g$ of this quantum of mass $m$ (and size its Compton wavelength $ \hbar /mc$) is given by
$$E_g = G m^2 / (\hbar / mc) = G m^3 c / \hbar
\label{e2}$$
This relation has been previously used in another context (@Siv [-@Siv]). Combining (\[e1\]) and (\[e2\]) we eliminate the mass $m$ to obtain
$$E_g \approx H \hbar
\label{e3}$$
Here $\hbar$ is Planck’s constant, usually interpreted as the smallest quantum of action (angular momentum). Since $H$ is of the order of $1/t$, $t$ the age of the universe ($t$ being a maximum time today), (\[e3\]) is the lowest quantum energy state that it may exist. It is equivalent to $\hbar c / \lambda$ with $\lambda$ of the order of the size of the visible universe (it is the lowest quantum energy state with $\lambda \approx ct$). We identify it with the quantum of the self gravitational potential energy of any quantum particle (@AAFa [-@AAFa]). We also identify it with the bit, the unit of information with minimum energy (@AAF2 [-@AAF2]). [@LLo], about 10 years ago, stated that [*Merely by existing, all physical systems register information*]{} and about 25 years ago [@Lan], as cited by [@LLo], stated [*Information is physical”. And today we say here: All physical systems of mass M (energy $Mc^2$) are equivalent to an amount of information in number of bits of the order of* ]{} $$Number \ of \ bits \ \approx M c^2 / E_g \approx Mc^2 / (H \hbar)
\label{e4}$$
The equivalence between information and energy, as implied by the above relation, can be interpreted as the result of a recent experiment (@Fun [-@Fun]) where it is shown that entanglement can produce a gain in thermodynamic work, the gain being determined by a change of information content. Also a link between information theory and thermodynamics has been experimentally verified (@Ber [-@Ber]). Previously, in 2010, an experimental demonstration of information-to-energy conversion was also published (@Toy [-@Toy]). Relation (\[e4\]) has general, universal validity. In this sense the unit of energy, that should naturally be taken as the minimum quantum of energy $H \hbar$, implies that the relativistic energy of any mass $M$ has $N$ times this minimum quantum of energy, $N H \hbar$, being $N$ its number of information bits. Therefore, $N H \hbar$ corresponds to the energy of all the information $N$ that carries the physical system. And as far as entropy $S$ is concerned, this number is also the same as $S/ (k_B \log 2)$, $k_B$ being Boltzmann constant, as we can talk about a generalized relation for entropy, in accordance with the ideas introduced by [@Lan2]:
$$S = N k_B \log 2 = \frac{Mc^2}{H \hbar} \ k_B \log 2
\label{e5}$$
We see that the extensive property of entropy is preserved because, in accordance with our proposal, it comes to be proportional to the relativistic energy of the system. In this work we propose a Cosmic Background Bose condensation (CBBC), present everywhere, where the particle with minimum energy $E_g$ and mass $m_g = E_g /c^2$ is defined in (\[e3\]). It is composed of very low energy and temperature components, with very high number density, and of course all in the same state.
[@Bek] found an upper bound for the ratio of the entropy $S_B$ to the energy $E = Mc^2$ of any bounded system with effective size $R$: $$S_B / E < 2 \pi k_B R / \hbar c
\label{eq.1}$$
About ten years later (@Hoo [-@Hoo]; @Sus [-@Sus]), a holographic principle was proposed giving a bound for the entropy $S_h$ of a bounded system of effective size $R$ as $$S_h \leq \pi k_B c^3 R^2 / \hbar G
\label{eq.2}$$
The Bekenstein bound (\[eq.1\]) is proportional to the product $MR$, while the holographic principle bound (\[eq.2\]) is proportional to the area $R^2$. If the two bounds are identical (hence $M$ is proportional to $R$) here we prove that the system obeys the Schwarzschild condition for a black hole. We analyze this conclusion for the case of a universe with finite mass $M$ and a Hubble size $R \approx ct$, $t$ being the age of the universe. $M$ and $R$ are obviously the maximum values they can have in our universe, the visible universe.
Also, for the case of a black hole we find that the Hawking and Unruh temperatures are the same. Then, for our universe we obtain the mass of the gravity quanta, of the order of $10^{-66} g$.
Consequences of the identification of the two bounds
====================================================
Identifying the bounds (\[eq.1\]) and (\[eq.2\]) we get $$2 M = c^2 R / G
\label{eq.3}$$ which is the condition for the system $(M, R)$ to be a black hole. Then, its entropy is given by the Hawking relation (@Haw [-@Haw]) $$S_H = \frac{4 \pi k_B}{\hbar c} \ G M^2
\label{eq.4}$$ that coincides with the two bounds (\[eq.1\]) and (\[eq.2\]). The conclusions here are exclusively related to the Schwarzschild radius within the context of Einstein’s general relativity.
The mass of the universe $M_u$ is a maximum. And so is its size $R$. A bounded system implies a finite value for both. Using present values for $M_u \approx 10^{56} g$ and $R \approx 10^{28} cm$ they fulfill the Schwarzschild condition (\[eq.3\]). This is an evidence for the Universe to be a black hole (@AAF1 [-@AAF1]). And its entropy today is about $10^{122} k_B \log 2$.
The case for the Unruh and Hawking temperatures
===============================================
The fact that a black hole has a temperature, and therefore an entropy (@Haw [-@Haw]) implies that an observer at its surface, or event horizon, sees a perfect blackbody radiation, a thermal radiation with temperature $T_H$ given by $$T_H = \hbar c^3 / (8 \pi GMk)
\label{eq.5}$$ where $\hbar$ is the Planck’s constant, $c$ the speed of light, $G$ the gravitational constant, $k$ Boltzmann constant and $M$ the mass of the black hole. This observer feels a surface gravitational acceleration $R''$. According to the [@Unr] effect an accelerated observer also sees a thermal radiation at a temperature $T_U$, proportional to the acceleration $R''$ and given by $$T_U = \hbar R'' / (2 \pi ck)
\label{eq.6}$$
Based upon the similarity between the mechanical and thermo dynamical properties of both effects, (\[eq.5\]) and (\[eq.6\]), we identify both temperatures and find the relation: $$R'' = c^4 / (4GM)
\label{eq.7}$$
Identifying the Unruh acceleration to the surface gravitational acceleration: $$R'' = GM/R^2
\label{eq.8}$$ and substituting in (\[eq.7\]) we finally get $$2GM/c^2 = R
\label{eq.9}$$
This is the condition for a black hole. Since the Hawking temperature refers to a black hole this result confirms the validity of the identification of the two temperatures, (\[eq.5\]) and (\[eq.6\]), as well as the interpretation of the Unruh acceleration in (\[eq.8\]).
Application of the cosmological principle
=========================================
The cosmological principle may be stated with the two special conditions of the universe: it is homogeneous and isotropic. This means that, on the average, all places in the universe are equivalent (at the same [*time*]{}) and that observing the universe at one location it looks the same in any direction. The cosmic microwave background radiation is a good example, a blackbody radiation at about $2.7 K$. This implies that there is no center of the universe, or equivalently, that any local place is a center.
We have seen that the universe may be taken as a black hole, and therefore that it makes sense to think that there must be an event horizon, a two dimensional bounding surface around each observer. If all places in the universe are equivalent then all places can say this, and the natural event horizon is the Hubble sphere, with radius $R$ about $c / H \approx 10^{28} cm$ today. Then at any point in the universe it can be interpreted as a two dimensional spherical surface, may be in a [*virtual*]{} sense. Following the holographic principle, all the information of the three dimensional world, as we see it, is contained in this spherical surface. And any observer, following the cosmological principle, can be seen as being at the center of the three dimensional [*sphere*]{}. To combine both principles, the cosmological and the holographic, we can think of an isotropic, spherically symmetric, acceleration present at each point in the universe given by (\[eq.8\]). This is a change of view from a 3 dimensional one to a 2 dimensional world. Also we have an isotropic temperature given by $$T = \hbar c^3 / (8 \pi GMk) = (1/4 \pi k) 1 / R \approx 10^{-29} K
\label{eq.10}$$
This is the temperature of the gravitational quanta (@AAF2 [-@AAF2]) at the present time. The equivalent mass of one quantum of gravitational potential energy is then from (\[eq.10\]) found to be about $10^{-66} g$. This may be interpreted as the ultimate quantum of mass. Its wavelength (in the Compton sense) is of the order of the size of the universe, $ct \approx 10^{28} cm$ and therefore it is a gravity quanta unlocalized in the universe, as the gravitational field (@Mis [-@Mis]). It is a boson and not a photon, the photon being the quantum of the electromagnetic field that should be localized in the universe.
The scale factor between the Planck scale and our universe today is about $10^{61}$. Multiplying the temperature found in (\[eq.10\]) by this numerical scale factor we get the Planck’s temperature $T_p \approx 10^{32} K$ at the Planck’s time $10^{-44} s$, when the universe had the Planck’s size $l_p \approx 10^{-33} cm$.
Cosmic Background Bose Condensation (CBBC)
==========================================
We now present the conditions favorable to have a Bose condensation as derived from the energy (@Eis [-@Eis]) per particle relation $E/N$:
$$E/N = \frac{3}{2} \ kT \left \{ 1 - \frac{1}{2^{5/2}} \ \frac{N h^3}{V (2 \pi m k_B T)^{3/2}} \right \}
\label{e5}$$
The term beyond 1 in the above bracket is the deviation of the Bose gas from the classical gas, the degeneracy effect. As we can see in the formulation (\[e5\]) this effect for bosons is more important the smaller the particle mass is ( $\sim 10^{-66} g$ in our case), the smaller the temperature is ( $\sim 10^{-29} K$ in our case) and the higher the number density is ($\sim 10^{36} \ particles/cm^3$ ). Then, the particle we are presenting here appears to be a good candidate for a universal Bose condensation background.
The CBBC and the expansion of the universe
==========================================
As we have seen in (\[e3\]) the gravity quanta proposed here does not depend on the related origin: baryons, dark matter or dark energy. Usually the expansion of the universe is considered to be related to the cosmological constant $\Lambda$. This constant was introduced by Einstein to avoid the collapse of the universe due to attractive gravitation by means of an outward pressure given by $\Lambda$. Within this context we know today that the percentages of baryonic content in the universe, dark matter and dark energy, are respectively about 4%, 27% and 69%. This is given in terms of the usual dimensionless $\Omega$: $$\Omega_b \approx 0.04, \ \ \ \ \ \Omega_{DM} \approx 0.27, \ \ \ \ \ \Omega_{\Lambda} \approx 0.69
\label{e6}$$ all of them adding up to 1. Given that our approach here does not discriminate between baryons, dark matter and dark energy, we may attribute the CBBC expansion effect as due to its pressure as related to the baryon component and possibly dark matter quanta or even dark energy quanta. We do not know today if there are quanta in the dark components. The maximum effect would correspond to consider that the total, critical density, is responsible for the gravity quanta presented here as the ground component of the CBBC, and then its effect would correspond to $\Omega_{CBBC} = 1$.
Conclusions
===========
The universe can be seen as a black hole. From each observer in it we can interpret that he/she is at the center of a sphere, with the Hubble radius. From this we interpret the spherical surface as the event horizon, a two dimensional surface that follows the physics of the holographic principle.
The isotropic acceleration present at each point in the universe, and given by (\[eq.8\]), implies that there is no distortion for the spherically distributed acceleration, as imposed by the cosmological principle. However, the presence of a nearby important mass, like the sun, will distort this spherically symmetric picture. With respect to the probes Pioneer 10/11, that detected an anomalous extra acceleration towards the sun of value ($8.74 \pm 1.33) \times 10^{-8} cm/s^2$ (@And [-@And]), we can see that this value is only a bit higher than the one predicted by (\[eq.8\]), which is about $7.7 \times 10^{-8}cm/s^2$. This difference is an effect that can be explained by the presence of the sun converting the isotropic acceleration to an anisotropic one.
Similarly there may be a factor, due to the influence of nearby masses (i.e. a massive black hole at the center of the galaxy), in the cases of the observed rotation curves in spiral galaxies. They imply that the speed of stars, instead of decreasing with the distance $r$ from the galactic center, is constant or even increases slowly when far from the central luminous object (@Dre [-@Dre]). For the case of globular clusters (@Sca [-@Sca]), where no dark matter is expected to be present, we have stronger evidence in support of the existence of the acceleration field. Also the escape velocity at the sun location, with respect to our galaxy, is higher than expected. The earth-moon distance increases with time and there is a residual part not explained by tidal effects. And the same occurs for the planets in the solar system. We present this evidence in support of the universal field of acceleration, $R''$ (@AAF3 [-@AAF3]).
Finally, for the case of a black hole we find that the Hawking and Unruh temperatures are the same. For our universe we obtain the mass of the gravity quanta, of the order of $10^{-66} g$, with a wavelength that corresponds to its size. It may be identified with the information bit, with entropy $k$. A Cosmic Background Bose Condensation fits well with the properties found here for the quantum of gravitational potential energy.
Alfonso-Faus, A., 2010a, “Universality of the self gravitational potential energy of any fundamental particle”, Astrophysics and Space Science, 337, 363
Alfonso-Faus, A., 2010b, “The case for the Universe to be a quantum black hole”, Astrophysics and Space Science, 325, 113
Alfonso-Faus, A., 2010c, “Galaxies: kinematics as a proof of the existence of a universal field of minimum acceleration”, preprint, arXiv:0708.0308
Alfonso-Faus, A., 2011, “Quantum gravity and information theories linked by the physical properties of the bit”, preprint, arXiv: 1105.3143
Anderson, J. D., et al., 1998, “Indication, from Pioneer 10/11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-Range Acceleration”, Physical Review Letters, 81, 2858
Bekenstein, J. D., 1981, Physical Review D, 23 (2), 287
Bérut, A. et al., 2012, “Experimental verification of Landauer’s principle linking information and thermodynamics”, Nature, 483, 187
Drees, M., & Chung-Lin, S., 2007, “Theoretical Interpretation of Experimental Data from Direct Dark Matter Detection”, Journal of Cosmology and Astroparticle Physics, 0706, 011
Eisberg, R. and Resnick, R., 1985, “Quantum Physics of atoms, molecules, solids, nuclei and particles” 1985, Second Edition, John Wiley & Sons Inc., New York
Funo, K., Watanabe, Y. & Ueda, M., 2012, “Thermodynamic Work Gain from Entanglement”, preprint, quant-ph/1207.6872
Hawking, S. W., 1974, “Black hole explosions?” , Nature, 248, 30
Landauer, R., 1961, “Irreversibility and Heat Generation in the Computing Process”, IBM Journal of Research and Development, 5, 183
Landauer, R., 1988, “Dissipation and noise immunity in computation and communication”, Nature, 335, 779
Lloyd, S., 2002, “Computational capacity of the universe”, Physical Review Letters 88, 237901
Misner, C.W., Thorne, K.S. & Wheeler, J.A., 1973, “Gravitation”, W.H. Freeman and Company, Reading (England), p.466 (“Why the energy of the gravitational field cannot be localized”)
Scarpa, R. & Falomo, R., 2010, “Testing Newtonian gravity in the low acceleration regime with globular clusters: the case of omega Centauri revisited”, Astronomy and Astrophysics, 523, A43
Sivaram, C., 1982, “Cosmological and quantum constraint on particle masses”, American Journal of Physics, 50, 279
Susskind, L., 1995, “The World as a hologram”, Journal of Mathematical Physics, 36, 6377
’t Hooft, G., 1993, “Dimensional Reduction in Quantum Gravity”, preprint, arXiv:gr-qc/9310026
Toyabe, S. et al., 2010, “Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality”, Nature Physics, 6, 988
Unruh, W.G., 1976, “Notes on black-hole evaporation”, Physical Review D, 14 (4), 870
Weinberg, S., 1972, “Gravitation and Cosmology: Principles and applications of the General Theory of Relativity”, John Wiley & Sons Inc., New York, p.619
|
---
abstract: |
In this paper we consider one particular mathematical problem of this large area of fractional powers of self-adjoined elliptic operators, defined either by Dunford-Taylor-like integrals or by the representation through the spectrum of the elliptic operator. Due to the mathematical modeling of various non-local phenomena using such operators recently a number of numerical methods for solving equations involving operators of fractional order were introduced, studied, and tested. Here we consider the discrete counterpart of such problems obtained from finite difference or finite element approximations of the corresponding elliptic problems. In short, these are linear equations for $ \tiluh \in {\mathbb{R}}^N$ of the type $\wcalAt^{\alpha} \tiluh = \tilfh$ or $\tiluh = \wcalAt^{-\alpha} \tilfh$, where $\alpha \in (0,1)$, and $\wcalAt \in {\mathbb{R}}^{N \times N}$ is an SPD matrix.
Among the existing methods is a method based on the best uniform rational approximation (BURA) introduced and analyzed in [@HLMMV18; @harizanov2019analysis]. In fact, the method of Bonito and Pasciak, [@bonito2017numerical; @BP15; @BP17], which uses exponentially convergent sinc-quadratures for the Dunford-Taylor integrals, results in a rational approximation of the corresponding kernel. Thus theoretically, the BURA approach should be as good as the method of Bonito-Pasciak.
In the simplest case, to implement the BURA method one needs to generate the best uniform rational approximation of $t^{-\alpha}$ on the spectrum of $\wcalAt$. In order to make the method feasible, instead we seek the BURA on the interval $[\lambda_1, \lambda_N]$, where $\lambda_1 \le \dots \le \lambda_N$ are the eigenvalues of $\wcalAt$. This is further simplified by rescaling the system so the solution is sought in the form $ \tiluh = (\lambda_1^{-1} \wcalAt)^{-\alpha} \lambda_1^{\alpha} \tilfh$, so we need the find the BURA of $t^{-\alpha}$ on $[1, \lambda_N/\lambda_1]$. If we introduce a parameter $0 \le \delta \le \lambda_1/\lambda_N$, an estimate of the ratio $ \lambda_1/\lambda_N$ from below, we can avoid the necessity to know the spectrum of $\wcalAt$.
In this report we provide all necessary information regarding the best uniform rational approximation (BURA) $r_{k,\alpha}(t) := P_k(t)/Q_k(t)$ of $t^{\alpha}$ on $[\delta, 1]$ for various $\alpha$, $\delta$, and $k$. The results are presented in 160 tables containing the coefficients of $P_k(t)$ and $Q_k(t) $, the zeros and the poles of $r_{k,\alpha}(t)$, the extremal point of the error $t^\alpha - r_{k,\alpha}(t)$, the representation of $r_{k,\alpha}(t)$ in terms of partial fractions, etc. In short, we provide all necessary data to compute efficiently the approximation $\tilwh = r_{k,\alpha}((\lambda_1^{-1} \wcalAt)^{-1}) \lambda_1^{-\alpha} \tilfh$ of the exact solution $ \tiluh = (\lambda_1^{-1} \wcalAt)^{-\alpha} \lambda_1^{\alpha} \tilfh$. Moreover, we provide links to the files with the data that characterize $r_{k,\alpha}(t)$ which are available with enough significant digits so one can use them in his/her own computations. The presented numerical results use Remez algorithm for computing BURA, see, e.g. [@Driscoll2014; @PGMASA1987]. It is well known that this method is highly sensitive to the computer arithmetics precision, so precomputing (or off-line computations) of BURA seems to be an important and necessary step, [@Driscoll2014; @PGMASA1987]. Here we report the results in performing this step and we provide the data associated with handling this problem.
Further, similar data (the poles, the extreme points, the the partial fraction representation) is generated and presented for the BURA of the function $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ (of three parameters $ q,\delta,\alpha$, such that $0\le q$, $0 \le \delta < 1$, $0 < \alpha <1$ and a variable $ \t \in [\delta,1]$). The information is organized in the way as the case BURA of $t^\alpha$. In the end on two examples of model problems we go through all necessary steps of extracting the necessary information in order to solve approximately the problems.
address:
- 'Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria'
- 'Department of Mathematics, Texas A&M University, College Station, TX 77843'
- 'Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria'
- 'Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria'
author:
- 'S. Harizanov, R.D. Lazarov, S. Margenov, P. Marinov'
bibliography:
- 'BURA\_refer\_exp.bib'
date: 'started July 13, 2019, today is '
title: |
The Best Uniform Rational Approximation:\
Applications to Solving Equations Involving Fractional powers of Elliptic Operators
---
Theoretical background {#chapt:theory}
======================
Introduction {#se:intro}
------------
### General comments
“Fractional” order differential operators appear naturally in many areas in mathematics and physics, e.g. trace theory of functions in Sobolev classes (Sobolev imbedding, elliptic type), the theory of special classes analytic functions (Dzhrbashyan, [@Djrbashian:1993], Riemann-Liouville fractional derivative, [@KilbasSrivastavaTrujillo:2006; @Podlubny:1999]), modeling various phenomena, e.g. particle movement in heterogeneous media, [@MetzlerKlafter:2004] and/or modeling dynamics in fractal media (Tarasov, [@tarasov2011fractional]), modeling materials with memory (e.g. viscoelasticity, Bagley-Torvik equation, [@torvik1984]) heavily tailed Levy flights of particles, [@HatanoHatano:1998], peridynamics (deformable media with fractures), image reconstruction using non-local operators, [@gilboa2008nonlocal], etc. The most important property of these operators is that they are non-local.
There are two main definitions of “fractional Laplacian” (and more general steady-state sub-diffusion problem) used modeling of various non-local diffusion-like problems. For a reveling and thorough discussion on this topic we refer to [@Karniadakis2018fractional]. Here we shall consider the case of so-called spectral fractional Laplacian.
In this work we consider numerical methods and algorithms for solving equations involving fractional powers of self-adjoined elliptic operators, defined either by Dunford-Taylor-like integrals or by the representation through the eigenvalues and the eigenfunctions of the elliptic operator. The numerical methods are done in two basic steps:\
(1) approximation of the corresponding elliptic operator by finite elements in a finite dimension space $V_h$ (of dimension $N$) or similar approximation by finite differences on a rectangular mesh leading to a matrix acting in the Euclidean space ${\mathbb{R}}^N$; this results in a semi-discrete scheme;\
(2) approximation of the fractional powers of the discretized elliptic operator using the best uniform rational approximation (BURA) of a certain function on $[0,1]$, which results in a fully discrete scheme.
### Semi-discrete and fully discrete approximations in the elliptic case
First, in Subsection \[ss:problem\] we consider systems of equations generated by fractional powers of elliptic operators of the type $ \calA^\alpha u = f $. The fractional powers of $\calA$ are defined by Dunford-Taylor integrals, which can be transformed when $\alpha \in (0, 1)$ to the Balakrishnan integral . Further we also consider other problems like $ \calA^\alpha u + \q u =f $ and initial value problem $\frac{\partial u (t)}{\partial t} + \calA^\alpha u(t) =f(t) $, $t >0$, with $ u(0)=v $.
The approximation is done in two steps. In the first step we discretize the elliptic operator $ \calA $. In the case of finite element approximation we get a symmetric and positive definite operator $\calAt: V_h \to V_h$, that results in an operator equation $\calAt^{\alpha} u_h = f_h$ for $f_h \in V_h$ given and $u_h \in V_h$ unknown. In the case of finite difference approximation we get a symmetric and positive definite matrix $\wcalAt \in {\mathbb{R}}^{N\times N}$ and a vector $\tilfh \in {\mathbb{R}}^N$, so that the approximate solution $\tiluh \in {\mathbb{R}}^N$ satisfies $\wcalAt^{\alpha} \tiluh = \tilfh$. The fractional powers of the operator $\calAt^\alpha$ and the matrix $\wcalAt^\alpha$ are defined by or (with finite summation), correspondingly. These equations generate the so-called semi-discrete solutions $$u_h = \calAt^{-\alpha} f_h \ \mbox{or/and} \ \ \tiluh = \wcalAt^{-\alpha} \tilfh.$$
In the second step we essentially approximate the Balakrishnan integral . This is done by introducing the rational function $r_{k,\alpha}(\t)$, which is the best uniform rational approximation (BURA) of $\t^{\alpha}$ on $[0,1]$ and apply it to produce the fully discrete approximations, see , $$w_h
=\lambda_1^{-\alpha} r_{\alpha,k} (\lambda_1 \calAt^{-1}) f_h
\quad \mbox{and} \quad
\tilwh
=\lambda_1^{-\alpha} r_{\alpha,k} (\lambda_1 \wcalAt^{-1}) \tilfh.$$ The paper is devoted to the characterization of the rational functions $r_{k,\alpha}(\t)$ and providing their extremal point, poles, partial fraction representation, etc.
##### *Semi-discrete and fully discrete approximations in the parabolic sub-diffusion case*
Further, in Sections \[ss:subdiffusion-react\] and \[ss:transient-subdiff\] we apply the same strategy to the sub-diffusion-reaction problem and time-stepping procedure for the transient sub-diffusion problem . Both results in solving the following type semi-discrete problem : $$u_h = (\calAt^\alpha + \q \calIt)^{-1} f_h.$$ In this case, we consider two possible fully discrete schemes. The first one is based on the BURA of the function $ t^\alpha/(1+q t^\alpha)$ on the interval $[\delta, 1]$. The second fully discrete scheme is based on a rational approximation of the same function, but NOT the best one, see Subsection \[ss:ura-res\] and Definition \[pdef0ura\] and call further URA-method.
Examples of problems involving fractional powers of elliptic operators {#s:ell-operators}
----------------------------------------------------------------------
### Spectral fractional powers of elliptic operators {#ss:problem}
In this paper we consider the following second order elliptic equation with homogeneous Dirichlet data: - ( a(x) v(x)) + (x) u&= f(x), x,\
v(x)&=0, x. \[strong\] Here $\Omega $ is a bounded domain in ${\mathbb{R}}^d$, $d\ge 1$. We assume that $0<a_0 \le a(x) $ and $\q(x) \ge 0$ for $x\in \Omega$.
The fractional powers of the elliptic operator associated with the problem are defined in terms of the weak form of , namely, $v(x)$ is the unique function in $V= H^1_0(\Omega)$ satisfying a(v,)= (f,)V. \[weak\] Here $$a(w,\theta):=\int_\Omega \left (a(x) \nabla w(x) \cdot \nabla \theta(x) + \q w \theta \right )\, dx \quad \hbox{ and } \quad
(w,\theta):=\int_\Omega w(x) \theta(x)\, dx.$$ For $f\in L^2(\Omega):= X$, defines a solution operator $T f :=v$. Following [@kato], we define an unbounded operator $\calA$ on $X$ as follows. The operator $\calA$ with domain $$D(\calA) = \{ Tf\,:\, f\in X\}$$ is defined by \[def-A\] v=g vD(), gX Tg=v. The operator $\calAt$ is well defined as $T$ is injective.
Thus, the focus of our work in this paper is numerical approximation and algorithm development for the equation: \[frac-eq\] \^u = f u =\^[-]{} f. Here $\calA^{-\alpha}=T^\alpha$ for $\alpha>0$ is defined by Dunford-Taylor integrals which can be transformed when $\alpha\in (0,1)$, to the Balakrishnan integral, e.g. [@balakrishnan]: for $f\in X$, u=\^[-]{} f = \_0\^\^[-]{} ([[I]{}]{}+)\^[-1]{}f d. \[bal\] This definition is sometimes referred to as the spectral definition of fractional powers. One can also use an equivalent definition through the expansion with respect to the eigenfunctions $\psi_j$ and the eigenvalues $\lambda_j$ of $\calA$, e.g. [@Karniadakis2018fractional; @Acosta-Borthagaray2017]: \[eq:spectral-0\] \^ u= \_[j=1]{}\^\_j\^ (u,\_j) \_j u = \_[j=1]{}\^\_j\^[-]{} (f,\_j) \_j Since the bilinear form $a(\cdot, \cdot)$ is symmetric on $ V \times V$ and $\calA$ is a unbounded operator we can show that $\lambda_j $ are real and positive and $\lim_{j \to \infty} \lambda_j =\infty$.
An operator $\calA$ is positivity preserving if $\calA f\ge 0$ when $f\ge 0$. We note that by the maximum principle, $(\mu {{\cal I}}+ \calA)^{-1}$ is a positivity preserving operator for $\mu\ge 0$ and the formula shows that $\calA^{-\alpha}$ is also. In many applications, it is important that the discrete approximations share this property.
### Sub-diffusion-reaction problems {#ss:subdiffusion-react}
Another possible model of sub-diffusion reaction is give by the operator equation: find $u \in V$ s.t. \[eq:sub-reaction\] \^u + u =f =const 0.
### Transient sub-diffusion-reaction problems {#ss:transient-subdiff}
For time dependent problems one can consider: find $ u(t) \in V$ for $t \in (0,t_{max}]$ such that \[eq:parabolic\] + \^u(t) =f(t) u(0)=v, with $v$ given initial data, $t_{max} >0$ is a given number, and the finite dimensional operator (matrix) $\calAt$ defined by .
Semi-discrete approximations of equations involving fractional powers of elliptic operators {#s:semi-discrete}
-------------------------------------------------------------------------------------------
We study approximations to $u= \calA^{-\alpha} f$ defined in terms of finite difference or finite element approximation of the operator $T$. We shall use the following convention regarding the approximate solutions by the finite element, [@ern-guermond], and finite difference, [@samarskii2001theory], methods.
The finite element solutions are functions in $V_h$, an $N$-dimensional space of continuous piece-wise linear functions over a partition ${\mathcal T}_h$ of the domain. Such functions will be denoted by $u_h$, $ v_h$, etc. Also we shall denote by $\calAt$, $\calIt$, etc operators acting on the elements $u_h, \theta_h$, etc in the finite dimensional space of functions $V_h$. When a nodal basis of the finite element space is introduced, then the vectors coefficients in this basis are denoted $\tiluh$, $\tilvh$, etc. Under this convention operator equations in $V_h$ such as $\calAt u_h=f_h$ will be written as a system of linear algebraic equations $ \wcalAt \tiluh=\tilfh$ in ${\mathbb{R}}^N$.
In the finite difference case, discrete solutions are vectors in ${\mathbb{R}}^N$ and are also denoted $\tiluh$, $\tilvh$, etc. Then the corresponding counterparts of operators action on these vectors are denoted by $\wcalAt, \wcalIt,$ etc.
### The finite difference approximation {#ss:FD}
In this case the approximation $\tiluh \in {\mathbb{R}}^N$ of $u$ is given by \^ = \_h f := , = \^[-]{} , \[fda\] where $\wcalAt$ is an $N\times N$ symmetric and positive definite matrix coming from a finite difference approximation to the differential operator appearing in , $\tiluh$ is the vector in ${\mathbb{R}}^N$ of the approximate solution at the interior $N$ grid points, and $\cI_h f:= \tilfh \in {\mathbb{R}}^N$ denotes the vector of the values of the data $f$ at the grid points.
##### *Example 1*
We first consider the one-dimensional equation with variable coefficient, namely, we study the following boundary value problem $ - ( a(x) u^{\prime})^{\prime} =f(x),$ $ u(0)=0, \ u(1)=0, \ $ for $ 0<x<1$, where $a(x)$ is uniformly positive function on $[0,1]$. On a uniform mesh $x_i =ih$, $i=0, \dots, N+1$, $ h=1/(N+1)$, we consider the three-point approximation of the second derivative $$\begin{split}
(a(x_i) u'(x_i))' &
\approx \frac{1}{h}\left (a_{i+\frac12}\frac{u(x_{i+1}) - u(x_i)}{h} - a_{i-\frac12}\frac{u(x_i) - u(x_{i-1})}{h} \right )
\end{split}$$ Here $a_{i-\frac12}=a(x_i - h/2)$ or $a_{i - \frac12}=\frac{1}{h}\int_{x_{i-1}}^{x_i} a(x) dx $. Note that the former is the standard finite difference approximation obtained from the balanced method (see, e.g. [@samarskii2001theory pp. 155–157]). Then the finite difference approximation of the differential equation $- (a(x) u'(x))' =f(x)$ is given by the matrix equation with $$\label{FD-matrix-1D-k}
\wcalAt= \frac{1}{h^2} \left[
\begin{array}{ccccc} a_{\frac12}+ a_{\frac32}& - a_{\frac32} &&&\\
- a_{\frac32} & a_{\frac32} + a_{\frac52} & - a_{\frac52}&&\\
\cdots &\cdots &\cdots &\cdots &\cdots\\
& - a_{i-\frac12}& a_{i-\frac12} + a_{i+\frac12} & a_{i+\frac12} &\\\cdots &\cdots &\cdots &\cdots &\cdots\\
&&& - a_{N-\frac12} & a_{N-\frac12} + a_{N +\frac12}
\end{array}\right], \ \
$$ The eigenvalues $\lambda_i$ of the matrix $\calAt $ are all real and positive and satisfy $$4 \pi^2 \min_x a(x) \le \lambda_i \le 4 \max_x a(x)/h^2, \ \ i=1, \dots, N.$$
### The finite element approximation {#ss:FEM}
The approximation in the finite element case is defined in terms of a conforming finite dimensional space $V_h\subset V$ of piece-wise linear functions over a quasi-uniform partition ${\mathcal T}_h$ of $\Omega$ into triangles or tetrahedrons. Note that the construction of negative fractional powers carries over to the finite dimensional case, replacing $V$ and $X$ by $V_h$ with $a(\cdot,\cdot)$ and $(\cdot,\cdot)$ unchanged.
The discrete operator $\calAt$ is defined to be the inverse of $T_h:V_h\rightarrow V_h$ with $T_h g_h:=v_h$ where $v_h\in V_h$ is the unique solution to a(v\_h,\_h) = (g\_h,\_h),\_hV\_h. \[T\_h\] The finite element approximation $ u_h \in V_h$ of $u$ is then given by \^ u\_h = \_h f, u\_h = \^[-]{} \_h f:=\^[-]{}f\_h, \[fea\] where $\pi_h$ denotes the $L^2(\Omega) $ projection into $V_h$. In this case, $N$ denotes the dimension of the space $V_h$ and equals the number of (interior) degrees of freedom. The operator $\calAt$ in the finite element case is a map of $V_h$ into $V_h$ so that $\calAt v_h:=g_h$, where $g_h\in V_h$ is the unique solution to (g\_h,\_h)=a(v\_h,\_h),\_hV\_h. \[calAt-f\] Let $\{\phi_j\}$ denote the standard “nodal" basis of $V_h$. In terms of this basis \[FEM-matrices\] = \^[-1]{} , \_[i,j]{} =a(\_i, \_j), \_[i,j]{} =(\_i, \_j). In the terminology of the finite element method, $\wcalMt$ and $ \wcalSt$ are the mass (consistent mass) and stiffness matrices, respectively. Obviously, if $\theta =\calAt \eta$ and $\widetilde \theta,\widetilde \eta \in {\mathbb{R}}^N$ are the coefficient vectors corresponding to $\theta,\eta\in V_h$, then $\widetilde \theta = \wcalAt
\widetilde \eta$. Now, for the coefficient vector $\tilfh$ corresponding to $f_h=\pi_hf$ we have $\tilfh = \wcalMt^{-1} \tilF$, where $\tilF$ is the vector with entries $${\tilF}_j=(f,\phi_j), \qquad \hbox{ for }j=1,2,\ldots,N.$$ Then using vector notation so that $\tiluh$ is the coefficient vector representing the solution $u_h$ through the nodal basis, we can write the finite element approximation of in the form of an algebraic system \[classic-FEM\] = \^[-1]{} = . Consequently, the finite element approximation of the sub-diffusion problem becomes \[mat-FEM\] \^= = \^[-]{} \^[-1]{} .
### The lumped mass finite element approximation {#ss:lumped-mass}
We shall also introduce the finite element method with “mass lumping" for two reasons. First, it leads to positivity preserving fully discrete methods. Second, it is well known that on uniform meshes lumped mass schemes for linear elements are equivalent to the simplest finite difference approximations. This will be used to study the convergence of the finite difference method for solving the problem , an outstanding issue in this area. We introduce the lumped mass (discrete) inner product $ (\cdot,\cdot)_h$ on $V_h$ in the following way (see, e.g. [@Thomee2006 pp. 239–242]) for $d$-simplexes in ${\mathbb{R}}^d$: \[mass-lumping\] (z,v)\_h = \_[\_h ]{} \_[i=1]{}\^[d+1]{} || z(P\_i) v(P\_i) \_h ={ (\_i,\_k)\_h}\_[i,k]{}\^N. Here $P_1, \dots, P_{d+1}$ are the vertexes of the $d$-simplex $\tau$ and $|\tau|$ is its $d$-dimensional measure. The matrix $\wcalMt_h$ is called lumped mass matrix. Simply, the “lumped mass" inner product is defined by replacing the integrals determining the finite element mass matrix by local quadrature approximation, specifically, the quadrature defined by summing values at the vertices of a triangle weighted by the area of the triangle. In this case, we define $\calAt$ by $\calAt v_h:=g_h$ where $g_h\in V_h$ is the unique solution to (g\_h,\_h)\_h=a(v\_h,\_h), \_hV\_h \[calAt-fm\] so that \[A-lumped\] = \_h\^[-1]{} , \_h ={ (\_i,\_k)\_h}\_[i,k]{}\^N. Here $ {\wcalMt}_h$ is the lumped mass matrix which is diagonal with positive entries. We also replace $\pi_h$ by ${{\cal I}}_h$ so that the lumped mass semi-discrete approximation is given by u\_h = \^[-]{} [[I]{}]{}\_h f := \^[-]{} f\_h u\_h = \^[-]{} F. \[lumped-semi\] Here $\widetilde F$ is the coefficient vector in the representation of the function ${{\cal I}}_hf$ with respect to the nodal basis in $V_h$. We shall call $\tiluh$ in and $ u_h$ in and of $u$.
### Discretization of sub-diffusion-reaction problem {#ss:sbreac-dscrete}
If we use , a finite difference approximation $\wcalAt$ of the operator $\calA$, then the corresponding discrete problem is: find $\tiluh \in {\mathbb{R}}^N$ such that $$\left ( \wcalAt^\alpha + \q \wcalIt \right ) \tiluh = \tilfh.$$
In a similar way one can introduce the corresponding finite element discretizations of the problem : \[FEM-sub-reaction\] ( \^+ ) u\_h = f\_h, where $\calAt$ is defined by and $\calIt: V_h \to V_h$ is the identity operator in $V_h$. Using consistent mass matrix evaluation the operator $ \calIt$ has matrix representation $\wcalMt$ defined by , while using lumped mass evaluation, the operator $ \calIt$ is represented by the lumped mass matrix $\wcalMt_h$ defined by .
By using these matrix representations this equation can be written as a system of linear algebraic equations in ${\mathbb{R}}^N$. Taking into account the matrix representation of the operator $\calAt$ we get the following systems, corresponding to the consistent mass and lumped mass finite elements approximations of the $L^2$-inner product in $V_h$: $$\wcalMt \left ( \wcalAt^\alpha + \q \wcalIt \right ) \tiluh = \tilF \quad \mbox{or} \quad
\wcalMt_h \left ( \wcalAt_h^\alpha + \q \wcalIt \right ) \tiluh = \tilfh.$$
### Time-dependent problems {#ss:parabolic}
Similarly, the discretization of the time-dependent problem with implicit Euler approximation in time, for $t_n=n \tau$, $n=1,2, \dots, M$, $\tau = t_{max}/M$ and $u_h^n \in V_h$ an approximation of $u(t_n)$, will lead to the operator equation \[eq:time-stepping\] ( + \^) u\_h\^[n]{} = u\_h\^[n-1]{} + f\_h\^n, n=1, …, M, where $ f_h^n$ is the $L^2$-projection of $f(t_n)$ on $V_h$. Denoting by $v_h^n = \frac{1}{\tau} u_h^{n-1} + f_h^n$, we have the following representation of the solution $ u_h^{n} $: \[eq:sol-time-n\] u\_h\^[n]{} = ( + \^)\^[-1]{} v\_h\^n.
Having in mind the matrix representations (for the consistent mass FEM), (for the lumped-mass FEM), and of the operator $\calAt: V_h \to V_h$ we get the following systems of algebraic equations: $$\left (\frac{1}{\tau} \wcalMt + \wcalSt^\alpha \right ) \widetilde u_h^{n} = \frac{1}{\tau} \wcalMt \widetilde u_h^{n-1} + \widetilde F^n
\quad
\mbox{and }
\quad
\left (\frac{1}{\tau} \wcalMt_h + \wcalSt^\alpha \right ) \widetilde u_h^{n} = \frac{1}{\tau} \wcalMt_h \widetilde u_h^{n-1} + \widetilde F^n,$$ for the standard and lumped mass FEM, correspondingly.
Brief review of methods for solving equations involving fractional powers of elliptic operators {#sec:other}
-----------------------------------------------------------------------------------------------
We note that computing the solutions of , , involves inverting fractional powers of elliptic operators or their shifts. This is a computationally intensive task and the aim of this paper is to provide a methodology that results in fast and efficient methods. Further in the paper these are called fully discrete schemes. Due to the serious interest of the computational mathematics and physics communities in modeling and simulations involving fractional powers of elliptic operators, a number of approaches and algorithms has been developed, studied and tested on various problems, [@aceto2018efficient; @nochetto2015pde; @nochetto2016pde; @HOFREITHER2019]. We survey some of these approaches by splitting them into four basic categories. These are methods based on:
1. An extension of the problem from $ \Omega \subset {\mathbb{R}}^d$ to a problem in $\Omega \times (0,\infty) \subset {\mathbb{R}}^{d+1}$, see, e.g. [@caffarelli2007extension]. Nochetto and co-authors in [@nochetto2015pde; @nochetto2016pde] developed efficient computational method based on finite element discretization of the extended problem and subsequent use of multi-grid technique. The main deficiency of the method is that instead of problem in ${\mathbb{R}}^d$ one needs to work in a domain in one dimension higher which adds to the complexity of the developed algorithms.
2. Reformulation of the problem as a pseudo-parabolic on the cylinder $(0,1) \times \Omega$ by adding a time variable $t \in (0,1)$. Such methods were proposed, developed, and tested by Vabishchevich in [@vabishchevich2015numerical; @vabishchevich2018numerical]. As shown in the numerical experiments in [@HOFREITHER2019], while using uniform time stepping, this method is very slow. However, the improvement made in [@DuanLazarovPasciak; @CIEGIS2019] makes the method quite competitive.
3. Approximation of the Dunford-Taylor integral representation of the solution of equations involving fractional powers of elliptic operators, proposed in the pioneering paper of Bonito and Pasciak [@BP15]. Further the idea was extended and augmented in various directions in [@BP17; @bonito2019sinc; @bonito2019numerical]. These methods use exponentially convergent sinc quadratures and are the most reliable and accurate in the existing literature.
4. Best uniform rational approximation of the function $t^\alpha$ on $[0,1]$, proposed in [@HMMV2016; @HLMMV18], and further developed in [@harizanov2017positive; @harizanov2019cmwa; @harizanov2019analysis] and called BURA methods.
As shown recently in [@HOFREITHER2019], these methods, though entirely different, are interrelated and all seem to involve certain rational approximation of the fractional powers of the underlying elliptic operator. As such, from mathematical point of view, those based on the best uniform rational approximation should be the best. However, one should realize that BURA methods involve application of the Remez method of finding the best uniform rational approximation, [@PGMASA1987; @Driscoll2014], a numerical algorithm for solving certain min-max problem that is highly nonlinear and sensitive to the precision of the computer arithmetic. For example, in [@varga1992some] the errors of the best uniform rational approximation of $t^\alpha$ for six values of $\alpha \in [0,1]$ are reported for degree $k\le 30$ by using computer arithmetic with $200$ significant digits.
Fully discrete schemes {#se:fully-discrete}
----------------------
### Explicit representation of the solution of $ u_h = \calAt^{-\alpha} f_h $ {#ss:explicit}
Now consider the spectral properties of the operator $\calAt$: $$\calAt \psi_j =\lambda_j \psi_j \ \ \mbox{or in matrix form} \ \ \wcalAt \psi_j = \lambda_j \psi_j, \ \ j=1, \dots, N.$$ Note that if $ \wcalAt $ is defined from the finite difference approximation, then this results in a standard matrix eigenvalue problem. In case of finite element approximation or this results in corresponding generalized eigenvalue problems \[eq:spectral\] \_j = \_j \_j \_j = \_j \_h \_j, j=1, …, N. Using the eigenvalues and the eigenfunctions we have explicit representation of the solution of the operator equation: \[eq:sol-spectral\] u\_h = \_[j=1]{}\^N \_j\^[-]{} (f\_h,\_j) \_j. This representation can be used as a direct method for solving the equation $ u_h = \calAt^{-\alpha} f_h $. Moreover, using FFT-like technique, in the cases when possible (rectangular domain and constant coefficients), this could be an efficient numerical method. However, this will limit substantially the applicability of such approach.
Similarly, the solution of the problem resulting in time-stepping method can be expressed through the eigenvalues and the eigenfunctions we have \[eq:sol-spectral-t\] u\_h\^n = \_[j=1]{}\^N (\_j\^ + )\^[-1]{} (v\_h\^n,\_j) \_j = \_[j=1]{}\^N \_j\^[-]{} (1 + \_j\^[-]{} )\^[-1]{} (v\_h\^n,\_j) \_j .
### The idea of the fully discrete schemes {#ss:fully-discrete-i}
To explain our approach we consider $P_k(t)$, the polynomial of degree $k$ that approximates $t^{\alpha}$ on the interval $[1/\lambda_N, 1/\lambda_1] $ in the maximum norm. Then the vector $w_h$ defined by $$w_h= \sum_{j=1}^N P_k(\lambda_j^{-1}) (f_h,\psi_j) \psi_j \ \ \mbox{is an approximation of}
\ \ u_h = \sum_{j=1}^N \lambda_j^{-\alpha} (f_h,\psi_j) \psi_j.$$ Moreover, we can express the error of this approximation through the approximation properties of $P_k(t)$ [@Lund17]: $$\| u_h - w_h \| \le \max_{t = \lambda_1, \dots, \lambda_N} |t^{-\alpha} - P_k(t^{-1}) | \| f_h\| =
\max_{t = 1/\lambda_N, \dots, 1/\lambda_1} |t^{\alpha} - P_k(t) | \| f_h\|.$$ Once we have the polynomial $P_k(t)$ then we can find its roots $\xi_i$, $i=1, \dots, k$, so that we have $$P_k(t^{-1})=c_0\prod_{i=1}^k (t^{-1} - \xi_i^{-1})$$ and consequently $$w_h= c_0 \prod_{i=1}^k (\calAt^{-1} - \xi_i^{-1} \calIt) f_h := c_0 \prod_{i=1}^k w_h^{(i)}.$$ Thus, to find $w_h$ we need to find $ w_h^{(i)} = (\calAt^{-1} - \xi_i^{-1} \calIt) f_h $, $i=1, \dots, k$ which results in solving $k$ systems $\calAt w_h^{(i)}=( \calIt - \xi_i^{-1} \calAt) f_h$. This idea will produce a computable approximate solution, but will not lead to an efficient method since the required polynomial degree $k$ will depend on the spectral condition number $\kappa (\calAt) = \lambda_N/\lambda_1$. Namely, the best uniform polynomial approximation of $t^\alpha$, $t\in [1/\lambda_N,1/\lambda_1]$, is given by the scaled and shifted Chebyshev polynomial $\widetilde{P}_k(t)$. Then, the error estimate $$\max_{t = \lambda_1, \dots, \lambda_N} |t^{-\alpha} - \widetilde{P}_k(t) | <
2\left (
\frac{\sqrt{\kappa^\alpha} +1} {\sqrt{\kappa^\alpha}-1}
\right )^{{k}}$$ holds true, and therefore a polynomial of degree $
k \approx 2 \kappa^{\frac{\alpha}{2}} \log\frac{2}{\epsilon}
$ will be needed to guarantee the the relative error less than $\epsilon$ of $\| u_h - w_h \|$. Note, that the matrix $ \calAt$ defined by has a condition number $\kappa (\calAt) = O\left (\max a(x)/\min a(x) h^{-2}\right )$ and the degree of the polynomial of best uniform approximation will grow as $1/h^{\alpha/2}$ as $h \to 0$.
Our aim is to produce a method that involves much smaller $k$, so we need to solve fewer systems of the type $( \calAt - \xi_i \calIt) v= f$ with $\xi_i \le 0$.
### Fundamental properties of BURA of $\t^\alpha$ on $(0,1]$ {#ss:BURA}
Instead of polynomial approximation, we shall seek a rational approximation of the function $t^{-\alpha}$. In order to make the computations uniform and to use the known results for the approximation theory, we first rewrite the solution of the in the form \[eq:sol-function\] u\_h = \_1\^[-]{} (\_1 \^[-1]{})\^f\_h. The scaling by $\lambda_1$ maps the eigenvalues of $ \lambda_1 \calAt^{-1}$ to the interval $(\lambda_1/\lambda_N,1] :=(\delta, 1] \subset (0,1]$. Here $\delta=\lambda_1/\lambda_N$ is a small parameter. Often below we shall take even $\delta =0$.
Similarly, the solution of the time dependent problem after the scaling by $\lambda_1$ maps the eigenvalues of $\lambda_1 \calAt^{-1}$ to the interval $(\lambda_1/\lambda_N, 1] :=[\delta,1] \subset (0,1]$. Then $$u_h^n = \sum_{j=1}^N \lambda_j^{-\alpha}(1 + \q \lambda_j^{-\alpha})^{-1} (v_h^n,\psi_j) \psi_j
= \Big (\q \lambda_1^{-\alpha} (\lambda_1^{-1} \calAt)^{-\alpha} + \calIt \Big )^{-1} (\lambda_1^{-1} \calAt)^{-\alpha} \lambda_1^{-\alpha} v_h^n.$$ Introducing the parameters $q=\q \lambda_1^{-1}$ and defining the function $
\f(\t):= \f(q,\delta, \alpha;\t ) =\frac{\t^{\alpha}}{ 1 + q \t^{\alpha} } \ \ \mbox{for} \ \ t \in (\lambda_1/\lambda_N, 1]
$ we get \[eq:sol-function-q\] u\_h\^n = (\_1 \^[-1]{}) \_1\^[-]{} v\_h\^n = \_[j=1]{}\^N (\_j) \_1\^[-]{} (v\_h\^n,\_j) \_j .
Now we consider BURA along the diagonal of the Walsh table and take $\rat k$ to be the set of rational functions $$\mathcal R_k= \bigl\{ r(\t): r(\t)=P_k(\t)/Q_k(\t),
\ P_k \in {\mathcal P}_k, \mbox{ and } \ Q_k \in {\mathcal P}_k, \ \mbox{monic} \bigr\}$$ with ${\mathcal P}_k$ set of algebraic polynomials of degree $k$. The best [*discrete*]{} uniform rational approximation (discrete BURA) of $t^\alpha$ is the rational function $R_{\alpha,k}\in \rat k$ satisfying $$\label{bura-discrete}
R_{\alpha,k}(t) := \argmin_{s(t)\in \rat k}\,
\max_{t \in \{\frac{\lambda_1}{\lambda_N}, \frac{\lambda_2}{\lambda_N}, \dots, 1\}} | s(t) - t^{\alpha} |.$$
Unfortunately, such approximation depends of the knowledge of the eigenvalues, something we would like to avoid. Now we shall show how to avoid in our computation such dependence for both solutions defined by and .
To find a computable approximation to we introduce the following best uniform rational approximation (BURA) $r_{\delta,\alpha,k}(t) $ of $t^\alpha$ on $[\delta,1]$ $$r_{\delta,\alpha,k}(t) := \argmin_{s(t)\in \rat k}\, \sup_{t \in [\delta,1]} | s(t) - t^{\alpha} |.$$
Obviously we have $$\max_{t \in \{\frac{\lambda_1}{\lambda_N}, \frac{\lambda_2}{\lambda_N}, \dots, 1\}} | s(t) - t^{\alpha} |
\le
\sup_{t \in [\delta,1]} | s(t) - t^{\alpha} |.$$
Often, for practical considerations, we would like to get rid of the parameter $\delta=\lambda_1/\lambda_N$ by using the best uniform rational approximation $ r_{\alpha,k}(t)$ of $t^\alpha$ on the whole interval $[0,1]$, namely $$\label{bura}
r_{\alpha,k}(t) := \argmin_{s(t)\in \rat k}\,
\max_{t \in [0,1]} | s(t) - t^{\alpha} | = \argmin_{s(t)\in \rat k}\, \| s(t) - t^{\alpha} \|_{L^\infty(0,1)}.$$
The problem has been studied extensively in the past, e.g. [@saff1992asymptotic; @Stahl93; @varga1992some]. Denoting the error by \[BURA-error\] E\_[,k]{}:=r\_[,k]{} (t) - t\^ \_[L\^]{}, and applying Theorem 1 of [@Stahl93] we conclude that there is a constant $C_\alpha>0$, independent of $k$, such that $$\label{error-bound}
E_{\alpha,k} \le C_\alpha e^{-2 \pi \sqrt{k \alpha}}.$$ Thus, the BURA error converges exponentially to zero as $k$ becomes large.
It is known that the best rational approximation $r_{\alpha,k}(t)=P_k(t)/Q_k(t)$ of $t^\alpha$ for $\alpha \in (0,1)$ is non-degenerate, i.e., the polynomials $P_k(t)$ and $Q_k(t)$ are of full degree $k$. Denote the roots of $P_k(t)$ and $Q_k(t)$ by $\zeta_1, \dots, \zeta_k$ and $d_1, \dots, d_k$, respectively. It is shown in [@saff1992asymptotic; @stahl2003] that the roots are real, interlace and satisfy $$\label{interlacing}
0 > \zeta_1 > d_1 > \zeta_2 > d_2 > \cdots > \zeta_k > d_k.$$ We then have r\_[,k]{} (t)=b\_[i=1]{}\^k\[prodr\] where, by and the fact that $r_{\alpha,k}(t)$ is a best approximation to a non-negative function, $b>0$ and $P_k(t)>0$ and $Q_k(t)>0$ for $t\ge 0$.
Knowing the poles $d_i$, $i=1, \dots, k$ we can give an equivalent representation of as a sum of partial fractions, namely r\_[,k]{}()=c\_0+\_[i=1]{}\^k \[trg\] where $c_0>0$ and $c_i<0$ for $i=1,\ldots,k$.
Now changing the variable $\tt =1/\t$ in $r_{\alpha,k}(t)$ we get a rational function $\trg(\tt)$ defined by \[eq:error-tilde\] ():=r\_[,k]{}(1/)=. Here $\widetilde P_k(\tt)=\t^k P_k(\t^{-1})$ and $\widetilde Q_k(\tt)= \t^k Q_k(\t^{-1})$ and hence their coefficients are defined by reversing the order of the coefficients in $P_k$ and $Q_k$ appearing in $r_{\alpha,k}(t)$. In addition, implies that we have the following properties for the roots of $\widetilde P_k$ and $\widetilde Q_k$, $\widetilde d_i=1/d_i$ and $\widetilde \zeta_i = 1/\zeta_i$, respectively. $$\label{interlacing1}
0 > \widetilde d_k > \widetilde \zeta_k > \widetilde d_{k-1} > \widetilde
\zeta_{k-1}\cdots > \widetilde d_1>\widetilde \zeta_1.$$
\[lemma:BURA\] For $\alpha\in (0,1)$, \[r-compute\] ()=c\_0+\_[i=1]{}\^k [c\_i]{}/[(-d\_i)]{} where $$\widetilde c_0 = c_0 - \sum_{i=1}^k c_i \widetilde d_i=r_{\alpha,k}(0)=E_{\alpha,k}>0 \ \ \mbox{with} \
\widetilde c_i = - c_i { d}_i^{-2} = - c_i {\widetilde d}_i^{2} >0, \ i=1,\ldots,k.$$
Indeed, $$\widetilde r_{\alpha,k}(\tt) = r_{\alpha,k}(1/\tt) = c_0+\sum_{i=1}^k \frac{\ c_i}{1/\tt- d_i}
= c_0 -\sum_{i=1}^k c_i d_i^{-1} - \sum_{i=1}^k \frac{c_i d_i^{-2}}{\tt - d_i^{-1}}$$ and having in mind that $\widetilde d_i=1/d_i$, we get .
### Fully discrete schemes based on BURA {#ss:fully-discrete}
Now we introduce the [**]{} approximations: $ w_h \in V_h$ of the finite element approximation $u_h \in V_h$, defined by , and $\tilwh \in {\mathbb{R}}^N$ of the finite difference approximation $\tiluh \in {\mathbb{R}}^N$ by w\_h =\_1\^[-]{} r\_[,k]{} (\_1 \^[-1]{}) f\_h =\_1\^[-]{} r\_[,k]{} (\_1 \^[-1]{}) . \[wh\] Here $\calAt$ and $f_h$ are as in or and $\wcalAt$ and $\tilfh$ are as in .
In the paper [@harizanov2019analysis], we studied the error of these fully discrete solutions. For the finite element case we obtain the error estimate u\_h - w\_h \_1\^[-]{} E\_[,k]{} f\_h \[pbest\] with $\|\cdot\|$ denoting the norm in $L^2(\Omega)$. In the finite difference case, we have - \_[\_2]{} \_1\^[-]{} E\_[,k]{} \_[\_2]{}, \[pbest-fd\] where the norm $\| \cdot \|_{\ell_2} $ denotes the Euclidean norm in ${\mathbb{R}}^N$.
### BURA approximation of ${\t^{\alpha}\over{1+q\,\t^{\alpha}}} \mbox{ on } [\delta,1] $ {#ss:bura-ura-res}
Now we consider the solution and introduce the function $\f(q,\delta,\alpha;\t)$, of the variable $\t$ on $[\delta, 1]$, $0 \le \delta < 1$ and two parameters $q \in [0, \infty)$ and $0 < \alpha <1$: $$\f_{q,\delta,\alpha}(\t):=\f(q,\delta,\alpha;\t)={\t^{\alpha}\over{1+q\,\t^{\alpha}}} \ \mbox{ on } \ \t \in [\delta,1].$$ Note that for $q=1/\tau$ we get the corresponding problem from time-discretization of sub-diffusion equation . The role of this function is clear from the representation of the solution by . As before, our goal is to approximate this function using the [*best uniform rational approximation*]{} (BURA). To find BURA of $\f(q,\delta,\alpha;\t)$ we employ Remez algorithm, cf. [@PGMASA1987].
\[pdef0bura\] The best uniform rational approximation ${r}_{q,\delta,\alpha,k}(\t) \in \mathcal R_k$ of $ \f(q,\delta,\alpha; \t)$ on $[\delta,1]$, called further $(q,\delta,\alpha,k)$-BURA, is the rational function \[BURA\] [r]{}\_[q,,,k]{}():= \_[ s R\_k]{} (q,,;)- s() \_[L\^]{}. Then the error-function is denoted by \[BURA-er-eps\] (q,,,k;) = [r]{}\_[q,,,k]{}()- (q,,;),\
and its $L^\infty$-norm is denoted by \[BURA-error-def\]
[E]{}\_[q,,,k]{} & = \_ | (q,,;)- [r]{}\_[q,,,k]{}() | = (q,,,k;) \_[L\^]{}.
We observe that the zeros and poles of ${r}_{q,\delta,\alpha,k}$ are again real, nonnegative, and interlacing for all considered choices of the four parameters $q,\delta,\alpha,k$. Furthermore, there seem to be $2k+2$ (the theoretically maximal possible number) points where ${\varepsilon}(q,\delta,\alpha,k;\t)$ achieves its extremal value $\pm {E}_{q,\delta,\alpha,k}$. However, we are not aware of a theoretical proof for any of the above observations in the general setting $q>0$ and/or $\delta>0$, which is not covered by Section \[ss:BURA\]. Obviously, for $\tt =1/ \t$, $0 < \t <1$, \[eq:tilde\] (q,,,k;):= (q,,,k;), we have $${E}_{q,\delta,\alpha,k}
= \left \| {\varepsilon}(q,\delta,\alpha,k;\t) \right \|_{L^\infty(\delta,1)}
= \left \| {\widetilde \varepsilon}(q,\delta,\alpha,k;\tt) \right \|_{L^\infty(1, 1/\delta)}.
$$
### URA approximation of $\f(q,\delta,\alpha;\t)={\t^{\alpha}\over{1+q\,\t^{\alpha}}} \mbox{ on } \t \in [\delta,1] $ {#ss:ura-res}
Now we shall introduce a rational approximations of function $\f(\t)$ that has simpler appearance and that is based on the BURA of $t^{\alpha}= \f(0,\delta,\alpha;\t)$.
\[pdef0ura\] The function \[0-URA\] |[r]{}\_[q,,,k]{}() := [[r]{}\_[0,,,k]{}()]{} R\_k is an approximation of $ \f(q,\delta,\alpha; \t)$ on $[\delta,1]$. Then the error-function is defined as $$\bar{\varepsilon}(q,\delta,\alpha,k;\t) = \f(q,\delta,\alpha;\t)- \bar{r}_{q,\delta,\alpha,k}( \t)$$ and $$\begin{aligned}
\bar{E}_{q,\delta,\alpha,k} & = \left \| \f(q,\delta,\alpha;\t)- \bar{r}_{q,\delta,\alpha,k}(\t) \right \|_{L^\infty[\delta,1)}
= \max_{\t\in [\delta,1]} \left| \bar{\varepsilon}(q,\delta,\alpha,k;\t) \right| .
\end{aligned}$$ The rational function $ \bar{r}_{q,\delta,\alpha,k}(\t)$ will be called $(q,\delta,\alpha,k)$-0-URA approximation of $ \f(q,\delta,\alpha;\t)$.
Further, we present this rational function as a sum of partial fractions \_[q,,,k]{}()=|c\_0+\_[i=1]{}\^k \[trg-bar\] where $\bar c_i>0$ and $\bar d_i$ are the poles, $i=0,1,\ldots,k$.
As shown in [@harizanov2019cmwa Theorem 2.4] the approximation error $ {E}_{q,\delta,\alpha,k} $ and $ \bar{E}_{q,\delta,\alpha,k}$ are related by $$\bar{E}_{q,\delta,\alpha,k}/( 1+q)^2 < {E}_{q,\delta,\alpha,k} < \bar{E}_{q,\delta,\alpha,k}.$$ The importance of this approximation is that by using corresponding BURA of $t^\alpha$ we reduce the number of the parameters involved.
\[prem0ura\] Now consider $q > q_0 >0$ and take $\bar{r}_{q_0,\delta,\alpha,k}(\t)$ as $(q_0,\delta,\alpha,k;\t)$-0-URA approximation of $\f(q_0,\delta,\alpha;\t)$. Then $$\bar{r}_{q,\delta,\alpha,k}(\t) ={\bar{r}_{q_0,\delta,\alpha,k}(\t)\over {1+(q - q_0)\bar{r}_{q_0,\delta,\alpha,k}(\t)}}
={{r}_{0,\delta,\alpha,k}(\t)\over {1+q \, {r}_{0,\delta,\alpha,k}(\t)}}\,.$$
\[pdef1ura\] Let $q =q_0 + q_1$, $q_0, q_1 >0$, and ${r}_{q_0,\delta,\alpha,k}(\t)$ be $(q_0,\delta,\alpha,k)$-BURA of $\f(q_0,\delta,\alpha;\t)$. A rational function $\bar{\bar{r}}_{q,\delta,\alpha,k}(\t) \in \mathcal R_k$ is an uniform approximation of $ \f(q,\delta,\alpha;\t)$ on $[\delta,1]$, called $(q,\delta,\alpha,k)$-1-URA approximation, and its error $\bar{\bar{E}}_{q,\delta,\alpha,k}$, are defined as: \[1-URA\] |[|[r]{}]{}\_[q,,,k]{}() := [[r]{}\_[q\_0,,,k]{}()]{} and $$\begin{aligned}
\bar{\bar{E}}_{q,\delta,\alpha,k} = \left \| \f(q,\delta,\alpha;\t) - \bar{\bar{r}}_{q,\delta,\alpha,k}(\t) \right \|_{L^\infty[\delta,1)}
= \sup_{\t\in [\delta,1]} \left| \bar{\bar{\varepsilon}}(q,\delta,\alpha,k;\t) \right| .
\end{aligned}$$
Finally, we present this rational function as a sum of partial fractions \[frac-1-URA\] |[|[r]{}]{}\_[q,,,k]{} ()=|[|c]{}\_0+\_[i=1]{}\^k
We remark that the $(q,\delta,\alpha,k)$-1-URA approximation gives a possibility to use a previously computed $(q_0,\delta,\alpha,k)$-BURA of $\f(q_0,\delta,\alpha;\t)$ for a fixed $q_0$ in order to find an acceptable approximation for $q=q_0 + q_1$ with $q_0, q_1 >0$.
Tables {#chapt:tabl}
=======
Description of the data provided by our numerical experiments {#ss:num-ex}
-------------------------------------------------------------
We provide all data for the uniform rational approximation of the function $\f(q,\delta,\alpha;\t)$ for various values of the parameters $q,\delta,\alpha,k$. The Tables and the corresponding files are named according the following encoding:
- $ q \in \{ 0,\,1,\,100,\,200,\,400\}$, coded by $qQQQ,\ QQQ\in \{000,001,100,200,400\}$, total 5 parameters;
- $\delta \in \{ 0.0,\, 10^{-6},\, 10^{-7},\,10^{-8}\}$, notation $dD,\ D\in\{0,6,7,8\}$ – 4 parameters;
- $\alpha \in \{ 0.250,\, 0.500,\, 0.750\}$, notation $aAA,\ AA\in\{25,50,75\}$ – 3 parameters;
- $ k \in \{ 3,\,4,\,5,\,6,\,7,\,8 \}$, notation $kK,\ k\in\{3,\ldots,8\}$ – 6 parameters;
- $q_0$ and $q_1$ go by pairs $(q_0,q_1)= (0,200)$, $(q_0,q_1)=(100,100)$, $(q_0,q_1)=(0,400)$, $(q_0,q_1)=(200,200) $, total of 4 cases, coded as $\{qq02,\, qq11,\, qq04,\, qq22\}$.
The computational data is presented in a number of Tables that contain:
1. the errors of $(q,\delta,\alpha,k)$-BURA;
2. the extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ defined by ;
3. the poles of ${r}_{q,\delta,\alpha,k}(\t)$ defined by ;
4. the coefficients of decomposition $\bigl\{{c}_j\bigr\}_{j=0}^{k}$ of ${r}_{q,\delta,\alpha,k}(\t)$ as a sum of partial fractions.
5. the poles of $\bar{r}_{q,\delta,\alpha,k}(\t)$ defined by ;
6. the coefficients of the decomposition $\bigl\{\bar{c}_j\bigr\}_{j=0}^{k}$ of $\bar{r}_{q,\delta,\alpha,k}(\t)$ as a sum of partial fractions.
7. the poles of $\bar{\bar{r}}_{q,\delta,\alpha,k}(\t)$ defined by ;
8. the coefficients of the decomposition $\bigl\{\bar{\bar{c}}_j\bigr\}_{j=0}^{k}$ of $\bar{\bar{r}}_{q,\delta,\alpha,k}(\t)$ as a sum of partial fractions.
Short description of the type-tables:
(a-b)
: These tables correspond to characterization of the BURA.
(c-d)
: These tables correspond to decomposition of BURA as a sum of partial fractions.
(e-f)
: These tables correspond to decomposition of 0-URA as a sum of partial fractions.
(g-h)
: These tables correspond to decomposition of 1-URA as a sum of partial fractions.
----------------- ----------- ---------- -------- ----------- ----------
Table & File Number of Rows Cols Folder Number
qQQQdDaAAkK.txt Tables ( q,k ) (d, a) name of Files
0-head-tabl.txt ( 1 ) ( 5\*6 ) (1+12) BURA-tabl 360
qQQQdDaAAkK.txt (q,k) ( ) (d, a)
BqQQQkK.txt (5\*6) (2\*K+2) (2+12) BURA-tabl 360
CqQQQkK.txt (5\*6) ( K ) (2+12) BURA-dcmp 360
DqQQQkK.txt (5\*6) ( K+1 ) (2+12) BURA-dcmp 360
EqQQQkK.txt (4\*6) ( K ) (2+12) 0URA-dcmp 288
FqQQQkK.txt (4\*6) ( K+1 ) (2+12) 0URA-dcmp 288
GqqQQkK.txt (2\*6) ( K ) (2+12) 1URA-dcmp 144
HqqQQkK.txt (2\*6) ( K+1 ) (2+12) 1URA-dcmp 144
----------------- ----------- ---------- -------- ----------- ----------
: Table of Tables and Data-files[]{data-label="p1ltabl"}
Total number of Tables is $(1+90+48+24)=163$.
More descriptions for the tables and files:
(a-b)
: The data for these tables in files of types (a),(b) correspond to characterization of the BURA. Folder with values of BURA and extreme points has $360$ files, and it is named `BURA-tabl`. Only for 5 cases the program did not finished with a solution.
(c-d)
: The files with normalized coefficients $A$ and $B$, poles $d_j$ (named $U0(j)$) and coefficients $c_j$ (named $E(j)$) are in the folder named `BURA-dcmp`. One sub-folder more was present (`BURA-dcmp/add/`) with more details about poles and coefficients – its Imaginary parts. Number of files is also $360$ and have names `qQQQdDaAAkK`.
(e-f)
: The files have the same names as in the previous item **(c-d)** and the same structure but folder is other. The folder is named `0URA-dcmp/`. One sub-folder more is given (`0URA-dcmp/add/`) with more details about poles and coefficients – its Imaginary parts. Number of files is $288$, because the cases $(0,\delta, \lambda)$-URA coincide with $(0,\delta, \lambda)$-BURA and files with $QQQ=000$ are not present.
(g-h)
: The folder with files is `1URA-dcmp/` and sub-folder `1URA-dcmp/add/`. The names of files are `qqQQdDaAAkK` and as the cases `qqQQ = qq02, qq04` are not present because coincide with the cases `qQQQdDaAAkK`, `qQQQ = q200, q400` from `0URA-dcmp`, and number of files is $144$ – for `qqQQ = qq11, qq22` only.
Tables type **(a-b)** for BURA-errors and BURA-extreme points
-------------------------------------------------------------
It is clear from Table \[tabl:A0BURAp\] that for fixed parameters $\alpha$, $q$, and $k$, the error is increasing, when $\delta \to 0$. However, the differences are not that pronounced, so for practical purposes one can use for all $\delta$ the approximations for $\delta=0$. The significance of using $\delta >0$ is in the performance of Remez algorithm for computing BURA.
One should realize that BURA-based methods involve Remez method of finding the best uniform rational approximation by solving the highly non-linear min-max problem . It is well known that Remez algorithm is very sensitive to the precision of the computer arithmetic, cf. [@PGMASA1987; @varga1992some; @Driscoll2014]. Various techniques for stabilization of the method have been used, mostly by using Tchebyshev orthogonal polynomials, cf. [@PGMASA1987; @Driscoll2014]. It seems that to achieve high accuracy one needs to use high arithmetic precision. For example, in [@varga1992some] the errors of best uniform rational approximation of $t^\alpha$ for six values of $\alpha \in [0,1]$ are reported for degree $k \le 30$ by using computer arithmetic with 200 significant digits. In short, for $\delta >0$ the Remez algorithm has substantially better stability and is significantly more reliable.
We also note that in Table \[tabl:A0BURAp\] there are 5 sets of parameters (all of them for $\alpha=0.25$) for which Remez algorithm dies not provide the needed information. In these cases the convergence in the iterative process for finding the extremal points of BURA for these parameters either does not converge or fail to produce equal absolute values at the extremal points with the desired accuracy.
d ($\delta$) $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
-------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------- ----------------- ---------------- -----------------
q,k $\alpha =0.25$ $\alpha =0.50$ $\alpha =0.75$ $\alpha =0.25$ $\alpha =0.50$ $\alpha =0.75$ $\alpha =0.25$ $\alpha =0.50$ $\alpha =0.75 $ $\alpha =0.25 $ $\alpha =0.50$ $\alpha =0.75 $
000,3 1.235E-2 2.282E-3 4.041E-4 7.364E-3 2.107E-3 3.993E-4 9.217E-3 2.223E-3 4.032E-4 1.048E-2 2.263E-3 4.039E-4
000,4 5.566E-3 7.366E-4 9.954E-5 2.270E-3 6.066E-4 9.591E-5 3.263E-3 6.880E-4 9.878E-5 4.082E-3 7.202E-4 9.939E-5
000,5 2.735E-3 2.690E-4 2.868E-5 6.972E-4 1.823E-4 2.607E-5 1.159E-3 2.309E-4 2.806E-5 1.619E-3 2.550E-4 2.855E-5
000,6 1.431E-3 1.075E-4 9.252E-6 2.134E-4 5.534E-5 7.548E-6 4.104E-4 8.015E-5 8.773E-6 6.424E-4 9.598E-5 9.146E-6
000,7 7.865E-4 4.604E-5 3.257E-6 6.522E-5 1.683E-5 2.245E-6 1.450E-4 2.811E-5 2.907E-6 2.545E-4 3.722E-5 3.168E-6
000,8 4.495E-4 2.085E-5 1.229E-6 1.990E-5 5.121E-6 6.751E-7 5.119E-5 9.881E-6 9.930E-7 1.007E-4 1.460E-5 1.159E-6
001,3 8.689E-3 1.696E-3 3.064E-4 4.495E-3 1.532E-3 3.018E-4 5.930E-3 1.640E-3 3.055E-4 6.991E-3 1.678E-3 3.062E-4
001,4 4.077E-3 5.669E-4 7.801E-5 1.386E-3 4.482E-4 7.458E-5 2.109E-3 5.209E-4 7.728E-5 2.756E-3 5.511E-4 7.787E-5
001,5 2.063E-3 2.122E-4 2.301E-5 4.255E-4 1.353E-4 2.058E-5 7.487E-4 1.769E-4 2.242E-5 1.095E-3 1.989E-4 2.288E-5
001,6 1.104E-3 8.643E-5 7.556E-6 1.302E-4 4.109E-5 5.999E-6 2.651E-4 6.165E-5 7.101E-6 4.346E-4 7.560E-5 7.453E-6
001,7 6.174E-4 3.759E-5 2.697E-6 3.979E-5 1.250E-5 1.789E-6 9.366E-5 2.164E-5 2.370E-6 1.721E-4 2.944E-5 2.612E-6
001,8 3.581E-4 1.723E-5 1.029E-6 1.214E-5 3.803E-6 5.385E-7 3.305E-5 7.609E-6 8.125E-7 6.806E-5 1.156E-5 9.628E-7
100,3 3.512E-4 1.011E-4 2.243E-5 1.944E-5 5.012E-5 1.997E-5 3.953E-5 7.387E-5 2.183E-5 6.991E-5 8.949E-5 2.231E-5
100,4 2.012E-4 4.290E-5 7.266E-6 5.995E-6 1.510E-5 5.722E-6 1.404E-5 2.557E-5 6.812E-6 2.756E-5 3.425E-5 7.163E-6
100,5 1.212E-4 1.943E-5 2.583E-6 1.841E-6 4.587E-6 1.693E-6 4.985E-6 8.962E-6 2.259E-6 1.095E-5 1.339E-5 2.498E-6
100,6 7.556E-5 9.242E-6 9.847E-7 5.634E-7 1.395E-6 5.071E-7 1.765E-6 3.150E-6 7.717E-7 4.346E-6 5.265E-6 9.187E-7
100,7 4.831E-5 4.570E-6 3.973E-7 1.721E-7 4.244E-7 1.527E-7 6.237E-7 1.108E-6 2.672E-7 1.721E-6 2.074E-6 3.487E-7
100,8 - - - 2.336E-6 1.680E-7 5.254E-8 1.291E-7 4.614E-8 2.201E-7 3.896E-7 9.305E-8 6.806E-7 8.170E-7 1.346E-7
200,3 1.788E-4 5.304E-5 1.218E-5 5.299E-6 1.978E-5 1.019E-5 1.121E-5 3.280E-5 1.164E-5 2.092E-5 4.328E-5 1.206E-5
200,4 1.033E-4 2.314E-5 4.095E-6 1.634E-6 5.911E-6 2.930E-6 3.976E-6 1.122E-5 3.711E-6 8.212E-6 1.644E-5 4.002E-6
200,5 6.295E-5 1.080E-5 1.506E-6 5.017E-7 1.793E-6 8.680E-7 1.412E-6 3.919E-6 1.246E-6 3.261E-6 6.406E-6 1.432E-6
200,6 3.979E-5 5.281E-6 5.917E-7 1.536E-7 5.451E-7 2.601E-7 5.000E-7 1.377E-6 4.280E-7 1.294E-6 2.517E-6 5.354E-7
200,7 - - - 2.679E-6 2.450E-7 4.693E-8 1.658E-7 7.836E-8 1.767E-7 4.840E-7 1.485E-7 5.125E-7 9.910E-7 2.051E-7
200,8 - - - 1.401E-6 1.060E-7 1.432E-8 5.045E-8 2.367E-8 6.236E-8 1.702E-7 5.176E-8 2.027E-7 3.904E-7 7.957E-8
400,3 9.025E-5 2.723E-5 6.417E-6 1.387E-6 7.014E-6 4.891E-6 3.005E-6 1.330E-5 5.957E-6 5.811E-6 1.953E-5 6.311E-6
400,4 5.239E-5 1.210E-5 2.226E-6 4.276E-7 2.076E-6 1.396E-6 1.065E-6 4.468E-6 1.914E-6 2.273E-6 7.271E-6 2.143E-6
400,5 3.211E-5 5.778E-6 8.448E-7 1.313E-7 6.284E-7 4.125E-7 3.781E-7 1.551E-6 6.456E-7 9.021E-7 2.807E-6 7.811E-7
400,6 2.046E-5 2.896E-6 3.416E-7 4.020E-8 1.910E-7 1.235E-7 1.339E-7 5.440E-7 2.222E-7 3.580E-7 1.100E-6 2.956E-7
400,7 - - - 1.505E-6 1.451E-7 1.228E-8 5.811E-8 3.719E-8 4.732E-8 1.912E-7 7.717E-8 1.418E-7 4.327E-7 1.139E-7
400,8 - - - 8.049E-7 6.421E-8 3.749E-9 1.768E-8 1.124E-8 1.670E-8 6.726E-8 2.691E-8 5.609E-8 1.704E-7 4.432E-8
: The error $E_{q, \delta,\alpha, k}$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ for $t \in [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, $k=3, \dots, 8$, and $q=0,1,100,200,400$ []{data-label="tabl:A0BURAp"}
----------------------- ---------------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j $\alpha =0.25$ 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.342E-6 1.037E-4 6.496E-4 1.645E-5 1.354E-4 6.702E-4 6.389E-6 1.134E-4 6.535E-4 3.403E-6 1.067E-4 6.503E-4
3 4.731E-5 1.612E-3 6.907E-3 2.440E-4 1.858E-3 7.007E-3 1.306E-4 1.690E-3 6.926E-3 8.650E-5 1.637E-3 6.911E-3
4 1.011E-3 1.263E-2 3.708E-2 2.682E-3 1.374E-2 3.739E-2 1.819E-3 1.299E-2 3.714E-2 1.423E-3 1.274E-2 3.709E-2
5 1.125E-2 6.819E-2 1.423E-1 2.100E-2 7.163E-2 1.429E-1 1.635E-2 6.931E-2 1.424E-1 1.397E-2 6.855E-2 1.423E-1
6 9.395E-2 2.688E-1 4.016E-1 1.311E-1 2.754E-1 4.024E-1 1.145E-1 2.710E-1 4.017E-1 1.053E-1 2.695E-1 4.016E-1
7 4.887E-1 7.013E-1 7.896E-1 5.472E-1 7.061E-1 7.900E-1 5.229E-1 7.029E-1 7.896E-1 5.082E-1 7.018E-1 7.896E-1
8 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ---------------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=0$[]{data-label="tabl:Bq000dDaAAk3p"}
------------------------ --------------- ---------------- ---------------- ---------------- --------------- --------------- ---------------- ---------------- --------------- ---------------- ---------------- -----------------
[**k=4** ]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j $\alpha=0.25$ $\alpha= 0.50$ $\alpha= 0.75$ $\alpha= 0.25$ $\alpha=0.50$ $\alpha=0.75$ $\alpha= 0.25$ $\alpha= 0.50$ $\alpha=0.75$ $\alpha= 0.25$ $\alpha= 0.50$ $\alpha= 0.75 $
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 5.542E-8 1.080E-5 1.004E-4 5.079E-6 2.285E-5 1.129E-4 1.213E-6 1.427E-5 1.029E-4 4.020E-7 1.186E-5 1.009E-4
3 1.954E-6 1.682E-4 1.073E-3 4.485E-5 2.581E-4 1.137E-3 1.621E-5 1.967E-4 1.086E-3 7.517E-6 1.772E-4 1.076E-3
4 4.176E-5 1.328E-3 5.879E-3 3.420E-4 1.756E-3 6.102E-3 1.655E-4 1.469E-3 5.925E-3 9.807E-5 1.373E-3 5.888E-3
5 4.680E-4 7.419E-3 2.406E-2 2.134E-3 8.969E-3 2.466E-2 1.251E-3 7.944E-3 2.418E-2 8.568E-4 7.590E-3 2.408E-2
6 4.099E-3 3.305E-2 8.003E-2 1.167E-2 3.752E-2 8.132E-2 8.011E-3 3.459E-2 8.030E-2 6.165E-3 3.355E-2 8.008E-2
7 2.769E-2 1.216E-1 2.213E-1 5.551E-2 1.316E-1 2.234E-1 4.315E-2 1.251E-1 2.217E-1 3.625E-2 1.228E-1 2.214E-1
8 1.525E-1 3.585E-1 4.932E-1 2.239E-1 3.729E-1 4.954E-1 1.947E-1 3.636E-1 4.937E-1 1.769E-1 3.602E-1 4.933E-1
9 5.753E-1 7.597E-1 8.327E-1 6.515E-1 7.681E-1 8.336E-1 6.231E-1 7.627E-1 8.329E-1 6.040E-1 7.607E-1 8.327E-1
10 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
------------------------ --------------- ---------------- ---------------- ---------------- --------------- --------------- ---------------- ---------------- --------------- ---------------- ---------------- -----------------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=0$[]{data-label="tabl:Bq000dDaAAk4p"}
----------------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- -----------------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j $\alpha=0.25$ $\alpha=0.50$ $\alpha=0.75$ $\alpha=0.25$ $\alpha=0.50$ $\alpha=0.75$ $\alpha=0.25$ $\alpha=0.50$ $\alpha=0.75$ $\alpha=0.25$ $\alpha=0.50$ $\alpha= 0.75 $
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 3.229E-9 1.440E-6 1.910E-5 2.756E-6 7.075E-6 2.676E-5 4.753E-7 2.875E-6 2.069E-5 1.049E-7 1.857E-6 1.942E-5
3 1.139E-7 2.242E-5 2.044E-4 1.525E-5 5.885E-5 2.437E-4 4.027E-6 3.373E-5 2.129E-4 1.327E-6 2.601E-5 2.061E-4
4 2.434E-6 1.772E-4 1.124E-3 8.372E-5 3.454E-4 1.265E-3 2.984E-5 2.338E-4 1.155E-3 1.302E-5 1.958E-4 1.130E-3
5 2.728E-5 9.942E-4 4.650E-3 4.129E-4 1.614E-3 5.057E-3 1.827E-4 1.213E-3 4.741E-3 9.609E-5 1.068E-3 4.669E-3
6 2.395E-4 4.496E-3 1.597E-2 1.864E-3 6.420E-3 1.697E-2 9.882E-4 5.200E-3 1.620E-2 6.054E-4 4.735E-3 1.602E-2
7 1.640E-3 1.739E-2 4.777E-2 7.730E-3 2.254E-2 4.988E-2 4.749E-3 1.933E-2 4.825E-2 3.275E-3 1.806E-2 4.787E-2
8 9.687E-3 5.904E-2 1.261E-1 2.972E-2 7.073E-2 1.298E-1 2.079E-2 6.352E-2 1.270E-1 1.587E-2 6.059E-2 1.263E-1
9 4.899E-2 1.748E-1 2.903E-1 1.043E-1 1.959E-1 2.954E-1 8.189E-2 1.830E-1 2.915E-1 6.826E-2 1.776E-1 2.905E-1
10 2.089E-1 4.301E-1 5.608E-1 3.172E-1 4.555E-1 5.655E-1 2.776E-1 4.402E-1 5.619E-1 2.510E-1 4.336E-1 5.610E-1
11 6.373E-1 7.990E-1 8.611E-1 7.260E-1 8.118E-1 8.629E-1 6.969E-1 8.042E-1 8.615E-1 6.754E-1 8.009E-1 8.612E-1
12 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- -----------------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=0$[]{data-label="tabl:Bq000dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 2.422E-10 2.299E-7 4.228E-6 1.971E-6 3.488E-6 9.101E-6 2.804E-7 9.260E-7 5.245E-6 4.709E-8 4.131E-7 4.435E-6
3 8.541E-9 3.581E-6 4.524E-5 7.510E-6 2.039E-5 6.892E-5 1.577E-6 8.448E-6 5.075E-5 3.966E-7 5.095E-6 4.640E-5
4 1.826E-7 2.830E-5 2.489E-4 3.133E-5 9.958E-5 3.337E-4 8.760E-6 5.191E-5 2.694E-4 2.926E-6 3.617E-5 2.532E-4
5 2.046E-6 1.589E-4 1.032E-3 1.252E-4 4.124E-4 1.281E-3 4.359E-5 2.499E-4 1.094E-3 1.787E-5 1.905E-4 1.046E-3
6 1.797E-5 7.200E-4 3.569E-3 4.734E-4 1.505E-3 4.205E-3 1.984E-4 1.018E-3 3.729E-3 9.646E-5 8.267E-4 3.604E-3
7 1.232E-4 2.806E-3 1.086E-2 1.692E-3 4.975E-3 1.231E-2 8.304E-4 3.666E-3 1.123E-2 4.636E-4 3.120E-3 1.094E-2
8 7.307E-4 9.740E-3 2.987E-2 5.754E-3 1.513E-2 3.283E-2 3.247E-3 1.195E-2 3.063E-2 2.044E-3 1.056E-2 3.003E-2
9 3.775E-3 3.067E-2 7.502E-2 1.861E-2 4.267E-2 8.039E-2 1.189E-2 3.574E-2 7.641E-2 8.300E-3 3.258E-2 7.532E-2
10 1.759E-2 8.786E-2 1.715E-1 5.707E-2 1.110E-1 1.798E-1 4.088E-2 9.790E-2 1.736E-1 3.132E-2 9.169E-2 1.719E-1
11 7.316E-2 2.247E-1 3.495E-1 1.621E-1 2.607E-1 3.598E-1 1.294E-1 2.406E-1 3.522E-1 1.081E-1 2.308E-1 3.501E-1
12 2.609E-1 4.879E-1 6.126E-1 4.033E-1 5.256E-1 6.210E-1 3.569E-1 5.050E-1 6.148E-1 3.234E-1 4.946E-1 6.130E-1
13 6.836E-1 8.273E-1 8.812E-1 7.801E-1 8.442E-1 8.844E-1 7.523E-1 8.351E-1 8.821E-1 7.303E-1 8.303E-1 8.814E-1
14 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=0$[]{data-label="tabl:Bq000dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 2.209E-11 4.219E-8 1.051E-6 1.618E-6 2.301E-6 4.350E-6 2.055E-7 4.423E-7 1.716E-6 2.914E-8 1.330E-7 1.189E-6
3 7.789E-10 6.570E-7 1.124E-5 4.642E-6 9.653E-6 2.558E-5 8.260E-7 2.977E-6 1.473E-5 1.699E-7 1.342E-6 1.202E-5
4 1.665E-8 5.192E-6 6.187E-5 1.545E-5 3.892E-5 1.116E-4 3.587E-6 1.568E-5 7.489E-5 9.675E-7 8.661E-6 6.485E-5
5 1.866E-7 2.915E-5 2.568E-4 5.160E-5 1.413E-4 4.016E-4 1.480E-5 6.830E-5 2.963E-4 4.905E-6 4.302E-5 2.660E-4
6 1.639E-6 1.322E-4 8.891E-4 1.673E-4 4.669E-4 1.262E-3 5.749E-5 2.590E-4 9.939E-4 2.269E-5 1.793E-4 9.136E-4
7 1.123E-5 5.158E-4 2.717E-3 5.227E-4 1.427E-3 3.586E-3 2.104E-4 8.830E-4 2.966E-3 9.624E-5 6.570E-4 2.776E-3
8 6.667E-5 1.797E-3 7.543E-3 1.578E-3 4.086E-3 9.407E-3 7.319E-4 2.765E-3 8.087E-3 3.811E-4 2.180E-3 7.672E-3
9 3.450E-4 5.722E-3 1.938E-2 4.604E-3 1.105E-2 2.306E-2 2.427E-3 8.064E-3 2.047E-2 1.416E-3 6.670E-3 1.964E-2
10 1.620E-3 1.687E-2 4.643E-2 1.300E-2 2.831E-2 5.311E-2 7.710E-3 2.208E-2 4.843E-2 4.983E-3 1.902E-2 4.691E-2
11 6.937E-3 4.633E-2 1.038E-1 3.543E-2 6.866E-2 1.147E-1 2.344E-2 5.681E-2 1.071E-1 1.663E-2 5.073E-2 1.046E-1
12 2.744E-2 1.177E-1 2.144E-1 9.230E-2 1.559E-1 2.298E-1 6.778E-2 1.362E-1 2.191E-1 5.246E-2 1.256E-1 2.156E-1
13 9.881E-2 2.705E-1 4.004E-1 2.240E-1 3.233E-1 4.176E-1 1.821E-1 2.967E-1 4.057E-1 1.532E-1 2.819E-1 4.017E-1
14 3.079E-1 5.354E-1 6.535E-1 4.792E-1 5.850E-1 6.666E-1 4.292E-1 5.606E-1 6.576E-1 3.911E-1 5.465E-1 6.544E-1
15 7.194E-1 8.485E-1 8.962E-1 8.202E-1 8.692E-1 9.009E-1 7.945E-1 8.592E-1 8.977E-1 7.731E-1 8.533E-1 8.966E-1
16 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=0$[]{data-label="tabl:Bq000dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 2.357E-12 8.655E-9 2.865E-7 1.429E-6 1.792E-6 2.706E-6 1.694E-7 2.780E-7 7.363E-7 2.163E-8 6.023E-8 3.800E-7
3 8.310E-11 1.348E-7 3.066E-6 3.315E-6 5.682E-6 1.207E-5 5.222E-7 1.378E-6 5.264E-6 9.249E-8 4.698E-7 3.584E-6
4 1.776E-9 1.065E-6 1.687E-5 9.172E-6 1.910E-5 4.625E-5 1.844E-6 6.137E-6 2.492E-5 4.208E-7 2.669E-6 1.886E-5
5 1.991E-8 5.981E-6 7.002E-5 2.630E-5 6.090E-5 1.534E-4 6.458E-6 2.387E-5 9.440E-5 1.799E-6 1.222E-5 7.622E-5
6 1.748E-7 2.712E-5 2.426E-4 7.474E-5 1.820E-4 4.549E-4 2.181E-5 8.316E-5 3.075E-4 7.208E-6 4.808E-5 2.594E-4
7 1.198E-6 1.058E-4 7.419E-4 2.078E-4 5.124E-4 1.238E-3 7.068E-5 2.654E-4 8.986E-4 2.709E-5 1.687E-4 7.830E-4
8 7.113E-6 3.691E-4 2.065E-3 5.642E-4 1.370E-3 3.141E-3 2.205E-4 7.878E-4 2.415E-3 9.647E-5 5.408E-4 2.158E-3
9 3.682E-5 1.178E-3 5.337E-3 1.495E-3 3.499E-3 7.525E-3 6.637E-4 2.201E-3 6.062E-3 3.269E-4 1.611E-3 5.531E-3
10 1.730E-4 3.493E-3 1.295E-2 3.872E-3 8.577E-3 1.714E-2 1.934E-3 5.836E-3 1.437E-2 1.060E-3 4.514E-3 1.334E-2
11 7.429E-4 9.731E-3 2.975E-2 9.796E-3 2.022E-2 3.724E-2 5.460E-3 1.476E-2 3.232E-2 3.304E-3 1.197E-2 3.045E-2
12 2.971E-3 2.561E-2 6.474E-2 2.419E-2 4.585E-2 7.717E-2 1.495E-2 3.564E-2 6.906E-2 9.912E-3 3.018E-2 6.592E-2
13 1.110E-2 6.360E-2 1.329E-1 5.798E-2 9.936E-2 1.516E-1 3.959E-2 8.190E-2 1.395E-1 2.858E-2 7.211E-2 1.347E-1
14 3.883E-2 1.476E-1 2.545E-1 1.333E-1 2.031E-1 2.787E-1 1.003E-1 1.769E-1 2.631E-1 7.861E-2 1.615E-1 2.568E-1
15 1.250E-1 3.123E-1 4.443E-1 2.860E-1 3.820E-1 4.695E-1 2.368E-1 3.501E-1 4.534E-1 2.012E-1 3.305E-1 4.468E-1
16 3.502E-1 5.750E-1 6.865E-1 5.447E-1 6.352E-1 7.048E-1 4.934E-1 6.085E-1 6.932E-1 4.527E-1 5.915E-1 6.884E-1
17 7.480E-1 8.651E-1 9.079E-1 8.507E-1 8.887E-1 9.141E-1 8.271E-1 8.785E-1 9.102E-1 8.069E-1 8.717E-1 9.085E-1
18 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=0$[]{data-label="tabl:Bq000dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 3.498E-7 5.726E-5 4.471E-4 9.698E-6 8.172E-5 4.657E-4 3.021E-6 6.467E-5 4.507E-4 1.317E-6 5.957E-5 4.478E-4
3 1.288E-5 8.925E-4 4.723E-3 1.229E-4 1.083E-3 4.813E-3 5.501E-5 9.531E-4 4.740E-3 3.130E-5 9.117E-4 4.726E-3
4 3.074E-4 7.073E-3 2.520E-2 1.280E-3 7.966E-3 2.549E-2 7.497E-4 7.363E-3 2.526E-2 5.237E-4 7.165E-3 2.521E-2
5 3.893E-3 3.986E-2 9.862E-2 1.037E-2 4.293E-2 9.929E-2 7.153E-3 4.087E-2 9.875E-2 5.591E-3 4.018E-2 9.864E-2
6 4.179E-2 1.777E-1 3.040E-1 7.497E-2 1.853E-1 3.051E-1 5.992E-2 1.802E-1 3.042E-1 5.172E-2 1.785E-1 3.040E-1
7 3.259E-1 5.902E-1 7.089E-1 4.194E-1 5.990E-1 7.097E-1 3.815E-1 5.931E-1 7.090E-1 3.580E-1 5.912E-1 7.089E-1
8 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=1$[]{data-label="tabl:Bq001dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.642E-8 6.397E-6 7.245E-5 3.852E-6 1.617E-5 8.378E-5 7.865E-7 9.141E-6 7.473E-5 2.161E-7 7.229E-6 7.288E-5
3 5.905E-7 9.964E-5 7.724E-4 2.867E-5 1.707E-4 8.306E-4 8.925E-6 1.222E-4 7.843E-4 3.538E-6 1.068E-4 7.747E-4
4 1.327E-5 7.878E-4 4.211E-3 1.987E-4 1.126E-3 4.414E-3 8.338E-5 9.004E-4 4.253E-3 4.310E-5 8.243E-4 4.219E-3
5 1.572E-4 4.430E-3 1.715E-2 1.192E-3 5.677E-3 1.770E-2 6.131E-4 4.857E-3 1.726E-2 3.732E-4 4.569E-3 1.717E-2
6 1.518E-3 2.012E-2 5.725E-2 6.554E-3 2.391E-2 5.848E-2 4.006E-3 2.145E-2 5.751E-2 2.794E-3 2.056E-2 5.730E-2
7 1.168E-2 7.818E-2 1.637E-1 3.299E-2 8.770E-2 1.659E-1 2.321E-2 8.156E-2 1.641E-1 1.796E-2 7.930E-2 1.638E-1
8 8.061E-2 2.607E-1 3.992E-1 1.523E-1 2.784E-1 4.021E-1 1.227E-1 2.671E-1 3.998E-1 1.049E-1 2.629E-1 3.993E-1
9 4.322E-1 6.732E-1 7.723E-1 5.553E-1 6.881E-1 7.740E-1 5.114E-1 6.787E-1 7.726E-1 4.808E-1 6.751E-1 7.723E-1
10 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=1$[]{data-label="tabl:Bq001dDaAAk4p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.060E-9 8.967E-7 1.424E-5 2.372E-6 5.717E-6 2.124E-5 3.725E-7 2.076E-6 1.569E-5 7.213E-8 1.233E-6 1.452E-5
3 3.775E-8 1.397E-5 1.522E-4 1.135E-5 4.360E-5 1.879E-4 2.686E-6 2.305E-5 1.600E-4 7.789E-7 1.684E-5 1.537E-4
4 8.273E-7 1.104E-4 8.348E-4 5.672E-5 2.450E-4 9.630E-4 1.800E-5 1.557E-4 8.637E-4 6.937E-6 1.253E-4 8.407E-4
5 9.531E-6 6.199E-4 3.443E-3 2.641E-4 1.114E-3 3.812E-3 1.042E-4 7.959E-4 3.528E-3 4.879E-5 6.794E-4 3.461E-3
6 8.766E-5 2.813E-3 1.178E-2 1.156E-3 4.360E-3 1.269E-2 5.481E-4 3.386E-3 1.199E-2 3.018E-4 3.010E-3 1.182E-2
7 6.370E-4 1.099E-2 3.519E-2 4.766E-3 1.524E-2 3.714E-2 2.636E-3 1.261E-2 3.564E-2 1.651E-3 1.155E-2 3.528E-2
8 4.120E-3 3.828E-2 9.395E-2 1.878E-2 4.855E-2 9.759E-2 1.194E-2 4.229E-2 9.480E-2 8.371E-3 3.969E-2 9.413E-2
9 2.367E-2 1.204E-1 2.253E-1 7.023E-2 1.415E-1 2.310E-1 5.072E-2 1.288E-1 2.267E-1 3.931E-2 1.234E-1 2.256E-1
10 1.242E-1 3.335E-1 4.744E-1 2.410E-1 3.654E-1 4.808E-1 1.980E-1 3.465E-1 4.759E-1 1.695E-1 3.382E-1 4.747E-1
11 5.139E-1 7.299E-1 8.140E-1 6.551E-1 7.515E-1 8.172E-1 6.114E-1 7.389E-1 8.148E-1 5.779E-1 7.332E-1 8.141E-1
12 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=1$[]{data-label="tabl:Bq001dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 8.636E-11 1.487E-7 3.227E-6 1.807E-6 3.078E-6 7.729E-6 2.433E-7 7.452E-7 4.164E-6 3.764E-8 3.006E-7 3.419E-6
3 3.061E-9 2.316E-6 3.452E-5 6.142E-6 1.650E-5 5.602E-5 1.192E-6 6.319E-6 3.957E-5 2.726E-7 3.550E-6 3.559E-5
4 6.631E-8 1.830E-5 1.897E-4 2.360E-5 7.678E-5 2.663E-4 6.037E-6 3.745E-5 2.086E-4 1.823E-6 2.469E-5 1.938E-4
5 7.541E-7 1.028E-4 7.859E-4 8.907E-5 3.078E-4 1.010E-3 2.823E-5 1.762E-4 8.426E-4 1.047E-5 1.285E-4 7.982E-4
6 6.784E-6 4.661E-4 2.710E-3 3.235E-4 1.098E-3 3.283E-3 1.232E-4 7.070E-4 2.857E-3 5.427E-5 5.532E-4 2.742E-3
7 4.795E-5 1.820E-3 8.221E-3 1.128E-3 3.569E-3 9.526E-3 5.033E-4 2.519E-3 8.561E-3 2.557E-4 2.079E-3 8.295E-3
8 2.978E-4 6.352E-3 2.255E-2 3.804E-3 1.076E-2 2.524E-2 1.956E-3 8.178E-3 2.326E-2 1.127E-3 7.041E-3 2.270E-2
9 1.634E-3 2.027E-2 5.679E-2 1.242E-2 3.041E-2 6.180E-2 7.265E-3 2.460E-2 5.812E-2 4.670E-3 2.193E-2 5.708E-2
10 8.305E-3 5.987E-2 1.320E-1 3.936E-2 8.086E-2 1.403E-1 2.599E-2 6.909E-2 1.343E-1 1.849E-2 6.346E-2 1.325E-1
11 3.909E-2 1.632E-1 2.816E-1 1.193E-1 2.003E-1 2.930E-1 8.867E-2 1.799E-1 2.846E-1 6.945E-2 1.698E-1 2.822E-1
12 1.690E-1 3.959E-1 5.345E-1 3.299E-1 4.438E-1 5.459E-1 2.770E-1 4.181E-1 5.376E-1 2.394E-1 4.048E-1 5.352E-1
13 5.771E-1 7.707E-1 8.433E-1 7.277E-1 7.982E-1 8.486E-1 6.869E-1 7.838E-1 8.447E-1 6.536E-1 7.760E-1 8.436E-1
14 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=1$[]{data-label="tabl:Bq001dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 8.427E-12 2.812E-8 8.172E-7 1.533E-6 2.136E-6 3.909E-6 1.884E-7 3.856E-7 1.433E-6 2.539E-8 1.060E-7 9.458E-7
3 2.980E-10 4.380E-7 8.743E-6 4.034E-6 8.309E-6 2.183E-5 6.776E-7 2.401E-6 1.194E-5 1.299E-7 1.006E-6 9.465E-6
4 6.417E-9 3.461E-6 4.809E-5 1.252E-5 3.197E-5 9.298E-5 2.715E-6 1.213E-5 6.000E-5 6.764E-7 6.307E-6 5.086E-5
5 7.252E-8 1.943E-5 1.995E-4 3.969E-5 1.123E-4 3.297E-4 1.056E-5 5.141E-5 2.356E-4 3.222E-6 3.076E-5 2.080E-4
6 6.454E-7 8.812E-5 6.899E-4 1.236E-4 3.620E-4 1.024E-3 3.923E-5 1.910E-4 7.857E-4 1.423E-5 1.265E-4 7.128E-4
7 4.499E-6 3.440E-4 2.104E-3 3.751E-4 1.085E-3 2.882E-3 1.391E-4 6.415E-4 2.332E-3 5.844E-5 4.594E-4 2.159E-3
8 2.738E-5 1.200E-3 5.828E-3 1.110E-3 3.059E-3 7.496E-3 4.738E-4 1.986E-3 6.326E-3 2.270E-4 1.515E-3 5.948E-3
9 1.463E-4 3.833E-3 1.493E-2 3.210E-3 8.183E-3 1.824E-2 1.558E-3 5.753E-3 1.593E-2 8.383E-4 4.620E-3 1.517E-2
10 7.188E-4 1.138E-2 3.575E-2 9.089E-3 2.090E-2 4.184E-2 4.972E-3 1.574E-2 3.762E-2 2.976E-3 1.321E-2 3.621E-2
11 3.267E-3 3.177E-2 8.044E-2 2.520E-2 5.109E-2 9.068E-2 1.544E-2 4.090E-2 8.362E-2 1.019E-2 3.567E-2 8.122E-2
12 1.406E-2 8.355E-2 1.697E-1 6.809E-2 1.191E-1 1.852E-1 4.658E-2 1.009E-1 1.746E-1 3.375E-2 9.106E-2 1.709E-1
13 5.704E-2 2.048E-1 3.320E-1 1.758E-1 2.606E-1 3.515E-1 1.345E-1 2.328E-1 3.382E-1 1.069E-1 2.171E-1 3.335E-1
14 2.126E-1 4.492E-1 5.831E-1 4.122E-1 5.125E-1 6.008E-1 3.536E-1 4.820E-1 5.888E-1 3.094E-1 4.640E-1 5.846E-1
15 6.271E-1 8.012E-1 8.649E-1 7.809E-1 8.335E-1 8.724E-1 7.441E-1 8.184E-1 8.673E-1 7.126E-1 8.091E-1 8.655E-1
16 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=1$[]{data-label="tabl:Bq001dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 9.514E-13 5.913E-9 2.263E-7 1.379E-6 1.711E-6 2.529E-6 1.601E-7 2.551E-7 6.464E-7 1.980E-8 5.166E-8 3.132E-7
3 3.360E-11 9.209E-8 2.421E-6 2.998E-6 5.106E-6 1.072E-5 4.525E-7 1.177E-6 4.437E-6 7.603E-8 3.761E-7 2.901E-6
4 7.213E-10 7.277E-7 1.332E-5 7.825E-6 1.645E-5 4.001E-5 1.491E-6 5.022E-6 2.065E-5 3.198E-7 2.063E-6 1.516E-5
5 8.123E-9 4.086E-6 5.528E-5 2.142E-5 5.082E-5 1.304E-4 4.952E-6 1.897E-5 7.744E-5 1.290E-6 9.234E-6 6.101E-5
6 7.188E-8 1.853E-5 1.914E-4 5.866E-5 1.482E-4 3.819E-4 1.603E-5 6.467E-5 2.503E-4 4.935E-6 3.571E-5 2.069E-4
7 4.976E-7 7.232E-5 5.849E-4 1.584E-4 4.088E-4 1.028E-3 5.025E-5 2.028E-4 7.271E-4 1.790E-5 1.237E-4 6.229E-4
8 2.996E-6 2.523E-4 1.626E-3 4.203E-4 1.075E-3 2.586E-3 1.528E-4 5.935E-4 1.943E-3 6.207E-5 3.927E-4 1.712E-3
9 1.580E-5 8.055E-4 4.193E-3 1.096E-3 2.708E-3 6.145E-3 4.520E-4 1.640E-3 4.851E-3 2.067E-4 1.161E-3 4.373E-3
10 7.613E-5 2.393E-3 1.015E-2 2.814E-3 6.570E-3 1.389E-2 1.305E-3 4.313E-3 1.144E-2 6.651E-4 3.236E-3 1.051E-2
11 3.379E-4 6.695E-3 2.328E-2 7.115E-3 1.540E-2 3.001E-2 3.685E-3 1.086E-2 2.563E-2 2.077E-3 8.572E-3 2.393E-2
12 1.413E-3 1.778E-2 5.072E-2 1.773E-2 3.496E-2 6.208E-2 1.020E-2 2.629E-2 5.475E-2 6.317E-3 2.170E-2 5.185E-2
13 5.596E-3 4.499E-2 1.051E-1 4.342E-2 7.669E-2 1.228E-1 2.769E-2 6.122E-2 1.115E-1 1.875E-2 5.263E-2 1.069E-1
14 2.125E-2 1.083E-1 2.061E-1 1.036E-1 1.614E-1 2.308E-1 7.324E-2 1.364E-1 2.151E-1 5.415E-2 1.218E-1 2.087E-1
15 7.675E-2 2.443E-1 3.770E-1 2.357E-1 3.197E-1 4.059E-1 1.851E-1 2.855E-1 3.877E-1 1.496E-1 2.645E-1 3.801E-1
16 2.542E-1 4.949E-1 6.232E-1 4.854E-1 5.719E-1 6.474E-1 4.243E-1 5.384E-1 6.323E-1 3.760E-1 5.167E-1 6.258E-1
17 6.672E-1 8.249E-1 8.815E-1 8.206E-1 8.605E-1 8.911E-1 7.878E-1 8.456E-1 8.851E-1 7.588E-1 8.354E-1 8.825E-1
18 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=1$[]{data-label="tabl:Bq001dDaAAk8p"}
---------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{}/$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.136E-12 2.045E-7 1.336E-5 2.013E-6 3.411E-6 1.999E-5 2.314E-7 8.732E-7 1.473E-5 2.793E-8 3.782E-7 1.363E-5
3 4.845E-11 3.280E-6 1.381E-4 9.434E-6 2.082E-5 1.703E-4 1.307E-6 8.162E-6 1.451E-4 1.933E-7 4.776E-6 1.395E-4
4 1.779E-9 2.874E-5 7.446E-4 6.248E-5 1.191E-4 8.627E-4 1.028E-5 5.675E-5 7.710E-4 1.789E-6 3.783E-5 7.499E-4
5 3.721E-8 2.112E-4 3.403E-3 4.963E-4 6.988E-4 3.824E-3 9.605E-5 3.706E-4 3.498E-3 1.909E-5 2.643E-4 3.422E-3
6 1.281E-6 1.721E-3 1.657E-2 5.355E-3 4.942E-3 1.833E-2 1.324E-3 2.818E-3 1.696E-2 3.195E-4 2.094E-3 1.665E-2
7 4.953E-5 2.233E-2 1.197E-1 7.502E-2 5.447E-2 1.296E-1 2.527E-2 3.408E-2 1.220E-1 7.590E-3 2.644E-2 1.201E-1
8 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
---------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=100$[]{data-label="tabl:Bq100dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.093E-13 3.667E-8 3.029E-6 1.629E-6 2.260E-6 7.384E-6 1.811E-7 4.220E-7 3.935E-6 2.080E-8 1.227E-7 3.215E-6
3 4.275E-12 5.752E-7 3.180E-5 5.177E-6 9.531E-6 5.191E-5 6.855E-7 2.811E-6 3.655E-5 9.535E-8 1.226E-6 3.281E-5
4 1.207E-10 4.676E-6 1.700E-4 2.203E-5 4.080E-5 2.397E-4 3.424E-6 1.534E-5 1.872E-4 5.567E-7 8.128E-6 1.737E-4
5 1.858E-9 2.848E-5 6.970E-4 1.071E-4 1.707E-4 9.056E-4 1.932E-5 7.445E-5 7.495E-4 3.579E-6 4.420E-5 7.084E-4
6 2.981E-8 1.559E-4 2.541E-3 5.870E-4 7.374E-4 3.154E-3 1.253E-4 3.548E-4 2.697E-3 2.679E-5 2.262E-4 2.575E-3
7 4.421E-7 8.860E-4 9.434E-3 3.609E-3 3.530E-3 1.138E-2 9.320E-4 1.836E-3 9.933E-3 2.320E-4 1.230E-3 9.543E-3
8 1.096E-5 6.101E-3 4.033E-2 2.655E-2 2.047E-2 4.745E-2 9.013E-3 1.160E-2 4.218E-2 2.826E-3 8.158E-3 4.073E-2
9 3.599E-4 6.466E-2 2.318E-1 2.199E-1 1.620E-1 2.589E-1 1.089E-1 1.070E-1 2.391E-1 4.626E-2 8.150E-2 2.334E-1
10 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=100$[]{data-label="tabl:Bq100dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.357E-14 7.520E-9 7.680E-7 1.432E-6 1.770E-6 3.790E-6 1.556E-7 2.695E-7 1.368E-6 1.730E-8 5.683E-8 8.935E-7
3 5.084E-13 1.173E-7 8.147E-6 3.505E-6 5.584E-6 2.065E-5 4.454E-7 1.311E-6 1.121E-5 5.898E-8 4.340E-7 8.841E-6
4 1.273E-11 9.346E-7 4.407E-5 1.118E-5 1.922E-5 8.592E-5 1.653E-6 5.873E-6 5.521E-5 2.544E-7 2.468E-6 4.667E-5
5 1.706E-10 5.381E-6 1.791E-4 4.021E-5 6.536E-5 2.983E-4 6.829E-6 2.377E-5 2.122E-4 1.195E-6 1.163E-5 1.869E-4
6 2.066E-9 2.600E-5 6.127E-4 1.560E-4 2.215E-4 9.241E-4 3.053E-5 9.108E-5 7.015E-4 6.058E-6 4.936E-5 6.339E-4
7 2.138E-8 1.164E-4 1.928E-3 6.452E-4 7.715E-4 2.727E-3 1.468E-4 3.478E-4 2.159E-3 3.313E-5 2.026E-4 1.984E-3
8 2.481E-7 5.254E-4 6.013E-3 2.878E-3 2.862E-3 8.151E-3 7.803E-4 1.397E-3 6.639E-3 2.050E-4 8.601E-4 6.164E-3
9 3.066E-6 2.607E-3 2.007E-2 1.389E-2 1.177E-2 2.624E-2 4.613E-3 6.244E-3 2.190E-2 1.442E-3 4.052E-3 2.052E-2
10 6.204E-5 1.576E-2 7.685E-2 7.355E-2 5.606E-2 9.593E-2 3.211E-2 3.323E-2 8.265E-2 1.284E-2 2.304E-2 7.827E-2
11 1.763E-3 1.343E-1 3.513E-1 3.850E-1 3.107E-1 3.995E-1 2.461E-1 2.249E-1 3.667E-1 1.386E-1 1.760E-1 3.551E-1
12 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=100$[]{data-label="tabl:Bq100dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.984E-15 1.700E-9 2.129E-7 1.316E-6 1.519E-6 2.479E-6 1.407E-7 2.032E-7 6.245E-7 1.530E-8 3.414E-8 2.979E-7
3 7.261E-14 2.649E-8 2.269E-6 2.681E-6 3.825E-6 1.030E-5 3.287E-7 7.514E-7 4.221E-6 4.177E-8 1.970E-7 2.735E-6
4 1.708E-12 2.098E-7 1.238E-5 6.953E-6 1.101E-5 3.775E-5 9.818E-7 2.822E-6 1.938E-5 1.440E-7 9.601E-7 1.415E-5
5 2.129E-11 1.187E-6 5.069E-5 2.025E-5 3.206E-5 1.208E-4 3.259E-6 9.905E-6 7.145E-5 5.409E-7 4.016E-6 5.609E-5
6 2.254E-10 5.499E-6 1.726E-4 6.285E-5 9.274E-5 3.479E-4 1.148E-5 3.290E-5 2.269E-4 2.141E-6 1.513E-5 1.870E-4
7 1.962E-9 2.256E-5 5.219E-4 2.040E-4 2.692E-4 9.355E-4 4.241E-5 1.062E-4 6.539E-4 8.876E-6 5.365E-5 5.573E-4
8 1.690E-8 8.709E-5 1.473E-3 6.910E-4 7.985E-4 2.433E-3 1.651E-4 3.435E-4 1.785E-3 3.911E-5 1.864E-4 1.558E-3
9 1.419E-7 3.343E-4 4.083E-3 2.446E-3 2.470E-3 6.356E-3 6.806E-4 1.149E-3 4.832E-3 1.843E-4 6.608E-4 4.287E-3
10 1.374E-6 1.348E-3 1.166E-2 9.106E-3 8.129E-3 1.729E-2 3.017E-3 4.096E-3 1.354E-2 9.555E-4 2.490E-3 1.217E-2
11 1.499E-5 6.049E-3 3.592E-2 3.560E-2 2.905E-2 5.059E-2 1.444E-2 1.610E-2 4.094E-2 5.487E-3 1.042E-2 3.731E-2
12 2.575E-4 3.236E-2 1.236E-1 1.439E-1 1.139E-1 1.615E-1 7.569E-2 7.206E-2 1.370E-1 3.669E-2 5.075E-2 1.273E-1
13 6.286E-3 2.210E-1 4.584E-1 5.251E-1 4.558E-1 5.243E-1 3.906E-1 3.585E-1 4.834E-1 2.657E-1 2.929E-1 4.656E-1
14 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=100$[]{data-label="tabl:Bq100dDaAAk6p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 3.253E-16 4.158E-10 6.353E-8 1.243E-6 1.373E-6 1.896E-6 1.312E-7 1.692E-7 3.618E-7 1.403E-8 2.440E-8 1.221E-7
3 1.174E-14 6.476E-9 6.786E-7 2.213E-6 2.908E-6 6.137E-6 2.634E-7 4.981E-7 1.936E-6 3.232E-8 1.082E-7 9.858E-7
4 2.664E-13 5.122E-8 3.719E-6 4.909E-6 7.207E-6 1.963E-5 6.659E-7 1.595E-6 8.059E-6 9.354E-8 4.523E-7 4.877E-6
5 3.191E-12 2.883E-7 1.533E-5 1.221E-5 1.849E-5 5.751E-5 1.872E-6 4.951E-6 2.803E-5 2.963E-7 1.692E-6 1.889E-5
6 3.141E-11 1.316E-6 5.249E-5 3.224E-5 4.746E-5 1.550E-4 5.559E-6 1.471E-5 8.551E-5 9.815E-7 5.787E-6 6.206E-5
7 2.490E-10 5.229E-6 1.583E-4 8.838E-5 1.215E-4 3.913E-4 1.711E-5 4.229E-5 2.369E-4 3.360E-6 1.856E-5 1.817E-4
8 1.846E-9 1.896E-5 4.362E-4 2.497E-4 3.130E-4 9.433E-4 5.450E-5 1.197E-4 6.129E-4 1.193E-5 5.729E-5 4.895E-4
9 1.276E-8 6.533E-5 1.135E-3 7.257E-4 8.199E-4 2.221E-3 1.797E-4 3.403E-4 1.522E-3 4.406E-5 1.745E-4 1.253E-3
10 9.100E-8 2.222E-4 2.880E-3 2.173E-3 2.211E-3 5.233E-3 6.168E-4 9.898E-4 3.734E-3 1.711E-4 5.387E-4 3.143E-3
11 6.687E-7 7.742E-4 7.389E-3 6.715E-3 6.210E-3 1.263E-2 2.213E-3 3.002E-3 9.322E-3 7.030E-4 1.728E-3 7.988E-3
12 5.713E-6 2.872E-3 1.978E-2 2.143E-2 1.836E-2 3.185E-2 8.358E-3 9.670E-3 2.433E-2 3.103E-3 5.909E-3 2.120E-2
13 5.613E-5 1.181E-2 5.679E-2 7.015E-2 5.751E-2 8.502E-2 3.323E-2 3.366E-2 6.775E-2 1.478E-2 2.211E-2 6.027E-2
14 8.290E-4 5.620E-2 1.764E-1 2.274E-1 1.884E-1 2.376E-1 1.372E-1 1.273E-1 2.013E-1 7.700E-2 9.241E-2 1.845E-1
15 1.715E-2 3.112E-1 5.467E-1 6.313E-1 5.748E-1 6.243E-1 5.144E-1 4.812E-1 5.808E-1 3.939E-1 4.099E-1 5.582E-1
16 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=100$[]{data-label="tabl:Bq100dDaAAk7p"}
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.087E-10 2.018E-8 1.192E-6 1.282E-6 1.596E-6 1.247E-7 1.496E-7 2.507E-7 1.318E-8 1.949E-8 6.184E-8
3 1.692E-9 2.158E-7 1.921E-6 2.374E-6 4.180E-6 2.231E-7 3.661E-7 1.054E-6 2.660E-8 6.855E-8 4.179E-7
4 1.338E-8 1.185E-6 3.773E-6 5.191E-6 1.169E-5 4.941E-7 1.018E-6 3.912E-6 6.684E-8 2.474E-7 1.930E-6
5 7.516E-8 4.902E-6 8.284E-6 1.196E-5 3.128E-5 1.218E-6 2.830E-6 1.267E-5 1.847E-7 8.323E-7 7.181E-6
6 3.416E-7 1.687E-5 1.932E-5 2.785E-5 7.882E-5 3.170E-6 7.646E-6 3.677E-5 5.334E-7 2.607E-6 2.301E-5
7 1.342E-6 5.114E-5 4.667E-5 6.478E-5 1.881E-4 8.519E-6 2.007E-5 9.803E-5 1.583E-6 7.710E-6 6.613E-5
8 4.747E-6 1.408E-4 1.156E-4 1.507E-4 4.293E-4 2.348E-5 5.167E-5 2.444E-4 4.816E-6 2.186E-5 1.748E-4
9 1.562E-5 3.607E-4 2.922E-4 3.529E-4 9.491E-4 6.627E-5 1.319E-4 5.794E-4 1.503E-5 6.041E-5 4.340E-4
10 4.918E-5 8.792E-4 7.536E-4 8.372E-4 2.062E-3 1.919E-4 3.378E-4 1.330E-3 4.838E-5 1.655E-4 1.032E-3
11 1.522E-4 2.088E-3 1.984E-3 2.029E-3 4.476E-3 5.711E-4 8.801E-4 3.017E-3 1.611E-4 4.574E-4 2.407E-3
12 4.757E-4 4.951E-3 5.339E-3 5.055E-3 9.856E-3 1.753E-3 2.360E-3 6.895E-3 5.587E-4 1.297E-3 5.625E-3
13 1.543E-3 1.201E-2 1.468E-2 1.304E-2 2.233E-2 5.564E-3 6.586E-3 1.619E-2 2.026E-3 3.835E-3 1.348E-2
14 5.340E-3 3.047E-2 4.113E-2 3.493E-2 5.254E-2 1.830E-2 1.933E-2 3.965E-2 7.752E-3 1.202E-2 3.373E-2
15 2.026E-2 8.193E-2 1.159E-1 9.694E-2 1.284E-1 6.198E-2 6.000E-2 1.020E-1 3.127E-2 4.042E-2 8.920E-2
16 8.635E-2 2.314E-1 3.136E-1 2.704E-1 3.169E-1 2.096E-1 1.941E-1 2.704E-1 1.315E-1 1.459E-1 2.459E-1
17 3.955E-1 6.172E-1 7.095E-1 6.656E-1 7.008E-1 6.118E-1 5.823E-1 6.585E-1 5.050E-1 5.130E-1 6.333E-1
18 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=100$[]{data-label="tabl:Bq100dDaAAk8p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 7.673E-14 5.630E-8 5.903E-6 1.922E-6 2.490E-6 1.106E-5 2.115E-7 4.920E-7 6.978E-6 2.376E-8 1.572E-7 6.120E-6
3 3.286E-12 9.051E-7 6.095E-5 8.531E-6 1.212E-5 8.523E-5 1.086E-6 3.697E-6 6.648E-5 1.413E-7 1.720E-6 6.209E-5
4 1.222E-10 7.992E-6 3.299E-4 5.497E-5 6.304E-5 4.196E-4 8.108E-6 2.357E-5 3.509E-4 1.199E-6 1.292E-5 3.342E-4
5 2.594E-9 5.980E-5 1.534E-3 4.338E-4 3.601E-4 1.866E-3 7.449E-5 1.501E-4 1.612E-3 1.236E-5 8.935E-5 1.550E-3
6 9.325E-8 5.073E-4 7.786E-3 4.723E-3 2.603E-3 9.290E-3 1.033E-3 1.166E-3 8.143E-3 2.060E-4 7.277E-4 7.861E-3
7 3.798E-6 7.250E-3 6.395E-2 6.800E-2 3.132E-2 7.437E-2 2.038E-2 1.541E-2 6.646E-2 5.061E-3 1.007E-2 6.448E-2
8 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=200$[]{data-label="tabl:Bq200dDaAAk3p"}
------------------------ ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4** ]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 7.638E-15 1.067E-8 1.408E-6 1.583E-6 1.869E-6 4.909E-6 1.712E-7 2.934E-7 2.123E-6 1.875E-8 6.576E-8 1.557E-6
3 2.997E-13 1.675E-7 1.476E-5 4.829E-6 6.463E-6 2.991E-5 6.006E-7 1.551E-6 1.843E-5 7.581E-8 5.396E-7 1.557E-5
4 8.537E-12 1.367E-6 7.883E-5 2.004E-5 2.489E-5 1.305E-4 2.866E-6 7.575E-6 9.211E-5 4.102E-7 3.277E-6 8.181E-5
5 1.327E-10 8.402E-6 3.247E-4 9.640E-5 9.927E-5 4.808E-4 1.584E-5 3.485E-5 3.659E-4 2.539E-6 1.712E-5 3.340E-4
6 2.176E-9 4.693E-5 1.202E-3 5.279E-4 4.250E-4 1.678E-3 1.022E-4 1.640E-4 1.329E-3 1.870E-5 8.699E-5 1.231E-3
7 3.319E-8 2.769E-4 4.627E-3 3.269E-3 2.081E-3 6.240E-3 7.655E-4 8.666E-4 5.062E-3 1.628E-4 4.846E-4 4.726E-3
8 8.824E-7 2.046E-3 2.121E-2 2.446E-2 1.277E-2 2.779E-2 7.595E-3 5.816E-3 2.301E-2 2.047E-3 3.421E-3 2.163E-2
9 3.160E-5 2.539E-2 1.455E-1 2.088E-1 1.146E-1 1.783E-1 9.618E-2 6.171E-2 1.548E-1 3.562E-2 3.961E-2 1.476E-1
10 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
------------------------ ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=200$[]{data-label="tabl:Bq200dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 9.914E-16 2.321E-9 3.740E-7 1.406E-6 1.571E-6 2.915E-6 1.499E-7 2.141E-7 8.567E-7 1.614E-8 3.740E-8 4.747E-7
3 3.723E-14 3.622E-8 3.963E-6 3.335E-6 4.217E-6 1.358E-5 4.043E-7 8.434E-7 6.325E-6 4.970E-8 2.300E-7 4.515E-6
4 9.388E-13 2.890E-7 2.140E-5 1.040E-5 1.308E-5 5.246E-5 1.440E-6 3.358E-6 2.988E-5 2.004E-7 1.174E-6 2.346E-5
5 1.267E-11 1.671E-6 8.689E-5 3.698E-5 4.200E-5 1.744E-4 5.824E-6 1.270E-5 1.121E-4 9.060E-7 5.198E-6 9.318E-5
6 1.558E-10 8.159E-6 2.985E-4 1.428E-4 1.388E-4 5.293E-4 2.578E-5 4.705E-5 3.670E-4 4.505E-6 2.140E-5 3.158E-4
7 1.643E-9 3.726E-5 9.524E-4 5.914E-4 4.836E-4 1.563E-3 1.239E-4 1.788E-4 1.137E-3 2.452E-5 8.751E-5 9.991E-4
8 1.974E-8 1.742E-4 3.058E-3 2.653E-3 1.833E-3 4.780E-3 6.642E-4 7.327E-4 3.584E-3 1.530E-4 3.795E-4 3.192E-3
9 2.554E-7 9.167E-4 1.076E-2 1.293E-2 7.858E-3 1.614E-2 3.991E-3 3.430E-3 1.243E-2 1.100E-3 1.877E-3 1.119E-2
10 5.733E-6 6.121E-3 4.509E-2 6.968E-2 4.009E-2 6.403E-2 2.859E-2 1.980E-2 5.114E-2 1.023E-2 1.165E-2 4.666E-2
11 1.856E-4 6.433E-2 2.511E-1 3.745E-1 2.533E-1 3.151E-1 2.294E-1 1.579E-1 2.729E-1 1.185E-1 1.069E-1 2.569E-1
12 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=200$[]{data-label="tabl:Bq200dDaAAk5p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.531E-16 5.552E-10 1.079E-7 1.300E-6 1.405E-6 2.101E-6 1.371E-7 1.751E-7 4.489E-7 1.457E-8 2.590E-8 1.767E-7
3 5.613E-15 8.652E-9 1.150E-6 2.585E-6 3.114E-6 7.534E-6 3.057E-7 5.413E-7 2.665E-6 3.666E-8 1.213E-7 1.518E-6
4 1.327E-13 6.857E-8 6.264E-6 6.573E-6 8.149E-6 2.549E-5 8.809E-7 1.813E-6 1.157E-5 1.190E-7 5.269E-7 7.649E-6
5 1.665E-12 3.887E-7 2.561E-5 1.892E-5 2.233E-5 7.742E-5 2.862E-6 5.913E-6 4.117E-5 4.311E-7 2.049E-6 2.984E-5
6 1.783E-11 1.809E-6 8.712E-5 5.838E-5 6.244E-5 2.157E-4 9.969E-6 1.877E-5 1.278E-4 1.671E-6 7.376E-6 9.847E-5
7 1.574E-10 7.499E-6 2.644E-4 1.892E-4 1.789E-4 5.700E-4 3.667E-5 5.934E-5 3.643E-4 6.863E-6 2.556E-5 2.927E-4
8 1.388E-9 2.951E-5 7.548E-4 6.419E-4 5.331E-4 1.482E-3 1.430E-4 1.919E-4 9.974E-4 3.024E-5 8.864E-5 8.242E-4
9 1.202E-8 1.169E-4 2.142E-3 2.283E-3 1.680E-3 3.941E-3 5.936E-4 6.535E-4 2.751E-3 1.437E-4 3.201E-4 2.317E-3
10 1.226E-7 4.955E-4 6.365E-3 8.560E-3 5.709E-3 1.111E-2 2.664E-3 2.418E-3 8.000E-3 7.587E-4 1.254E-3 6.839E-3
11 1.429E-6 2.398E-3 2.091E-2 3.382E-2 2.138E-2 3.436E-2 1.298E-2 1.006E-2 2.566E-2 4.476E-3 5.587E-3 2.230E-2
12 2.828E-5 1.448E-2 7.967E-2 1.388E-1 8.991E-2 1.196E-1 6.985E-2 4.918E-2 9.442E-2 3.125E-2 3.010E-2 8.407E-2
13 8.278E-4 1.258E-1 3.597E-1 5.167E-1 4.039E-1 4.512E-1 3.750E-1 2.873E-1 3.967E-1 2.418E-1 2.105E-1 3.712E-1
14 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=200$[]{data-label="tabl:Bq200dDaAAk6p"}
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.429E-10 3.334E-8 1.232E-6 1.302E-6 1.706E-6 1.287E-7 1.531E-7 2.895E-7 1.354E-8 2.029E-8 8.152E-8
3 2.226E-9 3.560E-7 2.153E-6 2.496E-6 4.868E-6 2.492E-7 3.894E-7 1.351E-6 2.922E-8 7.475E-8 5.986E-7
4 1.761E-8 1.950E-6 4.696E-6 5.685E-6 1.439E-5 6.104E-7 1.122E-6 5.268E-6 8.018E-8 2.789E-7 2.851E-6
5 9.917E-8 8.024E-6 1.154E-5 1.374E-5 3.992E-5 1.681E-6 3.236E-6 1.756E-5 2.452E-7 9.666E-7 1.078E-5
6 4.538E-7 2.744E-5 3.029E-5 3.395E-5 1.035E-4 4.931E-6 9.145E-6 5.196E-5 7.952E-7 3.134E-6 3.483E-5
7 1.811E-6 8.271E-5 8.279E-5 8.516E-5 2.543E-4 1.509E-5 2.550E-5 1.410E-4 2.691E-6 9.711E-6 1.008E-4
8 6.633E-6 2.284E-4 2.338E-4 2.180E-4 6.043E-4 4.799E-5 7.124E-5 3.605E-4 9.511E-6 2.946E-5 2.701E-4
9 2.326E-5 5.995E-4 6.808E-4 5.745E-4 1.422E-3 1.586E-4 2.029E-4 8.953E-4 3.521E-5 8.973E-5 6.941E-4
10 8.135E-5 1.551E-3 2.046E-3 1.575E-3 3.396E-3 5.477E-4 5.999E-4 2.227E-3 1.379E-4 2.813E-4 1.769E-3
11 2.957E-4 4.107E-3 6.357E-3 4.536E-3 8.422E-3 1.983E-3 1.874E-3 5.718E-3 5.744E-4 9.314E-4 4.631E-3
12 1.166E-3 1.154E-2 2.044E-2 1.388E-2 2.213E-2 7.587E-3 6.302E-3 1.559E-2 2.588E-3 3.342E-3 1.287E-2
13 5.231E-3 3.552E-2 6.753E-2 4.554E-2 6.258E-2 3.068E-2 2.329E-2 4.625E-2 1.267E-2 1.340E-2 3.912E-2
14 2.846E-2 1.228E-1 2.217E-1 1.591E-1 1.901E-1 1.296E-1 9.607E-2 1.511E-1 6.863E-2 6.209E-2 1.326E-1
15 2.028E-1 4.575E-1 6.251E-1 5.337E-1 5.664E-1 5.017E-1 4.181E-1 5.081E-1 3.716E-1 3.295E-1 4.759E-1
16 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=200$[]{data-label="tabl:Bq200dDaAAk7p"}
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 3.908E-11 1.091E-8 1.184E-6 1.234E-6 1.488E-6 1.229E-7 1.395E-7 2.160E-7 1.283E-8 1.718E-8 4.602E-8
3 6.087E-10 1.166E-7 1.881E-6 2.115E-6 3.519E-6 2.136E-7 3.036E-7 7.989E-7 2.456E-8 5.160E-8 2.773E-7
4 4.812E-9 6.404E-7 3.641E-6 4.298E-6 9.141E-6 4.604E-7 7.681E-7 2.765E-6 5.891E-8 1.674E-7 1.219E-6
5 2.704E-8 2.647E-6 7.909E-6 9.366E-6 2.319E-5 1.113E-6 1.991E-6 8.547E-6 1.575E-7 5.220E-7 4.399E-6
6 1.230E-7 9.100E-6 1.832E-5 2.096E-5 5.612E-5 2.861E-6 5.112E-6 2.398E-5 4.452E-7 1.546E-6 1.378E-5
7 4.841E-7 2.754E-5 4.409E-5 4.756E-5 1.299E-4 7.634E-6 1.295E-5 6.226E-5 1.304E-6 4.391E-6 3.894E-5
8 1.721E-6 7.576E-5 1.090E-4 1.092E-4 2.900E-4 2.097E-5 3.262E-5 1.522E-4 3.942E-6 1.212E-5 1.017E-4
9 5.715E-6 1.945E-4 2.757E-4 2.552E-4 6.336E-4 5.920E-5 8.254E-5 3.569E-4 1.229E-5 3.307E-5 2.507E-4
10 1.827E-5 4.772E-4 7.124E-4 6.094E-4 1.375E-3 1.719E-4 2.122E-4 8.182E-4 3.968E-5 9.065E-5 5.970E-4
11 5.794E-5 1.150E-3 1.881E-3 1.498E-3 3.017E-3 5.141E-4 5.609E-4 1.873E-3 1.331E-4 2.539E-4 1.407E-3
12 1.878E-4 2.796E-3 5.083E-3 3.809E-3 6.787E-3 1.590E-3 1.541E-3 4.375E-3 4.669E-4 7.389E-4 3.363E-3
13 6.407E-4 7.047E-3 1.405E-2 1.009E-2 1.587E-2 5.094E-3 4.448E-3 1.062E-2 1.720E-3 2.270E-3 8.348E-3
14 2.374E-3 1.884E-2 3.963E-2 2.796E-2 3.896E-2 1.696E-2 1.364E-2 2.726E-2 6.712E-3 7.489E-3 2.195E-2
15 9.900E-3 5.451E-2 1.126E-1 8.102E-2 1.007E-1 5.832E-2 4.489E-2 7.471E-2 2.778E-2 2.701E-2 6.215E-2
16 4.853E-2 1.713E-1 3.079E-1 2.393E-1 2.683E-1 2.012E-1 1.575E-1 2.167E-1 1.209E-1 1.077E-1 1.892E-1
17 2.850E-1 5.398E-1 7.049E-1 6.346E-1 6.567E-1 6.020E-1 5.311E-1 6.004E-1 4.865E-1 4.431E-1 5.651E-1
18 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=200$[]{data-label="tabl:Bq200dDaAAk8p"}
---------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{}/$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 4.989E-15 1.484E-8 2.509E-6 1.875E-6 1.997E-6 6.499E-6 2.014E-7 3.192E-7 3.334E-6 2.171E-8 7.570E-8 2.679E-6
3 2.141E-13 2.390E-7 2.589E-5 8.079E-6 7.922E-6 4.368E-5 9.784E-7 1.884E-6 3.008E-5 1.174E-7 6.794E-7 2.679E-5
4 8.012E-12 2.119E-6 1.405E-4 5.128E-5 3.763E-5 2.060E-4 7.090E-6 1.085E-5 1.565E-4 9.439E-7 4.713E-6 1.440E-4
5 1.714E-10 1.603E-5 6.613E-4 4.031E-4 2.089E-4 9.101E-4 6.447E-5 6.660E-5 7.228E-4 9.520E-6 3.170E-5 6.746E-4
6 6.305E-9 1.391E-4 3.454E-3 4.410E-3 1.527E-3 4.643E-3 8.962E-4 5.205E-4 3.749E-3 1.579E-4 2.601E-4 3.518E-3
7 2.638E-7 2.105E-3 3.106E-2 6.442E-2 1.942E-2 4.062E-2 1.799E-2 7.252E-3 3.347E-2 3.945E-3 3.791E-3 3.158E-2
8 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
---------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=400$[]{data-label="tabl:Bq400dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 5.056E-16 2.919E-9 6.241E-7 1.560E-6 1.630E-6 3.451E-6 1.659E-7 2.231E-7 1.179E-6 1.768E-8 4.000E-8 7.402E-7
3 1.987E-14 4.586E-8 6.535E-6 4.650E-6 4.751E-6 1.773E-5 5.574E-7 9.366E-7 9.280E-6 6.616E-8 2.597E-7 7.162E-6
4 5.685E-13 3.750E-7 3.487E-5 1.903E-5 1.663E-5 7.202E-5 2.591E-6 4.067E-6 4.476E-5 3.415E-7 1.414E-6 3.720E-5
5 8.888E-12 2.320E-6 1.442E-4 9.104E-5 6.357E-5 2.569E-4 1.414E-5 1.767E-5 1.752E-4 2.065E-6 6.990E-6 1.516E-4
6 1.474E-10 1.312E-5 5.404E-4 4.983E-4 2.699E-4 8.930E-4 9.091E-5 8.159E-5 6.388E-4 1.505E-5 3.486E-5 5.640E-4
7 2.282E-9 7.934E-5 2.139E-3 3.097E-3 1.343E-3 3.397E-3 6.841E-4 4.362E-4 2.493E-3 1.312E-4 1.961E-4 2.224E-3
8 6.301E-8 6.139E-4 1.035E-2 2.339E-2 8.575E-3 1.594E-2 6.883E-3 3.049E-3 1.195E-2 1.680E-3 1.441E-3 1.073E-2
9 2.363E-6 8.453E-3 8.187E-2 2.030E-1 8.414E-2 1.168E-1 8.945E-2 3.572E-2 9.232E-2 3.025E-2 1.844E-2 8.443E-2
10 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=400$[]{data-label="tabl:Bq400dDaAAk4p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 6.722E-17 6.647E-10 1.729E-7 1.392E-6 1.437E-6 2.331E-6 1.468E-7 1.794E-7 5.583E-7 1.550E-8 2.689E-8 2.521E-7
3 2.528E-15 1.038E-8 1.830E-6 3.245E-6 3.368E-6 9.153E-6 3.827E-7 5.787E-7 3.602E-6 4.486E-8 1.310E-7 2.256E-6
4 6.397E-14 8.288E-8 9.867E-6 1.000E-5 9.533E-6 3.246E-5 1.332E-6 2.046E-6 1.610E-5 1.735E-7 5.912E-7 1.145E-5
5 8.673E-13 4.808E-7 4.006E-5 3.532E-5 2.913E-5 1.027E-4 5.319E-6 7.229E-6 5.852E-5 7.664E-7 2.437E-6 4.491E-5
6 1.075E-11 2.365E-6 1.381E-4 1.361E-4 9.439E-5 3.044E-4 2.342E-5 2.591E-5 1.888E-4 3.766E-6 9.666E-6 1.517E-4
7 1.145E-10 1.096E-5 4.462E-4 5.641E-4 3.292E-4 8.978E-4 1.126E-4 9.780E-5 5.862E-4 2.044E-5 3.910E-5 4.838E-4
8 1.403E-9 5.255E-5 1.469E-3 2.539E-3 1.269E-3 2.799E-3 6.059E-4 4.067E-4 1.887E-3 1.281E-4 1.717E-4 1.582E-3
9 1.860E-8 2.883E-4 5.400E-3 1.245E-2 5.613E-3 9.844E-3 3.673E-3 1.971E-3 6.821E-3 9.323E-4 8.793E-4 5.787E-3
10 4.429E-7 2.070E-3 2.442E-2 6.765E-2 3.016E-2 4.187E-2 2.674E-2 1.211E-2 3.018E-2 8.894E-3 5.833E-3 2.601E-2
11 1.542E-5 2.547E-2 1.627E-1 3.688E-1 2.107E-1 2.395E-1 2.202E-1 1.100E-1 1.903E-1 1.073E-1 6.169E-2 1.706E-1
12 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=400$[]{data-label="tabl:Bq400dDaAAk5p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.042E-17 1.670E-10 5.185E-8 1.291E-6 1.322E-6 1.822E-6 1.351E-7 1.556E-7 3.347E-7 1.415E-8 2.078E-8 1.067E-7
3 3.926E-16 2.603E-9 5.520E-7 2.533E-6 2.635E-6 5.621E-6 2.933E-7 4.080E-7 1.707E-6 3.389E-8 7.887E-8 8.348E-7
4 9.332E-15 2.064E-8 3.005E-6 6.373E-6 6.344E-6 1.740E-5 8.279E-7 1.221E-6 6.913E-6 1.060E-7 3.025E-7 4.056E-6
5 1.174E-13 1.172E-7 1.227E-5 1.823E-5 1.651E-5 4.989E-5 2.657E-6 3.707E-6 2.355E-5 3.755E-7 1.087E-6 1.546E-5
6 1.264E-12 5.475E-7 4.173E-5 5.608E-5 4.494E-5 1.341E-4 9.196E-6 1.129E-5 7.113E-5 1.437E-6 3.723E-6 5.033E-5
7 1.125E-11 2.288E-6 1.271E-4 1.816E-4 1.277E-4 3.482E-4 3.376E-5 3.500E-5 2.000E-4 5.873E-6 1.258E-5 1.488E-4
8 1.006E-10 9.142E-6 3.668E-4 6.167E-4 3.824E-4 9.056E-4 1.318E-4 1.132E-4 5.481E-4 2.588E-5 4.341E-5 4.214E-4
9 8.865E-10 3.715E-5 1.064E-3 2.198E-3 1.225E-3 2.449E-3 5.491E-4 3.916E-4 1.537E-3 1.236E-4 1.589E-4 1.207E-3
10 9.310E-9 1.636E-4 3.278E-3 8.277E-3 4.268E-3 7.130E-3 2.481E-3 1.493E-3 4.622E-3 6.596E-4 6.423E-4 3.689E-3
11 1.128E-7 8.405E-4 1.140E-2 3.288E-2 1.657E-2 2.318E-2 1.221E-2 6.506E-3 1.563E-2 3.954E-3 3.011E-3 1.271E-2
12 2.432E-6 5.596E-3 4.761E-2 1.360E-1 7.355E-2 8.710E-2 6.671E-2 3.418E-2 6.255E-2 2.832E-2 1.762E-2 5.235E-2
13 7.922E-5 5.963E-2 2.611E-1 5.122E-1 3.624E-1 3.795E-1 3.662E-1 2.287E-1 3.113E-1 2.280E-1 1.451E-1 2.779E-1
14 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=400$[]{data-label="tabl:Bq400dDaAAk6p"}
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 4.510E-11 1.658E-8 1.226E-6 1.248E-6 1.555E-6 1.274E-7 1.411E-7 2.379E-7 1.326E-8 1.747E-8 5.592E-8
3 7.025E-10 1.770E-7 2.121E-6 2.200E-6 3.916E-6 2.414E-7 3.141E-7 9.565E-7 2.748E-8 5.370E-8 3.640E-7
4 5.558E-9 9.685E-7 4.582E-6 4.655E-6 1.063E-5 5.806E-7 8.170E-7 3.454E-6 7.297E-8 1.779E-7 1.649E-6
5 3.133E-8 3.982E-6 1.120E-5 1.070E-5 2.780E-5 1.580E-6 2.195E-6 1.095E-5 2.185E-7 5.688E-7 6.040E-6
6 1.436E-7 1.360E-5 2.927E-5 2.563E-5 6.910E-5 4.604E-6 5.923E-6 3.128E-5 6.995E-7 1.745E-6 1.909E-5
7 5.757E-7 4.099E-5 7.989E-5 6.334E-5 1.652E-4 1.404E-5 1.608E-5 8.283E-5 2.351E-6 5.220E-6 5.442E-5
8 2.126E-6 1.136E-4 2.256E-4 1.617E-4 3.877E-4 4.463E-5 4.446E-5 2.092E-4 8.292E-6 1.557E-5 1.449E-4
9 7.567E-6 3.008E-4 6.576E-4 4.292E-4 9.135E-4 1.478E-4 1.271E-4 5.196E-4 3.075E-5 4.738E-5 3.736E-4
10 2.711E-5 7.928E-4 1.981E-3 1.193E-3 2.213E-3 5.120E-4 3.815E-4 1.310E-3 1.210E-4 1.506E-4 9.663E-4
11 1.021E-4 2.164E-3 6.171E-3 3.510E-3 5.636E-3 1.863E-3 1.223E-3 3.455E-3 5.084E-4 5.123E-4 2.602E-3
12 4.238E-4 6.356E-3 1.992E-2 1.104E-2 1.538E-2 7.181E-3 4.263E-3 9.810E-3 2.319E-3 1.916E-3 7.543E-3
13 2.049E-3 2.089E-2 6.615E-2 3.754E-2 4.583E-2 2.931E-2 1.656E-2 3.084E-2 1.154E-2 8.146E-3 2.438E-2
14 1.253E-2 7.991E-2 2.186E-1 1.380E-1 1.503E-1 1.255E-1 7.349E-2 1.100E-1 6.397E-2 4.126E-2 9.083E-2
15 1.128E-1 3.610E-1 6.218E-1 4.998E-1 5.073E-1 4.945E-1 3.623E-1 4.320E-1 3.584E-1 2.578E-1 3.887E-1
16 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=400$[]{data-label="tabl:Bq400dDaAAk7p"}
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 0.000E0 0.000E0 1.000E-6 1.000E-6 1.000E-6 1.000E-7 1.000E-7 1.000E-7 1.000E-8 1.000E-8 1.000E-8
2 1.290E-11 5.594E-9 1.180E-6 1.196E-6 1.398E-6 1.220E-7 1.317E-7 1.894E-7 1.263E-8 1.547E-8 3.516E-8
3 2.009E-10 5.979E-8 1.859E-6 1.920E-6 2.990E-6 2.084E-7 2.575E-7 6.125E-7 2.340E-8 3.996E-8 1.853E-7
4 1.588E-9 3.281E-7 3.570E-6 3.660E-6 7.186E-6 4.421E-7 5.943E-7 1.960E-6 5.451E-8 1.161E-7 7.667E-7
5 8.928E-9 1.355E-6 7.710E-6 7.599E-6 1.722E-5 1.057E-6 1.440E-6 5.749E-6 1.428E-7 3.348E-7 2.660E-6
6 4.064E-8 4.654E-6 1.779E-5 1.647E-5 3.991E-5 2.697E-6 3.525E-6 1.551E-5 3.987E-7 9.367E-7 8.108E-6
7 1.603E-7 1.407E-5 4.274E-5 3.668E-5 8.945E-5 7.170E-6 8.658E-6 3.910E-5 1.159E-6 2.554E-6 2.243E-5
8 5.723E-7 3.869E-5 1.056E-4 8.360E-5 1.956E-4 1.967E-5 2.143E-5 9.363E-5 3.491E-6 6.875E-6 5.769E-5
9 1.916E-6 9.957E-5 2.671E-4 1.954E-4 4.233E-4 5.552E-5 5.396E-5 2.170E-4 1.088E-5 1.854E-5 1.412E-4
10 6.214E-6 2.462E-4 6.909E-4 4.701E-4 9.205E-4 1.614E-4 1.395E-4 4.972E-4 3.519E-5 5.090E-5 3.365E-4
11 2.015E-5 6.026E-4 1.828E-3 1.170E-3 2.044E-3 4.843E-4 3.739E-4 1.151E-3 1.186E-4 1.445E-4 8.021E-4
12 6.743E-5 1.503E-3 4.950E-3 3.027E-3 4.700E-3 1.504E-3 1.050E-3 2.747E-3 4.189E-4 4.305E-4 1.961E-3
13 2.406E-4 3.927E-3 1.372E-2 8.192E-3 1.133E-2 4.845E-3 3.121E-3 6.890E-3 1.557E-3 1.368E-3 5.034E-3
14 9.475E-4 1.104E-2 3.884E-2 2.331E-2 2.892E-2 1.624E-2 9.939E-3 1.847E-2 6.153E-3 4.723E-3 1.387E-2
15 4.303E-3 3.426E-2 1.108E-1 6.988E-2 7.868E-2 5.633E-2 3.435E-2 5.371E-2 2.585E-2 1.811E-2 4.188E-2
16 2.403E-2 1.196E-1 3.049E-1 2.160E-1 2.253E-1 1.965E-1 1.292E-1 1.698E-1 1.149E-1 7.890E-2 1.402E-1
17 1.807E-1 4.518E-1 7.024E-1 6.088E-1 6.108E-1 5.964E-1 4.844E-1 5.378E-1 4.753E-1 3.766E-1 4.900E-1
18 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0 1.000E0
----------------------- -------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The extreme points of ${\varepsilon}(q,\delta,\alpha,k;\t)$ of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=400$[]{data-label="tabl:Bq400dDaAAk8p"}
Tables type **(c)** for BURA-poles
----------------------------------
Now we provide the data about the poles of the BURA element $r_{q,\delta,\alpha,k}(\t)$:
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.36E-5 -8.81E-4 -5.45E-3 -9.29E-5 -1.04E-3 -5.53E-3 -4.51E-5 -9.32E-4 -5.46E-3 -2.78E-5 -8.97E-4 -5.45E-3
2 -4.78E-3 -4.93E-2 -1.49E-1 -1.01E-2 -5.24E-2 -1.50E-1 -7.49E-3 -5.03E-2 -1.50E-1 -6.21E-3 -4.97E-2 -1.50E-1
3 -4.14E-1 -1.77E0 -6.09E0 -5.48E-1 -1.82E0 -6.11E0 -4.88E-1 -1.79E0 -6.10E0 -4.55E-1 -1.78E0 -6.10E0
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=0$[]{data-label="tabl:Cq000dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -5.61E-7 -9.17E-5 -8.37E-4 -1.95E-5 -1.49E-4 -8.91E-4 -6.35E-6 -1.10E-4 -8.48E-4 -2.67E-6 -9.75E-5 -8.39E-4
2 -1.97E-4 -4.98E-3 -2.11E-2 -1.10E-3 -6.15E-3 -2.17E-2 -6.01E-4 -5.38E-3 -2.13E-2 -3.92E-4 -5.11E-3 -2.12E-2
3 -1.41E-2 -1.01E-1 -2.68E-1 -3.27E-2 -1.12E-1 -2.72E-1 -2.41E-2 -1.05E-1 -2.69E-1 -1.95E-2 -1.03E-1 -2.68E-1
4 -6.27E-1 -2.49E0 -8.40E0 -9.10E-1 -2.62E0 -8.47E0 -7.90E-1 -2.53E0 -8.42E0 -7.20E-1 -2.50E0 -8.41E0
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=0$[]{data-label="tabl:Cq000dDaAAk4p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -3.27E-8 -1.22E-5 -1.59E-4 -7.21E-6 -3.56E-5 -1.92E-4 -1.77E-6 -1.94E-5 -1.66E-4 -5.28E-7 -1.45E-5 -1.60E-4
2 -1.15E-5 -6.62E-4 -3.97E-3 -2.29E-4 -1.12E-3 -4.33E-3 -9.42E-5 -8.22E-4 -4.05E-3 -4.65E-5 -7.15E-4 -3.98E-3
3 -8.15E-4 -1.28E-2 -4.47E-2 -4.60E-3 -1.70E-2 -4.69E-2 -2.68E-3 -1.44E-2 -4.52E-2 -1.77E-3 -1.33E-2 -4.48E-2
4 -2.81E-2 -1.63E-1 -3.97E-1 -7.13E-2 -1.90E-1 -4.08E-1 -5.27E-2 -1.73E-1 -4.00E-1 -4.21E-2 -1.66E-1 -3.98E-1
5 -8.47E-1 -3.21E0 -1.08E1 -1.35E0 -3.52E0 -1.09E1 -1.15E0 -3.33E0 -1.08E1 -1.03E0 -3.25E0 -1.08E1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=0$[]{data-label="tabl:Cq000dDaAAk5p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.45E-9 -1.95E-6 -3.52E-5 -3.62E-6 -1.28E-5 -5.49E-5 -7.44E-7 -5.05E-6 -3.98E-5 -1.74E-7 -2.90E-6 -3.62E-5
2 -8.61E-7 -1.06E-4 -8.75E-4 -7.42E-5 -2.93E-4 -1.10E-3 -2.41E-5 -1.72E-4 -9.30E-4 -9.22E-6 -1.28E-4 -8.87E-4
3 -6.11E-5 -2.03E-3 -9.65E-3 -1.06E-3 -3.74E-3 -1.10E-2 -4.90E-4 -2.70E-3 -1.00E-2 -2.60E-4 -2.27E-3 -9.73E-3
4 -2.07E-3 -2.40E-2 -7.40E-2 -1.22E-2 -3.47E-2 -8.01E-2 -7.43E-3 -2.84E-2 -7.56E-2 -4.98E-3 -2.56E-2 -7.44E-2
5 -4.58E-2 -2.30E-1 -5.32E-1 -1.26E-1 -2.85E-1 -5.58E-1 -9.40E-2 -2.53E-1 -5.39E-1 -7.46E-2 -2.39E-1 -5.34E-1
6 -1.07E0 -3.95E0 -1.31E1 -1.86E0 -4.52E0 -1.36E1 -1.57E0 -4.20E0 -1.32E1 -1.38E0 -4.04E0 -1.32E1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=0$[]{data-label="tabl:Cq000dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.24E-10 -3.58E-7 -8.75E-6 -2.17E-6 -6.05E-6 -2.05E-5 -3.98E-7 -1.84E-6 -1.17E-5 -7.97E-8 -7.90E-7 -9.40E-6
2 -7.85E-8 -1.94E-5 -2.17E-4 -3.21E-5 -1.03E-4 -3.45E-4 -8.69E-6 -4.79E-5 -2.52E-4 -2.70E-6 -2.94E-5 -2.25E-4
3 -5.57E-6 -3.72E-4 -2.39E-3 -3.42E-4 -1.09E-3 -3.18E-3 -1.30E-4 -6.56E-4 -2.61E-3 -5.65E-5 -4.80E-4 -2.44E-3
4 -1.89E-4 -4.34E-3 -1.77E-2 -3.10E-3 -8.76E-3 -2.13E-2 -1.56E-3 -6.26E-3 -1.88E-2 -8.72E-4 -5.11E-3 -1.80E-2
5 -4.08E-3 -3.80E-2 -1.08E-1 -2.51E-2 -5.94E-2 -1.21E-1 -1.58E-2 -4.79E-2 -1.12E-1 -1.08E-2 -4.21E-2 -1.09E-1
6 -6.66E-2 -3.01E-1 -6.71E-1 -1.97E-1 -3.96E-1 -7.24E-1 -1.48E-1 -3.46E-1 -6.87E-1 -1.17E-1 -3.20E-1 -6.75E-1
7 -1.30E00 -4.69E00 -1.55E01 -2.45E00 -5.64E00 -1.64E01 -2.05E00 -5.15E00 -1.58E01 -1.78E00 -4.88E00 -1.56E01
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=0$[]{data-label="tabl:Cq000dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.39E-11 -7.35E-8 -2.38E-6 -1.45E-6 -3.45E-6 -9.54E-6 -2.47E-7 -8.67E-7 -4.20E-6 -4.45E-8 -2.86E-7 -2.82E-6
2 -8.37E-9 -3.98E-6 -5.93E-5 -1.69E-5 -4.52E-5 -1.33E-4 -3.97E-6 -1.71E-5 -8.07E-5 -1.04E-6 -8.49E-6 -6.47E-5
3 -5.95E-7 -7.62E-5 -6.50E-4 -1.41E-4 -3.97E-4 -1.10E-3 -4.56E-5 -2.00E-4 -7.91E-4 -1.66E-5 -1.24E-4 -6.87E-4
4 -2.01E-5 -8.89E-4 -4.79E-3 -1.04E-3 -2.79E-3 -6.84E-3 -4.41E-4 -1.72E-3 -5.47E-3 -2.08E-4 -1.24E-3 -4.97E-3
5 -4.34E-4 -7.65E-3 -2.79E-2 -6.99E-3 -1.67E-2 -3.55E-2 -3.73E-3 -1.19E-2 -3.05E-2 -2.17E-3 -9.55E-3 -2.86E-2
6 -6.88E-3 -5.44E-2 -1.44E-1 -4.40E-2 -9.13E-2 -1.69E-1 -2.85E-2 -7.28E-2 -1.53E-1 -1.97E-2 -6.28E-2 -1.47E-1
7 -8.99E-2 -3.75E-1 -8.13E-1 -2.83E-1 -5.23E-1 -9.06E-1 -2.13E-1 -4.51E-1 -8.46E-1 -1.69E-1 -4.10E-1 -8.22E-1
8 -1.53E00 -5.43E00 -1.79E01 -3.11E00 -6.88E00 -1.95E01 -2.58E00 -6.18E00 -1.85E01 -2.23E00 -5.78E00 -1.81E01
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=0$[]{data-label="tabl:Cq000dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -3.53E-6 -4.86E-4 -3.78E-3 -4.67E-5 -6.08E-4 -3.87E-3 -1.89E-5 -5.25E-4 -3.80E-3 -9.90E-6 -4.98E-4 -3.79E-3
2 -1.43E-3 -2.61E-2 -9.83E-2 -4.41E-3 -2.84E-2 -9.90E-2 -2.88E-3 -2.69E-2 -9.84E-2 -2.17E-3 -2.64E-2 -9.83E-2
3 -1.49E-1 -5.52E-1 -9.17E-1 -2.37E-1 -5.71E-1 -9.20E-1 -1.97E-1 -5.58E-1 -9.18E-1 -1.76E-1 -5.54E-1 -9.17E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=1$[]{data-label="tabl:Cq001dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.66E-7 -5.43E-5 -6.06E-4 -1.26E-5 -9.98E-5 -6.56E-4 -3.57E-6 -6.86E-5 -6.16E-4 -1.28E-6 -5.88E-5 -6.08E-4
2 -6.21E-5 -2.94E-3 -1.53E-2 -5.87E-4 -3.85E-3 -1.59E-2 -2.82E-4 -3.25E-3 -1.54E-2 -1.63E-4 -3.04E-3 -1.53E-2
3 -5.06E-3 -5.62E-2 -1.75E-1 -1.68E-2 -6.42E-2 -1.78E-1 -1.11E-2 -5.90E-2 -1.76E-1 -8.30E-3 -5.72E-2 -1.75E-1
4 -2.51E-1 -7.73E-1 -1.13E00 -4.39E-1 -8.23E-1 -1.13E00 -3.61E-1 -7.91E-1 -1.13E00 -3.14E-1 -7.79E-1 -1.13E00
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=1$[]{data-label="tabl:Cq001dDaAAk4p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.07E-8 -7.62E-6 -1.19E-4 -5.32E-6 -2.67E-5 -1.49E-4 -1.20E-6 -1.34E-5 -1.25E-4 -3.19E-7 -9.42E-6 -1.20E-4
2 -3.88E-6 -4.12E-4 -2.97E-3 -1.44E-4 -7.74E-4 -3.30E-3 -5.31E-5 -5.39E-4 -3.04E-3 -2.33E-5 -4.55E-4 -2.98E-3
3 -2.94E-4 -7.88E-3 -3.32E-2 -2.67E-3 -1.12E-2 -3.52E-2 -1.40E-3 -9.14E-3 -3.37E-2 -8.39E-4 -8.32E-3 -3.33E-2
4 -1.14E-2 -9.28E-2 -2.55E-1 -4.08E-2 -1.12E-1 -2.62E-1 -2.77E-2 -1.00E-1 -2.57E-1 -2.06E-2 -9.55E-2 -2.55E-1
5 -3.65E-1 -9.93E-1 -1.33E00 -6.96E-1 -1.10E00 -1.35E00 -5.67E-1 -1.04E00 -1.33E00 -4.87E-1 -1.01E00 -1.33E00
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=1$[]{data-label="tabl:Cq001dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -8.74E-10 -1.26E-6 -2.69E-5 -2.89E-6 -1.04E-5 -4.47E-5 -5.63E-7 -3.81E-6 -3.11E-5 -1.22E-7 -2.04E-6 -2.78E-5
2 -3.12E-7 -6.83E-5 -6.69E-4 -5.25E-5 -2.20E-4 -8.69E-4 -1.56E-5 -1.22E-4 -7.20E-4 -5.42E-6 -8.68E-5 -6.80E-4
3 -2.29E-5 -1.31E-3 -7.39E-3 -6.84E-4 -2.67E-3 -8.63E-3 -2.89E-4 -1.85E-3 -7.71E-3 -1.40E-4 -1.51E-3 -7.46E-3
4 -8.29E-4 -1.52E-2 -5.59E-2 -7.59E-3 -2.36E-2 -6.13E-2 -4.23E-3 -1.88E-2 -5.73E-2 -2.61E-3 -1.66E-2 -5.62E-2
5 -2.04E-2 -1.34E-1 -3.33E-1 -7.79E-2 -1.72E-1 -3.50E-1 -5.40E-2 -1.51E-1 -3.38E-1 -4.02E-2 -1.40E-1 -3.34E-1
6 -4.85E-1 -1.21E00 -1.53E00 -1.00E00 -1.41E00 -1.57E00 -8.13E-1 -1.30E00 -1.54E00 -6.91E-1 -1.25E00 -1.53E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=1$[]{data-label="tabl:Cq001dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -8.53E-11 -2.39E-7 -6.80E-6 -1.82E-6 -5.18E-6 -1.75E-5 -3.22E-7 -1.50E-6 -9.46E-6 -6.14E-8 -5.98E-7 -7.41E-6
2 -3.02E-8 -1.29E-5 -1.69E-4 -2.46E-5 -8.22E-5 -2.84E-4 -6.22E-6 -3.63E-5 -2.01E-4 -1.79E-6 -2.11E-5 -1.77E-4
3 -2.19E-6 -2.48E-4 -1.86E-3 -2.41E-4 -8.29E-4 -2.57E-3 -8.50E-5 -4.78E-4 -2.07E-3 -3.40E-5 -3.36E-4 -1.91E-3
4 -7.68E-5 -2.88E-3 -1.38E-2 -2.08E-3 -6.42E-3 -1.71E-2 -9.65E-4 -4.43E-3 -1.48E-2 -4.99E-4 -3.51E-3 -1.41E-2
5 -1.77E-3 -2.48E-2 -8.19E-2 -1.65E-2 -4.14E-2 -9.35E-2 -9.62E-3 -3.25E-2 -8.55E-2 -6.11E-3 -2.81E-2 -8.28E-2
6 -3.18E-2 -1.77E-1 -4.09E-1 -1.28E-1 -2.43E-1 -4.40E-1 -9.06E-2 -2.09E-1 -4.19E-1 -6.80E-2 -1.91E-1 -4.12E-1
7 -6.10E-1 -1.43E00 -1.73E00 -1.36E00 -1.75E00 -1.82E00 -1.10E00 -1.59E00 -1.76E00 -9.27E-1 -1.50E00 -1.74E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=1$[]{data-label="tabl:Cq001dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -9.64E-12 -5.02E-8 -1.88E-6 -1.25E-6 -3.06E-6 -8.44E-6 -2.09E-7 -7.40E-7 -3.54E-6 -3.64E-8 -2.31E-7 -2.28E-6
2 -3.40E-9 -2.72E-6 -4.68E-5 -1.36E-5 -3.79E-5 -1.13E-4 -3.05E-6 -1.37E-5 -6.63E-5 -7.56E-7 -6.44E-6 -5.18E-5
3 -2.44E-7 -5.21E-5 -5.14E-4 -1.07E-4 -3.18E-4 -9.17E-4 -3.23E-5 -1.53E-4 -6.42E-4 -1.09E-5 -9.15E-5 -5.48E-4
4 -8.43E-6 -6.07E-4 -3.79E-3 -7.45E-4 -2.16E-3 -5.63E-3 -2.94E-4 -1.28E-3 -4.41E-3 -1.29E-4 -8.90E-4 -3.96E-3
5 -1.89E-4 -5.20E-3 -2.21E-2 -4.85E-3 -1.25E-2 -2.89E-2 -2.41E-3 -8.65E-3 -2.44E-2 -1.31E-3 -6.75E-3 -2.27E-2
6 -3.18E-3 -3.61E-2 -1.10E-1 -3.01E-2 -6.48E-2 -1.31E-1 -1.83E-2 -5.05E-2 -1.18E-1 -1.19E-2 -4.28E-2 -1.12E-1
7 -4.54E-2 -2.23E-1 -4.83E-1 -1.92E-1 -3.24E-1 -5.33E-1 -1.37E-1 -2.75E-1 -5.01E-1 -1.04E-1 -2.48E-1 -4.88E-1
8 -7.38E-1 -1.65E00 -1.93E00 -1.76E00 -2.11E00 -2.08E00 -1.42E00 -1.89E00 -1.99E00 -1.19E00 -1.77E00 -1.95E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=1$[]{data-label="tabl:Cq001dDaAAk8p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.13E-11 -1.69E-6 -1.17E-4 -2.60E-6 -1.11E-5 -1.46E-4 -3.72E-7 -4.44E-6 -1.23E-4 -5.70E-8 -2.54E-6 -1.18E-4
2 -8.24E-9 -8.15E-5 -1.81E-3 -1.36E-4 -2.44E-4 -1.95E-3 -2.44E-5 -1.36E-4 -1.84E-3 -4.62E-6 -1.00E-4 -1.81E-3
3 -5.38E-6 -3.29E-3 -1.92E-2 -1.41E-2 -8.79E-3 -2.12E-2 -3.85E-3 -5.19E-3 -1.96E-2 -1.01E-3 -3.94E-3 -1.93E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=100$[]{data-label="tabl:Cq100dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.09E-12 -3.09E-7 -2.59E-5 -1.51E-6 -5.49E-6 -4.36E-5 -2.14E-7 -1.68E-6 -3.01E-5 -3.18E-8 -7.08E-7 -2.68E-5
2 -5.53E-10 -1.57E-5 -5.91E-4 -3.97E-5 -8.48E-5 -7.44E-4 -6.76E-6 -3.92E-5 -6.31E-4 -1.21E-6 -2.40E-5 -6.00E-4
3 -1.01E-7 -3.05E-4 -3.66E-3 -1.16E-3 -1.18E-3 -4.25E-3 -2.68E-4 -6.15E-4 -3.81E-3 -6.14E-5 -4.17E-4 -3.69E-3
4 -4.08E-5 -1.07E-2 -4.71E-2 -6.07E-2 -3.40E-2 -5.57E-2 -2.27E-2 -1.98E-2 -4.93E-2 -7.80E-3 -1.41E-2 -4.76E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=100$[]{data-label="tabl:Cq100dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.36E-13 -6.38E-8 -6.47E-6 -1.00E-6 -3.21E-6 -1.70E-5 -1.41E-7 -8.09E-7 -9.09E-6 -2.05E-8 -2.63E-7 -7.06E-6
2 -5.88E-11 -3.37E-6 -1.61E-4 -1.75E-5 -3.90E-5 -2.71E-4 -2.86E-6 -1.49E-5 -1.92E-4 -4.87E-7 -7.36E-6 -1.69E-4
3 -6.77E-9 -5.97E-5 -1.36E-3 -2.68E-4 -3.54E-4 -1.76E-3 -5.62E-5 -1.66E-4 -1.48E-3 -1.19E-5 -1.00E-4 -1.39E-3
4 -7.18E-7 -8.70E-4 -7.06E-3 -5.21E-3 -4.17E-3 -9.24E-3 -1.52E-3 -2.12E-3 -7.69E-3 -4.28E-4 -1.36E-3 -7.21E-3
5 -2.13E-4 -2.64E-2 -9.23E-2 -1.56E-1 -8.97E-2 -1.17E-1 -7.23E-2 -5.40E-2 -9.97E-2 -3.13E-2 -3.80E-2 -9.41E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${ r}(q,\delta,\alpha,k;\t)$, , of $r_{q,\delta,\alpha,k}(\t)$, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=100$[]{data-label="tabl:Cq100dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.00E-14 -1.44E-8 -1.78E-6 -7.16E-7 -2.10E-6 -8.23E-6 -1.01E-7 -4.61E-7 -3.41E-6 -1.45E-8 -1.23E-7 -2.17E-6
2 -7.94E-12 -7.75E-7 -4.47E-5 -9.60E-6 -2.11E-5 -1.10E-4 -1.52E-6 -6.78E-6 -6.39E-5 -2.49E-7 -2.74E-6 -4.96E-5
3 -7.40E-10 -1.42E-5 -4.68E-4 -9.83E-5 -1.50E-4 -7.97E-4 -1.92E-5 -6.25E-5 -5.78E-4 -3.83E-6 -3.27E-5 -4.98E-4
4 -4.60E-8 -1.60E-4 -2.35E-3 -1.12E-3 -1.13E-3 -3.30E-3 -2.84E-4 -5.24E-4 -2.67E-3 -7.14E-5 -3.06E-4 -2.44E-3
5 -3.61E-6 -2.06E-3 -1.29E-2 -1.54E-2 -1.13E-2 -1.90E-2 -5.45E-3 -5.86E-3 -1.49E-2 -1.84E-3 -3.65E-3 -1.35E-2
6 -8.23E-4 -5.27E-2 -1.54E-1 -3.01E-1 -1.83E-1 -2.08E-1 -1.60E-1 -1.15E-1 -1.73E-1 -8.17E-2 -8.14E-2 -1.59E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=100$[]{data-label="tabl:Cq100dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -3.29E-15 -3.53E-9 -5.30E-7 -5.41E-7 -1.48E-6 -4.68E-6 -7.58E-8 -2.95E-7 -1.56E-6 -1.08E-8 -6.78E-8 -7.86E-7
2 -1.24E-12 -1.91E-7 -1.32E-5 -6.04E-6 -1.28E-5 -5.16E-5 -9.28E-7 -3.55E-6 -2.48E-5 -1.47E-7 -1.20E-6 -1.65E-5
3 -1.04E-10 -3.59E-6 -1.45E-4 -4.69E-5 -7.71E-5 -3.61E-4 -8.65E-6 -2.83E-5 -2.19E-4 -1.64E-6 -1.27E-5 -1.67E-4
4 -4.98E-9 -3.95E-5 -9.42E-4 -3.76E-4 -4.43E-4 -1.62E-3 -8.67E-5 -1.89E-4 -1.20E-3 -2.01E-5 -1.00E-4 -1.02E-3
5 -2.20E-7 -3.56E-4 -3.73E-3 -3.34E-3 -2.94E-3 -6.03E-3 -1.00E-3 -1.38E-3 -4.56E-3 -2.94E-4 -7.87E-4 -3.99E-3
6 -1.40E-5 -4.18E-3 -2.17E-2 -3.45E-2 -2.48E-2 -3.51E-2 -1.42E-2 -1.34E-2 -2.67E-2 -5.60E-3 -8.32E-3 -2.33E-2
7 -2.48E-3 -9.00E-2 -2.30E-1 -4.93E-1 -3.13E-1 -3.29E-1 -2.87E-1 -2.05E-1 -2.69E-1 -1.63E-1 -1.47E-1 -2.42E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=100$[]{data-label="tabl:Cq100dDaAAk7p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -7.52E-16 -9.23E-10 -1.68E-7 -4.25E-7 -1.10E-6 -2.98E-6 -5.94E-8 -2.05E-7 -8.32E-7 -8.39E-9 -4.22E-8 -3.35E-7
2 3.96E-14 -4.99E-8 -4.19E-6 -4.15E-6 -8.40E-6 -2.78E-5 -6.24E-7 -2.08E-6 -1.11E-5 -9.63E-8 -6.07E-7 -6.22E-6
3 1.25E-11 -9.50E-7 -4.62E-5 -2.64E-5 -4.46E-5 -1.75E-4 -4.65E-6 -1.45E-5 -9.03E-5 -8.40E-7 -5.63E-6 -6.02E-5
4 -3.72E-9 -1.07E-5 -3.33E-4 -1.65E-4 -2.15E-4 -8.29E-4 -3.53E-5 -8.39E-5 -5.27E-4 -7.64E-6 -3.99E-5 -4.00E-4
5 3.72E-9 -8.78E-5 -1.54E-3 -1.09E-3 -1.10E-3 -2.78E-3 -2.92E-4 -4.76E-4 -2.05E-3 -7.74E-5 -2.51E-4 -1.72E-3
6 3.35E-6 -7.02E-4 -5.75E-3 -7.88E-3 -6.51E-3 -1.07E-2 -2.72E-3 -3.13E-3 -7.69E-3 -9.14E-4 -1.78E-3 -6.41E-3
7 2.46E-5 -7.55E-3 -3.36E-2 -6.42E-2 -4.66E-2 -5.90E-2 -2.97E-2 -2.61E-2 -4.40E-2 -1.32E-2 -1.65E-2 -3.73E-2
8 1.03E-2 -1.38E-1 -3.18E-1 -7.30E-1 -4.80E-1 -4.79E-1 -4.50E-1 -3.24E-1 -3.87E-1 -2.75E-1 -2.37E-1 -3.43E-1
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=100$[]{data-label="tabl:Cq100dDaAAk8p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -7.62E-13 -4.65E-7 -5.18E-5 -2.21E-6 -5.95E-6 -7.36E-5 -2.87E-7 -1.99E-6 -5.68E-5 -3.87E-8 -9.21E-7 -5.28E-5
2 -5.67E-10 -2.24E-5 -7.69E-4 -1.17E-4 -1.16E-4 -8.72E-4 -1.85E-5 -5.15E-5 -7.94E-4 -2.92E-6 -3.22E-5 -7.74E-4
3 -3.98E-7 -9.78E-4 -9.03E-3 -1.24E-2 -4.63E-3 -1.08E-2 -3.00E-3 -2.15E-3 -9.44E-3 -6.48E-4 -1.37E-3 -9.12E-3
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=200$[]{data-label="tabl:Cq200dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -7.64E-14 -9.00E-8 -1.21E-5 -1.31E-6 -3.41E-6 -2.54E-5 -1.72E-7 -9.12E-7 -1.53E-5 -2.30E-8 -3.17E-7 -1.28E-5
2 -3.91E-11 -4.53E-6 -2.69E-4 -3.51E-5 -4.49E-5 -3.74E-4 -5.42E-6 -1.72E-5 -2.98E-4 -8.33E-7 -8.90E-6 -2.76E-4
3 -7.43E-9 -9.08E-5 -1.65E-3 -1.04E-3 -6.54E-4 -2.11E-3 -2.16E-4 -2.71E-4 -1.78E-3 -4.20E-5 -1.54E-4 -1.68E-3
4 -3.37E-6 -3.68E-3 -2.49E-2 -5.61E-2 -2.14E-2 -3.28E-2 -1.92E-2 -1.00E-2 -2.71E-2 -5.71E-3 -6.03E-3 -2.54E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=200$[]{data-label="tabl:Cq200dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -9.97E-15 -1.97E-8 -3.15E-6 -8.77E-7 -2.20E-6 -1.12E-5 -1.16E-7 -5.07E-7 -5.18E-6 -1.55E-8 -1.41E-7 -3.63E-6
2 -4.33E-12 -1.04E-6 -7.83E-5 -1.58E-5 -2.28E-5 -1.56E-4 -2.38E-6 -7.56E-6 -1.01E-4 -3.58E-7 -3.21E-6 -8.41E-5
3 -5.11E-10 -1.83E-5 -6.27E-4 -2.43E-4 -2.06E-4 -8.91E-4 -4.67E-5 -7.88E-5 -7.11E-4 -8.63E-6 -4.03E-5 -6.48E-4
4 -5.77E-8 -2.87E-4 -3.50E-3 -4.79E-3 -2.64E-3 -5.30E-3 -1.29E-3 -1.10E-3 -4.04E-3 -3.18E-4 -5.89E-4 -3.64E-3
5 -2.05E-5 -1.05E-2 -5.40E-2 -1.48E-1 -6.45E-2 -7.77E-2 -6.47E-2 -3.27E-2 -6.15E-2 -2.53E-2 -1.96E-2 -5.60E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=200$[]{data-label="tabl:Cq200dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.54E-15 -4.71E-9 -9.03E-7 -6.32E-7 -1.54E-6 -5.94E-6 -8.37E-8 -3.19E-7 -2.16E-6 -1.12E-8 -7.57E-8 -1.21E-6
2 -6.16E-13 -2.53E-7 -2.27E-5 -8.77E-6 -1.35E-5 -7.07E-5 -1.29E-6 -3.90E-6 -3.71E-5 -1.91E-7 -1.38E-6 -2.66E-5
3 -5.84E-11 -4.59E-6 -2.32E-4 -9.01E-5 -9.19E-5 -4.49E-4 -1.63E-5 -3.23E-5 -3.09E-4 -2.90E-6 -1.47E-5 -2.55E-4
4 -3.79E-9 -5.29E-5 -1.12E-3 -1.03E-3 -7.30E-4 -1.83E-3 -2.44E-4 -2.79E-4 -1.36E-3 -5.46E-5 -1.38E-4 -1.19E-3
5 -3.27E-7 -7.57E-4 -7.00E-3 -1.44E-2 -7.94E-3 -1.22E-2 -4.81E-3 -3.46E-3 -8.77E-3 -1.47E-3 -1.84E-3 -7.51E-3
6 -9.57E-5 -2.42E-2 -9.77E-2 -2.89E-1 -1.43E-1 -1.51E-1 -1.48E-1 -7.87E-2 -1.17E-1 -7.01E-2 -4.89E-2 -1.03E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=200$[]{data-label="tabl:Cq200dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -9.85E-15 -1.21E-9 -2.78E-7 -4.79E-7 -1.14E-6 -3.61E-6 -6.37E-8 -2.18E-7 -1.08E-6 -8.54E-9 -4.61E-8 -4.80E-7
2 -4.28E-12 -6.55E-8 -6.96E-6 -5.57E-6 -8.79E-6 -3.61E-5 -8.07E-7 -2.26E-6 -1.57E-5 -1.17E-7 -6.85E-7 -9.46E-6
3 -5.07E-10 -1.23E-6 -7.61E-5 -4.34E-5 -4.96E-5 -2.28E-4 -7.50E-6 -1.59E-5 -1.30E-4 -1.28E-6 -6.38E-6 -9.29E-5
4 1.65E-9 -1.35E-5 -4.68E-4 -3.49E-4 -2.92E-4 -9.06E-4 -7.56E-5 -1.04E-4 -6.39E-4 -1.57E-5 -4.78E-5 -5.26E-4
5 -5.79E-8 -1.28E-4 -1.89E-3 -3.14E-3 -2.07E-3 -3.74E-3 -8.89E-4 -8.18E-4 -2.56E-3 -2.36E-4 -3.99E-4 -2.10E-3
6 -2.05E-5 -1.71E-3 -1.27E-2 -3.28E-2 -1.88E-2 -2.45E-2 -1.29E-2 -8.74E-3 -1.72E-2 -4.68E-3 -4.73E-3 -1.41E-2
7 1.51E-3 -4.63E-2 -1.55E-1 -4.78E-1 -2.60E-1 -2.54E-1 -2.70E-1 -1.53E-1 -1.95E-1 -1.45E-1 -9.90E-2 -1.69E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=200$[]{data-label="tabl:Cq200dDaAAk7p"}
----------------------- ---------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 6.67E-15 -3.32E-10 -9.09E-8 -3.77E-7 -8.79E-7 -2.41E-6 -5.03E-8 -1.59E-7 -6.21E-7 -6.75E-9 -3.08E-8 -2.23E-7
2 2.42E-13 -1.79E-8 -2.27E-6 -3.86E-6 -6.13E-6 -2.07E-5 -5.50E-7 -1.43E-6 -7.57E-6 -7.86E-8 -3.81E-7 -3.83E-6
3 2.39E-9 -3.41E-7 -2.50E-5 -2.46E-5 -3.02E-5 -1.21E-4 -4.09E-6 -8.86E-6 -5.78E-5 -6.76E-7 -3.12E-6 -3.57E-5
4 1.00E-8 -3.82E-6 -1.78E-4 -1.54E-4 -1.45E-4 -5.07E-4 -3.11E-5 -4.87E-5 -3.12E-4 -6.13E-6 -2.04E-5 -2.26E-4
5 4.96E-8 -3.16E-5 -7.72E-4 -1.03E-3 -7.80E-4 -1.66E-3 -2.60E-4 -2.86E-4 -1.12E-3 -6.29E-5 -1.30E-4 -8.99E-4
6 5.10E-7 -2.74E-4 -3.13E-3 -7.49E-3 -4.88E-3 -7.28E-3 -2.47E-3 -2.03E-3 -4.75E-3 -7.62E-4 -9.99E-4 -3.70E-3
7 3.81E-5 -3.40E-3 -2.08E-2 -6.18E-2 -3.73E-2 -4.38E-2 -2.75E-2 -1.85E-2 -3.03E-2 -1.15E-2 -1.03E-2 -2.43E-2
8 9.99E-3 -7.77E-2 -2.25E-1 -7.11E-1 -4.13E-1 -3.87E-1 -4.29E-1 -2.57E-1 -2.97E-1 -2.52E-1 -1.72E-1 -2.52E-1
----------------------- ---------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=200$[]{data-label="tabl:Cq200dDaAAk8p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -4.95E-14 -1.22E-7 -2.20E-5 -2.02E-6 -3.37E-6 -3.77E-5 -2.46E-7 -9.68E-7 -2.58E-5 -3.02E-8 -3.65E-7 -2.28E-5
2 -3.72E-11 -5.90E-6 -3.19E-4 -1.08E-4 -6.26E-5 -3.92E-4 -1.58E-5 -2.14E-5 -3.38E-4 -2.22E-6 -1.09E-5 -3.23E-4
3 -2.71E-8 -2.70E-4 -4.02E-3 -1.16E-2 -2.72E-3 -5.41E-3 -2.60E-3 -9.55E-4 -4.37E-3 -4.94E-4 -4.89E-4 -4.10E-3
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=400$[]{data-label="tabl:Cq400dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -5.06E-15 -2.46E-8 -5.36E-6 -1.21E-6 -2.15E-6 -1.51E-5 -1.50E-7 -5.23E-7 -7.80E-6 -1.85E-8 -1.53E-7 -5.92E-6
2 -2.61E-12 -1.24E-6 -1.17E-4 -3.29E-5 -2.62E-5 -1.86E-4 -4.78E-6 -8.07E-6 -1.38E-4 -6.66E-7 -3.47E-6 -1.22E-4
3 -5.04E-10 -2.52E-5 -7.21E-4 -9.75E-4 -4.05E-4 -1.07E-3 -1.91E-4 -1.30E-4 -8.17E-4 -3.34E-5 -5.94E-5 -7.44E-4
4 -2.44E-7 -1.12E-3 -1.23E-2 -5.37E-2 -1.45E-2 -1.90E-2 -1.75E-2 -5.33E-3 -1.42E-2 -4.71E-3 -2.57E-3 -1.27E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=400$[]{data-label="tabl:Cq400dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -6.76E-16 -5.63E-9 -1.46E-6 -8.11E-7 -1.50E-6 -7.48E-6 -1.02E-7 -3.26E-7 -2.98E-6 -1.27E-8 -8.01E-8 -1.83E-6
2 -2.95E-13 -2.96E-7 -3.61E-5 -1.49E-5 -1.43E-5 -8.95E-5 -2.14E-6 -4.00E-6 -5.26E-5 -2.97E-7 -1.45E-6 -4.05E-5
3 -3.53E-11 -5.24E-6 -2.77E-4 -2.30E-4 -1.33E-4 -4.54E-4 -4.20E-5 -4.03E-5 -3.35E-4 -7.10E-6 -1.69E-5 -2.93E-4
4 -4.12E-9 -8.64E-5 -1.66E-3 -4.58E-3 -1.83E-3 -3.08E-3 -1.18E-3 -6.04E-4 -2.10E-3 -2.66E-4 -2.63E-4 -1.78E-3
5 -1.62E-6 -3.66E-3 -2.94E-2 -1.43E-1 -4.89E-2 -5.10E-2 -6.07E-2 -2.03E-2 -3.65E-2 -2.21E-2 -1.00E-2 -3.14E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=400$[]{data-label="tabl:Cq400dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.08E-16 -1.42E-9 -4.34E-7 -5.86E-7 -1.12E-6 -4.33E-6 -7.42E-8 -2.23E-7 -1.39E-6 -9.35E-9 -4.82E-8 -6.73E-7
2 -4.33E-14 -7.59E-8 -1.09E-5 -8.35E-6 -8.95E-6 -4.53E-5 -1.18E-6 -2.29E-6 -2.14E-5 -1.63E-7 -7.17E-7 -1.39E-5
3 -4.14E-12 -1.37E-6 -1.09E-4 -8.60E-5 -6.14E-5 -2.47E-4 -1.49E-5 -1.76E-5 -1.60E-4 -2.45E-6 -6.77E-6 -1.25E-4
4 -2.75E-10 -1.61E-5 -5.15E-4 -9.90E-4 -5.15E-4 -1.04E-3 -2.24E-4 -1.60E-4 -6.91E-4 -4.64E-5 -6.48E-5 -5.68E-4
5 -2.51E-8 -2.51E-4 -3.60E-3 -1.40E-2 -5.95E-3 -7.87E-3 -4.48E-3 -2.14E-3 -5.08E-3 -1.28E-3 -9.42E-4 -4.06E-3
6 -8.52E-6 -9.61E-3 -5.81E-2 -2.83E-1 -1.17E-1 -1.09E-1 -1.41E-1 -5.52E-2 -7.70E-2 -6.39E-2 -2.91E-2 -6.41E-2
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=400$[]{data-label="tabl:Cq400dDaAAk6p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 9.05E-14 -3.83E-10 -1.38E-7 -4.45E-7 -8.62E-7 -2.79E-6 -5.67E-8 -1.62E-7 -7.58E-7 -7.20E-9 -3.20E-8 -2.93E-7
2 -1.37E-13 -2.06E-8 -3.47E-6 -5.32E-6 -6.13E-6 -2.52E-5 -7.43E-7 -1.45E-6 -9.86E-6 -1.01E-7 -3.97E-7 -5.34E-6
3 3.25E-13 -3.85E-7 -3.77E-5 -4.16E-5 -3.42E-5 -1.40E-4 -6.92E-6 -9.29E-6 -7.50E-5 -1.10E-6 -3.24E-6 -5.00E-5
4 4.90E-10 -4.22E-6 -2.21E-4 -3.36E-4 -2.10E-4 -5.11E-4 -6.99E-5 -6.17E-5 -3.32E-4 -1.36E-5 -2.36E-5 -2.60E-4
5 3.57E-9 -4.21E-5 -9.30E-4 -3.03E-3 -1.56E-3 -2.39E-3 -8.29E-4 -5.15E-4 -1.45E-3 -2.07E-4 -2.10E-4 -1.10E-3
6 7.90E-8 -6.29E-4 -7.01E-3 -3.20E-2 -1.50E-2 -1.72E-2 -1.22E-2 -5.95E-3 -1.09E-2 -4.21E-3 -2.73E-3 -8.34E-3
7 2.77E-4 -2.10E-2 -9.93E-2 -4.70E-1 -2.23E-1 -1.96E-1 -2.61E-1 -1.17E-1 -1.39E-1 -1.36E-1 -6.63E-2 -1.14E-1
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=400$[]{data-label="tabl:Cq400dDaAAk7p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 1.55E-13 -1.10E-10 -4.66E-8 -3.51E-7 -6.88E-7 -1.94E-6 -4.49E-8 -1.23E-7 -4.65E-7 -5.73E-9 -2.27E-8 -1.49E-7
2 2.42E-11 -5.92E-9 -1.16E-6 -3.70E-6 -4.47E-6 -1.53E-5 -5.10E-7 -9.81E-7 -5.13E-6 -6.91E-8 -2.42E-7 -2.34E-6
3 -1.05E-10 -1.12E-7 -1.28E-5 -2.37E-5 -2.14E-5 -8.14E-5 -3.81E-6 -5.51E-6 -3.64E-5 -5.92E-7 -1.74E-6 -2.07E-5
4 3.87E-10 -1.25E-6 -8.93E-5 -1.49E-4 -1.06E-4 -3.00E-4 -2.90E-5 -2.97E-5 -1.77E-4 -5.37E-6 -1.07E-5 -1.23E-4
5 -3.93E-9 -1.05E-5 -3.71E-4 -9.94E-4 -5.94E-4 -1.03E-3 -2.44E-4 -1.83E-4 -6.18E-4 -5.56E-5 -7.02E-5 -4.62E-4
6 -1.59E-7 -9.78E-5 -1.65E-3 -7.29E-3 -3.88E-3 -5.05E-3 -2.33E-3 -1.38E-3 -2.95E-3 -6.84E-4 -5.79E-4 -2.13E-3
7 -1.55E-5 -1.37E-3 -1.23E-2 -6.06E-2 -3.12E-2 -3.26E-2 -2.64E-2 -1.36E-2 -2.07E-2 -1.06E-2 -6.57E-3 -1.55E-2
8 -3.37E-3 -3.93E-2 -1.52E-1 -7.01E-1 -3.66E-1 -3.13E-1 -4.17E-1 -2.08E-1 -2.24E-1 -2.39E-1 -1.25E-1 -1.81E-1
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $r_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-BURA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=400$[]{data-label="tabl:Cq400dDaAAk8p"}
Tables type **(d)** for the coefficients of partial fractions representation of BURA
------------------------------------------------------------------------------------
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.23E00 2.28E00 6.15E00 1.29E00 2.31E00 6.17E00 1.27E00 2.29E00 6.16E00 1.25E00 2.28E00 6.15E00
1 -1.40E-6 -4.18E-5 -9.88E-5 -1.12E-5 -5.15E-5 -1.01E-4 -5.10E-6 -4.48E-5 -9.92E-5 -3.02E-6 -4.28E-5 -9.88E-5
2 -1.37E-3 -1.20E-2 -2.39E-2 -2.99E-3 -1.29E-2 -2.41E-2 -2.19E-3 -1.23E-2 -2.39E-2 -1.80E-3 -1.21E-2 -2.39E-2
3 -3.44E-1 -3.52E00 -3.64E01 -4.60E-1 -3.65E00 -3.66E01 -4.07E-1 -3.56E00 -3.65E01 -3.79E-1 -3.54E00 -3.64E01
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=0$[]{data-label="tabl:Dq000dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.32E00 2.64E00 7.74E00 1.42E00 2.70E00 7.78E00 1.38E00 2.66E00 7.75E00 1.36E00 2.65E00 7.74E00
1 -2.61E-8 -1.40E-6 -3.71E-6 -1.33E-6 -2.62E-6 -4.07E-6 -3.77E-7 -1.77E-6 -3.78E-6 -1.44E-7 -1.52E-6 -3.72E-6
2 -2.53E-5 -3.73E-4 -7.20E-4 -1.63E-4 -4.87E-4 -7.49E-4 -8.52E-5 -4.11E-4 -7.26E-4 -5.35E-5 -3.85E-4 -7.21E-4
3 -4.22E-3 -2.90E-2 -5.64E-2 -1.00E-2 -3.29E-2 -5.75E-2 -7.33E-3 -3.03E-2 -5.66E-2 -5.90E-3 -2.94E-2 -5.64E-2
4 -5.29E-1 -5.63E00 -6.29E01 -7.87E-1 -6.05E00 -6.38E01 -6.76E-1 -5.77E00 -6.31E01 -6.12E-1 -5.68E00 -6.30E01
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=0$[]{data-label="tabl:Dq000dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.40E00 2.96E00 9.24E00 1.54E00 3.08E00 9.36E00 1.49E00 3.01E00 9.27E00 1.45E00 2.98E00 9.25E00
1 -7.46E-10 -6.83E-8 -2.03E-7 -3.25E-7 -2.76E-7 -2.69E-7 -6.50E-8 -1.24E-7 -2.17E-7 -1.65E-8 -8.49E-8 -2.06E-7
2 -7.24E-7 -1.80E-5 -3.80E-5 -2.02E-5 -3.56E-5 -4.34E-5 -7.52E-6 -2.39E-5 -3.92E-5 -3.43E-6 -1.99E-5 -3.83E-5
3 -1.19E-4 -1.24E-3 -2.18E-3 -7.67E-4 -1.78E-3 -2.34E-3 -4.29E-4 -1.43E-3 -2.21E-3 -2.74E-4 -1.30E-3 -2.18E-3
4 -8.58E-3 -5.21E-2 -1.01E-1 -2.23E-2 -6.31E-2 -1.06E-1 -1.64E-2 -5.63E-2 -1.02E-1 -1.30E-2 -5.35E-2 -1.02E-1
5 -7.29E-1 -8.07E00 -9.60E01 -1.21E00 -9.17E00 -9.89E01 -1.02E00 -8.49E00 -9.66E01 -8.99E-1 -8.21E00 -9.61E01
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=0$[]{data-label="tabl:Dq000dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.47E00 3.25E00 1.07E01 1.65E00 3.46E00 1.10E01 1.59E00 3.35E00 1.08E01 1.54E00 3.29E00 1.07E01
1 -2.93E-11 -4.36E-9 -1.45E-8 -1.22E-7 -5.26E-8 -2.84E-8 -1.91E-8 -1.52E-8 -1.74E-8 -3.58E-9 -7.30E-9 -1.51E-8
2 -2.84E-8 -1.15E-6 -2.70E-6 -4.32E-6 -4.42E-6 -3.80E-6 -1.22E-6 -2.18E-6 -2.96E-6 -4.11E-7 -1.49E-6 -2.75E-6
3 -4.65E-6 -7.74E-5 -1.45E-4 -1.09E-4 -1.72E-4 -1.77E-4 -4.67E-5 -1.12E-4 -1.53E-4 -2.32E-5 -8.97E-5 -1.47E-4
4 -3.26E-4 -2.74E-3 -4.59E-3 -2.17E-3 -4.36E-3 -5.16E-3 -1.28E-3 -3.40E-3 -4.74E-3 -8.35E-4 -2.98E-3 -4.63E-3
5 -1.42E-2 -8.01E-2 -1.58E-1 -3.98E-2 -1.05E-1 -1.70E-1 -2.95E-2 -9.06E-2 -1.61E-1 -2.33E-2 -8.41E-2 -1.59E-1
6 -9.41E-1 -1.08E01 -1.35E02 -1.74E00 -1.31E01 -1.43E02 -1.44E00 -1.18E01 -1.37E02 -1.24E00 -1.12E01 -1.36E02
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=0$[]{data-label="tabl:Dq000dDaAAk6p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.53E00 3.52E00 1.21E01 1.75E00 3.84E00 1.26E01 1.68E00 3.68E00 1.22E01 1.63E00 3.59E00 1.21E01
1 -1.47E-12 -3.43E-10 -1.27E-9 -5.85E-8 -1.54E-8 -4.69E-9 -7.84E-9 -3.03E-9 -1.96E-9 -1.19E-9 -9.71E-10 -1.41E-9
2 -1.43E-9 -9.02E-8 -2.35E-7 -1.34E-6 -8.43E-7 -4.81E-7 -3.02E-7 -3.03E-7 -2.96E-7 -7.95E-8 -1.57E-7 -2.49E-7
3 -2.33E-7 -6.06E-6 -1.25E-5 -2.38E-5 -2.52E-5 -1.95E-5 -8.08E-6 -1.29E-5 -1.44E-5 -3.17E-6 -8.52E-6 -1.30E-5
4 -1.63E-5 -2.09E-4 -3.64E-4 -3.53E-4 -5.18E-4 -4.81E-4 -1.67E-4 -3.35E-4 -3.98E-4 -8.82E-5 -2.58E-4 -3.72E-4
5 -6.74E-4 -4.90E-3 -8.00E-3 -4.62E-3 -8.60E-3 -9.53E-3 -2.84E-3 -6.55E-3 -8.45E-3 -1.90E-3 -5.57E-3 -8.11E-3
6 -2.08E-2 -1.13E-1 -2.26E-1 -6.27E-2 -1.60E-1 -2.54E-1 -4.67E-2 -1.34E-1 -2.35E-1 -3.69E-2 -1.22E-1 -2.28E-1
7 -1.16E00 -1.38E01 -1.81E02 -2.38E00 -1.81E01 -1.99E02 -1.94E00 -1.58E01 -1.86E02 -1.66E00 -1.47E01 -1.82E02
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=0$[]{data-label="tabl:Dq000dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.58E00 3.77E00 1.34E01 1.84E00 4.22E00 1.43E01 1.77E00 4.01E00 1.37E01 1.71E00 3.89E00 1.35E01
1 -8.95E-14 -3.18E-11 -1.30E-10 -3.30E-8 -6.06E-9 -1.14E-9 -3.96E-9 -8.83E-10 -3.11E-10 -5.18E-10 -1.95E-10 -1.68E-10
2 -8.69E-11 -8.38E-9 -2.42E-8 -5.40E-7 -2.25E-7 -8.52E-8 -1.02E-7 -6.01E-8 -3.91E-8 -2.19E-8 -2.33E-8 -2.77E-8
3 -1.42E-8 -5.63E-7 -1.28E-6 -7.11E-6 -5.16E-6 -2.91E-6 -1.98E-6 -2.05E-6 -1.74E-6 -6.28E-7 -1.09E-6 -1.40E-6
4 -9.92E-7 -1.93E-5 -3.66E-5 -8.19E-5 -8.73E-5 -6.31E-5 -3.16E-5 -4.60E-5 -4.48E-5 -1.37E-5 -2.98E-5 -3.87E-5
5 -4.09E-5 -4.35E-4 -7.23E-4 -8.49E-4 -1.19E-3 -1.04E-3 -4.31E-4 -7.72E-4 -8.25E-4 -2.40E-4 -5.79E-4 -7.50E-4
6 -1.18E-3 -7.70E-3 -1.24E-2 -8.27E-3 -1.48E-2 -1.57E-2 -5.27E-3 -1.11E-2 -1.35E-2 -3.59E-3 -9.23E-3 -1.27E-2
7 -2.82E-2 -1.49E-1 -3.05E-1 -9.07E-2 -2.28E-1 -3.61E-1 -6.79E-2 -1.88E-1 -3.24E-1 -5.37E-2 -1.67E-1 -3.10E-1
8 -1.40E00 -1.71E01 -2.32E02 -3.14E00 -2.41E01 -2.68E02 -2.53E00 -2.06E01 -2.45E02 -2.13E00 -1.87E01 -2.36E02
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=0$[]{data-label="tabl:Dq000dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.28E-1 6.68E-1 8.42E-1 5.46E-1 6.71E-1 8.42E-1 5.39E-1 6.69E-1 8.42E-1 5.34E-1 6.68E-1 8.42E-1
1 -2.47E-7 -1.71E-5 -5.34E-5 -3.98E-6 -2.26E-5 -5.51E-5 -1.50E-6 -1.88E-5 -5.38E-5 -7.47E-7 -1.76E-5 -5.35E-5
2 -2.39E-4 -4.34E-3 -1.36E-2 -7.37E-4 -4.80E-3 -1.37E-2 -4.83E-4 -4.49E-3 -1.36E-2 -3.64E-4 -4.39E-3 -1.36E-2
3 -4.18E-2 -2.57E-1 -6.33E-1 -6.16E-2 -2.64E-1 -6.34E-1 -5.30E-2 -2.59E-1 -6.33E-1 -4.81E-2 -2.57E-1 -6.33E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=1$[]{data-label="tabl:Dq001dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.49E-1 7.02E-1 8.72E-1 5.73E-1 7.09E-1 8.73E-1 5.64E-1 7.05E-1 8.72E-1 5.58E-1 7.03E-1 8.72E-1
1 -5.56E-9 -6.39E-7 -2.12E-6 -6.63E-7 -1.40E-6 -2.39E-6 -1.63E-7 -8.61E-7 -2.17E-6 -5.23E-8 -7.08E-7 -2.13E-6
2 -5.37E-6 -1.67E-4 -4.34E-4 -5.84E-5 -2.36E-4 -4.57E-4 -2.70E-5 -1.90E-4 -4.39E-4 -1.51E-5 -1.75E-4 -4.35E-4
3 -8.43E-4 -1.07E-2 -3.38E-2 -2.66E-3 -1.25E-2 -3.47E-2 -1.81E-3 -1.14E-2 -3.40E-2 -1.36E-3 -1.10E-2 -3.39E-2
4 -6.48E-2 -3.41E-1 -7.29E-1 -1.03E-1 -3.60E-1 -7.32E-1 -8.72E-2 -3.48E-1 -7.30E-1 -7.78E-2 -3.44E-1 -7.29E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=1$[]{data-label="tabl:Dq001dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.64E-1 7.28E-1 8.92E-1 5.94E-1 7.38E-1 8.93E-1 5.84E-1 7.32E-1 8.92E-1 5.77E-1 7.29E-1 8.92E-1
1 -1.83E-10 -3.36E-8 -1.22E-7 -1.99E-7 -1.73E-7 -1.71E-7 -3.61E-8 -6.97E-8 -1.32E-7 -8.04E-9 -4.42E-8 -1.24E-7
2 -1.77E-7 -8.83E-6 -2.33E-5 -9.34E-6 -2.01E-5 -2.74E-5 -3.15E-6 -1.26E-5 -2.42E-5 -1.28E-6 -1.00E-5 -2.35E-5
3 -2.83E-5 -5.87E-4 -1.43E-3 -2.78E-4 -9.16E-4 -1.56E-3 -1.44E-4 -7.08E-4 -1.46E-3 -8.48E-5 -6.29E-4 -1.43E-3
4 -1.85E-3 -1.91E-2 -6.17E-2 -6.03E-3 -2.37E-2 -6.47E-2 -4.25E-3 -2.09E-2 -6.24E-2 -3.22E-3 -1.98E-2 -6.18E-2
5 -8.80E-2 -4.18E-1 -7.94E-1 -1.50E-1 -4.54E-1 -7.99E-1 -1.26E-1 -4.33E-1 -7.95E-1 -1.12E-1 -4.23E-1 -7.95E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=1$[]{data-label="tabl:Dq001dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.77E-1 7.47E-1 9.06E-1 6.12E-1 7.62E-1 9.08E-1 6.02E-1 7.54E-1 9.06E-1 5.94E-1 7.50E-1 9.06E-1
1 -8.02E-12 -2.27E-9 -9.03E-9 -8.37E-8 -3.74E-8 -1.96E-8 -1.24E-8 -9.72E-9 -1.13E-8 -2.13E-9 -4.23E-9 -9.49E-9
2 -7.77E-9 -5.97E-7 -1.70E-6 -2.39E-6 -2.81E-6 -2.53E-6 -6.22E-7 -1.28E-6 -1.90E-6 -1.91E-7 -8.20E-7 -1.74E-6
3 -1.25E-6 -4.00E-5 -9.45E-5 -4.91E-5 -1.02E-4 -1.20E-4 -1.97E-5 -6.28E-5 -1.01E-4 -9.01E-6 -4.81E-5 -9.59E-5
4 -8.38E-5 -1.35E-3 -3.22E-3 -7.88E-4 -2.32E-3 -3.73E-3 -4.42E-4 -1.75E-3 -3.35E-3 -2.72E-4 -1.50E-3 -3.25E-3
5 -3.20E-3 -2.87E-2 -9.47E-2 -1.07E-2 -3.79E-2 -1.02E-1 -7.75E-3 -3.28E-2 -9.67E-2 -5.96E-3 -3.03E-2 -9.51E-2
6 -1.11E-1 -4.89E-1 -8.38E-1 -2.03E-1 -5.47E-1 -8.46E-1 -1.70E-1 -5.15E-1 -8.40E-1 -1.49E-1 -4.99E-1 -8.39E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=1$[]{data-label="tabl:Dq001dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.88E-1 7.63E-1 9.16E-1 6.27E-1 7.81E-1 9.20E-1 6.16E-1 7.73E-1 9.17E-1 6.08E-1 7.68E-1 9.16E-1
1 -4.39E-13 -1.86E-10 -8.16E-10 -4.34E-8 -1.19E-8 -3.50E-9 -5.62E-9 -2.16E-9 -1.35E-9 -8.03E-10 -6.26E-10 -9.28E-10
2 -4.25E-10 -4.91E-8 -1.52E-7 -8.38E-7 -5.89E-7 -3.41E-7 -1.78E-7 -1.96E-7 -1.99E-7 -4.34E-8 -9.45E-8 -1.63E-7
3 -6.90E-8 -3.30E-6 -8.21E-6 -1.25E-5 -1.64E-5 -1.36E-5 -4.03E-6 -7.89E-6 -9.69E-6 -1.47E-6 -4.95E-6 -8.56E-6
4 -4.70E-6 -1.12E-4 -2.49E-4 -1.56E-4 -3.17E-4 -3.46E-4 -7.04E-5 -1.96E-4 -2.77E-4 -3.51E-5 -1.45E-4 -2.56E-4
5 -1.83E-4 -2.46E-3 -5.93E-3 -1.66E-3 -4.63E-3 -7.35E-3 -9.92E-4 -3.45E-3 -6.36E-3 -6.37E-4 -2.87E-3 -6.03E-3
6 -4.82E-3 -3.91E-2 -1.31E-1 -1.64E-2 -5.45E-2 -1.46E-1 -1.22E-2 -4.66E-2 -1.35E-1 -9.49E-3 -4.23E-2 -1.32E-1
7 -1.34E-1 -5.54E-1 -8.68E-1 -2.62E-1 -6.39E-1 -8.77E-1 -2.19E-1 -5.96E-1 -8.71E-1 -1.90E-1 -5.73E-1 -8.68E-1
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=1$[]{data-label="tabl:Dq001dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.97E-1 7.76E-1 9.24E-1 6.41E-1 7.98E-1 9.29E-1 6.29E-1 7.89E-1 9.26E-1 6.21E-1 7.83E-1 9.25E-1
1 -2.87E-14 -1.80E-11 -8.62E-11 -2.58E-8 -4.97E-9 -9.02E-10 -3.03E-9 -6.80E-10 -2.29E-10 -3.81E-10 -1.38E-10 -1.16E-10
2 -2.79E-11 -4.73E-9 -1.60E-8 -3.67E-7 -1.68E-7 -6.38E-8 -6.66E-8 -4.21E-8 -2.76E-8 -1.35E-8 -1.52E-8 -1.88E-8
3 -4.54E-9 -3.18E-7 -8.56E-7 -4.20E-6 -3.62E-6 -2.12E-6 -1.12E-6 -1.36E-6 -1.21E-6 -3.34E-7 -6.78E-7 -9.47E-7
4 -3.12E-7 -1.08E-5 -2.48E-5 -4.18E-5 -5.81E-5 -4.60E-5 -1.55E-5 -2.91E-5 -3.14E-5 -6.36E-6 -1.80E-5 -2.66E-5
5 -1.24E-5 -2.41E-4 -5.16E-4 -3.68E-4 -7.45E-4 -7.88E-4 -1.82E-4 -4.65E-4 -6.06E-4 -9.69E-5 -3.37E-4 -5.41E-4
6 -3.32E-4 -3.91E-3 -9.62E-3 -2.92E-3 -7.92E-3 -1.28E-2 -1.83E-3 -5.89E-3 -1.07E-2 -1.22E-3 -4.82E-3 -9.92E-3
7 -6.64E-3 -4.98E-2 -1.67E-1 -2.31E-2 -7.30E-2 -1.93E-1 -1.74E-2 -6.20E-2 -1.77E-1 -1.37E-2 -5.57E-2 -1.70E-1
8 -1.57E-1 -6.14E-1 -8.87E-1 -3.26E-1 -7.30E-1 -8.96E-1 -2.71E-1 -6.77E-1 -8.91E-1 -2.34E-1 -6.45E-1 -8.88E-1
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=1$[]{data-label="tabl:Dq001dDaAAk8p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.55E-3 9.81E-3 9.93E-3 9.89E-3 9.86E-3 9.93E-3 9.86E-3 9.84E-3 9.93E-3 9.83E-3 9.82E-3 9.93E-3
1 -2.87E-14 -3.41E-9 -1.37E-7 -4.26E-9 -3.69E-8 -1.96E-7 -8.28E-10 -1.16E-8 -1.50E-7 -1.56E-10 -5.71E-9 -1.40E-7
2 -3.27E-11 -4.27E-7 -1.22E-5 -1.03E-7 -1.07E-6 -1.32E-5 -2.83E-8 -6.74E-7 -1.24E-5 -7.76E-9 -5.15E-7 -1.22E-5
3 -1.44E-8 -8.08E-6 -3.78E-5 -4.72E-6 -1.30E-5 -3.81E-5 -1.96E-6 -1.01E-5 -3.79E-5 -7.62E-7 -8.84E-6 -3.78E-5
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=100$[]{data-label="tabl:Dq100dDaAAk3p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.70E-3 9.87E-3 9.95E-3 9.91E-3 9.91E-3 9.96E-3 9.89E-3 9.90E-3 9.95E-3 9.88E-3 9.88E-3 9.95E-3
1 -1.69E-15 -2.73E-10 -8.94E-9 -2.41E-9 -1.23E-8 -2.03E-8 -4.49E-10 -2.53E-9 -1.13E-8 -7.88E-11 -8.06E-10 -9.43E-9
2 -1.69E-12 -5.47E-8 -1.95E-6 -3.02E-8 -3.38E-7 -2.84E-6 -7.83E-9 -1.55E-7 -2.17E-6 -1.98E-9 -8.97E-8 -2.00E-6
3 -3.22E-10 -1.26E-6 -1.99E-5 -4.73E-7 -2.91E-6 -2.09E-5 -1.68E-7 -2.00E-6 -2.02E-5 -5.68E-8 -1.56E-6 -2.00E-5
4 -7.00E-8 -1.42E-5 -4.02E-5 -1.19E-5 -2.33E-5 -4.07E-5 -6.43E-6 -1.86E-5 -4.04E-5 -3.18E-6 -1.61E-5 -4.03E-5
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=100$[]{data-label="tabl:Dq100dDaAAk4p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.78E-3 9.90E-3 9.97E-3 9.92E-3 9.93E-3 9.97E-3 9.91E-3 9.92E-3 9.97E-3 9.90E-3 9.91E-3 9.97E-3
1 -1.32E-16 -2.57E-11 -7.63E-10 -1.57E-9 -5.18E-9 -3.47E-9 -2.83E-10 -7.79E-10 -1.29E-9 -4.71E-11 -1.69E-10 -8.73E-10
2 -1.28E-13 -6.22E-9 -1.68E-7 -1.31E-8 -1.25E-7 -4.02E-7 -3.22E-9 -4.09E-8 -2.25E-7 -7.63E-10 -1.70E-8 -1.81E-7
3 -2.02E-11 -2.40E-7 -7.01E-6 -1.19E-7 -1.06E-6 -9.55E-6 -3.86E-8 -6.05E-7 -7.82E-6 -1.20E-8 -3.93E-7 -7.21E-6
4 -1.71E-9 -2.46E-6 -2.26E-5 -1.30E-6 -5.44E-6 -2.28E-5 -5.69E-7 -3.95E-6 -2.27E-5 -2.35E-7 -3.14E-6 -2.26E-5
5 -2.43E-7 -2.11E-5 -4.22E-5 -2.08E-5 -3.39E-5 -4.29E-5 -1.32E-5 -2.80E-5 -4.24E-5 -7.90E-6 -2.44E-5 -4.23E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=100$[]{data-label="tabl:Dq100dDaAAk5p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.83E-3 9.92E-3 9.97E-3 9.93E-3 9.95E-3 9.98E-3 9.92E-3 9.94E-3 9.98E-3 9.91E-3 9.93E-3 9.97E-3
1 -1.22E-17 -2.77E-12 -7.88E-11 -1.12E-9 -2.59E-9 -8.81E-10 -1.95E-10 -3.09E-10 -2.16E-10 -3.12E-11 -4.93E-11 -1.07E-10
2 -1.18E-14 -7.12E-10 -1.60E-8 -7.04E-9 -5.23E-8 -6.97E-8 -1.65E-9 -1.26E-8 -2.86E-8 -3.68E-10 -3.83E-9 -1.90E-8
3 -1.82E-12 -3.85E-8 -1.03E-6 -4.55E-8 -4.54E-7 -2.55E-6 -1.37E-8 -2.03E-7 -1.48E-6 -3.99E-9 -1.02E-7 -1.14E-6
4 -1.20E-10 -5.87E-7 -1.25E-5 -3.20E-7 -2.10E-6 -1.52E-5 -1.24E-7 -1.36E-6 -1.37E-5 -4.60E-8 -9.56E-7 -1.29E-5
5 -6.37E-9 -3.88E-6 -2.28E-5 -2.59E-6 -8.36E-6 -2.28E-5 -1.34E-6 -6.33E-6 -2.28E-5 -6.49E-7 -5.12E-6 -2.28E-5
6 -6.57E-7 -2.78E-5 -4.38E-5 -3.03E-5 -4.36E-5 -4.47E-5 -2.11E-5 -3.71E-5 -4.41E-5 -1.42E-5 -3.27E-5 -4.39E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=100$[]{data-label="tabl:Dq100dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.85E-3 9.93E-3 9.98E-3 9.93E-3 9.96E-3 9.98E-3 9.93E-3 9.95E-3 9.98E-3 9.92E-3 9.94E-3 9.98E-3
1 -1.30E-18 -3.35E-13 -9.40E-12 -8.42E-10 -1.46E-9 -3.00E-10 -1.44E-10 -1.47E-10 -5.06E-11 -2.21E-11 -1.86E-11 -1.73E-11
2 -1.26E-15 -8.76E-11 -1.82E-9 -4.33E-9 -2.45E-8 -1.65E-8 -9.72E-10 -4.58E-9 -4.96E-9 -2.06E-10 -1.06E-9 -2.57E-9
3 -1.95E-13 -5.47E-9 -1.12E-7 -2.20E-8 -2.11E-7 -5.33E-7 -6.24E-9 -7.26E-8 -2.24E-7 -1.70E-9 -2.79E-8 -1.42E-7
4 -1.22E-11 -1.26E-7 -3.34E-6 -1.17E-7 -9.83E-7 -7.27E-6 -4.14E-8 -5.43E-7 -4.86E-6 -1.41E-8 -3.18E-7 -3.81E-6
5 -4.78E-10 -1.06E-6 -1.59E-5 -6.64E-7 -3.34E-6 -1.70E-5 -2.98E-7 -2.33E-6 -1.66E-5 -1.27E-7 -1.72E-6 -1.62E-5
6 -1.85E-8 -5.44E-6 -2.28E-5 -4.24E-6 -1.14E-5 -2.29E-5 -2.47E-6 -8.95E-6 -2.28E-5 -1.36E-6 -7.36E-6 -2.28E-5
7 -1.45E-6 -3.39E-5 -4.50E-5 -4.03E-5 -5.23E-5 -4.61E-5 -2.95E-5 -4.54E-5 -4.55E-5 -2.13E-5 -4.05E-5 -4.51E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=100$[]{data-label="tabl:Dq100dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.37E-3 9.94E-3 9.98E-3 9.94E-3 9.96E-3 9.99E-3 9.93E-3 9.96E-3 9.98E-3 9.93E-3 9.95E-3 9.98E-3
1 1.60E-19 -4.48E-14 -1.26E-12 -6.62E-10 -9.01E-10 -1.27E-10 -1.10E-10 -8.03E-11 -1.57E-11 -1.65E-11 -8.49E-12 -3.67E-12
2 4.31E-17 -1.18E-11 -2.38E-10 -2.93E-9 -1.27E-8 -5.02E-9 -6.31E-10 -1.94E-9 -1.13E-9 -1.27E-10 -3.59E-10 -4.46E-10
3 2.60E-20 -7.72E-10 -1.36E-8 -1.24E-8 -1.05E-7 -1.27E-7 -3.33E-9 -2.80E-8 -4.13E-8 -8.59E-10 -8.35E-9 -2.11E-8
4 3.54E-13 -2.25E-8 -4.64E-7 -5.36E-8 -5.04E-7 -2.28E-6 -1.77E-8 -2.28E-7 -1.04E-6 -5.65E-9 -1.07E-7 -6.37E-7
5 -1.36E-26 -2.83E-7 -6.85E-6 -2.41E-7 -1.64E-6 -1.18E-5 -9.79E-8 -1.03E-6 -9.36E-6 -3.81E-8 -6.71E-7 -7.82E-6
6 3.00E-25 -1.62E-6 -1.70E-5 -1.15E-6 -4.71E-6 -1.68E-5 -5.80E-7 -3.44E-6 -1.70E-5 -2.79E-7 -2.64E-6 -1.71E-5
7 8.45E-25 -7.06E-6 -2.29E-5 -6.11E-6 -1.44E-5 -2.32E-5 -3.88E-6 -1.16E-5 -2.30E-5 -2.36E-6 -9.73E-6 -2.29E-5
8 1.68E-23 -3.95E-5 -4.60E-5 -5.07E-5 -6.05E-5 -4.75E-5 -3.82E-5 -5.29E-5 -4.67E-5 -2.88E-5 -4.76E-5 -4.63E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=100$[]{data-label="tabl:Dq100dDaAAk8p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.80E-3 4.92E-3 4.97E-3 4.97E-3 4.96E-3 4.98E-3 4.96E-3 4.94E-3 4.97E-3 4.95E-3 4.93E-3 4.97E-3
1 -9.85E-16 -4.89E-10 -3.33E-8 -1.13E-9 -1.19E-8 -5.88E-8 -2.21E-10 -3.09E-9 -3.86E-8 -4.13E-11 -1.17E-9 -3.44E-8
2 -1.13E-12 -5.93E-8 -2.64E-6 -2.45E-8 -2.17E-7 -2.96E-6 -6.31E-9 -1.21E-7 -2.72E-6 -1.55E-9 -8.20E-8 -2.66E-6
3 -5.28E-10 -1.15E-6 -7.97E-6 -1.11E-6 -2.46E-6 -8.10E-6 -4.24E-7 -1.70E-6 -8.00E-6 -1.43E-7 -1.36E-6 -7.97E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=200$[]{data-label="tabl:Dq200dDaAAk3p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.87E-3 4.95E-3 4.98E-3 4.98E-3 4.97E-3 4.99E-3 4.97E-3 4.97E-3 4.99E-3 4.97E-3 4.96E-3 4.98E-3
1 -6.07E-17 -4.27E-11 -2.37E-9 -6.69E-10 -5.16E-9 -7.72E-9 -1.29E-10 -9.16E-10 -3.44E-9 -2.33E-11 -2.24E-10 -2.59E-9
2 -6.08E-14 -8.32E-9 -4.98E-7 -7.50E-9 -8.46E-8 -8.41E-7 -1.88E-9 -3.57E-8 -5.89E-7 -4.47E-10 -1.79E-8 -5.19E-7
3 -1.18E-11 -1.84E-7 -4.26E-6 -1.13E-7 -5.86E-7 -4.52E-6 -3.77E-8 -3.65E-7 -4.35E-6 -1.15E-8 -2.60E-7 -4.28E-6
4 -2.82E-9 -2.20E-6 -8.83E-6 -2.87E-6 -4.85E-6 -9.10E-6 -1.47E-6 -3.50E-6 -8.91E-6 -6.58E-7 -2.78E-6 -8.85E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=200$[]{data-label="tabl:Dq200dDaAAk4p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.91E-3 4.97E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.97E-3 4.97E-3 4.99E-3
1 -4.98E-18 -4.40E-12 -2.18E-10 -4.51E-10 -2.62E-9 -1.62E-9 -8.56E-11 -3.54E-10 -4.71E-10 -1.51E-11 -6.15E-11 -2.71E-10
2 -4.84E-15 -1.05E-9 -4.87E-8 -3.37E-9 -3.91E-8 -1.58E-7 -8.19E-10 -1.22E-8 -7.53E-8 -1.89E-10 -4.42E-9 -5.49E-8
3 -7.72E-13 -3.79E-8 -1.73E-6 -2.93E-8 -2.37E-7 -2.48E-6 -9.09E-9 -1.29E-7 -2.00E-6 -2.64E-9 -7.71E-8 -1.80E-6
4 -6.77E-11 -3.77E-7 -4.76E-6 -3.15E-7 -1.14E-6 -4.84E-6 -1.32E-7 -7.56E-7 -4.80E-6 -4.97E-8 -5.55E-7 -4.77E-6
5 -1.12E-8 -3.57E-6 -9.60E-6 -5.08E-6 -7.52E-6 -9.97E-6 -3.13E-6 -5.76E-6 -9.73E-6 -1.75E-6 -4.67E-6 -9.64E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=200$[]{data-label="tabl:Dq200dDaAAk5p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.94E-3 4.98E-3 4.99E-3 4.98E-3 4.99E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3
1 -4.96E-19 -5.16E-13 -2.41E-11 -3.29E-10 -1.49E-9 -4.82E-10 -6.17E-11 -1.65E-10 -9.42E-11 -1.06E-11 -2.22E-11 -3.80E-11
2 -4.80E-16 -1.32E-10 -4.95E-9 -1.86E-9 -1.99E-8 -3.29E-8 -4.40E-10 -4.74E-9 -1.11E-8 -9.84E-11 -1.26E-9 -6.42E-9
3 -7.40E-14 -6.82E-9 -3.12E-7 -1.14E-8 -1.17E-7 -9.23E-7 -3.36E-9 -5.26E-8 -5.07E-7 -9.34E-10 -2.50E-8 -3.65E-7
4 -4.95E-12 -9.57E-8 -2.91E-6 -7.88E-8 -4.60E-7 -3.48E-6 -2.94E-8 -2.81E-7 -3.20E-6 -1.02E-8 -1.86E-7 -3.01E-6
5 -2.79E-10 -6.30E-7 -4.88E-6 -6.35E-7 -1.83E-6 -4.98E-6 -3.16E-7 -1.28E-6 -4.92E-6 -1.43E-7 -9.64E-7 -4.89E-6
6 -3.55E-8 -5.10E-6 -1.02E-5 -7.47E-6 -1.00E-5 -1.06E-5 -5.09E-6 -8.09E-6 -1.04E-5 -3.28E-6 -6.76E-6 -1.03E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=200$[]{data-label="tabl:Dq200dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.91E-3 4.98E-3 4.99E-3 4.98E-3 4.99E-3 4.99E-3 4.98E-3 4.99E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3
1 -4.90E-18 -6.75E-14 -3.05E-12 -2.54E-10 -9.22E-10 -1.85E-10 -4.69E-11 -8.86E-11 -2.58E-11 -7.94E-12 -9.82E-12 -7.10E-12
2 -4.78E-15 -1.76E-11 -5.94E-10 -1.17E-9 -1.10E-8 -8.82E-9 -2.70E-10 -2.06E-9 -2.21E-9 -5.85E-11 -4.24E-10 -9.73E-10
3 -7.69E-13 -1.08E-9 -3.72E-8 -5.65E-9 -6.40E-8 -2.53E-7 -1.59E-9 -2.31E-8 -9.36E-8 -4.23E-10 -8.48E-9 -5.24E-8
4 2.47E-24 -2.28E-8 -9.82E-7 -2.91E-8 -2.32E-7 -2.14E-6 -1.01E-8 -1.27E-7 -1.50E-6 -3.26E-9 -7.26E-8 -1.16E-6
5 -6.79E-11 -1.78E-7 -3.50E-6 -1.64E-7 -7.41E-7 -3.62E-6 -7.11E-8 -4.85E-7 -3.60E-6 -2.87E-8 -3.40E-7 -3.55E-6
6 -1.12E-8 -9.33E-7 -4.99E-6 -1.05E-6 -2.58E-6 -5.18E-6 -5.92E-7 -1.90E-6 -5.07E-6 -3.10E-7 -1.47E-6 -5.02E-6
7 3.70E-24 -6.62E-6 -1.06E-5 -9.97E-6 -1.23E-5 -1.11E-5 -7.18E-6 -1.03E-5 -1.09E-5 -5.04E-6 -8.80E-6 -1.07E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=200$[]{data-label="tabl:Dq200dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.49E-3 4.98E-3 4.99E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 4.99E-3
1 3.41E-22 -9.66E-15 -4.29E-13 -2.03E-10 -6.10E-10 -8.50E-11 -3.71E-11 -5.25E-11 -9.06E-12 -6.18E-12 -5.03E-12 -1.74E-12
2 -3.82E-22 -2.54E-12 -8.17E-11 -8.08E-10 -6.52E-9 -2.97E-9 -1.81E-10 -1.00E-9 -5.67E-10 -3.81E-11 -1.67E-10 -1.89E-10
3 1.89E-26 -1.65E-10 -4.75E-9 -3.24E-9 -3.71E-8 -6.97E-8 -8.75E-10 -1.07E-8 -1.94E-8 -2.24E-10 -3.07E-9 -8.60E-9
4 -2.77E-27 -4.59E-9 -1.62E-7 -1.36E-8 -1.32E-7 -9.51E-7 -4.42E-9 -6.26E-8 -4.33E-7 -1.36E-9 -2.95E-8 -2.48E-7
5 4.60E-27 -5.17E-8 -1.85E-6 -6.02E-8 -3.77E-7 -2.85E-6 -2.38E-8 -2.29E-7 -2.47E-6 -8.83E-9 -1.46E-7 -2.13E-6
6 1.68E-26 -2.78E-7 -3.62E-6 -2.85E-7 -1.06E-6 -3.59E-6 -1.40E-7 -7.34E-7 -3.61E-6 -6.41E-8 -5.33E-7 -3.62E-6
7 6.86E-26 -1.27E-6 -5.13E-6 -1.51E-6 -3.32E-6 -5.37E-6 -9.41E-7 -2.56E-6 -5.25E-6 -5.51E-7 -2.04E-6 -5.17E-6
8 1.46E-23 -8.06E-6 -1.10E-5 -1.26E-5 -1.44E-5 -1.15E-5 -9.36E-6 -1.22E-5 -1.13E-5 -6.90E-6 -1.07E-5 -1.11E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=200$[]{data-label="tabl:Dq200dDaAAk8p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.40E-3 2.47E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.48E-3 2.49E-3 2.49E-3 2.47E-3 2.49E-3
1 -3.23E-17 -6.60E-11 -7.51E-9 -2.92E-10 -3.85E-9 -1.81E-8 -5.75E-11 -8.96E-10 -9.70E-9 -1.08E-11 -2.64E-10 -7.96E-9
2 -3.72E-14 -7.85E-9 -5.52E-7 -5.95E-9 -4.50E-8 -6.57E-7 -1.47E-9 -2.24E-8 -5.82E-7 -3.40E-10 -1.34E-8 -5.59E-7
3 -1.79E-11 -1.54E-7 -1.65E-6 -2.67E-7 -4.83E-7 -1.71E-6 -9.72E-8 -2.92E-7 -1.66E-6 -3.01E-8 -2.09E-7 -1.65E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=400$[]{data-label="tabl:Dq400dDaAAk3p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.44E-3 2.48E-3 2.49E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3
1 -2.03E-18 -6.11E-12 -5.74E-10 -1.77E-10 -2.06E-9 -3.05E-9 -3.49E-11 -3.48E-10 -1.04E-9 -6.49E-12 -6.85E-11 -6.71E-10
2 -2.04E-15 -1.17E-9 -1.17E-7 -1.87E-9 -2.05E-8 -2.38E-7 -4.58E-10 -8.20E-9 -1.52E-7 -1.05E-10 -3.65E-9 -1.25E-7
3 -3.99E-13 -2.51E-8 -8.87E-7 -2.76E-8 -1.21E-7 -9.55E-7 -8.88E-9 -6.74E-8 -9.14E-7 -2.54E-9 -4.34E-8 -8.94E-7
4 -1.01E-10 -3.16E-7 -1.90E-6 -7.03E-7 -1.03E-6 -2.02E-6 -3.49E-7 -6.60E-7 -1.94E-6 -1.46E-7 -4.70E-7 -1.91E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=400$[]{data-label="tabl:Dq400dDaAAk4p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.46E-3 2.49E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3
1 -1.72E-19 -6.74E-13 -5.67E-11 -1.21E-10 -1.23E-9 -7.79E-10 -2.39E-11 -1.65E-10 -1.74E-10 -4.41E-12 -2.41E-11 -8.04E-11
2 -1.67E-16 -1.59E-10 -1.28E-8 -8.54E-10 -1.11E-8 -6.09E-8 -2.06E-10 -3.54E-9 -2.44E-8 -4.67E-11 -1.16E-9 -1.55E-8
3 -2.68E-14 -5.50E-9 -3.97E-7 -7.25E-9 -5.25E-8 -6.01E-7 -2.19E-9 -2.67E-8 -4.80E-7 -6.07E-10 -1.48E-8 -4.21E-7
4 -2.40E-12 -5.39E-8 -9.91E-7 -7.75E-8 -2.44E-7 -1.03E-6 -3.14E-8 -1.46E-7 -1.00E-6 -1.12E-8 -9.73E-8 -9.95E-7
5 -4.31E-10 -5.55E-7 -2.15E-6 -1.25E-6 -1.69E-6 -2.31E-6 -7.57E-7 -1.19E-6 -2.21E-6 -4.05E-7 -8.76E-7 -2.17E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=400$[]{data-label="tabl:Dq400dDaAAk5p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.47E-3 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3
1 -1.78E-20 -8.52E-14 -6.69E-12 -9.01E-11 -7.92E-10 -2.69E-10 -1.76E-11 -8.90E-11 -4.17E-11 -3.23E-12 -1.05E-11 -1.32E-11
2 -1.72E-17 -2.17E-11 -1.39E-9 -4.79E-10 -6.66E-9 -1.55E-8 -1.14E-10 -1.69E-9 -4.27E-9 -2.54E-11 -4.16E-10 -2.07E-9
3 -2.66E-15 -1.08E-9 -8.50E-8 -2.87E-9 -2.83E-8 -2.97E-7 -8.29E-10 -1.27E-8 -1.60E-7 -2.23E-10 -5.83E-9 -1.07E-7
4 -1.79E-13 -1.43E-8 -6.42E-7 -1.95E-8 -1.01E-7 -7.50E-7 -7.11E-9 -5.74E-8 -7.09E-7 -2.35E-9 -3.54E-8 -6.68E-7
5 -1.05E-11 -9.51E-8 -1.04E-6 -1.57E-7 -4.07E-7 -1.10E-6 -7.64E-8 -2.62E-7 -1.06E-6 -3.32E-8 -1.80E-7 -1.04E-6
6 -1.51E-9 -8.60E-7 -2.35E-6 -1.85E-6 -2.34E-6 -2.53E-6 -1.25E-6 -1.77E-6 -2.43E-6 -7.81E-7 -1.38E-6 -2.38E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=400$[]{data-label="tabl:Dq400dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.39E-3 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3
1 3.10E-18 -1.20E-14 -8.99E-13 -7.03E-11 -5.39E-10 -1.15E-10 -1.37E-11 -5.30E-11 -1.33E-11 -2.49E-12 -5.32E-12 -2.90E-12
2 -4.31E-17 -3.12E-12 -1.77E-10 -3.06E-10 -4.25E-9 -4.77E-9 -7.12E-11 -8.75E-10 -9.75E-10 -1.57E-11 -1.69E-10 -3.56E-10
3 -1.24E-18 -1.87E-10 -1.12E-8 -1.43E-9 -1.71E-8 -1.11E-7 -3.99E-10 -6.63E-9 -3.74E-8 -1.04E-10 -2.43E-9 -1.83E-8
4 -3.53E-25 -3.71E-9 -2.57E-7 -7.27E-9 -5.33E-8 -5.46E-7 -2.47E-9 -2.82E-8 -4.11E-7 -7.74E-10 -1.56E-8 -3.17E-7
5 2.21E-27 -2.74E-8 -7.43E-7 -4.06E-8 -1.66E-7 -7.64E-7 -1.73E-8 -1.01E-7 -7.59E-7 -6.71E-9 -6.58E-8 -7.52E-7
6 2.50E-27 -1.49E-7 -1.09E-6 -2.60E-7 -5.92E-7 -1.18E-6 -1.45E-7 -4.07E-7 -1.13E-6 -7.33E-8 -2.92E-7 -1.11E-6
7 4.34E-25 -1.20E-6 -2.50E-6 -2.48E-6 -2.92E-6 -2.68E-6 -1.77E-6 -2.34E-6 -2.59E-6 -1.22E-6 -1.90E-6 -2.54E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=400$[]{data-label="tabl:Dq400dDaAAk7p"}
------------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[ **k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.49E-3 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 1.1E-16 -1.8E-15 -1.33E-13 -5.69E-11 -3.83E-10 -5.74E-11 -1.10E-11 -3.40E-11 -5.27E-12 -1.99E-12 -3.02E-12 -8.25E-13
2 -1.8E-16 -4.8E-13 -2.55E-11 -2.13E-10 -2.84E-9 -1.77E-9 -4.87E-11 -4.88E-10 -2.82E-10 -1.05E-11 -7.69E-11 -7.81E-11
3 -7.78E-14 -3.10E-11 -1.51E-9 -8.27E-10 -1.11E-8 -3.73E-8 -2.24E-10 -3.65E-9 -8.91E-9 -5.70E-11 -1.06E-9 -3.38E-9
4 -2.42E-16 -8.24E-10 -5.08E-8 -3.42E-9 -3.20E-8 -3.26E-7 -1.10E-9 -1.55E-8 -1.61E-7 -3.30E-10 -7.43E-9 -8.82E-8
5 -1.48E-12 -8.53E-9 -4.51E-7 -1.50E-8 -8.61E-8 -6.20E-7 -5.84E-9 -4.97E-8 -5.83E-7 -2.10E-9 -3.03E-8 -5.20E-7
6 -4.00E-11 -4.44E-8 -7.61E-7 -7.09E-8 -2.44E-7 -7.83E-7 -3.43E-8 -1.57E-7 -7.68E-7 -1.52E-8 -1.07E-7 -7.63E-7
7 -1.57E-9 -2.15E-7 -1.15E-6 -3.77E-7 -7.79E-7 -1.26E-6 -2.31E-7 -5.69E-7 -1.21E-6 -1.32E-7 -4.24E-7 -1.17E-6
8 -1.30E-7 -1.55E-6 -2.62E-6 -3.13E-6 -3.45E-6 -2.80E-6 -2.31E-6 -2.85E-6 -2.72E-6 -1.68E-6 -2.39E-6 -2.66E-6
------------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $c_j$, $ j=0, \dots, k$ of the partial fraction representation defined by of $(q,\delta,\alpha,k)$-BURA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=400$[]{data-label="tabl:Dq400dDaAAk8p"}
Tables type **(e)** for 0-URA-poles
-----------------------------------
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.23E-5 -8.41E-4 -5.35E-3 -8.32E-5 -9.92E-4 -5.43E-3 -4.06E-5 -8.89E-4 -5.36E-3 -2.51E-5 -8.57E-4 -5.35E-3
2 -3.80E-3 -3.98E-2 -1.27E-1 -8.05E-3 -4.21E-2 -1.27E-1 -5.96E-3 -4.05E-2 -1.27E-1 -4.93E-3 -4.00E-2 -1.27E-1
3 -2.60E-1 -7.04E-1 -1.02E00 -3.48E-1 -7.21E-1 -1.03E00 -3.09E-1 -7.10E-1 -1.02E00 -2.87E-1 -7.06E-1 -1.02E00
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=1$[]{data-label="tabl:Eq001dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -5.37E-7 -9.04E-5 -8.33E-4 -1.83E-5 -1.47E-4 -8.87E-4 -6.00E-6 -1.08E-4 -8.44E-4 -2.53E-6 -9.60E-5 -8.35E-4
2 -1.75E-4 -4.64E-3 -2.04E-2 -9.63E-4 -5.70E-3 -2.10E-2 -5.31E-4 -4.99E-3 -2.05E-2 -3.47E-4 -4.75E-3 -2.04E-2
3 -1.12E-2 -7.92E-2 -2.15E-1 -2.61E-2 -8.72E-2 -2.17E-1 -1.92E-2 -8.20E-2 -2.15E-1 -1.55E-2 -8.01E-2 -2.15E-1
4 -4.00E-1 -9.55E-1 -1.25E00 -5.87E-1 -1.00E00 -1.25E00 -5.08E-1 -9.71E-1 -1.25E00 -4.61E-1 -9.61E-1 -1.25E00
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the best uniform rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=1$[]{data-label="tabl:Eq001dDaAAk4p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -3.20E-8 -1.22E-5 -1.59E-4 -6.90E-6 -3.53E-5 -1.92E-4 -1.71E-6 -1.93E-5 -1.66E-4 -5.12E-7 -1.44E-5 -1.60E-4
2 -1.08E-5 -6.45E-4 -3.93E-3 -2.12E-4 -1.08E-3 -4.29E-3 -8.75E-5 -7.98E-4 -4.01E-3 -4.34E-5 -6.96E-4 -3.95E-3
3 -7.18E-4 -1.17E-2 -4.25E-2 -4.02E-3 -1.54E-2 -4.46E-2 -2.34E-3 -1.31E-2 -4.30E-2 -1.55E-3 -1.22E-2 -4.26E-2
4 -2.24E-2 -1.24E-1 -3.02E-1 -5.75E-2 -1.44E-1 -3.09E-1 -4.23E-2 -1.32E-1 -3.04E-1 -3.37E-2 -1.27E-1 -3.02E-1
5 -5.46E-1 -1.20E00 -1.47E00 -8.77E-1 -1.30E00 -1.49E00 -7.46E-1 -1.24E00 -1.47E00 -6.65E-1 -1.22E00 -1.47E00
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=1$[]{data-label="tabl:Eq001dDaAAk5p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.42E-9 -1.95E-6 -3.52E-5 -3.50E-6 -1.27E-5 -5.48E-5 -7.25E-7 -5.03E-6 -3.98E-5 -1.71E-7 -2.90E-6 -3.61E-5
2 -8.33E-7 -1.05E-4 -8.73E-4 -7.03E-5 -2.88E-4 -1.09E-3 -2.30E-5 -1.70E-4 -9.27E-4 -8.84E-6 -1.27E-4 -8.84E-4
3 -5.70E-5 -1.95E-3 -9.51E-3 -9.66E-4 -3.58E-3 -1.08E-2 -4.50E-4 -2.59E-3 -9.84E-3 -2.40E-4 -2.19E-3 -9.58E-3
4 -1.82E-3 -2.15E-2 -6.94E-2 -1.06E-2 -3.08E-2 -7.48E-2 -6.47E-3 -2.54E-2 -7.08E-2 -4.35E-3 -2.30E-2 -6.97E-2
5 -3.68E-2 -1.73E-1 -3.86E-1 -1.03E-1 -2.12E-1 -4.01E-1 -7.61E-2 -1.90E-1 -3.90E-1 -6.02E-2 -1.79E-1 -3.87E-1
6 -6.94E-1 -1.45E00 -1.68E00 -1.22E00 -1.63E00 -1.73E00 -1.02E00 -1.53E00 -1.70E00 -8.99E-1 -1.48E00 -1.69E00
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=1$[]{data-label="tabl:Eq001dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.22E-10 -3.58E-7 -8.74E-6 -2.11E-6 -6.03E-6 -2.05E-5 -3.90E-7 -1.84E-6 -1.16E-5 -7.85E-8 -7.89E-7 -9.40E-6
2 -7.71E-8 -1.93E-5 -2.17E-4 -3.09E-5 -1.02E-4 -3.44E-4 -8.41E-6 -4.76E-5 -2.52E-4 -2.62E-6 -2.92E-5 -2.25E-4
3 -5.35E-6 -3.66E-4 -2.37E-3 -3.21E-4 -1.06E-3 -3.16E-3 -1.23E-4 -6.43E-4 -2.60E-3 -5.36E-5 -4.72E-4 -2.43E-3
4 -1.75E-4 -4.14E-3 -1.74E-2 -2.82E-3 -8.27E-3 -2.08E-2 -1.42E-3 -5.95E-3 -1.84E-2 -7.99E-4 -4.87E-3 -1.76E-2
5 -3.56E-3 -3.37E-2 -9.93E-2 -2.19E-2 -5.20E-2 -1.11E-1 -1.38E-2 -4.22E-2 -1.03E-1 -9.39E-3 -3.72E-2 -1.00E-1
6 -5.37E-2 -2.23E-1 -4.66E-1 -1.61E-1 -2.90E-1 -4.95E-1 -1.20E-1 -2.55E-1 -4.75E-1 -9.51E-2 -2.37E-1 -4.68E-1
7 -8.45E-1 -1.69E00 -1.90E00 -1.60E00 -1.99E00 -1.98E00 -1.34E00 -1.83E00 -1.93E00 -1.16E00 -1.75E00 -1.91E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=1$[]{data-label="tabl:Eq001dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.38E-11 -7.35E-8 -2.38E-6 -1.41E-6 -3.44E-6 -9.54E-6 -2.43E-7 -8.66E-7 -4.20E-6 -4.40E-8 -2.86E-7 -2.82E-6
2 -8.29E-9 -3.97E-6 -5.92E-5 -1.64E-5 -4.50E-5 -1.32E-4 -3.88E-6 -1.71E-5 -8.06E-5 -1.02E-6 -8.47E-6 -6.47E-5
3 -5.81E-7 -7.56E-5 -6.48E-4 -1.35E-4 -3.92E-4 -1.10E-3 -4.38E-5 -1.98E-4 -7.89E-4 -1.60E-5 -1.23E-4 -6.85E-4
4 -1.92E-5 -8.70E-4 -4.76E-3 -9.72E-4 -2.70E-3 -6.77E-3 -4.14E-4 -1.67E-3 -5.42E-3 -1.96E-4 -1.21E-3 -4.93E-3
5 -3.99E-4 -7.24E-3 -2.72E-2 -6.34E-3 -1.56E-2 -3.45E-2 -3.39E-3 -1.12E-2 -2.97E-2 -1.98E-3 -9.00E-3 -2.79E-2
6 -6.00E-3 -4.78E-2 -1.31E-1 -3.84E-2 -7.88E-2 -1.53E-1 -2.48E-2 -6.33E-2 -1.39E-1 -1.72E-2 -5.49E-2 -1.33E-1
7 -7.27E-2 -2.75E-1 -5.43E-1 -2.32E-1 -3.78E-1 -5.90E-1 -1.75E-1 -3.28E-1 -5.60E-1 -1.38E-1 -3.00E-1 -5.47E-1
8 -9.97E-1 -1.92E00 -2.12E00 -2.03E00 -2.38E00 -2.27E00 -1.68E00 -2.16E00 -2.17E00 -1.45E00 -2.04E00 -2.14E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=1$[]{data-label="tabl:Eq001dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -2.41E-6 -1.69E-4 -1.55E-3 -2.60E-5 -2.02E-4 -1.57E-3 -1.09E-5 -1.79E-4 -1.55E-3 -6.03E-6 -1.72E-4 -1.55E-3
2 -1.44E-3 -9.07E-3 -1.70E-2 -3.64E-3 -9.91E-3 -1.71E-2 -2.52E-3 -9.34E-3 -1.70E-2 -2.00E-3 -9.16E-3 -1.70E-2
3 -1.39E-1 -2.70E-1 -3.19E-1 -1.99E-1 -2.80E-1 -3.20E-1 -1.72E-1 -2.73E-1 -3.19E-1 -1.57E-1 -2.71E-1 -3.19E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=100$[]{data-label="tabl:Eq100dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.40E-7 -3.64E-5 -5.45E-4 -7.16E-6 -5.61E-5 -5.73E-4 -2.11E-6 -4.27E-5 -5.51E-4 -8.11E-7 -3.84E-5 -5.46E-4
2 -6.34E-5 -1.17E-3 -4.22E-3 -4.56E-4 -1.50E-3 -4.33E-3 -2.30E-4 -1.28E-3 -4.24E-3 -1.41E-4 -1.20E-3 -4.23E-3
3 -5.44E-3 -2.54E-2 -4.39E-2 -1.48E-2 -2.92E-2 -4.48E-2 -1.03E-2 -2.67E-2 -4.40E-2 -8.03E-3 -2.59E-2 -4.39E-2
4 -2.36E-1 -4.32E-1 -5.15E-1 -3.71E-1 -4.63E-1 -5.21E-1 -3.13E-1 -4.43E-1 -5.16E-1 -2.79E-1 -4.36E-1 -5.15E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=100$[]{data-label="tabl:Eq100dDaAAk4p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.17E-8 -7.81E-6 -1.40E-4 -3.15E-6 -2.02E-5 -1.67E-4 -7.36E-7 -1.18E-5 -1.46E-4 -2.10E-7 -9.09E-6 -1.41E-4
2 -4.09E-6 -2.23E-4 -1.68E-3 -1.09E-4 -3.77E-4 -1.79E-3 -4.13E-5 -2.76E-4 -1.70E-3 -1.91E-5 -2.41E-4 -1.69E-3
3 -3.26E-4 -3.67E-3 -9.61E-3 -2.31E-3 -5.22E-3 -1.02E-2 -1.26E-3 -4.23E-3 -9.74E-3 -7.86E-4 -3.86E-3 -9.64E-3
4 -1.24E-2 -4.85E-2 -8.01E-2 -3.69E-2 -5.96E-2 -8.34E-2 -2.60E-2 -5.27E-2 -8.09E-2 -2.00E-2 -5.00E-2 -8.03E-2
5 -3.40E-1 -6.03E-1 -7.21E-1 -5.88E-1 -6.76E-1 -7.38E-1 -4.89E-1 -6.31E-1 -7.25E-1 -4.28E-1 -6.13E-1 -7.22E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=100$[]{data-label="tabl:Eq100dDaAAk5p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.20E-9 -1.59E-6 -3.38E-5 -1.77E-6 -9.04E-6 -5.21E-5 -3.59E-7 -3.88E-6 -3.81E-5 -8.39E-8 -2.32E-6 -3.47E-5
2 -3.53E-7 -5.32E-5 -6.32E-4 -3.94E-5 -1.34E-4 -7.57E-4 -1.20E-5 -8.24E-5 -6.64E-4 -4.31E-6 -6.34E-5 -6.39E-4
3 -2.57E-5 -7.00E-4 -3.21E-3 -5.77E-4 -1.36E-3 -3.58E-3 -2.50E-4 -9.53E-4 -3.31E-3 -1.26E-4 -7.91E-4 -3.23E-3
4 -9.41E-4 -7.95E-3 -1.83E-2 -6.86E-3 -1.25E-2 -2.03E-2 -3.94E-3 -9.79E-3 -1.89E-2 -2.52E-3 -8.63E-3 -1.85E-2
5 -2.21E-2 -7.66E-2 -1.23E-1 -7.12E-2 -1.01E-1 -1.31E-1 -5.08E-2 -8.70E-2 -1.25E-1 -3.89E-2 -8.05E-2 -1.23E-1
6 -4.50E-1 -7.80E-1 -9.33E-1 -8.48E-1 -9.19E-1 -9.73E-1 -6.99E-1 -8.40E-1 -9.43E-1 -6.04E-1 -8.03E-1 -9.35E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=100$[]{data-label="tabl:Eq100dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.39E-10 -3.27E-7 -8.62E-6 -1.14E-6 -4.82E-6 -2.00E-5 -2.12E-7 -1.58E-6 -1.15E-5 -4.35E-8 -7.04E-7 -9.26E-6
2 -3.80E-8 -1.33E-5 -1.94E-4 -1.86E-5 -5.98E-5 -2.97E-4 -4.77E-6 -3.01E-5 -2.23E-4 -1.42E-6 -1.94E-5 -2.01E-4
3 -2.53E-6 -1.68E-4 -1.37E-3 -2.01E-4 -4.79E-4 -1.68E-3 -7.19E-5 -2.90E-4 -1.46E-3 -2.96E-5 -2.14E-4 -1.39E-3
4 -8.85E-5 -1.61E-3 -5.57E-3 -1.85E-3 -3.54E-3 -6.73E-3 -8.80E-4 -2.42E-3 -5.91E-3 -4.69E-4 -1.93E-3 -5.65E-3
5 -2.02E-3 -1.40E-2 -3.00E-2 -1.52E-2 -2.41E-2 -3.51E-2 -9.12E-3 -1.85E-2 -3.15E-2 -5.96E-3 -1.58E-2 -3.04E-2
6 -3.41E-2 -1.08E-1 -1.70E-1 -1.18E-1 -1.53E-1 -1.88E-1 -8.51E-2 -1.29E-1 -1.75E-1 -6.53E-2 -1.17E-1 -1.71E-1
7 -5.64E-1 -9.60E-1 -1.15E00 -1.15E00 -1.19E00 -1.23E00 -9.41E-1 -1.07E00 -1.17E00 -8.06E-1 -1.01E00 -1.15E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=100$[]{data-label="tabl:Eq100dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.76E-11 -7.05E-8 -2.37E-6 -7.99E-7 -2.93E-6 -9.43E-6 -1.41E-7 -7.86E-7 -4.17E-6 -2.65E-8 -2.68E-7 -2.80E-6
2 -4.81E-9 -3.29E-6 -5.69E-5 -1.04E-5 -3.09E-5 -1.24E-4 -2.36E-6 -1.28E-5 -7.67E-5 -6.03E-7 -6.68E-6 -6.19E-5
3 -2.98E-7 -4.57E-5 -5.23E-4 -8.81E-5 -2.09E-4 -8.14E-4 -2.70E-5 -1.10E-4 -6.19E-4 -9.39E-6 -7.12E-5 -5.48E-4
4 -9.89E-6 -3.91E-4 -2.27E-3 -6.55E-4 -1.27E-3 -2.99E-3 -2.63E-4 -7.65E-4 -2.51E-3 -1.18E-4 -5.46E-4 -2.34E-3
5 -2.20E-4 -3.04E-3 -9.04E-3 -4.45E-3 -7.41E-3 -1.19E-2 -2.26E-3 -5.04E-3 -9.99E-3 -1.26E-3 -3.91E-3 -9.29E-3
6 -3.62E-3 -2.17E-2 -4.42E-2 -2.81E-2 -4.04E-2 -5.44E-2 -1.75E-2 -3.08E-2 -4.77E-2 -1.17E-2 -2.58E-2 -4.51E-2
7 -4.82E-2 -1.43E-1 -2.20E-1 -1.76E-1 -2.16E-1 -2.54E-1 -1.29E-1 -1.80E-1 -2.32E-1 -9.93E-2 -1.60E-1 -2.23E-1
8 -6.79E-1 -1.14E00 -1.37E00 -1.49E00 -1.50E00 -1.51E00 -1.21E00 -1.33E00 -1.42E00 -1.03E00 -1.23E00 -1.38E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=100$[]{data-label="tabl:Eq100dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.95E-6 -1.08E-4 -8.95E-4 -2.40E-5 -1.34E-4 -9.05E-4 -9.77E-6 -1.16E-4 -8.97E-4 -5.23E-6 -1.11E-4 -8.95E-4
2 -1.40E-3 -8.52E-3 -1.54E-2 -3.57E-3 -9.35E-3 -1.56E-2 -2.46E-3 -8.79E-3 -1.54E-2 -1.95E-3 -8.61E-3 -1.54E-2
3 -1.38E-1 -2.67E-1 -3.16E-1 -1.98E-1 -2.77E-1 -3.18E-1 -1.71E-1 -2.70E-1 -3.16E-1 -1.56E-1 -2.68E-1 -3.16E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=200$[]{data-label="tabl:Eq200dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.03E-7 -2.36E-5 -3.87E-4 -6.57E-6 -3.74E-5 -4.04E-4 -1.86E-6 -2.80E-5 -3.91E-4 -6.84E-7 -2.50E-5 -3.88E-4
2 -5.98E-5 -1.00E-3 -3.12E-3 -4.44E-4 -1.32E-3 -3.23E-3 -2.22E-4 -1.11E-3 -3.14E-3 -1.35E-4 -1.03E-3 -3.13E-3
3 -5.36E-3 -2.47E-2 -4.24E-2 -1.47E-2 -2.85E-2 -4.33E-2 -1.02E-2 -2.60E-2 -4.26E-2 -7.93E-3 -2.51E-2 -4.25E-2
4 -2.34E-1 -4.29E-1 -5.12E-1 -3.69E-1 -4.60E-1 -5.18E-1 -3.11E-1 -4.40E-1 -5.13E-1 -2.78E-1 -4.32E-1 -5.13E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=200$[]{data-label="tabl:Eq200dDaAAk4p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -8.24E-9 -5.76E-6 -1.23E-4 -2.88E-6 -1.46E-5 -1.45E-4 -6.49E-7 -8.62E-6 -1.28E-4 -1.77E-7 -6.68E-6 -1.24E-4
2 -3.70E-6 -1.71E-4 -1.10E-3 -1.06E-4 -3.06E-4 -1.17E-3 -3.95E-5 -2.17E-4 -1.12E-3 -1.80E-5 -1.86E-4 -1.10E-3
3 -3.16E-4 -3.41E-3 -8.55E-3 -2.27E-3 -4.93E-3 -9.14E-3 -1.23E-3 -3.96E-3 -8.68E-3 -7.69E-4 -3.60E-3 -8.57E-3
4 -1.23E-2 -4.76E-2 -7.88E-2 -3.67E-2 -5.87E-2 -8.20E-2 -2.58E-2 -5.18E-2 -7.95E-2 -1.98E-2 -4.91E-2 -7.89E-2
5 -3.39E-1 -6.00E-1 -7.18E-1 -5.85E-1 -6.72E-1 -7.35E-1 -4.87E-1 -6.28E-1 -7.22E-1 -4.26E-1 -6.09E-1 -7.19E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=200$[]{data-label="tabl:Eq200dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -8.56E-10 -1.34E-6 -3.24E-5 -1.62E-6 -7.10E-6 -4.94E-5 -3.17E-7 -3.16E-6 -3.64E-5 -7.10E-8 -1.93E-6 -3.33E-5
2 -3.04E-7 -3.93E-5 -4.74E-4 -3.81E-5 -1.05E-4 -5.55E-4 -1.13E-5 -6.23E-5 -4.95E-4 -4.01E-6 -4.72E-5 -4.79E-4
3 -2.44E-5 -6.05E-4 -2.40E-3 -5.67E-4 -1.23E-3 -2.75E-3 -2.44E-4 -8.43E-4 -2.49E-3 -1.22E-4 -6.90E-4 -2.42E-3
4 -9.22E-4 -7.61E-3 -1.73E-2 -6.80E-3 -1.21E-2 -1.94E-2 -3.89E-3 -9.44E-3 -1.79E-2 -2.49E-3 -8.29E-3 -1.75E-2
5 -2.19E-2 -7.56E-2 -1.21E-1 -7.09E-2 -1.00E-1 -1.30E-1 -5.05E-2 -8.60E-2 -1.24E-1 -3.87E-2 -7.95E-2 -1.22E-1
6 -4.48E-1 -7.76E-1 -9.30E-1 -8.45E-1 -9.15E-1 -9.71E-1 -6.96E-1 -8.36E-1 -9.41E-1 -6.02E-1 -7.99E-1 -9.32E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=200$[]{data-label="tabl:Eq200dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.05E-10 -3.00E-7 -8.50E-6 -1.04E-6 -4.03E-6 -1.95E-5 -1.88E-7 -1.39E-6 -1.13E-5 -3.72E-8 -6.34E-7 -9.12E-6
2 -3.15E-8 -1.04E-5 -1.73E-4 -1.79E-5 -4.70E-5 -2.55E-4 -4.51E-6 -2.33E-5 -1.96E-4 -1.31E-6 -1.50E-5 -1.78E-4
3 -2.33E-6 -1.34E-4 -9.56E-4 -1.97E-4 -4.17E-4 -1.19E-3 -6.99E-5 -2.42E-4 -1.02E-3 -2.85E-5 -1.75E-4 -9.72E-4
4 -8.55E-5 -1.47E-3 -4.74E-3 -1.83E-3 -3.36E-3 -5.91E-3 -8.67E-4 -2.26E-3 -5.08E-3 -4.60E-4 -1.78E-3 -4.82E-3
5 -1.99E-3 -1.36E-2 -2.91E-2 -1.51E-2 -2.36E-2 -3.41E-2 -9.05E-3 -1.81E-2 -3.06E-2 -5.91E-3 -1.54E-2 -2.94E-2
6 -3.39E-2 -1.07E-1 -1.68E-1 -1.17E-1 -1.52E-1 -1.87E-1 -8.47E-2 -1.28E-1 -1.74E-1 -6.50E-2 -1.16E-1 -1.70E-1
7 -5.61E-1 -9.56E-1 -1.15E00 -1.14E00 -1.19E00 -1.23E00 -9.38E-1 -1.07E00 -1.17E00 -8.04E-1 -1.00E00 -1.15E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=200$[]{data-label="tabl:Eq200dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.41E-11 -6.76E-8 -2.36E-6 -7.31E-7 -2.55E-6 -9.32E-6 -1.25E-7 -7.20E-7 -4.14E-6 -2.28E-8 -2.52E-7 -2.78E-6
2 -3.90E-9 -2.81E-6 -5.45E-5 -1.00E-5 -2.50E-5 -1.16E-4 -2.23E-6 -1.04E-5 -7.29E-5 -5.56E-7 -5.57E-6 -5.92E-5
3 -2.65E-7 -3.57E-5 -4.21E-4 -8.62E-5 -1.77E-4 -6.22E-4 -2.62E-5 -8.93E-5 -4.88E-4 -8.98E-6 -5.65E-5 -4.39E-4
4 -9.36E-6 -3.35E-4 -1.67E-3 -6.47E-4 -1.18E-3 -2.33E-3 -2.58E-4 -6.87E-4 -1.88E-3 -1.16E-4 -4.80E-4 -1.73E-3
5 -2.14E-4 -2.87E-3 -8.25E-3 -4.41E-3 -7.18E-3 -1.11E-2 -2.23E-3 -4.83E-3 -9.20E-3 -1.24E-3 -3.72E-3 -8.50E-3
6 -3.57E-3 -2.12E-2 -4.33E-2 -2.80E-2 -3.99E-2 -5.35E-2 -1.74E-2 -3.03E-2 -4.68E-2 -1.16E-2 -2.53E-2 -4.42E-2
7 -4.79E-2 -1.42E-1 -2.19E-1 -1.76E-1 -2.15E-1 -2.52E-1 -1.28E-1 -1.79E-1 -2.30E-1 -9.89E-2 -1.59E-1 -2.22E-1
8 -6.77E-1 -1.14E00 -1.36E00 -1.48E00 -1.49E00 -1.51E00 -1.21E00 -1.32E00 -1.41E00 -1.03E00 -1.22E00 -1.38E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=200$[]{data-label="tabl:Eq200dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.70E-6 -7.38E-5 -5.07E-4 -2.29E-5 -9.56E-5 -5.14E-4 -9.16E-6 -8.06E-5 -5.09E-4 -4.80E-6 -7.59E-5 -5.07E-4
2 -1.38E-3 -8.24E-3 -1.47E-2 -3.53E-3 -9.06E-3 -1.48E-2 -2.43E-3 -8.51E-3 -1.47E-2 -1.92E-3 -8.33E-3 -1.47E-2
3 -1.38E-1 -2.65E-1 -3.15E-1 -1.97E-1 -2.76E-1 -3.16E-1 -1.70E-1 -2.69E-1 -3.15E-1 -1.56E-1 -2.67E-1 -3.15E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q=400$[]{data-label="tabl:Eq400dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -8.26E-8 -1.49E-5 -2.42E-4 -6.25E-6 -2.52E-5 -2.51E-4 -1.73E-6 -1.81E-5 -2.44E-4 -6.14E-7 -1.59E-5 -2.42E-4
2 -5.80E-5 -9.14E-4 -2.60E-3 -4.38E-4 -1.22E-3 -2.70E-3 -2.18E-4 -1.01E-3 -2.62E-3 -1.32E-4 -9.46E-4 -2.60E-3
3 -5.31E-3 -2.43E-2 -4.17E-2 -1.46E-2 -2.81E-2 -4.27E-2 -1.01E-2 -2.56E-2 -4.19E-2 -7.87E-3 -2.48E-2 -4.18E-2
4 -2.34E-1 -4.27E-1 -5.11E-1 -3.68E-1 -4.58E-1 -5.17E-1 -3.10E-1 -4.38E-1 -5.12E-1 -2.77E-1 -4.31E-1 -5.11E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q=400$[]{data-label="tabl:Eq400dDaAAk4p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -6.07E-9 -3.85E-6 -9.78E-5 -2.74E-6 -1.01E-5 -1.13E-4 -5.99E-7 -5.82E-6 -1.01E-4 -1.57E-7 -4.48E-6 -9.85E-5
2 -3.49E-6 -1.42E-4 -7.46E-4 -1.04E-4 -2.68E-4 -8.13E-4 -3.85E-5 -1.84E-4 -7.61E-4 -1.74E-5 -1.56E-4 -7.49E-4
3 -3.11E-4 -3.28E-3 -8.06E-3 -2.26E-3 -4.78E-3 -8.66E-3 -1.22E-3 -3.83E-3 -8.20E-3 -7.61E-4 -3.47E-3 -8.09E-3
4 -1.22E-2 -4.72E-2 -7.81E-2 -3.66E-2 -5.82E-2 -8.13E-2 -2.57E-2 -5.14E-2 -7.88E-2 -1.97E-2 -4.87E-2 -7.82E-2
5 -3.38E-1 -5.98E-1 -7.17E-1 -5.84E-1 -6.70E-1 -7.34E-1 -4.86E-1 -6.26E-1 -7.21E-1 -4.25E-1 -6.08E-1 -7.18E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q=400$[]{data-label="tabl:Eq400dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -6.05E-10 -1.03E-6 -2.98E-5 -1.53E-6 -5.18E-6 -4.45E-5 -2.93E-7 -2.34E-6 -3.34E-5 -6.31E-8 -1.45E-6 -3.06E-5
2 -2.76E-7 -2.96E-5 -3.19E-4 -3.73E-5 -8.74E-5 -3.71E-4 -1.10E-5 -4.92E-5 -3.32E-4 -3.85E-6 -3.62E-5 -3.21E-4
3 -2.37E-5 -5.54E-4 -2.01E-3 -5.62E-4 -1.17E-3 -2.37E-3 -2.41E-4 -7.86E-4 -2.10E-3 -1.20E-4 -6.36E-4 -2.03E-3
4 -9.12E-4 -7.45E-3 -1.69E-2 -6.77E-3 -1.19E-2 -1.89E-2 -3.87E-3 -9.26E-3 -1.74E-2 -2.47E-3 -8.12E-3 -1.70E-2
5 -2.18E-2 -7.51E-2 -1.21E-1 -7.07E-2 -9.95E-2 -1.29E-1 -5.03E-2 -8.55E-2 -1.23E-1 -3.85E-2 -7.90E-2 -1.21E-1
6 -4.47E-1 -7.74E-1 -9.29E-1 -8.43E-1 -9.13E-1 -9.69E-1 -6.95E-1 -8.34E-1 -9.39E-1 -6.01E-1 -7.97E-1 -9.31E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q=400$[]{data-label="tabl:Eq400dDaAAk6p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -7.44E-11 -2.58E-7 -8.25E-6 -9.87E-7 -3.10E-6 -1.86E-5 -1.73E-7 -1.12E-6 -1.09E-5 -3.31E-8 -5.30E-7 -8.85E-6
2 -2.73E-8 -7.72E-6 -1.39E-4 -1.75E-5 -3.81E-5 -1.94E-4 -4.37E-6 -1.80E-5 -1.55E-4 -1.25E-6 -1.13E-5 -1.42E-4
3 -2.21E-6 -1.14E-4 -6.80E-4 -1.95E-4 -3.83E-4 -8.89E-4 -6.88E-5 -2.16E-4 -7.38E-4 -2.79E-5 -1.52E-4 -6.94E-4
4 -8.39E-5 -1.40E-3 -4.37E-3 -1.82E-3 -3.27E-3 -5.55E-3 -8.60E-4 -2.18E-3 -4.71E-3 -4.55E-4 -1.71E-3 -4.45E-3
5 -1.97E-3 -1.34E-2 -2.86E-2 -1.51E-2 -2.34E-2 -3.37E-2 -9.01E-3 -1.79E-2 -3.01E-2 -5.88E-3 -1.52E-2 -2.90E-2
6 -3.38E-2 -1.07E-1 -1.68E-1 -1.17E-1 -1.52E-1 -1.86E-1 -8.45E-2 -1.28E-1 -1.73E-1 -6.48E-2 -1.16E-1 -1.69E-1
7 -5.60E-1 -9.54E-1 -1.14E00 -1.14E00 -1.19E00 -1.23E00 -9.37E-1 -1.07E00 -1.17E00 -8.02E-1 -1.00E00 -1.15E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q=400$[]{data-label="tabl:Eq400dDaAAk7p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
1 -1.05E-11 -6.26E-8 -2.33E-6 -6.92E-7 -2.06E-6 -9.09E-6 -1.15E-7 -6.17E-7 -4.08E-6 -2.03E-8 -2.25E-7 -2.75E-6
2 -3.25E-9 -2.22E-6 -4.99E-5 -9.83E-6 -2.02E-5 -1.00E-4 -2.16E-6 -8.17E-6 -6.56E-5 -5.29E-7 -4.34E-6 -5.40E-5
3 -2.45E-7 -2.83E-5 -3.03E-4 -8.53E-5 -1.59E-4 -4.43E-4 -2.57E-5 -7.62E-5 -3.48E-4 -8.76E-6 -4.65E-5 -3.15E-4
4 -9.07E-6 -3.05E-4 -1.36E-3 -6.42E-4 -1.13E-3 -2.03E-3 -2.56E-4 -6.46E-4 -1.57E-3 -1.14E-4 -4.44E-4 -1.42E-3
5 -2.11E-4 -2.78E-3 -7.90E-3 -4.39E-3 -7.06E-3 -1.08E-2 -2.22E-3 -4.73E-3 -8.86E-3 -1.24E-3 -3.62E-3 -8.15E-3
6 -3.55E-3 -2.10E-2 -4.28E-2 -2.79E-2 -3.96E-2 -5.31E-2 -1.73E-2 -3.00E-2 -4.63E-2 -1.15E-2 -2.51E-2 -4.38E-2
7 -4.78E-2 -1.41E-1 -2.18E-1 -1.75E-1 -2.14E-1 -2.52E-1 -1.28E-1 -1.78E-1 -2.30E-1 -9.87E-2 -1.58E-1 -2.21E-1
8 -6.76E-1 -1.14E00 -1.36E00 -1.48E00 -1.49E00 -1.51E00 -1.21E00 -1.32E00 -1.41E00 -1.03E00 -1.22E00 -1.38E00
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of ${\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-0-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ of the type on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q=400$[]{data-label="tabl:Eq400dDaAAk8p"}
Tables type **(f)** for 0-URA-decomposition coefficients
--------------------------------------------------------
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.52E-1 6.95E-1 8.60E-1 5.64E-1 6.97E-1 8.60E-1 5.59E-1 6.96E-1 8.60E-1 5.56E-1 6.95E-1 8.60E-1
1 -1.13E-6 -3.83E-5 -9.73E-5 -8.38E-6 -4.70E-5 -9.96E-5 -3.94E-6 -4.10E-5 -9.77E-5 -2.38E-6 -3.92E-5 -9.74E-5
2 -7.03E-4 -7.62E-3 -2.11E-2 -1.43E-3 -8.14E-3 -2.13E-2 -1.08E-3 -7.79E-3 -2.11E-2 -9.02E-4 -7.67E-3 -2.11E-2
3 -6.86E-2 -3.21E-1 -6.91E-1 -8.66E-2 -3.27E-1 -6.92E-1 -7.86E-2 -3.23E-1 -6.91E-1 -7.42E-2 -3.22E-1 -6.91E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=1$[]{data-label="tabl:Fq001dDaAAk3p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.70E-1 7.25E-1 8.86E-1 5.87E-1 7.30E-1 8.86E-1 5.80E-1 7.27E-1 8.86E-1 5.76E-1 7.26E-1 8.86E-1
1 -2.36E-8 -1.36E-6 -3.70E-6 -1.10E-6 -2.53E-6 -4.06E-6 -3.25E-7 -1.71E-6 -3.77E-6 -1.27E-7 -1.47E-6 -3.71E-6
2 -1.83E-5 -3.23E-4 -7.14E-4 -1.06E-4 -4.19E-4 -7.43E-4 -5.77E-5 -3.55E-4 -7.20E-4 -3.72E-5 -3.34E-4 -7.16E-4
3 -1.96E-3 -1.67E-2 -4.82E-2 -4.23E-3 -1.86E-2 -4.91E-2 -3.20E-3 -1.73E-2 -4.84E-2 -2.64E-3 -1.69E-2 -4.82E-2
4 -9.68E-2 -4.10E-1 -7.76E-1 -1.32E-1 -4.25E-1 -7.78E-1 -1.17E-1 -4.15E-1 -7.77E-1 -1.09E-1 -4.12E-1 -7.76E-1
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=1$[]{data-label="tabl:Fq001dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.83E-1 7.48E-1 9.02E-1 6.06E-1 7.55E-1 9.03E-1 5.98E-1 7.51E-1 9.03E-1 5.93E-1 7.49E-1 9.02E-1
1 -7.10E-10 -6.76E-8 -2.03E-7 -2.85E-7 -2.72E-7 -2.69E-7 -5.87E-8 -1.22E-7 -2.16E-7 -1.52E-8 -8.40E-8 -2.05E-7
2 -6.13E-7 -1.71E-5 -3.80E-5 -1.52E-5 -3.35E-5 -4.34E-5 -5.91E-6 -2.26E-5 -3.92E-5 -2.77E-6 -1.89E-5 -3.82E-5
3 -7.88E-5 -1.02E-3 -2.19E-3 -4.45E-4 -1.45E-3 -2.36E-3 -2.61E-4 -1.18E-3 -2.23E-3 -1.72E-4 -1.08E-3 -2.20E-3
4 -3.69E-3 -2.74E-2 -8.26E-2 -8.48E-3 -3.22E-2 -8.56E-2 -6.49E-3 -2.92E-2 -8.33E-2 -5.31E-3 -2.81E-2 -8.27E-2
5 -1.24E-1 -4.89E-1 -8.31E-1 -1.83E-1 -5.20E-1 -8.34E-1 -1.60E-1 -5.01E-1 -8.32E-1 -1.46E-1 -4.93E-1 -8.31E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=1$[]{data-label="tabl:Fq001dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 5.95E-1 7.65E-1 9.14E-1 6.22E-1 7.76E-1 9.16E-1 6.13E-1 7.70E-1 9.15E-1 6.07E-1 7.67E-1 9.15E-1
1 -2.85E-11 -4.34E-9 -1.45E-8 -1.09E-7 -5.22E-8 -2.84E-8 -1.77E-8 -1.51E-8 -1.74E-8 -3.38E-9 -7.27E-9 -1.51E-8
2 -2.60E-8 -1.12E-6 -2.70E-6 -3.52E-6 -4.29E-6 -3.80E-6 -1.03E-6 -2.13E-6 -2.96E-6 -3.58E-7 -1.45E-6 -2.75E-6
3 -3.72E-6 -7.18E-5 -1.46E-4 -7.48E-5 -1.57E-4 -1.79E-4 -3.37E-5 -1.04E-4 -1.54E-4 -1.74E-5 -8.31E-5 -1.48E-4
4 -2.02E-4 -2.18E-3 -4.71E-3 -1.14E-3 -3.37E-3 -5.31E-3 -7.10E-4 -2.66E-3 -4.86E-3 -4.81E-4 -2.36E-3 -4.75E-3
5 -5.74E-3 -3.90E-2 -1.21E-1 -1.39E-2 -4.83E-2 -1.28E-1 -1.08E-2 -4.31E-2 -1.23E-1 -8.82E-3 -4.06E-2 -1.21E-1
6 -1.51E-1 -5.61E-1 -8.66E-1 -2.40E-1 -6.13E-1 -8.71E-1 -2.08E-1 -5.84E-1 -8.67E-1 -1.87E-1 -5.70E-1 -8.66E-1
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=1$[]{data-label="tabl:Fq001dDaAAk6p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 6.04E-1 7.79E-1 9.24E-1 6.36E-1 7.93E-1 9.26E-1 6.27E-1 7.86E-1 9.24E-1 6.20E-1 7.82E-1 9.24E-1
1 -1.45E-12 -3.42E-10 -1.27E-9 -5.35E-8 -1.54E-8 -4.69E-9 -7.37E-9 -3.02E-9 -1.96E-9 -1.14E-9 -9.69E-10 -1.41E-9
2 -1.36E-9 -8.94E-8 -2.35E-7 -1.14E-6 -8.30E-7 -4.81E-7 -2.67E-7 -2.99E-7 -2.96E-7 -7.21E-8 -1.55E-7 -2.49E-7
3 -2.05E-7 -5.87E-6 -1.26E-5 -1.80E-5 -2.40E-5 -1.95E-5 -6.42E-6 -1.24E-5 -1.45E-5 -2.60E-6 -8.23E-6 -1.30E-5
4 -1.24E-5 -1.90E-4 -3.70E-4 -2.23E-4 -4.59E-4 -4.91E-4 -1.11E-4 -3.01E-4 -4.05E-4 -6.14E-5 -2.33E-4 -3.78E-4
5 -3.96E-4 -3.75E-3 -8.35E-3 -2.25E-3 -6.30E-3 -1.00E-2 -1.46E-3 -4.91E-3 -8.84E-3 -1.01E-3 -4.23E-3 -8.47E-3
6 -7.99E-3 -5.10E-2 -1.61E-1 -2.03E-2 -6.65E-2 -1.76E-1 -1.59E-2 -5.85E-2 -1.65E-1 -1.30E-2 -5.42E-2 -1.62E-1
7 -1.77E-1 -6.27E-1 -8.88E-1 -3.02E-1 -7.04E-1 -8.94E-1 -2.59E-1 -6.65E-1 -8.90E-1 -2.31E-1 -6.43E-1 -8.89E-1
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=1$[]{data-label="tabl:Fq001dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 6.12E-1 7.91E-1 9.31E-1 6.48E-1 8.08E-1 9.35E-1 6.39E-1 8.00E-1 9.32E-1 6.31E-1 7.95E-1 9.31E-1
1 -8.88E-14 -3.18E-11 -1.30E-10 -3.05E-8 -6.04E-9 -1.14E-9 -3.76E-9 -8.82E-10 -3.11E-10 -4.99E-10 -1.94E-10 -1.68E-10
2 -8.45E-11 -8.35E-9 -2.42E-8 -4.73E-7 -2.22E-7 -8.52E-8 -9.23E-8 -5.97E-8 -3.91E-8 -2.03E-8 -2.31E-8 -2.77E-8
3 -1.32E-8 -5.55E-7 -1.29E-6 -5.71E-6 -5.03E-6 -2.91E-6 -1.66E-6 -2.01E-6 -1.75E-6 -5.45E-7 -1.07E-6 -1.40E-6
4 -8.45E-7 -1.85E-5 -3.68E-5 -5.79E-5 -8.20E-5 -6.37E-5 -2.36E-5 -4.36E-5 -4.51E-5 -1.06E-5 -2.84E-5 -3.90E-5
5 -2.97E-5 -3.88E-4 -7.43E-4 -4.99E-4 -1.03E-3 -1.08E-3 -2.68E-4 -6.76E-4 -8.52E-4 -1.56E-4 -5.12E-4 -7.72E-4
6 -6.59E-4 -5.69E-3 -1.31E-2 -3.74E-3 -1.03E-2 -1.68E-2 -2.52E-3 -7.95E-3 -1.44E-2 -1.80E-3 -6.72E-3 -1.35E-2
7 -1.04E-2 -6.31E-2 -2.00E-1 -2.75E-2 -8.63E-2 -2.24E-1 -2.17E-2 -7.52E-2 -2.08E-1 -1.79E-2 -6.88E-2 -2.02E-1
8 -2.03E-1 -6.88E-1 -9.02E-1 -3.69E-1 -7.95E-1 -9.07E-1 -3.15E-1 -7.45E-1 -9.04E-1 -2.78E-1 -7.15E-1 -9.02E-1
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=1$[]{data-label="tabl:Fq001dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.92E-3 9.96E-3 9.98E-3 9.92E-3 9.96E-3 9.98E-3 9.92E-3 9.96E-3 9.98E-3 9.92E-3 9.96E-3 9.98E-3
1 -8.88E-9 -1.13E-6 -1.17E-5 -3.94E-8 -1.27E-6 -1.18E-5 -2.26E-8 -1.17E-6 -1.17E-5 -1.55E-8 -1.14E-6 -1.17E-5
2 -7.98E-7 -1.10E-5 -3.24E-5 -1.36E-6 -1.12E-5 -3.23E-5 -1.10E-6 -1.10E-5 -3.24E-5 -9.62E-7 -1.10E-5 -3.24E-5
3 -2.16E-5 -5.53E-5 -5.19E-5 -2.59E-5 -5.58E-5 -5.19E-5 -2.40E-5 -5.55E-5 -5.19E-5 -2.29E-5 -5.54E-5 -5.19E-5
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=100$[]{data-label="tabl:Fq100dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.93E-3 9.96E-3 9.99E-3 9.93E-3 9.96E-3 9.99E-3 9.93E-3 9.96E-3 9.99E-3 9.93E-3 9.96E-3 9.99E-3
1 -6.77E-10 -2.10E-7 -2.13E-6 -1.13E-8 -3.15E-7 -2.30E-6 -4.71E-9 -2.44E-7 -2.16E-6 -2.37E-9 -2.21E-7 -2.13E-6
2 -6.90E-8 -3.26E-6 -2.20E-5 -2.39E-7 -3.63E-6 -2.21E-5 -1.56E-7 -3.39E-6 -2.21E-5 -1.14E-7 -3.31E-6 -2.20E-5
3 -1.71E-6 -1.46E-5 -2.93E-5 -3.00E-6 -1.52E-5 -2.92E-5 -2.45E-6 -1.48E-5 -2.92E-5 -2.13E-6 -1.47E-5 -2.93E-5
4 -2.82E-5 -6.25E-5 -5.15E-5 -3.58E-5 -6.37E-5 -5.15E-5 -3.27E-5 -6.29E-5 -5.15E-5 -3.08E-5 -6.26E-5 -5.15E-5
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=100$[]{data-label="tabl:Fq100dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.93E-3 9.97E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3 9.93E-3 9.97E-3 9.99E-3 9.93E-3 9.97E-3 9.99E-3
1 -5.91E-11 -2.82E-8 -1.79E-7 -4.95E-9 -8.32E-8 -2.36E-7 -1.62E-9 -4.53E-8 -1.91E-7 -6.06E-10 -3.36E-8 -1.81E-7
2 -7.40E-9 -9.46E-7 -9.98E-6 -6.77E-8 -1.31E-6 -1.06E-5 -3.57E-8 -1.09E-6 -1.01E-5 -2.12E-8 -9.95E-7 -1.00E-5
3 -1.94E-7 -5.08E-6 -2.24E-5 -6.42E-7 -5.73E-6 -2.22E-5 -4.45E-7 -5.34E-6 -2.23E-5 -3.34E-7 -5.17E-6 -2.24E-5
4 -2.71E-6 -1.73E-5 -2.78E-5 -4.93E-6 -1.82E-5 -2.77E-5 -4.08E-6 -1.77E-5 -2.78E-5 -3.53E-6 -1.74E-5 -2.78E-5
5 -3.42E-5 -6.87E-5 -5.19E-5 -4.60E-5 -7.10E-5 -5.19E-5 -4.15E-5 -6.96E-5 -5.19E-5 -3.86E-5 -6.90E-5 -5.19E-5
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=100$[]{data-label="tabl:Fq100dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.93E-3 9.97E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3
1 -5.37E-12 -2.95E-9 -1.40E-8 -2.75E-9 -2.57E-8 -2.74E-8 -7.58E-10 -8.96E-9 -1.68E-8 -2.26E-10 -4.70E-9 -1.46E-8
2 -8.93E-10 -2.24E-7 -2.02E-6 -2.64E-8 -5.04E-7 -2.69E-6 -1.16E-8 -3.35E-7 -2.18E-6 -5.65E-9 -2.65E-7 -2.05E-6
3 -2.60E-8 -1.84E-6 -1.60E-5 -1.96E-7 -2.49E-6 -1.65E-5 -1.15E-7 -2.13E-6 -1.61E-5 -7.40E-8 -1.95E-6 -1.60E-5
4 -3.73E-7 -6.58E-6 -2.08E-5 -1.22E-6 -7.57E-6 -2.06E-5 -8.83E-7 -7.03E-6 -2.08E-5 -6.77E-7 -6.75E-6 -2.08E-5
5 -3.73E-6 -1.95E-5 -2.71E-5 -7.00E-6 -2.09E-5 -2.70E-5 -5.85E-6 -2.01E-5 -2.70E-5 -5.07E-6 -1.97E-5 -2.71E-5
6 -3.97E-5 -7.42E-5 -5.25E-5 -5.66E-5 -7.82E-5 -5.27E-5 -5.07E-5 -7.60E-5 -5.26E-5 -4.67E-5 -7.49E-5 -5.25E-5
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=100$[]{data-label="tabl:Fq100dDaAAk6p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.93E-3 9.97E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3
1 -4.89E-13 -2.88E-10 -1.25E-9 -1.76E-9 -9.71E-9 -4.64E-9 -4.31E-10 -2.24E-9 -1.94E-9 -1.09E-10 -7.75E-10 -1.40E-9
2 -1.14E-10 -3.98E-8 -2.25E-7 -1.28E-8 -2.02E-7 -4.56E-7 -4.84E-9 -9.99E-8 -2.82E-7 -1.98E-9 -6.13E-8 -2.38E-7
3 -3.86E-9 -5.97E-7 -6.42E-6 -7.72E-8 -1.17E-6 -8.25E-6 -3.92E-8 -8.68E-7 -6.98E-6 -2.16E-8 -7.10E-7 -6.55E-6
4 -5.89E-8 -2.67E-6 -1.74E-5 -4.05E-7 -3.62E-6 -1.73E-5 -2.56E-7 -3.14E-6 -1.74E-5 -1.72E-7 -2.88E-6 -1.74E-5
5 -5.92E-7 -7.84E-6 -1.98E-5 -1.93E-6 -9.20E-6 -1.95E-5 -1.44E-6 -8.52E-6 -1.97E-5 -1.13E-6 -8.14E-6 -1.97E-5
6 -4.73E-6 -2.13E-5 -2.66E-5 -9.12E-6 -2.32E-5 -2.65E-5 -7.69E-6 -2.22E-5 -2.66E-5 -6.69E-6 -2.17E-5 -2.66E-5
7 -4.49E-5 -7.94E-5 -5.33E-5 -6.78E-5 -8.54E-5 -5.36E-5 -6.02E-5 -8.23E-5 -5.34E-5 -5.50E-5 -8.06E-5 -5.33E-5
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=100$[]{data-label="tabl:Fq100dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 9.94E-3 9.97E-3 9.99E-3 9.95E-3 9.98E-3 9.99E-3 9.94E-3 9.98E-3 9.99E-3 9.94E-3 9.97E-3 9.99E-3
1 -4.45E-14 -2.94E-11 -1.30E-10 -1.23E-9 -4.38E-9 -1.13E-9 -2.77E-10 -7.31E-10 -3.10E-10 -6.17E-11 -1.71E-10 -1.67E-10
2 -1.45E-11 -5.66E-9 -2.40E-8 -7.22E-9 -8.58E-8 -8.53E-8 -2.42E-9 -3.08E-8 -3.89E-8 -8.56E-10 -1.39E-8 -2.75E-8
3 -6.04E-10 -1.54E-7 -1.18E-6 -3.64E-8 -5.70E-7 -2.47E-6 -1.63E-8 -3.48E-7 -1.57E-6 -7.80E-9 -2.37E-7 -1.28E-6
4 -1.01E-8 -1.04E-6 -1.09E-5 -1.65E-7 -1.88E-6 -1.28E-5 -9.20E-8 -1.49E-6 -1.17E-5 -5.46E-8 -1.25E-6 -1.12E-5
5 -1.06E-7 -3.42E-6 -1.68E-5 -6.88E-7 -4.67E-6 -1.63E-5 -4.57E-7 -4.10E-6 -1.66E-5 -3.20E-7 -3.75E-6 -1.68E-5
6 -8.39E-7 -8.92E-6 -1.91E-5 -2.72E-6 -1.06E-5 -1.88E-5 -2.09E-6 -9.87E-6 -1.90E-5 -1.66E-6 -9.38E-6 -1.91E-5
7 -5.69E-6 -2.28E-5 -2.64E-5 -1.13E-5 -2.54E-5 -2.63E-5 -9.56E-6 -2.42E-5 -2.64E-5 -8.34E-6 -2.35E-5 -2.64E-5
8 -4.99E-5 -8.41E-5 -5.41E-5 -7.95E-5 -9.27E-5 -5.47E-5 -7.01E-5 -8.87E-5 -5.43E-5 -6.36E-5 -8.63E-5 -5.42E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=100$[]{data-label="tabl:Fq100dDaAAk8p"}
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3
1 -2.41E-9 -3.29E-7 -3.66E-6 -1.05E-8 -3.66E-7 -3.69E-6 -6.04E-9 -3.41E-7 -3.67E-6 -4.18E-9 -3.33E-7 -3.66E-6
2 -2.04E-7 -2.77E-6 -7.55E-6 -3.46E-7 -2.84E-6 -7.54E-6 -2.81E-7 -2.79E-6 -7.55E-6 -2.46E-7 -2.78E-6 -7.55E-6
3 -5.43E-6 -1.38E-5 -1.28E-5 -6.51E-6 -1.40E-5 -1.28E-5 -6.04E-6 -1.39E-5 -1.28E-5 -5.78E-6 -1.38E-5 -1.28E-5
----------------------- ---------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=200$[]{data-label="tabl:Fq200dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3
1 -2.00E-10 -7.79E-8 -1.15E-6 -3.10E-9 -1.11E-7 -1.22E-6 -1.32E-9 -8.89E-8 -1.17E-6 -6.76E-10 -8.15E-8 -1.16E-6
2 -1.82E-8 -8.62E-7 -5.41E-6 -6.19E-8 -9.49E-7 -5.41E-6 -4.06E-8 -8.93E-7 -5.41E-6 -2.99E-8 -8.72E-7 -5.41E-6
3 -4.35E-7 -3.66E-6 -6.95E-6 -7.61E-7 -3.80E-6 -6.94E-6 -6.23E-7 -3.71E-6 -6.95E-6 -5.42E-7 -3.68E-6 -6.95E-6
4 -7.10E-6 -1.56E-5 -1.28E-5 -8.99E-6 -1.59E-5 -1.28E-5 -8.22E-6 -1.57E-5 -1.28E-5 -7.75E-6 -1.56E-5 -1.28E-5
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=200$[]{data-label="tabl:Fq200dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3
1 -2.00E-11 -1.49E-8 -1.52E-7 -1.40E-9 -3.77E-8 -1.97E-7 -4.74E-10 -2.24E-8 -1.62E-7 -1.85E-10 -1.73E-8 -1.54E-7
2 -2.04E-9 -2.84E-7 -3.34E-6 -1.78E-8 -3.75E-7 -3.46E-6 -9.51E-9 -3.20E-7 -3.37E-6 -5.72E-9 -2.96E-7 -3.35E-6
3 -5.03E-8 -1.30E-6 -5.02E-6 -1.65E-7 -1.46E-6 -4.99E-6 -1.14E-7 -1.36E-6 -5.01E-6 -8.63E-8 -1.32E-6 -5.02E-6
4 -6.89E-7 -4.33E-6 -6.70E-6 -1.25E-6 -4.57E-6 -6.69E-6 -1.03E-6 -4.43E-6 -6.70E-6 -8.95E-7 -4.37E-6 -6.70E-6
5 -8.60E-6 -1.72E-5 -1.29E-5 -1.16E-5 -1.77E-5 -1.29E-5 -1.04E-5 -1.74E-5 -1.29E-5 -9.72E-6 -1.72E-5 -1.29E-5
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=200$[]{data-label="tabl:Fq200dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3
1 -2.17E-12 -2.10E-9 -1.34E-8 -7.98E-10 -1.45E-8 -2.61E-8 -2.32E-10 -5.75E-9 -1.61E-8 -7.35E-11 -3.21E-9 -1.40E-8
2 -2.67E-10 -8.62E-8 -1.21E-6 -7.08E-9 -1.66E-7 -1.48E-6 -3.17E-9 -1.20E-7 -1.28E-6 -1.58E-9 -9.86E-8 -1.22E-6
3 -6.97E-9 -4.99E-7 -4.05E-6 -5.09E-8 -6.54E-7 -4.04E-6 -3.02E-8 -5.68E-7 -4.05E-6 -1.95E-8 -5.26E-7 -4.05E-6
4 -9.62E-8 -1.66E-6 -4.73E-6 -3.12E-7 -1.91E-6 -4.70E-6 -2.26E-7 -1.77E-6 -4.72E-6 -1.74E-7 -1.70E-6 -4.73E-6
5 -9.46E-7 -4.87E-6 -6.58E-6 -1.77E-6 -5.22E-6 -6.56E-6 -1.48E-6 -5.03E-6 -6.57E-6 -1.28E-6 -4.93E-6 -6.58E-6
6 -9.98E-6 -1.85E-5 -1.30E-5 -1.42E-5 -1.95E-5 -1.31E-5 -1.27E-5 -1.90E-5 -1.31E-5 -1.17E-5 -1.87E-5 -1.31E-5
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=200$[]{data-label="tabl:Fq200dDaAAk6p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.98E-3 4.99E-3 5.00E-3 4.99E-3 4.99E-3 5.00E-3 4.99E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3
1 -2.41E-13 -2.44E-10 -1.24E-9 -5.22E-10 -6.42E-9 -4.57E-9 -1.37E-10 -1.70E-9 -1.91E-9 -3.74E-11 -6.28E-10 -1.38E-9
2 -3.82E-11 -2.11E-8 -2.01E-7 -3.49E-9 -7.95E-8 -3.83E-7 -1.36E-9 -4.53E-8 -2.48E-7 -5.77E-10 -3.03E-8 -2.11E-7
3 -1.09E-9 -1.90E-7 -2.55E-6 -2.03E-8 -3.29E-7 -2.88E-6 -1.04E-8 -2.57E-7 -2.66E-6 -5.81E-9 -2.19E-7 -2.57E-6
4 -1.55E-8 -6.97E-7 -3.88E-6 -1.04E-7 -9.24E-7 -3.79E-6 -6.61E-8 -8.09E-7 -3.85E-6 -4.47E-8 -7.45E-7 -3.87E-6
5 -1.52E-7 -1.97E-6 -4.58E-6 -4.91E-7 -2.31E-6 -4.55E-6 -3.67E-7 -2.14E-6 -4.57E-6 -2.88E-7 -2.04E-6 -4.58E-6
6 -1.20E-6 -5.32E-6 -6.51E-6 -2.30E-6 -5.81E-6 -6.50E-6 -1.94E-6 -5.56E-6 -6.51E-6 -1.69E-6 -5.42E-6 -6.51E-6
7 -1.13E-5 -1.98E-5 -1.33E-5 -1.70E-5 -2.13E-5 -1.33E-5 -1.51E-5 -2.06E-5 -1.33E-5 -1.38E-5 -2.01E-5 -1.33E-5
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=200$[]{data-label="tabl:Fq200dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.98E-3 4.99E-3 5.00E-3 4.99E-3 4.99E-3 5.00E-3 4.99E-3 4.99E-3 5.00E-3 4.99E-3 4.99E-3 5.00E-3
1 -2.65E-14 -2.72E-11 -1.29E-10 -3.73E-10 -3.22E-9 -1.13E-9 -9.09E-11 -6.08E-10 -3.08E-10 -2.23E-11 -1.52E-10 -1.66E-10
2 -5.69E-12 -3.96E-9 -2.35E-8 -2.00E-9 -4.04E-8 -8.25E-8 -6.98E-10 -1.74E-8 -3.81E-8 -2.59E-10 -8.79E-9 -2.70E-8
3 -1.86E-10 -6.42E-8 -8.48E-7 -9.69E-9 -1.79E-7 -1.44E-6 -4.41E-9 -1.22E-7 -1.05E-6 -2.16E-9 -9.01E-8 -9.02E-7
4 -2.79E-9 -2.99E-7 -3.19E-6 -4.29E-8 -4.96E-7 -3.25E-6 -2.41E-8 -4.04E-7 -3.23E-6 -1.44E-8 -3.49E-7 -3.21E-6
5 -2.78E-8 -8.77E-7 -3.67E-6 -1.76E-7 -1.18E-6 -3.58E-6 -1.17E-7 -1.04E-6 -3.63E-6 -8.24E-8 -9.56E-7 -3.66E-6
6 -2.15E-7 -2.24E-6 -4.50E-6 -6.91E-7 -2.67E-6 -4.46E-6 -5.30E-7 -2.47E-6 -4.48E-6 -4.22E-7 -2.35E-6 -4.49E-6
7 -1.44E-6 -5.71E-6 -6.48E-6 -2.85E-6 -6.35E-6 -6.47E-6 -2.41E-6 -6.05E-6 -6.47E-6 -2.11E-6 -5.87E-6 -6.47E-6
8 -1.25E-5 -2.10E-5 -1.35E-5 -2.00E-5 -2.31E-5 -1.36E-5 -1.76E-5 -2.21E-5 -1.35E-5 -1.60E-5 -2.16E-5 -1.35E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=200$[]{data-label="tabl:Fq200dDaAAk8p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 -6.27E-10 -8.91E-8 -1.02E-6 -2.70E-9 -9.86E-8 -1.03E-6 -1.56E-9 -9.22E-8 -1.02E-6 -1.08E-9 -9.01E-8 -1.02E-6
2 -5.17E-8 -6.96E-7 -1.80E-6 -8.75E-8 -7.13E-7 -1.80E-6 -7.10E-8 -7.01E-7 -1.80E-6 -6.22E-8 -6.97E-7 -1.80E-6
3 -1.36E-6 -3.46E-6 -3.18E-6 -1.63E-6 -3.49E-6 -3.18E-6 -1.52E-6 -3.47E-6 -3.18E-6 -1.45E-6 -3.46E-6 -3.18E-6
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q=400$[]{data-label="tabl:Fq400dDaAAk3p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 -5.46E-11 -2.44E-8 -4.53E-7 -8.14E-10 -3.35E-8 -4.72E-7 -3.50E-10 -2.75E-8 -4.57E-7 -1.81E-10 -2.54E-8 -4.54E-7
2 -4.66E-9 -2.20E-7 -1.25E-6 -1.58E-8 -2.41E-7 -1.25E-6 -1.04E-8 -2.28E-7 -1.25E-6 -7.64E-9 -2.23E-7 -1.25E-6
3 -1.10E-7 -9.18E-7 -1.69E-6 -1.92E-7 -9.52E-7 -1.69E-6 -1.57E-7 -9.30E-7 -1.69E-6 -1.37E-7 -9.22E-7 -1.69E-6
4 -1.78E-6 -3.90E-6 -3.17E-6 -2.26E-6 -3.98E-6 -3.17E-6 -2.06E-6 -3.93E-6 -3.17E-6 -1.94E-6 -3.91E-6 -3.17E-6
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q=400$[]{data-label="tabl:Fq400dDaAAk4p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 -5.91E-12 -6.10E-9 -1.06E-7 -3.73E-10 -1.36E-8 -1.31E-7 -1.29E-10 -8.68E-9 -1.11E-7 -5.13E-11 -6.95E-9 -1.07E-7
2 -5.38E-10 -7.71E-8 -9.12E-7 -4.58E-9 -9.89E-8 -9.23E-7 -2.46E-9 -8.56E-8 -9.15E-7 -1.49E-9 -8.01E-8 -9.12E-7
3 -1.28E-8 -3.27E-7 -1.16E-6 -4.17E-8 -3.66E-7 -1.15E-6 -2.90E-8 -3.42E-7 -1.16E-6 -2.19E-8 -3.32E-7 -1.16E-6
4 -1.74E-7 -1.08E-6 -1.64E-6 -3.14E-7 -1.14E-6 -1.64E-6 -2.60E-7 -1.11E-6 -1.64E-6 -2.25E-7 -1.09E-6 -1.64E-6
5 -2.16E-6 -4.29E-6 -3.21E-6 -2.90E-6 -4.43E-6 -3.21E-6 -2.62E-6 -4.34E-6 -3.21E-6 -2.44E-6 -4.31E-6 -3.21E-6
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q=400$[]{data-label="tabl:Fq400dDaAAk5p"}
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 -7.26E-13 -1.20E-9 -1.22E-8 -2.16E-10 -6.32E-9 -2.31E-8 -6.47E-11 -2.88E-9 -1.46E-8 -2.12E-11 -1.74E-9 -1.27E-8
2 -7.36E-11 -2.71E-8 -4.91E-7 -1.84E-9 -4.70E-8 -5.56E-7 -8.31E-10 -3.57E-8 -5.09E-7 -4.19E-10 -3.04E-8 -4.95E-7
3 -1.81E-9 -1.29E-7 -9.15E-7 -1.30E-8 -1.66E-7 -9.04E-7 -7.71E-9 -1.45E-7 -9.12E-7 -5.00E-9 -1.35E-7 -9.15E-7
4 -2.44E-8 -4.17E-7 -1.12E-6 -7.87E-8 -4.78E-7 -1.12E-6 -5.71E-8 -4.44E-7 -1.12E-6 -4.39E-8 -4.28E-7 -1.12E-6
5 -2.38E-7 -1.22E-6 -1.62E-6 -4.45E-7 -1.31E-6 -1.62E-6 -3.72E-7 -1.26E-6 -1.62E-6 -3.23E-7 -1.23E-6 -1.62E-6
6 -2.50E-6 -4.63E-6 -3.25E-6 -3.57E-6 -4.88E-6 -3.26E-6 -3.19E-6 -4.74E-6 -3.25E-6 -2.94E-6 -4.68E-6 -3.25E-6
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q=400$[]{data-label="tabl:Fq400dDaAAk6p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 -9.46E-14 -1.81E-10 -1.20E-9 -1.43E-10 -3.30E-9 -4.40E-9 -3.90E-11 -1.05E-9 -1.86E-9 -1.12E-11 -4.31E-10 -1.34E-9
2 -1.13E-11 -8.55E-9 -1.44E-7 -9.13E-10 -2.50E-8 -2.36E-7 -3.62E-10 -1.59E-8 -1.70E-7 -1.56E-10 -1.15E-8 -1.50E-7
3 -2.91E-10 -5.28E-8 -7.25E-7 -5.20E-9 -8.57E-8 -7.45E-7 -2.69E-9 -6.87E-8 -7.34E-7 -1.51E-9 -5.96E-8 -7.28E-7
4 -4.00E-9 -1.77E-7 -8.63E-7 -2.64E-8 -2.33E-7 -8.51E-7 -1.68E-8 -2.04E-7 -8.59E-7 -1.14E-8 -1.89E-7 -8.62E-7
5 -3.85E-8 -4.94E-7 -1.10E-6 -1.24E-7 -5.78E-7 -1.10E-6 -9.27E-8 -5.36E-7 -1.10E-6 -7.26E-8 -5.12E-7 -1.10E-6
6 -3.01E-7 -1.33E-6 -1.61E-6 -5.79E-7 -1.45E-6 -1.61E-6 -4.88E-7 -1.39E-6 -1.61E-6 -4.25E-7 -1.36E-6 -1.61E-6
7 -2.83E-6 -4.95E-6 -3.30E-6 -4.27E-6 -5.33E-6 -3.32E-6 -3.79E-6 -5.14E-6 -3.31E-6 -3.46E-6 -5.03E-6 -3.30E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q=400$[]{data-label="tabl:Fq400dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 -1.25E-14 -2.34E-11 -1.28E-10 -1.03E-10 -1.88E-9 -1.11E-9 -2.64E-11 -4.33E-10 -3.05E-10 -6.88E-12 -1.20E-10 -1.65E-10
2 -1.86E-12 -2.18E-9 -2.20E-8 -5.28E-10 -1.44E-8 -7.09E-8 -1.88E-10 -7.40E-9 -3.48E-8 -7.16E-11 -4.25E-9 -2.51E-8
3 -5.18E-11 -2.10E-8 -4.00E-7 -2.50E-9 -4.87E-8 -5.32E-7 -1.15E-9 -3.52E-8 -4.53E-7 -5.68E-10 -2.76E-8 -4.15E-7
4 -7.31E-10 -7.85E-8 -7.37E-7 -1.09E-8 -1.26E-7 -7.13E-7 -6.16E-9 -1.04E-7 -7.29E-7 -3.70E-9 -9.04E-8 -7.35E-7
5 -7.10E-9 -2.21E-7 -8.36E-7 -4.46E-8 -2.96E-7 -8.26E-7 -2.98E-8 -2.61E-7 -8.32E-7 -2.09E-8 -2.40E-7 -8.35E-7
6 -5.43E-8 -5.60E-7 -1.09E-6 -1.74E-7 -6.67E-7 -1.08E-6 -1.34E-7 -6.19E-7 -1.09E-6 -1.07E-7 -5.89E-7 -1.09E-6
7 -3.62E-7 -1.43E-6 -1.60E-6 -7.15E-7 -1.59E-6 -1.60E-6 -6.06E-7 -1.51E-6 -1.60E-6 -5.29E-7 -1.47E-6 -1.60E-6
8 -3.14E-6 -5.25E-6 -3.36E-6 -5.00E-6 -5.78E-6 -3.39E-6 -4.41E-6 -5.53E-6 -3.37E-6 -4.01E-6 -5.39E-6 -3.36E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $ \bar c_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-0-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q=400$[]{data-label="tabl:Fq400dDaAAk8p"}
Tables type **(g)** for 1-URA-poles
-----------------------------------
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.88E-3 4.95E-3 4.98E-3 4.97E-3 4.97E-3 4.98E-3 4.97E-3 4.96E-3 4.98E-3 4.96E-3 4.95E-3 4.98E-3
1 -1.73E-14 -2.35E-9 -1.18E-7 -1.21E-9 -1.92E-8 -1.64E-7 -2.51E-10 -7.07E-9 -1.28E-7 -5.17E-11 -3.76E-9 -1.20E-7
2 -1.15E-11 -1.41E-7 -3.74E-6 -2.68E-8 -3.10E-7 -3.93E-6 -7.54E-9 -2.08E-7 -3.79E-6 -2.12E-9 -1.65E-7 -3.75E-6
3 -3.78E-9 -2.02E-6 -8.77E-6 -1.19E-6 -3.24E-6 -8.85E-6 -4.97E-7 -2.53E-6 -8.79E-6 -1.94E-7 -2.21E-6 -8.78E-6
----------------------- ----------- ---------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q_0=q_1=100$[]{data-label="tabl:Gqq11dDaAAk3p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.92E-3 4.97E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.97E-3 4.97E-3 4.99E-3 4.97E-3 4.97E-3 4.99E-3
1 -1.23E-15 -2.31E-10 -8.64E-9 -7.02E-10 -7.79E-9 -1.94E-8 -1.42E-10 -1.88E-9 -1.09E-8 -2.77E-11 -6.48E-10 -9.11E-9
2 -7.73E-13 -2.67E-8 -1.14E-6 -8.06E-9 -1.19E-7 -1.49E-6 -2.16E-9 -6.32E-8 -1.23E-6 -5.73E-10 -4.03E-8 -1.16E-6
3 -9.92E-11 -3.57E-7 -4.87E-6 -1.21E-7 -7.57E-7 -4.94E-6 -4.35E-8 -5.37E-7 -4.89E-6 -1.49E-8 -4.31E-7 -4.88E-6
4 -1.80E-8 -3.55E-6 -9.58E-6 -3.00E-6 -5.82E-6 -9.74E-6 -1.62E-6 -4.65E-6 -9.62E-6 -8.05E-7 -4.02E-6 -9.59E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q_0=q_1=100$[]{data-label="tabl:Gqq11dDaAAk4p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.94E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.97E-3 4.98E-3 4.99E-3
1 -1.07E-16 -2.38E-11 -7.55E-10 -4.69E-10 -3.72E-9 -3.42E-9 -9.24E-11 -6.43E-10 -1.28E-9 -1.74E-11 -1.49E-10 -8.64E-10
2 -7.29E-14 -4.26E-9 -1.52E-7 -3.56E-9 -5.39E-8 -3.38E-7 -9.16E-10 -2.17E-8 -1.99E-7 -2.31E-10 -1.03E-8 -1.63E-7
3 -7.66E-12 -8.99E-8 -2.66E-6 -3.11E-8 -3.02E-7 -3.14E-6 -1.02E-8 -1.90E-7 -2.83E-6 -3.28E-9 -1.34E-7 -2.70E-6
4 -4.91E-10 -6.47E-7 -5.02E-6 -3.31E-7 -1.37E-6 -5.05E-6 -1.46E-7 -1.01E-6 -5.03E-6 -6.06E-8 -8.12E-7 -5.02E-6
5 -6.21E-8 -5.25E-6 -1.02E-5 -5.23E-6 -8.47E-6 -1.04E-5 -3.33E-6 -7.00E-6 -1.03E-5 -1.99E-6 -6.09E-6 -1.02E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q_0=q_1=100$[]{data-label="tabl:Gqq11dDaAAk5p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.96E-3 4.98E-3 4.99E-3 4.98E-3 4.99E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3
1 -1.07E-17 -2.67E-12 -7.85E-11 -3.40E-10 -2.01E-9 -8.77E-10 -6.57E-11 -2.71E-10 -2.15E-10 -1.20E-11 -4.57E-11 -1.07E-10
2 -8.01E-15 -5.91E-10 -1.57E-8 -1.95E-9 -2.70E-8 -6.76E-8 -4.83E-10 -8.16E-9 -2.80E-8 -1.16E-10 -2.83E-9 -1.86E-8
3 -8.51E-13 -2.06E-8 -7.47E-7 -1.20E-8 -1.48E-7 -1.45E-6 -3.73E-9 -7.90E-8 -9.88E-7 -1.13E-9 -4.59E-8 -8.13E-7
4 -4.02E-11 -1.85E-7 -3.54E-6 -8.25E-8 -5.51E-7 -3.72E-6 -3.23E-8 -3.74E-7 -3.64E-6 -1.22E-8 -2.77E-7 -3.57E-6
5 -1.75E-9 -9.93E-7 -5.10E-6 -6.58E-7 -2.10E-6 -5.18E-6 -3.41E-7 -1.60E-6 -5.12E-6 -1.66E-7 -1.30E-6 -5.10E-6
6 -1.67E-7 -6.93E-6 -1.07E-5 -7.62E-6 -1.09E-5 -1.10E-5 -5.31E-6 -9.26E-6 -1.08E-5 -3.58E-6 -8.17E-6 -1.07E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q_0=q_1=100$[]{data-label="tabl:Gqq11dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.96E-3 4.98E-3 4.99E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 4.99E-3
1 -1.19E-18 -3.29E-13 -9.39E-12 -2.61E-10 -1.19E-9 -3.00E-10 -4.95E-11 -1.34E-10 -5.06E-11 -8.77E-12 -1.77E-11 -1.72E-11
2 -9.61E-16 -8.00E-11 -1.81E-9 -1.22E-9 -1.46E-8 -1.65E-8 -2.92E-10 -3.39E-9 -4.95E-9 -6.74E-11 -8.79E-10 -2.57E-9
3 -1.10E-13 -3.90E-9 -1.07E-7 -5.90E-9 -8.05E-8 -4.55E-7 -1.73E-9 -3.51E-8 -2.08E-7 -4.97E-10 -1.61E-8 -1.35E-7
4 -4.90E-12 -5.41E-8 -1.72E-6 -3.04E-8 -2.77E-7 -2.61E-6 -1.10E-8 -1.71E-7 -2.14E-6 -3.84E-9 -1.12E-7 -1.87E-6
5 -1.47E-10 -3.03E-7 -3.74E-6 -1.70E-7 -8.53E-7 -3.70E-6 -7.68E-8 -6.06E-7 -3.73E-6 -3.31E-8 -4.62E-7 -3.74E-6
6 -4.96E-9 -1.38E-6 -5.21E-6 -1.07E-6 -2.85E-6 -5.35E-6 -6.26E-7 -2.24E-6 -5.27E-6 -3.47E-7 -1.85E-6 -5.23E-6
7 -3.66E-7 -8.48E-6 -1.11E-5 -1.01E-5 -1.31E-5 -1.14E-5 -7.41E-6 -1.13E-5 -1.12E-5 -5.36E-6 -1.01E-5 -1.11E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q_0=q_1=100$[]{data-label="tabl:Gqq11dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 3.49E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3
1 7.72E-20 -4.44E-14 -1.26E-12 -2.08E-10 -7.65E-10 -1.27E-10 -3.89E-11 -7.45E-11 -1.57E-11 -6.74E-12 -8.18E-12 -3.67E-12
2 1.80E-17 -1.12E-11 -2.38E-10 -8.36E-10 -8.45E-9 -5.05E-9 -1.94E-10 -1.56E-9 -1.14E-9 -4.31E-11 -3.16E-10 -4.46E-10
3 1.09E-20 -6.50E-10 -1.37E-8 -3.36E-9 -4.65E-8 -1.25E-7 -9.45E-10 -1.63E-8 -4.13E-8 -2.59E-10 -5.78E-9 -2.11E-8
4 -5.72E-27 -1.34E-8 -4.03E-7 -1.41E-8 -1.57E-7 -1.37E-6 -4.78E-9 -8.53E-8 -7.86E-7 -1.57E-9 -4.75E-8 -5.30E-7
5 1.50E-13 -1.02E-7 -2.55E-6 -6.23E-8 -4.34E-7 -3.02E-6 -2.56E-8 -2.88E-7 -2.87E-6 -1.01E-8 -2.02E-7 -2.69E-6
6 1.27E-25 -4.36E-7 -3.71E-6 -2.94E-7 -1.19E-6 -3.67E-6 -1.49E-7 -8.76E-7 -3.68E-6 -7.20E-8 -6.81E-7 -3.70E-6
7 3.58E-25 -1.78E-6 -5.34E-6 -1.55E-6 -3.59E-6 -5.52E-6 -9.82E-7 -2.91E-6 -5.42E-6 -5.99E-7 -2.44E-6 -5.37E-6
8 7.10E-24 -9.87E-6 -1.13E-5 -1.28E-5 -1.51E-5 -1.17E-5 -9.60E-6 -1.32E-5 -1.15E-5 -7.23E-6 -1.19E-5 -1.14E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q_0=q_1=100$[]{data-label="tabl:Gqq11dDaAAk8p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.45E-3 2.48E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.48E-3 2.49E-3
1 -5.88E-16 -3.33E-10 -2.80E-8 -3.03E-10 -5.31E-9 -4.72E-8 -6.15E-11 -1.70E-9 -3.22E-8 -1.22E-11 -7.25E-10 -2.89E-8
2 -3.94E-13 -1.92E-8 -7.86E-7 -6.25E-9 -5.92E-8 -8.46E-7 -1.63E-9 -3.54E-8 -8.02E-7 -4.09E-10 -2.54E-8 -7.90E-7
3 -1.37E-10 -2.87E-7 -1.86E-6 -2.78E-7 -6.13E-7 -1.90E-6 -1.07E-7 -4.24E-7 -1.87E-6 -3.61E-8 -3.40E-7 -1.86E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=3}$, and $q_0=q_1=200$[]{data-label="tabl:Gqq22dDaAAk3p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.47E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3
1 -4.37E-17 -3.58E-11 -2.27E-9 -1.82E-10 -2.74E-9 -7.28E-9 -3.67E-11 -6.13E-10 -3.29E-9 -7.15E-12 -1.70E-10 -2.49E-9
2 -2.74E-14 -3.92E-9 -2.70E-7 -1.94E-9 -2.61E-8 -3.85E-7 -4.95E-10 -1.29E-8 -3.03E-7 -1.21E-10 -7.39E-9 -2.78E-7
3 -3.60E-12 -5.10E-8 -1.01E-6 -2.87E-8 -1.49E-7 -1.03E-6 -9.62E-9 -9.51E-8 -1.02E-6 -2.96E-9 -6.97E-8 -1.02E-6
4 -7.23E-10 -5.50E-7 -2.12E-6 -7.20E-7 -1.21E-6 -2.20E-6 -3.69E-7 -8.72E-7 -2.15E-6 -1.65E-7 -6.92E-7 -2.13E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=4}$, and $q_0=q_1=200$[]{data-label="tabl:Gqq22dDaAAk4p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.48E-3 2.49E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3
1 -4.03E-18 -4.04E-12 -2.15E-10 -1.24E-10 -1.58E-9 -1.59E-9 -2.49E-11 -2.67E-10 -4.65E-10 -4.77E-12 -5.19E-11 -2.67E-10
2 -2.72E-15 -6.92E-10 -4.25E-8 -8.80E-10 -1.39E-8 -1.20E-7 -2.19E-10 -5.53E-9 -6.31E-8 -5.26E-11 -2.39E-9 -4.74E-8
3 -2.87E-13 -1.36E-8 -5.98E-7 -7.48E-9 -6.33E-8 -7.16E-7 -2.35E-9 -3.70E-8 -6.46E-7 -6.91E-10 -2.42E-8 -6.12E-7
4 -1.92E-11 -9.79E-8 -1.06E-6 -7.96E-8 -2.86E-7 -1.09E-6 -3.33E-8 -1.91E-7 -1.07E-6 -1.26E-8 -1.41E-7 -1.06E-6
5 -2.84E-9 -8.91E-7 -2.34E-6 -1.27E-6 -1.88E-6 -2.45E-6 -7.85E-7 -1.44E-6 -2.38E-6 -4.40E-7 -1.17E-6 -2.35E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=5}$, and $q_0=q_1=200$[]{data-label="tabl:Gqq22dDaAAk5p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.48E-3 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3
1 -4.32E-19 -4.96E-13 -2.40E-11 -9.16E-11 -9.85E-10 -4.79E-10 -1.82E-11 -1.34E-10 -9.38E-11 -3.45E-12 -1.98E-11 -3.78E-11
2 -3.19E-16 -1.07E-10 -4.83E-9 -4.91E-10 -8.19E-9 -3.07E-8 -1.20E-10 -2.58E-9 -1.08E-8 -2.80E-11 -8.37E-10 -6.26E-9
3 -3.37E-14 -3.43E-9 -2.03E-7 -2.95E-9 -3.36E-8 -4.16E-7 -8.78E-10 -1.76E-8 -2.86E-7 -2.49E-10 -9.76E-9 -2.28E-7
4 -1.62E-12 -2.89E-8 -7.60E-7 -2.00E-8 -1.17E-7 -7.85E-7 -7.50E-9 -7.37E-8 -7.77E-7 -2.62E-9 -5.10E-8 -7.67E-7
5 -7.59E-11 -1.60E-7 -1.11E-6 -1.60E-7 -4.57E-7 -1.16E-6 -7.97E-8 -3.21E-7 -1.13E-6 -3.63E-8 -2.42E-7 -1.11E-6
6 -8.98E-9 -1.27E-6 -2.51E-6 -1.87E-6 -2.51E-6 -2.62E-6 -1.28E-6 -2.02E-6 -2.56E-6 -8.24E-7 -1.69E-6 -2.52E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=6}$, and $q_0=q_1=200$[]{data-label="tabl:Gqq22dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.48E-3 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3
1 -3.97E-18 -6.61E-14 -3.04E-12 -7.13E-11 -6.54E-10 -1.85E-10 -1.41E-11 -7.54E-11 -2.58E-11 -2.63E-12 -9.04E-12 -7.08E-12
2 -2.69E-15 -1.58E-11 -5.92E-10 -3.12E-10 -5.15E-9 -8.76E-9 -7.44E-11 -1.30E-9 -2.20E-9 -1.70E-11 -3.21E-10 -9.69E-10
3 -2.86E-13 -7.27E-10 -3.45E-8 -1.46E-9 -2.01E-8 -1.81E-7 -4.19E-10 -9.13E-9 -8.03E-8 -1.15E-10 -4.15E-9 -4.76E-8
4 9.97E-25 -9.08E-9 -4.30E-7 -7.44E-9 -6.11E-8 -6.14E-7 -2.59E-9 -3.60E-8 -5.36E-7 -8.52E-10 -2.27E-8 -4.72E-7
5 -1.93E-11 -4.89E-8 -7.86E-7 -4.15E-8 -1.86E-7 -7.89E-7 -1.81E-8 -1.23E-7 -7.86E-7 -7.33E-9 -8.80E-8 -7.86E-7
6 -2.85E-9 -2.35E-7 -1.16E-6 -2.63E-7 -6.44E-7 -1.24E-6 -1.49E-7 -4.76E-7 -1.20E-6 -7.84E-8 -3.68E-7 -1.18E-6
7 9.43E-25 -1.65E-6 -2.63E-6 -2.50E-6 -3.07E-6 -2.75E-6 -1.80E-6 -2.56E-6 -2.69E-6 -1.26E-6 -2.20E-6 -2.65E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=7}$, and $q_0=q_1=200$[]{data-label="tabl:Gqq22dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.66E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 1.52E-22 -9.55E-15 -4.29E-13 -5.76E-11 -4.55E-10 -8.51E-11 -1.13E-11 -4.61E-11 -9.06E-12 -2.09E-12 -4.72E-12 -1.74E-12
2 -1.70E-22 -2.40E-12 -8.16E-11 -2.17E-10 -3.40E-9 -3.00E-9 -5.05E-11 -7.02E-10 -5.68E-10 -1.13E-11 -1.37E-10 -1.89E-10
3 8.43E-27 -1.34E-10 -4.73E-9 -8.44E-10 -1.30E-8 -6.40E-8 -2.33E-10 -5.01E-9 -1.90E-8 -6.20E-11 -1.82E-9 -8.55E-9
4 -1.24E-27 -2.52E-9 -1.28E-7 -3.49E-9 -3.65E-8 -4.15E-7 -1.15E-9 -1.98E-8 -2.64E-7 -3.60E-10 -1.10E-8 -1.78E-7
5 2.05E-27 -1.73E-8 -5.85E-7 -1.53E-8 -9.62E-8 -6.38E-7 -6.08E-9 -6.04E-8 -6.34E-7 -2.28E-9 -4.05E-8 -6.13E-7
6 7.47E-27 -7.30E-8 -7.87E-7 -7.20E-8 -2.67E-7 -8.09E-7 -3.55E-8 -1.85E-7 -7.94E-7 -1.63E-8 -1.35E-7 -7.89E-7
7 3.06E-26 -3.19E-7 -1.22E-6 -3.81E-7 -8.31E-7 -1.30E-6 -2.37E-7 -6.40E-7 -1.26E-6 -1.39E-7 -5.09E-7 -1.24E-6
8 6.51E-24 -2.01E-6 -2.72E-6 -3.15E-6 -3.59E-6 -2.86E-6 -2.35E-6 -3.06E-6 -2.79E-6 -1.73E-6 -2.67E-6 -2.75E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The poles of $\bar{\bar r}_{q,\delta,\alpha,k}(\t)$, , of $(q,\delta,\alpha,k)$-1-URA approximation, i.e. the rational approximation of $\f(q,\delta,\alpha;\t)=\t^{\alpha}/(1+q\,\t^{\alpha})$ with functions from $\mathcal R_k$ on $ [\delta, 1]$, $\alpha =0.25, 0.5, 0.75$, ${\bf k=8}$, and $q_0=q_1=200$[]{data-label="tabl:Gqq22dDaAAk8p"}
Tables type **(h)** for 1-URA-decomposition coefficients
--------------------------------------------------------
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.92E-3 4.97E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.97E-3 4.97E-3 4.99E-3 4.97E-3 4.97E-3 4.99E-3
1 -1.23E-15 -2.31E-10 -8.64E-9 -7.02E-10 -7.79E-9 -1.94E-8 -1.42E-10 -1.88E-9 -1.09E-8 -2.77E-11 -6.48E-10 -9.11E-9
2 -7.73E-13 -2.67E-8 -1.14E-6 -8.06E-9 -1.19E-7 -1.49E-6 -2.16E-9 -6.32E-8 -1.23E-6 -5.73E-10 -4.03E-8 -1.16E-6
3 -9.92E-11 -3.57E-7 -4.87E-6 -1.21E-7 -7.57E-7 -4.94E-6 -4.35E-8 -5.37E-7 -4.89E-6 -1.49E-8 -4.31E-7 -4.88E-6
4 -1.80E-8 -3.55E-6 -9.58E-6 -3.00E-6 -5.82E-6 -9.74E-6 -1.62E-6 -4.65E-6 -9.62E-6 -8.05E-7 -4.02E-6 -9.59E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q_0=q_1=100$ []{data-label="tabl:Hqq11dDaAAk4p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.94E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.97E-3 4.98E-3 4.99E-3
1 -1.07E-16 -2.38E-11 -7.55E-10 -4.69E-10 -3.72E-9 -3.42E-9 -9.24E-11 -6.43E-10 -1.28E-9 -1.74E-11 -1.49E-10 -8.64E-10
2 -7.29E-14 -4.26E-9 -1.52E-7 -3.56E-9 -5.39E-8 -3.38E-7 -9.16E-10 -2.17E-8 -1.99E-7 -2.31E-10 -1.03E-8 -1.63E-7
3 -7.66E-12 -8.99E-8 -2.66E-6 -3.11E-8 -3.02E-7 -3.14E-6 -1.02E-8 -1.90E-7 -2.83E-6 -3.28E-9 -1.34E-7 -2.70E-6
4 -4.91E-10 -6.47E-7 -5.02E-6 -3.31E-7 -1.37E-6 -5.05E-6 -1.46E-7 -1.01E-6 -5.03E-6 -6.06E-8 -8.12E-7 -5.02E-6
5 -6.21E-8 -5.25E-6 -1.02E-5 -5.23E-6 -8.47E-6 -1.04E-5 -3.33E-6 -7.00E-6 -1.03E-5 -1.99E-6 -6.09E-6 -1.02E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q_0=q_1=100$ []{data-label="tabl:Hqq11dDaAAk5p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.96E-3 4.98E-3 4.99E-3 4.98E-3 4.99E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3 4.98E-3 4.98E-3 4.99E-3
1 -1.07E-17 -2.67E-12 -7.85E-11 -3.40E-10 -2.01E-9 -8.77E-10 -6.57E-11 -2.71E-10 -2.15E-10 -1.20E-11 -4.57E-11 -1.07E-10
2 -8.01E-15 -5.91E-10 -1.57E-8 -1.95E-9 -2.70E-8 -6.76E-8 -4.83E-10 -8.16E-9 -2.80E-8 -1.16E-10 -2.83E-9 -1.86E-8
3 -8.51E-13 -2.06E-8 -7.47E-7 -1.20E-8 -1.48E-7 -1.45E-6 -3.73E-9 -7.90E-8 -9.88E-7 -1.13E-9 -4.59E-8 -8.13E-7
4 -4.02E-11 -1.85E-7 -3.54E-6 -8.25E-8 -5.51E-7 -3.72E-6 -3.23E-8 -3.74E-7 -3.64E-6 -1.22E-8 -2.77E-7 -3.57E-6
5 -1.75E-9 -9.93E-7 -5.10E-6 -6.58E-7 -2.10E-6 -5.18E-6 -3.41E-7 -1.60E-6 -5.12E-6 -1.66E-7 -1.30E-6 -5.10E-6
6 -1.67E-7 -6.93E-6 -1.07E-5 -7.62E-6 -1.09E-5 -1.10E-5 -5.31E-6 -9.26E-6 -1.08E-5 -3.58E-6 -8.17E-6 -1.07E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q_0=q_1=100$ []{data-label="tabl:Hqq11dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 4.96E-3 4.98E-3 4.99E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 4.99E-3
1 -1.19E-18 -3.29E-13 -9.39E-12 -2.61E-10 -1.19E-9 -3.00E-10 -4.95E-11 -1.34E-10 -5.06E-11 -8.77E-12 -1.77E-11 -1.72E-11
2 -9.61E-16 -8.00E-11 -1.81E-9 -1.22E-9 -1.46E-8 -1.65E-8 -2.92E-10 -3.39E-9 -4.95E-9 -6.74E-11 -8.79E-10 -2.57E-9
3 -1.10E-13 -3.90E-9 -1.07E-7 -5.90E-9 -8.05E-8 -4.55E-7 -1.73E-9 -3.51E-8 -2.08E-7 -4.97E-10 -1.61E-8 -1.35E-7
4 -4.90E-12 -5.41E-8 -1.72E-6 -3.04E-8 -2.77E-7 -2.61E-6 -1.10E-8 -1.71E-7 -2.14E-6 -3.84E-9 -1.12E-7 -1.87E-6
5 -1.47E-10 -3.03E-7 -3.74E-6 -1.70E-7 -8.53E-7 -3.70E-6 -7.68E-8 -6.06E-7 -3.73E-6 -3.31E-8 -4.62E-7 -3.74E-6
6 -4.96E-9 -1.38E-6 -5.21E-6 -1.07E-6 -2.85E-6 -5.35E-6 -6.26E-7 -2.24E-6 -5.27E-6 -3.47E-7 -1.85E-6 -5.23E-6
7 -3.66E-7 -8.48E-6 -1.11E-5 -1.01E-5 -1.31E-5 -1.14E-5 -7.41E-6 -1.13E-5 -1.12E-5 -5.36E-6 -1.01E-5 -1.11E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q_0=q_1=100$ []{data-label="tabl:Hqq11dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 3.49E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3 4.98E-3 4.99E-3 5.00E-3
1 7.72E-20 -4.44E-14 -1.26E-12 -2.08E-10 -7.65E-10 -1.27E-10 -3.89E-11 -7.45E-11 -1.57E-11 -6.74E-12 -8.18E-12 -3.67E-12
2 1.80E-17 -1.12E-11 -2.38E-10 -8.36E-10 -8.45E-9 -5.05E-9 -1.94E-10 -1.56E-9 -1.14E-9 -4.31E-11 -3.16E-10 -4.46E-10
3 1.09E-20 -6.50E-10 -1.37E-8 -3.36E-9 -4.65E-8 -1.25E-7 -9.45E-10 -1.63E-8 -4.13E-8 -2.59E-10 -5.78E-9 -2.11E-8
4 -5.72E-27 -1.34E-8 -4.03E-7 -1.41E-8 -1.57E-7 -1.37E-6 -4.78E-9 -8.53E-8 -7.86E-7 -1.57E-9 -4.75E-8 -5.30E-7
5 1.50E-13 -1.02E-7 -2.55E-6 -6.23E-8 -4.34E-7 -3.02E-6 -2.56E-8 -2.88E-7 -2.87E-6 -1.01E-8 -2.02E-7 -2.69E-6
6 1.27E-25 -4.36E-7 -3.71E-6 -2.94E-7 -1.19E-6 -3.67E-6 -1.49E-7 -8.76E-7 -3.68E-6 -7.20E-8 -6.81E-7 -3.70E-6
7 3.58E-25 -1.78E-6 -5.34E-6 -1.55E-6 -3.59E-6 -5.52E-6 -9.82E-7 -2.91E-6 -5.42E-6 -5.99E-7 -2.44E-6 -5.37E-6
8 7.10E-24 -9.87E-6 -1.13E-5 -1.28E-5 -1.51E-5 -1.17E-5 -9.60E-6 -1.32E-5 -1.15E-5 -7.23E-6 -1.19E-5 -1.14E-5
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q_0=q_1=100$ []{data-label="tabl:Hqq11dDaAAk8p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=3**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.45E-3 2.48E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.49E-3 2.48E-3 2.49E-3
1 -5.88E-16 -3.33E-10 -2.80E-8 -3.03E-10 -5.31E-9 -4.72E-8 -6.15E-11 -1.70E-9 -3.22E-8 -1.22E-11 -7.25E-10 -2.89E-8
2 -3.94E-13 -1.92E-8 -7.86E-7 -6.25E-9 -5.92E-8 -8.46E-7 -1.63E-9 -3.54E-8 -8.02E-7 -4.09E-10 -2.54E-8 -7.90E-7
3 -1.37E-10 -2.87E-7 -1.86E-6 -2.78E-7 -6.13E-7 -1.90E-6 -1.07E-7 -4.24E-7 -1.87E-6 -3.61E-8 -3.40E-7 -1.86E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=3}$, and $q_0=q_1=200$ []{data-label="tabl:Hqq22dDaAAk3p"}
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=4**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.47E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3
1 -4.37E-17 -3.58E-11 -2.27E-9 -1.82E-10 -2.74E-9 -7.28E-9 -3.67E-11 -6.13E-10 -3.29E-9 -7.15E-12 -1.70E-10 -2.49E-9
2 -2.74E-14 -3.92E-9 -2.70E-7 -1.94E-9 -2.61E-8 -3.85E-7 -4.95E-10 -1.29E-8 -3.03E-7 -1.21E-10 -7.39E-9 -2.78E-7
3 -3.60E-12 -5.10E-8 -1.01E-6 -2.87E-8 -1.49E-7 -1.03E-6 -9.62E-9 -9.51E-8 -1.02E-6 -2.96E-9 -6.97E-8 -1.02E-6
4 -7.23E-10 -5.50E-7 -2.12E-6 -7.20E-7 -1.21E-6 -2.20E-6 -3.69E-7 -8.72E-7 -2.15E-6 -1.65E-7 -6.92E-7 -2.13E-6
----------------------- ----------- ----------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=4}$, and $q_0=q_1=200$ []{data-label="tabl:Hqq22dDaAAk4p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=5**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.48E-3 2.49E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3 2.49E-3 2.49E-3 2.50E-3
1 -4.03E-18 -4.04E-12 -2.15E-10 -1.24E-10 -1.58E-9 -1.59E-9 -2.49E-11 -2.67E-10 -4.65E-10 -4.77E-12 -5.19E-11 -2.67E-10
2 -2.72E-15 -6.92E-10 -4.25E-8 -8.80E-10 -1.39E-8 -1.20E-7 -2.19E-10 -5.53E-9 -6.31E-8 -5.26E-11 -2.39E-9 -4.74E-8
3 -2.87E-13 -1.36E-8 -5.98E-7 -7.48E-9 -6.33E-8 -7.16E-7 -2.35E-9 -3.70E-8 -6.46E-7 -6.91E-10 -2.42E-8 -6.12E-7
4 -1.92E-11 -9.79E-8 -1.06E-6 -7.96E-8 -2.86E-7 -1.09E-6 -3.33E-8 -1.91E-7 -1.07E-6 -1.26E-8 -1.41E-7 -1.06E-6
5 -2.84E-9 -8.91E-7 -2.34E-6 -1.27E-6 -1.88E-6 -2.45E-6 -7.85E-7 -1.44E-6 -2.38E-6 -4.40E-7 -1.17E-6 -2.35E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=5}$, and $q_0=q_1=200$ []{data-label="tabl:Hqq22dDaAAk5p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=6**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.48E-3 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3
1 -4.32E-19 -4.96E-13 -2.40E-11 -9.16E-11 -9.85E-10 -4.79E-10 -1.82E-11 -1.34E-10 -9.38E-11 -3.45E-12 -1.98E-11 -3.78E-11
2 -3.19E-16 -1.07E-10 -4.83E-9 -4.91E-10 -8.19E-9 -3.07E-8 -1.20E-10 -2.58E-9 -1.08E-8 -2.80E-11 -8.37E-10 -6.26E-9
3 -3.37E-14 -3.43E-9 -2.03E-7 -2.95E-9 -3.36E-8 -4.16E-7 -8.78E-10 -1.76E-8 -2.86E-7 -2.49E-10 -9.76E-9 -2.28E-7
4 -1.62E-12 -2.89E-8 -7.60E-7 -2.00E-8 -1.17E-7 -7.85E-7 -7.50E-9 -7.37E-8 -7.77E-7 -2.62E-9 -5.10E-8 -7.67E-7
5 -7.59E-11 -1.60E-7 -1.11E-6 -1.60E-7 -4.57E-7 -1.16E-6 -7.97E-8 -3.21E-7 -1.13E-6 -3.63E-8 -2.42E-7 -1.11E-6
6 -8.98E-9 -1.27E-6 -2.51E-6 -1.87E-6 -2.51E-6 -2.62E-6 -1.28E-6 -2.02E-6 -2.56E-6 -8.24E-7 -1.69E-6 -2.52E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=6}$, and $q_0=q_1=200$ []{data-label="tabl:Hqq22dDaAAk6p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=7**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 2.48E-3 2.49E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.49E-3 2.50E-3 2.50E-3
1 -3.97E-18 -6.61E-14 -3.04E-12 -7.13E-11 -6.54E-10 -1.85E-10 -1.41E-11 -7.54E-11 -2.58E-11 -2.63E-12 -9.04E-12 -7.08E-12
2 -2.69E-15 -1.58E-11 -5.92E-10 -3.12E-10 -5.15E-9 -8.76E-9 -7.44E-11 -1.30E-9 -2.20E-9 -1.70E-11 -3.21E-10 -9.69E-10
3 -2.86E-13 -7.27E-10 -3.45E-8 -1.46E-9 -2.01E-8 -1.81E-7 -4.19E-10 -9.13E-9 -8.03E-8 -1.15E-10 -4.15E-9 -4.76E-8
4 9.97E-25 -9.08E-9 -4.30E-7 -7.44E-9 -6.11E-8 -6.14E-7 -2.59E-9 -3.60E-8 -5.36E-7 -8.52E-10 -2.27E-8 -4.72E-7
5 -1.93E-11 -4.89E-8 -7.86E-7 -4.15E-8 -1.86E-7 -7.89E-7 -1.81E-8 -1.23E-7 -7.86E-7 -7.33E-9 -8.80E-8 -7.86E-7
6 -2.85E-9 -2.35E-7 -1.16E-6 -2.63E-7 -6.44E-7 -1.24E-6 -1.49E-7 -4.76E-7 -1.20E-6 -7.84E-8 -3.68E-7 -1.18E-6
7 9.43E-25 -1.65E-6 -2.63E-6 -2.50E-6 -3.07E-6 -2.75E-6 -1.80E-6 -2.56E-6 -2.69E-6 -1.26E-6 -2.20E-6 -2.65E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=7}$, and $q_0=q_1=200$ []{data-label="tabl:Hqq22dDaAAk7p"}
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[**k=8**]{} /$\delta$ $0.00$ $0.00$ $0.00$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
j /$\alpha$ 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0 1.66E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3 2.50E-3
1 1.52E-22 -9.55E-15 -4.29E-13 -5.76E-11 -4.55E-10 -8.51E-11 -1.13E-11 -4.61E-11 -9.06E-12 -2.09E-12 -4.72E-12 -1.74E-12
2 -1.70E-22 -2.40E-12 -8.16E-11 -2.17E-10 -3.40E-9 -3.00E-9 -5.05E-11 -7.02E-10 -5.68E-10 -1.13E-11 -1.37E-10 -1.89E-10
3 8.43E-27 -1.34E-10 -4.73E-9 -8.44E-10 -1.30E-8 -6.40E-8 -2.33E-10 -5.01E-9 -1.90E-8 -6.20E-11 -1.82E-9 -8.55E-9
4 -1.24E-27 -2.52E-9 -1.28E-7 -3.49E-9 -3.65E-8 -4.15E-7 -1.15E-9 -1.98E-8 -2.64E-7 -3.60E-10 -1.10E-8 -1.78E-7
5 2.05E-27 -1.73E-8 -5.85E-7 -1.53E-8 -9.62E-8 -6.38E-7 -6.08E-9 -6.04E-8 -6.34E-7 -2.28E-9 -4.05E-8 -6.13E-7
6 7.47E-27 -7.30E-8 -7.87E-7 -7.20E-8 -2.67E-7 -8.09E-7 -3.55E-8 -1.85E-7 -7.94E-7 -1.63E-8 -1.35E-7 -7.89E-7
7 3.06E-26 -3.19E-7 -1.22E-6 -3.81E-7 -8.31E-7 -1.30E-6 -2.37E-7 -6.40E-7 -1.26E-6 -1.39E-7 -5.09E-7 -1.24E-6
8 6.51E-24 -2.01E-6 -2.72E-6 -3.15E-6 -3.59E-6 -2.86E-6 -2.35E-6 -3.06E-6 -2.79E-6 -1.73E-6 -2.67E-6 -2.75E-6
----------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The coefficients $\bar{\bar c}_j, j=0, \dots, k$ of the partial fraction representation of $(q,\delta,\alpha,k)$-1-URA approximation for $\delta=0, 10^{-6}, 10^{-7}, 10^{-8}$, $\alpha= 0.25, 0.50, 0.75$, ${\bf k=8}$, and $q_0=q_1=200$ []{data-label="tabl:Hqq22dDaAAk8p"}
How to access the BURA data from the repository
-----------------------------------------------
### Data generated by Remez algorithm {#data-soft}
Remez algorithm is implemented according to the results reported in [@PGMASA1987]. It is part of a software developed in IICT, Bulgarian Academy of Sciences, concerting the more general problem of the best uniform rational approximation of a given function of $ x \in [0,1]$ by $P_m(x)/Q_k(x)$, where $P_m$ and $Q_k$ are polynomials of degree $m \ge k$, correspondingly. In this work we use only the particular case when $m=k$. To make the writing more concise we use the following notations $$P_n(x) \quad \mbox{denoted by} \quad p(n,F;x) = \sum_{j=0}^{n} F(j)*x^j,$$ where $n$ is the degree of the polynomial and $F(j)$, $j=0,\dots, n$, are the coefficients. Further, we denote the zeros of $ Q_k(x)$ by $ U0(j))$, $j=0, \dots, k$, which in general are complex numbers $$U0(j)) = Re\{U0(j))\} + i Im\{ U0(j))\}, \ \ j=1, \dots, k, \ \ i^2=-1.$$ Accordingly, the best uniform rational approximation $P_m(x)/Q_k(x)= p(m,A;x) / p(k,B;x) $ in the data files will be denoted by and represented as a sum of partial fractions as $$\label{BURA-gen-data}
\begin{aligned}
p(m,A;x) / p(k,B;x)
& = p(m-k,C;x) + p(k-1,D;x)/p(k,B;x) \\
& = \sum_{j=0}^{m-k} C(j) * x^j+ \sum_{j=1}^{k} E(j) / (x-U0(j)).
\end{aligned}$$ where the coefficients $C(j)$ and $E(j)$ are in general complex numbers. Of course, when the poles are real, then the complex parts, since obtained by computational procedure for finding roots of a polynomial, will have very small imaginary parts.
In our case $k=m$ and the above representation becomes $$r(m,k,A,B, x) = C(0) + \sum_{j=1}^{k} E(j) / (x-U0(j)).$$
### Examples how to use the data in the repository {#example4use}
Now we shall illustrate on a couple of examples how to solve the system $\wcalAt^\alpha \tiluh = \tilfh$ arising in the approximation of the Example 1 that produces the matrix $ \wcalAt$ defined by by using BURA method . For definiteness and convenience, we assume that $h=10^{-3}$, $\min a(x)= 0.25/\pi^2$ and $\max a(x)= 1$, so that $ \lambda_1 \ge 1$ and $ \lambda_1/\lambda_N = h^2/4= 0.25*10^{-6}$. We assume also that the data $f$ is such that the step-size $h=10^{-3}$ allows the following estimate for the semi-discrete error $ \|u - u_h \| \le 10^{-3} \|f\|$.
The task we have is to find an approximation $\tilwh$ of the solution $\tiluh$ with relative error $10^{-3}$ by using so that (simplified due to $\lambda_1 =1$) $$\tilwh = r_{\alpha,k}(\wcalAt^{-1}) \tilfh = \left ( c_0 + \sum_{i=1}^k c_i( \wcalAt^{-1} -d_i)^{-1} \right ) \tilfh
= \left (\widetilde c_0+\sum_{i=1}^k {\widetilde c_i}{(\wcalAt -\widetilde d_i)^{-1}} \right ) \tilfh.$$ The coefficients $c_j$, $j=0, \dots, k$, and the poles $d_j$, $k=1, \dots, k$ will be obtained from the repository ` http://parallel.bas.bg/~pencho/BURA/`. Moreover, in the repository one can find more information about $r_{\alpha,k}(t)$, namely, the extremal points of the error $t^\alpha - r_{\alpha,k}(t)$, the zeros of the numerator, etc.
##### We need to solve $\wcalAt^\alpha \tiluh = \tilfh$ with relative accuracy $10^{-3}$ for $\delta=0$ and $\alpha=0.25$.
We visit Table \[tabl:A0BURAp\] and look for the corresponding error for $q=0$. It appears that error below $10^{-3}$ is achieved for $k=7$. In order to implement our algorithm we need to get the poles $d_j$ from the second row of Table \[tabl:Cq000dDaAAk7p\] (under $\delta=0$ and $\alpha=0.25$), and the coefficients $c_j$ from the second row of Table \[tabl:Dq000dDaAAk7p\] (under $\delta=0$ and $\alpha=0.25$) of BURA. According to Table \[p1ltabl\], the corresponding tables are encoded as $Cq000d0a25k7p$ and $ Dq000d0a25k7p$. To recover the data, in a browser open the site of the repository ` http://parallel.bas.bg/~pencho/BURA/ ` which will get you to the html-repository.
Index
mode size last-canged name
dr-x Jul 18 09:26 BURA-tabl/ - with extremal points
dr-x Jul 18 10:50 BURA-dcmp/ - with poles an decomposition coefficients for BURA
dr-x Jul 18 11:22 0URA-dcmp/ - with poles an decomposition coefficients for 0URA
dr-x Jul 18 11:35 1URA-dcmp/ - with poles an decomposition coefficients for 1URA
-r-- 672k Sep 21 11:51 BURA-data-report.pdf - 60 pages document
-r-- 3.2M Sep 21 11:51 all-in-one.zip - archive with all files (approx 2000)
-r-- 165k Sep 21 11:51 all-in-one_long.lst - long list of all folders and sub-folders
-r-- 720 Sep 21 11:51 all-in-one_short.lst - short list of the main folder
The coefficients in the partial fraction representation of BURA are in the folder `BURA-dcmp/`. Consequently, from ` http://parallel.bas.bg/~pencho/BURA/BURA-dcmp/` you get a long list of data files where you can extract the data from.
Index
mode size last-canged name
dr-x Aug 7 12:19 ../
dr-x Aug 7 13:16 add/
-r-- 229 Jul 18 07:26 0-data-info.txt
-r-- 20k Aug 18 07:26 0-data-info.pdf
-r-- 901 Jul 18 09:28 q000d0a25k3.tab
-r-- 1.1k Jul 18 09:28 q000d0a25k4.tab
-r-- 1.3k Jul 18 09:28 q000d0a25k5.tab
-r-- 1.5k Jul 18 09:28 q000d0a25k6.tab
-r-- 1.7k Jul 18 09:28 q000d0a25k7.tab
-r-- 1.9k Jul 18 09:28 q000d0a25k8.tab
. . .
Note that the name of the file corresponds for the data you need. In order to understand better how to recover the needed coefficient we recommend you read to file `0-data-info.pdf`. Now, if you need BURA for $q=0$, $\delta=0$, $\alpha=0.25$, and $k=7$ you open the file `q000d0a25k7.tab` to get the data shown on Figure \[data-example-1\].
........................
0 C(j), j=0,M-K
0, 1.5251198659461835471669134E+0000,
.......................
7 Re{U0(j)}, j=1,K
1, -2.2376872996078341567977828E-0010,
2, -7.8486161880478847863452285E-0008,
3, -5.5731465614475310569044313E-0006,
4, -1.8879914699444446592534226E-0004,
5, -4.0807806486154275568196059E-0003,
6, -6.6631003383177437492683783E-0002,
7, -1.3009123561707709448297558E+0000,
7 Re{E(j)}, j=1,K
1, -1.4685691492826336539266828E-0012,
2, -1.4250266060744046359349294E-0009,
3, -2.3302001273674185907801704E-0007,
4, -1.6270756180817065256760833E-0005,
5, -6.7439995512918544429211685E-0004,
6, -2.0780142734679021643365256E-0002,
7, -1.1636545994360130292202863E+0000.
Comparing this explained by formula with the representation of $r_{\alpha, k}( x)$ by we realize that the poles $d_j$ are in $ Re\{U0(j)\}$, while the coefficients $c_j$ are in $ Re\{E(j)\}$, $j=1, \dots,7$ except $c_0= C(0)$.
\[coeffs1\]
[0.47 ]{}
-------- ----------------------------------------------
$c_0$= $C(0)$ =$~\ ~ 1.5251198659461835$
$c_1$= $Re\{E(1)\}$=$-1.468569149282634*10^{-12} $,
$c_2$= $Re\{E(2)\}$=$-1.425026606074404*10^{-9}$,
$c_3$= $Re\{E(3)\}$=$-2.330200127367418*10^{-7}$,
$c_4$= $Re\{E(4)\}$=$-1.627075618081706*10^{-5} $,
$c_5$= $Re\{E(5)\}$=$-6.743999551291854*10^{-4}$,
$c_6$= $Re\{E(6)\}$=$-2.078014273467902*10^{-2}$,
$c_7$= $Re\{E(7)\}$=$-1.1636545994360130$,
-------- ----------------------------------------------
[0.47 ]{} (recall $d_j= Re\{U0(j)\}$, $j=1, \dots,7$)\
-------- ----------------------------------------------
$d_1$= $Re\{U0(1)\}$=$-2.237687299607834*10^{-10}$,
$d_2$= $Re\{U0(2)\}$=$-7.848616188047885*10^{-8}$,
$d_3$= $Re\{U0(3)\}$=$-5.573146561447531*10^{-6}$,
$d_4$= $Re\{U0(4)\}$=$-1.887991469944444*10^{-4}$,
$d_5$= $Re\{U0(5)\}$=$-4.080780648615427*10^{-3}$,
$d_6$= $Re\{U0(6)\}$=$-6.663100338317743*10^{-2}$,
$d_7$= $Re\{U0(7)\}$=$-1.300912356170771$.
-------- ----------------------------------------------
Note that the poles of the rational function $r_{\alpha,k}(t)$ are computed by finding the roots of its denominator. In the general case these are complex numbers with real and complex parts. However, in our case the roots are real, so the fact that the complex parts are near zero is another indication that our computations are correct. The same remark is valid also for the coefficients in the partial fraction representation. One can recover the data including the complex parts of the roots and coefficients from the file with the same name abut extension `.txt`. Thus, instead of file ` q000d0a25k7.tab` we need to access file `q000d0a25k7.txt` The full path is ` http://parallel.bas.bg/~pencho/BURA/BURA-dcmp/add/`. According Remark [\[lemma:BURA\]]{} and we obtain for the coefficients $\{\widetilde c_j\}_{j=0}^{k}$ and for the poles $\{\widetilde d_j\}_{j=1}^{k}$, shown on Figure \[coef-example-1-infty\].
j, coeffs \tilde c_j, j=0,...,k j, poles \tilde d_j, j=1,...,k
--------------------------------------- --------------------------------------
0, 7.8649908986141400766246111E-0004,
1, 6.8758756374255199468223446E-0001, 1, -7.6869129212016636558276595E-0001,
2, 4.6805384769874068750186188E+0000, 2, -1.5008028533643146339273005E+0001,
3, 4.0497762636194054740290213E+0001, 3, -2.4505115224443418015306629E+0002,
4, 4.5646520861012335445240629E+0002, 4, -5.2966340998851315319920697E+0003,
5, 7.5022631473305133482030654E+0003, 5, -1.7943185038727329359280164E+0005,
6, 2.3133257355683798168384452E+0005, 6, -1.2741099526854566986420878E+0007,
7, 2.9328888647334126024639520E+0007, 7, -4.4688996544568804592027580E+0009.
#####
We want to solve $\wcalAt^\alpha \tiluh = \tilfh$ with relative accuracy $10^{-4}$ for $\alpha=0.50$. Since $\lambda_N/\lambda_1 =2.5 * 10^{-7} > 10^{-7}$. Thus, we can choose any $\delta \le 10^{-7}$. We shall choose $\delta =10^{-8} $, which indeed is smaller than $\lambda_N/\lambda_1$. Visiting Table \[tabl:A0BURAp\] and looking for the corresponding error for $q=0$ and $\delta=10^{-8}$ we see that $k=6$ gives the desired accuracy. Then again, to implement the algorithm through the form we need to get the poles $d_j, j=1,\dots,6$ from Table \[tabl:Cq000dDaAAk6p\], row 6 (under $\delta=10^{-8}$ and $\alpha=0.50$), and the coefficients $c_j, j=0,1, \dots, 6$, from Table \[tabl:Dq000dDaAAk6p\], row 6 (under $\delta=10^{-8}$ and $\alpha=0.50$). In order to recover them from the html-repository with 25 significant digits we read the file `q000d8a50k6.tab`:
............................................
0 C(j), j=0,M-K
0, 3.2884110965902564982570418E+0000,
.............................................
6 Re{U0(j)}, j=1,K
1, -2.9026635633198611614392944E-0006,
2, -1.2844649138280910668994002E-0004,
3, -2.2714606179594160697493911E-0003,
4, -2.5644587248425397513332244E-0002,
5, -2.3875648347625806119814403E-0001,
6, -4.0426255143071180974523916E+0000,
6 Re{E(j)}, j=1,K
1, -7.3047263730915787436076796E-0009,
2, -1.4856610579547499079286817E-0006,
3, -8.9729276335619099229017533E-0005,
4, -2.9805865012026772176372313E-0003,
5, -8.4097511581798008760373107E-0002,
6, -1.1182633938371627571773423E+0001.
One can recover the data including the complex parts of the roots and coefficients from the file with the same name abut extension `.txt`. Thus, instead of file `q000d8a50k6.tab` we need to access file `q000d8a50k6.txt` to get (recall $m=k=6$). As we see, the complex parts of the poles $U0(j)$ and of the coefficients $E(j)$ are almost zero. Therefore, we ignore them in table `q000d8a50k6.tab` as seen on Figure \[data-example-2\].
\[coeffs2\]
[0.47 ]{}
-------- --------------------------------------------
$c_0$= $C(0)$ =$~\ ~ 3.288411096590256 $
$c_1$= $Re\{E(1)\}$=$-7.304726373091578*10^{-9}$
$c_2$= $Re\{E(2)\}$=$-1.485661057954749*10^{-6} $
$c_3$= $Re\{E(3)\}$=$-8.972927633561909*10^{-5} $
$c_4$= $Re\{E(4)\}$=$-2.980586501202677*10^{-3}$
$c_5$= $Re\{E(5)\}$=$-8.409751158179800*10^{-2} $
$c_6$= $Re\{E(6)\}$=$-1.118263393837162*10 $
-------- --------------------------------------------
[0.47 ]{} (recall $d_j= Re\{U0(j)\}$, $j=1, \dots,7$)\
-------- ---------------------------------------------
$d_1$= $Re\{U0(1)\}$=$-2.902663563319861*10^{-6} $
$d_2$= $Re\{U0(2)\}$=$-1.284464913828091*10^{-4}$
$d_3$= $Re\{U0(3)\}$=$-2.271460617959416*10^{-3} $
$d_4$= $Re\{U0(4)\}$=$-2.564458724842539*10^{-2} $
$d_5$= $Re\{U0(5)\}$=$-2.387564834762580*10^{-1} $
$d_6$= $Re\{U0(6)\}$=$-4.042625514307118*10 $
-------- ---------------------------------------------
........................
0 C(j), j=0,M-K
0, 3.2884110965902564982570418E+0000,
................................
6 U0(j), j=1,K
1, -2.9026635633198611614392944E-0006, -1.7520806294041483876573147E-0037
2, -1.2844649138280910668994002E-0004, 1.8242239649606423471743024E-0037
3, -2.2714606179594160697493911E-0003, -7.2143334828724323886476075E-0039
4, -2.5644587248425397513332244E-0002, -7.2776229715659618428404602E-0047
5, -2.3875648347625806119814403E-0001, -7.3384739154186567125843208E-0052
6, -4.0426255143071180974523916E+0000, -8.8654988085625587825735093E-0067
6 E(j), j=1,K
1, -7.3047263730915787436076796E-0009, 5.7573111695991243362896879E-0037
2, -1.4856610579547499079286817E-0006, -5.9992385756159082541115791E-0037
3, -8.9729276335619099229017533E-0005, 2.4155906802799248832468797E-0038
4, -2.9805865012026772176372313E-0003, 2.2938808341026098990357342E-0041
5, -8.4097511581798008760373107E-0002, 9.4285923737400470688723456E-0042
6, -1.1182633938371627571773423E+0001, 4.4663981643768036611005825E-0042
According Remark [\[lemma:BURA\]]{} and we obtain the coefficients $\{\widetilde c_j\}_{j=0}^{k}$ and the poles $\{\widetilde d_j\}_{j=1}^{k}$, shown on Figure \[example-2-infty\].
[0.45 ]{}
j, coeffs \tilde c_j, j=0,...,k j,
---------------------------------------
0, 1.8619525184162842197896828E-0004,
1, 6.8425358785323701384064700E-0001,
2, 1.4752743712419678037242149E+0000,
3, 4.5322129077686442621147161E+0000,
4, 1.7390967446300787239267527E+0001,
5, 9.0048244054276093870632027E+0001,
6, 8.6698293594856505069077514E+0002.
[0.45 ]{}
j, poles \tilde d_j, j=1,...,k
---------------------------------------
1, -2.4736399561644631729004006E-0001,
2, -4.1883679364017772326828007E+0000,
3, -3.8994583547504784590380748E+0001,
4, -4.4024536110969760472793286E+0002,
5, -7.7853430579096145080903339E+0003,
6, -3.4451116300101647683572511E+0005,
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work has been partly support by the Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014-2020) and co-financed by the EU through the European structural and Investment funds.
|
---
abstract: 'We present measurements of the magnetic susceptibility $\chi$ and the magnetization M of single crystals of metallic [Yb$_{2}$Pt$_{2}$Pb]{}, where localized Yb moments lie on the geometrically frustrated Shastry-Sutherland Lattice (SSL). Strong magnetic frustration is found in this quasi-two dimensional system, which orders antiferromagnetically (AF) at T$_{N}$=2.02 *K* from a paramagnetic liquid of Yb-dimers, having a gap $\Delta$=4.6 *K* between the singlet ground state and the triplet excited states. Magnetic fields suppress the AF order, which vanishes at a 1.25 *T* quantum critical point. The spin gap $\Delta$ persists to 1.5 *T*, indicating that the AF degenerates into a liquid of dimer triplets at T=0. Quantized steps are observed in M(B) within the AF state, a signature of SSL systems. Our results show that [Yb$_{2}$Pt$_{2}$Pb]{} is unique, both as a metallic SSL system that is close to an AF quantum critical point, and as a heavy fermion compound where geometrical frustration plays a decisive role.'
author:
- 'M. S. Kim'
- 'M. C. Aronson'
title: 'Spin Liquids and Antiferromagnetic Order in the Shastry-Sutherland-Lattice Yb$_{2}$Pt$_{2}$Pb'
---
Much interest has focused on systems with geometrical frustration, where conventional antiferromagnetic (AF) order is suppressed in favor of more exotic ground states. The Shastry Sutherland Lattice (SSL) is one of the simplest frustrated systems [@shastry1983], consisting of planes of orthogonal dimers of moments with interdimer coupling $J^{\prime}$ and the intradimer coupling *J*. The T=0 phase diagram has two limiting behaviors, depending on $J^{\prime}/J$. Nonordering dimers are found for small $J^{\prime}/J$, distinguished by an energy gap $\Delta$ between the singlet and triplet states of the dimer. Insulating [SrCu$_{2}$(BO$_{3}$)$_{2}$]{} exemplifies this disordered ‘spin liquid’(SL) regime [@kageyama1999; @miyahara2003; @onizuka2000]. Conversely, AF order with gapless magnetic excitations is favored for large $J^{\prime}/J$, and the RB$_{4}$ (R= Gd,Tb,Dy,Ho,Er) compounds may represent this limit [@etorneau1979; @michimura2006; @matas2010; @iga2007]. A T=0 transition between the SL and AF phases has been predicted for [@weihong1999; @miyahara2003; @alhajj2005; @isacsson2006], although symmetry-based arguments [@carpentier2002] suggest that an intermediate state is required, such as a helical magnet [@albrecht1996], a weak SDW [@carpentier2002], or a plaquet ordered solid [@koga2003]. The known SSL systems have so far not provided experimental access to this transitional regime.
Metallic SSL systems based on Ce or Yb moments have the potential for a more complex T=0 phase diagram [@vojta2008; @lacroix2009; @coleman2010; @aronson2010; @bernhard2011]. In the absence of frustration, both the Kondo temperature T$_{K}$ and the Neél temperature T$_{N}$ in systems of this sort can be tuned by pressure, magnetic fields, or doping, suppressing AF order among well-localized f-electrons to T=0 at a quantum critical point (QCP). The f-electrons may delocalize at or near this QCP, from which a strongly correlated paramagnetic phase with delocalized f-electrons emerges [@stewart2001; @vonlohneysen2007; @gegenwart2008]. A crucial ingredient of dimer formation in the SSL is a doublet ground state, and crystal fields can produce such a pseudo-spin S=1/2 ground state in several Ce and Yb based heavy fermion (HF) compounds based on the SSL. Complex magnetic order is found in Ce$_{2}$Pd$_{2}$Sn, where novel low temperature properties arise from ferromagnetic (FM) dimers with the S=1 ground state [@sereni2009]. Nonordering Yb$_{2}$Pd$_{2}$Sn can be driven AF via pressure [@bauer2008] and In doping [@kikuchi2009], but the T$_{K}$=17*K* of Yb$_{2}$Pd$_{2}$Sn remains large throughout. No evidence for dimer formation, such as a singlet-triplet gap, is found and instead the magnetic susceptibility $\chi$ becomes constant as T$\rightarrow$0, indicating that Kondo physics dominates in Yb$_{2}$Pd$_{2}$Sn [@bernhard2011]. In contrast, the Yb moments in [Yb$_{2}$Pt$_{2}$Pb]{} remain fully localized, ordering at T$_{N}$=2.07 *K*, with no indication of Kondo physics [@kim2008; @aronson2010].
We argue here that [Yb$_{2}$Pt$_{2}$Pb]{} is a SSL system where frustration dominates over Kondo physics. Measurements of the magnetic susceptibility $\chi$(T) find that [Yb$_{2}$Pt$_{2}$Pb]{} is a quasi-two dimensional system where magnetic interactions are highly frustrated. The dimer formation characteristic of the SSL is evidenced in [Yb$_{2}$Pt$_{2}$Pb]{} by a broad maximum in $\chi$(T), suggesting that AF order in [Yb$_{2}$Pt$_{2}$Pb]{} emerges from a paramagnetic dimer fluid with a singlet-triplet gap $\Delta$. Magnetic fields suppress AF order more quickly than the spin gap $\Delta$, indicating that the AF phase can only be entered from a spin liquid at higher temperatures and fields. Quantized magnetization steps are a signature of other SSL systems, such as [SrCu$_{2}$(BO$_{3}$)$_{2}$]{} and the RB$_{4}$ compounds, and they are observed as well within the AF phase of [Yb$_{2}$Pt$_{2}$Pb]{}. As a SSL system, [Yb$_{2}$Pt$_{2}$Pb]{} exemplifies a regime near AF instability that has not previously been experimentally accessible. As a HF, [Yb$_{2}$Pt$_{2}$Pb]{} is one of the first systems where the interplay of geometrical frustration and quantum criticality can be investigated.
All experiments were performed on single crystals of [Yb$_{2}$Pt$_{2}$Pb]{} that were prepared from Pb flux [@kim2008]. The electrical resistivity $\rho$ of [Yb$_{2}$Pt$_{2}$Pb]{} is metallic and approaches a residual value $\rho_{0}$=1.5 $\mu\Omega$-cm, attesting to low levels of crystalline disorder [@kim2008]. Measurements of the dc magnetization M were conducted at fixed fields ranging from 0.1 - 4 *T* in a Quantum Designs Magnetic Properties Measurement System (MPMS) for temperatures from 1.8 *K*-300 *K*, while a Hall sensor magnetometer was used for temperatures from 0.06 *K*-4 *K* [@wu2011].
The dc magnetic susceptibility $\chi$=M/B (Fig. 1) reveals both quasi-two dimensionality and strong frustration in [Yb$_{2}$Pt$_{2}$Pb]{}, where the Yb$^{3+}$ moments lie on the SSL (inset, Fig. 1a).
![(Color online) (a) Temperature dependencies of 1/$\chi$ for fixed fields B=2 *T*(T$\geq$ 300 *K*) and B=0.1 *T* (T$\leq$ 300 *K*) along \[001\] and \[110\]. Solid red lines are fits to Curie-Weiss expressions. Inset: the Shastry-Sutherland lattice has interdimer $J^{\prime}$ and intradimer *J* couplings as indicated, with moments directed along the dimer bonds. (b) $\chi$(T)=M/B for B=0.1 *T* (MPMS: $\bullet$, Hall sensor: $\bigcirc$). Solid line is fit to dimer expression (see text). Inset: expanded view of region near T$_{N}$=2.02 *K*. (vertical dashed line). $\chi$:$\bullet$, $\bigcirc$; d$\chi$/dT: $\vartriangle$. Solid line is a guide for the eye.](fig1.eps)
For 300 K$\leq$T$\leq$800 K, $\chi$ is well described by a Curie-Weiss temperature dependence $\chi$(T)=$\chi_{0}$+C/(T-$\theta$), where the fluctuating Yb moments are close to the 4.54 $\mu_{B}$/Yb$^{3+}$ Hund’s rule value (Fig. 1a). Weiss temperatures $\theta_{110}$=28 K (B$\|$\[110\]) and $\theta_{001}$=-217 K (B$\|$\[001\]), indicate weak FM correlations within the \[110\] SSL plane, but stronger AF coupling between the SSL planes. The Yb moments are likely directed along the \[110\] and equivalent easy directions (Fig. 1a, inset) [@kim2008]. A slope discontinuity in $\chi$ and the accompanying maximum in d$\chi$/dT marks the onset of AF order at T$_{N}$=2.02 *K*, slightly below the T$_{N}$=2.07 *K* that is found in specific heat measurements [@kim2008](Fig. 1b, inset). AF order occurs in [Yb$_{2}$Pt$_{2}$Pb]{} at a Neèl temperature T$_{N}$ that is much smaller than the mean field values indicated by the Weiss temperatures, a hallmark of frustration [@ramirez1994]. The in-plane frustration figure of merit f=$\theta_{110}$/T$_{N}$=14 and the interplanar f=$\theta_{001}$/T$_{N}$=105 reveal a profound interplanar frustration in [Yb$_{2}$Pt$_{2}$Pb]{} , indicating as does the large magnetic anisotropy $\chi_{110}$/$\chi_{001}$=30 (T=T$_{N}$), that the individual SSL planes remain magnetically uncorrelated at the lowest temperatures.
A broad peak is observed in $\chi$(T) for B$\|$\[110\] (Fig. 1b), indicating that the ground state of [Yb$_{2}$Pt$_{2}$Pb]{} is nonmagnetic. The magnetic susceptibility $\chi$(T) (B$\|$\[110\]) is well described using the mean field expression $\chi$(T)=$\chi_{D}$/(1-2n$J^{\prime}\chi_{D}$), where $J^{\prime}$ is the interdimer coupling, n the number of near neighbors, and $\chi_{D}$ is the susceptibility of a single dimer. Both of the Yb moments contribute two states, and coupling these moments into a dimer produces a singlet ground state and a triplet excited state, separated for B=0 by an energy $\Delta$= -2*J*. $\chi_{D}$ is readily calculated from this energy level scheme [@sasago1995], taking N to be the number of dimers, k$_{B}$ the Boltzmann constant, $\mu_{B}$ the Bohr magneton, and g the Landé g-factor: . Although there is a small upturn in $\chi$(T) at the lowest temperatures, perhaps indicating that a few Yb moments or even stray impurity moments do not participate in the magnetic dimers, the fit (Fig. 1b) provides an excellent account of the measured B=0.1 *T* susceptibility $\chi$(T) both above and below T$_{N}$ when $\Delta$=4.3$\pm$0.04 *K*, *J*=–2.3$\pm$0.01 *K*, $J^{\prime}$=-1.95$\pm$0.03 *K*, and g=5.43$\pm$0.02, the last consistent with observations in other systems where Yb$^{3+}$ is in a tetragonal crystal field [@reynolds1972].
Magnetic fields affect both T$_{N}$ and $\Delta$, fundamentally changing the balance of phases present for [Yb$_{2}$Pt$_{2}$Pb]{} at B=0.
![(Color online)(a) Magnetization M for different fields B (indicated). Arrows mark T$_{N}$, taken from the peak in d(M/B)/dT. (b) $\chi$(T)=M/B for different values of B$\|$\[110\] ($\bigcirc$: 0.5 *T*, $\bigtriangleup$: 0.75 *T*, $\blacklozenge$: 0.95 *T*). Solid red lines are mean field expression for $\chi$ (see text). Inset: arrows indicate peaks in d$\chi$/dT at T$_{N}$, obtained for B=0.5 *T*, 0.75 *T*, and 0.95 *(*T) (right to left).](fig2.eps)
Increasing magnetic fields B$\|$\[110\] shift both the slope discontinuity in M(T$_{N}$) (Fig. 2a) and its associated peak in d$\chi$/dT (Fig. 2b, inset), as well as the broad maximum in $\chi$(T)(Fig. 2b) to lower temperatures. T$_{N}$(B) is taken from the maximum in d$\chi$/dT (inset, Fig. 2b), and the values of T$_{N}$ determined for each field B are shown in Fig. 3a. T$_{N}$ vanishes for B$_{QCP}$=1.25$\pm$0.01 *T*, following with the XY class exponent $\nu$=0.46$\pm$0.03 [@kawashima2004]. This behavior resembles that of HFs like YbRh$_{2}$Si$_{2}$ [@gegenwart2007] and CeRhIn$_{5}$ [@park2006] near their AF-QCPs. In contrast, the Bose Einstein Condensation (BEC) exponent $\nu$=2/3 is found in quantum magnets like BaCuSi$_{2}$O$_{6}$ [@sebastian2006] and TlCuCl$_{3}$ [@yamada2008], where magnetic fields induce T=0 AF order by driving $\Delta\rightarrow$0, via the Zeeman splitting of excited triplet states [@giamarchi2008].
$\Delta$ and T$_{N}$ vanish at different fields in [Yb$_{2}$Pt$_{2}$Pb]{}. The analysis of the B=0 $\chi$(T) can be generalized for B$\neq$0, using the energy level scheme depicted in Fig. 3a (inset). Each dimer has a singlet ground state , and three excited states with energies ,, and . The dimer magnetization M$_{d}$ is derived from the partition function of these four states, yielding the expression:
$$\begin{aligned}
M_{d}=\frac{2g\mu_{B}\sinh\left(g\mu_{B}B/k_{B}T\right)}{1+\exp\left(-2J/k_{B}T\right)+2\cosh\left(g\mu_{B}B/k_{B}T\right)}\end{aligned}$$
The susceptibility $\chi$ of N interacting dimers, each with n neighbors, is given in turn by the mean field expression , where the dimer susceptibility . $\chi$(T) is calculated for each of the fields B represented in Fig. 2a, using the B=0.1 *T* values of $J^{\prime}$=-1.95 *K*, *J*=-2.34 *K*, and g=5.43. The resulting expressions agree well with the measured $\chi$(T), shown in Fig. 2b. The Zeeman splitting derived from this analysis gives , where$\Delta$ drops linearly from its B=0.1 *T* value of 4.3 *K* to zero for B$_{\Delta}$=1.5 *T* (Fig. 3a, inset). We deduce that $\Delta$(B=0)=4.6 *K*, by extrapolating the B=0.1 *T* value $\Delta$=4.3 *K* to B=0. The relative magnitudes of *J*=-2.3$\pm$0.01 *K* and $J^{\prime}$= -1.95$\pm$0.03 *K* extracted from the B=0.1 *T* fit give the ratio $J^{\prime}/J$=0.85, a value that is larger than the critical value ($J^{\prime}/J$)$_{C}$= 0.6 -0.7, placing [Yb$_{2}$Pt$_{2}$Pb]{} within the expected AF regime of the S=1/2 SSL [@shastry1983].
The phase diagram that is formed by comparing k$_{B}$T$_{N}$(B) and the energy scale $\Delta$(B)/k$_{B}$ (Fig. 3a) indicates that for B$_{QCP}\leq$B$\leq$B$_{\Delta}$, there is a nonzero singlet triplet gap $\Delta$, but no AF order. This regime can be considered a valence bond solid, where the ground state is a nonmagnetic singlet. The disappearance of $\Delta$ for B=B$_{\Delta}$ indicates that the singlet and triplet dimer states have become degenerate. In dimer systems like TlCuCl$_{3}$ [@ruegg2003] and BaCuSi$_{2}$O$_{6}$ [@sebastian2006], this gapless and magnetic state is unstable to AF order, and T$_{N}$ increases as field increases the population of dimer triplets, analogous to BEC. In [Yb$_{2}$Pt$_{2}$Pb]{} , the B=0 AF phase has already vanished when $\Delta\rightarrow$0, although it is possible that re-entrant AF order or another collective state may result for B$\geq$B$_{\Delta}$ [@yoshida2005].
Perhaps the most striking signature of the SSL is the observation of quantized steps in M(B), present either in the field-induced AF phase in [SrCu$_{2}$(BO$_{3}$)$_{2}$]{} [@kageyama1999; @sebastian2008], or in the AF phase that is present for B=0 in the RB$_{4}$ [@siemensmeyer2008; @yoshii2008]. We note that they may alternatively be metamagnetic transitions resulting from the interplay of exchange and magnetocrystalline anisotropy that is found in unfrustrated systems like CeSb [@rossat1983]. [Yb$_{2}$Pt$_{2}$Pb]{} is like the other SSL systems, as a sequence of magnetization plateaux are evident as broadened steps in M(B) or sharp peaks in dM/dB, measured at T=0.06 *K* (Fig. 3b). Increasing and decreasing field sweeps are hysteretic, indicating that [Yb$_{2}$Pt$_{2}$Pb]{} approaches full saturation with M$\rightarrow$M$_{S}$ via a series of intermediate phases that are separated by first order transitions, each with increasing fractions of dimer triplets aligned with the external field. Fig. 3b shows that the M(B) plateaux are only observed in [Yb$_{2}$Pt$_{2}$Pb]{} in the AF state with B$\leq$B$_{QCP}$. Unlike [SrCu$_{2}$(BO$_{3}$)$_{2}$]{}, TlCuCl$_{3}$, and BaCuSi$_{2}$O$_{6}$, where very large fields are required to approach saturation, in [Yb$_{2}$Pt$_{2}$Pb]{} M/M$_{S}\rightarrow$1 for B$\simeq$4 *T*, so it is straightforward to observe the entire magnetization process.
Our experiments on [Yb$_{2}$Pt$_{2}$Pb]{} provide new insight into AF order on the SSL. [Yb$_{2}$Pt$_{2}$Pb]{} is a conventional paramagnet when k$_{B}$T$\gg J, J^{\prime}$, but an increasing number of Yb moments form long-lived dimers as k$_{B}$T decreases towards $\Delta$=4.6 *K*. The stabilization of AF order requires a substantial occupancy of the excited moment-bearing triplet state, which is only possible when k$_{B}$T$_{N}$ is not much smaller than $\Delta$. [Yb$_{2}$Pt$_{2}$Pb]{} is the only known SSL system where this condition is met, and the apparent persistence of the singlet-triplet gap into the AF state suggests that AF order involves locking strongly bonded dimers together via weaker interdimer bonds. The phase diagram in Fig. 3a indicates that increasing either temperature or magnetic field breaks these fragile interdimer bonds, and [Yb$_{2}$Pt$_{2}$Pb]{} reverts to a liquid of uncoordinated dimers.
The unique characteristics of [Yb$_{2}$Pt$_{2}$Pb]{} are highlighted by comparing its properties to other SSL systems (Table 1). The magnitudes of the moments, as well as the Weiss temperatures for fields in the SSL plane $\theta_{ab}$ are similar for the RB$_{4}$ and [Yb$_{2}$Pt$_{2}$Pb]{}, and consequently [Yb$_{2}$Pt$_{2}$Pb]{} and the RB$_{4}$ might be expected to order at similar temperatures. However, AF order is only found in [Yb$_{2}$Pt$_{2}$Pb]{} when T$_{N}\ll\theta_{ab}$, resulting, in part, from the quasi-two dimensional character of $\chi$ in [Yb$_{2}$Pt$_{2}$Pb]{}, absent in the other SSL compounds. This suppression of k$_{B}$T$_{N}$ to a value that is comparable in magnitude to $J^{\prime}$ and *J* makes dimer formation an integral feature of AF [Yb$_{2}$Pt$_{2}$Pb]{}, making [Yb$_{2}$Pt$_{2}$Pb]{} the AF counterpart of the SL [SrCu$_{2}$(BO$_{3}$)$_{2}$]{}. As such, it is the only SSL system where the interplay of dimer formation and long-ranged AF order can be studied.
T$_{N}$ $\theta_{ab}$ f *J* $J^{\prime}$ $J^{\prime}/J$ $\%$ REF.
---------------------------- --------- --------------- ----- ------ -------------- ---------------- -------- -------------------------------------------
Yb$_{2}$Pt$_{2}$Pb 2.02 28 14 2.3 1.9 0.83 30 This work
GdB$_{4}$ 42 -68 1.6 8.9 0.68 0.076 1.05 [@fisk1981; @fernandez2005; @kikkawa2007]
TmB$_{4}$ 10 -63 6.3 0.85 0.3 0.36 1.5,20 [@matas2010]
TbB$_{4}$ 44 -27 0.6 1.55 0.33 0.21 0.88 [@fisk1981; @rhyee2007; @yoshii2008]
SrCu$_{2}$(BO$_{3}$)$_{2}$ – -103 – 85 54 0.64 1.28 [@kageyama1999b; @miyahara2003]
: \[tab:table1\]A comparison of the Neél temperature T$_{N}$, in-plane Weiss temperature $\theta_{ab}$, frustration figure of merit f=$\theta_{ab}$/T$_{N}$, interdimer exchange $J^{\prime}$, intradimer exchange *J*, $J^{\prime}/J$, and susceptibility anisotropy $\%$= $\chi_{ab}$/$\chi_{c}$, evaluated at T$_{N}$ in different SSL systems (2 *K* for [SrCu$_{2}$(BO$_{3}$)$_{2}$]{}, 250 *K* and 30 *K*, respectively, for TmB$_{4}$). T$_{N}$, $\theta_{ab}$ $J^{\prime}$, and *J* are all given in units of *K*.
The B=0 ground state for [Yb$_{2}$Pt$_{2}$Pb]{} is distinct among both HF and SSL compounds, with AF order developing from a liquid of dimers. The low Neél temperature, the persistence of the singlet-triplet dimer gap $\Delta$ in the AF state, and the suppression of AF order in a small magnetic field all place [Yb$_{2}$Pt$_{2}$Pb]{} very close to the AF-SL transition, a regime of the SSL that was previously only addressed theoretically. While there are other HFs that form on geometrically frustrated lattices [@lacroix2009], in these cases it is generally found that long ranged interactions such as the Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction replace the competing short ranged interactions that lead to frustration effects in insulating systems. Given that [Yb$_{2}$Pt$_{2}$Pb]{} is an excellent metal with substantial Yb moments, it is noteworthy that we observe the singlet-triplet gap, the dimer spin liquid, and the magnetization plateaux, all signatures of the SSL that were previously only observed in insulating [SrCu$_{2}$(BO$_{3}$)$_{2}$]{}. It is at present unknown whether the HF character of [Yb$_{2}$Pt$_{2}$Pb]{} will result in the same breakdown in normal metallic behavior and the stabilization of unconventional ordered phases that are found near AF quantum critical points in unfrustrated HF compounds, or if HFs with geometrical frustration have inherently different properties. [Yb$_{2}$Pt$_{2}$Pb]{} is one of a very small number of known compounds where these intriguing questions can be experimentally explored.
The authors thank J. Sereni for enlightening discussions. This work was supported by National Science Foundation grant NSF-DMR-0907457.
[10]{}
B. S. Shastry and B. Sutherland, Physica B $\bf{108}$, 1069 (1981). H. Kageyama, et al., Phys. Rev. Lett. $\bf{82}$, 3168 (1999). S. Miyahara and K. Ueda, J. Phys.: Cond. Matt. $\bf{15}$, R327 (2003). K. Onizuka, et al., J. Phys. Soc. Japan $\bf{69}$, 1016 (2000). J. Etourneau, et al., J. Less-Common Met. $\bf{67}$, 531 (1979). S. Michimura, et al., Physica B $\bf{378-380}$, 596 (2006). S. Matas, et al., J. Phys.: Conf. Ser. $\bf{200}$, 032041 (2010). F. Iga, et al., J. Magn. Magn. Mater. $\bf{310}$, e443 (2007). Z. Weihong, et al., Phys. Rev. B $\bf{60}$, 6608 (1999). M. Al Hajj and J. -P. Malrieu, Phys. Rev. B $\bf{72}$, 094436 (2005) A. Isacsson and O. F. Sylju[å]{}sen, Phys. Rev. E $\bf{74}$, 026701 (2006). D. Carpentier and L. Balents, Phys. Rev. B $\bf{65}$, 024427 (2002). M. Albrecht and F. Mila, Europhys. Lett. $\bf{34}$, 145 (1996). A. Koga, et al., J. Phys. Soc. Japan $\bf{72}$, 938 (2003). C. Lacroix, Physica B $\bf{404}$,3038 (2009). M. Vojta, Phys. Rev. B $\bf{78}$, 125109 (2008). P. Coleman and A. H. Nevidomskyy, J. Low Temp. Phys. $\bf{161}$, 182 (2010). M. C. Aronson, et al., J. Low Temp. Phys. $\bf{161}$, 98 (2010). B. H. Bernard, et al., Phys. Rev. B $\bf{83}$, 214427 (2011). G. R. Stewart, Rev. Mod. Phys. $\bf{73}$, 797(2001). H. v. Lohneysen, et al., Rev. Mod. Phys. $\bf{79}$, 1015 (2007). P. Gegenwart,et al., Nature Physics $\bf{4}$, 186 (2008). J. G. Sereni, et al., Phys. Rev. B $\bf{80}$, 024428 (2009). E. Bauer, et al., J. Optoelectron. Adv. Mater. $\bf{10}$, 1633 (2008). E. Kikuchi, et al., J. Phys. Soc. Japan $\bf{78}$, 083708 (2009). M. S. Kim, et al., Phys. Rev. B $\bf{77}$, 144425 (2008). L. S. Wu, et al., Phys. Rev. B $\bf{84}$, 134409 (2011). A. P. Ramirez, Ann. Rev. Mat. Sci, $\bf{24}$, 453 (1994). Y. Sasago, et al., Phys. Rev. B $\bf{52}$, 3533 (1995). R. W. Reynolds et al., J. Chem. Phys. $\bf{56}$, 5607 (1972). N. Kawashima, J. Phys. Soc. Japan $\bf{73}$, 3219 (2004). P. Gegenwart, et al., Science $\bf{315}$, 969 (2007). T. Park, et al., Nature $\bf{440}$, 63 (2006). S. E. Sebastian, et al., Nature $\bf{441}$, 617 (2006). F. Yamada, et al., J. Phys. Soc. Japan $\bf{77}$, 013701 (2008). T. Giamarchi, et al., Nature Physics $\bf{4}$, 196 (2008). C. Ruegg, et al., Nature $\bf{423}$, 62 (2003). Y. Yoshida, et al., J. Phys. Soc. Japan $\bf{74}$, 2917 (2005). S. E. Sebastian, et al., Proc. Nat. Acad. Sci. $\bf{105}$, 20157 (2008). K. Siemensmeyer, et al., Phys. Rev. Lett. $\bf{101}$, 177201 (2008). S. Yoshii, et al., Phys. Rev. Lett. $\bf{101}$, 087202 (2008). J. Rossat-Mignot et al., J. Magn. Magn. Mater. $\bf{31-34}$, 398 (1983). Z. Fisk, et al., Sol. St. Comm. $\bf{39}$, 1189(1981). J. Fernandez-Rodriguez, et al., Phys. Rev. B $\bf{72}$, 052407 (2005). A. Kikkawa, et al., J. Phys. Soc. Japan $\bf{76}$, 024711 (2007). J.-S. Rhyee, et al., J. Appl. Phys. $\bf{101}$, 09D509 (2007). H. Kageyama, et al., J. Phys. Soc. Japan $\bf{68}$, 1821 (1999).
|
---
author:
- Amit Seal Ami
- 'Md. Mehedi Hasan'
- 'Md. Rayhanur Rahman'
- Kazi Sakib
bibliography:
- 'mobisoft.bib'
title: 'MobiCoMonkey - Context Testing of Android Apps'
---
<ccs2012> <concept> <concept\_id>10010147.10010341.10010366.10010369</concept\_id> <concept\_desc>Computing methodologies Simulation tools</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10011007.10011006.10011066.10011069</concept\_id> <concept\_desc>Software and its engineering Integrated and visual development environments</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10011007.10011074.10011099.10011102.10011103</concept\_id> <concept\_desc>Software and its engineering Software testing and debugging</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10011007.10011074.10011099.10011693</concept\_id> <concept\_desc>Software and its engineering Empirical software validation</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction
============
Background Study and Related Works
==================================
The $MobiCoMonkey$ Tool
=======================
Conclusion
==========
|
---
address: 'School of Mathematics, Institute for Advanced Study, Einstein Drive, Fuld Hall, Princeton, NJ 08540'
author:
- Jun Yu
title: On the dimension datum problem and the linear dependence problem
---
Introduction
============
Given a compact Lie group $G$, the dimension datum $\mathscr{D}_{H}$ of a closed subgroup $H$ of $G$ is the map from $\widehat{G}$ to ${\mathbb{Z}}$, $$\mathscr{D}_{H}: V\mapsto\dim V^{H},$$ where $\widehat{G}$ is the set of equivalence classes of irreducible finite-dimensional complex linear representations of $G$ and $V^{H}$ is the subspace of $H$-invariant vectors in a complex linear representation $V$ of $G$. In number theory, dimension data arose from the determination of some monodromy groups (cf. [@Katz]). In the theory of automorphic forms, Langlands ([@Langlands]) has suggested to use dimension data as a key ingredient in his program “Beyond Endoscopy”. The idea is to use the dimension datum to identify the conjectural subgroup ${}^\lambda H_\pi\subset {}^L \mathscr{G}$ associated to an automorphic representation $\pi$ of $\mathscr{G}(\mathbb{A})$, where ${}^L\mathscr{G}$ is the $L$-group of $\mathscr{G}$ in the form of [@Borel 2.4(2)]. In differential geometry, the dimension datum $\mathscr{D}_H$ is related to the spectrum of the Laplace operator on the homogeneous Riemannian manifold $G/H$.
Let $H\subset G$ be compact Lie groups. The [*dimension datum problem*]{} asks the following question.
\[Q:dimension data\] To what extent is $H$ (up to $G$-conjugacy) determined by its dimension datum $\mathscr{D}_H$ ?
In [@Larsen-Pink], Larsen and Pink considered the dimension data of connected semisimple subgroups. They showed that the subgroups are determined up to isomorphism by their dimension data (cf. [@Larsen-Pink], Theorem 1), and not up to conjugacy in general (cf. [@Larsen-Pink], Theorem 3). Moreover, they showed that the subgroups $H$ are determined up to conjugacy if $G=\SU(n)$ and the inclusions $H\hookrightarrow\SU(n)$ give irreducible representations of $H$ on ${\mathbb{C}}^{n}$ (cf. [@Larsen-Pink], Theorem 2).
In [@Langlands], Subsections 1.1 and 1.6, Langlands raised another question (cf. [@An-Yu-Yu], Question 5.3) about dimension data. If this question has an affirmative answer, then it will facilitate a way of dealing with the dimension data of ${}^\lambda H_\pi$ using trace formulas. However, Langlands suspected (cf. [@Langlands], discussions following Equation (14)) that in general this question has an affirmative answer. As observed in [@An-Yu-Yu], this question proposed by Langlands is equivalent to the following question that we shall call the [*linear dependence problem*]{}.
\[Q:linear dependence\] Given a list of finitely many closed subgroups $H_1,\dots,H_{n}$ of $G$ with $\mathscr{D}_{H_{i}}\neq
\mathscr{D}_{H_{j}}$ for any $i\neq j$, are $\mathscr{D}_{H_1},\dots,\mathscr{D}_{H_{n}}$ linearly independent?
In [@An-Yu-Yu], jointly with Jinpeng An and Jiu-Kang Yu, we have given counter-examples to show that the affirmative answer to Question \[Q:dimension data\] (or \[Q:linear dependence\]) is not always positive. In this paper, we classify closed connected subgroups with equal dimension data or linearly dependent dimension data.The method of this classification is as follows.
Given a compact Lie group $G$ we choose a bi-invariant Riemannian metric $m$ on it. For a closed connected torus $T$ in $G$, we introduce root systems on $T$. Among the root systems on $T$, there is a maximal one, denoted by $\Psi_{T}$, which contains all other root systems on $T$. For each reduced root system $\Phi$ on $T$ and a finite group $W$ acting on ${\mathfrak{t}}_0=\Lie T$ satisfying some condition (cf. Definition \[D:characters\]), we define a character $F_{\Phi,W}$. In particular, this definition applies to the finite group $\Gamma^{\circ}:=N_{G}(T)/C_{G}(T)$. We define another root system $\Psi'_{T}$ on $T$ as the sub-root system of $\Psi'_{T}$ generated by the root systems of closed connected subgroups $H$ of $G$ with $T$ a maximal torus of $H$. Moreover we show that the Weyl group of $\Psi'_{T}$ is a sub-group of $\Gamma^{\circ}$. By Propositions \[P: ST support\] and \[P:group-root system\], we show that the dimension datum problem (or the linear dependence problem) reduces to comparing the characters $\{F_{\Phi,\Gamma^{\circ}}|\ \Phi\subset\Psi_{T}\}$ (or finding linear relations among them). We propose Questions \[Q:equal-character\] and \[Q:dependent-character\]. They are concerned with the characters associated to reduced sub-root systems of a given root system $\Psi$. In the next two sections we solve these two questions completely. We reduce both questions to the case where $\Psi$ is an irreducible root system. In the case that $\Psi$ is a classical irreducible root system, the classification of sub-root systems of $\Psi$ is well-known. In [@Larsen-Pink], Larsen and Pink defined an algebra isomorphism $E$ taking characters of sub-root systems to polynomials (cf. Lemma \[Isomorphism E\] and Definition \[D:abcd\]). In this case we solve Questions \[Q:equal-character\] and \[Q:dependent-character\] by getting all algebraic relations among polynomials in the image of $E$. The relations of some polynomials given in Propositions \[P:A=BB and A=CD\] and \[P:A-BB’CD\] are most important. In the case that $\Psi$ is an exceptional irreducible root system, Oshima classified sub-root systems of $\Psi$ (cf. [@Oshima]). In [@Larsen-Pink], the authors defined a weight $2\delta'_{\Phi}$ for each reduced sub-root system $\Phi$ of $\Psi$. We give the formulas of these weights in the Tables 2-9. We also define and calculate a generating function for each reduced sub-root system of $\Psi$ (cf. Definition \[D:f\]). With these, we are able to find and prove all linear relations among the characters of reduced sub-root systems of $\Psi$. Therefore we solve Questions \[Q:equal-character\] and \[Q:dependent-character\] completely.
Given a connected closed torus $T$ in $G$, the finite group $\Gamma^{\circ}$ palys an important role for the dimension data of closed connected subgroups $H$ of $G$ with $T$ a maximal torus of $H$. In Proposition \[P:Gamma0-Psi’\] we show that $\Gamma^{\circ}\sup W_{\Psi'_{T}}$, where $\Psi'_{T}$ is the sub-root system of $\Psi_{T}$ generated by root systems of closed connected subgroups $H$ of $G$ with $T$ a maximal torus of $H$. On the other hand, in Proposition \[P:Gamma0-2\] we give a construction showing that the finite group $\Gamma^{\circ}$ could be arbitrary providing that it containing $W_{\Psi'_{T}}$. These supplements show that our solution to Questions \[Q:equal-character\] and \[Q:dependent-character\] actually give all examples of closed connected subgroups with equal dimension data or linearly dependent dimension data.
In the last section, we study the equalities and linear relations among dimension data of closed subgroups of $\U(n)$ acting irreducibly on ${\mathbb{C}}^{n}$. For nonconnected irreducible subgroups, we give ineteresting examples with equal or linearly dependent dimension data for $n=16$ and $n=12$ respectively. We also show that the dimension data of closed connected irredicible subgroups are actually linearly independent. This strengthens a theorem of Larsen and Pink. The proof is identically theirs. The writing of this section is inspired by questions of Peter Sarnak.
The following theorem is a simple consequence of our classification of subgroups with equal dimension data. It follows from Proposition \[P:group-root system\], Theorem \[T:equal character-nonsimple\], Theorem \[T:character-classical\] and Theorem \[character-exceptional\].
\[T:dimension-algebra\] If two compact Lie groups $H,H'$ have inclusions to a compact Lie group $G$ with the same dimension data, then after replacing each ideal of $\Lie H$ and $\Lie H'$ isomorphic to $\mathfrak{u}(2n+1)$ ($n\geq 1$) by an ideal isomorphic to $\mathfrak{sp}(n)\oplus\mathfrak{so}(2n+2)$, the resulting Lie algebras are isomorphic.
The classification of subgroups with linearly dependent dimension data is more complicated. The following theorem is a simple consequence of this classification. It follows from Propositions \[P:linear-A\], \[P:linear-BCD\] and \[P:linear-exceptional\]. In the case of type $\B_2$, $\B_3$ or $\G_2$, we also have a linear independence result. The proof requires some consideration of possible connected full rank subgroups.
\[T:linear-type A\] Given $G=\SU(n)$ or a compact connected Lie group isogeneous to it, for any list $\{H_1,H_2,\dots,H_{s}\}$ of non-conjugate connected closed full rank subgroups of $G$, the dimension data $\mathscr{D}_{H_1},\mathscr{D}_{H_2},\cdots,\mathscr{D}_{H_s}$ are linearly independent. Given a comapct connected Lie group $G$ with a simple Lie algebra ${\mathfrak{g}}_0$, if ${\mathfrak{g}}_0$ is not of type $\A_{n}$, $\B_2$, $\B_3$ or $\G_2$, then there exist non-conjugate connected closed full rank subgroups $H_1$, $H_2$,...,$H_s$ of $G$ with linearly dependent dimension data.
The organization of this paper is as follows. In Section \[S:root systems\], after recalling the definitions of root datum and root system, we define root system in a lattice. In Proposition \[P:maximal root system\] we show that a given lattice contains a unique maximal root system in it. In Section \[S:characters\], given a compact Lie group $G$ with a biinvariant Riemannian metric $m$, we define root systems on $T$ and a maximal one $\Psi_{T}$ among them. For each reduced root system $\Phi$ on $T$ and a finite group $W$ acting on $T$ satisfying some condition, we define a character $F_{\Phi,W}$. Moreover, we discuss properties of these characters $\{F_{\Phi,W}\}$. Most importantly, in Propositions \[P: ST support\] and \[P:group-root system\], we reduce the dimension datum problem and the linear dependence problem to compare these characters and getting linear relations among them. In Section \[S:conjugacy\], we discuss the relations between several finite groups $\Gamma^{\circ}$, $\Gamma$, $W_{\Psi'_{T}}$, $W_{\Psi_{T}}$ and $\Aut(\Psi_{T})$. In particualr we prove that $W_{\Psi'_{T}}\subset\Gamma^{\circ}$. Moreover, we discuss the connection between conjugacy relations of root systems of subgroups with regard to some of these finite groups and relations of the subgroups. In Section \[S:formulation of questions\], we formulate two questions in terms of characters of reduced sub-root systems of a given root system. If the finite group $\Gamma^{\circ}$ contains $W_{\Psi_{T}}$, then these two questions are equivalent to the dimension datum problem and the linear dependence problem, respectively. In Section \[S:sub-root systems\], we discuss the classification of reduced sub-root systems of a given irreducible root system. In Section \[S:leading terms\], given an exceptional irreducible root system $\Psi_0$, we give the formulas of the weights $\{2\delta'_{\Phi}|\ \Phi\subset\Psi_0\}$. In Section \[S:dimension-equal\], given a root system $\Psi$, we classify reduced sub-root systems of $\Psi$ with equal characters $F_{\Phi,\Aut(\Psi)}$. This solves Question \[Q:equal-character\]. In Section \[S:dimension-dependent\], given a root system $\Psi$, we classify reduced sub-root systems of $\Psi$ with linearly dependent characters $F_{\Phi,W(\Psi)}$. This solves Question \[Q:dependent-character\]. In Section \[S:Gamma0\], given a root system $\Psi'$ and a finite group $W$ containing $W_{\Psi}$, we give examples of a compact connected simple Lie group $G$ and a connected closed torus $T$ in $G$ satisfying that: $\dim T=\rank\Psi'$, $\Psi'\subset\Psi_{T}$ and $\Psi'$ is stable under $\Gamma^{\circ}$, $\Gamma^{\circ}=W$ as groups acting on $\Psi'$ and each reduced sub-root system $\Phi$ of $\Psi'$ equals the root system of a closed connected subgprup $H$ with $T$ a maximal torus of $H$. In Section \[S:Irreducible\], we give interesting examples of irreducible subgroups of $\U(n)$ with equal dimension data or linearly dependent dimension data, and show that connected irreducible subgroups of $\U(n)$ actually have linearly independent dimension data.
[*Notation and conventions.*]{} Given a compact Lie group $G$,
1. denote by $G_0$ the subgroup of connected component of $G$ containing the identity element and $[G,G]$ the commutator subgroup.
2. Let ${\mathfrak{g}}_0=\Lie G$ be the Lie algebra of $G$.
3. Write $G^{\natural}$ for the set of conjugacy classes in $G$;
4. Denote by $\widehat{G}$ the set of equivalence classes of irreducible finite-dimensional complex linear representations of $G$.
5. Let $\mu_{G}$ be the unique Haar measure on $G$ with $\int_{G} 1\mu_{G}=1$.
6. Write $V^{G}$ for the subspace of $G$-invariant vectors in a complex linear representation $V$ of $G$.
7. Given a maximal torus $S$ of $G$, let $W_{G}=N_{G}(S)/C_{G}(S)$ be the Weyl group of $G$. Then the quotient space $S/W_{G}$ is a connected component of $G^{\natural}$. Moreover, in the case that $G$ is connected, they are identical.
8. Given a closed subgroup $H$ of $G$, denote by $\st_{H}$ the push-measure on $G^{\natural}$ of $\mu_{H}$ under the composition map $H\hookrightarrow G\longrightarrow G^{\natural}$. It is called the [*Sato-Tate measure*]{} of the subgroup $H$.
9. For any complex linear representation $\rho$ of $G$, $\mathscr{D}_{H}(\rho)=\int_{H}\chi_{\rho}(x)
\mu_{H}=\st_{H}(\chi_{\rho}^{\natural})$, where $\chi_{\rho}^{\natural}$ is a continuous function on $G^{\natural}$ induced from the character function $\chi_{\rho}$ of $\rho$. In this way, the dimension datum $\mathscr{D}_{H}$ and the Sato-Tate measure $\st_{H}$ determine each other.
Given a compact Lie group $G$ and a closed connected torus $T$, let $\Phi(G,T)$ be the set of non-zero weights of ${\mathfrak{g}}={\mathfrak{g}}_0\otimes_{{\mathbb{R}}}{\mathbb{C}}$ as a complex linear representation of $T$. In Section \[S:root systems\], we explain that in some cases $\Phi(G,T)$ is a root system.
Given a lattice $L$,
1. write $L_{\mathbb{Q}}=L\otimes_{\mathbb{Z}}\mathbb{Q}$ for the rational vector space generated by $L$.
2. Let ${\mathbb{Q}}[L]$ be the group ring of $L$ over the field ${\mathbb{Q}}$.
3. For any element $\lambda\in L$, denote by $[\lambda]$ the corresponding element in ${\mathbb{Q}}[L]$. Then, for any $\lambda,\mu\in L$ and $n\in{\mathbb{Z}}$, $[\lambda][\mu]=[\lambda+\mu]$ and $[\lambda]^{n}=[n\lambda]$.
Given an abstract root system $\Phi$ (cf. Definition \[D:abstract root system\]),
1. denote by ${\mathbb{Z}}\Phi$ the root lattice spanned by $\Phi$ and by ${\mathbb{Q}}\Phi={\mathbb{Z}}\Phi\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ the rational vector space spanned by $\Phi$.
2. Choosing a positive definite inner product $(\cdot,\cdot)_{m}$ on ${\mathbb{Q}}\Phi$ inducing the cusp product on $\Phi$ (which always exists and is unique if and only if $\Phi$ is irreducible), write $$\Lambda_{\Phi}=\big\{\lambda\in{\mathbb{Q}}\Phi|\ \frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\in{\mathbb{Z}},\
\forall\alpha\in\Phi\big\}$$ for the lattice of integral weights. One can show that $\Lambda_{\Phi}$ does not depend on the choice of the inner product $m$ on ${\mathbb{Q}}\Phi$ if it induces the cusp product on $\Phi$.
3. Let $\Phi^{\circ}$ be the subset of short vectors in $\Phi$: for any $\alpha\in\Phi$, $\alpha\in\Phi^{\circ}$ if and only if for any other $\beta\in\Phi$, $|\beta|\geq|\alpha|$ or $\langle\alpha,\beta\rangle=0$. One can show that $\Phi^{\circ}$ is a sub-root system of $\Phi$.
4. If $V$ is a rational (or real) vector space with a positive definite inner product $(\cdot,\cdot)_{m}$ containing $\Phi$, let $$\Lambda_{\Phi}(V)=\big\{\lambda\in V|\ \frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}
\in{\mathbb{Z}},\ \forall\alpha\in\Phi\big\}.$$ Then $\Lambda_{\Phi}(V)$ is the direct sum of $\Lambda_{\Phi}$ and the linear subspace of vectors in $V$ orthogonal to all vectors in $\Phi$.
5. Given a subset $X$ of $\Phi$, we call the minimal sub-root system of $\Phi$ containing $X$ (which exists and is unique) the sub-root system generated by $X$, and denote it by $\langle X\rangle$.
Write $$T_{k}=\{\diag\{z_1,z_2,\dots,z_{k}\}|\ |z_1|=|z_2|=\cdots=|z_{k}|=1\}.$$
We follow Bourbaki numbering to order the simple roots (cf. [@Bourbaki], Pages 265-300). Write $\omega_{i}$ for the $i$-th fundamental weight.
[**Acknowledgements.**]{} This article is a sequel of [@An-Yu-Yu]. The author is grateful to Jinpeng An and Jiu-Kang Yu for the collaboration. He is also grateful to Brent Doran, Richard Pink, Gopal Prasad, Peter Sarnak and Jiu-Kang for helpful discussions and suggestions.
Root system in a lattice {#S:root systems}
========================
In this section, after recalling the definitions of root datum and root system in a Euclidean vector space, we define root system in a lattice. Moreover, given a lattice, we show that it possesses a unique root system in it which contains all other root systems in it. Here lattice means a finite rank free abelian group with a positive definite inner product. Our discussion mostly follows [@Bourbaki], [@Knapp] and [@Springer]. However there exist minor differences between our definitions and definitions in each of them. For example, our definition of abstract root system is different from that in [@Knapp], and we allow our root systems to be neither reduced nor semisimple, in contrast to all of the above references.
Defined as in [@Springer], a root datum is a quadruple $(X,R,X^{\ast},R^{\ast})$ together with some additional structure (duality and the root-coroot correspondence).
\[D:root datum\] A root datum consists of a quadruple $(X,R,X^{\ast},R^{\ast})$, where
1. $X$ and $X^{\ast}$ are free abelian groups of finite rank together with a perfect pairing between them with values in ${\mathbb{Z}}$ which we denote by $(,)$ (in other words, each is identified with the dual lattice of the other).
2. $R$ is a finite subset of $X$ and $R^{\ast}$ is a finite subset of $X^{\ast}$ and there is a bijection from $R$ onto $R^{\ast}$, denoted by $\alpha\mapsto\alpha^{\ast}$.
3. For each $\alpha$, $(\alpha,\alpha^{\ast})=2$.
4. For each $\alpha$, the map taking $x$ to $x-(x,\alpha^{\ast})\alpha$ induces an automorphism of the root datum (in other words it maps $R$ to $R$ and the induced action on $X^{\ast}$ maps $R^{\ast}$ to $R^{\ast}$).
Two root data $(X_1,R_1,X_1^{\ast},R_1^{\ast})$ and $(X_2,R_2,X_2^{\ast},R_2^{\ast})$ are called isomorphic if there exists a linear isomorphism $f: X_1\rightarrow X_2$ such that $f(R_1)=f(R_2)$, $(f^{\ast})^{-1}(R_1^{\ast})=R_2^{\ast}$ and $f(\alpha)^{\ast}=((f^{\ast})^{-1})(\alpha^{\ast})$ for any $\alpha\in R_1$. Here, $f^{\ast}: X_2^{\ast}\rightarrow X_1^{\ast}$ is the dual linear map of $f$ and $(f^{\ast})^{-1}$ is its inverse.
The elements of $R$ are called [*roots*]{}, and the elements of $R^{\ast}$ are called [*coroots*]{}. If $R$ does not contain $2\alpha$ for any $\alpha$ in $R$, then the root datum is called [*reduced*]{}. To a connected compact Lie group $G$ with a maximal torus $S$, we can associate a reduced root datum $\RD(G,S)$, whose isomorphism class $\RD(G)$ depends only on $G$. Two connected compact Lie groups $G,G'$ are isomorphic if and only if $\RD(G)$ and $\RD(G')$ are isomorphic.
A root system in a Euclidean vector space is a finite set with some additional structure.
\[D:root system\] Let $V$ be a finite-dimensional Euclidean vector space, with an inner product $m$ denoted by $(\cdot,\cdot)_{m}$. A root system in $V$ is a finite set $\Phi$ of non-zero vectors (called roots) in $V$ that satisfy the following conditions:
1. For any two roots $\alpha$ and $\beta$, the element $\beta-\frac{2(\beta,\alpha)_{m}}
{(\alpha,\alpha)_{m}}\alpha\in\Phi$.
2. (**Integrality**) For any two roots $\alpha$ and $\beta$, the number $\frac{2(\beta,\alpha)_{m}}
{(\alpha,\alpha)_{m}}$ is an integer.
Given a root system $\Phi$ in a Euclidean vector space $V$ with an inner product $m$, we call $s_{\alpha}: V\rightarrow V$ defined by $$s_{\alpha}(\lambda)=\lambda-\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\alpha,\ \forall\lambda\in V$$ the reflection corresponding to the root $\alpha$.
Moreover, a root system $\Phi$ in a Euclidean vector space $V$ is called [*semisimple*]{} if the roots span $V$. It is called [*reduced*]{} if the only scalar multiples of a root $x\in\Phi$ that belong to $\Phi$ are $x$ and $-x$.
To a connected compact semisimple Lie group $G$ with a maximal torus $S$, we can associate a reduced semisimple root system $\R(G,S)$, whose isomorphism class $\RD(G)$ depends only on $G$. Two connected compact semisimple Lie groups $G,G'$ having isomorphic universal covers if and only if $\R(G)$ and $\R(G')$ are isomorphic.
There are several different notions of lattice in the literature.
\[D:lattice\] A lattice is a finite rank free abelian group with a positive definite inner product.
Now we define root systems in a lattice.
\[D:root system in a lattice\] Let $L$ be a lattice with a positive definite inner product $m$ denoted by $(\cdot,\cdot)_{m}$. A root system in $L$ is a finite set $\Phi$ of non-zero vectors (called roots) in $L$ that satisfy the following conditions:
1. For any two roots $\alpha$ and $\beta$, the element $\beta-\frac{2(\beta,\alpha)_{m}}{(\alpha,\alpha)_{m}}
\alpha\in\Phi$.
2. (**Strong integrality**) For any root $\alpha$ and any vector $\lambda\in L$, the number $\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}$ is an integer.
Given a root system $\Phi$ in a lattice $L$ with an inner product $m$, we call $s_{\alpha}: L\rightarrow L$ defined by $$s_{\alpha}(\lambda)=\lambda-\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\alpha,\
\forall\lambda\in L$$ the reflection corresponding to the root $\alpha$.
Moreover, a root system $\Phi$ in a lattice $L$ is called [*semisimple*]{} if the roots span $V=L\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. It is called [*reduced*]{} if the only scalar multiples of a root $x\in\Phi$ that belong to $\Phi$ are $x$ and $-x$.
If $\Phi\subset L$ is a root system in $L$, then it is a root system in $V=L\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. Conversely, if $\Phi\subset L$ is a root system in $V$, it is not necessarily a root system in $L$. For example, let $L={\mathbb{Z}}e_1$ be a rank $1$ lattice with $(e_1,e_1)>0$ and $V={\mathbb{R}}e_1$. Then for any $k\in{\mathbb{Z}}_{>0}$, $\{\pm{k}e_1\}$ is a root system in $V$. However $\{\pm{k}e_1\}$ is a root system in $L$ only if $k=1$ or $2$. This example indicates that strong integrality is strictly stronger than integrality.
\[D:abstract root system\] Given a root system $\Phi$ in a Euclidean vector space or a lattice with a positive definite inner product $m$, we call $$\langle y,x\rangle=\frac{2(x,y)_{m}}{(x,x)_{m}}$$ the cusp product on $\Phi$. Overlooking the Eulidean space or the lattice containg $\Phi$, we call $\Phi$ with the cusp product on it an abstract root system.
Different with the above definiton, an abstract root system as defined in the book [@Knapp] is simply a semisimple root system in a Euclidean vector space in our sense.
\[P:cusp product\] Given an abstract root system $\Phi$, for two roots $\alpha\in\Phi$ and $\beta\in\Phi$, let $k_1=\max\{k\in{\mathbb{Z}}|\ \beta+k\alpha\in\Phi\}$ and $k_2=\max\{k\in{\mathbb{Z}}|\ \beta-k\alpha\in\Phi\}$. Then $\langle\beta,\alpha\rangle=k_2-k_1$.
Applying the reflection $s_{\alpha}$, we get $\langle\beta+k_1\alpha,\alpha\rangle+\langle\beta-k_2\alpha,\alpha\rangle=0$. Thus $\langle\beta,\alpha\rangle=k_2-k_1$ follows from $\langle\alpha,\alpha\rangle=2$.
\[D:irreducible root system\] An abstract root system $\Phi$ with a cusp product $\langle\cdot,\cdot\rangle$ is called irreducible if there exist no non-trivial disjoint unions $\Phi=\Phi_1\bigcup\Phi_2$ such that $\langle\alpha_1,\alpha_2\rangle=0$ for any $\alpha_1\in\Phi_1$ and $\alpha_2\in\Phi_2$. A root system $\Phi$ in a Euclidean vector space or in a lattice is called irreducible if it is semisimple and irreducible as an abstract root system.
\[D:Automorphism group\] Given a root system $\Phi$ in a Euclidean vector space $V$ with a positive definite inner product $m$, denote by $\Aut(\Phi,m)$ the group of linear isomorphisms of $V$ stabilizing $\Phi$ and preserving the inner product $m$, and by $W_{\Phi}$ the group of linear isomorphisms of $V$ generated by $\{s_{\alpha}|\ \alpha\in\Phi\}$.
Given a root system $\Phi$ in a lattice $L$ with a positive definite inner product $m$, write $\Aut(\Phi,m)$ for the group of linear isomorphisms of $L$ stabilizing $\Phi$ and preserving the inner product $m$, and $W_{\Phi}$ for the group of linear isomorphisms of $L$ generated by $\{s_{\alpha}|\ \alpha\in\Phi\}$.
Given an abstract root system $\Phi$ with a cusp product $\langle\cdot,\cdot\rangle$, let $\Aut(\Phi)$ be the group of permutations on $\Phi$ preserving its cusp product, and by $W_{\Phi}$ the group of permutations on $\Phi$ generated by $\{s_{\alpha}|\ \alpha\in\Phi\}$.
By definitions we have $$W_{\Phi}\subset\Aut(\Phi,m),$$ and the restriction on $\Phi$ gives a homomorphism $$\pi: \Aut(\Phi,m)\rightarrow\Aut(\Phi).$$
Given a root system $\Phi$ in a Euclidean vector space $V$, we have the following characterization of lattices $L$ in $V$ such that $\Phi$ is also a root system in $L$.
\[P:root system-lattice\] Given a Euclidean vector space $V$ with a positive definite inner product $m$ denoted by $(\cdot,\cdot)$, let $\Phi$ be a root system in $V$. Then for any lattice $L\subset V$, $\Phi$ is a root system in $L$ if and only if ${\mathbb{Z}}\Phi\subset L\subset\Lambda_{\Phi}(V)$.
Suppose $\Phi$ is a root system in $L$. Since $\Phi\subset L$, ${\mathbb{Z}}\Phi\subset L$. On the other hand, by the condition strong integrality in Definition \[D:root system in a lattice\], $L\subset\Lambda_{\Phi}(V)$. Hence ${\mathbb{Z}}\Phi\subset L\subset\Lambda_{\Phi}(V)$.
Suppose ${\mathbb{Z}}\Phi\subset L\subset\Lambda_{\Phi}(V)$. Then $\Phi\subset L$ and strong integrality holds. On the other hand, since $\Phi$ is a root system in $V$, it is stable under the reflections $\{s_{x}|\ x\in\Phi\}$. Hence $\Phi$ is a root system in $L$.
Recall that a root system $\Phi$ in a Euclidean space (or in a lattice) with an inner product $m$ is called [*simply laced*]{} if it is irreducible and $(\alpha,\alpha)_{m}=(\beta,\beta)_{m}$ for any $\alpha,\beta\in\Phi$. Similarly, an abstrct root system $\Phi$ is called simply laced if it is irreducible and its cusp product takes values in the set $\{0,1,-1\}$.
An embedding (resp. isomorphism) of root systems in lattices $$f: (L,\Phi,m)\longrightarrow (L',\Phi',m')$$ is a ${\mathbb{Z}}$-linear bijection $f: L\rightarrow L'$ which is an isometry with respect to $(m,m')$ and satisfying that $f(\Phi)\subset\Phi'$ (resp. $f(\Phi)=\Phi'$). If $L = L'$, $m=m'$ and $f$ is the identity, we simply say that $\Phi$ is a sub-root system of $\Phi'$. Similarly, we define embeddings and sub-root systems for root systems in Euclidean vector spaces.
We remark that in the literature, root systems in a Euclidean vector space are often required to be semisimple and/or reduced (e.g. [@Humphreys] and [@Knapp]). We require neither. And it is essential for us to include the inner product in our definition of root systems in a Euclidean vector space or in a lattice.
For simplicity, we abbreviate as root system for a root system in a Euclidean vector space, a root system in a lattice or an abstract root system simply when it is clear from context which is intended. A root system in a lattice $(L,\Phi,m)$ (or a root system in a Euclidean vector space $(V,\Phi,m)$) will be denoted simply by $\Phi$ in the case that the lattice $L$ and the inner product $m$ (or the Euclidean vector space $V$ and the inner product $m$) are clear from the context.
\[P:maximal root system\] Given a lattice $L$ with a pisitve definite inner product $m$ denoted by $(\cdot,\cdot)_{m}$, there exists a root system $\Psi_{L}$ in $L$ containing all other root systems in $L$. The root system $\Psi_{L}$ is uniquely determined by this characterization.
Define $$\Psi_{L}=\big\{0\neq\alpha\in L|\ \frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\in{\mathbb{Z}},
\forall\lambda\in\Lambda\big\}.$$ We show that $\Psi_{L}$ satisfies the desired conclusion.
First we show that $\Psi_{L}$ is a root system in $L$. Obviously $\Psi_{L}$ satisfies the condition $(2)$ in Definition \[D:root system in a lattice\]. For any $\alpha,\beta\in\Psi$ and any $\lambda\in L$, $$\begin{aligned}
&&\frac{2(\lambda,s_{\alpha}(\beta))_{m}}
{(s_{\alpha}(\beta),s_{\alpha}(\beta))_{m}}
\\&=&\frac{2(s_{\alpha}(\lambda),\beta)_{m}}{(\beta,\beta)_{m}}
\\&=&\frac{2(\lambda-\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\alpha,\beta)_{m}}{(\beta,\beta)_{m}}
\\&=&\frac{2(\lambda,\beta)_{m}}{(\beta,\beta)_{m}}-
\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\frac{2(\alpha,\beta)_{m}}{(\beta,\beta)_{m}}.\end{aligned}$$ Since $\alpha,\beta\in\Psi_{L}$ and $\alpha,\lambda\in L$, the number in the last line is an integer. Hence the vector $s_{\alpha}(\beta)$ is contained in $\Psi_{L}$. This proves the condition $(1)$. Therefore $\Psi_{L}$ is a root system in $L$.
By the condition strong integrality in Definition \[D:root system in a lattice\], $\Psi_{L}$ contains all other root systems in $L$. This indicates that $\Psi_{L}$ is unique with this characterization.
From Proposition \[P:maximal root system\] and its proof, we see that the condition strong integrality really matters. In the following proposition, we give some examples of root systems in lattices. These root systems will be used in the proof of Proposition \[P:construction-simple\].
\[P:root system\] Let $G$ be a connected compact Lie group with a biinvariant Riemannian metric $m$, $T$ a connected closed torus in $G$, $L=\Hom(T,\U(1))$ the weight lattice, and $m$ the induced positive definite inner product on $L$. Then the set $\Phi(G,T)$ of non-zero $T$-weights in ${\mathfrak{g}}={\mathfrak{g}}_0\otimes_{{\mathbb{R}}}{\mathbb{C}}$ is a root system in $L$ in the case that
1. $T$ is a maximal torus of $G$,
2. there exists an involutive automorphism $\theta$ of ${\mathfrak{g}}_0$ and $\Lie T$ is a maximal abelian subspace of ${\mathfrak{g}}_0^{-\theta}=\{X\in{\mathfrak{g}}_0|\theta(X)=-X\}$, or
3. ${\mathfrak{g}}_0=\mathfrak{so(8)}$ and $\Lie T$ is conjugate to a Cartan subalgebra of a (unique up to conjugacy) subalgebra of ${\mathfrak{g}}_0$ isomorphic to ${\mathfrak{g}}_2$.
We prove $(1)$ first. It is well-known that $\Phi(G,T)$ is a root system in $L\otimes_{{\mathbb{Z}}}{\mathbb{R}}=
\Hom_{{\mathbb{R}}}(\Lie T,i{\mathbb{R}})$. Thus it satisfies the condition $(1)$ in Definition \[D:root system in a lattice\]. We need to show that it satisfies the condition $(2)$. Write $r=\dim T=\rank G$. For each $\alpha\in\Phi(G,T)$, there exists a homomorphism (cf. [@Knapp], Pages 143-149) $$f_{\alpha}:\SU(2)\times\U(1)^{r-1}\longrightarrow G$$ such that $f_{\alpha}(\U(1)^{r-1})=(\ker\alpha)_0$, $f_{\alpha}(T'\times \U(1)^{r-1})=T$, and the complexified Lie algebra of $f_{\alpha}(\SU(2))$ is the $\mathfrak{sl}_2$ subalgebra corresponding to the root $\alpha$. Here $T'$ is the subgroup of $\SU(2)$ of diagonal matrices. We have $T'\cong\U(1)$ and this gives a linear character $\alpha_0$ of $T'$. We can normalize $\alpha_0$ such that $\alpha\circ (f_{\alpha}|_{T'})$ equals $2\alpha_0$ . Let $$W=N_{G}(T)/C_{G}(T),$$ $$n_{\alpha}=f_{\alpha}(\left(\begin{array}{cc}0&1\\-1&0\\\end{array}\right))$$ and $$s_{\alpha}=\Ad(n_{\alpha})\in W.$$ For any $\lambda\in L$, let $k\alpha_0=\lambda\circ(f_{\alpha}|_{T'})$. Then $k\in{\mathbb{Z}}$. We prove that $(2\lambda-k\alpha,\alpha)_{m}=0$. This is equivalent to $s_{\alpha}(2\lambda-k\alpha)=2\lambda-k\alpha$, and also equivalent to $$\label{Eq:H} (2\lambda-k\alpha)(s_{\alpha}(H))=(2\lambda-k\alpha)(H)$$ for any $H\in\Lie T$. Write $H$ as $H=H_1+H_2$ where $H_1\in\Lie (f_{\alpha}(T'))$ and $H_2\in\Lie(\ker\alpha)$. Then we have $s_{\alpha}(H_1)=-H_1$ and $s_{\alpha}(H_2)=H_2$. By this Equation (\[Eq:H\]) is equivalent to $(2\lambda-k\alpha)(H_1)=0$. This follows from $k\alpha_0=\lambda\circ(f_{\alpha}|_{T'})$ and $\alpha\circ(f_{\alpha}|_{T'})=2\alpha_0$. Thus $$\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}
=\frac{2(\frac{k}{2}\alpha,\alpha)_{m}}{(\alpha,\alpha)_{m}}=k\in{\mathbb{Z}}.$$ Hence $\Phi(G,T)$ satisfies the condition $(2)$ in Definition \[D:root system in a lattice\].
The proof for $(2)$ is similar as the proof for $(1)$. In [@Knapp], Pages 379-380, it is proven that $\Phi(G,T)$ is a root system in $L\otimes_{{\mathbb{Z}}}{\mathbb{R}}=\Hom_{{\mathbb{R}}}(\Lie T,i{\mathbb{R}})$. Thus it satisfies the condition $(1)$ in Definition \[D:root system in a lattice\]. Similar as in the above proof, there is an reflection $s_{\alpha}\in N_{G}(T)/C_{G}(T)$. Using it we are able to show that $\Phi(G,T)$ satisfies the condition $(2)$ in Definition \[D:root system in a lattice\].
For $(3)$, let $H$ denote a closed subgroup of $H$ isomorphic to $\G_2$ and with $T$ a maximal torus of $H$. Then we have $\Phi(G,T)=\Phi(H,T)$. In this way $(3)$ follows from $(1)$.
In the thory of integral lattice, there is a notion of root system of a lattice, which is different with our discussion here.
\[R:lattice-root system\] Given a lattice $L$ with a positive definite inner product $m$ denoted by $(\cdot,\cdot)$, it is called an integral lattice if $(\lambda_1,\lambda_2)\in{\mathbb{Z}}$ for any $\lambda_1,\lambda_2\in L$. In this case the set $X=\{\lambda\in L|\ (\lambda,\lambda)=1\textrm{ or }2\}$ is called the root system of $L$. In general $X$ is a proper subset of $\Psi_{L}$.
\[R:RD-AR\] Root systems (in Euclidean vector spaces) classify compact connected Lie groups up to isogeny, however root data classify them up to isomorphism. Hence the difference between root datum and root system in a Euclidean vector space is clear. In another direction, we think that root datum and root system in a lattice should be equivalent objects. That is to say, there should exist a construction from a root datum to a root system in a lattice, and vice versa.
The Larsen-Pink method {#S:characters}
======================
In this section, given a compact Lie group $G$ with a biinvariant Riemannian metric $m$ and a closed connected torus $T$ in $G$, we define root systems on $T$ and get two specific root systems $\Psi_{T}$ and $\Psi'_{T}$. Moreover, we define a character $F_{\Phi,W}$ for each root system $\Phi$ on $T$ and a finite group $W$ acting on the Lie algebra of $T$ and satisfying some condition (cf. Definition \[D:characters\]). We study properties of the character $F_{\Phi,W}$ and their relation with dimension data. In this way, we reduce the dimension datum problem and the linear dependence problem to comparing these characters and getting linear relations among them.
Let $G$ be a compact Lie group with a biinvariant Riemannian metric $m$. For a closed connected torus $T$ contained in $G$, let $$\Lambda_{T}=\Hom(T,U(1))$$ be the integral weight lattice of $T$ and $\Lambda_{{\mathbb{Q}},T}=\Lambda_{T}\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ the ${\mathbb{Q}}$-weight space of $T$. By restriction the Riemannian metric $m$ induces an inner product on the Lie algebra ${\mathfrak{t}}_0=\Lie T$ of $T$. Denote it by $m$ as well. Write $({\mathfrak{t}}_0)^{\ast}=\Hom_{{\mathbb{R}}}({\mathfrak{t}}_0,{\mathbb{R}})$ and $({\mathfrak{t}}_0)_{{\mathbb{C}}}^{\ast}=\Hom_{{\mathbb{R}}}({\mathfrak{t}}_0,{\mathbb{C}})$. Then $({\mathfrak{t}}_0)^{\ast}\subset({\mathfrak{t}}_0)_{{\mathbb{C}}}^{\ast}$ and $({\mathfrak{t}}_0)_{{\mathbb{C}}}^{\ast}=({\mathfrak{t}}_0)^{\ast}\oplus i({\mathfrak{t}}_0)^{\ast}$. Since $m$ is positive definite, in particular non-degenerate, the induced linear map $$p: {\mathfrak{t}}_0\longrightarrow({\mathfrak{t}}_0)^{\ast}=\Hom_{{\mathbb{R}}}({\mathfrak{t}}_0,{\mathbb{R}})$$ is a linear isomorphism. For any $\lambda,\mu\in({\mathfrak{t}}_0)^{\ast}$, let $$(\lambda,\mu)_{m}=-(p^{-1}\lambda,p^{-1}\mu)_{m}.$$ Then $(\cdot,\cdot)_{m}$ is a negative definite inner product on $({\mathfrak{t}}_0)^{\ast}$. For any $\lambda_1,\lambda_2,\mu_1,\mu_2\in({\mathfrak{t}}_0)^{\ast}$, let $$(\lambda_1+i\lambda_2,\mu_1+i\mu_2)_{m}=((\lambda_1,\mu_1)_{m}-(\lambda_2,\mu_2)_{m})+
i((\lambda_1,\mu_2)_{m}+(\lambda_2,\mu_1)_{m}).$$ Then $(\cdot,\cdot)_{m}$ is a non-degenerate symmetric bilinear form on $({\mathfrak{t}}_0)_{{\mathbb{C}}}^{\ast}$. It is positive definite on $i({\mathfrak{t}}_0)^{\ast}$ by restriction. Since $\Lambda\subset\Lambda_{{\mathbb{Q}}}\subset i{\mathfrak{t}}_0^{\ast}$, they inherit this inner product.
\[D:Gamma-Gamma0\] Let $\Gamma$ be the group of automorphisms of $T$ preserving the Riemannian metric $m$ on it, and let $$\Gamma^{\circ}=N_{G}(T)/C_{G}(T).$$
By this definiton we have $\Gamma^{\circ}\subset\Gamma$. The group $\Gamma$ has an equivalent definition as the group of automorphisms of the lattie $\Lambda_{T}$ preversing the inner product $m$ on $\Lambda_{T}$. Choose a maximal torus $S$ of $G$ containing $T$ and let $W_{G}=N_{G}(S)/C_{G}(S)$ be the Weyl group of $G$. Then $\Gamma^{\circ}$ has an equivalent definition as the image of $\Stab _{W_{G}}(T)$ in $\Aut(T)$. Given a non-zero element $\alpha\in\Lambda_{T}$, define $s_{\alpha}:i{\mathfrak{t}}_0^{\ast}\longrightarrow
i{\mathfrak{t}}_0^{\ast}$ by $$s_{\alpha}(\lambda)=\lambda-\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\alpha,\
\forall\lambda\in i{\mathfrak{t}}_0^{\ast}.$$ Then $s_{\alpha}$ is a reflection on $i{\mathfrak{t}}_0^{\ast}$.
\[D:subroot\] A subset $\Phi\subset\Lambda_{T}$ is called a root system on $T$ if for any $\alpha\in\Phi$ and $\lambda\in\Lambda_{T}$, $$s_{\alpha}(\Phi)=\Phi$$ and $$\frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}
\in{\mathbb{Z}}.$$ In other words, a subset $\Phi\subset\Lambda_{T}$ is a root system on $T$ if it is a root system in $\Lambda_{T}$.
\[D:maxroot\] Let $$\Psi_{T}=\big\{0\neq\alpha\in\Lambda_{T}|\ \frac{2(\lambda,\alpha)_{m}}{(\alpha,\alpha)_{m}}\in{\mathbb{Z}},\
\forall\lambda\in\Lambda_{T}\big\}.$$
By Proposition \[P:maximal root system\], $\Psi_{T}$ is the unique maximal root system in the lattice $\Lambda_{T}$. By the definition $\Psi_{T}$ is stable under the action of $\Gamma=\Aut(\Lambda_{T},m)$. Thus we have $$\Gamma=\Aut(\Psi_{T},m),$$ where $\Aut(\Psi_{T},m)$, the automorphism of the root system $\Psi_{T}$ in the lattice $\Lambda_{T}$, is defined in Definition \[D:Automorphism group\].
\[P:Psi\] For any closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$, the root system of $H$, $\Phi(H)=\Phi(H,T)$ is a root system on $T$. Moreover, it is a reduced root system. Defined as above, $\Psi_{T}$ is a root system on $T$ and it contains all root systems on $T$. In particular $\Phi(H,T)\subset\Psi_{T}$ for any closed subgroup $H$ of $G$ with $T$ a maximal torus of $H$.
The fact that subset $\Phi(H,T)$ is a root system on $T$ is proven in Proposition \[P:root system\]. The fact that the subset $\Psi_{T}$ is a root system on $T$ and contains all other root systems on $T$ is proved in Proposition \[P:maximal root system\].
In general the rank of $\Psi_{T}$ can be any integer between $0$ and $\dim T$. It happens that $\rank\Psi_{T}=\dim T$ if and only if there exists a root system on $T$ of rank equal to $\dim T$. In particular if there exists a semisimple closed subgroup $H$ of $G$ with $T$ a maximal torus of $H$, then $\rank\Psi_{T}$ is equal to $\dim T$. We choose and fix an ordering on $\Lambda_{T}$ so that we get a positive system $(\Psi_{T})^{+}$.
\[R:compare Psi\] In [@Larsen-Pink], Larsen-Pink defined a root system $\Psi$ from a dimension datum. In general this $\Psi$ is a root system on $T$ and is strictly contained in our $\Psi_{T}$.
The root system $\Psi_{T}$ depends on the torus $T$ and a biinvariant Riemannian metric $m|_{T}$. If we replace $G$ by a group isogenous of it, then the lattice $\Lambda_{T}$ becomes another lattice isogeneous to it. The root system $\Psi_{T}$ will change probably after this modification. On the other hand, $\Psi_{T}$ is also sensitive with the biinvariant Riemannian metric $m|_{T}$. In a different direction, we can define another root system $\Psi'_{T}$ which depends on $T$ and the the isogeny class of $G$. However, if we do not know $G$ and $T$ explicitly, it is hard to determine $\Psi'_{T}$.
\[D:Psi prime\] Define $\Psi'_{T}$ as the sub-root system of $\Psi_{T}$ generated by $\{\Phi(H,T)\}$, where $H$ runs through all closed connected subgroups $H$ of $G$ with $T$ a maximal torus of $H$.
It is clear that $\Psi'_{T}$ is stable under $\Gamma^{\circ}$. However it is not stable under $
W_{\Psi_{T}}$, $\Gamma$ or $\Aut(\Psi_{T})$ in general.
\[R:Psi-simple group\] With $\Psi_{T}$, we are able to consider all root systems $$\{\Phi(H,T)|\ T \textrm{ is a maximal
torus of } H\}$$ together by viewing them as sub-root systems of $\Psi_{T}$. In the case that $G$ is a connected simple Lie group and $T$ is a maximal torus of $G$, let $\Psi_0=\Phi(G,T)$ be the root system of $G$. Then $\Psi_0\subset\Psi_{T}$. If $\Psi_0$ is simply laced, then for any sub-root system $\Phi$ of $\Psi_0$ there exists a closed subgroup $H$ of $G$ with $T$ a maximal torus of $H$ such that $\Phi(H,T)=\Phi$. If $\Psi_0$ is not simply laced, then not every sub-root system $\Phi$ of $\Psi_0$ is of the form $\Phi(H,T)=\Phi$ where $H$ is a closed subgroup of $G$ with $T$ a maximal torus of $H$. In the case that $\Psi_0$ is of type $\A_{n}$ ($n\geq 4$), $\E_6$, $\E_7$, $\E_8$, $\F_4$ or $\G_2$, it must be that $\Psi_{T}=\Psi_0$. However, if $\Psi_0$ is of type $\A_2$, $\B_n$, $\C_{n}$ or $\D_{n}$ ($n\geq 3$), then $\Psi_{T}$ may be strictly larger than $\Psi_0$. Precisely to say, in the case that $\Psi_0=\A_{2}$, it is possible that $\Psi_{T}=\G_{2}$. In the case that $\Psi_0=\B_{n}$ or $\C_{n}$, it is possible that $\Psi_{T}=\BC_{n}$. In the case that $\Psi_0=\D_{n}$, it is possible that $\Psi_{T}=\B_{n}$, $\C_{n}$ or $\BC_{n}$.
\[L:reflection group\] Given a root system $\Psi$ and its Weyl group $W_{\Psi}$, let $X$ be a non-empty subset of $\Psi$ and $W$ be the subgroup of $W_{\Psi}$ generated by reflections corresponding to elements in $X$. If there exist no proper sub-root systems of $\Psi$ containing $X$, then $W=W_{\Psi}$.
Let $X'=\{\alpha\in\Psi|\ s_{\alpha}\in W\}$. For any two roots $\alpha,\beta$ contained in $X'$, since $$s_{-\alpha}=s_{\alpha}\in W$$ and $$s_{s_{\alpha}(\beta)}=s_{\alpha}s_{\beta}s_{\alpha}\in W,$$ we get $-\alpha\in X'$ and $s_{\alpha}(\beta)\in X'$. That means, $X'$ is a sub-root system of $\Psi$. On the other hand, we have $X\subset X'$. By the condition that there exist no proper sub-root systems of $\Psi$ containing $X$, we get $X'=\Psi$. Therefore $W=W_{\Psi}$.
\[P:Gamma0-Psi’\] Given a compact Lie group $G$ and a biinvariant Riemannian metric $m$ on $G$, for any clsoed connected torus $T$ in $G$, we have $W_{\Psi'_{T}}\subset\Gamma^{\circ}$
Recall that $\Psi'_{T}$ is the sub-root system of $\Psi_{T}$ generated by $\{\Phi(H,T)\subset\Psi_{T}\}$, where $H$ run over closed connected subgroups of $G$ with $T$ a maximal torus of $H$. Let $X$ be the subset of $\Psi'_{T}$ defined as: $\alpha\in X$ if and only if there exists a closed connected subgroup $H$ of $G$ with $T$ a maiximal tours of $H$ and the root system of $H$ being $\{\pm{\alpha}\}$. Let $W$ be the subgroup of $W_{\Psi'_{T}}$ generated by reflections corresponding to elements in $X$.
We show that $W\subset\Gamma^{\circ}$ and there exist no proper sub-root systems of $\Psi'_{T}$ containing $X$. For any $\alpha\in X$, there exists a closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and the root system of $H$ being $\{\pm{\alpha}\}$. Thus there exists a finite surjection $$p:\SU(2)\times\U(1)^{r-1}\longrightarrow H.$$ Let $$n_{\alpha}=p(\left(\begin{array}{cc}0&1\\-1&0\\\end{array}\right),1),$$ where $\left(\begin{array}{cc}0&1\\-1&0\\\end{array}\right)\in\SU(2)$ and $1\in\U(1)^{r-1}$. Then $n_{\alpha}\in N_{G}(T)$ and $\Ad(n_{\alpha})|_{T}=s_{\alpha}$. Hence $s_{\alpha}\in\Gamma^{\circ}$. Therefore $W\subset\Gamma^{\circ}$. Suppose there exists a proper sub-root system $\Psi'$ of $\Psi'_{T}$ containing $X$. For any closed connected subgroup $H$ of $G$ with $T$ a maximla torus of $H$, one has that each root of $\Phi(H,T)$ is contained in $X$. Hence $\Phi(H,T)\subset X\subset\Psi'$. This contradicts to the condition that the root systems $\Phi(H,T)$ generate $\Psi'_{T}$.
By the above two facts and Lemma \[L:reflection group\] we get $W_{\Psi'_{T}}=W\subset\Gamma^{\circ}$.
\[C:Psi’-A1\] For any root $\alpha$ in $\Psi'_{T}$, there exists a closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and having root system equal to $\{\pm{\alpha}\}$. The abstract root system $\Psi'_{T}$ depends only on the group $G$ and the connected torus $T$, not on the biinvariatn Riemannian metric $m$.
Conisdering the subset $X$ defined in the above proof for Proposition \[P:Gamma0-Psi’\], it is obviously $\Gamma^{\circ}$ invariant. Since $\Gamma^{\circ}\supset W_{\Psi'_{T}}$ as the above proof showed, $X$ is a union of some $W_{\Psi'_{T}}$ on $\Psi'_{T}$. Thus $X$ is a sub-root system of $\Psi'_{T}$. On the other hand, the above proof showed that $X$ generates $\Psi'_{T}$. Therefore $X=\Psi'_{T}$. This is just the first statement in the conclusion of the corollary. Moreover, by Proposition \[P:cusp product\] the cusp product on $\Psi'_{T}$ is determined. Therefore, the abstract root system $\Psi'_{T}$ is determined by $G$ and $T$, indpendent with the biinvariant Riemannian metric $m$.
\[R:Psi’-Phi\] Suppose the abstract root system $\Psi'_{T}$ is given, an interesting question is to determine the sub-root systems $\Phi$ of $\Psi'_{T}$ so that there exists a closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and having root system equal to $\Phi$. From the above corollary, we know that each rank-$1$ sub-root system of $\Psi'_{T}$ is the root system of a closed connected subgroup. For sub-root systems of higher rank, it is unclear to the author which sub-root systems of $\Psi'_{T}$ are root systems of closed connected subgroups. Moreover, is it possible that any sub-root system of $\Psi'_{T}$ of rank larger than one is not the root system of a closed connected subgroup?
Given a [*reduced root system*]{} $\Phi$ on $T$, let $$\delta_{\Phi}=\frac{1}{2}\sum_{\alpha\in\Phi^{+}}
\alpha.$$ Let $$\delta'_{\Phi}$$ the unique dominant weight with respect to $(\Psi_{T})^{+}$ in the orbit $W_{\Psi_{T}}\delta_{\Phi}=\{\gamma\delta_{\Phi}:\gamma\in W_{\Psi_{T}}\}$. Here $W_{\Psi_{T}}$ is the Weyl group of the root system $\Psi_{T}$ and $\Phi^{+}=\Phi\cap(\Psi_{T})^{+}$.
\[D:characters\] Define a character on $T$, $$A_{\Phi}=\sum_{w\in W_{\Phi}}\sign(w)[\delta_{\Phi}-w\delta_{\Phi}].$$
Moreover, for any finite group $W$ between $W_{\Phi}$ and $\Aut(\Lambda_{{\mathbb{Q}},T},m)$, define another character on $T$ $$F_{\Phi,W}=\frac{1}{|W|}\sum_{\gamma\in W}\gamma(A_{\Phi}).$$
\[D:chi-lambda\] Given a finite subgroup $W$ of $\Aut(\Lambda_{{\mathbb{Q}},T},m)$ and an integral weight $\lambda\in\Lambda_{T}$, let $$\chi^{\ast}_{\lambda,W}=\frac{1}{|W|}\sum_{\gamma\in W}[\gamma\lambda]\in{\mathbb{Q}}[\Lambda_{{\mathbb{Q}},T}].$$
The character $\chi^{\ast}_{\lambda,W}$ depends only on the orbit $W\lambda=\{\gamma\lambda:\gamma\in W\}$. If $W\subset\Aut(\Lambda_{T},m)$, then $\chi^{\ast}_{\lambda,W}\in{\mathbb{Q}}[\Lambda_{T}]$ for any $\lambda\in\Lambda_{T}$. Moreover, the set $\big\{\chi^{\ast}_{\lambda,W}|\lambda\in\Lambda_{T}^{1}\big\}$ is a basis of ${\mathbb{Q}}[\Lambda_{T}]^{W}$, where $\Lambda_{T}^{1}$ is a set of representatives of $W$ orbits in $\Lambda_{T}$. In the case that $\rank\Psi_{T}=\dim T$ and $W=W_{\Psi_{T}}$, we can choose $\Lambda_{T}^{1}=\Lambda_{T}^{+}$. Here, $\Lambda_{T}^{+}$ is the set of dominant integral weights in $\Lambda_{T}$ with respect to $(\Psi_{T})^{+}$.
The characters $F_{\Phi,W}$ have the following property.
\[P:characters\] Each character $F_{\Phi,W}$ is a linear combination of $\{\chi^{\ast}_{\lambda,W}|\ \lambda\in
\Lambda_{\mathbb{Q},T}\}$ with integer coefficients and the constant term of $F_{\Phi,W}$ is $1$. With $\{\chi^{\ast}_{\lambda,W}|\ \lambda\in\Lambda_{\mathbb{Q},T}\}$ as a basis, the minimal length terms of $F_{\Phi,W}-1$ are of the form $\{c_{\alpha}\cdot\chi^{\ast}_{\alpha,W}|\ c_{\alpha}\in{\mathbb{Z}},\alpha
\in\Phi^{\circ}\}$ and the unique longest term of $F_{\Phi,W}$ is of the form $c\cdot\chi^{\ast}_{2\delta_{\Phi},W}$ where $c=\pm{1}$.
Since $W$ contains $W_{\Phi}$, $$\begin{aligned}
&&F_{\Phi,W}\\&=&\frac{1}{|W|}\sum_{\gamma\in W}
\gamma(A_{\Phi})\\&=&\frac{1}{|W|}\sum_{\gamma\in W}\sum_{w\in W_{\Phi}}\sign(w)(\gamma[\delta_{\Phi}-
w\delta_{\Phi}])\\&=&\sum_{w\in W_{\Phi}}\sign(w)\chi^{\ast}_{\delta_{\Phi}-w\delta_{\Phi},W}.\end{aligned}$$
One has $|\delta_{\Phi}-w\delta_{\Phi}|^{2}=\langle 2\delta_{\Phi},\delta_{\Phi}-w\delta_{\Phi}\rangle$ and $$\delta_{\Phi}-w\delta_{\Phi}=\frac{1}{2}\sum_{\alpha\in\Phi^{+}}\alpha-\frac{1}{2}\sum_{\alpha\in
\Phi^{+}}w\alpha=\sum_{\alpha\in\Phi^{+}\cap w^{-1}\Phi^{-}}\alpha.$$ Then for $w\neq 1$, $\delta_{\Phi}-w\delta_{\Phi}$ is of shortest length exactly when $w=s_{\alpha}$ for $\alpha$, a short root of minimal length; and it is of longest length exactly when $w\Phi^{+}=\Phi^{-}$, i.e., $w=w_0$ is the unique longest element in $W$.
If $W_{\Psi_{T}}\subset W$, then $\chi^{\ast}_{2\delta_{\Phi},W}=\chi^{\ast}_{2\delta'_{\Phi},W}$. The weight $2\delta'_{\Phi}$ is defined in [@Larsen-Pink] and it is observed there that $\chi^{\ast}_{2\delta'_{\Phi},W}$ is the unique leading term of $F_{\Phi,W}$ if $W_{\Psi_{T}}\subset W$.
The following two propositions connect Sato-Tate measures $\st_{H}$ and the characters $\{F_{\Phi,\Gamma^{\circ}}\}$. The importance of them is: we are able to tackle dimension data by studying the algebraic objects $F_{\Phi,\Gamma^{\circ}}$. Proposition \[P:group-root system\] reduces the dimension datum problem and the linear dependence problem to comparing the characters $\{F_{\Phi,\Gamma^{\circ}}|\ \Phi\subset\Psi_{T}\}$ and getting linear relations among them. After this, combinatorial classification of reduced sub-root systems and algebraic method treating these characters come into force.
The proof of the following proposition can be found in [@Larsen-Pink], Section 1 or [@An-Yu-Yu], Section 4. For completeness, we give a sketch of the proof.
\[P: ST support\] For a closed subgroup $H$ of $G$, the support of the Sato-Tate measure $\st_{H}$ is the set of conjugacy classes contained in the set $\{gxg^{-1}|\ g\in G, x\in H\}$. If $H$ is connected, then for a maximal torus $T$ of $H$ contained in a maximal torus $S$ of $G$, oen has $$\big(\bigcup_{w=nS\in W_{G}}nTn^{-1}\big)/W_{G}\subset S/W_{G}
\subset G^{\natural},$$ $S/W_{G}$ being a connected component of $G^{\natural}$ and $\big(\bigcup_{w=nS\in W_{G}}nTn^{-1}\big)/W_{G}$ being equal to the support of $\st_{H}$. Again if $H$ is connected, then for the natural map $\pi: T\rightarrow \supp(\st_{H})$, oen has $$\pi^{\ast}(\st_{H})=F_{\Phi,\Gamma^{\circ}}(t),$$ where $\Phi=\Phi(H)$.
The first and the second statements are clear.
Given $H$ being connected, for a conjugation invariant continuous function $f$ on $G$, one has $$\int_{H} fd\mu_{H}=\int_{H} fd\mu_{H}.$$ By the Weyl integration formula, $$\int_{H} fd\mu_{H}=\frac{1}{|W_{H}|}\int_{T}f(t)F_{\Phi}(t)dt,$$ where $F_{\Phi}(t)$ is the Weyl product. Writing this expression $\Gamma^{\circ}$ invariant, we get $$\begin{aligned}
&&\frac{1}{|W_{H}|}\int_{T}f(t)F_{\Phi}(t)dt\\&=&
\frac{1}{|W_{H}|}\frac{1}{|Stab_{W_{G}}(T)|}\sum_{\gamma=nS\in W_{G},nTn^{-1}=T}
\int_{T}f(n^{-1}tn)F_{\Phi}(n^{-1}tn)dt\\&=&
\frac{1}{|W_{H}|}\frac{1}{|Stab_{W_{G}}(T)|}\sum_{\gamma=nS\in W_{G}, nTn^{-1}=T}
\int_{T}f(t)F_{\Phi}(n^{-1}tn)dt\\&=&\frac{1}{|W_{H}|}\frac{1}{|\Gamma^{\circ}|}
\int_{T}f(t)\big(\sum_{\gamma\in\Gamma^{\circ}}F_{\Phi}(\gamma^{-1}(t))\big)dt.\end{aligned}$$
Moreover, we have $$\begin{aligned}
&&F_{\Phi}(t)\\&=&\prod_{\alpha\in\Phi}\big(1-[\alpha]\big)
\\&=&\prod_{\alpha\in\Phi^{+}}\big([\frac{-\alpha}{2}]-[\frac{\alpha}{2}]\big)\big([\frac{\alpha}{2}]-
[\frac{-\alpha}{2}]\big)\\&=&\big(\sum_{w\in W_{\Phi}}\sgn(w)[-w\delta]\big)\big(\sum_{w\in W_{\Phi}}
\sgn(w)[w\delta]\big)\\&=&\sum_{w,\tau\in W}\sgn(w)\sgn(\tau)[-w\delta+\tau\delta]\\&=&
\sum_{w,\tau\in W_{\Phi}}\sgn(w)[-\tau w\delta+\tau\delta]\quad (\textrm{use }
w\rightarrow\tau w)\\&=&\sum_{\tau\in W_{\Phi}}\tau\big(\sum_{w\in W_{\Phi}}\sgn(w)
[\delta-w\delta]\big)\end{aligned}$$ and $$\begin{aligned}
&&\frac{1}{|W_{H}|}\frac{1}{|\Gamma^{\circ}|}
\sum_{\gamma\in\Gamma^{\circ}}F_{\Phi}(\gamma^{-1}(t))\\&=&\frac{1}{|W_{H}|}
\frac{1}{|\Gamma^{\circ}|}\sum_{\gamma\in\Gamma^{\circ}}\gamma(F_{\Phi}(t))
\\&=&\frac{1}{|W_{H}|}\frac{1}{|\Gamma^{\circ}|}\sum_{\gamma\in\Gamma^{\circ}}
\gamma\big(\sum_{\tau\in W_{\Phi}}\tau(\sum_{w\in W_{\Phi}}\sgn(w)[\delta-w\delta])\big)
\\&=&\frac{1}{|\Gamma^{\circ}|}\sum_{\gamma\in\Gamma^{\circ}}\gamma\big(\sum_{w\in
W_{\Phi}}\sgn(w)[\delta-w\delta]\big)\\&=& F_{\Phi,\Gamma^{\circ}}(t).\end{aligned}$$ Hence the last statement follows.
The proof of the following proposition is a bit complicated. The basic idea is as follows. Given a maximal torus $S$ of a connected compact Lie group $G$ and $W_{G}=N_{G}(S)/C_{G}(S)$, then the set $G^{\natural}$ of conjugacy classes in $G$ can be identified with $S/W_{G}$. For any connected closed subgroup $H$ with a maximal torus $T$ contained in $S$, the support of the Sato-Tate measure $\st_{H}$ of $H$ is $p(T)$ where $p: S\longrightarrow S/W_{G}$ be the natural projection. This indicates that: not only $\supp(\st_{H})$ has dimension equal to $\dim T$, but also the integral of $\st_{H}$ against a continuous function on $S/W_{G}$ with a given bound and support near a given set of dimension less than $\dim T$ can be made arbitrarily small if the distance of the support of the function to that given set is suffiently small. With this fact, by choosing test functions on $S/W_{G}$ appropriately we are able to distinguish different Sato-Tate measures by their supports.
\[P:group-root system\] Given a list $\{H_{1},H_2,\dots,H_{s}\}$ of connected closed subgroups of a compact Lie group $G$ and non-zero real numbers $c_1,\cdots,c_{s}$, $$\sum_{1\leq i\leq s} c_{i}\mathscr{D}_{H_i}=0$$ if and only if for any closed connected torus $T$ in $G$, $$\sum_{1\leq j\leq t} c_{i_{j}}F_{\Phi_{i_{j}},\Gamma^{\circ}}=0,$$ where $\{H_{i_{j}}|i_1<i_2<\cdots<i_{t}\}$ are all subgroups among $\{H_{i}|1\leq i\leq s\}$ with maximal tori conjugate to $T$ in $G$, $\Phi_{i_{j}}$ is the root system of $H_{i_{j}}$ regarded as a root system on $T$ and $\Gamma^{\circ}=N_{G}(T)/C_{G}(T)$.
By the relation between dimension data and Sato-Tate measures, the equality $$]\sum_{1\leq i\leq s}c_{i}\mathscr{D}_{H_i}=0$$ is equivalent to $\sum_{1\leq i\leq s} c_{i}\st_{H_{i}}=0$.
Choosing a maximal torus $T_{i}$ of $G_{i}$, for any $1\leq i\leq s$, we may and do assume that $\{T_{i}: 1\leq i\leq s\}$ are all contained in a maximal torus $S$ of $G$. The set of conjugacy classes of $G$ contained in $G_0$, $G_0/\Ad(G)\subset G^{\natural}$ can be identified with $S/W_{G}$. Let $p: S\longrightarrow S/W_{G}$ be the natural projection. Under the above identification, the support of $\st_{H_{i}}$ is $p(T_{i})$. Among $\{T_{i}: 1\leq i\leq s\}$, we may assume that $T=T_1=T_2=\cdots=T_{t}$, and any other $T_{i}$ ($i\geq t+1$) is not conjugate to $T$ and has dimension $\geq\dim T$.
The Riemannian metric $m$ on $G$ induces a Riemannian metric on $S$. It gives $S$ the structure of a metric space. Since $W_{G}$ acts on $S$ by isometries, $S/W_{G}$ inherits a metric structure. For any $i\geq t+1$, since $T_{i}$ is not conjugate to $T$ and has dimension $\geq\dim T$, $gT_{i}g^{-1}\not\subset T$ for any $g\in G$. Moreover each $T_{i}$ is a closed torus, thus $p(T)\cap p(T_{i})\subset p(T_{i})$ is a union of the images under $p$ of finitely many closed tori of $S$ of strictly lower dimension than $\dim T_{i}$.
Given any $\epsilon>0$, let $\mathcal{U}_{\epsilon}$ be the subset of $S/W_{G}$ consisting of points with distance to $p(T)$ within $\epsilon$. For any continuous function $f$ on $S/W_{G}$ with absolute value bounded by a positive number $K/2$, there exists a continuous function $f_{\epsilon}$ on $S/W_{G}$ with support contained in $\mathcal{U}_{\epsilon}$ and absolute value bounded by $K$, and having restriction to $p(T)$ equal to $f|_{T}$. Since $\sum_{1\leq i\leq s} c_{i}\st_{H_{i}}=0$, $$\label{A}\sum_{1\leq i\leq s} c_{i}\st_{H_{i}}(f_{\epsilon})=0.$$ For any $i\geq t+1$, since $p(T)\cap p(T_{i})\subset p(T_{i})$ is a union of the images under $p$ of finitely many closed tori of $S$ of strictly lower dimension than $\dim T_{i}$ and $\st_{H_{i}}$ has support equal to $p(T_{i})$, we get $$\lim_{\epsilon\rightarrow 0}\st_{H_{i}}(f_{\epsilon})=0.$$ Taking the limit as $\epsilon$ approaches $0$ in Equation (\[A\]), we get $$\sum_{1\leq i\leq t} c_{i}\st_{H_{i}}(f)=0.$$ Hence $$\sum_{1\leq i\leq t} c_{i}\st_{H_i}=0.$$ Therefore an inductive argument on $s$ finishes the proof.
\[R:DDPLDP-character\] In Proposition \[P:group-root system\], if $G$ is a connected compact simple Lie group with a root system $\Psi$, then $\Gamma^{\circ}=W_{\Psi}$. Moreover, in this case the conjugacy class of a full rank subgroup connected closed subgroup is determined by its root system, which can be regarded as a sub-root system of $\Psi$. In this way, finding linear relations among dimension data of connected full rank subgroups is equivalent to finding linear relations among the characters $\{F_{\Phi,W_{\Psi}}|\ \Phi\subset\Psi\}.$
Comparison of different conjugacy conditions {#S:conjugacy}
============================================
In this section we discuss relations among several finite groups $\Gamma^{\circ}$, $\Gamma$, $W_{\Psi'_{T}}$, $W_{\Psi_{T}}$ and $\Aut(\Psi_{T})$ defined in previous sections. Moreover, we discuss the connection between various relations of two subgroups and conjugacy relations of their root-systems with respect to these finite groups.
Given a compact Lie group $G$ with a biinvariant Riemannian metric $m$ and a closed connected torus $T$, denote by $\Lambda_{T}=\Hom(T,\U(1))$. We have defined two finite groups $\Gamma^{\circ}$, $\Gamma$ by $$\Gamma^{\circ}=N_{G}(T)/C_{G}(T)$$ and $$\Gamma=\Aut(\Lambda_{T},m|_{\Lambda_{T}}).$$ Moreover, we get a root system $\Psi_{T}$ on $T$. Thus we have the finite groups $W_{\Psi_{T}}$ and $\Aut(\Psi_{T})$. By Definition \[D:maxroot\] the action of $W_{\Psi_{T}}$ stabilizes $\Lambda_{T}$ and preserves $m|_{\Lambda_{T}}$. Thus $W_{\Psi_{T}}$ can be regarded as a subgroup of $\Gamma=\Aut(\Lambda_{T},m|_{\Lambda_{T}})$. By the definition of $\Psi_{T}$ it is stable under the action of $\Gamma$. Thus there is a group homomorphism $$\pi:\Gamma\longrightarrow\Aut(\Psi_{T}).$$ In the case that $\rank\Psi_{T}=\dim T$, $\pi$ is an injective map. In general it is not injective, however its restriction to $W_{\Psi_{T}}$ is injective.
\[P:finite groups\] In general the inclusions $\Gamma^{\circ}\subset\Gamma$, $W_{\Psi_{T}}\subset\Gamma$ and $\pi(\Gamma)\subset\Aut(\Psi_{T})$ are proper inclusions and neither $\Gamma^{\circ}$ nor $W_{\Psi_{T}}$ contain the other one.
Given $n\geq 5$, denote by $G=\Aut(\mathfrak{su(n)})$ and $T$ a maximal torus of $G$. Then we have $\Psi_{T}=\A_{n-1}$, $\rank\Psi_{T}=\dim T$, $\Gamma^{\circ}=\Gamma=\Aut(\Psi_{T})=
S_{n}\times\{-1\}$ and $W_{\Psi_{T}}=S_{n}$. In this case $\Gamma\neq W_{\Psi_{T}}$ and $\Gamma^{\circ}\not\subset W_{\Psi_{T}}$.
In Example \[E:LP\], by choosing $W$ appropriately we have $\Gamma\neq\Gamma^{\circ}$ and $W_{\Psi_{T}}\not\subset\Gamma^{\circ}$.
Let $G=\big(\SU(2)^{3}\big)/\langle(-I,-I,I)\rangle$ and $T$ a maximal torus of $G$. In this case $\Psi_{T}=3\A_1$ and $\rank\Psi_{T}=\dim T$. Thus $\pi$ is injective. Moreover, we have $$\Gamma=\Aut(T,m|_{T})=\{\pm{1}\}^{3}\rtimes\langle\sigma_{12}\rangle$$ and $$\Aut(\Psi_{T})=\{\pm{1}\}^{3}\rtimes S_3,$$ where $\sigma_{12}$ is the transposition on the first and the second positions. Therefore $\pi(\Gamma)\neq\Aut(\Psi_{T})$.
\[D:local conjugacy\] Given two compact Lie groups $H$ and $G$, two homomorphisms $\phi_1,\phi_2: H\longrightarrow G$ are said element-conjugate if $\phi_1(x)\sim\phi_2(x)$ for any $x\in H$. Similarly, we call two closed subgroups $H_1,H_2$ of $G$ element-conjugate if there exists an isomorphism $\phi:H_1\longrightarrow H_2$ such that $x\sim\phi(x)$ for any $x\in H$.
Element-conjugate homomorphisms are defined and studied in [@Larsen2]. It is proved in [@Larsen2] and [@Larsen3] that for $G$ equal to $\SU(n)$, $\SO(2n+1)$, $\Sp(n)$ or $\G_2$, element-conjugate homomorphisms to $G$ are actually conjugate. In the converse direction, for $G$ equal to $\SO(2n)$ ($n\geq 4$) or a connected simple group of type $\E_6$, $\E_7$, $\E_8$ or $\F_4$, there exist element-conjugate homomorphisms to $G$ which are not conjugate. In [@Wang], S. Wang considered homomorphisms from connected groups and used the name locally conjugate instead of element-conjugate. He gave some examples of element-conjugate homomorphisms from connected groups to $\SO(2n)$ which are not conjuagte. In the 1950s Dynkin classified semisimple subalgebras of complex simple Lie algebras up to [*linear conjugacy*]{} (cf. [@Dynkin]). Moreover, Minchenko distinguished the conjugacy classes among linear conjugacy classes of semisimple subalgebras (cf. [@Minchenko]). By [@Minchenko] Theorem 1 and Proposition \[P:local-root system\] below, two connected closed subgroups are element-conjugate if and only if their subalgebras are linear conjugate. From this, connected closed subgroups of connected compact simple Lie group which are locally conjugate but not globally conjugate can be classified as well.
\[P:local-root system\] Given two compact Lie groups $H$, $G$ with $H$ connected, and a maximal torus $T$ of $H$, two homomorphisms $\phi_1,\phi_2: H\longrightarrow G$ are locally conjugate if and only if there exists $g\in G$ such that $\phi_2|_{T}=(\Ad(g)\circ\phi_1)|_{T}$ and the root systems $\Phi(H_1,\phi_1(T))=\Phi(H_2,\phi_1(T))$, where we denote by $H_{1}=(\Ad(g)\circ\phi_1)(H)$ and $H_{2}=\phi_{2}(H)$.
The “if” part is clear. We prove the “only if” part. Choosing a maximal torus $S$ of $G$, write $W(G,S)=N_{G}(S)/C_{G}(S)$. Since $\phi_1,\phi_2$ are element-conjugate, we have $\ker\phi_1=\ker\phi_2$. Without loss of generality we may assume that $\phi_1,\phi_2$ are injective and $\phi_1(T),\phi_2(T)\subset S$.
For any $x\in T$, since $\phi_1(x)\sim\phi_2(x)$, we have $\phi_2=n(\phi_1(x))n^{-1}$ for some $w=nS\in W(G,S)$. As $W(G,S)$ is finite, there exists $w\in W(G,S)$ such that $\phi_2=n(\phi_1(x))n^{-1}$ holds for a set of $x\in T$ generating $T$. Hence $\phi_2=n(\phi_1(x))n^{-1}$ for any $x\in T$. This means that $(\Ad(n)\circ\phi_1)|_{T}=\phi_2|_{T}$. For simplicity, we assume that $\phi_1|_{T}=\phi_2|_{T}$. We still denote by $T$ the image in $S$ of $T$ under $\phi_1$ and $\phi_2$. In this way $\phi_1|_{T}$ and $\phi_2|_{T}$ are both the identity map. Write $${\mathfrak{h}}={\mathfrak{t}}\oplus(\bigoplus_{\alpha\in\Phi}{\mathfrak{h}}_{\alpha})$$ for the root space decomposition of ${\mathfrak{h}}=(\Lie H)\otimes_{{\mathbb{R}}}{\mathbb{C}}$ with respect to the $T$ action and $${\mathfrak{g}}={\mathfrak{g}}^{T}\oplus(\bigoplus_{\lambda\in\Lambda_{T}}{\mathfrak{g}}_{\lambda})$$ the generalized root space decomposition of ${\mathfrak{g}}=(\Lie G)\otimes_{{\mathbb{R}}}{\mathbb{C}}$ with respect to the $T$ action, where $\Phi=\Phi(H,T)$ is root system of $H$ and $\Lambda_{T}=\Hom(T,\U(1))$ is the integral weight lattice. As $\phi_1|_{T}$ and $\phi_2|_{T}$ are both the identity map, $$\phi_1({\mathfrak{h}}_{\alpha}),\phi_2({\mathfrak{h}}_{\alpha})\subset{\mathfrak{g}}_{\alpha}$$ for any $\alpha\in\Phi$. That just means $\Phi(H_1,\phi_1(T))=\Phi(H_2,\phi_1(T))$.
Given a compact Lie group $G$ with a biinvariant Riemannian metric $m$ and a connected closed torus $T$, denote by $H_1$, $H_2$ two connected closed subgroups of $G$, we could consider different conjugacy relations between their root systems. Precisely, write $\Phi_{i}=\Phi(H_{i},T)$ for the root system of $H_{i}$, $i=1,2$. We may consider if $\Phi_1,\Phi_2\subset\Psi_{T}\subset\Lambda_{T}$ are conjugate under $\Gamma^{\circ}$, $\Gamma$, $W_{\Psi}$ or $\Aut(\Psi_{T})$. Given two root systems $\Phi_1,\Phi_2\subset\Psi_{T}$, since $\Gamma^{\circ}\subset\Gamma$ and $\pi(\Gamma)\subset\Aut(\Psi_{T})$, we have $$\Phi_1\sim_{\Gamma^{\circ}}\Phi_2\Longrightarrow\Phi_1\sim_{\Gamma}\Phi_2\Longrightarrow\Phi_1
\sim_{\Aut(\Psi_{T})}\Phi_2.$$
We have the following connections between the conjugacy relations between the root systems and the relations between the subgroups.
\[P:root system-group\] Given a compact Lie group $G$ with a biinvariant Riemannian metric $m$ and a connected closed torus $T$, for two closed subgroups $H_1,H_2$ with $T$ a common maximal torus of them, denote by $\Phi_{i}=
\Phi(H_{i},T)$, $i=1,2$. We have:
- [if $\Phi_1\sim_{\Aut(\Psi_{T})}\Phi_2$, then the Lie algebras of $H_1$ and $H_2$ are isomorphic.]{}
- [If $\Phi_1\sim_{\Gamma}\Phi_2$, then the Lie groups $H_1$ and $H_2$ are isomorphic.]{}
- [$\Phi_1\sim_{\Gamma^{\circ}}\Phi_2$ if and only if $H_1$ and $H_2$ are locally conjugate.]{}
Write $${\mathfrak{g}}={\mathfrak{g}}^{T}\oplus(\bigoplus_{\lambda\in\Lambda_{T}}{\mathfrak{g}}_{\lambda})$$ for the generalized root space decomposition of ${\mathfrak{g}}=(\Lie G)\otimes_{{\mathbb{R}}}{\mathbb{C}}$ with respect to the $T$ action and $${\mathfrak{h}}_{i}={\mathfrak{t}}\oplus(\bigoplus_{\alpha\in\Phi_{i}}{\mathfrak{h}}_{i,\alpha})$$ the root space decomposition of ${\mathfrak{h}}_{i}=(\Lie H_{i})\otimes_{{\mathbb{R}}}{\mathbb{C}}$ with respect to the $T$ action, $i=1,2$. We have $\Phi_1,\Phi_2\subset\Psi_{T}\subset\Lambda_{T}$.
Suppse $\Phi_1\sim_{\Aut(\Psi_{T})}\Phi_2$. This means that there exists $\gamma\in\Aut(\Psi_{T})$ such that $\Phi_2=\gamma\Phi_1$. By the classification of complex reductive Lie algebras (cf. [@Knapp]), $\gamma$ can be lifted to an isomorphism $f:{\mathfrak{h}}_1\longrightarrow{\mathfrak{h}}_2$ such that $f({\mathfrak{h}}_{1,\alpha})={\mathfrak{h}}_{2,\gamma\alpha}$ for any $\alpha\in\Phi_1$. This proves $(1)$.
Suppose $\Phi_1\sim_{\Gamma}\Phi_2$. Recall that $\Gamma=\Hom(T,m|_{T})$. For the automorphism $f$ in the above proof, furthermore we may assume that $f|_{\Lie T}$ is induced by an automorphism of $T$. Thus $f$ can be lifted to an isomorphism $f: H_1\longrightarrow H_2$. This proves $(2)$.
Recall that $\Gamma^{\circ}=N_{G}(T)/C_{G}(T)$. If $\Phi_1\sim_{\Gamma^{\circ}}\Phi_2$, then $\Phi_1=\Phi'_2$ for some $H'_2=gH_2g^{-1}$, $g\in N_{G}(T)$ and $\Phi'_2=\Phi(H'_2,T)$. Therefore $(3)$ follows from Proposition \[P:local-root system\].
\[R:local-maximal torus\] Given a compact Lie group $G$ and a connected closed torus $T$, as indicated in the proof of Proposition \[P:local-root system\], a trivial observation is: the existence of locally-conjugate but not globally conjugate connected closed subgroups with $T$ a maximal torus of them is due to the fact the generalized root spaces $\{{\mathfrak{g}}_{\lambda}:\lambda\in\Lambda_{T}\}$ may have dimension larger than one and the different choices of eigenvectors really matter for the conjugacy classes of the subgroups. Hence there is no local-global issue if each ${\mathfrak{g}}_{\alpha}$ having dimension one. In particular, two full rank connected subgroups are conjugate if and only if they are locally conjugate.
Formulation of the problems in terms of root systems {#S:formulation of questions}
====================================================
We formulate the following two questions, analoguous to the dimension datum problem and the linear dependence problem respectively.
\[Q:equal-character\] Given a root system $\Psi$, for which pairs $(\Phi_1,\Phi_2)$ of reduced sub-root systems of $\Psi$, we have $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$?
\[Q:dependent-character\] Given a root system $\Psi$ and a finite group $W$ such that $W_{\Psi}\subset W\subset\Aut(\Psi)$, for which reduced sub-root systems $\Phi_1,\Phi_2,\cdots,\Phi_{s}$ of $\Psi$, the characters $$F_{\Phi_1,W},F_{\Phi_2,W},\cdots,F_{\Phi_{s},W}$$ are linearly dependent?
\[R:linear\] We could take $(\Psi,W)=(\Psi_{T},\pi(\Gamma))$ or $(\Psi,W)=(\Psi'_{T},\pi(\Gamma^{\circ}))$ in Question \[Q:dependent-character\]. The property $W_{\Psi_{T}}\subset\pi(\Gamma)\subset\Aut(\Psi_{T})$ follows from the defintions of $\Gamma$ and of $\Psi_{T}$. The property $W_{\Psi'_{T}}\subset\pi(\Gamma^{\circ})\subset
\Aut(\Psi'_{T})$ is proved in Proposition \[P:Gamma0-Psi’\]. Taking the pair $(\Psi'_{T},\Gamma^{\circ})$, the solution to Question \[Q:dependent-character\] gives a solution to Question \[Q:linear dependence\].
Classification of sub-root systems {#S:sub-root systems}
==================================
In this section, given an irreducible root system $\Psi_0$, we discuss the classification of sub-root systems of $\Psi_0$ up to $W_{\Psi_0}$ conjugation. In the case that $\Psi_0$ is a classical irreducible root system, classification of sub-root systems of it seems to have been known to experts la ong time ago. A detailed exposition of this classification is given in [@Oshima]. In the case that $\Psi_0$ is an exceptional irreducible root system, classification of sub-root systems of it was first achieved by Oshima in [@Oshima] except that the classification of sub-root systems of $\F_4$ or $\G_2$ was known before. Our discussion follows [@Larsen-Pink] and [@Oshima].
Before discussing the classification, we fix notations. Let $V={\mathbb{R}}^{n}$ be an $n$-dimensional Euclidean vector space with an orthonormal basis $\{e_1,e_2,...,e_{n}\}$. Denote by the root systems $$\begin{aligned}
&&\A_{n-1}=\big\{\pm{(e_{i}-e_{j})}|\ 1\leq i<j\leq n\big\},\\&&
\B_{n}=\big\{\pm{e_{i}}\pm{e_{j}}|\ 1\leq i<j\leq n\}\cup\{\pm{e_{i}|1\leq i\leq n}\big\},\\&&
\C_{n}=\big\{\pm{e_{i}}\pm{e_{j}}|\ 1\leq i<j\leq n\big\}\bigcup\big\{\pm{2e_{i}|\ 1\leq i\leq n}\big\},\\&&
\BC_{n}=\big\{\pm{e_{i}}\pm{e_{j}}|\ 1\leq i<j\leq n\big\}\bigcup\big\{\pm{e_{i}},\pm{2e_{i}|\ 1\leq i\leq n}
\big\},\\&& \D_{n}=\big\{\pm{e_{i}}\pm{e_{j}}|\ 1\leq i<j\leq n\big\}. \end{aligned}$$ Thus $$\A_{n-1}\subset\D_{n}\subset\B_{n},\C_{n}\subset\BC_{n}.$$
Note that $\A_0=\D_1=\emptyset$ and we regard $\A_1,\B_1,\C_1$ different from each other though they are isomorphic as abstract root systems. Each of the following pairs $(\B_2,\C_2)$, $(\D_2, 2\A_1)$, $(\A_3,\D_3)$ are also regarded different.
[*Type $\A$.*]{} Given $\Psi_0=\A_{n-1}$ ($n\geq 2$), let $\Phi\subset\Psi_0$ be a sub-root system. Define a relation on $\{1,2,...,n\}$ by $$i\sim j\Leftrightarrow e_{i}-e_{j}\in\Phi.$$ Thus $\sim$ is an equivalence relation. It divides $\{1,2,...,n\}$ into $l$ subsets of cardinalities $n_1,n_2,...,n_{l}$ where $n_1\geq n_2\geq\cdots\geq n_{l}$ and $n_1+n_2+\cdots+n_{l}=n$. In this way we can show $$\Phi\sim\bigcup_{1\leq k\leq l}
\big\{\pm{(e_{i}-e_{j})}|\ n_1+n_2+\cdots+n_{k-1}+1\leq i<j\leq n_1+n_2+\cdots+n_{k}\big\}.$$ We denote by $$\Phi_{n_1,n_2,...,n_{l}}=\bigcup_{1\leq k\leq l}\big\{\pm{(e_{i}-e_{j})}|\ n_1+n_2+\cdots+n_{k-1}+
1\leq i<j\leq n_1+n_2+\cdots+n_{k}\big\}.$$ It is isomorphic to $\bigcup_{1\leq k\leq l}\A_{n_{k}-1}$.
[*Type $\B$.*]{} Given $\Psi_0=\B_{n}$ ($n\geq 1$), let $\Phi\subset\Psi_0$ be a sub-root system. Define a relation on $\{1,2,...,r\}$ by $$i\sim j\Leftrightarrow e_{i}-e_{j}\in\Phi\textrm{ or }e_{i}+e_{j}\in\Phi.$$ Thus $\sim$ is an equivalent relation. It divides $\{1,2,...,n\}$ into $l$ subsets of cardinarities $n_1,n_2,...,n_{l}$ where $n_1+n_2+\cdots+n_{l}=n$. Write $n'_{k}=n_1+n_2+\cdots+n_{k}$ for $1\leq k\leq l$. Without loss of generality we may assume that $\{i|\ n'_{k-1}+1\leq i\leq n'_{k}\}$ is an equivalence set for any $1\leq k\leq l$ and $$\big\{\pm{(e_{i}-e_{j})}|\ n'_{k-1}+1\leq i<j\leq n'_{k}\big\}\subset\Phi.$$ If $e_{i}\in\Phi$ for some $n'_{k-1}+1\leq i\leq n_{k}$, we have $$\B_{n_{k}}\cong\big\{\pm{e_{i}}\pm{e_{j}}|\ n'_{k-1}+1\leq i<j\leq n'_{k}\big\}\bigcup
\big\{\pm{e_{i}}|\ n'_{k-1}+1\leq i\leq n'_{k}\big\}\subset\Phi.$$ If $e_{i}\not\in\Phi$ for any $n'_{k-1}+1\leq i\leq n_{k}$ and $e_{i}+e_{j}\in\Phi$ for some $n'_{k-1}+1\leq i<j\leq n'_{k}$, we have $$\D_{n_{k}}\cong\big\{\pm{e_{i}}\pm{e_{j}}|\ n'_{k-1}+1\leq i<j\leq n'_{k}\big\}\subset\Phi.$$ In summary we have $$\Phi\cong\big(\bigcup_{1\leq i\leq u}\B_{r_{i}}\big)\bigcup\big(\bigcup_{1\leq j\leq v}
\D_{s_{j}}\big)\bigcup\big(\bigcup_{1\leq k\leq w}\A_{t_{k}-1}\big)$$ where $\sum_{1\leq i\leq u} r_{i}+\sum_{1\leq j\leq v} s_{j}+\sum_{1\leq k\leq w} t_{k}=n$, $r_{i}\geq 1$, $s_{j}\geq 2$ and $t_{k}\geq 1$. Moreover, the conjugacy class of $\Phi$ is uniquely determined by these indices $\{r_{i},s_{j},t_{k}\}$. We denote by $\Phi_{\{r_{i}\},\{s_{j}\},\{t_{k}\}}$ a sub-root system with the indices $\{r_{i},s_{j},t_{k}\}$.
[*Type $\C$.*]{} Given $\Psi_0=\C_{n}$, let $\Phi\subset\Psi_0$ be a sub-root system. Similar as in the $\B_{n}$ case, we can show $$\Phi\cong(\bigcup_{1\leq i\leq u}\C_{r_{i}})\bigcup(\bigcup_{1\leq j\leq v}
\D_{s_{j}})\bigcup(\bigcup_{1\leq k\leq w}\A_{t_{k}-1})$$ where $\sum_{1\leq i\leq u} r_{i}+\sum_{1\leq j\leq v} s_{j}+\sum_{1\leq k\leq w} t_{k}=n$, $r_{i}\geq 1$, $s_{j}\geq 2$ and $t_{k}\geq 1$. Moreover, the conjugacy class of $\Phi$ is uniquely determined by these indices $\{r_{i},s_{j},t_{k}\}$. We denote by $\Phi_{\{r_{i}\},\{s_{j}\},\{t_{k}\}}$ a sub-root system with the indices $\{r_{i},s_{j},t_{k}\}$.
[*Type $\BC$.*]{} Given $\Psi_0=\BC_{n}$, let $\Phi\subset\Psi_0$ be a sub-root system. Similar as in the $\B_{n}$ case, we can show $$\Phi\cong(\bigcup_{1\leq i\leq u}\BC_{r_{i}})\bigcup(\bigcup_{1\leq j\leq v}
\B_{s_{j}})\bigcup(\bigcup_{1\leq k\leq w}\C_{t_{k}})\bigcup(\bigcup_{1\leq e\leq x}\D_{p_{e}})\bigcup
(\bigcup_{1\leq f\leq y}\A_{q_{f}-1})$$ where $\sum_{1\leq i\leq u} r_{i}+\sum_{1\leq j\leq v} s_{j}+
\sum_{1\leq k\leq w} t_{k}+\sum_{1\leq e\leq x} p_{e}+\sum_{1\leq f\leq y}q_{f}=n$, $r_{i}\geq 1$, $s_{j}\geq 1$, $t_{k}\geq 1$, $p_{e}\geq 2$ and $q_{f}\geq 1$. Moreover, the conjugacy class of $\Phi$ is uniquely determined by the indices $\{r_{i},s_{j},t_{k},p_{e},q_{f}\}$. We denote by $\Phi_{\{r_{i}\},\{s_{j}\},\{t_{k}\},\{p_{e}\},\{q_{f}\}}$ a sub-root system with the indices $\{r_{i},s_{j},t_{k},p_{e},q_{f}\}$.
[*Type $\D$.*]{} Given $\Psi_0=\D_{n}$ ($n\geq 5$), in this case $\Aut(\Psi_0)=W_{n}=\{\pm{1}\}^{n}\rtimes S_{n}$ and $W_{\Psi_0}$ is a subgroup of index $2$ in $W_{n}$. Let $\Phi\subset\Psi_0$ be a sub-root system. Similar as in $\B_{n}$ case, we have show $$\Phi\cong(\bigcup_{1\leq j\leq v}\D_{s_{j}})\bigcup(\bigcup_{1\leq k\leq w}
\A_{t_{k}-1})$$ where $\sum_{1\leq j\leq v} s_{j}+\sum_{1\leq k\leq w} t_{k}=n$, $s_{j}\geq 2$ and $t_{k}\geq 1$. Moreover, the $\Aut(\Psi_0)$ conjugacy class of $\Phi$ is uniquely determined by the indices $\{s_{j},t_{k}\}$. Given a set of indices $\{s_{j},t_{k}:1\leq j\leq v,1\leq k\leq w\}$, there are at most two conjugacy classes of sub-root systems up to $W_{\Psi_0}$ conjugation. In the case that there is a unique $W_{\Psi_0}$ conjugacy class of sub-root systems with this set of indices, we denote by $\Phi_{\{s_{j}\},
\{t_{k}\}}$ one of them. Otherwise, we can show that each $s_{j}=0$ and each $t_{k}$ is even. Write $$\Phi_{\{t_{k}\}}=\bigcup_{1\leq k\leq l}\big\{\pm{(e_{i}-e_{j})}|\ t_1+t_2+\cdots+t_{k-1}+1\leq i<j\leq
t_1+t_2+\cdots+t_{k}\big\}$$ and $\Phi'_{\{t_{k}\}}=s_{e_1}\Phi_{\{t_{k}\}}$. They represent these two conjugacy classes.
[*Type $\D_4$.*]{} Given $\Psi_0=\D_{4}=\{\pm{e_{i}}\pm{e_{j}}|1\leq i<j\leq 4\}$, we denote by $$\begin{aligned}
&&\A_1=\langle e_1-e_2\rangle,\\&&\A_2=\langle e_1-e_2, e_2-e_3\rangle,\\&&
\A_3=\langle e_1-e_2,e_2-e_3,e_3-e_4\rangle,\\&&\A'_3=\langle e_1-e_2,e_2-e_3,e_3+e_4\rangle,\\&&
2\A_1=\langle e_1-e_2,e_3-e_4\rangle,\\&& 2\A'_1=\langle e_1-e_2,e_3+e_4\rangle, \\&& \D_2=
\langle e_1-e_2,e_1+e_2\rangle,\\&& 3\A_1=\langle e_1-e_2,e_1+e_2,e_3-e_4\rangle,\\&& 2\D_2
=\langle e_1-e_1,e_1+e_2,e_3-e_4,e_3+e_4\rangle,\\&&\D_3=\langle e_1-e_2,e_2-e_3,e_2+e_3\rangle,
\\&&\D_4=\langle e_1-e_1,e_2-e_3,e_3-e_4,e_3+e_4\rangle.\end{aligned}$$ We can show that any sub-root system of $\D_4$ is conjugate to one of them up to $W_{\Psi_0}$ conjugation. On the other hand, we have $$\begin{aligned}
&&\A_3\sim_{\Aut(\D_4)}\A'_3\sim_{\Aut(\D_4)}\D_3, \\&& 2\A_1\sim_{\Aut(\D_4)}2\A'_1
\sim_{\Aut(\D_4)}\D_2.\end{aligned}$$ These are all $\Aut(\D_4)$ conjugacy relations among them.
[*Type $\E_6$, $\E_7$ and $\E_8$.*]{} Given $\Psi_0=\E_6$, $\E_7$ or $\E_8$, the classification of sub-root systems of $\Psi_0$ is hard to describe. The readers can refer [@Oshima] for the details.
[*Type $\F_4$.*]{} Given $\Psi_0=\F_4$, we denote by $\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$ a simple system. The long roots and short roots in $\Psi_0$ consist in the sub-root systems $\D_4^{L}$ and $\D_4^{S}$, respectively. There is a unique conjugacy class of sub-root systems of $\F_4$ isomorphic to $\B_4$ (or $\C_4$). We denote by $$\B_4=\langle\alpha_2,\alpha_1,\alpha_2+2\alpha_3,\alpha_4\rangle$$ and $$\C_4=\langle\alpha_3,\alpha_4,\alpha_2+\alpha_3,\alpha_1\rangle,$$ which are representatives of them. The sub-root systems $\D_4^{L}$ and $\D_4^{S}$ are stable under $W_{\F_4}$. Hence there exist homomorphisms $$W_{\F_4}\longrightarrow\Aut(\D_4^{L})$$ and $$W_{\F_4}\longrightarrow\Aut(\D_4^{S}).$$
Each of these two homomorphisms is an isomorphism.
We show $W_{\F_4}\longrightarrow\Aut(\D_4^{L})$ is an isomorphism. The proof for $W_{\F_4}
\longrightarrow\Aut(\D_4^{S})$ is an isomorphism is similar. Suppose $w\in W_{\F_4}$ is an element with trivial restriction on $\D_4^{L}$. Choose an element $\alpha$ in $\D_4^{S}$. Since $\alpha$ and $\D_4^{L}$ generate a sub-root system isomorphic to $\B_{4}$, we have $\alpha=\frac{\beta_1+\beta_2}{2}$ for some $\beta_1,\beta_2\in D_4^{L}$. Thus $$w(\alpha)=\frac{w\beta_1+w\beta_2}{2}=
\frac{\beta_1+\beta_2}{2}=\alpha.$$ Hence $w=1$. Therefore $W_{F_4}\longrightarrow\Aut(D_4^{L})$ is injective. On the other hand, by calculating stabilizers we get $$|W_{\F_4}|=24|W_{\B_3}|=24\times 3!\times 2^3=3\times 4!\times 2^4=|\Aut(\D_4)|.$$ Hence $W_{\F_4}\longrightarrow\Aut(\D_4^{L})$ is an isomorphism.
By the above lemma and the classification in $\D_4$ case, we get $$\begin{aligned}
&&\D_2^{L}\sim
2\A_1^{L}\sim 2\A'^{L}_{1},\\&&\D_3^{L}\sim\A_3^{L}\sim\A'^{L}_{3},\\&&\D_2^{S}\sim 2\A_1^{S}\sim
2\A'^{S}_{1},\\&&\D_3^{S}\sim\A_3^{S}\sim\A'^{S}_{3}.\end{aligned}$$ The following sub-root systems of $\F_4$ are contained in $\B_4$: $$\D_4^{L},\ \A_3^{L}+\A_1^{S},\ 4\A_1^{L},\ \B_2+2\A_1^{L},\ 2\A_1^{L}+2\A_1^{S},$$ $$\B_4,\ \B_3+\A_1^{S},\ 2\B_2,\ \B_2+2\A_1^{S},\ 4\A_1^{S},\ \A_3^{L},$$ $$2\A_1^{L}+\A_1^{S},\ \B_3,\ \B_2+\A_1^{S},\ 3\A_1^{S},\ 3\A_1^{L},$$ $$\B_2+\A_1^{L},\ \A_1^{L}+2\A_1^{S},\ \A_2^{L}+\A_1^{S},\ 2\A_1^{L},$$ $$\B_2,\ 2\A_1^{S},\ \A_1^{L}+\A_1^{S},\ \A_2^{L},\ \A_1^{S},\ \A_1^{L}.$$ By duality, we get sub-root systems of $\F_4$ contained in some $\C_4$. In particular, the following are those contained in $\C_4$ and not contained in any $\B_4$: $$\D_4^{S},\ \A_1^{L}+\A_3^{S},\ \C_4,\ \C_3+\A_1^{L},$$ $$\A_3^{S},\ \C_3,\ \A_1^{L}+\A_2^{S},\ \A_2^{S}.$$
Denote by $\Phi\subset\F_4$ a sub-root system not contained in any $\B_4$ or $\C_4$. If $\Phi$ is simple, then $\Phi=\F_4$. If $\Phi$ is not simple, it can not contain a factor $\A_1^{L}$, $\A_1^{S}$ or $\B_2$ since otherwise $\Phi$ is contained in a sub-root system isomorphic to $\B_4$ or $\C_4$. Hence $\Phi=\A_2^{L}+\A_2^{S}$. Moreover, there exists a unique conjugacy class of sub-root systems in each of the above types.
[*Type $\G_2$.*]{} Given $\Psi_0=\G_2$, denote by $\{\alpha,\beta\}$ a simple system of $\Psi_0$. We denote by $$\A_1^{L}=\langle\alpha\rangle,\ \A_1^{S}=\langle\beta\rangle,\ \A_2^{L}
=\langle\alpha,\alpha+3\beta\rangle,\ \A_2^{S}=\langle\alpha+\beta,\beta\rangle$$ and $$\A_1^{L}+\A_1^{S}=\langle\alpha,\alpha+2\beta\rangle,\ \G_2=\langle\alpha,\beta\rangle.$$ Therefore each sub-root system of $\G_2$ is conjugate to one of them.
Formulas of the leading terms {#S:leading terms}
=============================
Given an irreducible root system $\Psi_0$ with a positive system $\Psi_0^{+}$, we normalize the inner product on $\Psi_0$ by letting the short roots having length $1$.
Given a reduced sub-root system $\Phi$ of $\Psi_0$, write $$\delta_{\Phi}=\frac{1}{2}\sum_{\alpha\in\Phi\cap\Psi_0^{+}}\alpha.$$ Let $\delta'_{\Phi}$ be the unique dominant weight with respect to $\Psi_0^{+}$ in the orbit $W_{\Psi_0}\delta_{\Phi}$. Denote by $e_{\Psi_0}(\Phi)=|2\delta'_{\Phi}|^{2}$ and $e_{\Psi_0}=e_{\Psi_0}(\Psi_0)$.
In the case that $\Psi_0$ is clear from the contest, we simply write $e(\Phi)$ for $e_{\Psi_0}(\Phi)$. Oshima classified sub-root systems $\Phi$ of any irreducible root system $\Psi_0$ up to $W_{\Psi}$ conjugation. In this section, we give the formulas of $2\delta'_{\Phi}$ and $e(\Phi)$ for the reduced sub-root systems.
First we have formulas for $e_{\Psi_0}=|2\delta'_{\Psi_0}|^{2}$ as in Table $1$.
\[Ta:1\]
[|c |c |c |c |c |c |]{} $\Psi_0$ & $\A_{n-1}$ & $\B_{n} $& $\C_{n}$ & $\D_{n}$ &\
\[0.3ex\]
$e_{\Psi_0}$ & $\frac{(n-1)n(n+1)}{6}$ & $\frac{(2n-1)n(2n+1)}{3}$ & $\frac{n(n+1)(2n+1)}{3}$ & $\frac{n(n-1)(2n-1)}{3}$ &\
\[0.3ex\] $\Psi_0$ & $\E_6$ & $\E_7$ & $\E_8$ & $\F_4$ & $\G_2$\
\[0.3ex\]
$e_{\Psi_0}$ & $156$ & $399$ & $1240$ & $156$ & $28$\
\[0.3ex\]
Given a reduced sub-root system $\Phi\subset\Psi_0$, denote by $$\Phi=\bigsqcup_{1\leq i\leq s}\Phi_{i}$$ the decomposition of $\Phi$ into a disjoint union of irreducible sub-root systems and $\sqrt{k_{i}}$ the ratio of the length of short roots of $\Phi_{i}$ and $\Psi$. Then we have $$e_{\Psi_0}(\Phi)=\sum_{1\leq i\leq s}
k_{i}e_{\Phi_{i}}.$$ With this formula we can calculate $e_{\Psi_0}(\Phi)$ quickly from Table $1$. On the other hand, from the expression of $2\delta'_{\Phi}$ into a linear combination of fundamental weights, we can also calculate $e_{\Psi_0}(\Phi)=|2\delta'_{\Phi}|^{2}$. This helps on checking whether a formula for $2\delta'_{\Phi}$ is correct or not. Given a classical irreducible root system $\Psi$, the calculation of $2\delta'_{\Phi}$ for sub-root systems $\Phi$ of $\Psi$ is easy. We omit it here. For an exceptional irreducible root system $\Psi$, in Oshima’s classification (cf. [@Oshima]), given an abstract root system $\Phi$, sometimes there exist two conjugacy classes of sub-root systems of $\Psi$ isomorphic to $\Phi$. In the following we give representatives of sub-root systems for all the cases when the above ambiguity occurs.
Given $\Psi_0=\E_7$, denote by $\beta=2\alpha_1+2\alpha_2+3\alpha_3+4\alpha_4+3\alpha_5+2\alpha_6+
\alpha_7$ and $\beta'=\alpha_2+\alpha_3+2\alpha_4+\alpha_5$. We set
- [$A_5:\langle\alpha_2,\alpha_4,\alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$(A_5)': \langle\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$3A_1: \langle\alpha_2,\alpha_5,\alpha_7\rangle$,]{}
- [$(3A_1)':\langle\alpha_3,\alpha_5,\alpha_7\rangle$,]{}
- [$4A_1: \langle\alpha_2,\alpha_3,\alpha_5,\alpha_7\rangle$,]{}
- [$(4A_1)': \langle\alpha_2,\alpha_3,\alpha_5,\beta'\rangle$,]{}
- [$A_3+A_1: \langle\alpha_2,\alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$(A_3+A_1)': \langle\alpha_3,\alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$2A_1+A_3: \langle\alpha_2,\alpha_3,\alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$(2A_1+A_3)': \langle\beta,\alpha_3,\alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$A_1+A_5: \langle\beta,\alpha_2,\alpha_4, \alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$(A_1+A_5)': \langle\beta,\alpha_3,\alpha_4, \alpha_5,\alpha_6,\alpha_7\rangle$.]{}
Given $\Psi_0=\E_8$, let $\beta=2\alpha_1+3\alpha_2+4\alpha_3+6\alpha_4+5\alpha_5+4\alpha_6+3\alpha_7+2\alpha_8$ $\beta'=\alpha_1+2\alpha_2+2\alpha_3+4\alpha_4+4\alpha_5+3\alpha_6+2\alpha_7+\alpha_8$, and $\beta''=2\alpha_1+2\alpha_2+4\alpha_3+5\alpha_4+4\alpha_5+3\alpha_6+2\alpha_7+\alpha_8$. We set
- [$A_7: \langle\beta,\alpha_1,\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7\rangle$,]{}
- [$(A_7)':\langle\alpha_1,\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7,\alpha_8\rangle$,]{}
- [$4A_1:\langle\alpha_2,\alpha_3,\alpha_5,\alpha_2+\alpha_3+2\alpha_4+\alpha_5\rangle$,]{}
- [$(4A_1)':\langle\alpha_2,\alpha_3,\alpha_5, \alpha_8\rangle$,]{}
- [$2A_1+A_3: \langle\beta',\alpha_1,\alpha_6,\alpha_7,\alpha_8\rangle$,]{}
- [$(2A_1+A_3)':\langle\alpha_2,\alpha_4,\alpha_3,\alpha_6,\alpha_8\rangle$,]{}
- [$2A_3:\langle\beta',\alpha_3,\alpha_1,\alpha_6,\alpha_7,\alpha_8\rangle$,]{}
- [$(2A_3)':\langle\alpha_2,\alpha_3,\alpha_4,\alpha_6,\alpha_7,\alpha_8\rangle$,]{}
- [$A_1+A_5:\langle\beta'',\alpha_4,\alpha_5,\alpha_6,\alpha_7,\alpha_8\rangle$,]{}
- [$(A_1+A_5)':\langle\alpha_1,\alpha_4,\alpha_5,\alpha_6,\alpha_7,\alpha_8\rangle$]{}
Given $\Psi_0=\D_4$, we set
- [$2A_1:\langle\alpha_1,\alpha_3\rangle$,]{}
- [$2A'_1: \langle\alpha_1,\alpha_4\rangle$,]{}
- [$D_2: \langle\alpha_3,\alpha_4\rangle$,]{}
- [$A_3:\langle\alpha_1,\alpha_2,\alpha_3\rangle$,]{}
- [$A'_3: \langle\alpha_1,\alpha_2,\alpha_4\rangle$,]{}
- [$D_3: \langle\alpha_2,\alpha_3,\alpha_4\rangle$.]{}
\[Ta:D4\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- --------- ------------------------------------------ ----------- -- -- -- -- --
$\D_4$ $A_1$ $\omega_2$ 1
$\D_4$ $2A_1$ $2\omega_4$ 2
$\D_4$ $2A'_1$ $2\omega_3$ 2
$\D_4$ $D_2$ $2\omega_1$ 2
$\D_4$ $A_2$ $2\omega_2$ 4
$\D_4$ $3A_1$ $\omega_1+\omega_3+\omega_4$ 3
$\D_4$ $A_3$ $2(\omega_2+\omega_4)$ 10
$\D_4$ $A'_3$ $2(\omega_2+\omega_3)$ 10
$\D_4$ $D_3$ $2(\omega_2+\omega_1)$ 10
$\D_4$ $4A_1$ $2\omega_2$ 4
$\D_4$ $D_4$ $2(\omega_1+\omega_2+\omega_3+\omega_4)$ 28
: Formulas of $2\delta'_{\Phi}$
\[Ta:E6\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- ------------ ------------------------------------------------------------ -----------
$\E_6$ $A_1$ $\omega_2$ 1
$\E_6$ $A_2$ $2\omega_2$ 4
$\E_6$ $A_3$ $\omega_1+2\omega_2+\omega_6$ 10
$\E_6$ $A_4$ $2\omega_1+2\omega_2+2\omega_6$ 20
$\E_6$ $A_5$ $2\omega_1+\omega_2+\omega_3+\omega_5+2\omega_6$ 35
$\E_6$ $D_4$ $2\omega_2+2\omega_4$ 28
$\E_6$ $D_5$ $2\omega_1+2\omega_2+2\omega_4+2\omega_6$ 60
$\E_6$ $E_6$ $2(\omega_1+\omega_2+\omega_3+\omega_4+\omega_5+\omega_6)$ 156
$\E_6$ $2A_1$ $\omega_1+\omega_6$ 2
$\E_6$ $3A_1$ $\omega_4$ 3
$\E_6$ $4A_1$ $2\omega_2$ 4
$\E_6$ $A_2+A_1$ $\omega_1+\omega_2+\omega_6$ 5
$\E_6$ $A_2+2A_1$ $\omega_3+\omega_5$ 6
$\E_6$ $2A_2$ $2\omega_1+2\omega_6$ 8
$\E_6$ $2A_2+A_1$ $\omega_1+\omega_4+\omega_6$ 9
$\E_6$ $3A_2$ $2\omega_4$ 12
$\E_6$ $A_3+A_1$ $\omega_2+\omega_3+\omega_5$ 11
$\E_6$ $A_3+2A_1$ $2\omega_4$ 12
$\E_6$ $A_4+A_1$ $2\omega_1+2\omega_4+2\omega_6$ 21
$\E_6$ $A_5+A_1$ $\omega_1+\omega_2+\omega_3+\omega_5+\omega_6$ 36
: Formulas of $2\delta'_{\Phi}$
\[Ta:E7\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- --------------- --------------------------------------------------------------------- ----------- -- -- --
$\E_7$ $A_{1}$ $\omega_1$ 1
$\E_7$ $A_2$ $2\omega_1$ 4
$\E_7$ $A_3$ $2\omega_1+\omega_6$ 10
$\E_7$ $A_4$ $2\omega_1+2\omega_6$ 20
$\E_7$ $A_5$ $2\omega_1+2\omega_6+2\omega_7$ 35
$\E_7$ $(A_5)'$ $\omega_1+\omega_4+2\omega_6$ 35
$\E_7$ $A_6$ $2\omega_4+2\omega_6$ 56
$\E_7$ $A_7$ $2\omega_1+2\omega_4+2\omega_6$ 84
$\E_7$ $D_4$ $2\omega_1+2\omega_3$ 28
$\E_7$ $D_5$ $\omega_1+2\omega_3+2\omega_6$ 60
$\E_7$ $D_6$ $2\omega_1+\omega_2+\omega_3+\omega_5+2\omega_6+2\omega_7$ 110
$\E_7$ $E_6$ $\omega_1+\omega_3+\omega_4+\omega_6$ 156
$\E_7$ $E_7$ $2(\omega_1+\omega_2+\omega_3+\omega_4+\omega_5+\omega_6+\omega_7)$ 399
$\E_7$ $2A_1$ $\omega_6$ 2
$\E_7$ $3A_1$ $2\omega_7$ 3
$\E_7$ $(3A_1)'$ $\omega_3$ 3
$\E_7$ $4A_1$ $\omega_2+\omega_7$ 4
$\E_7$ $(4A_1)'$ $2\omega_1$ 4
$\E_7$ $5A_1$ $\omega_1+\omega_6$ 5
$\E_7$ $6A_1$ $\omega_4$ 6
$\E_7$ $7A_1$ $2\omega_2$ 7
$\E_7$ $A_2+A_1$ $\omega_1+\omega_6$ 5
$\E_7$ $A_2+2A_1$ $\omega_4$ 6
$\E_7$ $A_2+3A_1$ $2\omega_2$ 7
$\E_7$ $2A_2$ $2\omega_6$ 8
$\E_7$ $2A_2+A_1$ $\omega_3+\omega_6$ 9
$\E_7$ $3A_2$ $2\omega_3$ 12
$\E_7$ $A_3+A_1$ $2\omega_1+2\omega_7$ 11
$\E_7$ $(A_3+A_1)'$ $\omega_1+\omega_4$ 11
$\E_7$ $A_3+2A_1$ $\omega_1+\omega_5+\omega_7$ 12
$\E_7$ $(A_3+2A_1)'$ $2\omega_3$ 12
$\E_7$ $A_3+3A_1$ $\omega_2+\omega_3+\omega_7$ 13
$\E_7$ $A_3+A_2$ $\omega_4+\omega_6$ 14
$\E_7$ $A_3+A_2+A_1$ $2\omega_5$ 15
$\E_7$ $2A_3$ $2\omega_1+2\omega_6$ 20
$\E_7$ $2A_3+A_1$ $\omega_1+\omega_4+\omega_6$ 21
$\E_7$ $A_4+A_1$ $\omega_1+\omega_4+\omega_6$ 21
$\E_7$ $A_4+A_2$ $2\omega_4$ 24
$\E_7$ $A_5+A_1$ $\omega_1+\omega_4+\omega_6+2\omega_7$ 36
$\E_7$ $(A_5+A_1)'$ $2\omega_3+2\omega_6$ 36
$\E_7$ $A_5+A_2$ $2\omega_4+2\omega_7$ 39
$\E_7$ $D_4+A_1$ $2\omega_1+\omega_2+\omega_3+\omega_7$ 29
$\E_7$ $D_4+2A_1$ $2\omega_1+\omega_4+\omega_6$ 30
: Formulas of $2\delta'_{\Phi}$
\[Ta:E7-II\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- ------------ ------------------------------------------------- ----------- -- -- --
$\E_7$ $D_4+3A_1$ $2\omega_1+2\omega_5$ 31
$\E_7$ $D_5+A_1$ $2\omega_1+\omega_2+\omega_3+\omega_5+\omega_6$ 61
$\E_7$ $D_6+A_1$ $2\omega_1+2\omega_4+2\omega_6+2\omega_7$ 111
: Formulas of $2\delta'_{\Phi}$
\[Ta:E8\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- ------------- ------------------------------------------------------------------------------ ----------- --
$\E_8$ $A_{1}$ $\omega_8$ 1
$\E_8$ $A_2$ $2\omega_8$ 4
$\E_8$ $A_3$ $\omega_1+2\omega_8$ 10
$\E_8$ $A_4$ $2\omega_1+2\omega_8$ 14
$\E_8$ $A_5$ $2\omega_1+\omega_6+\omega_8$ 35
$\E_8$ $A_6$ $2\omega_1+2\omega_6$ 56
$\E_8$ $A_7$ $2\omega_1+2\omega_6+2\omega_8$ 84
$\E_8$ $(A_7)'$ $\omega_1+\omega_4+\omega_6+\omega_7$ 84
$\E_8$ $A_8$ $2\omega_4+2\omega_7$ 120
$\E_8$ $D_4$ $2\omega_7+2\omega_8$ 28
$\E_8$ $D_5$ $2\omega_1+2\omega_7+2\omega_8$ 60
$\E_8$ $D_6$ $2\omega_1+\omega_2+\omega_3+\omega_7+2\omega_8$ 110
$\E_8$ $D_7$ $2\omega_1+\omega_2+\omega_3+\omega_5+\omega_6+\omega_8$ 182
$\E_8$ $D_8$ $2\omega_1+2\omega_4+2\omega_6+2\omega_8$ 280
$\E_8$ $E_6$ $\omega_1+\omega_6+\omega_7+\omega_8$ 156
$\E_8$ $E_7$ $2\omega_1+\omega_2+\omega_3+\omega_5+2\omega_6+2\omega_7+2\omega_8$ 399
$\E_8$ $E_8$ $2(\omega_1+\omega_2+\omega_3+\omega_4+\omega_5+\omega_6+\omega_7+\omega_8)$ 1240
$\E_8$ $2A_1$ $\omega_1$ 2
$\E_8$ $3A_1$ $\omega_7$ 3
$\E_8$ $4A_1$ $2\omega_8$ 4
$\E_8$ $(4A_1)'$ $\omega_2$ 4
$\E_8$ $5A_1$ $\omega_1+\omega_8$ 5
$\E_8$ $6A_1$ $\omega_6$ 6
$\E_8$ $7A_1$ $\omega_3$ 7
$\E_8$ $8A_1$ $2\omega_1$ 8
$\E_8$ $A_2+A_1$ $\omega_1+\omega_8$ 5
$\E_8$ $A_2+2A_1$ $\omega_6$ 6
$\E_8$ $A_2+3A_1$ $\omega_3$ 7
$\E_8$ $A_2+4A_1$ $2\omega_1$ 8
$\E_8$ $2A_2$ $2\omega_1$ 8
$\E_8$ $2A_2+A_1$ $\omega_1+\omega_7$ 9
$\E_8$ $2A_2+2A_1$ $\omega_5$ 10
$\E_8$ $3A_2$ $2\omega_7$ 12
$\E_8$ $3A_2+A_1$ $\omega_2+\omega_7$ 13
$\E_8$ $4A_2$ $2\omega_2$ 16
$\E_8$ $A_3+A_1$ $\omega_6+\omega_8$ 11
: Formulas of $2\delta'_{\Phi}$
\[Ta:E8-II\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- ---------------- ----------------------------------------------------- ----------- --
$\E_8$ $A_3+2A_1$ $2\omega_7$ 12
$\E_8$ $(A_3+2A_1)'$ $\omega_3+\omega_8$ 12
$\E_8$ $A_3+3A_1$ $\omega_2+\omega_7$ 13
$\E_8$ $A_3+4A_1$ $\omega_1+\omega_6$ 14
$\E_8$ $A_3+A_2$ $\omega_1+\omega_6$ 14
$\E_8$ $A_3+A_2+A_1$ $\omega_4$ 15
$\E_8$ $A_3+A_2+2A_1$ $2\omega_2$ 16
$\E_8$ $2A_3$ $2\omega_1+2\omega_8$ 20
$\E_8$ $(2A_3)'$ $\omega_1+\omega_5$ 20
$\E_8$ $2A_3+A_1$ $\omega_1+\omega_6+\omega_8$ 21
$\E_8$ $2A_3+2A_1$ $\omega_4+\omega_8$ 22
$\E_8$ $A_4+A_1$ $\omega_1+\omega_6+\omega_8$ 21
$\E_8$ $A_4+2A_1$ $\omega_4+\omega_8$ 22
$\E_8$ $A_4+A_2$ $2\omega_6$ 24
$\E_8$ $A_4+A_2+A_1$ $\omega_3+\omega_6$ 25
$\E_8$ $A_4+A_3$ $\omega_4+\omega_7$ 30
$\E_8$ $2A_4$ $2\omega_5$ 40
$\E_8$ $A_5+A_1$ $2\omega_1+2\omega_7$ 36
$\E_8$ $(A_5+A_1)'$ $\omega_1+\omega_4+\omega_8$ 36
$\E_8$ $A_5+2A_1$ $\omega_1+\omega_5+\omega_7$ 37
$\E_8$ $A_5+A_2$ $\omega_4+\omega_6$ 39
$\E_8$ $A_5+A_2+A_1$ $2\omega_5$ 40
$\E_8$ $A_6+A_1$ $\omega_1+\omega_4+\omega_6$ 57
$\E_8$ $A_7+A_1$ $\omega_1+\omega_4+\omega_6+2\omega_8$ 85
$\E_8$ $D_4+A_1$ $\omega_2+\omega_7+2\omega_8$ 29
$\E_8$ $D_4+2A_1$ $\omega_1+\omega_6+2\omega_8$ 30
$\E_8$ $D_4+3A_1$ $\omega_4+2\omega_8$ 31
$\E_8$ $D_4+4A_1$ $2\omega_2+2\omega_8$ 32
$\E_8$ $D_4+A_2$ $2\omega_2+2\omega_8$ 32
$\E_8$ $D_4+A_3$ $\omega_2+\omega_3+\omega_7$ 38
$\E_8$ $2D_4$ $2\omega_1+2\omega_6$ 56
$\E_8$ $D_5+A_1$ $\omega_1+\omega_5+\omega_7+2\omega_8$ 61
$\E_8$ $D_5+2A_1$ $\omega_2+\omega_3+\omega_7+2\omega_8$ 62
$\E_8$ $D_5+A_2$ $2\omega_5+2\omega_8$ 64
$\E_8$ $D_5+A_3$ $\omega_1+\omega_4+\omega_6+\omega_8$ 70
$\E_8$ $D_6+A_1$ $2\omega_1+\omega_4+\omega_6+2\omega_8$ 111
$\E_8$ $D_6+2A_1$ $2\omega_1+2\omega_5+2\omega_8$ 112
$\E_8$ $E_6+A_1$ $\omega_1+\omega_4+\omega_6+2\omega_7+2\omega_8$ 157
$\E_8$ $E_6+A_2$ $\omega_4+\omega_7+\omega_8$ 160
$\E_8$ $E_7+A_1$ $2\omega_1+2\omega_4+2\omega_6+2\omega_7+2\omega_8$ 400
: Formulas of $2\delta'_{\Phi}$
\[Ta:F4\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- ----------------- ------------------------------------------- ----------- -- -- -- -- --
$\F_4$ $A_1^L$ $\omega_1$ 2
$\F_4$ $A_1^S$ $\omega_4$ 1
$\F_4$ $A_2^L$ $2\omega_1$ 8
$\F_4$ $A_2^S$ $2\omega_4$ 4
$\F_4$ $A_3^L$ $2\omega_1+2\omega_4$ 20
$\F_4$ $A_3^S$ $\omega_1+2\omega_4$ 10
$\F_4$ $D_4^L$ $2\omega_1+2\omega_2$ 56
$\F_4$ $D_4^S$ $2\omega_3+2\omega_4$ 28
$\F_4$ $B_2$ $\omega_1+2\omega_4$ 10
$\F_4$ $B_3$ $2\omega_1+\omega_2+\omega_4$ 35
$\F_4$ $C_3$ $2\omega_3+2\omega_4$ 28
$\F_4$ $B_4$ $2\omega_1+2\omega_2+2\omega_4$ 84
$\F_4$ $C_4$ $2\omega_1+2\omega_3+2\omega_4$ 60
$\F_4$ $F_4$ $2\omega_1+2\omega_2+2\omega_3+2\omega_4$ 156
$\F_4$ $2A_1^L$ $2\omega_4$ 4
$\F_4$ $2A_1^S$ $\omega_1$ 2
$\F_4$ $A_1^S+A_1^L$ $\omega_3$ 3
$\F_4$ $3A_1^L$ $\omega_2$ 6
$\F_4$ $3A_1^S$ $\omega_3$ 3
$\F_4$ $A_1^S+2A_1^L$ $\omega_1+\omega_4$ 5
$\F_4$ $2A_1^S+A_1^L$ $2\omega_4$ 4
$\F_4$ $4A_1^L$ $2\omega_1$ 8
$\F_4$ $4A_1^S$ $2\omega_4$ 4
$\F_4$ $2A_1^S+2A_1^L$ $\omega_2$ 6
$\F_4$ $A_2^L+A_1^S$ $\omega_1+\omega_3$ 9
$\F_4$ $A_2^S+A_1^L$ $\omega_2$ 6
$\F_4$ $A_2^S+A_2^L$ $2\omega_3$ 12
$\F_4$ $B_2+A_1^L$ $2\omega_3$ 12
$\F_4$ $B_2+A_1^S$ $\omega_2+\omega_4$ 11
$\F_4$ $B_2+2A_1^L$ $\omega_1+\omega_4$ 14
$\F_4$ $B_2+2A_1^S$ $2\omega_3$ 12
$\F_4$ $2B_2$ $2\omega_1+2\omega_4$ 20
$\F_4$ $A_3^S+A_1^L$ $2\omega_3$ 12
$\F_4$ $A_3^L+A_1^S$ $\omega_1+\omega_2+\omega_4$ 21
$\F_4$ $C_3+A_1^L$ $\omega_1+\omega_2+2\omega_4$ 30
$\F_4$ $B_3+A_1^S$ $2\omega_1+2\omega_3$ 36
: Formulas of $2\delta'_{\Phi}$
\[Ta:G2\]
$\Psi_0$ $\Phi$ $2\delta'_{\Phi}$ $e(\Phi)$
---------- --------------- ------------------------ ----------- -- -- -- -- --
$\G_2$ $A_1^L$ $\omega_1$ 3
$\G_2$ $A_1^S$ $\omega_2$ 1
$\G_2$ $A_2^L$ $2\omega_1$ 12
$\G_2$ $A_2^S$ $2\omega_2$ 4
$\G_2$ $G_2$ $2(\omega_1+\omega_2)$ 28
$\G_2$ $A_1^S+A_1^L$ $2\omega_2$ 4
: Formulas of $2\delta'_{\Phi}$
Equalities among dimension data {#S:dimension-equal}
===============================
In this section, we solve Question \[Q:equal-character\]. The strategy is as follows. First we reduce it to the case that the root system $\Psi$ is an irreducible root system. In the case that $\Psi$ if of type $\BC$, there is an algebra isomorphism $E$ from the ring of characters to the polynomial ring ${\mathbb{Q}}[x_0,x_1,\dots]$ defined in [@Larsen-Pink]. Moreover, the polynomials in the image are generated by certain polynomials $a_{n}$, $b_{n}$, $c_{n}$, $d_{n}$, corresponding to the sub-root systems $\A_{n-1}$, $\B_{n}$, $\C_{n}$ and $\D_{n}$, respectively. We are able to get all multiplicative relations among these polynomials $\{a_{n},b_{n},c_{n},d_{n}|\ n\geq 1\}$. In the case that $\Psi$ is of type $\B$, $\C$ or $\D$, the solution in type $\BC$ case solves the question in this case as well, with a little more consideration if $\Psi=\D_{n}$. In the case that $\Psi$ is of type $\A$, we show that the characters of non-conjugate sub-root systems have different leading terms. In the case that $\Psi$ is an exceptional irreducible root system, the formulas of the leading terms $\{2\delta'_{\Phi}|\ \Phi\subset\Psi\}$ have been given in Section \[S:leading terms\]. We give a case by case discussion of sub-root systems with equal leading terms in their characters.
\[T:equal character-nonsimple\] Given a root system $\Psi$ and two reduced sub-root systems $\Phi_1$ and $\Phi_2$, $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$ if and only if there exists $\gamma\in\Aut(\Psi)$ such that $$F_{\Phi_{i}^{(1)},\Aut(\Psi_{i})}=F_{\Phi_{i}^{(2)},\Aut(\Psi_{i})}$$ for any $1\leq i\leq m$, where $\Psi=\bigsqcup_{1\leq i\leq m}\Psi_{i}$ with each $\Psi_{i}$ an irreducible root system, $$\gamma\Phi_1=\bigsqcup_{1\leq i\leq m}\Phi_{i}^{(1)}$$ and $$\Phi_2=\bigsqcup_{1\leq i\leq m}\Phi_{i}^{(2)}$$ with $\Phi_{i}^{(1)},\Phi_{i}^{(2)}\subset\Psi_{i}$.
Since $\Aut(\Psi)$ permutes simple factors of $\Psi$ and it permutes two simple factors if and only if they are isomorphic abstract irreducible root systems, we may assume that $\Psi=m\Psi_0$ and $\Aut(\Psi)=\Aut(\Psi_0)^{m}\rtimes S_{m}$ where $\Psi_0$ is an irreducible root system and $m\Psi_0$ denotes the direct sum of $m$ copies of $\Psi_0$.
Denote by $\Lambda={\mathbb{Z}}\Psi_0$ the root lattice, by ${\mathbb{Q}}[\Lambda]$ the character ring and by $U={\mathbb{Q}}[\Lambda]^{\Aut(\Psi_0)}$ the invariant characters. Thus $U$ is a ${\mathbb{Q}}$ vector space with a basis $\Lambda$. Write $$S(U)=\bigoplus_{n\geq 0} S^{n}(U)$$ for the symmetric tensor algebra over $U$. It is a polynomial algebra with symmetric tensor product as multiplication. Hence it is a unique factorization domain.
Write $\Phi_{j}=\bigoplus_{1\leq i\leq m}\Phi_{i}^{(j)},j=1,2$. In $S(U)$, we have $$F_{\Phi_{j},
\Aut(\Psi)}=F_{\Phi_1^{(j)},\Aut(\Psi_0)}\cdot F_{\Phi_2^{(j)},\Aut(\Psi_0)}\cdots F_{\Phi_{m-1}^{(j)},
\Aut(\Psi_0)}\cdot F_{\Phi_{m}^{(j)},\Aut(\Psi_0)}.$$ As each $F_{\Phi_{i}^{(j)},\Aut(\Psi_0)}$ being of degree one and having constant term $1$, $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$ if and only if $\{F_{\Phi_{i}^{(1)},\Aut(\Psi_0)}|\ 1\leq i\leq m\}$ and $\{F_{\Phi_{i}^{(2)},\Aut(\Psi_0)}|\ 1\leq i
\leq m\}$ differ by a permutation. Therefore the conclusion follows.
\[R:equal character-nonsimple\] In the case of $\Psi_0=\BC_1=\{\pm{e_1},\pm{2e_1}\}$, $$\Lambda={\mathbb{Z}}\Psi_0=\{ne_1:n\in{\mathbb{Z}}\}.$$ Write $x_{n}=
\frac{[ne_1]+[-ne_1]}{2}$ for any $n\in{\mathbb{Z}}_{\geq 0}$. In this case $U=\span_{{\mathbb{Q}}}\{x_{n}|\ n\geq 0\}$ and $$S(U)={\mathbb{Q}}[x_0,x_1,\dots,x_{n},\dots]$$ is the usual polynomial algebra. The identification is given by the map $E$ defined in [@Larsen-Pink].
Theorem \[T:equal character-nonsimple\] reduces Question \[Q:equal-character\] to the case that $\Psi$ is an irreducible root system. We introduce some notation now. Given a root system $\Psi$, recall that we have defined the integral weight lattice $\Lambda_{\Psi}$ (which is a subset of the rational vector space spanned by roots of $\Psi$) in the “notation and conventions“ part.
Given an integral weight $\lambda$, let $$\chi_{\lambda,\Psi}^{\ast}=\frac{1}{|W_{\Psi}|}
\sum_{\gamma\in W_{\Psi}}[\gamma\lambda].$$
In the case that the root system $\Psi$ is clear from the context, we simply write $\chi_{\lambda}^{\ast}$ for $\chi_{\lambda,\Psi}^{\ast}$. As discussed in Section \[S:characters\], the characters $\{\chi_{\lambda}^{\ast}|\ \lambda\in\Lambda_{\Psi}^{+}\}$ is a basis of of the vector space ${\mathbb{Q}}[\Lambda_{\Psi}]^{W_{\Psi_0}}$, where $\Lambda_{\Psi}^{+}$ is the set of dominant weights in $\Lambda_{\Psi}$ (with respect to a positive system determined by a chosen order).
Combining Theorems \[T:character-classical\] and \[character-exceptional\], we get the following theorem. It answers Question \[Q:equal-character\] completely in the case that $\Psi$ is an irreducible root system.
\[T:equal character-simple\] Given an irreducible root system $\Psi$, if there exist two non-conjugate reduced sub-root systems $\Phi_1,\Phi_2\subset\Psi$ with $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$, then $\Psi\cong\C_{n}$, $\BC_{n}$ or $\F_4$.
In the case that $\Psi=\C_{n}$ or $\BC_{n}$, $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$ if and only if $$b_{m}(\Phi_1)-b_{m}(\Phi_2)=a_{2m}(\Phi_1)-a_{2m}(\Phi_2)=0$$ and $$a_{2m-1}(\Phi_1)-a_{2m-1}(\Phi_2)=c_{m-1}(\Phi_2)-c_{m-1}(\Phi_1)=d_{m}(\Phi_2)-d_{m}(\Phi_1)$$ for any $m\geq 1$. Here $a_{m}(\Phi_{i})$, $b_{m}(\Phi_{i})$, $c_{m}(\Phi_{i})$, $d_{m}(\Phi_{i})$ is the number of simple factors of $\Phi_{i}\subset\Psi\subset\BC_{n}$ isomorphic to $\A_{m-1}$, $\B_{m}$, $\C_{m}$ or $\D_{m}$, respectively.
In the case that $\Psi=\F_4$, $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$ if and only if $$\Phi_1\sim\Phi_2,$$ $$\{\Phi_1,\Phi_2\}\sim\{A_2^{S},A_1^{L}+2A_1^{S}\}$$ or $$\{\Phi_1,\Phi_2\}\sim\{A_1^{L}+A_2^{S}, 2A_1^{L}+2A_1^{S}\}.$$
Classical irreducible root systems {#SS:Character-classical}
----------------------------------
As in Section 3 of [@Larsen-Pink], let $$\begin{aligned}
&& {\mathbb{Z}}^{n}:={\mathbb{Z}}\BC_{n}=\Lambda_{\BC_{n}}=\span_{{\mathbb{Z}}}\{e_1,e_2,...,e_{n}\},
\\&& W_{n}:=\Aut(\BC_{n})=W_{\BC_{n}}=\{\pm{1}\}^{n}\rtimes S_{n},\\&& {\mathbb{Z}}_{n}:={\mathbb{Q}}[{\mathbb{Z}}^{n}],
\\&& Y_{n}:={\mathbb{Z}}_{n}^{W_{n}}. \end{aligned}$$ For $m\leq n$, the injection $${\mathbb{Z}}^{m}\hookrightarrow{\mathbb{Z}}^{n}:(a_1,...,a_{m})\mapsto(a_1,...,,a_{m},0,...,0)$$ extends to an injection $i_{m,n}:{\mathbb{Z}}_{m}\hookrightarrow{\mathbb{Z}}_{n}$. Define $\phi_{m,n}:{\mathbb{Z}}_{m}\rightarrow{\mathbb{Z}}_{n}$ by $$\phi_{m,n}(z)=\frac{1}{|W_{n}|}\sum_{w\in W_{n}}w(i_{m,n}(z)).$$ Thus $\phi_{m,n}\phi_{k,m}=\phi_{k,n}$ for any $k\leq m\leq n$ and the image of $\phi_{m,n}$ lies in $Y_n$. Hence $\{Y_{m}:\phi_{m,n}\}$ forms a direct system and we define $$Y=\lim_{\longrightarrow_{n}} Y_{n}.$$ Define the map $j_{n}: {\mathbb{Z}}_{n}\rightarrow Y$ by composing $\phi_{n,p}$ with the injection $Y_{p}\hookrightarrow Y$. The isomorphism ${\mathbb{Z}}^{m}\oplus{\mathbb{Z}}^{n}\longrightarrow{\mathbb{Z}}^{m+n}$ gives a canonical isomorphism $M: {\mathbb{Z}}_{m}\otimes_{{\mathbb{Q}}}{\mathbb{Z}}_{n}\longrightarrow{\mathbb{Z}}_{m+n}$. Given two elements of $Y$ represented by $y\in Y_{m}$ and $y'\in Y_{n}$ we define $$yy'=j_{m+n}(M(y\otimes y')).$$ This product is independent of the choice of $m$ and $n$ and makes $Y$ a commutative associative algebra. The monomials $[e_1]^{a_1}\cdots[e_{n}]^{a_{n}}$ ($a_1,a_2,\cdots,a_{n}\in{\mathbb{Z}}$) form a ${\mathbb{Q}}$ basis of ${\mathbb{Z}}_{n}$, where $[e_i]^{a_i}=[a_ie_i]\in{\mathbb{Z}}_1$ is a linear character. Hence $Y$ has a ${\mathbb{Q}}$ basis $$e(a_1,a_2,...,a_{n})=j_{n}([e_1]^{a_1}\cdots[e_{n}]^{a_{n}})$$ indexed by $n\geq 0$ and $a_1\geq a_2\geq\cdots\geq a_{n}\geq 0$. Mapping $e(a_1,a_2,...,a_{n})$ to $x_{a_1}x_{a_2}\cdots x_{a_{n}}$, we get a ${\mathbb{Q}}$ linear map $$E: Y\longrightarrow {\mathbb{Q}}[x_0,x_1,...,x_{n},...].$$
([@Larsen-Pink] Page 390)\[Isomorphism E\] The above map $E$ is an algebra isomorphism.
The bijectivity is clear. It is an isomorphism follows from the equality $$e(a_1,a_2,...,a_{m})e(b_1,b_2,...,b_{n})=e(c_1,c_2,...,c_{m+n}),$$ where $(c_1,c_2,...,c_{m+n})$ is the re-permutation of $\{a_1,...,a_{m},b_1,...,b_{n}\}$ in deceasing order. This equlity can be proved by a combinatorial calculation.
\[D:abcd\] Write $a_{n}$, $b_{n}$, $c_{n}$, $d_{n}$ for the image of $j_{n}(F_{\Phi,W_{n}})$ under $E$ for $\Phi=\A_{n-1}$, $\B_{n}$, $\C_{n}$ or $\D_{n}$, respectively.
Observe that $a_{n}$, $b_{n}$, $c_{n}$, $d_{n}$ are homogeneous polynomials of degree $n$ with integer coefficients and a term $x_0^{n}$.
\[R:x0=1\] Here, $E$ maps linear characters to homogeneous polynomials. Letting $x_0=1$, then the definition here becomes that in [@Larsen-Pink]. In this terminology, each of $a_{n}$, $b_{n}$, $c_{n}$, $d_{n}$ has integer coefficients and constant term $1$.
Note that our $b_{n},c_{n},d_{n}$ are $b'_{n},c'_{n},d'_{n}$ in [@Larsen-Pink]. For any $a_1\geq a_2\geq \cdots\geq a_{n}\geq 0$ and $\lambda=a_1e_1+a_2e_2+\cdots+a_{n}e_{n}$, one sees that $$\chi^{\ast}_{\lambda,W_{n}}=[e_1]^{a_1}\cdots[e_{n}]^{a_{n}}$$ in $Y$. Thus $E(\chi^{\ast}_{\lambda,W_{n}})=x_{a_1}x_{a_2}\cdots x_{a_{n}}$.
For small $n$, we have $a_{1}=d_1=x_0$, $b_1=x_0-x_1$, $c_1=x_0-x_2$, $$a_2=x_0^{2}-x_1^{2},$$ $$b_2=x_0^{2}-x_0x_1-x_0x_3+x_1x_3-x_1^{2}+2x_1x_2-x_2^{2},$$ $$c_2=x_0^{2}-x_0x_2-x_0x_4+x_2x_4-x_1^{2}+2x_1x_3-x_3^{2},$$ $$d_2=x_0^{2}-2x_1^{2}+x_0x_2$$ and $a_3=x_0^{3}-2x_0x_1^{2}+2x_1^{2}x_2-x_0x_2^{2}$.
For the convenience in writing notations, we define $a_0=b_0=c_0=d_0=1$ and $c_{-1}=x_0^{-1}$.
([@Larsen-Pink], Page 390)\[basic properties\]
- [We have $c_{n},d_{n+1}\in{\mathbb{Q}}[x_1,x_2,...,x_{2n}]-{\mathbb{Q}}[x_1,x_2,...,x_{2n-1}]$,\
$b_{n}\in{\mathbb{Q}}[x_1,x_2,...,x_{2n-1}]-{\mathbb{Q}}[x_1,x_2,...,x_{2n-2}]$.]{}
- [Each of $b_{n},c_{n},d_{n+1}$ is a prime in ${\mathbb{Q}}[x_1,x_2,...]$ and any two of them are different.]{}
- [Each of the subsets $\{b_1,...,b_{n},c_1,...,c_{n}\}$, $\{b_1,...,b_{n},d_2,...,d_{n+1}\}$,\
$\{c_1,...,c_{n},d_2,...,d_{n+1}\}$ is algebraically independent.]{}
Given $f\in{\mathbb{Q}}[x_0,x_1,...]$, let $$\sigma(f)(x_0,x_1,...,x_{2n},x_{2n+1},...)=
f(x_0,-x_1,...,x_{2n},-x_{2n+1},...).$$ Then $\sigma$ is an involutive automorphism of ${\mathbb{Q}}[x_0,x_1,...]$.
\[P:abcd-symmetry\] We have $\sigma(a_{n})=a_{n}$, $\sigma(c_{n})=c_{n}$, $\sigma(d_{n})=d_{n}$ for any $n$ and but $\sigma(b_{n})\neq b_{n}$ when $n\geq 1$.
This follows from the formula $$F_{\Phi,W_{n}}=\sum_{w\in W_{\Phi}}\epsilon(w)\chi^{\ast}_{\delta_{\Phi}-w\delta_{\Phi},W_{n}}$$ and the expression of $\delta_{\Phi}$ for $\Phi=\A_{n-1},\B_{n},\C_{n},\D_{n}$.
\[D:bn\] Define $b'_{n}=\sigma(b_{n})$.
\[P:A=BB and A=CD\] For any $n\geq 1$, we have $a_{2n}=b_{n}b'_{n}$ and $a_{2n+1}=c_{n}d_{n+1}$.
Given $n\geq 1$, define the matrices $A_{n}=(x_{|i-j|})_{n\times n}$, $$\begin{aligned}
&&
B_{n}=(x_{|i-j|}-x_{i+j-1})_{n\times n},\\&&
C_{n}=(x_{|i-j|}-x_{i+j})_{n\times n},\\&& B'_{n}=(x_{|i-j|}+x_{i+j-1})_{n\times n},\\&&
D_{n}=(x_{|i-j|}+x_{i+j-2})_{n\times n},\\&& D'_{n}=(a_{i,j})_{n\times n}, \end{aligned}$$ where $a_{i,j}=x_{|i-j|}+x_{i+j-2}$ if $i,j\geq 2$, $a_{1,j}=a_{j,1}=\sqrt{2}x_{j-1}$, $a_{1,1}=1$. Then we have the following equalities: $$\label{Eq:an}
a_{n}=\det A_{n},$$ $$\label{Eq:bn}
b_{n}=\det B_{n},$$ $$\label{Eq:bn2}
b'_{n}=\det B'_{n},$$ $$\label{Eq:cn}
c_{n}=\det C_{n},$$ $$\label{Eq:dn}
d_{n}=\frac{1}{2}\det D_{n}=\det D'_{n}.$$
We prove the equality (\[Eq:an\]). The others can be proved similarly. Recall that $$F_{\A_{n-1},W_{n}}=\sum_{w\in S_{n}}\epsilon(w)\chi^{\ast}_{\delta-w\delta,W_{n}},$$ where $$\begin{aligned}
\delta &=&\frac{(n-1)e_1+(n-3)e_2+\cdots+(1-n)e_{n}}{2}\\&=&
(ne_1+(n-1)e_2+\cdots+e_{n})-\frac{n+1}{2}(e_1+e_2+\cdots+e_{n}).\end{aligned}$$ Let $\delta'=ne_1+(n-1)e_2+\cdots+e_{n}$. Then $$F_{\A_{n-1},W_{n}}=\sum_{w\in S_{n}}\epsilon(w)\chi^{\ast}_{\delta'-w\delta',W_{n}}.$$ Expanding $\det A_{n}=\det (x_{|i-j|})_{n\times n}$ into the sum of terms according to permutations, a term corresponding to a permutation $w\in S_{n}$ is equal to the polynomial $E(j_{n}(\chi^{\ast}_{\delta'-w\delta',W_{n}}))$. Summing up all terms, we get $a_{n}=\det A_{n}$.
Let $$L_{n}=\left(\begin{array}{ccc}&&1\\&\ddots&\\1&&\end{array}\right),$$ $$J_{2n}=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}I_{n}&-\frac{1}{\sqrt{2}}L_{n}\\
\frac{1}{\sqrt{2}}L_{n}&\frac{1}{\sqrt{2}}I_{n} \end{array}\right)$$ and $$J_{2n+1}=\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}I_{n}&\vdots&-\frac{1}{\sqrt{2}}L_{n}
\\\cdots&1&\vdots\\\frac{1}{\sqrt{2}}L_{n}&\vdots&\frac{1}{\sqrt{2}}I_{n} \end{array}\right).$$
By matrix calculation, we have $$\begin{aligned}
&&J_{2n}A_{2n}J_{2n}^{-1}=\left(\begin{array}
{cc}B_{n}&\\&B'_{n}\end{array}\right),\\&& J_{2n+1}A_{2n+1}J_{2n+1}^{-1}=\left(\begin{array}
{cc}C_{n}&\\&D'_{n+1}\end{array} \right).\end{aligned}$$ Taking determinants, we get $a_{2n}=b_{n}b'_{n}$ and $a_{2n+1}=c_{n}d_{n+1}$.
\[multiplicative relation, polynomials\] Any multiplicative relation among $\{a_{n+1},b_{n},c_{n},d_{n+1}|n\geq 1\}$ is generated by $\{a_{2n+1}=c_{n}d_{n+1}: n\geq 1\}$.
This follows from Proposition \[basic properties\] and Proposition \[P:A=BB and A=CD\].
\[T:character-classical\] Given a classical irreducible root system $\Psi$, if there exists non-conjugate sub-root systems $\Phi_1$ and $\Phi_2$ of $\Psi$ such that $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$, then $\Psi\cong\C_{n}$ or $\BC_{n}$.
In the case that $\Psi=\C_{n}$ or $\BC_{n}$, $F_{\Phi_1,\Aut_{\Psi}}=F_{\Phi_2,\Aut(\Psi)}$ if and only if $$\forall m\leq n, b_{m}(\Phi_1)-b_{m}(\Phi_2)=a_{2m}(\Phi_1)-a_{2m}(\Phi_2)=0$$ $$\textrm{and }a_{2m+1}(\Phi_1)-a_{2m+1}(\Phi_2)=c_{m}(\Phi_2)-c_{m}(\Phi_1)=d_{m+1}(\Phi_2)-
d_{m+1}(\Phi_1).$$ Here $a_{m}(\Phi_{i}),b_{m}(\Phi_{i}),c_{m}(\Phi_{i}),d_{m}(\Phi_{i})$ is the number of simple factors of $\Phi_{i}\subset\BC_{n}$ isomorphic to $\A_{m-1},\B_{m},\C_{m},\D_{m}$ respectively.
Note that, we embed $\A_{m-1},\B_{m},\C_{m},\D_{m}$ into $\BC_{n}$ ($n\geq m$) in the standard way. This means that the sub-root systems $\A_1$, $\B_1$, $\C_1$ are non-isomorphic to each other, as well as each of the pairs $(\B_2,\C_2)$, $(\D_2,\A_1\bigsqcup\A_1)$ and $(\A_3,\D_3)$.
In the case of $\Psi=\C_{n}$ or $\BC_{n}$, since $j_{n}: Y_{n}\longrightarrow Y$ is an injection, $F_{\Phi_1,W_{n}}=F_{\Phi_2,W_{n}}$ if and only if $E(F_{\Phi_1,W_{n}})=E(F_{\Phi_2,W_{n}})$, i.e. $$\prod_{1\leq i\leq n}(a_i^{r_{i}^{(1)}}b_{i}^{s_{i}^{(1)}}c_{i}^{u_{i}^{(1)}}d_{i}^{v_{i}^{(1)}})
=\prod_{1\leq i\leq n}(a_{i}^{r_{i}^{(2)}}b_{i}^{s_{i}^{(2)}}c_{i}^{u_{i}^{(2)}}d_{i}^{v_{i}^{(2)}}).$$ Here we write $r_{m}^{(j)}=a_{m}(\Phi_{j})$, $s_{m}^{(j)}=b_{m}(\Phi_{j})$, $u_{m}^{(j)}=c_{m}(\Phi_{j})$, $v_{m}^{(j)}=d_{m}(\Phi_{j})$. Therefore the conclusion follows from Corollary \[multiplicative relation, polynomials\].
In the case that $\Psi=\B_{n}$ ($n\geq 1$) or $\D_{n}$ ($n\geq 5$), $\Aut(\Psi_0)=W_{n}$. Since any $\C_{k}$ is not contained in $\B_{n}$ or $\D_{n}$, the conclusion follows from the conclusion for $\BC_{n}$ case.
In the case that $\Psi=\D_4$, only the characters of the non-conjugate sub-root systems $A_2,4A_1$ have the equal leading term, which is $\chi_{2\omega_2}$. We have $$F_{A_2,W_{\Psi}}=
1-2\chi^{\ast}_{\omega_2}+2\chi^{\ast}_{\omega_1+\omega_3+\omega_4}-\chi^{\ast}_{2\omega_2}$$ and $$F_{4A_1,W_{\Psi}}=1-4\chi^{\ast}_{\omega_2}+2(\chi^{\ast}_{2\omega_1}+\chi^{\ast}_{2\omega_3}+
\chi^{\ast}_{2\omega_4})-4\chi^{\ast}_{\omega_1+\omega_3+\omega_4}+\chi^{\ast}_{2\omega_2}.$$ Thus $F_{A_2,W_{\Psi}}\neq F_{4A_1,W_{\Psi}}$. Therefore the conclusion follows.
In the case that $\Psi=\A_{n-1}$, $F_{\Phi_1,\Aut_{\Psi}}=F_{\Phi_2,\Aut_{\Psi}}$ implies that $2\delta_{\Phi_1}\sim_{\Aut(\Psi)}2\delta_{\Phi_2}$. And the latter implies that $\Phi_1\sim_{\Aut(\Psi)}\Phi_2$. Therefore the conclusion follows.
Exceptional irreducible root systems {#SS:Character-exceptional}
------------------------------------
In Section \[S:leading terms\], we give the formulas of $2\delta'_{\Phi}$ and their modulus squares for reduced sub-root systems of the exceptional simple root systems $\Psi=\E_6$, $\E_7$, $\E_8$, $\F_4$ or $\G_2$. Using these formulas, in this section we classify non-conjugate sub-root systems $\Phi_1,\Phi_2$ of any exceptional irreducible root system $\Psi$ such that $$F_{\Phi_1,W_{\Psi}}=F_{\Phi_2,W_{\Psi}}.$$
\[character-exceptional\] Given an irreducible root system $\Psi=\E_6$, $\E_7$, $\E_8$, $\F_4$ or $\G_2$, if there exists non-conjugate reduced sub-root systems $\Phi_1$ and $\Phi_2$ of $\Psi$ such that $F_{\Phi_1,\Aut(\Psi)}=F_{\Phi_2,\Aut(\Psi)}$, then $\Psi\cong F_4$.
In the case that $\Psi=\F_4$, $F_{\Phi_1,\Psi}=F_{\Phi_2,\Psi}$ and $\Phi_1\not\sim\Phi_2$ if and only if $$\{\Phi_1,\Phi_2\}\sim\{A_2^{S},A_1^{L}+2A_1^{S}\}\textrm{ or }\{A_1^{L}+A_2^{S}, 2A_1^{L}+2A_1^{S}\}.$$
In the case that $\Psi=\E_6$, among the dominant integral weights appearing in $\{2\delta'_{\Phi}|\ \Phi\subset\E_6\}$, those weights that appearing more than once include $\{2\omega_2,2\omega_4\}$ and the sub-root systems $\Phi$ with $2\delta_{\Phi}$ conjugate to them are
- [$2\omega_2$: $A_2$, $4A_1^{S}$, appears 2 times.]{}
- [$2\omega_4$: $3A_2$, $A_3+2A_1$, appears 2 times.]{}
The coefficients of $\chi^{\ast}_{\omega_2}$ in $F_{A_2,\Aut(\E_6)},F_{4A_1,\Aut(\E_6)}$ are different and the coefficients of $\chi^{\ast}_{\omega_2}$ in $F_{3A_2,\Aut(\E_6)},F_{A_3+2A_1,\Aut(\E_6)}$ are also different. Therefore the conclusion in the $\E_6$ case follows.
In the case that $\Psi=\E_7$, among the dominant integral weights appearing in $\{2\delta'_{\Phi}|\ \Phi\subset\E_7\}$, those weights that appearing more than once include $$\{2\omega_1, \omega_1+\omega_6, \omega_4, 2\omega_2, 2\omega_3, 2\omega_1+2\omega_6,
\omega_1+\omega_4+\omega_6\}$$ and the sub-root systems $\Phi$ with $2\delta_{\Phi}$ conjugate to them are
- [$2\omega_1$: $A_2$, $4A'_1$, appears two times.]{}
- [$\omega_1+\omega_6$: $A_2+A_1$, $5A_1$, appears two times.]{}
- [$\omega_4$: $A_2+2A_1$, $6A_1$, appears two times.]{}
- [$2\omega_2$: $A_2+3A_1$, $7A_1$, appears two times.]{}
- [$2\omega_3$: $3A_2$, $A_3+2A_1$, appears two times.]{}
- [$2\omega_1+2\omega_6$: $A_4$, $2A_3$, appears two times.]{}
- [$\omega_1+\omega_4+\omega_6$: $A_4+A_1$, $2A_3+A_1$, appears two times.]{}
Given a weight $\lambda$, there are at most two conjugacy classes of sub-root systems $\Phi_1,\Phi_2$ of $\E_7$ such that $2\delta'_{\Phi}=\lambda$. For each of such $\lambda$, the numbers of simple roots in $\Phi_1,\Phi_2$ are non-equal. Thus the coefficients of $\chi^{\ast}_{\omega_1}$ in $F_{\Phi_1,W_{\E_7}}$ and $F_{\Phi_2,W_{\E_7}}$ are different. Therefore the conclusion in the $\E_7$ case follows.
In the case that $\Psi=\E_8$, among the dominant integral weights appearing in $\{2\delta'_{\Phi}|\ \Phi\subset\E_8\}$, those weights that appearing more than once include $$\begin{aligned}
&& \{2\omega_8, \omega_1+\omega_8,
\omega_6, \omega_3, 2\omega_1, 2\omega_7, \omega_2+\omega_7, \omega_1+\omega_6, 2\omega_2,
2\omega_1+2\omega_8, \omega_1+\omega_6+\omega_8, \\&& \omega_4+\omega_8, 2\omega_2+2\omega_8,
2\omega_5, 2\omega_1+2\omega_6\}\end{aligned}$$ and the sub-root systems $\Phi$ with $2\delta_{\Phi}$ conjugate to them are
- [$2\omega_8$: $A_2$, $4A'_1$, appears two times.]{}
- [$\omega_1+\omega_8$: $A_2+A_1$, $5A_1$, appears two times.]{}
- [$\omega_6$: $A_2+2A_1$, $6A_1$, appears two times.]{}
- [$\omega_3$: $A_2+3A_1$, $7A_1$, appears two times.]{}
- [$2\omega_1$: $2A_2$, $A_2+4A_1$, $8A_1$, appears three times.]{}
- [$2\omega_7$: $A_3+2A_1$, $3A_2$, appears two times.]{}
- [$\omega_2+\omega_7$: $A_3+3A_1$, $3A_2+A_1$, appears two times.]{}
- [$\omega_1+\omega_6$: $A_3+4A_1$, $A_3+A_2$, appears two times.]{}
- [$2\omega_2$: $A_3+A_2+2A_1$, $4A_2$, appears two times.]{}
- [$2\omega_1+2\omega_8$: $A_4$, $2A_3$, appears two times.]{}
- [$\omega_1+\omega_6+\omega_8$: $A_4+A_1$, $2A_3+A_1$, appears two times.]{}
- [$\omega_4+\omega_8$: $A_4+2A_1$, $2A_3+2A_1$, appears two times.]{}
- [$2\omega_2+2\omega_8$: $D_4+A_2$, $D_4+4A_1$, appears two times.]{}
- [$2\omega_5$: $A_5+A_2+A_1$, $2A_4$, appears two times.]{}
- [$2\omega_1+2\omega_6$: $A_6$, $2D_4$, appears two times.]{}
One sees: for any two non-conjugate sub-root systems $\Phi_1,\Phi_2\subset\E_8$ with $2\delta'_{\Phi_1}=2\delta'_{\Phi_2}$, the numbers of simple roots of $\Phi_1,\Phi_2$ are non-equal, so the coefficients of $\chi^{\ast}_{\omega_1}$ in $F_{\Phi_1,W_{\E_8}},F_{\Phi_2,W_{\E_8}}$ are different. Thus $F_{\Phi_1,W_{\E_8}}\neq F_{\Phi_2,W_{\E_8}}$. Therefore the conclusion in the $\E_8$ case follows.
In the case that $\Psi=\F_4$, among the dominant integral weights appearing in $\{2\delta'_{\Phi}:\Phi\subset\F_4\}$, those weights that appearing more than once include $$\{\omega_1,\omega_3,2\omega_4,\omega_2,
2\omega_1,\omega_1+2\omega_4,2\omega_3,2\omega_1+2\omega_4,2\omega_3+2\omega_4\}$$ and the sub-root systems $\Phi$ with $2\delta_{\Phi}$ conjugate to them are
- [$\omega_1$: $A_1^{L}$, $2A_1^{S}$, appears 2 times.]{}
- [$\omega_3$: $A_1^{L}+A_1^{S}$, $3A_1^{S}$, appears 2 times.]{}
- [$2\omega_4$: $A_2^{S}$, $2A_1^{L}$, $A_1^{L}+2A_1^{S}$, $4A_1^{S}$, appears 4 times.]{}
- [$\omega_2$: $3A_1^{L}$, $2A_1^{L}+2A_1^{S}$, $A_1^{L}+A_2^{S}$, appears 3 times.]{}
- [$2\omega_1$: $A_2^{L}$, $4A_1^{L}$, appears 2 times.]{}
- [$\omega_1+2\omega_4$: $A_3^{S}$, $B_2$, appears 2 times.]{}
- [$2\omega_3$: $A_2^{L}+A_2^{S}$, $A_1^{L}+B_2$, $2A_1^{S}+B_2$, $A_1^{L}+A_3^{S}$, appears 4 times.]{}
- [$2\omega_1+2\omega_4$: $A_3^{L}$, $2B_2$, appears 2 times.]{}
- [$2\omega_3+2\omega_4$: $D_4^{S}$, $C_3$, appears 2 times.]{}
The non-conjugate pairs of sub-root system $\Phi_1,\Phi_2\subset\F_4$ with conjugate leading terms $2\delta'_{\Phi_{i}}$ and the same number of short simple roots are $(A_2^{S},A_1^{L}+2A_1^{S})$, $(2A_1^{L}+2A_1^{S},A_1^{L}+A_2^{S})$, $(A_2^{L},4A_1^{L})$, $(2A_1^{S}+B_2,A_1^{L}+A_3^{S})$. The coefficients of shortest terms in $F_{A_2^{L},F_4},F_{4A_1^{L},F_4}$ are non-equal, and $F_{2A_1^{S}+B_2}$ is not equal to $F_{A_1^{L}+A_3^{S}}$ since $$\begin{aligned}
F_{2A_1^{S}+B_2,W_{\F_4}}&=&1-3\chi^{\ast}_{\omega_4}+
2\chi^{\ast}_{\omega_1}+\chi^{\ast}_{\omega_3}-
\chi^{\ast}_{2\omega_4}+2\chi^{\ast}_{\omega_1+\omega_4}-4\chi^{\ast}_{\omega_2}+
2\chi^{\ast}_{\omega_3+\omega_4}-\\&&\chi^{\ast}_{2\omega_1}
+2\chi^{\ast}_{\omega_1+\omega_3}-\chi^{\ast}_{3\omega_4}+2\chi^{\ast}_{\omega_1+2\omega_4}
-3\chi^{\ast}_{\omega_2+\omega_4}+\chi^{\ast}_{2\omega_3}\end{aligned}$$ and $$\begin{aligned}
F_{A_1^{L}+A_3^{S},W_{\F_4}}&=&1-3\chi^{\ast}_{\omega_4}+7\chi^{\ast}_{\omega_3}-
3\chi^{\ast}_{2\omega_4}-6\chi^{\ast}_{\omega_1+\omega_4}+6\chi^{\ast}_{\omega_3+\omega_4}
+3\chi^{\ast}_{2\omega_1}-\\&&6\chi^{\ast}_{\omega_1+\omega_3}-\chi^{\ast}_{3\omega_4}+
3\chi^{\ast}_{\omega_2+\omega_4}-\chi^{\ast}_{2\omega_3}. \end{aligned}$$ Calculation shows that $$F_{A_2^{S},W_{\F_4}}=F_{A_1^{L}+2A_1^{S},W_{\F_4}}=1-2\chi^{\ast}_{\omega_4}+
2\chi^{\ast}_{\omega_3}-\chi^{\ast}_{2\omega_4}$$ and $$F_{A_1^{L}+A_2^{S},W_{\F_4}}=
F_{2A_1^{L}+2A_1^{S},W_{\F_4}}=1-2\chi^{\ast}_{\omega_4}-\chi^{\ast}_{\omega_1}+4\chi^{\ast}_{\omega_3}-
\chi^{\ast}_{2\omega_4}-2\chi^{\ast}_{\omega_1+\omega_4}+\chi^{\ast}_{\omega_2}.$$ Therefore the conclusion in the $\F_4$ case follows.
In the case that $\Psi=\G_2$, the only non-conjugate pair $(\Phi_1,\Phi_2)$ of sub-root systems such that $2\delta'_{\Phi_1}=2\delta'_{\Phi_2}$ is $(A_2^{S},A_1^{L}+A_1^{S})$. The numbers of short simple roots of $A_2^{S}$ and $A_1^{L}+A_1^{S}$ are different, so the coefficients of $\chi^{\ast}_{\omega_2}$ in $F_{A_2^{S},W_{G_2}}, F_{A_1^{L}+A_1^{S},W_{G_2}}$ are non-equal. Thus $F_{\Phi_1,W_{\G_2}}\neq F_{\Phi_2,W_{\G_1}}$. Therefore the conclusion in the $\G_2$ case follows.
Linear relations among dimension data {#S:dimension-dependent}
=====================================
In this section, we solve Question \[Q:dependent-character\].
First, once we know all linear relations among $\{F_{\Phi,W_{\Psi}}|\ \Phi\subset\Psi\}$, we also know all linear relations among $\{F_{\Phi,W}|\ \Phi\subset\Psi\}$ for any finite group $W$ between $W_{\Psi}$ and $\Aut(\Psi)$. So we just need to consider the linear relations among $\{F_{\Phi,W_{\Psi}}|\ \Phi\subset\Psi\}$.
If $\Psi$ is not an irreducible root system, let $$\Psi=\bigsqcup_{1\leq i\leq s}\Psi_{i}$$ be the decomposition of $\Psi$ into a direct sum of simple root systems. For a reduced sub-root system $\Phi$ of $\Psi$, $\Phi$ can be written as $\Phi=\bigsqcup_{1\leq i\leq s}\Phi_{i}$, where $\Phi_{i}\subset\Psi_{i}$ for any $1\leq i\leq s$. Thus we have $$F_{\Phi,W_{\Psi}}=F_{\Phi_1,W_{\Psi_1}}\otimes\cdots\otimes F_{\Phi_{s},W_{\Psi_{s}}}.$$ From this, we see that linear relations among $\{F_{\Phi,W_{\Psi}}|\ \Phi\subset\Psi\}$ arise from linear relations among $\{F_{\Phi_{i},W_{\Psi_{i}}}|\ \Phi_{i}\subset\Psi_{i}, 1\leq i\leq s\}$. Hence it is sufficient to consider $\{F_{\Phi,W_{\Psi}}|\ \Phi\subset\Psi\}$ for reduced sub-root systems $\{\Phi|\ \Phi\subset\Psi\}$ of an irreducible root system $\Psi$.
\[R:linear2\] The passing from $\{F_{\Phi,W}|\Phi\subset\Psi\}$ to $\{F_{\Phi,W_{\Psi}}|\Phi\subset\Psi\}$ is like that: any linear relation among the former is also a linear relation among the latter; and any linear relation among the latter gives a linear relation among the former after the $W$-averaging process, that is to replace a character $F_{\Phi,W_{\Psi}}\in{\mathbb{Q}}[\Lambda]$ by $$F_{\Phi,W}=\frac{1}{|W|}\sum_{\gamma\in w}\gamma F_{\Phi,W_{\Psi}}.$$ But the relation between the two sets of linear relations might not be described explicitly, there are at least two reasons for this. The first reason is the number of distinct characters in $\{\gamma F_{\Phi,W_{\Psi}}:\gamma\in W/W_{\Psi}\}$ may vary; the second and the more serious reason is different linear relations among $\{F_{\Phi,W_{\Psi}}|\Phi\subset\Psi\}$ may give the same linear relation among $\{F_{\Phi,W}|\Phi\subset\Psi\}$ after the $W$-averaging process.
The passing from $\{F_{\Phi,W_{\Psi}}|\Phi\subset\Psi\}$ to $\{F_{\Phi_{i},W_{\Psi_{i}}}|\Phi_{i}
\subset\Psi_{i}\}$ is reasonably well, since by Linear Algebra all linear relations among the former can be explicitly expressed in terms of linear relations among the latter.
\[R:equal2\] Another way of getting all linear relations among $\{F_{\Phi,W}|\Phi\subset\Psi\}$ is to express each $F_{\Phi,W}$ in terms of $\{F_{\Phi_{i},W_{\Psi_{i}}}\}$ by the tensor operation and the $W$-averaging operation. Starting from all linear linear relations among $\{F_{\Phi_{i},W_{\Psi_{i}}}:\Phi_{i}\subset\Psi_{i}\}$, in this way we can get all linear relations among $\{F_{\Phi,W}|\Phi\subset\Psi\}$.
For two reduced sub-root systems $\Phi_1,\Phi_2\subset\Psi$ with $\Phi_2\not\sim_{W}\Phi_1$, the above way is useful for checking if $F_{\Phi_1,W}=F_{\Phi_2,W}$ or not. We express $F_{\Phi_1,W}-F_{\Phi_2,W}$ in terms of $\{F_{\Phi_{i},W_{\Psi_{i}}}\}$ by tensor operation, addition and subtraction. Then the linear relations among $\{F_{\Phi_{i,j},W_{\Psi_{i}}}:\Phi_{i,j}\subset\Psi_{i}\}$ tells us if such an expression is $0$ or not. For example, in the case that $\Psi=m\Psi_0$ and $W=(W_{\Psi_0})^{m}\rtimes A_{m}$, if $\Phi_1,\dots,\Phi_{m}\subset\Psi_0$ are reduced sub-root systems such that $F_{\Phi_1,W_{\Psi_0}},\dots,F_{\Phi_m,W_{\Psi_0}}$ are linearly dependent, then for $$\Phi=\Phi_1\sqcup\Phi_2\sqcup\cdots\sqcup\Phi_{m}$$ and $$\Phi=\Phi_{2}\sqcup\Phi_{1}\sqcup\Phi_3\sqcup\Phi_4\sqcup\cdots\sqcup\Phi_{m},$$ we have $F_{\Phi_1,W}=F_{\Phi_2,W}$.
Algebraic relations among $\{a_{n},b_{n},b'_{n},c_{n},d_{n}|\ n\geq 1\}$ {#SS:abcd}
------------------------------------------------------------------------
The following proposition can be proven in a similar way as the proof of Proposition \[P:A=BB and A=CD\]. The method is: after showing equalities (\[Eq:an\])-(\[Eq:dn\]), we are led to prove some identities for determinants.
\[P:A-BB’CD\] For any $n\geq 0$, we have $a_{2n}=b_{n}b'_{n}$, $a_{2n+1}=c_{n}d_{n+1}$, $2a_{2n}=c_{n}d_{n}+c_{n-1}d_{n+1}$ and $2a_{2n+1}=b_{n}b'_{n+1}+b'_{n}b_{n+1}$.
The equalties $$\label{Eq:a1} a_{2n}=b_{n}b'_{n}$$ and $$\label{Eq:a2} a_{2n+1}=c_{n}d_{n+1}$$ are proven in Proposition \[P:A=BB and A=CD\]. We prove $$\label{Eq:a3} 2a_{2n}=c_{n}d_{n}+c_{n-1}d_{n+1}$$ and $$\label{Eq:a4} 2a_{2n+1}=b_{n}b'_{n+1}+b'_{n}b_{n+1}$$ here. In the proof of Proposition \[P:A=BB and A=CD\], we have introduced the matrices $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, $D'_{n}$ and expressed their determinants in terms of the polynomials $a_{n}$, $b_{n}$, $c_{n}$, $d_{n}$.
Let $$A'_{2n}=\left(\begin{array}{ccccc}x_0&x_1&\ldots&x_{2n-2}&x_{2n-1}\\x_1&x_0&\ldots&x_{2n-3}&x_{2n-2}\\
\vdots&\vdots&&\vdots&\vdots\\x_{2n-2}&x_{2n-3}&\ldots&x_0&x_1\\x_{2n-1}+x_{1}&x_{2n-2}+x_{2}&\ldots
&x_1+x_{2n-1}&x_0+x_{2n}\\\end{array}\right)$$ and $$A''_{2n}=\left(\begin{array}{ccccc}x_0&x_1&\ldots&x_{2n-2}&x_{2n-1}\\x_1&x_0&\ldots&x_{2n-3}&x_{2n-2}\\
\vdots&\vdots&&\vdots&\vdots\\x_{2n-2}&x_{2n-3}&\ldots&x_0&x_1\\x_{2n-1}-x_{1}&x_{2n-2}-x_{2}&\ldots
&x_1-x_{2n-1}&x_0-x_{2n}\\\end{array}\right).$$ Then $$\det A'_{2n}+\det A''_{2n}=2\det A_{2n}=2a_{2n}.$$
Define $$J'_{2n}=\left(\begin{array}{cccc}\frac{1}{\sqrt{2}}I_{n-1}&0_{(n-1)\times 1}&-\frac{1}{\sqrt{2}}L_{n-1}
&0_{(n-1)\times 1}\\0_{1\times(n-1)}&1&0_{1\times(n-1)}&0\\\frac{1}{\sqrt{2}}L_{n-1}&0_{(n-1)\times 1}&\frac{1}
{\sqrt{2}}I_{n-1}&0_{(n-1)\times 1}\\0_{1\times(n-1)}&0&0_{1\times(n-1)}&1\end{array}\right)$$ and $$J''_{2n}=\left(\begin{array}{cccc}\frac{1}{\sqrt{2}}I_{n-1}&0_{(n-1)\times 1}&\frac{1}{\sqrt{2}}L_{n-1}
&0_{(n-1)\times 1}\\0_{1\times(n-1)}&1&0_{1\times(n-1)}&0\\-\frac{1}{\sqrt{2}}L_{n-1}&0_{(n-1)\times 1}&\frac{1}
{\sqrt{2}}I_{n-1}&0_{(n-1)\times 1}\\0_{1\times(n-1)}&0&0_{1\times(n-1)}&1\end{array}\right).$$ By matrix calculation, we get $$J'_{2n}A'_{2n}(J'_{2n})^{-1}=\left(\begin{array}{cc}X_{1}&\ast_1\\0_{(n+1)\times(n-1)}&X_2\end{array}
\right)$$ with $\det X_1=c_{n-1}$ and $\det X_2=d_{n+1}$. Similarly we get $$J''_{2n}A''_{2n}(J''_{2n})^{-1}=\left(\begin{array}{cc}Y_{2}&\ast_2\\0_{n\times n}&Y_1\end{array}
\right)$$ with $\det Y_1=c_{n}$ and $\det Y_2=d_{n}$. Taking determinants, we get $$2a_{2n}=c_{n}d_{n}+c_{n-1}d_{n+1}.$$
Let $$A'_{2n+1}=\left(\begin{array}{ccccc}x_0&x_1&\ldots&x_{2n-2}&x_{2n}\\x_1&x_0&\ldots&x_{2n-2}&x_{2n-1}\\
\vdots&\vdots&&\vdots&\vdots\\x_{2n-1}&x_{2n-2}&\ldots&x_0&x_1\\x_{2n}+x_{1}&x_{2n-1}+x_{2}&\ldots
&x_1+x_{2n}&x_0+x_{2n+1}\\\end{array}\right)$$ and $$A''_{2n+1}=\left(\begin{array}{ccccc}x_0&x_1&\ldots&x_{2n-1}&x_{2n}\\x_1&x_0&\ldots&x_{2n-2}&x_{2n-1}\\
\vdots&\vdots&&\vdots&\vdots\\x_{2n-1}&x_{2n-2}&\ldots&x_0&x_1\\x_{2n}-x_{1}&x_{2n-1}-x_{2}&\ldots
&x_1-x_{2n}&x_0-x_{2n+1}\\\end{array}\right).$$ Then we have $$\det A'_{2n+1}+\det A''_{2n+1}=2\det A_{2n+1}=2a_{2n+1}.$$
Let $$J'_{2n+1}=\left(\begin{array}{cccc}\frac{1}{\sqrt{2}}I_{n-1}&-\frac{1}{\sqrt{2}}L_{n-1}
&0_{(n-1)\times 1}\\\frac{1}{\sqrt{2}}L_{n-1}&\frac{1}{\sqrt{2}}I_{n-1}&0_{(n-1)\times 1}\\
0_{1\times(n-1)}&0_{1\times(n-1)}&1\end{array}\right)$$ and $$J''_{2n+1}=\left(\begin{array}{cccc}\frac{1}{\sqrt{2}}I_{n-1}&\frac{1}{\sqrt{2}}L_{n-1}
&0_{(n-1)\times 1}\\-\frac{1}{\sqrt{2}}L_{n-1}&\frac{1}{\sqrt{2}}I_{n-1}&0_{(n-1)\times 1}\\
0_{1\times(n-1)}&0_{1\times(n-1)}&1\end{array}\right).$$ By matrix calculation, we get $$J'_{2n+1}A'_{2n+1}(J'_{2n+1})^{-1}=\left(\begin{array}{cc}X_{1}&\ast_1\\0_{(n+1)\times(n-1)}&X_2\end{array}
\right)$$ with $\det X_1=b_{n}$ and $\det X_2=b'_{n+1}$. Similarly we get $$J''_{2n+1}A''_{2n+1}(J''_{2n+1})^{-1}=\left(\begin{array}{cc}Y_{2}&\ast_2\\0_{n\times n}&Y_1\end{array}
\right)$$ with $\det Y_1=b_{n+1}$ and $\det Y_2=b'_{n}$. Taking determinants, we get $$2a_{2n+1}=b_{n}b'_{n+1}+b_{n+1}b'_{n}.$$
\[R:A-BB’CD\] Here, we remark that, the equations (\[Eq:an\])-(\[Eq:dn\]) are discovered in [@An-Yu-Yu] and the equalities $$\det A_{2n}=\det B_{n}\det B'_{n},$$ $$\det A_{2n+1}=\det C_{n}\det D'_{n+1},$$ $$2\det A_{2n}=\det C_{n}\det D'_{n}+\det C_{n-1}\det D'_{n+1}$$ and $$2\det A_{2n+1}=\det B_{n}\det B'_{n+1}+\det B_{n+1}\det B'_{n}$$ are proven in the book [@Vein-Dale], Page 88.
\[P:algebraic relations\] The equalities $a_{2n}=b_{n}b'_{n}$, $2a_{2n}=c_{n}d_{n}+c_{n-1}d_{n+1}$, $a_{2n+1}=c_{n}d_{n+1}$, $2a_{2n+1}=b_{n}b'_{n+1}+b'_{n}b_{n+1}$ generate all algebraic relations among $$\{a_{n},b_{n},b'_{n},c_{n},d_{n}|n\geq 1\}.$$
By Proposition \[basic properties\], we know that $\{b_{n},c_{n}|n\geq 1\}$ are algebraically independent. It is clear that, from these equalities we can express $a_{n},b'_{n},d_{n}$ in terms of rational functions of $\{b_{m},c_{m}|1\leq m\leq n\}$, so these equalities generate all algebraic relations among $\{a_{n},b_{n},b'_{n},c_{n},d_{n}|n\geq 1\}$.
More important than Proposition \[P:algebraic relations\] itself is: we can use the four identities to derive many other algebraic relations. Moreover, by expressing $a_{n},b'_{n},d_{n}$ in terms of rational functions of $\{b_{m},c_{m}|1\leq m\leq n\}$, we are able to check whether any given polynomial function of $\{a_{n},b_{n},b'_{n},c_{n},d_{n}|n\geq 1\}$ is identical to 0 or not.
**Type $\BC$.** From the equations $b_{n+1}b'_{n}+b_{n}b_{n+1}=2c_{n}d_{n+1}$, $2b_{n}b'_{n}=c_{n}d_{n}+c_{n-1}d_{n+1}$, $2b_{n+1}b'_{n+1}=c_{n+1}d_{n+1}+c_{n}d_{n+2}$, we get $$\label{Eq:BC} b_{n+1}^{2}(c_{n}d_{n}+c_{n-1}d_{n+1})+b_{n}^{2}(c_{n+1}d_{n+1}+c_{n}d_{n+2})-
4b_{n+1}b_{n}c_{n}d_{n+1}=0.$$ When $n=0$, this equation is $2b_1^{2}+(c_1d_1+d_2)-4b_1d_1=0$. Since $\{d_{n},c_{n}|n\geq 1\}$ are algebraically independent, so these equations generate all algebraic relations among $\{b_{n},c_{n},d_{n}|\ n\geq 1\}$. Together with the identities $a_{2n-1}-c_{n-1}d_{n}=0$ and $2a_{2n}-c_{n}d_{n}-c_{n-1}d_{n+1}=0$, they generate all algebraic relations among $\{a_{n},b_{n},c_{n},d_{n}|\ n\geq 1\}$.
**Type $\B$.** From $2a_{2n+1}=b_{n+1}b'_{n}+b_{n}b_{n+1}$, $a_{2n+2}=b_{n+1}b'_{n+1}$, $a_{2n}=b_{n}b'_{n}$, we get $$\label{Eq:B1}a_{2n+2}b_{n}^{2}+a_{2n}b_{n+1}^{2}=
2a_{2n+1}b_{n}b_{n+1}.$$ From $2a_{2n}=c_{n}d_{n}+c_{n-1}d_{n+1}$, $a_{2n+1}=c_{n}d_{n+1}$, $a_{2n-1}=c_{n-1}d_{n}$, we get $$\label{Eq:B2} a_{2n+1}d_{n}^{2}+a_{2n-1}d_{n+1}^{2}=
2a_{2n}d_{n}d_{n+1}.$$ The equalities $$\{a_{2n+2}b_{n}^{2}+a_{2n}b_{n+1}^{2}-2a_{2n+1}b_{n}b_{n+1}=0|\ n\geq 0\}$$ and $$\{a_{2n+1}d_{n}^{2}+a_{2n-1}d_{n+1}^{2}-2a_{2n}d_{n}d_{n+1}=0|\ n\geq 1\}$$ generate all algebraic relations among $\{a_{n},b_{n},d_{n}|\ n\geq 1\}$. Here we regard $a_{-1}=d_1^{-1}=x_0^{-1}$, $a_0=1$.
**Type $\C$.** We have $$\label{Eq:C1} a_{2n+1}=c_{n}d_{n+1}$$ and $$\label{Eq:C2} 2a_{2n+2}=c_{n+1}d_{n+1}+c_{n}d_{n+2}$$ As $\{d_{n},c_{n}|n\geq 1\}$ are algebraically independent, the equalities $$\{a_{2n+1}-c_{n}d_{n+1}=0|\ n\geq 1\}$$ and $$\{2a_{2n+2}-c_{n+1}d_{n+1}-c_{n}d_{n+2}=0|\ n\geq 0\}$$ generate all algebraic relations among $\{a_{n},c_{n},d_{n}|\ n\geq 1\}$.
**Type $\D$.** The equations $$\{a_{2n+1}d_{n}^{2}+a_{2n-1}d_{n+1}^{2}-2a_{2n}d_{n}d_{n+1}=0|\ n\geq 1\}$$ generate all algebraic relations among $\{a_{n},d_{n}|\ n\geq 1\}$.
\[R:ideal\] Let $$R={\mathbb{Q}}[\{y_{1,n},y_{2,n},y_{3,n},y_{4,n},y_{5,n}|\ n\geq 1\}]$$ be the free polynomial algebra over the rational field with infinitely many indeterminates $$\{y_{1,n},y_{2,n},y_{3,n},y_{4,n},y_{5,n}|\ n\geq 1\}.$$ Let $$\Eva: R\longrightarrow{\mathbb{Q}}[\{x_{n}|\ n\geq 1\}]$$ be the homomorphism defined by $$\Eva(x_{1,n})=a_{n},\ \Eva(x_{2,n})=b_{n},\ \Eva(x_{3,n})=b'_{n}$$ and $$\Eva(x_{4,n})=c_{n},\ \Eva(x_{5,n})=d_{n}.$$ Let $I$ be the kernel of $I$. Let $$\label{Eq:I1} f_{1,n}=y_{1,2n}-y_{2,n}y_{3,n},$$ $$\label{Eq:I2} f_{2,n}=y_{1,2n+1}-y_{4,n}y_{5,n+1},$$ $$\label{Eq:I3} f_{3,n}=2y_{1,2n}-y_{4,n}y_{5,n}-y_{4,n-1}y_{5,n+1}$$ and $$\label{Eq:I4} f_{4,n}=2y_{1,2n+1}-y_{2,n}y_{3,n+1}-y_{3,n}y_{2,n+1}.$$ Then $f_{1,n},f_{2,n},f_{3,n},f_{4,n}\in I$ for any $n\geq 1$. Moreover, let $R_1$ be the lozalization of $R$ with respect to the multiplicative system generated by $\{y_{2,n},y_{4,n}|\ n\geq 1\}$. Then we have a homomorphism $\Eva_1: R_1\longrightarrow{\mathbb{Q}}(\{x_{n}|\ n\geq 1\}])$. One can show that, the elements $\{f_{1,n},f_{2,n},f_{3,n},f_{4,n}|\ n\geq 1\}$ genetate the kernel of $\Eva_1$. Similarly, they generate the corresponding kernels if we consider the localization with respect to the multiplicative system generated by $\{y_{3,n},y_{4,n}|\ n\geq 1\}$ (or $\{y_{2,n},y_{4,n}|\ n\geq 1\}$, etc) and the similar homomorphism. Does $\{f_{1,n},f_{2,n},f_{3,n},f_{4,n}|\ n\geq 1\}$ generate $I$? It seems to the author that the answer to this is no. In that case, can one find a system of gneerators of $I$? One could consider a subset of $\{a_{n},b_{n},b'_{n},c_{n},d_{n}|\ n\geq 1\}$ (e.g, $\{a_{n},b_{n},c_{n},d_{n}|\ n\geq 1\}$), the similar homomorphism as the above $\Eva$ and ask about the generators of the kernel.
Classical irreducible root systems {#SS:classical root system}
----------------------------------
\[P:gamma-character\] Given a root system $\Psi$ and an automorphism $\gamma\in\Aut(\Psi)$, if $\gamma\Phi=\Phi$ for a reduced sub-root system $\Phi$ of $\Psi$, then $\gamma F_{\Phi,W_{\Psi}}=F_{\Phi,W_{\Psi}}$.
Replacing $\gamma$ by some $w\gamma$ ($w\in W_{\Phi}$) if necessary, we may assume that $\gamma\Phi^{+}=\Phi^{+}$. Then $\gamma\delta_{\Phi}=\delta_{\Phi}$ and $\gamma$ maps simple roots of $\Phi$ to simple roots. By the latter, we get that $\gamma W_{\Phi}\gamma^{-1}=W_{\Phi}$.
Hence $$\begin{aligned}
\gamma F_{\Phi,W_{\Psi}}&=&\gamma(\sum_{w\in W_{\Phi}}
\chi^{\ast}_{\delta_{\Phi}-w\delta_{\Phi},W_{\Psi}})\\&=&\sum_{w\in W_{\Phi}}
\chi^{\ast}_{\gamma\delta_{\Phi}-(\gamma w\gamma^{-1})\gamma\delta_{\Phi},W_{\Psi}}
\\&=&\sum_{w\in W_{\Phi}}\chi^{\ast}_{\delta_{\Phi}-(\gamma w\gamma^{-1})\delta_{\Phi},W_{\Psi}}
\\&=&\sum_{w\in W_{\Phi}}\chi^{\ast}_{\delta_{\Phi}-w\delta_{\Phi},W_{\Psi}}\\&=
& F_{\Phi,W_{\Psi}}.\end{aligned}$$
Given a classical irreducible root system $\Psi$ of rank $n$, in the case that $\Psi=\A_{n}$ ($n\geq 1$), we have the following simple statement.
\[P:linear-A\] For any sub-root system $\Phi\subset\A_{n}$, $F_{\Phi,W_{\A_{n}}}=F_{\Phi,\Aut(\A_{n})}$.
For any two sub-root systems $\Phi_1,\Phi_2\subset\A_{n}$, $2\delta'_{\Phi_1}=
2\delta'_{\Phi_2}$ if and only if $\Phi_1\sim\Phi_2$.
For any pairwise non-conjugate sub-root systems $\Phi_1,\dots,\Phi_{s}\subset\A_{n}$, the characters $\{F_{\Phi_{i},W_{\A_{n}}}|1\leq i\leq s\}$ are linearly independent.
By Proposition \[P:gamma-character\], we have $F_{\Phi,W_{\A_{n}}}=F_{\Phi,\Aut(\A_{n})}$ since there exists $\gamma\in\Aut(\A_{n})-W_{\A_{n}}$ such that $\gamma\Phi=\Phi$ (the index of $W_{\A_{n}}$ in $\Aut(\A_{n})$ is $2$).
For a sub-root system $$\Phi\cong\bigsqcup_{1\leq i\leq s}\A_{n_{i}-1}\subset\A_{n}$$ with $n_{i}\geq 1$ and $\sum_{1\leq i\leq s}n_{i}=n+1$, $$2\delta_{\Phi}\sim(\underbrace{n_1-1,n_1-3,...,1-n_1},....,
\underbrace{n_{s}-1,n_{s}-3,...,1-n_{s}}).$$ From this one sees that the weights $2\delta_{\Phi}$ are non-conjugate for any two non-conjugate sub-root systems. Therefore the remaining conclusions in the proposition follow.
In the case that $\Psi=\BC_{n}$, $\B_{n}$ or $\C_{n}$, we have $W_{\Psi}=W_{\BC_{n}}=W_{n}$. Therefore, the question of getting linear relations among $\{F_{\Phi,W_{\Psi}}|\ \Phi\subset\Psi\}$ reduces to the question of getting algebraic relations of homogeneous degree $n$ among the polynomials $\{a_{m},b_{m},c_{m},d_{m}|\ 1\leq m\leq n\}$, $\{a_{m},b_{m},d_{m}|\ 1\leq m\leq n\}$, $\{a_{m},c_{m},d_{m}|\ 1\leq m\leq n\}$, respectively. These algebraic relations are discussed in the last subsection.
In the case of $\Psi=\BC_{n}$, we make an interesting observation. We have showed the equations $b_{n+1}^{2}(c_{n}d_{n}+c_{n-1}d_{n+1})+b_{n}^{2}(c_{n+1}d_{n+1}+c_{n}d_{n+2})-
4b_{n+1}b_{n}c_{n}d_{n+1}=0$ for any $n\geq 0$. When $n=0$ and $n=1$, they are $2b_1^{2}+(c_1d_1+d_2)-4b_1d_1=0$ and $$b_{2}^{2}(c_{1}d_{1}+d_{2})+b_{1}^{2}(c_{2}d_{2}+c_{1}d_{3})-
4b_{2}b_{1}c_{1}d_{2}=0.$$ Eliminating $d_1$, we get $$\begin{aligned}
0&=&(4b_1-c_1)(b_{2}^{2}(c_{1}d_{1}+d_{2})+b_{1}^{2}(c_{2}d_{2}+c_{1}d_{3})-
4b_{2}b_{1}c_{1}d_{2})\\&&+(b_2^{2}c_1)(2b_1^{2}+(c_1d_1+d_2)-4b_1d_1)
\\&=&b_1(2b_1b_2^{2}c_1+4b_1^2c_2d_2+4b_1^2c_1d_3+4b_2c_1^2d_2+
4b_2^{2}d_2\\&&-b_1c_1c_2d_2-b_1c_1^2d_3-16b_1b_2c_1d_2).\end{aligned}$$ Since $b_1=x_0-x_1$ is irreducible, we get $$2b_1b_2^{2}c_1+4b_1^2c_2d_2+
4b_1^2c_1d_3+4b_2c_1^2d_2+4b_2^{2}d_2-b_1c_1c_2d_2-b_1c_1^2d_3-16b_1b_2c_1d_2=0.$$ This gives 8 rank-6 semisimple subgroups of $\SU(15)$ with linearly dependent dimension data. This equation is also given in [@An-Yu-Yu], Example 5.6 and implicit in [@Larsen-Pink], Page 393.
In the case of $\Psi=\D_{n}$ ($n\geq 4$), algebraic relations of homogeneous degree $n$ among $\{a_{m},c_{m},d_{m}|1\leq m\leq 1\}$ correspond to the linear relations among the characters $$\{F_{\Phi,W_{n}}|\ \Phi\subset\D_{n}\}.$$ Note that $$W_{D_{n}}=\Gamma_{n}\rtimes S_{n}$$ is a subgroup of $W_{n}$ of index 2, where $$\Gamma_{n}=\{(a_1,a_2,\dots,a_{n})\subset\{\pm{1}\}^{n}:a_1a_2\cdots a_{n}=1\}.$$
\[P:Character-D\] Given $\Psi=\D_{n}$ ($n\geq 4$) and a sub-root system $\Phi\subset\Psi$, the following conditions are equivalent to each other:
- [$F_{\Phi,W_{\Psi}}\neq F_{\Phi,W_{n}}$.]{}
- [$2\delta'_{\Phi}$ is not $W_{n}$ invariant.]{}
- [$$\Phi\cong\bigsqcup_{1\leq i\leq s}\A_{n_{i}-1}$$ with $2\leq n_1\leq n_2\cdots\leq n_{s}\leq n$, each $n_{i}$ is even, and $n_1+n_2+\cdots+n_{s}=n$.]{}
- [$\Phi\not\sim_{W_{\D_{n}}}\gamma\Phi$ for some (equivalently, for any) $\gamma\in W_{n}-W_{\D_{n}}$.]{}
$(2)\Rightarrow(1)$ is clear. We prove $(1)\Rightarrow(3)$, $(3)\Rightarrow(2)$ and $(3)\Leftrightarrow(4)$ in the below.
Any sub-root system $\Phi$ of $\D_{n}$ (cf. [@Larsen-Pink] and [@Oshima]) is conjugate to one of $$(\bigsqcup_{1\leq i\leq s}\D_{n_{i}})\bigsqcup(\bigsqcup_{1\leq j\leq t}\A_{m_{j}}),$$ where $n_{i}\geq 1$, $m_{j}\geq 2$, $\sum_{1\leq i\leq s}n_{i}+\sum_{1\leq j\leq t}m_{j}=n$. Thus $(3)\Leftrightarrow(4)$.
If $(3)$ does not hold, then $s\geq 1$ or $s=0$ and some $m_{i}$ is odd. In the case that $s\geq 1$, we may assume that $\D_{n_1}=\langle e_1-e_2,\dots,e_{n_{1}-1}-e_{n_1},e_{n_{1}-1}+e_{n_1}\rangle\subset\Phi$. Thus $s_1\Phi=\Phi$. By Proposition \[P:gamma-character\] we get $s_1 F_{\Phi,W_{\Psi}}=F_{\Phi,W_{\Psi}}$. Since $s_1,W_{\D_{n}}$ generate $\Aut(\D_{n})$ ($n\geq 5$), we have $$F_{\Phi,W_{\Psi}}=F_{\Phi,W_{n}}.$$ In the case that $s=0$ and some $m_{j}$ is odd, we may and do assume that $m_1$ is odd and $A_{m_1}=\langle e_1-e_2,\dots,
e_{m_{1}-1}-e_{m_1}\rangle$. Then $s_1s_2\cdots s_{m_1}\Phi=\Phi$. By Proposition \[P:gamma-character\], $$s_1s_1\cdots s_{m_1} F_{\Phi,W_{\Psi}}=F_{\Phi,W_{\Psi}}.$$ Since $s_1s_2\cdots s_{m_1},W_{\D_{n}}$ generate $\Aut(\D_{n})$ ($n\geq 5$), we get $F_{\Phi,W_{\Psi}}=F_{\Phi,W_{n}}$. This proves $(1)\Rightarrow(3)$.
If $(3)$ holds, by calculation we get $$2\delta_{\Phi}=(n_{1}-1,n_{1}-3,\dots,1-n_1,\dots,n_{s}-1,\dots,3-n_{s},1-n_{s}).$$ Thus $2\delta'_{\Phi}$ is not $W_{n}$ invariant. This proves $(3)\Rightarrow(2)$.
By Proposition \[P:Character-D\], for any sub-root system $\Phi$ of $\D_n$, either $W_{\D_{n}}\Phi=W_{n}\Phi$ and $F_{\Phi,W_{\D_{n}}}=F_{\Phi,W_{n}}$, or $W_{\D_{n}}\Phi\neq W_{n}\Phi$ and $F_{\Phi,W_{\D_{n}}}$, $F_{\gamma\Phi,W_{\D_{n}}}$ are linearly independent. Here $\gamma$ is any element in $W_{n}-W_{\D_{n}}$. In this way we get all linear relations among $\{F_{\Phi,W_{\D_{n}}}: \Phi\subset\D_{n}\}$ from the linear relations among $\{F_{\Phi,W_{n}}: \Phi\subset\D_{n}\}$.
In the case of $\Psi=\D_4$, its automorphism group is larger than $W_{n}$. We discuss more on this case. Firstly the $W_{\Psi}$-conjugacy classes of sub-root systems are $$\emptyset,\ A_1,\ A_2,\
D_2,\ 2A_1,\ (2A_1)',\ A_3$$ and $$(A_3)',\ D_3,\ D_2+A_1,\ D_4,\ 2D_2.$$ Here $$D_2=\langle e_1-e_2,e_1+e_2\rangle,$$ $$2A_1=\langle e_1-e_2,e_3-e_4\rangle,$$ $$(2A_1)'=\langle e_1-e_2,e_3+e_4\rangle,$$ $$A_3=\langle e_1-e_2,e_2-e_3,e_3-e_4\rangle,$$ $$(A_3)'=\langle e_1-e_2,e_2-e_3,e_3+e_4\rangle$$ and $$D_3=\langle e_2-e_3,e_3-e_4,e_3+e_4\rangle.$$ The sub-root systems $D_2$, $2A_1$, $(2A_1)'$ are conjugate to each other under $\Aut(\D_4)$, and the sub-root systems $A_3$, $(A_3)'$, $D_3$ are conjugate to each other under $\Aut(\D_4)$.
Only the characters of sub-root systems $A_2$ and $4A_1$ have equal leading term, which is $2\omega_2$. We have $$\begin{aligned}
&&F_{A_2,W_{\Psi}}=1-2\chi^{\ast}_{\omega_2}+
2\chi^{\ast}_{\omega_1+\omega_3+\omega_4}-\chi^{\ast}_{2\omega_2},\\&& F_{4A_1,W_{\Psi}}
=1-4\chi^{\ast}_{\omega_2}+2(\chi^{\ast}_{2\omega_1}+\chi^{\ast}_{2\omega_3}+
\chi^{\ast}_{2\omega_4})-4\chi^{\ast}_{\omega_1+\omega_3+\omega_4}+\chi^{\ast}_{2\omega_2}.\end{aligned}$$ A little more calculation shows that $$F_{A_2,W_{\Psi}}+F_{4A_1,W_{\Psi}}-
2F_{D_2+A_1,W_{\Psi}}=0.$$ This equality also follows from the equality $a_3+d_2^{2}=d_2(c_1+d_2)=2a_2d_2$ by noting that $F_{\Phi,W_{\D_4}}=F_{\Phi,W_4}$ if $\Phi=A_2$, $4A_1$ or $D_2+A_1$. This is the only linear relation among $\{F_{\Phi,W_{\D_4}}|\ \Phi\subset\D_4\}$.
\[P:linear-BCD\] Given a compact connected simple Lie group $G$ of type $\B_{n}$ ($n\geq 4$), $\C_{n}$ ($n\geq 3$) or $\D_{n}$ ($n\geq 4$), there exist non-isomorphic closed connected subgroups of $G$ with linearly dependent dimension data.
Taking $n=1$ in Equation (\[Eq:B2\]) and using $a_1=d_1=x_0$, we get $$a_3d_1+d_2^2-2a_2d_2=0.$$ This gives us a linear relation of three non-isomorphic subgroups of $G$ in the case that $G$ is of type $\B_{n}$ or $\D_{n}$ with $n\geq 4$. In the $\D_n$ case, by Proposition \[P:Character-D\] we know that the $W_{\D_{n}}$ trace and the $W_{n}$ trace of characters are equal for the sub-root systems corresponding to the polynomials $a_3d_1$, $d_2^2$ and $a_2d_2$.
By Equations (\[Eq:C1\]) and (\[Eq:C2\]), we get $$2a_{2n+2}c_{n}c_{n+1}-c_{n+1}^{2}a_{2n+1}-
c_{n}^{2}a_{2n+3}=0.$$ Taking $n=0$, we get $2a_2c_1-a_1c_1^{2}-a_3=0$, which gives us a linear relation of three non-isomorphic subgroups of $G$ in the case that $G$ is of type $\C_{n}$ with $n\geq 3$.
A generating function {#SS:generating function}
---------------------
Given an irreducible root system $\Psi$ with a positive system $\Psi^{+}$, a root $\alpha\in\Psi$ is called a short root if for any other root $\beta\in\Psi$, either $(\alpha,\beta)=0$ or $|\beta|\geq|\alpha|$. We normalize the inner product on $\Psi$ (or to say, on $\Lambda_{\Psi}$) by letting all the short roots of $\Psi$ have length 1. For a reduced sub-root system $\Phi$ of $\Psi$, recall that $$\delta_{\Phi}=\frac{1}{2}\sum_{\alpha\in\Phi\cap\Psi^{+}}\alpha$$ and $\delta'_{\Phi}$ be the unique dominant weight in the Weyl group (of $\Psi$) orbit of $\delta_{\Phi}$. Let $e(\Phi)=|2\delta'_{\Phi}|^{2}$ be the square of the length of $2\delta'_{\Phi}$.
\[D:f\] For a reduced sub-root system $\Phi$ of $\Psi$, let $$f_{\Phi,\Psi}(t)=\sum_{w\in W_{\Phi}}\epsilon(w)t^{|\delta_{\Phi}-w\delta_{\Phi}|^2}.$$
In particular let $f_{\Psi}(t)=f_{\Psi,\Psi}(t)$.
As $|\delta_{\Phi}-w\delta_{\Phi}|^2=(2\delta_{\Phi},\delta_{\Phi}-w\delta_{\Phi})$, an equivalent definition for $f_{\Phi}$ is $$f_{\Phi}(t)=
\sum_{w\in W_{\Phi}}\epsilon(w)t^{(2\delta_{\Phi},\delta_{\Phi}-w\delta_{\Phi})}.$$
If $\Phi=\bigsqcup_{1\leq i\leq s}\Phi_{s}$ is an orthogonal decomposition of $\Phi$ into irreducible sub-root systems and $\sqrt{r_{i}}$ is the shortest length of roots in $\Phi_{i}\subset\Psi$, then $$f_{\Phi,\Psi}=\prod_{1\leq i\leq s}f_{\Phi_{i}}(t^{r_{i}}).$$ Thus the calculation of the polynomials $f_{\Phi,\Psi}(t)$ reduces to the calculation of the polynomials $f_{\Psi}(t)$ for irreducible root systems.
\[D:chi-lambda and E’\] Let $\Lambda_{\Psi}\subset{\mathbb{Q}}\Psi$ be the set of integral weights of $\Psi$. For any weight $\lambda\in\Lambda_{\Psi}$, define $$\chi^{\ast}_{\lambda}=\frac{1}{|W_{\Psi}|}\sum_{\gamma\in W_{\Psi}}\gamma\lambda.$$ We define a linear map $$E'={\mathbb{Q}}[\Lambda_{\Psi}]^{W_{\Psi}}\longrightarrow{\mathbb{Q}}[t]$$ by $E'(\chi^{\ast}_{\lambda})=t^{|\lambda|^2}$.
\[P:character-1-polynomial\] Given a reduced sub-root system $\Phi$ of $\Psi$, we have $E'(F_{\Phi,W_{\Psi}})=f_{\Phi,\Psi}$.
This follows from the formulas $$F_{\Phi,W_{\Psi}}=\sum_{w\in W_{\Phi}}\epsilon(w)\chi^{\ast}_{\delta_{\Phi}-w\delta_{\Phi}}$$ and $$f_{\Phi,\Psi}(t)=\sum_{w\in W_{\Phi}}\epsilon(w)t^{|\delta_{\Phi}-w\delta_{\Phi}|^2}.$$
Let $\psi:{\mathbb{Q}}[x_0,x_1,\dots,x_{n},\dots]\longrightarrow{\mathbb{Q}}[t]$ be an algebra homomorphism defined by $$\psi(x_{n})=t^{n^2},\forall n\geq 0.$$
\[P:n-polynomial-1-polynomial\] Given $\Psi=\BC_{n}$ and a reduced sub-root system $\Phi$ of $\BC_{n}$, we have $$f_{\Phi,\Psi}=\psi(E(j_{n}(F_{\Phi,W_{n}})).$$
This follows from the definitions of $E$, $\psi$ and $f_{\Phi,\Psi}$.
Recall that $E(j_{n}(F_{\Phi,W_{n}}))\in{\mathbb{Q}}[x_0,x_1,\dots,x_{n},\dots]$ is the multi-variable polynomial associated to reduced sub-root systems of $\BC_{n}$ in Section \[S:dimension-equal\]. Proposition \[P:n-polynomial-1-polynomial\] connects two polynomials by a simple relation.
For irreducible reduced root systems of small rank, calculation shows that $f_{\A_1}=1-t$, $$f_{\A_2}=(1-t)^{2}(1-t^2),$$ $$f_{\B_2}=(1-t)(1-t^{2})(1-t^{3})(1-t^4),$$ $$f_{\G_2}=(1-t)(1-t^{3})(1-t^4)(1-t^{5})(1-t^6)(1-t^{9}),$$ and $f_{\A_3}=(1-t)^3(1-t^{2})^{2}(1-t^3)$. By calculation, we also have $$f_{\A_4}=1-4t+3x^2+6x^3-7x^4-2x^5-4x^{6}+\textrm{higher terms},$$ $$f_{\D_4}=1-4t+3x^2+5x^3-3x^4-6x^5-6x^{6}+\textrm{higher terms},$$ $$f_{\A_5}=1-5t+6x^2+7x^3-16x^4+0x^5+2x^{6}+\textrm{higher terms}.$$
\[P:f-basic\] $f_{\Phi,\Psi}$ has constant term 1 and leading term $(-1)^{|\Phi^{+}|}t^{|2\delta_{\Phi}|^{2}}$, it satisfies $$f_{\Phi,\Psi}(t)=(-1)^{|\Phi^{+}|}t^{|2\delta_{\Phi}|^{2}}f_{\Phi,\Psi}(t^{-1}).$$
Recall that, $W_{\Phi}$ has a longest element $\omega_0$ which maps $\Phi^{+}$ to $-\Phi^{+}$, so $\epsilon_{w_0}=(-1)^{|\Phi^{+}|}$ and $w_0(\delta_{\Phi})=-\delta_{\Phi}$. Moreover, For any $w\in W$, we have $$(2\delta_{\Phi},\delta_{\Phi}-w\delta_{\Phi})+
(2\delta_{\Phi},\delta_{\Phi}-w_0w^{-1}\delta_{\Phi})=|2\delta_{\Phi}|^2.$$ Therefore the proposition follows.
\[P:Weyl-generating function\] Given an irreducible reduced root system $\Psi$, we have $$f_{\Psi}=\prod_{\alpha\in\Psi^{+}}
(1-t^{(2\delta_{\Psi},\alpha)}).$$
Let $E'':{\mathbb{Q}}[\Lambda_{\Psi}]\longrightarrow{\mathbb{Q}}[t]$ be a linear map defined by $$E''([\lambda])=t^{\langle2\delta_{\Psi},\lambda\rangle},\ \forall \lambda\in\Lambda_{\Psi}.$$ Then $E''$ is an algebra homomorphism and we have $$f_{\Psi}(t)=E''(\sum_{w\in W_{\Psi}}\sign(w)[\delta_{\Psi}-w\delta_{\Psi}]).$$
By the Weyl denominator formula, we have $$\sum_{w\in W_{\Psi}}\sign(w)[w\delta_{\Psi}]=
\prod_{\alpha\in\Psi^{+}}([\frac{\alpha}{2}]-[-\frac{\alpha}{2}]).$$ Then we have $$\sum_{w\in W_{\Psi}}\sign(w)[\delta_{\Psi}-w\delta_{\Psi}]=
\prod_{\alpha\in\Psi^{+}}(1-[\alpha]).$$ Taking the map $E''$ on both sides, we get $$f_{\Psi}=\prod_{\alpha\in\Psi^{+}}(1-t^{(2\delta_{\Psi},\alpha)}).$$
Exceptional irreducible root systems {#SS:linear-exceptional}
------------------------------------
In this subsection, for any exceptional irreducible root system $\Psi$, we consider the linear relations among the characters $\{F_{\Phi,W_{\Psi}}|\ \Phi\subset\Psi\}$.
We start with some observations.
Firstly, if $$\sum_{1\leq i\leq s} c_{i}F_{\Phi_{i},W_{\Psi}}=0$$ for some reduced sub-root systems $\{\Phi_{i}\subset\Psi:1\leq i\leq s\}$ and some non-zero coefficients $\{c_{i}\in{\mathbb{R}}: 1\leq i\leq s\}$, then for any $i$ with $$|\delta'_{\Phi_{i}}|=\max\{|\delta'_{\Phi_{j}}|:1\leq j\leq i\},$$ there exits $j\neq i$ such that $\delta'_{\Phi_{j}}=\delta'_{\Phi_{i}}$. Since otherwise, $\chi_{2\delta_{\Phi_{i}},W_{\Psi}}^{\ast}$ has a non-zero coefficient in the character $\sum_{1\leq i\leq s} c_{i}F_{\Phi_{i},W_{\Psi}}$.
Secondly, if $\Phi_1,\dots,\Phi_{s}$ are all contained in another reduced sub-root system $\Psi'$ of $\Psi$, and $\sum_{1\leq i\leq s} c_{i}F_{\Phi_{i},W_{\Psi'}}=0$ for some constants $c_1,\dots,c_{s}\in{\mathbb{R}}$, then $\sum_{1\leq i\leq s} c_{i}F_{\Phi_{i},W_{\Psi}}=0$. This is due to $$F_{\Phi_{i},W_{\Psi}}=
\frac{1}{|W_{\Psi}|}\sum_{\gamma\in W_{\Psi}}\gamma F_{\Phi_{i},W_{\Psi'}}$$ for each $i$, $1\leq i\leq s$.
Thirdly, if $\Phi'_1,\dots,\Phi'_{s}$ are all contained in another reduced sub-root system $\Psi'$ of $\Psi$, and $$\sum_{1\leq i\leq s} c_{i}F_{\Phi'_{i},W_{\Psi'}}=0$$ for some constants $c_1,\dots,c_{s}$. Let $$\Phi'\subset\Psi'^{\perp}=\{\alpha\in\Psi|(\alpha,\beta)=0,\forall\beta\in\Psi'\}$$ and $\Phi_{i}=\Phi'_{i}\sqcup\Phi'$. Then $$\sum_{1\leq i\leq s} c_{i}F_{\Phi_{i},W_{\Psi}}=0.$$ This follows from the second observation.
Fourthly, if $\sum_{1\leq i\leq s} c_{i}F_{\Phi_{i},W_{\Psi}}=0$, then $$\sum_{1\leq i\leq s}
c_{i}f_{\Phi_{i},\Psi}=0.$$ This follows by applying the map $E'$.
**Type $\E_6$.** In the proof of Theorem \[character-exceptional\], we have observed that only the two weights $2\omega_2, 2\omega_4$ are of the form $2\delta'_{\Phi}$ for at least two non-conjugate reduced sub-root systems of $\E_6$. It happens that $2\delta'_{\Phi}=2\omega_2$ for $\Phi=A_2,4A_1$, and $2\delta'_{\Phi}=2\omega_4$ for $\Phi=3A_2,A_3+2A_1$.
Since $\D_4\subset\E_6$, from the conclusion in $\D_4$ case and the second observation, we get $$F_{A_2,W_{\E_6}}+F_{4A_1,W_{\E_6}}-2F_{3A_1,W_{\E_6}}=0.$$ Moreover, we have $$F_{3A_2,W_{\E_6}}+F_{A_3+2A_1,W_{\E_6}}+F_{A_3+A_1,W_{\E_6}}-3F_{2A_2+A_1,W_{\E_6}}=0.
\label{Eq:E6-identity}$$ The proof of this equality is given below. These two relations generate all linear relations among $\{F_{\Phi,W_{\E_6}}|\Phi\subset\E_6\}$.
Let $\theta$ be a linear map on weights defined by $\theta(\omega_1)=\omega_6$, $\theta(\omega_6)=
\omega_1$, $\theta(\omega_3)=\omega_5$, $\theta(\omega_5)=\omega_3$, $\theta(\omega_2)=\omega_2$, $\theta(\omega_4)=\omega_4$. Then $\theta$ acts as an isometry and it maps dominant integral weights to dominant integral weights. We have $\Aut(\E_6)=W_{\E_6}\rtimes\langle\theta\rangle$ as groups acting on the weights.
\[L:E6-small length\] For a positive integer $k\leq 11$, if $k\neq 4,5,7,8,9,10$, then there exists a unique $\Aut(\E_6)$-orbit of weights $\lambda$ in the root lattice such that $|\lambda|^{2}=k$.
When $k=4,5,7,8,9$, there exist two $\Aut(\E_6)$-orbits of weights in the root lattice with $|\lambda|^{2}=k$. The representatives are $$k=4:\{\omega_1+\omega_3,2\omega_2\};$$ $$k=5:\{\omega_3+\omega_5,\omega_1+\omega_2+\omega_6\};$$ $$k=7:\{2\omega_1+\omega_5,\omega_2+\omega_4\};$$ $$k=8:\{2\omega_1+2\omega_6,\omega_1+\omega_2+\omega_3\};$$ $$k=9:\{3\omega_2,\omega_1+\omega_4+\omega_6\}.$$
When $k=10$, there exist four $\Aut(\E_6)$-orbits of weights in the root lattice with $|\lambda|^{2}=10$. The representatives are $$\omega_1+2\omega_5,
3\omega_1+\omega_2,\omega_2+\omega_3+\omega_5,\omega_1+2\omega_2+\omega_6.$$
The inverse to the Cartan matrix of $\E_6$ is $$\frac{1}{3}\times\left(\begin{array}{cccccc}4&3&5&6&4&2\\3&6&6&9&6&3
\\5&6&10&12&8&4\\6&9&12&18&12&6\\4&6&8&12&10&5\\2&3&4&6&5&4\\\end{array}\right).$$ Given a dominant integral weight $\lambda=\sum_{1\leq i\leq 6}a_{i}\omega_{i}$, $a_{i}\in{\mathbb{Z}}_{\geq 0}$, we have $$\begin{aligned}
|\lambda|^2&=&\frac{1}{3}(2a_{1}^{2}+2a_6^2+5a_3^2+5a_5^2)+\\&&
\frac{1}{3}(2a_1a_6+8a_3a_5+5a_1a_3+5a_5a_6+4a_1a_5+4a_3a_6)+\\&&(a_2^2+3a_4^2+3a_2a_4)+
(2a_3+2a_5+a_1+a_6)(a_2+2a_4).\end{aligned}$$ We also have: $\lambda$ is in the root lattice if and only if $3|a_1+a_5-a_3-a_6$.
If $|\lambda|^2\leq 11$, let $\lambda_1=a_1\omega_1+a_3\omega_3+a_5\omega_5+a_6\omega_6$ and $\lambda_2=a_2\omega_2+a_4\omega_4$. Then, $|\lambda_1|^2\leq 11$ and $|\lambda_2|^2\leq 11$.
Consider the weight $\lambda_1$. Let $k=a_1+a_5$ and $l=a_3+a_6$, $\lambda$ is in the root lattice implies that $3|k-l$. From $|\lambda_1|^2\leq 11$, we get $k,l\leq 3$. When $|k-l|=3$, we have $$\lambda_1=3\omega_1 (6),\ 3\omega_6 (6),\ 2\omega_1+\omega_5 (7),$$ $$\omega_3+2\omega_6 (7),\
\omega_1+2\omega_5 (10),\ 2\omega_3+\omega_6 (10).$$ Here, the numbers in the brackets mean the squares of modulus of the weights. Similar notation will be used in the remaining part of this proof and the proof for Lemma \[L:F4-short\]. When $|k-l|=0$, we have $k=l\leq 2$. Moreover, we have $$\lambda_1=2\omega_1+\omega_3+\omega_6 (11),\ \omega_1+\omega_5+2\omega_6 (11),\
2\omega_1+2\omega_6 (8),$$ $$\omega_3+\omega_5 (5),\ \omega_1+\omega_3 (4),\omega_5+\omega_6 (4),\
\omega_1+\omega_6 (2).$$
Consider the weight $\lambda_2$. From $|\lambda_2|^2\leq 11$, we get $$\lambda_2=\omega_4+\omega_2 (7),\
\omega_4 (3),3\omega_2 (9),\ 2\omega_2 (4),\ \omega_2 (1).$$
In the case that $\lambda_1\neq 0$ and $\lambda_2\neq 0$, we have $$\lambda=\omega_1+2\omega_2+\omega_6 (10),\ 3\omega_1+\omega_2 (10), \omega_2+3\omega_6 (10),$$ $$\omega_2+\omega_3+\omega_5 (10),\ \omega_1+\omega_4+\omega_6 (9),\ \omega_1+\omega_2+\omega_3 (8),$$ $$\omega_2+\omega_5+\omega_6 (8),\ \omega_1+\omega_2+\omega_6 (5).$$
We finish the proof of the lemma.
Since any weight appearing in $F_{\Phi,W_{\E_6}}$ is an integral linear combination of roots, so any term $\chi^{\ast}_{\lambda}$ appearing in $F_{\Phi,W_{\E_6}}$ having $\lambda$ in the root lattice. A case by case calculation enables us to show that any reduced sub-root system $\Phi\subset\E_6$ is stable under the action of some $\gamma\in\Aut(\E_6)-W_{\E_6}$, so $\theta F_{\Phi,W_{\E_6}}=F_{\Phi,W_{\E_6}}$ (cf. Proposition \[P:gamma-character\]). Thus the coefficient of any term $\chi^{\ast}_{\lambda}$ in $F_{\Phi,W_{\E_6}}$ is equal to the coefficient of the term $\chi^{\ast}_{\theta\lambda}$ in $F_{\Phi,W_{\E_6}}$. By Lemma \[L:E6-small length\], to prove equality (\[Eq:E6-identity\]) it is enough to prove the corresponding equality about $f_{\Phi,\E_6}$ and to calculate the coefficients of terms $\chi^{\ast}_{\lambda}$ with $|\lambda|^2=4,5,7,8,9,10$.
For the functions $f_{\Phi,\E_6}$, we have $$\begin{aligned}
&& f_{3A_2,\E_6}+f_{A_3+2A_1,\E_6}+f_{A_3+A_1,\E_6}-3f_{2A_2+A_1,\E_6}
\\&=&(1-t)^{6}(1-t^{2})^{3}+(1-t)^{5}(1-t^2)^{2}(1-t^3)\\&&+
(1-t)^{4}(1-t^2)^{2}(1-t^3)-3(1-t)^{5}(1-t^2)^{2}\\&=&(1-t)^{4}(1-t^{2})^{2}
((1-2t+2t^{3}-t^4)+(1-t-t^{3}+t^{4})+(1-t^3))\\&&-3(1-t)^{5}(1-t^2)^{2}
\\&=&(1-t)^{4}(1-t^2)^{2}(3-3t)-3(1-t)^{5}(1-t^2)^{2}\\&=&
0.\end{aligned}$$
The terms $\chi^{\ast}_{\lambda}$ with $|\lambda|^2=4,5,7,8,9,10$ in $$F_{3A_2,W_{\E_6}}+F_{A_3+2A_1,W_{\E_6}}+F_{A_3+A_1,W_{\E_6}}-
3F_{2A_2+A_1,W_{\E_6}}$$ are $$\begin{aligned}
&&(-12\chi^{\ast}_{\omega_1+\omega_3}-12\chi^{\ast}_{\omega_5+\omega_6}
-3\chi^{\ast}_{2\omega_2})+
(-4\chi^{\ast}_{\omega_1+\omega_3}-4\chi^{\ast}_{\omega_5+\omega_6}
-\chi^{\ast}_{2\omega_2})+\\&&
(-2\chi^{\ast}_{\omega_1+\omega_3}-2\chi^{\ast}_{\omega_5+\omega_6}
-2\chi^{\ast}_{2\omega_2})+
(-6\chi^{\ast}_{\omega_1+\omega_3}-6\chi^{\ast}_{\omega_5+\omega_6}
-2\chi^{\ast}_{2\omega_2})=0, \end{aligned}$$ $$(36\chi^{\ast}_{\omega_1+\omega_2+\omega_6})+
(6\chi^{\ast}_{\omega_1+\omega_2+\omega_6})+
(0\chi^{\ast}_{\omega_1+\omega_2+\omega_6})-
3(14\chi^{\ast}_{\omega_1+\omega_2+\omega_6})=0,$$ $$\begin{aligned}
&&(-12\chi^{\ast}_{2\omega_1+\omega_5}-12\chi^{\ast}_{\omega_3+2\omega_6}
-12\chi^{\ast}_{\omega_2+\omega_4})+(2\chi^{\ast}_{2\omega_1+\omega_5}+
2\chi^{\ast}_{\omega_3+2\omega_6}+2\chi^{\ast}_{\omega_2+\omega_4})+\\&&
(\chi^{\ast}_{2\omega_1+\omega_5}+\chi^{\ast}_{\omega_3+2\omega_6}+
4\chi^{\ast}_{\omega_2+\omega_4})-3(-3\chi^{\ast}_{2\omega_1+\omega_5}-
3\chi^{\ast}_{\omega_3+2\omega_6}-2\chi^{\ast}_{\omega_2+\omega_4})=0,\end{aligned}$$ $$\begin{aligned}
&&
(3\chi^{\ast}_{2\omega_1+2\omega_6}+12\chi^{\ast}_{\omega_1+\omega_2+\omega_3}
+12\chi^{\ast}_{\omega_2+\omega_5+\omega_6})+
(-\chi^{\ast}_{2\omega_1+2\omega_6}-4\chi^{\ast}_{\omega_1+\omega_2+\omega_3}-
\\&&4\chi^{\ast}_{\omega_2+\omega_5+\omega_6})+(\chi^{\ast}_{2\omega_1+2\omega_6}
-2\chi^{\ast}_{\omega_1+\omega_2+\omega_3}-2\chi^{\ast}_{\omega_2+\omega_5+\omega_6})
-\\&&3(\chi^{\ast}_{2\omega_1+2\omega_6}+2\chi^{\ast}_{\omega_1+\omega_2+\omega_3}
+2\chi^{\ast}_{\omega_2+\omega_5+\omega_6})=0,\end{aligned}$$ $$(2\chi^{\ast}_{3\omega_2})+(-\chi^{\ast}_{3\omega_2})+
(-3\chi^{\ast}_{\omega_1+\omega_4+\omega_6}-\chi^{\ast}_{3\omega_2})-
3(-\chi^{\ast}_{\omega_1+\omega_4+\omega_6})=0$$ and $$\begin{aligned}
&&
(-3\chi^{\ast}_{\omega_1+2\omega_5}-3\chi^{\ast}_{2\omega_3+\omega_6}-
6\chi^{\ast}_{\omega_1+2\omega_2+\omega_6})+(2\chi^{\ast}_{\omega_1+2\omega_5}+
2\chi^{\ast}_{2\omega_3+\omega_6}+4\chi^{\ast}_{\omega_1+2\omega_2+\omega_6})\\&&+
(\chi^{\ast}_{\omega_1+2\omega_5}+\chi^{\ast}_{2\omega_3+\omega_6}+
2\chi^{\ast}_{\omega_1+2\omega_2+\omega_6})=0\end{aligned}$$ respectively.
Therefore the equality (\[Eq:E6-identity\]) follows.
**Type $\E_7$.** As in the proof of Theorem \[character-exceptional\], we have observed that those weights that appearing more than once in $\{2\delta'_{\Phi}|\ \Phi\subset\E_7\}$ and the reduced sub-root systems for which they appeared in are as follows,
- [$2\omega_1$: $A_2$, $4A'_1$, appears two times.]{}
- [$\omega_1+\omega_6$: $A_2+A_1$, $5A_1$, appears two times.]{}
- [$\omega_4$: $A_2+2A_1$, $6A_1$, appears two times.]{}
- [$2\omega_2$: $A_2+3A_1$, $7A_1$, appears two times.]{}
- [$2\omega_3$: $3A_2$, $A_3+2A_1$, appears two times.]{}
- [$2\omega_1+2\omega_6$: $A_4$, $2A_3$, appears two times.]{}
- [$\omega_1+\omega_4+\omega_6$: $A_4+A_1$, $2A_3+A_1$, appears two times.]{}
By the conclusion from the $\E_6$ case and the second and the third observations in the beginning of this subsection, we get $$F_{A_2,W_{\E_7}}+F_{(4A_1)',W_{\E_7}}-2F_{3A_1,W_{\E_7}}=0,$$ $$F_{A_2+A_1,W_{\E_7}}+F_{5A_1,W_{\E_7}}-2F_{4A_1,W_{\E_7}}=0,$$ $$F_{A_2+2A_1,W_{\E_7}}+F_{6A_1,W_{\E_7}}-2F_{5A_1,W_{\E_7}}=0,$$ $$F_{A_2+3A_1,W_{\E_7}}+F_{7A_1,W_{\E_7}}-2F_{6A_1,W_{\E_7}}=0,$$ $$F_{3A_2,W_{\E_7}}+F_{A_3+2A_1,W_{\E_7}}+F_{A_3+A_1,W_{\E_7}}-3F_{2A_2+A_1,W_{\E_7}}=0.$$
We prove that these relations generate all linear relations among $\{F_{\Phi,W_{\E_7}}|\ \Phi\subset\E_7\}$. By the first observation, to show this, we just need to show: any non-trivial linear combination $$c_1 F_{A_4,W_{\E_7}}+c_2 F_{2A_3,W_{\E_7}}$$ is not a linear combination of the characters $\{F_{\Phi,W_{\E_7}}:\ |\delta'_{\Phi}|^2<
|\delta'_{A_4}|^2=20\}$ and any non-trivial linear combination $$c_1 F_{A_4+A_1,W_{\E_7}}+
c_2 F_{2A_3+A_1,W_{\E_7}}$$ is not a linear combination of the characters $\{F_{\Phi,W_{\E_7}}||\delta'_{\Phi}|^2<|\delta'_{A_4+A_1}|^2=21\}$.
Suppose $$c_1 F_{A_4,W_{\E_7}}+c_2 F_{2A_3,W_{\E_7}}=\sum_{1\leq i\leq s, |2\delta'_{\Phi_{i}}|
\leq 19}c_{i}F_{\Phi_{i},W_{\E_7}}.$$ Then $$c_1 f_{A_4,\E_7}+c_2 f_{2A_3,\E_7}=
\sum_{1\leq i\leq s, |2\delta'_{\Phi_{i}}|\leq 19}c_{i}f_{\Phi_{i},\E_7}$$ by the fourth observation. Since there is no $\Phi\subset\E_7$ with $|2\delta'_{\Phi}|^2=19$, the RHS of the above equation has a degree at most $18$. Comparing the coefficients of the terms $t^{20}$ and $t^{19}$, we get $c_1+c_2=0$ and $-4c_1-6c_2=0$. Hence $c_1=c_2=0$.
After calculating the highest (=longest) terms, we get $$\begin{aligned}
F_{A_4+A_1,W_{\E_7}}&=&-\chi^{\ast}_{\omega_1+\omega_4+\omega_6}+
2\chi^{\ast}_{\omega_2+\omega_5+\omega_6}+\chi^{\ast}_{2\omega_1+2\omega_6}
\\&&+2\chi^{\ast}_{\omega_1+\omega_2+\omega_3+\omega_7}+\textrm{lower terms},\end{aligned}$$ $$\begin{aligned}
F_{2A_3+A_1,W_{\E_7}}&=&-\chi^{\ast}_{\omega_1+\omega_4+\omega_6}+
2\chi^{\ast}_{\omega_2+\omega_5+\omega_6}+\chi^{\ast}_{2\omega_1+2\omega_6}
\\&&+4\chi^{\ast}_{\omega_1+\omega_2+\omega_3+\omega_7}+\textrm{lower terms}.\end{aligned}$$ Considering the coefficients of the terms $\chi^{\ast}_{\omega_1+\omega_4+\omega_6}$ and $\chi^{\ast}_{\omega_1+\omega_2+\omega_3+\omega_7}$, we see any non-trivial linear combination $c_1 F_{A_4+A_1,W_{\E_7}}+c_2 F_{2A_3+A_1,W_{\E_7}}$ is not a linear combination of the characters $\{F_{\Phi,W_{\E_7}}:\ |\delta'_{\Phi}|^2<|\delta'_{A_4+A_1}|^2=21\}$.
**Type $\E_8$.** As in the proof of Theorem \[character-exceptional\], we have observed that those weights that appearing more than once in $\{2\delta'_{\Phi}|\ \Phi\subset\E_8\}$ and the sub-root systems for which they appeared in are as follows,
- [$2\omega_8$: $A_2$, $4A'_1$, appears two times.]{}
- [$\omega_1+\omega_8$: $A_2+A_1$, $5A_1$, appears two times.]{}
- [$\omega_6$: $A_2+2A_1$, $6A_1$, appears two times.]{}
- [$\omega_3$: $A_2+3A_1$, $7A_1$, appears two times.]{}
- [$2\omega_1$: $2A_2$, $A_2+4A_1$, $8A_1$, appears three times.]{}
- [$2\omega_7$: $A_3+2A_1$, $3A_2$, appears two times.]{}
- [$\omega_2+\omega_7$: $A_3+3A_1$, $3A_2+A_1$, appears two times.]{}
- [$\omega_1+\omega_6$: $A_3+4A_1$, $A_3+A_2$, appears two times.]{}
- [$2\omega_2$: $A_3+A_2+2A_1$, $4A_2$, appears two times.]{}
- [$2\omega_1+2\omega_8$: $A_4$, $2A_3$, appears two times.]{}
- [$\omega_1+\omega_6+\omega_8$: $A_4+A_1$, $2A_3+A_1$, appears two times.]{}
- [$\omega_4+\omega_8$: $A_4+2A_1$, $2A_3+2A_1$, appears two times.]{}
- [$2\omega_2+2\omega_8$: $D_4+A_2$, $D_4+4A_1$, appears two times.]{}
- [$2\omega_5$: $A_5+A_2+A_1$, $2A_4$, appears two times.]{}
- [$2\omega_1+2\omega_6$: $A_6$, $2D_4$, appears two times.]{}
By the conclusion from the $\E_7$ case and the second and the third observations, we get $$F_{A_2,W_{\E_8}}+F_{(4A_1)',W_{\E_8}}-2F_{3A_1,W_{\E_8}}=0,$$ $$F_{A_2+A_1,W_{\E_8}}+F_{5A_1,W_{\E_8}}-2F_{4A_1,W_{\E_8}}=0,$$ $$F_{A_2+2A_1,W_{\E_8}}+F_{6A_1,W_{\E_8}}-2F_{5A_1,W_{\E_8}}=0,$$ $$F_{A_2+3A_1,W_{\E_8}}+F_{7A_1,W_{\E_8}}-2F_{6A_1,W_{\E_8}}=0,$$ $$F_{A_2+4A_1,W_{\E_8}}+F_{8A_1,W_{\E_8}}-2F_{7A_1,W_{\E_8}}=0,$$ $$F_{2A_2,W_{\E_8}}+F_{A_2+4A_1,W_{\E_8}}-2F_{A_2+3A_1,W_{\E_8}}=0,$$ $$F_{3A_2,W_{\E_8}}+F_{A_3+2A_1,W_{\E_8}}+F_{A_3+A_1,W_{\E_8}}-3F_{2A_2+A_1,W_{\E_8}}=0,$$ $$F_{3A_2+A_1,W_{\E_8}}+F_{A_3+3A_1,W_{\E_8}}+F_{A_3+2A_1,W_{\E_8}}-3F_{2A_2+2A_1,W_{\E_8}}=0,$$ $$F_{4A_2,W_{\E_8}}+F_{A_3+A_2+2A_1,W_{\E_8}}+F_{A_3+A_2+A_1,W_{\E_8}}-3F_{3A_2+A_1,W_{\E_8}}=0,$$ $$F_{D_4+A_2,W_{\E_8}}+F_{D_4+4A_1,W_{\E_8}}-2F_{D_4+3A_1,W_{\E_8}}=0.$$ In the below we prove that these relations generate all linear relations among $\{F_{\Phi,W_{\E_8}}|\ \Phi\subset\E_8\}$.
Since $\Psi=\E_8$ has no sub-root systems $\Phi$ with $e(\Phi)=19$, one can show similarly as in the $\E_7$ case that any non-trivial linear combination $c_1 F_{A_4,W_{\E_8}}+
c_2 F_{2A_3,W_{\E_8}}$ is not a linear combination of the characters $\{F_{\Phi,W_{\E_8}}:\ |\delta'_{\Phi}|^2<|\delta'_{A_4+A_1}|^2=20\}$.
By calculating the highest (=longest) terms, we get $$\begin{aligned}
F_{A_4+A_1,W_{\E_8}}&=&-\chi^{\ast}_{\omega_1+\omega_6+\omega_8}+
2\chi^{\ast}_{\omega_1+\omega_5}+\chi^{\ast}_{2\omega_1+2\omega_8}
\\&&+2\chi^{\ast}_{\omega_2+\omega_7+\omega_8}+\textrm{lower terms},\end{aligned}$$ $$\begin{aligned}
F_{2A_3+A_1,W_{\E_8}}&=&-\chi^{\ast}_{\omega_1+\omega_6+\omega_8}+
2\chi^{\ast}_{\omega_1+\omega_5}+\chi^{\ast}_{2\omega_1+2\omega_8}
\\&&+4\chi^{\ast}_{\omega_2+\omega_7+\omega_8}+\textrm{lower terms},\end{aligned}$$ Considering the coefficients of the terms $\chi^{\ast}_{\omega_1+\omega_6+\omega_8}$ and $\chi^{\ast}_{\omega_2+\omega_7+\omega_8}$, we see that any non-trivial linear combination $c_1 F_{A_4+A_1,W_{\E_8}}+c_2 F_{2A_3+A_1,W_{\E_8}}$ is not a linear combination of the characters $\{F_{\Phi,W_{\E_8}}:\ |\delta'_{\Phi}|^2<|\delta'_{A_4+A_1}|^2=21\}$.
We have $$\begin{aligned}
F_{A_4+2A_1,W_{\E_8}}&=&\chi^{\ast}_{\omega_4+\omega_8}-
4\chi^{\ast}_{\omega_1+\omega_6+\omega_8}-2\chi^{\ast}_{\omega_2+\omega_3}\\&&+\textrm{lower terms},\end{aligned}$$ $$\begin{aligned}
F_{2A_3+2A_1,W_{\E_8}}&=&\chi^{\ast}_{\omega_4+\omega_8}-
4\chi^{\ast}_{\omega_1+\omega_6+\omega_8}-4\chi^{\ast}_{\omega_2+\omega_3}\\&&+\textrm{lower terms}.\end{aligned}$$ Considering the coefficients of the terms $\chi^{\ast}_{\omega_4+\omega_8}$ and $\chi^{\ast}_{\omega_2+\omega_3}$, we see that any non-trivial linear combination $c_1 F_{A_4+2A_1,W_{\E_8}}+c_2 F_{2A_3+2A_1,W_{\E_8}}$ is not a linear combination of the characters $\{F_{\Phi,W_{\E_8}}:\ |\delta'_{\Phi}|^2<|\delta'_{A_4+A_1}|^2=21\}$.
For the weight $2\omega_5$ and sub-root systems $A_5+A_2+A_1$, $2A_4$, suppose some non-trivial linear combination $c_1 F_{A_5+A_2+A_1,W_{\E_8}}+c_2 F_{2A_4,W_{\E_8}}$ is a linear combination of the characters $\{F_{\Phi,W_{\E_8}}:\ |\delta'_{\Phi}|^2<|2\omega_5|^2=40\}$. The sub-root systems with $33\leq |2\delta'_{\Phi}|^{2}\leq 39$ include $\{A_5+A_2,D_4+A_3,A_5+2A_1,A_5+A_1,(A_5+A_1)',A_5\}$, so there exists constants $c_3,c_4,c_5,c_6,c_7,c_8$ such that $$\begin{aligned}
&& c_1f_{A_5+A_2+A_1,\E_8}+c_2 f_{2A_4,\E_8}+c_3 f_{A_5+A_2,\E_8}+c_4 f_{D_4+A_3,\E_8}+\\&&
c_5 f_{A_5+2A_1,\E_8}+c_6 f_{A_5+A_1,\E_8}+c_7 f_{(A_5+A_1)',\E_8}+c_8 f_{A_5,\E_8}\end{aligned}$$ is a polynomial of degree $\leq 32$.
By formulas in Subsection \[SS:generating function\], we have $$f_{A_5+A_2+A_1}(t)=-t^{40}+8t^{39}-23t^{38}+19t^{37}+38t^{36}-90t^{35}+39t^{34}+\textrm{lower terms},$$ $$f_{2A_4}(t)=t^{40}-8t^{39}+22t^{38}-12t^{37}-53t^{36}+88t^{35}+2t^{34}+\textrm{lower terms},$$ $$f_{D_4+A_3}(t)=t^{38}-7t^{37}+16t^{36}-4t^{35}-33t^{34}+\textrm{lower terms},$$ $$f_{A_5+A_1}(t)=f_{(A_5+A_1)'}(t)=t^{36}-6t^{35}+11t^{34}+\textrm{lower terms}.$$ Moreover, $$f_{A_5+A_2}(t)=t^{39}+\textrm{lower terms},$$ $$f_{A_5+2A_1}(t)=-t^{37}+\textrm{lower terms},$$ $$f_{A_5}(t)=-t^{35}+\textrm{lower terms}.$$
Considering the coefficients of the terms $t^{40}$ and $t^{39}$, we get $$-c_1+c_2=0,\quad 8c_1-8c_2+c_3=0.$$ Thus $c_2=c_1$ and $c_3=0$. Considering the coefficients of the terms $t^{38}$, $t^{37}$, we get $$-23c_1+22c_2+c_4=0,\quad 19c_1-12c_2-7c_4-c_5=0.$$ Thus $c_4=c_1$ and $c_5=0$. Considering the coefficients of $t^{36}$, $t^{35}$, we get $$38c_1-53c_2+16c_4+(c_6+c_7)=0$$ and $$-90c_1+88c_2-4c_4-6(c_6+c_7)-c_8=0.$$ Hence $c_6+c_7=-c_1$ and $c_8=0$. Finally, considering the coefficient of $t^{34}$, we get $$0=39c_1+2c_2-33c_3+11(c_6+c_7)=-3c_1.$$ Therefore $c_1=c_2=0$.
For the weight $2\omega_1+2\omega_6$ and sub-root systems $A_6$, $2D_4$, since $\Psi=\E_8$ has no sub-root systems $\Phi$ with $e(\Phi)=55$, we can show any non-trivial linear combination $$c_1 F_{A_6,W_{\E_8}}+c_2 F_{2D_4,W_{\E_8}}$$ is not a linear combination of the characters $\{F_{\Phi,W_{\E_8}}:\ |\delta'_{\Phi}|^2<|2\omega_1+2\omega_6|^2=56\}$.
This proves the conclusion in the $\E_8$ case.
**Type $\F_4$.** As in the proof of Theorem \[character-exceptional\], we have observed that those weights appearing more than once in $\{2\delta'_{\Phi}|\ \Phi\subset\F_4\}$ and the sub-root systems for which they appeared in are as follows:
- [$\omega_1$: $A_1^{L}$, $2A_1^{S}$, appears 2 times.]{}
- [$\omega_3$: $A_1^{L}+A_1^{S}$, $3A_1^{S}$, appears 2 times.]{}
- [$2\omega_4$: $A_2^{S}$, $2A_1^{L}$, $A_1^{L}+2A_1^{S}$, $4A_1^{S}$, appears 4 times.]{}
- [$\omega_2$: $3A_1^{L}$, $2A_1^{L}+2A_1^{S}$, $A_1^{L}+A_2^{S}$, appears 3 times.]{}
- [$2\omega_1$: $A_2^{L}$, $4A_1^{L}$, appears 2 times.]{}
- [$\omega_1+2\omega_4$: $A_3^{S}$, $B_2$, appears 2 times.]{}
- [$2\omega_3$: $A_2^{L}+A_2^{S}$, $A_1^{L}+B_2$, $2A_1^{S}+B_2$, $A_1^{L}+A_3^{S}$, appears 4 times.]{}
- [$2\omega_1+2\omega_4$: $A_3^{L}$, $2B_2$, appears 2 times.]{}
- [$2\omega_3+2\omega_4$: $D_4^{S}$, $C_3$, appears 2 times.]{}
Since $\C_4\subset\F_4$, $\B_4\subset\F_4$, using the following equalities $$c_1+d_2=2a_2,\quad c_1a_2+d_2a_2=a_2^{2},\quad c_1d_2+d_2^{2}=2a_2d_2,$$ $$c_1^{2}+c_1d_2=2a_2c_1,\quad a_3=c_1d_2,\quad c_1^{3}+c_1^{2}d_2=2a_2c_1^{2},$$ $$c_1a_3=c_1^{2}d_2,\quad a_3+d_2^{2}=2a_2d_2,\quad c_1c_2+d_2c_2=2a_2c_2,$$ and by the second and the third observations, we get $$F_{A_1^{L},W_{\F_4}}+F_{2A_1^{S},W_{\F_8}}-2F_{A_1^{S},W_{\F_4}}=0,$$ $$F_{A_1^{L}+A_1^{S},W_{\F_4}}+F_{3A_1^{S},W_{\F_4}}-2F_{2A_1^{S},W_{\F_4}}=0,$$ $$F_{A_1^{L}+2A_1^{S},W_{\F_4}}+F_{4A_1^{S},W_{\F_4}}-2F_{3A_1^{S},W_{\F_4}}=0,$$ $$F_{2A_1^{L},W_{\F_4}}+F_{A_1^{L}+2A_1^{S},W_{\F_4}}-2F_{A_1^{L}+A_1^{S},W_{\F_4}}=0,$$ $$F_{A_2^{S},W_{\F_4}}-F_{A_1^{L}+2A_1^{S},W_{\F_4}}=0,$$ $$F_{3A_1^{L},W_{\F_4}}+F_{2A_1^{L}+2A_1^{S},W_{\F_4}}-2F_{2A_1^{L}+A_1^{S},W_{\F_4}}=0,$$ $$F_{A_1^{L}+A_2^{S},W_{\F_4}}-F_{2A_1^{L}+2A_1^{S},W_{\F_4}}=0,$$ $$F_{A_2^{L},W_{\F_4}}+F_{4A_1^{L},W_{\F_4}}-2F_{3A_1^{L},W_{\F_4}}=0,$$\] $$F_{A_1^{L}+B_2,W_{\F_4}}+F_{2A_1^{S}+B_2,W_{\F_4}}-2F_{A_1^{S}+B_2,W_{\F_4}}=0.$$ Moreover, we will prove three more equalities $$F_{A_3^{S},W_{\F_4}}-F_{B_2,W_{\F_4}}+
2F_{A_2^{L}+A_1^{S},W_{\F_4}}-2F_{2A_1^{L}+2A_1^{S},W_{\F_4}}=0,$$ $$F_{A_2^{L}+A_2^{S},W_{\F_4}}+F_{A_1^{L}+B_2,W_{\F_4}}-F_{A_1^{S}+B_2,W_{\F_4}}
+2F_{B_2,W_{\F_4}}-3F_{A_2^{L}+A_1^{S},W_{\F_4}}=0,$$ $$F_{A_1^{L}+A_3^{S},W_{\F_4}}-F_{A_1^{L}+B_2,W_{\F_4}}+2F_{A_1^{S}+B_2,W_{\F_4}}-
2F_{A_3^{S},W_{\F_4}}=0.$$
We will show these relations generate all linear relations among $\{F_{\Phi,W_{\F_8}}|\ \Phi\subset\F_8\}$.
\[L:F4-short\] Given an integer $k$ with $1\leq k\leq 12$ and $k\neq 9$, there exists a unique dominant integral weight $\lambda$ for the root system $\Psi=\F_4$ such that $|\lambda|^{2}=k$.
Given $k=9$, there exist two dominant integral weights $\lambda$ for the root system $\Psi=\F_4$ such that $|\lambda|^{2}=k$. They are $3\omega_4$ and $\omega_1+\omega_3$.
The inverse to the Cartan matrix of $\F_4$ is $\left(\begin{array}{cccc}2&3&4&2\\
3&6&8&4\\2&4&6&3\\1&2&3&2\\\end{array}\right)$, so for a dominant integral weight $\lambda=\sum_{1\leq i\leq 4}a_{i}\omega_{i}$ ($a_{i}\in{\mathbb{Z}}_{\geq 0}$), we have $$|\lambda|^{2}=2a_1^{2}+6a_2^{2}+3a_3^{2}+a_4^{2}+(6a_1a_2+4a_1a_3+2a_1a_4+
8a_2a_3+4a_2a_4+3a_3a_4).$$
Suppose $|\lambda|^2\leq 12$. First we must have $a_2=0,1$. If $a_2=1$, we have $$\lambda=\omega_2\ (6),\omega_2+\omega_4\ (11).$$ We also have $a_3\leq 2$. When $a_3\geq 1$, we have $$\lambda=2\omega_3\ (12),\omega_3+\omega_1\ (9),
\omega_3+\omega_4\ (7),\omega_3\ (3).$$ When $a_2=a_3=0$, we have $$\lambda=2\omega_1\ (8),\omega_1+2\omega_4\ (10), \omega_1+\omega_4\ (5),
\omega_1\ (2),3\omega_4\ (9), 2\omega_4\ (4),\omega_4 (1).$$
This proves the lemma.
First any term $\chi^{\ast}_{\lambda}$ appearing in these three equalities has $|\lambda|^2\leq 12$. By Lemma \[L:F4-short\], to prove these equalities, we just need to prove the corresponding equalities about generating functions $f_{\Phi,\F_4}$ and to calculate the coefficients of the terms $\chi^{\ast}_{\lambda}$ with $|\lambda|^2=9$.
For the functions $f_{\Phi,\F_4}$, we have $$\begin{aligned}
&& f_{A_3^{S},\F_4}-f_{B_2,\F_4}+
2f_{A_2^{L}+A_1^{S},\F_4}-2f_{2A_1^{L}+2A_1^{S},\F_4}\\&=&(1-t)^{3}(1-t^2)^{2}(1-t^3)-
(1-t)(1-t^2)(1-t^3)(1-t^4)\\&&+2(1-t)(1-t^2)^{2}(1-t^4)-2(1-t)^{2}(1-t^{2})^{2}\\&=&
(1-t)(1-t^2)(1-t^3)(-2t+2t^3)+2(1-t)(1-t^2)^{2}(t-t^4)\\&=&0,\end{aligned}$$ $$\begin{aligned}
&& f_{A_2^{L}+A_2^{S},\F_4}+F_{A_1^{L}+B_2,\F_4}-F_{A_1^{S}+B_2,\F_4}+2F_{B_2,\F_4}
-3F_{A_2^{L}+A_1^{S},\F_4}\\&=&(1-t)^{2}(1-t^{2})^{3}(1-t^4)+(1-t)(1-t^2)^{2}(1-t^3)(1-t^4)\\&&-
(1-t)^{2}(1-t^2)(1-t^3)(1-t^4)+2(1-t)(1-t^2)(1-t^3)(1-t^4)\\&&-3(1-t)(1-t^2)^{2}(1-t^4)\\&=&
(1-t)(1-t^{2})^{2}(1-t^4)(2-t-t^2)+(1-t^2)^{2}(1-t^3)(1-t^4)\\&&-3(1-t)(1-t^2)^{2}(1-t^4)\\&=&
(1-t^{2})^{2}(1-t^4)(3-3t)-3(1-t)(1-t^2)^{2}(1-t^4)\\&=&0,\end{aligned}$$ $$\begin{aligned}
&&f_{A_1^{L}+A_3^{S},\F_4}-f_{A_1^{L}+B_2,\F_4}+2f_{A_1^{S}+B_2,\F_4}-2f_{A_3^{S},\F_4}
\\&=&(1-t)^{3}(1-t^2)^{3}(1-t^3)-(1-t)(1-t^2)^{2}(1-t^3)(1-t^4)\\&&+2(1-t)^{2}(1-t^2)(1-t^3)(1-t^4)-
2(1-t)^{3}(1-t^{2})^{2}(1-t^3)\\&=&(1-t)(1-t^2)^{2}(1-t^3)(-2t+2t^3)\\&&+
2(1-t)^{2}(1-t^2)(1-t^3)(t+t^2-t^3-t^4)\\&=&0.\end{aligned}$$
The terms $\chi^{\ast}_{\lambda}$ with $|\lambda|^2=9$ in $$F_{A_3^{S},W_{\F_4}}-F_{B_2,W_{\F_4}}+2F_{A_2^{L}+A_1^{S},W_{\F_4}}-
2F_{2A_1^{L}+2A_1^{S},W_{\F_4}},$$ $$F_{A_2^{L}+A_2^{S},W_{\F_4}}+F_{A_1^{L}+B_2,W_{\F_4}}-F_{A_1^{S}+B_2,W_{\F_4}}
+2F_{B_2,W_{\F_4}}-3F_{A_2^{L}+A_1^{S},W_{\F_4}},$$ $$F_{A_1^{L}+A_3^{S},W_{\F_4}}-F_{A_1^{L}+B_2,W_{\F_4}}+2F_{A_1^{S}+B_2,W_{\F_4}}-
2F_{A_3^{S},W_{\F_4}}$$ are $$(-2\chi^{\ast}_{\omega_1+\omega_3}-\chi^{\ast}_{3\omega_4})-
(-\chi^{\ast}_{3\omega_4})+2\chi^{\ast}_{\omega_1+\omega_3}=0,$$ $$(4\chi^{\ast}_{\omega_1+\omega_3}+2\chi^{\ast}_{3\omega_4})+(-\chi^{\ast}_{3\omega_4})
-(\chi^{\ast}_{\omega_1+\omega_3}-\chi^{\ast}_{3\omega_4})+
2(-\chi^{\ast}_{3\omega_4})-3\chi^{\ast}_{\omega_1+\omega_3}=0,$$ $$(-6\chi^{\ast}_{\omega_1+\omega_3}-\chi^{\ast}_{3\omega_4})-
(-\chi^{\ast}_{3\omega_4})+2(\chi^{\ast}_{\omega_1+\omega_3}-\chi^{\ast}_{3\omega_4})
-2(-2\chi^{\ast}_{\omega_1+\omega_3}-\chi^{\ast}_{3\omega_4})=0$$ respectively.
Therefore we get the three equalities.
For the weights $2\omega_1+2\omega_4$ and $2\omega_3+2\omega_4$, since $\Psi=\F_4$ has no sub-root systems $\Phi$ with $e(\Phi)=19$ or $e(\Phi)=27$, we can show any non-trivial linear combination $c_1 F_{A_3^{L},W_{\F_4}}+c_2 F_{2B_2,W_{\F_4}}$ is not a linear combination of the characters $\{F_{\Phi,W_{\F_4}}:\ |\delta'_{\Phi}|^2<|\delta'_{A_4+A_1}|^2=20\}$ and any non-trivial linear combination $c_1 F_{D_4^{S},W_{\F_4}}+c_2 F_{C_3,W_{\F_4}}$ is not a linear combination of the characters $\{F_{\Phi,W_{\F_4}}:\ |\delta'_{\Phi}|^2<|\delta'_{A_4+A_1}|^2=28\}$. Thus the relations as listed above generate all linear relations among $\{F_{\Phi,W_{\F_4}}|\ \Phi\subset\F_4\}$.
**Type $\G_2$.** For $\Psi=G_2$, the only non-conjugate reduced sub-root systems with conjugate leading terms are $A_2^{S}$ and $A_1^{L}+A_1^{S}$. We have $$F_{A_2^{S},W_{\G_2}}+
F_{A_1^{L}+A_1^{S},W_{\G_2}}+F_{A_1^{L},W_{\G_2}}-3F_{A_1^{S},W_{\G_2}}=0.$$ This is the unique linear relation between the characters $\{F_{\Phi,W_{\G_2}}|\ \Phi\subset\G_2\}$.
\[R:LP-equal-linear relation\] Given an irreducible root system $\Psi$, if $\Psi_0\not\cong\BC_{n}$, then the characters $\{F_{\Phi,W_{\Psi}}:\Phi\subset\Psi_0,\rank\Phi=\Psi_0\}$ are linearly independent. If $\Psi=\BC_{n}$, for two sub-root systems $\Phi_1,\Phi_2$ of $\Psi$ with $\rank\Phi_1=\rank\Phi_2=n$, $F_{\Phi_1,W_{n}}=F_{\Phi_2,W_{n}}$ if and only if $\Phi_1\sim_{W_{n}}\Phi_2$. These follow from the results we showed above. Note that, these were also proved by Larsen-Pink. Actually this is the essential part to the proof of Theorem 1 in [@Larsen-Pink]. On the other hand, Larsen-Pink have proved the existence of algebraic relations among the polynomials $\{b_{n},c_{n},d_{n+1}|n\geq 1\}$. And they used this to construct non-conjugate closed subgroups with equal dimension data.
\[P:linear-exceptional\] Given a compact connectee Lie group $G$ of type $\E_6$, $\E_7$, $\E_8$ or $\F_4$, there exist non-isomorphic closed connected full rank subgroups with linearly dependent dimension data.
In the case that $G$ is of type $\E_6$, $\E_7$, $\E_8$, $G$ possesses a Levi subgroup of type $\D_4$. Hence the conclusion follows from Proposition \[P:linear-BCD\]. In the case that $G$ is of type $\F_4$, $G$ possesses a subgroup isomorphic to $\Spin(8)$. Therefore the conclusion follows from Proposition \[P:linear-BCD\].
Comparison of Question \[Q:linear dependence\] and Question \[Q:dependent-character\] {#S:Gamma0}
=====================================================================================
In this section we give constructions which show that Question \[Q:dependent-character\] (or \[Q:equal-character\]) is not an excessive generalization of Question \[Q:linear dependence\] (or \[Q:dimension data\]), as we promised after introducing Questions \[Q:equal-character\] and \[Q:dependent-character\]. The significance of these constructions is that each equality (or linear relation) we found in Question \[Q:equal-character\] (or \[Q:dependent-character\]) indeed corresponds to an equality (or a linear relation) of dimension data of closed connected subgroups in a suitable group. Another consequence of these constructions is showing that the group $\Gamma^{\circ}$ could be quite arbitrary.
Precisely, what we do is as follows. Given a root system $\Psi'$ and a finite group $W$ acting faithfully on $\Psi'$ and containing $W_{\Psi'}$, we construct some pair $(G,T)$ with $G$ a compact Lie group and $T$ a closed connected torus in $G$ such that:
- [$\rank\Psi_{T}=\dim T=\rank\Psi'$.]{}
- [$\Psi'\subset\Psi_{T}$ and is stable under $\Gamma^{\circ}$.]{}
- [$\Gamma^{\circ}=W$ as groups acting on $\Psi'$.]{}
- [For each reduced sub-root system $\Phi$ of $\Psi'$, there exists a connected closed subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and with root system $\Phi(H,T)=\Phi$.]{}
Sub-root systems of any given irreducible root system $\Psi_0$ are classified in [@Oshima], in Section \[S:sub-root systems\] we have discussed this classification in the case that $\Psi_0$ is a classical irreducible root system or an exceptional irreducible root system of type $\F_4$ or $\G_2$.
\[P:construction-simple\] Given an irreducible root system $\Psi_0$, there exists a compact connected simple Lie group $G$ and a closed connected torus $T$ in $G$ such that:
- [$\rank\Psi_{T}=\dim T=\rank\Psi_0$.]{}
- [$\Psi_0\subset\Psi_{T}$ and is stable under $\Gamma^{\circ}$.]{}
- [$\Gamma^{\circ}=W_{\Psi_0}$ as groups acting on $\Psi_0$.]{}
- [For each reduced sub-root system $\Phi$ of $\Psi_0$, there exists a connected closed subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and with root system $\Phi(H,T)=\Phi$.]{}
[*Simply laced case.*]{} In the case that $\Psi_0$ is a simply laced irreducible root system, let ${\mathfrak{u}}_0$ be a compact simple Lie algebra with root system isomorphic to $\Psi_0$. Taking $G=\Int({\mathfrak{u}}_0)$ and $T$ a maximal torus of $G$, the conclusion in the proposition is satisfied for $(G,T)$.
[*Type $\C_{n}$.*]{} In the case that $\Psi_0=\C_n$, let $G=\SU(2n)$, $$T=\{\diag\{z_1,\dots,z_{n},z_1^{-1},\dots,z_{n}^{-1}\}|\ |z_1|=
|z_2|=\cdots=|z_{n}|=1\}$$ and $\theta=\Ad\left(\begin{array}{cc}0&I_{n}\\I_{n}&0\\\end{array}\right)$. Then $\theta$ is an involutive automorphism of $G$ and $\Lie T$ is a maximal abelian subspace of ${\mathfrak{p}}_0=\{X\in\Lie G|\ \theta(X)=-X\}$. In this example, the restricted root system $\Phi(G,T)\cong\C_{n}$ (cf. [@Knapp], Page 424), each long root occurs with multiplicity one, each short root occurs with multiplicity two and $C_{G}(T)$ is the subgroup of diagonal matrices in $G$. We have $$\Gamma^{\circ}=N_{G}(T)/C_{G}(T)=\{-1\}^{n}\rtimes S_{n}=W_{C_{n}}.$$ Denote by $$J_{k}=\left(\begin{array}{cccc}\frac{1}{\sqrt{2}}I_{k}&&\frac{1}{\sqrt{2}}iI_{k}
&\\&I_{n-k}&&\\\frac{1}{\sqrt{2}}iI_{k}&&\frac{1}{\sqrt{2}}I_{k}&\\&&&I_{n-k}
\\\end{array}\right),$$ which is an $2n\times 2n$ matrix. Define the subgroups $$A_{k-1}=\big\{\diag\{A,K,\overline{A},\overline{K}\}|\ A\in U(k), K\in T_{n-k}\big\},$$ $$C_{k}=\big\{\left(\begin{array}{cccc}A&0&B&0\\0&K&0&0\\C&0&D&0\\0&0&0&\overline{K}
\\\end{array}\right)|\ \left(\begin{array}{cc}A&B\\C&D\\\end{array}\right)\in\Sp(k),K\in T_{n-k}\big\},$$ $$D'_{k}=\big\{\left(\begin{array}{cccc}A&0&B&0\\0&K&0&0\\C&0&D&0\\0&0&0&\overline{K}
\\\end{array}\right)|\left(\begin{array}{cc}A&B\\C&D\\\end{array}\right)\in
\SO(2k),K\in T_{n-k}\big\}$$ and $$D_{k}=J_{k}D'_{k}J_{k}^{-1}.$$ Thus $T$ is a maximal torus of $D_{k}$, $C_{k}$ and $A_{k-1}$, and the root systems $$\Phi(D_{k},T)=\D_{k},$$ $$\Phi(C_{k},T)=\C_{k}$$ and $$\Phi(A_{k-1},T)=\A_{k-1}.$$ By [@Oshima], we know that any sub-root system $\Phi\subset\C_{n}$ is of the form $$\A_{r_1-1}+\cdots+\A_{r_{i}-1}+\C_{s_{1}}+\cdots+\C_{s_{j}}+\D_{t_1}+\cdots+\D_{t_{k}},$$ where $r_{1},..,r_{i},s_1,...,s_{j},t_1,...,t_{k}\geq 1$ and $$r_1+\cdots+r_{i}+s_1+\cdots+
s_{j}+t_1+\cdots+t_{k}=n.$$ Here we regard $\A_{0}=\emptyset$. Using subgroups of block-form, we see that each sub-root system $\Phi$ of $\Psi_0$ is of the form $\Phi(H,T)$ for some closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$.
[*Type $\BC_{n}$.*]{} In the case that $\Psi_0=\BC_{n}$, let $G=\SU(2n+t)$, $t\geq n$, $$T=\big\{\diag\{I_{t},z_1,\dots,z_{n},z_1^{-1},\dots,z_{n}^{-1}: |z_1|=|z_2|=\cdots=|z_{n}|=1\big\}$$ and $\theta=\Ad\left(\begin{array}{ccc}I_{t}&0&0\\0&0&I_{n}\\0&I_{n}&0\\\end{array}\right)$. Then $\theta$ is an involutive automorphism of $G$ and $\Lie T$ is a maximal abelian subspace of ${\mathfrak{p}}_0=\{X\in\Lie G|\ \theta(X)=-X\}$. Moreover, the restricted root system $$\Phi(G,T)=\BC_{n}=
\{\pm{e_{i}}\pm{e_{j}}|1\leq i<j\leq n\}\cup\{\pm{e_{i}},\pm{2e_{i}|1\leq i\leq n}\}$$ (cf. [@Knapp], Page 424), each root $2e_{i}$ occurs with multiplicity one, each root $\pm{e_{i}}\pm{e_{j}}$ occurs with multiplicity two and each root $e_{i}$ occurs with multiplicity $2t$. We have $$\Gamma^{\circ}=N_{G}(T)/C_{G}(T)=W_{\BC_{n}}=\Aut(\BC_{n})=\Gamma.$$ Denote by $$J_{k}=\left(\begin{array}{ccccc}I_{t}&&&&\\&\frac{1}{\sqrt{2}}I_{k}&0&0
\frac{1}{\sqrt{2}}iI_{k}&0\\&0&I_{n-k}&0&0\\&\frac{1}{\sqrt{2}}iI_{k}&0&\frac{1}
{\sqrt{2}}I_{k}&0\\&0&0&0&I_{n-k}\\\end{array}\right),$$ which is a $2n+t$ square matrix. Define the subgroups $$A_{k-1}=\big\{\left(\begin{array}{ccccc}I_{t}&&&&\\&A&0&0&0\\&0&K&0&0\\&0&0&
\overline{A}&0\\&0&0&0&\overline{K}\\\end{array}\right)|\ A\in\U(k),K\in T_{n-k}\big\},$$ $$B'_{k}=\big\{\left(\begin{array}{ccccc}I_{t-1}&&&&\\&A&0&B&0\\&0&K&0&0
\\&C&0&D&0\\&0&0&0&\overline{K}\\\end{array}\right)|\ \left(\begin{array}{cc}
A&B\\C&D\\\end{array}\right)\in\SO(2k+1),K\in T_{n-k}\big\},$$ $$C_{k}=\big\{\left(\begin{array}{ccccc}I_{t}&&&&\\&A&0&B&0\\&0&K&0&0\\&C&0&D&0\\&0&0&0&\overline{K}
\\\end{array}\right)|\ \left(\begin{array}{cc}A&B\\C&D\\\end{array}\right)\in\Sp(k),K\in T_{n-k}\big\},$$ $$D'_{k}=\big\{\left(\begin{array}{ccccc}I_{t}&&&&\\&A&0&B&0\\&0&K&0&0\\&C&0&D&0
\\&0&0&0&\overline{K}\\\end{array}\right)|\ \left(\begin{array}{cc}A&B\\C&D
\\\end{array}\right)\in\SO(2k),K\in T_{n-k}\big\},$$ $$B_{k}=J_{k}B'_{k}J_{k}^{-1}$$ and $$D_{k}=J_{k}D'_{k}J_{k}^{-1}.$$ Thus $T$ is a maximal torus of $D_{k}$, $C_{k}$, $A_{k-1}$ and $B_{k}$, and the root systems $$\Phi(D_{k},T)=\D_{k},$$ $$\Phi(C_{k},T)=\C_{k},$$ $$\Phi(A_{k-1},T)=\A_{k-1}$$ and $$\Phi(B_{k},T)=\B_{k}.$$ Using subgroups of block form, we see that each reduced root system $\Phi$ of $\Psi_0$ is of the form $\Phi(H,T)$ for some connected closed subgroup $H$ of $G$ with $T$ a maximal torus of $H$. Note that the condition $t\geq n$ makes sure that the sub-root system $n\B_1$ corresponds to a subgroup.
[*Type $\B_{n}$*]{} In the case that $\Psi_0=\B_{n}$, let $(G,T)$ be the same as the above for $\BC_{n}$. Then this pair satisfies the desired conclusion.
[*Type $\F_4$.*]{} In the case that $\Psi_0=\F_4$. Recall that the complex simple Lie algebra ${\mathfrak{e}}_7({\mathbb{C}})$ has a real form with a restricted root system isomorphic to $\F_4$ (cf. [@Knapp], Page 425). Let $G=\Aut({\mathfrak{e}}_7)$ be the automorphism group of a compact simple Lie algebra of type $\E_7$. Then we can choose an involution $\theta_0$ in $G$ and a closed connected torus $T$ in $G$ such that $\theta|_{T}=-1$, $\Lie T$ is a maximal abelian subspace of ${\mathfrak{e}}_7^{-\theta}=\{X\in{\mathfrak{e}}_7|\Ad(\theta)(X)=-X\}$ and the restricted root system $\Phi(G,T)\cong\F_4$. Write $\Psi_0=\Phi(G,T)$. Then $\Psi_0\cong\F_4$ and we show that any sub-root system $\Phi$ of $\Psi_0$ is of the form $\Phi=\Phi(H,T)$ for a closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$. To show this, we use some results from [@Huang-Yu]. Starting with the Klein four subgroup $\Gamma_6$ of $G$ (cf. [@Huang-Yu], Table 4), we have $G^{\Gamma_6}=(G^{\Gamma_6})_0\times\Gamma$ and $(G^{\Gamma_6})_0\cong\F_4$. From [@Huang-Yu], we know that some involutions in $(G^{\Gamma_6})_0$ are conjugate to $\theta_0$ in $G$. Without loss of generality we may assume that $\theta_0\in(G^{\Gamma_6})_0$. Choosing any $1\neq\theta_1\in\Gamma_6$, we have (cf. [@Huang-Yu]) $$(G^{\theta_1})_0\cong(\E_6\times\U(1))/\langle(c,e^{\frac{2\pi i}{3}})\rangle,$$ $$(G^{\theta_0\theta_1})_0\cong\SU(8)/\langle iI\rangle$$ and $$\theta_0\in(G^{\theta_1})_0\cap(G^{\theta_0\theta_1})_0.$$ Here $1\neq c\in Z(\E_6)$ with $o(c)=3$. By these, we get three closed connected subgroups $H_1,H_2,H_3$ of $G$ containing $T$ and $\theta_0$, and such that $$\Lie H_1\cong\mathfrak{su}(8),$$ $$\Lie H_2\cong\mathfrak{e}_6$$ and $$\Lie H_3\cong\mathfrak{f}_4.$$ Moreover, $\Lie T$ is a maximal abelian subspace of each of $(\Lie H_{i})^{-\theta_0}$, $i=1,2,3$. By the classification of sub-root systems of $\F_4$ (cf. Section \[S:sub-root systems\]), we can show that any sub-root system $\Phi$ of $\Psi_0\cong\F_4$ is of the form $\Phi(H,T)$ for a connected closed subgroup $H$ contained in one of $H_1,H_2,H_3$.
[*Type $\G_2$.*]{} In the case that $\Psi_0=\G_2$, let $G=\Spin(8)$. There exist (cf. [@Helgason], Page 517) two order three outer automorphisms $\theta,\theta'$ of $\G$ such that $$H_1=G^{\theta}\cong\G_2,$$ $$H_2=G^{\theta'}\cong\PSU(3)$$ and $H_1,H_2$ share a common maximal torus $T$. Thus $\Phi(G,T)\cong\G_2$, each long root occurs with multiplicity $1$ and each short root occurs with multiplicity $3$. By the classification of sub-root systems of $\F_4$ (cf. Section \[S:sub-root systems\]), one can show that any sub-root system $\Phi$ of $\Psi_0$ is of the form $\Phi(H,T)$ for some connected closed subgroup $H$ of $H_1$ or $H_2$.
\[R:construction-simple\] When $\Psi_0$ is an irreducible root system not of type $\B_{n}$ ($n\geq 3$), in the above construction we actually have $\Psi'_{T}=\Psi_0$.
\[P:Gamma0\] Given a root system $\Psi'$ and a finite group $W$ between $W_{\Psi'}$ and $\Aut(\Psi')$, there exists a compact (not necessarily connected) Lie group $G$ with a bi-invariant Riemannian metric $m$ and a connected closed torus $T$ in $G$ such that:
- [$\rank\Psi_{T}=\dim T=\rank\Psi'$.]{}
- [$\Psi'\subset\Psi_{T}$ and is stable under $\Gamma^{\circ}$.]{}
- [$\Gamma^{\circ}=W$ as groups acting on $\Psi'$.]{}
- [For each reduced sub-root system $\Phi$ of $\Psi'$, there exists a connected closed subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and with root system $\Phi(H,T)=\Phi$.]{}
Decomposing $\Psi'$ into a disjoint union of (orthogonal) simple root systems and jointing those isomorphic simple factors, we may write $\Psi'$ as the form $$\Psi'=\Psi'_1\bigsqcup\cdots\bigsqcup\Psi_{s}$$ where each $\Psi'_{i}=m_{i}\Psi_{i,0}$ is a union of $m_{i}$ root systems all isomorphic to an irreducible root system $\Psi_{i,0}$.
For each $i$, by Proposition \[P:construction-simple\], we get $(G_{i},T_{i})$ corresponds to the irreducible root system $\Psi_{i,0}$ and the finite group $W_{\Psi_{i,0}}$ satisfying all conditions in the Proposition. Let $$G'=G_{1}^{m_{1}}\times\cdots\times G_{s}^{m_{s}}$$ and $$T=T_{1}^{m_{1}}\times\cdots\times T_{s}^{m_{s}}.$$ Then $(G',T)$ corresponds to the root system $\Psi'$ and the finite group $W_{\Psi'}$ with all conditions in the conclusion satisfied.
Among $\{\Psi_{i,0}|\ 1\leq i\leq s\}$, we may assume that $\Aut(\Psi_{i,0})\neq W_{\Psi_{i,0}}$ happens exactly when $1\leq i\leq t$. For each $i\leq t$, since $\Aut(\Psi_{i,0})\neq W_{\Psi_{i,0}}$, $\Psi_{i,0}$ must be simply laced. In this case we have $G_{i}=\Int({\mathfrak{u}}_{i})$ for a compact simple Lie algebra ${\mathfrak{u}}_{i}$ with root system $\Psi_{i,0}$. Let $$G''=(\Aut({\mathfrak{u}}_{1})^{m_{1}}\rtimes S_{m_1})\times\cdots\times(\Aut({\mathfrak{u}}_{t})^{m_{t}}
\rtimes S_{m_{t}})\times (G_{t+1}^{m_{t+1}}\rtimes S_{m_{t+1}})\times\cdots\times
(G_{s}^{m_{s}}\rtimes S_{m_{s}})$$ and $$T=T_{1}^{m_{1}}\times\cdots\times T_{s}^{m_{s}}.$$ Hence $(G'',T)$ corresponds to the root system $\Psi'$ and the finite group $\Aut(\Psi')$ with all conditions in the conclusion satisfied.
We have $G''/G'\cong\Aut(\Psi')/W_{\Psi'}$. Corresponding to the subgroup $W/W_{\Psi'}$ of $\Aut(\Psi')/W_{\Psi'}$, we get a subgroup $G$ of $G''$ containing $G'$ and with $G/G'=W/W_{\Psi'}$ in the identification $G''/G'=\Aut(\Psi')/W_{\Psi'}$. Therefore $(G,T)$ corresponds to $(\Psi',W)$ with all conditions in the conclusion satisfied.
As in the proof of Proposition \[P:construction-simple\], given an irreducible root system $\Psi_0=\BC_{n}$, let $G_1=\SU(2n+t)$ for $t\geq n$ and $$T_1=\{\diag\{z_1,\dots,z_{n},z_1^{-1},\dots,
z_{n}^{-1},\underbrace{1,...,1}_{t}\}|\ |z_1|=|z_2|=\cdots=|z_{n}|=1\}.$$ Denote by $S_1$ the group of diagonal matrices in $G_1$. It is a maximal torus of $G_1$. Let $$\epsilon_{i}
(\diag\{z_1,\cdots,z_{2n+t}\})=z_{i}$$ for any $\diag\{z_1,\cdots,z_{2n+t}\}\in S_1$ and $$e_{i}(\diag\{z_1,\dots,z_{n},z_1^{-1},\dots,z_{n}^{-1},\underbrace{1,...,1}_{t}\})=z_{i}$$ for any $\diag\{z_1,\dots,z_{n},z_1^{-1},\dots,z_{n}^{-1},\underbrace{1,...,1}_{t}\}\in T$. Thus $$\epsilon_{i}|_{T}=\left\{\begin{array}{rcl} e_{i} &\mbox{if} &1\leq i\leq n\\
e_{i-n}^{-1}&\mbox{if} &n+1\leq i\leq 2n\\ 1 &\mbox{if} &2n+1\leq i\leq 2n+t.
\end{array}\right.$$
The following example is a modification of an example in [@Larsen-Pink], Page 392.
\[E:LP\] Given $r\geq n$, denote by $G_1=\SU(2n+t)$, $T_1\subset G_1$ the subgroup of diagonal matrices, $G_2=(G_1)^{r}$ and $T_2=(T_1)^{r}$. Write $$\lambda_{j}=\sum_{1\leq i\leq n}(nj-i+1)\epsilon_{i}$$ for $1\leq j\leq r$. Let $V_{j}=V_{\lambda_{j}}$ be an irreducible representation of $G_1$ with highest weight $\lambda_{j}$ and $$V=\bigoplus_{\sigma\in A_{r}}V_{\sigma(1)}\otimes\cdots\otimes V_{\sigma(r)}.$$ Denote by $G=\SU(k)$, where $$k=\frac{r!}{2}\prod_{1\leq j\leq r}\dim V_{j}.$$ The representation $V$ gives us an embedding $G_2\subset G$. Write $T$ for the image of $T_2$ under this embedding. Since $G=\SU(k)$ is simple, a biinvariant Riemannian metric $m$ on $G$ is unique up to a scalar multiple. Hence $\Psi_{T}$ does not depend on the choice of $m$. In this example, we have:
- [$\rank\Psi_{T}=\dim T=rn$.]{}
- [$\Psi'=r\BC_{n}\subset\Psi_{T}$ and it is stable under $\Gamma^{\circ}$.]{}
- [$\Gamma^{\circ}=W_{\Psi'}\rtimes A_{r}=(W_{\BC_{n}})^{r}\rtimes A_{r}$ as groups acting on $\Psi'$.]{}
- [For each reduced sub-root system $\Phi$ of $\Psi'$, there exists a closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and with root system $\Phi(H,T)=\Phi$.]{}
First we have $(W_{\BC_{n}})^{r}\rtimes A_{r}\subset\Gamma^{\circ}$ and $\Psi_{T}=\BC_{rn}$ by our construction of the pair $(G,T)$.
We have $\Psi'=r\BC_{n}\subset\Psi_{T}$ since $r\BC_{n}$ is the root system of $(G_2,T)$. By Proposition \[P:construction-simple\], any reduced sub-root system $\Phi$ of $r\BC_{n}$ is of the form $\Phi(H,T)$ for a closed connected subgroup $H$ of $G_2$.
The equality $\Gamma^{\circ}=(W_{\BC_{n}})^{r}\rtimes A_{r}$ can be proven using the idea in [@Larsen-Pink], Page 392. It proceeds as follows. By the construction of $G_2$ and $V$, the character $$\chi_{V_1\otimes\cdots\otimes V_{r}}|_{T}$$ has $\lambda=(nr,nr-1,\cdots,1)$ as a leading term. It is regular with respect to $\Aut(T,m)=W_{\BC_{rn}}$. The weights appearing in $\chi_{V}|_{T}$ of maximal length are in the orbit $((W_{\BC_{n}})^{r}\rtimes A_{r})\lambda$. Hence $\Gamma^{\circ}=(W_{\BC_{n}})^{r}\rtimes A_{r}$.
\[R:LP\] In Example \[E:LP\], moreover we have $\Psi'_{T}=r\BC_{n}=\Psi'$ since otherwise $\Gamma^{\circ}$ must be larger than $(W_{\BC_{n}})^{r}\rtimes A_{r}$.
\[P:Gamma0-2\] Given a root system $\Psi'$ and a finite group $W$ between $W_{\Psi'}$ and $\Aut(\Psi')$, there exists some $G=\SU(k)$ and a closed connected torus $T$ in $G$ such that:
- [$\rank\Psi_{T}=\dim T=\rank\Psi'$.]{}
- [$\Psi'\subset\Psi_{T}$ and it is stable under $\Gamma^{\circ}$.]{}
- [$\Gamma^{\circ}=W$ as groups acting on $\Psi'$.]{}
- [For each reduced sub-root system $\Phi$ of $\Psi'$, there exists a closed connected subgroup $H$ of $G$ with $T$ a maximal torus of $H$ and with root system $\Phi(H,T)=\Phi$.]{}
As in the proof of Proposition \[P:Gamma0\], we write $\Psi'$ as the form $$\Psi'=\Psi'_1\bigsqcup\cdots\bigsqcup\Psi_{s},$$ where each $\Psi'_{i}=m_{i}\Psi_{i,0}$ is a union of $m_{i}$ root systems all isomorphic to an irreducible root system $\Psi_{i,0}$.
For each $i$, by Proposition \[P:construction-simple\], we get $(G_{i},T_{i})$ corresponds to the root system $\Psi_{i,0}$ and the finite group $W_{\Psi_{i,0}}$ satisfying all conditions in the Proposition. Moreover, if $\Psi_{i,0}$ is simply laced, then $G_{i}=\Int({\mathfrak{u}}_{i})$ for a compact simple Lie algebra ${\mathfrak{u}}_{i}$ with root system isomorphic to $\Psi_{i,0}$. Let $$G'=G_{1}^{m_{1}}\times\cdots\times G_{s}^{m_{s}}$$ and $$T'=T_{1}^{m_{1}}\times\cdots\times T_{s}^{m_{s}}.$$
Among $\Psi_{i,0}$, we assume that $\Aut(\Psi_{i,0})\neq W_{\Psi_{i,0}}$ happens exactly when $1\leq i\leq t$. We have $$\Aut(\Psi')/W_{\Psi'}=(\Out(\Psi_{1,0})^{m_1}\rtimes S_{m_1})
\times\cdots\times(\Out(\Psi_{t,0})^{m_t}\rtimes S_{m_t})\times S_{m_{t+1}}\times\cdots\times
S_{m_{s}}.$$ Here $\Out(\Psi_{i,0})=\Aut(\Psi_{i,0})/W_{\Psi_{i,0}}$. The group $\Aut(\Psi')/W_{\Psi'}$ acts on $\Rep(G')$ (here $G_{i}=\Int({\mathfrak{u}}_{i})$ plays a role) through its action on the dominant integral weights.
For any $1\leq i\leq s$, choose a maximal torus $S_{i}$ of $G_{i}$ containing $T_{i}$. If $i\leq t$, we choose $m_{i}$ dominant integral weights $\lambda_{i,1},\cdots,\lambda_{i,m_{i}}$ of $S_{i}=T_{i}$ such that the set $\{\gamma\lambda_{i,j}|\gamma\in\Aut(\Psi_{i,0}),1\leq j\leq m_{i}\}$ has cardinality exactly $m_{i}|\Aut(\Psi_{i,0})|$. If $i\geq t+1$, the restriction map of weight lattices $$p_{i}: \hat{S_{i}}\longrightarrow\hat{T_{i}}$$ is surjective and it is an orthogonal projection. Choose $m_{i}$ regular dominant integral weights $\lambda_{i,1},\cdots,\lambda_{i,m_{i}}$ of $S_{i}$ with an additional property: each $\lambda_{i,j}$ is orthogonal to the weights in $\ker p_{i}$ and their images under $p_{i}$ are regular and distinct to each other. Thus the weights of maximal length in $V_{\lambda_{i,j}}|_{T_{i}}$ are those in the orbit $W_{\Psi_{i,0}}\lambda_{i,j}$, and each occurs with multiplicity one. Here $V_{\lambda_{i,j}}$ is an irreducible representation of $G_{i}$ with highest weight $\lambda_{i,j}$. Denote by $$V=\bigoplus_{\sigma\in W/W_{\Psi'}}\sigma(V_{\lambda_1}\otimes\cdots\otimes V_{\lambda_r}).$$ and $G=\SU(k)$, where $$k=\dim V=\frac{|W|}{|W_{\Psi'}|}\prod_{1\leq j\leq r}\dim V_{j}.$$ The representation $V$ gives us an embedding $G'\subset G=\SU(k)$. Write $T$ for the image of $T'$ under this embedding. Arguing similarly as in the proof of Example \[E:LP\], we get $\Gamma^{\circ}=W$. By Proposition \[P:construction-simple\], the other conditions in the conclusion are also satisfied.
In the above construction, we know $\Psi'_{T}\supset\Psi'$ by Property $(4)$ of Proposition \[P:Gamma0-2\] and $W_{\Psi'_{T}}\subset\Gamma^{\circ}$ by Proposition \[P:Gamma0-Psi’\]. From this $\Psi'_{T}$ is almost determined, which should be close to being equal to $\Psi'$. On the other hand, $\Psi_{T}$ is probably much larger than $\Psi'$. For example it may happen that $\Psi'=r\BC_{n}$ and $\Psi'_{T}=\BC_{nr}$.
Can we make an example with $\Psi'_{T}=\Psi'$ in Proposition \[P:Gamma0-2\]? Moreover, can we make an example with $\Psi_{T}=\Psi'$?
Irreducible subgroups {#S:Irreducible}
=====================
Let $G=\U(n)$. In this section, we study dimension data of closed subgroups $H$ of $G$ acting irreducibly on ${\mathbb{C}}^{n}$.
Choose a prime $p$, an integer $m\geq 4$ and a prime $q>p^{m}$. Let $n=p^{m}$, $T\subset G$ be the subgroup of diagonal unitary matrices and $$A=\{\diag\{a_1,a_2,\dots,a_{n}\}|\ a_1^{q}=a_2^{q}=\cdots=a_{n}^{q}=1\}.$$
\[L:GA-normalizer\] $C_{G}(A)=T$ and $N_{G}(A)=T\rtimes S_{n}$, where $S_{n}$ is the subgroup of permutation matrices in $G$.
There exists a unique conjugacy class of subgroups $\overline{N}$ of $S_{n}$ isomorphic to $(C_{p})^{m}$ and with non-identity elements all conjugate to $(1,2,\dots,p)(p+1,p+2,\dots,2p)\cdots(n-p+1,n-p+2,\dots,n)$.
For any subgroup $\overline{N}$ of $S_{n}$ as above, $C_{S_{n}}(\overline{N})=\overline{N}$ and $$N_{S_{n}}(\overline{N})/\overline{N}\cong GL(m,\mathbb{F}_{p}).$$
The first statement is clear. For the second statement, one can prove the uniqueness by induction on $m$. Again, one can show $C_{S_{n}}(\overline{N})=\overline{N}$ by induction on $m$. Finally, by the uniqueness of $\overline{N}$, we get $$N_{S_{n}}(\overline{N})/\overline{N}\cong\Aut(\overline{N})\cong GL(m,\mathbb{F}_{p}).$$
Now we specify $p=2$ and $m=4$. Let $\overline{H}=N_{S_{n}}(\overline{N})$. Then $\overline{H}/\overline{N}
\cong GL(4,\mathbb{F}_{2})$. Choose a subgroup $\overline{H}_1$ of $\overline{H}$ with $\overline{H}_1/\overline{N}$ corresponding to $$\langle\left(\begin{array}{cccc}1&0&1&0\\0&1&0&1\\0&0&1&0\\0&0&0&1\\\end{array}\right),
\left(\begin{array}{cccc}1&0&0&1\\0&1&1&1\\0&0&1&0\\0&0&0&1\\\end{array}\right)\rangle$$ and a subgroup $\overline{H}_2$ of $\overline{H}$ with $\overline{H}_2/\overline{N}$ corresponding to $$\langle\left(\begin{array}{cccc}1&0&1&0\\0&1&0&1\\0&0&1&0\\0&0&0&1\\\end{array}\right),
\left(\begin{array}{cccc}1&1&0&1\\0&1&0&0\\0&0&1&1\\0&0&0&1\\\end{array}\right)\rangle$$ Let $K=A\rtimes S_{n}$ and $N\subset K$ with $N/A=\overline{N}$. Write $H=N_{K}(N)$. Then $H/N=\overline{H}$. Let $H_1, H_2\subset H$ with $H_1/N=\overline{H}_1$ and $H_2/N=\overline{H}_2$.
\[P:U16-equal\] The two subgroups $H_1$ and $H_2$ act irreducibly on ${\mathbb{C}}^{16}$, have the the same dimension data and are non-isomorphic.
Since $N$ acts irreducibly on ${\mathbb{C}}^{16}$ and $H_1,H_2$ contain $N$, $H_1$ and $H_2$ act irreducibly on ${\mathbb{C}}^{16}$.
To show $H_1$ and $H_2$ having the same dimension data in $G$, it is sufficient to show they have the same dimension data in $H$. Since $N$ is a normal subgroup of $H$, $H_1$ and $H_2$, it is sufficient to show $\overline{H}_1=H_1/N$ and $\overline{H}_2=H_2/N$ have the same dimension data in $\overline{H}=H/N\cong GL(4,\mathbb{F}_2)$. We have $\overline{H}_1\cong\overline{H}_2\cong C_2\times C_2$. On the other hand one can show that non-identity elements of $\overline{H}_1$ and $\overline{H}_2$ are all conjugate to $$\left(\begin{array}{cccc}1&0&1&0\\0&1&0&1\\0&0&1&0\\0&0&0&1\\\end{array}\right).$$ Hence, $\overline{H}_1$ and $\overline{H}_2$ have the same dimension data in $\overline{H}\cong GL(4,\mathbb{F}_2)$.
We show that the groups $H_1$ and $H_2$ are non-isomorphic. Since $A$ is a characteristic subgroup of $H_1$ and $H_2$, it is sufficient to show $|Z(H_1/A)|\neq |Z(H_2/A)|$. Obviously $A\lhd N\lhd H_1,H_2$. Moroever we have $$\overline{N}=N/A\cong (\mathbb{F}_2)^4$$ as an abelian group and $H_1$, $H_2$ act on $N/A$ through the action of $\overline{H}_1$, $\overline{H}_2$ on $(\mathbb{F}_2)^4$. By Lemma \[L:GA-normalizer\], $\overline{N}$ is a maximal abelian subgorup of $S_{n}$. Therefore, $$Z(H_{i}/A)=\overline{N}^{\overline{H}_{i}},$$ the latter means the fixed point subgroup of the $\overline{H}_{i}$ action on $\overline{N}$. By the definition of $\overline{H}_{1}$ and $\overline{H}_{2}$, we get $$\overline{N}^{\overline{H}_{1}}\cong(\mathbb{F}_2)^2$$ and $$\overline{N}^{\overline{H}_{1}}\cong\mathbb{F}_2.$$ Hence, $H_1\not\cong H_2$.
Choose $n=12$ and a prime $q>3$. Let $T\subset G$ be the subgroup of diagonal unitary matrices and $$A=\{\diag\{a_1,a_2,\dots,a_{n}\}|\ a_1^{q}=a_2^{q}=\cdots=a_{n}^{q}=1\}.$$ Write $$N=A\rtimes\langle\left(
\begin{array}{ccccc}0&1&0&\cdots&0\\0&0&1&\cdots&0\\\vdots&\vdots&\vdots&\vdots&\vdots\\0&0&0&\cdots&1\\
1&0&0&\cdots&0\\\end{array}\right)\rangle$$ and $K=A\rtimes S_{n}$.
\[L:GA-normalizer2\] $N_{K}(N)/N\cong({\mathbb{Z}}/12{\mathbb{Z}})^{\times}\cong C_2\times C_2$.
By Lemma \[L:GA-normalizer\], $C_{G}(A)=T$ and $N_{G}(A)=T\rtimes S_{n}$. Since $N/A\cong C_{12}$ is a maximal abelian subgroup of $S_{12}$, $$N_{K}(N)/N\cong N_{S_{12}}(C_{12})/C_{12}\cong({\mathbb{Z}}/12{\mathbb{Z}})^{\times}.$$ By elementary number theory, $({\mathbb{Z}}/12{\mathbb{Z}})^{\times}\cong C_2\times C_2$.
Let $H=N_{K}(N)$ and $\overline{H}=H/N$. Then $\overline{H}\cong C_2\times C_2$. Write $H_1$, $H_2$, $H_3$, $H_4$, $H_5$ for the subgroups of $H$ containing $N$ and with $\overline{H}_1=H_1/N$, $\overline{H}_2=H_2/N$, $\overline{H}_3=H_3/N$, $\overline{H}_4=H_4/N$, $\overline{H}_5=H_5/N$ all the subgroups of $\overline{H}$, where $|\overline{H}_1|=1$, $|\overline{H}_2|=|\overline{H}_3|=|\overline{H}_4|=2$ and $|\overline{H}_5|=4$.
\[P:U12-linear\] We have $$\mathscr{D}_{H_1}+2\mathscr{D}_{H_5}-(\mathscr{D}_{H_2}+\mathscr{D}_{H_3}+\mathscr{D}_{H_4})=0.$$
Since $N$ is a normal subgroup of $H_1$, $H_2$, $H_3$, $H_4$ and $H_5$, it is sufficient to show the dimension data of the subgroups $\overline{H}_1$, $\overline{H}_2$, $\overline{H}_3$, $\overline{H}_4$, $\overline{H}_5$ of $\overline{H}$ have the corresponding linear relation. The latter follows from a consideration on irreducible representations of $C_2\times C_2$.
There exists an example as in Proposition \[P:U12-linear\] when $n=4$. For $n=4$, one has $N_{G}(A)/C_{G}(A)\cong S_4$. Chooose a normal subgroup $\overline{N}$ of $S_4$ isomorphic to $C_2\times C_2$ (it is unique) and the subgroups $\overline{H}_1$, $\overline{H}_2$, $\overline{H}_3$, $\overline{H}_4$ of $S_4$ (in the order of increasing orders) containing $\overline{N}$ Let $H_1$, $H_2$, $H_3$, $H_4$ be the corresponding subgroups of $A\rtimes S_4$. One can show that $$\mathscr{D}_{H_1}+2\mathscr{D}_{H_4}-(2\mathscr{D}_{H_2}+\mathscr{D}_{H_3})=0.$$
Given a closed connected torus $T$ in $G$, recall that we have a weight lattice $\Lambda_{T}=\Hom(T,\U(1))$ and a finite group $\Gamma^{\circ}=N_{G}(T)/C_{G}(T)$. Let $\Phi$ be the root system of $H$, $\Lambda$ be the root lattice of $H$ and $\rho_{T}$ be the character of the representation of $T$ on ${\mathbb{C}}^{n}$. Write $X=\Lambda_{T}\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. The following lemma is identical to Theorem 4 in [@Larsen-Pink]. We state their theorem in a form we needed and sketch the proof.
\[L:torus sharing\] Let $T$ be a closed connected torus in $G$. If there are more than one conjugacy classes of closed connected subgroups $H$ of $G$ with $T$ a maximal rotus and acting irreducibly on ${\mathbb{C}}^{n}$, then these representations $(H,{\mathbb{C}}^{n})$ are tensor products of the following list:
1. $n=2^{m}$, $H=(\Spin(2m_1+1)\times\cdots\times\Spin(2m_{s}+1)/Z)$ and ${\mathbb{C}}^{n}=M_{m_1}\otimes\cdots M_{m_{s}}$, where $$m=m_1+m_2+\cdots+m_{s},$$ $$Z=\{(\epsilon_1,\dots,
\epsilon_{s})|\ \epsilon_{i}=\pm{1}, \epsilon_1\epsilon_2\cdots\epsilon_{s}=1\}$$ and $M_{m}$ is Spinor representation of $\Spin(2m+1)$. In this case, $\{\pm{1}\}^{m}\rtimes S_{m}\subset\Gamma^{\circ}$.
2. $n=2^{k^2+k}\prod_{1\leq i\leq k}\frac{i!}{(2i)!}\frac{(m-k-1+2i)!}{(m-k-1+i)!}$, $H=\Sp(m)$ or $\SO(2m)$, ${\mathbb{C}}^{n}=V_{(k,k-1,\dots,1,0,\dots,0)}$, $1\leq k\leq m-1$ and $\frac{k(k+1)}{2}$ is odd; or, $H=\Sp(m)/\langle-I\rangle$ or $\SO(2m)/\langle-I\rangle$, ${\mathbb{C}}^{n}=V_{(k,k-1,\dots,1,0,\dots,0)}$, $1\leq k\leq m-1$ and $\frac{k(k+1)}{2}$ is even. In this case, $\{\pm{1}\}^{m}\rtimes S_{m}\subset\Gamma^{\circ}$.
3. $n=27$, $H=\G_2$ or $\PSU(3)$, ${\mathbb{C}}^{n}=V_{\lambda}$, $\lambda=2(e_1-e_3)$. The weight $\lambda=2\omega_2$ for $\G_2$ and $\lambda=2\omega_1+2\omega_2$ for $\A_2$. In this case, $W_{\G_2}\subset\Gamma^{\circ}$.
4. $n=2^{12}$, $H=\F_4$, $\Sp(4)/\langle-I\rangle$ or $\SO(8)/\langle-I\rangle$, ${\mathbb{C}}^{n}=V_{\lambda}$, $\lambda=3e_1+2e_2+e_3$. The weight $\lambda=\omega_3+\omega_4$ for $\F_4$, $\lambda=\omega_1+\omega_2+\omega_3$ for $\C_4$ and $\lambda=\omega_1+\omega_2+\omega_3+\omega_4$ for $\D_4$, In this case, $W_{\F_4}\subset\Gamma^{\circ}$.
5. Any $n\geq 1$ and irreducible semisimple subgroup $H$ of $\U(n)$. In this case, $W_{\Phi}\subset\Gamma^{\circ}$, where $\Phi$ is the root system of $H$.
Since $H$ acts irreducibly on ${\mathbb{C}}^{n}$, the center if $H$ is equal to $H\cap Z(\U(n))$. Hence, it is either finite or 1-dimensional. Therefore, the action of $W_{\Phi}$ on $X$ is multiplicity free. As $\Gamma^{\circ}\supset W_{\Phi}$, $\Gamma^{\circ}$ acts on $X$ multiplicity freely. Let $X=\bigoplus_{1\leq i\leq m} X_{i}$ be the decomposition of $X$ into a sum of irredicible summands. Then we have $\Phi=\bigsqcup_{1\leq i\leq m}\Phi\cap X_{i}$ and $$\rho_{T}=\bigotimes_{1\leq i\leq m}\rho_{i},$$ where $\rho_{i}$ is the character of a representation of the Lie algebra with root system $\Phi\cap X_{i}$. Since the character ring ${\mathbb{Q}}[\Lambda_{T}]$ is a unque factorization domain and each $\rho_{i}$ has dominat terms with coefficient 1, the characters $\{\rho_{i}\}$ are determined by $\rho_{T}$. By considering each $\rho_{i}$, we may assume that $\Gamma^{\circ}$ acts irreducibly and non-trivially on $X$, which forces $H$ being semisimple. We suppose this since now on.
Since $H$ is assumed to be a closed connected subgroup of $\U(n)$, the root lattice $\Lambda$ is generated by the differences of elements in $\rho_{T}$ and the integral weight lattice $\Lambda_{T}$ is generated by elements in $\rho_{T}$. This indicates, both $\Lambda$ and $\Lambda_{T}$ are determined by $T$. Endowing $G$ with a biinvariant Riemannian metric, this gives $\Lambda$ a positive definite inner product. We show that $\Phi^{\circ}$ is determined by $\Lambda$. This can be proved by induction on the rank $l$ of $\Lambda$ (which is equal to the rank of $\Phi^{\circ}$). If $l=1$, then $\Phi^{\circ}=\A_1$ and it consists of the non-zero elements of $\Lambda$ of shortest length. Hence, the statement is clear in this case. If $l>1$, one can show that the shortest non-zero elements of $\Lambda$ are contained in $\Phi^{\circ}$. For each element $\lambda$, let $$\lambda=\sum_{1\leq i\leq m}\alpha_{i},$$ $\alpha_{i}\in\Phi^{\circ}$ be an expression of this form with $m$ minimal. We have $(\alpha_{i},\alpha_{j})\geq 0$ for any $1\leq i\leq j\leq s$ since otherwise $\alpha_i+\alpha_{j}\in\Phi^{\circ}$ and we could find an expression with $m$ smaller. Hence, $|\lambda|\geq|\alpha_{i}|$ for each $1\leq i\leq m$. Therefore, $m=1$ and $\lambda\in\Phi^{\circ}$. Let $\Phi_1$ be the set of shortest non-zero elements of $\Lambda$. By the above, $\Phi_1$ is a root system and is contained in $\Phi^{\circ}$. Let $\Lambda'$ be the sublattice of elements in $\Lambda$ orthogonal to elements in $\Phi_1$ and $\Phi'$ be the sub-root system of elements in $\Phi^{\circ}$ orthogonal to elements in $\Phi_1$. Then, $\Lambda'$ is determined by $\Lambda$ and $\Phi'$ generates $\Lambda'$. By induction, $\Phi'$ is determined by $\Lambda'$. Therfore, $\Phi^{\circ}$ is determined by $\Lambda$.
Now we have $\Phi^{\circ}$ determined by $\Lambda$. Since $\Gamma^{\circ}$ acts irreducibly on $X$, it acts transitively on the irreducible factors of $\Phi^{\circ}$. Hence, $\Phi^{\circ}$ is an orthogonal sum of isomorphic irredicible root systems. Write $\Phi^{\circ}=m\Omega$ where $\Omega$ is an irredicible root system. Note that $(B_{l})^{\circ}=lB_1$, $(C_{l})^{\circ}=D_{l}$, $(F_{4})^{\circ}=D_{4}$, $(G_{2})^{\circ}=A_{2}$ and $\Phi^{\circ}=\Phi$ if $\Phi$ being of type $ADE$. Hence, if $\Omega\neq\A_1$, then $\Phi=\sum_{1\leq i\leq m}\Phi_{i}$ with $(\Phi_{i})^{\circ}=\Omega$ for each $i$. In this case, $\rho_{T}$ has a decomposition $$\rho_{T}=\bigotimes_{1\leq i\leq m}\rho_{i}$$ where $\rho_{i}$ is the character of an irreducible representation of a simple Lie algebra of type $\Phi_{i}$. Since each $\rho_{i}$ has dominant terms of coefficient 1, $\{\rho_{i}|\ 1\leq i\leq m\}$ are determined by $\rho_{T}$. Therefore, we may assume that $m=1$. That is, we need to consider the ambiguity arsing from the pairs $D_l\subset C_l$ ($l\geq 3$), $D_4\subset F_4$, $C_4\subset F_4$ and $A_2\subset G_2$. This follows from some detailed study of the characters of highest weight modules (cf. [@Larsen-Pink], Page 395-396). We omit the details here but list the results precisely in the items $(2)$-$(4)$ of the conclusion of the theorem. If $\Omega\neq\A_1$, then $$\{\pm{1}\}^{m}\subset\Gamma^{\circ}\subset\{\pm{1}\}^{m}\rtimes S_{m}.$$ The action of $\Gamma^{\circ}$ on $\Phi^{\circ}$ gives a partition of $\{1,2,\dots,m\}$. This in turn gives a canonical tensor decomposition of $\rho_{T}$ similar as the above for $\Omega\neq\A_1$ case. By this it is enough to consider the case of $\Gamma^{\circ}=\{\pm{1}\}^{m}\rtimes S_{m}$. In this case the ambiguity is either as listed in item $(1)$ of the conclusion, or there is no ambiguity as in item $(5)$.
\[R:torus sharing\] In each item $(1)$-$(4)$, from the above proof, $\Gamma^{\circ}$ contains the Weyl group of the largest dimensional Lie group appearing in the ambiguity, which is $\Spin(2m+1)$, $\Sp(m)$, $\F_4$, $\G_2$ respectively,
\[T:irreducible-independence\] Let $H_1,H_2,\dots,H_{s}$ be a list of closed connected subgroups of $G$ acting irreducibly on ${\mathbb{C}}^{n}$. If they are non-conjugate to each other, then their dimension data are linearly independent.
By Proposition \[P:group-root system\], we may assume that $H_1$, $H_2$,..., $H_{s}$ have a common maximal torus $T$. Applying Lemma \[L:torus sharing\], we get a canonical tensor decompostion of $\rho_{T}$ (the character of the representation of $T$ on ${\mathbb{C}}^{n}$), which in turn gives a decompostion of each $H_{i}$. Observe that if each item $(1)$-$(4)$ occurs, the group $\Gamma^{\circ}$ contains a large finite group, i.e., the Weyl group of the largest connected compact Lie group appearing in the ambiguity. By calculation one can show that the dominant weights $2\delta_{\Phi}$ are non-conjugate to each other under this Weyl group, where $\{\Phi\}$ are the root systems of groups appearing in each item $(1)$-$(4)$. Therefore, the dominant weights $2\delta_{\Phi_i}$ are non-conjugate to each other under $\Gamma^{\circ}$, where $\Phi_{i}$ is root system of $H_{i}$. Hence, the dimension data of $H_1,H_2,\dots,H_{s}$ are linearly independent.
[999]{} J. An; J.-K. Yu; J. Yu, *On the dimension datum of a subgroup and its application to isospectral manifolds.* Journal of Differential Geometry. To appear.
A. Borel, *Automorphic $L$-functions,* Automorphic forms, representations and $L$-functions, Part 2, pp. 27-61, Proc. Sympos. Pure Math. [**33**]{}, Amer. Math. Soc., Providence, R.I. (1979).
N. Bourbaki, *Lie groups and Lie algebras.* Chapters 4–6. Translated from the 1968 French original by Andrew Pressley. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002.
E.-B. Dynkin , *Semisimple subalgebras of semisimple Lie algebras*, AMS translation, (2) **6** (1957) 111-254.
C. Gordon, *Survey of isospectral manifolds,* Handbook of differential geometry, Vol. I, 747-778, North-Holland, Amsterdam, 2000.
S. Helgason, *Differential geometry, Lie groups, and symmetric spaces*. Pure and Applied Mathematics, 80. Academic Press, Inc. \[Harcourt Brace Jovanovich, Publishers\], New York-London, 1978.
J.E. Humphreys, *Linear algebraic groups.* Graduate Texts in Mathematics, No. **21**. Springer-Verlag, New York-Heidelberg, 1975.
J.S. Huang; J. Yu, *Klein four subgroups of Lie algebra automorphisms.* Pacific Journal of Mathematics. To appear.
N. Katz, *Larsen’s alternative, moments, and the monodromy of Lefschetz pencils.* Contributions to automorphic forms, geometry, and number theory, 521-560, Johns Hopkins Univ. Press (2004).
A. W. Knapp, *Lie groups beyond an introduction.* Second edition. Progressin Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 2002.
R. Langlands, *Beyond endoscopy*, Contributions to automorphic forms, geometry, and number theory, 611-697, Johns Hopkins Univ. Press, Baltimore, MD (2004), MR2058622, Zbl 1078.11033.
M. Larsen, *On the conjugacy of element-conjugate homomorphisms*. Israel J. Math. **88** (1994), no. 1-3, 253-277.
M. Larsen, *On the conjugacy of element-conjugate homomorphisms*. II. Quart. J. Math. Oxford Ser. (2) **47** (1996), no. 185, 73-85.
M. Larsen; R. Pink, *Determining representations from invariant dimensions.* Invent. Math. **102** (1990), no. 2, 377-398.
A.N. Minchenko, *Semisimple subalgebras of exceptional Lie algebras.* (Russian) Tr. Mosk. Mat. Obs. **67** (2006), 256-293; translation in Trans. Moscow Math. Soc. 2006, ISBN: 5-484-00146-X 225-259.
T. Oshima, *A classification of subsystems of a root system.* arXiv: math.RT/0611904.
T.A. Springer, *Reductive groups.* Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 3-27, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.
T.A. Springer, *Linear algebraic groups.* Second edition. Progress in Mathematics, **9**. Birkhäuser Boston, Inc., Boston, MA, 1998.
T. Sunada, [*Riemannian coverings and isospectral manifolds,*]{} Ann. of Math. (2) [**121**]{} (1985), no. 1, 169-186, MR0782558, Zbl 0585.58047.
C. Sutton, *Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions,* Comment. Math. Helv. **77** (2002), 701-717.
R. Vein; P. Dale, *Determinants and their applications in mathematical physics,* Springer-Verlag, New York, 1999.
S. Wang, *Dimension data, local conjugacy and global conjugacy in reductive groups.* Preprint (2007).
|
---
author:
- 'Fedor V. Fomin[^1]'
- 'Serge Gaspers[^2] [^3]'
- Daniel Lokshtanov
- 'Saket Saurabh [^4]'
bibliography:
- 'book\_pc.bib'
- 'references.bib'
title: Exact Algorithms via Monotone Local Search
---
Introduction
============
Combining Random Sampling with FPT Algorithms {#sec:randalgo}
=============================================
Efficient Construction of Set-Inclusion-Families {#sec:derandomization}
================================================
Conclusion and Discusison {#sec:concl}
=========================
[ **Acknowledgements.** Many thanks to Russell Impagliazzo and Meirav Zehavi for insightful discussions. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreements n. 267959 and no. 306992. NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council (ARC) through the ICT Centre of Excellence Program. Serge Gaspers is the recipient of an ARC Future Fellowship (project number FT140100048) and acknowledges support under the ARC’s Discovery Projects funding scheme (project number DP150101134). Lokshtanov is supported by the Beating Hardness by Pre-processing grant under the recruitment programme of the of Bergen Research Foundation. ]{}
[^1]: University of Bergen, Norway. `{fomin|daniello}@ii.uib.no`
[^2]: The University of New South Wales, Sydney, Australia. `sergeg@cse.unsw.edu.au`
[^3]: Data61 (formerly: NICTA), CSIRO, Sydney, Australia
[^4]: The Institute of Mathematical Sciences, Chennai, India. `saket@imsc.res.in`.
|
---
abstract: 'Single particle distribution function of plasma particles has been derived from the first member of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy utilising the pair correlation function evaluated in [@kn:ab1] from the second member of the BBGKY hierarchy. This distribution function may be employed to probe the thermodynamic properties of the weakly inhomogeneous plasma systems.'
author:
- Anirban Bose
title: Determination of the single particle distribution function in a weakly correlated weakly inhomogeneous plasma
---
Pair correlation function plays a significant role in studying the thermodynamic properties of plasma. Therefore, the determination and application of pair correlation function in finding out the properties of plasma has been the topic of interest for a long time. The method proposed by Dupree [@kn:dup] in case of slowly time and space varying system has been sucessfully applied to demonstrate the pair correlation function in a dilute, uniform, quiescent plasma [@kn:wolf]. By using path integral technique the quantum pair correlation function has been calculated in a one component plasma [@kn:jan]. The thermodynamic property of strongly coupled classical plasma has been investigated by nodal expansion [@kn:fur]. interaction potential and thermodynamic functions of dusty plasma has been investigated by measured correlation functions [@kn:fortov]. Study of pair correlation function using molecular dynamics simulation of the strongly coupled two temperature plasma is reported [@kn:nrs]. parametrization of the pair correlation function and the static structure factor of the Coulomb one component plasma(OCP) is presented from the weakly coupled regime to the strongly coupled regime [@kn:des]. The pair distribution function of both strongly coupled and strongly degenerate Fermi systems have been calculated using a novel path integral representation [@kn:fil] of the many-particle density operator. Recently, Monte Carlo simulations have been utilized to study distribution functions of the warm dense uniform electron gas in the thermodynamic limit [@kn:do].
The statistical mechanics of plasma by using the Bogoliubov-Born-Green-Kirkwood-Yvon chain of equations has been attempted by many researchers in the past. In this connection O’neil and Rostoker [@kn:neil] have considered the plasma system in thermal equilibrium and evaluated the two and three body correlation function in homogeneous plasma. Using [@kn:da] the BBGKY hierarchy, the quantum binary and triplet distribution functions for a neutral many-component plasma has been determined. Previously, in a series of papers [@kn:ab; @kn:ab1] an equation of pair correlation function was established from the first two members of BBGKY hierarchy in the weakly correlated limit and subsequently an expression of pair correlation function was derived in the weakly inhomogeneous limit. In this article we shall discuss the effect of the pair correlation function on the single particle distribution function. In order to do this we should insert the pair correlation function in the first member of BBGKY hierarchy and try to identify the contribution as some measureable macroscopic quantity to the potential energy of the system.
We have considered a plasma system of N electrons and N infinitely massive ions which are randomly distributed in a volume V. The plasma is in thermal equilibrium.
In this article, the system is considered to be weakly inhomogeneous. The inhomogeneous density of electrons is considered as $$\begin{aligned}
n(3)= n_{0}+n_{1}(3) \end{aligned}$$ $n_{0}$ is the homogeneous part and the inhomogeneous part is $n_{1}(3)=B\cos (\textbf{p}\cdot \textbf{x}_{3})$.
The average interaction potential between two electrons separated by a distance $r= \mid {\bf {x_1-x_2}} \mid $ is assumed to be of the Debye-Hückel type[@kn:akh]: $$\phi_{12}({\bf {x_1-x_2}})= \frac{e^2}{r}{\rm exp}(-k_Dr)\label{v1}$$ The electronic charge is e and $T$ is the absolute temperature of the electrons. $$k_D^2 =n e^2/k_B T,$$ As we are interested in the inhomogeneous case, the density $n$ is space dependent and for a pair of particles with the space coordinates $\textbf{x}_{1}$ and $\textbf{x}_{2}$ we may choose the density $n$ in the expression of $k_{D}$ at the mid position $\textbf{r}_{m}$ ($\textbf{r}_{m}=\frac{\textbf{x}_{i}+\textbf{x}_{j}}{2}$) of the concerned particles in the first approximation.
Hence, $$n(\textbf{r}_{m})=n_{0}+B\cos[\frac{\mathbf{p}}{2}\cdot(\mathbf{x}_{1}+\mathbf{x}_{2})]$$ In the weakly inhomogeneous case ($B\ll n_{0}$), $\phi_{12}$ as given by eq.(\[v1\]) may be expressed as $$\phi_{12}= \frac{e^2}{r}{\rm exp}(-k_{D0}r)(1-B\frac{k_{D0}r}{2n_{0}}\cos[\frac{\mathbf{p}}{2}\cdot(\mathbf{x}_{1}+\mathbf{x}_{2})])\label{v2}$$ In the above expression we have neglected terms containing $(\frac{B}{n_{0}})^{s}$ when $s\geq 2$.
The first member of BBGKY hierarchy is given by [@kn:akh] $$\frac{\partial f_1}{\partial t} + {\bf v_1}\cdot\frac{\partial f_1}{\partial {\bf x_1}}
+ n_0\int d{\bf X_2} {\bf a_{12}}\cdot\frac{\partial}{\partial{\bf
v_1}}\left [ f_1({\bf X_1})f_1({\bf X_2})+g_{12}\right ] = 0$$ For the electrons in equilibrium ($\frac{\partial f_1}{\partial t}=0$), the first member of BBGKY hierarchy is $${\bf v_{1}} \cdot \frac{\partial f_{1}}{\partial {\bf
x_{1}}}+n_{0}\int d{\bf X_2} f_{1}({\bf X_2}){\bf a_{12}}\cdot
\frac{\partial f_{1}({\bf X_1})}{\partial {\bf v_{1}}}+n_{0}\int
d{\bf X_2}{\bf a_{12}}\cdot \frac{\partial g_{12}}{\partial
{\bf v_{1}}}=0\label{v3}$$ where ${\bf a_{12}}
=-(1/m){\partial\phi_{12}}/{\partial {\bf x_1}} $ with $\phi_{12}$ given by eq.(\[v2\]). The single particle distribution functions are written in the follwing form: $$f_1 ({\bf X_1}) = f_M({\bf v_1})F_1({\bf x_1})$$ where $f_M$ is a Maxwellian distribution and $F_1$ denotes the spatially dependent part of the single particle distribution function. For a plasma in equilibrium, the pair correlation function $g_{12}$ is written in the form $$\begin{aligned}
g_{12}({\bf {X_1,X_2}})=f_{1}({\bf X_1})f_{1}({\bf X_2})\chi_{12}({\bf x_1,x_2})\end{aligned}$$ $\chi_{12}({\bf x_1,x_2})$ is a symmetric function of $\bf x_1$ and $\bf x_2$. The expression of $\chi_{12}$ is obtained in [@kn:ab1] and it can be expressed as $$\chi_{12}= -\frac{\phi_{12}}{k_B T}$$ where $\phi_{12}$ is given by eq.(\[v2\]).
The last term of eq.(\[v3\]) is given by
$$\begin{aligned}
&& n_{0}\int
d{\bf X_2}{\bf a_{12}}\cdot \frac{\partial g_{12}}{\partial
{\bf v_{1}}} = n_0 \int d{\bf X_{2}} \left (
-\frac{1}{m}\frac{\partial}{\partial {\bf x_1}}{\phi_{12}}\right )
\cdot \frac{\partial} {\partial{\bf v_{1}}}\left (
-\frac{\phi_{12}}{k_B T}f_{1}({\bf X_1})f_{1}({\bf X_2}) \right )
\nonumber \\
&=& \frac{n_0}{mk_B T} \int d{\bf X_2}\nabla_{\bf x_1}
\phi_{12}^2 \cdot \frac{\partial f_{1}({\bf X_1})}{\partial {\bf
v_1}}f_{1}({\bf X_2})
\nonumber \\
&=& \frac{n_0}{mk_B T} \int d{\bf X_2}\nabla_{\bf x_1}
[\frac{q^4}{r^{2}}{\rm exp}(-2k_{D0}r)(1-B\frac{k_{D0}r}{n_{0}}\cos[\frac{\mathbf{p}}{2}\cdot(\mathbf{x}_{1}+\mathbf{x}_{2})])]\cdot \frac{\partial f_{1}({\bf X_1})}{\partial {\bf
v_1}}f_{1}({\bf X_2})
\nonumber \\\end{aligned}$$
where $$\phi_{12}^{2}= \frac{q^4}{r^{2}}{\rm exp}(-2k_{D0}r)(1-B\frac{k_{D0}r}{n_{0}}\cos[\frac{\mathbf{p}}{2}\cdot(\mathbf{x}_{1}+\mathbf{x}_{2})])$$ In the above expression we have neglected terms containing $(\frac{B}{n_{0}•})^{s}$ when $s\geq 2$. The contribution of the first term of the $\phi_{12}^{2}$ has been calculated in [@kn:ab3]. The value is given by $$\begin{aligned}
&&\frac{n_{0}}{mk_B T} \int d{\bf X_2}\nabla_{\bf x_1}
[\frac{e^4}{r^{2}}{\rm exp}(-2k_{D0}r)]\cdot \frac{\partial f_{1}({\bf X_1})}{\partial {\bf
v_1}}f_{1}({\bf X_2})
\nonumber \\
&=&
\frac{e^{4}}{2mk_{B}Tk_{D}^{0}•}\frac{\partial }{\partial {\bf x_{1}}}[n(\textbf{x}_{1})]\cdot
\frac{\partial f_{1}({\bf X_1})}{\partial {\bf v_{1}}}
\nonumber \\ \end{aligned}$$
The calculation of the contribution of the second part is $$\begin{aligned}
&-&\frac{Bk_{D0}}{mk_B T} \int d{\bf X_2}\nabla_{\bf x_1}
[\frac{e^4}{r}{\rm exp}(-2k_{D0}r)(\cos[\frac{\mathbf{p}}{2}\cdot(\mathbf{x}_{1}+\mathbf{x}_{2})])]\cdot \frac{\partial f_{1}({\bf X_1})}{\partial {\bf
v_1}}f_{1}({\bf X_2})
\nonumber \\
&=&G\int d{\bf X_{2}}
\nabla_{\bf x_1}[\cos[\frac{\mathbf{p}}{2}\cdot(\mathbf{x}_{1}+\mathbf{x}_{2})] \int d{\bf k} \frac{e^{-i{\bf k}\cdot{(
{\bf{x_1-x_2}}) }}}{k^2+4 k_{D0}^2} ]\cdot \frac{\partial f_{1}({\bf
X_1})}{\partial {\bf v_1}}f_{1}({\bf X_2})
\nonumber \\
&=&G\int d{\bf X_{2}}
\nabla_{\bf x_1} [\cos[\frac{\mathbf{p}}{2}\cdot(\mathbf{x}_{1}+\mathbf{x}_{2})] \int d{\bf k} \frac{e^{-i{\bf k}\cdot{(
{\bf{x_1-x_2}}) }}}{k^2+4 k_{D0}^2} ] \cdot \frac{\partial f_{1}({\bf
X_1})}{\partial {\bf v_1}}\int d{\bf k_2} {\tilde f_{1}}({\bf
{k_2,v_2}})\exp(-i{\bf{k_2\cdot x_2}})
\nonumber \\
&=&\frac{G}{2}\int d{\bf v_{2}}
\nabla_{\bf x_1} \int d{\bf k} \frac{e^{-i{\bf
k}\cdot{\bf{x_1}}}}{k^2+4 k_{D0}^2} \cdot \frac{\partial f_{1}({\bf
X_1})}{\partial {\bf v_1}}\int d{\bf k_2} {\tilde f_{1}}({\bf
{k_2,v_2}})[e^{i{\bf
\frac{p}{2}}\cdot{\bf{x_1}}}\delta {\bf (k+\frac{p}{2}-k_2)}+e^{-i{\bf
\frac{p}{2}}\cdot{\bf{x_1}}}\delta {\bf (k-\frac{p}{2}-k_2)}]
\nonumber \\\end{aligned}$$ where $G=-\frac{Bk_{D0}e^{4}}{mk_B T}$, $\tilde f_1({\bf
{k_2,v_2}}) $ is the Fourier transform of $f_1({\bf X_2})$. For long wavelength perturbations such that $k \ll k_{D} $ , we can approximate $$\frac{1}{k^{2} + 4k_{D0}^{2}} \simeq
\frac{1}{4k_{D0}^2}$$ $$\begin{aligned}
&=&\frac{G}{2}\int d{\bf v_{2}}
\nabla_{\bf x_1} \int d{\bf k} \frac{e^{-i{\bf
k}\cdot{\bf{x_1}}}}{4 k_{D0}^{2}} \cdot \frac{\partial f_{1}({\bf
X_1})}{\partial {\bf v_1}} [e^{i{\bf
\frac{p}{2}}\cdot{\bf{x_1}}}{\tilde f_{1}}({\bf {k+\frac{p}{2},v_2}})+e^{-i{\bf
\frac{p}{2}}\cdot{\bf{x_1}}}{\tilde f_{1}}({\bf {k-\frac{p}{2},v_2}})]
\nonumber \\
&=&
-\frac{e^{4}}{mk_{B}Tk_{D0}}\frac{\partial }{\partial {\bf x_{1}}}[n(\textbf{x}_{1})[\frac{B}{4n_{0}}\cos(\mathbf{p}\cdot\mathbf{x}_{1})]\cdot
\frac{\partial f_{1}({\bf X_1})}{\partial {\bf v_{1}}}
\nonumber \\\end{aligned}$$
Finally, adding the contributions of eq.(11) and eq.(13) $$n_{0}\int
d{\bf X_2}{\bf a_{12}}\cdot \frac{\partial g_{12}}{\partial
{\bf v_{1}}}=\frac{e^{4}}{2mk_{B}Tk_{D0}}\frac{\partial }{\partial {\bf x_{1}}}[n(\textbf{x}_{1})[1-\frac{B}{2n_{0}}\cos(\mathbf{p}\cdot\mathbf{x}_{1})]\cdot
\frac{\partial f_{1}({\bf X_1})}{\partial {\bf v_{1}}}\label{v4}$$ We may define $$\theta(\textbf{x}_{1})=\frac{1}{2n_{0}}[n(\textbf{x}_{1})[1-\frac{B}{2n_{0}}\cos(\mathbf{p}\cdot\mathbf{x}_{1})]$$
The kinetic equation containing effects of weak correlations is given by the following :
$${\bf v_{1}} \cdot \frac{\partial f_1({\bf X_1})}{\partial {\bf
x_{1}}}-\frac{\partial \phi({\bf
x_1})}{\partial {\bf x_1}}\cdot\frac {\partial f_1({\bf
X_1})}{\partial {\bf v_{1}}}+g\frac{\partial \theta({\bf
x_1})}{\partial {\bf x_1}}\cdot\frac{\partial f_{1}({\bf
X_1})}{\partial {\bf v_{1}}}=0\label{v5}$$
In eq.(\[v5\]), the variables $x,n,v,\phi$ are normalized by $\lambda_{D0} (=n_{0}e^{2}/k_B T)$, $n_0$, $v_{th} (=\sqrt{k_B T/m})$, $k_B T$ respectively and $$g= \frac{1}{n_{0}\lambda_{D0}^{3}}$$
The equilibrium solution of eq.(\[v5\]) is $$f_{1}({\bf
X_1})=e^{-(v_{1}^{2}/2+\phi({\bf
x_1})-g\theta({\bf
x_1}))}\label{v6}$$
It is a Maxwell-Boltzmann distribution with an additional contribution of $-g\theta(\textbf{x}_{1})$ to the potential energy due to the correlation of the particles.
In an earlier attempt [@kn:ab3] we had obtained an expression of single particle distribution function in the context of weakly correlated inhomogeneos plasma system. Therefore, the difference between this work and [@kn:ab3] should be mentioned clearly. In [@kn:ab3] we had made an Ansatz that the correlation function which depends only on the separation of the particles and the single particle distribution function in the homogenous plasma is also applicable to the Inhomogeneos systems. Therefore, in that case the inhomogeneity was observed in the single particle distribution function. The assumption is now lifted in this article as we have chosen the most general form of the correlation function obtained in [@kn:ab1] and the correlation function depends not only on the separation of the particles and the single particle distribution function but also on the average position of the particles. Finally, it may be concluded that inclusion of correlation contributes to the potential energy of the system and the reults seems to be different atleast of the order of $B/n_{0}$ from the result obtained in [@kn:ab3]. At the end, it may be concluded that we have been able to incorporate the effect of inhomogeneity in a more proper way as it has been derived from a more accurate expression of the pair correlation function in the weakly inhomogeneous regime. This single particle distribution function may be utilized to study the thermodynamic properties of the concerned systems.
[99]{} T. H. Dupree, Phys. Fluids. [**[4]{}**]{}, 696 (1961). P. A. Wolff, Phys. Fluids. [**[5]{}**]{}, 316 (1962).
B. Jancovici, J. Stat. Phys. [**[17]{}**]{}, 357 (1977). C. Deutsch, Y. Furutani and M.M. Gombert, Phys. Reports, [**[69]{}**]{},85 (1981). V.E. Fortov, A.V. Gavrikov, O.F. Petrov, I.A. Shakhova and V.S. Vorobev, Phys. Plasmas [**[14]{}**]{}, 040705, (2007). N. R. Shaffer, S. K. Tiwari and S. D. Baalrud, Phys. Plasmas [**[24]{}**]{}, 092703 (2017). N. Desbiens, P. Arnault, and J. Clérouin, Phys. Plasmas, [**[23]{}**]{}, 092120 (2016). V. S. Filinov, V. E. Fortov, M. Bonitz, and D. Kremp, Phys. Lett. A [**[274]{}**]{}, 228 (2000) T. Dornheim, S. Groth, T. Sjostrom, F. D. Malone, W. Foulkes, and M. Bonitz, Phys. Rev. Lett. [**[117]{}**]{}, 156403 (2016) Thomas O’Neil and N. Rostoker, Phys. Fluids, [**[8]{}**]{}, 1109 (1965). D. A. Eisa, Phys. Plasmas [**[25]{}**]{}, 033507 (2018) A. Bose, Phys. Plasmas [**[23]{}**]{}, 104505 (2016). A. Bose, Phys. Plasmas [**[26]{}**]{}, 064501 (2019). A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko and K. N. Stepanov, Plasma Electrodynamics Volume I (Pergamon, Oxford, 1975) A. Bose and M. S. Janaki, Phys. Plasmas [**[14]{}**]{}, 063701 (2007).
|
---
abstract: 'Using the intertwining relation we construct a pseudosuperpartner for a (non-Hermitian) Dirac-like Hamiltonian describing a two-level system interacting in the rotating wave approximation with the electric component of an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry generators close a quadratic pseudosuperalgebra. A class of time dependent electric fields for which the equation of motion for a two level system placed in this field can be solved exactly is obtained. New interesting phenomenon is observed. There exists such a time-dependent detuning of the field frequency from the resonance value that the probability to populate the excited level ceases to oscillate and becomes a monotonically growing function of time tending to $3/4$. It is shown that near this fixed excitation regime the probability exhibits two kinds of oscillations. The oscillations with a small amplitude and a frequency close to the Rabi frequency (fast oscillations) take place at the background of the ones with a big amplitude and a small frequency (slow oscillations). During the period of slow oscillations the minimal value of the probability to populate the excited level may exceed $1/2$ suggesting for an ensemble of such two-level atoms the possibility to acquire the inverse population and exhibit lasing properties.'
address: ' Department of Physics, Tomsk State University, 36 Lenin Avenue, 634050 Tomsk, Russia '
author:
- Boris F Samsonov and V V Shamshutdinova
title: 'Quadratic pseudosupersymmetry in two-level systems'
---
**Corresponding Author**: B F Samsonov
E-mail: [*samsonov@phys.tsu.ru*]{}
Introduction
============
The supersymmetry in physics has been introduced in the quantum field theory for unifying different interactions in a unique construct [@r1]. Supersymmetric formulation of quantum mechanics is due to the problem of spontaneous supersymmetry breaking [@r2]. Ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems since, and now there are no doubts that the supersymmetric quantum mechanics (SUSY QM) has its own right to exist (for recent developments see a special issue of Journal of Physics A, vol. 34, No 43, 2004). It is worth noticing that most papers in this field deal with the Hermitian Hamiltonians.
Differential equation of Schrödinger-like type with a non-Hermitian Hamiltonian appears in many physical models. One can cite quantum systems coupled to the environment like a hydrogen “atom" in an interacting medium subject to a dissipative force [@ND] (see also [@Wong]) or different decay or collision reactions (see e. g. [@Baz]; for more recent developments see [@RDM]; in [@Baye] the method of SUSY QM is involved). Physical needs initiated a deep mathematical study of spectral problems with non-Hermitian Hamiltonians in 50th and 60th of the previous century. The most essential result was first obtained by Keldysh [@Keldysh] who proved the completeness of the set of eigenfunctions and associated functions for a regular Sturm-Liouville problem with a non-Hermitian Hamiltonian. In the books by Naimark [@Naimark] and Marchenko [@Marchenko] one can find good reviews of these studies.
A new impact to studying different properties of non-Hermitian Hamiltonians is due to the discovery that the real character of the spectrum of a non-Hermitian Hamiltonian may be in particular related with so-called -symmetry [@Bender1] and suggestion to generalize the quantum mechanics by accepting non-Hermitian Hamiltonians with a real spectrum to describe physical observables [@Bender2] (see also the review [@Bender3]). The necessary condition for such a generalization consists in the possibility to define a Hilbert space with a positive definite metric which is intimately related with the property of a Hamiltonian to be diagonalizable (for recent discussions see e.g. [@Bogdan; @my1]). This apparently may be assured in many cases since non-diagonalizable Hamiltonians may be transformed into diagonalizable ones by SUSY transformations [@my2]. The latter property permits us to suppose that the method of SUSY QM may become an essential ingredient of the complex quantum mechanics. This conjecture is also supported by established properties of this method not only to offer possibility for obtaining new exactly solvable complex potentials from known ones [@Csusy1] but also to help deeper understanding different properties of complex potentials [@Csusy1; @Csusy2]. In particular, an explicit construction of a superalgebra involving non-Hermitian Hamiltonians, which may be useful in different contexts i.e. integrability, quantization, different quantum field models etc, is shown to be possible [@susyAlg] and even is now developed till the notion of pseudosupersymmetry [@Mostafa] and nonlinear pseudosupersymmetry [@Pl].
The relation of the general two-level model described by a non-Hermitian Hamiltonian acting in the two-dimensional Hilbert space $\Bbb C^2$ with the pseudosupersymmetry is discussed by Mostafazadeh [@Mostafa]. In contrast to the approach of this author we reduce the time-dependent Schrödinger equation for the two level system, interacting in the rotating wave approximation with the electric component of an electromagnetic field, with a Hermitian Hamiltonian (see e.g. [@books]) to the one-dimensional stationary Dirac equation with an effective non-Hermitian Hamiltonian where the time plays the role of the space variable. If we considered the spectral properties of the latter Hamiltonian we would define it in the Hilbert space $L^2(0,T)\otimes \Bbb C^2$. But as we shall see in our approach the spectral parameter in the Dirac equation is not related with spectral properties of the two-level system. Therefore we will not discuss any spectral features of this Hamiltonian and in particular its diagonalizability. Of course, the obtained Dirac equation is completely equivalent to the initial Schrödinger equation and if one studied it by usual means one would not get any new information about the two-level system. From this point of view the method of SUSY QM we are using proves its extreme efficiency once again.
To find a pseudosuperpartner for the given Dirac-like Hamiltonian we are using the technique of intertwining operators developed in [@Annals] for the one-dimensional stationary Dirac equation. We have to notice that the application of results of this paper to our particular problem is not straightforward since transformation operators of the general form do not preserve the very peculiar form of the effective Dirac Hamiltonian corresponding to the two-level system. So, below we show how from the wide variety of possible transformations one can choose the necessary ones. In our approach in contrast to [@Mostafa] the two pseudosuperpartners and pseudosupersymmetry generators constructed with the help of first order intertwiners close a quadratic pseudosuperalgebra. As it usually happens for the method of intertwining operators [@BSTMF] if one of the two Hamiltonians is exactly solvable the same property takes place for the other. In this way starting from the simplest case corresponding to the famous Rabi oscillations we have found new electric fields having time-dependent frequencies for which the equation of motion of the two-level system has exact solutions. While analyzing solutions of the [Schrödinger ]{}equation we have found a new interesting physical phenomenon. We show that there exists such a time-dependent detuning of the field frequency from the resonance value that the probability to populate the excited level ceases to oscillate and becomes a monotonically growing function of time tending to $3/4$. Of course this is a strictly fixed excitation regime similar to resonance. We also study how the above probability behaves under small deviations from this specific regime. We have found that when the parameters of the model are close enough to the specific values the probability exhibits two kinds of oscillations. The oscillations with a small amplitude and a frequency close to the Rabi frequency (fast oscillations) take place at the background of the ones with a big amplitude and a small frequency (slow oscillations). During the period of slow oscillations, which grows when the parameters of the model approach the above specific values, the minimal value of the probability to populate the excited level may exceed $1/2$ suggesting for an ensemble of such two-level atoms the possibility to acquire the inverse population and exhibit lasing properties.
We have to notice that some of the results we expose below are known from the previous paper [@Bagrov]. These authors also use a similar intertwining technique but they do not relate it with the pseudosupersymmetry and do not give any analysis of solutions this method can provide with. Moreover, we give a deeper analysis of restrictions imposed on transformation operators by the features of the two-level system. In particular, we show that both the new Hamiltonian and solutions of the new Dirac equation can be expressed in terms of a real-valued function which is a solution of a second order differential equation with real coefficients. Since such equations have real solutions always our analysis opens the direct possibility to realize chains of transformations preserving the form of the Dirac-like Hamiltonian imposed by the features of the two-level system.
Preliminary
===========
The two-level model in the rotating wave approximation with a possibly time-dependent detuning is described by the following system of equations (see e.g. [@books]): \[Asist1\] iA\_1-fA\_1=A\_2 iA\_2+fA\_2=A\_1 where $\xi=\frac{1}{2\hbar}E_0d_{12}$, $d_{12}$ is the matrix element of the dipole interaction operator, $E_0$ is the amplitude of the electric component of an external electromagnetic field; $f=\frac 12\frac{d}{dt}(\d t)$, $\d(t)=\o_{12}-\o(t)$, $\o_{12}=\frac{1}{\hbar}(\e_1-\e_2)$, $\e_1$ and $\e_2$ are energy levels of the free atom and $\o(t)$ is the field frequency; the dot over the symbol means the derivative with respect to time. While normalized properly the functions $|A_1(t)|^2$ and $|A_2(t)|^2$ give occupation probabilities for the ground and excited states respectively. If $\o$ does not depend on time (hence $f=\frac 12\d=\mbox{const}$) solutions of the system (\[Asist1\]) are well-known. For instance, with the initial condition $A_2=0$ and $A_1=1$ at $t=0$ we get the well-known formula [@books] for the excited state occupation probability if initially the system is in the ground state \[C2\] P(t)= |A\_2(t)|\^2=\^2=f\^2+\^2 with $2\xi$ known as the Rabi frequency. The probability (\[C2\]) is an oscillating function of time (so called Rabi oscillations). At the resonance ($f=\frac 12 \d =0$) it oscillates with the Rabi frequency. Therefore the value $\d(t)$ characterizes the detuning of $\o(t)$ from its resonance value equal $\o_{12}$. In Section 5 using the formalism developed in Section 4 we shall get time-dependent functions $f=f(t)$ (and hence $\d(t)$) for which system (\[Asist1\]) permits exact solutions. As we show below (Section 5) time-dependent corrections to the detuning that we will consider although may change crucially the time-dependent behavior of the solutions of system (\[Asist1\]) but they essentially keep oscillating character of the probability to populate the excited level with the frequency close to $2\Omega$. Yet, the absence of the Rabi oscillations may be considered as oscillations with the same frequency but with the zero amplitude since they may be obtained as corresponding limiting case of oscillations with a non-zero amplitude. So, in our approach the rotating wave approximation is as good as it is in the classical case of the electric field of a constant frequency.
Let us rewrite system (\[Asist1\]) in the matrix form \[MS\] h\_0=Eh\_0=\_t+V\_0 where \[V0\] V\_0= i f\_0\_y $\g=i\s_x$, $E=\xi$, $\Psi= (A_1, A_2)^T$ (the superscript “$T$" denotes the transposition) and we replaced $f$ (which we will call the “potential") in (\[Asist1\]) by $f_0$; $\s_{x,y,z}$ denote the standard Pauli matrices. Equation (\[MS\]) is the one-dimensional stationary Dirac equation with the non-Hermitian Hamiltonian $h_0$ defined by the potential (\[V0\]) where $t$ plays the role of the space variable. By construction the parameters $f_0$ and $E$ are real. For a fixed value of the dipole momentum of the irradiated system the parameter $E=\xi$ is defined by the amplitude of the electric field and, hence, is not related with spectral properties of the system. A useful comment is that since the Hamiltonian of the system (\[Asist1\]) is Hermitian, $ H_{sch}=\left(
\begin{array}{cc}f & \xi \\ \xi & -f \end{array}
\right)
$, the evolution of the two-level system is unitary even for a time-dependent function $f=f(t)$. This means that the $\Bbb C^2$ inner product, $|A_1(t)|^2+|A_2(t)|^2$, for the Dirac equation (\[MS\]) is $t$-independent.
SUSY algebra with non-Hermitian Hamiltonians
============================================
Let us have a non-Hermitian Hamiltonian $h_0$. We will not consider it as a Hamiltonian acting in a Hilbert space but to construct a SUSY algebra we need adjoint operators which we will introduce in a formal way. Denote by $h_0^+$ the operator formally adjoint to $h_0$. As usual the adjoint operation consists in taking the complex conjugation and transposition, the operator of the first derivative is skew-Hermitian and $(AB)^+=B^+A^+$.
Let $h_1$ be a “transformed Hamiltonian" which should be found together with the transformation operator $L$ by solving the intertwining relation $Lh_0=h_1L$ and $h_1^+$ be its adjoint. The later participates in the adjoint intertwining relation $h_0^+L^+=L^+h_1^+$. It means that the operator $L^+$ transforms eigenfunctions of $h_1^+$ into eigenfunctions of $h_0^+$.
Let us suppose that there exists an operator $J$ such that $h_{0,1}^+=Jh_{0,1}J$ and $J^2=\pm 1$, $J^+=\pm J$ (in general both signs may be accepted). Then from the adjoint intertwining relation it follows that $JL^+Jh_1=h_0JL^+J$ meaning that the operator $JL^+J$ realizes the backward transformation from $h_1$ to $h_0$ and the operator $JLJ$ transforms from $h_0^+$ to $h_1^+$. From here we infer that the superposition $JL^+JL$ transforms solutions of the equation (\[MS\]) into solutions of the same equation meaning that this is a symmetry operator for this equation. In the simplest case when $L$ is a differential operator that we would like to consider this symmetry operator may be a function of $h_0$, so we will suppose that $JL^+JL=F_1(h_0)$. By the same reason the superposition $LJL^+J$ may be a function of $h_1$ leading to $LJL^+J=F_2(h_1)$. Moreover, we will also suppose that $F_2(x)=F_1(x)\equiv F(x)$ is an analytic function. These properties generalize the known factorization (polynomial factorization if $F(x)$ is a polynomial, see e.g. [@Annals; @BSTMF]) properties taking place for the Hermitian case.
It follows from (\[MS\]) and (\[V0\]) that in our case $J=\sigma_x$.
Keeping in mind the properties of the operators $L$ and $J$ let us introduce the following matrix operators: H=(
[cc]{} h\_0 & 0\
0 & h\_1
) Q\_1=(
[cc]{} 0 & 0\
L & 0
) Q\_2=(
[cc]{} 0 & JL\^+J\
0 & 0
). It follows from the intertwining relations that the operators $Q_1$ and $Q_2$ commute with $H$ and they apparently are nilpotent. The above factorization properties are equivalent to the following anticommutation relation: $Q_1Q_2+Q_2Q_1=F(H)$.
Now if we identify our $J$ operator with $\eta_-=\eta_+^{-1}$ introduced in [@Mostafa], $J=\eta_-=\eta_+^{-1}$, our $L$ operator with $D$ and $JL^+J$ with $D^{\sharp}$, we conclude that the operator $Q_2$ becomes pseudoadjoint to $Q_1$, the operators $H$,$Q_1$ and $Q_2$ close a nonlinear superalgebra and one can associate a nonlinear pseudosupersymmetry with quantum system described by the Hamiltonian $H$. In the next Section we shall show that a quadratic pseudosupersymmetry may be associated with the two-level system.
Intertwining operators for two-level Hamiltonians
=================================================
To be able to associate a pseudosupersymmetry with the Hamiltonian given in (\[MS\]) and (\[V0\]) we have to find an intertwining operator and a partner Hamiltonian $h_1$. According to Ref. [@Annals] the intertwining operator $L$ for a matrix equation such as (\[MS\]) is defined with the help of a matrix-valued function $\cU=\cU(t)$ satisfying the equation \[U\] h\_0=ŁŁ=(ł\_1,ł\_2) called the “transformation function", as follows: \[L\] L=\_t-W W=\^[-1]{}. Here $\l_1$ and $\l_2$ are arbitrary constants. The operator $L$ transforms a solution $\Psi$ of equation (\[MS\]) into a solution $\Phi$ of the same equation where the matrix $V_0$ is replaced by \[V1\] V\_1=V\_0+V V=W -W. Here and in the following the subscript $0$ marks quantities before the transformation and $1$ marks these after the transformation. It is not difficult to see that to preserve the form (\[V0\]) of the potential so that $V_1=if_1\s_y$ it is sufficient to take the transformation function of the form \[cU\] =(
[cc]{} u\_[11]{} & u\_[11]{}\
u\_[21]{} & -u\_[21]{}
). In this case the column-vector $U_1=(u_{11},u_{21})^T$ is a solution to the initial equation (\[MS\]) corresponding to the eigenvalue $\l$ and the column-vector $U_2=(u_{11},-u_{21})^T$ is a solution to the same equation with the eigenvalue $-\l$ (note that this symmetry is built into the system (\[MS\])!) so that $\L$ in (\[U\]) has the form $\L=\mbox{diag}(\l,-\l)$. After some simple algebra one finds from (\[V1\]) that $f_1=f_0+\Delta f$ where \[Df\] f=ł( - ) -2f\_0.
In general, solutions $U_{1,2}$ of equation (\[MS\]) from which the matrix $\cU$ is composed, $\cU=(U_1,U_2)$, are complex, leading to a complex-valued potential difference $\Delta f$. For physical reasons we require real potentials. A necessary condition for $\Delta f$ to be real is that the eigenvalue $\l$ be purely imaginary. Indeed, it is easy to show that $\l$ cannot be real. According to (\[Df\]) $\Delta f$ is defined by the expression $\frac{u_{11\vphantom{I_i}}}{u_{21}}-\frac{u_{21\vphantom{I_i}}}{u_{11}}$. Putting $\frac{u_{11\vphantom{I_i}}}{u_{21}}=\varrho\exp(i\vfi)$ one finds \[u1121\] -= (-)+i(+)and our claim follows from the fact that $\varrho+\frac{1}{\varrho}$ is never equal to zero. Finally one can prove that $\l^2$ is real (cf. [@Bagrov]).
Now when the imaginary character of $\l$ is established we see from (\[Df\]) that the left hand side of (\[u1121\]) must be purely imaginary, which is possible only if $\varrho=1$, meaning that $u_{11}$ and $u_{21}$ have the same absolute value. Therefore one can put $u_{11}=\rho\exp(i\vfi_1)$ and $u_{21}=\rho\exp(i\vfi_2)$. Using the fact that $U_1=(u_{11},u_{21})^T$ satisfies equation (\[MS\]) with $E=\l$ and setting $\l=iR$, where $R$ is real, one gets from (\[MS\]) a system of equations for $\rho$, $\vfi_1$ and $\vfi_2$. Of these equations we need only \[vfi\] \_2-\_1-2f\_0+2R(\_2-\_1)=0. If $R=0$ (\[vfi\]) can be readily integrated. Suppose $R\ne 0$. The change of the dependent variable in equation (\[vfi\]), $\vfi_2-\vfi_1=2\arctan q$, yields for $q$ the Riccati equation q+2Rq-f\_0(1+q\^2)=0. If $f_0=0$ the equation for $q$ is readily integrated: $q=\exp(-2Rt)$. Considering $f_0\ne 0$ one can linearize (13) by putting $q=-{\dot u}/({uf_0})$, so $u$ is a solution to the second order equation \[qE\] u+(2R- [f\_0]{}/[f\_0]{})u+f\_0\^2u=0. Introducing the new variable $\psi$ by putting $u=\exp(-Rt)\sqrt {f_0}\psi$ one eliminates the first derivative term from (\[qE\]) thus obtaining \[psieq\] +=0. This equation has two linearly independent real solutions and, hence, $\psi$ is defined up to one real constant. Once $\psi$ is fixed one calculates $q$: \[q\] q=-- and the potential difference $\Delta f=2R\sin(\vfi_2-\vfi_1)-2f_0$: f=-2f\_0.
Solution $\Phi$ of the equation $h_1\Phi=E\Phi$ with $h_1=\g\partial_t+V_1$, $V_1=V_0+\Delta V$, $\Delta V=i\Delta f
\s_y$ can be found by applying the transformation operator (\[L\]) to solution $\Psi$ of the equation (\[MS\]), $\Phi=L\Psi$. It is easy to see that the matrix $W$ is diagonal \[w12\] W=(w\_1,w\_2)w\_1=-if\_0+Ru\_[21]{}/u\_[11]{} w\_2=w\_1\^\*. and the ratio of the components of the spinor $U_1$ defining $w_1$ in (\[w12\]) is also expressible in terms of the function $q$: \[fracu\] =.
Finally skipping calculational details but noticing that just in the same way as it was done in [@Annals] one can find the following factorizations: JL\^+JL=h\_0\^2-ł\^2,LJL\^+J=h\_1\^2-ł\^2 with $J=\sigma_1$. This means that the function $F$ from Section 3 is $F(x)=x^2-\l^2$, the operators $H$, $Q_1$ and $Q_2$ close the quadratic superalgebra and the quadratic pseudosupersymmetry underlies the two-level system interacting with the electric component of an electromagnetic field.
Application: SUSY transformations of the Rabi oscillations
==========================================================
In this Section we show a new physical phenomenon we observed while analyzing solutions of the system (\[Asist1\]) obtained using the above developed technique.
We start with $\d_0=2f_0=\o_{12}-\o_0 =constant$ (this corresponds to the Rabi oscillations (\[C2\])) to get a time-dependent “potential" $f_1(t)=f_0+\Delta f(t)=\frac 12 \frac d{dt}[\d_1(t) t]$. Once $f_1(t)$ is found we calculate the detuning $\d_1(t)=\o_{12}-\o_1(t)$ by integrating the previous equation \[deltat\] \_1(t)=2t\_0\^tf\_1(t)dt. We have found that relatively small but time-dependent perturbations of the field frequency $\o_1(t)$ from its resonance value equal $\o_{12}$ may influence essentially the time behavior of the probability $P_{1}(t)$ to populate the excited state level with respect to the constant frequency case.
If $f_0=\mbox{const}$ equation (\[psieq\]) for $\psi$ reduces to \[psidd\] +\^2=0 \^2 = f\_0\^2-R\^2=. Solutions of this equation have different properties depending on whether the value $\varpi^2$ is positive, negative or zero. We have found that the oscillating behavior of the probability $P_1(t)$ disappears when $\varpi=0$. In this case the general solution to equation (\[psidd\]) is a linear function of time $\psi=At+B$ which according to (\[q\]) gives the following time dependence of the function $q$: $q(t)=1-A/(Atf_0+Bf_0)$. Once $q(t)$ is found one calculates the “potential difference" with the help of formula (\[Df\]) and finally the new “potential" $f=f_1(t)$: $$f_1(t)=f_0-\frac{2A^2f_0}{2 A^2f_0^2t^2-
2 Af_0(A-2Bf_0)t+A^2-2ABf_0+2B^2f_0^2}\,.$$ Another restriction leading to the desirable result is $A=2Bf_0$ which reduces the previous equation to a simpler form $$\label{f1t}
f_1(t)=f_0- \frac{4f_0}{1+4f_0^2t^2}\,.$$
Since solutions $A_{10}(t)$ and $A_{20}(t)$ of the system (\[Asist1\]) for $f=f_0=\mbox{const}$ are known one can find solutions $A_{11}(t)$ and $A_{21}(t)$ of the same system with $f=f_1(t)$ by applying the transformation operator $L$ defined by formulas (\[L\]), (\[w12\]) and (\[fracu\]) to the previous solution. In this way imposing the initial condition $A_{11}(0)=1$ and $A_{21}(0)=0$ one finds the probability $P_1(t)$ to populate the excited level at the time moment $t$ if at $t=0$ only the ground state level is populated P\_1(t)=|A\_[21]{}(t)|\^2= . Here $\O=\sqrt{f_0^2+\xi^2}$ and $2\O$ is the frequency of oscillations of the probability $P_0(t)$ (\[C2\]) at $f=f_0$. It is clearly seen that $P_1(t)$ is an oscillating function provided $\xi^2\ne 3f_0^2$. For $\xi^2=3f_0^2$ ($\O=2f_0$) the probability becomes equal \[P1t\] P\_1(t)= which is a function monotonically growing from zero at the initial time moment till the value $3/4$ at $t\to\infty$. We have to notice that for a fixed $\xi$ the parameter $f_0$ is fixed also, $f_0=\xi/\sqrt 3$, which by means of formulas (\[f1t\]) and (\[deltat\]) fixes the frequency of the electric field in the unique way. So, for the given dipole momentum this excitation regime is fixed by the amplitude of the electric field. Let us analyze now what is happening with the probability $P_1(t)$ when the parameters of the model are close to this exceptional point.
Suppose now $\varpi^2>0$ and we will consider it to be close to zero. In this case the general solution to equation (\[psidd\]) may be written as $\psi=\frac{A}{\varpi}\sin(\varpi t+a+b)$. The function $q$ as given in (\[q\]) does not depend on the value of the coefficient $\frac{A}{\varpi}$ but we need this coefficient to realize the limit $\varpi\to 0$ thus recovering the previously obtained solution. Choosing $b$ such that $\sin 2b=\varpi/f_0$ and $\cos 2b=R/f_0$ but keeping $a$ arbitrary one gets = -[R-f\_0(2t+2a)]{}. This leads to the following expression for $q$: q=[f\_0(2t+2a)-R]{} and finally to the “potential difference" of the form \[dfBagr1\] f(t)=. This formula has been previously derived by V.G. Bagrov et. al. by other means [@Bagrov]. Putting $a=\mbox{arctg}\frac{\varpi}{2f_0}-
\frac 12\mbox{arctg}\frac{\varpi}{R}$ one recovers for $f_1(t)=f_0+\Delta f(t)$ the previous result (\[f1t\]) as the limit $\varpi\to0$. This means that for $\varpi$ close to zero the probability $P_1(t)$ corresponding to the potential difference (\[dfBagr1\]) should be close to the previous value (\[P1t\]). The analytic expression for $P_1(t)$ is rather complicated and we will restrict ourselves by graphical illustrations.
Let us fix the Rabi frequency $2\xi$. The function $\Delta f(t)$ (\[dfBagr1\]) contains three parameters $\varpi$, $f_0$ and $a$. The parameter $f_0$ defines the value $2\O=2\sqrt{f_0^2+\xi^2}$, which is the frequency of oscillations of the function $P_0(t)$ given by (\[C2\]) to which $P_1(t)$ is reduced when the time dependent correction $\Delta f(t)$ is absent. As it was already mentioned when $f_0=0$ (resonance case) the function $P_0(t)$ oscillates with the Rabi frequency $2\xi$. The parameter $\varpi$ defines the frequency of the time dependent correction $\Delta f(t)$ (\[dfBagr1\]) for $f_1=f_0+\Delta f$ and the parameter $a$ is responsible for the initial value of $f_1(t)$. The probability $P_1(t)$ is a periodical function if $\O$ is commensurable with $\varpi$. In this case it exhibits two kinds of oscillations, namely, fast oscillations with the frequency $2\O$, which is close to the Rabi frequency when $f_0$ is close to zero, taking place at the background of slow oscillations with the frequency $2\varpi$.
For our numerical illustrations we choose $f_0=1$. If in standard units this is $1\cdot 10^{11}$ c$^{-1}$ this corresponds to $10^{-11}$ c as the unity of time in our figures.
Fig. 1a shows the probability $P_1(t)$ for $\O=2$, $a=0.015$ and $\varpi=1/4$ (solid line) and $\varpi=1/6$ (dotted line).
\[fig1\]
Fig. 1b illustrates the time behavior of the detuning $\delta_1(t)$ calculated according to (\[deltat\]) for $a=0.015$, $\varpi=1/4$ and $\varpi=1/6$ (solid and dotted lines respectively) together with its limiting value corresponding to $\varpi=10^{-3}$ and $a=10^{-6}$ (dashed line). It is clearly seen from Fig. 1a that the period of slow oscillations grows when $\varpi$ decreases and fast oscillations go around the limiting value $0.75$ with the amplitude increasing with $\varpi$ decreasing. Moreover, Fig. 1b says that oscillating behavior of $P_1(t)$ is transformed into monotonically growing one when for $\varpi=0$ the detuning becomes a monotone function of time (dotted line on Fig 1b). If it acquires some oscillating perturbations the probability starts to oscillate also.
\[fig1\]
The next two figures show the dependence of the same quantities on the parameter $a$ which is responsible for the phase shift in formula (\[dfBagr1\]) at the fixed value $\varpi=1/5$. Dotted, dashed and solid lines (figure 2a) correspond to $a=0$, $a=0.02$ and $a=0.08$ respectively. Figure 2b shows the time dependence of the detuning $\d(t)$ for $a=0$ (dotted line) and $a=0.08$ (solid line). From Fig. 2b we can conclude that the parameter $a$ defines mainly the maximum of the absolute value of the detuning which it takes at $t=0$. Fig. 2a says that the amplitude of fast oscillations grows together with $a$.
\[fig3\]
The next figure shows the dependence of $P_1(t)$ from the frequency of fast oscillations $\O$ at $a=0$ and $\varpi=0.2$. Dotted, solid and dashed lines correspond to $\O=2$, $\O=1.6$ and $\O=1.2$ respectively. More it differs from the critical value equal $2$ corresponding to $\xi^2=3f_0^2$, when the oscillations in formula (\[Aoscil\]) disappear, bigger the amplitude of the fast oscillations becomes.
Conclusion
==========
Using the technique of intertwining operators for a Dirac-like system developed in [@Annals] we have found time dependent electric fields for which the equation of motion for a two-level system placed in this field obtained after the rotating wave approximation can be solved exactly. Pseudosupersymmetry generators constructed with the help of intertwining operators together with the super-Hamiltonian close a quadratic deformation of the superalgebra constructed in [@Mostafa]. We conclude, hence, that two-level systems in external electromagnetic fields may have hidden quadratic pseudosupersymmetry which is responsible for the new phenomenon consisting in disappearance of the Rabi oscillations.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work is partially supported by the President Grant of Russia 1743.2003.2 and the Spanish MCYT and European FEDER grant BFM 2002-03773. Authors are grateful to V.G. Bagrov for attracting their attention to this problem. BFS is grateful to P. Roy, M. Znojil and M. Ioffe for pointing out some useful publications.
References {#references .unnumbered}
==========
[99]{}
Golfand Y A and Likhtman E P 1971 [*JETP Lett.*]{} [**13**]{} 323;\
Ramond P 1971 [*Phys. Rev.*]{} [**D3**]{} 2415;\
Neveu A and Schwarz J 1971 [*Nucl. Phys.*]{} [**B31**]{} 86
Witten E 1981 [*Nucl. Phys.*]{} [**B188**]{} 513; 1982 [*Nucl. Phys.*]{} [**B202**]{} 253
Nakayama T. and DeWitt H 1964 [*J. Quant. Spectr. Radiative Transfer*]{} [**4**]{} 623
Wong J 1967 [*J. Math. Phys.*]{} [**8**]{} 2039
Baz’ A I, Zel’dovich Ya B and Perelomov A M 1969 [*Scattering, reactions and decay in non-relativistic quantum mechanics*]{} (Jerusalem: Israel Programm for Scientific Translations)
Baker H C 1984 [*Phys. Rev.*]{} [**A30**]{} 773\
Ruschhaupt A, Delgado F and Muga J G 2005 [*J. Phys.*]{} [**A38**]{} L171
Sparenberg J-M and Baye D 1996 [*Phys. Rev.*]{} [**C54**]{} 1309
Keldysh M V 1951 [*DAN USSR (Doklady Akademii Nauk SSSR)*]{} [**77**]{} 11
Naimark M A 1969 [*Linear differential operators*]{} (Moscow: Nauka)
Marchenko V A 1977 [*Sturm-Liouvulle operators and their applications*]{} (Kiev: Naukova Dumka)
Bender C M and Boettcher S 1998 [*Phys. Rev. Lett.*]{} [**24**]{} 5243
Bender C M, Brody D C and Jones H F, 2002 [*Phys. Rev. Lett.*]{} [**89**]{} 270401 and 2004 [**92**]{} 119902 (erratum)
Bender C M, Brod J, Refig A and Reuter M 2004 [*J. Phys.*]{} [**A37**]{} 10139
Ramírez A and Mielnik B 2003 [*Rev. Mex. Fis*]{} [**49**]{} (S2) 130
Samsonov B F and Roy P 2005 [*J. Phys.*]{} [**A38**]{} L249
Samsonov B F 2005 [*SUSY transformations between diagonalizable and non-diagonalizable Hamiltonians*]{} preprint quant/ph 0503075
Cannata F, Junker G and Trost J 1998 [*Phys. Lett.*]{} [**A246**]{} 219\
Andrianov A, Cannata F, Dedonder J P and Ioffe M V 1999 [*Int. J. Mod. Phys.*]{} [**A14**]{} 2675\
Bagchi B, Mallik S and Quesne C 2001 [*Int.J.Mod.Phys.*]{} [**A16**]{} 2859\
Petrović J S, Milanović V and Ikonić Z 2002 [*Phys. Lett.*]{} [**A300**]{} 595\
Fernández D J, Muños R and Ramos A 2003 [*Phys.Lett.*]{} [**A308**]{} 11\
Rosas-Ortiz O and Muñoz R 2003 [*J. Phys*]{} [**A36**]{} 8497\
Bagchi B, Bíla H, Jakubský V, Mallik S, Quesne C and Znojil M 2005 [*PT -symmetric supersymmetry in a solvable short-range model*]{} preprint quant-ph/0503035
Znojil M 2001 [*Czech. J. Phys.*]{} [**51**]{} 420\
Dorey P, Dunning C and Tateo R 2001 [*J. Phys.*]{} [**A 34**]{} L391\
Cannata F, Ioffe M V, Roychoudhury R and Roy P 2001 [*Phys.Lett.*]{} [**A281**]{} 305\
Levai G and Znojil M 2002 [*J. Phys.*]{} [**A35**]{} 8793\
Cannata F, Ioffe M V and Nishnianidze D N 2003 [*Phys.Lett.*]{} [**A310**]{} 344\
Znojil M 2003 [*PT-symmetry and supersymmetry*]{} in “GROUP 24: Physical and Mathematical Aspects of Symmetries" (IOP Publishing, Bristol). pp. 629 - 632 (proceedings of the XXIV International Colloquium on Group Theoretical Methods in Physics, Paris, July 15-20, 2002, Institute of Physics Conference Series Nr. 173, Sect. 7, Ed. Jean-Pierre Gazeau, Richard Kerner, Jean-Pierre Antoine, Stephane Metens and Jean-Yves Thibon, preprint hep-th/0209062)\
Bagchi B, Banerjee A, Caliceti E, Cannata F, Geyer H B Quesne C and Znojil M 2004 [*CPT -conserving Hamiltonians and their nonlinear supersymmetrization using differential charge-operators C*]{} Preprint hep-th/0412211
Znojil M, Cannata F, Bagchi B and Roychoudhury R 2000 [*Phys. Lett.*]{} [**B483**]{} 284
Mostafazadeh A 2002 [*J. Math. Phys.*]{} [**43**]{} 205; 2002 [*Nuclear Physics*]{} [**B640**]{} 419
Klishevich S M and Plyushchay M S 2002 [*Nucl. Phys.*]{} [**B628**]{} 217
Orszag M 2000 [*Quantum optics*]{} (Berlin: Springer-Verlag)
Nieto L M, Pecheritsin A A and Samsonov B F 2003 [*Ann. Phys. (NY)*]{} [**305/2**]{} 151
Bagrov V G and Samsonov B F 1995 [*Theor. Math. Phys.*]{} [**104**]{} 1051
Bagrov V G, Baldiotti M C, Gitman D M and Shamshutdinova V V 2004 [*Darboux transformations of two-level systems*]{} preprint math-ph/0404078
|
---
abstract: 'We present 888 visual-wavelength spectra of 122 nearby type II supernovae (SNe II) obtained between 1986 and 2009, and ranging between 3 and 363 days post explosion. In this first paper, we outline our observations and data reduction techniques, together with a characterization based on the spectral diversity of SNe II. A statistical analysis of the spectral matching technique is discussed as an alternative to non-detection constraints for estimating SN explosion epochs. The time evolution of spectral lines is presented and analysed in terms of how this differs for SNe of different photometric, spectral, and environmental properties: velocities, pseudo-equivalent widths, decline rates, magnitudes, time durations, and environment metallicity. Our sample displays a large range in ejecta expansion velocities, from $\sim9600$ to $\sim1500$ km s$^{-1}$ at 50 days post explosion with a median H$_{\alpha}$ value of 7300 km s$^{-1}$. This is most likely explained through differing explosion energies. Significant diversity is also observed in the absolute strength of spectral lines, characterised through their pseudo-equivalent widths. This implies significant diversity in both temperature evolution (linked to progenitor radius) and progenitor metallicity between different SNe II. Around 60% of our sample show an extra absorption component on the blue side of the H$_{\alpha}$ P-Cygni profile (“Cachito” feature) between 7 and 120 days since explosion. Studying the nature of Cachito, we conclude that these features at early times (before $\sim35$ days) are associated with $\lambda6355$, while past the middle of the plateau phase they are related to high velocity (HV) features of hydrogen lines.'
author:
- 'Claudia P. Gutiérrez, Joseph P. Anderson, Mario Hamuy, Nidia Morrell, Santiago González-Gaitan, Maximilian D. Stritzinger, Mark M. Phillips, Lluis Galbany, Gastón Folatelli, Luc Dessart, Carlos Contreras, Massimo Della Valle, Wendy L. Freedman, Eric Y. Hsiao, Kevin Krisciunas, Barry F. Madore, José Maza, Nicholas B. Suntzeff, Jose Luis Prieto, Luis González, Enrico Cappellaro, Mauricio Navarrete, Alessandro Pizzella, Maria T. Ruiz, R. Chris Smith, Massimo Turatto'
title: 'Type II supernova spectral diversity I: Observations, sample characterization and spectral line evolution[^1]'
---
Introduction
============
Supernovae (SNe) exhibiting prevalent Balmer lines in their spectra are known as Type II SNe (SNe II henceforth, @Minkowski41). They are produced by the explosion of massive ($>8$ M$_\odot$) stars, which have retained a significant part of their hydrogen envelope at the time of explosion. Red supergiant (RSG) stars have been found at the position of SN II explosion sites in pre-explosion images [e.g. @VanDyk03; @Smartt04; @Smartt09; @Maund05; @Smartt15], suggesting that they are the direct progenitors of the vast majority of SNe II.\
Initially SNe II were classified according to the shape of the light curve: SNe with faster ‘linear’ declining light curves were cataloged as SNe IIL, while SNe with a plateau (quasi-constant luminosity for a period of a few months) as SNe IIP [@Barbon79]. Years later, two spectroscopic classes and one photometric were added within the SNe II group: SNe IIn and SNe IIb, and SN 1987A-like, respectively. SNe IIn show long-lasting narrow emission lines in their spectra [@Schlegel90], attributed to interaction with circumstellar medium (CSM), while SNe IIb are thought to be transitional objects, between SNe II and SNe Ib [@Filippenko93]. On the other hand, the 1987A-like events, following the prototype of SN 1987A [e.g. @Blanco87; @Menzies87; @Hamuy88; @Phillips88; @Suntzeff88], are spectrotroscopically similar to the typical SNe II, however their light curves display a peculiar long rise to maximum ($\sim100$ days), which is consistent with a compact progenitor. The latter three sub-types (IIn, IIb and 87A-like) are not included in the bulk of the analysis for this paper.\
Although it has been shown that SNe II[^2] are a continuous single population [e.g. @Anderson14; @Sanders15; @Valenti16], a large spectral and photometric diversity is observed. @Pastorello04 and @Spiro14 studied a sample of low luminosity SNe II. They show that these events present, in addition to low luminosities (M$_V\geq -15.5$ at peak), narrow spectral lines. Later, [@Inserra13] analyzed a sample of moderately luminous SNe II, finding that these SNe, in contrast to the low luminosity events, are relatively bright at peak (M$_V\leq -16.95$).\
In addition to these samples, many individual studies have been published showing spectral line identification, evolution and parameters such as velocities and pseudo-equivalent widths (pEWs) for specific SNe. Examples of very well studied SNe include SN 1979C [e.g. @Branch81; @Immler05], SN 1980K [e.g. @Buta82; @Dwek83; @Fesen99], SN 1999em [e.g. @Baron00; @Hamuy01; @Leonard02b; @Dessart06], SN 1999gi [e.g. @Leonard02a], SN 2004et [e.g. @Li05; @Sahu06; @Misra07; @Maguire10], SN 2005cs [e.g. @Pastorello06; @Dessart08b; @Pastorello09], and SN 2012aw [e.g. @Bose13; @Dallora14; @Jerkstrand14]. The first two SNe (1979C and 1980K) are the prototypes of fast declining SNe II (SNe IIL), together with unusually bright light curves and high ejecta velocities. On the other hand, the rest of the objects listed are generally referred to as SNe IIP as they display relatively slowly declining light curves. For faint SNe similar to SN 2005cs, the expansion velocity and luminosity are even lower, probably due to low energy explosions (see @Pastorello09).\
In recent years, the number of studies of individual SNe II has continued to increase, however there are still only a handfull of statistical analyses of large samples [e.g. @Patat94; @Arcavi10; @Anderson14; @Gutierrez14; @Faran14a; @Faran14b; @Sanders15; @Pejcha15; @Pejcha15a; @Valenti16; @Galbany16; @Muller17]. Here we attempt to remedy this situation. The purpose of this paper is to present a statistical characterization of the optical spectra of SNe II, as well as an initial analysis of their spectral features. We have analyzed 888 spectra of 122 SNe II ranging between 3 and 363 days since explosion. We selected 11 features in the photospheric phase with the aim of understanding the overall evolution of visual-wavelength spectroscopy of SNe II with time.\
The paper is organized as follows. In section \[data\] we describe the data sample. The spectroscopic observations and data reduction techniques are presented in section \[obs\]. In section \[explo\] the estimation of the explosion epoch is presented. In section \[prop\] we describe the sample properties, while in section \[line\] we identify spectral features. The spectral measurements are presented in section \[measure\] while the line evolution analysis and the conclusions are in section \[ana\] and \[conc\], respectively.\
In Paper II, we study the correlations between different spectral and photometric parameters, and try to understand these in terms of the diversity of the underlying physics of the explosions and their progenitors.
Data sample {#data}
===========
Our dataset was obtained between 1986 and 2009 from a variety of different sources. This sample consists of 888 optical spectra of 122 SNe II[^3], of which four were provided by the Cerro Tololo Supernova Survey (CTSS), seven were obtained by the Calán/Tololo survey (CT, @Hamuy93, PI: Hamuy 1989-1993), five by the Supernova Optical and Infrared Survey (SOIRS, PI: Hamuy, 1999-2000), 31 by the Carnegie Type II Supernova Survey (CATS, PI: Hamuy, 2002-2003) and 75 by the Carnegie Supernova Project (CSP-I, @Hamuy06, 2004-2009). These follow-up campaigns concentrated on obtaining well-sampled and high-cadence light curves and spectral sequences of nearby SNe, based mainly on 2 criteria: 1) that the SN was brighter than V$\sim17$ mag at discovery, and 2) that those discovered SNe were classified as being relatively young, i.e., less than one month from explosion.\
The redshift distribution of our sample is shown in Figure \[snz\]. The figure shows that the majority of the sample have a redshift $\leq0.03$. SN 2002ig has the highest redshift in the sample with a value of 0.077, while the nearest SN (SN 2008bk) has a redshift of 0.00076. The mean redshift value of the sample is 0.0179 and the median is 0.0152. The redshift information comes from the heliocentric recession velocity of each host galaxy as published in the NASA/IPAC extragalactic Database (NED)[^4]. These NED values were compared with those obtained through the measurement of narrow emission lines observed within SN spectra and originating from host regions. In cases of discrepancy between the two sources, we give priority to our spectral estimations. Two of our objects (SN 2006Y and SN 2007ld) occur in unknown host galaxies. Their redshifts were obtained from the Asiago supernova catalog[^5] and from the narrow emission lines within SN spectra originating from the underlying host galaxy, respectively. Table \[t\_info\] lists the sample of SNe II selected for this work, their host galaxy information, and the campaign to which they belong.\
From our SNe II sample, SNe IIn, SNe IIb and SN 1987A-like events (SN 2006au and SN 2006V; @Taddia12) were excluded based on photometric information. Details of the SNe IIn sample can be found in @Taddia13, while those of the SNe IIb in @Stritzinger17 and @Taddia17. The photometry of our sample in the $V-$band was published by @Anderson14. More recently, @Galbany16 released the UBVRIz photometry of our sample obtained by CATS between 1986 and 2003. Around 750 spectra of $\sim100$ objects are published here for the first time. Now we briefly discuss each of the surveys providing SNe for our analysis.
![Distribution of heliocentric redshifts for the 122 SN II in our sample.[]{data-label="snz"}](snz.pdf){width="8.5cm"}
The Cerro Tololo Supernova Survey
---------------------------------
A total of 4 SNe II (SN 1986L, SN 1988A, SN 1990E, and SN 1990K) were extensively observed at CTIO by the Cerro Tololo SN program (PIs: Phillips & Suntzeff, 1986-2003). These SNe have been analyzed in previous works [e.g @Schmidt93; @Turatto93a; @Cappellaro95; @Hamuy01T].
The Calán/Tololo survey (CT)
----------------------------
The Calán/Tololo survey was a program of both discovery and follow-up of SNe. A total of 50 SNe were obtained between 1989 and 1993. The analysis of SNe Ia was published by @Hamuy96. Spectral and photometric details of six SNe II were presented by @Hamuy01T. In this analysis we include these SNe II and an additional object, SN 1993K.
The Supernova Optical and Infrared Survey (SOIRS)
-------------------------------------------------
The Supernova Optical and Infrared Survey carried out a program to obtain optical and IR photometry and spectroscopy of nearby SNe ($z<0.08$). In the course of 1999-2000, 20 SNe were observed, six of which are SNe II. Details of these SNe were published by @Hamuy01T [@Hamuy01], @Hamuy02L, and @Hamuy03.
The Carnegie Type II Supernova Survey (CATS)
--------------------------------------------
Between 2002 and 2003 the Carnegie Type II Supernova Survey observed 34 SNe II. While optical spectroscopy and photometry of these SNe II have been previously used to derive distances [@Olivares08; @Jones09], the spectral observations have not been officially released until now.
The Carnegie Supernova Project I (CSP-I)
----------------------------------------
The Carnegie Supernova Project I (CSP-I) was a five year follow-up program to obtain high quality optical and near infrared light curves and optical spectroscopy. The data obtained by the CSP-I between 2004 and 2009 consist of $\sim250$ SNe of all types, of which 75 correspond to SNe II. The first SN Ia photometry data were published in @Contreras10, while their analysis was done by @Folatelli10. A second data release was provided by @Stritzinger11. A spectroscopy analysis of SNe Ia was published by @Folatelli13. Recently, @Stritzinger17 and @Taddia17 published the photometry data release of stripped-envelope supernovae. The CSP-I spectral data for SNe II are published here for the first time, while the complete optical and near-IR photometry will be published by Contreras et al. (in prep).
Observations and data reduction {#obs}
===============================
In this section we summarize our observations and the data reduction techniques. However, a detailed description of the CT methodology is presented in @Hamuy93, in the case of SOIRS is described in @Hamuy01 and for CSP-I can be found in @Hamuy06 and @Folatelli13.
Observations
------------
The data presented here were obtained with a large variety of instruments and telescopes, as shown in Table \[t\_spec\]. The majority of the spectra were taken in long-slit spectroscopic mode with the slit placed along the parallactic angle. However, when the SN was located close to the host, it was necessary to pick a different and more convenient angle to avoid contamination from the host. The majority of our spectra cover the range of $\sim3800$ to $\sim9500$ Å. The observations were performed with the Cassegrain spectrographs at 1.5-m and 4.0-m telescopes at Cerro Tololo, with the Wide Field CCD Camera (WFCCD) at the 2.5m du Pont Telescope, the Low Dispersion Survey Spectrograph (LDSS2; @Allington-Smith94) on the Magellan Clay 6.5-m telescope and the Inamori Magellan Areal Camera and Spectrograph (IMACS; @Dressler11) on the Magellan Baade 6.5-m telescope at Las Campanas Observatory. At La Silla, the observations were carried out with the ESO Multi-Mode Instrument (EMMI; @Dekker86) in medium resolution spectroscopy mode (at the NTT) and the ESO Faint Object Spectrograph and Camera (EFOSC; @Buzzoni84) at the NTT and 3.6-m telescopes. We also have 3 spectra for SN 2006ee obtained with the Boller & Chivens CCD spectrograph at the Hiltner 2.4 m Telescope of the MDM Observatory. Table \[t\_spec\] displays a complete journal of the 888 spectral observations, listing for each spectrum the UT and Julian dates, phases, wavelength range, FWHM resolution, exposure time, airmass, and the telescope and instrument used.\
The distribution of the number of spectra per object for our sample is shown in Figure \[spec\]. Seven SNe (SN 1993A, SN 2005dt, SN 2005dx, SN 2005es, SN 2005gz, SN2005me, SN 2008H) only have one spectrum, while 90% of the sample have between two and twelve spectra. SN 1986L is the object with the most spectra (31), followed by SN 2008bk with 26. On average we have 7 spectra per SN and a median of 6. There are 87 SNe II for which we have five or more spectra, 32 that have ten or more, and 6 objects with over 15 spectra (SN 1986L, SN 1993K, SN 2007oc, SN 2008ag, SN 2008bk and SN 2008if). In the current work, 4% of our obtained spectra are not used for analysis. 3% correspond to spectra with low S/N that does not allow useful extraction of our defined parameters, while 1% are related with peculiarities in the spectra (see Section \[prop\] for more details). Despite this, these spectra are still included in the data release, and are noted in Table \[t\_spec\].
![Histogram of the number of spectra per SN. The distribution peaks at 4 spectra.[]{data-label="spec"}](spec.pdf){width="8.5cm"}
Data reduction
--------------
Spectral reduction was achieved in the same manner for all data, using IRAF and employing standard routines, including: bias subtraction, flat-fielding correction, one-dimensional (1D) spectral extraction and sky subtraction, wavelength correction, and flux calibration. Telluric corrections have only been applied to data obtained after October 2004.\
In Appendix \[spectra\] (spectral series) we show plots with the spectral series for all SNe of our sample.
Explosion epoch estimations {#explo}
===========================
Before discussing the properties of our sample, in this section we outline our methods for estimating explosion epochs. The non-detection of SNe on pre-discovery images with high cadence is the most accurate method for determining the explosion epoch for any given SN. Explosion epochs based on non-detections are set to the mid-point between SN discovery and non-detection. The representative uncertainty on this epoch is then (MJD$_{disc}-$MJD$_{non-det})/2$. However within our sample (and for many other current SN search campaigns) many SNe do not have such accurate constraints from this method due to the low cadence of the observations.\
Over the last decade several tools have been published, enabling explosion epoch estimations through matching of observed SN spectra to libraries of spectral templates. Programs such as the Supernova Identification (SNID) code [@Blondin07], the GEneric cLAssification TOol (Gelato) [@Harutyunyan08], and superfit [@Howell05] allow the user to estimate the type of supernova and its epoch by providing an observed spectrum. All perform classifications by comparison using different methods. In our analysis we used only the first two methods: SNID and Gelato. We find that Gelato gives a large percentage of their quality of fit to H$_{\alpha}$ P-Cygni profile. However, based on our analysis (see Section \[ana\]), the most significant changes with time are observed in the blue part of the spectra (i.e. between 4000 and 6000 Å). Moreover, according to @Gutierrez14, the H$_{\alpha}$ P-Cygni profile shows a wide diversity and there is no clear, consistent evolution with time. In addition, SNID provides the possibility of adding additional templates to improve the accuracy of explosion epoch determinations. We take advantage of this attribute in the following sections by adding new spectral templates, which aid in obtaining more accurate explosion epochs for our sample.\
While for many SNe this spectral matching is required to obtain a reliable explosion epoch, a significant fraction of our sample do have explosion epoch constraining SN non-detections before discovery. In cases where the non-detection is $<20$ days before discovery, we use that information to estimate our final values. In cases where this difference is larger than 20 days, we use the spectral matching technique. As a test of our methodology, for non-detection SNe we also estimate explosion epochs using spectral matching to check the latter’s validity (see below for more details).
SNID implementation
-------------------
To constrain the explosion epoch for our sample, we compare the first spectrum of each SN II with a library of spectral templates provided by SNID and then, we choose the best match. For each SN we examined multiple matches putting emphasis on the fit of the blue part of the spectrum between 4000 and 6000 Å. This region contains many spectral lines that display a somewhat consistent evolution with time, unlike the dominant H$_{\alpha}$ profile at redder wavelengths. Explosion epoch errors from this spectral matching are obtained by taking the standard deviation of several good matches of the observed spectrum of our selected object with those from the SNID library. H$_{\alpha}$ is the dominant feature in SN II spectra, however its evolution and morphology varies greatly between SNe in a manner that does not aid in the spectral matching technique. We therefore ignore this wavelength region.\
The red part of the spectrum can be ignored during spectral matching in a variety of ways. 1) using the SNID options; or 2) checking only the match in the blue part. For the former, SNID gives to the user the alternative to modifying some parameters. In our case, we can constrain the wavelength range using *wmin* and *wmax*. Hence, the structure used is: *“snid wmin=3500 wmax=6000 spec.dat”*. For the latter, we just need to ignore visually the red part of the spectra and explore the matches obtained by SNID until find a good fit in the blue part[^6].\
From the SNID library we use those template SNe that have well constrained explosion epochs, meaning SNe II with explosion epoch errors of less than five days (see Table \[t\_expl\]). Specifically, we used SN 1999em [@Leonard02b], SN 1999gi [@Leonard02a], SN 2004et [@Li05], SN 2005cs [@Pastorello06], and SN 2006bp [@Dessart08b]. In the database of SNID there are a total of 166 spectra. However, these templates do not provide a good coverage of the overall diversity of SNe II within our sample/the literature. Most of the SNe in the library are relatively ‘normal’, with only one sub-luminous event (SN 2005cs). This means that any non-normal event within our sample will probably have poor constraints on its explosion epoch using these templates. For this reason we decided to use some of our own well-observed SNe II to complement the SNID database.
New SNID templates
------------------
We created a new set of spectral templates using our own SNe II non-detection limits. SNe II are included as new SNID templates if they have errors on explosion epochs (through non-detection constraints) of less than 5 days. Given this criterion, we included 22 SNe, which show significant spectral and photometric diversity. In this manner, the new SNID templates were constructed using $\sim150$ spectra and prepared using the *logwave* program included in the SNID packages. Adding our own template SNe to the SNID database we can now use a total of 27 template SNe II to estimate the explosion epoch. Table \[t\_expl\] shows the explosion epoch and the maximum dates in $V-$band for the reference SNe, as well as the explosion epoch for our new templates. We note an important difference between our templates and previous ones in SNID: for the newer templates epochs are labelled with respect to the explosion epoch, while for the older templates epochs are labelled with respect to maximum light (meaning that one then has to add the “rise time” to obtain the actual explosion date, see Table \[t\_expl\]).\
Explosion epochs for the current sample
---------------------------------------
With the inclusion of these 22 SNe to SNID we estimated the explosion epoch for our full sample. An example of the best match is shown in Figure \[expl\]. We can see that first spectrum of SN 2003iq (October 16th) is best matched with SN 2006bp, SN 2004et, SN 1999em and SN 2004fc 12, 13, 7 and 9 days from explosion, respectively. Taking the average, we conclude that the spectrum was obtained at 10$\pm7$ days since explosion. Table \[t\_info\] shows the explosion epoch for each SN as well as the method employed to derive it, while Table \[table\_explosion\] shows all the details of spectral matching and non-detection techniques. Appendix \[snid\] (SNID matches) shows the plots with the best matches for each SN in our sample.
{width="6.5cm"} {width="6.5cm"} {width="6.5cm"} {width="6.5cm"}
To check the validity of spectral matching we compare the explosion epoch estimated with this technique and those with non-detections. These two estimations are displayed in Table \[table\_explosion\]. From the second to the seventh column, the spectral matching details are shown (spectrum date, best match found, days from maximum –from the SNID templates– days from explosion, average, and explosion date), while from eighth to tenth, those obtained from the non-detection (non-detection date, discovery date and explosion date). The differences between both methods are presented in the last column. Such an analysis was previously performed by @Anderson14 where good agreement was found. With the use of our new templates we are able to improve the agreement between different explosion epoch constraining methods, thus justifying their inclusion. Figure \[explosion\] shows a comparison between both methods, where the mean absolute error between them diminishes from 4.2 [@Anderson14] to 3.9 days. Also the mean offset decreases from 1.5 days in @Anderson14 to 0.5 days in this work. Cases where explosion epochs have changed between @Anderson14 and the current work are noted in Table \[t\_info\]. Nevertheless, although this method works well as a substitute for non-detections, exact constraints for any particular object are affected by any peculiarities inherent to the observed (or indeed template) SN. For example, differences in the colour (and therefore temperature) evolution of events can mimic differences in time evolution, while progenitor metallicity differences can delay/hasten the onset of line formation. Further improvements of this technique can only be obtained by the inclusion of additional, well observed SNeII in the future.\
![Comparison between spectral matching and non-detection methods.[]{data-label="explosion"}](explosion.pdf){width="8.5cm"}
Sample properties {#prop}
=================
As mentioned in Section \[data\] we have 888 optical spectra of 122 SNe II, however due to low signal-to-noise ($S/N$) we remove 26 spectra of 12 SNe for our analysis. We also remove nine spectra of SN 2005lw because they contain peculiarities that we expect are not intrinsic to the SN (most probably defects resulting from the observing procedure or data reduction). In total, we remove 35 spectra ($\sim4\%$). Figure \[nspec\] shows the epoch distribution of our spectra since explosion to 370 days. One can see the majority (86%) of the spectra were observed between 0 and 100 days since explosion, with a total of 738 spectra. Our earliest spectrum corresponds to SN 2008il at $3\pm3$ days and SN 2008gr at $3\pm6$ days from explosion, while the oldest spectrum is at $363\pm9$ days for SN 1993K. 53% of the spectra were taken prior to 50 days, 3.8% of which were observed before 10 days for 23 SNe. Between $\sim30$ to 84 days there are 441 spectra of 114 SNe. There are 115 spectra older than 100 days and 27 older than 200 days, corresponding to 45 and 4 SNe, respectively. The average of spectra as a function of epoch from explosion is 60 days, while its median is 46 days.
![Distribution of the number of spectra as a function of epoch from explosion. The inset on the right shows the same distribution between 100 and 370 days. []{data-label="nspec"}](nspec.pdf){width="8.5cm"}
Figure \[puspec\] shows the epoch distribution of the first and last spectrum for each SN in our sample. The majority of SNe have their first spectra within 40 days from explosion. There are 31 SNe with their first spectra around 10 days (the peak of the distribution). On the other hand, the peak of the distribution of the last spectrum is around 100 days. Almost all SNe have their last spectra between 30 and 120 days, i.e., in the photospheric phase. There are 11 SNe with their last spectrum after 140 days, while only 4 SNe (SN 1993K, 2003B, SN 2007it, SN 2008bk) have their last spectrum in the nebular phase ($\geq200$ days).
![*Top:* Epoch from explosion of first spectrum. *Bottom:* Epoch from explosion of last spectrum. []{data-label="puspec"}](puspec.pdf){width="8.5cm"}
The photometric behaviour of our sample in terms of their plateau decline rate (s$_2$; defined in @Anderson14) in the $V$ band is shown in Figure \[s2\]. For our sample of 117 SNe II, we measure $s_2$ values ranging between $-0.76$ and 3.29 mag 100d$^{-1}$. Higher $s_2$ values mean that the SN has a faster declining light curve. We can see a continuum in the s$_2$ distribution, which shows that the majority of the SNe (83) have a s$_2$ value between 0 and 2. There are 8 objects with s$_2$ values smaller than 0, while 3 SNe show a value larger than 3. The average of s$_2$ in our sample is 1.20. We are unable to estimate the s$_2$ value for 5 SNe as there is insufficient information from their light curves. The s$_2$ distribution for the 22 SNe II used as new templates in SNID is also shown in Figure \[s2\]. Although the diversity in the SNID templates increased with the inclusion of these SNe, the template distribution is still biased to low $s_2$ values.
{width="8.5cm"}
. \[s2\]
Spectral line identification {#line}
============================
We identified 20 absorption features within our photospheric spectra, in the observed wavelength range of 3800 to 9500 Å. Their identification was performed using the Atomic Spectra Database[^7] and theoretical models [e.g. @Dessart05; @Dessart06; @Dessart11]. Early spectra exhibit lines of H$_{\alpha}$ $\lambda6563$, H$_{\beta}$ $\lambda4861$, H$_{\gamma}$ $\lambda4341$, H$_{\delta}$ $\lambda4102$, and $\lambda5876$, with the latter disappearing at $\sim20-25$ days past explosion. An extra absorption component on the blue side of H$_{\alpha}$ (hereafter “Cachito”[^8] is present in many SNe). That line has previously been attributed to high velocity (HV) features of hydrogen or $\lambda6533$. Figure \[linesear\] shows the main lines in early spectra of SNe II at 3 and 7 days from explosion. We can see that SN 2008il shows the Balmer lines and , while SN 2007X, in addition to these lines, also shows Cachito on the blue side of H$_{\alpha}$.
![Line identification in the early spectrum of SN 2008il (top) and SN 2007X (bottom).[]{data-label="linesear"}](lines.pdf "fig:"){width="7cm"} ![Line identification in the early spectrum of SN 2008il (top) and SN 2007X (bottom).[]{data-label="linesear"}](lines0.pdf "fig:"){width="7cm"}
In Figure \[linesph\] we label the lines present in the spectra of SNe II during the photospheric phase at 31, 70 and 72 days from explosion. Later than $\sim15$ days the iron group lines start to appear and dominate the region between 4000 and 6000 Å. We can see Fe-group blends near $\lambda4554$, and between 5200 and 5450 Å (where we refer to the latter as “Fe II blend” throughout the rest of the text). Strong features such as Fe II $\lambda4924$, $\lambda5018$, $\lambda5169$, / $\lambda5531$, the multiplet $\lambda5663$ (hereafter “ M”), $\lambda6142$, $\lambda6247$, $\lambda7774$, $\lambda9263$ and the triplet $\lambda\lambda8498,8662$ ($\lambda8579$) are also present from $\sim20$ days to the end of the plateau. At 31 days, SN 2003hn shows all these lines, except , while at 70 and 72 days, SN 2003bn and SN 2007W show all the lines. Unlike SN 2003bn, SN 2007W shows Cachito and the “Fe line forest”[^9]. The Fe line forest is visible in a small fraction of SNe from 25-30 days (see the analysis in section \[ana\]). As we can see there are significant differences between two different SNe at almost the same epoch. Later we analyze and discuss how these differences can be understood in terms of overall diversity of SN II properties.\
![Line identification in the photospheric phase for SNe II 2003hn at 31 days (top), 2003bn at 70 days (middle), and 2007W at 72 days (bottom).[]{data-label="linesph"}](lines1.pdf "fig:"){width="9.2cm" height="8.8cm"} ![Line identification in the photospheric phase for SNe II 2003hn at 31 days (top), 2003bn at 70 days (middle), and 2007W at 72 days (bottom).[]{data-label="linesph"}](lines2.pdf "fig:"){width="9.2cm" height="8.8cm"} ![Line identification in the photospheric phase for SNe II 2003hn at 31 days (top), 2003bn at 70 days (middle), and 2007W at 72 days (bottom).[]{data-label="linesph"}](lines3.pdf "fig:"){width="9.2cm" height="8.8cm"}
In the nebular phase, later than 200 days post explosion, the forbidden lines $\lambda\lambda7291,$ 7323, $\lambda\lambda6300,$ 6364 and $\lambda7155$ emerge in the spectra. At this epoch H$_{\alpha}$, H$_{\beta}$, D, the triplet, and the Fe group lines between 4800 and 5500 Å, and 6000-6500 Å are also still present. Figure \[linesneb\] shows a nebular spectrum of SN 2007it at 250 days from explosion.\
![Line identification in the nebular spectrum of SN II 2007it at 250 days from explosion. []{data-label="linesneb"}](nebular.pdf){width="9.2cm"}
The H$_{\alpha}$ P-Cygni profile {#halpha}
--------------------------------
H$_{\alpha}$ $\lambda6563$ is the dominant spectral feature in SNe II. It is usually used to distinguish different SN types using the initial spectral observation. This line is present from explosion until nebular phases, showing, in the majority of cases, a P-Cygni profile. Although the P-Cygni profile has an absorption and emission component, SNe display a huge diversity in the absorption feature.
{width="16cm"}
. \[distvel\]
@Gutierrez14 showed that SNe with little absorption of H$_{\alpha}$ (smaller absorption to emission ($a/e$) values) appear to have higher velocities, faster declining light curves and tend to be more luminous. Here we show H$_{\alpha}$ displays a large range of velocities in the photospheric phase, from 9500 km s$^{-1}$ to 1500 km s$^{-1}$ at 50 days (see the first two panels in Figure \[distvel\], which correspond to the H$_{\alpha}$ velocity derived from the FWHM of the emission component and from the minimum flux of the absorption, respectively).\
The diversity of H$_{\alpha}$ in the photospheric phase is also observed through the blueshift of the emission peak at early times [@Dessart08a; @Anderson14a] and the boxy profile [@Inserra11; @Inserra12b]. The former is associated with differing density distributions of the ejecta, while the latter with an interaction of the ejecta with a dense CSM. In the nebular phase this shift in H$_{\alpha}$ emission peak has been interpreted as evidence of dust production in the SN ejecta. Despite the fact that this is an important issue in SNe II, only a few studies [e.g. @Sahu06; @Kotak09; @Fabbri11] have focussed on these features.\
In Figure \[Ha\] we show an example of the evolution of H$_{\alpha}$ P-Cygni profile in SN 1992ba. We can see in early phases a normal profile which evolves to a complicated profile around 65 days. Cachito on the blue side of H$_{\alpha}$ is present from 65 to 183 days.\
![H$_{\alpha}$ P-Cygni profile evolution in SN 1992ba. The epochs are labeled on the right.[]{data-label="Ha"}](Ha92ba.pdf){width="9cm"}
H$_{\beta}$, H$_{\gamma}$ and H$_{\delta}$ absorption features {#hbeta}
--------------------------------------------------------------
H$_{\beta}$ $\lambda4861$, H$_{\gamma}$ $\lambda4341$ and H$_{\delta}$ $\lambda4102$ like H$_{\alpha}$ are present from the first epochs. In earlier phases, these lines show a P-Cygni profile, however, from $\sim15$ days the spectra only display the absorption component, giving space to Fe group lines. The range of velocities of H$_{\beta}$, H$_{\gamma}$ and H$_{\delta}$ at 50 days post explosion vary from 8000 to 1000 km s$^{-1}$ (see Figure \[distvel\]).\
Although H$_{\delta}$ is a common line in SNe II, we do not include a detailed analysis of this line because in many cases the spectra are noisy in the blue part of the spectrum. Besides, like other lines in the blue, this line is blended with Fe-group lines later than 30 days.\
Around 30 days from explosion H$_{\gamma}$ starts to blend with other lines, such as and . Meanwhile in a few SNe, the H$_{\beta}$ absorption feature is surrounded by the Fe line forest. Our later analysis shows that SNe displaying this behaviour are generally dimmer and lower velocity events (see Section \[ana\] for more details).
$\lambda5876$ and D $\lambda5893$
-----------------------------------
$\lambda5876$ is present in very early phases when the temperature of the ejecta is high enough to excite the ground state of helium. As the temperature decreases, the line starts to disappear due to low excitation of ions (around 15 days; @Roy11 [@Dessart10a]). At $\sim30$ days the D $\lambda5893$ absorption feature arises in the spectrum at a similar position where was located. This new line evolves with time to a strong P-Cygni profile, displaying velocities between 8000 km s$^{-1}$ to 1500 km s$^{-1}$ at 50 days from explosion (Figure \[distvel\]).\
In many SNe II (or indeed SNe of all types), at these wavelengths one often observes narrow absorption features arising from slow-moving line of sight material from the interstellar medium, ISM (or possibly from circumstellar material, CSM). Such material can constrain the amount of foreground reddening suffered by SNe, however we do not discuss this here.
Fe-group lines
--------------
When the SN ejecta has cooled sufficiently, features start to dominate SNe II spectra between 4000 to 6500 Å. The first line that appears is $\lambda5169$ on top of the emission component of H$_{\beta}$. With time $\lambda5018$ and $\lambda4924$ emerge between H$_{\beta}$ and $\lambda5169$. $\lambda5169$ becomes a blend later than $\sim30-40$ days. At $\sim50$ days the 4000-5500 Å region is completely filled with these lines and the continuum is diminished due to line-blanketing. The H$_{\gamma}$ and H$_{\delta}$ absorption features are blended with Fe-group lines, such as , , and . Between $\sim5400$ and 6500 Å other metal lines appear in the spectra. Lines such as / $\lambda5531$, M, $\lambda6142$ and $\lambda6247$ get stronger with time.\
As we can see in Figure \[distvel\], the Fe-group lines show a range of velocities between 7000 km s$^{-1}$ to 500 km s$^{-1}$ at 50 days. The peak of the distribution of the group lines velocities is around 4000 km s$^{-1}$. In the case of , the peak is lower (around 3000 km s$^{-1}$).\
Although lines always appear at late phases, few SNe show the iron line forest at 30 days. This feature appears earlier in low velocity/luminosity SNe (See section \[ana\]).\
The NIR triplet
---------------
The NIR triplet is a strong feature in the spectra of SNe II. This line appears at $\sim20-30$ days as an absorption feature, but with time it starts to show an emission component. The NIR triplet results in a blend of $\lambda$8498 and $\lambda$8542 in the bluer part and a distinct component, $\lambda$8662 on the red part. In SNe II with higher velocities these lines are blended producing a broad absorption and emission profile, however, in low-velocity SNe, we see two absorption components and one emission in the red part. The velocities of the NIR triplet range between 9000 to 1000 km s$^{-1}$ at 50 days. In the nebular phase the NIR triplet is also present, however at this epoch it only exhibits the emission component.\
Although in the majority of our spectra we can not see H & K $\lambda3945$, due to the poor signal to noise in this region, this line is present in the photospheric phase of SNe II.\
While the NIR triplet is a prominent feature in SNe II, we do not include its analysis in the subsequent discussion, given that the overlap of lines makes a consistent comparison of velocities and pseudo-equivalent widths (pEWs) difficult.
lines {#oi}
------
The $\lambda\lambda7772$, 7775 doublet (hereafter $\lambda7774$) and $\lambda9263$ are the oxygen lines in the optical spectra of SNe II. These lines are mainly driven by recombination and they appear when the temperature decreases sufficiently. The $\lambda7774$ line is relatively strong and emerges around 20 days from explosion, however in the majority of cases it is contaminated by the telluric A-band absorption ($\sim7600-7630$ Å), which hinders detailed analysis. $\lambda9263$ is weaker and appears one month later than $\lambda7774$. These lines are present until the nebular phase and their expansion velocity at 50 days post explosion goes from $\sim7000$ km s$^{-1}$ to 500 km s$^{-1}$, as can be seen in Figure \[distvel\].\
Cachito: Hydrogen High Velocity (HV) Features or the Si II $\lambda$6355 line? {#cachito}
------------------------------------------------------------------------------
The extra absorption component on the blue side of H$_{\alpha}$ P-Cygni profile, called here “Cachito”, is seen in early phases in some SNe (e.g. SN 2005cs, @Pastorello06; SN 1999em, @Baron00) as well as in the plateau phase (e.g. SN 1999em, @Leonard02b SN 2007od, @Inserra11). However, its shape and strength is completely different in the two phases. @Baron00 assigned the term “complicated P-Cygni profle" to explain the presence of this component on the blue side of the Balmer series. They concluded that these features are due to velocity structures in the expanding ejecta of the SNe II. A few years later, @Pooley02 and @Chugai07 argued that this extra component might originate from ejecta – circumstellar (CS) interactions, while @Pastorello06 earmarked this feature as $\lambda6355$.\
In general, Cachito appears around 5-7 days between 6100 and 6300 Å, and disappears at $\sim35$ days after explosion. Later than 40 days the Cachito feature emerges closer to H$_{\alpha}$ (between 6250-6450 Å) and it can be seen until 100-120 days. Figure \[cach\] shows this component in SN 2007X. In early phases this feature is marked with letter A and later with letter B. If attributed to H$_{\alpha}$ the derived velocities are 18000 km s$^{-1}$ and 10000 km s$^{-1}$, respectively. A detailed analysis of this feature is presented in \[cacho\].
![H$_{\alpha}$ P-Cygni profile of the SN 2007X. The epochs are labeled on the right. The dashed lines indicate the velocities for the A and B features, which we call ‘Cachito’.[]{data-label="cach"}](ha07x.pdf){width="9cm"}
Nebular Features {#nebularf}
----------------
As mentioned above, H$_{\alpha}$, H$_{\beta}$, the NIR triplet, D , and are also present in the nebular phase (later than 200 days since explosion), however in the case of the NIR triplet, its appearance changes, passing from absorption and emission components to only emission components when the nebular phase starts. The rest of the lines have the same behaviour but at much later epochs. The emergence of forbidden emission lines signifies that the spectrum is now forming in regions of low density. At this phase, the ejecta has become transparent, allowing us to peer into the inner layers of the rapidly expanding ejecta. Lines such as $\lambda\lambda7291,$ 7323, $\lambda\lambda6300,$ 6364 and H$_{\alpha}$ are the strongest features visible in the spectra.\
The doublet observed at nebular times is one of the most important diagnostic lines of the helium-core mass [@Fransson87; @Jerkstrand12]. Usually the doublet is blended, however in SNe with low velocities these lines can be resolved (see e.g. SN 2008bk). On the other hand, $\lambda7155$ is easily detectable, but in most cases it is blended with $\lambda\lambda7291,$ 7323 and $\lambda7065$, which may hinder their analysis. In Figure \[neb\] we can see the diversity found in the nebular spectra in our sample.\
![Nebular spectra of seven different SNe of our sample. The spectra are organized according to epoch.[]{data-label="neb"}](neb.pdf){width="9cm"}
Spectral measurements {#measure}
=====================
As discussed previously, SNe II spectra evolve from having a blue continuum with a few lines (Balmer series and ) to redder spectra with many lines: , , D, , , and . To analyze the spectral properties of SNe II, we measure the expansion velocities and pEWs of eleven features in the photospheric phase (see in Table \[table\_features\] the features used), the ratio of absorption to emission ($a/e$) of H$_{\alpha}$ P-Cygni profile before 120 days, and the velocity decline rate of H$_{\beta}$.
Expansion ejecta velocities
---------------------------
The expansion velocities of the ejecta are commonly measured from the minimum flux of the absorption component of the P-Cygni line profile. Using the Doppler relativistic equation and the rest wavelength of each line, we can derive the velocity. To obtain the position of the minimum line flux (in wavelength), a Gaussian fitting was employed, which was performed with IRAF using the *splot* package. As the absorption component presents a wide diversity (e.g. asymmetries, flat shape, extra absorption components) we repeat the process many times (changing the pseudo-continuum), and the mean of the measurements was taken as the minimum flux wavelength. As our measurement error we take the standard deviation on the measurements. This error is added in quadrature to errors arising from the spectral resolution of our observations (measured in Å and converted to km s$^{-1}$) and from peculiar velocities of host galaxies with respect to the Hubble flow (200 km s$^{-1}$). This means that, in addition to the standard deviation error, which realizes the width of the line and $S/N$, we take into account the spectral resolution, that in our case is the most dominant parameter to determine the error.\
The particular case of the H$_{\alpha}$ velocity was explored in @Gutierrez14. Due to the difficulty of measuring the minimum flux in a few SNe with little or extremely weak absorption component, we derive the expansion velocity of H$_{\alpha}$ using both the minimum flux of the absorption component and the full-width-at-half-maximum (FWHM) of the emission line.\
In the case of $\lambda$ 7774 where the telluric lines can affect our measurement of its absorption minimum, we only use SNe with a clear separation between the two features. This means that the number of SNe with measurements is significantly smaller (only 47 SNe) compared to the other measured features.
Velocity decline rate
---------------------
To calculate the time derivative of the expansion velocity in SNe II, we select the H$_{\beta}$ absorption line. It is present from the early spectra, it is easy to identify and it is relatively isolated. To analyze quantitatively our sample, we introduce the $\Delta v($H$_{\beta})$ as the mean velocity decline rate in a fixed phase range \[t$_0$,t$_1$\]:\
$\Delta v($H$_{\beta})=\frac{\Delta v_{abs}}{\Delta t}=\frac{v_{abs}(t_1)-v_{abs}(t_0)}{t_1-t_0}$.\
This parameter was measured over the interval \[+15,+30\] d, \[+15,+50\] d, \[+30,+50\] d, \[+30,+80\] d, and \[+50,+80\] d.
Pseudo-equivalent widths
------------------------
To quantify the spectral properties of SNe II, another avenue for investigation is the measurement and characterization of spectral line pEWs. The prefix “pseudo” is used to indicate that the reference continuum level adopted does not represent the true underlying continuum level of the SN, given that in many regions the spectrum is formed from a superposition of many spectral lines. The pEW basically defines the strength of any given line (with respect to the pseudo-continuum) at any given time. The simplest and most often used method is to draw a straight line across the absorption feature to mimic the continuum flux. Figure \[features\] shows an example of this technique applied to SN 2003bn. We do not include analysis of spectral lines where it is difficult to define the continuum level, due to complicated line morphology, such as significant blending between lines. For example, later than 20 - 25 days all absorption features bluer than H$_{\beta}$ are produced by blends of Fe-group lines plus other strong lines, such as H & K and H$_{\gamma}$. On the other hand, the NIR triplet $\lambda\lambda8498$, 8662 shows a profile that depends on the SN velocity (higher velocity SNe show a single broad absorption, while low velocity SNe show two absorption characteristics). These attributes make a consistent analysis between SNe difficult, and therefore we do not include this line in our analysis.
![Examples of pEWs used in this work for eleven features in the photospheric phase of SN 2003bn (at 70 days).[]{data-label="features"}](featuresnew11.pdf){width="9cm"}
We measure the ratio of absorption to emission ($a/e$) in H$_{\alpha}$ until 120 days. In the same way the pEWs of the absorption lines are measured, we evaluate the pEWs for the emission in H$_{\alpha}$, thus we have:\
$a/e=\frac{pEW(H_{\alpha(abs)})}{pEW(H_{\alpha(emis)})}$.
Line Evolution analysis {#ana}
=======================
Here we study the time of appearance of different lines within different SNe, and make a comparison of those SNe with/without specific lines at different epochs. For all lines included in our analysis, we search for their presence in each observed spectrum. Then, at any given epoch we obtain the percentage of SNe that display each line. This enables an analysis of the overall line evolution of our sample and whether the speed of this evolution changes between different SNe of different light curve, spectral, and environment (metallicity) characteristics.\
In Figure \[lines\] we show the percentage of SNe displaying specific spectral features as a function of time. As discussed previously, H$_{\alpha}$ and H$_{\beta}$ are permanently present in all the SNe II spectra from the first days, so we do not include them in the plot. We can see that:
- The feature located in the position of / D is visible in all epochs, however around 15-25 days fewer SNe show the line respect to either earlier or later spectrum. We suggest that in this epoch the transition from to D happens. Therefore, after 30 days we refer to this line as D. It is present in 96% of the spectra from $\sim35$ days. Later than 43 days it is present in all spectra.
- The NIR triplet is present in $50\%$ of the sample at $\sim20$ days. Before 20 days it is present in $\sim12\%$ of the sample, while later than 25 days is visible in almost all the sample, but with one exception at 38 days. The latter is SN 2009aj, which shows signs of CS interaction in the early phases.
- H$_{\gamma}$ blend with Fe-group lines starts at $\sim20$ days from explosion, growing dramatically at 35-45 days. Only one spectrum at $\sim46$ days does not show the blend (SN 2008bp).
- The Fe-group lines start to appear at around 10 days (see Figure \[lines\]). The first line that emerges is $\lambda5169$. We can see that few SNe exhibit the absorption feature before 15 days, however later at 15 days around 50% of SNe show the line and at 30 days all objects have it. The next line that arises is $\lambda5018$. This line is seen from 15 days, being present in all SNe later than 40 days. Meanwhile, $\lambda4924$ is seen in one spectrum at 13 days (SN 2008br). From 30 days it is visible in more than 50% of the spectra. The / $\lambda5531$, multiplet $\lambda5668$, $\lambda6142$ and $\lambda6246$ are detectable later than 30 days. The emergence of the / $\lambda5531$ and multiplet $\lambda5668$ happens at similar epochs, as well as $\lambda6142$ and $\lambda6246$.
{width="5.8cm"} {width="5.8cm"} {width="5.8cm"} {width="5.8cm"} {width="5.8cm"} {width="5.8cm"}
In order to further understand the differences in line-strength evolution of SNe II, we separate the sample into those SNe that do/do not display a certain spectral feature at some specific epoch. We then investigate whether these different samples also display differences in their light-curves and spectra. This is done by using the Kolmogorov-Smirnov (KS) test. Presented in Table \[table\_kstest\] are all the results obtained with KS test: SNe with/without a given line as a function of $a/e$ and H$_{\alpha}$ velocity at t$_{tran+10}$[[^10]]{}, M$_{max}$, s$_2$, and metallicity (derived from the ratio of H$_{\alpha}$ to \[\] $\lambda6583$, henceforth M13 N2 diagnostic; @Marino13) in a particular epoch. The values of the first four parameters can be found in Table 1 in @Gutierrez14, while the metallicity information was obtained from @Anderson16. We find that:
- SNe II that never display the line forest are distinctly different from those that do display the feature. Specifically, those that do show this feature have slow declining light curves (smaller s$_2$), are dimmer, and are found to explode in higher metallicity regions within their hosts (see Table 4 for exact statistics).
- There is less than a 2% probability that those SNe II where the line is detected between 18 and 22 days post explosion arise from the same underlying parent population of $a/e$. This suggests that temperature differences between SNe II affect the morphology of the H$_{\alpha}$ feature.
- $\lambda$6142 and $\lambda$6247 are both more likely to be detected at around 40 days post explosion in dimmer SNe II, with only a 2% probability that the two populations (with and without these lines) are drawn from the same M$_{max}$ distribution.
- Finally, when splitting the SNe II sample into those that do and do not display: / $\lambda5531$, multiplet $\lambda5668$, $\lambda$6142 and $\lambda$6247 at around 40 days post explosion we find that there is only around a 1% probability that the two samples are drawn from the same distribution of metallicity: those SNe that do not display these lines at this epoch are found to generally explode in regions of lower metallciity within their hosts
Figure \[ksplot\] presents the cumulative distributions of the most significative findings obtained with the KS-test analysis.
{width="\textwidth"}
This analysis was also performed with synthetic spectra for seven different models from @Dessart13a. Four models (m15z2m3, m15z8m3, m15z8m3, m15z4m2) show differences in the metallicity, while the rest of the properties are almost the same. The three remaining models have the same metallicity (solar metallicity), however the other parameters are different: m15mlt1 has a bigger radius (twice times the radius of the two other models), m15mlt3 has higher kinetic energy, while m12mlt3 displays a smaller final progenitor mass and less kinetic energy (1/5 E$_{kin}$ compared with the other models). More details are shown in Table \[table\_models\].\
In general, the synthetic spectra show the same behaviour (in relation to the appearance of the lines) as observed spectra. However, some differences are found, the majority of which are probably related with the low area of parameter space covered by the models that currently exist as compared to the parameter space covered by real events. The transition between to D is more evident, and it happens between 18 and 40 days. Although the transition in the models is unambiguously identified by knowing the optical depth of specific lines, in these synthetic spectra this happens a little bit later than in observed ones. This suggests the temperature in specific models stays higher for a longer time than the average for observed SNe II. It is also likely that the observed SNe II span a smaller range in progenitor metallicity than the models (that go down to a tenth solar). The D is visible in 100% of the sample after 50 days, only 5 days later than the observed spectra. shows the same behaviour in both synthetic and observed spectra, however H$_{\gamma}$ is blended in all of the sample later than 90 days, unlike the observed spectra that show it from 45 days. On the other hand, the line forest is visible from 55 days, in contrast to the observed spectra that show this characteristic from 30 days. This behaviour is only present in the spectra of higher metallicity model (2 times solar) and in the lower explosion energy model. The iron lines ( $\lambda4924$, $\lambda5018$, $\lambda5169$, and the blended) are present from $\sim10$ days. $\lambda5169$ is visible in 50% of the spectra at $\sim15$ days, while $\lambda4924$ is only visible in $\sim10$%. From 20 days $\lambda5169$ is present in all the synthetic spectra, 10 days before than in the observed ones. The behaviour of $\lambda5018$ is similar in both synthetic and observed spectra, whereas $\lambda4924$ starts faster in the models and it is visible in 85% of the spectra from 30 days. We can see differences in the blend, which is visible in 100% of the sample from 50 days in the models, however in the observed spectra that never happens. More differences are also appreciable between models and observation in / $\lambda5531$, the multiplet $\lambda5668$, $\lambda6142$ and $\lambda6246$. These lines in models arise from 20 days, but in the observations it occurs from 38-40 days. Nevertheless, the evolution of the distribution is similar from 50 days. In conclusion, while in general the models produce a time evolution of spectral lines that is quite similar to the observations - supporting the robustness of the models - we observe small differences, suggesting a wider range of explosion and progenitor properties is required to explain the full diversity of observed SNe II.
Expansion velocity evolution
----------------------------
{width="18cm"}
Figure \[vel\] shows the velocity evolution of eleven spectral features as a function of time. The first two panels of the plot show the expansion velocity of the H$_{\alpha}$ feature: on the left the velocity derived from the FWHM and on the right that derived from the minimum absorption flux. As we can see, the behaviour is similar, however the velocity obtained from the minimum absorption flux is offset between 10 and 20% to higher velocities. Figure \[hadist\] shows this shift at 50 days. Velocities obtained from the minimum absorption flux are higher around $\sim1000$ km s$^{-1}$. However, it is possible to see few SNe (with higher H$_{\alpha}$ velocities) showing higher values from the FWHM. Using the Pearson correlation test we find a weak correlation, with a value of $\rho=0.37$. SNe II with narrower emission components display a larger offset between the velocity from the FWHM and that from the minimum of the absorption. In contrast, those SNe II displaying the highest FWHM velocities present comparatively lower minimum absorption velocities. We note also the presence of two outliers (extreme cases, the lowest and highest value). Figure \[distvel\] shows the velocity distribution for the eleven features at 50 days post explosion. We can see that H$_{\alpha}$ shows higher velocities than the other lines, followed by H$_{\beta}$. The lowest velocities are presented by the iron-group lines. In Figure \[vel\] it is possible to see that the H$_{\beta}$ expansion velocity shows the typical evolution for a homologous expansion and like H$_{\alpha}$, it is possible to see it from early phases. The iron lines display lower velocities than the Balmer lines. So, the highest velocity in SNe II is found in H$_{\alpha}$, which implies that it is formed in the outer layers of the SN ejecta. Meanwhile, based on the lower velocities, the iron-group lines form in the inner part, closer to the photosphere. The line does not show a strong evolution. As we can see, its velocity evolution is almost flat.\
The lowest velocities are found in SN 2008bm, SN 2009aj and SN 2009au. However, these SNe are distinct from the rest of the population. Unlike sub-luminous SNe II (such as SN 2008bk and SN 1999br) – that also display low expansion velocities – these events are relatively bright. They also show early signs of CS interactions, e.g., narrow emission lines. By contrast, SN 2007ab, SN 2008if and SN 2005Z have the largest velocities.
![Shifts of the H$_{\alpha}$ velocity obtained form the FWHM of the emission and from the minimum of the absorption at 50 days post explosion.[]{data-label="hadist"}](hadist.pdf){width="9cm"}
Velocity decline rate of H$_{\beta}$ analysis
---------------------------------------------
The velocity decline rate of SNe II, denoted $\Delta v($H$_{\beta}$), has not been previously analyzed. We estimate $\Delta v($H$_{\beta})$ in five different epochs (outlined above) to understand their behaviour. We find that SNe with a higher decline rate at early times continue to show such behaviour at later times. The median velocity decline rate for our sample between 15 and 30 days is 105 km $s^{-1}$ d$^{-1}$, while between 50 and 80 days is 29 km $s^{-1}$ d$^{-1}$. This results show an evident decrease in the velocity decline rate at two different intervals, which is consitent with homologous expansion.
pEWs evolution
--------------
{width="18cm"}
The temporal evolution of pEWs for each of the eleven spectral features is shown in Figure \[pews\]. In general the pEWs increase quickly in the first 1-2 months then level off. The first two panels show the pEW evolution of H$_{\alpha}$. On the left is displayed the absorption, while on the right the emission component. The absorption component monotonically increases from 0 increasing to $\sim100$ Å, however in a few SNe its evolution is different: from 70 days the pEW decreases significantly. This behaviour is observed in low and intermediate velocity SNe (e.g., SN 2003bl, SN 2006ee, SN 2007W, SN 2008bk, SN 2008in and SN 2009N). Generally, these SNe show a very narrow H$_{\alpha}$ P-cygni profile, and at around 70 days from explosion $\lambda6497$ appears in the spectra as a dominant feature (see @Roy11 [@Lisakov17] for more details). In Figure \[ba2\] we can see the H$_{\alpha}$ P-Cygni profile with the presence of $\lambda6497$, and the HV feature of hydrogen line (see section \[cacho\] for more details) on the blue side of .\
Figure \[pews\] also shows the H$_{\alpha}$ emission component evolution. An increment in the pEW in the majority of SNe is appreciable. There are a couple cases (e.g. SN 2006Y), displaying a quasi-constant evolution. The range of pEW of H$_{\alpha}$ emission goes up 400 Å. In the case of H$_{\beta}$, we can see that from 60 days there are few SNe with low pEW values, which show a quasi-constant evolution. SNe with this behaviour are those that show the line forest. The remaining SNe show an increase. The pEWs of iron-group lines grow with time, however there is a group of SNe with pEW$=0$. This indicates that some specific SNe do not have the line yet. For /, the multiplet, , and this is more obvious. On the other hand, the shows a quasi-constant behaviour and D a steady increase. Comparing the values, we can see that the absorption of H$_{\alpha}$, H$_{\beta}$ and D have the highest values (from 0 to $\sim120$), while $\lambda4924$, $\lambda5018$, /, the multiplet, , and have the lowest ones (from 0 to $\sim50$).
![H$_{\alpha}$ P-Cygni profile of low and intermediate velocity SNe II: 2003bl, 2006ee, 2007W, 2008bk, 2008in and 2009N around 95 days post-explosion. []{data-label="ba2"}](BaII.pdf){width="8.8cm"}
The $a/e$ evolution is displayed in Figure \[ae\]. One can see an increase until $\sim60$ days and then, the quantity remains constant or slightly decreases.
![Evolution of the ratio absorption to emission ($a/e$) of H$_{\alpha}$ between explosion and 120 days.[]{data-label="ae"}](ae.pdf){width="9cm"}
Cachito: Hydrogen HV features or line {#cacho}
-------------------------------------
{width="18cm"}
The nature of Cachito has recently been studied. Its presence on the blue side of H$_{\alpha}$ has given rise to multiple interpretations, such as HV features of hydrogen [e.g. @Leonard02b; @Baron00; @Chugai07; @Inserra11] or $\lambda6355$ [e.g. @Pastorello06; @Valenti13a; @Tomasella13]. Seventy SNe from our sample show Cachito in the photospheric phase, between 7 and 120 days post-explosion, however its behaviour, shape and evolution is different depending on the phase. To investigate the nature of Cachito we examine the following possibilities:
- If Cachito is produced by its velocity should be similar to those presented by other metal lines.
- If Cachito is related to HV features of hydrogen, its velocity should be almost the same as those obtained from H$_{\alpha}$ at early phases. In addition, if it is present, a counterpart should be visible on the blue side of H$_{\beta}$.
Analyzing our sample we can detect Cachito in 50 SNe at early phases (before 40 days). Because of the high temperatures at these epochs, the presence of $\lambda6497$ is discarded. Assuming that Cachito is produced by , we find that 60% of SNe present a good match with $\lambda5018$ and $\lambda5169$ velocities[^11]. Conversely, the rest of the sample shows velocities comparable to those measured at very early phases for H$_{\alpha}$. Curiously the Cachito shape is different between the two SN groups. In the former, the line is deeper and broader, while in the latter, the line is shallow. In Figure \[velsi2\] we present the velocity comparison for the former group, where a good agreement is found between Cachito, assumed as $\lambda6355$ (blue), and the iron lines, $\lambda5018$ (green) and $\lambda5169$ (red).
![Spectral evolution of H$_{\alpha}$ and H$_{\beta}$ lines of SN 2004fc. The dotted lines correspond to the HV features seen on the blue side of H$_{\alpha}$ and H$_{\beta}$ from 50 to 120 days. We can see that the HV features show a velocity evolution from $\sim9000$ to $\sim8000$ km s$^{-1}$.[]{data-label="hv04fc"}](spectrahahb.pdf){width="9cm"}
Later than 40 days we detect Cachito in 43 SNe. Proceeding with the velocity comparison, we can discard its identification as or $\lambda6497$ (the latter, visible in few SNe from 60 days, see Figure \[ba2\]), which suggests that Cachito is associated to hydrogen. During the plateau it is possible to see Cachito as a shallow absorption feature only in H$_{\alpha}$ and/or as a narrow and deeper absorption on the blue side of both H$_{\alpha}$ and H$_{\beta}$ (see an example in Figure \[hv04fc\]). According to @Chugai07, the interaction between the SN ejecta and the RSG wind should result in the emergence of these HV absorption features. They argue that the existence of a shallow absorption feature is the result of the enhanced excitation of the outer unshocked ejecta, which is visible on the blue side of H$_{\alpha}$ (and $\lambda$10830). At early times the H$_{\beta}$ Cachito feature is not predicted by @Chugai07, who argue that the optical depth is too low at the line forming region. They also discuss that in addition to the HV shallow absorption, a HV notch is formed in the cool dense shell (CDS) located behind the reverse shock. Given the relatively high H$_{\alpha}$ optical depth of the CDS, a counterpart could be seen in H$_{\beta}$ as well. We found that 63% of the SNe with Cachito during the plateau show a counterpart in H$_{\beta}$ with the same velocity as that presented on H$_{\alpha}$, which favours the interpretation as CS interaction. The HV notch of is found in 27 SNe, however in the low velocity/luminosity SNe, it is only present in H$_{\alpha}$. After 50 days the blue part of the spectrum (<5000 Å) is dominated by metal lines, which may hinder its detection. Nonetheless, we argue that these can be HV because at least one low velocity/luminosity SN, SN 2006ee shows a Cachito feature on the blue side of both H$_{alpha}$ and H$_{beta}$, at around 50 days with consistent velocities. A summary of the analysis is displayed in Figure \[hvfs\], where the H$_{\alpha}$ (red), HV H$_{\alpha}$ (blue), H$_{\beta}$ (cyan), and HV H$_{\beta}$ (green) velocity evolution is presented for 20 SNe.
{width="18cm"}
In addition to the 70 SNe where Cachito can be identified either with or HV features of , we find six SNe II that display Cachito at certain epochs, however its exact properties do not align with the above interpretations (because of differences in shape and/or velocity). These are SN 2003bl, SN 2005an, SN 2007U, SN 2008br, SN 2002gd and SN 2004fb. In summary 59% of the full SNe sample show Cachito at some epoch, while 41% never show this feature. Soon after shock-break out all SNe II have extremely high temperature ejecta. Therefore, if we were able to obtain spectral sequences shortly after explosion, the feature would always be observed. However, observationally this is not the case as there are many SNe II within our sample without detections. This is simply an observational bias, due to the lack of data at very early times. Nevertheless, for SNe II that stay hotter for longer the probability of detecting becomes larger. We therefore speculate that SNe II that have detected at early times have larger radii, which leads to a slower cooling of the ejecta and hence facilitates detection. Interestingly, when we split the sample into those SNe II that do and do not display the line, those where the line is detected are found to have lower $a/e$ values, with only a 4% chance that the two populations are drawn from the same underlying distribution. This is also consistent with the previous finding that those SNe II with evident detections at around 20 days post explosion are also found to have lower $a/e$ values, suggesting that the value of $a/e$ is related to ejecta temperature evolution.\
In the case of those SNe II displaying Cachito consistent with HV features, these are most likely produced by the interaction of the SN ejecta with the RSG wind, where the exact shape and persistence of Cachito is related to the wind density @Chugai07. In Figure \[compc\] one can observe the significant diversity in the different detection of Cachito.
![Cachito’s shape according to its nature. **On left panel:** The line in SN 2007X. **In the middle panel:** HV features of as a shallow absorption in SN 2008ag. **On right panel:** HV features of as narrow and deeper absorption component in SN 2003hl.[]{data-label="compc"}](cachito.pdf){width="6.5cm"}
Conclusions {#conc}
===========
In this paper we have presented optical spectra of 122 nearby SNe II observed between 1986 and 2009. A total of 888 spectra ranging between 3 and 363 days post explosion have been analyzed. The spectral matching technique was discussed as an alternative to non-detection constraints for estimating SN explosion epochs.\
In order to quantify the spectral diversity we analyze the appearance of the photospheric lines and their time evolution in terms of the $a/e$ and H$_{\alpha}$ velocity at the $B$-band transition time plus 10 days (t$_{tran+10}$; see @Gutierrez14 for more details), the magnitude at maximum (M$_{max}$), the plateau decline (s$_2$), and metallicity (M13 N2). We analyzed the velocity decline rate of H$_{\beta}$, the $a/e$ evolution, the expansion ejecta velocities and the pEWs for eleven features: H$_{\alpha}$, H$_{\beta}$, / D, $\lambda4924$, $\lambda5018$, $\lambda5169$, blend, /, multiplet, , , and . We find a large range in velocities and pEWs, which may be related with a diversity in the explosion energy, radius of the progenitor, and metallicity. The evolution of line strengths was analyzed and compared to that of spectral models. SNe II displaying differences in spectral line evolution were also found to have other different spectral, photometric and environmental properties. Finally, we discuss the detection and origin of Cachito on the blue side of H$_{\alpha}$.\
The main results obtained with our analysis are summarized as follows:
- The line evolution indicates differences in temperatures and/or metallicity. Thus, SNe with slower temperature gradients show the appearance of the iron lines later, while SNe in environments with higher metallicities show them earlier. In fact, the line forest is present in faint SNe with low ejecta temperatures and/or in high metallicity environments. Comparing this result with the synthetic spectra, we find that indeed this feature is only present in higher metallicity (2 times solar) and lower explosion energy models, which is consistent with our observations.
- SNe II display a significant variety of expansion velocities, suggesting a large range in explosion energies.
- At early phases (before 25 days), SNe II with a weak H$_{\alpha}$ absorption component show $\lambda5876$ and the $\lambda6355$ features. We speculate that this occurs because of higher temperatures at these epochs.
- Around 60% of our SNe II show the Cachito feature between 7 and 120 days since explosion. When Cachito is detected less than 30 days post explosion then it is identified with . The epochs of early detection can thus inform us to the temperature evolution: SNe II with detections at later epochs have higher temperatures, and this may be related to higher-radius progenitors. At later epochs, during the recombination phase, we suggest that Cachito is related to HV of hydrogen lines. Such HV features are most likely related to the interaction of the SN ejecta with the RSG wind.
All data analyzed in this work are available on http://csp.obs.carnegiescience.edu/, as well as the additional SNID templates (22 SNe), for the SNe II comparison.
C.P.G., S.G.G. acknowledge support by projects IC120009 “Millennium Institute of Astrophysics (MAS)" and P10-064-F “Millennium Center for Supernova Science" of the Iniciativa Científica Milenio del Ministerio Economía, Fomento y Turismo de Chile. C.P.G. acknowledges support from EU/FP7-ERC grant No. \[615929\]. M.D.S. is supported by the Danish Agency for Science and Technology and Innovation realized through a Sapere Aude Level 2 grant and by a research grant (13261) from the VILLUM FONDEN. We gratefully acknowledge support of the CSP by the NSF under grants AST–0306969, AST–0908886, AST–0607438, AST–1008343, AST-1613472, AST-1613426 and AST-1613455. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics.
[98]{} natexlab\#1[\#1]{}
, J., [Breare]{}, M., [Ellis]{}, R., [et al.]{} 1994, , 106, 983
, J. P., [Dessart]{}, L., [Gutierrez]{}, C. P., [et al.]{} 2014, , 441, 671
, J. P., [et al.]{} 2014, , 786, 67
, J. P., [Guti[é]{}rrez]{}, C. P., [Dessart]{}, L., [et al.]{} 2016, , 589, A110
, I., [Gal-Yam]{}, A., [Kasliwal]{}, M. M., [et al.]{} 2010, , 721, 777
, R., [Buond[í]{}]{}, V., [Cappellaro]{}, E., & [Turatto]{}, M. 1999, , 139, 531
, R., [Ciatti]{}, F., & [Rosino]{}, L. 1979, , 72, 287
, E., [Branch]{}, D., [Hauschildt]{}, P. H., [et al.]{} 2000, , 545, 444
, V. M., [Gregory]{}, B., [Hamuy]{}, M., [et al.]{} 1987, , 320, 589
, S., & [Tonry]{}, J. L. 2007, , 666, 1024
, S., [Kumar]{}, B., [Sutaria]{}, F., [et al.]{} 2013, , 433, 1871
, D., [Falk]{}, S. W., [Uomoto]{}, A. K., [et al.]{} 1981, , 244, 780
, R. J. 1982, , 94, 578
, B., [Delabre]{}, B., [Dekker]{}, H., [et al.]{} 1984, The Messenger, 38, 9
, E., [Danziger]{}, I. J., [della Valle]{}, M., [Gouiffes]{}, C., & [Turatto]{}, M. 1995, , 293, 723
, N. N., [Chevalier]{}, R. A., & [Utrobin]{}, V. P. 2007, , 662, 1136
, C., [Hamuy]{}, M., [Phillips]{}, M. M., [et al.]{} 2010, , 139, 519
, M., [Botticella]{}, M. T., [Pumo]{}, M. L., [et al.]{} 2014, , 787, 139
, H., [Delabre]{}, B., & [Dodorico]{}, S. 1986, in , Vol. 627, Instrumentation in astronomy VI, ed. D. L. [Crawford]{}, 339–348
, L., & [Hillier]{}, D. J. 2005, , 437, 667
—. 2006, , 447, 691
—. 2008, , 383, 57
—. 2010, , 405, 2141
—. 2011, , 410, 1739
, L., [Hillier]{}, D. J., [Waldman]{}, R., & [Livne]{}, E. 2013, , 433, 1745
, L., [Blondin]{}, S., [Brown]{}, P. J., [et al.]{} 2008, , 675, 644
, A., [Bigelow]{}, B., [Hare]{}, T., [et al.]{} 2011, , 123, 288
, E. 1983, , 274, 175
, J., [Otsuka]{}, M., [Barlow]{}, M. J., [et al.]{} 2011, , 418, 1285
, T., [Poznanski]{}, D., [Filippenko]{}, A. V., [et al.]{} 2014, , 445, 554
—. 2014, , 442, 844
, R. A., [Gerardy]{}, C. L., [Filippenko]{}, A. V., [et al.]{} 1999, , 117, 725
, A. V., [Matheson]{}, T., & [Ho]{}, L. C. 1993, , 415, L103
, G., [Phillips]{}, M. M., [Burns]{}, C. R., [et al.]{} 2010, , 139, 120
, G., [Morrell]{}, N., [Phillips]{}, M. M., [et al.]{} 2013, , 773, 53
, C., & [Chevalier]{}, R. A. 1987, , 322, L15
, L., [Hamuy]{}, M., [Phillips]{}, M. M., [et al.]{} 2016, , 151, 33
, C. P., [et al.]{} 2014, , 786, L15
, M. 2003, , 582, 905
, M., [Phillips]{}, M. M., [Suntzeff]{}, N. B., [et al.]{} 1996, , 112, 2438
, M., & [Pinto]{}, P. A. 2002, , 566, L63
, M., [Suntzeff]{}, N. B., [Gonzalez]{}, R., & [Martin]{}, G. 1988, , 95, 63
, M., [Maza]{}, J., [Phillips]{}, M. M., [et al.]{} 1993, , 106, 2392
, M., [Pinto]{}, P. A., [Maza]{}, J., [et al.]{} 2001, , 558, 615
, M., [Folatelli]{}, G., [Morrell]{}, N. I., [et al.]{} 2006, , 118, 2
, M. A. 2001, PhD thesis, The University of Arizona
, A. H., [et al.]{} 2008, , 488, 383
, D. A., [et al.]{} 2005, , 634, 1190
, S., [Fesen]{}, R. A., [Van Dyk]{}, S. D., [et al.]{} 2005, , 632, 283
, C., [Turatto]{}, M., [Pastorello]{}, A., [et al.]{} 2011, , 417, 261
—. 2012, , 422, 1122
, C., [Pastorello]{}, A., [Turatto]{}, M., [et al.]{} 2013, , 555, A142
, A., [Fransson]{}, C., [Maguire]{}, K., [et al.]{} 2012, , 546, A28
, A., [Smartt]{}, S. J., [Fraser]{}, M., [et al.]{} 2014, , 439, 3694
, M. I., [Hamuy]{}, M., [Lira]{}, P., [et al.]{} 2009, , 696, 1176
, R., [Meikle]{}, W. P. S., [Farrah]{}, D., [et al.]{} 2009, , 704, 306
, D. C., [Filippenko]{}, A. V., [Li]{}, W., [et al.]{} 2002, , 124, 2490
, D. C., [Filippenko]{}, A. V., [Gates]{}, E. L., [et al.]{} 2002, , 114, 35
, W., [Van Dyk]{}, S. D., [Filippenko]{}, A. V., & [Cuillandre]{}, J.-C. 2005, , 117, 121
, S. M., [Dessart]{}, L., [Hillier]{}, D. J., [Waldman]{}, R., & [Livne]{}, E. 2017, , 466, 34
, K., [Di Carlo]{}, E., [Smartt]{}, S. J., [et al.]{} 2010, , 404, 981
, R. A., [Rosales-Ortega]{}, F. F., [S[á]{}nchez]{}, S. F., [et al.]{} 2013, , 559, A114
, J. R., & [Smartt]{}, S. J. 2005, , 360, 288
, J. W., [Catchpole]{}, R. M., [van Vuuren]{}, G., [et al.]{} 1987, , 227, 39P
, R. 1941, , 53, 224
, K., [et al.]{} 2007, , 381, 280
, T., [Prieto]{}, J. L., [Pejcha]{}, O., & [Clocchiatti]{}, A. 2017, , 841, 127
, F. 2008, ArXiv e-prints
, A., [Zampieri]{}, L., [Turatto]{}, M., [et al.]{} 2004, , 347, 74
, A., [Sauer]{}, D., [Taubenberger]{}, S., [et al.]{} 2006, , 370, 1752
, A., [Valenti]{}, S., [Zampieri]{}, L., [et al.]{} 2009, , 394, 2266
, F., [Barbon]{}, R., [Cappellaro]{}, E., & [Turatto]{}, M. 1994, , 282, 731
, O., & [Prieto]{}, J. L. 2015, , 799, 215
—. 2015, , 806, 225
, M. M., [Heathcote]{}, S. R., [Hamuy]{}, M., & [Navarrete]{}, M. 1988, , 95, 1087
, D., [Lewin]{}, W. H. G., [Fox]{}, D. W., [et al.]{} 2002, , 572, 932
, R., [Kumar]{}, B., [Benetti]{}, S., [et al.]{} 2011, , 736, 76
, D. K., [Anupama]{}, G. C., [Srividya]{}, S., & [Muneer]{}, S. 2006, , 372, 1315
, N. E., [Soderberg]{}, A. M., [Gezari]{}, S., [et al.]{} 2015, , 799, 208
, E. F., & [Finkbeiner]{}, D. P. 2011, , 737, 103
, E. M. 1990, , 244, 269
, B. P., [Kirshner]{}, R. P., [Schild]{}, R., [et al.]{} 1993, , 105, 2236
, S. J. 2015, Publications of the Astronomical Society of Australia, 32, e016
, S. J., [Eldridge]{}, J. J., [Crockett]{}, R. M., & [Maund]{}, J. R. 2009, , 395, 1409
, S. J., [Maund]{}, J. R., [Hendry]{}, M. A., [et al.]{} 2004, Science, 303, 499
, S., [Pastorello]{}, A., [Pumo]{}, M. L., [et al.]{} 2014, ArXiv e-prints
, M. D., [et al.]{} 2011, , 142, 156
, M. D., [Anderson]{}, J. P., [Contreras]{}, C., [et al.]{} 2017, ArXiv e-prints
, N. B., [Hamuy]{}, M., [Martin]{}, G., [Gomez]{}, A., & [Gonzalez]{}, R. 1988, , 96, 1864
, F., [Stritzinger]{}, M. D., [Sollerman]{}, J., [et al.]{} 2012, , 537, A140
, F., [et al.]{} 2013, , 555, A10
, F., [Stritzinger]{}, M. D., [Bersten]{}, M., [et al.]{} 2017, ArXiv e-prints
, L., [Cappellaro]{}, E., [Fraser]{}, M., [et al.]{} 2013, , 434, 1636
, M., [Cappellaro]{}, E., [Benetti]{}, S., & [Danziger]{}, I. J. 1993, , 265, 471
, S., [et al.]{} 2013, ArXiv e-prints
, S., [Howell]{}, D. A., [Stritzinger]{}, M. D., [et al.]{} 2016, , 459, 3939
, S. D., [Li]{}, W., & [Filippenko]{}, A. V. 2003, , 115, 1289
, W. M., [Aldering]{}, G., [Lee]{}, B. C., [et al.]{} 2004, New Astronomy Reviews, 48, 637
Spectral series {#spectra}
===============
SNID matches {#snid}
============
[^1]: T M T L C O C G O C P C (G P GS-2008B-Q-56). B E O A R S H C (ESO P 076.A-0156, 078.D-0048, 080.A-0516, 082.A-0526).
[^2]: Throughout the remainder of the manuscript we use SN II to refer to all SNe which would historically have been classified as SN IIP or SN IIL. In general we will differentiate these events by referring to their specific light curve or spectral morphology, and we only return to this historical separation if clarification and comparison with previous works is required.
[^3]: In the data release we include eight spectra of the SN 2000cb, a SN 1987A-like event, which is not analyzed in this work.
[^4]: http://ned.ipac.caltech.edu
[^5]: http://sngroup.oapd.inaf.it
[^6]: Note that the results obtained from the spectral matching are not altered if you use either all visible wavelength spectrum or just the region between 4000 and 6000 Å.
[^7]: http://physics.nist.gov/asd3
[^8]: Cachito is a Hispanic word that means small piece of something (like a notch). We use this name to refer to the small absorption components blue ward of H$_{\alpha}$, giving its (until now) previously ambiguous nature.
[^9]: We label “Fe line forest” to that region around H$_{\beta}$ where a series of Fe-group (e.g. $\lambda4629$, $\lambda4670$, $\lambda4924$) absorption lines emerge.
[^10]: t$_{tran+10}$ is defined as the transition time (in $V-band$) between the initial and the plateau decline, plus 10 days. In other words, [t$_{tran}$]{} marks the start of the recombination phase. (See A14 and [@Gutierrez14] for more details.)
[^11]: Four SNe show a good match with in very early phases, but between 30 and 40 days they do not show it. They also show a different shape.
|
Introduction
============
In the past decade or so, several scenarios have been discovered where the gravitational collapse of a massive matter cloud results in the development of a naked singularity [@1]. The final outcome of gravitational collapse in general relativity is an issue of great importance and interest from the perspective of black hole physics as well as its astrophysical implications. When there is a continual collapse without any final equilibrium, either a black hole forms when the super-dense regions of matter are hidden from the outside observer within an event horizon of gravity, or a naked singularity results as the end product, depending on the nature of the initial data from which the collapse develops.
The theoretical and observational properties of a naked singularity would be quite different from those of a black hole (see [@jdm] for further discussion of this). Thus it is of crucial importance to understand what are the key physical characteristics and dynamical features in collapse that give rise to a naked singularity, rather than a black hole. While many models of naked singularity formation within dynamically developing collapse scenarios have been found and analyzed [@hm], not much attention has been given to understanding this important aspect. We begin here an investigation of this question.
The main purpose of this paper is to identify the physical process which exposes the singularity. We find that it is shearing effects which, if sufficiently strong near the central worldline of the collapsing cloud, would delay the formation of the apparent horizon so that the singularity becomes visible and communication from the very strong gravity regions to outside observers becomes possible. When the shear is weak (and in the extreme case of no shear), the collapse necessarily ends in a black hole, because an early formation of the apparent horizon leads to the singularity being hidden behind an event horizon.
For spherical gravitational collapse of a massive matter cloud, the interior metric in comoving coordinates is $$\label{1}
ds^2 = - e^{2\nu(t,r)} dt^2 + e^{2\psi(t,r)} dr^2 + R^2(t,r)
d\Omega^2\,.$$ The matter shear is $$\label{17}
\sigma_{ab} = e^{-\nu} \left({\dot R \over R} - \dot
\psi\right)({\textstyle{1\over3}}h_{ab} -n_an_b)\,,$$ where $h_{ab}=g_{ab}+u_au_b$ is the induced metric on 3-surfaces orthogonal to the fluid 4-velocity $u^a$, and $n^a$ is a unit radial vector.
The initial data for collapse are the values on $t=t_{\rm i}$ of the three metric functions, the density, the pressures, and the mass function that arises from integrating the Einstein equations (for details see e.g. [@3]), $$\label{4}
F(t_{\rm i},r) = \int \rho(t_{\rm i},r) r^2 dr\,,$$ where $4\pi F(t_{\rm i},r_{\rm b})=M$, the total mass of the collapsing cloud, and where $r>r_{\rm b}$ is a Schwarzschild spacetime. We use the rescaling freedom in $r$ to set $$\label{3}
R(t_{\rm i},r) = r\,,$$ so that the physical area radius $R$ increases monotonically in $r$, and with $R_{\rm i}'=1$ there are no shell-crossings on the initial surface. (We will be interested here only in the central shell-focusing singularity at $R=0,r=0$ which is a gravitationally strong singularity, as opposed to the shell-crossing ones which are weak, and through which the spacetime may sometimes be extended.) The evolution of the density and radial pressure are given by $$\label{5}
\rho = {F'\over R^2 R'}, \quad p_{\rm r} = {\dot F\over R^2 \dot
R}\,.$$ The central singularity at $r=0$, where density and curvature are infinite, is naked if there are outgoing nonspacelike geodesics which reach outside observers in the future and terminate at the singularity in the past. Outgoing radial null geodesics of Eq. (\[1\]) are given by $$\label{6}
{dt\over dr} = e^{\psi - \nu}\,.$$
Consider first the case of homogeneous-density collapse, $\rho=\rho(t)$. Writing $f= e^{-2\psi} R'^2-1$, the Einstein equations give $f - e^{-2\nu} \dot R^2 = - {F/R}$. Then Eq. (\[6\]) can be written as [@4] $$\label{8}
{dR\over du} = \left( 1 - \sqrt{ {f + {F/ R}\over 1 +f}}\right)
{R'\over \alpha r^{\alpha -1}}\,,$$ where $u=r^\alpha$ ($\alpha >1$). If there are outgoing radial null geodesics terminating in the past at the singularity with a definite tangent, then at the singularity we have $dR/du > 0$. For homogeneous density, the entire mass of the cloud collapses to the singularity simultaneously at the event $(t=t_{\rm s},r=0)$, so that $F/ R \to \infty$. By Eq. (\[8\]), ${dR/ du} \to -\infty$, so that no radial null geodesics can emerge from the central singularity. It can be similarly shown that all the later epochs $t>t_{\rm s}$ are similarly covered.
We have thus shown that [*for spherical gravitational collapse with homogeneous density (and arbitrary pressures), the final outcome is necessarily a black hole*]{}. We note that this conclusion does not require homogeneity of the pressures $p_{\rm r}$ and $p_\perp$, and is independent of their behavior. The result generalizes the well-known Oppenheimer-Snyder result for the special case of dust, where the homogeneous cloud collapses to form a black hole always.
An immediate consequence is that [*if the final outcome of spherical gravitational collapse is not a black hole, then the density must be inhomogeneous*]{}. In any physically realistic scenario, the density will be typically higher at the center, so that generically collapse is inhomogeneous.
Inhomogeneous dust
==================
Consider now a collapsing inhomogeneous dust cloud ($p=0$), with density higher at the center. The metric is Tolman-Bondi-Lema[î]{}tre, given by Eq. (\[1\]) with $\nu=0$ and $e^{2\psi}=R'^2/(1+f)$, and $$\label{12}
\dot R^2 = f(r) + {F(r)\over R}\,.$$ These models are fully characterized by the initial data, specified on an initial surface $t=t_{\rm i}$ from which the collapse develops, which consist of two free functions: the initial density $\rho_{\rm i}(r)= \rho(t_{\rm i}, r)$ (or equivalently, the mass function $F(r)$), and $f(r)$, which describes the initial velocities of collapsing matter shells. At the onset of collapse the spacetime is singularity-free, so that by Eq. (\[5\]), $$\label{12'}
F(r)=r^3\bar{F}(r)\,,~~~ 0<\bar{F}(0)<\infty \,.$$ The initial density $\rho_{\rm i}(r)$ is $$\label{14}
\rho_{\rm i}(r)=r^{-2}F'(r)\,.$$
The shell-focusing singularity appears along the curve $t=t_{\rm
s}(r)$ defined by $$\label{s}
R(t_{\rm s}(r),r)=0\,.$$ As the density grows without bound, trapped surfaces develop within the collapsing cloud. These can be traced explicitly via the outgoing null geodesics, and the equation of the apparent horizon, $t=t_{\rm ah}(r)$, which marks the boundary of the trapped region, is given by $$\label{a}
R(t_{\rm ah}(r),r)=F(r)\,.$$ If the apparent horizon starts developing earlier than the epoch of singularity formation, then the event horizon can fully cover the strong gravity regions including the final singularity, which will thus be hidden within a black hole. On the other hand, if trapped surfaces form sufficiently later during the evolution of collapse, then it is possible for the singularity to communicate with outside observers.
For the sake of clarity, we consider marginally bound collapse, $f=0$, although the conclusions can be generalized to hold for the general case. Then Eq. (\[12\]) can be integrated to give $$\label{r}
R^{3/2}(t,r) = r^{3/2} - {\textstyle{3\over2}}(t-t_{\rm i})
F^{1/2}(r)\,,$$ and Eqs. (\[s\]) and (\[a\]) lead to $$\begin{aligned}
t_{\rm s}(r)&=&t_{\rm i}+{2\over3}\left[ {r^3\over
F(r)}\right]^{1/2}\,,\label{s'} \\ t_{\rm ah}(r) &=& t_{\rm s}(r)-
{2\over3}F(r)\,.\label{a'}\end{aligned}$$ The central singularity at $r=0$ appears at the time $$t_0=t_{\rm s}(0)=t_{\rm i}+{2\over\sqrt{3\rho_{\rm c}}}\,,$$ where where $\rho_{\rm c}=\rho_{\rm i}(0)$. Unlike the homogeneous dust case (Oppenheimer-Snyder), the collapse is not simultaneous in comoving coordinates, and the singularity is described by a curve, the first point being $(t=t_0,r=0)$.
For inhomogeneous dust, Eqs. (\[17\]) and (\[r\]) give $$\label{21}
\sigma^2\equiv {1\over2}\sigma_{ab}\sigma^{ab} = {r\over 6R^4
{R'}^2 F} \left(3F - rF'\right)^2\,.$$ A generic (inhomogeneous) mass profile has the form $$\label{19}
F(r)= F_0r^3 + F_1 r^4 + F_2 r^5 +\cdots\,,$$ near $r=0$, where $F_0=\rho_{\rm c}/3$. Homogeneous dust (Oppenheimer-Snyder) collapse has $F_n=0$ for $n>0$, and Eq. (\[21\]) implies $\sigma=0$. The converse is also true in this case: if we impose vanishing shear $\sigma=0$, we get $F_n=0$. Whenever there is a negative density gradient, e.g., when there is higher density at the center, then $F_n\neq0$ for some $n>0$, and it follows from Eq. (\[21\]) that the shear is then necessarily nonzero. Note that if we want the density profile to be analytic, we can set all odd terms $F_{2n-1}$ to zero; however, we note that this is not as such required by our own analysis, which is independent of any assumptions on $F_n$.
The important question is: what is the effect of such a shear on the evolution and development of the trapped surfaces? In other words, we want to determine the behavior of the apparent horizon in the vicinity of the central singularity at $R=0,r=0$. To this end, let the first non-vanishing derivative of the density at $r=0$ be the $n$-th one ($n>0$), i.e., $$F(r) = F_0 r^3 + F_n r^{n+3}+\cdots\,,~~ F_n<0\,,$$ near the center. By Eqs. (\[21\]) and (\[a’\]), $$\begin{aligned}
\sigma^2(t,r) &=& {n^2 {F_n}^2 \over 6 F_0}\left[1-3F_0^{1/2}
(t-t_{\rm i}) +{9\over4}F_0 (t-t_{\rm
i})^2\right]r^{2n}\nonumber\\ &&~~~{}+ O(r^{2n+1}) \label{d}\,,
\\ t_{\rm ah}(r) &=& t_0 - {2 \over 3}F_0r^3 - {F_n \over
3{F_0}^{3/2}} r^n+O(r^{n+1})\,.\label{a''}\end{aligned}$$ The time-dependent factor in square brackets on the right of Eq. (\[d\]) decreases monotonically from 1 at $t=t_{\rm i}$ to 0 at $t=t_0$. Thus the qualitative role of the shear in singularity formation can be seen by looking at the initial shear. The initial shear $\sigma_{\rm i}=\sigma(t_{\rm i},r)$ on the surface $t=t_{\rm i}$ grows as $r^{n}$, $n\geq1$, near $r=0$. A dimensionless and covariant measure of the shear is the relative shear, $|\sigma/\Theta|$, where $$\Theta=2{\dot R\over R}+{\dot R'\over R'}\,,$$ is the volume expansion. It follows that $$\label{d'}
\left|{\sigma\over\Theta}\right|_{\rm i}={-nF_n\over3\sqrt{6}
F_0}\, r^n\left[1+O(r)\right] \,.$$
It is now possible to see how such an initial shear distribution determines the growth and evolution of the trapped surfaces, as prescribed by the apparent horizon curve $t_{\rm ah}(r)$, given by Eq. (\[a”\]). If we assume the initial density profile is smooth at the center, then $\rho_{\rm i}(r)=\rho_{\rm
c}+\rho_2r^2+\cdots$, with $\rho_2\leq0$, which corresponds to $F(r)=F_0r^3+F_2r^5+\cdots$, with $F_2\leq0$. Now suppose that $\rho_2$ (and hence $F_2$) is nonzero. Then Eq. (\[a”\]) implies that the apparent horizon curve initiates at $r=0$ at the epoch $t_0$, and increases near $r=0$ with increasing $r$, moving to the future. Note that as soon as $F_2$ is nonzero, even with very small magnitude, the behavior of the apparent horizon changes qualitatively. Rather than going back into the past from the center, as would happen in the homogeneous case with $F_2=0$, it is future pointed. This is what leads to a locally naked singularity. The singularity may be globally naked, i.e. visible to faraway observers, depending on the nature of the density function at large $r$.
A naked singularity occurs when a comoving observer (at fixed $r$) does not encounter any trapped surfaces until the time of singularity formation, whereas for a black hole, trapped surfaces form before the singularity. Thus for a black hole, we require $$\label{bh}
t_{\rm ah}(r)\leq t_0~\mbox{for}~r>0\,,~\mbox{near}~r=0\,.$$ In the general case (not necessarily smooth initial density), this condition is violated for $n=1,2$, as follows from Eq. (\[a”\]). The apparent horizon curve initiates at the singularity $r=0$ at the epoch $t_0$, and increases with increasing $r$, moving to the future, i.e. $t_{\rm ah}>t_0$ for $r>0$ near the center. The behavior of the outgoing families of null geodesics has been analyzed in detail in these cases, and it is known that the geodesics terminate at the singularity in the past [@3], which results in a naked singularity. In such cases the extreme strong gravity regions can communicate with outside observers. For the case $n=3$, Eq. (\[bh\]) shows that we can have a black hole if $F_3\geq -2 F_0^{5/2}$, or a naked singularity, if $F_3 <-2 F_0^{5/2}$. This is illustrated in Fig. 1. For $n\geq4$, Eq. (\[bh\]) is always satisfied, and a black hole forms.
{height="8cm" width="8cm"}
When the dust density is homogeneous, the apparent horizon starts developing earlier than the epoch of singularity formation, which is then fully hidden within a black hole. There is no density gradient, and no shear. On the other hand, if a density gradient is present at the center, then the trapped surface development is delayed via shear, and, depending on the “strength" of the density gradient/shear at the center, this may expose the singularity. It is the rate of decrease of shear as we approach the center $r=0$ on the initial surface $t=t_i$, given by Eq. (\[d’\]), that determines the end-state of collapse. [*When the shear falls rapidly to zero at the center, the result is necessarily a black hole; if shear falls more slowly, there is a naked singularity.*]{} It is thus seen that naked singularities are caused by the sufficiently strong shearing forces near the singularity, as generated by the inhomogeneities in density distribution of the collapsing configuration. When shear decays rapidly near the singularity, the situation is effectively like the shear-free (and homogeneous density) case, with a black hole end-state.
It provides a useful insight to note that when a black hole forms, the apparent horizon typically springs into being as a finite-sized surface, at a finite $r$, then moving to the center $r=0$. This is what happens, for example, in the Oppenheimer-Snyder black hole formation in homogeneous dust collapse. In such cases, the event horizon, which does typically start at a point, could have formed earlier than the apparent horizon. On the other hand, in the case of a naked singularity, it follows from Eqs. (\[a’\]) and (\[a”\]), that the apparent horizon starts at $r=0$, and then is future directed in time, i.e. $t_{\rm ah}$ grows with increasing coordinate radius $r$ along the apparent horizon curve $R=F$. These two behaviors of the apparent horizon curve are very different, and governed by shearing effects. A comoving observer will [*not*]{} encounter any trapped surfaces until the time of singularity formation in the naked singularity case, whereas in the black hole case, the apparent horizon typically develops [*before*]{} the epoch of singularity formation. This is what we mean by delayed formation of the apparent horizon, caused by shearing effects.
The relation between density gradients and shear may be understood via the nonlocal (or free) gravitational field. Density gradients act as a source for the electric Weyl tensor [@7] $${\rm D}^bE_{ab} = {\textstyle{1\over3}}{\rm D}_a\rho\,,$$ where ${\rm D}_a$ is the covariant spatial derivative. (The magnetic Weyl tensor vanishes for spherical symmetry.) In turn, the gravito-electric field is a source for shear (equivalently, the shear is a gravito-electric potential [@7]): $$\label{se}
u^c\nabla_c{\sigma}_{ab}+{2\over3} \Theta\sigma_{ab}+
\sigma_{ac}\sigma^c{}_b -{2\over3}\sigma^2 h_{ab}=-E_{ab}\,.$$ Thus density gradients may be directly related to shear: $$\begin{aligned}
\label{sd}
{\rm D}_a\rho &=& -4\sigma{\rm D}_a \sigma -2\Theta {\rm
D}^b\sigma_{ab} -3{\rm D}^b\left(u^c\nabla_c
{\sigma}_{ab}\right)\nonumber\\ &&~~{} -3\sigma_a{}^b{\rm D}^c
\sigma_{bc}-3 {\rm D}^b\left(\sigma_{ac}\sigma^c{}_b\right)\,,\end{aligned}$$ where we have used the shear constraint ${\rm
D}^b\sigma_{ab}={2\over3} {\rm D}_a\Theta$. Equation (\[sd\]) makes explicit the link between the behavior of density gradients and shear near the center, which was discussed above. The free gravitational field, which mediates this link, can also provide a covariant characterization of singularity formation. By Eqs. (\[d’\]) and (\[se\]), the relative gravito-electric field $E/\Theta^2$ (where $E^2={1\over2} E^{ab}E_{ab}$) near $r=0$ is given at $t=t_{\rm i}$ by $$\label{ee}
\left({E\over\Theta^2}\right)_{\rm i}={-7nF_n\over 18 \sqrt{6}
F_0}\, r^n\left[1+O(r)\right] \,.$$ Thus naked singularities in spherical dust collapse are signalled by a less rapid fall-off of the relative gravito-electric field as we approach the singularity. Equations (\[d’\]) and (\[ee\]) provide two equivalent ways of expressing the result. This specifies how much shear is sufficient to create a (locally) naked singularity.
For the case of dust collapse, the role of shear in deciding the end-state of collapse is fairly transparent. To understand how shear affects the formation of the apparent horizon for general matter fields with pressures included is much more complicated, in particular since $F=F(t,r)$, whereas $\dot F=0$ for dust. In fact, even in some general classes of non-dust models (with nonzero pressure), it is possible to characterize collapse covariantly. Above we showed that [*homogeneous density*]{} implies a black hole end-state. The next logical step would be to consider models for which the [*initial*]{} density is homogeneous. For example, if the mass function is $$\label{mass}
F(t,r)= f(r)-R^3(t,r)\,,~f(r)=2r^3\,,$$ then Eq. (\[5\]) shows that $\rho_{\rm i}$ and $(p_{\rm r})_{\rm
i}$ are constants. The density and pressure may however develop inhomogeneities as the collapse proceeds, depending on the choice of the remaining functions, including in particular the initial velocities of the collapsing shells, and the collapse may then end up in either a black hole or a naked singularity, depending on that (for a discussion on this for the case of dust collapse, we refer to [@4]). In fact, we can show that [*zero shear implies a black hole*]{} for these models. By Eqs. (\[17\]), (\[5\]) and (\[mass\]), the shear-free condition leads to $R'/R=1/r$, and Eq. (\[5\]) then shows that $\rho= \rho(t)$, i.e. the density evolution is necessarily homogeneous. As shown above, the collapse thus necessarily ends in a black hole. For the class of models given by Eq. (\[mass\]), whenever the collapse ends in a naked singularity, the shear must necessarily be nonvanishing. Although this class of models is somewhat special, the result indicates that the behavior of the shear remains a crucial factor even when pressures are nonvanishing.
Conclusions
===========
Since black holes and naked singularities are of great interest in gravitation theory and astrophysics, it is important to understand [*why*]{} these objects develop. The physics of this needs to be probed carefully in order to make further progress towards cosmic censorship, or to understand the physical implications of naked singularities.
It would appear that the only way a singularity can be laid bare is by distorting the apparent horizon surface and so delay trapped surface formation suitably. As we have shown here, the shear provides a rather natural explanation for the occurrence of (locally) naked singularities. Our main result is that sufficiently strong shearing effects in spherical collapsing dust delay the formation of the apparent horizon, thereby exposing the strong gravity regions to the outside world and leading to a (locally) naked singularity. When shear decays rapidly near the singularity, the situation is effectively like the shear-free case, with a black hole end-state. An important point is that naked singularities can develop in quite a natural manner, very much within the standard framework of general relativity, governed by shearing effects.
In the case of spherical dust collapse, shear and density inhomogeneity are equivalent, i.e., the one implies the other. Although shear contributes positively to the focusing effect via the Raychaudhuri equation, $$\dot\Theta+{1\over3}\Theta^2=-{1\over 2}\rho-2\sigma^2\,,$$ its dynamical action can make the collapse incoherent and dispersive. (It is this feature which also plays the crucial role in avoidance of the big-bang singularity in singularity-free cosmological models [@d].) [*Depending on the rate of fall-off of shear near the singularity, its dispersive effect can play the critical role of delaying formation of the apparent horizon, without directly hampering the process of collapse.*]{} The dispersive effect of shear always tends to delay formation of the apparent horizon, but is only able to expose the singularity when the shear is strong enough near the singularity.
We have considered here spherical collapse. Very little is known about nonspherical collapse, either analytically or numerically, towards determining the outcome in terms of black holes and naked singularities. However, phenomena such as trapped surface formation and apparent horizon are independent of any spacetime symmetries, and it is also clear that a naked singularity will not develop in general unless there is a suitable delay of the apparent horizon. This suggests that the shear will continue to be pivotal in determining the final fate of gravitational collapse, independently of any spacetime symmetries. In any case, our main purpose here has been to try to understand and find the physical mechanism which leads the collapse to the development of a naked singularity rather than a black hole in some of the well-known classes exhibiting such behavior. What we find is that the shear provides a covariant dynamical explanation of the phenomenon of naked singularity formation in spherical gravitational collapse.
\
We thank Shrirang Deshingkar for help with the figure. This work arose out of discussions when PSJ and ND were visiting the University of Natal, and they thank Sunil Maharaj for discussions and warm hospitality. Part of the work was done during visits by RM and ND to the Tata Institute, Mumbai, by PSJ and RM to IUCAA, Pune, and by ND to the University of Portsmouth (supported by PPARC).
For recent reviews, see, e.g., P. S. Joshi, Pramana [**55**]{}, 529 (2000); C. Gundlach, Living Rev. Rel. [**2**]{}, 4 (1999); A. Krolak, Prog. Theor. Phys. Suppl. [**136**]{}, 45 (1999); R. Penrose, in [*Black holes and relativistic stars*]{}, ed. R. M. Wald (University of Chicago Press, 1998); T. P. Singh, gr-qc/9805066.
T. Harada, H. Iguchi, and K.I. Nakao, Phys. Rev. D [**61**]{}, 101502 (2000); P. S. Joshi, N. Dadhich, and R. Maartens, Mod. Phys. Lett. [**A15**]{}, 991 (2000); T. Harada, H. Iguchi, K.I. Nakao, T. P. Singh, T. Tanaka, and C. Vaz, Phys. Rev. D [**64**]{}, 041501 (2001).
For recent examples, see, e.g., H. Iguchi, K.I. Nakao, and T. Harada, Phys. Rev. D [**57**]{}, 7262 (1998); S. S. Deshingkar, P. S. Joshi, and I. H. Dwivedi, Phys. Rev. D [**59**]{}, 044018 (1999); R. V. Saraykar and S. H. Ghate, Class. Quantum Grav. [**16**]{}, 281 (1999); B. J. Carr and A. Coley, Class. Quantum Grav. [**16**]{}, R31 (1999); S. M. Wagh and S. D. Maharaj, Gen. Rel. Grav. [**31**]{}, 975 (1999); B.C. Nolan, Phys. Rev. D [**60**]{}, 024014 (1999); F. C. Mena, R. Tavakol, and P. S. Joshi, Phys. Rev. D [**62**]{}, 044001 (2000); S. Jhingan and G. Magli, Phys. Rev. D [**61**]{}, 124006 (2000); S. M. C. V. Goncalves, Phys. Rev. D [**63**]{}, 124017 (2001); S. Jhingan, N. Dadhich, and P. S. Joshi, Phys. Rev. D [**63**]{}, 044010 (2001); T. Harada and H. Maeda, Phys. Rev. D [**63**]{}, 084022 (2001); S. M. C. V. Goncalves and S. Jhingan, gr-qc/0107054; F. Mena and B.C. Nolan, Class. Quantum Grav. [**18**]{}, 4531 (2001) ; S. G. Ghosh and A. Beesham, Phys. Rev. D [**64**]{}, 124005 (2001).
P. S. Joshi and I. H. Dwivedi, Class. Quantum Grav. [**16**]{}, 41 (1999).
P. S. Joshi and I. H. Dwivedi, Phys. Rev. D [**47**]{}, 5357 (1993).
R. Maartens and B. A. Bassett, Class. Quantum Grav. [**15**]{}, 705 (1998).
N. Dadhich, J. Astrophys. Astr. [**18**]{}, 343 (1997); N. Dadhich and A. K. Raychaudhuri, Mod. Phys. Lett. [**A14**]{}, 2135 (1999).
|
---
abstract: 'Motivated by the recent discovery of high temperature antiferromagnet SrRu$_2$O$_6$ [@Hiley2014; @Tian2015] and its potential to be the parent of a new superconductor upon doping, we construct a minimal $t_{2g}$-orbital model on a honeycomb lattice to simulate its low energy band structure. Local Coulomb interaction is taken into account through both random phase approximation and mean field theory. Experimentally observed antiferromagnetic order is obtained in both approximations. In addition, our theory predicts that the magnetic moments on three $t_{2g}$-orbitals are non-collinear as a result of the strong spin-orbit coupling of Ru atoms.'
author:
- Da Wang
- 'Wan-Sheng Wang'
- 'Qiang-Hua Wang'
bibliography:
- 'Ru126.bib'
title: '$t_{2g}$-orbital model on a honeycomb lattice: application to antiferromagnet SrRu$_2$O$_6$'
---
Introduction
============
Magnetism and superconductivity are closely related to each other, as a common thread in several families of unconventional superconductors. [@Scalapino2012] Singlet Cooper pairing is expected to be mediated by antiferromagnetic (AF) fluctuations near the AF phase boundary like in most cuprates [@Bednorz1986], iron-based [@Kamihara2008], and heavy fermion [@Steglich1979] superconductors. While triplet Cooper pairing is widely believed to be triggered by ferromagnetic fluctuations as in Sr$_2$RuO$_4$ [@Maeno1994; @Huo2013; @Wang2013]. Therefore, looking for unconventional superconductivity in materials with strong magnetic fluctuations is a guiding principle in the community. SrRu$_2$O$_6$ is synthesized recently [@Hiley2014] and reported as an antiferromagnet with a Neel temperature as high as $565$K [@Tian2015]. Similar to other ruthenate Sr$_{1+n}$Ru$_n$O$_{1+3n}$[@Grigera2001], SrRu$_2$O$_6$ is also a layered material. Due to the layered property of the sample, the AF order may be easily destroyed by introducing quantum fluctuations via doping or high pressure. As a result, SrRu$_2$O$_6$ may be a good parent compound to realize high temperature superconductivity.
Different from ruthenate Sr$_{1+n}$Ru$_n$O$_{1+3n}$ in which the RuO$_6$ octahedra are point-sharing and form a square lattice [@Singh2001], in SrRu$_2$O$_6$ the RuO$_6$ octahedra are edge-sharing and the Ru atoms are arranged on a honeycomb lattice. According to first-principle calculations, [@Tian2015; @Singh2015] the $t_{2g}$-orbitals of Ru are found to dominate the low energy states. This suggests that a $t_{2g}$-orbital model on a honeycomb lattice is a relevant minimal model for further studies on the correlation effect. On the other hand, even though the spin and charge degrees of freedom have been broadly studied in the honeycomb lattice (such as graphene) with only $\pi$-electrons [@CastroNeto2009], the orbital degrees of freedom would bring us new features. For instance, the studies of ($p_x,p_y$)-orbital models on the honeycomb lattice revealed Wigner crystallization [@Wu2007] and anomalous quantum Hall effect [@Wu2008; @Zhang2014]. In parallel, SrRu$_2$O$_6$ provides us a natural realization of the $t_{2g}$ d-orbital on the honeycomb lattice.
In this paper, we first derive an effective $t_{2g}$-orbital tight-binding Hamiltonian as a minimal model for SrRu$_2$O$_6$. We then consider the correlation effect through both random phase approximation (RPA) and mean field theory. We obtain the experimentally observed AF order and estimate the Neel moment and transition temperature within the mean field theory. Furthermore, we find the orbital-resolved AF moments on three $t_{2g}$-orbitals are non-collinear as a result of the strong spin-orbit coupling (SOC) on Ru atoms. Our minimal model provides the basis for further theoretical studies, and the magnetic structure we uncovered would trigger further experimental interests in this new member of the ruthenate family.
$t_{2g}$-orbital tight-binding hamiltonian
==========================================
![Schematic discription of the lattice structure and all hopping elements up to the 3rd nearest neighbour. Three $t_{2g}$-orbitals of Ru are placed on the honeycomb lattice. O atoms (small circles) are distributed above (yellow) or below (red) the Ru-plane. Each RuO$_6$ octahedron is slightly distorted.[]{data-label="fig:hopping"}](hopping){width="50.00000%"}
In SrRu$_2$O$_6$, the low energy bands mainly come from the $t_{2g}$-orbitals of Ru atoms as reported by first-principle calculations. [@Tian2015; @Singh2015] The Ru atoms form a honeycomb lattice in each RuO$_6$ layer. Therefore, to mimic the system by a minimal model, we only consider the $t_{2g}$-orbital electrons on a honeycomb lattice as shown in Fig. \[fig:hopping\]. The coordinate is set up with the origin at Ru-site and three axes point to three oxygen atoms above the Ru-plane in the undistorted RuO$_6$ octahedron. In this coordinate, the c-axis perpendicular to Ru-plane is along (1,1,1)-direction, and the $d_{xy}$, $d_{xz}$ and $d_{yz}$ orbitals forms an exact $t_{2g}$ multiplet, as schematically represented in Fig. \[fig:hopping\]. The blue/yellow/red lobes lie within/above/below the Ru-plane. Although in SrRu$_2$O$_6$, the RuO$_6$ octahedron is slightly twisted around and stretched along the c-axis, the $t_{2g}$-orbital degeneracy is protected by the $C_{3v}$ symmetry. As a result, the coordinate established in the undistorted octahedron will be used in this work.
To construct a tight-binding Hamiltonian, we keep nearest neighbor hopping $t_{1\sim4}$, next nearest neighbor hopping $t_{5\sim8,5',8'}$, and the third-neighbor hopping $t_{1'\sim4'}$. All these hopping elements are schematically shown in Fig. \[fig:hopping\]. We notice that $t_5\neq t_5'$ and $t_8\neq t_8'$ due to the distortion of the RuO$_6$ octahedra. Furthermore, we add intra- and inter-orbital on-site energies $V_0$ and $V_0'$, respectively. In a pure two dimensional model, $V_0'$ should be exactly zero due to the orthogonality of different orbitals. But here we are looking for an effective two dimensional model in the $k_c=0$ plane and thus the inter-layer hopping could lead to an effective on-site inter-orbital mixing $V_0'$. In addition, we add an SOC term $H_{\rm SOC}=-\lambda \sum_{i\mu} \psi_i^\dag L_\mu \otimes\sigma_\mu \psi_i$, where $\psi_i^t=[d_{i,xy,\uparrow},d_{i,xz,\uparrow},d_{i,yz,\uparrow},d_{i,xy,\downarrow},d_{i,xz,\downarrow},d_{i,yz,\downarrow}]$, $\sigma_\mu=[\sigma_x,\sigma_y,\sigma_z]$ are three Pauli’s matrices, and $L_\mu=[L_{x},L_{y},L_{z}]$ are rank-3 angular momentum matrices acting on orbital space, with all nonzero elements given by $$\begin{aligned}
&& L_{x,12}=-L_{x,21}=-i, \quad L_{y,13}=-L_{y,31}=-i, \nonumber\\
&& L_{z,23}=-L_{z,32}=i.\end{aligned}$$ With the above ingredients and for a given inter-layer momentum $k_c=0$, we arrive at an effective two-dimensional tight-binding Hamiltonian $H=\sum_{{\bf k}}\Psi_{{\bf k}}^\dag H_{{{\bf k}}} \Psi_{{\bf k}}$ in the basis $\Psi_{{\bf k}}^t=[\psi_{A{{\bf k}}\uparrow},\psi_{B{{\bf k}}\uparrow},\psi_{A{{\bf k}}\downarrow},\psi_{B{{\bf k}}\downarrow}]$ where $\psi_{s,{{\bf k}}\sigma}^t=[d_{s,xy,{{\bf k}}\sigma},d_{s,xz,{{\bf k}}\sigma},d_{s,yz,{{\bf k}}\sigma}]$ for $s=A,B$ on the two sublattices. The matrix $H_{{{\bf k}}}$ is explicitly written as
$$\begin{aligned}
H_{{\bf k}}=\left[\begin{array}{cccc}
{\mathcal{T}}_{AA}({{\bf k}})+\mathcal{V}-\lambda L_z & {\mathcal{T}}_{AB}({{\bf k}}) & -\lambda L_x+i\lambda L_y & 0 \\
{\mathcal{T}}_{AB}({{\bf k}})^\dag & {\mathcal{T}}_{BB}({{\bf k}})+\mathcal{V}-\lambda L_z & 0 & -\lambda L_x+i\lambda L_y \\
-\lambda L_x-i\lambda L_y & 0 & {\mathcal{T}}_{AA}({{\bf k}})+\mathcal{V}+\lambda L_z & {\mathcal{T}}_{AB}({{\bf k}}) \\
0 & -\lambda L_x-i\lambda L_y & {\mathcal{T}}_{AB}({{\bf k}})^\dag & {\mathcal{T}}_{BB}({{\bf k}})+\mathcal{V}+\lambda L_z
\end{array}
\right] ,\nonumber\\
\label{eq:Hk}\end{aligned}$$
in which $$\begin{aligned}
T_{AB}({{\bf k}})&=&\left[\begin{array}{ccc}t_1&t_4&t_4\\t_4&t_2&t_3\\t_4&t_3&t_2 \end{array}\right]{\rm e}^{i{{\bf k}}\cdot {{\bf a}}_1}
+ \left[\begin{array}{ccc}t_2&t_3&t_4\\t_3&t_2&t_4\\t_4&t_4&t_1 \end{array}\right]{\rm e}^{i{{\bf k}}\cdot {{\bf a}}_2}
+ \left[\begin{array}{ccc}t_2&t_4&t_3\\t_4&t_1&t_4\\t_3&t_4&t_2 \end{array}\right]{\rm e}^{i{{\bf k}}\cdot {{\bf a}}_3} ,\nonumber \\
&+&\left[\begin{array}{ccc}t_1'&t_4'&t_4'\\t_4'&t_2'&t_3'\\t_4'&t_3'&t_2' \end{array}\right]{\rm e}^{-2i{{\bf k}}\cdot {{\bf a}}_1}
+ \left[\begin{array}{ccc}t_2'&t_3'&t_4'\\t_3'&t_2'&t_4'\\t_4'&t_4'&t_1' \end{array}\right]{\rm e}^{-2i{{\bf k}}\cdot {{\bf a}}_2}
+ \left[\begin{array}{ccc}t_2'&t_4'&t_3'\\t_4'&t_1'&t_4'\\t_3'&t_4'&t_2' \end{array}\right]{\rm e}^{-2i{{\bf k}}\cdot {{\bf a}}_3} ,\nonumber\\
{\mathcal{T}}_{AA}({{\bf k}})&=&\left[\begin{array}{ccc}t_7&t_8&t_8'\\t_8'&t_6&t_5'\\t_8&t_5&t_6 \end{array}\right]{\rm e}^{i{{\bf k}}\cdot ({{\bf a}}_2-{{\bf a}}_3)}
+ \left[\begin{array}{ccc}t_6&t_5'&t_8'\\t_5&t_6&t_8\\t_8&t_8'&t_7 \end{array}\right]{\rm e}^{i{{\bf k}}\cdot ({{\bf a}}_3-{{\bf a}}_1)}
+ \left[\begin{array}{ccc}t_6&t_8&t_5\\t_8'&t_7&t_8\\t_5'&t_8'&t_6 \end{array}\right]{\rm e}^{i{{\bf k}}\cdot ({{\bf a}}_1-{{\bf a}}_2)} +h.c. ,\nonumber\\
{\mathcal{T}}_{BB}({{\bf k}})&=&\left[\begin{array}{ccc}t_7&t_8&t_8'\\t_8'&t_6&t_5'\\t_8&t_5&t_6 \end{array}\right]{\rm e}^{-i{{\bf k}}\cdot ({{\bf a}}_2-{{\bf a}}_3)}
+ \left[\begin{array}{ccc}t_6&t_5'&t_8'\\t_5&t_6&t_8\\t_8&t_8'&t_7 \end{array}\right]{\rm e}^{-i{{\bf k}}\cdot ({{\bf a}}_3-{{\bf a}}_1)}
+ \left[\begin{array}{ccc}t_6&t_8&t_5\\t_8'&t_7&t_8\\t_5'&t_8'&t_6 \end{array}\right]{\rm e}^{-i{{\bf k}}\cdot ({{\bf a}}_1-{{\bf a}}_2)} +h.c. ,\nonumber\\\end{aligned}$$
and $$\begin{aligned}
\mathcal{V}=\left[\begin{array}{ccc}V_0&V_0'&V_0'\\V_0'&V_0&V_0'\\V_0'&V_0'&V_0\end{array}\right] ,\end{aligned}$$ where $({{\bf a}}_1,{{\bf a}}_2,{{\bf a}}_3)$ are three displacements from an A-site to its nearest neighbour B-sites. All the model parameters are determined by fitting the first-principle band structure [@Singh2015]: (in unit of eV) $t_1=0.16$, $t_2=-0.01$, $t_3=0.30$, $t_4=-0.02$, $t_5=-0.10$, $t_5'=0$, $t_6=-0.01$, $t_7=0.04$, $t_8=0.11$, $t_8'=0.02$, $t_1'=-0.04$, $t_2'=-0.01$, $t_3'=-0.01$, $t_4'=-0.01$, $V_0=-0.09$, $V_0'=-0.07$, $\lambda=0.16$. In particular, we obtain the SOC strength $\lambda=0.16$eV, which is in agreement with the literature [@Mizokawa2001]. Based on these parameters, the band structure of our minimal model is plotted in Fig. \[fig:band\].
![Band dispersion of our $t_{2g}$-orbital tight-binding model along high symmetry lines. The local-density-approximation result (from Ref. ) is also shown (red symbols) for comparison.[]{data-label="fig:band"}](bandstructure){width="45.00000%"}
In the paramagnetic state SrRu$_2$O$_6$ is a band insulator, as seen in our band structure and the first-principle result [@Singh2015; @Tian2015]. However, a strong Hund’s coupling would bind up the electrons to form a spin-$3/2$ state. Due to the bipartite lattice an AF order with moment $3\mu_B$/Ru is expected. Such an AF order has already been observed by Neutron scattering experiment in SrRu$_2$O$_6$,[@Tian2015] but the observed moment is only $1.3\mu_B$/Ru, much smaller than $3\mu_B$. This indicates the inadequacy of a naive local moment picture. Instead, the itinerant property of electrons and SOC may play important roles. In the following, we will investigate the effect of correlation and SOC on the AF order in the itinerant picture of the $t_{2g}$-orbital model.
Interaction and antiferromagnetism
==================================
We adopt general multi-orbital local Coulomb interactions as follows, $$\begin{aligned}
H_{\rm I}&=&\sum_{i}\left[\right. U\sum_{a} n_{ia\uparrow}n_{ia\downarrow}
+V\sum_{a > b} n_{ia}n_{ib} \nonumber\\
&+&J\sum_{a > b, \sigma\sigma'} a_{i\sigma}^\dag b_{i\sigma} b_{i\sigma'}^\dag a_{i\sigma'} +J'\sum_{a \neq b} a_{i\uparrow}^\dag a_{i\downarrow}^\dag
b_{i\downarrow}b_{i\uparrow} \left.\right] ,\end{aligned}$$ where $n_{ia}=\sum_\sigma n_{ia\sigma}=\sum_\sigma a_{i\sigma}^\dag a_{i\sigma}$. $U$ is the Hubbard interaction, $V$ is the inter-orbital charge interaction, $J$ is the Hund’s coupling and $J'$ is the pair hopping term. These four interactions satisfies the relation $J'=J$ and $U=V+2J$. [@Castellani1978] Among these four terms, only $U$ and $J$ are responsible for magnetic channel instabilities. [@Raghu2008] So in the following discussions, we will only retain the $U$ and $J$ terms.
Since the non-interacting model is a band insulator, the bare susceptibility only depends on momentum very weakly. So we perform an RPA level calculation instead, since RPA will pick out relevant channels and strongly enhance their susceptibilities.
![(a)Vertex function $\hat{\Gamma}(a\mu n;b\nu m)$ defined in Eq. \[eq:vertex\] in the magnetic channel. (b)The Feynman diagrammatic representation of the magnetic susceptibility matrix (Eq. \[eq:rpa\]) within the RPA.[]{data-label="fig:rpascheme"}](vertex "fig:"){width="35.00000%"} ![(a)Vertex function $\hat{\Gamma}(a\mu n;b\nu m)$ defined in Eq. \[eq:vertex\] in the magnetic channel. (b)The Feynman diagrammatic representation of the magnetic susceptibility matrix (Eq. \[eq:rpa\]) within the RPA.[]{data-label="fig:rpascheme"}](rpascheme "fig:"){width="45.00000%"}
The interactions ($U$ and $J$ terms) are first written in magnetic channels: $-S_I \hat{\Gamma}(I;J) S_J$, where $I=(a\mu n)$ and $J=(b\nu m)$ with the orbital index ($a/b$), spin-direction index ($\mu/\nu$), and sublattice index ($n/m$). The spin operator is $S_{I=(a\mu n)}=\frac{1}{2} \sum_{\alpha\beta} a_{n\alpha}^\dag \sigma^\mu_{\alpha\beta} a_{n\beta}$, and the vertex function $\hat{\Gamma}(a\mu n;b\nu m)$ represented by Fig. \[fig:rpascheme\](a) is diagonal in both spin-direction and sublattice subspaces and is given by $$\begin{aligned}
\hat{\Gamma}(a\mu n;b\nu m)=\left\{
\begin{array}{ll}
2U \delta_{\mu\nu}\delta_{nm},&a=b \\
2J \delta_{\mu\nu}\delta_{nm},&a\neq b
\end{array} \right. .
\label{eq:vertex}\end{aligned}$$
The RPA is a bubble summation as represented in Fig. \[fig:rpascheme\](b). After solving the iterate equation we obtain the magnetic susceptibility matrix as $$\begin{aligned}
\hat{\chi}({{\bf q}},i\nu_n)=\left[\hat{
{\rm I}}-\hat{\Gamma}\hat{\chi}^{(0)}({{\bf q}},i\nu_n)\right]^{-1}\hat{\chi}^{(0)}({{\bf q}},i\nu_n).
\label{eq:rpa}\end{aligned}$$ $\hat{\chi}^{(0)}({{\bf q}},i\nu_n)$ is the bare susceptibility whose matrix element is defined as $$\begin{aligned}
\hat{\chi}_{IJ}^{(0)}({{\bf q}},i\nu_n)=\int_0^{1/T} \langle
S_{I}(-{{\bf q}},\tau)S_{J}({{\bf q}},0) \rangle \mathrm{e}^{i\nu_n\tau} \mathrm{d}\tau .\nonumber \\\end{aligned}$$ Here, we have used Matsubara frequency with $T$ the temperature.
![The largest eigenvalue $\Lambda$ of $\hat{\chi}({{\bf q}},\omega=0)$ along high symmetry cuts. Here $U=1.35$eV, $J=0.14$eV, $k_BT=0.3$eV.[]{data-label="fig:rpa"}](chirpa){width="40.00000%"}
We use the interaction parameters $U=1.35$eV and $J=0.14$eV, being half of the values obtained by first principle calculation [@Tian2015], since RPA is known to overestimate the magnetic instability. We plot the leading (largest) eigenvalue $\Lambda$ of the hermitian susceptibility matrix $\hat{\chi}({{\bf q}},0)$ as a function of ${{\bf q}}$ in Fig. \[fig:rpa\]. The peak at ${{\bf q}}=0$ implies a magnetic instability that is periodic across the unit cell. The corresponding leading eigenvector decides the form factor of the magnetic order, namely the magnetic structure within a unit cell. This is shown in Fig. \[fig:meanfield\](a). We find the total moment (of the three orbitals) is along the $c$-axis and changes sign from one to the other sublattice within the unit cell. This is exactly the AF order seen in the neutron scattering experiment. More interestingly, the moments on the three orbitals are non-colinear about the $c$-axis. This is a prediction of the present work.
Next, we employ mean field theory to quantitatively investigate the AF moment and the transition temperature. The interactions are decoupled in magnetic channels: $$\begin{aligned}
&&Un_{ia\uparrow}n_{ia\downarrow}\rightarrow -\frac{2U}{3}{\bf S}_{ia}\cdot {\bf S}_{ia} \nonumber \\
&&\rightarrow -\frac{4U}{3}\langle {\bf S}_{ia}\rangle \cdot {\bf S}_{ia} + \frac{2U}{3}\langle {\bf S}_{ia}\rangle\cdot \langle {\bf S}_{ia}\rangle ,\end{aligned}$$ and $$\begin{aligned}
&&J\sum_{\sigma\sigma'} a_{i\sigma}^\dag b_{i\sigma} b_{i\sigma'}^\dag a_{i\sigma'}\rightarrow -2J{\bf S}_{ia}\cdot {\bf S}_{ib} \nonumber \\
&&\rightarrow -2J\langle{\bf S}_{ia}\rangle\cdot {\bf S}_{ib} -2J{\bf S}_{ia}\cdot \langle{\bf S}_{ib}\rangle +2J\langle{\bf S}_{ia}\rangle\cdot \langle{\bf S}_{ib}\rangle . \nonumber\\\end{aligned}$$
Then we obtain the mean field Hamiltonian $$\begin{aligned}
H_{MF}&=&H_{0} - \frac{4U}{3}\sum_{ia}\langle {\bf S}_{ia} \rangle \cdot {\bf S}_{ia} \nonumber\\
&& - 2J\sum_{i,a\neq b}\langle {\bf S}_{ia} \rangle \cdot {\bf S}_{ib} ,\end{aligned}$$ where $H_0$ is the non-interacting part, which is the same as Eq. \[eq:Hk\] but written in real space. The order parameters $\langle{\bf S}_{ia}\rangle=\frac{1}{2}\sum_{\alpha\beta}\langle a_{i\alpha}^\dag {\bf \sigma}_{\alpha\beta} a_{i\beta} \rangle$ are then numerically solved iteratively until convergence is achieved.
![(a)The AF configuration is schematically shown within a unit cell, with three orbital resolved moment represented by three different colors. The total moment is along c-axis \[($1,1,1$)-direction\]. (b)The total AF moment $M$ and the deviation angle $\theta$ vs temperature within the mean field approximation.[]{data-label="fig:meanfield"}](AFpattern "fig:"){width="30.00000%"} ![(a)The AF configuration is schematically shown within a unit cell, with three orbital resolved moment represented by three different colors. The total moment is along c-axis \[($1,1,1$)-direction\]. (b)The total AF moment $M$ and the deviation angle $\theta$ vs temperature within the mean field approximation.[]{data-label="fig:meanfield"}](meanfield "fig:"){width="45.00000%"}
Our mean field result confirms the non-collinear AF configuration \[Fig. \[fig:meanfield\](a)\] revealed by RPA in the normal state. We have performed [*unrestricted*]{} mean field calculations starting from random initial spin configurations. No translation symmetry is assumed in advance. However, the results all converge to the same non-collinear AF configuration up to a shift of the sublattice. The magnetic unit cell is always found to be equal to the lattice unit cell, which is in agreement with the unique peak ${{\bf q}}=0$ in the leading eigenvalue of the RPA magnetic susceptibility in the momentum space as shown in Fig. \[fig:rpa\]. The non-collinear AF configuration is the result of the strong SOC on Ru atoms. The values of the total moment $M$ and the deviation angle $\theta$ of each orbital moment relative to c-axis \[shown in Fig. \[fig:meanfield\](a)\] vs temperature are plotted in Fig. \[fig:meanfield\](b). Two sets of parameters are used. One is $(U,J)=(2.7,0.28)$eV from first principle calculation [@Tian2015], and the other is weakened by a factor of two, $(U,J)=(1.35,0.14)$eV. The latter leads to reduced $M$ and transition temperature. Since mean field theory overestimates the ordering, further considerations of quantum/thermal fluctuations beyond the mean field theory are necessary for quantitative comparison to experiments. Interestingly, we find $\theta$ becomes smaller with increasing interaction strength. This is because a larger $J$ tends to align the spins on different orbitals, while a larger SOC breaks the Hund’s rule more significantly. Since neutron scattering only ’sees’ the total moment, we expect this particular kind of orbital-resolved AF order can be observed in more delicate experiments like orbital-selective nuclear magnetic resonance[@Shimizu2012; @Shimizu2015] through the anisotropic hyperfine interaction [@Kiyama2003] or orbital-resolved angle-resolved photoemission spectroscopy[@Cao2013] through the photon polarization selection rule [@Damascelli2003].
summary and future works
========================
In summary, we have constructed a $t_{2g}$-orbital model on a honeycomb lattice. Local Coulomb interaction was investigated in both RPA and mean field theory. Experimentally observed Neel order is obtained. Furthermore, our theory predicts that the magnetic moments on three orbitals are non-collinear as a result of the strong spin-orbit coupling of Ru atoms. This particular kind of orbital-resolved AF order is expected to be observed in future experiments.
For future works, possible superconductivity in this compound after doping or under pressure is an interesting direction. Our $t_{2g}$-orbital model can be used as a minimal model to study possible superconductivity upon doping. The AF fluctuation may induce singlet Cooper pairing between the nearest neighbours. However, due to the strong SOC, triplet pairing may coexist with the singlet pairing. This part of work is being in progress.
|
---
abstract: 'In this paper we present a method to derive an exact master equation for a bosonic system coupled to a set of other bosonic systems, which plays the role of the reservoir, under the strong coupling regime, i.e., without resorting to[ either]{} the rotating-wave or secular approximations.[ Working with phase-space distribution functions, ]{}we verify that the dynamics [are separated in the evolution of its center, which follows classical mechanics, and its shape, which becomes distorted. This is the generalization of a result by Glauber, who stated that coherent states remain coherent under certain circumstances, specifically when the rotating-wave approximation and a zero-temperature reservoir are used. We show that the counter-rotating terms generate fluctuations that distort the vacuum state, much the same as thermal fluctuations. Finally, we discuss conditions for non-Markovian dynamics. ]{}'
author:
- 'T. B. Batalhão$^{1,2}$, G. D. de Moraes Neto$^{2}$, M. A. de Ponte$^{3}$, and M. H. Y. Moussa$^{2}$'
title: 'An exact master equation for the system-reservoir dynamics under the strong coupling regime and non-Markovian dynamics'
---
Introduction
============
The subject of open quantum systems has undergone substantial growth in the last three decades, starting with contributions to the field of fundamental quantum physics with the aim of understanding the process of decoherence. Based on the von Neumann approach to the reduction of the state vector [@Neumann], these contributions were mainly driven by the pioneering work of Zurek [@Zurek], Caldeira and Leggett [@CL], and Joos and Zeh [@JZ]. The repercussions of their work, together with the advent of the field of quantum information theory, led to renewed interest in open quantum systems, the focus now shifting from fundamental issues to practical applications in circuits to implement quantum logic operations.
The master equation approach has long been used to derive system-reservoir dynamics, to account for energy loss under a weak coupling regime [@Walls]. Its effectiveness comes from the fact that the energy loss of most quantum mechanical systems, especially within quantum and atomic optics, can be handled by the single-pole Wigner-Weisskopf approximation [@WW], where a perturbative expansion is performed in the system-reservoir coupling. Following developments by Caldeira and Leggett [@CL], more sophisticated methods to deal with the system-reservoir strong coupling regime have been advanced, such as the Hu-Paz-Zhang [@HPZ] master equation, with time-dependent coefficients, which allows for non-Markovian dynamics. Halliwell and Yu [@HY] have published an alternative derivation of the Hu-Paz-Zhang equation, in which the dynamics is represented by the Wigner function, and an exact solution of this equation was given by Ford and O’Connell [@FO].
Recently, the non-Markovian dynamics of open quantum systems has been studied with renewed interest, especially in connection with quantum information theory, as in Refs. [@Nori; @Wu]. However, in these studies, as well as in most of the derivations of master equations with time-dependent coefficients, the authors assume either the rotating-wave approximation (RWA) or the secular approximation (SA) for the system-reservoir coupling [@Makela]. Since non-Markovian behavior is sensitive to the counter-rotating terms in the interaction Hamiltonian, important features of the dynamics are missing under the RWA in the strong-coupling regime. It is worth mentioning that a study of the effect of the RWA and the SA on the non-Markovian behavior in the spin-boson model at zero temperature has already been advanced [@Makela], without, however, deriving a master equation.
Our goal in this work is to derive [and investigate the consequences of]{} a master equation within the strong-coupling regime, which prevents us resorting to either the RWA or the SA in the system-reservoir coupling. Moreover, instead of the path integrals approach [@FH], we use the formalism of quasi-probability distributions, thus enabling us to cast the problem as the solution of a linear system of equations. Our results follow from the general treatment of a bosonic dissipative network we have previously presented in Ref. [@MickelGeral], where the network dynamics were investigated, and further used for quantum information purposes [@MickelBunch]. However, differently from our previous developments, we first consider the general model for a network of bosonic non-dissipative oscillators and, subsequently, we focus on some of these oscillators (or in just one of them) as our system of interest, and treat all the others as a (structured) reservoir. The exact dynamics of the network allows us to obtain an exact dynamics of the system-reservoir interaction. Moreover, we present a simple inequality to distinguish between Markovian and non-Markovian dynamics.
Finally, this development enables us to generalize an earlier result by Glauber [@GlauberBook].[ When using the RWA and a zero-temperature reservoir, it was shown that the quasi-probability functions maintain their shape while they are displaced in phase space; in particular, coherent states remain coherent states]{}. We find that, for a general Gaussian state, the center of its phase space distribution follows classical dynamics (as in Ref. [@GlauberBook]), but its shape is changed. Furthermore, this change can be derived from the evolution of the vacuum state, which is no longer stationary, because of the counter-rotating terms. The change in shape is affected by both quantum and thermal fluctuations, and these contributions can be distinguished, at least in theory. Our developments can be straightforwardly translated to the derivation of an exact master equation for fermionic systems, using the reasoning in Ref. [@Glauber].
Unitary dynamics of the universe {#sec:model}
================================
The universe considered here consists of a set of $M+N$ harmonic oscillators, which are linearly coupled to each other in an arbitrary network. We consider $M$ of them to be part of our system of interest, and the remaining $N$ to be part of a reservoir. However, at this stage, we are concerned with the full dynamics of the universe, and there is actually no difference between system and reservoir modes. The oscillators are described by mass $m_{k}$ and natural, isolated frequencies $\varpi_{k}$; the coupling between modes $k$ and $j$, which occurs via their position coordinates, has strength $\lambda_{kj}$ (which, without loss of generality, is symmetric in its indices). Before we write the Hamiltonian that describes such a universe, we note that it must be positive-definite, in order to be bounded from below and have a well-defined ground state. Then, the Hamiltonian which is compatible with this model is $$H=\frac{1}{2}\sum_{k=1}^{M+N}\left( \frac{1}{m_{k}}\hat{p}_{k}^{2}+m_{k}\varpi_{k}^{2}\hat{q}_{k}^{2}\right) +\frac{1}{4}\sum_{kj=1}^{M+N}\lambda_{kj}\left( \hat{q}_{k}-\hat{q}_{j}\right) ^{2},
\label{eq:hamiltonqp}$$ where t[he coefficients $\lambda_{kj}$ form a real, symmetric matrix. We do not assume any particular form for them, so as to generate an arbitrary network, as depicted in Fig. \[fig:fig1\] ]{}The coupling term induces a change in the natural frequency of each mode, that is now represented by $$\omega_{k}=\sqrt{\varpi_{k}^{2}+\frac{1}{m_{k}}\sum_{j=1}^{N}\lambda_{kj}}.$$
[fig1.eps]{}
\[fig:fig1\]
Using this renormalized frequency, we can define annihilation operators $a_{k}$ and rewrite the Hamiltonian as $$H=\sum_{k=1}^{M+N}\omega_{k}a_{k}^{\dagger}a_{k}+\frac{1}{2}\sum_{kj=1}^{M+N}g_{kj}\left( a_{k}+a_{k}^{\dagger}\right) \left( a_{j}+a_{j}^{\dagger}\right) , \label{eq:hamiltona}$$ the coupling in this picture being given by $$g_{kj}=\frac{\lambda_{kj}}{2\sqrt{m_{k}m_{j}\omega_{k}\omega_{j}}}.
\label{eq:grenorm}$$
From here on, we will focus on $\omega_{k}$ and $g_{kj}$, the latter forming a real, symmetric matrix.
Characteristic function
-----------------------
The dynamics given by the Hamiltonian of Eq. (\[eq:hamiltona\]) is best understood in terms of the characteristic function of a state, which is just the expected value of the multimode displacement operator in the symmetric ordering,[ $$\chi\left( \left\{ \beta_{k}\right\} \right) =\left\langle \prod
_{k=1}^{M+N}\exp\left( \beta_{k}a_{k}^{\dagger}-\beta_{k}^{\ast}a_{k}\right)
\right\rangle \;,$$ where $\left\{ \beta_{k}\right\} $ represents all coordinates $\beta_{k}$ with $k=1,\dots,N$, as well as their complex conjugates. ]{}
The characteristic function carries the complete information about the state, and in particular information about moments of all orders; this is one of the reasons it is a better approach than using the Heisenberg equations of motion directly. The von Neumann equation in Hilbert space is mapped to a differential equation in dual phase space (where the characteristic function is defined):$$\frac{\partial\chi}{\partial t}=i\sum_{k=1}^{M+N}\left( \omega_{k}\beta
_{k}-\sum_{j=1}^{N}g_{kj}\left( \beta_{j}+\beta_{j}^{\ast}\right) \right)
\frac{\partial\chi}{\partial\beta_{k}}+\text{ H.c.}.$$
Being linear and of first order, this equation admits a simple ansatz, $$\chi\left( \left\{ \beta_{k}\right\} ,t\right) =\chi\left( \left\{
\beta_{k}\left( t\right) \right\} ,0\right) , \label{eq:ansatz}$$ which implies that the characteristic function maintains its shape, but the underlying (dual) phase space undergoes a linear transformation, given by $$\beta_{k}\left( t\right) =\sum_{j=1}^{M+N}\left( U_{j,k}\left( t\right)
\beta_{j}-V_{j,k}\left( t\right) \beta_{j}^{\ast}\right) .
\label{eq:linear}$$ This transformation is defined by the solution to a system of differential equations,
$$\begin{aligned}
\frac{dU_{kj}}{dt} & =i\omega_{j}U_{kj}-i\sum_{n=1}^{M+N}\left(
U_{k,n}-V_{k,n}\right) g_{n,j},\label{s1}\\
\frac{dV_{kj}}{dt} & =-i\omega_{j}V_{kj}-i\sum_{n=1}^{M+N}\left(
U_{k,n}-V_{k,n}\right) g_{n,j}. \label{s2}$$
The Heisenberg equations of motion for the first moments have a similar structure. However, since they refer only to first moments, they do not represent a complete solution of the problem, which can be obtained from the characteristic function with the same computational effort.
Reduced dynamics of the system
==============================
From this point on, we shall be interested only in the behavior of a subset of $M$ oscillators (the ones labeled $1$ to $M$), which form our system of interest, while the oscillators labeled $M+1$ to $M+N$ play the role of a (structured) reservoir. The complete solution to the dynamics is given by Eq.(\[eq:ansatz\]); in order to eliminate the reservoir degrees of freedom, all we need to do is set $\beta_{k}=0$ if $k>M$ (i.e., evaluate the characteristic function at the origin of the phase space of the modes we want to eliminate from the description). Before continuing, we observe that although not strictly necessary in our method, for the sake of simplicity we assume the usual sudden-coupling hypothesis, i.e., that the states of system and reservoir are initially uncorrelated:
$$\chi_{SR}\left( \left\{ \beta_{k}\right\} ,0\right) =\chi_{S}\left(
\left\{ \beta_{k}\right\} _{k\leq M},0\right) \chi_{R}\left( \left\{
\beta_{m}\right\} _{m>M}\right) . \label{eq:initial}$$
Tracing out the reservoir degrees of freedom, following the procedure above, leads to $$\chi_{S}\left( \left\{ \beta_{k}\right\} ,t\right) =\chi_{S}\left(
\left\{ \beta_{k}\left( t\right) \right\} ,0\right) \chi_{\text{in}}\left( \left\{ \beta_{k}\right\} ,t\right) \;, \label{eq:reducedsolution}$$ where the indices run only through the degrees of freedom of the system (i.e., $k$ runs from $1$ to $M$). Therefore, we must use Eq.(\[eq:linear\]) with $\beta_{k}=0$ for $k>M$, and it follows that we only need $U_{kj}$ and $V_{kj}$ for $k\leq M$. Eqs. (\[s1\],\[s2\]), although written as a matrix equation, are actually a set of $N$ independent vector equations and we conclude that only a few of these need to be solved. In fact, if our system of interest were a single oscillator, we would reduce the problem of finding its exact dynamics to a single vector equation of dimension $2N$.
The two terms of Eq. (\[eq:reducedsolution\]) are called the homogeneous (because it depends on the initial state of the system) and inhomogeneous terms (because it is independent of it, depending only on the initial state of the reservoir). The homogeneous part of the solution is just the linear transformation of phase space induced only by the elements $U_{kj}$ and $V_{kj}$ for which both $k,j\leq M$. These elements can be arranged in two general complex $M\times M$ matrices, resulting in $4M^{2}$ real parameters.
At this point, we make an additional assumption that the initial state of the reservoir is Gaussian [@Gaussian], i.e., its characteristic function has the Gaussian form. Moreover, the reservoir is unbiased (i.e., $\left\langle
a_{m}\right\rangle =0$ for $m>M$). These are reasonable hypotheses, since the Gaussian states include the thermal states of quadratic Hamiltonians. The inhomogeneous characteristic function is then also a Gaussian function: $$\begin{aligned}
\chi_{in}\left( \left\{ \beta_{k}\right\} ,t\right) & =\exp\left(
-\frac{1}{2}\sum_{kj=1}^{M}A_{kj}\left( t\right) \beta_{k}\beta_{j}^{\ast
}\right) \nonumber\\
& \times\exp\left( \sum_{kj=1}^{M}B_{kj}\left( t\right) \beta_{k}\beta
_{j}+\text{c.c}\right) \text{.}$$ The time-dependent functions $A_{kj}$ and $B_{kj}$ may be divided into two terms, in the form $A_{kj}=A_{kj}^{\left( 0\right) }+A_{kj}^{\left(
th\right) }$ (and similarly for $B$), the first of which is the solution for a zero-temperature reservoir,
\[eq:pqzero\]$$\begin{aligned}
A_{kj}^{\left( 0\right) } & =\frac{1}{2}\sum_{m=M+1}^{M+N}\left(
U_{km}U_{jm}^{\ast}+V_{km}V_{jm}^{\ast}\right) \\
B_{kj}^{\left( 0\right) } & =\frac{1}{2}\sum_{m=M+1}^{M+N}\left(
U_{km}V_{jm}+V_{km}U_{jm}\right) \;,\end{aligned}$$ while the second incorporates the effects of the reservoir initial state, which is completely characterized by the second-order moments $\left\langle
a_{m}^{\dagger}a_{n}\right\rangle _{0}$ and $\left\langle a_{m}a_{n}\right\rangle _{0}$,
\[eq:pqtemp\]$$\begin{aligned}
A_{kj}^{\left( th\right) }= & \sum_{m=M+1}^{M+N}\left\langle
a_{m}^{\dagger}a_{n}\right\rangle _{0}\left( U_{km}U_{jn}^{\ast}+V_{kn}V_{jm}^{\ast}\right) \\
& +\sum_{m=M+1}^{M+N}\left( \left\langle a_{m}a_{n}\right\rangle _{0}V_{km}U_{jn}^{\ast}+\text{c.c.}\right) \nonumber\\
B_{kj}^{\left( th\right) } & =\sum_{m=M+1}^{M+N}\left\langle
a_{m}^{\dagger}a_{n}\right\rangle _{0}\left( U_{kn}V_{jm}+V_{km}U_{jn}^{\ast
}\right) \nonumber\\
& +\sum_{m=M+1}^{M+N}\left( \left\langle a_{m}a_{n}\right\rangle _{0}V_{km}V_{jn}+\text{c.c.}\right) \;.\end{aligned}$$ Both $A$ and $B$ form complex $M\times M$ matrices; however, $A$ must be Hermitian, while $B$ is not. This represents an additional $3M^{2}$ real parameters, giving a total of $7M^{2}$ that completely specifies a given Gaussian evolution map (so called because, if the initial state of the system is Gaussian, it will remain Gaussian).
The functions $A_{kj}^{\left( 0\right) }$ and $B_{kj}^{\left( 0\right) }$ represent the solution for a zero-temperature reservoir; therefore, they represent the quantum, or zero-point fluctuations. The functions $A_{kj}^{\left( th\right) }$ and $B_{kj}^{\left( th\right) }$ represent the thermal fluctuations (when the reservoir is assumed to be in a thermal state), and other effects that may arise due to, e.g., squeezing in the reservoir modes.
Single-mode Dynamics
====================
The above result may be written in a simpler fashion for the case of a single oscillator taken as the system of interest:
$$\begin{aligned}
\chi\left( \beta,t\right) = & \chi\left( U\beta-V\beta^{\ast},0\right)
\nonumber\\
& \times\exp\left( -A\left\vert \beta\right\vert ^{2}+\frac{1}{2}B\beta
^{2}+\frac{1}{2}B^{\ast}\beta^{\ast2}\right) \;, \label{eq:solution}$$
where the indices $1,1$ are dropped. The single-mode Gaussian map is completely characterized by $7$ real parameters (since $A$ is real, and $U$, $V$ and $B$ are complex).
When a single mode is considered as the system of interest, we can perform a diagonalization of the reservoir part of the Hamiltonian, and consider the interaction of the system with each of the reservoir normal modes, as depicted in Fig. \[fig:fig2\] (normal modes of the reservoir do not interact with each other, but interact with the system).
[fig2.eps]{}
In order to get physical results in the limit $N\rightarrow\infty$, it is essential to keep track of the oscillator masses ($m_{k}$ in Eq. (\[eq:hamiltonqp\])). Essentially, the central oscillator must be much more massive than the reservoir modes. This is the case with Brownian motion, where the observed particle, though mesoscopic, is still much larger than the bath of fluid molecules it interacts with. It is also the case in Quantum Optics, where the mode inside a cavity has a much smaller mode volume (i.e., it is concentrated in a small region) than the vacuum modes outside the cavity. We shall consider then that the central oscillator has mass $M$ and the reservoir modes have mass $\mu$, with $M\gg\mu$, and the renormalized frequencies and couplings are
$$\begin{aligned}
\omega_{1} & =\sqrt{\varpi_{1}^{2}+\frac{1}{M}\sum_{j=2}^{N+1}\lambda_{1j}}\\
\omega_{j} & =\sqrt{\varpi_{j}^{2}+\frac{1}{\mu}\lambda_{1j}}\quad\left(
2\leq j\leq N+1\right) \\
g_{j} & =\frac{1}{2\sqrt{\mu M}}\frac{\lambda_{1j}}{\sqrt{\omega_{1}\omega_{j}}}\quad\left( 2\leq j\leq N+1\right)\end{aligned}$$
Dropping the first index, Eqs.(\[s1\],\[s2\]) become
$$\begin{aligned}
\frac{dU_{1}}{dt} & =i\omega_{1}U_{1}-i\sum_{j=2}^{N}g_{j}\left(
U_{j}-V_{j}\right) \\
\frac{dV_{1}}{dt} & =-i\omega_{1}V_{1}-i\sum_{j=2}^{N}g_{j}\left(
U_{j}-V_{j}\right) \\
\frac{dU_{j}}{dt} & =i\omega_{j}U_{j}-ig_{j}\left( U_{1}-V_{1}\right)
\quad\quad\left( j\neq1\right) \\
\frac{dV_{j}}{dt} & =-i\omega_{j}V_{j}-ig_{j}\left( U_{1}-V_{1}\right)
\quad\quad\left( j\neq1\right) \;.\end{aligned}$$
The bottom two equations can be solved by considering $U_{1}$ and $V_{1}$ as external parameters. Then, by substituting them into the top two equations, we get a pair of coupled integro-differential equations:
$$\begin{aligned}
\frac{dU_{1}}{dt} & =i\omega_{1}U_{1}+i\int_{0}^{t}d\tau h\left(
t-\tau\right) \left( U_{1}\left( \tau\right) -V_{1}\left( \tau\right)
\right) \label{eq:u1integro}\\
\frac{dV_{1}}{dt} & =-i\omega_{1}V_{1}+i\int_{0}^{t}d\tau h\left(
t-\tau\right) \left( U_{1}\left( \tau\right) -V_{1}\left( \tau\right)
\right) \;, \label{eq:v1integro}$$
which depends on the reservoir topology only through the function
$$h\left( t\right) =\sum_{j=2}^{N+1}g_{j}^{2}\sin\left( \omega_{j}t\right)
=\frac{1}{4\mu M\omega_{1}}\sum_{j=2}^{N+1}\frac{\lambda_{j}^{2}}{\omega_{j}}\sin\left( \omega_{j}t\right) \;,$$
which in turn is related to the Fourier transform of the reservoir spectral density $$J\left( \omega\right) =\sum_{j=2}^{N+1}g_{j}^{2}\delta\left( \omega
-\omega_{j}\right) =\frac{1}{4\mu M\omega_{1}}\sum_{j=2}^{N+1}\frac
{\lambda_{j}^{2}}{\omega_{j}}\delta\left( \omega-\omega_{j}\right)$$
This is the homogeneous part of the solution. To obtain the inhomogeneous one, we need to use the solution found previously for $U_{k}$ and $V_{k}$ in terms of the now known $U_{1}$ and $V_{1}$, and then use Eqs. (\[eq:pqzero\]) and (\[eq:pqtemp\]).
Master Equation
===============
The complete solution for single-mode dynamics is Eq. (\[eq:solution\]), with time-dependent functions $U$, $V$, $A$ and $B$. It was derived by assuming an explicit microscopic model for the reservoir as a set of other modes, which are coupled to the mode of interest, but over which the experimenter has little control (except for macroscopic parameters such as temperature). In this section, our goal is to find a dynamical equation (in fact, a master equation) whose solution is precisely Eq. (\[eq:solution\]), but which does not need to involve any other degrees of freedom, besides those of the system.
We start by differentiating Eq. (\[eq:solution\]) with respect to time, and then mapping it from phase space back to Hilbert space:$$\frac{d\rho}{dt}=-i\left[ H_{S}\left( t\right) ,\rho\left( t\right)
\right] +\mathcal{D}_{t}\left( \rho\left( t\right) \right) ,
\label{eq:master}$$ where we have a time-dependent effective Hamiltonian $$H_{S}\left( t\right) =\omega\left( t\right) a^{\dagger}a+\xi\left(
t\right) a^{\dagger2}+\xi^{\ast}\left( t\right) a^{2}\;,
\label{eq:masterham}$$ and a time-dependent dissipation super-operator, $$\begin{aligned}
\mathcal{D}_{t}\left( \rho\right) = & \frac{\gamma_{1}\left( t\right)
+\gamma_{2}\left( t\right) }{2}\left( \left[ a\rho,a^{\dagger}\right]
+\left[ a,\rho a^{\dagger}\right] \right) \nonumber\\
& +\frac{\gamma_{2}\left( t\right) }{2}\left( \left[ a^{\dagger}\rho,a\right] +\left[ a^{\dagger},\rho a\right] \right) \nonumber\\
& -\frac{1}{2}\left( \eta\left( t\right) \left( \left[ a^{\dagger}\rho,a^{\dagger}\right] +\left[ a^{\dagger},\rho a^{\dagger}\right]
\right) +\text{H.c.}\right) \;. \label{eq:masterdiss}$$
This master equation depends on $7$ real time-dependent parameters, which in turn depend on the $7$ real parameters that define solution Eq.(\[eq:solution\]); the three real parameters
$$\omega\left( t\right) =\frac{1}{\left\vert U\right\vert ^{2}-\left\vert
V\right\vert ^{2}}\Im\left( U^{\ast}\frac{dU}{dt}-V^{\ast}\frac{dV}{dt}\right) \;,$$
$$\begin{aligned}
\gamma_{1}\left( t\right) = & \frac{-2}{\left\vert U\right\vert
^{2}-\left\vert V\right\vert ^{2}}\Re\left( U^{\ast}\frac{dU}{dt}-V^{\ast
}\frac{dV}{dt}\right) \nonumber\\
= & -\frac{d}{dt}\log\left( \left\vert U\right\vert ^{2}-\left\vert
V\right\vert ^{2}\right) \;, \label{eq:gammafrommapa}$$
$$\gamma_{2}\left( t\right) =\frac{dA}{dt}+\gamma_{1}\left( A-\frac{1}{2}\right) +2\Im\left( \xi^{\ast}B\right) \;, \label{eq:gamma2}$$
and the two complex parameters $$\xi\left( t\right) =\frac{-i}{\left\vert U\right\vert ^{2}-\left\vert
V\right\vert ^{2}}\left( U\frac{dV}{dt}-V\frac{dU}{dt}\right) ,$$$$\eta\left( t\right) =\frac{dB}{dt}+\left( \gamma_{1}+2i\omega\right)
B+2i\xi A. \label{eq:eta}$$ The time-dependent functions $\omega\left( t\right) $, $\gamma_{1}\left(
t\right) $ and $\xi\left( t\right) $ are independent of the initial state of the reservoir, while $\gamma_{2}\left( t\right) $ and $\eta\left(
t\right) $ depend on it.
The dissipator, Eq. (\[eq:masterdiss\]), is not explicitly in Lindblad-like form, but can be put into it,
$$\mathcal{D}_{t}\left( \rho\right) =\sum_{n=1}^{2}\frac{\lambda_{n}\left(
t\right) }{2}\left( \left[ L_{n}\left( t\right) \rho,L_{n}^{\dagger
}\left( t\right) \right] +\left[ L_{n}\left( t\right) ,\rho
L_{n}^{\dagger}\left( t\right) \right] \right) \label{eq:masterdisslind}$$
by defining the Lindblad operators
$$\begin{aligned}
L_{1}\left( t\right) & =\cos\left( \frac{\theta}{2}\right) a-\sin\left(
\frac{\theta}{2}\right) \frac{\eta}{\left\vert \eta\right\vert }a^{\dagger
}\label{1}\\
L_{2}\left( t\right) & =\cos\left( \frac{\theta}{2}\right) a^{\dagger
}+\sin\left( \frac{\theta}{2}\right) \frac{\eta^{\ast}}{\left\vert
\eta\right\vert }a\;, \label{2}$$
and Lindblad rates
$$\begin{aligned}
\lambda_{1}\left( t\right) & =\frac{\gamma_{1}}{2}+\frac{\gamma_{1}}{\left\vert \gamma_{1}\right\vert }\sqrt{\frac{\gamma_{1}^{2}}{4}+\left\vert
\eta\right\vert ^{2}}+\gamma_{2}\\
\lambda_{2}\left( t\right) & =\frac{\gamma_{1}}{2}-\frac{\gamma_{1}}{\left\vert \gamma_{1}\right\vert }\sqrt{\frac{\gamma_{1}^{2}}{4}+\left\vert
\eta\right\vert ^{2}}+\gamma_{2}\;,\end{aligned}$$
with the auxiliary definition
$$\theta=\arctan\left( \frac{2\left\vert \eta\right\vert }{\gamma_{1}}\right)
\quad\left( -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}\right)$$
The standard master equation derived with the Born-Markov approximation has the same form as equations Eq. (\[eq:master\])-(\[eq:masterdiss\]), but with constant-in-time parameters. In it, each term has a physical meaning:
- The first term in Eq. (\[eq:masterham\]), with $\omega\left(
t\right) =\omega_{1}+\Delta\omega\left( t\right) $, accounts for the free dynamics of the system, modified by a frequency shift due to its interaction with the reservoir.
- The second term in Eq. (\[eq:masterham\]) is a squeezing term, arising from an asymmetry between position and momentum variables in the coupling Hamiltonian. However, in the weak-coupling regime, this term is small (being exactly zero in the RWA), leading to a negligible squeezing effect.
- $\gamma_{1}\left( t\right) $ is a decay rate, that drives the center of the system wave-packet towards its equilibrium at the origin of phase space.
- $\gamma_{2}\left( t\right) $ is a diffusion coefficient, related to injection of extra noise into the system due to non-zero reservoir temperature and counter-rotating terms, which only spreads the wave-packet without affecting the trajectory of its center.
- $\eta\left( t\right) $ is a coefficient of anomalous diffusion, which injects different levels of noise in position and momentum. From Eqs. (\[1\],\[2\]), we see that, when $\eta\neq0$, the Lindblad operators are not given by $a$ and $a^{\dagger}$, but by linear combinations of the two, giving rise to anomalous diffusion.
Markovian and non-Markovian behavior
------------------------------------
An interesting discussion in the current literature (see Ref. [@NonMarkovian] and references therein) concerns non-Markovian behavior. The Born-Markov approximation always leads to a Lindblad equation with a dissipator written in the form of Eq.(\[eq:masterdisslind\]), with rates $\lambda_{n}\left( t\right) $, which are positive but may vary in time (in which case it can be called a *time-dependent Markovian process*). If, at any given time, one of these rates assumes a negative value, then it is said to be a *non-Markovian process*, according to the divisibility criterion of Rivas-Huelga-Plenio [@NonMarkovian; @RHP].
The model we have developed allows us to compute these rates exactly from the solution, obtained through the system-reservoir interaction Hamiltonian. We can thus describe the system as *Markovian* if the following conditions hold for all times $t$:
$$\begin{aligned}
\gamma_{1}\left( t\right) +2\gamma_{2}\left( t\right) & \geq0\\
\gamma_{1}\left( t\right) \gamma_{2}\left( t\right) +\gamma_{2}^{2}\left(
t\right) -\left\vert \eta\left( t\right) \right\vert ^{2} & \geq0\;,\end{aligned}$$
where the functions are defined in Eq. (\[eq:gammafrommapa\]), Eq. (\[eq:gamma2\]) and Eq. (\[eq:eta\]).
Rotating Wave Approximation
===========================
In many physical systems described by the Hamiltonian of Eq. (\[eq:hamiltona\]), the typical coupling intensity, $\left\vert
g_{kj}\right\vert $, is many orders of magnitude smaller than the frequencies $\omega_{k}$, characterizing the *weak coupling regime*. It is then a good approximation to drop the counter-rotating terms ($a_{k}a_{j}$ and $a_{k}^{\dagger}a_{j}^{\dagger}$), a procedure which is known as the *rotating wave approximation* (*RWA*). Eqs. (\[s1\],\[s2\]) are greatly simplified, with $V_{kj}=0$ and $U_{kj}$ obeying:
$$\frac{dU_{kj}}{dt}=i\omega_{j}U_{kj}-i\sum_{n=1}^{N}U_{kn}g_{nj}\;.$$
The condition $V_{kj}=0$ (for all $kj$) implies both $\xi\left( t\right) =0$ (no squeezing term in the effective system Hamiltonian) and $B^{\left(
0\right) }=0$ and, unless the reservoir initial state has some degree of squeezing (i.e., $\left\langle a_{m}a_{n}\right\rangle _{0}\neq0$ for some $m,n$), then also $B^{\left( th\right) }=0$. Together, this implies that $\eta\left( t\right) =0$. The condition $\xi\left( t\right) =\eta\left(
t\right) =0$ is required to maintain the symmetry between position and momentum variables (the exchange $\left( \hat{q},\hat{p}\right)
\leftrightarrow\left( \hat{p},-\hat{q}\right) $ leaves the RWA Hamiltonian unchanged, while it changes the one in Eq. (\[eq:hamiltonqp\])). Therefore, in RWA, the squeezing term in Eq. (\[eq:masterham\]) and the last term in Eq. (\[eq:masterdiss\]) both vanish at all times, leading to the usual three terms (frequency shift, dissipation and diffusion) in the expression. The Markovianity condition is then simplified to
$$\begin{aligned}
\gamma_{1}\left( t\right) +2\gamma_{2}\left( t\right) & \geq0\\
\gamma_{2}\left( t\right) & \geq0\end{aligned}$$
Natural Basis For System Evolution
==================================
It is a well known result [@GlauberBook] that a coherent state remains coherent when in contact with a reservoir at absolute zero, if one assumes RWA. This makes coherent states a natural basis to analyze system dynamics, ultimately motivating Glauber and Sudarshan to define the normal-order quasi-probability $P$ function:
$$\rho\left( t\right) =\int d^{2M}\left\{ \alpha\right\} P\left( \left\{
\alpha\right\} ,t\right) \left\vert \left\{ \alpha\right\} \right\rangle
\left\langle \left\{ \alpha\right\} \right\vert .$$
[We have returned to the general case, where the system is composed of $M$ modes.]{} The coherent state follows a dynamics in phase space that can be written $\left\vert \left\{ \alpha\right\} \right\rangle \rightarrow
\left\vert \left\{ \alpha\left( t\right) \right\} \right\rangle $, where $\left\{ \alpha\left( t\right) \right\} $ is given by (compare with Eq. (\[eq:linear\])) $$\alpha_{k}\left( t\right) =\sum_{j=1}^{M}\left( U_{kj}\alpha_{j}+V_{kj}\alpha_{j}^{\ast}\right) \quad\left( 1\leq k\leq M\right) \;.
\label{eq:lineardirect}$$ Combining these two equations, we have the familiar result $$\rho\left( t\right) =\int d^{2M}\left\{ \alpha\right\} P\left( \left\{
\alpha\right\} ,0\right) \left\vert \left\{ \alpha\left( t\right)
\right\} \right\rangle \left\langle \left\{ \alpha\left( t\right)
\right\} \right\vert . \label{eq:glauberevolution}$$
The fact that coherent states remain coherent is intimately connected with the fact that the vacuum is a stationary state of this non-unitary evolution. However, for non-zero temperature, or when one includes the counter-rotating terms, this is no longer true: coherent states do not maintain their coherence, and we must resort to another basis, formed by Gaussian states. In the same way that the coherent states are generated by displacing the vacuum, the time-dependent Gaussian basis states are generated by displacing a squeezed thermal state: $$\rho_{B}\left( \left\{ \alpha\right\} ,t\right) =D\left( \left\{
\alpha\right\} \right) \rho_{o}\left( t\right) D^{\dagger}\left( \left\{
\alpha\right\} \right) ,$$ where $\rho_{o}\left( t\right) $ is obtained by allowing an initial vacuum state to evolve in accordance with the solution presented in Eq. (\[eq:solution\]): $$\left\vert 0\right\rangle \left\langle 0\right\vert \rightarrow\rho_{o}\left(
t\right) =\int d^{2M}\left\{ \alpha\right\} P_{o}\left( \left\{
\alpha\right\} ,t\right) \left\vert \left\{ \alpha\right\} \right\rangle
\left\langle \left\{ \alpha\right\} \right\vert \label{eq:evolvacuum}$$
Adopting then this natural Gaussian basis, we can write the evolution of any initial state as: $$\rho\left( t\right) =\int d^{2M}\left\{ \alpha\right\} P\left( \left\{
\alpha\right\} ,0\right) \rho_{B}\left( \left\{ \alpha\left( t\right)
\right\} ,t\right) . \label{eq:evolany}$$
Combining Eq. (\[eq:evolvacuum\]) and Eq. (\[eq:evolany\]), we can rewrite the evolution of an arbitrary initial state (albeit one with a reasonably well-defined $P$ function) as $$\begin{aligned}
\rho\left( t\right) = & \int d^{2M}\left\{ \alpha\right\} \int
d^{2M}\left\{ \eta\right\} P\left( \left\{ \alpha\right\} ,0\right)
P_{o}\left( \left\{ \eta\right\} ,t\right) \nonumber\\
& \times\left\vert \left\{ \eta+\alpha\left( t\right) \right\}
\right\rangle \left\langle \left\{ \eta+\alpha\left( t\right) \right\}
\right\vert , \label{eq:naturalbasis}$$ where $\left\{ \alpha\left( t\right) \right\} $ describe the evolution of the *center* of the wavepacket (which obeys a classical equation of motion, as required by the Ehrenfest theorem, and is independent of the state of the reservoir) and $P_{o}\left( \left\{ \eta\right\} ,t\right) $ describe the evolution of the *shape* of the wavepacket.
When the RWA and an absolute-zero reservoir are assumed, the wavepacket is not distorted, and $P_{o}\left( \left\{ \eta\right\} ,t\right) $ reduces to a delta function at the origin, making Eq. (\[eq:naturalbasis\]) identical to Eq. (\[eq:glauberevolution\]). Therefore, Eq. (\[eq:naturalbasis\]) is a generalization of Eq. (\[eq:glauberevolution\]) and we have obtained a generalization of the dynamics described in Ref. [@GlauberBook].
Another way to look at this result is that the displaced phase-space quasi-probability function is convoluted with another function, which accounts for the change in shape. $$P\left( \left\{ \alpha\right\} ,t\right) =\int d^{2M}\left\{
\gamma\right\} P\left( \left\{ \gamma\right\} ,0\right) P_{o}\left(
\left\{ \alpha-\gamma\left( t\right) \right\} ,t\right)$$ For a single mode, the center path follows $\alpha\left( t\right)
=U_{1}\alpha+V_{1}\alpha^{\ast}$, $U_{1}$ and $V_{1}$ being given by the solutions to Eqs. (\[eq:u1integro\]) and (\[eq:v1integro\]). The function $P_{o}\left( \left\{ \alpha\right\} ,t\right) $ is just the solution when the initial state is the vacuum, i.e., it satisfies the initial condition $P_{o}\left( \left\{ \alpha\right\} ,0\right) = \delta^{\left( 2\right)
}\left( \alpha\right) $. Under the RWA, this continues to be true at all times, $P_{o}^{\text{RWA}}\left( \left\{ \alpha\right\} ,t\right) =
\delta^{\left( 2\right) }\left( \alpha\right) $.
Conclusions
===========
We have presented a technique to derive an exact master equation for the system-reservoir dynamics under the strong coupling regime, where neither the rotating-wave-approximation nor the secular approximation apply. To this end, we adopted the strategy of considering a network of bosonic systems coupled to each other, picking out one of them as the system of interest and leaving the rest to play the role of the reservoir. Working [with phase-space distribution functions and Gaussian states, we generalize an earlier result by Glauber, that a coherent state remains coherent despite dissipation when coupled to a zero temperature reservoir. We demonstrate that t]{}here is a class of Gaussian states which serves as a generalization of the coherent state basis of the Glauber-Sudarshan $P$ representation. This class of Gaussian states follows from the distortion of the vacuum state which, in the strong-coupling regime, is no longer a stationary state, even for a zero temperature reservoir. We have also presented an investigation of the conditions that lead to a non-completely-divisible map, and thus non-Markovian dynamics. So far, conditions for non-Markovianity have been studied for finite Hilbert spaces under the rotating-wave and/or secular approximations. We remark that a master equation similar to the one derived here has been obtained using the Path Integrals approach [@HPZ]. The simplicity of our development, using phase-space distribution functions, offers the significant advantage of enabling us to cast the problem as the solution of a linear system of equations.
The authors acknowledge financial support from PRP/USP within the Research Support Center Initiative (NAP Q-NANO) and FAPESP, CNPQ and CAPES, Brazilian agencies.
[99]{}
J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1955).
W. H. Zurek, Phys. Rev. D **24**, 1516 (1981); *ibid.* **26**, 1862 (1982).
A. O. Caldeira and A. J. Leggett, Physica **121A**, 587 (1993), *ibid.*, ** Ann. Phys. (N.Y.) 149, **374** (1983), *ibid.*, ** Phys. Rev. A **31**, 1059 (1985).
E. Joos and H. D. Zeh, Z. Phys. B: Condens. Matter **59**, 223 (1985).
E. B. Davies, Quantum Theory of Open Systems (Academic Press, New York, 1976); D. Walls, G. Milburn, Quantum Optics (Spinger-Verlag, Berlin, 1994); M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge Press, London, 1997).
E. P. Wigner and V. F. Weisskopf, Z. Physik **63**, 54 (1930).
B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D **45**, 2843 (1992).
J. J. Halliwell and T. Yu, Phys. Rev. D **53**, 2012 (1996).
G. W. Ford and R. F. O’Connell, Phys. Rev. D **64**, 105020 (2001).
W.-M. Zhang, P.-Y. Lo, H.-N. Xiong, M. W.-Y. Tu, and F. Nori, Phys. Rev. Lett. **109**, 170402 (2012).
H.-N. Xiong, W.-M, Zhang, X. Wang, and M.-H. Wu, Phys. Rev. A **82**, 012105 (2010).
H. Mäkelä and M. Möttönen, Phys. Rev. A **88**, 052111 (2013).
R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
M. A. de Ponte, S. S. Mizrahi, and M. H. Y. Moussa, Phys. Rev. A **76**, 032101 (2007); M. A. de Ponte, M. C. de Oliveira, and M. H. Y. Moussa, Phys. Rev. A **70**, 022324 (2004); *ibid*. Phys. Rev. A **70**, 022325 (2004); *ibid*. Ann. Phys. (N.Y.) **317**, 72 (2005).
M. A. de Ponte, S. S. Mizrahi, and M. H. Y. Moussa, Ann. Phys **322**, 2077 (2007); *ibid*, Phys. Rev. A **84**, 012331 (2011).
R. Glauber, Quantum theory of optialoherene: seleted papers and letures (Berlin, Germany: Wiley-VCH, 2007).
K. E. Cahill and R. J. Glauber, Phys. Rev. A **59**, 1538 (1999).
C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. **84**, 621 (2012).
C. Addis, B. Bylicka, D. Chru[ś]{}ci[ń]{}ski and S. Maniscalco, arXiv:1402.4975.
A. Rivas, S. F. Huelga and M. B. Plenio, Phys. Rev. Lett. **105**, 050403 (2010).
|
---
abstract: 'By using the scaling method we derive the virial theorem for the relativistic mean field model of nuclei treated in the Thomas–Fermi approach. The Thomas–Fermi solutions statisfy the stability condition against scaling. We apply the formalism to study the excitation energy of the breathing mode in finite nuclei with several relativistic parameter sets of common use.'
---
[**Scaling in Relativistic Thomas–Fermi\
Approach for Nuclei**]{}\
S.K. Patra[^1], M. Centelles, X. Viñas and M. Del Estal\
[*Departament d’Estructura i Constituents de la Matèria, Facultat de Física,\
Universitat de Barcelona, Diagonal [*647*]{}, E-[*08028*]{} Barcelona, Spain*]{}
[*PACS:*]{} 21.60.-n; 24.30.Cz; 21.30.Fe; 21.65.+f
[*Keywords:*]{} relativistic mean field; scaling; virial theorem; giant resonances; nuclear incompressibility; Thomas–Fermi theory
[*E-mail address:*]{} mario@ecm.ub.es (M. Centelles)
The relativistic mean field (RMF) treatment of Quantum Hadrodynamics [@serot86; @serot97] has proven to be very useful for describing different properties of nuclei along the periodic table. The simplest model, the linear $\sigma-\omega$ model of Walecka [@Wa74], explains the nuclear force in terms of the exchange of $\sigma$ and $\omega$ mesons. It is known that the value of the nuclear matter incompressibility is unreasonably high in this linear model ($K_{\rm
nm} \sim 550$ MeV). The problem can be cured by introducing cubic and quartic self-interactions of the $\sigma$ meson [@boguta77], and the model can be refined by adding an isovector $\rho$ meson. Current non-linear parameter sets, such as the NL3 set [@lalaz97], give ground-state energies and densities in excellent agreement with the experimental data, not only for magic nuclei but also for deformed nuclei and for nuclei far from the stability line.
The scaling method has been often employed to derive the virial theorem in the non-relativistic framework [@parr89], e.g., for nuclear effective interactions such as the Skyrme force [@bohigas79]. It has also been applied in calculations of nuclear collective excitations like the breathing mode (isoscalar giant monopole resonance) [@bohigas79]. Relativistic generalizations of the virial theorem obtained by use of the scaling method exist for particles in external potentials [@brack83; @lucha90]. In the RMF model of nuclei the mean field potentials are generated self-consistently. Owing to the meson-exchange nature of the relativistic model one has to deal with finite range forces, which renders the scaling more involved than for zero-range Skyrme forces. Moreover, in contrast to the non-relativistic situation, there exist two different densities, namely the baryon and the scalar density, in accordance with the fact that one has two types of fields, the vector and the scalar field.
In this letter we shall make use of the principle of scale invariance to obtain the virial theorem for the RMF theory by working in the Thomas–Fermi approximation. We shall include non-linear self-couplings of the scalar field and shall deal with spherical finite nuclei. The second derivative of the scaled energy with respect to the scaling parameter, the so-called restoring force, turns out to be positive (stability condition) in the Thomas–Fermi calculations. Thus we are able to apply the method to compute the excitation energy of the isoscalar giant monopole resonance in finite nuclei with realistic parameter sets of the relativistic model.
The meson field equations of the non-linear $\sigma-\omega$ model are [@serot86; @boguta77] $$\begin{aligned}
(\Delta- m_{\rm s}^2)\phi & = & -g_{\rm s} \rho_{\rm s}
+b\phi^2 +c\phi^3
\label{eqFN4} \\[1.mm]
(\Delta - m_{\rm v}^2) V & = & -g_{\rm v} \rho
\label{eqFN5} \\[1.mm]
(\Delta - m_\rho^2) R & = & - g_\rho \rho_3
\label{eqFN6} \\[1.mm]
\Delta {\cal A} & = & -e \rho_{\rm p} .
\label{eqFN7}\end{aligned}$$ Here $\rho= \rho_{\rm p}+\rho_{\rm n}$ is the baryon density, $\rho_3=
{\textstyle{1\over2}} (\rho_{\rm p}-\rho_{\rm n})$ is the isovector density, and $\rho_{\rm s}$ is the scalar density. The meson fields $\phi$, $V$ and $R$ are associated with the $\sigma$, $\omega$ and $\rho$ mesons, respectively, and $\cal A$ is the Coulomb field. It is understood that the densities and fields are local quantities that depend on position, even if we do not make it explicit. Units are $\hbar= c = 1$.
Taking into account the above field equations, the relativistic energy density of a finite nucleus in Thomas–Fermi approximation can be written as [@serot86; @boguta77] $${\cal H} = {\cal E} +\frac{1}{2}g_{\rm s}\phi \rho^{\rm eff}_{\rm s}
+ \frac{1}{3}b\phi^3+\frac{1}{4} c \phi^4
+\frac{1}{2} g_{\rm v} V \rho +\frac{1}{2} g_\rho R \rho_3
+\frac{1}{2} e {\cal A} \rho_{\rm p} ,
\label{eqFN8c}$$ in terms of the nucleon energy density $${\cal E} =
\sum_{q} \frac{1}{8\pi^2} \left[k_{{\rm F}q}\epsilon^{3}_{{\rm F}q}
+k^{3}_{{\rm F}q}\epsilon_{{\rm F}q}
-{m^*}^{4}\ln\frac{k_{{\rm F}q}+\epsilon_{{\rm F}q}}{m^*}\right]
\label{eqFN2b}$$ and of $g_{\rm s} \rho^{\rm eff}_{\rm s} = g_{\rm s} \rho_{\rm s}
-b \phi^2 - c \phi^3$, where $$\rho_{\rm s} = \frac{\partial {\cal E}}{\partial m^*} =
\sum_{q} \frac{m^*}{2\pi^2}\left[k_{{\rm F}q}\epsilon_{{\rm
F}q}-{m^*}^2 \ln\frac{k_{{\rm F}q}
+\epsilon_{{\rm F}q}} {m^*}\right]
\label{eqFN8}$$ is the scalar density and $m^* = m - g_{\rm s}\phi$ is the nucleon effective mass. For each kind of nucleon ($q= {\rm n}, {\rm p}$) the local Fermi momentum $k_{{\rm F}q}$ is defined by $k_{{\rm F}q}=
(3\pi^2 \rho_{q})^{1/3}$, while $\epsilon_{{\rm F}q}=\sqrt{k^2_{{\rm
F}q}+{m^*}^2}$.
The virial theorem relates the kinetic and potential energy components of the energy in certain circumstances. This theorem results from homogeneity properties of the kinetic and potential energy components of $\cal H$ with respect to a scaling transformation that preserves the normalization. One such normalized scaled version of the baryon density is $$\rho_\lambda ({\mbox{\boldmath$r$}}) = \lambda^3 \rho(\lambda{\mbox{\boldmath$r$}}),
\label{eqFN10}$$ where $\lambda$ is an arbitrary scaling parameter. Accordingly, the local Fermi momentum changes as $$k_{{\rm F}q\lambda} ({\mbox{\boldmath$r$}}) =
[ 3\pi^2\rho_{q\lambda}({\mbox{\boldmath$r$}}) ]^{1/3}=
\lambda k_{{\rm F}q} (\lambda{\mbox{\boldmath$r$}}) .
\label{eqFN11}$$ The meson fields and the Coulomb field are also modified by the scaling due to the self-consistent equations (\[eqFN4\])–(\[eqFN7\]), which will relate the scaled fields to the scaled densities. Unfortunately, the meson fields do not scale according to simple power laws of $\lambda$ because of the finite-range character of the meson interactions. This is most apparent for the scalar field $\phi$, since the scalar density in the source term of Eq. (\[eqFN4\]) transforms not only due to the scaling of $k_{{\rm F}q}$ but also of $\phi$ itself (or $m^*$), see Eq. (\[eqFN8\]) for $\rho_{\rm s}$. For reasons that will become clear immediately, we shall write the scaled effective mass $m^*_\lambda({\mbox{\boldmath$r$}})= m - g_{\rm s}
\phi_\lambda({\mbox{\boldmath$r$}})$ in the form $$m^*_\lambda({\mbox{\boldmath$r$}}) \equiv \lambda {\tilde m}^* (\lambda{\mbox{\boldmath$r$}}) .
\label{eqFN13}$$ The quantity ${\tilde m}^*$ carries an implicit dependence on $\lambda$ apart from the parametric dependence on $\lambda{\mbox{\boldmath$r$}}$.
On account of Eqs. (\[eqFN11\]) and (\[eqFN13\]) the scaled form of $\cal E$ reads ${\cal E}_\lambda({\mbox{\boldmath$r$}}) =
\lambda^4 {\cal E} [ k_{{\rm F}q} (\lambda{\mbox{\boldmath$r$}}),
{\tilde m}^* (\lambda{\mbox{\boldmath$r$}}) ]
\equiv \lambda^4 {\tilde{\cal E}} (\lambda{\mbox{\boldmath$r$}})$, while the scaled scalar density reads $\rho_{\rm s\lambda} ({\mbox{\boldmath$r$}}) =
\lambda^3 \rho_{\rm s} [ k_{{\rm F}q} (\lambda{\mbox{\boldmath$r$}}),
{\tilde m}^* (\lambda{\mbox{\boldmath$r$}}) ]
\equiv \lambda^3 {\tilde\rho}_{\rm s} (\lambda{\mbox{\boldmath$r$}})$. The tilded quantities ${\tilde{\cal E}}$ and ${\tilde\rho}_{\rm s}$ are given by Eqs. (\[eqFN2b\]) and (\[eqFN8\]) after replacing $m^*$ by ${\tilde m}^*$. Note the usefulness of (\[eqFN13\]) to be able to put the transformed densities ${\cal E}_\lambda$ and $\rho_{\rm s\lambda}$ into the above compact form. This way, for the scaled total energy density ${\cal H_\lambda}$ we obtain $${\cal H}_\lambda = \lambda^3 \bigg[ \lambda {\tilde{\cal E}}
+\frac{1}{2}g_{\rm s} \phi_\lambda {\tilde\rho}^{\rm eff}_{\rm s}
+\frac{1}{3} \frac{b}{\lambda^3}\phi_\lambda^3
+\frac{1}{4} \frac{c}{\lambda^3} \phi_\lambda^4
+\frac{1}{2} g_{\rm v} V_\lambda \rho
+\frac{1}{2}g_\rho R_\lambda \rho_3
+\frac{1}{2} e {\cal A}_\lambda\rho_{\rm p} \bigg],
\label{eqFN17}$$ with the definition $g_{\rm s}{\tilde\rho}^{\rm eff}_{\rm s} =
g_{\rm s} {\tilde\rho}_{\rm s}
-b \phi_\lambda^2/\lambda^3 - c \phi_\lambda^3/\lambda^3$.
The scaled energy is stationary for $\lambda=1$ (which leads to the virial theorem): $$\begin{aligned}
0 & = & \left[ \frac{\partial}{\partial \lambda}
\int \frac{d (\lambda {\mbox{\boldmath$r$}})}{\lambda^3}
{\cal H}_\lambda ({\mbox{\boldmath$r$}}) \right]_{\lambda=1}
\nonumber \\[1.mm]
& = &
\int d {\mbox{\boldmath$r$}} \left[ {\tilde{\cal E}}
- {\tilde m}^* {\tilde\rho}_{\rm s}
-\frac{b}{\lambda^4}\phi_\lambda^3
-\frac{3}{4} \frac{c}{\lambda^4}\phi_\lambda^4
-\frac{1}{2}g_{\rm s}{\tilde\rho}^{\rm eff}_{\rm s}
\frac{\partial \phi_\lambda}{\partial \lambda}
+\frac{1}{2}g_{\rm s} \phi_\lambda
\frac{\partial {\tilde\rho}^{\rm eff}_{\rm s}}{\partial \lambda}
\right.
\nonumber \\[1.mm]
& & \left. \mbox{}
+\frac{1}{2}g_{\rm v}\rho\frac{\partial V_\lambda}{\partial \lambda}
+\frac{1}{2}g_\rho \rho_3 \frac{\partial R_\lambda}
{\partial \lambda}+\frac{1}{2}e\rho_{\rm p} \frac{\partial
{\cal A}_\lambda} {\partial \lambda} \right]_{\lambda=1} .
\label{eqFN19}\end{aligned}$$ Here we have used $\partial {\tilde{\cal E}}/\partial \lambda=
{\tilde\rho}_{\rm s} \, \partial {\tilde m}^*/\partial \lambda$ (as ${\tilde\rho}_{\rm s}= \partial {\tilde{\cal E}}/\partial {\tilde
m}^*$) and, from the definition of ${\tilde m}^*$, $$\frac{\partial m^*_\lambda}{\partial \lambda} =
{\tilde m}^* + \lambda \frac{\partial {\tilde m}^*}
{\partial \lambda}=-g_{\rm s} \frac{\partial \phi_\lambda}{\partial
\lambda}.
\label{eqFN20}$$ Let us exemplify the calculation of the derivatives of the scaled fields with respect to $\lambda$ with the omega field $V_\lambda$. It fulfils the scaled Klein–Gordon equation $(
\Delta_{{\mbox{\boldmath$\scriptstyle u$}}} - m_{\rm v}^2/\lambda^2 )
V_\lambda({\mbox{\boldmath$u$}}) = - \lambda g_{\rm v} \rho({\mbox{\boldmath$u$}})$, where we have used Eq. (\[eqFN10\]) for $\rho_\lambda$ and have switched to the coordinate ${\mbox{\boldmath$u$}}= \lambda{\mbox{\boldmath$r$}}$. On differenciating this equation with respect to $\lambda$ we have $$\left( \Delta_{{\mbox{\boldmath$\scriptstyle u$}}}
- \frac{m_{\rm v}^2}{\lambda^2} \right)
\frac{\partial V_\lambda}{\partial\lambda} =
- g_{\rm v} \rho - \frac{2 m_{\rm v}^2}{\lambda^3} V_\lambda .
\label{eqv2}$$ If one now sets $\lambda= 1$ the solution of this equation provides $\partial V_\lambda /\partial\lambda |_{\lambda=1}$. Nevertheless, for our purposes it is more useful to multiply both sides of (\[eqv2\]) by $V_\lambda$, integrate over the space and then use Green’s identity on the left hand side. This way it is straightforward to get $$\frac{1}{2} \int d {\mbox{\boldmath$u$}} \, g_{\rm v} \rho
\frac{\partial V_\lambda}{\partial \lambda}
= \int d {\mbox{\boldmath$u$}} \left[
\frac{1}{2\lambda} g_{\rm v} \rho V_\lambda +
\frac{1}{\lambda^4} m_{\rm v}^2 V_\lambda^2 \right] ,
\label{eqv3}$$ which at $\lambda=1$ is just one of the contributions we need in Eq.(\[eqFN19\]). Analogous results are found for the rho and Coulomb fields (with a zero mass for the latter). In the case of the scalar field additional terms appear due to the fact that the scalar density itself is a function of the scalar field. Following the same steps as above, from the scaled field equation $( \Delta_{{\mbox{\boldmath$\scriptstyle
u$}}} - m_{\rm s}^2/\lambda^2 ) \phi_\lambda({\mbox{\boldmath$u$}}) = - \lambda
g_{\rm s} {\tilde\rho}^{\rm eff}_{\rm s}({\mbox{\boldmath$u$}})$ one easily arrives at $$\left( \Delta_{{\mbox{\boldmath$\scriptstyle u$}}}
- \frac{m_{\rm s}^2}{\lambda^2} \right)
\frac{\partial \phi_\lambda}{\partial\lambda} =
- g_{\rm s} {\tilde\rho}^{\rm eff}_{\rm s} - \lambda g_{\rm s}
\frac{\partial {\tilde\rho}^{\rm eff}_{\rm s}}{\partial \lambda}
- \frac{2 m_{\rm s}^2}{\lambda^3} \phi_\lambda ,
\label{eqv4}$$ whence $$\int d {\mbox{\boldmath$u$}} \left[ -\frac{1}{2} g_{\rm s} {\tilde\rho}^{\rm
eff}_{\rm s} \frac{\partial \phi_\lambda}{\partial \lambda} +
\frac{1}{2} g_{\rm s} \phi_\lambda \frac{\partial
{\tilde\rho}^{\rm eff}_{\rm s}}{\partial \lambda} \right] =
\int d {\mbox{\boldmath$u$}} \left[ - \frac{1}{2\lambda} g_{\rm s}
{\tilde\rho}^{\rm eff}_{\rm s} \phi_\lambda - \frac{1}{\lambda^4}
m_{\rm s}^2 \phi_\lambda^2 \right] .
\label{eqv5}$$
From substitution of Eqs. (\[eqv3\]) and (\[eqv5\]) (and of the corresponding results for the rho and Coulomb fields) into Eq.(\[eqFN19\]) the virial theorem for the non-linear $\sigma-\omega$ model becomes $$\begin{aligned}
0 & = & \int d {\mbox{\boldmath$r$}}
\left[ {\cal E} - m^* \rho_{\rm s}
- \frac{1}{2}g_{\rm s} \phi \rho_{\rm s} - m_{\rm s}^2 \phi^2
- \frac{1}{2}b\phi^3 - \frac{1}{4} c \phi^4
+\frac{1}{2} g_{\rm v} V \rho + m_{\rm v}^2 V^2 \right.
\nonumber \\[1.mm]
& & \left. \mbox{}
+\frac{1}{2} g_\rho R \rho_3 + m_\rho^2 R^2
+\frac{1}{2} e {\cal A} \rho_{\rm p} \right] .
\label{eqFN21d}\end{aligned}$$ Actually, introducing the kinetic energy density $\tau$ we have ${\cal
E} - m^* \rho_{\rm s} = \tau + m \rho- m \rho_{\rm s}$, which makes more obvious the kinetic energy component in the virial theorem. Using Eq. (\[eqFN21d\]) to eliminate $\cal E$ from the expression of the relativistic energy density $\cal H$, the RMF energy of a nucleus takes the remarkably simple form $$\int d {\mbox{\boldmath$r$}} [ {\cal H} - m \rho ] =
\int d {\mbox{\boldmath$r$}} \left[
m (\rho_{\rm s} - \rho) + m_{\rm s}^2 \phi^2
- m_{\rm v}^2 V^2 - m_\rho^2 R^2 + \frac{1}{3} b\phi^3 \right] ,
\label{eqFN21h}$$ where we have subtracted the nucleon rest mass contribution. This expression shows very clearly the relativistic mechanism for nuclear binding. It stems from the cancellation between the scalar and vector potentials and from the difference between the scalar and the baryon density (i.e., from the small components of the wave functions). Equations (\[eqFN21d\]) and (\[eqFN21h\]) are satisfied not only by the Thomas–Fermi solutions, but also by the ground-state densities and meson fields obtained from a quantal Hartree calculation. Of course, the energy stationarity condition against dilation of the RMF problem must be fulfilled by any approximation scheme utilized to solve it.
As a further application of the method we shall use it in calculations of the isoscalar giant monopole resonance (ISGMR). It is customary to write the excitation energy of the ISGMR as $$E_{\rm M} = \sqrt{ \frac{C_{\rm M}}{B_{\rm M}} } ,
\label{eqFN36b}$$ where $C_{\rm M}$ and $B_{\rm M}$ are called, respectively, the restoring force (or incompressibility of the finite nucleus) and the mass parameter of the monopole vibration. To study $E_{\rm M}$ in the RMF the authors of Refs. [@nishizaki87; @zhu91] resorted to a local Lorentz boost and the scaling method. Following these works one has $$C_{\rm M} = \frac{1}{A} \left[ \frac{\partial^2}{\partial \lambda^2}
\int d {\mbox{\boldmath$r$}} {\cal H}_\lambda ({\mbox{\boldmath$r$}}) \right]_{\lambda=1} ,
\label{eqFN36}$$ where the scaling parameter $\lambda$ now plays the role of the collective coordinate of the monopole vibration, and $$B_{\rm M} = \frac{1}{A} \int d{\mbox{\boldmath$r$}} r^2 {\cal H}({\mbox{\boldmath$r$}}) ,
\label{eqFN37}$$ with $A$ being the mass number of the nucleus. The investigations of Refs. [@nishizaki87; @zhu91] were restricted to the linear $\sigma-\omega$ model, either for nuclear matter [@nishizaki87] or for symmetric and uncharged finite nuclei (with the densities solved in Thomas–Fermi approximation) [@zhu91]. It is well known that the surface properties of nuclei cannot be described within the linear model, and hence nor can the overall properties of nuclei.
To compute $C_{\rm M}$ it is easiest to replace the relations (\[eqv3\]) and (\[eqv5\]) into the expression of $\partial [ \int
d {\mbox{\boldmath$r$}} {\cal H}_\lambda ({\mbox{\boldmath$r$}})] / \partial\lambda$ and derive again with respect to $\lambda$. After some algebra we obtain the restoring force as $$\begin{aligned}
C_{\rm M} & = &
\frac{1}{A} \int d {\mbox{\boldmath$r$}} \left[
- m \frac{\partial {\tilde\rho}_{\rm s}}{\partial \lambda}
+ 3 \left( m_{\rm s}^2 \phi^2 + \frac{1}{3} b \phi^3
- m_{\rm v}^2 V^2 - m_\rho^2 R^2 \right) \right.
\nonumber \\[1.mm]
& & \left. \mbox{}
- (2 m_{\rm s}^2 \phi + b\phi^2)
\frac{\partial \phi_\lambda}{\partial\lambda}
+ 2 m_{\rm v}^2 V
\frac{\partial V_\lambda}{\partial\lambda}
+ 2 m_\rho^2 R
\frac{\partial R_\lambda}{\partial\lambda} \right]_{\lambda=1} ,
\label{eqFN30}\end{aligned}$$ where $$\left. \frac{\partial{\tilde\rho}_{\rm s}}
{\partial \lambda} \right|_{\lambda=1} =
\left. \frac{\partial{\tilde\rho}_{\rm s}}{\partial {\tilde m}^*}
\frac{\partial {\tilde m}^*}{\partial \lambda} \right|_{\lambda=1} =
- \frac{\partial \rho_{\rm s}}{\partial m^*}
\left[ m^* + g_{\rm s} \frac{\partial \phi_\lambda}
{\partial \lambda} \right]_{\lambda=1} .
\label{eqFN245}$$ The derivatives of the scaled meson fields with respect to $\lambda$ are computed by solving Eqs. (\[eqv2\]) and (\[eqv4\]) at $\lambda=1$. We have found $C_{\rm M}$ to be positive for all of the (linear and non-linear) parameter sets we have tested in the Thomas–Fermi calculations. A large part of the final value of $C_{\rm
M}$ (usually far more than a half) is due to the contribution of the term $\partial {\tilde\rho}_{\rm s}/\partial \lambda|_{\lambda=1}$.
In Table 1 we display the calculated Thomas–Fermi excitation energies of the ISGMR, together with the empirical estimate $E_{\rm M} \sim
80/A^{1/3}$ MeV [@woude87], for $^{40}$Ca, $^{90}$Zr, $^{116}$Sn, $^{144}$Sm and $^{208}$Pb. Recent experimental data on the centroid energy of the ISGMR are available for these nuclei [@young99]. We have employed the non-linear parameter sets NL-Z2 ($K_{\rm nm}=172$ MeV) [@bender99], NL1 ($K_{\rm nm}=212$ MeV) [@reinhard86], NL3 ($K_{\rm nm}=272$ MeV) [@lalaz97], NL-SH ($K_{\rm nm}=355$ MeV) [@sharma93] and NL2 ($K_{\rm nm}=399$ MeV) [@lee86]. These parameter sets have been determined by least-squares fits to ground-state properties of a few spherical nuclei and are of common use in RMF calculations. From the table one can see that the smaller the mass number, the larger is the monopole energy. The energy of the ISGMR increases with increasing $K_{\rm nm}$ in the various parameter sets. For example, the monopole energy in $^{208}$Pb is 12.3 MeV for NL-Z2 while it is 18.1 MeV for NL2. The dependence on $K_{\rm nm}$ is roughly linear for each nucleus.
In assuming nuclear matter within a certain volume the authors of Ref. [@nishizaki87] estimated the monopole excitation energy of a finite nucleus as $$E_{\rm M} =
\sqrt{ \frac{K_{\rm nm}}{ \langle r^2 \rangle
(\epsilon_{\rm Fnm} + g_{\rm v} V_{\rm nm})} } ,
\label{eqFN55}$$ with $\langle r^2 \rangle= \frac{3}{5} R^2$ and $R= 1.2 A^{1/3}$ fm. They evaluated (\[eqFN55\]) for the linear model of Walecka and found $E_{\rm M}=160/A^{1/3}$ MeV, which has the correct dependence on the mass number but is twice as large as the empirical value. From Eq.(\[eqFN55\]) one finds $E_{\rm M}= 92$, 102, 115, 132 and $140/A^{1/3}$ MeV for the non-linear sets NL-Z2, NL1, NL3, NL-SH and NL2, respectively. Comparing with Table 1, the finite size effects reduce the prediction obtained from nuclear matter by a factor ranging from $\sim 1.4$ in $^{40}$Ca to $\sim 1.3$ in $^{208}$Pb, rather independently of the parameter set.
The ISGMR has been studied in the time-dependent RMF (TDRMF) theory by Vretenar et al. [@vretenar97]. We include in Table 1 their results for the energy of the main peaks that appear in the monopole strength distributions of $^{90}$Zr and $^{208}$Pb. Our scaling results compare very well in the case of $^{208}$Pb for all parameter sets, but give somewhat larger excitation energies for $^{90}$Zr. It should be mentioned that the Fourier spectrum of $^{90}$Zr in the TDRMF calculation is considerably fragmented (specially for the sets with higher $K_{\rm nm}$) and then the determination of the centroid energy is more uncertain [@vretenar97]. Very recently, it has been demonstrated that the relativistic random phase approximation (RRPA) is equivalent to the small amplitude limit of the TDRMF theory in the no-sea approximation, when pairs formed from the empty Dirac sea states and the occupied Fermi sea states are included in the RRPA [@ma01].
Microscopic calculations of ISGMR energies in nuclei are a valuable source of information on the nuclear compression modulus $K_{\rm nm}$ [@blaizot95; @farine97], which is an important ingredient not only for finite nuclei but also for heavy ion collisions, supernovae and neutron stars. A further inspection of Table 1 shows that the empirical law $E_{\rm M} \sim 80/A^{1/3}$ MeV lies between the predictions of the NL1 and NL3 sets, as expected from the reasonable value of $K_{\rm nm}$ in these interactions. On the contrary, $K_{\rm
nm}$ is too high in NL-SH and NL2 and we see that these sets overestimate the empirical curve and the experimental data for all nuclei of Table 1. No RMF parameter set seems capable of reproducing the mass-number dependence of the experimental data over the whole analyzed region, particularly in the lighter nuclei. One should note, however, that our calculation provides a prediction for the mean value or centroid of the excitation energy of the resonance. To establish a link between $K_{\rm nm}$ and the measured energies the most favourable situation is then met in heavy nuclei, where the experimental strength is less fragmented than in medium and light nuclei [@young99]. If we only take into account the data of $^{144}$Sm and $^{208}$Pb, our results of Table 1 suggest that $K_{\rm
nm}$ of a RMF interaction should belong to the range 225–255 MeV. (If we disregard the set NL-Z2, as in Refs. [@vretenar97; @ma01], the range is 230–260 MeV.) From their TDRMF and RRPA calculations the authors of Refs. [@vretenar97; @ma01] conclude that the value of $K_{\rm nm}$ should be close to 250–270 MeV. Non-relativistic Hartree–Fock plus RPA analyses using Skyrme and Gogny interactions determine $K_{\rm nm}$ to be $215\pm15$ MeV [@blaizot95; @farine97], thus lower than in the RMF model.
We have derived the virial theorem for the relativistic nuclear mean field model on the basis of the scaling method and the Thomas–Fermi approximation. In this approach we have calculated for realistic parameter sets of the RMF theory the breathing-mode energy of finite nuclei fully self-consistently (i.e., we did not use a leptodermous expansion of the finite nucleus incompressibility as in some previous studies with the scaling method [@stoitsov94]). The present calculations extend earlier work performed with the linear $\sigma-\omega$ model [@nishizaki87; @zhu91].
The excitation energies of the monopole oscillation turn out to be in good agreement with the outcome of dynamical time-dependent RMF calculations. It has been shown very recently that the relativisitc RPA, with the inclusion of Dirac sea states, amounts to the limit of small amplitude oscillations of the TDRMF theory [@ma01]. From the present Thomas–Fermi analysis one can thus conclude that, similarly to the non-relativistic case, also in the relativistic framework the excitation energies obtained with the scaling method simulate the results of the random phase approximation.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank J. Navarro and Nguyen Van Giai for valuable discussions. Support from the DGICYT (Spain) under grant PB98-1247 and from DGR (Catalonia) under grant 2000SGR-00024 is acknowledged. S.K.P. thanks the Spanish grant SB97-OL174874 for financial support.
[99]{} = -1.5mm B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. [**16**]{} (1986) 1. B.D. Serot and J.D. Walecka, Int. J. of Mod. Phys. [**E6**]{} (1997) 515. J.D. Walecka, Ann. Phys. (N.Y.) [**83**]{}, 491 (1974). J. Boguta and A.R. Bodmer, Nucl. Phys. [ **A292**]{} (1977) 413. G.A. Lalazissis, J. Köning and P. Ring, Phys. Rev. [**C55**]{} (1997) 540. For scaling relations and the virial theorem see, e.g., R.G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989), and references therein. O. Bohigas, A. Lane and J. Martorell, Phys. Rep. [**51**]{} (1979) 267. M. Brack, Phys. Rev. [**D27**]{} (1983) 1950. W. Lucha and F.F. Schöberl, Phys. Rev. Lett. [**64**]{} (1990) 2733. S. Nishizaki, H. Kurasawa and T. Suzuki, Nucl. Phys. [**A462**]{} (1987) 689. Chaoyuan Zhu and Xi-Jun Qiu, J. Phys. [**G17**]{} (1991) L11. A. van der Woude, Prog. Part. Nucl. Phys. [**18**]{} (1987) 217. D.H. Youngblood, H.L. Clark and Y.-W. Lui, Phys. Rev. Lett. [**82**]{} (1999) 691; D.H. Youngblood, Y.-W. Lui and H.L. Clark, Phys. Rev. [**C63**]{} (2001) 067301. M. Bender, K. Rutz, P.-G. Reinhard, J.A. Maruhn and W. Greiner, Phys. Rev. [**C60**]{} (1999) 034304. P.-G. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. [**A323**]{} (1986) 13. M.M. Sharma, M.A. Nagarajan and P. Ring, Phys. Lett. [**B312**]{} (1993) 377. S.J. Lee, J. Fink, A.B. Balantekin, M.R. Strayer, A.S. Umar, P.-G. Reinhard, J.A. Maruhn and W. Greiner, Phys. Rev. Lett. [**57**]{} (1986) 2916. D. Vretenar, G.A. Lalazissis, R. Behnsch, W. Pöschl and P. Ring, Nucl. Phys. [**A621**]{} (1997) 853. Zhong-yu Ma, Nguyen Van Giai, A. Wandelt, D. Vretenar and P. Ring, Nucl. Phys. [**A686**]{} (2001) 173; P. Ring, Zhong-yu Ma, Nguyen Van Giai, D. Vretenar, A. Wandelt and Li-gang Cao, Nucl. Phys. [****]{} (2001), in press. J.P. Blaizot, J.F. Berger, J. Dechargé and M. Girod, Nucl. Phys. [**A591**]{} (1995) 435. M. Farine, J.M. Pearson and F. Tondeur, Nucl. Phys. [**A615**]{} (1997) 135. D. Von-Eiff, J.M. Pearson, W. Stocker and M.K. Weigel, Phys. Rev. [**C50**]{} (1994), 831; M.V. Stoitsov, M.L. Cescato, P. Ring and M.M. Sharma, J. Phys. [**G20**]{} (1994) L149; T. v. Chossy and W Stocker, Phys. Rev. [**C56**]{} (1997) 2518.
NL-Z2 NL1 NL3 NL-SH NL2 $80A^{-1/3}$ Exp.
--------------- -- ------- -- ------ -- ----------- -- ------- -- ------ -- -------------- -- --------------
$^{40}$Ca 20.5 21.2 23.5 26.6 29.5 23.4 $19.2\pm0.4$
$^{90}$Zr 16.4 17.2 19.2 21.9 24.0 17.9 $17.9\pm0.2$
[@vretenar97] 15.7 $\sim 18$
$^{116}$Sn 15.1 15.9 17.7 20.3 22.3 16.4 $16.1\pm0.1$
$^{144}$Sm 14.1 14.9 16.6 19.0 20.8 15.3 $15.4\pm0.3$
$^{208}$Pb 12.3 12.9 14.5 16.6 18.1 13.5 $14.2\pm0.3$
[@vretenar97] 12.4 14.1 16.1 17.8
: Excitation energy of the monopole state (in MeV) obtained in the scaling approach by using various relativistic parameter sets (in order of increasing value of the compression modulus $K_{\rm nm}$). The energies of the main peaks found in the time-dependent RMF calculations of Ref. [@vretenar97] are also shown for $^{90}$Zr and $^{208}$Pb. The experimental centroid energies are from Ref.[@young99].
[^1]: Present address: [*Institute of Physics, Sachivalaya Marg, Bhubaneswar-[*751 005*]{}, India*]{}
|
---
abstract: |
A statistician designing an experiment wants to get as much information as possible from the data gathered. Often this means the most precise estimate possible (that is, an estimate with minimum possible variance) of the unknown parameters. If there are several parameters, this can be interpreted in many ways: do we want to minimize the average variance, or the maximum variance, or the volume of a confidence region for the parameters?
In the case of block designs, these optimality criteria can be calculated from the concurrence graph of the design, and in many cases from its Laplacian eigenvalues. The Levi graph can also be used. The various criteria turn out to be closely connected with other properties of the graph as a network, such as number of spanning trees, isoperimetric number, and the sum of the resistances between pairs of vertices when the graph is regarded as an electrical network.
In this chapter, we discuss the notions of optimality for incomplete-block designs, explain the graph-theoretic connections, and prove some old and new results about optimality.
author:
- 'R. A. Bailey and Peter J. Cameron'
title: Using graphs to find the best block designs
---
R. A. Bailey obtained a DPhil in group theory from the University of Oxford. She worked at the Open University, and then held a post-doctoral research fellowship in Statistics at the University of Edinburgh. This was followed by ten years in the Statistics Department at Rothamsted Experimental Station, which at that time came under the auspices of the Agriculture and Food Research Council. She returned to university life as Professor of Mathematical Sciences at Goldsmiths’ College, University of London, and has been Professor of Statistics at Queen Mary, University of London, since 1994.
Peter J. Cameron is Professor of Mathematics at Queen Mary, University of London, where he has been since 1986, following a position as tutorial fellow at Merton College, Oxford. Since his DPhil in Oxford, he has been interested in a variety of topics in algebra and combinatorics, especially their interactions. He has held visiting positions at the University of Michigan, California Institute of Technology, and the University of Sydney. He is currently chair of the British Combinatorial Committee.
What makes an incomplete-block design good for experiments? {#sec:intro}
===========================================================
Experiments are designed in many ways: for example, Latin squares, block designs, split-plot designs. Combinatorialists, on the other hand, have a much more specialized usage of the term “design”, as we remark later. We are concerned here with incomplete-block designs, more special than the statistician’s designs and more general than the mathematician’s.
To a statistician, a *block design* has two components. There is an underlying set of experimental units, partitioned into $b$ blocks of size $k$. There is a further set of $v$ treatments, and also a function $f$ from units to treatments, specifying which treatment is allocated to which experimental unit; that is, $f(\omega)$ is the treatment allocated to experimental unit $\omega$. Thus each block defines a subset, or maybe a multi-subset, of the treatments.
In a *complete-block design*, we have $k=v$ and each treatment occurs once in every block. Here we assume that blocks are *incomplete* in the sense that $k<v$. We assume that the purpose of the experiment is to find out about the treatments, and differences between them. The blocks are an unavoidable nuisance, an inherent feature of the experimental units. In an agricultural experiment the experimental units may be field plots and the blocks may be fields or plough-lines; in a clinical trial the experimental units may be patients and the blocks hospitals; in process engineering the experimental units may be runs of a machine that is recalibrated each day and the blocks days. See [@rabbook] for further examples.
In all of these situations, the values of $b$, $k$ and $v$ are given. Given these values, not all incomplete-block designs are equally good. This chapter describes some criteria that can be used to choose between them.
For example, Fig. \[fig:queen\] shows two block designs with $v=15$, $b=7$ and $k=3$. We use the convention that the treatments are labelled $1$, …, $v$, that columns represent blocks, and that the order of the entries in each column is not significant. Where necessary, we use the notation $\Gamma_j$ to refer to the block which is shown as the $j$th column, for $j=1$, …, $b$.
[c@c]{} $\begin{array}{|c|c|c|c|c|c|c|}
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6\\
2 & 4 & 5 & 6 & 10 & 11 & 12\\
3 & 7 & 8 & 9 & 13 & 14 & 15\\
\hline
\end{array}
$ & $\begin{array}{|c|c|c|c|c|c|c|}
\hline
1 & 1 & 1 & 1 & 1 & 1 & 1\\
2 & 4 & 6 & 8 & 10 & 12 & 14\\
3 & 5 & 7 & 9 & 11 & 13 & 15\\
\hline
\end{array}
$\
\
(a) & (b)
The *replication* $r_i$ of treatment $i$ is defined to be ${\left|f^{-1}(i)\right|}$, which is the number of experimental units to which it is allocated. For the design in Fig. \[fig:queen\](a), $r_i \in {\left\{1,2\right\}}$ for all $i$. As we see later, statisticians tend to prefer designs in which all the replications are as equal as possible. If $r_i=r_j$ for $1\leq i<j\leq v$ then the design is *equireplicate*: then the common value of $r_i$ is usually written as $r$, and $vr=bk$.
The design in Fig. \[fig:queen\](b) is a *queen-bee design* because there is (at least) one treatment that occurs in every block. Scientists tend to prefer such designs because they have been taught to compare every treatment to one distinguished treatment, which may be called a *control treatment*.
[c@c]{} $\begin{array}{|c|c|c|c|c|c|c|}
\hline
1 & 1 & 1 & 1 & 2 & 2 & 2\\
2 & 3 & 3 & 4 & 3 & 3 & 4\\
3 & 4 & 5 & 5 & 4 & 5 & 5\\
\hline
\end{array}
$ & $\begin{array}{|c|c|c|c|c|c|c|}
\hline
1 & 1 & 1 & 1 & 2 & 2 & 2\\
1 & 3 & 3 & 4 & 3 & 3 & 4\\
2 & 4 & 5 & 5 & 4 & 5 & 5\\
\hline
\end{array}
$\
\
(a) & (b)
Fig. \[fig:binary\] shows two block designs with $v=5$, $b=7$ and $k=3$. The design in Fig. \[fig:binary\](b) shows a new feature: treatment $1$ occurs on two experimental units in block $\Gamma_1$. A block design is *binary* if $f(\alpha)\ne f(\omega)$ whenever $\alpha$ and $\omega$ are experimental units in the same block. The design in Fig. \[fig:binary\](a) is binary. It seems to be obvious that binary designs must be better than non-binary ones, but we shall see later that this is not necessarily so. However, if there is any block on which $f$ is constant, then that block provides no information about treatments, so we assume from now on that there are no such blocks.
[c@c]{} $\begin{array}{|c|c|c|c|c|c|c|}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7\\
2 & 3 & 4 & 5 & 6 & 7 & 1\\
4 & 5 & 6 & 7 & 1 & 2 & 3\\
\hline
\end{array}
$ & $\begin{array}{|c|c|c|c|c|c|c|}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7\\
2 & 3 & 4 & 5 & 6 & 7 & 1\\
3 & 4 & 5 & 6 & 7 & 1 & 2\\
\hline
\end{array}
$\
\
(a) & (b)
Fig. \[fig:fano\] shows two equireplicate binary block designs with $v=7$, $b=7$ and $k=3$. A binary design is *balanced* if every pair of distinct treatments occurs together in the same number of blocks. If that number is $\lambda$, then $r(k-1) = (v-1)\lambda$. Such designs are also called *$2$-designs* or *BIBDs*. The design in Fig. \[fig:fano\](a) is balanced with $\lambda=1$; the design in Fig. \[fig:fano\](b) is not balanced.
Pure mathematicians usually assume that, if they exist, balanced designs are better than non-balanced ones. (Indeed, many do not call a structure a ‘design’ unless it is balanced.) As we shall show in Section \[sec:bibd\], this assumption is correct for all the criteria considered here. However, for given values of $v$ and $k$, a non-balanced design with a larger value of $b$ may produce more information than a balanced design with a smaller value of $b$.
Graphs from block designs
=========================
The Levi graph
--------------
A simple way of representing a block design is its *Levi graph*, or *incidence graph*, introduced in [@levi]. This graph has $v+b$ vertices, one for each block and one for each treatment. There are $bk$ edges, one for each experimental unit. If experimental unit $\omega$ is in block $j$ and $f(\omega) = i$, then the corresponding edge $\tilde e_\omega$ joins vertices $i$ and $j$. Thus the graph is bipartite, with one part consisting of block vertices and the other part consisting of treatment vertices. Moreover, the graph has multiple edges if the design is not binary. Fig. \[fig:levibin\] gives the Levi graph of the design in Fig. \[fig:binary\](b).
(12,3.5)(0,0.5) (0,3)(2,0)[7]{} (2,1)(2,0)[5]{} (0,3.3)[(0,0)\[b\][$\Gamma_1$]{}]{} (2,3.3)[(0,0)\[b\][$\Gamma_2$]{}]{} (4,3.3)[(0,0)\[b\][$\Gamma_3$]{}]{} (6,3.3)[(0,0)\[b\][$\Gamma_4$]{}]{} (8,3.3)[(0,0)\[b\][$\Gamma_5$]{}]{} (10,3.3)[(0,0)\[b\][$\Gamma_6$]{}]{} (12,3.3)[(0,0)\[b\][$\Gamma_7$]{}]{} (2,0.7)[(0,0)\[t\][1]{}]{} (4,0.7)[(0,0)\[t\][2]{}]{} (6,0.7)[(0,0)\[t\][3]{}]{} (8,0.7)[(0,0)\[t\][4]{}]{} (10,0.7)[(0,0)\[t\][5]{}]{} (2,0.9)[(-1,1)[2]{}]{} (2,1.1)[(-1,1)[2]{}]{} (4,1)[(-2,1)[4]{}]{} (2,1)[(0,1)[2]{}]{} (6,1)[(-2,1)[4]{}]{} (8,1)[(-3,1)[6]{}]{} (2,1)[(1,1)[2]{}]{} (6,1)[(-1,1)[2]{}]{} (10,1)[(-3,1)[6]{}]{} (2,1)[(2,1)[4]{}]{} (8,1)[(-1,1)[2]{}]{} (10,1)[(-2,1)[4]{}]{} (4,1)[(2,1)[4]{}]{} (6,1)[(1,1)[2]{}]{} (8,1)[(0,1)[2]{}]{} (4,1)[(3,1)[6]{}]{} (6,1)[(2,1)[4]{}]{} (10,1)[(0,1)[2]{}]{} (4,1)[(4,1)[8]{}]{} (8,1)[(2,1)[4]{}]{} (10,1)[(1,1)[2]{}]{}
We regard two block designs as the same if one can be obtained from the other by permuting the experimental units within each block. Since the vertices of the Levi graph are labelled, there is a bijection between block designs and their Levi graphs.
Let $n_{ij}$ be the number of edges from treatment-vertex $i$ to block-vertex $j$; that is, treatment $i$ occurs on $n_{ij}$ experimental units in block $j$. The $v\times b$ matrix $\mathbf{N}$ whose entries are the $n_{ij}$ is the *incidence matrix* of the block design. If the rows and columns of $\mathbf{N}$ are labelled, we can recover the block design from its incidence matrix.
The concurrence graph
---------------------
In a binary design, the *concurrence* $\lambda_{ij}$ of treatments $i$ and $j$ is $r_i$ if $i=j$ and otherwise is the number of blocks in which $i$ and $j$ both occur. For non-binary designs we have to count the number of occurrences of the pair ${\left\{i,j\right\}}$ in blocks according to multiplicity, so that $\lambda_{ij}$ is the $(i,j)$-entry of $\boldsymbol{\Lambda}$, where $\boldsymbol{\Lambda} = \mathbf{N}
\mathbf{N}^\top$. The matrix $\boldsymbol{\Lambda}$ is called the *concurrence matrix* of the design.
The *concurrence graph* of the design has the treatments as vertices. There are no loops. If $i\ne j$, then there are $\lambda_{ij}$ edges between vertices $i$ and $j$. Each such edge corresponds to a pair ${\left\{\alpha,\omega\right\}}$ of experimental units in the same block, with $f(\alpha)=i$ and $f(\omega)=j$: we denote this edge by $e_{\alpha\omega}$. (This edge does not join the experimental units $\alpha$ and $\omega$; it joins the treatments applied to these units.) It follows that the degree $d_i$ of vertex $i$ is given by $$d_i = \sum_{j\ne i} \lambda_{ij}.
\label{eq:valency}$$ Figs. \[fig:cgqueen\] and \[fig:cgbin\] show the concurrence graphs of the designs in Figs. \[fig:queen\] and \[fig:binary\], respectively.
[c@c]{}
(5,6)(0.8,0) (0.4,4.6) (0.4,4.9)[(0,0)\[b\][13]{}]{} (2.4,4.6) (2.4,4.9)[(0,0)\[b\][7]{}]{} (3.6,4.6) (3.6,4.9)[(0,0)\[b\][8]{}]{} (5.6,4.6) (5.6,4.9)[(0,0)\[b\][11]{}]{} (0.4,3.4) (0.4,3.1)[(0,0)\[t\][10]{}]{} (2.4,3.4) (2.1,3.1)[(0,0)\[rt\][1]{}]{} (3.6,3.4) (3.9,3.1)[(0,0)\[tl\][2]{}]{} (5.6,3.4) (5.6,3.1)[(0,0)\[t\][14]{}]{} (1.2,1.4) (0.9,1.4)[(0,0)\[r\][12]{}]{} (2.4,1.4) (2.6,1.2)[(0,0)\[tl\][6]{}]{} (3.6,1.4) (3.9,1.4)[(0,0)\[l\][9]{}]{} (1.8,0.4) (1.8,0.1)[(0,0)\[t\][15]{}]{} (1.4,4.0) (1.4,4.3)[(0,0)\[b\][4]{}]{} (4.6,4.0) (4.6,4.3)[(0,0)\[b\][5]{}]{} (3.0,2.4) (3.3,2.4)[(0,0)\[l\][3]{}]{} (1.2,1.4)[(1,0)[2.4]{}]{} (1.8,0.4)[(3,5)[1.2]{}]{} (1.8,0.4)[(-3,5)[0.6]{}]{} (2.4,1.4)[(3,5)[1.2]{}]{} (3.6,1.4)[(-3,5)[1.2]{}]{} (2.4,3.4)[(1,0)[1.2]{}]{} (2.4,3.4)[(0,1)[1.2]{}]{} (2.4,3.4)[(-5,3)[2.0]{}]{} (0.4,3.4)[(0,1)[1.2]{}]{} (0.4,3.4)[(5,3)[2.0]{}]{} (5.6,3.4)[(0,1)[1.2]{}]{} (5.6,3.4)[(-5,3)[2.0]{}]{} (3.6,3.4)[(5,3)[2]{}]{} (3.6,3.4)[(0,1)[1.2]{}]{}
&
(5,6)(0,0) (2.5,2.5) (2.5,2.2)[(0,0)\[t\][1]{}]{} (2.5,2.5)[(5,1)[2.5]{}]{} (2.5,2.5)[(3,2)[2.1]{}]{} (2.5,2.5)[(2,3)[1.4]{}]{} (2.5,2.5)[(1,5)[0.5]{}]{} (2.5,2.5)[(-5,1)[2.5]{}]{} (2.5,2.5)[(-3,2)[2.1]{}]{} (2.5,2.5)[(-2,3)[1.4]{}]{} (2.5,2.5)[(-1,5)[0.5]{}]{} (2.5,2.5)[(-3,-2)[2.1]{}]{} (2.5,2.5)[(-2,-3)[1.4]{}]{} (2.5,2.5)[(-5,-1)[2.5]{}]{} (2.5,2.5)[(5,-1)[2.5]{}]{} (2.5,2.5)[(3,-2)[2.1]{}]{} (2.5,2.5)[(2,-3)[1.4]{}]{} (5,3) (5.3,3)[(0,0)\[l\][12]{}]{} (4.6,3.9) (4.9,3.9)[(0,0)\[l\][11]{}]{} (3.9,4.6) (3.9,4.9)[(0,0)\[b\][10]{}]{} (3,5) (3,5.3)[(0,0)\[b\][9]{}]{} (0,3) (-0.3,3)[(0,0)\[r\][5]{}]{} (0.4,3.9) (0.1,3.9)[(0,0)\[r\][6]{}]{} (1.1,4.6) (1.1,4.9)[(0,0)\[b\][7]{}]{} (2,5) (2,5.3)[(0,0)\[b\][8]{}]{} (4.6,1.1) (4.9,1.1)[(0,0)\[l\][14]{}]{} (3.9,0.4) (3.9,0.1)[(0,0)\[t\][15]{}]{} (5,2) (5.3,2)[(0,0)\[l\][13]{}]{} (0,2) (-0.3,2)[(0,0)\[r\][4]{}]{} (0.4,1.1) (0.1,1.1)[(0,0)\[r\][3]{}]{} (1.1,0.4) (1.1,0.1)[(0,0)\[t\][2]{}]{} (3.9,0.4)[(1,1)[0.7]{}]{} (5,2)[(0,1)[1]{}]{} (4.6,3.9)[(-1,1)[0.7]{}]{} (2,5)[(1,0)[1]{}]{} (1.1,0.4)[(-1,1)[0.7]{}]{} (0,2)[(0,1)[1]{}]{} (0.4,3.9)[(1,1)[0.7]{}]{}
\
\
(a) & (b)
[c@c]{}
(4,3.6)(-2,0) (-1,0) (1,0) (-2,2) (2,2) (0,3) (-1,-0.1)[(1,0)[2]{}]{} (-1,0.1)[(1,0)[2]{}]{} (-2,2)[(1,0)[4]{}]{} (-2,2)[(2,1)[2]{}]{} (2,2)[(-2,1)[2]{}]{} (-1.1,0.1)[(1,3)[1]{}]{} (-0.9,-0.1)[(1,3)[1]{}]{} (1.1,0.1)[(-1,3)[1]{}]{} (0.9,-0.1)[(-1,3)[1]{}]{} (-1.1,-0.1)[(-1,2)[1]{}]{} (-0.9,0.1)[(-1,2)[1]{}]{} (1.1,-0.1)[(1,2)[1]{}]{} (0.9,0.1)[(1,2)[1]{}]{} (-2.1,2.1)[(2,1)[2]{}]{} (-1.9,1.9)[(2,1)[2]{}]{} (2.1,2.1)[(-2,1)[2]{}]{} (1.9,1.9)[(-2,1)[2]{}]{} (-1.1,0.1)[(3,2)[3]{}]{} (-0.9,-0.1)[(3,2)[3]{}]{} (1.1,0.1)[(-3,2)[3]{}]{} (0.9,-0.1)[(-3,2)[3]{}]{} (0,3.3)[(0,0)\[b\][3]{}]{} (2.3,2)[(0,0)\[l\][2]{}]{} (1.3,0)[(0,0)\[l\][5]{}]{} (-2.3,2)[(0,0)\[r\][1]{}]{} (-1.3,0)[(0,0)\[r\][4]{}]{}
&
(4,3)(-2,0) (-1,0) (1,0) (-2,2) (2,2) (0,3) (-1,-0.1)[(1,0)[2]{}]{} (-1,0.1)[(1,0)[2]{}]{} (-2,2.1)[(1,0)[4]{}]{} (-2,1.9)[(1,0)[4]{}]{} (-1.1,0.1)[(1,3)[1]{}]{} (-0.9,-0.1)[(1,3)[1]{}]{} (1.1,0.1)[(-1,3)[1]{}]{} (0.9,-0.1)[(-1,3)[1]{}]{} (-1.1,-0.1)[(-1,2)[1]{}]{} (-0.9,0.1)[(-1,2)[1]{}]{} (1.1,-0.1)[(1,2)[1]{}]{} (0.9,0.1)[(1,2)[1]{}]{} (-2.1,2.1)[(2,1)[2]{}]{} (-1.9,1.9)[(2,1)[2]{}]{} (2.1,2.1)[(-2,1)[2]{}]{} (1.9,1.9)[(-2,1)[2]{}]{} (-1.1,0.1)[(3,2)[3]{}]{} (-0.9,-0.1)[(3,2)[3]{}]{} (1.1,0.1)[(-3,2)[3]{}]{} (0.9,-0.1)[(-3,2)[3]{}]{} (0,3.3)[(0,0)\[b\][3]{}]{} (2.3,2)[(0,0)\[l\][2]{}]{} (1.3,0)[(0,0)\[l\][5]{}]{} (-2.3,2)[(0,0)\[r\][1]{}]{} (-1.3,0)[(0,0)\[r\][4]{}]{}
\
\
(a) & (b)
[c@c]{} $\left[
\begin{array}{rrrrr}
8 & -1 & -3 & -2 & -2\\
-1 & 8 & {-3} & -2 & -2\\
-3 & {-3} & 10 & -2 & -2\\
-2 & -2 & -2 & 8 & -2\\
-2 & -2 & -2 & -2 & 8
\end{array}
\right]$ & $\left[
\begin{array}{rrrrr}
8 & {-2} & -2 & -2 & -2\\
{-2} & 8 & {-2} & -2 & -2\\
-2 & {-2} & 8 & -2 & -2\\
-2 & -2 & -2 & 8 & -2\\
-2 & -2 & -2 & -2 & 8
\end{array}
\right]
$\
\
(a) & (b)
If $k=2$, then the concurrence graph is effectively the same as the block design. Although the block design cannot be recovered from the concurrence graph for larger values of $k$, we shall see in Section \[sec:crit\] that the concurrence graphs contain enough information to decide between two block designs on any of the usual statistical criteria. They were introduced as *variety concurrence graphs* in [@hdperw], but are so useful that they may have been considered earlier.
The Laplacian matrix of a graph
-------------------------------
Let $H$ be an arbitrary graph with $n$ vertices: it may have multiple edges, but no loops. The *Laplacian matrix* $\mathbf{L}$ of $H$ is defined to be the square matrix with rows and columns indexed by the vertices of $H$ whose $(i,i)$-entry $L_{ii}$ is the valency of vertex $i$ and whose $(i,j)$-entry $L_{ij}$ is the negative of the number of edges between vertices $i$ and $j$ if $i\ne j$. Then $L_{ii} = \sum_{j\ne i} L_{ij}$ for $1\leq i \leq n$, and so the row sums of $\mathbf{L}$ are all zero. It follows that $\mathbf{L}$ has eigenvalue $0$ on the all-$1$ vector; this is called the *trivial eigenvalue* of $\mathbf{L}$. We show below that the multiplicity of the zero eigenvalue is equal to the number of connected components of $H$. Thus the multiplicity is $1$ if and only if $H$ is connected.
Call the remaining eigenvalues of $\mathbf{L}$ *non-trivial*. They are all non-negative, as we show in the following theorem (see [@bcc09]).
\[thm:ledge\]
1. If $\mathbf{L}$ is a Laplacian matrix, then $\mathbf{L}$ is positive semi-definite.
2. If $\mathbf{L}$ is a Laplacian matrix of order $n$ and $\mathbf{x}$ is any vector in ${\mathbb{R}}^n$, then $$\mathbf{x}^\top \mathbf{L x} = \sum_{\mathrm{edges\ }ij} (x_i - x_j)^2.$$
3. If $\mathbf{L}_1$ and $\mathbf{L}_2$ are the Laplacian matrices of graphs $H_1$ and $H_2$ with the same vertices, and if $H_2$ is obtained from $H_1$ by inserting one extra edge, then $\mathbf{L}_2- \mathbf{L}_1$ is positive semi-definite.
4. If $\mathbf{L}$ is the Laplacian matrix of the graph $H$, then the multiplicity of the zero eigenvalue of $\mathbf{L}$ is equal to the number of connected components of $H$.
Each edge between vertices $i$ and $j$ defines a $v\times v$ matrix whose entries are all $0$ apart from the following submatrix: $$\begin{array}{cc}
& \begin{array}{cc}
\hphantom{-}i & \hphantom{-}j
\end{array}\\
\begin{array}{c}i\\j\end{array}
&
\left[
\begin{array}{rr}
1 & -1\\
-1 & 1\rlap{\qquad .}
\end{array}
\right]
\end{array}
$$ The Laplacian is the sum of these matrices, which are all positive semi-definite. This proves (a), (b) and (c).
From (b), the vector $\mathbf{x}$ is in the null space of the Laplacian if and only if $\mathbf{x}$ takes the same value on both vertices of each edge, which happens if and only if it takes a constant value on each connected component. This proves (d).
Theorem \[thm:ledge\] shows that the smallest non-trivial eigenvalue of a connected graph is positive. This eigenvalue is sometimes called the *algebraic connectivity* of the graph. The statistical importance of this is shown in Section \[sec:crit\].
In Section \[sec:estvar\] we shall need the Moore–Penrose generalized inverse of $\mathbf{L}^-$ of $\mathbf{L}$ (see [@geninv]). Put $\mathbf{P}_0 = n^{-1}\mathbf{J}_n$, where $\mathbf{J}_n$ is the $n\times n$ matrix whose entries are all $1$, so that $\mathbf{P}_0$ is the matrix of orthogonal projection onto the space spanned by the all-$1$ vector. If $H$ is connected then $\mathbf{L}+\mathbf{P}_0$ is invertible, and $$\mathbf{L}^- = \left(\mathbf{L}+\mathbf{P}_0 \right)^{-1} -\mathbf{P}_0,$$ so that $\mathbf{L}\mathbf{L}^- = \mathbf{L}^-\mathbf{L} =
\mathbf{I}_n -\mathbf{P}_0$, where $\mathbf{I}_n$ is the identity matrix of order $n$.
Laplacians of the concurrence and Levi graphs
---------------------------------------------
There is a relationship between the Laplacian matrices of the concurrence and Levi graphs of a block design $\Delta$. Let $\mathbf{N}$ be the incidence matrix of the design, and $\mathbf{R}$ the diagonal matrix (with rows and columns indexed by treatments) whose $(i,i)$ entry is the replication $r_i$ of treatment $i$. If the design is equireplicate, then $\mathbf{R}=r\mathbf{I}_v$, where $r$ is the replication number.
For the remainder of the paper, we will use $\mathbf{L}$ for the Laplacian matrix of the concurrence graph $G$ of $\Delta$, and $\tilde{\mathbf{L}}$ for the Laplacian matrix of the Levi graph $\tilde{G}$ of $\Delta$. Then it is straightforward to show that $$\mathbf{L}=k\mathbf{R}-\mathbf{NN}^\top,\qquad
\tilde{\mathbf{L}}=\left[
\matrix{\mathbf{R} & -\mathbf{N}\cr
-\mathbf{N}^\top & k\mathbf{I}\cr}
\right]
.$$
The Levi graph is connected if and only if the concurrence graph is connected; thus $0$ is a simple eigenvalue of $\tilde\mathbf{L}$ if and only if it is a simple eigenvalue of $\mathbf{L}$, which in turn occurs if and only if all contrasts between treatment parameters are estimable (see Section \[sec:estvar\]). A block design with this property is itself called *connected*: we consider only connected block designs.
In the equireplicate case, the above expressions for $\mathbf{L}$ and $\tilde\mathbf{L}$ give a relationship between their Laplacian eigenvalues, as follows. Let $\mathbf{x}$ be an eigenvector of $\mathbf{L}$ with eigenvalue $\phi\ne rk$. Then, for each of the two solutions $\theta$ of the quadratic equation $$rk-\phi = (r-\theta)(k-\theta),$$ there is a unique vector $\mathbf{z}$ in ${\mathbb{R}}^b$ such that $[\begin{array}{cc}
\mathbf{x}^\top & \mathbf{z}^\top
\end{array}]^\top$ is an eigenvector of $\tilde{\mathbf{L}}$ with eigenvalue $\theta$. Conversely, any eigenvalue $\theta\ne k$ of $\tilde{\mathbf{L}}$ arises in this way.
The Laplacian matrices of the concurrence graphs in Fig. \[fig:cgbin\] are shown in Table \[tab:binary\].
Statistical issues
==================
Estimation and variance {#sec:estvar}
-----------------------
As part of the experiment, we measure the response $Y_\omega$ on each experimental unit $\omega$. If $\omega$ is in block $\Gamma$, then we assume that $$Y_\omega = \tau_{f(\omega)} + \beta_\Gamma + \varepsilon_\omega;
\label{eq:linmod}$$ here, $\tau_i$ is a constant depending on treatment $i$, $\beta_\Gamma$ is a constant depending on block $\Gamma$, and $\varepsilon_\omega$ is a random variable with expectation $0$ and variance $\sigma^2$. Furthermore, if $\alpha\ne\omega$, then $\varepsilon_\alpha$ and $\varepsilon_\omega$ are uncorrelated.
It is clear that we can add a constant to every block parameter, and subtract that constant from every treatment parameter, without changing (\[eq:linmod\]). It is therefore impossible to estimate the individual treatment parameters. However, if the design is connected, then we can estimate all *contrasts* in the treatment parameters: that is, all linear combinations of the form $\sum_i x_i \tau_i$ for which $\sum_i x_i = 0$. In particular, we can estimate all the simple treatment differences $\tau_i - \tau_j$.
An *estimator* is a function of the responses $Y_\omega$, so it is itself a random variable. An estimator of a value is *unbiased* if its expectation is equal to the true value; it is *linear* if it is a linear function of the responses. Amongst linear unbiased estimators, the *best* one (the so-called BLUE), is the one with the least variance. Let $V_{ij}$ be the variance of the BLUE for $\tau_i - \tau_j$.
If all the experimental units form a single block, then the BLUE of ${\tau_1 - \tau_2}$ is just the difference between the average responses for treatments $1$ and $2$. It follows that $$V_{12} =\left(\frac{1}{r_1} + \frac{1}{r_2}\right)\sigma^2.$$ When $v=2$, this variance is minimized (for a given number of experimental units) when $r_1=r_2$. Moreover, if the responses are normally distributed then the length of the $95$% confidence interval for $\tau_1 - \tau_2$ is proportional to $\mathrm{t}(r_1 + r_2 -2,0.975)\sqrt{V_{12}}$, where $\mathrm{t}(d,p)$ is the $100p$-th percentile of the $\mathrm{t}$ distribution on $d$ degrees of freedom. The smaller the confidence interval, the more likely is our estimate to be close to the true value. This length can be made smaller by increasing $r_1 + r_2$, decreasing ${\left|r_1 - r_2\right|}$, or decreasing $\sigma^2$.
However, matters are not so simple when $k<v$ and $v>2$. The following result can be found in any statistical textbook about block designs (see the section on further reading for recommendations).
\[thm:var\] Let $\mathbf{L}$ be the Laplacian matrix of the concurrence graph of a connected block design. If $\sum_i x_i=0$, then the variance of the BLUE of $\sum_ix_i\tau_i$ is equal to $(\mathbf{x}^\top \mathbf{L}^{-} \mathbf{x})k\sigma^2$. In particular, the variance $V_{ij} $ of the BLUE of the simple difference $\tau_i - \tau_j$ is given by $V_{ij} = \left(L_{ii}^- +L_{jj}^{-} -2L_{ij}^{-}\right)k\sigma^2$.
Optimality criteria {#sec:crit}
-------------------
We want all of the $V_{ij}$ to be as small as possible, but this is a multi-dimensional problem if $v>2$. Let $\bar V$ be the average of the variances $V_{ij}$ over all treatments $i$, $j$ with $i\ne j$. Theorem \[thm:var\] shows that, for each fixed $i$, $$\begin{aligned}
\sum_{j\ne i} V_{ij} &=&
\sum_{j\ne i} (L^-_{ii} + L^-_{jj} - 2L^-_{ij}) k\sigma^2\\
& = & [(v-1)L_{ii}^- + ({\mathop{\mathrm{Tr}}}(\mathbf{L}^-) - L^-_{ii}) + 2L_{ii}^-]k\sigma^2\\
& = & [vL^-_{ii} + {\mathop{\mathrm{Tr}}}(\mathbf{L}^{-})]k\sigma^2,\end{aligned}$$ because the row sums and column sums of $L$ are all $0$. It follows that $\bar V =
2k\sigma^2 {\mathop{\mathrm{Tr}}}(\mathbf{L}^-)/(v-1)$.
Let $\theta_1$, …, $\theta_{v-1}$ be the non-trivial eigenvalues of $\mathbf{L}$, now listed according to multiplicity and in non-decreasing order. Then $${\mathop{\mathrm{Tr}}}(\mathbf{L}^-) = \frac{1}{\theta_1} + \cdots + \frac{1}{\theta_{v-1}},$$ and so $$\bar V = 2k\sigma^2 \times
\frac{1}{\mbox{harmonic mean of }\theta_1, \dots, \theta_{v-1}}.$$
A block design is defined to be *A-optimal* (in some given class of designs with the same values of $b$, $k$ and $v$) if it minimizes the value of $\bar V$; here ‘A’ stands for ‘average’. Thus a design is A-optimal if and only if it maximizes the harmonic mean of $\theta_1$, …, $\theta_{v-1}$.
For $v>2$, the generalization of a confidence interval is a confidence ellipsoid centered at the point $(\hat \tau_1, \ldots, \hat \tau_v)$ which gives the estimated value of $(\tau_1, \ldots, \tau_v)$ in the $(v-1)$-dimensional subspace of ${\mathbb{R}}^v$ for which $\sum \tau_i = 0$. A block design is called *D-optimal* if it minimizes the volume of this confidence ellipsoid. Since this volume is proportional to $\sqrt{\det(\mathbf{L}^- + \mathbf{P}_0)}$, a design is D-optimal if and only if it maximizes the geometric mean of $\theta_1$, …, $\theta_{v-1}$. Here ‘D’ stands for ‘determinant’.
Rather than looking at averages, we might consider the worst case. If all the entries in the vector $\mathbf{x}$ are multiplied by a constant $c$, then the variance of the estimator of $\sum x_i\tau_i$ is multiplied by $c^2$. Thus, those contrast vectors $\mathbf{x}$ which give the largest variance relative to their own length are those which maximize $\mathbf{x}^\top \mathbf{L}^- \mathbf{x}/\mathbf{x}^\top \mathbf{x}$; these are precisely the eigenvectors of $\mathbf{L}$ with eigenvalue $\theta_1$. A design is defined to be *E-optimal* if it maximizes the value of $\theta_1$; here ‘E’ stands for ‘extreme’.
More generally, for $p$ in $(0, \infty)$, a design is called *$\Phi_p$-optimal* if it minimizes $$\left(\frac{\sum_{i=1}^{v-1} \theta_i^{-p}}{v-1}\right)^{1/p}.$$ Thus A-optimality corresponds to $p=1$, D-optimality corresponds to the limit as $p\rightarrow 0$, and E-optimality corresponds to the limit as $p\rightarrow \infty$.
Let $\mathbf{L}_1$ and $\mathbf{L}_2$ be the Laplacian matrices of the concurrence graphs of block designs $\Delta_1$ and $\Delta_2$ for $v$ treatments in blocks of size $k$. If $\mathbf{L}_2 -
\mathbf{L}_1$ is positive semi-definite, then $\Delta_2$ is at least as good as $\Delta_1$ on all the $\Phi_p$-criteria. Theorem \[thm:ledge\](c) shows that adding an extra block to a design cannot decrease its performance on any $\Phi_p$-criterion.
There are even more general classes of optimality criteria (see [@harman] and [@ss] for details). Here we concentrate on A-, D- and E-optimality.
Questions and an example
------------------------
A first obvious question to ask is: do these criteria agree with each other?
Our optimality properties are all functions of the concurrence graph. What features of this graph should we look for if we are searching for optimal, or near-optimal, designs? Symmetry? (Nearly) equal degrees? (Nearly) equal numbers of edges between pairs of vertices? Distance-regularity? Large girth (ignoring cycles within a block)? Small numbers of short cycles (ditto)? High connectivity? Non-trivial automorphism group?
Is it more useful to look at the Levi graph rather than the concurrence graph?
\[eg:cube\]
Fig. \[fig:cube\] shows the values of the A- and D-criteria for all equireplicate block designs with $v=8$, $b=12$ and $k=2$: of course, these are just regular graphs with $8$ vertices and degree $3$. The harmonic mean is shown on the $A$-axis, and the geometric mean on the $D$-axis. (Note that this figure includes some designs that were omitted from Figure 3 of [@rabpaper].) The rankings on these two criteria are not exactly the same, but they do agree at the top end, where it matters. The second-best graph on both criteria is the cube; the best is the Möbius ladder, whose vertices are the elements of ${\mathbb{Z}}_8$ and whose edges are ${\left\{i,i+1\right\}}$ and ${\left\{i,i+4\right\}}$ for $i$ in ${\mathbb{Z}}_8$. These two graphs are so close on both criteria that, for practical purposes, they can be regarded as equally good.
(5.4,3.2)(0.4,2.6) (1,3)[(1,0)[4.5]{}]{} (1,3)[(0,1)[2.5]{}]{} [(1,2.9)[(0,1)[0.2]{}]{} (1,2.6)[(0,0.3)[$0.6$]{}]{}]{} [(2,2.9)[(0,1)[0.2]{}]{} (2,2.6)[(0,0.3)[$1.2$]{}]{}]{} [(3,2.9)[(0,1)[0.2]{}]{} (3,2.6)[(0,0.3)[$1.8$]{}]{}]{} [(4,2.9)[(0,1)[0.2]{}]{} (4,2.6)[(0,0.3)[$2.4$]{}]{}]{} [(5,2.9)[(0,1)[0.2]{}]{} (5,2.6)[(0,0.3)[$3.0$]{}]{}]{} [(0.9,3)[(1,0)[0.2]{}]{} (0.4,3)[(0.4,0)\[r\][$1.8$]{}]{}]{} [(0.9,4)[(1,0)[0.2]{}]{} (0.4,4)[(0.4,0)\[r\][$2.4$]{}]{}]{} [(0.9,5)[(1,0)[0.2]{}]{} (0.4,5)[(0.4,0)\[r\][$3.0$]{}]{}]{} (5.7,2.8)[(0,0)[$A$]{}]{} (0.8,5.7)[(0,0)[$D$]{}]{} (4.83,5.25)[(0,0)[$\boldsymbol{\times}$]{}]{} (2.98,4.31)[(0,0)[$\boldsymbol{+}$]{}]{} (3.35,4.66)[(0,0)[$\boldsymbol{+}$]{}]{} (3.32,4.51)[(0,0)[$\boldsymbol{+}$]{}]{} (3.89,4.95)[(0,0)[$\boldsymbol{+}$]{}]{} (4.88,5.26)[(0,0)[$\boldsymbol{\times}$]{}]{} (4.54,5.15)[(0,0)[$\boldsymbol{\times}$]{}]{} (4.70,5.21)[(0,0)[$\boldsymbol{\times}$]{}]{} (1.75,3.92)[(0,0)[$\boldsymbol{\circ}$]{}]{} (3.60,4.81)[(0,0)[$\boldsymbol{+}$]{}]{} (3.43,4.63)[(0,0)[$\boldsymbol{+}$]{}]{} (4.01,4.91)[(0,0)[$\boldsymbol{+}$]{}]{} (2.31,4.30)[(0,0)[$\boldsymbol{\circ}$]{}]{} (4.08,4.95)[(0,0)[$\boldsymbol{+}$]{}]{} (2.30,4.20)[(0,0)[$\boldsymbol{\circ}$]{}]{} (2.44,4.45)[(0,0)[$\boldsymbol{\circ}$]{}]{} (3.71,4.77)[(0,0)[$\boldsymbol{+}$]{}]{} (4.19,4.99)[(0,0)[$\boldsymbol{+}$]{}]{} (4.9,5.4)[(0,0)\[l\][Möbius ladder]{}]{} (4.8,5.1)[(0,0)\[l\][cube]{}]{} (4.18,4.75) (4.3,4.7)[(0,0)\[l\][$K_{2,6}$]{}]{}
The plotting symbols show the edge-connectivity of the graphs: edge-connectivity $3$, $2$, $1$ is shown as $\boldsymbol{\times}$, $\boldsymbol{+}$, $\boldsymbol{\circ}$ respectively. This does suggest that the higher the edge-connectivity the better is the design on the A- and D- criteria. This is intuitively reasonable: if $k=2$, then the edge-connectivity is the minimum number of blocks whose removal disconnects the design. In this context, it has been called *breakdown number*: see [@mahbub].
The four graphs with edge-connectivity $3$ have no double edges, so concurrences differ by at most $1$. The only other regular graph with no double edges is ranked eighth (amongst regular graphs) by the A-criterion. This suggests that (near-)equality of concurrences is not sufficient to give a good design.
The symbol shows the non-regular graph $K_{2,6}$, which also has eight vertices and twelve edges. It is not as good as the regular graphs with edge-connectivity $3$, but it beats many of the other regular graphs.
This pattern is typical of the block designs investigated by statisticians for most of the 20th century. The A- and D-criteria agree closely at the top end. High edge-connectivity appears to show good designs. Many of the best designs have a high degree of symmetry.
Highly patterned block designs
==============================
Balanced incomplete-block designs {#sec:bibd}
---------------------------------
BIBDs are intuitively appealing, as they seem to give equal weight to all treatment comparisons. They were introduced for agricultural experiments by Yates in [@fy:ibd]. In [@ksh], Kshirsagar proved that, if there exists a BIBD for given values of $v$, $b$ and $k$, then it is A-optimal. Kiefer generalized this in [@kiefer] to cover $\Phi_p$-optimality for all $p$ in $(0,\infty)$, including the limiting cases of D- and E-optimality. The core of Kiefer’s proof is as follows: binary designs maximize ${\mathop{\mathrm{Tr}}}(\mathbf{L})$, which is equal to $\sum_{i=1}^{v-1} \theta_i$; for any fixed value $T$ of this sum of positive numbers, $\sum \theta_i^{-p}$ is minimized at $[T/(v-1)]^{-p}$ when $\theta_1 = \cdots = \theta_{v-1} = T/(v-1)$; and $T^{-p}$ is minimized when $T$ is maximized.
Other special designs
---------------------
Of course, it frequently occurs that the values of $b,v,k$ available for an experiment are such that no BIBD exists. (Necessary conditions for the existence of a BIBD include the well-known divisibility conditions $v\mid bk$ and $v(v-1)\mid bk(k-1)$, which follow from the elementary results in Section \[sec:intro\], and *Fisher’s inequality* asserting that $b\ge v$.)
In the absence of a BIBD, various other special types of design have been considered, and some of these have been proved optimal. Here is a short sample.
A design is *group-divisible* if the treatments can be partitioned into “groups” all of the same size, so that the number of blocks containing two treatments is $\lambda_1$ if they belong to the same group and $\lambda_2$ otherwise. Chêng [@cheng1; @cheng2] showed that if there is a group-divisible design with two groups and $\lambda_2=\lambda_1+1$ in the class of designs with given $v$, $b$, $k$, then it is $\Phi_p$-optimal for all $p$, and in particular is A-, D- and E-optimal.
A *regular-graph design* is a binary equireplicate design with two possible concurrences $\lambda$ and $\lambda+1$. It is easily proved that, in such a design, the number of treatments lying in $\lambda+1$ blocks with a given treatment is constant; so the graph $H$ whose vertices are the treatments, two vertices joined if they lie in $\lambda+1$ blocks, is regular.
Now Chêng [@cheng2] showed that a group-divisible design with $\lambda_2 = \lambda_1 + 1 = 1$, if one exists, is $\Phi_p$-optimal in the class of regular-graph designs for all $p$. Cheng and Bailey [@cheng_bailey] showed that a regular-graph design for which the graph is *strongly regular* (see [@pjc:sr]) and which has singular concurrence matrix is $\Phi_p$-optimal, for all $p$, among binary equireplicate designs with given $v$, $b$, $k$.
Designs with the property described here are particular examples of *partially balanced designs* with respect to an association scheme: see Bailey [@rab:as].
Another class which has turns out to be optimal in many cases, but whose definition is less combinatorial, consists of the *variance-balanced designs*, which we consider later in the chapter.
Graph concepts linked to D-optimality {#sec:span}
=====================================
Spanning trees of the concurrence graph
---------------------------------------
Let $G$ be the concurrence graph of a connected block design, and let $\mathbf{L}$ be its Laplacian matrix. A *spanning tree* for $G$ is a spanning subgraph which is a tree. Kirchhoff’s famous Matrix-tree theorem in [@old] states the following:
\[thm:span\] If $G$ is a connected graph with $v$ vertices and Laplacian matrix $\mathbf{L}$, then the product of the non-trivial eigenvalues of $\mathbf{L}$ is equal to $v$ multiplied by the number of spanning trees for $G$.
Thus we have a test for D-optimality:
> *A design is D-optimal if and only if its concurrence graph has the maximal number of spanning trees.*
Note that Theorem \[thm:span\] gives an easy proof of Cayley’s theorem on the number of spanning trees for the complete graph $K_v$. The non-trivial eigenvalues of its Laplacian matrix are all equal to $v$, so Theorem \[thm:span\] shows that it has $v^{v-2}$ spanning trees.
If $G$ is sparse, it may be much easier to count the number of spanning trees than to compute the eigenvalues of $\mathbf{L}$. For example, if $G$ has a single cycle, which has length $s$, then the number of spanning trees is $s$, irrespective of the remaining edges in $G$.
In the context of optimal block designs, Gaffke discovered the importance of Kirchhoff’s theorem in [@gaff]. Cheng followed this up in papers such as [@cheng1; @cheng2; @cheng3]. Particularly intriguing is the following theorem from [@cmw].
Consider block designs with $k=2$ (connected graphs). For each given $v$ there is a threshold $b_0$ such that if $b\geq b_0$ then any D-optimal design for $v$ treatments in $b$ blocks of size $2$ is *nearly balanced* in the sense that
- no pair of replications differ by more than $1$;
- for each fixed $i$, no pair of concurrences $\lambda_{ij}$ differ by more than $1$.
In fact, there is no known example with $b_0 >v-1$, which is the minimal number of blocks required for connectivity.
Spanning trees of the Levi graph
--------------------------------
In [@gafLevi] Gaffke stated the following relationship between the numbers of spanning trees in the concurrence graph and the Levi graph.
\[thm:gaff\] Let $G$ and $\tilde G$ be the concurrence graph and Levi graph for a connected incomplete-block design for $v$ treatments in $b$ blocks of size $k$. Then the number of spanning trees for $\tilde G$ is equal to $k^{b-v+1}$ times the number of spanning trees for $G$.
Thus, an alternative test for D-optimality is to count the number of spanning trees in the Levi graph. For binary designs, the Levi graph has fewer edges than the concurrence graph if and only if $k\geq 4$.
Graph concepts linked to A-optimality
=====================================
The concurrence graph as an electrical network {#sec:concelec}
----------------------------------------------
We can consider the concurrence graph $G$ as an electrical network with a 1-ohm resistance in each edge. Connect a $1$-volt battery between vertices $i$ and $j$. Then current flows in the network, according to these rules.
Ohm’s Law:
: In every edge, the voltage drop is the product of the current and the resistance.
Kirchhoff’s Voltage Law:
: The total voltage drop from one vertex to any other vertex is the same no matter which path we take from one to the other.
Kirchhoff’s Current Law:
: At each vertex which is not connected to the battery, the total current coming in is equal to the total current going out.
We find the total current from $i$ to $j$, and then use Ohm’s Law to define the effective resistance $R_{ij}$ between $i$ and $j$ as the reciprocal of this current. It is a standard result of electrical network theory that the linear equations implicitly defined above for the currents and voltage differences have a unique solution.
Let $\mathcal{T}$ be the set of treatments and $\Omega$ the set of experimental units. Current flows in each edge $e_{\alpha\omega}$, where $\alpha$ and $\omega$ are experimental units in the same block which receive different treatments; let $I(\alpha,\omega)$ be the current from $f(\alpha)$ to $f(\omega)$ in this edge. Thus $I$ is a function $I \colon \Omega \times \Omega \mapsto {\mathbb{R}}$ such that
1. $I(\alpha,\omega) = 0$ if $\alpha=\omega$ or if $f(\alpha) =
f(\omega)$ or if $\alpha$ and $\omega$ are in different blocks.
2. $I(\alpha,\omega) = -I(\omega,\alpha)$ for $(\alpha,\omega)$ in $\Omega\times \Omega$.
This defines a further function $I_\mathrm{out} \colon \mathcal{T} \mapsto {\mathbb{R}}$ by $$I_\mathrm{out} (l) = \sum_{\alpha: f(\alpha)=l} \ \sum_{\omega\in\Omega}
I(\alpha,\omega) \quad \mbox{for $l$ in $\mathcal{T}$}.$$ Voltage is another function $V\colon \mathcal{T} \mapsto {\mathbb{R}}$. The following two conditions ensure that Ohm’s and Kirchhoff’s Laws are satisfied.
1. If there is any edge in $G$ between $f(\alpha)$ and $f(\omega)$, then $$I(\alpha,\omega) = V(f(\alpha)) - V(f(\omega)).$$
2. If $l\notin{\left\{i,j\right\}}$, then $I_{\mathrm{out}}(l)=0$.
If $G$ is connected and different voltages $V(i)$ and $V(j)$ are given for a pair of distinct treatments $i$ and $j$, then there are unique functions $I$ and $V$ satisfying conditions (a)–(d). Moreover, $I_\mathrm{out}(j)
= -I_{\mathrm{out}} (i) \ne 0$. Then $R_{ij}$ is defined by $$R_{ij} = \frac{V(i) - V(j)}{I_{\mathrm{out}}(i)}.$$ It can be shown that the value of $R_{ij}$ does not depend on the choice of values for $V(i)$ and $V(j)$, so long as these are different. In practical examples, it is usually convenient to take $V(i)=0$ and let $I$ take integer values.
What has all of this got to do with block designs? The following theorem, which is a standard result from electrical engineering, gives the answer.
\[thm:resist\] If $\mathbf{L}$ is the Laplacian matrix of a connected graph $G$, then the effective resistance $R_{ij}$ between vertices $i$ and $j$ is given by $$R_{ij} = \left(L_{ii}^- +L_{jj}^{-} -2L_{ij}^{-}\right).$$
Comparing this with Theorem \[thm:var\], we see that $V_{ij} = R_{ij} \times k\sigma^2$. Hence we have a test for A-optimality:
> *A design is A-optimal if and only if its concurrence graph, regarded as an electrical network, minimizes the sum of the pairwise effective resistances between all pairs of vertices.*
Effective resistances are easy to calculate without matrix inversion if the graph is sparse.
(15,14)(-6,-7) (0,6.67) (0,-6.67) (-6,-3.33) (-2,-3.33) (6,-3.33) (2,-3.33) (-6,3.33) (-2,3.33) (6,3.33) (2,3.33) (-4,0) (4,0) (-6,-3.33)[(1,0)[12]{}]{} (-6,3.33)[(1,0)[12]{}]{} (-6,-3.33)[(3,5)[6]{}]{} (6,-3.33)[(-3,5)[6]{}]{} (-6,3.33)[(3,-5)[6]{}]{} (6,3.33)[(-3,-5)[6]{}]{} (0,-3.33)[(1,0)[0]{}]{} (0,-2.9)[(0,0)[14]{}]{} (4,-3.33)[(1,0)[0]{}]{} (4,-2.9)[(0,0)[7]{}]{} (-4,-3.33)[(-1,0)[0]{}]{} (-4,-2.9)[(0,0)[5]{}]{} (-1,-5)[(4,-3)[0]{}]{} (-1.4,-5.2)[(0,0)[7]{}]{} (-3,-1.67)[(-4,3)[0]{}]{} (-2.6,-1.4)[(0,0)[10]{}]{} (-5,1.67)[(-4,3)[0]{}]{} (-5.4,1.7)[(0,0)[5]{}]{} (1,-5)[(4,3)[0]{}]{} (1.4,-5.2)[(0,0)[7]{}]{} (3,-1.67)[(4,3)[0]{}]{} (2.6,-1.4)[(0,0)[14]{}]{} (5,1.67)[(4,3)[0]{}]{} (5.4,1.7)[(0,0)[19]{}]{} (-5,-1.67)[(4,3)[0]{}]{} (-5.4,-1.4)[(0,0)[5]{}]{} (-3,1.67)[(4,3)[0]{}]{} (-2.6,1.4)[(0,0)[10]{}]{} (-1,5)[(4,3)[0]{}]{} (-1.4,5.2)[(0,0)[5]{}]{} (0,3.33)[(1,0)[0]{}]{} (0,2.9)[(0,0)[10]{}]{} (4,3.33)[(1,0)[0]{}]{} (4,2.9)[(0,0)[17]{}]{} (-4,3.33)[(1,0)[0]{}]{} (-4,2.9)[(0,0)[5]{}]{} (5,-1.67)[(-4,3)[0]{}]{} (5.4,-1.4)[(0,0)[7]{}]{} (3,1.67)[(-4,3)[0]{}]{} (2.6,1.4)[(0,0)[2]{}]{} (1,5)[(4,-3)[0]{}]{} (1.4,5.2)[(0,0)[5]{}]{} (-1.7,-2.83)[(0,0)[$i$]{}]{} (-2.6,-4)[(0,0)[$[0]$]{}]{} (2.6,-4)[(0,0)[$[-14]$]{}]{} (-2.6,4)[(0,0)[$[-20]$]{}]{} (2.6,4)[(0,0)[$[-30]$]{}]{} (-5.1,0)[(0,0)[$[-10]$]{}]{} (5.1,0)[(0,0)[$[-28]$]{}]{} (5.4,2.9)[(0,0)[$j$]{}]{} (6.6,4)[(0,0)[$[-47]$]{}]{} (0,-7.33)[(0,0)[$[-7]$]{}]{} (0,7.33)[(0,0)[$[-25]$]{}]{} (-6.6,-4)[(0,0)[$[-5]$]{}]{} (-6.6,4)[(0,0)[$[-15]$]{}]{} (6.6,-4)[(0,0)[$[-21]$]{}]{}
Figure \[fig:star\] shows the concurrence graph of a block design with $v=12$, $b=6$ and $k=3$. Only vertices $i$ and $j$ are labelled. Otherwise, numbers beside arrows denote current and numbers in square brackets denote voltage. It is straightforward to check that conditions (a)–(d) are satisfied. Now $V(i) - V(j) = 47$ and $I_{\mathrm{out}}(i) = 36$, and so $R_{ij} = 47/36$. Therefore $V_{ij} = (47/12)\sigma^2$. Moreover, for graphs consisting of $b$ triangles arranged in a cycle like this, it is clear that average effective resistance, and hence the average pairwise variance, can be calculated as a function of $b$.
The Levi graph as an electrical network {#sec:levielec}
---------------------------------------
The Levi graph $\tilde G$ of a block design can also be considered as an electrical network. Denote by $\mathcal{B}$ the set of blocks. Now current is defined on the ordered edges of the Levi graph. Recall that, if $\omega$ is an experimental unit in block $\Gamma$, then the edge $\tilde{e}_{\omega}$ joins $\Gamma$ to $f(\omega)$. Thus current is defined on $(\Omega \times \mathcal{B}) \cup (\mathcal{B} \times \Omega)$ and voltage is defined on $\mathcal{T} \cup \mathcal{B}$. Conditions (a)–(d) in Section \[sec:concelec\] need to be modified appropriately.
The next theorem shows that a current–voltage pair $(I,V)$ on the concurrence graph $G$ can be transformed into a current–voltage pair $(\tilde I, \tilde V)$ on the Levi graph $\tilde G$. In $\tilde G$, the current $\tilde I(\alpha,\Gamma)$ flows in edge $\tilde{e}_\alpha$ from vertex $f(\alpha)$ to vertex $\Gamma$, where $\alpha\in\Gamma$. Hence the pairwise variance $V_{ij}$ can also be calculated from the effective resistance $\tilde R_{ij}$ in the Levi graph.
\[thm:tjur\] Let $G$ be the concurrence graph and $\tilde G$ be the Levi graph of a connected block design with block size $k$. If $i$ and $j$ are two distinct treatments, let $R_{ij}$ and $\tilde R_{ij}$ be the effective resistance between vertices $i$ and $j$ in the electrical networks defined by $G$ and $\tilde G$, respectively. Then $\tilde R_{ij} = kR_{ij}$, and so $V_{ij} = \tilde R_{ij} \sigma^2$.
Let $(I,V)$ be a current–voltage pair on $G$. For $(\alpha,\Gamma) \in \Omega \times \mathcal{B}$, put $$\tilde I(\alpha,\Gamma) = -\tilde I(\Gamma,\alpha) =
\sum_{\omega\in\Gamma} I(\alpha,\omega)$$ if $\alpha\in\Gamma$; otherwise, put $\tilde I(\alpha,\Gamma) = \tilde I(\Gamma,\alpha) = 0$. Put $\tilde V(i) = kV(i)$ for all $i$ in $\mathcal{T}$, and $$\tilde V(\Gamma) = \sum_{\omega\in\Gamma} V(f(\omega))$$ for all $\Gamma$ in $\mathcal{B}$. It is clear that $\tilde I$ satisfies the analogues of conditions (a) and (b).
If $\alpha\in\Gamma$, then $$\begin{aligned}
\tilde I(\alpha,\Gamma) &=&
\sum_{\omega\in\Gamma} I(\alpha,\omega)\\
& = & \sum_{\omega\in\Gamma}[V(f(\alpha)) - V(f(\omega))]\\
& = & kV(f(\alpha)) - \tilde V(\Gamma)
= \tilde V(f(\alpha)) - \tilde V(\Gamma),\end{aligned}$$ so the analogue of condition (c) is satisfied.
If $\Gamma\in\mathcal{B}$, then $$\tilde I_{\mathrm{out}}(\Gamma) =
\sum_{\alpha\in\Gamma} \tilde I(\Gamma,\alpha)
= -\sum_{\alpha\in\Gamma} \sum_{\omega\in\Gamma} I(\alpha,\omega) = 0,$$ because $I(\alpha,\alpha)=0$ and $I(\alpha,\omega) = -I(\omega,\alpha)$. If $l\in \mathcal{T}$ then $$\tilde I_{\mathrm{out}}(l) =
\sum_{\alpha: f(\alpha)=l} \ \sum_{\Gamma\in\mathcal{B}}
\tilde I(\alpha,\Gamma) =
\sum_{\alpha: f(\alpha)=l}\ \sum_{\omega\in\Omega} I(\alpha,\omega) =
I_{\mathrm{out}}(l).$$ In particular, $\tilde I_{\mathrm{out}}(l) = 0$ if $l\notin{\left\{i,j\right\}}$, which shows that the analogue of condition (d) is satisfied. It follows that $(\tilde I,\tilde V)$ is the current–voltage pair on $\tilde G$ defined by $\tilde V(i)$ and $\tilde V(j)$.
Now $$\tilde R_{ij} = \frac{\tilde V(i) - \tilde V(j)}{\tilde I_{\mathrm{out}}(i)}
= \frac{k(V(i)-V(j))}{I_{\mathrm{out}}(i)} = kR_{ij}.$$ Then Theorems \[thm:var\] and \[thm:resist\] show that $V_{ij} = R_{ij}\sigma^2$.
When $k=2$ it seems to be easier to use the concurrence graph than the Levi graph, because it has fewer vertices, but for larger values of $k$ the Levi graph may be better, as it does not have all the within-block cycles that the concurrence graph has. Fig. \[fig:levistar\] gives the Levi graph of the block design whose concurrence graph is in Fig. \[fig:star\], with the same two vertices $i$ and $j$ attached to the battery. This gives $\tilde R_{ij} = 47/12$, which is in accordance with Theorem \[thm:tjur\].
(10,6)(-0.5,0) (0,1)(2,0)[6]{} (0,3)(2,0)[6]{} (0,5)(2,0)[6]{} (-0.5,1)[(0,0)\[r\][treatments]{}]{} (-0.5,3)[(0,0)\[r\][blocks]{}]{} (-0.5,5)[(0,0)\[r\][treatments]{}]{} (0,1)(2,0)[6]{}[(0,1)[4]{}]{} (0,1)[(5,1)[10]{}]{} (2,1)(2,0)[5]{}[(-1,1)[2]{}]{} (2,2.1)(2,0)[3]{}[(0,1)[0]{}]{} (6,4.4)[(0,1)[0]{}]{} (0,1.9)[(0,-1)[0]{}]{} (8,1.9)[(0,-1)[0]{}]{} (10,1.9)[(0,-1)[0]{}]{} (3.1,1.9)[(1,-1)[0]{}]{} (4.7,2.3)[(1,-1)[0]{}]{} (0.9,2.1)[(-1,1)[0]{}]{} (6.9,2.1)[(-1,1)[0]{}]{} (8.9,2.1)[(-1,1)[0]{}]{} (4.7,1.95)[(4,1)[0]{}]{} (0,0.7)[(0,0)\[t\][$[-10]$]{}]{} (2,0.7)[(0,0)\[t\][$[0]$]{}]{} (2,0)[(0,0)\[t\][$i$]{}]{} (4,0.7)[(0,0)\[t\][$[-14]$]{}]{} (6,0.7)[(0,0)\[t\][$[-28]$]{}]{} (8,0.7)[(0,0)\[t\][$[-30]$]{}]{} (10,0.7)[(0,0)\[t\][$[-20]$]{}]{} (0.7,3.3)[(0,0)[$[-5]$]{}]{} (2.7,3.3)[(0,0)[$[-7]$]{}]{} (4.7,3.3)[(0,0)[$[-21]$]{}]{} (6.7,3.3)[(0,0)[$[-35]$]{}]{} (8.7,3.3)[(0,0)[$[-25]$]{}]{} (10.7,3.3)[(0,0)[$[-15]$]{}]{} (0,5.3)[(0,0)\[b\][$[-5]$]{}]{} (2,5.3)[(0,0)\[b\][$[-7]$]{}]{} (4,5.3)[(0,0)\[b\][$[-21]$]{}]{} (6,5.3)[(0,0)\[b\][$[-47]$]{}]{} (6,6)[(0,0)\[b\][$j$]{}]{} (8,5.3)[(0,0)\[b\][$[-25]$]{}]{} (10,5.3)[(0,0)\[b\][$[-15]$]{}]{} (-0.3,2)[(0,0)[5]{}]{} (0.7,2)[(0,0)[5]{}]{} (1.7,2)[(0,0)[7]{}]{} (2.7,2)[(0,0)[7]{}]{} (3.7,2)[(0,0)[7]{}]{} (4.6,1.6)[(0,0)[5]{}]{} (4.8,2.5)[(0,0)[7]{}]{} (6.3,2)[(0,0)[7]{}]{} (7.3,2)[(0,0)[5]{}]{} (8.3,2)[(0,0)[5]{}]{} (9.3,2)[(0,0)[5]{}]{} (10.3,2)[(0,0)[5]{}]{} (5.5,4.3)[(0,0)[12]{}]{}
Here is another way of visualizing Theorem \[thm:tjur\]. From the block design we construct a graph $G_0$ with vertex-set $\mathcal{T} \cup \Omega
\cup\mathcal{B}$: the edges are ${\left\{\alpha,\Gamma\right\}}$ for $\alpha\in\Gamma
\in \mathcal{B}$ and ${\left\{\alpha,f(\alpha)\right\}}$ for $\alpha\in\Omega$. Let $(I_0,V_0)$ be a current–voltage pair on $G_0$ for which both battery vertices are in $\mathcal{T}$. We obtain the Levi graph $\tilde G$ from $G_0$ by becoming blind to the vertices in $\Omega$. Thus the resistance in each edge of $\tilde G$ is twice that in each edge in $G_0$, so this step multiplies each effective resistance by $2$.
Because none of the battery vertices is in $\mathcal{B}$, we can now obtain $G$ from $\tilde G$ by replacing each path of the form $(i,\Gamma,j)$ by an edge $(i,j)$. There is no harm in scaling all the voltages by the same amount, so we can obtain $(I,V)$ on $G$ from $(\tilde I, \tilde V)$ on $\tilde G$ by putting $V(i) = \tilde V(i)/k$ for $i$ in $\mathcal{T}$, and $I(\alpha,\omega) = V(f(\alpha)) - V(f(\omega))$ for $\alpha$, $\omega$ in the same block. If $\Gamma$ is a block, then $$0 = \sum_{\alpha\in\Gamma} \tilde I(\alpha,\Gamma)
= \sum_{\alpha\in\Gamma}[\tilde V(f(\alpha)) - \tilde V(\Gamma)]
= k\sum_{\alpha\in\Gamma} V(f(\alpha)) - k\tilde V(\Gamma),$$ and so $\tilde V(\Gamma) = \sum_{\alpha\in\Gamma} V(f(\alpha))$. Also, if $\alpha\in\Gamma$, then $$\begin{aligned}
\sum_{\omega\in\Gamma} I(\alpha,\omega)& =&
\sum_{\omega\in\Gamma}[V(f(\alpha)) - V(f(\omega))]\\
&=& kV(f(\alpha)) - \sum_{\omega\in\Gamma}V(f(\omega)) \\
&=& \tilde V(f(\alpha)) - \tilde V (\Gamma)\\& =& \tilde I(\alpha,\Gamma).\end{aligned}$$ Therefore, this transformation reverses the one used in the proof of Theorem \[thm:tjur\].
There is yet another way of obtaining Theorem \[thm:tjur\]. If we use the responses $Y_\omega$ to estimate the block parameters $\beta_\Gamma$ in (\[eq:linmod\]) as well as the treatment parameters $\tau_i$, then standard theory of linear models shows that, if the design is connected, then we can estimate linear combinations of the form $\sum_{i=1}^v x_i \tau_i + \sum_{j=1}^b z_j\beta_j$ so long as $\sum x_i = \sum z_j$. Moreover, the variance of the BLUE of this linear combination is $$[\begin{array}{cc}\mathbf{x}^\top & \mathbf{z}^\top\end{array}] \mathbf{C}^-
\left[
\begin{array}{c}\mathbf{x} \\ \mathbf{z}\end{array}
\right]
\sigma^2,
\qquad\mbox{where} \qquad
\mathbf{C} =
\left[
\begin{array}{lc}
\mathbf{R} & \mathbf{N}\\
\mathbf{N}^\top & k\mathbf{I}_b
\end{array}
\right]$$ and $\mathbf{R}$ is the diagonal matrix of replications.
If we reparametrize equation (\[eq:linmod\]) by replacing $\beta_j$ by $-\gamma_j$ for $j=1$, …, $b$, then the estimable quantities are the contrasts in $\tau_1$, …, $\tau_v$, $\gamma_1$, …, $\gamma_b$. The so-called *information matrix* $\mathbf{C}$ must be modified by multiplying the last $b$ rows and the last $b$ columns by $-1$: this gives precisely the Laplacian $\tilde \mathbf{L}$ of the Levi graph $\tilde G$. Just as for $\mathbf{L}$, but unlike $\mathbf{C}$, the null space is spanned by the all-$1$ vector.
Spanning thickets
-----------------
We have seen that the value of the D-criterion is a function of the number of spanning trees of the concurrence graph $G$. It turns out that the closely related notion of a spanning thicket enables us to calculate the A-criterion; more precisely, the value of each pairwise effective resistance in $G$.
A *spanning thicket* for the graph is a spanning subgraph that consists of two trees (one of them may be an isolated vertex).
\[thm:thick\] If $i$ and $j$ are distinct vertices of $G$ then $$R_{ij} =
\frac{\mbox{\normalfont{number of spanning thickets with $i$, $j$ in different
parts}}}{\mbox{\normalfont{number of spanning trees}}}
\ .$$
This is also rather easy to calculate directly when the graph is sparse.
Summing all the $R_{ij}$ and using Theorem \[thm:thick\] gives the following result from [@shap].
If $F$ is a spanning thicket for the concurrence graph $G$, denote by $F_1$ and $F_2$ the sets of vertices in its two trees. Then $$\sum_{i< j} R_{ij} =\frac{\displaystyle
\sum_{\mathrm{spanning\ thickets}\ F} \left| F_1 \right|
\left| F_2\right|}{\mbox{\normalfont{number of spanning trees}}}$$
Random walks and electrical networks
------------------------------------
It was first pointed out by Kakutani in 1945 that there is a very close connection between random walks and electrical networks. In a simple random walk, a single step works as follows: starting at a vertex, we choose an edge containing the vertex at random, and move along it to the other end. This definition accommodates multiple edges, and is easily adapted to graphs with edge weights (where the probability of moving along an edge is proportional to the weight of the edge).
If we are thinking of an edge-weighted graph as an electrical network, we take the weights to be the conductances of the edges (the reciprocals of the resistances).
The connection is simple to state:
Let $i$ and $j$ be distinct vertices of the connected edge-weighted graph $G$. Apply voltages of $1$ at $i$ and $0$ at $j$. Then the voltage at a vertex $l$ is equal to the probability that the random walk, starting at $l$, reaches $i$ before it reaches $j$.
From this theorem, it is possible to derive a formula for the effective resistance between two vertices. Here are two such formulas. Given two vertices $i$ and $j$, let $P_\mathrm{esc}(i\to j)$ be the probability that a random walk starting at $i$ reaches $j$ before returning to $i$; and let $S_i(i,j)$ be the expected number of times that a random walk starting at $i$ visits $i$ before reaching $j$. Then the effective resistance between $i$ and $j$ is given by either of the two expressions $$\frac{1}{d_iP_\mathrm{esc}(i\to j)}\qquad\mbox{and}\qquad
\frac{S_i(i,j)}{d_i},$$ where $d_i$ is the degree of $i$. (If the edge resistances are not all $1$, then the term $d_i$ should be replaced by the sum of the reciprocals of the resistances of all edges incident with vertex $i$.)
The random walk approach gives alternative proofs of some of the main results about electrical networks. We discuss this further in the guide to the literature.
Foster’s formula and generalizations
------------------------------------
In 1948, Foster [@foster] discovered that the sum of the effective resistances between all *adjacent* pairs of vertices of a connected graph on $v$ vertices is equal to $v-1$. Thirteen years later, he found a similar formula for pairs of vertices at distance $2$: $$\sum_{i\sim h\sim j}\frac{R_{ij}}{d_h} = v-2.$$ Further extensions have been found, but require a stronger condition on the graph. The sum of resistances between all pairs of vertices at distance at most $m$ can be written down explicitly if the graph is *walk-regular up to distance $m$*; this means that the number of closed walks of length $k$ starting and finishing at a vertex $i$ is independent of $i$, for $k\le m$. The formula was discovered by Emil Vaughan, to whom this part of the chapter owes a debt.
In particular, if the graph is distance-regular (see [@bcn]), then the value of the A-criterion can be written down in terms of the so-called *intersection array* of the graph.
Distance
--------
At first sight it seems obvious that pairwise variance should decrease as concurrence increases, but there are many counter-examples to this. However, the following theorem is proved in [@rab:as].
If the Laplacian matrix $\mathbf{L}$ has precisely two distinct non-trivial eigenvalues, then pairwise variance is a decreasing linear function of concurrence.
It does appear that effective resistance, and hence pairwise variance, generally increases with distance in the concurrence graph. In [@bcc09 Question 5.1] we pointed out that this is not always exactly so, and asked if it is nevertheless true that the maximal value of $R_{ij}$ is achieved for some pair of vertices ${\left\{i,j\right\}}$ whose distance apart in the graph is maximal. Here is a counter-example.
Let $k=2$, so that the block design is the same as its concurrence graph. Take $v=10$ and $b=14$. The graph consists of a cube, with two extra vertices $1$ and $2$ attached as leaves to vertex $3$. The vertex antipodal to $3$ in the cube is labelled $4$. It is straightforward to check (either using an electrical network, or by using the fact that the cube is distance-regular) that the effective resistance between a pair of cube vertices is $7/12$, $3/4$ and $5/6$ for vertices at distances $1$, $2$ and $3$. Hence $R_{1j}\leq 11/6$ for all cube vertices $j$, while $R_{12}=2$. On the other hand, the distance between vertices $1$ and $2$ is only $2$, while that between either of them and vertex $4$ is $4$.
There are some ‘nice’ graphs where pairwise variance does indeed increase with distance. The following result is proved in [@rab:dgh]. Biggs gave the equivalent result for effective resistances in [@biggselec].
Suppose that a block design has just two distinct concurrences, and that the pairs of vertices corresponding to the larger concurrence form the edges of a distance-regular graph $H$. Then pairwise variance increases with distance in $H$.
Graph concepts linked to E-optimality
=====================================
Measures of bottlenecks
-----------------------
A ‘good’ graph (for use as a network) is one without bottlenecks: any set of vertices should have many edges joining it to its complement. So, for any subset $S$ of vertices, we let $\partial(S)$ (the *boundary* of $S$) be the set of edges which have one vertex in $S$ and the other in its complement, and then define the *isoperimetric number* $\iota(G)$ by $$\iota(G) = \min\left\{\frac{|\partial S|}{|S|}:
S\subseteq V(G),\ 0<|S|\le \frac{v}{2}
\right\}.$$
The next result shows that the isoperimetric number is related to the E-criterion. It is useful not so much for identifying the E-optimal designs as for easily showing that large classes of designs cannot be E-optimal: any design whose concurrence graph has low isoperimetric number performs poorly on the E-criterion.
Let $G$ have an edge-cutset of size $c$ whose removal separates the graph into parts $S$ and $G\setminus S$ with $m$ and $n$ vertices respectively, where $0 < m\leq n$. Then $$\theta_1\le c\left(\frac{1}{m}+\frac{1}{n}\right)
\leq \frac{2\left|\partial S\right|}{\left|S\right|}.$$
We know that $\theta_1$ is the minimum of $\mathbf{x}^\top \mathbf{L}\mathbf{x}/\mathbf{x}^\top \mathbf{x}$ over real vectors $\mathbf{x}$ with $\sum_i x_i = 0$. Put $$x_i= \left\{\begin{array}{rl}n & \mbox{ if $i\in S$}\\
-m & \mbox{otherwise.}
\end{array}\right.$$ Then $\mathbf{x}^\top \mathbf{x} = nm(m+n)$ and $$\mathbf{x}^\top \mathbf{L} \mathbf{x} =
\sum_{\mathrm{edges\ }ij} (x_i-x_j)^2 = c(m+n)^2.$$ Hence $$\theta_1 \leq
\frac{\mathbf{x}^\top \mathbf{L} \mathbf{x}}{\mathbf{x}^\top \mathbf{x}}
= \frac{c(m+n)^2}{nm(m+n)}
= c\left(\frac{1}{m} + \frac{1}{n}\right) \leq \frac{2c}{m} =
\frac{2\left|\partial S\right|}{\left|S\right|}.
{\protect\nolinebreak\mbox{\quad\rule{1ex}{1ex}}}$$
\[thm:iso\] Let $\theta_1$ be the smallest non-trivial eigenvalue of the Laplacian matrix $\mathbf{L}$ of the connected graph $G$. Then $\theta_1 \leq 2
\iota(G)$.
There is also an upper bound for the isoperimetric number in terms of $\theta_1$, which is loosely referred to as a ‘Cheeger-type inequality’; for details, see the further reading.
We also require a second cutset lemma, phrased in terms of vertex cutsets.
Let $G$ have a vertex-cutset $C$ of size $c$ whose removal separates the graph into parts $S$ and $T$ with $m$, $n$ vertices respectively (so $nm>0$). Let $m'$ and $n'$ be the number of edges from vertices in $C$ to vertices in $S$, $T$ respectively. Then $$\theta_1 \leq \frac{m'n^2 + n'm^2}{nm(m+n)}.$$ In particular, if there are no multiple edges at any vertex of $C$ then $\theta_1\leq c$, with equality if and only if every vertex in $C$ is joined to every vertex in $S\cup T$.
Put $$x_i= \left\{\begin{array}{rl}n & \mbox{ if $i\in S$}\\
-m & \mbox{ if $i\in T$}\\
0 & \mbox{ otherwise.}
\end{array}\right.$$ Then $\mathbf{x}^\top \mathbf{x} = nm(m+n)$ and $\mathbf{x}^\top \mathbf{L}\mathbf{x} = m'n^2+n'm^2$, and so $$\theta_1 \leq \frac{m'n^2 + n'm^2}{nm(m+n)}.$$ If there are no multiple edges at any vertex in $C$ then $m'\leq cm$ and $n'\leq cn$ and the result follows.
Variance balance
----------------
A block design is *variance-balanced* if all the concurrences $\lambda_{ij}$ are equal for $i\ne j$. In such a design, all of the pairwise variances $V_{ij}$ are equal. Morgan and Srivastav proved the following result in [@ms].
If the constant concurrence $\lambda$ of a variance-balanced design satisfies $(v-1)\lambda = \lfloor(bk/v)\rfloor (k-1)$ then the design is E-optimal.
A block with $k$ different treatments contributes $k(k-1)/2$ edges to the concurrence graph. Let us define the *defect* of a block to be $$\frac{k(k-1)}{2} - \mbox{the number of edges it contributes to the graph.}$$
The following result is proved in [@bcc09].
\[thm:defect\] If $k<v$, then a variance-balanced design with $v$ treatments is E-optimal if the sum of the block defects is less than $v/2$.
Table \[tab:binary\](b) shows that the design in Fig. \[fig:binary\](b) is variance-balanced. Block $\Gamma_1$ has defect $1$, and each other block has defect $0$, so the sum of the block defects is certainly less than $5/2$, and Theorem \[thm:defect\] shows that the design is E-optimal. It is rather counter-intuitive that the non-binary design in Fig. \[fig:binary\](b) can be better than the design in Fig. \[fig:binary\](a); in fact, in his contribution to the discussion of Tocher’s paper [@tocher], which introduced this design, David Cox said
> I suspect that … balanced ternary designs are of no practical value.
Computation shows that the design in Fig. \[fig:binary\](a) is $\Phi_p$-better than the one in Fig. \[fig:binary\](b) if $p<5.327$. In particular, it is A- and D-better.
Some history
============
As we have seen, if the experimental units form a single block and there are only two treatments then it is best for their replications to be as equal as possible. Statisticians know this so well that it is hard for us to imagine that more information may be obtained, about *all* treatment comparisons, if replications differ by more than $1$.
In agriculture, or in any area with qualitative treatments, A-optimality is the natural criterion. If treatments are quantities of different substances, then D-optimality is preferable, as the ranking on this criterion is invariant to change of measurement units. Thus industrial statisticians have tended to prefer D-optimality, although E-optimality has become popular among chemical process engineers. Perhaps the different camps have not talked to each other as much as they should have.
For most of the 20th century, it was normal practice in field experiments to have all treatments replicated three or four times. Where incomplete blocks were used, they typically had size from $3$ to $20$. Yates introduced his square lattice designs with $v=k^2$ in [@fy:lattice]. He used uniformity data and two worked examples to show that these designs can give lower average pairwise variance than a design using a highly replicated control treatment, but both of his examples were equireplicate with $r\in\{3,4\}$.
In the 1930s, 1940s and 1950s, analysis of the data from an experiment involved inverting the Laplacian matrix without a computer: this is easy for BIBDs, and only slightly harder if the Laplacian matrix has only two distinct non-trivial eigenvalues. The results in [@kiefer] and [@ksh] encouraged the beliefs that the optimal designs, on all $\Phi_p$-criteria, are as equireplicate as possible, with concurrences as equal as possible, and that the same designs are optimal, or nearly so, on all of these criteria.
Three short papers in the same journal in 1977–1982 demonstrate the beliefs at that time. In [@JAJM:rgd], John and Mitchell did not even consider designs with unequal replication. They conjectured that, if there exist any regular-graph designs for given values of $v$, $b$ and $k$, then the A- and D-optimal designs are regular-graph designs. For the parameter sets which they had examined by computer search, the same designs were optimal on the A- and D-criteria. In [@BJJAE], Jones and Eccleston reported the results of various computer searches for A-optimal designs without the constraint of equal replication. For $k=2$ and $b=v\in\{10,11,12\}$ (but not $v=9$) their A-optimal design is almost a queen-bee design, and their designs are D-worse than those in [@JAJM:rgd]. The belief in equal replication was so ingrained that some readers assumed that there was an error in their program.
John and Williams followed this with the paper [@yyy] on conjectures for optimal block designs for given values of $v$, $b$ and $k$. Their conjectures included:
- the set of regular-graph designs always contains one that is optimal without this restriction;
- among regular-graph designs, the same designs are optimal on the A- and D-criteria.
They endorsed Cox’s dismissal of non-binary designs, strengthening it to the statement that they “are inefficient”, and declared that the three unequally replicated A-optimal designs in [@BJJAE] were “of academic rather than of practical interest”. These conjectures and opinions seemed quite reasonable to people who had been finding good designs for the sizes needed in agricultural experiments.
At the end of the 20th century, there was an explosion in the number of experiments in genomics, using microarrays. Simplifying the story greatly, these are effectively block designs with $k=2$, and biologists wanted A-optimal designs, but they did not know the vocabulary ‘block’ or ‘A-optimal’, ‘graph’ or ‘cycle’. Computers were now much more powerful than in 1980, and researchers in genomics could simply undertake computer searches without the benefit of any statistical theory. In 2001, Kerr and Churchill [@KCh] published the results of a computer search for A-optimal designs with $k=2$ and $v=b\leq 11$. For $v\in\{10,11,12\}$, their results were completely consistent with those in [@BJJAE], which they did not cite. They called cycles *loop designs*.
Mainstream statisticians began to get involved. In 2005, Wit, Nobile and Khanin published the paper [@Wit] giving the results of a computer search for A- and D-optimal designs with $k=2$ and $v=b$. The results are shown in Fig. \[fig:wit\]. The A-optimal designs differ from the D-optimal designs when $v\geq 9$, but are consistent with those found in [@KCh].
[|c|c|c|]{}
(200,200)(-100,-100) (85.1,0) (42.6,73.7) (-42.6,73.7) (-85.1,0) (-42.6,-73.7) (42.6,-73.7) (85.1,0)(63.85,36.85)(42.6,73.7) (42.6,73.7)(0,73.7)(-42.6,73.7) (-85.1,0)(-63.85,36.85)(-42.6,73.7) (85.1,0)(63.85,-36.85)(42.6,-73.7) (42.6,-73.7)(0,-73.7)(-42.6,-73.7) (-85.1,0)(-63.85,-36.85)(-42.6,-73.7)
&
(200,200)(-100,-100) (85.1,0) (53.1,66.5) (-18.9,83) (-76.7,36.9) (-76.7,-36.9) (-18.9,-83) (53.1,-66.5) (85.1,0)(69.1,33.25)(53.1,66.5) (53.1,66.5)(17.1,74.75)(-18.9,83) (-18.9,83)(-47.8,59.95)(-76.7,36.9) (-76.7,36.9)(-76.7,0)(-76.7,-36.9) (-18.9,-83)(-47.8,-59.95)(-76.7,-36.9) (53.1,-66.5)(17.1,-74.75)(-18.9,-83) (85.1,0)(69.1,-33.25)(53.1,-66.5)
&
(200,200)(-100,-100) (85.1,0) (60.2,60.2) (0,85.1) (-60.2,60.2) (-85.1,0) (-60.2,-60.2) (0,-85.1) (60.2,-60.2) (85.1,0)(72.75,30.1)(60.2,60.2) (60.2,60.2)(30.1,72.75)(0,85.1) (-85.1,0)(-72.75,30.1)(-60.2,60.2) (-60.2,60.2)(-30.1,72.75)(0,85.1) (85.1,0)(72.75,-30.1)(60.2,-60.2) (60.2,-60.2)(30.1,-72.75)(0,-85.1) (-85.1,0)(-72.75,-30.1)(-60.2,-60.2) (-60.2,-60.2)(-30.1,-72.75)(0,-85.1)
\
(200,200)(-100,-100) (85.1,0) (65.2,54.7) (14.8,83.8) (-42.6,73.7) (-80,29.1) (-80,-29.1) (-42.6,-73.7) (14.8,-83.8) (65.2,-54.7) (85.1,0)(75.15,27.35)(65.2,54.7) (65.2,54.7)(40,69.25)(14.8,83.8) (14.8,83.8)(-13.9,78.75)(-42.6,73.7) (-42.6,73.7)(-62.3,51.4)(-80,29.1) (-80,29.1)(-80,0)(-80,-29.1) (85.1,0)(75.15,-27.35)(65.2,-54.7) (65.2,-54.7)(40,-69.25)(14.8,-83.8) (14.8,-83.8)(-13.9,-78.75)(-42.6,-73.7) (-42.6,-73.7)(-62.3,-51.4)(-80,-29.1)
&
(200,200)(-100,-31.2) (50,0) (80.9,95.1) (0,153.9) (-80.9,95.1) (-50,0) (0,-16.3) (80.9,42.5) (50,137.6) (-50,137.6) (-80.9,42.5) (50,0)(65.45,21.25)(80.9,42.5) (-50,0)(-65.45,21.25)(-80.9,42.5) (80.9,95.1)(65.45,118.35)(50,137.6) (-80.9,95.1)(-65.45,118.35)(-50,137.6) (0,153.9)(-25,145.75)(-50,137.6) (0,153.9)(25,145.75)(50,137.6) (-80.9,95.1)(-80.9,68.8)(-80.9,42.5) (80.9,95.1)(80.9,68.8)(80.9,42.5) (0,-16.3)(25,-8.15)(50,0) (0,-16.3)(-25,-8.15)(-50,0)
&
(200,200)(-100,-100) (85.1,0) (71.6,46) (35.4,77.4) (-12.1,84.2) (-55.7,64.3) (-81.7,24) (71.6,-46) (35.4,-77.4) (-12.1,-84.2) (-55.7,-64.3) (-81.7,-24) (85.1,0)(78.35,23)(71.6,46) (71.6,46)(53.5,61.7)(35.4,77.4) (35.4,77.4)(11.65,80.8)(-12.1,84.2) (-12.1,84.2)(-33.9,74.25)(-55.7,64.3) (-55.7,64.3)(-68.7,44.15)(-81.7,24) (-81.7,24)(-81.7,0)(-81.7,-24) (85.1,0)(78.35,-23)(71.6,-46) (71.6,-46)(53.5,-61.7)(35.4,-77.4) (35.4,-77.4)(11.65,-80.8)(-12.1,-84.2) (-12.1,-84.2)(-33.9,-74.25)(-55.7,-64.3) (-55.7,-64.3)(-68.7,-44.15)(-81.7,-24)
\
\
(200,200)(-100,-100) (85.1,0) (42.6,73.7) (-42.6,73.7) (-85.1,0) (-42.6,-73.7) (42.6,-73.7) (85.1,0)(63.85,36.85)(42.6,73.7) (42.6,73.7)(0,73.7)(-42.6,73.7) (-85.1,0)(-63.85,36.85)(-42.6,73.7) (85.1,0)(63.85,-36.85)(42.6,-73.7) (42.6,-73.7)(0,-73.7)(-42.6,-73.7) (-85.1,0)(-63.85,-36.85)(-42.6,-73.7)
&
(200,200)(-100,-100) (85.1,0) (53.1,66.5) (-18.9,83) (-76.7,36.9) (-76.7,-36.9) (-18.9,-83) (53.1,-66.5) (85.1,0)(69.1,33.25)(53.1,66.5) (53.1,66.5)(17.1,74.75)(-18.9,83) (-18.9,83)(-47.8,59.95)(-76.7,36.9) (-76.7,36.9)(-76.7,0)(-76.7,-36.9) (-18.9,-83)(-47.8,-59.95)(-76.7,-36.9) (53.1,-66.5)(17.1,-74.75)(-18.9,-83) (85.1,0)(69.1,-33.25)(53.1,-66.5)
&
(200,200)(-100,-100) (85.1,0) (60.2,60.2) (0,85.1) (-60.2,60.2) (-85.1,0) (-60.2,-60.2) (0,-85.1) (60.2,-60.2) (85.1,0)(72.75,30.1)(60.2,60.2) (60.2,60.2)(30.1,72.75)(0,85.1) (-85.1,0)(-72.75,30.1)(-60.2,60.2) (-60.2,60.2)(-30.1,72.75)(0,85.1) (85.1,0)(72.75,-30.1)(60.2,-60.2) (60.2,-60.2)(30.1,-72.75)(0,-85.1) (-85.1,0)(-72.75,-30.1)(-60.2,-60.2) (-60.2,-60.2)(-30.1,-72.75)(0,-85.1)
\
(130,130)(-65,-65) (0,0) (-50,0) (50,0) (-50,-50) (0,-50) (0,50) (43.3,25) (43.3,-25) (25,43.3) (0,0)(-25,0)(-50,0) (0,0)(0,-25)(0,-50) (0,0)(0,25)(0,50) (0,0)(25,0)(50,0) (-50,-50)(-50,-25)(-50,0) (-50,-50)(-25,-50)(0,-50) (0,0)(21.65,-12.5)(43.3,-25) (0,0)(21.65,12.5)(43.3,25) (0,0)(12.5,21.65)(25,43.3)
&
(130,130)(-65,-65) (0,0) (-50,0) (50,0) (-50,-50) (0,-50) (0,50) (43.3,25) (43.3,-25) (25,43.3) (-25,43.3) (0,0)(-25,0)(-50,0) (0,0)(0,-25)(0,-50) (0,0)(0,25)(0,50) (0,0)(25,0)(50,0) (-50,-50)(-50,-25)(-50,0) (-50,-50)(-25,-50)(0,-50) (0,0)(21.65,-12.5)(43.3,-25) (0,0)(21.65,12.5)(43.3,25) (0,0)(12.5,21.65)(25,43.3) (0,0)(-12.5,21.65)(-25,43.3)
&
(130,130)(-65,-65) (0,0) (-50,0) (50,0) (-50,-50) (0,-50) (0,50) (43.3,25) (43.3,-25) (25,43.3) (-25,43.3) (-43.3,25) (0,0)(-25,0)(-50,0) (0,0)(0,-25)(0,-50) (0,0)(0,25)(0,50) (0,0)(25,0)(50,0) (-50,-50)(-50,-25)(-50,0) (-50,-50)(-25,-50)(0,-50) (0,0)(21.65,-12.5)(43.3,-25) (0,0)(21.65,12.5)(43.3,25) (0,0)(12.5,21.65)(25,43.3) (0,0)(-12.5,21.65)(-25,43.3) (0,0)(-21.65,12.5)(-43.3,25)
\
\
What is going on here? Why are the designs so different when $v\geq 9$? Why is there such a sudden, large change in the A-optimal designs? We explain this in the next section.
Block size two
==============
Least replication {#sec:least}
-----------------
If $k=2$, then the design is the same as its concurrence graph, and connectivity requires that $b\geq v-1$. If $b=v-1$, then all connected designs are trees, such as those in Fig. \[fig:tree\]. Theorem \[thm:span\] shows that the D-criterion does not differentiate between them.
In a tree, the effective resistance $R_{ij}$ is just the length of the unique path between vertices $i$ and $j$. Theorems \[thm:var\] and \[thm:resist\] show that the only A-optimal designs are the stars, such as the graph on the right of Fig. \[fig:tree\].
[c@c]{}
(100,100) (50,50) (50,100) (50,0) (0,50) (100,50) (83.3,83.3) (16.7,83.3) (83.3,16.7) (16.7,16.7) (50,0)[(-2,1)[33.3]{}]{} (0,50)[(1,0)[50]{}]{} (0,50)[(1,-2)[16.7]{}]{} (100,50)[(-1,-2)[16.7]{}]{} (100,50)[(-1,2)[16.7]{}]{} (50,100)[(-2,-1)[33.3]{}]{} (16.7,83.3)[(1,-1)[67]{}]{}
&
(100,100) (50,50) (50,100) (50,0) (0,50) (100,50) (85,85) (15,85) (85,15) (15,15) (0,50)[(1,0)[100]{}]{} (50,0)[(0,1)[100]{}]{} (15,15)[(1,1)[70]{}]{} (15,85)[(1,-1)[70]{}]{}
In a star with $v$ vertices, the contrast between any two leaves is an eigenvector of the Laplacian matrix $\mathbf{L}$ with eigenvalue $1$, while the contrast between the central vertex and all the other vertices is an eigenvector with eigenvalue $v$. If $v\geq5$ and $G$ is not a star then there is an edge whose removal splits the graph into two components of sizes at least $2$ and $3$. Cutset Lemma 1 then shows that $\theta_1 \leq 5/6 <1$. The only other tree which is not a star is the path of length $3$, for which direct calculation shows that $\theta_1 = 2 - \sqrt{2} <1$. Hence the E-optimal designs are also the stars.
One fewer treatment {#sec:loop}
-------------------
If $b=v$ and $k=2$, then the concurrence graph $G$ contains a single cycle: such graphs are called *unicyclic*. Let $s$ be the length of the cycle. All the remaining vertices are in trees attached to various vertices of the cycle. Fig. \[fig:move\] shows two unicyclic graphs with $v=12$ and $s=6$.
[@c@c@]{}
(8,7.5)(-2,-2) (2,0) (-2,0) (1,1.67) (3,1.67) (2,3.33) (4,3.33) (4,5) (6,3.33) (-1,1.67) (-2,3.33) (1,-1.67) (-1,-1.67) (0.7,1.37)[(0,0)[[[$6$]{}]{}]{}]{} (-0.7,1.37)[(0,0)[[[$1$]{}]{}]{}]{} (-1.6,0)[(0,0)[[$2$]{}]{}]{} (-0.7,-1.37)[(0,0)[[$3$]{}]{}]{} (0.7,-1.37)[(0,0)[[$4$]{}]{}]{} (1.6,0)[(0,0)[[$5$]{}]{}]{} (-1.4,3.33)[(0,0)[[$12$]{}]{}]{} (3.4,1.67)[(0,0)[[[$7$]{}]{}]{}]{} (1.6,3.33)[(0,0)[[[$8$]{}]{}]{}]{} (4,2.83)[(0,0)[[$9$]{}]{}]{} (3.4,5)[(0,0)[[$10$]{}]{}]{} (6,2.83)[(0,0)[[$11$]{}]{}]{} (1,1.67)[(-1,0)[2]{}]{} (1,-1.67)[(-1,0)[2]{}]{} (3,1.67)[(-1,0)[2]{}]{} (4,3.33)[(-1,0)[2]{}]{} (6,3.33)[(-1,0)[2]{}]{} (4,3.33)[(0,1)[1.63]{}]{} (-1,1.67)[(-3,5)[1]{}]{} (-1,-1.67)[(-3,5)[1]{}]{} (2,0)[(-3,5)[1]{}]{} (1,1.67)[(3,5)[1]{}]{} (1,-1.67)[(3,5)[1]{}]{} (-2,0)[(3,5)[1]{}]{}
&
(6,7.5)(-2,-2) (2,0) (-2,0) (1,1.67) (3,1.67) (2,3.33) (4,3.33) (2,5) (0,3.33) (-1,1.67) (-2,3.33) (1,-1.67) (-1,-1.67) (0.7,1.37)[(0,0)[[[$6$]{}]{}]{}]{} (-0.7,1.37)[(0,0)[[[$1$]{}]{}]{}]{} (-1.6,0)[(0,0)[[$2$]{}]{}]{} (-0.7,-1.37)[(0,0)[[$3$]{}]{}]{} (0.7,-1.37)[(0,0)[[$4$]{}]{}]{} (1.6,0)[(0,0)[[$5$]{}]{}]{} (-1.4,3.33)[(0,0)[[$12$]{}]{}]{} (3.4,1.67)[(0,0)[[[$7$]{}]{}]{}]{} (0.4,3.33)[(0,0)[[[$8$]{}]{}]{}]{} (2,2.83)[(0,0)[[$9$]{}]{}]{} (1.4,5)[(0,0)[[$10$]{}]{}]{} (4,2.83)[(0,0)[[$11$]{}]{}]{} (1,1.67)[(-1,0)[2]{}]{} (1,-1.67)[(-1,0)[2]{}]{} (3,1.67)[(-1,0)[2]{}]{} (4,3.33)[(-1,0)[2]{}]{} (2,3.33)[(0,1)[1.63]{}]{} (-1,1.67)[(-3,5)[1]{}]{} (-1,-1.67)[(-3,5)[1]{}]{} (2,0)[(-3,5)[1]{}]{} (1,1.67)[(3,5)[1]{}]{} (1,1.67)[(-3,5)[1]{}]{} (1,-1.67)[(3,5)[1]{}]{} (-2,0)[(3,5)[1]{}]{}
\
\
(a) & (b)
As we remarked in Section \[sec:span\], the number of spanning trees in a unicyclic graph is equal to the length of the cycle. Hence, Theorem \[thm:span\] gives the following result.
If $k=2$ and $b=v\geq3$, then the D-optimal designs are precisely the cycles.
For A-optimality, we first show that no graph like the one in Fig. \[fig:move\](a) can be optimal. If vertex $12$ is moved so that it is joined to vertex $6$, instead of vertex $1$, then the sum of the variances $V_{i,12}$ for $i$ in the cycle is unchanged and the variances $V_{i,12}$ for the remaining vertices $i$ are all decreased. This argument shows that all the trees must be attached to the same vertex of the cycle.
Now consider the tree on vertices $6$, $8$, $9$, $10$ and $11$ in Fig. \[fig:move\](a). If the two edges incident with vertex $8$ are modified to those in Fig. \[fig:move\](b), then the set of variances between these five vertices are unchanged, as are all others involving vertex $8$, but those between vertices $9$, $10$, $11$ and vertices outside this tree are all decreased. This argument shows that, for any given length $s$ of the cycle, the only candidate for an A-optimal design has $v-s$ leaves attached to a single vertex of the cycle.
The effective resistance between a pair of vertices at distance $d$ in a cycle of length $s$ is $d(s-d)/s$, while that between a leaf and the cycle vertex to which it is attached is $1$. Hence a short calculation shows that the sum of the pairwise effective resistances is equal to $g(s)/12$, where $$g(s) = -s^3 +2vs^2 + 13s -12sv +12v^2 -14v.$$ Now $\bar V/\sigma^2 = g(s)/[3v(v-1)]$ and we seek the minimum of $g(s)$ for integers $s$ in the interval $[2,v]$.
(7,3.6)(-0.6,1.8) (0,2)[(1,0)[5.5]{}]{} (0,2)[(0,1)[3.2]{}]{} [(1,1.95)[(0,1)[0.1]{}]{} (1,1.7)[(0,0.3)[$3$]{}]{}]{} [(2,1.95)[(0,1)[0.1]{}]{} (2,1.7)[(0,0.3)[$6$]{}]{}]{} [(3,1.95)[(0,1)[0.1]{}]{} (3,1.7)[(0,0.3)[$9$]{}]{}]{} [(4,1.95)[(0,1)[0.1]{}]{} (4,1.7)[(0,0.3)[$12$]{}]{}]{} [(5,1.95)[(0,1)[0.1]{}]{} (5,1.7)[(0,0.3)[$15$]{}]{}]{} [(-0.05,2)[(1,0)[0.1]{}]{} (-0.5,2)[(0.4,0)\[r\][$2$]{}]{}]{} [(-0.05,3)[(1,0)[0.1]{}]{} (-0.5,3)[(0.4,0)\[r\][$3$]{}]{}]{} [(-0.05,4)[(1,0)[0.1]{}]{} (-0.5,4)[(0.4,0)\[r\][$4$]{}]{}]{} [(-0.05,5)[(1,0)[0.1]{}]{} (-0.5,5)[(0.4,0)\[r\][$5$]{}]{}]{} (5.8,2)[(0,0)[$s$]{}]{} (0,5.35)[(0,0)[$\bar V/\sigma^2$]{}]{} (0.67,3)[(0,0)[$\boldsymbol{\star}$]{}]{} (1,2.8)[(0,0)[$\boldsymbol{\star}$]{}]{} (1.33,2.67)[(0,0)[$\boldsymbol{\star}$]{}]{} (1.67,2.53)[(0,0)[$\boldsymbol{\star}$]{}]{} (2,2.33)[(0,0)[$\boldsymbol{\star}$]{}]{} (0.67,3.14)[(0,0)[$\boldsymbol{\odot}$]{}]{} (1,2.98)[(0,0)[$\boldsymbol{\odot}$]{}]{} (1.33,2.90)[(0,0)[$\boldsymbol{\odot}$]{}]{} (1.67,2.86)[(0,0)[$\boldsymbol{\odot}$]{}]{} (2,2.79)[(0,0)[$\boldsymbol{\odot}$]{}]{} (2.33,2.67)[(0,0)[$\boldsymbol{\odot}$]{}]{} (0.67,3.25)[(0,0)[$\boldsymbol{\ast}$]{}]{} (1,3.12)[(0,0)[$\boldsymbol{\ast}$]{}]{} (1.33,3.07)[(0,0)[$\boldsymbol{\ast}$]{}]{} (1.67,3.07)[(0,0)[$\boldsymbol{\ast}$]{}]{} (2,3.083)[(0,0)[$\boldsymbol{\ast}$]{}]{} (2.33,3.07)[(0,0)[$\boldsymbol{\ast}$]{}]{} (2.67,3)[(0,0)[$\boldsymbol{\ast}$]{}]{} (0.67,3.33)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (1,3.22)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (1.33,3.19)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (1.67,3.22)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (2,3.28)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (2.33,3.33)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (2.67,3.36)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (3,3.33)[(0,0)[$\boldsymbol{\diamond}$]{}]{} (0.67,3.4)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (1,3.3)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (1.33,3.29)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (1.67,3.33)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (2,3.41)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (2.33,3.51)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (2.67,3.6)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (3,3.66)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (3.33,3.67)[(0,0)[$\boldsymbol{\bullet}$]{}]{} (0.67,3.45)[(0,0)[$\boldsymbol{\circ}$]{}]{} (1,3.37)[(0,0)[$\boldsymbol{\circ}$]{}]{} (1.33,3.36)[(0,0)[$\boldsymbol{\circ}$]{}]{} (1.67,3.42)[(0,0)[$\boldsymbol{\circ}$]{}]{} (2,3.52)[(0,0)[$\boldsymbol{\circ}$]{}]{} (2.33,3.64)[(0,0)[$\boldsymbol{\circ}$]{}]{} (2.67,3.76)[(0,0)[$\boldsymbol{\circ}$]{}]{} (3,3.88)[(0,0)[$\boldsymbol{\circ}$]{}]{} (3.33,3.96)[(0,0)[$\boldsymbol{\circ}$]{}]{} (3.67,4)[(0,0)[$\boldsymbol{\circ}$]{}]{} (0.67,3.5)[(0,0)[$\boldsymbol{\times}$]{}]{} (1,3.424)[(0,0)[$\boldsymbol{\times}$]{}]{} (1.33,3.424)[(0,0)[$\boldsymbol{\times}$]{}]{} (1.67,3.48)[(0,0)[$\boldsymbol{\times}$]{}]{} (2,3.59)[(0,0)[$\boldsymbol{\times}$]{}]{} (2.33,3.73)[(0,0)[$\boldsymbol{\times}$]{}]{} (2.67,3.88)[(0,0)[$\boldsymbol{\times}$]{}]{} (3,4.03)[(0,0)[$\boldsymbol{\times}$]{}]{} (3.33,4.17)[(0,0)[$\boldsymbol{\times}$]{}]{} (3.67,4.27)[(0,0)[$\boldsymbol{\times}$]{}]{} (4,4.33)[(0,0)[$\boldsymbol{\times}$]{}]{} (0.67,3.54)[(0,0)[$\boldsymbol{+}$]{}]{} (1,3.419)[(0,0)[$\boldsymbol{+}$]{}]{} (1.33,3.47)[(0,0)[$\boldsymbol{+}$]{}]{} (1.67,3.54)[(0,0)[$\boldsymbol{+}$]{}]{} (2,3.65)[(0,0)[$\boldsymbol{+}$]{}]{} (2.33,3.79)[(0,0)[$\boldsymbol{+}$]{}]{} (2.67,3.96)[(0,0)[$\boldsymbol{+}$]{}]{} (3,4.14)[(0,0)[$\boldsymbol{+}$]{}]{} (3.33,4.31)[(0,0)[$\boldsymbol{+}$]{}]{} (3.67,4.46)[(0,0)[$\boldsymbol{+}$]{}]{} (4,4.59)[(0,0)[$\boldsymbol{+}$]{}]{} (4.33,4.67)[(0,0)[$\boldsymbol{+}$]{}]{} (4.83,3.73)
Fig. \[fig:cubic\] plots $g(s)/[3v(v-1)]$ for $s$ in $[2,v]$ and $6\leq v\leq 13$. When $v\leq 7$, the function $g$ is monotonic decreasing, so it attains its minimum on $[2,v]$ at $s=v$. For all larger values of $v$, the function $g$ has a local minimum in the interval $[3,5]$: when $v\geq 9$, the value at this local minimum is less than $g(v)$. This change from the upper end of the interval to the local minimum explains the sudden change in the A-optimal designs. Detailed examination of the local minimum gives the following result.
\[thm:aopt\] If $k=2$ and $b=v\geq3$ then the A-optimal designs are:
- a cycle, if $v\leq 8$;
- a square with $v-4$ leaves attached to one vertex, if $9\leq v \leq 11$;
- a triangle with $v-4$ leaves attached to one vertex, if $ v \geq 13$;
- either of the last two, if $v=12$.
What about E-optimality? The smallest eigenvalue of the Laplacian matrix of the triangle with one or more leaves attached to one vertex is 1, as is that of the digon with two or more leaves attached to one vertex. We now show that almost all other unicyclic graphs have at least one non-trivial eigenvalue smaller than this.
Suppose that vertex $i$ in the cycle has a non-empty tree attached to it, so that $\{i\}$ is a vertex cutset. If $s\geq 3$ then there are no double edges, so Cutset Lemma 2 shows that $\theta_1 <1$ unless all vertices are joined to $i$, in which case $s=3$. If $s=2$ and there are trees attached to both vertices of the digon, then applying Cutset Lemma 2 at each of these vertices shows that $\theta_1<1$ unless $v=4$ and there is one leaf at each vertex of the digon: for this graph, $\theta_1 = 2-\sqrt{5} <1$. A digon with leaves attached to one vertex is just a star with one edge doubled.
The cycle of size $v$ is a cyclic design. The smallest eigenvalue of its Laplacian matrix is $2(1 - \cos(2\pi/v))$, which is greater than $1$ when $v\leq 5$, is equal to $1$ when $v=6$, and is less than $1$ when $v\leq 7$. When $v=3$ it is equal to $3$, which is greater than $3-\sqrt{3}$, which is the smallest Laplacian eigenvalue of the digon with one leaf.
Putting all of this together proves the following result.
\[thm:eopt\] If $k=2$ and $b=v\geq3$, then the E-optimal designs are:
- a cycle, if $v\leq 5$;
- a triangle with $v-3$ leaves attached to one vertex, or a star with one edge doubled, if $v\geq 7$;
- either of the last two, if $v=6$.
Thus, for $v\geq 9$, the ranking on the D-criterion is essentially the opposite of the ranking on the A- and E-criteria. The A- and E-optimal designs are far from equireplicate. The change is sudden, not gradual. These findings were initially quite shocking to statisticians.
More blocks
-----------
What happens when $b$ is larger than $v$ but still has the same order of magnitude? The following theorems show that the A- and E-optimal designs are very different from the D-optimal designs when $v$ is large. The proofs of Theorems \[thm:thresh\] and \[thm:eleaf\] are in [@rabpaper] and [@bcc09] respectively.
Let $G$ be the concurrence graph of a connected block design $\Delta$ with $k=2$ and $b\geq v$. If $\Delta$ is D-optimal then $G$ does not contain any bridge (an edge cutset of size one): in particular, $G$ contains no leaves.
Suppose that $\{i,j\}$ is an edge-cutset for $G$. Let $H$ and $K$ be the parts of $G$ containing $i$ and $j$, respectively.
Since $G$ is not a tree, we may assume that $H$ is not a tree, and so there is some edge $e$ in $H$ that is not in every spanning tree for $H$. Let $n_1$ and $n_2$ be the numbers of spanning trees for $H$ that include and exclude $e$, respectively, and let $m$ be the number of spanning trees for $K$. Every spanning tree for $G$ consists of spanning trees for $H$ and $K$ together with the edge $\{i,j\}$. Hence $G$ has $(n_1+n_2)m$ spanning trees.
Let $\ell$ be a vertex on $e$ with $\ell \ne i$. Form $G'$ from $G$ by removing edge $e$ and inserting the edge $e'$, where $e' =
\{\ell,j\}$.
Let $T$ and $T'$ be spanning trees for $H$ and $K$ respectively. If $T$ does not contain $e$ then $T \cup \{\{i,j\}\} \cup T'$ and $T \cup
\{e'\} \cup T'$ are both spanning trees for $G'$. If $T$ contains $e$ then $(T\setminus \{e\}) \cup \{\{i,j\}\} \cup \{e'\} \cup T'$ is a spanning tree for $G'$. Hence the number of spanning trees for $G'$ is at least $(2n_2 +n_1)m$, which is greater than $(n_1+n_2)m$ because $n_2\geq 1$. Hence $G$ does not have the maximal number of spanning trees and so $\Delta$ is not D-optimal.
\[thm:thresh\] Let $c$ be a positive integer. Then there is a positive integer $v_c$ such that if $b-v=c$ and $v\geq v_c$ then all A-optimal designs with $k=2$ contain leaves.
\[thm:eleaf\] If $20 \leq v \leq b\leq 5v/4$ then the concurrence graph for any E-optimal design with $k=2$ contains leaves.
Of course, to obtain a BIBD when $k=2$, $b$ needs to be a quadratic function of $v$. What happens if $b$ is merely a linear function of $v$? In [@bcc09] we conjectured that if $b=cv$ for some constant $c$ then there is a threshold result like the one in Theorem \[thm:thresh\]. However, current work by Robert Johnson and Mark Walters [@JRJMW] suggests something much more interesting—that there is a constant $C$ with $3<C<4$ such that if $b\geq Cv$ and $k=2$ then all A-optimal designs are (nearly) equireplicate, and that random such graphs (in a suitable model) are close to A-optimal with high probability. On the other hand if $b\leq Cv$ then a graph consisting of a large almost equireplicate part (all degrees 3 and 4 with average degree close to $Cv$) together with a suitable number of leaves joined to a single vertex is strictly better than any queen-bee design.
A little more history
---------------------
The results on D- and A-optimality in Sections \[sec:least\] and \[sec:loop\] were proved in [@rabpaper], partly to put to rest mutterings that the results of [@BJJAE; @KCh; @Wit] found by computer search were incorrect. The results on E-optimality are in [@bcc09].
In spite of the horror with which these results were greeted, it transpired that they were not new. The D- and E-optimal designs for $b=
(v-1)/(k-1)$ were identified in [@bap] in 1991. The A-optimal designs for $k=2$ and $b=v-1$ had been given in [@mandal] in 1991. Also in 1991, Tjur gave the A-optimal designs for $k=2$ and $b=v$ in [@tj1]: his proof used the Levi graph as an electrical network.
A fairly common response to these unexpected results was ‘It seems to be just block size $2$ that is a problem.’ Perhaps those of us who usually deal with larger blocks had simply not thought that it was worth while to investigate block size $2$ before the introduction of microarrays.
However, as we sketch in the next section, the problem is not block size $2$ but very low average replication. The proofs there are similar to those in this section; they are given in more detail in [@rabalia; @alia]. Once again, it turns out that these results are not all new. The D-optimal designs for $v/(k-1)$ blocks of size $k$ were given by Balasubramanian and Dey in [@baldey] in 1996—but their proof uses a version of Theorem \[thm:gaff\] with the wrong value of the constant. The A-optimal designs for $v/(k-1)$ blocks of size $k$ were published by Krafft and Schaefer in [@krafft] in 1997—but those authors are not blameless either, because they apparently had not read [@tj1]!
Our best explanation is that agricultural statisticians are so familiar with average replication being at least $3$ that when we saw these papers we decided that they had no applicability and so forgot them.
Very low average replication
============================
In this section we once again consider general block size $k$. A block design is connected if and only if its Levi graph is connected. The Levi graph has $v+b$ vertices and $bk$ edges, so connectivity implies that $bk \geq b+v-1$; that is, $b(k-1) \geq v-1$.
Least replication {#least-replication}
-----------------
If $b(k-1) = v-1$ and the design is connected, then the Levi graph $\tilde G$ is a tree and the concurrence graph $G$ looks like those in Fig. \[fig:cgqueen\]. Hypergraph-theorists do not seem to have an agreed name for such designs.
For both D- and A-optimality, it turns out to be convenient to use the Levi graph. Since all the Levi graphs are trees, Theorem \[thm:gaff\] shows that the D-criterion does not distinguish among connected designs.
By Theorem \[thm:tjur\], $V_{ij} = \tilde R_{ij} \sigma^2$. When $\tilde G$ is a tree, $\tilde R_{ij}=2$ when $i$ and $j$ are in the same block; otherwise, $\tilde R_{ij} = 4$ if any block containing $i$ has a treatment in common with any block containing $j$; and otherwise, $\tilde R_{ij} \geq 6$. The queen-bee designs are the only ones for which $\tilde R_{ij}\leq 4$ for all $i$ and $j$, and so they are the A-optimal designs.
The non-trivial eigenvalues of a queen-bee design are $1$, $k$ and $v$, with multiplicities $b-1$, $b(k-2)$ and $1$, respectively. If the design is not a queen-bee design, then there is a treatment $i$ that is in more than one block but not in all blocks. Thus vertex $i$ forms a cutset for the concurrence graph $G$ which is not joined to every other vertex of $G$. Cutset Lemma 2 shows that $\theta_1<1$. Hence the E-optimal designs are also the queen-bee designs.
One fewer treatment {#one-fewer-treatment}
-------------------
If $b(k-1)=v$, then the Levi graph $\tilde G$ has $bk$ edges and $bk$ vertices, and so it contains a single cycle, which must be of some even length $2s$. If $2\leq s \leq b$, then the design is binary; if $s=1$, then there is a single non-binary block, whose defect is $1$. In this case, $k\geq 3$, because each block must have more than one treatment.
For $2\leq s \leq b$, let $\mathcal{C}(b,k,s)$ be the class of designs constructed as follows. Start with a loop design for $s$ treatments. Insert $k-2$ extra treatments into each block. The remaining $b-s$ blocks all contain the same treatment from the loop design, together with $k-1$ extra treatments. Figs. \[fig:star\] and \[fig:levistar\] show the concurrence graph and Levi graph, respectively, of a design in $\mathcal{C}(6,3,6)$.
For $k\geq 4$, the designs in $\mathcal{C}(b,k,1)$ have one treatment which occurs twice in one block and once in all other blocks, with the remaining treatments all replicated once. The class $\mathcal{C}(b,3,1)$ contains all such designs, and also those in which the treatment in every block is the one which occurs only once in the non-binary block.
If $b(k-1)=v$, then the D-optimal designs are those in $\mathcal{C}(b,k,b)$.
The Levi graph $\tilde G$ is unicyclic, so its number of spanning trees is maximized when the cycle has maximal length. Theorem \[thm:gaff\] shows that the D-optimal designs are precisely those with $s=b$.
\[thm:abig\] If $b(k-1)=v$ then the A-optimal designs are those in $\mathcal{C}(b,k,s)$, where the value of $s$ is given in Table \[tab:sval\].
$$\begin{array}{cc|ccccccccccccc}
k & b & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10& 11 & 12 & 13\\
\hline
2 & & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 4 & 4 & 4 & 3 \mbox{ or } 4& 3\\
3 & & 2 & 3 & 4 & 5 & 6 & 3 & 3 & 3 & 3 & 3 & 2 & 2\\
4 & & 2 & 3 & 4 & 5 & 3 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
5 & & 2 & 3 & 4 & 5 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\
6 & & 2 & 3 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2
\end{array}$$
The Levi graph $\tilde G$ has one cycle, whose length is $2s$, where $1\leq s\leq b$. A similar argument to the one used at the start of the proof of Theorem \[thm:aopt\] shows that this cannot be A-optimal unless the design is in $\mathcal{C}(b,k,s)$. If $s\geq 2$ or $k\geq 4$, then each block-vertex in the cycle has $k-2$ treatment-vertices attached as leaves; all other block-vertices are joined to the same single treatment-vertex in the cycle, and each has $k-1$ treatment vertices attached as leaves. In $\mathcal{C}(b,3,1)$ the first type of design has a Levi graph like this, and the other type has the same multiset of effective resistances between treatment-vertices, because their concurrence graphs are identical. The following calculations use the first type.
Let $\mathcal{V}_1$ be the set of treatment-vertices in the cycle, $\mathcal{V}_2$ the set of other treatment-vertices joined to blocks in the cycle, and $\mathcal{V}_3$ the set of remaining treatment-vertices. For $1\leq i\leq j\leq 3$, denote by $\mathcal{R}_{ij}$ the sum of the pairwise resistances between vertices in $\mathcal{V}_i$ and $\mathcal{V}_j$.
Put $$R_1 = \sum_{d=1}^{s-1}\frac{2d(2s-2d)}{2s} = \frac{s^2-1}{3}$$ and $$R_2 = \sum_{d=0}^{s-1}\frac{(2d+1)(2s-2d-1)}{2s} = \frac{2s^2+1}{6}.$$ Then $\mathcal{R}_{11} = sR_1/2$, $\mathcal{R}_{12} = s(k-2)(R_2+s)$, $\mathcal{R}_{13} = (b-s)(k-1)(R_1+2s)$, $\mathcal{R}_{22} = s(k-2)(k-3) + s(k-2)^2[R_1 + 2(s-1)]/2$, $\mathcal{R}_{23} = (b-s)(k-1)(k-2)(R_2+3s)$, and $\mathcal{R}_{33} = (b-s)(k-1)(k-2) + 2(b-s)(b-s-1)(k-1)^2$. Hence the sum of the pairwise effective resistances between treatment-vertices in the Levi graph is $g(s)/6$, where $$g(s) = -(k-1)^2s^3 +2b(k-1)^2 s^2 -[6bk(k-1) -4k^2 +2k -1]s +c$$ and $c=b(k-1)[12b(k-1) -5k-4]$.
If $s=1$ then the design is non-binary. However, $$g(1)-g(2) =
(3k-9+6b)(k-1) -3,$$ which is positive, because $k\geq 2$ and $b\geq
2$. Therefore the non-binary designs are never A-optimal.
Direct calculation shows that $g(2) > g(3)$ when $b=3$, and that $g(2)> g(3)> g(4)$ when $b=4$. These inequalities hold for all values of $k$, even though $g$ is not decreasing on the interval $[2,4]$ for large $k$ when $b=4$.
If $b=5$ and $k\geq 6$, then $g(3)>g(2)$ and $g(5)>g(2)$. Thus the local minimum of $g$ occurs in the interval $(1,3)$ and is the overall minimum of $g$ on the interval $[1,5]$.
Differentiation gives $$g'(b) = b(k-1)[(b-6)(k-1) - 6] +4k^2 - 2k +1.$$ If $g'(b)>0$ then $g$ has a local minimum in the interval $(1,b)$. If, in addition, $g(3)>g(2)$, then the minimal value for integer $s$ occurs at $s=2$. These conditions are both satisfied if $k=3$ and $b\geq 12$, $k= 4$ and $b\geq 8$, $k\geq 5$ and $b\geq 7$, or $k\geq 9$ and $b\geq 6$.
Given Theorem \[thm:aopt\], there remain only a finite number of pairs $(b,k)$ to be checked individually to find the smallest value of $g(s)$. The results are in Table \[tab:sval\].
If $b(k-1)=v$, $b\geq 3$ and $k\geq 3$, then the E-optimal designs are those in $\mathcal{C}(b,k,b)$ if $b\leq4$, and those in $\mathcal{C}(b,k,2)$ and $\mathcal{C}(b,k,1)$ if $b\geq 5$.
If $2<s<b$ then the concurrence graph $G$ has a vertex which forms a vertex-cutset and which is not joined to all other vertices; moreover, $G$ has no multiple edges. Thus Cutset Lemma 2 shows that $\theta_1<1$.
Direct calculation shows that $\theta_1=1$ if $s=1$ or $s=2$. For $k\geq 4$, all contrasts between singly replicated treatments in the same block are eigenvectors of the Laplacian matrix $L$ with eigenvalue $k$. When $k\geq 3$ and $s=b$ the contrast between singly and doubly replicated treatments has eigenvalue $2(k-1)$. For $s=b$, a straightforward calculation shows that the remaining eigenvalues of $L$ are $$k - \cos \left ( \frac{2\pi n}{b} \right) \pm
\sqrt{(k-1)^2 - \sin^2 \left( \frac{2\pi n}{b}\right)}$$ for $1\leq n\leq b-1$. The smallest of these is $$k - \cos(2\pi/b) - \sqrt{(k-1^2) - \sin^2(2\pi/b)}:$$ this is greater than $1$ if $b=3$ or $b=4$, but less than $1$ if $k\geq 3$ and $b\geq 5$.
Further reading
===============
The Laplacian matrix of a graph, and its eigenvalues, are widely used, especially in connection with network properties such as connectivity, expansion, and random walks. A good introduction to this material can be found in the textbook by Bollobás [@bollobas], especially Chapters II (electrical networks) and IX (random walks). Connection between the smallest non-zero eigenvalue and connectivity is described in surveys by de Abreu [@survey] and by Mohar [@mohar]. In this terminology, a version of Theorem \[thm:eopt\] is in [@algconn].
The basic properties of electrical networks can be found in textbooks of electrical engineering, for example Balabanian and Bickart [@wag]. A treatment connected to the multivariate Tutte polynomial appears in Sokal’s survey [@sokal]. Bollobás describes several approaches to the theory, including the fact (which we have not used) that the current flow minimises the power consumed in the network, and explains the interactions between electrical networks and random walks in the network. See also Deo [@deo].
The connection with optimal design theory was discussed in detail by the authors in their survey [@bcc09]. Further reading on optimal design can be found in John and Williams [@JAJERW:book], Schwabe [@Schwab], or Shah and Sinha [@ss]. For general principles of experimental design, see Bailey [@rabbook].
#### Acknowledgement
This chapter was written at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, during the 2011 programme on Design and Analysis of Experiments.
[99]{}
Nair Maria Maia de Abreu, Old and new results on algebraic connectivity of graphs. *Linear Algebra and its Applications*, **423** (2007), 53–73.
N. Alon and V. Milman, $\lambda_1$, isoperimetric inequalities for graphs, and superconcentrators. *J. Combinatorial Theory Series B* **38** (1985), 73–88.
R. A. Bailey, *Association Schemes: Designed Experiments, Algebra and Combinatorics*, Cambridge Studies in Advanced Mathematics **84**, Cambridge University Press, Cambridge, 2004.
R. A. Bailey, Designs for two-colour microarray experiments, *Applied Statistics* **56** (2007), 365–394.
R. A. Bailey, *Design of Comparative Experiments*. Cambridge Series in Statistical and Probabilistic Mathematics **25**, Cambridge University Press, Cambridge, 2008.
R. A. Bailey, Variance and concurrence in block designs, and distance in the corresponding graphs, *Michigan Mathematical Journal*, **58**, (2009), 105–124.
R. A. Bailey and Peter J. Cameron, Combinatorics of optimal designs. In *Surveys in Combinatorics 2009* (ed. S. Huczynska, J. D. Mitchell and C. M. Roney-Dougal), London Math. Soc. Lecture Notes **365**, Cambridge University Press 2009, pp. 19–73.
R. A. Bailey and Alia Sajjad, Optimal inccomplete-block designs with nearly minimal replication, in preparation.
N. Balabanian and T. A. Bickart, *Electrical Network Theory*, Wiley, 1969.
K. Balasubramanian and Aloke Dey, D-optimal designs with minimal and nearly minimal number of units, *Journal of Statistical Planning and Inference* **52** (1996), 255–262.
R. B. Bapat and A. Dey, Optimal block designs with minimal number of observations, *Statist. Probab. Lett.* **11** (1991), 399–402.
N. L. Biggs, Potential theory on distance-regular graphs, *Combinatorics, Probability and Computing*, **2** (1993), 107–119.
B. Bollobás, *Modern Graph Theory*. Graduate Texts in Mathematics **184**, Springer, New York, 1998.
A. E. Brouwer, A. M. Cohen and A. Neumaier, *Distance-Regular Graphs*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), **18**, Springer-Verlag, Berlin, 1989.
Peter J. Cameron, Strongly regular graphs. In *Topics in Algebraic Graph Theory* (ed. L. W. Beineke and R. J. Wilson), Cambridge University Press, Cambridge, 2004, pp. 203–221.
C.-S. Chêng, Optimality of certain asymmetrical experimental designs, *Annals of Statistics* **6** (1978), 1239–1261.
C.-S. Chêng, Maximizing the total number of spanning trees in a graph: two related problems in graph theory and optimum design theory, *J. Combinatorial Theory Series B* **31** (1981), 240–248.
C.-S. Cheng, Graph and optimum design theories—some connections and examples, *Bull. Internat. Statistical Institute* **49 (1)** (Proceedings of the 43rd Session, Buenos Aires) (1981), 580–590.
C.-S. Cheng, On the optimality of (M.S)-optimal designs in large systems, *Sankhyā* **54** (1992), 117–125.
C.-S. Cheng and R. A. Bailey, Optimality of some two-associate-class partially balanced incomplete-block designs, *Annals of Statistics* **19** (1991), 1667–1671.
C.-S. Cheng, J. C. Masaro and C. S. Wong, Do nearly balanced multigraphs have more spanning trees?, *J. Graph Theory* **8** (1985), 342–345.
N. Deo, *Graph Theory with Applications to Engineering and Computer Science*, Prentice Hall, New Delhi, 1980.
J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, *Trans. American Math. Soc.* **284** (1984), 787–794.
Shaun M. Fallat, Steve Kirkland and Sukanta Pati, Maximizing algebraic connectivity over unicyclic graphs. *Linear and Multilinear Algebra* **3** (2003), 221–241.
R. M. Foster, The average impedance of an electrical network, in *Reissner Anniversary Volume, Contributions to Applied Mathematics* (ed. J. W. Edwards), Ann Arbor, Michigan, 1948, pp. 333–340.
N. Gaffke, *Optimale Versuchsplanung für linear Zwei-Faktor Modelle*, PhD thesis, Rheinisch-Westfälische Technische Hochschule, Aachen, 1978.
N. Gaffke, D-optimal block designs with at most six varieties, *Journal of Statistical Planning and Inference* **6** (1982), 183–200.
Radoslav Harman, Minimal efficiency of designs under the class of orthogonally invariant information criteria, *Metrika* **60** (2004), 137–153.
J. A. John and T. J. Mitchell, Optimal incomplete block designs, *J. Royal Statistical Soc. Series B* **39** (1977), 39–43.
J. A. John and E. R. Williams, Conjectures for optimal block designs, *J. Royal Statistical Soc. Series B* **44** (1982), 221–225.
J. A. John and E. R. Williams, *Cyclic and Computer Generated Designs* (2nd edition), Monographs on Statistics and Applied Probability **38**, Chapman and Hall, London, 1995.
J. R. Johnson and M. Walters, Optimal resistor networks, in preparation.
B. Jones and J. A. Eccleston, Exchange and interchange procedures to search for optimal designs, *J. Royal Statistical Soc. Series B* **42** (1980), 238–243.
M. K. Kerr and G. A. Churchill, Experimental design for gene expression microarrays, *Biostatistics* **2** (2001), 183–201.
J. Kiefer, Construction and optimality of generalized Youden designs. In *A Survey of Statistical Design and Linear Models* (ed. J. N. Srivastava), North-Holland, Amsterdam (1975), pp. 333–353.
G. Kirchhoff, Über die Auflösung der Gleichenung, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme gefürht wird, *Annals of Physical Chemistry* **72** (1847), 497–508.
Olaf Krafft and Martin Schaefer, A-optimal connected block designs with nearly minimal number of observations, *Journal of Statistical Planning and Inference* **65** (1997), 357–386.
A. M. Kshirsagar, A note on incomplete block designs, *Annals of Mathematical Statistics* **29** (1958), 907–910.
A. H. M. M. Latif, F. Bretz and E. Brunner, Robustness considerations in selecting efficient two-color microarray designs, *Bioinformatics* **25** (2009), 2355–2361.
F. W. Levi, *Finite Geometrical Systems*, University of Calcutta, Calcutta, 1942. iii + 51 pp. N. K. Mandal, K. R. Shah and B. K. Sinha, Uncertain resources and optimal designs: problems and perspectives, *Calcutta Statistical Association Bulletin* **40** (1991), 267–282.
Bojan Mohar, The Laplacian spectrum of graphs. *Graph Theory, Combinatorics and Applications*, Vol. 2 (Kalamazoo, MI, 1988), 871–898, Wiley-Interscience, New York, 1991.
J. P. Morgan and S. K. Srivastav, The completely symmetric designs with blocksize three, *J. Statistical Planning and Inference* **106** (2002), 21–30.
H. D. Patterson and E. R. Williams, Some theoretical results on general block designs, *Congressus Numerantium* **15** (1976), 489–496.
R. M. Pringle and A. A. Rayner, *Generalized Inverse Matrices with Applications to Statistics*, Griffin’s Statistical Monographs and Courses, **28**, Griffin, London, 1971.
Alia Sajjad, *Optimality in sparse block designs*, PhD thesis, Quaid-i-Azam University, Islamabad, 2011.
R. Schwabe, *Optimum Designs for Multi-Factor Models*, Lecture Notes in Statistics **113**, Springer, New York, 1996.
K. R. Shah and B. K. Sinha, *Theory of Optimal Designs*, Lecture Notes in Statistics **54**, Springer-Verlag, New York, 1989.
L. W. Shapiro, An electrical lemma, *Math. Magazine* **60** (1987) 36–38.
Alan D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. *Surveys in Combinatorics 2005* (ed. B. S. Webb), London Mathematical Society Lecture Notes **327**, Cambridge University Press, 2005, pp. 173–226.
K. D. Tocher, The design and analysis of block experiments, *J. Royal Statistical Soc. Series B* **14** (1952), 45–100.
T. Tjur, Block designs and electrical networks, *Annals of Statistics* **19** (1991), 1010–1027.
E. Wit, A. Nobile and R. Khanin, Near-optimal designs for dual channel microarray studies, *Applied Statistics* **54** (2005), 817–830.
F. Yates, Incomplete randomized designs, *Annals of Eugenics* **7** (1936), 121–140.
Frank Yates, A new method of arranging variety trials involving a large number of varieties, *Journal of Agricultural Science* **26** (1936), 424–455.
|
---
abstract: |
Wormholes that meet the flare-out condition violate the weak energy condition in classical general relativity. The purpose of this note is to show that even a slight modification of the gravitational theory could, under certain conditions, avoid this violation. The first part discusses some general criteria based on the field equations, while the second part assumes a specific equation of state describing normal matter, together with a particular type of shape function. The analysis is confined to wormholes with zero tidal forces.
PACS numbers: [04.20.Jb, 04.20.Gz, 04.50.-h]{}
address: |
Department of Mathematics\
Milwaukee School of Engineering\
Milwaukee, Wisconsin 53202-3109
author:
- 'Peter K.F. Kuhfittig'
title: A note on wormholes in slightly modified gravitational theories
---
Introduction
============
Interest in modified theories of gravity has increased greatly since the discovery that our Universe is undergoing an accelerated expansion. In particular, $f(R)$ modified gravity replaces the Ricci scalar $R$ in the Einstein-Hilbert action $$\label{action1}
S_{\text{EH}}=\int\sqrt{-g}\,R\,d^4x$$ by a nonlinear function $f(R)$: $$\label{action2}
S_{f(R)}=\int\sqrt{-g}\,f(R)\,d^4x.$$ (For a review, see Refs. [@SF08; @NO07; @fL08].) Wormhole geometries in $f(R)$ modified gravitational theories are discussed in Ref. [@LO09]. Ref. [@J] assumes a noncommutative-geometry background in constructing wormhole geometries in $f(R)$ gravity.
It is well known that wormholes in classical general relativity (GR) require a violation of the null energy condition, usually calling for the use of “exotic matter" [@MT88]. Such matter must be confined to a very narrow band around the throat [@FR96]. Moreover, it is shown in Refs. [@Kuhf1] and [@Kuhf2] that for a wormhole to be compatible with quantum field theory, it is necessary to strike a balance between reducing the size of the exotic region and the degree of fine-tuning of the metric coefficients required to achieve this reduction. So dealing with a very small region suggests that a small modification of the gravitational theory may take the place of exotic matter, analogous to the way that the smearing effect in noncommutative geometry can replace such exotic matter [@pK13]. So in studying the effect of the slightly modified gravity, we concentrate mainly on the vicinity of the throat.
A sufficiently small modification of the gravitational theory is likely to be consistent with observation.
Wormhole geometries in slightly modified $f(R)$ gravity
=======================================================
To describe a spherically symmetric wormhole spacetime, we take the metric to be [@MT88] $$ds^2=-e^{\Phi(r)}dt^2+\frac{dr^2}{1-b(r)/r}
+r^2(d\theta^2+\text{sin}^2\theta\,d\phi^2).$$ Here we recall that $b=b(r)$ is called the *shape function* and $\Phi=\Phi(r)$ the *redshift function*. For the shape function we must have $b(r_0)=r_0$, where $r=r_0$ is the radius of the *throat* of the wormhole. In addition, $b'(r_0)<1$ and $b(r)<r$ to satisfy the *flare-out condition* [@MT88]. These restrictions result in the violation of the weak energy condition in classical general relativity, especially in the vicinity of the throat.
Regarding the redshift function, we normally require that $\Phi(r)$ remain finite to prevent an event horizon. In the present study involving $f(R)$ gravity, we need to assume that $\Phi(r)\equiv
\text{constant}$, so that $\Phi'\equiv 0$. Otherwise, according to Lobo [@LO09], the analysis becomes intractable.
Our next task is to define what is meant by slightly modified gravity. To this end, we list the gravitational field equations in the form used by Lobo [@LO09]: $$\label{FE1}
\rho(r)=F(r)\frac{b'(r)}{r^2},$$ $$\label{FE2}
p_r(r)=-F(r)\frac{b(r)}{r^3}+F'(r)
\frac{rb'(r)-b(r)}{2r^2}-F''(r)
\left[1-\frac{b(r)}{r}\right].$$ and $$\label{FE3}
p_t(r)=-\frac{F'(r)}{r}\left[1-\frac{b(r)}{r}
\right]+\frac{F(r)}{2r^3}[b(r)-rb'(r)],$$ where $F=\frac{df}{dR}$. The curvature scalar $R$ is given by $$\label{Ricci}
R(r)=\frac{2b'(r)}{r^2}.$$ While it is possible in principle to obtain $f(R)$ from $F(r)$, our goal is more modest: how to define *slightly modified $f(R)$ gravity*. To this end, we observe that the above field equations reduce to the Einstein equations for $\Phi'\equiv 0$ whenever $F\equiv 1$. Consequently, comparing Eqs. (\[FE1\]) and (\[Ricci\]), a slight change in $F$ results in a slight change in $R$, which, referring to Eqs. (\[action1\]) and (\[action2\]), characterizes $f(R)$ modified gravity. So we may quantify the notion of slightly modified gravity by assuming that $F(r)$ remains close to unity and relatively $``\text{flat},"$ i.e., both $F'(r)$ and $F''(r)$ remain relatively small in absolute value. To discuss wormholes, we need the additional assumption that $F'(r_0)$ is negative. (We will see in the next section that this condition is actually met when $F(r)$ is computed from a known shape function.) $F''(r)$ will be discussed later. Observe that in Eq. (\[FE1\]), $F$ behaves like a dimensionless scale factor.
Suppose the shape function meets the flare-out condition $b'(r_0)<1$. By continuity, $b(r)<r$ in the immediate vicinity of the throat. Our goal is to show that in this region, we may have $\rho+p_r \ge 0$, as well as $\rho+p_t\ge 0$, thereby satisfying the weak energy condition. From Eqs. (\[FE1\]) and (\[FE2\]) $$\label{NE1}
\rho+p_r=F\frac{b'}{r^2}-F\frac{b}{r^3}+F'
\frac{rb'-b}{2r^2}-F''\left(1-\frac{b}{r}
\right)=(rb'-b)\left(\frac{F}{r^3}
+\frac{F'}{2r^2}\right)-F''\left(1-\frac{b}{r}
\right)\ge 0.$$ Since $1-b/r$ is close to zero near the throat, let us disregard the last term for now. The flare-out condition implies that $rb'(r)-b(r)<0$; so we must have $$\frac{F}{r^3}+\frac{F'}{2r^2}\le 0.$$ Given that $F(r)$ is very close to unity near the throat and that $F'(r)<0$ and relatively small in absolute value, $F/r_0^3+F'/2r_0^2$ can only be negative if $r_0$ is sufficiently large. Consider a simple example. In the vicinity of the throat, suppose that $$F=2-e^{a(r-r_0)};\quad \text{then}\quad
F'=-e^{a(r-r_0)}a\quad \text{and}\quad
F''=-e^{a(r-r_0)}a^2,$$ where $a$ is a small positive constant. At $r=r_0$, $F(r_0)=1$, $F'(r_0)=-a$, and $F''(r_0)=-a^2$. Substituting in Eq. (\[NE1\]), we get $$(rb'-b)\frac{2-ar_0}{2r_0^3}+a^2
\left(1-\frac{b}{r}\right)\ge 0,$$ provided that $2-ar_0\le 0$. We conclude that $$\label{large1}
r_0\ge \frac{2}{a}.$$ For example, if $a=0.001\, \text{m}^{-1}$, then $r_0\ge 2\,\text{km}$. The general conclusion is that $$\label{large2}
r_0\ge \frac{2}{-F'(r_0)},\,\,\,\text{where}
\,\,\,F'(r_0)<0.$$ Observe that $F''(r_0)$ must either be negative or negligibly small. (In the above example, it is actually both.) Finally, by Eq. (\[FE3\]), we also have $\rho+p_t\ge 0$.
The closer $F'(r_0)$ is to zero, the larger $r_0$ has to be to meet the condition $\rho +p_r\ge 0$. Returning to Einstein gravity, if $F'(r_0)
\rightarrow 0$, then $r_0\rightarrow\infty$, and we do not get a wormhole. In this case, then, the existence of a wormhole requires that $\rho+p_r<0$, the usual violation of the null energy condition in GR. As we have seen, we also get $\rho+p_r<0$ whenever $F'(r_0)\ge 0$.
A known shape function
======================
The above analysis assumes a small change in the Ricci scalar, induced by a small change in $F$. An alternative approach is to assume a certain equation of state and a known shape function and then determine $F$ in the vicinity of the throat. To make the analysis tractable, $b(r)$ must be relatively simple, yet typical enough to yield a reasonably general conclusion.
The equation of state to be used for now assumes normal matter: $$\label{EoS}
p_r=\omega\rho,\quad 0<\omega <1.$$ For the shape function we use the form $$\label{shape}
b(r)=r_0^{1-\alpha}r^{\alpha},\quad 0<\alpha <1.$$ Observe that $b(r_0)=r_0$ and $b'(r_0)=\alpha<1$, so that the flare-out condition has been met.
As before, we want $F(r_0)\approx 1$, but for computational convenience, we assume that $F(r_0)=1$. From Eqs. (\[FE1\]) and (\[FE2\]), $$\label{NE2}
\omega F\frac{r_0^{1-\alpha}\alpha r^{\alpha-1}}{r^2}
=-F\frac{r_0^{1-\alpha}r^{\alpha}}{r^3}
+F'\frac{rr_0^{1-\alpha}\alpha r^{\alpha-1}-
r_0^{1-\alpha}r^{\alpha}}{2r^2}-F''
\left(1-\frac{r_0^{1-\alpha}r^{\alpha}}{r}\right).$$ According to Lobo [@LO09], the existence of $F''$ makes it virtually impossible in most cases to get an exact solution. While the last term is once again zero at the throat, we still need to be concerned with the vicinity of the throat. To this end, we need to assume that $\alpha$ is close to unity, so that the last term is close to zero. What remains is easy enough to solve. After simplifying, we obtain the linear differential equation $$F'(r)+\frac{2(1+\alpha\omega)}{1-\alpha}\frac{1}{r}
F(r)=0,$$ where $r$ is near the throat. The solution is $$\label{solution1}
F(r)=\left(\frac{r}{r_0}\right)^
{-2(1+\alpha\omega)/(1-\alpha)},$$ which satisfies the condition $F(r_0)=1$ and so $F(r)\approx 1$ in the vicinity of the throat. Having satisfied the normal-matter equation, Eq. (\[EoS\]), in the vicinity of the throat, as well as the flare-out condition, we conclude that the wormhole is sustained due to the modified gravity.
Even though the last term in Eq. (\[NE2\]) was neglected, we are still dealing with a second-order equation; so we need to consider $F'(r)$ in the vicinity of the throat:
$$\label{derivative1}
\left. F'(r_0)=\frac{-2(1+\alpha\omega)}{1-\alpha}
\left(\frac{r}{r_0}\right)^{-2(1+\alpha\omega)
/(1-\alpha)-1}\frac{1}{r_0}\right|_{r=r_0}=
\frac{-2(1+\alpha\omega)}{1-\alpha}\frac{1}
{r_0}<0,$$
i.e., $F'(r_0)<0$, as in the previous section, but we still want $|F'(r_0)|$ to be relatively small in order to remain close to Einstein gravity. So once again, $r_0$ has to be sufficiently large: $$\label{large3}
r_0\ge \frac{2}{-F'(r_0)}\frac{1+\alpha\omega}
{1-\alpha}.$$ For example, if $\alpha=0.99$, $F'(r_0)=-0.3$, and $\omega=0.5$, we obtain $r_0\ge 1\,\text{km}$. (It is readily checked that if $|F'(r_0)|$ is small, then so is $|F''(r_0)|$.)
The parameter $\omega$ was chosen to describe normal matter. However, solution (\[solution1\]) is valid for any $\omega$. Thus if $\omega=-1$, which is equivalent to assuming Einstein’s cosmological constant [@mC01], inequality (\[large3\]) reduces to inequality (\[large2\]). If $\omega<-1$, we are dealing with phantom energy, which is known to support wormholes in classical GR [@pK09; @sS05; @fL05; @oZ05]. In the present situation, we still need to consider $\alpha$. So if we assume that $$-\frac{1}{\omega}<\alpha<1,$$ then $(1+\alpha\omega)/(1-\alpha)$ becomes negative and condition (\[large3\]) is automatically satisfied for all $r_0$. We conclude that phantom energy can also support wormholes in our slightly modified gravitational theory.
*A remark concerning $F''$:* Retaining $F''$ is possible if $b(r)$ is sufficiently simple. Suppose $b(r)=ar$, $a>0$. Then $b(r_0)=ar_0 \approx r_0$, provided that $r_0$ is large compared to $a$. Then Eq. (\[NE1\]) becomes $$F''+\frac{1}{r^2}\frac{a(1+\omega)}{1-a}F=0,
\quad F(r_0)=1,\quad 0<\omega<1.$$ The solution is $$F(r)=\left(\frac{r}{r_0}\right)
^{\frac{1}{2}[1+\sqrt{1-4a(1+\omega)/(1-a)}]}.$$ Given that $a$ is a small constant, $F'(r_0)\approx 1/r_0$ and $F''(r_0)\approx -1/r_0^2$. Since $r_0$ is assumed to be large, $F'(r)$ and $|F''(r)|$ are small in the vicinity of the throat.
Conclusion
==========
Wormholes that meet the flare-out condition violate the weak energy condition in classical general relativity. It is shown in the first part of this note that a slight modification of the field equations, and hence of the gravitational theory, could avoid this violation. The modification calls for a change in $F(r)$, which induces a change in $R$ in the Einstein-Hilbert action: for every $r$, $F(r)$ remains close to unity, while $F'(r)$ and $F''(r)$ are relatively small in absolute value. The weak energy condition is met provided that $r_0$, the radius of the throat, is sufficiently large and that $F'(r_0)<0$; $F''(r_0)$ must be either negative or negligibly small.
The second part assumes the equation of state $p_r=\omega\rho$, $0<\omega <1$, thereby representing normal matter, as well as a shape function of the form $b(r)=r_0^{1-\alpha}r^{\alpha}$, $0<\alpha <1$; hence $b'(r_0)=\alpha <1$. The equation of state yields the solution $F(r)=(r/r_0)^{-2(1+\alpha\omega)/(1-\alpha)}$ in the vicinity of the throat, where $\alpha$ is close to unity and $r_0$ is sufficiently large. Since the wormhole consists of ordinary matter, its survival must be attributed to the modified gravity. Moreover, a small modification would likely be consistent with observation.
The assumption that $\omega$ is positive is not actually necessary: in particular, phantom dark energy can support a wormhole in slightly modified gravitational theory, as well as in GR. Zero tidal forces are assumed throughout.
[99]{} T.P. Sotiriou and V. Faraoni, “$f(R)$ theories of gravity," Rev. Mod. Phys. **82**, 451 (2010). S. Nojiri and S.D. Odintsov, “Introduction to modified gravity and gravitational alternative for dark energy," Int. J. Geom. Meth. Mod. Phys. **4**, 115 (2007). F.S.N. Lobo, “The dark side of gravity: Modified theories of gravity," arXiv: 0807.1640. F.S.N. Lobo and M.A. Oliveira, “Wormhole geometries in $f(R)$ modified theories of gravity," Phys. Rev. D **80**, 104012 (2009). M. Jamil, F. Rahaman, R. Myrzakulov, P.K.F. Kuhfittig, N. Ahmed, and U. F. Mondal, “Noncommutative wormholes in $f(R)$ gravity," submitted. M.S. Morris and K.S. Thorne, “Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity," Am. J. Phys. **56**, 395 (1988). L.H. Ford and T.A. Roman, “Quantum field theory constrains traversable wormhole geometries," Phys. Rev. D **53**, 5496 (1996). P.K.F. Kuhfittig, “Viable models of traversable wormholes supported by small amounts of exotic matter," IJPAM **44**, 467 (2008). P.K.F. Kuhfittig, “Theoretical construction of Morris-Thorne wormholes compatible with quantum field theory," arXiv: 0908.4233. P.K.F. Kuhfittig, “Macroscopic wormholes in noncommutative geometry," Int. J. Pure Appl. Math. **89**, 401 (2013). M. Carmeli, “Accelerating universe, cosmological constant and dark energy," arXiv: astro-ph/0111259. P.K.F. Kuhfittig, “A single model of traversable wormholes supported by generalized phantom energy or Chaplygin gas," Gen. Rel. Grav. **41**, 1485 (2009). S.V. Sushkov, “Wormholes supported by a phantom energy," Phys. Rev. D **71**, 043520 (2005). F.S.N. Lobo, “Phantom energy traversable wormholes," Phys. Rev. D **71** 084011 (2005). O.B. Zaslavskii, “Exactly solvable model of a wormhole supported by phantom energy," Phys. Rev. D **72**, 061303(R) (2005).
|
ø
[FTUV/96-36 IFIC/96-44]{}\
[hep-ph/9606356]{}
.25cm
[Simple decay-lepton asymmetries in polarized $e^+e^- \ra
\tt$ and $CP$-violating dipole couplings of the top quark ]{} .5cm [P. Poulose]{}\
[*Theory Group, Physical Research Laboratory\
Navrangpura, Ahmedabad 380009, India*]{} .3cm [Saurabh D. Rindani[^1]]{}\
[*Institut de F' isica Corpuscular\
Departament de Física Te\` orica , Universitat de Val\` encia\
Av. Dr. Moliner 50, 46100 Burjassot, Val\` encia, Spain* ]{}
.2cm We study two simple $CP$-violating asymmetries of leptons coming from the decay of $t$ and $\o t$ in $e^+e^- \ra \tt$, which do not need the full reconstruction of the $t$ or $\o t$ for their measurement. They can arise when the top quark possesses nonzero electric and weak dipole form factors in the couplings to the photon and $Z$, respectively. Together, these two asymmetries can help to determine the electric and weak dipole form factors independently. If longitudinal beam polarization is available, independent determination of form factors can be done by measuring only one of the asymmetries. We obtain estimates of 90% confidence limits that can be put on these form factors at a future linear $\ee$ collider operating at $\sqrt{s}=500$ GeV.
Experiments at the Tevatron have seen evidence for the top qark with mass in the range of about 170-200 GeV [@expt]. Future runs of the experiment will be able to determine the mass more precisely and also determine other properties of the top quark. $\tt$ pairs will be produced more copiously at proposed $e^+ e^-$ linear colliders operating above threshold. It would then be possible to investigate these properties further.
While the standard model (SM) predicts $CP$ violation outside the $D$- and $B$-meson systems to be unobservably small, in some extensions of SM, $CP$ violation might be considerably enhanced, especially in the presence of a heavy top quark. In particular, $CP$-violating electric dipole form factor of the top quark, and the analogous $CP$-violating “weak" dipole form factor in the $\tt$ coupling to $Z$, could be enhanced. These $CP$-violating form factors could be determined in a model-independent way at high energy $\ee$ linear colliders, where $\ee \ra \tt$ would proceed through virtual $\g$ and $Z$ exchange.
Since a heavy top quark ($m_t \ge 120$ GeV) is expected to decay before it hadronizes [@heavytop], it has been suggested [@toppol] that top polarization asymmetry in $\ee \ra \tt$ can be used to determine the $CP$-violating dipole form factors, since polarization information would be retained in the decay product distribution. Experiments have been proposed in which the $CP$-violating dipole couplings could be measured in decay momentum correlations$^1$ [@bern; @atwood; @cuypers] or asymmetries [@asymm; @PP], even with beam polarization [@atwood; @PP]. These suggestions on the measurement of asymmetries have concentrated on experiments requiring the reconstruction of the top-quark momentum (with the exception of lepton energy asymmetry [@toppol; @asymm; @PP]). In this note we look at very simple lepton angular asymmetries in $e^+e^- \ra \tt$ which do not require the experimental determination of the $t$ or $\o{t}$ momentum. Being single-lepton asymmetries, they do not require both $t$ and $\o{t}$ to decay leptonically. Since either $t$ or $\o{t}$ is also allowed to decay hadronically, there is a gain in statistics.
The two asymmetries we study here are as follows. We look at the angular distributions of the charged leptons arising from the decay of $t$ and $\o t$ in $e^+e^- \ra \tt$. In terms of the polar angle distribution of the leptons with respect to the $e^-$ beam direction in the centre-of-mass (cm) frame, we can define two $CP$-violating asymmetries. One is simply the total lepton-charge asymmetry, with a cut-off $\theta_0$ on the forward and backward polar angles of the leptons, with respect to the beam direction as $z$ axis. The other is the leptonic forward-backward asymmetry combined with charge asymmetry, again with the angles within $\theta_0$ of the forward and backward directions excluded. (See later for details).
Our results are based on a fully analytical calculation of single lepton distributions in the production and subsequent decay of $\tt$. We present here only the leptonic asymmetries obtained by an integration of these distributions. The details of the fully differential distribution as well as the distribution in the polar angle of the lepton with respect to the beam direction in the centre-of-mass (cm) frame can be found elsewhere [@longpaper].
We have also included the effect of electron longitudinal polarization, likely to be easily available at linear colliders. In an earlier paper [@PP], we had shown how polarization helps to put independent limits on electric and weak dipole couplings, while providing greater sensitivity in the case of asymmetries. We also demonstrate these advantages for the present case, strengthening the case for polarization studies.
We first describe the calculation of these asymmetries in terms of the electric and weak dipole couplings of the top quarks. These $CP$-violating couplings give rise to top polarization asymmetries in the production of $\tt$ in $\ee \ra \tt$ which in turn give rise to angular asymmetries in the subsequent decay $t \ra b l^+ \nu_l\;
(\overline{t} \ra
\overline{b} l^- \overline{\nu_l})$. We adopt the narrow-width approximation for $t$ and $\overline{t}$, as well as for $W^{\pm}$ produced in $t,\;\overline{t}$ decay.
We assume the top quark couplings to $\g$ and $Z$ to be given by the vertex factor $ie\Gamma_\mu^j$, where \_\^j=c\_v\^j\_+c\_a\^j\_\_5+ i\_5 (p\_t-p\_)\_,j=,Z, with &=&,=0,\
&=&,\
&=&-, and $x_w=sin^2\theta_w$, $\theta_w$ being the weak mixing angle. We have assumed in (1) that the only addition to the SM couplings $c^{{\g},Z}_{v,a}$ are the $CP$-violating electric and weak dipole form factors, $e\cdg/m_t$ and $e\cdz/m_t$, which are assumed small. Use has also been made of the Dirac equation in rewriting the usual dipole coupling $\sigma_{\mu\nu}(p_t+p_{\overline{t}})^{\nu}\g_5$ as $i\g_5(p_t-p_{\overline{t}})_{\mu}$, dropping small corrections to the vector and axial-vector couplings. We assume that there is no $CP$ violation in $t$, $\overline{t}$ decay$^2$.
The helicity amplitudes for $\ee \ra \g^*,Z^* \ra \tt$ in the cm frame, including $\cdgz$ couplings, have been given in [@asymm] (see also Kane [*et al.*]{}, ref. [@toppol]). We have calculated the helicity amplitudes for $$t \ra b W^+,\;
\;W^+ \ra l^+ \nu_l$$ and$$\overline{t} \ra \overline{b}W^-,\;\; W^- \ra
l^-\overline{\nu_l}$$ in the respective rest frames of $t$, $\overline{t}$, assuming standard model couplings and neglecting all masses except $m_t$, the top mass. The expressions for these can be found in [@longpaper].
Combining the production and decay amplitudes in the narrow-width approximation for $t,\overline{t},W^+,W^-$, and using appropriate Lorentz boosts to calculate everything in the $\ee$ cm frame, we obtained the $l^+$ and $l^-$ distributions for the case of $e^-$, $e^+$ with polarization $P_e$, $P_{\o e}$, the expressions can again be found in [@longpaper]. We further carry out the necessary integrations to obtain only the polar angle distributions for the leptons, which we use to write down the expressions for the $CP$-violating asymmetries defined below.
We define two independent -violating asymmetries, which depend on different linear combinations of Im$\cdg$ and Im$\cdz$ . (It is not possible to define -odd quantities which determine Re$\cdgz$ using single-lepton distributions [@longpaper]). One is simply the total lepton-charge asymmetry, with a cut-off of $\theta_0$ on the forward and backward directions: \_[ch]{}(\_0)=. The other is the leptonic forward-backward asymmetry combined with charge asymmetry, again with the angles within $\theta_0$ of the forward and backward directions excluded: \_[fb]{}(\_0)= .
In the above equations, $\sigma^+$ and $\sigma^-$ refer respectively to the cm $l^+$ and $l^-$ distributions. $\thetal$ is used to represent the polar angle angle of either $l^+$ or $l^-$, with the $z$ axis chosen along the $e^-$ momentum.
These asymmetries are a measure of $CP$ violation in the unpolarized case and in the case when polarization is present, but $P_e=-P_{\overline{e}}$. When $P_e\neq -P_{\overline{e}}$, the initial state is not invariant under $CP$, and therefore $CP$-invariant interactions can contribute to the asymmetries. However, to the leading order in $\alpha$, these $CP$-invariant contributions vanish in the limit $m_e=0$. Order-$\alpha$ collinear helicity-flip photon emission can give a $CP$-even contribution. However, this background can be suppressed by a suitable cut on the visible energy.
The expressions for ${\cal A}_{ch}(\theta_0)$ and ${\cal
A}_{fb}(\theta_0)$ and are given below.
$$\begin{aligned}
\lefteqn{{\cal A}_{ch}(\theta_0)=\frac{1}{2\sigma
(\theta_0)}\frac{3\pi\alpha^2} {4s}B_tB_{\o
t}\,2\cos\theta_0\sin^2\theta_0
\left(
(1-\beta^2)\log\frac{1+\beta}{1-\beta}-2\beta\right)}\nonumber\\
&&\times\left( {\rm Im}\cdg \left\{\left[ 2 c_v^
{\gamma}+(r_L+r_R)c_v^Z\right](1-P_eP_{\overline
e})+(r_L-r_R)c_v^Z(P_{\overline e}-P_e)
\right\} \right.\nonumber \\&&\left.
+ {\rm Im}\cdz \left\{\left[ (r_L+r_R) c_v^
{\gamma}+(r_L^2+r_R^2)c_v^Z\right](1-P_eP_{\overline e})+
\left[(r_L-r_R)c_v^{\gamma}\right.\right.\right.\nonumber\\
&&\left.\left.\left.+(r_L^2-r_R^2)c_v^Z\right] (P_{\overline
e}-P_e)\right\} \right);\end{aligned}$$
$$\begin{aligned}
{\cal A}_{fb}(\theta_0)&=&\frac{1}{2\sigma
(\theta_0)}\frac{3\pi\alpha^2} {2s}B_tB_{\o t}\,\cos^2\theta_0
\left(
(1-\beta^2)\log\frac{1+\beta}{1-\beta}-2\beta\right)c_a^Z\nonumber\\
&&\times \left\{ {\rm Im}\cdg \left[ (r_L-r_R)(1-P_eP_{\overline
e})+(r_L+r_R)(P_{\overline e}-P_e)
\right] \right.\nonumber \\&&\left.
+ {\rm Im}\cdz \left[ (r_L^2-r_R^2)(1-P_eP_{\overline e})
+(r_L^2+r_R^2)(P_{\overline e}-P_e)\right]\right\}.\end{aligned}$$
Here $\sigma(\theta_0)$ is the cross section for $l^+$ or $l^-$ production with a cut-off $\theta_0$, and is given by $$\begin{aligned}
\sigma(\theta_0)&=& \frac{3\pi\alpha^2}{8s}B_tB_{\o t} \,
2\cos\theta_0\left(
\left\{(1-\beta^2)\log\frac{1+\beta}{1-\beta}\sin^2\theta_0
\right. \right. \nonumber \\
&& \left. \left. + 2\beta\left[1+(1-\frac{2}{3}
\beta^2)\cos^2\theta_0\right]\right\} \right.\nonumber\\
&& \times \left.
\left\{\left[2{c_v^{\g}}^2+2c_v^{\g}c_v^Z(r_L+r_R)+{c_v^Z}^2
(r_L^2+r_R^2)\right](1-P_eP_{\overline e})\right.\right.\nonumber \\
&&\left. \left. +c_v^Z\left[(r_L-r_R)c_v^{\g} +
(r_L^2-r_R^2)c_v^Z\right](P_{\overline e}-P_e)\right\}\right.
\nonumber \\
&&+\left.\left\{(1-\beta^2)\log\frac{1+\beta}{1-\beta}\sin^2\theta_0
+ 2\beta\left[2\beta^2-1+(1-\frac{2}{3}
\beta^2)\cos^2\theta_0\right]\right\}\right.\nonumber \\
&&\times\left. {c_a^Z}^2 \left\{(r_L^2+r_R^2)(1-P_eP_{\overline e}) +
(r_L^2-r_R^2) (P_{\overline e}-P_e)\right\} -2(1-\beta^2)\right.
\nonumber \\ &&\left. \times\left( \log\frac{ 1+\beta}{1-\beta} - 2
\right)\sin^2 \theta_0
c_a^Z\left\{\left[(r_L+r_R)c_v^{\g} + (r_L^2+r_R^2) c_v^Z \right]
\right.\right. \nonumber \\ &&\times\left. \left.(1-P_eP_{\overline
e}) + \left[ (r_L-r_R)c_v^{\g}+ (r_L^2-r_R^2) c_v^Z
\right] (P_{\overline e}-P_e)\right\}
\right).\end{aligned}$$
In these equations, $\beta$ is the $t$ (or $\o t$) velocity: $\beta=\sqrt{1-4m_t^2/s}$, and $\gamma = 1/\sqrt{1-\beta^2}$, and $B_t$ and $B_{\o t}$ are respectively the branching ratios of $t$ and $\o t$ into the final states being considered. $-er_{L,R}/s$ is the product of the $Z$-propagator and left-handed (right-handed) electron couplings to $Z$, with r\_L&=&,\
r\_R&=&.
We note the curious fact that ${\cal A}_{ch}(\theta_0)$ vanishes for $\theta_0=0$. This implies that the $CP$-violating charge asymmetry does not exist unless a cut-off is imposed on the lepton production angle. ${\cal A}_{fb}(\theta_0)$, however, is nonzero for $\theta_0=0$.
We now describe the numerical results for the calculation of 90% confidence level (CL) limits that could be put on Im$\cdgz$ using the asymmetries described earlier, as well as the $CP$-odd part of the angular distribution in eq. (9).
We look at only semileptonic final states. That is to say, when $t$ decays leptonically, we assume $\o t$ decays hadronically, and [*vice versa*]{}. We sum over the electron and muon decay channels. Thus, $B_tB_{\o t}$ is taken to be $2/3\times2/9$. The number of events for various relevant $\theta_0$ and for beam polarizations $P_e=0$, $\pm 0.5$ are listed in Table 1.
In each case we have derived simultaneous 90% CL limits on Im$c_d^{\g}$ and Im$c_d^Z$ that could be put in an experiment at a future linear collider with $\sqrt{s}=500$ GeV and an integrated luminosity of 10 fb$^{-1}$. We do this by equating the asymmetry (${\cal A}_{ch}$ or ${\cal A}_{fb}$) to $2.15/\sqrt{N}$, where $N$ is the total number of expected events. In the unpolarized case, each of ${\cal A}_{ch}$ and ${\cal A}_{fb}$ gives a band of allowed values in the Im$c_d^{\g}-$Im$c_d^Z$ plane. If both ${\cal A}_{ch}$ and ${\cal
A}_{fb}$ are looked for in an experiment, the intersection region of the corresponding bands determines the best 90% CL limits which can be put simultaneously on Im$c_d^{\g}$ and Im$c_d^Z$. These best results are obtained for $\theta_0=35^\circ$ and are shown in Fig. 1(a) and Fig. 1(b), for two values of the top mass, $m_t=174$ GeV, and $m_t=200$ GeV respectively.
We see from Fig. 1 that the 90% CL limits that could be put on Im$\cdg$ and Im$\cdz$ simultaneously are, respectively, 2.4 and 17, for $m_t=174$ GeV. The same limits are 4.0 and 28 for $m_t=200$ GeV.
In the case where the $e^-$ beam is longitudinally polarized, we have assumed the degree of polarization $P_e=\pm 0.5$, and determined 90% CL limits which can be achieved. In this case, the use of $P_e=+0.5$ and $P_e=-0.5$ is sufficient to constrain Im$c_d^{\g}$ and Im$c_d^Z$ simultaneously even though only one asymmetry (either ${\cal A}_{ch}$ or ${\cal A}_{fb}$) is determined. The 90% CL bands corresponding to $P_e=\pm0.5$ are shown in Figs. 2 and 3, for ${\cal A}_{ch}$ with $\theta_0=60^\circ$, and for ${\cal A}_{fb}$ with $\theta_0=10^\circ$, respectively. Again, these values of $\theta_0$ are chosen to maximize the sensitivity$^3$.
It can be seen from these figures that the simultaneous limits expected to be obtained on Im$\cdg$ and Im$\cdz$ are, respectively, about 0.45 and 1.5 for $m_t=174$ GeV from both the types of asymmetries. These limits are about 0.78 and 2.5 for $m_t=200$ GeV. We see thus that the use of polarization leads to an improvement of by a factor of about 5 in the sensitivity to the measurement of Im$\cdg$, and by a factor of at least 10 in the case of Im$\cdz$. Moreover, with polarization, either of ${\cal A}_{fb}$ and ${\cal
A}_{ch}$, with a suitably chosen cut-off, suffices to get the same improvement in sensitivity.
Apart from simultaneous limits on Im$\cdgz$, we have also found out the sensitivities of one of Im$\cdgz$, assuming the other to be zero, using the $CP$-odd combination of angular distributions $\f{d\sigma^+}{d\cos\theta} (\theta_l) - \f{d\sigma^-}{d\cos\theta}
(\pi-\theta_l)$. We assume that the data is collected over bins in $\theta_l$, and add the 90% CL limits obtained from individual bins in inverse quadrature. We find that the best individual limits are respectively 0.12 and 0.28 for Im$\cdg$ and Im$\cdz$, both in the case of $P_e=-0.5$, for $m_t=174$ GeV. The corresponding limits for $m_t=200$ GeV are 0.18 and 0.43. As expected, these limits are better than simultaneous ones. Even here, there is an improvement due to polarization, but it is not as dramatic as in the case of simultaneous limits.
Our limits on Im$\cdgz$ are summarized in Table 2.
To conclude, we have obtained expressions for certain simple $CP$-violating angular asymmetries in the production and subsequent decay of $\tt$ in the presence of electric and weak dipole form factors of the top quark. These asymmetries are specially chosen so that they do not require the reconstruction of the $t$ or $\o t$ directions or energies. We have also included the effect of longitudinal electron beam polarization. We have analyzed these asymmetries to obtain simultaneous 90% CL limits on the imaginary parts of the electric and weak dipole couplings which would be possible at future linear $\ee$ collider operating at $\sqrt{s}= 500$ GeV and with a luminosity of 10 fb$^{-1}$. Figs. 1-3 show the allowed regions in the Im$\cdg$–Im$\cdz$ plane at the 90% CL. Table 2 summarizes the 90% CL limits on Im$\cdgz$ in various cases.
Our general conclusion is that the sensitivity to the measurement of dipole couplings is improved considerably if the electron beam is polarized, a situation which might easily obtain at linear colliders. Another general observation is that the sensitivity is better for a lower top mass than a higher one.
If we compare these results for sensitivities with those obtained in [@PP], where we studied asymmetries requiring the top momentum determination, we find that while the sensitivities with the asymmetries studied here are worse by a factor of about 3 in the unpolarized case, the limits in the polarized case are higher by a factor of about 2 as compared to those in [@PP]. It is likely that since in the experiments suggested here, only the lepton charges and direction need be determined, improvement in experimental accuracy can easily compensate for these factors. A detailed simulation of experimental conditions is needed to reach a definite conclusion on the exact overall sensitivities.
We have also compared our results with those of [@cuypers], where CP-odd momentum correlations are studied in the presence of $e^-$ polarization. With comparable parameters, the sensitivities we obtain are comparable to those obtained in [@cuypers]. In some cases our sensitivities are slightly worse because we require either $t$ or $\overline{t}$ to decay leptonically, leading to a reduced event rate. However, the better experimental efficiencies in lepton momentum measurement may again compensate for this loss.
As mentioned earlier, since we consider only the electron beam to be polarized, the asymmetries considered here can have backgrounds from order-$\alpha$ collinear initial-state photon emission, which, in principle, have to be calculated and subtracted. However, in case of correlations, it was found in [@back] that the background contribution can be neglected for the luminosity we assume here. This is likely to be the case in the asymmetries we consider here.
The theoretical predictions for $c_d^{\g,Z}$ are at the level of $10^{-2}-10^{-3} $, as for example, in the Higgs-exchange and supersymmetric models of CP violation [@bern; @asymm; @new]. Hence the measurements suggested here cannot exclude these modes at the 90% C.L. However, as simultaneous model-independent limits on both $c_d^{Z}$ and $c_d^{\g}$, the ones obtainable from the experiments we suggest, are an improvement over those obtainable from measurements in unpolarized experiments.
Increase in polarization beyond $\pm 0.5$ can increase the asymmetries in some cases we consider. Also, a change in the $e^+\,e^-$ cm energy also has an effect on the asymmetries. However, we have tried to give here only the salient features of the outcome of a possible experiment in the presence of longitudinal beam polarization.
It is obvious that the success of our proposal depends crucially on proper identification of the $\tt$ events and measurements of the polar angles and the charges of leptons. This will require cuts, and will lead to experimetnal detection efficiencies less than one as assumed here. Our results are quantitatively exact under ideal experimental conditions. Inclusion of experimental detection efficiencies may change our results somewhat. However, the main thrust of our conclusions, that we have identified rather simple observables for measurement of dipole form factors, and that longitudinal beam polarization improves the sensitivity, would still remain valid.
[99]{} F. Abe [*et al.*]{} (CDF Collab.), Phys. Rev. Lett. 74 (1995) 2626; S. Abachi [*et al.*]{} (D0 Collab.), Phys. Rev. Lett. 74 (1995) 2632. I. Bigi and H. Krasemann, Z. Phys. C 7 (1981) 127; J. Kühn, Acta Phys. Austr. Suppl. XXIV (1982) 203; I. Bigi [*et al.*]{}, Phys. Lett. B 181 (1986) 157. J.F. Donoghue and G. Valencia, Phys. Rev. Lett. 58 (1987) 451; C.A. Nelson, Phys. Rev. D 41 (1990) 2805; G.L. Kane, G.A. Ladinsky and C.-P. Yuan, Phys. Rev. D 45 (1991) 124; C.R. Schmidt and M.E. Peskin, Phys. Rev. Lett. 69 (1992) 410; C.R. Schmidt, Phys. Lett. B 293 (1992) 111; T. Arens and L.M. Sehgal, Phys. Rev. D 50 (1994) 4372. W. Bernreuther, T. Schröder and T.N. Pham, Phys. Lett. B 279 (1992) 389; W. Bernreuther and P. Overmann, Nucl. Phys. B 388 (1992) 53; Z. Phys. C 61 (1994) 599; W. Bernreuther and A. Brandenburg, Phys. Lett. B 314 (1993) 104; Phys. Rev. D 49 (1994) 4481; J.P. Ma and A. Brandenburg, Z. Phys. C 56 (1992) 97; A. Brandenburg and J.P. Ma, Phys. Lett. B 298 (1993) 211. D. Atwood and A. Soni, Phys. Rev. D 45 (1992) 2405. F. Cuypers and S.D. Rindani, Phys. Lett. B 343 (1994) 333. D. Chang, W.-Y. Keung and I. Phillips, Nucl. Phys. B 408 (1993) 286; 429 (1994) 255 (E). P. Poulose and S.D. Rindani, Phys. Lett. B 349 (1995) 379. P. Poulose and S.D. Rindani, PRL Ahmedabad preprint PRL-TH-95/17 (1995), hep-ph/9509299. B. Grzadkowski and W.-Y. Keung, Phys. Lett. B 316 (1993) 137; E. Christova and M. Fabbrichesi, Phys. Lett. B 320 (1994) 299; A. Bartl, E. Christova and W. Majerotto, Preprint HEPHY-PUB 624/95, UWThPh-1995-9, hepph/9507445 (1995). B. Ananthanarayan and S.D. Rindani, Phys. Rev. D 52 (1995) 2684. A. Bartl [*et al.*]{}, ref. [@decay].
**Footnotes**
1.5cm
1. The paper of Atwood and Soni [@atwood] introduces optimal variables whose expectation values maximize the statistical sensitivity.
2. CP violation in top decays has been considered, for example, in [@decay].
3. In case of ${\cal A}_{fb}$, the choice $\theta_0=0$ also gives very similar sensitivities. However, since this condition would be impossible to achieve in a practical situation, we choose a nonzero value of $\theta_0$.
.5cm
**Figure Captions**
1.5cm Bands showing simultaneous 90% CL limits on Im $c_d^{\gamma}$ and Im $c_d^Z$ using ${\cal A}_{fb}$ and ${\cal A}_{ch}$ with unpolarized electron beam at cm energy 500 GeV and cut-off angle $35^{\circ}$. Mass of the top quark is taken to be (a) 174 GeV and (b) 200 GeV. v
Bands showing simultaneous 90% CL limits on Im $c_d^{\gamma}$ and Im $c_d^Z$ using ${\cal A}_{ch}$ with different beam polarizations, and at a cm energy of 500 GeV and cut-off angle $60^{\circ}$. Mass of the top quark is taken to be (a) 174 GeV and (b) 200 GeV. v
Bands showing simultaneous 90% CL limits on Im $c_d^{\gamma}$ and Im $c_d^Z$ using ${\cal A}_{fb}$ with different beam polarizations, and at a cm energy of 500 GeV and cut-off angle $10^{\circ}$. Mass of the top quark is taken to be (a) 174 GeV and (b) 200 GeV.
.5cm
v
**Table Captions**
1.5cm Number of $t \bar t$ events, with either or decaying leptonically, for c.m. energy 500 GeV and integrated luminosity $10\;{\rm fb}^{-1}$ for two different top masses with polarized and unpolarized electron beams at different cut-off angles $\theta_0$. v Limits on dipole couplings obtainable from different asymmetries. In case (a) limits are obtained from ${\cal A}_{ch}$ and ${\cal A}_{fb}$ using unpolarized beams (Fig. 1), and in case (b) from either of ${\cal A}_{ch}$ (Fig. 2) and ${\cal A}_{fb}$ (Fig. 3) with polarizations $P_e=0,\;\pm 0.5$. Charge-asymmetric angular distribution is used in case (c) where 0 and $\pm 0.5$ polarizations are considered separately. All the limits are at 90% CL.
[||c|c|c|c||c|c|c||]{} &&\
$\theta_0$& $P_e=-0.5$& $P_e=~0$&$P_e=+0.5$&$P_e=-0.5$&$P_e=~0$&$P_e=+0.5$\
$0^\circ$&1003&845&687 &862&723&585\
$10^\circ$&988&832&675 &849&712&576\
$35^\circ$&826&689&553 &711&593&475\
$60^\circ$&507&419&330 &438&362&286\
\
\
[|ll|c|c|c|c|]{} &&\
&$|{\rm Im}c_d^{\gamma}|$&$|$Im$
c_d^Z|$&$|$Im$c_d^{\gamma}|$&$|$Im$c_d^Z|$\
(a) unpolarized&&2.4&17& 4.0&28\
(b) polarized($P_e= 0,\;\pm$0.5)&&0.45&1.5&0.78&2.5\
(c) angular distribution:&$P_e=+0.5$&0.13&0.74&0.21&1.21\
&$P_e=~~0.0$&0.13&0.81&0.20&1.30\
&$P_e=-0.5$&0.12&0.28&0.18&0.43\
\
\
[^1]: On sabbatical leave from Theory Group, Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India, until September 19, 1996
|
---
abstract: 'The size of the smallest dark matter collapsed structures, or protohalos, is set by the temperature at which dark matter particles fall out of kinetic equilibrium. The process of kinetic decoupling involves elastic scattering of dark matter off of Standard Model particles in the early universe, and the relevant cross section is thus closely related to the cross section for dark matter scattering off of nuclei (direct detection) but also, via crossing symmetries, for dark matter pair production at colliders and for pair-annihilation. In this study, we employ an effective field theoretic approach to calculate constraints on the kinetic decoupling temperature, and thus on the size of the smallest protohalos, from a variety of direct, indirect and collider probes of particle dark matter.'
author:
- 'Jonathan M. Cornell'
- Stefano Profumo
- William Shepherd
title: |
Kinetic Decoupling and Small-Scale Structure\
in Effective Theories of Dark Matter
---
Introduction
============
In the paradigm of weakly interacting massive particles (WIMPs) as dark matter candidates, the abundance of dark matter observed in the Universe stems from thermal decoupling of the dark matter particles in the early universe. This process involves the pair-annihilation and pair-creation of WIMPs going out of [*chemical*]{} equilibrium, with the resulting number density freezing out and remaining approximately constant per comoving volume to the present age. WIMP models possess the right range of masses and pair-annihilation/creation cross sections to produce a thermal relic density in the same ballpark as the observed dark matter density, a feat often dubbed the “WIMP miracle” [@wimp_miracle].
After chemical decoupling, WIMPs do not cease to interact with the surrounding thermal bath. It is simply their number density which is no longer affected by particle-number-changing processes. WIMPs ($\chi$) continue to scatter off of (Standard Model) particles in the thermal bath ($f$), thus remaining in [*kinetic*]{} equilibrium with the thermal bath, up until the relevant elastic processes ($\chi f\leftrightarrow\chi f$) go out of equilibrium, i.e. the rate for such processes falls below the Hubble expansion rate. At that point, WIMPs completely decouple from the thermal bath, free-streaming and slowing down as the Universe keeps expanding. To a first approximation, this is the age when the first gravitationally collapsed dark matter structures form, with typical size on the same order as an Hubble length at that epoch. WIMP [*kinetic*]{} decoupling thus sets the small-scale cut-off to the dark matter power spectrum (for a recent review see e.g. Ref. [@review]).
Given a WIMP model, it is thus in principle a well-posed question to ask what the small-scale cutoff to dark matter halos (which we hereafter refer to with the symbol $M_{\rm cut}$) is. The cut-off scale is an important quantity in cosmology: if large enough, it could affect significantly how many “visible” small-scale structures, such as dwarf galaxies, form, perhaps being relevant to the question of the “missing satellite problem” [@Strigari:2007ma] or to other issues associated with small scales in cold dark matter cosmologies [@Primack:2009jr]. In principle, the small-scale cutoff sets the size of the most numerous dark matter “mini-halos”, or protohalos, which might be detectable today either with direct [@directhalos] or indirect [@indirecthalos] dark matter search experiments. Finally, the cutoff scale is highly relevant to the question of the so-called “boost factor”, as it literally sets the integration cutoff in the calculation of this factor (in practice, the enhancement to the annihilation rate from a given dark matter halo from sub-structure within the halo).
The calculation of the kinetic decoupling temperature $T_{\rm kd}$, and thus of the small-scale cutoff $M_{\rm cut}$ has been carried out in a variety of model-dependent contexts, including supersymmetry [@Hofmann:2001bi; @Bringmann:2006mu; @Profumo:2006bv], universal extra-dimensions [@Profumo:2006bv; @Hooper:2007qk], and models with Sommerfeld enhancement [@vandenAarssen:2012ag; @Aarssen:2012fx]. It has become clear that WIMP models accommodate a broad variety of kinetic decoupling temperatures, with resulting cutoff scales ranging from $10^3\ M_\odot$ to much less than $10^{-6}\ M_{\odot}$ even only within the limited framework of the minimal supersymemtric extension of the Standard Model [@Profumo:2006bv], where the symbol $M_\odot$ indicates the mass of the Sun. Particle physics details of the WIMP model affect in a highly model-dependent way the kinetic decoupling, producing a wide array of outcomes, but for many particle physics models there is still a decent correlation between certain experimentally accessible quantities such as the direct detection scattering cross section, as explored in Ref. [@Cornell:2012tb], and $M_{\rm cut}$.
A possible model-independent route to evaluating ranges for the expected small-scale cutoff is to consider an effective theory description of interactions between Standard Model and dark matter particles, as pursued, recently, in Ref. [@Gondolo:2012vh; @Shoemaker:2013tda]. For example, assuming the dark matter is a spin 1/2 fermion, it is simple to write down the complete set of lowest dimensional operators that mediate such interactions. In turn, by assuming that only one single operator is dominating the relevant dark matter interactions, crossing symmetry allows to draw stringent constraints on the allowed effective energy scale associated with the operator, for example from direct dark matter detection or from collider searches. As a result we can robustly set [*upper limits*]{} to the size of the small-scale cutoff, for each class of operators, as a function of the relevant operator’s effective energy scale. This upper limit is quite significant, as cosmologically relevant effects occur only for sufficiently large such cutoffs.
While rather sophisticated codes now exist to reliably calculate $T_{\rm kd}$ (see e.g. [@review]), two potentially important ingredients have been only marginally studied thus far:
\(i) scattering off of quarks only, for example in “lepto-phobic” theories with suppressed couplings to leptons (this was first partly addressed in Ref. [@Gondolo:2012vh]), and
\(ii) the role of scattering off of pions, for the same class of theories, for kinetic decoupling temperatures below the QCD confinement phase transition.
In addition, a third aspect that remains entirely unexplored to date is (iii) the relevance of loop-mediated scattering off of leptons, again notably in leptophobic theories.
In the present study, in addition to the general program of setting upper limits to the small-scale cutoff in the context of the mentioned effective theory description of interactions between Standard Model and dark matter particles, we address in detail the three novel issues listed above. We show that for leptophobic theories there exists an interesting interplay between loop-mediated scattering off of leptons and scattering off of pions, and that the two effects are generically comparable. We find that for WIMP models that can be described to a good approximation by an effective operator belonging to the class we consider here, there are stringent upper limits on the cut-off scale to the matter power spectrum, typically on the order of $10^{-3}\ M_\odot$. This scale hints at the fact that WIMP effective theories are not likely to have any impact on small-scale structure issues in cold dark matter cosmology. On the other hand, since the predicted protohalos are typically very small, sizable boost factors from substructure enhancements are a rather generic prediction of effective theories of dark matter.
The reminder of this paper is organized as follows: we outline the class of effective operators we consider in the following section \[sec:operators\]; we then discuss how we calculate the kinetic decoupling and how we estimate the size of the small-scale halo size cutoff in section \[sec:protohalos\]; section \[sec:results\] presents all of our results; and the final section \[sec:concl\] summarizes our findings and concludes.
Classification of effective operators {#sec:operators}
=====================================
The effective operator framework has been explored as a method for comparing experimental bounds coming from various types of experiments on dark matter couplings to Standard Model fields [@EffDM; @Fox:2011fx]. Within this framework, one writes down higher-dimensional operators which couple dark matter to quarks, leptons, or Standard Model bosons, requiring that (i) the operator contain at least two dark matter particles to ensure stability, and that (ii) Standard Model gauge symmetries are respected. One operator from the list of possible operators is then assumed to be the dominant one for the physics being investigated, and its effects are explored assuming the other operators are suppressed and, thus, do not contribute to the observables in question. Each operator of interest is investigated separately in this way, and any interference effects from having multiple operators active simultaneously are assumed to be small. Generally these interference effects are equivalent to changing the assumed chirality structure of the operator in question, e.g. interfering a vector and an axial operator with equal suppression scales is equivalent to considering an operator which only couples to one chirality.
The basic assumption of this parametrization of dark matter interactions is that dark matter is the only new field light enough to be kinematically relevant, and these operators are suppressed by a mass scale which is related to the expected mass of the additional particles which mediate the interactions in some more complete model underlying the effective theory. Within the region of parameter space where this assumption is valid, a given complete model can be mapped into the space of these operators by integrating out the additional heavy fields. This assumption is a fairly weak one for elastic scattering of dark matter off of Standard Model particles, where the momentum exchange is typically on the order of the MeV, but is a fairly strong assumption for LHC searches, where the center of mass energy of the created dark matter pair can be quite large compared to the dark matter mass. We therefore encourage caution when considering the collider bounds on these operators, but expect that the results for kinetic decoupling and the bounds arising from direct detection should be robust.
We also calculate the thermal relic density of WIMPs under the assumption that the same operator dominates dark matter interactions with Standard Model particles in the early as well as in the late universe. Of course, this is a rather strong assumption, as it entails for example the absence of processes such as coannihilation, the presence of thresholds or resonances that could exist at finite temperature but not in the late universe, and the absence of temperature-suppressed operators that might dominate the chemical freeze-out while being irrelevant at the later kinetic freeze-out. We note, however, that this assumption is largely equivalent to other assumptions discussed above, where it is presumed that dark matter is the only kinematically relevant new particle in the theory and that one operator is dominant in all of the observables being searched for.
We consider here a subset of all possible operators which conserve parity in addition to the Standard Model gauge symmetries. The operators of interest are $$\begin{aligned}
\cal{O}_S&=&\frac{m_f}{\Lambda_S^3}\bar\chi\chi\bar ff \label{eq:scalarop}\\
\cal{O}_P&=&\frac{m_f}{\Lambda_P^3}\bar\chi\gamma^5\chi\bar f\gamma^5f \label{eq:pscalarop}\\
\cal{O}_V&=&\frac{1}{\Lambda_V^2}\bar\chi\gamma^\mu\chi\bar f\gamma_\mu f \label{eq:vectorop}\\
\cal{O}_A&=&\frac{1}{\Lambda_V^2}\bar\chi\gamma^\mu\gamma^5\chi\bar f\gamma_\mu\gamma^5f \label{eq:pvectorop} \\
\cal{O}_T&=&\frac{m_f}{\Lambda_T^3}\bar\chi\sigma^{\mu\nu}\chi\bar f\sigma_{\mu\nu}f \label{eq:tensorop},\end{aligned}$$ where $\Lambda_I$ is the suppression scale for operator $\cal{O}_I$. Note that the operators which are chirality-violating are assumed to be proportional to the fermion mass to preserve $SU(2)_L$ and avoid inducing large effects in low-energy flavor observables. The first four operator normalizations are standard within the effective dark matter literature, but previous searches for contact operators have not included the mass suppression for the tensor operator to better make contact with direct detection bounds. We choose to consider the theoretically better motivated normalization of the tensor operator which does include a quark mass suppression, as the operator is chirality-violating and thus would require an insertion of the Higgs field to respect the SM gauge symmetries. Previous analyses have considered the operator without a quark mass dependence to make better contact with direct detection searches, as the unsuppressed tensor induces a coupling to the spin of the quarks composing the nucleon minus the spin of the antiquarks in the nucleon, but it is not clear how a tensor operator with that normalization would be alligned with the mass basis of the quarks so well as to avoid inducing unacceptably large corrections to flavor observables. The choice to include the quark mass suppression of the tensor operator leaves us without collider and direct detection bounds to compare to, and therefore we will only plot the early universe curves for these operators.
For each operator, we specify which Standard Model fermions the dark matter particle couples to. Generically, leptons are the most significant contributors to keeping the dark matter in kinetic equilibrium with the Standard Model thermal bath, while many of the key experimental searches constrain primarily the couplings to quarks. We choose here to consider explicitly three cases, wherein the dark matter couples only to leptons, only to quarks, or to both with equal suppression scales. For cases including quark couplings we plot the strongest available experimental bounds from LHC searches and direct detection searches, and in cases including lepton couplings we will additionally plot LEP search bounds. In the special case of the lepton-only vector operator we will in addition plot the direct detection bounds induced at one-loop order, as discussed in Ref. [@ZupanKoppetal; @Fox:2011fx].
The formation of protohalos {#sec:protohalos}
===========================
Temperature of kinetic decoupling {#sec:tkd}
---------------------------------
To calculate the temperature of kinetic decoupling, we use the numerical routine described in Ref. [@review], which has been integrated into the DarkSUSY code [@darksusy]. An effective WIMP temperature parameter is defined in the following form: $$\label{eq:tchi}
T_\chi \equiv \frac{2}{3} \left< \frac{{\bf p}^2}{2 m_\chi} \right> = \frac{1}{3 m_\chi n_\chi} \int \frac{{\rm d}^3 p}{(2 \pi)^3} {\bf p}^2 f({\bf p}).$$ In the equation above $m_\chi$ is the WIMP mass and $n_\chi$ is its number density. To determine the time evolution of this parameter, we consider the Boltzmann equation for a flat Friedmann-Robertson-Walker metric: $$\label{eq:Boltzmann}
E(\partial_t - H {\bf p} \cdot \nabla_{\bf p}) f = C[f].$$ Here $f$ is the WIMP phase space density, $E$ and [**p**]{} are the comoving energy and 3-momentum respecitively, and $H$ is the Hubble parameter. $C[f]$ is the collision term for a scattering process between a non-relativistic WIMP and a relativistic Standard Model scattering partner. This was shown in Ref. [@review] to be of the form $$C[f] = c(T) m_\chi^2 \left[m_\chi T \nabla^2_{\bf p} + {\bf p} \cdot \nabla_{\bf p} + 3 \right] f ({\bf p}),$$ where $$\label{cTdef}
c(T) = \sum_i\frac{g_\mathrm{SM}}{6(2\pi)^3m_\chi^4T} \int dk\,k^5 \omega^{-1}\,g^\pm\left(1\mp g^\pm\right)\mathop{\hspace{-11ex}\overline{\left|\mathcal{M}\right|}^2_{t=0}}_{\hspace{4ex}s=m_\chi^2+2m_\chi\omega+m_\ell^2}\,.$$ In Equation (\[cTdef\]), the sum is taken over all possible Standard Model scattering partners, $g_{\rm SM}$ is the number of associated spin degrees of freedom, $\omega$ is the energy of the Standard Model particle and $k$ its momentum, and $g^\pm$ is the distribution for Fermi or Bose statistics, $g^\pm(\omega) = (e^{\omega/T} \pm 1)^{-1}$. In all expressions above, the upper sign is for fermions and the lower is for bosons. $\overline{\left|\mathcal{M}\right|}^2$ represents the scattering amplitude squared, summed over final and averaged over initial spin states. Detailed calculations of $\overline{\left|\mathcal{M}\right|}^2$ for all relevant cases for our results are included in the appendices.
As kinetic decoupling can take place either before or after the QCD phase transition at $T_c \approx 170 ~{\rm MeV}$, we need to consider carefully the effects of quark confinement on the above sum. At temperatures before $4 \, T_c$, we follow the convention of Ref. [@review], where the WIMPs scatter off leptons and, to be conservative, the three lightest quarks. After $4 \, T_c$, we no longer consider scattering off quarks. We however extend the treatment of Ref. [@review] by including scattering of the dark matter off pions after the QCD phase transitions for the cases in which this process occurs at leading order, i.e. for the scalar, Eq. (\[eq:scalarop\]), and vector, Eq. (\[eq:vectorop\]), operator cases. Also, it is important to note in the above expression for $c(T)$, the scattering amplitude is evaluated in the $t= 0$ limit, where $t$ is the squared difference between the incoming and outgoing 4-momenta of a scattering particle. This limit is reasonable because the average momentum transfer in a scattering event between a relativistic particle and a heavy WIMP should be quite small. However, for the pseudoscalar case, Eq. (\[eq:pscalarop\]), the scattering amplitude vanishes for forward scattering, so we need to consider the scattering amplitude when the momentum transfer is not zero. Ref. [@Gondolo:2012vh] introduced a method to average over all possible values of $t$, in which $c(T)$ now takes the form: $$c(T) = \sum_i\frac{g_\mathrm{SM}}{6(2\pi)^3m_\chi^4T} \int dk\, k^5 \omega^{-1}\,g^\pm\left(1\mp g^\pm\right) \frac{1}{\left(4 k^2 \right)^2} \int_{-4k^2}^0 dt (-t) \mathop{\overline{\left|\mathcal{M}\right|}^2_{\hspace{0ex}s=m_\chi^2+2m_\chi\omega+m_\ell^2}}.$$
Returning now to $T_\chi$ in Eq. (\[eq:tchi\]), to find the differential equation which describes its evolution, we multiply Eq. \[eq:Boltzmann\] by ${\bf p}^2/E$ and integrate over ${\bf p}$ to find $$\label{eq:diff}
\left( \partial_t + 5 H \right) T_\chi = 2 m_\chi c(T) \left(T-T_\chi \right).$$ The behavior of $T_\chi$ has two limiting cases: at high temperatures when $T_\chi = T$ and at low temperatures when $T_\chi$ changes only because of the expansion of the universe, i.e. $T_\chi \propto a^{-2}$, and the kinetic decoupling temperature is when there is a rapid change between these two behaviors. As described in Ref. [@review], a code has been developed to numerically integrate Eq. (\[eq:diff\]) and find this transition temperature, and we use this routine to calculate $T_{\rm kd}$.
Protohalo Size
--------------
There are two mechanisms which independently set a limit on the smallest possible protohalo mass, $M_{\rm cut}$: (i) the free streaming of WIMPs after kinetic decoupling and (ii) the coupling of the WIMP fluid to acoustic oscillations in the SM particle heat bath. In determining our limit on the protohalo mass, we use the outcome of these two processes giving the largest (hence dominant) ${M_{\rm cut}}$.
### Viscosity and Free Streaming
At kinetic decoupling, the decoupling of the WIMP fluid from the SM particle fluid leads to viscosity between the two fluids that cause density perturbations in the WIMP fluid to be damped out [@Hofmann:2001bi]. After $T_{\rm kd}$, the WIMPs free stream from areas of high to low density, causing further damping of the perturbations. The net result of these processes is an exponential damping of the perturbations with a characteristic comoving wavenumber [@green; @Green:2005fa]: $$\label{eq:fs}
k_\mathrm{fs} \approx \left( \dfrac{m}{T_\mathrm{kd}} \right)^{1/2} \dfrac{a_\mathrm{eq} / a_\mathrm{kd}}{\ln (4 a_\mathrm{eq} / a_\mathrm{kd})} \dfrac{a_\mathrm{eq}}{a_\mathrm{0}} H_\mathrm{eq}.$$ In the Equation above the “eq” subscript signifies that the quantity should be evaluated at matter-radiation equality. To find the resulting mass cutoff from these effects, one just calculates the mass contained in a sphere of radius $\pi/k_\mathrm{fs}$, i.e. [@review]: $$M_\mathrm{fs} \approx \dfrac{4 \pi}{3} \rho_\chi \left( \dfrac{\pi}{k_\mathrm{fs}} \right)^3 = 2.9 \times 10^{-6} M_{\odot} \left( \frac{1 + \ln \left(g_\mathrm{eff}^{1/4} T_\mathrm{kd}/ \mathrm{30 \ MeV} \right)/18.56} {\left(m_\chi / \mathrm{100 \ GeV} \right)^{1/2} g_\mathrm{eff}^{1/4} \left(T_\mathrm{kd}/\mathrm{50 \ MeV} \right)^{1/2}} \right)^3.$$ In the above equation $g_{\rm eff}$ is the number of effective degrees of freedom in the early universe evaluated at $T_{\rm kd}$.
### Acoustic Oscillations
It has also been noted that the density perturbations in the WIMP fluid, coupled to the SM particle fluid before $T_{\rm kd}$, should oscillate with the acoustic oscillations in the heat bath. At kinetic decoupling, modes of oscillation with $k$ values large enough that they have entered the horizon are damped out, while modes with $k$ values corresponding to scales larger than the horizon size grow logarithmically [@loebzalda; @Bertschinger:2006nq]. Therefore, the characteristic damping scale is just the size of the horizon at kinetic decoupling ($k_{\rm ao} \approx \pi H_{\rm kd}$), and there is another cutoff mass from this process of the form [@review]: $$\label{eq:mao}
M_\mathrm{ao} \approx \dfrac{4 \pi}{3} \dfrac{\rho_\chi}{H^3} \bigg|_{T=T_\mathrm{kd}} = 3.4 \times 10^{-6} M_\odot \left( \dfrac{T_\mathrm{kd} g_\mathrm{eff}^{1/4}}{50 \; \mathrm{MeV}} \right)^{-3}.$$
Results {#sec:results}
=======
As discussed above, for each effective operator in Eq. (\[eq:scalarop\]-\[eq:tensorop\]) we consider three cases as far as the relevant Standard Model particle class the dark matter couples to:
1. Universal couplings to all SM fermions;
2. Couplings to leptons only;
3. Couplings to quarks only.
Each case presents distinct behaviors in the early as well as in the late universe, and leads to different constraints and conclusions for the effective cutoff scale. Leptonic couplings, when present, tend to dominate the process of kinetic decoupling, as a simple result of the fact that at the relevant temperatures leptons (especially electron/positron and neutrinos) are in a relativistic state and the number densities are not Boltzmann-suppressed. On the other hand, quark couplings lead to stronger bounds from colliders and direct detection. If lepton couplings are absent then the contributions of quark couplings to kinetic decoupling must be treated with care due to the QCD confinement phase transition. Before the phase transition there is a thermal bath of quarks and the calculation of the scattering rate proceeds analogously with that for the leptonic couplings, but after the phase transition pions are the dominant hadrons and the matrix element of the quark bilinear in the pion must be evaluated. In addition, loop-induced scattering off of leptons arises generically even for vanishing direct couplings to leptons. This effect, which has never been considered in this context before, competes with scattering off of pions, and becomes more and more relevant as pions become less and less abundant at decreasing temperatures due to Boltzmann suppression. We will discuss each operator’s coupling to pions individually in presenting our results.
For each case we also present all relevant bounds on effective dark matter interactions from collider searches both at the LHC [@atlasconf; @cmspas] and at LEP [@Fox:2011fx]. These constraints are subject to the concerns described in section \[sec:operators\] regarding the possibility of probing additional particles at colliders due to the large center of mass energies involved. For all operators which lead to appreciable direct detection cross sections we also plot the current leading bounds from those experiments. For spin-independent scattering the current leading bounds come from the Xenon 100 experiment [@Aprile:2012nq], while for spin-dependent scattering they are set by the SIMPLE [@Felizardo:2011uw] and PICASSO [@Archambault:2012pm] experiments.
For all plots we also present relic density constraints. The line on the plots corresponds to when $\Omega_\chi h^2 = 0.1189$, the best fit value quoted by the Planck collboration [@planck] when combining their CMB results, WMAP polarization results, high-$\ell$ CMB data from ground telescopes and baryon acoustic oscillation measurements. For all operators except the tensor case (which has no simple, tree-level UV completion) we use the micrOMEGAs code [@Belanger:2010pz; @Belanger:2006is] to calculate the relic density. This was checked analytically to correspond with setting the annihilation cross section to the appropriate value $\langle\sigma v\rangle\approx3\times10^{-26}~{\rm cm^3/s}$, and this analytical requirement was used to calculate the relic density requirement for the tensor operator case.
Scalar Operator
---------------
The scalar-type coupling of dark matter to SM fermions contributes to direct detection in the case of quark couplings, and has been constrained by collider searches for both quark and lepton couplings. The collider constraints are relatively weak in this case, however, because of the mass-suppression of this chirality-violating operator. While pair annihilation, direct detection, and scattering responsible for kinetic decoupling all have access to the heavier SM fermion generations, the initial state, for collider studies, is dominated by the lighter states, and therefore collider bounds are weakened relative to the other dark matter probes.
For the case of universal coupling to SM fermions through the scalar operator the results are presented in figure \[fig:Scalar\_univ\]. We note that for dark matter above approximately 10 GeV in mass the bounds from direct detection, which are strongest in that region, indicate that the kinetic decoupling temperature must be on the order of 1 GeV. The resulting cutoff scale for the smallest protohalos is on the order of the Earth mass (about $10^{-6}\ M_{\odot}$) for WIMP masses above 10 GeV. The relic density matches the observed dark matter only for masses above 200 GeV. Models that possess the right thermal relic density have extremely suppressed cutoff scales, smaller than $10^{-9}\ M_\odot$ (see the right panel of figure \[fig:Scalar\_univ\]).
For the case of lepton-only couplings, the only relevant bound on this operator is from LEP, and the resulting bound is weak enough to not significantly constrain the process of kinetic decoupling. The corresponding results are shown in figure \[fig:Scalar\_leptons\], which indicates that kinetic decoupling temperatures below 10 MeV are possible in this case, resulting in small-scale cutoffs exceeding the Earth mass. We estimate in this case that the largest possible cutoff mass scale is of about $M_{\rm cut}\sim 10^{-3}\ M_\odot$. We also note that the LEP limits do not impact the cutoff scales $\Lambda_S$ needed to produce the correct thermal relic density. For WIMP masses of about 10 GeV we find that models that have the correct thermal relic density produce a small scale cutoff of $10^{-5}\ M_\odot$, while those with a mass of 100 GeV of about $10^{-7}\ M_\odot$ and those with a mass of 1 TeV of approximately $10^{-9}\ M_\odot$.
Finally, for quark-only couplings the matrix element $\langle\pi|\bar qq|\pi\rangle$, implicitly summed over quark flavors, is needed to evaluate the coupling to pions after the QCD phase transition. This has been evaluated previously in the context of contributions to direct detection by [@kamionkowskietal] using soft-pion techniques to be $$\langle\pi^a|\bar qq|\pi^a\rangle=\frac{m_\pi^2}{2}\langle\pi^a|\vec\pi\cdot\vec\pi|\pi^a\rangle,$$ where $\vec\pi$ is a pion iso-vector, Eq. (\[eq:pionvec\]). We have implemented this scattering amplitude for interactions after the QCD phase transition with pions, which are the dominant components of the thermal bath at the relevant temperatures. The results for quark-only couplings are shown in figure \[fig:Scalar\_quarks\]. Direct detection forces in this case the size of the smallest protohalos to values well below $10^{-9}\ M_{\odot}$ for dark matter particle masses larger than 20 GeV. Models with the correct relic density must have masses above 200 GeV, and small scale cutoff smaller than $10^{-11}\ M_{\odot}$ in this case.
Vector Operator
---------------
{width="100.00000%"}
{width="100.00000%"}
Vector operators are generically better constrained by collider searches than scalar operators are, but are also very tightly constrained by direct dark matter detection. For universal couplings, direct detection is again the dominant constraint for dark matter masses above about 10 GeV, and those constraints again force us to conclude that the kinetic decoupling temperature must be of order 1 GeV. We show our results in figure \[fig:Vector\_univ\]. We do not find any sub-TeV WIMP models with a viable thermal relic density, if this operator is the only important contributor to chemical freeze-out. For masses above 100 GeV, we find that the cutoff scale is always smaller than approximately $10^{-9}\ M_\odot$.
Our results for vector interactions only with leptons are shown in figure \[fig:Vector\_leptons\]. We have plotted bounds from LEP and from direct detection, which arise at one-loop level by effectively inducing a mixing between the integrated-out heavy vector boson and the SM photon. This mechanism was first discussed by Fox et. al. [@Fox:2011fx], and the bounds we plot are updates of those they derived from the first release of Xenon 100 data to take in to account the full 2012 data set. Even with a loop suppression, direct detection is still the dominant bound on dark matter models with masses above about 8 GeV, and the decoupling temperature is required to be of order 100 MeV. The resulting smallest possible protohalos are smaller than about $10^{-5}\ M_\odot$ for WIMP masses below 100 GeV, and are generically of order $10^{-7}\ M_\odot$ or smaller for masses above 100 GeV.
Considering couplings to quarks only below the QCD phase transition, we now must evaluate $\langle\pi^a|\bar q\gamma^\mu q|\pi^a\rangle$. This also was shown in [@kamionkowskietal], using the conservation of the vector current, to be $$\langle\pi^a|\bar q\gamma^\mu q|\pi^a\rangle=\left(a_u-a_d\right)\langle\pi^a|\vec\pi\times\partial^\mu\vec\pi|\pi^a\rangle,$$ where $a_q$ is the coupling to quarks of type $q$. This clearly vanishes for the coupling structure we have chosen of universal couplings to all quark flavors.
For a vector interaction coupling only to quarks, the induced direct detection amplitude in the former case can be effectively inverted to give an induced, loop-level coupling to leptons, which can be important as leptons are generically greater contributors to kinetic decoupling than quarks. Since kinetic decoupling is dominated by scatterings at low dark matter velocities, the loop induced coupling to leptons from quarks can be considered as a simple rescaling of the suppression scale involved, and can be calculated in the “leading-log” approximation as discussed in Ref. [@ZupanKoppetal]. The formula for this rescaling in the case where we consider identical couplings to all quark flavors is $$\label{eq:loop}
\Lambda_\ell=\sqrt{\frac{3\pi}{2\alpha}}\frac{\Lambda_q}{\sqrt{\sum_{d,s,b}\ln\left(m_q/\Lambda_q\right)-2\sum_{u,c,t}\ln\left(m_q/\Lambda_q\right)}},$$ where $\alpha$ is the electromagnetic fine structure constant and all running quantities are evaluated at the renormalization scale of $\Lambda_q$, the scale of the effective operator. While this does not minimize the logarithms involved, it does allow us to neglect renormalization running and mixing of different operators, such that it is well-defined to assume one operator is dominant. We enforce perturbativity of this loop expansion by truncating our results when the induced coupling to leptons is not weaker than the initial coupling to quarks. The results for this coupling structure are presented in figure \[fig:Vector\_quarks\]. We find, as in fig. \[fig:Vector\_univ\], that no models with sub-TeV masses have the right thermal relic density, and that the predicted cutoff for masses above 10 GeV is always smaller than $10^{-7}\ M_\odot$, while it is smaller than $10^{-9}\ M_\odot$ for masses above 100 GeV.
To explore the relative importance of pion scattering to that of loop-induced lepton scattering, we also consider a quarks-only vector-like operator which couples with opposite sign to up- and down-type quarks. This doesn’t change the bounds from colliders or the scattering amplitudes above the QCD phase transition, but it allows for pion scattering below the QCD phase transition and alters the bounds from direct detection and the loop-induced coupling to leptons by changing the relative sign of the up- and down-type contributions in Eq. (\[eq:loop\]). We have presented the results for this coupling structure in figure \[fig:Vector\_quarks2\]. This plot only shows results for including the coupling to pions or the loop coupling to leptons, but including both contributions leads to a curve that lies along the curve of larger suppression scale: e.g. for $T_{\rm kd} = 100 \, {\rm MeV}$ the curve with both effects included would lie along the pion only curve. From this plot, we observe that following the QCD phase transition, pion scattering is the dominant process in setting $T_{\rm kd}$, but as $T$ decreases, the loop coupling to leptons becomes more important as the pions become non-relativistic and their interaction rate with dark matter is Boltzmann suppressed. We thus note that the relative importance of scattering off of pions versus loop-mediated scattering off of leptons below the QCD phase transition is generically comparable, with one process dominating over the other depending upon the kinetic decoupling temperature: for large decoupling temperatures, hence closer to the QCD confinement phase transition, scattering off of pions dominates, while lepton loop-induced scattering dominates as the number density of pions declines at lower temperatures.
Pseudoscalar Operator
---------------------
Pseudoscalar operators present a unique complication among all parity-conserving operators, in that the scattering amplitude vanishes in the limit $t\to0$ even when the center-of-mass velocity is large. This necessitates a summation over angles which can be neglected in the case of the other operators, as described in section \[sec:tkd\].
Pseudoscalar operators lead to strongly suppressed direct detection scattering, so the only relevant bounds are from collider searches. Here, when the coupling is universal, the largest possible value for $M_{\rm cut}$ is $10^{-6} M_\odot$ when $m_\chi \geq 20 \, {\rm MeV}$, as shown in figure \[fig:PS\_univ\]. Models with the correct relic density have masses of a few GeV and higher and increasingly suppressed cutoff scales as a function of mass: from $10^{-5} M_\odot$ at 5 GeV to $10^{-7} M_\odot$ at 20 GeV, and downward to $10^{-9} M_\odot$ and smaller for any mass larger than 200 GeV.
With lepton only couplings constrained by just LEP data, $M_{\rm cut}$ is again much less constrained, as the next figure, \[fig:PS\_leptons\], shows. Focusing again on models with the correct relic density, we find kinetic decoupling temperatures from slightly more than 10 MeV at WIMP masses in the GeV range, up to 1 GeV for 400 GeV WIMPs. The inferred cutoff mass scale ranges from $10^{-5} M_\odot$ at 6 GeV to $10^{-7} M_\odot$ at 30 GeV, to $10^{-9} M_\odot$ and smaller for any mass larger than 200 GeV, again for models with the correct thermal relic density.
For quark-only pseudoscalar couplings there exists a minimum value of $T_{\rm kd}$, irrespective of how strongly the dark matter couples, which is the QCD phase transition temperature. After the phase transition the only hadronic state available with cosmologically-relevant number densities are the pions. Since QCD is a parity-conserving theory, we can require that the parity behavior of the quark bilinear match that of the pion state which the dark matter would couple to. However, with only two pions it is impossible to construct any pseudoscalar invariant. This indicates that elastic scattering off of two pions is completely forbidden by the symmetries of the theory for this operator. Other scattering processes are possible, however. Inelastic scatterings, whether changing the number of particles or changing, for example, a pion into a sigma meson, are allowed by the symmetries of the problem. These nonetheless do not contribute efficiently to the continued thermalization of the dark matter kinematics, because the thermal bath is not energetic enough to produce the more exotic (i.e. higher-mass) QCD states or to provide sufficient energy to produce additional pions in scattering. Thus, the leading contribution is a one-loop-suppressed process requiring two insertions of the operator, which is very strongly suppressed.
Axial-Vector Operator
---------------------
Axial-vector couplings are constrained at levels comparable to vector couplings by colliders, but lead to spin-dependent rather than -independent scattering in direct detection, so the collider bounds are generically stronger than the direct detection bounds for these interactions. Universal couplings to SM fermions are presented in figure \[fig:Axial\_univ\]. We find that with universal couplings existing bounds generically require $T_{\rm kd}$ to be above 10 MeV, and the cutoff is smaller than $10^{-5}\ M_\odot$ for any mass above 20 GeV. This class of operators produces the right thermal relic density for WIMPs above 100 GeV, leading in all cases to cutoff masses smaller than $10^{-5}\ M_\odot$
For couplings to leptons only there are no appreciable direct detection bounds, as any loop-induced scattering akin to that in the vector case would have to proceed through $Z$-boson exchange, and the additional suppression of $t/M_Z^2$ makes such contributions negligible. Thus, only LEP bounds are shown along with our results in figure \[fig:Axial\_leptons\]. The figure indicates that cutoff scales as small as about $10^{-3}\ M_\odot$ are in principle possible for very light WIMPs. The thermal relic density and LEP bounds put the dark matter mass in the 100 GeV and up range, with cutoff scales at most of $10^{-4}\ M_\odot$, as before suppressed with increasing WIMP mass.
For the same reason that there are no bounds from direct detection on leptonic axial-vector couplings, there is no induced lepton coupling in the case of a quark-only interaction. Additionally, elastic scattering of dark matter off of pions vanishes in this model, as there is no axial invariant which can be constructed from the kinematics of two pions. Once again, inelastic scattering, whether producing or destroying an additional pion or scattering a pion into a different QCD state, is possible, but the low temperature below the QCD phase transition makes these possibilities contribute negligibly to the kinetic decoupling. Thus axial interactions with quarks only also have a minimum $T_{\rm kd}=T_c$, analogously with the case of pseudoscalar couplings.
Tensor Operator
---------------
The tensor operator normalization which we consider preserves the SM gauge group where other normalizations do not, but is not particularly well studied because it does not correspond to the QCD matrix element which is probed in direct detection. Thus, we cannot present bounds from direct detection on this operator. Additionally, current collider searches have been normalized to correspond to direct detection, so we can’t compare directly to those results either. The closest approximation to collider searches which could be considered would be the constraints on the other chirality-suppressed operators, in this paper the scalar and pseudoscalar cases. For the direct detection comparison the theoretical picture is a bit more muddled, as the quark mass which appears in this operator should be taken to be related to the yukawa coupling, which will be affected by renormalization running in the strong phase of QCD, and is therefore nontrivial to factor out of the operator and find a meaningful bound. Since neither comparison technique yields a perfect mapping, we will only discuss the early-universe behavior of the operator.
The results for universal couplings are given in figure \[fig:Tensor\_univ\], while those for leptons only are in figure \[fig:Tensor\_leptons\]. As a tensor mediated interaction cannot be implemented in CalcHEP, we do not use micrOMEGAs to calculate the relic density, but rather require the velocity averaged cross section to equal the canonical value for s-wave annihilation that gives the correct relic density, i.e $$\left<\sigma v_{\rm rel} \right> = \sum_f \frac{9}{2 \pi} \frac{m_f^2 m_\chi^2}{\Lambda_T^6} \left(1 - \frac{m_f^2}{m_\chi^2} \right)^{1/2} \approx 3 \times 10^{-26} \, \frac{\rm cm^3}{\rm s} \, ,$$ where the sum is over all kinematically accessible fermion annihilation products. Once again, after QCD confines in the early universe there is no pion configuration which has the Lorentz transformation properties of a tensor, and thus quark couplings become irrelevant below $T_c$.
Fig. \[fig:Tensor\_univ\] indicates that for good thermal relics, the expected kinetic decoupling temperatures are of 10 MeV in the few GeV mass range, up to 100 MeV for a 30 GeV WIMP, and to 1 GeV for a 300 GeV WIMP. The resulting small-scale cutoff masses are of $10^{-5}\ M_\odot$ for a 10 GeV WIMP mass, decreasing to below $10^{-9}\ M_\odot$ for masses above 200 GeV. For the exclusively leptonic coupling tensor case, we find a qualitatively similar behavior, with good thermal relics producing slightly lower kinetic decoupling temperatures, and slightly larger cutoff scales.
Discussion and Conclusions {#sec:concl}
==========================
In this study, we addressed the question of establishing the small-scale cutoff of the cosmological matter power spectrum in a variety of particle dark matter models where WIMP coupling to Standard Model fermions is described by effective operators. We included cases where the dark matter separately couples exclusively to leptons, exclusively to quarks, or universally to both leptons and quarks. We also used collider searches and dark matter direct detection to set model-independent limits on the largest experimentally viable value of the small-scale cutoff resulting from kinetic decoupling for each class of operators, and we calculated the dark matter thermal relic abundance on the same parameter space.
The largest possible cutoffs are found for theories where the dark matter exclusively couples to leptons, as a result of the absence of limits from hadron colliders. For the case of coupling to quarks, in some cases direct dark matter searches squeeze the maximal cutoff scale for protohalos to very small values, in some instances much smaller than the Earth mass. Insisting on setups that produce a thermal relic density in accordance with the observed dark matter density we universally find increasingly suppressed small scale cutoffs with increasing dark matter particle masses.
For theories with quark-only couplings, if the kinetic decoupling falls below the QCD confinement phase transition, two effects exhibit an interesting interplay: scattering off of the lightest available hadronic bound states (pions) and loop-mediated scattering off of leptons, this latter process never having been considered before. We showed that depending on the operator’s effective energy scale one or the other effect can dominate the kinetic decoupling process.
While there exist many instances of dark matter models where the effective operator description we adopted here does not apply, the present study has wide applicability to a broad range of WIMP models. In addition, our findings provide a model-independent framework where the relevant range for the small-scale cutoff to the matter power spectrum can effectively be predicted. Finally, this study highlights the complementarity of collider and direct detection of dark matter with questions pertaining to the cosmology of dark matter and the formation of structure in the universe.
JMC is supported by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No. (DGE-0809125) and by the Swedish Research Council (VR) and the NSF through the NSF Nordic Research Opportunity. WS and SP are partly supported by the US Department of Energy under contract DE-FG02-04ER41268.
Standard Model Fermion Scattering Matrix Elements {#sec:fermionm}
=================================================
Scalar
------
For the effective operator describing the scalar interaction between a SM fermion $f$ and a Dirac fermion $\chi$ $$\mathcal{O}_S = \frac{m_f}{\Lambda_S^3} \bar\chi\chi\bar ff \, ,$$ the matrix element for the scattering process between $f$ and $\chi$ squared and summed over initial spin states and averaged over final spin states is of the form $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 = 4 \frac{m_f^2}{\Lambda_S^6} \left( {\bf p} \cdot {\bf p'} + m_{\chi}^2 \right) \left( {\bf k} \cdot {\bf k'} + m_f^2 \right) \, ,$$ where ${\bf p}$ and ${\bf p'}$ are the incoming and outgoing 4-momentum of the $\chi$ particle respectively and ${\bf f}$ and ${\bf f'}$ are the same for the SM fermion. Setting $t = 0$, this becomes $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right| = 16 \frac{m_f^4 m_\chi^2}{\Lambda_S^6} \, .$$
Psuedoscalar
------------
We now consider the effective operator describing pseudoscalar interactions, $$\mathcal{O}_P = \frac{m_f}{\Lambda_P^3} \bar\chi \gamma^5 \chi \bar f \gamma^5 f \, .$$ For a scattering process when the interaction is described by this operator, we find $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 = 4 \frac{m_f^2}{\Lambda_P^6} \left(m_\chi^2 - {\bf p} \cdot {\bf p'} \right) \left( m_f^2- {\bf f} \cdot {\bf f'} \right) = \frac{m_f^2}{\Lambda_P^6} t^2 \, .$$
Vector
------
Now considering the operator $$\mathcal{O}_V = \frac{1}{\Lambda_V^2}\bar\chi\gamma^\mu\chi\bar f\gamma_\mu f \, , \label{eq:apvectop}$$ $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 = \frac{8}{\Lambda_V^4} \left[ \left({\bf p} \cdot {\bf k} \right) \left( {\bf p'} \cdot {\bf k'} \right) + \left( {\bf p} \cdot {\bf k'} \right) \left( {\bf p'} \cdot {\bf k} \right) - \left( {\bf p} \cdot {\bf p'} \right) m_f^2 - \left( {\bf k} \cdot {\bf k'} \right) m_\chi^2 + 2m_\chi^2m_f^2 \right] \, .$$ As before, we consider only forward scattering, so $t = 0$. Working in the frame where the dark matter particle is stationary, $s = m_\chi^2+2m_\chi \omega + m_f^2$ and the matrix element becomes: $$\frac{1}{4} \sum_{\rm Spin \; States}\left| \mathcal{M} \right|^2 = 16 \frac {m_\chi^2}{\Lambda_V^4} \omega^2 \, .$$
Pseudovector
------------
The axial vector operator is of the form $$\mathcal{O}_A = \frac{1}{\Lambda_V^2}\bar\chi\gamma^\mu\gamma^5\chi\bar f\gamma_\mu\gamma^5f \, ,$$ $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 = \frac{8}{\Lambda_A^4} \left[ \left({\bf p} \cdot {\bf k} \right) \left( {\bf p'} \cdot {\bf k'} \right) + \left( {\bf p} \cdot {\bf k'} \right) \left( {\bf p'} \cdot {\bf k} \right) + \left( {\bf p} \cdot {\bf p'} \right) m_f^2 + \left( {\bf k} \cdot {\bf k'} \right) m_\chi^2 + 2m_\chi^2m_f^2 \right] \, .$$ Once again, taking the two limits $t = 0$ and $s = m_\chi^2+2m_\chi \omega + m_f^2$, this becomes $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 = 16 \frac{m_\chi^2}{\Lambda_A^4} \left( \omega^2 + 2 m_f^2 \right) \, .$$ As $m_f \approx 0$ for relativistic fermions, this is essentially the same as the result for the vector operator.
Tensor
------
Finally, the tensor operator takes the form $$\mathcal{O}_T = \frac{m_f}{\Lambda_T^3}\bar\chi\sigma^{\mu\nu}\chi\bar f\sigma_{\mu\nu}f$$ where $\sigma^{\mu \nu} = (i / 2) [\gamma^\mu, \gamma^\nu]$. $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 = 32 \frac{ m_f^2}{\Lambda_T^6} \left(2 \left({\bf p'} \cdot {\bf k} \right) \left( {\bf p} \cdot {\bf k'} \right) - \left( {\bf p} \cdot {\bf p'} \right) \left( {\bf k} \cdot {\bf k'} \right) +2 \left( {\bf p} \cdot {\bf k} \right) \left({\bf p'} \cdot {\bf k'} \right) + 3 m_\chi^2 m_f^2 \right) \, ,$$ and then when $t = 0$ and $s = m_\chi^2+2m_\chi \omega + m_f^2$, this becomes $$\frac{1}{4} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 = 64 \frac{m_f^2 m_\chi^2}{\Lambda_T^6} \left(2 \omega^2 + m_f^2 \right) \, .$$
Pion Scattering Matrix Elements {#sec:pionm}
===============================
Scalar
------
For the scalar pion coupling we have a Lagrangian term [@kamionkowskietal] $$\mathcal{L}\supset \frac{m_\pi^2}{2\Lambda_S^3}\bar\chi\chi\vec\pi\cdot\vec\pi \, ,$$ where $$\vec{\pi} =
\left( \begin{array}{c}
\frac{1}{\sqrt{2}} \left(\pi^+ + \pi^- \right) \\
\frac{i}{\sqrt{2}} \left(\pi^+ - \pi^- \right) \\
\pi^0 \\
\end{array} \right) \, . \label{eq:pionvec}$$ Simplifying the dot product, this gives $$\mathcal{L}\supset \frac{m_\pi^2}{2\Lambda^3}\bar\chi\chi\left(\pi^0\pi^0+2\pi^+\pi^-\right) \, .$$ Note that this leads to a Feynman rule which is identical for all pion charges, and the scattering amplitude which we calculate is $$i \mathcal{M} = \frac{m_\pi^2}{\Lambda_S^3}\bar\chi\chi \, .$$ Squaring and averaging over initial spins, then choosing the zero relative velocity limit, gives the final result $$\frac{1}{2} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2=\frac{4m_\pi^4m_\chi^2}{\Lambda_S^6} \, .$$
Vector
------
The coupling from the vector operator has the form [@kamionkowskietal] $$\mathcal{L}\supset \frac{2i}{\Lambda_V^2}\bar\chi\gamma_\mu\chi\left(\vec\pi\times\partial^\mu\vec\pi\right)_3$$ when we introduce a negative sign in front of the operator in Eq. \[eq:apvectop\] for down type quark interactions, as otherwise this term is zero. The relevant component of the cross product is $\pi_1\partial^\mu\pi_2-\pi_2\partial^\mu\pi_1$, which can be rewritten in terms of the physical fields to give $$\mathcal{L}\supset \frac{2i}{\Lambda_V^2}\bar\chi\gamma_\mu\chi\left(\pi^+\partial^\mu\pi^--\pi^-\partial^\mu\pi^+\right) \, .$$ Thus, this operator does not couple to neutral pions, and the scattering amplitude off of a charged pion is equal to $$i \mathcal{M}=\frac{2}{\Lambda_V^2}\bar\chi\gamma_\mu\chi\left({\bf k}+{\bf k^\prime}\right)^\mu \, .$$ Squaring and averaging over incoming spins, we have $$\frac{1}{2} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2 =\frac{8}{\Lambda_V^4}\left( \left({\bf k}+{\bf k'}\right)^2 \left(m_\chi^2-{\bf p} \cdot {\bf p^\prime}\right)+2{\bf p}\cdot\left({\bf k}+{\bf k^\prime} \right){\bf p^\prime} \cdot\left({\bf k}+{\bf k^\prime}\right)\right) \, .$$ Simplifying this in terms of Mandelstam variables we find $$\frac{1}{2} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2=\frac{4}{\Lambda_V^4}\left(4m_\pi^2t-t^2+s^2+u^2-2su\right) \, ,$$ and working in the limit where $t \rightarrow 0$, this becomes $$\frac{1}{2} \sum_{\rm Spin \; States} \left| \mathcal{M} \right|^2=\frac{64m_\chi^2\omega^2}{\Lambda_V^4} \, .$$
[300]{}
Ya. B. Zeldovich, Adv. Astron. Astrophys. [**3**]{}, 241 (1965); H.Y. Chiu, Phys. Rev. Lett. [**17**]{}, 712 (1966); G. Steigman, Ann. Rev. Nucl. Part. Sci. [**29**]{}, 313 (1979); R.J. Scherrer and M.S. Turner, Phys. Rev. D [**33**]{}, 1585 (1986).
T. Bringmann, New J. Phys. [**11**]{}, 105027 (2009). \[arXiv:0903.0189 \[astro-ph.CO\]\].
L. E. Strigari, J. S. Bullock, M. Kaplinghat, J. Diemand, M. Kuhlen and P. Madau, Astrophys. J. [**669**]{}, 676 (2007) \[arXiv:0704.1817 \[astro-ph\]\]. J. R. Primack, New J. Phys. [**11**]{}, 105029 (2009) \[arXiv:0909.2247 \[astro-ph.CO\]\].
J. Diemand, M. Kuhlen, P. Madau, M. Zemp, B. Moore, D. Potter and J. Stadel, Nature [**454**]{}, 735 (2008) \[arXiv:0805.1244 \[astro-ph\]\]; M. Kuhlen, N. Weiner, J. Diemand, P. Madau, B. Moore, D. Potter, J. Stadel and M. Zemp, JCAP [**1002**]{}, 030 (2010) \[arXiv:0912.2358 \[astro-ph.GA\]\]. J. Diemand, M. Kuhlen and P. Madau, Astrophys. J. [**657**]{}, 262 (2007) \[astro-ph/0611370\]. S. Hofmann, D. J. Schwarz, H. Stoecker, Phys. Rev. [**D64**]{}, 083507 (2001). \[astro-ph/0104173\].
T. Bringmann and S. Hofmann, JCAP [**0407**]{}, 016 (2007) \[hep-ph/0612238\]. S. Profumo, K. Sigurdson, M. Kamionkowski, Phys. Rev. Lett. [**97**]{}, 031301 (2006). \[astro-ph/0603373\].
D. Hooper and S. Profumo, Phys. Rept. [**453**]{}, 29 (2007) \[hep-ph/0701197\].
L. G. van den Aarssen, T. Bringmann and Y. CGoedecke, Phys. Rev. D [**85**]{}, 123512 (2012) \[arXiv:1202.5456 \[hep-ph\]\]. L. G. van den Aarssen, T. Bringmann and C. Pfrommer, Phys. Rev. Lett. [**109**]{}, 231301 (2012) \[arXiv:1205.5809 \[astro-ph.CO\]\]. J. M. Cornell and S. Profumo, JCAP [**1206**]{}, 011 (2012) \[arXiv:1203.1100 \[hep-ph\]\]. P. Gondolo, J. Hisano and K. Kadota, Phys. Rev. D [**86**]{}, 083523 (2012) \[arXiv:1205.1914 \[hep-ph\]\].
I. M. Shoemaker, arXiv:1305.1936 \[hep-ph\].
P. J. Fox, R. Harnik, R. Primulando and C. -T. Yu, Phys. Rev. D [**86**]{}, 015010 (2012) \[arXiv:1203.1662 \[hep-ph\]\]. A. Rajaraman, W. Shepherd, T. M. P. Tait and A. M. Wijangco, Phys. Rev. D [**84**]{}, 095013 (2011) \[arXiv:1108.1196 \[hep-ph\]\], P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, Phys. Rev. D [**85**]{}, 056011 (2012) \[arXiv:1109.4398 \[hep-ph\]\]. J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. -B. Yu, Phys. Rev. D [**82**]{}, 116010 (2010) \[arXiv:1008.1783 \[hep-ph\]\], Y. Bai, P. J. Fox and R. Harnik, JHEP [**1012**]{}, 048 (2010) \[arXiv:1005.3797 \[hep-ph\]\]. J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. -B. Yu, Phys. Lett. B [**695**]{}, 185 (2011) \[arXiv:1005.1286 \[hep-ph\]\]. P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, Phys. Rev. D [**84**]{}, 014028 (2011) \[arXiv:1103.0240 \[hep-ph\]\].
J. Kopp, V. Niro, T. Schwetz and J. Zupan, Phys. Rev. D [**80**]{}, 083502 (2009) \[arXiv:0907.3159 \[hep-ph\]\]. P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke and E. A. Baltz, JCAP [**0407**]{}, 008 (2004) \[astro-ph/0406204\]. A. M. Green, S. Hofmann, D. J. Schwarz, Mon. Not. Roy. Astron. Soc. [**353**]{}, L23 (2004). \[astro-ph/0309621\].
A. M. Green, S. Hofmann, D. J. Schwarz, JCAP [**0508**]{}, 003 (2005). \[arXiv:astro-ph/0503387 \[astro-ph\]\].
A. Loeb, M. Zaldarriaga, Phys. Rev. [**D71**]{}, 103520 (2005). \[astro-ph/0504112\].
E. Bertschinger, Phys. Rev. [**D74**]{}, 063509 (2006). \[astro-ph/0607319\].
ATLAS-CONF-2012-147.
CMS-PAS-EXO-12-048.
E. Aprile [*et al.*]{} \[XENON100 Collaboration\], Phys. Rev. Lett. [**109**]{}, 181301 (2012) \[arXiv:1207.5988 \[astro-ph.CO\]\]. M. Felizardo, T. A. Girard, T. Morlat, A. C. Fernandes, A. R. Ramos, J. G. Marques, A. Kling and J. Puibasset [*et al.*]{}, Phys. Rev. Lett. [**108**]{}, 201302 (2012) \[arXiv:1106.3014 \[astro-ph.CO\]\]. S. Archambault [*et al.*]{} \[PICASSO Collaboration\], Phys. Lett. B [**711**]{}, 153 (2012) \[arXiv:1202.1240 \[hep-ex\]\]. P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], arXiv:1303.5076 \[astro-ph.CO\]. G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, arXiv:1005.4133 \[hep-ph\]. G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. [**176**]{} (2007) 367 \[arXiv:hep-ph/0607059\]
G. Prezeau, A. Kurylov, M. Kamionkowski and P. Vogel, Phys. Rev. Lett. [**91**]{}, 231301 (2003) \[astro-ph/0309115\].
|
---
abstract: 'We present a probabilistic variant of the recently introduced *maxout* unit. The success of deep neural networks utilizing *maxout* can partly be attributed to favorable performance under dropout, when compared to rectified linear units. It however also depends on the fact that each maxout unit performs a pooling operation over a group of linear transformations and is thus partially invariant to changes in its input. Starting from this observation we ask the question: Can the desirable properties of *maxout* units be preserved while improving their invariance properties ? We argue that our probabilistic maxout (*probout*) units successfully achieve this balance. We quantitatively verify this claim and report classification performance matching or exceeding the current state of the art on three challenging image classification benchmarks (CIFAR-10, CIFAR-100 and SVHN).'
author:
- |
Jost Tobias Springenberg and Martin Riedmiller\
Department of Computer Science\
University of Freiburg\
79110, Freiburg im Breisgau, Germany\
`{springj,riedmiller}@cs.uni-freiburg.de`\
bibliography:
- 'papers.bib'
title: Improving Deep Neural Networks with Probabilistic Maxout Units
---
Introduction
============
Regularization of large neural networks through stochastic model averaging was recently shown to be an effective tool against overfitting in supervised classification tasks. Dropout [@Hinton2012] was the first of these stochastic methods which led to improved performance on several benchmarks ranging from small to large scale classification problems [@Krizhevsky2012; @Hinton2012]. The idea behind dropout is to randomly drop the activation of each unit within the network with a probability of $50 \%$. This can be seen as an extreme form of bagging in which parameters are shared among models, and the number of trained models is exponential in the number of these model parameters. During testing an approximation is used to average over this large number of models without instantiating each of them. When combined with efficient parallel implementations this procedure opened the possibility to train large neural networks with millions of parameters via back-propagation [@Krizhevsky2012; @Zeiler2013] .
Inspired by this success a number of other stochastic regularization techniques were recently developed. This includes the work on dropconnect[@WanLi2013], a generalization of dropout, in which connections between units rather than their activation are dropped at random. Adaptive dropout [@Frey2013] is a recently introduced variant of dropout in which the stochastic regularization is performed through a binary belief network that is learned alongside the neural network to decrease the information content of its hidden units. Stochastic pooling [@ZeilerStochastic2013] is a technique applicable to convolutional networks in which the pooling operation is replaced with a sampling procedure.
Instead of changing the regularizer the authors in [@Goodfellow2013] searched for an activation function for which dropout performs well. As a result they introduced the maxout unit, which can be seen as a generalization of rectified linear units (ReLUs) [@Nair2010; @Glorot2011], that is especially suited for the model averaging performed by dropout. The success of maxout can partly be attributed to the fact that maxout aids the optimization procedure by partially preventing units from becoming inactive; an artifact caused by the thresholding performed by the rectified linear unit. Additionally, similar to ReLUs, they are piecewise linear and – in contrast to e.g. sigmoid units – typically do not saturate, which makes networks containing maxout units easier to optimize. We argue that an equally important property of the maxout unit however is that its activation function can be seen as performing a pooling operation over a subspace of $k$ linear feature mappings (in the following referred to as subspace pooling). As a result of this subspace pooling operation each maxout unit is partially invariant to changes within its input. A natural question arising from this observation is thus whether it could be beneficial to replace the maximum operation used in maxout units with other pooling operations, such as L2 pooling. The utility of different subspace pooling operations has already been explored in the context of unsupervised learning where e.g. L2-pooling is known give rise to interesting invariances [@Hyvarinnen2009; @Bergstra2009; @Zou2012]. While work on generalizing maxout by replacing the max-operation with general $Lp$-pooling exists [@Gulcere2013], a deviation from the standard maximum operation comes at the price of discarding some of the desirable properties of the maxout unit. For example abandoning piecewise linearity, restricting units to positive values and the introduction of saturation regimes, which potentially worsen the accuracy of the approximate model averaging performed by dropout.
Based on these observations we propose a stochastic generalization of the maxout unit that preserves its desirable properties while improving the subspace pooling operation of each unit. As an additional benefit when training a neural network using our proposed probabilistic maxout units the gradient of the training error is more evenly distributed among the linear feature mappings of each unit. In contrast, a maxout network helps gradient flow through each of the maxout units but not through their k linear feature mappings. Compared to maxout our probabilistic units thus learn to better utilize their full k-dimensional subspace. We evaluate the classification performance of a model consisting of these units and show that it matches the state of the art performance on three challenging classification benchmarks.
Model Description
=================
Before defining the probabilistic maxout unit we briefly review the notation used in the following for defining deep neural network models. We adopt the standard feed-forward neural network formulation in which given an input $\mathbf{x}$ and desired output $y$ (a class label) the network realizes a function computing a $C$-dimensional vector $\mathbf{o}$ – where $C$ is the number of classes – predicting the desired output. The prediction is computed by first sequentially mapping the input to a hierarchy of $N$ hidden layers $\mathbf{h}^{(1)}, \dots, \mathbf{h}^{(N)}$. Each unit $h_i^{(l)}$ within hidden layer $l \in [1,N]$ in the hierarchy realizes a function $h_i^{(l)}(\mathbf{v}; \mathbf{w}^{(l)}_i,
b^{(l)}_i)$ mapping its inputs $\mathbf{v}$ (given either as the input $\mathbf{x}$ or the output of the previous layer $h^{(l-1)}$) to an activation using weight and bias parameters $\mathbf{w}^{(l)}_i$ and $b^{(l)}_i$. Finally the prediction is computed based on the last layer output $\mathbf{h}^{N}$. This prediction is realized using a softmax layer $\mathbf{o} = softmax(\mathbf{W}^{N+1} \mathbf{h}^{(N)}
+ \mathbf{b}^{N+1})$ with weights $\mathbf{W}^{N+1}$ and bias $\mathbf{b}^{N+1}$. All parameters $\theta = \{ W^{(1)}, b^{(1)},
\dots, W^{(N+1)}, b^{(N+1)} \}$ are then learned by minimizing the cross entropy loss between output probabilities $\mathbf{o}$ and label $y$ : $ \mathcal{L}(o, y; \mathbf{x}) = - \sum_{i = 1}^C y_i \log(o_i)
+ (1 - y_i) log(1 - o_i)$.
Probabilistic Maxout Units
--------------------------
![Schematic of different pooling operations. a) An exemplary input image taken from the ImageNet dataset together with the depiction of a spatial pooling region (cyan) as well as the input to one maxout / probout unit (marked in magenta). b) Spatial max-pooling proceeds by computing the maximum of one filter response at the four different positions from a). c) Maxout computes a pooled response of two linear filter mappings applied to one input patch. d) The activation of a probout unit is computed by sampling one of the linear responses according to their probability.[]{data-label="pooling_fig"}](owl_pooling.pdf){width="\columnwidth"}
The maxout unit was recently introduced in [@Goodfellow2013] and can be formalized as follows: Given the units input $\mathbf{v} \in
\mathbb{R}^d$ (either the activation from the previous layer or the input vector) the activation of a maxout unit is computed by first computing k linear feature mappings $\mathbf{z} \in
\mathbb{R}^{k}$ where $$z_i = \mathbf{w}_i \mathbf{v} + b_i,$$ and k is the number of linear sub-units combined by one maxout unit. Afterwards the output $h_{maxout}$ of the maxout hidden unit is given as the maximum over the k feature mappings: $$h_{maxout}(\mathbf{v}) = \max [z_1, \dots, z_k].
\label{maxout_act}$$ When formalized like this it becomes clear that (in contrast to conventional activation functions) the maxout unit can be interpreted as performing a pooling operation over a k-dimensional subspace of linear units $[z_1, \dots, z_k]$ each representing one transformation of the input $\mathbf{v}$. This is similar to spatial max-pooling which is commonly employed in convolutional neural networks. However, unlike in spatial pooling the maxout unit pools over a subspace of k different linear transformations applied to the same input $\mathbf{v}$. In contrast to this, spatial max-pooling of linear feature maps would compute a pooling over one linear transformation applied to k different inputs. A schematic of the difference between several pooling operations is given in Fig. \[pooling\_fig\] .
As such maxout is thus more similar to the subspace pooling operations used for example in topographic ICA [@Hyvarinnen2009] which is known to result in partial invariance to changes within its input. On the basis of this observation we propose a stochastic generalization of the maxout unit that preserves its desirable properties while improving gradient propagation among the $k$ linear feature mappings as well as the invariance properties of each unit. In the following we call these generalized units *probout units* since they are a direct probabilistic generalization of maxout.
We derive the probout unit activation function from the maxout formulation by replacing the maximum operation in Eq. with a probabilistic sampling procedure. More specifically we assume a Boltzmann distribution over the $k$ linear feature mappings and sample the activation $h(\mathbf{v})$ from the activation of the corresponding subspace units. To this end we first define a probability for each of the k linear units in the subspace as: $$p_i = \frac{e^{\lambda z_i}}{\sum_{j=1}^k e^{\lambda z_j}},$$ where $\lambda$ is a hyperparameter (referred to as an inverse temperature parameter) controlling the variance of the distribution. The activation $h_{probout}(\mathbf{x})$ is then sampled as $$h_{probout}(\mathbf{v}) = z_i, \text{ where } i \sim Multinomial\{p_1, \dots, p_k\}.
\label{probout_act}$$ Comparing Eq. to Eq. we see that both, are not bounded from above or below and their activation is always given as one of the linear feature mappings within their subspace. The probout unit hence preserves most of the properties of the maxout unit, only replacing the sub-unit selection mechanism.
We can further see that Eq. reduces to the maxout activation for $\lambda \rightarrow \infty$. For other values of $\lambda$ the probout unit will behave similarly to maxout when the activation of one linear unit in the subspace dominates. However, if the activation of multiple linear units differs only slightly they will be selected with almost equal probability. Futhermore, each active linear unit will have a chance to be selected. The sampling approach therefore ensures that gradient flows through each of the $k$ linear subspace units of a given probout unit for some examples (given that $\lambda$ is sufficiently small). We hence argue that probout units can learn to better utilize their full k-dimensional subspace. In practice we want to combine the probout units described by Eq. with dropout for regularizing the learned model. To achieve this we directly include dropout in the probabilistic sampling step by re-defining the probabilities as: $$\begin{aligned}
\hat{p}_0 &= 0.5 \\
\hat{p}_i &= \frac{e^{\lambda z_i}}{2 \cdot \sum_{j=1}^k e^{\lambda z_j}}.\end{aligned}$$ Consequently, we sample the probout activation function including dropout $\hat{h}_{probout}(\mathbf{v})$ as $$\hat{h}_{probout}(\mathbf{v}) = \begin{cases}
0 \text{ if } i = 0 \\
z_i \text{ else }
\end{cases}, \text{ where } i \sim Multinomial\{\hat{p}_0 , \hat{p}_1, \dots,
\hat{p}_k \}.
\label{probout_actdo}$$
Relation to other pooling operations
------------------------------------
The idea of using a stochastic pooling operation has been explored in the context of spatial pooling within the machine learning literature before. Among this work the approach most similar to ours is [@Lee2009]. There the authors introduced a probabilistic pooling approach in order to derive a convolutional deep believe network (DBN). They also use a Boltzmann distribution based on unit activations to calculate a sampling probability. The main difference between their work and ours is that they calculate the probability of sampling one unit at different spatial locations whereas we calculate the probability of sampling a unit among k units forming a subspace at one spatial location. Another difference is that we forward propagate the sampled activation $z_i$ whereas they use the calculated probability to activate a binary stochastic unit.
Another approach closely related to our work is the stochastic pooling presented in [@ZeilerStochastic2013]. Their stochastic pooling operation samples the activation of a pooling unit $p_i$ proportionally to the activation $a$ of a rectified linear unit [@Nair2010] computed at different spatial positions. This is similar to Eq. in the sense that the activation is sampled from a set of different activations. Similar to [@Lee2009] it however differs in that the sampling is performed over spatial locations rather than activations of different units.
It should be noted that our work also bears some resemblance to recent work on training stochastic units, embedded in an autoencoder network, via back-propagation [@BengioStochastic2013; @BengioGSN2013]. In contrast to their work, which aims at using stochastic neurons to train a generative model, we embrace stochasticity in the subspace pooling operation as an effective means to regularize a discriminative model.
Inference {#sect_inference}
---------
At test time we need to account for the stochastic nature of a neural network containing probout units. During a forward pass through the network the value of each probout unit is sampled from one of $k$ values according to their probability. The output of such a forward pass thus always represents only one of $k^M$ different instantiations of the trained probout network; where $M$ is the number of probout units in the network. When combined with dropout the number of possible instantiations increases to ${(k+1)}^M$. Evaluating all possible models at test time is therefore clearly infeasible. The Dropout formulation from [@Hinton2012] deals with this large amount of possible models by removing dropout at test time and halving the weights of each unit. If the network consists of only one softmax layer then this modified network performs exact model averaging [@Hinton2012]. For general models this computation is merely an approximation of the true model average which, however, performs well in practice for both deep ReLU networks [@Krizhevsky2012] and the maxout model [@Goodfellow2013].
We adopt the same procedure of halving the weights for removing the influence of dropout at test-time and rescale the probabilities such that $\sum_{i=1}^k \hat{p}_i = 1$ and $\hat{p}_0 = 0$, effectively replacing the sampling from Eq . with Eq. . We further observe that from the $k^M$ models remaining after removing dropout only few models will be instantiated with high probability. We therefore resort to sampling a small number of outputs $\mathbf{o}$ from the networks softmax layer and average their values. An evaluation of the exact effect of this model averaging can be found in Section \[sect\_prelim\_eval\] .
Evaluation
==========
We evaluate our method on three different image classification datasets (CIFAR-10, CIFAR-100 and SVHN) comparing it against the basic maxout model as well as the current state of the art on all datasets. All experiments were performed using an implementation based on Theano and the pylearn2 library [@GoodfellowPylearn] using the fast convoltion code of [@Krizhevsky2012]. We use mini-batch stochastic gradient descent with a batch size of 100. For each of the datasets we start with the same network used in [@Goodfellow2013] – retaining all of their hyperparameter choices – to ensure comparability between results. We replace the maxout units in the network with probout units and choose one $\lambda^{(l)}$ via crossvalidation for each layer $l$ in a preliminary experiment on CIFAR-10.
Experiments on CIFAR-10
-----------------------
We begin our experiments with the CIFAR-10 [@Krizhevsky2009] dataset. It consists of $50,000$ training images and $10,000$ test images that are grouped into $10$ categories. Each of these images is of size $32\times32$ pixels and contains $3$ color channels. Maxout is known to yield good performance on this dataset, making it an ideal starting point for evaluating the difference between maxout and probout units.
### Effect of replacing maxout with probout units {#sect_prelim_eval}
![Visualization of pairs of first layer linear filters learned by the maxout model (left) as well as the probout model (right). In contrast to the maxout filters the filter pairs learned by the probout model appear to mostly be transformed versions of each other.[]{data-label="filters_fig"}](maxout_filters_nice.pdf "fig:") ![Visualization of pairs of first layer linear filters learned by the maxout model (left) as well as the probout model (right). In contrast to the maxout filters the filter pairs learned by the probout model appear to mostly be transformed versions of each other.[]{data-label="filters_fig"}](probout_filters_nice.pdf "fig:")
We conducted a preliminary experiment to evaluate the effect of the probout parameters $\lambda^{(l)}$ on the performance and compare it to the standard maxout model. For this purpose we use a five layer model consisting of three convolutional layers with 48, 128 and 128 probout units respectively which pool over 2 linear units each. The penultimate layer then consists of 240 probout units pooling over a subspace of 5 linear units. The final layer is a standard softmax layer mapping from the 240 units in the penultimate layer to the 10 classes of CIFAR-10. The receptive fields of units in the convolutional layers are 8, 8 and 5 respectively. Additionally, spatial max-pooling is performed after each convolutional layer with pooling size of $4 \times 4$, $4 \times
4$ and $2 \times 2$ using a stride of 2 in all layers. We split the CIFAR-10 training data retaining the first 40000 samples for training and using the last 10000 samples as a validation set.
We start our evaluation by using probout units everywhere in the network and cross-validate the choice of the inverse-temperature parameters $\lambda^{(l)} \in \{ 0.1, 0.5, 1, 2, 3, 4 \}$ keeping all other hyperparameters fixed. We find that annealing the $\lambda^{(l)}$ parameter during training to a lower value improved performance for all $\lambda^{(l)} > 0.5$ and hence linearly decrease $\lambda^{(l)}$ to a value that is $0.9$ lower than the initial $\lambda$ in these cases. As shown in Fig. \[lambda\_plot\] the best classification performance is achieved when $\lambda$ is set to allow higher variance sampling for the first two layers, specifically when $\lambda^{(1)} = 1$ and $\lambda^{(2)} = 2$. For the third as well as the fully connected layer we observe a performance increase when $\lambda^{(3)}$ is chosen as $\lambda^{(3)} = 3$ and $\lambda^{(4)} =
4$, meaning that the sampling procedure selects the maximum value with high probability. This indicates that the probabilistic sampling is most effective in lower layers. We verified this by replacing the probout units in the last two layers with maxout units which did not significantly decrease classification accuracy.
[0.48]{}
+\[error bars/.cd,y dir=both\] table\[x=lambda,y=lambda1\] [lamdadata.dat]{};
+\[error bars/.cd,y dir=both\] table\[x=lambda,y=lambda2\] [lamdadata.dat]{};
[0.5]{}
+\[error bars/.cd,y dir=both, y explicit\] table\[x=X,y=Maxout,y error=MaxoutSTD\] [dataaveraging.dat]{};
+\[error bars/.cd,y dir=both, y explicit\] table\[x=X,y=Probout,y error=ProboutSTD\] [dataaveraging.dat]{};
;
We hypothesize that increasing the probability of sampling a non maximal linear unit in the subspace pulls the units in the subspace closer together and forces the network to become “more invariant” to changes within this subspace. This is a property that is desired in lower layers but might turn to be detrimental in higher layers where the model averaging effect of maxout is more important than achieving invariance. Here sampling units with non-maximal activation could result in unwanted correlation between the “submodels”. To qualitatively verify this claim we plot the first layer linear filters learned using probout units alongside the filters learned by a model consisting only of maxout units in Fig. \[filters\_fig\]. When inspecting the filters we can see that many of the filters belonging to one subspace formed by a probout unit seem to be transformed versions of each other, with some of then resembling “quadrature pairs” of filters. Among the linear filters learned by the maxout model some also appear to encode invariance to local transformations. Most of the filters contained in a subspace however are seemingly unrelated. To support this observation empirically we probed for changes in the feature vectors of different layers (extracted from both maxout and probout models) when they are applied to translated and rotated images from the validation set. Similar to [@Koray_CVPR2009; @Zeiler2013] we calculate the normalized Euclidean distance between feature vectors extracted from an unchanged image and a transformed version. We then plot these distances for several exemplary images as well as the mean over 100 randomly sampled images. The result of this experiment is given in Fig. \[plots\_invariance\], showing that introducing probout units into the network has a moderate positive effect on both invariance to translation and rotations.
Finally, we evaluate the computational cost of the model averaging procedure described in Section \[sect\_inference\] at test time. As depicted in Fig. \[model\_average\_plot\] the classification error for the probout model decreases with more model evaluations saturating when a moderate amount of 50 evaluations is reached. Conversely, using sampling at test time in conjunction with the standard maxout model significantly decreases performance. This indicates that the maxout model is highly optimized for the maximum responses and cannot deal with the noise introduced through the sampling procedure. We additionally also tried to replace the model averaging mechanism with cheaper approximations. Replacing the sampling in the probout units with a maximum operation at test time resulted in a decrease in performance, reaching $14.13 \%$. We also tried to use probability weighting during testing [@ZeilerStochastic2013] which however performed even worse, achieving $15.21 \%$.
### Evaluation of Classification Performance
As the next step, we evaluate the performance of our model on the full CIFAR-10 benchmark. We follow the same protocol as in [@Goodfellow2013] to train the probout model. That is, we first preprocess all images by applying contrast normalization followed by ZCA whitening. We then train our model using the first 40000 examples from the training set using the last 10000 examples as a validation set. Training then proceeds until the validation error stops decreasing. We then retrain the model on the complete training set for the same amount of epochs it took to reach the best validation error.
0.15in
Method Error
-------------------------------------------------- ----------------
Conv. Net + Spearmint [@Snoek2012] 14.98 $\%$
Conv. Net + Maxout [@Goodfellow2013] 11.69 $\%$
Conv. Net + Probout **11.35 $\%$**
12 $\times$ Conv. Net + dropconnect [@WanLi2013] **9.32 $\%$**
Conv. Net + Maxout [@Goodfellow2013] 9.38 $\%$
Conv. Net + Probout 9.39 $\%$
: Classification error of different models on the CIFAR-10 dataset.
-0.1in \[cifar10\_results\]
To comply with the experiments in [@Goodfellow2013] we used a larger version of the model from Section \[sect\_prelim\_eval\] in all experiments. Compared to the preliminary experiment the size of the convolutional layers was increased to $96$, $192$ and $192$ units respectively. The size of the fully connected layer was increased to 500 probout units pooling over a 5 dimensional subspace.
The top half of Table \[cifar10\_results\] shows the result of training this model as well as other recent results. We achieve an error of $11.35 \%$, slightly better than – but statistically tied to – the previous state of the art given by the maxout model. We also evaluated the performance of this model when the training data is augmented with additional transformed training examples. For this purpose we train our model using the original training images as well as add randomly translated and horizontally flipped versions of the images. The bottom half of Table \[cifar10\_results\] shows a comparison of different results for training on CIFAR-10 with additional data augmentation. Using this augmentation process we achieve a classification error of $9.39 \%$, matching, but not outperforming the maxout result.
CIFAR-100
---------
The images contained in the CIFAR-100 dataset [@Krizhevsky2009] are – just as the CIFAR-10 images – taken from a subset of the 10-million images database. The dataset contains $50,000$ training and $10,000$ test examples of size $32\times32$ pixels each. The dataset is hence similar to CIFAR-10 in both size and image content. It, however, differs from CIFAR-10 in its label distribution. Concretely, CIFAR-100 contains images of 100 classes grouped into 20 “super-classes”. The training data therefore contains 500 training images per class – 10 times less examples per class than in CIFAR-10 – which are accompanied by 100 examples in the test-set.
We do not make use of the 20 super-classes and train a model using a similar setup to the experiments we carried out on CIFAR-10. Specifically, we use the same preprocessing and training procedure (determining the amount of epochs using a validation set and then retraining the model on the complete data). The same network as in Section \[sect\_prelim\_eval\] was used for this experiment (adapted to classify 100 classes). Again, this is the same architecture used in [@Goodfellow2013] thus ensuring comparability between results. During testing we use 50 model evaluations to average over the sampled *probout* units.
The result of this experiment is given in Table \[cifar100\_results\]. In agreement with the CIFAR-10 results our model performs marginally better than the maxout model (by $0.45 \%$[^1]). As also shown in the table the current best method on CIFAR-100 achieves a classification error of $36.85 \%$ [@Nitish2013], using a larger convolutional neural network together with a tree-based prior on the classes formed by utilizing the super-classes. A similar performance increase could potentially be achieved by combining their tree-based prior with our model.
SVHN
----
The street view house numbers dataset [@Netzer2011] is a collection of images depicting digits which were obtained from google street view images. The dataset comes in two variants of which we restrict ourselves to the one containing cropped $32 \times 32$ pixel images. Similar to the well known MNIST dataset [@LeCun1998] the task for this dataset is to classify each image as one of 10 digits in the range from 0 to 9. The task is considerably more difficult than MNIST since the images are cropped out of natural image data. The images thus contain color information and show significant contrast variation. Furthermore, although centered on one digit, several images contain multiple visible digits, complicating the classification task.
0.15in
Method Error
-------------------------------------------------------- ----------------
Receptive Field Learning [@Jia2012] 45.17 $\%$
Learned Pooling [@Malinowski2013] 43.71 $\%$
Conv. Net + Stochastic Pooling [@ZeilerStochastic2013] 42.51 $\%$
Conv. Net + dropout + tree [@Nitish2013] **36.85 $\%$**
Conv. Net + Maxout [@Goodfellow2013] 38.57 $\%$
Conv. Net + Probout 38.14 $\%$
: Classification error of different models on the CIFAR-100 dataset.
-0.1in \[cifar100\_results\]
The training and test set contain $73,257$ and $20,032$ labeled examples respectively. In addition to this data there is an “extra” set of $531,131$ labeled digits which are somewhat less difficult to differentiate and can be used as additional training data. As in [@Goodfellow2013] we build a validation set by selecting 400 examples per class from the training and 200 examples per class from the extra dataset. We conflate all remaining training images to a large set of $598,388$ images which we use for training.
The model trained for this task consists of three convolutional layers containing 64, 128 and 128 units respectively, pooling over a 2 dimensional subspace. These are followed by a fully connected and a softmax layer of which the fully connected layer contains 400 units pooling over a 5 dimensional subspace. This yields a classification error of $2.39 \%$ (using 50 model evaluations at test-time), matching the current state of the art for a model trained on SVHN without data augmentation achieved by the maxout model ($2.47 \%$). A comparison to other results can be found in Table \[svhn\_results\] . This includes the current best result with data augmentation which was obtained using a generalization of dropout in conjunction with a large network containing rectified linear units [@WanLi2013].
Method Error
-------------------------------------------------------- ---------------
Conv. Net + Stochastic Pooling [@ZeilerStochastic2013] 2.80 $\%$
Conv. Net + dropout [@Nitish2013Mas] 2.78 $\%$
Conv. Net + Maxout [@Goodfellow2013] 2.47 $\%$
Conv. Net + Probout **2.39 $\%$**
Conv. Net + dropout [@Nitish2013Mas] 2.68 $\%$
5 $\times$ Conv. Net + dropconnect [@WanLi2013] **1.93 $\%$**
: Classification error of different models on the SVHN dataset. The top half shows a comparison of our result with the current state of the art achieved without data augmentation. The bottom half gives the best performance achieved with data augmentation as additional reference.
-0.1in \[svhn\_results\]
Conclusion
==========
We presented a probabilistic version of the recently introduced maxout unit. A model built using these units was shown to yield competitive performance on three challenging datasets (CIFAR-10, CIFAR-100, SVHN). As it stands, replacing maxout units with probout units is computationally expensive at test time. This problem could be diminished by developing an approximate inference scheme similar to [@Krizhevsky2012; @Zeiler2013] which we see as an interesting possibility for future work.
We see our approach as part of a larger body of work on exploring the utility of learning “complex cell like” units which can give rise to interesting invariances in neural networks. While this paradigm has extensively been studied in unsupervised learning it is less explored in the supervised scenario. We believe that work towards building activation functions incorporating such invariance properties, while at the same time designed for use with efficient model averaging techniques such as dropout, is a worthwhile endeavor for advancing the field.
### Acknowledgments {#acknowledgments .unnumbered}
The authors want to thank Alexey Dosovistkiy for helpful discussions and comments, as well as Thomas Brox for generously providing additional computing resources.
[0.3]{} ![Analysis of the impact of vertical translation and rotation on features extracted from a maxout and probout network. We plot the distance between normalized feature vectors extracted on transformed images and the original, unchanged, image. The distances for the probout model are plotted using thick lines. The distances for the maxout model are depicted using dashed lines. (a,b) 4 exemplary images undergoing different vertical translations and rotations respectively. (c,d) Euclidean distance between feature vectors from the original 4 images depicted in (a,b) and transformed images for Layer 1 (convolutional) and Layer 4 (fully connected) respectively. (e,f) Euclidean distance between feature vectors from the original 4 images and transformed versions for Layer 2 (convolutional) and Layer 4 (fully connected) respectively. (g,h) Mean Euclidean distance between feature vectors extracted from 100 randomly selected images and their transformed versions for different layers in the network.[]{data-label="plots_invariance"}](example_translations.png "fig:"){width="\columnwidth"}
[0.3]{} ![Analysis of the impact of vertical translation and rotation on features extracted from a maxout and probout network. We plot the distance between normalized feature vectors extracted on transformed images and the original, unchanged, image. The distances for the probout model are plotted using thick lines. The distances for the maxout model are depicted using dashed lines. (a,b) 4 exemplary images undergoing different vertical translations and rotations respectively. (c,d) Euclidean distance between feature vectors from the original 4 images depicted in (a,b) and transformed images for Layer 1 (convolutional) and Layer 4 (fully connected) respectively. (e,f) Euclidean distance between feature vectors from the original 4 images and transformed versions for Layer 2 (convolutional) and Layer 4 (fully connected) respectively. (g,h) Mean Euclidean distance between feature vectors extracted from 100 randomly selected images and their transformed versions for different layers in the network.[]{data-label="plots_invariance"}](example_rotations.png "fig:"){width="\columnwidth"}
[0.492]{}
+\[mark=none, blue\] table\[x=px,y=conv1\] [output\_probout\_trans\_single\_10\_new.val]{}; +\[mark=none, red\] table\[x=px,y=conv1\] [output\_probout\_trans\_single\_20\_new.val]{}; +\[mark=none, green\] table\[x=px,y=conv1\] [output\_probout\_trans\_single\_40\_new.val]{}; +\[mark=none, black\] table\[x=px,y=conv1\] [output\_probout\_trans\_single\_66\_new.val]{};
+\[mark=none, blue, dashed, forget plot\] table\[x=px,y=conv1\] [output\_maxout\_trans\_single\_10\_new.val]{}; +\[mark=none, red, dashed, forget plot\] table\[x=px,y=conv1\] [output\_maxout\_trans\_single\_20\_new.val]{}; +\[mark=none, green, dashed, forget plot\] table\[x=px,y=conv1\] [output\_maxout\_trans\_single\_40\_new.val]{}; +\[mark=none, black, dashed, forget plot\] table\[x=px,y=conv1\] [output\_maxout\_trans\_single\_66\_new.val]{};
[0.492]{}
+\[mark=none, blue\] table\[x=px,y=fcon\] [output\_probout\_trans\_single\_10\_new.val]{}; +\[mark=none, red\] table\[x=px,y=fcon\] [output\_probout\_trans\_single\_20\_new.val]{}; +\[mark=none, green\] table\[x=px,y=fcon\] [output\_probout\_trans\_single\_40\_new.val]{}; +\[mark=none, black\] table\[x=px,y=fcon\] [output\_probout\_trans\_single\_66\_new.val]{};
+\[mark=none, blue, dashed, forget plot\] table\[x=px,y=fcon\] [output\_maxout\_trans\_single\_10\_new.val]{}; +\[mark=none, red, dashed, forget plot\] table\[x=px,y=fcon\] [output\_maxout\_trans\_single\_20\_new.val]{}; +\[mark=none, green, dashed, forget plot\] table\[x=px,y=fcon\] [output\_maxout\_trans\_single\_40\_new.val]{}; +\[mark=none, black, dashed, forget plot\] table\[x=px,y=fcon\] [output\_maxout\_trans\_single\_66\_new.val]{};
[0.48]{}
+\[mark=none, blue\] table\[x=px,y=conv2\] [output\_probout\_rot\_single\_10\_new.val]{}; +\[mark=none, red\] table\[x=px,y=conv2\] [output\_probout\_rot\_single\_20\_new.val]{}; +\[mark=none, green\] table\[x=px,y=conv2\] [output\_probout\_rot\_single\_40\_new.val]{}; +\[mark=none, black\] table\[x=px,y=conv2\] [output\_probout\_rot\_single\_66\_new.val]{};
+\[mark=none, blue, dashed, forget plot\] table\[x=px,y=conv2\] [output\_maxout\_rot\_single\_10\_new.val]{}; +\[mark=none, red, dashed, forget plot\] table\[x=px,y=conv2\] [output\_maxout\_rot\_single\_20\_new.val]{}; +\[mark=none, green, dashed, forget plot\] table\[x=px,y=conv2\] [output\_maxout\_rot\_single\_40\_new.val]{}; +\[mark=none, black, dashed, forget plot\] table\[x=px,y=conv2\] [output\_maxout\_rot\_single\_66\_new.val]{};
[0.48]{}
+\[mark=none, blue\] table\[x=px,y=conv3\] [output\_probout\_rot\_single\_10\_new.val]{}; +\[mark=none, red\] table\[x=px,y=conv3\] [output\_probout\_rot\_single\_20\_new.val]{}; +\[mark=none, green\] table\[x=px,y=conv3\] [output\_probout\_rot\_single\_40\_new.val]{}; +\[mark=none, black\] table\[x=px,y=conv3\] [output\_probout\_rot\_single\_66\_new.val]{};
+\[mark=none, blue, dashed, forget plot\] table\[x=px,y=conv3\] [output\_maxout\_rot\_single\_10\_new.val]{}; +\[mark=none, red, dashed, forget plot\] table\[x=px,y=conv3\] [output\_maxout\_rot\_single\_20\_new.val]{}; +\[mark=none, green, dashed, forget plot\] table\[x=px,y=conv3\] [output\_maxout\_rot\_single\_40\_new.val]{}; +\[mark=none, black, dashed, forget plot\] table\[x=px,y=conv3\] [output\_maxout\_rot\_single\_66\_new.val]{};
[0.492]{}
+\[mark=none, red\] table\[x=px,y=conv1\] [output\_probout\_trans\_mean\_new.val]{}; +\[mark=none, brown\] table\[x=px,y=conv3\] [output\_probout\_trans\_mean\_new.val]{}; +\[mark=none, black\] table\[x=px,y=fcon\] [output\_probout\_trans\_mean\_new.val]{};
+\[mark=none, red, dashed, forget plot\] table\[x=px,y=conv1\] [output\_maxout\_trans\_mean\_new.val]{}; +\[mark=none, brown, dashed, forget plot\] table\[x=px,y=conv3\] [output\_maxout\_trans\_mean\_new.val]{}; +\[mark=none, black, dashed, forget plot\] table\[x=px,y=fcon\] [output\_maxout\_trans\_mean\_new.val]{};
[0.48]{}
+\[mark=none, red\] table\[x=px,y=conv1\] [output\_probout\_rot\_mean\_new.val]{}; +\[mark=none, brown\] table\[x=px,y=conv3\] [output\_probout\_rot\_mean\_new.val]{}; +\[mark=none, black\] table\[x=px,y=fcon\] [output\_probout\_rot\_mean\_new.val]{};
+\[mark=none, red, dashed, forget plot\] table\[x=px,y=conv1\] [output\_maxout\_rot\_mean\_new.val]{}; +\[mark=none, brown, dashed, forget plot\] table\[x=px,y=conv3\] [output\_maxout\_rot\_mean\_new.val]{}; +\[mark=none, black, dashed, forget plot\] table\[x=px,y=fcon\] [output\_maxout\_rot\_mean\_new.val]{};
[^1]: While we were writing this manuscript it came to our attention that the experiments on CIFAR-100 in [@Goodfellow2013] were carried out using a different preprocessing than mentioned in the original paper. To ensure that this does not substantially effect our comparison we ran their experiment using the same preprocessing used in our experiments. This resulted in a slightly improved classification error of $38.50 \%$.
|
---
author:
- |
Patrick J. Orlando[^1], Felix A. Pollock[^2], Kavan Modi[^3]\
[*School of Physics and Astronomy, Monash University, Victoria 3800, Australia*]{}
bibliography:
- 'ResearchRef.bib'
title: 'How does interference fall?'
---
[<span style="font-variant:small-caps;">Abstract</span>]{}
> We study how single- and double-slit interference patterns fall in the presence of gravity. First, we demonstrate that universality of free fall still holds in this case, *i.e.*, interference patterns fall just like classical objects. Next, we explore lowest order relativistic effects in the Newtonian regime by employing a recent quantum formalism which treats mass as an operator. This leads to interactions between non-degenerate internal degrees of freedom (like spin in an external magnetic field) and external degrees of freedom (like position). Based on these effects, we present an unusual phenomenon, in which a falling double slit interference pattern periodically decoheres and recoheres. The oscillations in the visibility of this interference occur due to correlations built up between spin and position. Finally, we connect the interference visibility revivals with non-Markovian quantum dynamics.
Since the days of Galileo and Newton, it has been known that acceleration under the influence of gravity is independent of an object’s mass [@Galilei:1953ua; @Newton:821668]. This peculiarity has led to the proposition of various gravitational equivalence principles which, if broken, represent a departure from our current understanding of the theory of gravity. Einstein’s theory of general relativity is fundamentally classical, describing gravity on large length scales in terms of curvature of the underlying spacetime metric. Although it is possible to formulate quantum field theories on a static curved metric, it remains unclear how existing theory should be modified to describe gravity on the quantum mechanical scale [@birrell1984quantum]. Whilst the work we present here does not attempt to quantise gravity, it demonstrates that there is much insight to be gained from exploring non-relativistic quantum mechanics in weak-field gravity.
In the weak-field limit, a Newtonian description of gravity provides a satisfactory approximation and is, advantageously, compatible with the Hamiltonian formulation of quantum mechanics; however, its disadvantage lies in the concealment of relativistic effects, such as gravitational time dilation and the gravitational redshift of photons. Fortunately, one need not utilise the complete machinery of general relativity to take these effects into account. In fact, lowest order relativistic effects can be introduced by simply considering the mass contributions of different energy states, as given by the mass-energy relation $E=mc^2$ of special relativity [@Einstein:1911wk].
This is true even in the case of internal energy and becomes particularly interesting for quantum systems, whose internal energy can exist in superposition. Recent work by Zych and Brukner [@Zych:2015vm] treats this by promoting mass to an operator, the purpose of which is to account for the effective mass of quantised internal energy. In addition to introducing lowest order relativistic effects, this construction provides a new quantum mechanical generalisation of the Einstein equivalence principle to superpositions of energy eigenstates.
The role that Newtonian gravity plays in quantum theory was perhaps best highlighted by the famous experiment of Colella, Overhauser and Werner (COW), who demonstrated interference of cold neutrons due to a relative phase acquired due to the difference in gravitational potential between two arms of an interferometer. We include details of the COW experiment in Appendix \[sec:COW\].
More recently, the theory of ultra-cold atom condensates has provided a way to test gravitational equivalence principles with quantum systems, by using optically trapped atomic gases as an integrated interferometer [@Inguscio:1602444; @Berrada:2013bn; @Peters:1999iz; @Bonnin2015; @Zhou2015]. The short de Broglie wavelength of an atom makes atomic interferometers highly sensitive, whilst the macroscopic nature of the condensate allows for a high degree of control. Proposals for tests on board the international space station have been put forward which, if performed, are expected to surpass the best classical tests by a factor of 100 [@QTEST2015]. Finally, tests of the uniquely quantum mechanical equivalence principle for superpositions have also been proposed [@PJORLANDO2015].
In this article, we study how single- and double-slit interference patterns fall due to gravity. Initially, we ignore the lowest order relativistic effects introduced by internal degrees of freedom and find (unsurprisingly) that the interference patterns fall just as classical objects do; in other words, the universality of free fall holds for spatially delocalised quantum systems.
We then pedagogically introduce the mass operator and use it to explore non-Newtonian effects on quantum systems with quantised internal energy. One such system is a particle with intrinsic spin incident on a double slit in a gravitation field. We demonstrate that when placed in a uniform magnetic field, the internal energy results in periodic decoherence and re-coherence of the double-slit pattern. This result is an example of decoherence due to gravitational time dilation presented by Pikovski *et al.* [@Pikovski:2015du] and other related works [@Zych:2011jz; @Zych:2012kq].
The decoherence occurs due to the buildup of correlations between the spin and position degrees of the particle. We identify the oscillations in the visibility of the interference fringes as a signature of non-Markovian quantum dynamics [@arXiv:1512.00589], and demonstrate explicitly how memory effects play a role in the evolution of these fringes. This illustrates that the tools of open quantum systems theory can help us clearly understand Newtonian gravity in a quantum mechanical context.
Dropping a quantum interference pattern {#dropping-a-quantum-interference-pattern .unnumbered}
=======================================
General relativity arose from the concept that gravitational effects are a result of the underlying spacetime geometry. Whilst three fundamental forces of nature: electromagnetism, the strong force and the weak force; all depend on the internal properties of matter, gravity, in the Newtonian regime, depends only on the mass of the particle. Further, its dependence on the mass is such that the dynamics are completely independent of the particle itself. This is often attributed to Galileo in a famous thought experiment, devised to refute Aristotle’s claim that the gravitational acceleration of a body is proportional to its mass. His very elegant thought experiment, described in Figure \[fig:galileo\], led to the conclusion that all objects must fall at the same rate, regardless of their mass. This is known as the *universality of free fall*, and has profound consequences for theories of gravity.
![[**Galileo’s thought experiment.**]{} Galileo considered three spheres composed of the same material. Two of the spheres had identical mass, whilst the third sphere was much lighter. He then imagined attaching a rope between the small mass and one of the larger masses, and wondered what would happen if all three were simultaneously dropped from the leaning tower of Pisa. According to Aristotle, the small mass should fall slower than the large mass, pulling the rope taught and impeding the acceleration of the larger mass. One would then expect to see the solitary large mass hit the ground before the attached pair. However, one could also consider the pair of attached masses as a single body, whose mass exceeds that of the large mass alone. In this case, the attached pair of masses would be expected to hit the ground before the solitary large mass. This results in a logical contradiction, from which the only escape is to conclude that both the small and large masses fall with the same acceleration.[]{data-label="fig:galileo"}](Galileo.pdf){width="60.00000%"}
In this section we study how quantum interference patterns fall due to gravity. We imagine that massive quantum particles (say neutrons) are ejected towards a single or double slit. Once the particle passes through the slit, it falls freely under the influence of gravity, while simultaneously interfering with itself.
Here, we are concerned with the possibility of interesting gravitational effects appearing in a single slit diffraction or double slit interference experiment. In accordance with the Einstein equivalence principle, we have come to expect that all objects should fall identically under the influence of gravity, and this by no means excludes quantum particles exhibiting their wave-like nature. However, this does not discount the possibility of COW-like phases [@Colella:1975jc] skewing the wavefunction at the screen to give *apparent* violations.
This leads us to our first result, which is to explore how the phase generated by the gravitational potential results in an interference pattern that appears to fall like a classical object. It also provides the foundation for a more sophisticated problem explored in a later Section. From a conceptual point of view, this is an interesting scenario to investigate, especially when one considers the path integral formulation of quantum mechanics.
In simple terms, the Feynman propagator is the Green’s function for the Schrödinger equation, the solution resulting from the initial spatial wavefunction being a dirac-delta distribution. It represents the amplitude for a particle at position $x$ and time $t$ to be found at $cx{\null^{\prime}}$ a later time $t{\null^{\prime}}$. Once the propagator is known, the evolution for any initial wavefunction can be found by convolution with the propagator. The one dimensional propagator is often expressed as $$\begin{gathered}
{\left\langle{x{\null^{\prime}}}\middle|{U(t{\null^{\prime}}- t)}\middle|{x}\right\rangle} = K_0(x{\null^{\prime}},t{\null^{\prime}};x,t) = \bigintsss{\mathcal{D}}\left(x(t)\right) \exp\left[\frac{i}{\hbar}\int_t^{t{\null^{\prime}}} L\left(x(s),\dot{x}(s),s\right)ds\right],{\refstepcounter{equation}\tag{\theequation}}\label{eq:FeynmanPropDef}\end{gathered}$$ where $U(\delta t ) = \exp(-i{\hat{H}}\delta t/\hbar)$ is the time evolution operator, $x(t)$ is a parametrised path in space, ${\mathcal{D}}\left(x(t)\right)$ is the Feynman measure over all possible paths and $L\left(x,\dot{x},t\right) $ is the Lagrangian describing the system.
The path-integral formulation of quantum mechanics is conceptually very appealing, since it can be interpreted as a statement about how quantum mechanical objects may deviate from the laws of classical dynamics. In fact, even in the presence of a gravitational field, there is a non-zero amplitude which corresponds to the quantum system not falling at all: ${\left\langle{x,t{\null^{\prime}}}\middle|{x,t}\right\rangle}>0 $ for some $t{\null^{\prime}}>t$. Thus, from a foundational point of view, we would like to use the path integral approach to examine the way in which the gravitational potential affects a single-particle, double-slit interference pattern.
A short outline of the derivation is shown here, with full details available in Appendix \[App:PathIntGravity\]. The Lagrangian for a particle in a Newtonian gravitational potential is $L= \frac12 m \dot{x}^2 - m g x$. With reference to the propagator defined in Eq. , we parametrise the path $x(t)$ in terms of deviations $\delta x(t)$ from the classical trajectory, $x_c(t)$, between the two points. This gives $x(t) = x_c(t) + \delta x(t)$, with $\delta x(t)=\delta x(t{\null^{\prime}}) = 0$. This parametrisation leaves the Feynman measure unchanged, as a sum over all paths is equivalent to a sum over all deviations from a specific path. We are then left with two terms: a phase dependent on the action of the classical trajectory and a Feynman integral over the deviations that has a form identical to that of a free particle. We substitute the integral with the free particle propagator, but acknowledge that, since this is sum over deviations, we must set $x=x{\null^{\prime}}=0$. The propagator for a particle in a Newtonian gravitation potential is then $$\begin{gathered}
K_g(x{\null^{\prime}},t{\null^{\prime}};x,t)= \frac{\exp\left[\frac{i}{\hbar} S\left[x_c(t)\right]\right]}{\sqrt{2\pi i \hbar (t{\null^{\prime}}- t)/m}},{\refstepcounter{equation}\tag{\theequation}}\label{eq:GgProp}\end{gathered}$$ where $S\left[x_c(t)\right]$ is the functional that gives action associated with the classical trajectory between the points. We can express it as a function of $(x,t,x{\null^{\prime}},t{\null^{\prime}})$ by solving the equations of motion for the boundary conditions $x_c(t)= x$ and $x_c(t{\null^{\prime}})= x{\null^{\prime}}$. With complete details in Appendix \[app:PropDeriv\], the general form for the classical action is given by, $$\begin{gathered}
S[x_c(t)] = \frac{m}{2}\left\{\frac{(x{\null^{\prime}}- x)^2}{t{\null^{\prime}}-t} - g(x+x{\null^{\prime}})(t{\null^{\prime}}-t) - \frac{g^2}{12}(t{\null^{\prime}}-t)^3\right\}.{\refstepcounter{equation}\tag{\theequation}}\label{eq:Sclassical}\end{gathered}$$
Single and double slit interference {#single-and-double-slit-interference .unnumbered}
-----------------------------------
We now consider applying this propagator to the problem at hand. Let’s begin by assuming that the slits are long enough to ignore diffraction effects in the $y$-direction (perpendicular to the gravitational field – which is in the negative $x$-direction – but in the plane of the screen), this allows us to effectively reduce the problem to two dimensions. Consider a source of particles at the origin $(0,0)$ and let a double slit be located at distance $D$ from the source in the $z$-direction. Each slit has width $2a$ with centre located at $x=\pm b$. The screen is then a further distance $L$ away from the slits. The two-dimensional propagator required for this problem is given by a free particle propagator in the $z$-direction, multiplied by the gravitational propagator for the $x$ direction, as calculated in Eq. . This propagator allows us to ask the question: *If a particle initially starts at position ${\vec{r}}= (0,0)$, what is the probability of finding it at position ${\vec{r}\, {\null^{\prime}}}=(x,D+L)$ on the screen?* This distribution in $x$ will be the the two slit interference pattern that we seek.
When computing this amplitude, we consider a semi-classical approach. We assume that the ‘trajectory’ of the neutron can be separated into two parts: (a) the path from the source to the slits, followed by (b) the path from the slits to the screen. Quantum mechanically, the particles need not pass through the slits and there even exists the possibility of them passing through the slits multiple times before hitting the screen. That being said, the probabilities associated with these events are negligible under certain conditions: The semi-classical approach is valid, provided that the majority of the particle’s momentum is in the $z$ direction, such that the wavelength is approximately the $z$-direction wavelength, $\lambda \approx \frac{2\pi \hbar}{m v_z}$. We assume that this wavelength is much smaller than the relevant $z$-direction length scales, $D$ and $L$, in conjunction with the assumption that these are much larger than the relevant $x$ direction length scales, $b$ and $a$. Within this regime, the problem reduces to a single dimension. After a rather tedious calculation (included in Appendix \[sec:SlitWF\] for completeness), the wavefunction at the screen due to a single slit centred at $x=b$, the instant the particle hits it in the semi-classical approximation ($\tau=L/v_z$), is given by $$\begin{gathered}
\psi^\text{(1)}(x) = \frac{e^{i\phi(x)}}{i2\sqrt{\eta a}} \bigg\{C[\sigma_+(x)] - C[\sigma_-(x)] + iS[\sigma_+(x)]-iS[\sigma_-(x)]\bigg\},{\refstepcounter{equation}\tag{\theequation}}\label{eq:psi1}\end{gathered}$$ where $C[u] \equiv \int_0^u \cos\left(\frac{\pi}{2}x^2\right)dx$ is the Fresnel cosine function, $S[u] \equiv \int_0^u \sin\left(\frac{\pi}{2}x^2\right)dx$ is the Fresnel sine function and $\eta = 1 + \frac{L}{D}$. Above $$\begin{aligned}
&\sigma_{\pm}(x) = \sqrt{\frac{2}{\lambda L}\eta}\left\{(b\pm a) -\frac{x}{\eta} - \frac12g \frac{m^2 \lambda^2}{h^2}DL\right\} {\refstepcounter{equation}\tag{\theequation}}\label{eq:sigmapm}
\quad \quad \mbox{and}{\\[3mm]}&\phi(x) = \pi \left\{\frac{x^2}{\lambda(D+L)} - m g x\frac{\lambda(D+L)}{h^2} - \frac{g^2}{12}\frac{m^4\lambda^3}{h^4}(D+L)(D-L)^2 \right\}{\refstepcounter{equation}\tag{\theequation}}\label{eq:phix}.\end{aligned}$$ If $b$ is set to zero, then this gives single slit diffraction. Extension to double slit or even $N$-slit interference is given by taking a normalised superposition of the wavefunctions corresponding to the different slit positions.
The square of this wavefunction will give the observed probability distribution for the position at which the particle hits the screen; this is plotted for a single slit in Figure \[fig:FallingDiff\]. The pattern clearly appears to shift towards the negative $x$ direction as the screen is moved further from the slit. In general, this is far easier to identify in single slit diffraction, as the spreading of the pattern is less noticeable than in the double slit case.
![**Single Slit Diffraction in a Gravitational field.** In the top row, the magnitude squared of the wavefunction in Eq. $|\psi^{(1)}(x)|^2$ is plotted for source to screen distance $D=2\, {\rm m}$ and slit to screen distances $z=\{1\, {\rm m},3\, {\rm m},8\, {\rm m}\}$. The second row shows the same information as a two-dimensional probability density on the screen. The particle was chosen to be a neutron with wavelength $\lambda \sim 10^{-9}\, {\rm m}$, and the gravitational field strength $g=9.8\, {\rm m s}^{-2}$. In addition to the typical spreading of the pattern, we observe an apparent translation of the pattern, which we can interpret as falling. []{data-label="fig:FallingDiff"}](Fig2.png){width="96.00000%"}
The location of the central maximum is indicative of the position at which a classical point particle would arrive. If we consider a single particle incident on the slit, then there exists a possibility that it will be detected above the central maximum. We could interpret this as the particle having fallen less than expected classically. Similarly we could detect a particle below the central maximum, indicating that it fell faster than expected classically. Although it would be appealing to label this as violation of the equivalence principle, to do so would be incorrect. The easiest way to verify this is to transform to an accelerated coordinate system, taking us to the freely falling frame, in which gravitational effects should completely vanish.
When calculating the propagator in Eq. \[eq:GgProp\], we made use of the general form for the classical trajectory, which satisfies the Euler-Lagrange equations of motion. Using the solution, we find that the classical parabolic trajectory an object takes from the source at $(x=0,t=0)$ to the slit at $(x=0,t=T)$ requires an initial upward speed $v_x(t=0) = \frac12 g T$ and final vertical speed of $v_x(t=T)= - \frac12gT$. The position of the object $\tau$ seconds later would then be, $$\begin{gathered}
x_c(T+\tau) = -\frac{g\tau}{2}g(\tau+T)
= -\frac{g m^2\lambda^2 }{2h^2}\left(L^2 + LD\right). {\refstepcounter{equation}\tag{\theequation}}\label{eq:ClassicalDisp}\end{gathered}$$ We would expect that, by performing the coordinate transformation $x = \xi + x_c(T+\tau)$, the pattern should become identical to the case where $g=0$, in accordance with the equivalence principle. Since the $x$ dependence in the wavefunction $\psi^\text{(1)}$ appears only through the function $\sigma_{\pm}(x)$, we can work directly with the expression given in Eq. , $$\begin{aligned}
\sigma_{\pm}(\xi) &= \sqrt{\frac{2}{\lambda L}\eta}\left\{(b\pm a) -\frac{\xi - \frac12g\frac{m^2\lambda^2 D^2 }{h^2}\left(L^2 + LD\right)}{\eta} - \frac12g \frac{m^2 \lambda^2}{h^2}DL\right\}{\\[3mm]}&= \sqrt{\frac{2}{\lambda L}\eta}\left\{(b\pm a) -\frac{\xi}{\eta} + \frac12g\frac{m^2\lambda^2 D^2 }{h^2}L\left[\frac{D\left(L + D\right)}{L+D}-D\right]\right\}{\\[3mm]}&= \sqrt{\frac{2}{\lambda L}\eta}\left\{(b\pm a) -\frac{\xi}{\eta} \right\}.{\refstepcounter{equation}\tag{\theequation}}\label{eq:sigmaXI}\end{aligned}$$ The result above shows that the gravitational effect on the interference can be eliminated by transforming to an accelerated coordinate system. It is now clear that there are no equivalence principle violations; if we detect a particle away from the central maximum, it is interpreted as the usual deviations of a quantum particle from its classically expected trajectory.
We don’t need to calculate the double slit pattern to identify the absence of a COW phase. Since the multi-slit wavefunction is simply a superposition of single slit wavefunctions, the coordinate transform above extends to the general case; the only effect gravity will have is a translation of the entire pattern. The reason for this is that in the COW experiment, the Mach-Zehnder interferometer constrains the path of the particle to an approximate binary. Whilst confined to these paths, the relative Aharonov-Bohm-like phase is accumulated. In the case of single-slit diffraction, there is no path confinement, and even for multiple slits, where there are discrete variations between paths, there is no relative phase accumulated; this is because the slits are effectively infinitely thin in our scenario.
Therefore, it would appear that there are no peculiar quantum effects that appear in a freely falling interference pattern, beyond what one would expect in the absence of gravity. The quantum mechanical deviations from the classically expected trajectory represent a departure from the laws of classical physics, and, although the deviations might seem to constitute a violation of the universality of free fall, the effects are completely consistent with quantum behaviour as viewed from an accelerating coordinate system. In other words, an interference pattern falls like a classical object.
Effects of internal degrees of freedom {#effects-of-internal-degrees-of-freedom .unnumbered}
======================================
In this Section, we examine some of the gravitational effects that appear at leading relativistic order for particles with internal degrees of freedom. These effects, which can be seen as arising from relative time dilation of different internal levels, were first investigated in detail by Zych *et al.* [@Zych:2011jz; @Zych:2012kq] and Pikovski *et al.* [@Pikovski:2015du], and were further discussed by Zych and Brukner in the context of the equivalence principle [@Zych:2015vm].
The Hamiltonian formulation {#the-hamiltonian-formulation .unnumbered}
---------------------------
According to the Einstein equivalence principle, all internal energy acts as a mass from the perspective of both general and special relativity. That is, the mass terms appearing in the kinetic and potential energy of a system in a gravitational field should depend on the internal energy state. When the internal state corresponds to a dynamically varying degree of freedom, with its own Hamiltonian $H^\text{int}$, then all terms involving the mass should couple it to the external degree of freedom. In other words, the mass is promoted to an operator on the internal degree of freedom: $$\begin{gathered}
m\rightarrow M = m \, {\mathds{1}}^\text{int} + \frac{H^\text{int}}{c^2}.\end{gathered}$$ The full Hamiltonian for a particle in a uniform gravitational field, including the newly defined mass operator is then (to leading relativistic order) [@Zych:2015vm] $$\begin{aligned}
H &= M c^2+ \frac{P^2}{2M} + M g x{\\[3mm]}&= \left(m{\mathds{1}}^\text{int} + \frac{H^\text{int}}{c^2}\right)c^2+ \frac{P^2}{2\left(m {\mathds{1}}^\text{int} + \frac{H^\text{int}}{c^2}\right)} + \left(m {\mathds{1}}^\text{int} + \frac{H^\text{int}}{c^2}\right) g x{\\[3mm]}&= m c^2 + H^\text{int} + \frac{P^2}{2m } + m g x + \frac{1}{m c^2}\left\{-\frac{P^2}{2m}H^\text{int} + g x H^\text{int} \right\} + \mathcal{O}\left(c^{-4}\right),{\refstepcounter{equation}\tag{\theequation}}\label{FOHam}\end{aligned}$$ where, in the last line, we have expanded the $M^{-1}$ in a Taylor expansion. This is valid, provided that the largest eigenvalue of the internal Hamiltonian, denoted by $\|{H^\text{int}}\|$, satisfies $\|{H^\text{int}}\|/m c^2\ll 1$, *i.e.*, the internal energy is small compared to the rest mass. The additional terms introduced by the mass operator give lowest order relativistic effects. The first effect is introduced by the coupling of the internal energy to the kinetic energy operator, which represents lowest order special relativistic time dilation. The other interaction term, coupling the internal energy to the Newtonian potential, represents lowest order gravitational time dilation effects.
We can verify this by looking at the evolution of the internal degree of freedom. Provided that the internal evolution is not trivial, i.e., that it is not in an eigenstate of the internal Hamiltonian, it can be considered operationally as a clock [@Pikovski:2015du]. If we denote $q$ to be an observable of the internal degree of freedom, then the evolution given in the Heisenberg picture is, as described in Ref. [@Zych:2015vm], $$\begin{aligned}
\dot{q} = \frac{1}{i\hbar}[q,H] &= \frac{1}{i\hbar}\left\{[q,{H^\text{int}}]{\mathds{1}}^\text{ext} - [q,{H^\text{int}}]\frac{P^2}{2m^2c^2} + [q,{H^\text{int}}]\frac{gx}{c^2}\right\}{\\[3mm]}&= \dot{q}_\text{loc}\left(1 - \frac{P^2}{2m^2c^2} + \frac{g \,x}{c^2}\right).{\refstepcounter{equation}\tag{\theequation}}\label{eq:internalEvol}\end{aligned}$$ Here $\dot{q}_\text{loc}$, is the normal rate of internal evolution as given in the system’s rest frame. Recalling that the rate of change of proper time, in the non-relativistic, weak-field limit, is $d\tau = (1 -\frac{v^2}{2c^2} - \frac{\phi(x)}{c^2})dt$, we can easily identify these additional terms as a result of lowest order time dilation. For semi-classical evolution of the external degrees of freedom, the evolution of the internal degree of freedom is affected in a manner that is consistent with our understanding of relativistic effects. Interestingly this equation is valid not just for semi-classical systems, but also for non-local systems or systems with momentum that is not well defined. In these cases however, we cannot apply any of our classical intuition [@Pikovski:2015du].
This result can be interpreted in the following way; general relativity provides a description for the evolution of clocks which are attached to observers evolving according to the laws of classical mechanics. On the other hand, the mass operator, has in a sense, allowed us to describe the evolution of a clock attached to an observer who evolves according to the laws of quantum mechanics.
Though this intuition can be applied to the internal evolution, we will present results that show this is not true when observing the external evolution. The evolution of the position degree of freedom is given by, $$\begin{gathered}
\dot{x} = \frac{1}{i \hbar} [x,H] = \frac{1}{i\hbar}\frac{[x,P^2]}{2m\left(1 + \frac{{H^\text{int}}}{mc^2}\right)}
= \frac{P}{m}\left(1 - \frac{{H^\text{int}}}{mc^2}\right) + {\mathcal{O}}\left(m^{-2}c^{-4}\right).{\refstepcounter{equation}\tag{\theequation}}\label{eq:Vel}\end{gathered}$$ Again, if we consider a semi-classical wavepacket, and take the expectation value of the equation above, we find that the velocity of this wavepacket depends on the state of the internal degree of freedom. In particular, a particle in an excited state will have a slower expected velocity than one in its ground state. If the particle is prepared in a superposition state of internal energy, then its position at a later time will be entangled with the internal degree of freedom. Thus, the mass operator introduces spatial decoherence, which even appears in the case of a free particle [@Pikovski:2015du].
The path integral formulation {#the-path-integral-formulation .unnumbered}
-----------------------------
To examine these effects further, we will investigate the mass operator from the perspective of the path integral formalism. We motivate the work here with the question: *How does the mass operator affect the falling interference presented in the last Section?* In order for there to be any effect, the particle must have some non-degenerate internal energy levels. We will restrict ourselves to the simple case discussed in Ref. [@PJORLANDO2015], where the particle is spin-$\frac{1}{2}$ with a Zeeman splitting induced by an external magnetic field[^4]. Before we can answer the above question, we need to find the form for the propagator in this scenario.
We begin with the newly defined Lagrangian for this problem, $$\begin{gathered}
L(x,\dot{x}) = \frac12M\dot{x}^2 - Mg x - Mc^2, {\refstepcounter{equation}\tag{\theequation}}\label{eq:LMOP}\end{gathered}$$ For a particle with magnetic moment $\mu$ in a uniform magnetic field of strength $B$, the mass operator is given by $$\begin{gathered}
M = \left[\begin{array}{cc} m - \frac{\mu B}{2c^2} & 0 \\ 0 & m + \frac{\mu B}{2c^2}\end{array}\right].{\refstepcounter{equation}\tag{\theequation}}\label{eq:MOP}\end{gathered}$$ From this point, we can construct the Feynman propagator, in accordance with Eq. . $$\begin{gathered}
K^{\chi{\null^{\prime}}\!,\chi}(x{\null^{\prime}},t{\null^{\prime}};x,t) = {\left\langle{x{\null^{\prime}},\chi{\null^{\prime}}}\middle|{U(t{\null^{\prime}}- t)}\middle|{x,\chi}\right\rangle}{\refstepcounter{equation}\tag{\theequation}}\label{eq:SpinProp}\end{gathered}$$ This is still a matrix element of the time-evolution operator; however, the evolution operator now contains an index for spin, accounting for the two-dimensional internal Hilbert space. This also naturally leads to a matrix representation: $$\begin{gathered}
K^{\chi{\null^{\prime}}\!,\chi}(x{\null^{\prime}},t{\null^{\prime}};x,t) = \left[\begin{array}{cc}K^{{\scalebox{0.9}{$\uparrow$}}{\scalebox{0.9}{$\uparrow$}}}(x{\null^{\prime}},t{\null^{\prime}};x,t) & K^{{\scalebox{0.9}{$\uparrow$}}{\scalebox{0.9}{$\downarrow$}}}(x{\null^{\prime}},t{\null^{\prime}};x,t) {\\[3mm]}K^{{\scalebox{0.9}{$\downarrow$}}{\scalebox{0.9}{$\uparrow$}}}(x{\null^{\prime}},t{\null^{\prime}};x,t) & K^{{\scalebox{0.9}{$\downarrow$}}{\scalebox{0.9}{$\downarrow$}}}(x{\null^{\prime}},t{\null^{\prime}};x,t) \end{array}\right]. {\refstepcounter{equation}\tag{\theequation}}\label{eq:DefMatrixProp}\end{gathered}$$ Since the interaction terms appearing in Eq. \[eq:LMOP\] all commute with the mass operator in Eq. \[eq:MOP\], the propagator can be greatly simplified, as it is then diagonal in the internal energy eigenbasis: $$\begin{gathered}
{\mathbf{K}}(x{\null^{\prime}},t{\null^{\prime}};x,t) = \left[\begin{array}{cc}K^{m_-}(x{\null^{\prime}},t{\null^{\prime}};x,t) &0 {\\[3mm]}0& K^{m_+}(x{\null^{\prime}},t{\null^{\prime}};x,t)\end{array}\right],{\refstepcounter{equation}\tag{\theequation}}\label{eq:DefMatrixProp2}\end{gathered}$$ where $K^{m_\pm}(x{\null^{\prime}},t{\null^{\prime}};x,t)$ is the propagator for a particle of mass $m_\pm=m \pm \mu B/(2c^2)$; from the perspective of the propagator, the different internal energy states just appear as modified masses. Typically, the rest mass energy is excluded from the Lagrangian; it has no effect on the dynamics, and merely leads to an unmeasurable global phase $\exp(-{i mc^2 t}/{\hbar})$. When promoting mass to an operator, a relative phase of $\exp({-i \mu B t }/{\hbar})$ is introduced between the two internal states, which could in principle have a measurable effect.
We wish to use this propagator to calculate the wavefunctions for particles which initially have spin in the superposition $\ket{\chi_{_0}} = \alpha\ket{{\scalebox{0.9}{$\uparrow$}}} + \beta\ket{{\scalebox{0.9}{$\downarrow$}}}$, with $|\alpha|^2 + |\beta|^2 =1$. However, if we are only interested in the interference pattern observed on the screen, and do not measure the spin state of the particle, then we need to trace out the spin information. We review how to do this in Appendix \[app:PropagatorTrace\]. We find that the spatial probability distribution for an initial state $\ket{\chi_{_0}}\otimes\ket{\psi_0}$ is then $$\begin{gathered}
{\left\langle{x}\right\rangle} = |\alpha|^2\left|\int dx K^{m_-}(x,t;x,0)\psi_0(x)\right|^2 +|\beta|^2\left|\int dx K^{m_+}(x,t;x,0)\psi_0(x)\right|^2, {\refstepcounter{equation}\tag{\theequation}}\label{eq:ConvexSum}\end{gathered}$$ which is a convex sum of the contributions coming from each internal state. This is immediately identifiable as decoherence, which is consistent with our interpretation of Eq. . Additionally, this demonstrates that, when tracing out the spin degree of freedom, the phase introduced by the rest mass operator becomes irrelevant.
The form of Eq. allows for easy calculation of the decohered spatial distribution, which we will illustrate with an example. Take the propagator to be that of a free particle and choose the initial wavefunction to be a Gaussian wavepacket with momentum $p$. This wavefunction is given by $$\begin{gathered}
\psi_0(x) = {\left(\pi \sigma\right)^{-\frac14}} \exp\left[{-\frac{x^2}{2\sigma^2}+\frac{ip x}{\hbar}}\right].{\refstepcounter{equation}\tag{\theequation}}\label{eq:initialWF}\end{gathered}$$ After convolving this with the free space propagator $K_0^{m_\pm}(x,t,x_0,0)$, we have $$\begin{gathered}
\psi_{m_\pm}(x,t) = \frac{\exp\left[i\phi-2\frac{(z-p/m_\pm t)^2}{\sigma^2(1+\gamma_{m_\pm}^2)}\right]}{(\pi \sigma^2)^{1/4}\sqrt{i - \gamma_{m_\pm}}},\end{gathered}$$ where $\gamma_{m_\pm} = \frac{\hbar t}{m_\pm\sigma^2}$ and $\phi$ is an irrelevant phase factor. We notice that the mean of this Gaussian moves with speed $p/m_\pm$. If the particle is prepared in a superposition state of internal energy then Eq. states that the probability distribution will be given by $$\begin{gathered}
P(x,t)= |\alpha|^2\big|\psi_{m_-}(x,t)\big|^2 + |\beta|^2\big|\psi_{m_+}(x,t)\big|^2.\end{gathered}$$ In other words, the spatial distribution is given by a mixture of Gaussian wavepackets propagating with different speeds.
Given the initial state $\ket{\Psi} = (\alpha\ket{{\scalebox{0.9}{$\uparrow$}}} + \beta\ket{{\scalebox{0.9}{$\downarrow$}}}) \otimes \ket{\psi_0}$, the coupling introduced by the mass operator evolves the state to $\ket{\Psi(t)} = \alpha\ket{{\scalebox{0.9}{$\uparrow$}}}\otimes\ket{\psi_{m_-}(t)} + \beta\ket{{\scalebox{0.9}{$\downarrow$}}} \otimes \ket{\psi_{m_+}(t)}$. If a detector is placed at a distance from the source far enough for the Gaussian distributions described by ${\left\langle{x}\middle|{\psi_{m_-}(t)}\right\rangle}$ and ${\left\langle{x}\middle|{\psi_{m_+}(t)}\right\rangle}$ to become distinct, then the arrival time of the particle will be bimodal. Again, if the position degree of freedom is considered to be a ‘clock’ – its non-trivial evolution permits this – then this may be considered to be a special relativistic time dilation effect [@Pikovski:2015du].
Gravitational decoherence in double slit interference {#gravitational-decoherence-in-double-slit-interference .unnumbered}
=====================================================
We have now developed the tools to explore falling double slit interference with an internal degree of freedom. Again, we consider a particle incident on slits of width $2a$ centred at $x=\pm b$. Our earlier calculation in the first Section used the semi-classical approximation for the $z$-direction, to replace arrival times $T$ and $\tau$ with the classically expected times, $D/v$ and $L/v$. We have just shown, however, that the arrival time of the particle will no longer be well defined when the internal degree of freedom plays a dynamical role.
We also saw, in the previous Section, that the effect of using the gravitational propagator was equivalent to performing a mass-independent coordinate transformation. This means that, for a wavepacket with zero initial average momentum, the mass operator has no effect on the position expectation value under the influence of a gravitational potential[^5]. This leads to an interesting effect if we consider a two dimensional Gaussian wavepacket with zero average momentum in the $x$-direction (in the direction of the gravitational field) and a non-zero average momentum in the $z$-direction. At some fixed distance along $z$ from the particle’s initial location, the difference in expected arrival times will mean that, depending on the internal state, gravity will have displaced the wavepacket for different amounts of time. As a result, the higher energy state will fall further than the lower energy state, causing gravity to act, in some sense, like an asymmetric Stern-Gerlach device. This effect will be very small, as it depends on the magnitude of $\|{H^\text{int}}\|/(mc^2)$, but can be sensitively detected by introducing an interference pattern along $x$.
We calculate the pattern produced by the double slit by returning to a two dimensional propagator, and simplifying the problem to a Gaussian particle distribution incident on the slits which is then detected at a screen $L$ metres away. In this case, the wavefunction just beyond the slits, for a particle of mass $m$, is given by $$\begin{gathered}
\psi^\text{(1)}\left(x,z,t\right) = \frac{\int_{b-a}^{b+a}\int_{{-\infty}}^\infty dx_0 dz_0 K_0(z,t;z_0,t)K_g(x,t;x_0,0)\psi_0(x_0,z_0)}{\int_{b-a}^{b+a}dx \psi(x_0,0)}.{\refstepcounter{equation}\tag{\theequation}}\label{eq:Gauss2DDiff}\end{gathered}$$ This avoids the semi-classical approximations made in the previous calculation, but leads to a time-dependent wavefunction. However, we are only interested in the spatial distribution observed at the screen, with no measurement performed regarding the time of arrival. The simplest way to account for this is to average the distribution over some length of time so that $$\begin{gathered}
\bar{\psi}(x,z=L) = \frac{1}{2\Delta t}\int_{-\Delta t}^{\Delta t} |\psi(x,L,t)|^2dt,{\refstepcounter{equation}\tag{\theequation}}\label{eq:TimeAveraged}\end{gathered}$$ where $\Delta t$ will have some relationship with the spatial spread of the Gaussian packet in the $z$ direction, such that the majority of the probability lies within $\pm\Delta t$. Figure \[fig:IntDec\] shows a plot of the resulting two-slit interference, calculated for a neutron in equal superposition of its internal energy states. The energy splitting is $\Delta E \approx 10^{-14}\,{\rm J}$, corresponding to a magnetic field on the order of $10^{12}\, {\rm T}$. Even with this infeasibly large energy splitting, the decoherence effect occurs over tens of metres. If a more reasonable value for the energy splitting is used, then spreading of the wavepacket delocalises the particle before the decoherence is even detectable.
![**Decoherence of Double Slit Interference in a Gravitational Field.** The internal energy splitting leads to different expected arrival times for the wavepacket at the screen. The pattern corresponding to the higher energy spin state will fall further than its counterpart, resulting in periodic reductions in the visibility of the interference. The intensity at the screen for distances of $10\,{\rm m}$, $30\,{\rm m}$ and $50\,{\rm m}$ is shown for a neutron with wavelength $\lambda \sim 10^{-8}\,{\rm m}$ and internal energy splitting of $\Delta E = 10^{-14}\,{\rm J}$. The top row shows the time-averaged spatial probability distribution for the up (green, solid) and down (blue, dotted) spin components, while the bottom row plots the spin-averaged probability distribution as it would be observed on the screen. All $x$ positions are relative to the position of a classical particle with mass $m_-$.[]{data-label="fig:IntDec"}](Fig3.png){width="96.00000%"}
This decoherence is suggestive of the effects described in Ref. [@Zych:2011jz]; where Zych *et al.* demonstrate that the interference pattern in a Mach-Zehnder interferometer decoheres as a result of proper time. The work we present here is a free space interference analogue, which similarly exhibits periodic decoherence effects. This gives strength to the argument in Ref. [@Pikovski:2015du] that the effects on external evolution, introduced by the mass operator, are complementary to the interpretation of time dilation, occurring for evolution of the internal degrees of freedom. To put it more elegantly, embedding an operational clock in a system which behaves quantum mechanically results in an evolution which destroys this quantum nature.
Coherence, correlations, and non-Markovian dynamics {#coherence-correlations-and-non-markovian-dynamics .unnumbered}
===================================================
The decoherence of the interference fringes discussed above can be better understood in terms of correlations between internal and external degrees of freedom. It also turns out that spin coherence is necessary for the generation of non-classical correlations between the internal and external degrees of freedom. We discuss each of these ideas successively below, beginning with coherence theory, before providing a further interpretation in terms of non-Markovian open dynamics.
*Incoherent operations.* The total Hamiltonian in Eq. is diagonal in the spin basis and can therefore be expressed as $H = {\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} \otimes H_{m_{-}} + {\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|} \otimes H_{m_{+}}.$ Therefore, the unitary operator for the joint dynamics has the form of a controlled-unitary on the external degree of freedom: $$\begin{gathered}
\label{eq:incoherent-U}
U = {\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} \otimes U_{m_{-}}
+ {\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|} \otimes U_{m_{+}},\end{gathered}$$ which has the potential to generate correlations between the internal and external degrees of freedom. However, when considering the reduced dynamics of the spin alone, evolution between two points in time is described by an incoherent operation (one which maps incoherent states to incoherent states) in the energy eigenbasis [@arXiv:1609.02439], *i.e.*, $$\begin{gathered}
U\ket{s\, \psi(0)} = \ket{s\, \psi_{m_r}(t)},\end{gathered}$$ where $s\in\{{\scalebox{0.9}{$\downarrow$}},{\scalebox{0.9}{$\uparrow$}}\}$, $r=+$ if $s={\scalebox{0.9}{$\downarrow$}}$, and $r=-$ if $s={\scalebox{0.9}{$\uparrow$}}$. More specifically, $U$ is incoherent since it will map a mixed state of the form $\sigma_{IC}=q {\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} + (1-q) {\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|}$ to itself: $$\begin{gathered}
\operatorname{tr}_{ext}[U \sigma_{IC} \otimes \rho U^\dag] = \sigma_{IC}, \end{gathered}$$ where $\rho$ is any state of the external degree and $IC$ stands for incoherent.
On the other hand, due to the non-vanishing commutator $[P^2,x] \ne 0$, the dynamics of the external degree of freedom alone is not described by an incoherent operation in the position basis. That is, $U\ket{s\, x} = \ket{s\, \psi_{m_r}(t)}$, and coherence of the wavefunction can increase. For the same reason, both conditional unitary operations $U_{m_{-}}$ and $U_{m_{+}}$ will also look like incoherent operations from the perspective of the spin.
*Entanglement and discord.* In recent years, researchers studying coherence theory have shown that incoherent operations can lead to generation of entanglement and quantum discord when the initial spin state possesses coherence. The generation of entanglement is easily checked by taking the spin to initially be in the pure state $\alpha \ket{{\scalebox{0.9}{$\downarrow$}}} + \beta \ket{{\scalebox{0.9}{$\uparrow$}}}$ for $\alpha,\beta \ne 0$ and the position state to be $\ket{\psi}$: $$\begin{gathered}
U\left( \alpha \ket{{\scalebox{0.9}{$\downarrow$}}\, \psi(0)} + \beta \ket{{\scalebox{0.9}{$\uparrow$}}\, \psi(0)}\right) = \alpha \ket{{\scalebox{0.9}{$\downarrow$}}\, \psi_{m_{+}}(t)} + \beta \ket{{\scalebox{0.9}{$\uparrow$}}\, \psi_{m_{-}}(t)},\end{gathered}$$ which is an entangled state, since the marginal states are not pure.
It is known that any coherence can be turned into entanglement via some incoherent operation and a pure ancilla [@PhysRevLett.115.020403]. However, in our setup we are limited to a specific incoherent operation, which may not be able to generate entanglement for all coherent initial states. Consider the case when the initial spin state is the mixed state $$\begin{gathered}
\sigma_{CO} = w {\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} + (1-w) {\left|{+}\middle\rangle\middle\langle{+}\right|},\end{gathered}$$ where $\ket{+}=(\ket{{\scalebox{0.9}{$\uparrow$}}}+\ket{{\scalebox{0.9}{$\downarrow$}}})/\sqrt{2}$ and $CO$ stands for coherent. For $0 \le w \le 1$, the time-evolved state will have quantum correlations, but for some values of $w$ will have no entanglement; the future state will be fully separable for $w=1$ and entangled for $w=0$. Therefore, there must be a critical value for $w=w_c$ where the transition from entangled state to separable state occurs. In the regime where the state is separable, it will necessarily have quantum discord [@arXiv:1107.3428; @RevModPhys.84.1655; @arXiv:1605.00806] as measured by the internal or external degree of freedom.
In fact, the only time quantum discord vanishes for $t>0$ is when the initial spin state has the form $\sigma_{IC} = q {\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} + (1-q) {\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|}$. Let us further suppose that the initial external state is given by a density matrix $\rho$. After evolving for some time $t$, the system will be in state $$\begin{gathered}
\label{eq:clstate}
U\sigma_{IC} \otimes \rho \, U^\dag =
q {\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} \otimes \rho_{m_-}(t)+ (1-q) {\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|} \otimes \rho_{m_+}(t).\end{gathered}$$ This clearly becomes a classical mixture of the states $\rho_{m_-}(t)$ and $\rho_{m_+}(t)$ with weighting $w$ when the internal degree of freedom is measured (whichever measurement basis is chosen). That is, the unitary operation in Eq. , being an incoherent operation on the internal degree of freedom, will not generate any non-classical correlations when the initial spin state is a classical mixture of energy eigenstates.
On the other hand, with the exception of pathological cases where the initial wavefunction does not have support everywhere, the two spin-conditioned external states will never be exactly orthogonal, *i.e.*, $\operatorname{tr}[\rho_{m_+}(t)\rho_{m_-}(t)]\ne 0$. This means that the spin state after a measurement on the external degree of freedom will, in general, depend on the choice of measurement basis; in other words, there are non-classical correlations (discord) in one direction.
Let us now consider the case where both initial states can be arbitrary mixed states. Then the time evolved states have form $$\begin{gathered}
\label{eq:totalstate}
U
\begin{pmatrix} a & b \\
b^* & 1-a \end{pmatrix}
\otimes \rho \, U^\dag =
\begin{pmatrix} a \, \rho_{m_-} & b\, U_{m_-} \, \rho \, U_{m_+}^\dag\\
b^* U_{m_+} \, \rho \, U_{m_-}^\dag & (1-a) \rho_{m_+}.
\end{pmatrix} \end{gathered}$$ If we can make (non-unitary) operations on the spin degree of freedom, such as projections, we will see different interference patterns corresponding to different outcomes. By making strong measurements on the spin by, *e.g.*, introducing a Stern-Gerlach apparatus, the correlations could be used to steer the interference pattern.
For example, to see how correlations affect the reduced dynamics, consider the middle column of Figure \[fig:IntDec\]. If the magnetic field, and hence the effective coupling, was turned off for $z>30m$, the interference pattern would subsequently evolve unitarily with a single mass-$m$ propagator. It would continue to fall as if it were a classical object, and the fringe visibility would never return. This is because the spin degree of freedom, which ‘remembers’ the original two-slit pattern, is no longer interacting with the position degree of freedom. However, if the spin components were filtered out at a later time using a Stern-Gerlach apparatus, the visibility could be recovered in full; the spin acts as a memory, hiding information about the particle’s earlier trajectory. In other words, the periodic re-coherence of the spatial wavefunction is indicative of non-Markovian behaviour, which we will now discuss further.
*Reduced non-Markovian dynamics.* In order to see the effects of these correlations from another perspective, suppose we only look at the position of the particle. From the perspective of an observer who cannot measure the internal degree of freedom, the evolution of the particle appears to be open, with the spin acting as an environment. We immediately see that the same features are seen whether the initial spin state possesses coherence or not. The external state is obtained by tracing over the spin in Eq. to get $$\begin{gathered}
\label{eq:marginalstate}
\rho_{ext} = a \rho_{m_-} + (1-a) \rho_{m_+}.\end{gathered}$$ That is, the observed interference pattern is indistinguishable and independent of $b$. When $b=0$ the internal and external degrees of freedom become classically correlated, and both entanglement and discord are vanishing. Conversely, when $b \ne 0$ discord (and possibly entanglement) will be present.
Whenever the past state of a system directly affects its future evolution, a process is called non-Markovian. While there have been several mathematical definitions of ‘non-Markovianity’ proposed for quantum processes [@NMRev; @breuer-rev] (with variable levels of descriptive success), the operational meaning of the term is clear cut [@arXiv:1512.00589]: If the causal continuity of a system’s evolution is broken at some time $t$ by, for example, making a measurement and re-preparing the system in a fixed state $\ket{\phi}$, independently of the measurement outcome, then the process is non-Markovian if the system’s density operator $\rho_\tau(x,x{\null^{\prime}})$ at a later time $\tau$ depends on the measurement outcome $k$ or on the system’s history $h$ prior to $t$. Formally, $$\begin{gathered}
\rho_\tau(x,x{\null^{\prime}}\,|\ket{\phi},k,h)\neq\rho_\tau(x,x{\null^{\prime}}\,|\ket{\phi},k{\null^{\prime}},h{\null^{\prime}}) \Rightarrow\; \text{Non-Markovian}.\end{gathered}$$ This kind of behaviour implies that there is some sort of memory transmitting information from the past across the causal break. We will see that this is the case for the falling particle described earlier in this Section.
![**Non-Markovian Behaviour of an Interfering Particle with Spin.** A grating filter is introduced at $z=30\,{\rm m}$, allowing particles to pass through either the white or black regions in the top-left panel. The particle is then subsequently re-prepared in a Gaussian state, whose probability density is shown in the bottom-left panel; this state is the same for either choice of filter. The plots on the right show the probability distribution to find the particle at different positions on a screen placed at $z=80\,{\rm m}$ for the two choices of filter; top curves show spin-up (green, solid) and spin-down (blue, dotted) projections, the lower plots show the spin-averaged position distribution. Distinguishability of the two cases indicates non-Markovian behaviour. All $x$ positions are relative to the corresponding position of a classical particle with mass $m_-$.[]{data-label="fig:CausalBreak"}](Fig4.png){width="100.00000%"}
In order to introduce a causal break in the evolution, we will put the spatial filter shown in the left column of Figure \[fig:CausalBreak\] at $z=30m$. This can be set to either allow particles through the white region (which has the greatest overlap with the spin-up wavefunction) or the black region (which has the greatest overlap with the spin-down wavefunction). After the filter, the particle is rapidly (effectively instantaneously) collimated into a Gaussian state along $x$, which does not depend on whether the black or white filter is chosen.
Since the overlap of the white (black) filter function $f_{w(b)}(x)$ with the spin-up and spin-down wavefunctions at $z=30m$ is different, the subsequent spin state will be conditioned on the choice of filter. The post-filter spin density operator is given by $$\begin{aligned}
\rho_{w(b)} =& |\alpha|^2\left|\int {\rm d} x\,f_{w(b)}(x)\psi_{m_-}(x)\right| {\left|{\uparrow}\middle\rangle\middle\langle{\uparrow}\right|} + |\beta|^2\left|\int {\rm d} x{\null^{\prime}}\,f_{w(b)}(x)\psi_{m_+}(x)\right| {\left|{\downarrow}\middle\rangle\middle\langle{\downarrow}\right|} \nonumber \\
&+\alpha\beta^*\int {\rm d} x {\rm d} x{\null^{\prime}}\,f_{w(b)}(x)\psi_{m_-}(x)f_{w(b)}(x{\null^{\prime}})\psi_{m_+}^*(x{\null^{\prime}}){\left|{\uparrow}\middle\rangle\middle\langle{\downarrow}\right|}+ {\rm h.c.},\end{aligned}$$ where $\psi_{m_\pm}(x)$ is the time-averaged wavefunction for the relevant spin branch at the $z$ position the filter is applied. For $\alpha=\beta=1/\sqrt{2}$, the post-filter probabilities for the spin-up and down states are $\sim \frac{4}{5}$ and $\sim\frac{1}{5}$ respectively for the white filter, and *vice versa* for the black filter.
The right hand side of Figure \[fig:CausalBreak\] shows the probability distribution further from the slits after each of the filters is applied (note that the two spin components have already begun to separate again). Since the two conditional distributions are clearly different, the dynamics of the spatial distribution must be non-Markovian; the only way the post re-preparation evolution can depend on which filter was applied is through the spin state, which is acting as a memory.
Conclusion {#conclusion .unnumbered}
==========
The universality of free fall is a pervasive phenomenon, and one which has inspired more fundamental gravitational equivalence principles. This includes Einstein’s famous equivalence between mass and energy which, ultimately, forms part of the foundation for our current understanding of gravity. Here, we have explored how a self-interfering quantum particle falls under the influence of Newtonian gravity. We have shown that the universality of free fall holds even in this case, as the interference pattern itself fall just like a classical particle.
We have also considered interference of falling neutrons in the presence of a strong magnetic field. The presence of the magnetic field leads to splitting of the internal energy of the neutrons which, according to the Einstein equivalence principle, should make spin-down neutrons more massive than spin-up neutrons. Moreover, if a neutron is prepared in a spin-superposition state (with respect to the internal energy eigenbasis), this seemingly leads to the violation of a super-selection rule, *i.e.*, superposition of masses. However, we use the mass operator formalism [@Zych:2015vm] to show that, if the energy splitting of the internal spin contributes to the mass of the neutron, then the visibility of the interference pattern periodically decreases and increases.
Our results indicate that these decoherence effects are a consequence of an operational clock embedded within a quantum mechanical rest frame. That is, the internal degree of freedom keeps track of the time the particle spends being in different mass states. Finally, we have shown that this accounting of the internal energy (mass) state can be understood as non-Markovian dynamics for the position degree of freedom, with the spin acting as a memory. We show the non-Markovian behaviour by operational methods using the notion of causal break introduced in Ref. [@arXiv:1512.00589]. In particular, we have given an operational recipe to witness the non-Markovian memory by solely acting on the external degree of freedom.
In Ref. [@Pikovski:2015du] it is argued that gravity may be the culprit for quantum decoherence. This mechanism does not depart from how we think of decoherence in open systems theory more generally. This view is fundamentally different from that posited by the proponents of collapse theory who claim that gravity leads to fundamentally irreversible dynamics, *cf.* Ref. [@Snadden:1998bj]. Thankfully, one can differentiate between the two hypotheses by checking whether the decoherence can be reversed [@arenz_distinguishing_2015], which we do here demonstrating that coherence-information loss due to gravity can be recovered.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Lucas Celeri and Robert Mann for insightful discussions, German Valencia for pointing out an error in an earlier version of this work.
Appendices {#appendices .unnumbered}
==========
The COW Experiment {#sec:COW}
==================
The Colella-Overhauser-Werner experiment provided the first evidence of a gravitational effect that is purely quantum mechanical [@Colella:1975jc]. In this experiment, Colella *et al.* used a silicon crystal interferometer to split a beam of neutrons, placing one of the beam paths in a higher gravitational potential (see Figure \[fig:COWsetup\]). The difference in the gravitational potential between each arm results in a relative phase shift, which, when recombined, can be measured as modulated intensity. On the length scale of the interferometer, the gravitational field is approximately constant. This allows the relative phase difference between the beams to be calculated using the Wentzel-Kramers-Brillouin (WKB) approximation; that is to integrate the potential difference between the classical trajectories over time [@hall2013quantum]. The two vertical paths of the interferometer contribute phases which cancel out, leaving only the horizontal paths. The phase shift is found to be $$\begin{gathered}
\Delta\Phi = \frac{2\pi m_{{{\protect\scalebox{0.9}{$_I$}}}}m_{{{\protect\scalebox{0.9}{$_G$}}}}g A \lambda}{h^2}\sin\phi. {\refstepcounter{equation}\tag{\theequation}}\label{COWPHASE}\end{gathered}$$
![**COW Interferometer**| Left: Schematic of the apparatus used in the COW experiment, taken from Ref. [@Colella:1975jc]. The interferometer is rotated about the axis of the first Bragg angle of diffraction. Right: Simplified diagram used to derive the induced phase shift.[]{data-label="fig:COWsetup"}](COWsetup.png "fig:") ![**COW Interferometer**| Left: Schematic of the apparatus used in the COW experiment, taken from Ref. [@Colella:1975jc]. The interferometer is rotated about the axis of the first Bragg angle of diffraction. Right: Simplified diagram used to derive the induced phase shift.[]{data-label="fig:COWsetup"}](COWPhaseDerivation.pdf "fig:")
A phase shift of this form would be predicted for a quantum mechanical particle in the presence of any scalar potential; in this case, it is the Newtonian gravitational potential. A full description of this effect requires only regular quantum mechanics and Newtonian theory, needing no metric description of gravity, but being unexplainable by classical Newtonian gravity alone. It represents the first evidence of gravity interacting in a truly quantum mechanical way. However, from the perspective of quantum theory, this effect is well understood as a scalar Aharanov-Bohm effect and manifests similarly for charged particles in electric potentials [@Allman:1999fh; @Zych:2011jz].
Single slit diffraction in a Newtonian gravitational potential {#App:PathIntGravity}
==============================================================
Derivation of the propagator {#app:PropDeriv}
----------------------------
First consider the Lagrangian for a free particle $L = \frac12m\dot{x}^2$. The Feynman propagator is given by $$\begin{aligned}
{\left\langle{x{\null^{\prime}},t{\null^{\prime}}}\middle|{x,t}\right\rangle} = K_0(x{\null^{\prime}},t{\null^{\prime}};x,t) &= \bigintsss{\mathcal{D}}\left(x(t)\right) \exp\left[\frac{i}{\hbar}\int_t^{t{\null^{\prime}}}\frac12 m \dot{x}(s)^2 ds\right]{\refstepcounter{equation}\tag{\theequation}}\label{eq:FreeFInt}
{\\[3mm]}&= \frac{\exp\left[\frac{im}{2\hbar}\frac{(x{\null^{\prime}}- x)^2}{t{\null^{\prime}}- t}\right]}{\sqrt{2\pi i \hbar (t{\null^{\prime}}- t)/m}}{\refstepcounter{equation}\tag{\theequation}}\label{eq:FreeProp}.\end{aligned}$$ This result will be needed when we consider the propagator for a particle in a linear gravitational potential. In this case the Lagrangian is given by $L = \frac12 m \dot{x}^2 - m g x$, which gives the Feynman propagator $$\begin{gathered}
K_g(x{\null^{\prime}},t{\null^{\prime}};x,t) = \bigintsss{\mathcal{D}}\left(x(t)\right) \exp\left[\frac{i}{\hbar}\int_t^{t{\null^{\prime}}} ds \left\{ \frac12 m \dot{x}(s)^2 - m g x(s)\right\}\right].{\refstepcounter{equation}\tag{\theequation}}\label{eq:GFP-1}\end{gathered}$$ To simplify this calculation we express the path $x(t)$ in terms of deviations from the classical trajectory $x_c(t)$ which satisfies the Euler-Lagrange equations of motion. The Feynman measure which sums over all possible paths then becomes a sum over all possible deviations from the classical path. The action expressed in terms of this new parametrisation is $$\begin{aligned}
S[x_c(t) + \delta x(t)] =& \int_{t}^{t{\null^{\prime}}} dt \left\{ \frac12 m \Big(\dot{x} + \delta \dot{x}\Big)^2 - mg\Big(x_c + \delta x\Big) \right\} {\\[3mm]}=&\int_{t}^{t{\null^{\prime}}} dt \left\{
\frac12m\dot{x}_c^2 - m g x_c + m\dot{x}_c\delta \dot{x}
+ \frac12 m(\delta\dot{x})^2 - mg \delta x \right\},\end{aligned}$$ where $\frac12m\dot{x}_c^2 - m g x_c = {\textstyle}s[x_c(t)]$, which is evidently the extremised action given by the classical trajectory. The term containing $\dot{x}_c\delta\dot{x}$ can be integrated by parts, realising that the deviations are zero at the endpoints of the path: $$\begin{gathered}
S[x_c(t) + \delta x(t)] = S[x_c(t)] + \int_{t}^{t{\null^{\prime}}} \frac12 m (\delta \dot{x})^2 \, dt
+ [ \ddot{x} \delta x]_{t}^{t{\null^{\prime}}}
- \int_{t}^{t{\null^{\prime}}}m\delta x \left(\ddot{x}_c + g\right) dt.
$$ The last two terms vanish ($\ddot{x}_c = -g$) and substituting the rest into Eq. and factoring out the classical action we arrive at $$\begin{gathered}
K_g(x{\null^{\prime}},t{\null^{\prime}};x,t) = \exp\left[\frac{i}{\hbar} S\left[x_c(t)\right]\right]\bigintsss{\mathcal{D}}\left(\delta x(t)\right) \exp\left[\frac{i}{\hbar}\int_t^{t{\null^{\prime}}}\frac12 m \delta \dot{x}(s)^2ds\right].{\refstepcounter{equation}\tag{\theequation}}\label{eq:GFP-2}\end{gathered}$$ The remaining Feynman integral over the deviations is recognised as the free particle propagator in Eq. , but with the subtle difference being that $x=x{\null^{\prime}}=0$. Substituting the integral with the expression from Eq. the propagator becomes $$\begin{gathered}
K_g(x{\null^{\prime}},t{\null^{\prime}};x,t)= \frac{\exp\left[\frac{i}{\hbar} S\left[x_c(t)\right]\right]}{\sqrt{2\pi i \hbar (t{\null^{\prime}}- t)/m}}.{\refstepcounter{equation}\tag{\theequation}}\label{eq:GFP-3}\end{gathered}$$ Now, all that remains is to find the explicit form for the classical action. We begin with the classical equation of motion, $\ddot{x}_c = -g$, and solve to find the general solution, $$\begin{gathered}
x_c(t)= -\frac12 g t^2 + a t + b.\end{gathered}$$ We now impose the boundary conditions $x(t)= x$ and $x(t{\null^{\prime}})= x{\null^{\prime}}$ and solve for the constants $a$ and $b$: $$\begin{gathered}
x = -\frac12 g t^2 + at + b \quad\mbox{and}\quad
x{\null^{\prime}}= -\frac12gt{\null^{\prime\raisebox{1pt}{$\scriptstyle{2}$}}}+ a t{\null^{\prime}}+ b{\refstepcounter{equation}\tag{\theequation}}\label{eq:con2}.\end{gathered}$$ Solving for $a$ and $b$ gives $$\begin{gathered}
a= \frac{x{\null^{\prime}}-x}{t{\null^{\prime}}-t} - \frac{g(t^2 - t{\null^{\prime\raisebox{1pt}{$\scriptstyle{2}$}}})}{2(t{\null^{\prime}}- t)}
= \frac{x{\null^{\prime}}-x}{t{\null^{\prime}}-t} + \frac12 g(t + t{\null^{\prime}}),{\refstepcounter{equation}\tag{\theequation}}\label{a}\end{gathered}$$ and $$\begin{aligned}
b&= \frac12\left(x{\null^{\prime}}+ x + \frac12 g (t^2 + t{\null^{\prime\raisebox{1pt}{$\scriptstyle{2}$}}}) - a(t+t{\null^{\prime}})\right){\\[3mm]}&= \frac12\left(x{\null^{\prime}}+ x - (x{\null^{\prime}}- x)\frac{t + t{\null^{\prime}}}{t{\null^{\prime}}-t} + \frac12 g (t^2 + t{\null^{\prime\raisebox{1pt}{$\scriptstyle{2}$}}}) -\frac12 g (t+t{\null^{\prime}})^2\right){\\[3mm]}&= \frac12\left(\frac{(x{\null^{\prime}}+ x)(t{\null^{\prime}}-t) - (x{\null^{\prime}}- x)(t + t{\null^{\prime}})}{t{\null^{\prime}}-t} - gtt{\null^{\prime}}\right){\\[3mm]}&= \frac12\left(\frac{2(xt{\null^{\prime}}- x{\null^{\prime}}t)}{t{\null^{\prime}}-t} - gtt{\null^{\prime}}\right)
= \frac{x t{\null^{\prime}}- x{\null^{\prime}}t}{t{\null^{\prime}}-t} - \frac12 g t t{\null^{\prime}}.{\refstepcounter{equation}\tag{\theequation}}\label{b}\end{aligned}$$ Thus, the action of the path taken from $(x,t)$ to $(x{\null^{\prime}}, t{\null^{\prime}})$ is $$\begin{aligned}
S[x_c(t)] &= \int_{t}^{t{\null^{\prime}}} dt \left\{ \frac12m(-gt + a)^2 - m g(-\frac12 g t^2 + at + b) \right\} {\\[3mm]}&= \frac{m}{2}\int_{t}^{t{\null^{\prime}}} dt \left\{2g^2t^2 -2 g a t+ a^2 - 2g b \right\}{\\[3mm]}&= \frac{m}{2}\left(a^2(t{\null^{\prime}}-t) - 2g\left(\rule{0cm}{.4cm}a(t{\null^{\prime}}-t) + b\right)(t{\null^{\prime}}-t) + \frac{2 g^2}{3}(t{\null^{\prime}}-t{\null^{\prime}})^3 \right)\\
&= \frac{m}{2}\left\{\frac{(x{\null^{\prime}}- x)^2}{t{\null^{\prime}}-t} - g(x+x{\null^{\prime}})(t{\null^{\prime}}-t) - \frac{g^2}{12}(t{\null^{\prime}}-t)^3\right\}.{\refstepcounter{equation}\tag{\theequation}}\label{eq:clAc}\end{aligned}$$ Finally, substituting this into Eq. , we arrive at the complete expression for the propagator for a particle in a gravitational potential, $$\begin{aligned}
K_g(x{\null^{\prime}},t{\null^{\prime}};x,t) &=\; \frac{\exp\left[\frac{im}{2\hbar}\left\{\frac{(x{\null^{\prime}}- x)^2}{t{\null^{\prime}}-t} - g(x+x{\null^{\prime}})(t{\null^{\prime}}-t) - \frac{g^2}{12}(t{\null^{\prime}}-t)^3\right\}\right]}{\sqrt{2\pi i \hbar (t{\null^{\prime}}- t)/m}}{\refstepcounter{equation}\tag{\theequation}}\label{eq:GFP}\;.\end{aligned}$$
Calculating the single slit wavefunction {#sec:SlitWF}
----------------------------------------
We now consider applying this propagator to the problem at hand. Let’s begin by assuming that the slits are long enough to ignore diffraction effects in the $y$ direction. Consider a source of particles at the origin $(0,0)$ and let a double slit be located at distance $z=D$ metres from the source. Each slit has width $2a$ with centre located at $x=\pm b$. The screen is then a further $L$ metres away from the slits. The two dimensional propagator required for this problem is given by a free particle propagator in the $z$-direction multiplied by the gravitational propagator for the $x$ direction as calculated in Eq. . This propagator allows us to ask the question of *If a particle initially starts at position ${\vec{r}}= (0,0)$, what is the probability of finding the it at position ${\vec{r}\, {\null^{\prime}}}=(x,D+L)$ on the screen?* This distribution in $x$ will be the the two slit interference pattern that we seek. When computing this amplitude we consider a semi-classical approach. We assume that the ‘trajectory’ of the neutron can be separated into two parts: (a) the path from the source to the slits, followed by (b) the path from the slits to the screen. Quantum mechanically the particles need not pass through the slits and there even exists the possibility of them passing through the slits multiple times before hitting the screen. That being said the probabilities associated with these events are negligible.
The semi-classical approach is valid provided that the majority of the particle’s momentum is in the $z$ direction, such that the wavelength is approximately the $z$-direction wavelength, $\lambda \approx \frac{2\pi \hbar}{m v_z}$. We assume that this wavelength is much smaller than the $z$-direction scale lengths $D$ and $L$ in conjunction with the assumption that these are much larger than the $x$ direction scale lengths. This allows us to consider the particles motion in the $z$-direction as approximately classical and allows for the motion to be partitioned about the slits. The specific propagator $K_g^{(1)}(x,T+\tau;0,0)$, for the process of starting at point $(x=0,z=0)$ at time $t=0$, passing through position $({\omega}\in[b-a,b+a],D)$ at time $T$ and then arriving at position $(x,D+L)$ at time $T+\tau$ will simply be a product of propagator for each independent component of the path, integrated over the slit distribution $\Omega({\omega})$, $$\begin{aligned}
\Omega({\omega}) &= \left\{\begin{array}{cc}
1 & b-a < {\omega}<b+a \\
0 & \text{otherwise}
\end{array}\right.,\end{aligned}$$ $$\begin{aligned}
K_g^{(1)}(x,T+\tau;0,0) =& K_0(D,T;0,0)K_0(D+L,T+\tau;D,T) \notag{\\[3mm]}& \quad \quad \times \int_{b-a}^{b+a} K_g({\omega},T;0,0)K_g(x,T+\tau;{\omega},T)d{\omega}{\refstepcounter{equation}\tag{\theequation}}\label{eq:G1Slit-1}.\end{aligned}$$ Evidently for any particular choice of $D$ and $L$, the two $z$ propagators will only give global phase which is identical for all $x$. This global phase has no measurable effects, allowing us to discard the $z$ propagators. This integral is performed in below giving the result $$\begin{aligned}
& K_g^{(1)}(x,T+\tau;0,0) = \frac{e^{i\phi(x)}} {i\sqrt{2 \lambda (D+L)}} \notag {\\[3mm]}& \hspace{4cm} \times
\bigg\{C[\sigma_+(x)] - C[\sigma_-(x)] + iS[\sigma_+(x)]-iS[\sigma_-(x)]\bigg\},{\refstepcounter{equation}\tag{\theequation}}\label{eq:G1Slit}\end{aligned}$$ where $C[u] \equiv \int_0^u \cos\left(\frac{\pi}{2}x^2\right)dx$ is the Fresnel cosine function, $S[u] \equiv \int_0^u \sin\left(\frac{\pi}{2}x^2\right)dx$ is the Fresnel sine function and $\eta = 1 + \frac{L}{D}$ and $$\begin{aligned}
&\sigma_{\pm}(x) = \sqrt{\frac{2}{\lambda L}\eta}\left\{(b\pm a) -\frac{x}{\eta} - \frac12g \frac{m^2 \lambda^2}{h^2}DL\right\},{\refstepcounter{equation}\tag{\theequation}}\label{eq:sigmapm2}{\\[3mm]}&\phi(x) = \pi\left\{\frac{x^2}{\lambda(D+L)} - m g x\frac{\lambda(D+L)}{h^2} - \frac{g^2}{12}\frac{m^4\lambda^3}{h^4}(D+L)(D-L)^2 \right\}.{\refstepcounter{equation}\tag{\theequation}}\label{eq:phix2}\end{aligned}$$
The propagator obtains its name for good reason. An initial wavefunction $\psi_0(x)$ convoluted with the propagator will give the future state of the wavefunction for all time $\psi(x,t) = \int G(x,t;s,0) \psi_0(s)ds$. For the purposes of this calculation we can assume a point source of particles such that the initial spacial distribution of the particle is a $\delta$-function. This however means that the wavefunction is the ‘square root of a $\delta$-function’, which is not guaranteed to be defined. That aside, we can calculate the spatial distribution of the particle at the screen, but in order to have this distribution be normalised, we must account for the fact that a large portion of the wavefunction does not pass through the slit. Thus in actual fact the distribution at the screen is given by the conditional probability to be at position $x$ and time $T+\tau$ given that it was at position $x{\null^{\prime}}\in [-a,a]$ at time $T$. Fortunately as outlined below, the normalised wavefunction is simply the propagator in Eq. multiplied by a factor $\sqrt{\frac{\lambda D}{2a}}$. Finally we arrive at the normalised wavefunction at the screen, for a particle passing through a single slit of width $2a$, centred at $x=b$, $$\begin{aligned}
\psi^\text{(1)}(x) &= \frac{e^{i\phi(x)}}{i2\sqrt{\eta a}} \bigg\{C[\sigma_+(x)] - C[\sigma_-(x)] + iS[\sigma_+(x)]-iS[\sigma_-(x)]\bigg\},{\refstepcounter{equation}\tag{\theequation}}\label{eq:psi12}\end{aligned}$$ with $\phi(x)$ and $\sigma_\pm(x)$ defined in Eqs. and , and $\eta = 1+L/D$. The square of this wavefunction will give the probability distribution for the particle at the slit, which is plotted in Figure \[fig:FallingDiff\] for various distances between slit and screen. The pattern clearly appears to shift towards the negative $x$ direction as the screen is moved further from the slit.
### Integrating over the slit profile {#app:SlitIntegral .unnumbered}
The propagator to arrive at $x$ having passed through a single slit of width $2a$ with centre at $x=b$ is found by integrating over the slit distribution, $\Omega({\omega})$, which is $1$ for $b-a < {\omega}<b+a$ and 0 otherwise: $$\begin{aligned}
K_g^{(1)}(x;a,b) =& \int_{-\infty}^\infty A(x,{\omega})\Omega({\omega})d{\omega}{\\[3mm]}=& \int_{b-a}^{b+a} K_g({\omega},T;0,0)K_g(x,T+\tau;{\omega},T)d{\omega}{\\[3mm]}=&\int_{-\infty}^\infty d{\omega}\, \Omega({\omega}) \sqrt{\frac{m}{2\pi i \hbar T}} \sqrt{\frac{m}{2\pi i \hbar \tau}}
\exp\left[\frac{im}{2\hbar}\left\{\frac{{\omega}^2}{T} - g{\omega}T - \frac{g^2}{12}T^3\right\}\right]\\
\notag
&\qquad \times \exp\left[\frac{im}{2\hbar}\left\{\frac{(x-{\omega})^2}{\tau} - g(x+{\omega}) \tau - \frac{g^2}{12}\tau^3\right\}\right].
{\refstepcounter{equation}\tag{\theequation}}\label{eq:G1Slit-3}\end{aligned}$$ Completing the square in ${\omega}$, $$\begin{aligned}
\frac{{\omega}^2}{T} - g{\omega}T - & \frac{g^2}{12}T^3 + \frac{(x-{\omega})^2}{\tau} - g(x+{\omega}) \tau - \frac{g^2}{12}\tau^3{\\[3mm]}&= \frac{{\omega}^2}{T} + \frac{{\omega}^2}{\tau} - \frac{2 x {\omega}}{\tau} - g{\omega}(T + \tau) +\frac{x^2}{\tau} - gx\tau +\frac{g^2}{4}(T^3 + \tau^3){\\[3mm]}&= {\omega}^2\frac{T + \tau}{T\tau} - 2{\omega}\left(\frac{x}{\tau} + \frac12g(T+\tau)\right) +\frac{x^2}{\tau} - gx\tau - \frac{g^2}{12}(T^3 + \tau^3){\\[3mm]}&=\zeta\left({\omega}- \frac{\kappa}{\zeta}\right)^2 - \frac{\kappa^2}{\zeta} +\frac{x^2}{\tau} - gx\tau - \frac{g^2}{12}(T^3 + \tau^3),\end{aligned}$$ where $\zeta = \frac{T + \tau}{T\tau}$ and $\kappa = \frac{x}{\tau} + \frac12g(T+\tau)$. Returning to Eq. , $$\begin{aligned}
K_g^{(1)}(x;a,b) &= e^{i\phi(x,T,\tau)}\sqrt{\frac{m}{2\pi i \hbar T}} \sqrt{\frac{m}{2\pi i \hbar \tau}} \int_{-\infty}^\infty d{\omega}\exp\left[\frac{im\zeta}{2\hbar}\left({\omega}- \frac{\kappa}{\zeta}\right)^2\right],\end{aligned}$$ where $\phi(x,T,\tau) = \frac{m}{2\hbar}\left(\frac{x^2}{\tau} -\frac{\kappa^2}{\zeta} - gx\tau - \frac{g^2}{12}(T^3 + \tau^3)\right)$ is the phase produced by terms not dependent on ${\omega}$. We make the substitution ${\textstyle}v = \sqrt{\frac{m\zeta}{\pi \hbar}}({\omega}- \frac{\kappa}{\zeta})$, and define new limits of integration $\sigma_{\pm}(x) =\sqrt{\frac{m\zeta}{\pi \hbar}}\left((b\pm a) - x \frac{T}{T+\tau} - \frac12gT\tau\right)$: $$\begin{aligned}
K_g^{(1)}(x;a,b) &=e^{i\phi(x,T,\tau)}\sqrt{\frac{2 \pi i \hbar}{m\zeta}}\sqrt{\frac{m^2}{(2\pi i \hbar)^2 T\tau}}\int_{\sigma_-}^{\sigma_+}\exp\left[\frac{i\pi}{2} v^2\right]dv {\\[3mm]}&= \frac{e^{i\phi(x,T,\tau)}}{\sqrt{(2i)^2 \pi \hbar (T+\tau)/m}} \int_{\sigma_-}^{\sigma_+}\left\{\cos\left(\frac{i\pi}{2}v^2 \right)+ i \sin\left(\frac{i\pi}{2}v^2\right)\right\}\, dv{\\[3mm]}&= \frac{e^{i\phi(x,T,\tau)}}{2i\sqrt{\pi \hbar (T+\tau)/m}}\bigg\{C[\sigma_+(x)] - C[\sigma_-(x)] + iS[\sigma_+(x)]-iS[\sigma_-(x)]\bigg\},{\refstepcounter{equation}\tag{\theequation}}\label{eq:G1Slit-4}\end{aligned}$$ where $C[u] \equiv \int_0^u \cos\left(\frac{\pi}{2}x^2\right)dx$ is the Fresnel cosine function, $S[u] \equiv \int_0^u \sin\left(\frac{\pi}{2}x^2\right)dx$ is the Fresnel sine function. Now to simplify $\phi(x,T,\tau)$ we first have $$\begin{aligned}
\kappa^2 &= \frac{x^2}{\tau^2} + \frac{x}{\tau}g(T+\tau) - \frac{g^2}{12}(T+\tau)^2,{\\[3mm]}\frac{\kappa^2}{\zeta} &= \frac{x^2}{\tau^2}\frac{T \tau}{T+\tau} + \frac{x}{\tau}g(T+\tau)\frac{T \tau}{T+\tau} - \frac{g^2}{12}(T+\tau)^2\frac{T \tau}{T+\tau}{\\[3mm]}&= \frac{x^2 T}{\tau(T+\tau)} + gxT - \frac{g^2}{12}T \tau(T+\tau)\end{aligned}$$ to get $$\begin{aligned}
\phi(x,T,\tau)&= \frac{m}{2\hbar}\left\{\frac{x^2}{\tau} -\frac{\kappa^2}{\zeta} - gx\tau - \frac{g^2}{12}(T^3 + \tau^3)\right\}{\\[3mm]}&=\frac{m}{2}\left\{\frac{x^2(T+\tau) - x^2 T}{\tau (T+\tau)} - gx(T+\tau) - \frac{g^2}{12}(T^3 + \tau^3 - T\tau(T+\tau))\right\}{\\[3mm]}&= \frac{m}{2}\left\{\frac{x^2}{T +\tau} - gx(T+\tau) - \frac{g^2}{12}(T+\tau)(T-\tau)^2\right\}.\end{aligned}$$ We can make use of the approximation $v_z\gg v_x$ and that $\lambda \approx \frac{h}{mv_z}$ to find expressions $T=\frac{m \lambda D}{h}$ and $\tau=\frac{m\lambda L}{h}$. Thus, we have $$\begin{gathered}
T \pm \tau = \frac{m\lambda(D\pm L)}{h}
\quad \mbox{and} \quad
T\tau = \frac{m^2\lambda^2}{h^2}DL.\end{gathered}$$ Using these we get $$\begin{gathered}
\frac{m\zeta}{\pi \hbar}= \frac{2}{\lambda}\left(\frac{1}{D} + \frac{1}{L}\right) \qquad \mbox{where} \qquad
\zeta = \frac{T+\tau}{T\tau} = \frac{h}{m\lambda}\frac{D+L}{DL} =\frac{h}{m\lambda}\left(\frac{1}{D} + \frac{1}{L}\right).\end{gathered}$$ Next, let $$\begin{gathered}
\eta = \frac{T}{T+\tau} = \frac{D}{D+L} = \frac{1}{1+L/D},\end{gathered}$$ allowing us to express $\phi$ and $\sigma_{\pm}$ in terms of $L$, $D$ and $\lambda$: $$\begin{aligned}
\notag
&\sigma_{\pm}(x) =\sqrt{\frac{m\zeta}{\pi \hbar}}\left((b\pm a) - x \frac{T}{T+\tau} - \frac12gT\tau\right) \notag{\\[3mm]}&\phantom{\sigma_{\pm}(x)}
= \sqrt{\frac{2}{\lambda L}\eta}\left\{(b\pm a) -\frac{x}{\eta} - \frac12g \frac{m^2 \lambda^2}{h^2}DL\right\}, {\\[3mm]}&\phi(x,T,\tau) = \frac{m}{2}\left\{\frac{x^2}{T +\tau} - gx(T+\tau) - \frac{g^2}{12}(T+\tau)(T-\tau)^2\right\},{\\[3mm]}&\phi(x)=\pi\left\{\frac{x^2}{\lambda(D+L)} - m g x\frac{\lambda(D+L)}{h^2} - \frac{g^2}{12}\frac{m^4\lambda^3}{h^4}(D+L)(D-L)^2 \right\}. \end{aligned}$$ This is the form of the propagator given in Eq. .
### Normalisation of the distribution at the screen {#app:Re-normalisation .unnumbered}
As derived in the first Section the propagator for the process of starting at position ${\vec{r}}= (0,0)$, passing through the point $(x{\null^{\prime}}\in[b-a,b+a],D)$ and finally being detected at position ${\vec{r}\, {\null^{\prime}}}=(x,D+L)$ on the screen is not the same as the wave-function at the screen. To obtain this we must first convolve the propagator with a initial wavefunction whose square magnitude is a $\delta$-function giving the wavefunction as seen at the other side of the slit. This wavefunction however will not be normalised due to the fact that only a portion of the initially normalise wavefunction has been propagated beyond the slits. It can be renormalised however by scaling by the probability of passing through the slit. Unfortunately the ‘square root of a $\delta$-function’ is not always well defined as is the case for operators acting on any distribution. We can however attempt to use a Gaussian with variance $\sigma$ as the initial wavefunction, compute the quantity of interest and take the limit $\sigma \rightarrow 0$. Under suitable circumstances the limit will be defined giving the desired result.
We will begin with an initial wavefunction that is the square root of a Gaussian $$\begin{aligned}
\psi_\sigma(x) = g_\sigma(x) &= \frac{1}{(2\pi \sigma^2)^{\frac14}} e^{-\frac{x^2}{4\sigma^2}}{\refstepcounter{equation}\tag{\theequation}}\label{Gaussianwavefunction}.\end{aligned}$$ However we notice that the square root of a Gaussian is simply another Gaussian of variance $\rho =\sigma\sqrt{2}$ multiplied by the factor $(8\pi \sigma^2)^\frac{1}{4}$. So the initial function can represented as $$\begin{aligned}
\psi_\rho(x) &= (4\pi \rho^2)^\frac{1}{4} \frac{e^{-\frac{x^2}{2\rho^2}}}{\sqrt{2\pi \rho^2}} = (4\pi \rho^2)^\frac{1}{4} g_\rho(x){\refstepcounter{equation}\tag{\theequation}}\label{redefinedGaussianWF}.\end{aligned}$$
Convolving this with the propagator $K_g^{(1)}(x,T+\tau;x_0,0)$ will give the un-normalised wavefunction at the screen: $$\begin{gathered}
\psi_\rho(x,T+\tau) = \int_{\infty}^\infty dx_0 K_g^{(1)}(x,T+\tau;x_0,0)\psi_\rho(x_0).{\refstepcounter{equation}\tag{\theequation}}\label{IntegrateInitialWF1}\end{gathered}$$ To renormalise this, we scale by the probability of the particle passing through the slit. The probability of the particle being in $x\in[b-a,b+a]$ at time $T$ is $$\begin{gathered}
P(x\in[b-a,b+a];T) = \int_{b-a}^{b+a} \, \left|\int_{-\infty}^{\infty}K_g(x{\null^{\prime}},T;x_0,0)\psi_\rho(x_0)dx_0\right|^2dx{\null^{\prime}},\end{gathered}$$ which gives that the renormalised wavefunction at the screen is $$\begin{aligned}
\psi_\rho{\null^{\prime}}(x,T+\tau) &= \frac{\psi(x,T+\tau)}{\sqrt{\int_{-a}^a \, |\int_{-\infty}^{\infty}K_g(x{\null^{\prime}},T;x_0,0)\psi_\rho(x_0)dx_0|^2dx{\null^{\prime}}}}{\\[3mm]}&=\frac{\int_{\infty}^\infty K_g^{(1)}(x,T+\tau;x_0,0)\psi_\rho(x_0)dx_0}{\sqrt{\int_{b-a}^{b+a} \, |\int_{-\infty}^{\infty}K_g(x{\null^{\prime}},T;x_0,0)\psi_\rho(x_0)dx_0|^2dx{\null^{\prime}}}}{\\[3mm]}&= \frac{(4\pi \rho^2)^\frac{1}{4}\int_{\infty}^\infty K_g^{(1)}(x,T+\tau;x_0,0)g_\rho(x_0)dx_0}{(4\pi \rho^2)^\frac{1}{4}\sqrt{\int_{b-a}^{b+a} \, |\int_{-\infty}^{\infty}K_g(x{\null^{\prime}},T;x_0,0)g_\rho(x_0)dx_0|^2dx{\null^{\prime}}}}.\end{aligned}$$ Now the limit $\sigma\rightarrow 0$ can equivalently be taken as $\rho\rightarrow 0$. The Gaussians $g_\rho(x)$ then become delta functions $\delta(x)$: $$\begin{aligned}
\psi{\null^{\prime}}(x,T+\tau)&= \lim_{\rho-\rightarrow 0}\frac{\int_{\infty}^\infty K_g^{(1)}(x,T+\tau;x_0,0)g_\rho(x_0)dx_0}{\sqrt{\int_{b-a}^{b+a} \, |\int_{-\infty}^{\infty}K_g(x{\null^{\prime}},T;x_0,0)g_\rho(x_0)dx_0|^2dx{\null^{\prime}}}}{\\[3mm]}&= \frac{K_g^{(1)}(x,T+\tau;0,0)}{\sqrt{\int_{b-a}^{b+a} \, |K_g(x{\null^{\prime}},T;0,0)|^2dx{\null^{\prime}}}}
= \frac{K_g^{(1)}(x,T+\tau;0,0)}{\sqrt{\int_{b-a}^{b+a} \,(2\pi \hbar T/m)^{-1}dx{\null^{\prime}}}}{\\[3mm]}&= K_g^{(1)}(x,T+\tau;0,0) \sqrt{\frac{\frac{h}{m} T}{2a}}
= K_g^{(1)}(x,T+\tau;0,0) \sqrt{\frac{\frac{m\lambda D}{h} \frac{h}{m}}{2a}}{\\[3mm]}&= K_g^{(1)}(x,T+\tau;0,0) \sqrt{\frac{\lambda D}{2a}},\end{aligned}$$ where $|K_g(x{\null^{\prime}},T;0,0)|^2$ was taken from Eq. . Thus we see that the normalised wavefunction at the screen is given by multiplying the propagator by the factor $\sqrt{\frac{\lambda D}{2a}}$.
Tracing out spin from a matrix propagator {#app:PropagatorTrace}
=========================================
This is best achieved using the density operator prescription. The pure density operator $\rho$ for a quantum state $\ket{\psi}$ is $\rho = \ket{\psi}\bra{\psi}$. For a state comprised of two subsystems, we can ignore the state of a subsystem by tracing it out. This is given by the operation $\operatorname{tr}_B[X_{AB}] = \sum_k \bra{k}_B X_{AB} \ket{k}_B$, where $X_{AB}$ is an operator on the composite system $AB$ and $\{\ket{k}_B\}$ forms a complete basis for subsystem $B$.
Here, we would like to trace out the spin state. The initial density operator is $\rho_0 = {\left|{\chi_0}\middle\rangle\middle\langle{\chi_0}\right|}\otimes{\left|{\psi_0}\middle\rangle\middle\langle{\psi_0}\right|}$, where ${\left\langle{x}\middle|{\psi_0}\right\rangle} = \psi_0(x)$, is the spatial distribution of the particle, and it is assumed that, initially, the spatial location of the particle is uncorrelated with the spin state. The state of the system at later time $t$ is given by $\rho(t) = U(t)\rho_0U^\dagger(t)$. We can represent the time evolution operator in terms of the propagator by making use of the resolution of the identity $\sum_{\{\chi,\chi{\null^{\prime}}\}\in\{{\scalebox{0.9}{$\downarrow$}},{\scalebox{0.9}{$\uparrow$}}\}}\int dx \,dx{\null^{\prime}}{\left|{x{\null^{\prime}},\chi{\null^{\prime}}}\middle\rangle\middle\langle{x,\chi}\right|} = {\mathds{1}}$: $$\begin{gathered}
U(t)=\sum_{\{\chi,\chi{\null^{\prime}}\}\in\{{\scalebox{0.9}{$\downarrow$}},{\scalebox{0.9}{$\uparrow$}}\}}\int dx \,dx {\null^{\prime}}K_g^{\chi{\null^{\prime}}\!,\chi}(x{\null^{\prime}},t;x,0) {\left|{x{\null^{\prime}},\chi{\null^{\prime}}}\middle\rangle\middle\langle{x,\chi}\right|}{\refstepcounter{equation}\tag{\theequation}}\label{eq:UnitaryProp},\end{gathered}$$ which gives that, $$\begin{aligned}
\rho(t) =& \sum \int dx \, dx{\null^{\prime}}dy\, dy{\null^{\prime}}K_g^{\chi{\null^{\prime}}\!,\chi}(x{\null^{\prime}},t;x,0)K_g^{^*\phi{\null^{\prime}}\!,\phi}(y{\null^{\prime}},t;y,0)\ket{x{\null^{\prime}},\chi{\null^{\prime}}}\notag{\\[3mm]}& \qquad \times
{\left\langle{\vphantom{\chi{\null^{\prime}}}x,\chi}\middle|{\rho_0}\middle|{\vphantom{\chi{\null^{\prime}}}y,\phi}\right\rangle}\bra{y{\null^{\prime}},\phi{\null^{\prime}}},\end{aligned}$$ where the sum is over all spin variables. We can then take the trace over the spin subspace to give, $$\begin{aligned}
\tilde{\rho}(t) = \operatorname{tr}_\text{spin}[\rho(t)] =& \sum \int dx \, dx{\null^{\prime}}dy\, dy{\null^{\prime}}K_g^{\chi{\null^{\prime}}\!,\chi}(x{\null^{\prime}},t;x,0)K_g^{^*\phi{\null^{\prime}}\!,\phi}(y{\null^{\prime}},t;y,0)
\notag{\\[3mm]}& \qquad \times{\left\langle{\vphantom{\chi{\null^{\prime}}}x,\chi}\middle|{\rho_0}\middle|{\vphantom{\chi{\null^{\prime}}}y,\phi}\right\rangle} {\left\langle{\chi{\null^{\prime}}}\middle|{\phi{\null^{\prime}}}\right\rangle} {\left|{x{\null^{\prime}}}\middle\rangle\middle\langle{y{\null^{\prime}}}\right|}.\end{aligned}$$ The spatial probability distribution is then given by the expectation of the position operator ${\left\langle{\hat{x}}\right\rangle} = \operatorname{tr}[\tilde{\rho}(t)\hat{x}]$. Making use of the fact that ${\left\langle{\chi{\null^{\prime}}}\middle|{\phi{\null^{\prime}}}\right\rangle} = \delta_{\chi{\null^{\prime}},\phi{\null^{\prime}}}$ and ${\left\langle{y{\null^{\prime}}}\middle|{x{\null^{\prime}}}\right\rangle} = \delta(x{\null^{\prime}}- y{\null^{\prime}})$, we find $$\begin{gathered}
{\left\langle{\hat{x}}\right\rangle} = \sum \int dx \, dy K_g^{\chi{\null^{\prime}}\!,\chi}(x{\null^{\prime}},t;x,0)K_g^{^*\chi{\null^{\prime}}\!,\phi}(x{\null^{\prime}},t;y,0){\left\langle{\vphantom{\chi{\null^{\prime}}}x,\chi}\middle|{\rho_0}\middle|{\vphantom{\chi{\null^{\prime}}}y,\phi}\right\rangle}.{\refstepcounter{equation}\tag{\theequation}}\label{eq:exofx}\end{gathered}$$ At this point, we work with the term ${\left\langle{\vphantom{\chi{\null^{\prime}}}x,\chi}\middle|{\rho_0}\middle|{\vphantom{\chi{\null^{\prime}}}y,\phi}\right\rangle}$ $$\begin{aligned}
{\left\langle{\vphantom{\chi{\null^{\prime}}}x,\chi}\middle|{\rho_0}\middle|{\vphantom{\chi{\null^{\prime}}}y,\phi}\right\rangle} =& {\left\langle{\vphantom{\chi{\null^{\prime}}}x,\chi}\middle|{\Big({\left|{\chi_0}\middle\rangle\middle\langle{\chi_0}\right|}\otimes{\left|{\psi_0}\middle\rangle\middle\langle{\psi_0}\right|}\Big)}\middle|{\vphantom{\chi{\null^{\prime}}}y,\phi}\right\rangle}{\\[3mm]}=& \bra{\chi}{\Big(\alpha\ket{{\scalebox{0.9}{$\uparrow$}}} + \beta\ket{{\scalebox{0.9}{$\downarrow$}}}\Big)\Big(\alpha^*\bra{{\scalebox{0.9}{$\uparrow$}}} + \beta^*\bra{{\scalebox{0.9}{$\downarrow$}}}\Big)}\ket{\phi}{\left\langle{x}\middle|{\psi_0}\right\rangle}{\left\langle{\psi_0}\middle|{y}\right\rangle}{\\[3mm]}=& \bra{\chi}{\Big(|\alpha|^2{\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} + |\beta|^2{\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|} + \alpha^*\beta{\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|} + \beta*\alpha{\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|}\Big)}\ket{\phi}
{\refstepcounter{equation}\tag{\theequation}}\label{eq:ProjectionInitialRho}{\\[3mm]}\notag
& \quad \times \psi_0(x)\psi_0^*(y).\end{aligned}$$ Since the matrix propagator in Eq. is diagonal, we can immediately discard the ${\left|{{\scalebox{0.9}{$\downarrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\uparrow$}}}\right|}$ and ${\left|{{\scalebox{0.9}{$\uparrow$}}}\middle\rangle\middle\langle{{\scalebox{0.9}{$\downarrow$}}}\right|}$ terms when substituting into the expression for the spatial distribution in Eq. , $$\begin{aligned}
{\left\langle{\hat{x}}\right\rangle} =& \int dx \, dy |\alpha|^2K_g^{{\scalebox{0.9}{$\uparrow$}},{\scalebox{0.9}{$\uparrow$}}}(x{\null^{\prime}},t;x,0)\psi_0(x)\left(K_g^{{\scalebox{0.9}{$\uparrow$}},{\scalebox{0.9}{$\uparrow$}}}(x{\null^{\prime}},t;y,0)\psi_0(y)\right)^*\\
& + |\beta|^2K_g^{{\scalebox{0.9}{$\downarrow$}},{\scalebox{0.9}{$\downarrow$}}}(x{\null^{\prime}},t;x,0)\psi_0(x)\left(K_g^{{\scalebox{0.9}{$\downarrow$}},{\scalebox{0.9}{$\downarrow$}}}(x{\null^{\prime}},t;y,0)\psi_0(y)\right)^*
\notag{\\[3mm]}=& |\alpha|^2\left|\int dx K_g^{{\scalebox{0.9}{$\uparrow$}},{\scalebox{0.9}{$\uparrow$}}}(x{\null^{\prime}},t;x,0)\psi_0(x)\right|^2 +|\beta|^2\left|\int dx K_g^{{\scalebox{0.9}{$\downarrow$}},{\scalebox{0.9}{$\downarrow$}}}(x{\null^{\prime}},t;x,0)\psi_0(x)\right|^2. {\refstepcounter{equation}\tag{\theequation}}\label{appeq:ConvexSum}\end{aligned}$$ This result is simply a convex sum of the initial spatial distribution evolved by each propagator.
[^1]: patrick.james.orlando@gmail.com
[^2]: felix.pollock@monash.edu
[^3]: kavan.modi@monash.edu
[^4]: We will, however, still consider the particle to be neutral, so there is no coupling to the electromagnetic field beyond its spin interaction.
[^5]: It will however affect the spreading of the wavepacket and therefore the variance in the position.
|
---
abstract: |
In the presence of suitable power spaces, compactness of $\mathbf{X}$ can be characterized as the singleton $\{X\}$ being open in the space $\mathcal{O}(\mathbf{X})$ of open subsets of $\mathbf{X}$. Equivalently, this means that universal quantification over a compact space preserves open predicates.
Using the language of represented spaces, one can make sense of notions such as a $\Sigma^0_2$-subset of the space of $\Sigma^0_2$-subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g. , investigate the spaces $\mathbf{X}$ where $\{X\}$ is a $\Delta^0_2$-subset of the space of $\Delta^0_2$-subsets of $\mathbf{X}$. Call this notion $\nabla$-compactness. As $\Delta^0_2$ is self-dual, we find that both universal and existential quantifier over $\nabla$-compact spaces preserve $\Delta^0_2$ predicates.
Recall that a space is called Noetherian iff every subset is compact. Within the setting of Quasi-Polish spaces, we can fully characterize the $\nabla$-compact spaces: A Quasi-Polish space is Noetherian iff it is $\nabla$-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples.
author:
- Matthew de Brecht
- Arno Pauly
bibliography:
- '../../spieltheorie.bib'
title: 'Noetherian Quasi-Polish Spaces[^1]'
---
\[theorem\][Definition]{} \[theorem\][Problem]{} \[theorem\][Assumption]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Observation]{} \[theorem\][Fact]{} \[theorem\][Question]{} \[theorem\][Example]{}
Introduction
============
### Noetherian spaces {#noetherian-spaces .unnumbered}
\[def:noetherian\] A topological space $\mathbf{X}$ is called *Noetherian*, iff every strictly ascending chain of open sets is finite.
Noetherian spaces were first studied in algebraic geometry. Here, the prime motivation is that the Zariski topology on the spectrum of a Noetherian commutative ring is Noetherian (which earns the Noetherian spaces their name).
The relevance of Noetherian spaces for computer science was noted by [<span style="font-variant:small-caps;">Goubault-Larrecq</span>]{} [@goubault2], based on their relationship to well quasiorders. Via well-structured transition systems [@finkel], well quasiorders are used in verification to prove decidability of termination and related properties. Unfortunately, well quasiorders lack some desirable closure properties (the standard counterexample is due to [<span style="font-variant:small-caps;">Rado</span>]{} [@rado]), which led to the introduction of better quasiorders by [<span style="font-variant:small-caps;">Nash-Williams</span>]{} [@nashwilliams], which is a more restrictive notion avoiding the shortcomings of well quasiorders.
Noetherian spaces generalize well-quasi orders: The Alexandrov topology on a quasi-order is Noetherian iff the quasi-order is a well-quasi order. As shown by [<span style="font-variant:small-caps;">Goubault-Larrecq</span>]{} [@goubault], results on the preservation of well-quasi orders under various constructions (such as Higman’s Lemma or Kruskal’s Tree Theorem [@goubault3]) extend to Noetherian spaces; furthermore, Noetherian spaces exhibit some additional closure properties, e.g. the Hoare space of a Noetherian space is Noetherian again [@goubault2]. The usefulness of Noetherian spaces for verification is detailed by [<span style="font-variant:small-caps;">Goubault-Larrecq</span>]{} in [@goubault4].
### Quasi-Polish spaces {#quasi-polish-spaces .unnumbered}
A countably-based topological space is called quasi-Polish if its topology can be derived from a Smyth-complete quasi-metric. Quasi-Polish spaces were introduced by [<span style="font-variant:small-caps;">dB.</span>]{} in [@debrecht6] as a joint generalization of Polish spaces and $\omega$-continuous domains in order to satisfy the desire for a unified setting for descriptive set theory in those areas (expressed e.g. by [<span style="font-variant:small-caps;">Selivanov</span>]{} [@selivanov3]).
### Synthetic DST {#synthetic-dst .unnumbered}
Synthetic descriptive set theory as proposed by the authors in [@paulydebrecht2-lics] reinterprets descriptive set theory in a category-theoretic context. In particular, it provides notions of lifted counterparts to topological concepts such as open sets (e.g. $\Sigma$-classes from descriptive set theory), compactness, and so on.
### Our contributions {#our-contributions .unnumbered}
In the present paper, we will study Noetherian quasi-Polish spaces. As our main result, we show that in the setting of quasi-Polish spaces, being Noetherian is the $\Delta^0_2$-analogue to compactness. We present the result in two different incarnations: Theorem \[theo:finitecover\] states the result in the language of traditional topology. Theorem \[theo:nablacompact\] then restates the main result in the language of synthetic topology, which first requires us to define a computable version of being Noetherian (Definition \[def:real\]). The second instance in particular has as a consequence that universal and existential quantification over Noetherian spaces preserves $\Delta^0_2$-predicates – and this characterizes Noetherian spaces (Proposition \[prop:quantifierelimination\]).
### Structure of the article {#structure-of-the-article .unnumbered}
In Section \[sec:quasipolish\] we recall some results on Noetherian spaces and on quasi-Polish spaces, and then prove some observations on Noetherian quasi-Polish spaces. In particular, Theorem \[theo:finitecover\] shows that for quasi-Polish spaces, being Noetherian is equivalent to any $\Delta^0_2$-cover admitting a finite subcover. This section requires only some basic background from topology.
Section \[prop:quantifierelimination\] introduces the additional background material we need for the remainder of the paper, in particular from computable analysis and synthetic topology.
In Section \[sec:computablynoetherian\] we investigate how Noetherian spaces ought to be defined in synthetic topology ([<span style="font-variant:small-caps;">Escardó</span>]{} [@escardo]), specifically in the setting of the category of represented spaces ([<span style="font-variant:small-caps;">P.</span>]{} [@pauly-synthetic]). As an application, we show that computable well-quasiorders give rise to $\nabla$-computably Noetherian spaces.
Our main result will be presented in Section \[sec:nablacompact\]: The Noetherian spaces can be characterized amongst the quasi-Polish spaces as those allowing quantifier elimination over $\Delta^0_2$-statements (Theorem \[theo:nablacompact\] and Corollary \[corr:quantifier\]). The core idea is that just as compact spaces are characterized by $\{X\}$ being an open subset of the space $\mathcal{O}(\mathbf{X})$ of open subsets, the Noetherian spaces are (amongst the quasi-Polish) characterized by $\{X\}$ being a $\Delta^0_2$-subset of the space of $\Delta^0_2$-subsets.
Looking onwards, we briefly discuss potential future extensions of characterizations of higher-order analogues to compactness and overtness in Section \[sec:othernotions\].
Initial observations on Noetherian quasi-Polish spaces {#sec:quasipolish}
======================================================
Background on Quasi-Polish spaces
---------------------------------
Recall that a quasi-metric on $\mathbf{X}$ is a function $d : \mathbf{X} \times \mathbf{X} \to [0,\infty)$ such that $x = y \Leftrightarrow d(x,y) = d(y,x) = 0$ and $d(x,z) \leq d(x,y) + d(y,z)$. A quasi-metric induces a topology via the basis $(B(x,2^{-k}) := \{y \in \mathbf{X} \mid d(x,y) < 2^{-k}\})_{x \in \mathbf{X},k \in \mathbb{N}}$. A topological space is called quasi-Polish, if it is countably-based and the topology can be obtained from a Smyth-complete quasi-metric (from [<span style="font-variant:small-caps;">Smyth</span>]{} [@smyth]). For details we refer to [@debrecht6], and only recall some select results to be used later on here.
\[prop:quasisubspace\] A subspace of a quasi-Polish space is a quasi-Polish space iff it is a $\Pi^0_2$-subspace.
\[corr:quasisingleton\] In a quasi-Polish space each singleton is $\Pi^0_2$.
A space is quasi-Polish iff it is homoeomorphic to a $\Pi^0_2$-subspace of the Scott domain $\mathcal{P}(\omega)$.
\[theo:bairecategory\] Let $\mathbf{X}$ be quasi-Polish. If $\mathbf{X} = \bigcup_{i \in \mathbb{N}} A_i$ with each $A_i$ being $\Sigma^0_2$, then there is some $i_0$ such that $A_{i_0}$ has non-empty interior.
Recall that a closed set is called *irreducible*, if it is not the union of two proper closed subsets. A topological space is called *sober*, if each non-empty irreducible closed set is the closure of a singleton.
\[prop:soberquasipolish\] A countably-based locally compact sober space is quasi-Polish. Conversely, each quasi-Polish space is sober.
Background on Noetherian spaces
-------------------------------
\[theo:glnoethcharac\] The following are equivalent for a topological space $\mathbf{X}$:
1. $\mathbf{X}$ is Noetherian, i.e. every strictly ascending chain of open sets is finite (Definition \[def:noetherian\]).
2. Every strictly descending chain of closed sets is finite.
3. Every open set is compact.
4. Every subset is compact.
As being Noetherian is preserved by sobrification[^2], we do not lose much by restricting our attention to sober Noetherian spaces. These admit a useful characterization as the upper topologies for certain well-founded partial orders. In the following we use the notation $\downarrow x := \{y \in X \mid y \prec x\}$.
\[theo:glsobernoethcharac\] The following are equivalent for a topological space $\mathbf{X} = (X, \mathcal{T})$:
1. $\mathbf{X}$ is a sober Noetherian space.
2. There is some well-founded partial order $\prec$ on $X$ such that $\mathcal{T}$ is the upper topology induced by $\prec$ and for any finite $F \subseteq \mathbf{X}$ there is a finite $G \subseteq \mathbf{X}$ such that: $$\bigcap_{x \in F} \downarrow x = \bigcup_{y \in G} \downarrow y$$
\[lem:closedfinitelygenerated\] Every closed subset of a sober Noetherian space is the closure of a finite set.
Some new observations
---------------------
\[theo:characnoethquasi\] The following are equivalent for a sober Noetherian space $\mathbf{X}$:
1. $\mathbf{X}$ is countable.
2. $\mathbf{X}$ is countably-based.
3. $\mathbf{X}$ is quasi-Polish.
$1. \Rightarrow 2.$
: By Theorem \[theo:glsobernoethcharac\], we can consider $\mathbf{X}$ to be equipped with the upper topology for some partial order. If $\mathbf{X}$ is countable, then any upper topology is countable, too.
$2. \Rightarrow 1.$
: By Theorem \[theo:glnoethcharac\], every open subset is compact, hence a finite union of basic open sets. Thus, a countably-based Noetherian topology is countable. A sober space with a countable topology has only countably many points.
$2. \Rightarrow 3.$
: As a Noetherian space is compact, we know $\mathbf{X}$ to be a countably-based sober compact space. Proposition \[prop:soberquasipolish\] then implies $\mathbf{X}$ to be quasi-Polish.
$3. \Rightarrow 2.$
: By definition.
A subspace of a quasi-Polish Noetherian space is sober iff it is a $\Pi^0_2$-subspace.
Combine Theorem \[theo:characnoethquasi\] with Proposition \[prop:quasisubspace\].
The following theorem already showcases the link between being Noetherian and a $\Delta^0_2$-analogue to compactness. Its proof is split into Lemmata \[lemma:converingcoverse\],\[lemma:delta02coveringfinite\] and Observation \[obs:sigmadelta\].
\[theo:finitecover\] The following are equivalent for a quasi-Polish space $\mathbf{X}$:
1. $\mathbf{X}$ is Noetherian.
2. Every $\Delta^0_2$-cover of $\mathbf{X}$ has a finite subcover.
3. Every $\Sigma^0_2$-cover of $\mathbf{X}$ has a finite subcover.
\[lemma:converingcoverse\] If a topological space $\mathbf{X}$ is not Noetherian, then it admits a countably-infinite $\Delta^0_2$-partition.
If $\mathbf{X}$ is not Noetherian, then there must be an infinite strictly ascending chain $(U_i)_{i \in \mathbb{N}}$ of open sets. Then $\{U_{i+1} \setminus U_i \mid i \in \mathbb{N}\} \cup \{U_0, \left (\bigcup_{i \in \mathbb{N}} U_i\right )^C\}$ constitutes a $\Delta^0_2$-partition with countably-infinitely many non-trivial pieces.
\[lemma:delta02coveringfinite\] Any $\Delta^0_2$-cover of a Noetherian quasi-Polish space has a finite subcover.
Since $\mathbf{X}$ is countable we can assume the covering is countable. By the Baire category theorem for quasi-Polish spaces (Theorem \[theo:bairecategory\]), there is a $\Delta^0_2$-set $A_0$ in the covering such that its interior, $U_0$ is non-empty.
For $n \geq 0$, if $\mathbf{X} \neq U_n$, then we repeat the same argument with respect to $\mathbf{X} \setminus U_n$ to get a $\Delta^0_2$-set $A_{n+1}$ in the covering with non-empty interior relative to $\mathbf{X} \setminus U_n$. Define $U_{n+1}$ to be the union of $U_n$ and the relative interior of $A_{n+1}$. Then $U_{n+1}$ is an open subset of $\mathbf{X}$ which strictly contains $U_n$. Since $\mathbf{X}$ is Noetherian, eventually $\mathbf{X} = U_n$, and $A_0,\ldots,A_n$ will yield a finite subcovering of $\mathbf{X}$.
\[obs:sigmadelta\] Any $\Sigma^0_2$-cover of a quasi-Polish space can be refined into a $\Delta^0_2$-cover, and any $\Delta^0_2$-cover is a $\Sigma^0_2$-cover.
Let $\mathbf{X}$ be a Noetherian quasi-Polish space, and let $\mathbf{X}^\delta$ be the topology induced by the $\Delta^0_2$-subsets of $\mathbf{X}$. Then $\mathbf{X}^\delta$ is a compact Hausdorff space.
That $\mathbf{X}^\delta$ is compact follows from Lemma \[lemma:delta02coveringfinite\]. To see that it is Hausdorff, we just note that in any $T_0$-space, two distinct points can be separated by a disjoint pair of an open and a closed set – hence by $\Delta^0_2$-sets.
Recall that a topological space satisfies the $T_D$-separation axiom (cf. [@aull]) iff every singleton is a $\Delta^0_2$-set.
A Noetherian quasi-Polish space is $T_D$ iff it is finite.
If $\mathbf{X}$ is a $T_D$ space, then $\mathbf{X} = \bigcup_{x \in \mathbf{X}} \{x\}$ is a $\Delta^0_2$ covering of it. By Lemma \[lemma:delta02coveringfinite\], it then follows that there is a finite subcovering, which can only be identical to the original covering – hence, $\mathbf{X}$ is finite. For the converse direction, by Corollary \[corr:quasisingleton\] every singleton in a quasi-Polish space is $\Pi^0_2$. In a finite space, it follows that they are even $\Delta^0_2$.
An infinite Noetherian quasi-Polish space contains a $\Pi^0_2$-complete singleton.
We can also obtain the following special case of [<span style="font-variant:small-caps;">Goubault-Larrecq</span>]{}’s Lemma \[lem:closedfinitelygenerated\] as a corollary of Lemma \[lemma:delta02coveringfinite\]:
\[corr:closedfinitelygenerated\] Every closed subset of a quasi-Polish Noetherian space is the closure of a finite set.
Given some closed subset $A \subseteq \mathbf{X}$, consider the $\Delta^0_2$-cover $\mathbf{X} = A^C \cup \bigcup_{x \in A} \textrm{cl} \{x\}$. By Lemma \[lemma:delta02coveringfinite\] there is some finite subcover $\mathbf{X} = A^C \cup \bigcup_{x \in F} \textrm{cl} \{x\}$, but then it follows that $A = \textrm{cl} F$.
Neither being sober nor being quasi-Polish is preserved by continuous images in general. However, being Noetherian is not only preserved itself, but in its presence, so are the other properties:
Let $\mathbf{X}$ be a Noetherian sober (quasi-Polish) space and $\sigma : \mathbf{X} \to \mathbf{Y}$ a continuous surjection. Then $\mathbf{Y}$ is Noetherian sober (quasi-Polish) space, too.
Let $C \subseteq \mathbf{Y}$ be irreducible closed. Then $\sigma^{-1}(C)$ is closed, so by Lemma \[lem:closedfinitelygenerated\] (or Corollary \[corr:closedfinitelygenerated\]) there is finite $F \subseteq \mathbf{X}$ such that $\textrm{cl}(F) = \sigma^{-1}(C)$. Continuity implies $\textrm{cl}(\sigma(F)) \supseteq \sigma(\textrm{cl}(F)) = C$, hence $\textrm{cl}(\sigma(F)) = C$. Since $\sigma(F)$ is finite and $C$ is irreducible, $C$ must be equal to the closure of some element of $\sigma(F)$. Therefore, $\mathbf{Y}$ is sober.
By Theorem \[theo:characnoethquasi\], for Noetherian sober spaces being quasi-Polish is equivalent to being countable, which is clearly preserved by (continuous) surjections.
Background {#sec:background}
==========
### Computable analysis {#computable-analysis .unnumbered}
In the remainder of this article, we wish to explore the uniform or effective aspects of the theory of Noetherian Quasi-Polish spaces. The basic framework for this is provided by *computable analysis* [@weihrauchd]. Here the core idea is to introduce notions of continuity and in particular continuity on a wide range of spaces by translating them from those on Baire space via the so-called representations. Our notation and presentation follows closely that of [@pauly-synthetic], which in turn is heavily influenced by [<span style="font-variant:small-caps;">Escardó</span>]{}’s *synthetic topology* [@escardo], and by work by [<span style="font-variant:small-caps;">Schröder</span>]{} [@schroder5].
A represented space is a pair $\mathbf{X} = (X, \delta_\mathbf{X})$ where $X$ is a set and $\delta_\mathbf{X} : \subseteq {\mathbb{N}^\mathbb{N}}\to X$ is a partial surjection. A function between represented spaces is a function between the underlying sets.
For $f : \subseteq \mathbf{X} \to \mathbf{Y}$ and $F : \subseteq {\mathbb{N}^\mathbb{N}}\to {\mathbb{N}^\mathbb{N}}$, we call $F$ a realizer of $f$ (notation $F \vdash f$), iff $\delta_Y(F(p)) = f(\delta_X(p))$ for all $p \in {\operatorname{dom}}(f\delta_X)$, i.e. if the following diagram commutes: $$\begin{CD}
{\mathbb{N}^\mathbb{N}}@>F>> {\mathbb{N}^\mathbb{N}}\\
@VV\delta_\mathbf{X}V @VV\delta_\mathbf{Y}V\\
\mathbf{X} @>f>> \mathbf{Y}
\end{CD}$$ A map between represented spaces is called computable (continuous), iff it has a computable (continuous) realizer.
Two represented spaces of particular importance are the integers $\mathbb{N}$ and Sierpiński space $\mathbb{S}$. The represented space $\mathbb{N}$ has as underlying set $\mathbb{N}$ and the representation $\delta_\mathbb{N} : {\mathbb{N}^\mathbb{N}}\to \mathbb{N}$ defined by $\delta_\mathbb{N}(p) = p(0)$. The Sierpiński space $\mathbb{S}$ has the underlying set $\{\top,\bot\}$ and the representation $\delta_\mathbb{S}$ with $\delta_\mathbb{S}(0^\omega) = \top$ and $\delta_\mathbb{S}(p) = \bot$ for $p \neq 0^\omega$.
Represented spaces have binary products, defined in the obvious way: The underlying set of $\mathbf{X} \times \mathbf{Y}$ is $X \times Y$, with the representation $\delta_{\mathbf{X} \times \mathbf{Y}}(\langle p, q\rangle) = (\delta_\mathbf{X}(p),\delta_\mathbf{Y}(q))$. Here $\langle \ , \ \rangle : {\mathbb{N}^\mathbb{N}}\times {\mathbb{N}^\mathbb{N}}\to {\mathbb{N}^\mathbb{N}}$ is the pairing function defined via $\langle p, q\rangle(2n) = p(n)$ and $\langle p, q\rangle(2n+1) = q(n)$.
A central reason for why the category of represented space is such a convenient setting lies in the fact that it is cartesian closed: We have available a function space construction $\mathcal{C}(\cdot, \cdot)$, where the represented space $\mathcal{C}(\mathbf{X},\mathbf{Y})$ has as underlying set the continuous functions from $\mathbf{X}$ to $\mathbf{Y}$, represented in such a way that the evaluation map $(f, x) : \mathcal{C}(\mathbf{X},\mathbf{Y}) \times \mathbf{X} \to \mathbf{Y}$ becomes computable. This can be achieved, e.g., by letting $nq$ represent $f$, if the $n$-th Turing machine equipped with oracle $q$ computes a realizer of $f$. This also makes currying, uncurrying and composition all computable maps.
Having available to us the space $\mathbb{S}$ and the function space construction, we can introduce the spaces $\mathcal{O}(\mathbf{X})$ and $\mathcal{A}(\mathbf{X})$ of open and closed subsets respectively of a given represented space $\mathbf{X}$. For this, we identity an open subset $U$ of $\mathbf{X}$ with its (continuous) characteristic function $\chi_U : \mathbf{X} \to \mathbb{S}$, and a closed subset with the characteristic function of the complement. As countable join (or) and binary meet (and) on $\mathbb{S}$ are computable, we can conclude that open sets are uniformly closed under countable unions, binary intersections and preimages under continuous functions by merely using elementary arguments about function spaces. The space $\mathcal{A}(\mathbf{X})$ corresponds to the upper Fell topology [@fell] on the hyperspace of closed sets.
Note that neither negation $\mathalpha{\neg} : \mathbb{S} \to \mathbb{S}$ (i.e. mapping $\top$ to $\bot$ and $\bot$ to $\top$) nor countable meet (and) $\bigwedge : \mathcal{C}(\mathbb{N},\mathbb{S}) \to \mathbb{S}$ (i.e. mapping the constant sequence $(\top)_{n \in \mathbb{N}}$ to $\top$ and every other sequence to $\bot$) are continuous or computable operations. They will play the role of fundamental counterexamples in the following. Both operations are equivalent to the *limited principle of omniscience* (${\textrm{LPO}}$) in the sense of Weihrauch reducibility [@weihrauchc].
We need two further hyperspaces, which both will be introduced as subspaces of $\mathcal{O}(\mathcal{O}(\mathbf{X}))$. The space $\mathcal{K}(\mathbf{X})$ of saturated compact sets identifies $A \subseteq \mathbf{X}$ with $\{U \in \mathcal{O}(\mathbf{X}) \mid A \subseteq U\} \in \mathcal{O}(\mathcal{O}(\mathbf{X}))$. Recall that a set is saturated, iff it is equal to the intersection of all open sets containing it (this makes the identification work). The saturation of $A$ is denoted by ${\uparrow}{A} := \bigcap \{U \in \mathcal{O}(\mathbf{X}) \mid A \subseteq A\}$. Compactness of $A$ corresponds to $\{U \in \mathcal{O}(\mathbf{X}) \mid A \subseteq U\}$ being open itself. The dual notion to compactness is *overtness*[^3]. We obtain the space $\mathcal{V}(\mathbf{X})$ of overt set by identifying a closed set $A$ with $\{U \in \mathcal{O}(\mathbf{X}) \mid A \cap U \neq \emptyset\} \in \mathcal{O}(\mathcal{O}(\mathbf{X}))$. The space $\mathcal{V}(\mathbf{X})$ corresponds to the lower Fell (equivalently, the lower Vietoris) topology.
Aligned with the definition of the compact and overt subsets of a space, we can also define when a space itself is compact respectively overt:
A represented space $\mathbf{X}$ is (computably) compact, iff $\textrm{isFull} : \mathcal{O}(\mathbf{X}) \to \mathbb{S}$ mapping $X$ to $\top$ and any other open set to $\bot$ is continuous (computable). Dually, it is (computably) overt, iff $\textrm{isNonEmpty} : \mathcal{O}(\mathbf{X}) \to \mathbb{S}$ mapping $\emptyset$ to $\bot$ and any non-empty open set to $\top$ is continuous (computable).
The relevance of $\mathcal{K}(\mathbf{X})$ and $\mathcal{V}(\mathbf{X})$ is found in particular in the following characterizations, which show that compactness just makes universal quantification preserve open predicates, and dually, overtness makes existential quantification preserve open predicates. We shall see later that being Noetherian has the same role for $\Delta^0_2$-predicates.
\[prop:exists\] The map $\exists : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{V}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ defined by $\exists(R, A) = \{y \in Y \mid \exists x \in A \ (x, y) \in R\}$ is computable. Moreover, whenever $\exists : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{S}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ is computable for some hyperspace $\mathcal{S}(\mathbf{X})$ and some space $\mathbf{Y}$ containing a computable element $y_0$, then $\overline{\phantom{A}} : \mathcal{S}(\mathbf{X}) \to \mathcal{V}(\mathbf{X})$ is computable.
\[prop:forall\] The map $\forall : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{K}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ defined by $\forall(R, A) = \{y \in Y \mid \forall x \in A \ (x, y) \in R\}$ is computable. Moreover, whenever $\forall : \mathcal{O}(\mathbf{X} \times \mathbf{Y}) \times \mathcal{S}(\mathbf{X}) \to \mathcal{O}(\mathbf{Y})$ is computable for some hyperspace $\mathcal{S}(\mathbf{X})$ and some space $\mathbf{Y}$ containing a computable element $y_0$, then ${\uparrow}{{\textnormal{id}}} : \mathcal{S}(\mathbf{X}) \to \mathcal{K}(\mathbf{X})$ is computable.
### Connecting computable analysis and topology {#connecting-computable-analysis-and-topology .unnumbered}
Calling the elements of $\mathcal{O}(\mathbf{X})$ the *open sets* is justified by noting that they indeed form a topology, namely the final topology $X$ inherits from the subspace topology of ${\operatorname{dom}}(\delta_\mathbf{X})$ along $\delta_\mathbf{X}$. The notion of a continuous map between the represented spaces $\mathbf{X}$, $\mathbf{Y}$ however differs from that of a continuous map between the induced topological spaces. For a large class of spaces, the notions do coincide after all, as observed originally by [<span style="font-variant:small-caps;">Schröder</span>]{} [@schroder].
Call $\mathbf{X}$ admissible, if the map $x \mapsto \{U \in \mathcal{O}(\mathbf{X}) \mid x \in U\} : \mathbf{X} \to \mathcal{O}(\mathcal{O}(\mathbf{X}))$ admits a continuous partial inverse.
A represented space $\mathbf{X}$ is admissible iff any map $f : \mathbf{Y} \to \mathbf{X}$ is continuous as a map between represented spaces iff it is continuous as a map between the induced topological spaces.
The admissible represented spaces are themselves cartesian closed (in fact, it suffices for $\mathbf{Y}$ to be admissible in order to make $\mathcal{C}(\mathbf{X},\mathbf{Y})$ admissible). They can be seen as a joint subcategory of the sequential topological spaces and the represented spaces, and thus form the natural setting for computable topology. They have been characterized by [<span style="font-variant:small-caps;">Schröder</span>]{} as the $\textrm{QCB}_0$-spaces [@schroder], the $\mathrm{T}_0$ quotients of countably based spaces.
[<span style="font-variant:small-caps;">Weihrauch</span>]{} [@weihrauchd; @weihrauchh] introduced the standard representation of a countably based space: Given some enumeration $(U_n)_{n \in \mathbb{N}}$ of a basis of a topological space $\mathbf{X}$, one can introduce the representation $\delta_\mathbf{B}$ where $\delta_\mathbf{B}(p) = x$ iff $\{n \in \mathbb{N} \mid \exists i \ p(i) = n+1\} = \{n \in \mathbb{N} \mid x \in U_n\}$. This yields an admissible representation, which in turn induces the original topology on $\mathbf{X}$.
Amongst the countably based spaces, the quasi-Polish spaces are distinguished by a completeness properties. We will make use of the following characterization:
A topological space $\mathbf{X}$ is quasi-Polish, iff its topology is induced by an open admissible total representation $\delta_\mathbf{X} : {\mathbb{N}^\mathbb{N}}\to \mathbf{X}$.
### Synthetic descriptive set theory {#synthetic-descriptive-set-theory .unnumbered}
The central addition of synthetic descriptive set theory (as proposed by the authors in [@pauly-descriptive; @pauly-descriptive-lics]) is the notion of a computable endofunctor:
An endofunctor $d$ on the category of represented spaces is called *computable*, if for any represented spaces $\mathbf{X}$, $\mathbf{Y}$ the induced morphism $d : \mathcal{C}(\mathbf{X},\mathbf{Y}) \to \mathcal{C}(d\mathbf{X},d\mathbf{Y})$ is computable.
To keep things simple, we will restrict our attention here to endofunctors that do not change the underlying set of a represented spaces, but may only modify the representation. Such endofunctors can in particular be derived from certain maps on Baire space, called *jump operators* by [<span style="font-variant:small-caps;">dB.</span>]{} in [@debrecht5]. Here, we instead adopt the terminology *transparent map* introduced in [@gherardi4]. Further properties of transparent maps were studied in [@pauly-nobrega-arxiv].
Call $T : \subseteq {\mathbb{N}^\mathbb{N}}\to {\mathbb{N}^\mathbb{N}}$ *transparent* iff for any computable (continuous) $g : \subseteq {\mathbb{N}^\mathbb{N}}\to {\mathbb{N}^\mathbb{N}}$ there is a computable (continuous) $f : \subseteq {\mathbb{N}^\mathbb{N}}\to {\mathbb{N}^\mathbb{N}}$ with $T \circ f = g \circ T$.
If the relationship between $g$ and $f$ establishing $T$ to be transparent is uniform, then $T$ will induce a computable endofunctor $t$ by setting $t\mathbf{X}$ to be $(X,\delta_\mathbf{X} \circ T)$, and extending to functions in the obvious way.
By applying a suitable endofunctor to Sierpiński space, we can define further classes of subsets; in particular those commonly studied in descriptive set theory. This idea and its relationship to universal sets is further explored in [@pauly-gregoriades]. Basically, we introduce the space $\mathcal{O}^d(\mathbf{X})$ of $d$-open subsets of $\mathbf{X}$ by identifying a subset $U$ with its continuous characteristic function $\chi_U : \mathbf{X} \to d\mathbb{S}$. If $d$ preserves countable products, it automatically follows that the $d$-open subsets are effectively closed under countable unions, binary intersections and preimages under continuous maps. The complements of the $d$-opens are the $d$-closed sets, denoted by $\mathcal{A}^d(\mathbb{S})$.
We will use the endofunctors to generate lifted versions of compactness and overtness:
A represented space $\mathbf{X}$ is (computably) $d$-compact, iff $\textrm{isFull} : \mathcal{O}^d(\mathbf{X}) \to d\mathbb{S}$ mapping $X$ to $\top$ and any other open set to $\bot$ is continuous (computable). Dually, it is (computably) $d$-overt, iff $\textrm{isNonEmpty} : \mathcal{O}^d(\mathbf{X}) \to d\mathbb{S}$ mapping $\emptyset$ to $\bot$ and any non-empty open set to $\top$ is continuous (computable).
A fundamental example of a computable endofunctor linked to notions from descriptive set theory is the limit or jump endofunctor;
\[def:jump\] Let $\lim : \subseteq {\mathbb{N}^\mathbb{N}}\to {\mathbb{N}^\mathbb{N}}$ be defined via $\lim(p)(n) = \lim_{i \to \infty} p(\langle n, i\rangle)$, where $\langle \ , \ \rangle : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is a standard pairing function. Define the computable endofunctor $'$ by $(X,\delta_\mathbf{X})' = (X,\delta_\mathbf{X} \circ \lim)$ and the straight-forward lift to functions.
The map $\lim$ and its relation to the Borel hierarchy and Weihrauch reducibility was studied by [<span style="font-variant:small-caps;">Brattka</span>]{} in [@brattka]. The jump of a represented spaces was studied in [@ziegler3; @gherardi4]. The $'$-open sets are just the $\Sigma^0_2$-sets, and the further levels of the Borel hierarchy can be obtained by iterating the endofunctor.
### Computability with finitely many mindchanges {#computability-with-finitely-many-mindchanges .unnumbered}
The most important endofunctor for our investigation of Noetherian Quasi-Polish spaces is the finite mindchange endofunctor $\nabla$:
Define $\Delta : \subseteq {\mathbb{N}^\mathbb{N}}\to {\mathbb{N}^\mathbb{N}}$ via $\Delta(p)(n) = p(n + 1 + \max \{i \mid p(i) = 0\}) - 1$. Let the finite mindchange endofunctor be defined via $(X, \delta_X)^\nabla = (X, \delta_X \circ \Delta)$ and $(f : \mathbf{X} \to \mathbf{Y})^\nabla = f : \mathbf{X}^\nabla \to \mathbf{Y}^\nabla$.
We find that $\nabla$ is a monad, and moreover, that $f : \mathbf{X} \to \mathbf{Y}^\nabla$ is computable (continuous) iff $f : \mathbf{X}^\nabla \to \mathbf{Y}^\nabla$ is. The computable maps from $\mathbf{X}$ to $\mathbf{Y}^\nabla$ can equivalently be understood as those maps from $\mathbf{X}$ to $\mathbf{Y}$ that are computable with finitely many mindchanges.
A machine model for computation with finitely many mindchanges is obtained by adding the option of resetting the output tape to the initial state. To ensure that the output is well-defined, such a reset may only be used finitely many times. In the context of computable analysis, this model was studied by a number of authors [@ziegler3; @debrecht; @paulyoracletypetwo; @paulybrattka; @paulybrattka2; @paulyneumann]. For our purposes, an equivalent model based on non-deterministic computation turns out to be more useful. We say that a function from $\mathbf{X}$ to $\mathbf{Y}$ is non-deterministically computable with advice space $\mathbb{N}$, if on input $p$ (a name for some $x \in \mathbf{X}$) the machine can guess some $n \in \mathbb{N}$ and then either continue for $\omega$ many steps and output a valid name for $f(x)$, or at some finite time reject the guess. We demand that for any $p$ there is some $n \in \mathbb{N}$ that is not rejected. The equivalence of the two models is shown in [@paulybrattka].
The interpretation of $\nabla$ in descriptive set theory is related to the $\Delta^0_2$-sets. In particular, the $\nabla$-open sets are the $\Delta^0_2$-sets, the continuous functions from $\mathbf{X}$ to $\mathbf{Y}^\nabla$ are the piecewise continuous functions for Polish $\mathbf{X}$, and the lifted version of admissibility under $\nabla$ corresponds to the Jayne-Rogers theorem (cf. [@jaynerogers; @ros; @kihara4]). This was explored in detail by the authors in [@paulydebrecht].
$\nabla$-computably Noetherian spaces {#sec:computablynoetherian}
=====================================
In this section, we want to investigate the notion of being Noetherian in the setting of synthetic topology. We will see that the naive approach fails, but then provide a well-behaved definition. That it is adequate will be substantiated by providing a computable counterpart to the relationship between Noetherian spaces and well-quasiorders. First, however, we will explore a prototypical example.
A case study on computably Noetherian spaces
--------------------------------------------
Let $\mathbb{N}_<$ be the natural numbers with the topology $\mathcal{T}_< := \{L_n := \{i \in \mathbb{N} \mid i \geq n\} \mid n \in \mathbb{N}\} \cup \{\emptyset\}$. Then let $\overline{\mathbb{N}}_<$ be the result of adjoining $\infty$, which is contained in all non-empty open sets. In $\overline{\mathbb{N}}_<$ we find a very simple yet non-trivial example of a quasi-Polish Noetherian space.
Similarly, let $\mathbb{N}_>$ be the natural numbers with the topology $\mathcal{T}_> := \{U_n := \{i \in\mathbb{N} \mid i < n\} \mid n \in \mathbb{N}\} \cup \{\mathbb{N}\}$. By $\overline{\mathbb{N}}_>$ I denote the space resulting from adjoining an element $\infty$, which is only contained in one open set. In terms of representations, we can conceive of an element in $\mathbb{N}_<$ as being given as the limit of an increasing sequence, and of an element in $\mathbb{N}_>$ as the limit of a decreasing sequence.
Looking at the way how we defined $\mathcal{T}_<$, we see that we have a countable basis, and given indices of open sets, can e.g. decide subset inclusion. The indexing is fully effective, in the sense that this is a computable basis as follows:
An effective countable base for $\mathbf{X}$ is a computable sequence $(U_i)_{i \in \mathbb{N}} \in \mathcal{C}(\mathbb{N}, \mathcal{O}(\mathbf{X}))$ such that the multivalued partial map $\textrm{Base} :\subseteq \mathbf{X} \times \mathcal{O}(\mathbf{X}) {\rightrightarrows}\mathbb{N}$ is computable. Here ${\operatorname{dom}}(\textrm{Base}) = \{(x, U) \mid x \in U\}$ and $n \in \textrm{Base}(x, U)$ iff $x \in U_n \subseteq U$.
Even though all open sets are basis elements, we should still distinguish computability on the open sets themselves, and computability on the indices. For example, the map $\bigcup : \mathcal{O}(\mathbf{X})^\mathbb{N} \to \mathcal{O}(\mathbf{X})$, i.e. the countable union of open sets, should always be a computable operation. This, however, cannot be done on the indices. More generally, in the synthetic topology framework the space of open subsets of a given space automatically comes with its own natural topology. This topology is obtained by demanding that given a point and an open set, we can recognize (semidecide) membership. In the case of $\mathbb{N}_<$, we can establish a quite convenient characterization of its open subsets:
\[prop:opensets\] The map $n \mapsto \{i \in \mathbb{N} \mid i \geq n\} : \overline{\mathbb{N}}_> \to \mathcal{O}(\mathbb{N}_<)$ is a computable isomorphism.
1. The map is computable.
Given $m \in \mathbb{N}_<$ and $n \in \overline{\mathbb{N}}_>$, we can semidecide $m \geq n$ (just wait until the increasing and the decreasing approximations pass each other).
2. The map is surjective.
At the moment some number $m$ is recognized to be an element of some open set $U \in \mathcal{O}(\mathbb{N}_<)$, we have only learned some lower bound on $m$ so far. Thus, any number greater than $m$ is contained in $U$, too. Hence all open subsets of $\mathbb{N}_>$ are final segments.
3. The inverse of the map is computable.
Given $U \in \mathcal{O}(\mathbb{N}_<)$, we can simultaneously begin testing $i \in U?$ for all $i \in \mathbb{N}$. Any positive test provides an upper bound for the $n$ such that $U = \{i \in \mathbb{N} \mid i \geq n\}$.
The space of (saturated) compact subsets likewise comes with its own topology, in this case obtained by demanding that given a compact $K$ and an open $U$, we can recognize if $K \subseteq U$. Similarly to the preceding proposition, we can also characterize the compact subsets of $\mathbb{N}_>$:
The map $n \mapsto \{i \in \mathbb{N} \mid i \geq n\} : \overline{\mathbb{N}}_< \to \mathcal{K}(\mathbb{N}_<)$ is a computable isomorphism.
1. The map is computable.
We need to show that given $n \in \overline{\mathbb{N}}_<$ and $U \in \mathcal{O}(\mathbb{N}_>)$ we can recognize that $\{i \in \mathbb{N} \mid i \geq n\} \subseteq U$. By Proposition \[prop:opensets\], we can assume that $U$ is of the form $U = \{i \in \mathbb{N} \mid i \geq m\}$ with $m \in \overline{\mathbb{N}}_>$. Now for such $n, m$, we can indeed semidecide $m \leq n$ – again, just wait until the approximating sequences reach the same value.
2. The map is surjective.
While any subset of $\mathbb{N}_<$ is compact, only the saturated compact sets appear in $\mathcal{K}(\mathbb{N}_>)$, and these are the given ones.
3. The inverse map is computable.
Given a compact set $K \in \mathcal{K}(\mathbb{N}_<)$, we simultaneously test if it is covered by open sets of the form $\{i \mid i \geq m\}$. Any such $m$ we find provides a lower bound for the $n$ for which $K = \{i \mid i \geq n\}$ holds.
So we see that while the [**spaces**]{} $\mathcal{O}(\mathbb{N}_<)$ and $\mathcal{K}(\mathbb{N}_<)$ contain the same points, their topologies differ – and are, in fact, incomparable. There are two potential ways to capture the idea that *opens are compact* in a synthetic way:
We could work with open and compact sets when in a Noetherian space, i.e. with the space $\mathcal{O}(\mathbb{N}_<) \wedge \mathcal{K}(\mathbb{N}_<)$ carrying the join of the topologies. As $\mathbb{N}_< \wedge \mathbb{N}_> \cong \mathbb{N}$, in this special cases we would end up in the same situation as using computability on base indices straightaway. In general though it is not even obvious if $\cap : \left (\mathcal{O}(\mathbf{X}) \wedge \mathcal{K}(\mathbf{X})\right ) \times \left (\mathcal{O}(\mathbf{X}) \wedge \mathcal{K}(\mathbf{X})\right ) \to \left (\mathcal{O}(\mathbf{X}) \wedge \mathcal{K}(\mathbf{X})\right )$ should be computable.
The second approach relies on the observation that $\mathbb{N}_<$ and $\mathbb{N}_>$ do not differ by *that much*. We can consider *computability with finitely many mindchanges* – and the distinction between $\mathbb{N}_<$, $\mathbb{N}_>$ and $\mathbb{N}$ disappears, as we find $\mathbb{N}_<^\nabla \cong \mathbb{N}_>^\nabla \cong\mathbb{N}^\nabla$. As the next subsection shows, computability with finitely many mindchanges seems adequate to give *opens are compact* a computable interpretation.
The abstract approach
---------------------
The straightforward approach to formulate a synthetic topology version of Noetherian would be the following:
\[def:hypo\] Call a space $\mathbf{X}$ *computably Noetherian*, iff ${\textnormal{id}}_{\mathcal{O},\mathcal{K}} : \mathcal{O}(\mathbf{X}) \to \mathcal{K}(\mathbf{X})$ is well-defined and computable.
This fails entirely, though:
Let $\mathbf{X}$ be non-empty. Then $\mathbf{X}$ is not computably Noetherian according to Definition \[def:hypo\].
Note that $\mathalpha{\subseteq} : \mathcal{K}(\mathbf{X}) \times \mathcal{O}(\mathbf{X}) \to \mathbb{S}$ is by definition of $\mathcal{K}$ a computable map, i.e. inclusion of a compact in an open set is semidecidable. Furthermore, $\iota : \mathbb{S} \to \mathbf{X}$ defined via $\iota(\top) = X$ and $\iota(\bot) = \emptyset$ is a always a computable injection for non-empty $\mathbf{X}$. Now if $\mathbf{X}$ were computably Noetherian, then the map $t \mapsto \mathalpha{\subseteq}({\textnormal{id}}_{\mathcal{O},\mathcal{K}}(\iota(t)), \emptyset)$ would be computable and identical to $\neg : \mathbb{S} \to \mathbb{S}$, but the latter is non-computable.
We can avoid this problem by relaxing the computability-requirement to computability with finitely many mindchanges. Now we can try again:
\[def:real\] Call a space $\mathbf{X}$ $\nabla$-*computably Noetherian*, iff ${\textnormal{id}}_{\mathcal{O},\mathcal{K}} : \mathcal{O}(\mathbf{X}) \to \left (\mathcal{K}(\mathbf{X}) \right )^\nabla$ is well-defined and computable.
Say that an effective countable base is *nice*, if $\{\langle u, v\rangle \mid \left (U_{u(1)} \cup \ldots \cup U_{u(|u|)} \right ) \subseteq \left (U_{v(1)} \cup \ldots \cup U_{v(|u|)} \right )\} \subseteq \mathbb{N}^* \times \mathbb{N}^*$ is decidable. Clearly any effective countable base is nice relative to some oracle, hence this requirement is unproblematic from the perspective of continuity.
We can now state and prove the following theorem, which can be seen as a uniform counterpart to Theorem \[theo:glnoethcharac\]:
Let $\mathbf{X}$ be quasi-Polish, and in particular have a nice effective countable base. Then the following are equivalent:
1. $\mathbf{X}$ is $\nabla$-computably Noetherian
2. ${\textnormal{id}}_{\mathcal{O},\mathcal{K}} : \mathcal{O}(\mathbf{X}) \to \left (\mathcal{K}(\mathbf{X}) \right )^\nabla$ is well-defined and computable.
3. $\mathalpha{\subseteq} : \mathcal{O}(\mathbf{X}) \times \mathcal{O}(\mathbf{X}) \to \mathbb{S}^\nabla$ is computable.
4. $\operatorname{Stabilize} : \mathcal{C}(\mathbb{N},\mathcal{O}(\mathbf{X})) {\rightrightarrows}\mathbb{N}^\nabla$ is well-defined and computable, where $N \in \operatorname{Stabilize}((V_i)_{i \in \mathbb{N}})$ iff $\left (\bigcup_{i = 0}^N V_i \right ) = \left (\bigcup_{i \in \mathbb{N}} V_i \right )$.
5. $\operatorname{Stabilize} : \mathcal{C}(\mathbb{N},\mathcal{A}(\mathbf{X})) {\rightrightarrows}\mathbb{N}^\nabla$ is well-defined and computable, where $N \in \operatorname{Stabilize}((A_i)_{i \in \mathbb{N}})$ iff $\left (\bigcap_{i = 0}^N A_i \right ) = \left (\bigcup_{i \in \mathbb{N}} A_i \right )$.
6. The computable map $u \mapsto \left (U_{u(1)} \cup \ldots \cup U_{u(|u|)} \right ) : \mathbb{N}^* \to \mathcal{O}(\mathbf{X})$ is a surjection and has a $\nabla$-computable right-inverse.
Note that the forward implications hold for arbitrary represented spaces, as long as they make sense.
$1. \Leftrightarrow 2.$
: This is the definition.
$2. \Rightarrow 3.$
: By taking into account the definition of $\mathcal{K}$, we have ${\textnormal{id}}_{\mathcal{O},\mathcal{K}} : \mathcal{O}(\mathbf{X}) \to \left (\mathcal{C}(\mathcal{O}(\mathbf{X}), \mathbb{S} ) \right )^\nabla$. Moreover, ${\textnormal{id}}: \mathcal{C}(\mathbf{Y}, \mathbf{Z})^\nabla \to \mathcal{C}(\mathbf{Y}, \mathbf{Z}^\nabla)$ is always computable, so currying yields the claim.
$3. \Rightarrow 4.$
: First, we prove that $\operatorname{Stabilize}$ is well-defined. Assume that it is not, then there is a family $(V_i)_{i \in \mathbb{N}}$ of open sets such that $V := \bigcup_{i \in \mathbb{N}} V_i \neq \bigcup_{i = 0}^N V_i$ for all $N \in \mathbb{N}$. Consider the computable map $q \mapsto \mathalpha{\subseteq}\left (V, \bigcup_{i \in \mathbb{N}} V_{q(i)} \right ) : {\mathbb{N}^\mathbb{N}}\to \mathbb{S}^\nabla$. If the range of $q$ is finite, then the output must be $\bot$, if the range of $q$ is $\mathbb{N}$, then the output must be $\top$. However, these two cases cannot be distinguished in a $\Delta^0_2$-way, thus the $(V_i)_{i \in \mathbb{N}}$ cannot exist, and $\operatorname{Stabilize}$ is well-defined..
To see that we can compute the (multivalued) inverse, we employ the equivalence to $\nabla$-computability and non-deterministic computation with advice space $\mathbb{N}$ from [@paulybrattka]. Given $(V_i)_{i \in \mathbb{N}}$, we guess $N \in \mathbb{N}$ together with an upper bound $b$ on the number of mindchanges happening in verifying that $\mathalpha{\subseteq}(\bigcup_{i = 0}^N V_i, \bigcup_{i \in \mathbb{N}} V_i ) = \top$. Any correct guess contains a valid solution, and any wrong guess can be rejected.
$4. \Leftrightarrow 5.$
: By de Morgan’s law.
$4. \Rightarrow 6.$
: In a quasi-Polish space $\mathbf{X}$ with effectively countable basis $(U_i)_{i \in \mathbb{N}}$, any $U \in \mathcal{O}(\mathbf{X})$ can be effectively represented by $p \in {\mathbb{N}^\mathbb{N}}$ with $U = \bigcup_{i \in \mathbb{N}} U_{p(i)}$. Applying stabilize to the family $(U_{p(i)})_{i \in \mathbb{N}}$ shows subjectivity and computability of the multivalued inverse.
$6. \Rightarrow 2.$
: We will argue that $u \mapsto \left (U_{u(1)} \cup \ldots \cup U_{u(|u|)} \right ) : \mathbb{N}^* \to \mathcal{K}(\mathbf{X})$ is computable, provided that $(U_n)_{n \in \mathbb{N}}$ is a nice basis. For this, note that given $u \in \mathbb{N}^*$ and $p \in {\mathbb{N}^\mathbb{N}}$, we can semidecide whether $\left (U_{u(1)} \cup \ldots \cup U_{u(|u|)} \right ) \subseteq \bigcup_{n \in \mathbb{N}} U_{p(n)}$.
All finite spaces containing only computable points are $\nabla$-computably Noetherian; any quasi-Polish Noetherian space is $\nabla$-computably Noetherian relative to some oracle (which is not vacuous). $\nabla$-computably Noetherian spaces are closed under finite products and finite coproducts, and computable images of $\nabla$-computably Noetherian spaces are $\nabla$-computably Noetherian.
Well-quasiorders and $\nabla$-computably Noetherian spaces
----------------------------------------------------------
A quasiorder $(X,\preceq)$ can be seen as a topological space via the Alexandrov topology, which consists of the upper sets regarding $\preceq$. The quasiorder is recovered from the topology as the specialization order (i.e. $x \preceq y$ iff $x \in \overline{\{y\}}$). As mentioned in the introduction, the Alexandrov topology of a quasiorder is Noetherian iff the quasiorder is a well-quasiorder. Here, we shall investigate the computability aspects of this connection in the case of countable quasiorders, more precisely, quasiorders over $\mathbb{N}$. We first consider arbitrary quasiorders over $\mathbb{N}$, before coming to the special case of well-quasiorders.
### Arbitrary quasiorders over $\mathbb{N}$ and their Alexandrov topologies {#arbitrary-quasiorders-over-mathbbn-and-their-alexandrov-topologies .unnumbered}
Given some quasiorder $(\mathbb{N},\preceq)$ we define the represented space $\mathrm{Av}(\preceq)$ to have the underlying set $\mathbb{N}$ and the representation $\psi_\preceq : \subseteq {\mathbb{N}^\mathbb{N}}\to \mathbb{N}$ defined via $\psi_\preceq(p) = \mathbf{n}$ iff: $$\{ k \in \mathbb{N} \mid k \preceq \mathbf{n}\} = \{p(i) \mid i \in \mathbb{N}\}$$
The represented space $\mathrm{Av}(\preceq)$ corresponds to the Alexandrov-topology induced by $\preceq$. This is seen by the following proposition, which also establishes some basic observations on how computability works in this setting.
\[prop:quasiorder-opens\] Let $\preceq$ be computable. Then
1. $\uparrow_\preceq : \mathcal{O}(\mathbb{N}) \to \mathcal{O}(\mathbb{N})$ is computable.
2. $\uparrow_\preceq : \mathcal{O}(\mathbb{N}) \to \mathcal{O}(\mathrm{Av}(\preceq))$ is a computable surjection.
3. ${\textnormal{id}}: \mathbb{N} \to \mathrm{Av}(\preceq)$ is computable.
4. ${\textnormal{id}}: \mathcal{O}(\mathrm{Av}(\preceq)) \to \mathcal{O}(\mathbb{N})$ is a computable embedding.
<!-- -->
1. Straight-forward.
2. To show that the map is computable, by (1) it suffices to show that given some $\preceq$-upwards closed set $U \in \mathcal{O}(\mathbb{N})$ and $\mathbf{n} \in \mathrm{Av}(\preceq)$, we can semidecide if $\mathbf{n} \in U$. But given the definition of $\mathrm{Av}(\preceq)$, we find that $\psi_\preceq(p) \in U$ iff $\exists i \ p(i) \in U$, hence the semidecidability follows.
It remains to argue that map is surjective, i.e. that any $U \in \mathcal{O}(\mathrm{Av}(\preceq))$ is $\preceq$-upwards closed. Assume for the sake of a contradiction that $U \in \mathcal{O}(\mathrm{Av}(\preceq))$ is not upwards-closed, i.e. that there are $\mathbf{n} \in U$, $\mathbf{m} \notin U$ with $\mathbf{n} \preceq \mathbf{m}$. Pick some $\psi_\preceq$-name $p$ of $\mathbf{n}$ and a realizer $u$ of $\chi_U : \mathrm{Av}(\preceq) \to \mathbb{S}$. Now $u$ will accept the input $p$ after having read some finite prefix $w$ of $p$. Let $q$ be a $\psi_\preceq$-name of $\mathbf{m}$. Now $\psi_\preceq(wq) = \mathbf{m}$, and $u$ will accept $wq$, hence $\mathbf{m} \in U$ follows.
3. Straight-forward.
4. That ${\textnormal{id}}: \mathcal{O}(\mathrm{Av}(\preceq)) \to \mathcal{O}(\mathbb{N})$ is computable follows from (3). Its computable inverse is given by $\uparrow_\preceq : \mathcal{O}(\mathbb{N}) \to \mathcal{O}(\mathrm{Av}(\preceq)$ from (2).
The terminology *Alexandrov topology* goes back to an observation by Pavel Alexandrov [@alexandrov] that certain topological spaces correspond to partial orders, namely those characterized by the property that arbitrary intersections of open sets are open again. Further characterizations are provided in [@arenas]. Similar to the failure of the naive Definition \[def:hypo\], one can readily check that e.g. $\bigcap : \mathcal{O}(\mathbf{X})^\mathbb{N} \to \mathcal{O}(\mathbf{X})$ is never a continuous well-defined map, as long as $\mathbf{X}$ is non-empty. Thus, this characterization does not extend to a uniform statement in a straight-forward manner.
Next, we shall explore the compact subsets of $\mathrm{Av}(\preceq)$. As usual in the study of represented spaces, we restrict our attention to the *saturated* compact subsets. Recall that $A \subseteq \mathbf{X}$ is called saturated, iff $A = \bigcap {\{U \in \mathcal{O} \mid A \subseteq U\}}$; the saturation of a set $A$ is $\bigcap {\{U \in \mathcal{O} \mid A \subseteq U\}}$. As a set is compact iff its saturation is, this restriction is without loss of generality. In $\mathrm{Av}(\preceq)$, a set is saturated iff it is upwards closed.
\[prop:quasiordercompacts\] The map $n_0\ldots n_k \mapsto \uparrow \{n_0,\ldots,n_k\} : \mathbb{N}^* \to \mathcal{K}(\mathrm{Av}(\preceq))$ is a computable surjection.
To show that the map is computable, we need to argue that given $n_0\ldots n_k \in \mathbb{N}^*$ and $U \in \mathcal{O}(\mathrm{Av}(\preceq))$, we can semidecide if $\uparrow \{n_0,\ldots,n_k\} \subseteq U$. Since $U$ itself is upwards closed, this is equivalent to $\{n_0,\ldots,n_k\} \subseteq U$. It follows from Proposition \[prop:quasiorder-opens\] (4) that it is.
To see that the map is surjective, consider some $A \in \mathcal{K}(\mathrm{Av}(\preceq))$. As $A$ is upwards closed, we find that in particular also $A \in \mathcal{O}(\mathrm{Av}(\preceq))$ (in a non-uniform way of course). As $\mathalpha{\subseteq} : \mathcal{K}(\mathbf{X}) \times \mathcal{O}(\mathbf{X}) \to \mathbb{S}$ is computable, we can given the compact set $A$ and the open set $A$ semidecide that indeed $A \subseteq A$. At the moment of the decision, only finite information about the sets has been read. In particular, by Proposition \[prop:quasiorder-opens\] (4) we can assume that all we have learned about the open set $A$ is $\{n_0,\ldots,n_k\} \subseteq A$ for some finite set $\{n_0,\ldots,n_k\}$. As the semidecision procedure would also accept the compact set $A$ and the open set $\uparrow \{n_0,\ldots,n_k\}$, it follows that $A = \uparrow \{n_0,\ldots,n_k\}$.
Except for trivial cases, the map from the preceding proposition cannot be computably invertible: A compact set $A \in \mathcal{K}(\mathrm{Av}(\preceq))$ can always shrink, whereas each $n_0\ldots n_k \in \mathbb{N}^*$ is completely determined at some finite time. However, moving to computability with finitely many mindchanges suffices to bridge the gap:
\[prop:basecompactcomputable\] If $\preceq$ is computable, then the multivalued map $\operatorname{Base} : \mathcal{K}(\mathrm{Av}(\preceq)) {\rightrightarrows}\left (\mathbb{N}^* \right )^\nabla$ where $n_0\ldots n_k \in \operatorname{Base}(A)$ iff $\uparrow \{n_0,\ldots,n_k\} = A$, is computable.
We utilize the equivalence between computability with finitely many mindchanges and non-deterministic computation with discrete advice. Given $A \in \mathcal{K}(\mathrm{Av}(\preceq))$, $n_0\ldots n_k \in \mathbb{N}^*$ and a parameter $t \in \mathbb{N}$ we proceed as follows: If $A \subseteq \uparrow \{n_0,\ldots,n_k\}$ is not confirmed within $t$ steps, reject. If we can find some $m_0,\ldots,m_j$ such that $A \subseteq \uparrow \{m_0,\ldots,m_j\}$ but not $\{n_0,\ldots,n_k\} \subseteq \uparrow \{m_0,\ldots,m_j\}$, then reject.
For fixed $A$ and $n_0\ldots n_k \in \mathbb{N}^*$ , there is a parameter $t \in \mathbb{N}$ not leading to a rejection iff $A = \uparrow \{n_0,\ldots,n_k\}$.
Before we move on to well-quasiorders, we shall consider sobriety for Alexandrov topologies, in light of Proposition \[prop:soberquasipolish\] and the overall usefulness of sobriety for the results in Section \[sec:quasipolish\]. Recall that a countable quasiorder $(X, \preceq)$ is a dcpo if for any sequence $(a_i)_{i \in \mathbb{N}}$ in $X$ such that $a_i \preceq a_{i+1}$ we find that there is some $b \in X$ such that for all $c \in X$: $$b \preceq c \Leftrightarrow \forall n \in \mathbb{N} \quad a_n \preceq c$$
$\mathrm{Av}(\preceq)$ is sober iff $(\mathbb{N},\preceq)$ is a dcpo.
### Well-quasiorders and their Alexandrov topologies {#well-quasiorders-and-their-alexandrov-topologies .unnumbered}
\[prop:wqo-opens\]
1. If $\preceq$ is computable, then $n_0\ldots n_k \mapsto \uparrow \{n_0,\ldots,n_k\} : \mathbb{N}^* \to \mathcal{O}(\mathrm{Av}(\preceq))$ is computable.
2. $\preceq$ is a well-quasiorder iff $n_0\ldots n_k \mapsto \uparrow \{n_0,\ldots,n_k\} : \mathbb{N}^* \to \mathcal{O}(\mathrm{Av}(\preceq))$ is a surjection.
3. If $\preceq$ is a computable well-quasiorder, then $\operatorname{Base} : \mathcal{O}(\mathrm{Av}(\preceq)) {\rightrightarrows}\left (\mathbb{N}^* \right)^\nabla$ where $n_0\ldots n_k \in \operatorname{Base}(U)$ iff $\uparrow \{n_0,\ldots,n_k\} = U$, is well-defined and computable.
<!-- -->
1. This is straight-forward, using Proposition \[prop:quasiorder-opens\] (4).
2. Let us assume that $U \in \mathcal{O}(\mathrm{Av}(\preceq))$ is not of the form $\uparrow \{n_0,\ldots,n_k\}$. Then in particular, $U \neq \emptyset$. Pick some $a_0 \in U$. As $U \neq \uparrow \{a_0\}$, there is some $a_1 \in U \setminus \{a_0\}$. Subsequently, always pick $a_{n+1} \in U \setminus \uparrow \{a_0,\ldots,a_n\}$. Now $(a_n)_{n \in \mathbb{N}}$ satisfies by construction that for $n < m$ never $a_n \preceq a_m$ holds, i.e. $(a_n)_{n \in \mathbb{N}}$ is a bad sequence, contradicting the hypothesis $\preceq$ were a wqo.
Conversely, let $(a_n)_{n \in \mathbb{N}}$ be a bad sequence witnessing that $\preceq$ is not a wqo. Assume that $\uparrow \{a_i \mid i \in \mathbb{N}\} = \uparrow \{n_0,\ldots,n_k\}$. As $n_j \in \uparrow \{a_i \mid i \in \mathbb{N}\}$ for $j \leq k$, there is some $i_j$ such that $n_j \succeq a_{i_j}$. Pick $i_\infty > \max_{j \leq k} i_j$. As $a_{i_\infty} \in \uparrow \{n_0,\ldots,n_k\}$ there is some $j_\infty \leq k$ such that $a_{i_\infty} \succeq n_{j_\infty}$. But then $a_{i_{j_\infty}} \preceq a_{i_\infty}$ follows, and since $i_{j_\infty} < i_\infty$ by construction, this contradicts $(a_i)_{i \in \mathbb{N}}$ being a bad sequence.
3. That the map is well-defined follows from (2). The proof that it is computable is similar to the proof of Proposition \[prop:basecompactcomputable\]. Given some $U \in \mathcal{O}(\mathrm{Av}(\preceq))$, some $n_0\ldots n_k \in \mathbb{N}^*$ and a parameter $t \in \mathbb{N}$, we test whether for all $j \leq k$ it can be verified in at most $t$ steps that $n_j \in U$, otherwise we reject. In addition, we search for some $a \in U$ such that $n_j \npreceq a$ for all $j \leq k$, if we find one, we reject. For fixed $U$, $n_0\ldots n_k$ there is a value of the parameter $t$ not leading to a rejection iff $\uparrow \{n_0,\ldots,n_k\} = U$.
The following is the computable counterpart to [@goubault2 Proposition 3.1]. It serves in particular as evidence that our definition of $\nabla$-computably Noetherian is not too restrictive:
Let $\preceq$ be a computable well-quasiorder. Then $\mathrm{Av}(\preceq)$ is $\nabla$-computably Noetherian.
By combining Proposition \[prop:wqo-opens\] (3) with Proposition \[prop:quasiordercompacts\], we see that for a computable well-quasiorder $\preceq$ the map ${\textnormal{id}}: \mathcal{O}(\mathrm{Av}(\preceq)) \to \left (\mathcal{K}(\mathrm{Av}(\preceq)) \right )^\nabla$ is computable.
In future work, one should investigate the hyperspace constructions explored in [@goubault2] for whether or not they preserve $\nabla$-computable Noetherianess. Research in reverse mathematics has revealed that the preservation of being Noetherian is already equivalent to $\textrm{ACA}_0$ [@shafer3], which typically indicates that $'$ or some iteration thereof is needed, not merely $\nabla$. However, the computational hardness is found in the reverse direction: Showing that if the hyperspace is not Noetherian, then the original well-quasiorder is not computable. Thus, these results merely provide an upper bound on this question in our setting.
Noetherian spaces as $\nabla$-compact spaces {#sec:nablacompact}
============================================
For some hyperspace $P(\mathbf{X})$ of subsets of a represented space $\mathbf{X}$, and a space $B$ of truth values $\bot$, $\top$, we define the map $\textrm{isFull} : P(X) \to B$ by $\textrm{isFull}(X) = \top$ and $\textrm{isFull}(A) = \bot$ for $A \neq X$. We recall from [@pauly-synthetic] that a represented space is (computably) compact iff $\textrm{isFull} : \mathcal{O}(\mathbf{X}) \to \mathbb{S}$ is continuous (computable).
The space $\mathbb{S}^\nabla \cong \mathbf{2}^\nabla$ can be considered as the space of $\Delta^0_2$-truth values. In particular, we can identify $\Delta^0_2$-subsets of $\mathbf{X}$ with their continuous characteristic functions into $\mathbf{2}^\nabla$, just as the open subsets are identifiable with their continuous characteristic functions into $\mathbb{S}$. By replacing both occurrences of $\mathbb{S}$ in the definition of compactness (one is hidden inside $\mathcal{O}$) by $\mathbb{S}^\nabla$, we arrive at:
A represented space $\mathbf{X}$ is called $\nabla$-compact, iff $\textrm{isFull} : \Delta^0_2(\mathbf{X}) \to \mathbb{S}^\nabla$ is computable.
\[theo:nablacompact\] A Quasi-Polish space is $\nabla$-compact iff it is $\nabla$-computably Noetherian (relative to some oracle).
The proof is provided in the following lemmata and propositions.
Recall that construcible subsets of a topological space are finite boolean combinations of open subsets. For a represented space $\mathbf{X}$, there is an obvious represented space ${\mathfrak{C}}(\mathbf{X})$ of constructible subsets of $\mathbf{X}$: A set $A \in {\mathfrak{C}}(\mathbf{X})$ is given by a (Goedel-number of a) boolean expression $\phi$ in $n$ variables, and an $n$-tuple of open sets $U_1,\ldots,U_n$ such that $A = \phi(U_1,\ldots,U_n)$. Straight-forward calculation shows that we can always assume that $\phi(x_1,\ldots,x_{2n}) = (x_1 \setminus x_2) \cup \ldots \cup (x_{2n-1} \setminus x_{2n})$ without limitation of generality.
\[lemma:noetherianconstructible\] Let $\mathbf{X}$ be a $\nabla$-computably Noetherian Quasi-Polish space. Then ${\textnormal{id}}: {{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2(\mathbf{X}) \to {\mathfrak{C}}(\mathbf{X})^\nabla$ is well-defined and computable.
As $\mathbf{X}$ is Quasi-Polish, we can take it to be represented by an effectively open representation $\delta_\mathbf{X} : {\mathbb{N}^\mathbb{N}}\to \mathbf{X}$.
We can consider our input $A \in {{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2(\mathbf{X})$ to be given by a realizer $f : {\mathbb{N}^\mathbb{N}}\to \{0,1\}$ of a finite mindchange computation. We consider the positions where a mindchange happens, i.e. those $w \in \mathbb{N}^*$ which if read by $f$ will cause a mindchange to happen before reading any more of the input. W.l.o.g. we may assume that the realizer makes at most one mindchange at a given position $w \in \mathbb{N}^*$, and the realizer initially outputs $0$ before reading any of the input.
Let $W \subseteq \mathbb{N}^*$ be the set of mindchange positions. To simplify the following, we will view $\varepsilon$ (the empty string in $\mathbb{N}^*$) as being an element of $W$ (this assumption can be justified formally by viewing the initial output of $0$ as being a mindchange from “undefined” to $0$). Note that $W$ is decidable by simply observing the computation of $f$. If we denote the prefix relation on $\mathbb{N}^*$ by $\sqsubseteq$, we see that there are no infinite strictly ascending sequences in $W$ with respect to $\sqsubseteq$, since any such sequence would correspond to an input that induces infinitely many mindchanges. It follows that $(W, \preceq)$ is a computable total well-order with maximal element $\varepsilon$, where $\preceq$ is the (restriction of the) Kleene-Brouwer order and defined as $v \preceq w$ if and only if (i) $w \sqsubseteq v$, or (ii) $v(n) < w(n)$, where $n$ is the least position where $v$ and $w$ are both defined and disagree.
We first note that $\min :\subseteq {{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2(W) \to W$ is $\nabla$-computable, where $\min$ is the function mapping each non-empty $S\in {{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2(W)$ to the $\preceq$-minimal element of $S$. A realizer for $\min$ on input $S$ can test in parallel whether each element of $W$ is in $S$, and output as a guess the $\preceq$-minimal element which it currently believes to be in $S$. Since $\preceq$ is a well-order and it only takes finitely many mindchanges to determine whether or not a given element is in $S$, this computation is guaranteed to converge to the correct answer.
For each $w\in W$, define $U_w := \bigcup_{v \in W, v \preceq w} \delta_\mathbf{X}[v{\mathbb{N}^\mathbb{N}}]$, which is an effectively open subset of $\mathbf{X}$ and a uniform definition because $\preceq$ is decidable. Next, let $\mathbf{1} = \{*\}$ be the totally represented space with a single point, and define $h \colon W \to ({{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2(W) + \mathbf{1})$ as $h(w) = *$ if $U_w = \mathbf{X}$ and $h(w) = \{v\in W \mid U_v \subsetneq U_w \}$, otherwise. The computability of the mapping $w \mapsto U_w$ and the assumption that $\mathbf{X}$ is $\nabla$-computably Noetherian implies that it is $\nabla$-decidable whether $U_w = \mathbf{X}$, and also that the characteristic function of the set $\{v\in W \mid U_v \subsetneq U_w \}$ is $\nabla$-computable given $w\in W$. It follows that $h$ is well-defined and $\nabla$-computable.
We construct a finite sequence $v_0 \prec \ldots \prec v_k$ in $W$ by defining $v_0= \min(W)$ and $v_{n+1}=\min(h(v_n))$ whenever $h(v_n)\not=*$. This sequence is necessarily finite because the $U_{v_n}$ form a strictly increasing sequence of open sets and $\mathbf{X}$ is Noetherian. Note that the last element $v_k$ in the sequence satisfies $h(v_k)=*$. It follows that the sequence $\langle v_0,\ldots, v_k\rangle \in W^*$ can be $\nabla$-computed from the realizer $f$ because it only involves a finite composition of $\nabla$-computable functions, and it can be $\nabla$-decided when the sequence terminates.
Define $\eta \colon W \to\{0,1\}$ to be the computable function mapping each $w\in W$ to the output of the realizer $f$ after the mindchange upon reading $w$ (thus $\eta(\varepsilon)=0$). For $n\leq k$ define $V_n := U_{v_n} \setminus \bigcup_{m<n} U_{v_m}$. We claim that $A = \bigcup\{V_n \mid 0\leq n \leq k \text{ \& } \eta(v_n)=1\}$, from which it will follow that we can $\nabla$-compute a name for $A \in {\mathfrak{C}}(\mathbf{X})$ from the realizer $f$.
Fix $x\in\mathbf{X}$, and let $w \in W$ be $\preceq$-minimal such that $x\in \delta_\mathbf{X}[w{\mathbb{N}^\mathbb{N}}]$. It follows that $x\in A$ if and only if $\eta(w) = 1$, because $w$ is a prefix of some name $p$ for $x$, and the $\preceq$-minimality of $w$ implies that the realizer $f$ does not make any additional mindchanges on input $p$ after reading $w$. Next, let $n \in \{0,\ldots, k\}$ be the least number satisfying $x\in V_n$. It is clear that $w \preceq v_n$. Conversely, if $n=0$ then $v_n = v_0 \preceq w$ by the $\preceq$-minimality of $v_0$. If $n>0$, then $w \not\preceq v_{n-1}$ hence $x$ is a witness to $U_{v_{n-1}} \subsetneq U_w$, which implies $v_n =h(v_{n-1}) \preceq w$. Thus $w=v_n$, and it follows that $x\in A$ if and only if $x \in \bigcup\{V_n \mid 0\leq n \leq k \text{ \& } \eta(v_n)=1\}$, which completes the proof.
\[prop:constructibleisfull\] Let $\mathbf{X}$ be $\nabla$-computably Noetherian. Then $\textrm{isFull} : {\mathfrak{C}}(\mathbf{X}) \to {\mathbf{2}}^\nabla$ is computable.
It is well-known that the sets in ${\mathfrak{C}}(\mathbf{X})$ have a normal form $A = (U_0 \setminus V_0) \cup \ldots \cup (U_n \setminus V_n)$, and this is obtainable uniformly. Now $A = X$ iff $\forall I \subseteq \{0,\ldots,n\} \ \left ( \bigcap_{j \notin I} V_j \right ) \subseteq \left ( \bigcup_{i \in I} U_i \right )$. To see this, first note that the special case $I = \{0,\ldots,n\}$ yields $X = \bigcup_{i \in I} U_i$. Now consider for each $x \in X$ the statement for $I = \{i \mid x \notin V_i\}$.
In a $\nabla$-computably Noetherian space, we can compute $\left ( \bigcap_{j \notin I} V_j \right ) $ as a compact set, and decide its inclusion in $\left ( \bigcup_{i \in I} U_i \right )$ with finitely many mindchanges. Doing this for the finitely many choices of $I$ is unproblematic, thus yielding the claim.
\[prop:notnablacompact\] Let $\mathbf{X}$ admit a partition $(A_n)_{n \in \mathbb{N}}$ into non-empty ${{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2$-sets. Then $\mathbf{X}$ is not $\nabla$-compact.
Given some $(t_i)_{i \in \mathbb{N}} \in ({\mathbf{2}}^\nabla)^\mathbb{N}$, we can compute the set $A := \{x \in \mathbf{X} \mid \exists n \in \mathbb{N} \ x \in A_n \wedge t_n = 1\} \in {{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2(\mathbf{X})$. If $\mathbf{X}$ were $\nabla$-compact, then applying $\textrm{isFull} : {{\protect\raisebox{0pt}[0pt][0pt]{$\underset{\widetilde{}}{\boldsymbol{\Delta}}$}\mbox{\hskip 1pt}}}^0_2(\mathbf{X}) \to \mathbb{S}^\nabla \cong {\mathbf{2}}^\nabla$ to $A$ would yield a computable realizer of $\bigwedge : ({\mathbf{2}}^\nabla)^\mathbb{N} \to {\mathbf{2}}^\nabla$.
By combining Lemma \[lemma:noetherianconstructible\] and Proposition \[prop:constructibleisfull\], we see that for a $\nabla$-computably Noetherian quasi-Polish space $\mathbf{X}$ the map $\textrm{isFull} : \Delta^0_2(\mathbf{X}) \to \mathbb{S}^\nabla$ is computable, i.e. it is $\nabla$-compact. Conversely, if $\mathbf{X}$ is not Noetherian, then by Lemma \[lemma:converingcoverse\] there is a countably-infinite $\Delta^0_2$-partition of $\mathbf{X}$, so by Proposition \[prop:notnablacompact\], it cannot be $\nabla$-compact.
The significance of $\nabla$-compactness and Theorem \[theo:nablacompact\] lies in the following proposition that supplies the desired quantifier-elimination result. The proof is a straight-forward adaption of the corresponding result for compact spaces and open predicates from [@pauly-synthetic] (recalled here as Propositions \[prop:exists\],\[prop:forall\]), which in turn has [@escardo2] and [@nachbin] as intellectual predecessors. Note that as $\neg : \mathbb{S}^\nabla \to \mathbb{S}^\nabla$ is computable, it follows that $\nabla$-compactness and $\nabla$-overtness coincide:
\[prop:quantifierelimination\] The following are equivalent for a represented space $\mathbf{X}$:
1. $\mathbf{X}$ is $\nabla$-compact.
2. For any represented space $\mathbf{Y}$, the map $\mathalpha{\forall} : \Delta^0_2(\mathbf{X} \times \mathbf{Y}) \to \Delta^0_2(\mathbf{Y})$ mapping $R$ to $\{y \in \mathbf{Y} \mid \forall x \in \mathbf{X} \ (x,y) \in R\}$ is computable.
3. For any represented space $\mathbf{Y}$, the map $\mathalpha{\exists} : \Delta^0_2(\mathbf{X} \times \mathbf{Y}) \to \Delta^0_2(\mathbf{Y})$ mapping $R$ to $\{y \in \mathbf{Y} \mid \exists x \in \mathbf{X} \ (x,y) \in R\}$ is computable.
\[corr:quantifier\] A formula built from $\Delta^0_2$-predicates, boolean operations and universal and existential quantification over Noetherian quasi-Polish spaces defines itself a $\Delta^0_2$-predicate.
Let $\mathbf{X} = \mathbf{X}_0 \times \ldots \times \mathbf{X}_n$ be a Noetherian Quasi-Polish space. If a subset $U \subseteq \mathbf{X}_0$ is definable using a finite expression involving open predicates in $\mathbf{X}$, boolean operations, and existential and universal quantification, then $U$ is definable using a finite expression involving open predicates in $\mathbf{X}_0$ and boolean operations.
Combine Corollary \[corr:quantifier\] and Lemma \[lemma:noetherianconstructible\].
Other compactness and overtness notions {#sec:othernotions}
=======================================
It is a natural question whether further lifted counterparts of compactness and overtness might coincide with familiar notions from topology. We will in particular explore this for the $'$-endofunctor from Definition \[def:jump\]. One such result was already obtained before:
A Polish space is $\sigma$-compact iff it is $'$-overt.
For our remaining investigations, we will rely on the following lemma:
\[lem:genericounterexample\] Let $d$ be a computable endofunctor such that $\bigvee : \mathcal{C}(\mathbb{N},d\mathbb{S}) \to \mathbb{S}$ and $\wedge : d\mathbb{S} \times d\mathbb{S} \to d\mathbb{S}$ are computable, but $\bigwedge : \mathcal{C}(\mathbb{N},d\mathbb{S}) \to \mathbb{S}$ is not continuous. Then if $\mathbf{X}$ admits a partition into countably-infinitely many non-empty $d$-open subsets, $\mathbf{X}$ is not $d$-compact.
Let $(U_n)_{n \in \mathbb{N}}$ be a partition into countably-many $d$-open sets. From the computability of $\bigvee : \mathcal{C}(\mathbb{N},d\mathbb{S}) \to \mathbb{S}$ and and $\wedge : d\mathbb{S} \times d\mathbb{S} \to d\mathbb{S}$ we can conclude that $(b_n)_{n \in \mathbb{N}} \mapsto \bigcup_{\{i \in \mathbb{N} \mid b_i = \top\}} U_i : \mathcal{C}(\mathbb{N},d\mathbb{S}) \to \mathcal{O}^d(\mathbf{X})$ is continuous. If $\mathbf{X}$ were $d$-compact, then we could apply $\textrm{isFull} : \mathcal{O}^d(\mathbf{X}) \to d\mathbb{S}$ to the resulting set, and would obtain $\bigwedge_{n \in \mathbb{N}} b_n \in d\mathbb{S}$, in contradiction to our assumption.
A quasi-Polish space is $\nabla$-compact iff it is $'$-compact.
Assume that a quasi-Polish space $\mathbf{X}$ is $\nabla$-compact. Let $U \in \mathcal{O}'(\mathbf{X})$ be a $\Sigma^0_2$-set. This can be effectively written as $U = \bigcup_{n \in \mathbb{N}} U_n$ with disjoint $\Delta^0_2$-sets $U_n$. As $\bigvee : \mathcal{C}(\mathbb{N},\mathbb{S}^\nabla) \to \mathbb{S}'$ is computable, we can compute $\bigvee_{N \in \mathbb{N}} \textrm{isFull}(\bigcup_{n \leq N} U_n)$ using $\textrm{isFull} : \mathcal{O}^\nabla(\mathbf{X}) \to \mathbb{S}^\nabla$. If this yields $\top$, then clearly $U = X$. Conversely, if $U = X$, then by Theorem \[theo:finitecover\], already $U = U_N = X$ for sufficiently large $N$, hence the procedure yields $\top$. It follows that $\mathbf{X}$ is $'$-compact.
Now assume that a quasi-Polish space $\mathbf{X}$ is not $\nabla$-compact. Then by Theorem \[theo:nablacompact\] it is not Noetherian, hence by Theorem \[theo:finitecover\] there is an infinite $\Delta^0_2$-cover $(U_n)_{n \in \mathbb{N}}$ without a finite subcover. We can refine this into a $\Delta^0_2$-partition (which of course is also a $\Sigma^0_2$-partition). Note that Lemma \[lem:genericounterexample\] applies to $'$, hence $\mathbf{X}$ is not $'$-compact.
A quasi-Polish space is $''$-compact relative to some oracle iff it is finite.
A finite quasi-Polish space is $''$-compact relative to an oracle enumerating the points, as $\wedge : \mathbb{S}'' \times \mathbb{S}'' \to \mathbb{S}''$ is computable. Conversely, any singleton $\{x\}$ in a quasi-Polish space is $\Pi^0_2$, hence also $\Sigma^0_3$. If $\mathbf{X}$ is an infinite quasi-Polish space, we can thus find a proper countably-infinite $\Sigma^0_3$-partition. By Lemma \[lem:genericounterexample\], it can then not be $''$-compact.
One could also start the search from the other direction, by exploring some variations on compactness from topology. We conclude by listing some potentially promising examples.
A topological space $\mathbf{X}$ is called *Menger*, if for any sequence $(\mathcal{U}_{n \in \mathbb{N}})$ of open covers of $\mathbf{X}$ there exists finite subsets $\mathcal{V}_n \subseteq \mathcal{U}_n$ such that $\bigcup_{n \in \mathbb{N}} \mathcal{V}_n$ is an open cover of $\mathbf{X}$.
It had been asked by [<span style="font-variant:small-caps;">Hurewicz</span>]{} whether Menger spaces might coincide with the $\sigma$-compact ones [@hurewicz3]. The two notions were conditionally separated by [<span style="font-variant:small-caps;">Miller</span>]{} and [<span style="font-variant:small-caps;">Fremlin</span>]{} [@fremlin], and then unconditionally by Bartoszynski and Tsaban [@tsaban]. A similar property is named after Hurewicz: In a Hurewicz space, the cover $\bigcup_{n \in \mathbb{N}} \mathcal{V}_n$ needs to have the property that any point belongs to all but finitely many sets from the cover. Both the Menger and the Hurewicz property are special cases of selection principles as identified by [<span style="font-variant:small-caps;">Scheepers</span>]{} [@scheepers]. These might provide a fruitful hunting ground for further topological properties corresponding to relativized compactness or overtness notions.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to the participants of the Dagstuhl-seminar *Well-quasi orders in Computer Science* for valuable discussions and inspiration (cf. [@dagstuhlwqo]). In particular we would like to thank Jean Goubault-Larrecq and Takayuki Kihara. Boaz Tsaban suggested the investigation of Menger and Hurewicz spaces in this context to us. The second author thanks Paul Shafer for explaining results pertaining to Noetherian spaces in reverse mathematics.
[^1]: This work was supported by JSPS Core-to-Core Program, A. Advanced Research Networks. The first author was supported by JSPS KAKENHI Grant Number 15K15940. The second author was supported by the ERC inVEST (279499) project.
[^2]: Sobrification only adds points, not open sets, and being Noetherian is only about open sets.
[^3]: This notion is much less known than compactness, as it is classically trivial. It is crucial in a uniform perspective, though. The term *overt* was coined by [<span style="font-variant:small-caps;">Taylor</span>]{} [@taylor], based on the observation that these sets share several closure properties with the open sets.
|
---
abstract: 'We develop a strategy to determine the cosmic birefringence and miscalibrated polarization angles simultaneously using the observed $EB$ polarization power spectra of the cosmic microwave background and the Galactic foreground emission. We extend the methodology of Y. Minami et al. (Prog. Theor. Exp. Phys. [**2019**]{}, 083E02, 2019), which was developed for auto frequency power spectra, by including cross frequency spectra. By fitting one global birefringence angle and independent miscalibration angles at different frequency bands, we determine both angles with significantly smaller uncertainties (by more than a factor of two) compared to the auto spectra.'
author:
- Yuto Minami
- Eiichiro Komatsu
bibliography:
- 'references.bib'
title: 'Simultaneous determination of the cosmic birefringence and miscalibrated polarization angles II: Including cross frequency spectra'
---
Introduction {#sec:Introduction}
============
Methodology {#sec:Methodology}
===========
Sky simulations {#sec:SkySim}
===============
Results {#sec:Results}
=======
Simultaneous determination of alpha and beta {#sec:Birefringence}
--------------------------------------------
Discussion and conclusion {#sec:Conclusion}
=========================
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Y. Chinone, K. Ichiki, N. Katayama, T. Matsumura, H. Ochi, S. Takakura, and Joint Study Group of the LiteBIRD collaboration for useful discussions. This work was supported in part by the Japan Society for the Promotion of Science (JSPS) KAKENHI, Grant Number JP20K1449 and JP15H05896, and the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311. The Kavli IPMU is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.
|
---
abstract: |
\[sec:Abstract\]
We report the creation of ultracold bosonic $^{23}$Na$^{87}$Rb Feshbach molecules via magneto-association. By ramping the magnetic field across an interspecies Feshbach resonance, at least 4000 molecules can be produced out of the near degenerate ultracold mixture. Fast loss due to inelastic atom-molecule collisions is observed, which limits the pure molecule number, after residual atoms removal, to 1700. The pure molecule sample can live for 21.8(8) ms in the optical trap, long enough for future molecular spectroscopy studies toward coherently transferring to the singlet ro-vibrational ground state, where these molecules are stable against chemical reaction and have a permanent electric dipole moment of 3.3 Debye. We have also measured the Feshbach molecule’s binding energy near the Feshbach resonance by the oscillating magnetic field method and found these molecules have a large closed-channel fraction.
address: |
Department of Physics, the Chinese University of Hong Kong,\
Shatin, Hong Kong SAR, China
author:
- 'Fudong Wang, Xiaodong He, Xiaoke Li, Bing Zhu, Jun Chen and Dajun Wang'
title: Formation of Ultracold NaRb Feshbach Molecules
---
The creation and manipulation of ultracold heteronuclear molecules have received intensive attentions in recent years due to the versatile and promising potential applications [@Carr2009; @Krems2009] of these molecules. With controllable, anisotropic and long range dipole-dipole interactions, they could be used in quantum computation [@DeMille2000; @Andre2006], quantum simulation [@Micheli2006], precision measurement [@Zelevinsky2008; @Hudson2011] and controlled cold chemistry [@Krems2008]. So far, the most successful scheme for producing ultracold ground-state dipolar molecule is by associating ultracold atoms near Feshbach resonances [@Kohler2006; @Chin2010] to form weakly-bound molecules first, followed by a stimulated Raman adiabatic rapid passage (STIRAP) [@Bergmann1998] to transfer them to a deeply bound state [@Winkler2007; @Ospelkaus2008]. This has been successfully applied to the $^{40}$K$^{87}$Rb system [@Ni2008], where near degenerate ground-state dipolar fermionic molecules are created. However, the chemical reaction 2 KRb$\rightarrow$ K$_2$ + Rb$_2$ is an exoergic process which results in a large inelastic loss, severely limiting the trap lifetime of the KRb molecular gas [@Ospelkaus2010; @Ni2010; @Miranda2011].
Currently, there is a great effort in generalizing the KRb production scheme to other heteronuclear alkali dimers. Creation of Feshbach molecules of RbCs [@Takekoshi2012], LiNa [@Heo2012] and NaK [@Wu2012] were already reported in 2012, and very recently ground-state RbCs molecules were successfully produced [@Takekoshi2014; @Molony2014]. In this work, we focus on the bosonic NaRb molecule, which in the absolute ground state is stable against chemical reactions [@Zuchowski2010] and has a permanent electric dipole moment as large as 3.3 Debye [@Aymar2005]. The ground-state NaRb molecule can be readily polarized with a moderate electric field. For instance, at 5 kV/cm the induced dipole moment is already more than 2 Debye. Therefore, it is an appealing system for studying the bosonic quantum gas with strong dipolar interactions. Recently, the double species Bose-Einstein condensates (BECs) of $^{23}$Na and $^{87}$Rb atoms have been produced [@Xiong2013] and their interspecies Feshbach resonances(FRs) [@Wang2013] were also investigated in our group. One of the s-wave resonances between atoms in their lowest hyperfine Zeeman state locates conveniently at a magnetic field $B_0$ = 347.7 G with a width of $\Delta$ = 4.9 G. Here we report the creation and characterization of $^{23}$Na$^{87}$Rb Feshbach molecules by magneto-association with this FR.
We prepare the ultracold mixture of $^{23}$Na and $^{87}$Rb atoms in the same setup for producing the double species BEC [@Xiong2013]. Here we only describe it briefly. The initial evaporative cooling is performed in a hybrid trap [@Lin2009] with $^{87}$Rb as the coolant and the minority $^{23}$Na is sympathetically cooled. The sympathetic cooling is quite efficient thanks to the ideal heteronuclear scattering length of 66.8 $a_0$ [@Wang2013], with $a_0$ the Bohr radius. The atoms are then transferred to a crossed optical dipole trap (ODT) where the final evaporative cooling is performed. In our previous work, the ODT was formed by a 1070 nm laser which generates very different trapping potential depths for Na and Rb, with a ratio [U$_\textrm{Na}$/U$_\textrm{Rb}$]{} $\approx$ 1/3. Thus when lowering the ODT power, Na always evaporates faster than Rb, limiting the double BEC numbers to about 4$\times 10^4$ for each species. Here, we add in another single beam trap created by a 660 nm laser (Cobolt Flamenco), which generates a repulsive potential for Rb and an attractive one for Na. In this dichromatic ODT, we can tune the relative trap depths almost at will to allow the majority Rb atoms to always act as the coolant in the ODT and minimize the loss of Na atoms. With this upgrade, the double BEC atom numbers are increased to more than 10$^5$ for both species.
For the best atom-to-molecule conversion, it is crucial for the two atomic clouds to have the maximum overlap in phase space [@Hodby2005], which ideally should be achieved with both species condensed. However, we have found empirically in our experiment that the largest pure Feshbach molecule sample can only be obtained with a near-degenerate mixture. This is believed to be a result of the very fast inelastic losses caused by collisions with the high density residual atoms inherent from the condensates. We also observe that better results can be obtained by turning off the 660 nm beam adiabatically at the end of the evaporative cooling. This is understood as a result of the smaller differential gravitational sag and thus better spatial overlap between the two clouds, as trap oscillation frequencies for the two species are much more similar in the pure 1070 nm ODT than in the dichromatic trap.
During evaporation, both atoms are transferred to their $|F = 1,m_{F} = 1\rangle$ hyperfine ground state with a radio-frequency adiabatic rapid passage in a 2 G magnetic field. The magnetic field is then ramped up to 364 G, far above the 347.7 G FR. Evaporation continues at this field and stops near the critical temperature $T_c$ of the Na BEC. We then jump the magnetic field down to 350 G, the starting point of the magneto-association. After 5 ms holding for the magnetic field to stabilize, we typically have an ultracold mixture with $8 \times 10^{4}$ Na and $1 \times 10^{5}$ Rb atoms at a temperature of $350$ nK. The average final trap oscillation frequency is measured to be $\overline{\omega} = 2\pi \times 170$(148) [Hz]{} for Na(Rb). The calculated peak number density and phase space density are $4 \times 10^{12}$($ 2.6 \times 10^{13}$) cm$^{-3}$ and 1.0(0.84) for Na(Rb), respectively.
![\[fig:Association\] Magneto-association and dissociation. (a) Remaining atoms after the magnetic field is swept downward and stopped at various values near the FR. (b) Reverse the sweeping direction without holding after the association, an atom recovery shows up as a result of molecule dissociation. Due to limited reverse ramp rate, we stop the association close to resonance at 347.2 G to shorten the dissociation sweep duration. Red solid lines are fit to the hyperbolic tangent function. Error bars are from statistics of typically 3 shots and represent one standard deviation.](FigAssociation){width=".7\linewidth"}
To perform magneto-association, we adiabatically sweep the magnetic field toward the FR and across it at a fixed rate of 5.2 G/ms. As shown in Fig. \[fig:Association\](a), we stop at various field values and then turn the field off rapidly without holding before measuring the number of remaining atoms. Huge losses are observed for both species after the FR is crossed, amount to about 50$\%$ or $4 \times 10^{4}$ atoms for the minority Na atom. We note that these losses are the combined result of Feshbach molecule creation and enhanced three-body recombination. To confirm Feshbach molecules are indeed created, we ramp the field back up right after the association with a fixed rate of 3.9 G/ms. During this process, any Feshbach molecules will be dissociated to free atoms and show up as an atomic number recovery. A typical dissocaition curve is shown in Fig. \[fig:Association\](b). An increase in the number of atoms is clearly observed after the field is ramped up across the FR, which verifys that Feshbach molecules were created and then dissociated. However, only 10$\%$ or 4000 of the lossed atoms were recovered. We believe that this is a lower bound of the Feshbach molecule number as fast atom-molecule collisions during the finite ramp up time can kill these molecules.
![\[fig:Image\] Creation of pure Feshbach molecule samples by removing residual atoms with a high magnetic field gradient after association. (a) shows the magnetic field sequence and (b) shows images of Na and Rb at different stages of the association and dissociation procedure with the removal gradient pulse always present(see text for details). Color bar denotes optical density. By varying the holding time before or after the gradient pulse, we can measure the lifetime of Feshbach molecules with or without residual atoms.](FigImage){width=".7\linewidth"}
In future molecular spectroscopy studies and quest for absolute ground-state NaRb molecules, it is desirable to have a pure sample of Feshbach molecules. To this end, we have to remove residual Na and Rb atoms quickly after the association. According to the previous coupled-channel modeling [@Wang2013], Feshbach molecules created with the 347.7 G FR have a nearly zero magnetic dipole moment when the binding energy is more than 11 MHz which corresponds to magnetic fields $\sim$7.5 G below resonance. As residual atoms in ${\left|{1,1}\right\rangle}$ state are high field seekers, we can thus remove them with a magnetic field gradient without affecting molecules. As illustrated in Fig. \[fig:Image\], this is accomplished by a 3 ms gradient pulse of 150 G/cm with the Feshbach magnetic field at 335.0 G. After this, molecules are dissociated and the magnetic field is turned off before images are taken. With the high magnetic gradient applied, we have confirmed that both Na and Rb atoms can be removed completely without the association and dissociation ramps, or with the association ramp only, as evident by blank images in the first two columns of Fig. \[fig:Image\] (b). Signals appeared in the “Association & dissociation” column of Fig. \[fig:Image\] (b) which are obtained after completing the full magnetic field sequence, can then only come from pure Feshbach molecules created by the association and survived the magnetic field gradient. In the optimized condition, we can produce a pure sample of 1700 NaRb Feshbach molecules which corresponds to a overall atom-to-molecule conversion efficiency (including contributions from both association and inelastic loss) of $2\%$.
![\[fig:Lifetime\] Lifetime of Feshbach molecules. Black solid circles are measured number of Feshbach molecules vs. holding time with residual Rb and Na atoms, while red open circles are without. Solid curves are exponential fitting for extracting lifetimes. Error bars represent one standard deviation from typically 3 shots. All measurements are performed at a magnetic field of 335 G, where Feshbach molecules are almost of pure closed-channel character (see discussions below).](FigLifetime){width=".5\linewidth"}
We notice that similar overall conversion efficiencies were also reported by other groups working on different bosonic heteronuclear molecular species [@Takekoshi2012; @Koppinger2014]. Nevertheless, we want to emphasize that this low molecule number is not due to the association step only, which actually could have a much higher conversion efficiency. It is also a result of inelastic atom-molecule collisions. Even with the high field gradient, we estimate that about 1 ms is still needed for separating molecules from residual atoms completely, which is still ample time for inelastic collisions. As shown in Fig. \[fig:Lifetime\], we have measured the Feshbach molecule lifetime with and without the residual atoms by varying the holding time before and after the gradient pulse. With residual atoms, the lifetime is 1.1(1) ms. In 1 ms, almost 60$\%$ of the population will be destroyed. With the residual atoms removed, the lifetime increases by a factor of twenty to 21.8(8) ms.
This lifetime and the current signal to noise ratio are good enough for further experiments towards ground state molecules. Faster residual atom removal with light blasting [@Thalhammer2006; @Ni2008] should be able to increase the number of pure molecules. Ultimately, inelastic losses may be eliminated by first loading both atoms into three dimensional optical lattices and making a double species Mott insulator before the association [@Thalhammer2006; @Ospelkaus2006; @Chotia2012; @Damski2003]. Currently, our numbers of atoms in the atomic mixture are also on the low side, but we cannot see any fundamental limitations which will prevent us from increasing them significantly by technical improvements.
![\[fig:rfSpectroscopy\] Feshbach molecule binding energy measurement by the oscillating $B$ field method. Inset of (a): an oscillating field can transfer atom pairs to a molecular state when its frequency matches with the Feshbach molecule’s binding energy which is determined by the Feshbach field detuning. When scanning the Feshbach field in the presence of a 1 MHz oscillation field, besides the bare resonance at 347.7 G, additional loss features show up for both Rb (a) and Na (b) when the resonant condition is met. Solid dots are experimental data and solid curves are fitting results (see text). The dashed vertical lines indicate the resonance magnetic field found from the fit.](FigrfSpectroscopy){width=".7\linewidth"}
To further characterize the Feshbach molecule, we measure its binding energy vs. the magnetic field detuning with the oscillating field method [@Thompson2005]. The oscillating field is produced by a single loop coil driven by a 2 W radio frequency amplifier. It is placed coaxially with the Helmholtz coils producing the Feshbach field. As illustrated in the inset of Fig. \[fig:rfSpectroscopy\](a), when the oscillation frequency matches with the Feshbach molecules’ binding energy at the selected magnetic field detuning, transition between the free atom and Feshbach molecule states will be induced. This coupling can associate pairs of free atoms to Feshbach molecules, which will get lost from the trap quickly due to inelastic collisions.
As the single loop coil has a rather narrow bandwidth, maintaining a constant field amplitude over all relevant frequencies is difficult. Similar to reference [@Takekoshi2012], for the binding energy measurement, we instead fix frequency of the oscillating field, and scan the Feshbach magnetic field. As shown in Fig. \[fig:rfSpectroscopy\](a) for Na and (b) for Rb are such a measurement in the presence of a 1 MHz oscillating field. Besides the main broad loss feature centering at 347.7 G, another narrower dip shows up on the lower field side when the Feshbach molecule’s binding energy matches with the oscillation frequency. To extract the resonant magnetic field at this oscillating frequency, the apparently asymmetric lineshapes are fitted with a Gaussian function convolved with the Boltzmann distribution due to the finite temperature of our sample [@Hanna2007; @Weber2008]. Typically, the center of the measured lineshape is shifted away from the resonance, as shown by the vertical lines in Fig. \[fig:rfSpectroscopy\](a) and (b). The same measurements are repeated for oscillation frequencies from 100 kHz to 3.5 MHz, beyond which the coupling becomes too weak for reliable measurement due to reduced free-bound Franck-Condon factors [@Chin2005].
The results are summarized in Fig. \[fig:FigBinding\](a). We have also performed the same characterization for another s-wave FR at 478.8 G [@Wang2013], as shown in Fig. \[fig:FigBinding\](b). We notice immediately that binding energies change quickly with magnetic field detuning for both resonances. Within 2 G, the Feshbach molecules are already bound by more than 3 MHz, which is a strong evidence of large closed-channel fraction or that these molecules are “non-universal” [@Kohler2006]. This finding is not surprising as the background scattering length is a positive and moderate value of 66.8 $a_0$, which indicates that closed channels for both FRs are real molecular states. We note that this behavior agrees qualitatively with the prediction from the coupled channel calculation [@Wang2013].
![\[fig:FigBinding\] Binding energy vs. magnetic field near (a) 347.7 G and (b) 478.8 G FRs. Open circles are experimental data. Red solid lines are fitting results with the square well model and dashed lines indicate the thresholds. Error bars on magnetic field are from fitting and magnetic field calibration. The inset of (a) shows the closed-channel fraction of the Feshbach molecule vs. magnetic field near the 347.7 G FR calculated from the fitting parameters. ](FigBinding){width=".7\linewidth"}
To extract quantitative information from the binding energy measurement, we fit our data with the square well model developed by Lange *et al* [@Lange2009]. Near a Feshbach resonance with intermediate width, the binding energy $E_b=\frac{\hbar^2 k_m^2}{2\mu}$ can be determined from the magnetic dipole moment difference $\delta\mu$ between the open and closed channels and the Feshbach coupling strength $\Gamma$ as $$k_m(B)=\frac{1}{a_{bg}-\overline{a}}+\frac{\Gamma/2}{\overline{a} \left(E_b+\delta\mu\left(B-B_c\right)\right)}.$$ Here $k_m$ is the wave number, $B_c$ is the magnetic field where the bare molecular state is tunned to the open-channel threshold and $\overline{a}$ is the mean scattering length which can be calculated from the van der Waals $C_6$ coefficient. The solid curves in Fig. \[fig:FigBinding\] are the fitting results with $\delta\mu$, $\Gamma$, and $B_c$ as the fitting parameters. Here C$_6$ = 1.2946 $\times$ 10$^7$ cm$^{-1}$$^{6}$ is used for calculating $\overline{a}$ and $a_{bg}$ is taken to be 66.77 $a_0$ [@Wang2013].
-------- ------ ----------- ------------ ----------
347.75 4.89 347.64(3) 5.20(0.27) 2.66(29)
478.79 3.80 478.83(3) 4.81(0.27) 2.52(26)
-------- ------ ----------- ------------ ----------
\[table1\]
The resonance width $\Delta=\frac{1}{\delta\mu}\frac{\left(a_{bg}-\overline{a}\right)^2}{a_{bg}\overline{a}}\frac{\Gamma}{2}$, and resonance magnetic field $B_0=B_c-\frac{a_{bg}}{a_{bg}-\overline{a}}\Delta$ can then be obtained from combination of these parameters, as summarized in Table. \[table1\]. While $B_0$ for both resonances agree well with results from the global coupled channel modeling, the resonance widths show some discrepancy. We also want to mention that binding energies predicted from coupled channel calculations also have some disagreement with our measurement. We hope to resolve these problems by adding more data points to the Feshbach spectroscopy with improved resolutions later.
From the fitting parameters, the widely used dimensionless resonance strength parameter s$_{res}$ [@Chin2005] for the two FRs are calculated to be 0.72 and 0.63, respectively, which confirms that their coupling strengths are both in the intermediate regime. We have also deduced the closed channel fraction $\frac{1}{\delta\mu}\frac{\partial{E_B}}{\partial{B}}$ of the Feshbach state. As shown in the inset of Fig. \[fig:FigBinding\](a), for the 347.64 G FR, the closed-channel character increases rapidly with the magnetic field detuning. At about 346 G, the closed-channel fraction is already 50$\%$ and reaches over 80$\%$ at 340 G.
In conclusion, we have successfully produced ultracold $^{23}$Na$^{87}$Rb Feshbach molecules by Feshbach magneto-association. After removing residual atoms, the small pure molecular sample lives long enough for further work. These Feshbach molecules have a large closed-channel fraction near the Feshbach resonance, advantageous for finding high efficiency STIRAP routes for population transfer. We note that a promising path for population transfer via the $2^1\Sigma^+/1^3\Pi$ excited state admixture was already investigated in detail with conventional molecular spectroscopy[@Docenko2007]. We are now in a good starting point toward chemically stable ground-state bosonic molecules with a large electric dipole moment of 3.3 Debye for investigating ultracold gas with dominating dipolar interactions.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank E. Tiemann, O. Dulieu and G. Quéméner for valuable discussions and Mingyang Guo for technical assistance. This work is supported by Hong Kong Research Grants Council (General Research Fund Projects 403111, 404712 and the ANR/RGC Joint Research Scheme ACUHK403/13).
References {#references .unnumbered}
==========
[38]{}
Carr L D, Demille D, Krems R V and Ye J 2009 [*New J. Phys.*]{} [**11**]{} 055049 , edited by Krems R V, Stwalley W C and Friedrich B (CRC Press, Boca Raton, 2009) Demille D 2002 [*Phys. Rev. Lett.*]{} [**88**]{} 067901 André A, Demille D, Doyle J M, Lukin M D, Maxwell S E, Rabl P, Schoelkopf R J and Zoller P 2006 [*Nat. Phys.*]{} [**2**]{} 636 Micheli A, Brennen G K and Zoller P 2006 [*Nat. Phys.*]{} [**2**]{} 341 Zelevinsky T, Kotochigova S, and Ye J 2008 [*Phys. Rev. Lett.*]{} [**100**]{} 043201 Hudson J J, Kara D M, Smallman I J, Sauer B E, Tarbutt M R and Hinds E A 2011 [*Nature*]{} [**473**]{} 493 Krems R V 2008 [*Phys. Chem. Chem. Phys.*]{} [**10**]{} 4079 Köhler T, Góral K, and Julienne P S 2006 [*Rev. Mod. Phys.*]{} [**78**]{} 1311 Chin C, Grimm R, Julienne P, and Tiesinga E 2010 [*Rev. Mod. Phys.*]{} [**82**]{} 1225 Bergmann K, Theuer H, and Shore B W 1998 [*Rev. Mod. Phys.*]{} [**70**]{} 1003 Winkler K, Lang F, Thalhammer G, van der Straten P, Grimm R, and Denschlag J H 2007 [*Phys. Rev. Lett.*]{} [**98**]{} 043201 Ospelkaus S, Pe’er A, Ni K K, Zirbel J J, Neyenhuis B, Kotochigova S, Julienne P S, Ye J and Jin D S 2008 [*Nat. Phys.*]{} [**4**]{} 622 Ni K K, Ospelkaus S, Miranda M H G de, Pe’er A, Neyenhuis A, Zirbel J J, Kotochigova S, Julienne P S, Jin D S, and Ye J 2008 [*Science*]{} [**322**]{} 231 Ospelkaus S, Ni K K, Wang D, Miranda M H G de, Neyenhuis B, Quéméner G, Julienne P S, Bohn J L, Jin D S and Ye J 2010 [*Science*]{} [**327**]{} 853 Ni K K, Ospelkaus S, Wang D, Quéméner G, Neyenhuis B, Miranda M H G de, Bohn J L, Ye J and Jin D S 2010 [*Nature*]{} [**464**]{} 1324 Miranda M H G de, Chotia A, Neyehuis B, Wang D, Quéméner G, Ospelkaus S, Bohn J L, Ye J and Jin D S 2011 [*Nat. Phys.*]{} [**7**]{} 502 Takekoshi T, Debatin M, Rameshan R, Ferlaino F, Grimm R, Nägerl H C, Le Sueur C R, Hutson J M, Julienne P S, Kotochigova S, and Tiemann E 2012 [*Phys. Rev. A*]{} [**85**]{} 032506 Heo M S, Wang T T, Christensen C A, Rvachov T M, Cotta D A, Choi J H, Lee Y R and Ketterle W 2012 [*Phys. Rev. A*]{} [**86**]{} 021602(R) Wu C H, Park J W, Ahmadi P, Will S, and Zwierlein M W 2012 [*Phys. Rev. Lett.*]{} [**109**]{} 085301 Takekoshi T, Reichsöllner L, Schindewolf A, Hutson J M, Le Sueur C R, Dulieu O, Ferlaino F, Grimm R and Nägerl H C 2014 [*Phys. Rev. Lett.*]{} [**113**]{} 205301 Molony P K, Gregory P D, Ji Z, Lu B, Köppinger M P, Sueur C R L, Blackley C L, Hutson J M and Cornish S L 2014 [*Phys. Rev. Lett.*]{} [**113**]{} 255301 Żuchowski P S and Hutson J M 2010 [*Phys. Rev. A*]{} [**81**]{} 060703(R) Aymar M and Dulieu O 2005 [*J. Chem. Phys.*]{} [**122**]{} 204302 Xiong D Z, Li X K, Wang F D and Wang D J arXiv:1305.7091 Wang F D, Xiong D Z, Li X K, Wang D J and Tiemann E 2013 [*Phys. Rev. A*]{} [**87**]{} 050702(R) Lin Y J, Perry A R, Compton R L, Spielman I B and Porto J V 2009 [*Phys. Rev. A*]{} [**79**]{} 063631 Hodby E, Thompson S T, Regal C A, Greiner M, Wilson A C, Jin D S, Cornell E A and Wieman C E 2015 [*Phys. Rev. Lett.*]{} [**94**]{} 120402 Köppinger M P, McCarron D J, Jenkin D L, Molony P K, Cho H W, Cornish S L, Sueur C R L, Blackley C L and Hutson J M 2014 [*Phys. Rev. A*]{} [**89**]{} 033604 Thalhammer G, Winkler K, Lang F, Schmid S, Grimm R, and Denschlag J H 2006 [*Phys. Rev. Lett.*]{} [**96**]{} 050402 Ospelkaus C, Ospelkaus S, Humbert L, Ernst P, Sengstock K and Bongs K 2006 [*Phys. Rev. Lett.*]{} [**97**]{} 120402 Chotia A, Neyenhuis B, Moses S A, Yan B, Covey J P, Foss-Feig M, Rey A M, Jin D S, and Ye J 2012 [*Phys. Rev. Lett.*]{} [**108**]{} 080405 Damski B, Santos L, Tiemann E, Lewenstein M, Kotochigova S, Julienne P S and Zoller P 2003 [*Phys. Rev. Lett.*]{} [**90**]{} 110401 Thompson S T, Hodby E, and Wieman C E 2005 [*Phys, Rev. Lett.*]{} [**95**]{} 190404 Hanna T M, Köhler T, and Burnett K 2007 [*Phys. Rev. A*]{} [**75**]{} 013606 Weber C, Barontini G, Catani J, Thalhammer G, Inguscio M and Minardi F 2008 [*Phys. Rev. A*]{} [**78**]{} 061601(R) Chin C and Julienne P S 2005 [*Phys. Rev. A*]{} [**71**]{} 012713 Lange A D, Pilch K, Prantner A, Ferlaino F, Engeser B, Nägerl H C, Grimm R, and Chin C 2009 [*Phys. Rev. A*]{} [**79**]{} 013622 Docenko O, Tamanis M, Ferber R, Pazyuk E A, Zaitsevskii A, Stolyarov A V, Pashov A, Knöckel H and Tiemann E 2007 [*Phys. Rev. A*]{} [**75**]{} 042503
|
---
abstract: 'We demonstrate that first-principles based adiabatic continuation approach is a very powerful and efficient tool for constructing topological phase diagrams and locating non-trivial topological insulator materials. Using this technique, we predict that the ternary intermetallic series Li$_2M''X$ where $M''$=Cu, Ag, Au, or Cd, and $X$=Sb, Bi, or Sn, hosts a number of topological insulators with remarkable functional variants and tunability. We also predict that several III-V semimetallic compounds are topologically non-trivial. We construct a topological phase diagram in the parameter space of the atomic numbers of atoms in Li$_2M''X$ compounds, which places a large number of topological materials presented in this work as well as in earlier studies within a single unified topological framework. Our results demonstrate the efficacy of adiabatic continuation as a useful tool for exploring topologically nontrivial alloying systems and for identifying new topological insulators even when the underlying lattice does not possess inversion symmetry, and the approaches based on parity analysis are not viable.'
author:
- 'Hsin Lin$^{1}$, Tanmoy Das$^{2}$, Yung Jui Wang$^{1}$, L.A. Wray$^{3,4}$, S.-Y. Xu$^{3}$, M. Z. Hasan$^{3}$, A. Bansil$^{1}$'
title: 'Adiabatic transformation as a search tool for new topological insulators: distorted ternary Li$_2$AgSb-class semiconductors and related compounds'
---
The recent discovery of topological insulators has opened up a new research direction in condensed-matter and materials science communities with tremendous potential for practical applications as well as for conceptual novelties.[@reviewHasan; @reviewZhang; @reviewMoore; @wilchek] The unusual properties of topological insulators become manifest at their surfaces through the appearance of spin-polarized metallic states even though the bulk remains an insulator. These surface states have been predicted to host a large number of interesting topological phenomena.[@ZhangDyon; @essinaxion; @exciton; @Zhangmonopole; @wilchek; @Majorana; @WrayCuBiSe] For instance, the electrodynamics of a topological insulator entails additional terms in Maxwell’s equations yielding novel Faraday-Kerr effect, magnetoelectric effect, and axion physics.[@ZhangDyon; @essinaxion] Other exciting possibilities with topological surface states involve exciton condensation[@exciton], presence of an image magnetic monopole induced by an electric charge, and Majorana fermions induced by the proximity effect from a superconductor.[@Zhangmonopole; @Majorana; @WrayCuBiSe] In this way, topological insulators may be thought of as unifying condensed matter physics with particle physics and cosmology.[@wilchek]
The practical realization of many of the aforementioned opportunities with topological insulators has not been possible to date because the choice of available topological materials is quite limited. The currently known topological insulators mainly belong to the $Z_2$ class, which includes Bi$_2$Te$_3$ family [@MatthewNatPhys; @BiTeSbTe; @tetradymite], the ternary half-Heusler compounds [@heuslerhasan; @heuslerZhang], thallium-based chalcogenides [@TlBiTe2; @TlBiTe2Zhang; @SuYangThallium], Ge$_m$Bi$_{2n}$Te$_{m+3n}$-class [@SuYangGBT], quaternary chalcogenide compounds, and ternary famatinite compounds [@Ray2011]. The search has also been extended to other classes of materials where many-body physics and/or crystalline symmetries conspire to produce non-trivial topological phases.[@Tkondo; @TAFM; @TMott; @THO; @topocrystal] Despite this progress however the realization of wide classes of emergent topological properties has remained a challenge.
In connection with the materials discovery effort, adiabatic continuation is an elegant and powerful tool for obtaining a handle on the Z$_2$ time reversal invariant even when the lattice does not possess a center of inversion symmetry. The idea is to start with a known topological insulator and exploit the fact that if the Hamiltonian of this system can be transformed into that of the new system of interest through a continuous series of transformations without inducing a band inversion, then the new material will also be a topological insulator. In what follows, the adiabatic continuation approach will be seen to be not only a method for identifying the topological property of a material, but to also provide a pathway for finding new classes of topologically interesting materials. We specifically discuss how gray Sn, which is known to be topologically non-trivial, can be mutated to obtain new families of topologically non-trivial phases in zinc-blende-like materials and their superlattice variations. In particular, we predict that distorted-Li$_2$AgSb, a lightweight compound, is adiabatically connected to Sn, and thus harbors a 3D topological insulator state and that its cubic ground state lies near a critical point. In contrast, the related Li$_2$CuSb-type compounds are found to be topologically trivial. A topological phase diagram in the parameter space of the atomic numbers of the constituent elements in Li$_2$AgSb structure is also adduced, which places a number of topological materials within a uniform topological framework. The ternary Li$_2M'X$ series (with a number of variants) is thus identified as a new platform for deriving a host of topological compounds, alloys and nanoscale heterostructures via the flexibility of their lattice parameters and spin-orbit interaction strength. We also predict several III-V semimetallic compounds to be topologically nontrivial.
![\[fig:sketch\] (Color online) [Crystal structure and topological band inversion.]{} (a) Crystal structure of Li$_2$AgSb. Li, Ag, and Sb are denoted by green, red, and blue balls, respectively. Sb and Ag form the zinc-blende sublattice. (b) The zinc-blende structure of InSb. In and Sb are denoted by gold and blue balls, respectively. (c) The diamond structure of gray tin. Diagrams in (d) (e), and (f) illustrate band structures near the $\Gamma$-point for trivial insulator, non-trivial semimetal, and non-trivial insulator, respectively. Blue dots denote the s-like orbitals at $\Gamma$. Band inversion occurs in the non-trivial case where the s-like orbitals at the $\Gamma$-point fall below the four-fold degenerate $j=3/2$ states. The degeneracy of the $j=3/2$ states is lifted by the lattice distortion in the non-cubic case.](fig1){width="8.5cm"}
We start our discussion of band inversion in the electronic structure with reference to gray tin[@FuKane]. It is useful to understand first how a non-trivial topological phase emerges in Sn, whereas its isostructural counterparts such as silicon (Si) and germanium (Ge) are topologically trivial, see Fig. 1(c). As shown in Fig. 1(d), the $s$-like conduction bands (blue dots) lie well above the $p$-like valence bands in Si and Ge, indicating the absence of band inversion in Si and Ge. As the lattice constant increases upon replacing Si or Ge with Sn, the conduction band moves down to touch the valence band, resulting in a band inversion at only the $\Gamma$-point. This pushes the twofold degenerate $s-$like bands, which were originally lying in the conduction band in Si or Ge, below the valence bands of the fourfold degenerate $p-$states of total angular momentum $j=3/2$, and the Fermi level ($E_F$) goes through the $p-$states as shown in Fig. 1(e). As the $s-$states are now occupied at $\Gamma-$point, the $Z_2$ topological invariant picks up an extra time-reversal factor of -1 compared to Si or Ge. Therefore, Sn is a topologically nontrivial zero gap semiconductor or semi-metal in its pristine phase.[@FuKane]
If we define band-inversion strength (BIS) as the energy difference between the $j=3/2$ states and the twofold degenerate $s$-states at the $\Gamma$-point, compounds with positive values of BIS will be topologically non-trivial, while those with negative values will be topologically trivial. The discussion in the preceding paragraph suggests a general route for realizing a topologically insulating phase where one lifts the 4-fold degeneracy of the $j=3/2$ states at the $\Gamma-$point by breaking the cubic symmetry of the crystal without inducing a negative band-inversion strength. We explore this possibility by showing how interplay of effects of finite distortion, heavy element substitution, and/or formation of superlattice of cubic structure, can either remove the band inversion if the band-inversion strength is not sufficiently large, or conversely, it can enhance the band-inversion strength.
In carrying out adiabatic transformations, it is important to consider the crystal structure first. Here, we begin with the ternary compositions Li$_2M'X$ ($M'$=Cu, Ag, and Au, $X$=Sb and Bi) whose crystal structure belongs to the space group $F\bar{4}3m$, with the atomic arrangement presented in Fig. 1(a). $M'$ and $X$ atoms occupy the Wyckoff 4$d$ and 4$a$ positions, respectively. Li atoms fill the remaining empty space in the Wyckoff 4$b$ and 4$c$ positions. Because $M'$ and $X$ atoms form a zincblende type sublattice, these materials resemble InSb if Li atoms are removed \[Fig. 1(b)\]. Note that In and Sb precede and follow Sn in the periodic table and InSb resembles gray Sn with diamond structure as shown in Fig. 1(c). These observations suggest that Li$_2M'X$ compounds could be candidates for Z$_2$ topological insulators if their electronic band structures resemble the band structure of structurally similar Sn. As already noted above, since there is no spatial inversion symmetry in Li$_2M'X$ or in zinc-blend InSb, parity methods[@FuKane] of computing $Z_2$ topological number are not viable.
![ (Color online) [Electronic band structures of topological insulating state in Li$_2$AgSb with lattice distortion.]{} (a) Li$_2$AgSb with experimental lattice constants. (b) Li$_2$AgSb with 3% expansion of lattice constants. (c) Rhombohedral Li$_2$AgSb with hexagonal lattice constants $a=a_0+3\%a_0$ and $c=c_0$, where $a_0$ and $c_0$ correspond to the experimental values. (d) A sketch of band structure near the $\Gamma$-point for topologically non-trivial Li$_2$AgSb with a lattice distortion. The s-like states are marked with a blue dot. Lattice distortion causes a gap to open at $E_F$, resulting in a topological insulating state in which surface bands will span the bulk band gap, resembling the dispersion plotted with red lines.](fig2){width="8.5cm"}
First-principles band structure calculations were performed with the linear augmented-plane-wave (LAPW) method using the WIEN2K package [@wien2k] within the framework of the density functional theory (DFT). The generalized gradient approximation (GGA)[@PBE96] was used to describe the exchange-correlation potential. Spin orbital coupling (SOC) was included as a second variational step using a basis of scalar-relativistic eigenfunctions.
The computed band structure of Li$_2$AgSb along high symmetry lines in the Brillouin zone is presented in Fig. 2(a). Away from the $\Gamma-$ point, the Fermi level is completely gapped and thus the topological properties can be determined from observations of band structure only near the $\Gamma$-point. Focusing on the band structure very close to the Fermi level, we find that the orbital angular momentum symmetries of these compounds are identical to those of the low energy spectrum in Figs. 1(d) and 1(e). The $s-$type and $p-$type states are nearly degenerate at the $\Gamma$-point around which the conduction bands and one of the valence band have almost linear dispersion, indicating that the system is at the critical point of the topological phase \[Fig. 2(a)\]. With a small expansion of lattice, the $s/p$ band inversion occurs as shown in Fig. 2(b). Conversely, the topological band inversion in Li$_2$AgSb can be removed altogether by uniformly decreasing all lattice constants, demonstrating chemical tunability of the system.
Since Li$_2$AgSb is at the critical point of the topological phase, it is possible to achieve a 3D topological phase by a finite distortion which lifts the degeneracy of the $j=3/2$ states and simultaneously induces a band inversion. We find that a rhombohedral distortion with expansion along the hexagonal ab-plane with $a=a_0+3\%a_0$ and $c=c_0$, where $a_0,~c_0$ denote experimental lattice constants, does the job as shown in Fig. 2(c). The lattice expansion is seen to push the $s-$states well below the $p-$states, and to also open an insulating gap by lifting the four-fold degeneracy of $j=3/2$ states through the loss of cubic symmetry. Since the band inversion occurs only at one point in the Brillouin zone, non-cubic Li$_2$AgSb is a 3D strong topological insulator. We return to this point below to show more rigorously that the aforementioned band inversion correctly indicates that distorted non-cubic Li$_2$AgSb is topologically nontrivial.
![(Color online) [Band inversion strength and topological phase diagram.]{} Band inversion strength is plotted as a function of the atomic numbers of $M'$ and $X$ elements in the zinc-blende sublattice. The calculation is carried out at $a$ = 12.8 Bohr which is 3% greater than the experimental lattice constant of Li$_2$AgSb. The black line, which defines the topological critical points by the zero value of the band inversion strength, separates the trivial and non-trivial topological phases.](fig3){width="8.5cm"}
Next, we apply the adiabatic continuity principle to a larger range of atomic compositions of Li$_2M'X$ by systematically changing the nuclear charge $Z$ of the atoms. Let us assume that atoms at the Li, Ag, and Sb positions possess hypothetical nuclear charges $Z_{M}=3-0.5 x+0.5 y$, $Z_{M'}=47+x$, and $Z_X=51-y$, respectively, where $x$ and $y$ are adjustable parameters. This choice guarantees that the system remains neutral for all values of $x$ and $y$. Note that $x$ and $y$ need not be integers and that non-integral nuclear charges are easily accommodated in first-principles all-electron computations in order to change the Hamiltonian continuously, while maintaining charge self-consistency throughout the transformation process. This mapping can be started with $x=0$ and $y=0$, which corresponds to Li$_2$AgSb, and end with $x=3$ and $y=1$, which corresponds to the artificial compound He$_2$SnSn. He$_2$SnSn possess inversion symmetry and the wavefunction parity analysis can be used to obtain the $Z_2$ topological invariant[@FuKane]. Since He is chemically inert, He$_2$SnSn has similar band structure as Sn and is a topologically non-trivial semimetal with $Z_2=1$.
A topological phase diagram can now be constructed based on the band inversion strength as a function of atomic numbers, as shown in Fig. 3 in the region of $0\leq x\leq 3$ and $0\leq y\leq 1$ at the lattice constant $a$=12.8 Bohr, which is 3% longer than the experimental value for Li$_2$AgSb. A borderline is drawn at the zero of the band inversion strength (thick black line). A phase transition occurs when one crosses this borderline in that the two regions separated by the borderline are not topologically equivalent. Li$_2$AgSb with lattice expansion and gray tin are seen to lie on the same side of the thick black line, and are thus adiabatically connected, implying that they are topologically equivalent. Since $Z_2=1$ for gray tin (actually, He$_2$SnSn with lattice expansion or distortion), Li$_2$AgSb with lattice expansion \[Fig. 2(b)\] or distortion \[Fig. 2(c)\] is also topologically nontrivial with $Z_2=1$. The borderline itself marks topologically critical phases. It would move toward positive $y$ direction if the lattice constants decreased. Li$_2$AgSb with experimental lattice constants \[Fig. 2(a)\] lies very close to this critical line. Note that the variation of $Z$s can also be considered as a doping effect in the spirit of the virtual crystal approximations.[@new1; @new2; @new3] Therefore, our topological phase diagram in Fig. 3 could be used also to guide search of topologically interesting electronic structures through alloying.
![(Color online) [Electronic structure of Li$_2M'X$ series and III-V binary compounds.]{} Bulk electronic structures of TlP in (a), TlAs in (b), Li$_2$AgBi in (c), Li$_2$AuBi in (d), Li$_2$CdSn in (e), and Li$_2$CuSb in (f). The size of blue data points is proportional to the probability of s-orbital occupation on the anion site. Grey shaded areas in panels (a)-(e) highlight band inversion corresponding to the band-structure in Fig. 1(e).](fig4){width="8.5cm"}
To further establish the usefulness of the adiabatic continuation approach in extending the search for topologically non-trivial materials, we carefully examine the phase diagram of Fig. 3 in which we can identify two new candidates for topologically nontrivial materials, namely, InSb and Li$_2$CdSn. InSb is a topologically trivial semiconductor in its native phase with the experimental gap of 0.26 eV [@InSbgap]. Here we predict that in InSb with adequate lattice expansion, the gap can be closed and a band inversion can occur. It is well known that the band gap in the III-V zinc-blende structure decreases for heavier constituent atoms and the lattice constants are larger. Earlier first-principles calculations have predicted that among heavy-atom III-V compounds, TlP and TlAs are stable in the zinc-blende structures[@TlAs]. We take the total-energy optimized lattice constants[@TlAslattice] and obtain the band structures for TlP and TlAs in Fig. 4. Both compounds possess the band inversion feature at the $\Gamma$ point. Therefore, thallium-based III-V compounds are predicted to be topologically nontrivial. In addition, the II-VI compound CdTe lies at $x=1$ and $y=-1$ in the phase diagram of Fig. 3. While CdTe is topologically trivial, the related compound HgTe is a known topologically nontrivial semimetal[@HgTe]. Li$_2$CdSn is also reported to be a cubic crystal[@Li2CdSn]. If we assume Li$_2$CdSn has the same crystal structure as Li$_2$AgSb, the optimized lattice constant is obtained as $a_0=12.85$ Bohr. Our band structures of Li$_2$CdSn in Fig. 4 show that a band inversion occurs at the $\Gamma$ point and the conduction bands are below the $E_F$ around the $X$-point, allowing us to predict that Li$_2$CdSn is a topologically nontrivial metal.
Along the preceding lines, we can theoretically predict still other topological insulators. If we replace Li$_2$AgSb with heavier atoms, the band inversion can be achieved at the $\Gamma-$point (Fig. 4). We predict thus that Li$_2$AgBi and Li$_2$AuBi are non-trivial topological metals in their pristine phases whereas Li$_2$CuSb is a trivial band insulator. Another novel way of breaking the cubic symmetry of the zinc-blende structure to attain the topological insulating state is to double the unit cell, which leads to a tetragonal crystal structure. The chalcopyrite semiconductors of I-III-VI$_2$ and II-IV-V$_2$ group compositions which are described by this tetragonal structure, are shown to be non-trivial topological insulators in Ref. , which is consistent with our predictions. Furthermore, we have found that quaternary chalcogenides and ternary famatinites, which are tetragonal with zinc-blende sublattice, are candidates for 3D topological insulators.[@Ray2011] We note that the adiabatic continuation method used in the present calculations also predicts that half-Heusler compounds are topologically non-trivial.[@heuslerhasan]
Concerning the practical realization of topologically nontrivial phases in Li$_2$AgSb class of compounds, as shown in Fig. 2(c), the nontrivial topological insulating phase is predicted to exist when a uniaxial tensile strain is applied. This could be achieved by growing the materials on suitable semiconducting or insulating substrates. Notably, all the materials discussed in the present study display electronic structures similar to those in HgTe/CdTe, which were used to achieve the two dimensional topological insulating phase or the quantum spin Hall (QSH) phase in a quantum well system.[@HgTe] Therefore, our work also expands the list of candidate materials for realizing QSH phases.
In conclusion, we have predicted new classes of topological insulator materials in III-V semiconductors and Li-based compounds Li$_2M'X$. Small-gap semiconductors such as InSb could be turned into nontrivial phases with a lattice expansion. With the example of Li$_2$AgSb, we show how the nontrivial insulating phase could be obtained by a uniaxial strain. The exciting opportunities present within the ternary Li$_2M'X$ series (with a number of variants) offer a new platform for realizing multi-functional topological devices for spintronics and fault-tolerant quantum computing applications. Undergirding our study is the demonstration that a first-principles approach based on arguments of adiabatic continuation provides a powerful tool for materials discovery in search for new topologically interesting materials encompassing centrosymmetric as well as non-centrosymmetric crystal structures. The topological phase diagrams of the sort we have obtained for the Li-based compounds in this work, which encode global materials characteristics extending from single element to binary, ternary and possibly quaternary compositions would provide systematic opportunities for materials discovery and tunability between trivial and non-trivial topological phases, and serve to motivate further research for both fundamental and applied purposes.
[**Acknowledgements.**]{} The work at Northeastern and Princeton is supported by the Division of Materials Science and Engineering, Basic Energy Sciences, U.S. Department of Energy Grants DE-FG02-07ER46352, DE-FG-02-05ER46200 and AC02-05CH11231, and benefited from theory support at the Advanced Light Source, Berkeley, and the allocation of supercomputer time at NERSC and Northeastern University’s Advanced Scientific Computation Center (ASCC). M.Z.H is supported by NSF-DMR-1006492 and DARPA-N66001-11-1-4110.
[99]{} M. Z. Hasan, and C. L. Kane, [Rev. Mod. Phys.]{} [**82**]{}, 3045-3067 (2010). X.-L. Qi, and S.-C. Zhang, [Rev. Mod. Phys.]{} [**83**]{}, 1057 (2011). M. Z. Hasan, and J. E. Moore, [Annual Review of Condensed Matter Physics]{} [**2**]{}, 5578 (2011). F. Wilczek, Journal Club. [Nature]{} [**458**]{}, 129 (2009). X.-L. Qi, T. L. Hughes, and S.-C. Zhang, [Phys. Rev. B]{} **78**, 195424 (2008). A. M. Essin, J. E. Moore, and D. Vanderbilt, [Phys. Rev. Lett.]{} [**102**]{}, 146805 (2009) B. Seradjeh, J. E. Moore, and M. Franz,[Phys. Rev. Lett.]{} [**103**]{}, 066402 (2009). X.-L. Qi, R. Li, J. Zhang, and S.-C. Zhang, [Science]{} [**323**]{}, 1184 (2009). L. Fu, and C. L. Kane, [Phys. Rev. Lett.]{} **100**, 096407 (2008). L. A. Wray [*et al.*]{}, [Nat. Phys.]{} **6**:855-859. Y. Xia *et al.*, [Nat. Phys.]{} **5**, 398-402 (2009). D. Hsieh *et al.*, [Phys. Rev. Lett.]{} **103**, 146401 (2009). H. Lin [*et al.*]{}, [New Journal of Physics]{} [**13**]{}, 095005 (2011). H. Lin [*et al.*]{} [Nature Materials]{} [**9**]{}, 546 (2010) S. Chadov [*et al.*]{} Nature Materials **9**, 541 (2010). H. Lin [*et al.*]{}, [Phys. Rev. Lett.]{} **105**, 036404 (2010). B. Yan [*et al.*]{}, EPL (Europhysics Letters) **90**, 37002 (2010).
S.-Y. Xu [*et al.*]{}, [Science]{} [**332**]{}, 560-564 (2011). S.-Y. Xu [*et al.*]{}, arXiv:1007.5111. Y. J. Wang [*et al.*]{} [New Journal of Physics]{} [**13**]{}, 085017 (2011). M. Dzero, K. Sun, V. Galitski, and P. Coleman, [Phys. Rev. Lett.]{} [**104**]{}, 106408 (2010). R. S. K. Mong, A. M. Essin, and J. E. Moore, [Phys. Rev. B]{} [**81**]{}, 245209 (2010). S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, [Phys. Rev. Lett.]{} [**100**]{}, 156401 (2008). T. Das, Sci. Rep. 2, 596 (2012). L. Fu, [Phys. Rev. Lett.]{} [**106**]{}, 106802 (2011). L. Fu, and C. L. Kane, [Phys. Rev. B]{} **76**, 045302 (2007). P. Blaha [*et al.*]{} *WIEN2k, An augmented plane wave plus local orbitals program for calculating crystal properties.* (Karlheinz Schwarz, Techn. University Wien, Austria) (2001). J. P. Perdew, K. Burke, M. Ernzerhof, [Phys. Rev. Lett.]{} **77**, 3865-3868 (1996). H. Lin, S. Sahrakorpi, R.S. Markiewicz, and A. Bansil, Phys. Rev. Lett. **96**, 097001 (2006); A. Bansil, Phys. Rev. B **20**, 4025 (1979); A. Bansil, Phys. Rev. B **20**, 4035 (1979). It will be interesting to examine effects of alloying and doping using first-principles approaches beyond the virtual crystal or rigid band type approximations, see Ref. . A. Bansil, R. S. Rao, P. E. Mijnarends and L. Schwartz, Phys. Rev. B **23**, 3608 (1981); P. E. Mijnarends and A. Bansil, Phys. Rev. B **13**, 2381 (1976); L. Huisman, D. Nicholson, L. Schwartz and A. Bansil, Phys. Rev. B **24**, 1824 (1981); L. Schwartz and A. Bansil, Phys. Rev. B **10**, 3261 (1974). Sher van Schilfgaarde, and A.-B.Chen, [Appl. Phys. Lett.]{} **62**, 1857-1859 (1993). Y. O. Ciftci, K. Colakoglu, E. A. Deligoz, [Cent. Eur. J. Phys.]{} **6**, 802-807 (2008). N. Saidi-Houat, A. Zaoui, and M. Ferhat, [J. Phys.: Condens. Mat.]{} [**19**]{}, 106221 (2007). B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, [Science]{} [**314**]{}, 1757 (2006). H.-U. Schuster, [*Naturwissenschaften*]{} **53**, 361 (1966). W. Feng, D. Xiao, J. Ding, and Y. Yao, [Phys. Rev. Lett.]{} [**106**]{}, 016402 (2011).
|
---
abstract: 'In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences $q_n$ and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index of determinacy. It is also established that Yafaev’s result holds if the moments satisfy $\root{2n}\of{q_{2n}}=o(n)$.'
author:
- Christian Berg and Ryszard Szwarc
title: Closable Hankel operators and moment problems
---
[**Mathematics Subject Classification**]{}: Primary 47A05; Secondary 47B25, 47B35
[**Keywords**]{}. Hankel operators, moment problems.
Introduction
============
In [@Y] Yafaev considers Hankel operators associated with Hamburger moment sequences $$\label{eq:mom}
q_n=\int_{-\infty}^\infty x^n\,dM(x), \quad n=0,1,\ldots,$$ where $M$ is a positive measure on the real line such that the set of polynomials ${\mathbb{C}}[x]$ is contained in the Hilbert space $L^2(M)$.
We use the notation of [@Y] and let $\mathcal D$ denote the dense subspace of $\ell^2=\ell^2({\mathbb{Z}}_+)$ of complex sequences with only finitely many non-zero terms. The standard orthonormal basis in $\ell^2$ is denoted $e_n, n=0,1,\ldots$.
Furthermore, we let $A: \mathcal D\to {\mathbb{C}}[x]$ denote the operator $$\label{eq:pol}
Ag(x)= \sum_{n\ge 0} g_nx^n,\quad g=(g_0,g_1,\ldots)\in\mathcal D,$$ considered as a densely defined operator from the Hilbert space $\ell^2$ to $L^2(M)$.
The Hankel form $q[g,g]$ defined on $\mathcal D$ by $$\label{eq:han}
q[g,g]:=\sum_{n,m\ge 0} q_{n+m}g_n\overline{g_m},\quad g\in\mathcal D$$ clearly satisfies $$\label{eq:fu}
q[g,g]=||Ag||^2_{L^2(M)},$$ which gives the following result, see [@Y Lemma 2.1].
The form $q[g,g]$ is closable in $\ell^2$ if and only if $A$ is closable.
Because of this result we shall only consider closability of $A$ and leave aside closability of the form $q$.
The main result [@Y Theorem 1.2] can be stated like this.
\[thm:Yafaev\] Let $q_n$ denote the moments . Then the following conditions are equivalent:
1. The operator $A$ in is closable.
2. $\lim_{n\to\infty} q_n=0$.
3. The measure satisfies $M({\mathbb{R}}\setminus (-1,1))=0$, in other words $ \operatorname{supp}(M)\subseteq[-1,1]$ and $M(\{\pm 1\})=0$.
It is elementary that (ii) and (iii) are equivalent and that these conditions imply that (i) holds. However, (i) does not imply (ii). We shall come back to where the proof in [@Y] breaks down, but start by giving our main results:
\[thm:indet\] If the measure $M$ is indeterminate, then $A$ is closable.
\[thm:det\] There exist determinate measures with unbounded support such that $A$ is closable. This holds in particular for all determinate measures with finite index of determinacy.
\[thm:strongdet\] Suppose the moments satisfy $\root{2n}\of{q_{2n}}=o(n)$. Then the moment problem is determinate and if $A$ is closable, then condition (iii) holds.
The proof of the last theorem follows the proof of Yafaev, but as the first two theorems show, some kind of “strong” determinacy condition is necessary for $(i)\implies (iii)$ to hold. We do not know if the condition of Theorem \[thm:strongdet\] is optimal.
Let us give some background material for these theorems, see [@Ak] for details. Associated with the moments we have the orthonormal polynomials $(P_n)$, which are uniquely determined by the conditions $$\label{eq:orpo}
\int_{-\infty}^\infty P_n(x)P_m(x)\,dM(x)=\delta_{n,m},$$ when we assume that all $P_n$ have positive leading coefficients.
If the moment problem is indeterminate, there exists an infinite convex set $V$ of measures $M$ satisfying . All measures $M\in V$ have unbounded support. Among the solutions are the Nevalinna extremal or in short the N-extremal, which are precisely the measures $M\in V$ for which ${\mathbb{C}}[x]$ is dense in $L^2(M)$ by a theorem of M. Riesz. The N-extremal measures are discrete measures supported by the zero set $\Lambda$ of certain entire functions of minimal exponential type, i.e., of the form $$M=\sum_{{\lambda}\in\Lambda} c_{\lambda}\delta_{{\lambda}}, \quad c_{{\lambda}}>0.$$
By a theorem going back to Stieltjes in special cases, the following remarkable fact holds: If one mass is removed from $M$, then the new measure becomes determinate, i.e., $$\widetilde{M}:=M-c_{{\lambda}_0}\delta_{{\lambda}_0},\quad {\lambda}_0\in\Lambda$$ is determinate. For details see e.g. [@B:C], where this result was exploited. The measure $\widetilde{M}$ is a so-called determinate measure of index of determinacy 0 and if further $n\ge 1$ masses are removed we arrive at a determinate measure $M'$ of index of determinacy $n$, in symbols $\operatorname{ind}(M')=n$. See [@B:D], which contains an intrinsic definition of such measures $M$ by the study of an index $\operatorname{ind}_z(M)$ associated to a point $z\in{\mathbb{C}}$. Finally, in [@B:D1 Equation (1.5)] we define $\operatorname{ind}(M):=\operatorname{ind}_z(M)$ for $z\in{\mathbb{C}}\setminus\operatorname{supp}(M)$ because $\operatorname{ind}_z(M)$ is independent of $z$ outside the support of $M$.
In the indeterminate case the polynomials $P_n$ form an orthonormal basis in $L^2(M)$ for all the N-extremal solutions $M$, and for the other solutions $M$ they form an orthonormal basis in the closure $\overline{{\mathbb{C}}[x]}^{L^2(M)}$.
It is known that this closure is isomorphic as Hilbert space with the space $\mathcal E$ of entire functions of the form $$\label{eq:ent}
u(z)=\sum_{k=0}^\infty g_kP_k(z),\quad g\in\ell^2, \;z\in{\mathbb{C}}.$$ By Parseval’s Theorem $$\int_{-\infty}^\infty u(x)\overline{v(x)}\,dM(x)=\sum_{k=0}^\infty g_k\overline{h_k},\quad
u=\sum_{k=0}^\infty g_kP_k,\; v=\sum_{k=0}^\infty h_kP_k,\quad g,h\in\ell^2.$$ Note that we have the orthogonal decomposition $$\label{eq:ortdec}
L^2(M)=\mathcal E \oplus {\mathbb{C}}[x]^\perp,\quad M\in V.$$
Proofs
======
[*Proof of Theorem \[thm:indet\]*]{}
Assume $g^{(n)}\in\mathcal D\to 0$ in $\ell^2$ and that $Ag^{(n)}\to f$ in $L^2(M)$. We have to prove that $f=0$. Clearly $f\in\mathcal E$.
Since $\mathcal E$ is a reproducing kernel Hilbert space of entire functions, we know that convergence in the Hilbert norm implies locally uniform convergence in the complex plane, not only for the functions but also for derivatives of any order. Therefore $$g^{(n)}_k=\frac{D^k(Ag^{(n)})(0)}{k!}\to \frac{D^k f(0)}{k!},$$ so the Taylor series of $f$ vanishes because in particular $g^{(n)}_k\to 0$ for $n\to\infty$ for any fixed $k$. $\quad\square$
Let us for simplicity first consider an N-extremal measure $M$ with mass $c>0$ at 0 and consider $\widetilde{M}:=M-c\delta_0$, which is a discrete determinate measure with unbounded support. A concrete example is studied in [@B:S1 p. 128]. The measure $\widetilde{M}$ does not satisfy condition (iii) of Theorem \[thm:Yafaev\]. Let $A$ and $\widetilde{A}$ denote the operators with values in $L^2(M)$ and $L^2(\widetilde{M})$ respectively. We know that the operator $A:\mathcal D\to L^2(M)$ is closable by Theorem \[thm:indet\].
Assume that $g^{(n)}\to 0$ in $\ell^2$, where $g^{(n)}\in\mathcal D$, and that $\widetilde{A}g^{(n)}\to f$ in $L^2(\widetilde{M})$. We have $\widetilde{A}g^{(n)}(0)=g^{(n)}_0\to 0$, and therefore $$Ag^{(n)}(x)\to \begin{cases} f(x),\quad & x\in \operatorname{supp}(\widetilde{M})\\
0,\quad &x=0
\end{cases}$$ in $L^2(M)$ because $M=\widetilde{M}+c\delta_0$. Since $A$ is closable, we conclude that $f=0$.
Let us next modify the proof just given by removing one or finitely many masses one by one at mass-points ${\lambda}_0$ satisfying $|{\lambda}_0|<1$ of an N-extremal measure $M$. In fact, for $n\to\infty$ also $$\widetilde{A}g^{(n)}({\lambda}_0)=\sum_{k\ge 0} g^{(n)}_k{\lambda}_0^k \to 0,$$ because $$|\sum_{k\ge 0} g^{(n)}_k{\lambda}_0^k|\le ||g^{(n)}||_{\ell^2} \left(1-|{\lambda}_0|^2\right)^{-1/2}.$$
We finally claim that if $M$ is an arbitrary determinate measure with $\operatorname{ind}(M)=n\ge 0$, then the corresponding operator $A$ is closable. In fact let $\Lambda\subset (-1,1)$ denote a set of $n+1$ points disjoint with $\operatorname{supp}(M)$. Such a choice is clearly possible since the support is discrete in ${\mathbb{R}}$. By [@B:D Theorem 3.9] the measure $$M^+:=M+\sum_{{\lambda}\in\Lambda}\delta_{\lambda}$$ is N-extremal and the corresponding operator $A^+$ is closable by Theorem \[thm:indet\]. By removing the masses $\delta_{\lambda}$ for ${\lambda}\in\Lambda$ one by one we obtain that the operator $A$ associated with $M$ is closable. $\quad\square$
Yafaev’s proof is based on a study of the set $\mathcal D_*\subset L^2(M)$ for an arbitrary positive measure $M$ with moments of any order as in , namely $$\label{eq:D*}
\mathcal D_*:=\left\{u\in L^2(M) : u_n:=\int_{-\infty}^\infty u(t) t^n\,dM(t) \in \ell^2\right\}.$$
Lemma 2.2 in [@Y] states that the adjoint $A^*$ of the operator $A$ from is given by $\operatorname{dom}(A^*)=\mathcal D_*$ and $$\label{eq:A*}
(A^*u)_n=\int_{-\infty}^\infty u(t) t^n\,dM(t),\;n=0,1,\ldots,\quad u\in\mathcal D_*.$$
Yafaev uses the following result, Theorem 2.3 in [@Y], which is not true:
The following conditions are equivalent:
1. of Theorem \[thm:Yafaev\],
2. $\mathcal D_*$ is dense in $L^2(M)$.
While it is correct that (iii) implies (iv), the converse is not true. In Theorem \[thm:2\] we prove that (iv) holds, if $M$ is an indeterminate measure and hence (iii) does not hold.
For $u\in L^2(M)$ we consider the complex Fourier transform $$\label{eq:Fou}
f(z)=\int_{-\infty}^\infty e^{izt}u(t)\,dM(t),\quad z=x+iy\in{\mathbb{C}},$$ which is an entire function under the assumption $\root{2n}\of{q_{2n}}=o(n)$. In fact, $$\begin{aligned}
|f(z)| &\le& \int_{-\infty}^\infty e^{|t||y|}|u(t)|\,dM(t)=\sum_{n=0}^\infty \frac{|y|^n}{n!}\int_{-\infty}^\infty |t|^n|u(t)|\,dM(t)\\
&\le& \sum_{n=0}^\infty \frac{|y|^n}{n!} \sqrt{q_{2n}}||u||_{L^2(M)}<\infty,\end{aligned}$$ because $(\sqrt{q_{2n}}/n!)^{1/n}\to 0$ by Stirling’s formula and the assumption on the moments. In particular $\root{2n}\of{q_{2n}}\le Kn$ for a suitable constant, and therefore the Carleman condition $$\sum_{n=0}^\infty \frac{1}{\root{2n}\of{q_{2n}}}=\infty$$ secures that the moment problem is determinate, cf. [@Ak].
The function $f$ is considered in [@Y formula (2.6)] as a $C^{\infty}$-function on the real line, and it is claimed that it is equal to its Taylor series. This need not be the case under the assumptions in [@Y], but holds true in the present case. Therefore the argument in Yafaev’s paper can be carried through. $\quad\square$
Additional results
==================
We use the following notation for the orthonormal polynomials . $$\begin{aligned}
P_n(x)&=&b_{n,n}x^n+b_{n-1,n}x^{n-1}+\ldots + b_{1,n}x+b_{0,n},
\label{eq:p}\\
x^n&=&c_{n,n}P_n(x)+c_{n-1,n}P_{n-1}(x)+\ldots + c_{1,n}P_1(x)+c_{0,n}P_0(x).\label{eq:x}\end{aligned}$$
The matrices $\mathcal B=\{b_{i,j}\}$ and $\mathcal C=\{c_{i,j}\}$ with the assumption $$b_{i,j}=c_{i,j}=0\qquad{\rm for} \ i>j$$ are upper-triangular. Since $\mathcal B$ and $\mathcal C$ are transition matrices between two sequences of linearly independent systems of functions, we have $$\label{eq:BC}
\mathcal B\mathcal C=\mathcal C\mathcal B=\mathcal I.$$
Both matrices define operators in $\ell^2$ with domain $\mathcal D$ by defining the image of $e_n\in\mathcal D$ to be the $n$’th column of the matrix. We use the same symbol for these operators as their matrices.
In the following we assume the moment problem to be indeterminate. In this case $\mathcal B$ extends to a bounded operator on $\ell^2$ which is Hilbert-Schmidt by [@B:S1 Proposition 4.2]. We denote it here $\overline{{\mathcal B}}$, since it is the closure of $\mathcal B$. We know that $\overline{{\mathcal B}}$ is one-to-one by [@B:S1 Proposition 4.3], and then it is easy to see that $\mathcal C$ is closable and $$\label{eq:oC}
\operatorname{dom}(\overline{\mathcal C})=\overline{\mathcal B}(\ell^2),\quad\overline{\mathcal C}=\overline{\mathcal B}^{-1}.$$
\[thm:2\] Suppose $M$ is indeterminate. Then the set $\mathcal D _*$ is dense in $L^2(M)$.
For $u\in {\mathbb{C}}[x]^\perp$ we have $$u_n = \int_{-\infty}^\infty u(x) x^n\,dM(x)=0,\quad n=0,1,\ldots,$$ and for $u\in \mathcal E$ given by we find $$\begin{aligned}
u_n &=& \int_{-\infty}^\infty u(x) x^n\,dM(x)=\sum_{k=0}^\infty g_k\int_{-\infty}^\infty P_k(x)x^n\,dM(x)\\
&=&\sum_{k=0}^n c_{k,n}g_k = (\mathcal C^t g)_n,\end{aligned}$$ where we have used .
By the orthogonal decomposition we find $$\mathcal D_*=\left\{u=\sum_{k=0}^\infty g_kP_k \mid g\in\ell^2,\;\mathcal C^t g\in\ell^2\right\}\oplus {\mathbb{C}}[x]^\perp,$$ so $\mathcal D_*$ is dense in $L^2(M)$ if and only if $$X:=\{g\in\ell^2 \mid \mathcal C^t g\in\ell^2\} \;\mbox{ is dense in }\; \ell^2.$$
However, $\{\mathcal B^t\eta \mid \eta\in\mathcal D\}\subset X$ and the subset is already dense in $\ell^2$.
In fact, for $\eta\in\mathcal D$ we have $\mathcal B^t \eta\in\ell^2$ because the matrix $\mathcal B$ is Hilbert-Schmidt. Furthermore, $\mathcal C^t(\mathcal B^t \eta)=\eta\in \ell^2$ because of .
Finally, since $\overline{\mathcal B}$ is a bounded operator and one-to-one on $\ell^2$, the set $\{\mathcal B^t\eta\, \mid \,\eta\in \mathcal D\}$ is dense in $\ell^2$.
By Theorem \[thm:indet\] we know that the operator $A$ given by is closable, when $M$ is indeterminate. We shall now describe the closure $\overline{A}$ in this case. For this we need the unitary operator $U:\ell^2\to\mathcal E$ given by $U(e_n)=P_n,\,n=0,1,\ldots$.
\[thm:oA\] Suppose $M$ is indeterminate. Then $$\label{eq:oA}
\operatorname{dom}(\overline{A})=\overline{\mathcal B}(\ell^2), \quad\overline{A}=U\overline{\mathcal C}.$$ For $\xi\in\operatorname{dom}(\overline{A})$ we have $\xi=\overline{\mathcal B}y$ for a unique $y\in\ell^2$ and the following series expansions hold $$\label{eq:oA1}
\overline{A}\xi(z)=\sum_{k=0}^\infty \xi_k z^k=\sum_{n=0}^\infty y_nP_n(z),\quad z\in{\mathbb{C}},$$ uniformly for $z$ in compact subsets of ${\mathbb{C}}$.
We clearly have $A=U\mathcal C$, hence $\overline{A}=U\overline{\mathcal C}$, and therefore $\operatorname{dom}(\overline{A})=\operatorname{dom}(\overline{\mathcal C})=\overline{\mathcal B}(\ell^2)$.
For $\xi=\overline{\mathcal B}y$ for $y\in\ell^2$, we have $\overline{A}\xi=Uy$ and $$f(z):=Uy(z)=\sum_{n=0}^\infty y_nP_n(z),$$ uniformly for $z$ in compact subsets of ${\mathbb{C}}$. By Cauchy’s integral formula we therefore get $$\begin{aligned}
\frac{f^{(k)}(0)}{k!}&=&\frac{1}{2\pi i}\int_{|z|=1}\frac{f(z)}{z^{k+1}}\,dz
=\sum_{n=0}^\infty y_n\frac{1}{2\pi i}\int_{|z|=1}\frac{P_n(z)}{z^{k+1}}\,dz\\
&=&\sum_{n=0}^\infty y_nb_{k,n}=\xi_k.\end{aligned}$$ This shows the first expression in .
We end with an example related to Yafaev’s condition (iii).
Let $M$ be a positive measure on $[-1,1]$ with $M(\{1\})=c>0$. The operator $A$ is not closable.
In fact, define $$g^{(n)}_k=\begin{cases} 1/n, \quad & 0\le k\le n-1,\\
0, \quad & k\ge n.
\end{cases}$$ Then $g^{(n)}\to 0$ in $\ell^2$. We have, $$Ag^{(n)}(x)=\begin{cases} 1, \quad & x=1,\\
\frac{1}{n}\frac{1-x^n}{1-x}, \quad & -1\le x<1.
\end{cases}$$ Hence $Ag^{(n)}(x)\to \chi_{1}(x)$ pointwise and also in $L^2(M)$, where $\chi_B$ denotes the indicator function of a subset $B$ of the real line. Thus $A$ is not closable.
[12]{} N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis. English translation, Oliver and Boyd, Edinburgh, 1965.
C. Berg and J. P. R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier [**31**]{}, no. 3 (1981), 99–114.
C. Berg and A. J. Dur[á]{}n, The index of determinacy for measures and the $\ell^2$-norm of orthonormal polynomials. Trans. Amer. Math. Soc. [**347**]{} (1995), 2795–2811.
C. Berg and A. J. Dur[á]{}n, When does a discrete differential perturbation of a sequence of orthonormal polynomials belong to $\ell^2$? Journal of Functional Analysis [**136**]{} (1996), 127–153.
C. Berg and R. Szwarc, The smallest eigenvalue of Hankel matrices, Constr. Approx. [**34**]{} (2011), 107–133.
C. Berg and R. Szwarc, Inverse of infinite Hankel moment matrices, SIGMA [**14**]{} (2018), 109, 48 pages
D. R. Yafaev, Unbounded Hankel operators and moment problems, Integr. Equ. Oper. Theory [**85**]{} (2016), 289–300. DOI 10.1007/s00020-016-2289-y.
Christian Berg\
Department of Mathematical Sciences, University of Copenhagen\
Universitetsparken 5, DK-2100 Copenhagen, Denmark\
e-mail:
Ryszard Szwarc\
Institute of Mathematics, University of Wroc[ł]{}aw\
pl. Grunwaldzki 2/4, 50-384 Wroc[ł]{}aw, Poland\
e-mail:
|
---
abstract: |
Let $(\ger g,\ger k)$ be a reductive symmetric superpair of even type, *i.e.* so that there exists an even Cartan subspace $\ger a\subset\ger p$. The restriction map $S(\ger p^*)^{\ger k}\to S(\ger a^*)^W$ where $W=W(\ger g_0:\ger a)$ is the Weyl group, is injective. We determine its image explicitly.
In particular, our theorem applies to the case of a symmetric superpair of group type, *i.e.* $(\ger k\oplus\ger k,\ger k)$ with the flip involution where $\ger k$ is a classical Lie superalgebra with a non-degenerate invariant even form (equivalently, a finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new proof of the generalisation of Chevalley’s restriction theorem due to Sergeev and Kac, Gorelik.
For general symmetric superpairs, the invariants exhibit a new and surprising behaviour. We illustrate this phenomenon by a detailed discussion in the example $\ger g=C(q+1)=\ger{osp}(2|2q,{\mathbb{C}})$, endowed with a special involution. Here, the invariant algebra defines a singular algebraic curve.
author:
- 'Alexander Alldridge, Joachim Hilgert, and Martin R. Zirnbauer'
bibliography:
- 'ahz-chevalley.bib'
title: |
Chevalley’s restriction theorem\
for reductive symmetric superpairs
---
Introduction
============
The physical motivation for the development of supermanifolds stems from quantum field theory in its functional integral formulation, which describes fermionic particles by anticommuting fields. In the 1970s, pioneering work by Berezin strongly suggested that commuting and anticommuting variables should be treated on equal footing. Several theories of supermanifolds have been advocated, among which the definition of Berezin, Kostant, and Leites is one of the most commonly used in mathematics.
Our motivation for the study of supermanifolds comes from the study of certain nonlinear $\sigma$-models with supersymmetry. Indeed, it is known from the work of the third named author [@zirnbauer-rsss] that Riemannian symmetric superspaces occur naturally in the large $N$ limit of certain random matrix ensembles, which correspond to Cartan’s ten infinite series of symmetric spaces. In spite of their importance in physics, the mathematical theory of these superspaces is virtually non-existent. (But compare [@duflo-petracci-formalss; @zirnbauer-superbos; @goertsches-diss].) We intend to initiate the systematic study of Riemannian symmetric superspaces, in order to obtain a good understanding of, in particular, the invariant differential operators, the spherical functions, and the related harmonic analysis. The present work lays an important foundation for this endeavour: the generalisation of Chevalley’s restriction theorem to the super setting.
To describe our results in detail, let us make our assumptions more precise. Let $\ger g$ be a complex Lie superalgebra with even centre [such that ]{}$\ger g_0$ is reductive in $\ger g$ and $\ger g$ carries an even invariant supersymmetric form. Let $\theta$ be an involutive automorphism of $\ger g$, and denote by $\ger g=\ger k\oplus\ger p$ the decomposition into $\theta$-eigenspaces. We say that $(\ger g,\ger k)$ is a *reductive superpair*, and it is of *even type* if there exists an even Cartan subspace $\ger a\subset\ger p_0$.
Assume that $(\ger g,\ger k)$ is a reductive symmetric superpair of even type. Let $\bar\Sigma_1^+$ denote the set of positive roots of $\ger g_1:\ger a$ [such that ]{}$\lambda,2\lambda$ are no roots of $\ger g_0:\ger a$. To each $\lambda\in\bar\Sigma_1^+$, one associates a set $\mathcal R_\lambda$ of differential operators with rational coefficients on $\ger a$.
Our main results are as follows.
Let $I(\ger a^*)$ be the image of the restriction map $S(\ger p^*)^{\ger k}\to S(\ger a^*)$ (which is injective). Then $I(\ger a^*)$ is the set of $W$-invariant polynomials on $\ger a$ which lie in the common domain of all operators in $\mathcal R_\lambda$, $\lambda\in\bar\Sigma_1^+$. Here, $W$ is the Weyl group of $\ger g_0:\ger a$.
For $\lambda\in\bar\Sigma_1^+$, let $A_\lambda\in\ger a$ be the corresponding coroot, and denote by $\partial(A_\lambda)$ the directional derivative operator in the direction of $A_\lambda$. Then the image $I(\ger a^*)$ can be characterised in more explicit terms, as follows.
We have $I(\ger a^*)=\bigcap_{\lambda\in\bar\Sigma_1^+}S(\ger a^*)^W\cap I_\lambda$ where $$I_\lambda=\textstyle\bigcap_{j=1}^{\frac12m_{1,\lambda}}\operatorname{\mathrm{dom}}\lambda^{-j}\partial(A_\lambda)^j{\quad\text{{if}}\quad}\lambda(A_\lambda)=0\ ,$$ and if $\lambda(A_\lambda)\neq0$, then $I_\lambda$ consists of those $p\in{\mathbb{C}}[\ger a]$ [such that ]{} $$\partial(A_\lambda)^kp|_{\ker\lambda}=0{\quad\text{{for all }{odd integers} }} k\ ,\ 1{\leqslant}k{\leqslant}m_{1,\lambda}-1\ .$$ Here, $m_{1,\lambda}$ denotes the multiplicity of $\lambda$ in $\ger g_1$ (and is an even integer).
If the symmetric pair $(\ger g,\ger k)$ is of *group type*, *i.e.* $\ger g=\ger k\oplus\ger k$ with the flip involution, then [for all ]{}$\lambda\in\bar\Sigma_1^+$, $\lambda(A_\lambda)=0$, and the multiplicity $m_{1,\lambda}=2$. In this case, Theorem (B) reduces to $I(\ger a^*)=\bigcap_{\lambda\in\bar\Sigma_1^+}S(\ger a^*)^W\cap\operatorname{\mathrm{dom}}\lambda^{-1}\partial(A_\lambda)$. The situation where $\lambda(A_\lambda)\neq0$ [for some ]{}$\lambda\in\bar\Sigma_1^+$ occurs if and only if $\ger g$ contains symmetric subalgebras $\ger s\cong C(2)=\ger{osp}(2|2)$ where $\ger s_0\cap\ger k=\operatorname{\ger{sl}}(2,{\mathbb{C}})$. This is case for $\ger g=C(q+1)$ with a special involution, and in this case, the invariant algebra $I(\ger a^*)$ defines the singular curve $z^{2q+1}=w^2$ ().
Let us place our result in the context of the literature. The Theorems (A) and (B) apply to the case of classical Lie superalgebras with non-degenerate invariant even form (equivalently, finite-dimensional contragredient Lie superalgebras), considered as symmetric superspaces of group type. In this case, the result is due to Sergeev [@sergeev-invpol], Kac [@kac-laplace], and Gorelik [@gorelik-kacconstruction], and we simply furnish a new (and elementary) proof. (The results of Sergeev are also valid for basic Lie superalgebras which are not contragredient.) For some particular cases, there are earlier results by Berezin [@berezin].
Sergeev’s original proof involves case-by-case calculations. The proof by Gorelik—which carries out in detail ideas due to Kac in the context of Kac–Moody algebras—is classification-free, and uses so-called Shapovalov determinants. Moreover, the result of Kac and Gorelik actually characterises the image of the Harish-Chandra homomorphism rather than the image of the restriction map on the symmetric algebra, and is therefore more fundamental than our result.
Still in the case of symmetric superpairs of group type, Kac [@kac-typical] and Santos [@santos-zuckerman] describe the image of the restriction morphism in terms of supercharacters of certain (cohomologically) induced modules (instead of a characterisation in terms of a system of differential equations). This approach cannot carry over to the case of symmetric pairs, as is known in the even case from the work of Helgason [@helgason-fundamental].
Our result also applies in the context of Riemannian symmetric superspaces, where one has an even non-degenerate $\mathcal G$-invariant supersymmetric form on $\mathcal G/\mathcal K$ whose restriction to the base $G/K$ is Riemannian. In this setting, it is to our knowledge completely new and not covered by earlier results. We point out that a particular case was proved in the PhD thesis of Fuchs [@fuchs-diss], in the framework of the ‘supermatrix model’, using a technique due to Berezin.
In the context of harmonic analysis of even Riemannian symmetric spaces $G/K$, Chevalley’s restriction theorem enters crucially, since it determines the image of the Harish-Chandra homomorphism, and thereby, the spectrum of the algebra $\mathbb D(G/K)$ of $G$-invariant differential operators on $G/K$. It is an important ingredient in the proof of Harish-Chandra’s integral formula for the spherical functions. In a series of forthcoming papers, we will apply our generalisation of Chevalley’s restriction theorem to obtain analogous results in the context of Riemannian symmetric superspaces.
Let us give a brief overview of the contents of our paper. We review some basic facts on root decompositions in sections 2.1-2.2. In section 2.3, we introduce our main tool in the proof of Theorem (A), a certain twisted action $u_z$ on the supersymmetric algebra $S(\ger p)$. In section 3.1, we define the ‘radial component’ map $\gamma_z$ via the twisted action $u_z$. The proofs of Theorems (A) and (B) are contained in sections 3.2 and 3.3, respectively. The former comes down to a study of the singularities of $\gamma_z$ as a function of the semi-simple $z\in\ger p_0$, whereas the latter consists in an elementary and explicit discussion of the radial components of certain differential operators. In sections 4.1 and 4.2, we discuss the generality of the ‘even type’ condition, and study an extreme example in some detail.
The first named author wishes to thank C. Torossian (Paris VII) for his enlightening remarks on a talk given on an earlier version of this paper. The first and second named author wish to thank M. Duflo (Paris VII) for helpful discussions, comments, and references. The second named author wishes to thank K. Nishiyama (Kyoto) for several discussions on the topic. Last, not least, we wish to thank an anonymous but diligent referee whose suggestions greatly improved the presentation of our main technical device.
This research was partly funded by the IRTG “Geometry and Analysis of Symmetries”, supported by Deutsche Forschungsgemeinschaft (DFG), Ministère de l’Éducation Nationale (MENESR), and Deutsch-Französische Hochschule (DFH-UFA), and by the SFB/Transregio 12 “Symmetry and Universality in Mesoscopic Systems”, supported by Deutsche Forschungsgemeinschaft (DFG).
Some basic facts and definitions
================================
In this section, we mostly collect some basic facts concerning (restricted) root decompositions of Lie superalgebras, and the (super-) symmetric algebra, along with some definitions which we find useful to formulate our main results. As general references for matters super, we refer the reader to [@kostant-supergeom; @deligne-morgan-susy; @kac-liesuperalgs; @scheunert-liesuperalgs]
Roots of a basic quadratic Lie superalgebra
-------------------------------------------
Let $\ger g=\ger g_0\oplus\ger g_1$ be a Lie superalgebra over ${\mathbb{C}}$ and $b$ a bilinear form $b$. Recall that $b$ is *supersymmetric* if $b(u,v)=(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{u}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{v}\r@m{}\r@r\rvert}
}}b(v,u)$ [for all ]{}homogeneous $u,v$. We shall call $(\ger g,b)$ *quadratic* if $b$ is a non-degenerate, $\ger g$-invariant, even and supersymmetric form on $\ger g$. We shall say that $\ger g$ is *basic* if $\ger g_0$ is reductive in $\ger g$ (*i.e.* $\ger g$ is a semi-simple $\ger g_0$-module) and $\ger z(\ger g)\subset\ger g_0$ where $\ger z(\ger g)$ denotes the centre of $\ger g$.
Let $(\ger g,b)$ be a basic quadratic Lie superalgebra, and $\ger b$ be a Cartan subalgebra of $\ger g_0$.
As usual [@scheunert-liesuperalgs Chapter II, § 4.6], we define $$V^\alpha={ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx10 \def\r@m{} \fi \r@l\{}{x\in V}{|}{\exists\,n\in{\mathbb{N}}\,:\,(h-\alpha(h))^n(x)=0\ \text{for all }h\in\ger b}{\}{}}\r@m\quad{,}\r@r\quad}\alpha\in\ger b^*$$ for any $\ger b$-module $V$. Further, the sets of even resp. odd roots for $\ger b$ are $$\Delta_0(\ger g:\ger b)
={ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx10 \def\r@m{} \fi \r@l\{}{\alpha\in\ger b^*\setminus0}{|}{\ger g_0^\alpha\neq0}{\}{}}\r@m{\quad\text{{{and}}}\quad}{\Delta}\r@r_}1(\ger g:\ger b)
={ \Size{1}{\{}{\alpha\in\ger b^*}{|}{\ger g_1^\alpha\neq0}{\}}}\ .$$ We also write $\Delta_j=\Delta_j(\ger g:\ger b)$. Let $\Delta=\Delta(\ger g:\ger b)=\Delta_0\cup\Delta_1$. The elements of $\Delta$ are called *roots*. We have $$\ger g=\ger b\oplus\textstyle\bigoplus_{\alpha\in\Delta}\ger g^\alpha
=\ger b\oplus\bigoplus_{\alpha\in\Delta_0}\ger g_0^\alpha
\oplus\bigoplus_{\alpha\in\Delta_1}\ger g_1^\alpha\ .$$ It is obvious that $\Delta_0=\Delta(\ger g_0:\ger b)$, so in particular, it is a reduced abstract root system in its real linear span. Also, since $\ger g_0$ is reductive in $\ger g$, the root spaces $\ger g_i^\alpha$ are the joint eigenspaces of $\operatorname{\mathrm{ad}}h$, $h\in\ger b$ (and not only generalised ones).
We collect the basic statements about $\ger b$-roots. The results are known (*e.g.* [@scheunert-liesuperalgs; @benayadi-root]), so we omit their proofs.
Let $\ger g$ be a basic quadratic Lie superalgebra with invariant form $b$, and $\ger b$ a Cartan subalgebra of $\ger g_0$.
1. For $\alpha,\beta\in\Delta\cup0$, we have $b(\ger g_j^\alpha,\ger g_k^\beta)=0$ unless $j=k$ and $\alpha=-\beta$.
2. The form $b$ induces a non-degenerate pairing $\ger g_j^\alpha\times\ger g_j^{-\alpha}\to{\mathbb{C}}$. In particular, we have $\dim\ger g_j^\alpha=\dim\ger g_j^{-\alpha}$ and $\Delta_j=-\Delta_j$ for $j\in{\mathbb{Z}}/2{\mathbb{Z}}$.
3. The form $b$ is non-degenerate on $\ger b$, so for any $\lambda\in\ger b^*$, there exists a unique $h_\lambda\in\ger b$ [such that ]{}$b(h_\lambda,h)=\lambda(h)$ [for all ]{}$h\in\ger b$.
4. If $\alpha(h_\alpha)\neq0$, $\alpha\in\Delta_1$, then $2\alpha\in\Delta_0$. In particular, $\Delta_0\cap\Delta_1={\varnothing}$.
5. We have $\ger g_1^0=\ger z_1(\ger g)=\{x\in\ger g_1\mid[x,\ger g]=0\}=0$, so $0\not\in\Delta_1$.
6. All root spaces $\ger g^\alpha$, $\alpha\in\Delta$, $\alpha(h_\alpha)\neq0$, are one-dimensional.
Restricted roots of a reductive symmetric superpair
---------------------------------------------------
Let $(\ger g,b)$ be a complex quadratic Lie superalgebra, and $\theta:\ger g\to\ger g$ an involutive automorphism leaving the form $b$ invariant. If $\ger g=\ger k\oplus\ger p$ is the $\theta$-eigenspace decomposition, then we shall call $(\ger g,\ger k)$ a *symmetric superpair*. We shall say that $(\ger g,\ger k)$ is *reductive* if, moreover, $\ger g$ is basic.
Note that for any symmetric superpair $(\ger g,\ger k)$, $\ger k$ and $\ger p$ are $b$-orthogonal and non-degenerate. It is also useful to consider the form $b^\theta(x,y)=b(x,\theta y)$ which is even, supersymmetric, non-degenerate and $\ger k$-invariant.
Let $(\ger g,\ger k)$ be a reductive symmetric superpair. For arbitrary subspaces $\ger c,\ger d\subset\ger g$, let $\ger z_{\ger d}(\ger c)={ \Size{0}{\{}{d\in\ger d}{|}{[d,\ger c]=0}{\}}}$ denote the centraliser of $\ger c$ in $\ger d$. Any linear subspace $\ger a=\ger z_{\ger p}(\ger a)\subset\ger p_0$ consisting of semi-simple elements of $\ger g_0$ is called an *even Cartan subspace*. If an even Cartan subspace exists, then we say that $(\ger g,\ger k)$ is of *even type*.
We state some generalities on even Cartan subspaces. These are known and straightforward to deduce from standard texts such as [@dixmier-envalg; @borel-rss].
Let $\ger a\subset\ger g$ be an even Cartan subspace.
1. $\ger a$ is reductive in $\ger g$, *i.e.* $\ger g$ is a semi-simple $\ger a$-module.
2. $\ger z_{\ger g_0}(\ger a)$ and $\ger z_{\ger g_1}(\ger a)$ are $b$-non-degenerate.
3. $\ger z_{\ger g_0}(\ger a)=\ger m_0\oplus\ger a$ and $\ger z_{\ger g_1}(\ger a)=\ger m_1$ where $\ger m_i=\ger z_{\ger k_i}(\ger a)$, and the sum is $b$-orthogonal.
4. $\ger m_0$, $\ger m_1$, and $\ger a$ are $b$-non-degenerate.
5. There exists a $\theta$-stable Cartan subalgebra $\ger b$ of $\ger g_0$ containing $\ger a$.
Let $\ger k$ be a classical Lie superalgebra with a non-degenerate invariant even form $B$ [@kac-repnclassical]. Then $\ger k_0$ is reductive in $\ger k$, and $\ger z(\ger k)$ is even. We may define $\ger g=\ger k\oplus\ger k$, and $b(x,y,x',y')=B(x,x')+B(y,y')$. Then $(\ger g,b)$ is basic quadratic. The flip involution $\theta(x,y)=(y,x)$ turns $(\ger g,\ger k)$ into a reductive symmetric superpair (where $\ger k$ is, as is customary, identified with the diagonal in $\ger g$). We call such a pair of *group type*.
Moreover, any Cartan subalgebra $\ger a$ of $\ger k_0$ yields an even Cartan subspace for the superpair $(\ger g,\ger k)$. Indeed, $\ger p={ \Size{1}{\{}{(x,-x)}{|}{x\in\ger k}{\}}}$, and the assertion follows from (v).
In what follows, let $(\ger g,\ger k)$ be a reductive symmetric superpair of even type, $\ger a\subset\ger p$ an even Cartan subspace, and $\ger b\subset\ger g_0$ a $\theta$-stable Cartan subalgebra containing $\ger a$. The involution $\theta$ acts on $\ger b^*$ by $\theta\alpha=\alpha\circ\theta$ [for all ]{}$\alpha\in\ger b^*$. Let $\alpha_\pm=\frac12(1\pm\theta)\alpha$ [for all ]{}$\alpha\in\ger b^*$, and set $$\Sigma_j=\Sigma_j(\ger g:\ger a)={ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx10 \def\r@m{} \fi \r@l\{}{\alpha_-}{|}{\alpha\in\Delta_j\ ,\ \alpha\neq\theta\alpha}{\}{}}\r@m\ {,}\r@r\ }\Sigma=\Sigma(\ger g:\ger a)=\Sigma_0\cup\Sigma_1\ .$$ (The union might not be disjoint.) Identifying $\ger a^*$ with the annihilator of $\ger b\cap\ger k$ in $\ger b^*$, these may be considered as subsets of $\ger a^*$. The elements of $\Sigma_0$, $\Sigma_1$, and $\Sigma$ are called *even restricted roots*, *odd restricted roots*, and *restricted roots*, respectively. For $\lambda\in\Sigma$, let $$\Sigma_j(\lambda)={ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx10 \def\r@m{} \fi \r@l\{}{\alpha\in\Delta_j}{|}{\lambda=\alpha_-}{\}{}}\r@m\ {,}\r@r\ }\Sigma(\lambda)=\Sigma_0(\lambda)\cup\Sigma_1(\lambda)\ .$$ In the following lemma, observe that $\lambda\in\Sigma_j(\lambda)$ means that $\lambda\in\Delta_j$. We omit the simple proof, which is exactly the same as in the even case [@warner-vol1 Chapter 1.1, Appendix 2, Lemma 1].
Let $\lambda\in\Sigma_j$, $j=0,1$. The map $\alpha\mapsto-\theta\alpha$ is a fixed point free involution of $\Sigma_j(\lambda)\setminus\lambda$. In particular, the cardinality of this set is even.
For $\lambda\in\Sigma$, let $$\ger g_{j,\ger a}^\lambda={ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx10 \def\r@m{} \fi \r@l\{}{x\in\ger g_j}{|}{\forall h\in\ger a\,:\,[h,x]=\lambda(h)\cdot x}{\}{}}\r@m\ {,}\r@r\ }\ger g_{\ger a}^\lambda=\ger g_{0,\ger a}^\lambda\oplus\ger g_{1,\ger a}^\lambda\ ,$$ and $m_{j,\lambda}=\dim_{\mathbb{C}}\ger g^\lambda_{j,\ger a}$, the *even* or *odd multiplicity* of $\lambda$, according to whether $j=0$ or $j=1$. It is clear that $$\textstyle\ger g_{j,\ger a}^\lambda=\bigoplus_{\alpha\in\Sigma_j(\lambda)}\ger g^\alpha_j\ ,\ m_{j,\lambda}=\sum_{\alpha\in\Sigma_j(\lambda)}\dim_{\mathbb{C}}\ger g_j^\alpha\ ,{\ \text{{and}}\ }\ger g=\ger z_{\ger g}(\ger a)\oplus\bigoplus_{\lambda\in\Sigma}\ger g^\lambda_{\ger a}\ .$$
The following facts are certainly well-known. Lacking a reference, we give the short proof.
Let $\alpha,\beta\in\Delta$, $\lambda\in\Sigma$, and $j,k\in\{0,1\}$.
1. The form $b^\theta$ is zero on $\ger g^\alpha_j\times\ger g_k^\beta$, unless $j=k$ and $\alpha=-\theta\beta$, in which case it gives a non-degenerate pairing.
2. There exists a unique $A_\lambda\in\ger a$ [such that ]{}$b(A_\lambda,h)=\lambda(h)$ [for all ]{}$h\in\ger a$.
3. We have $\dim_{\mathbb{C}}\ger g_j^\alpha=\dim_{\mathbb{C}}\ger g_j^{-\theta\alpha}$.
4. The subspace $\ger g_j(\lambda)=\ger g_{j,\ger a}^\lambda\oplus\ger g_{j,\ger a}^{-\lambda}$ is $\theta$-invariant and decomposes into $\theta$-eigenspaces as $\ger g_j(\lambda)=\ger k_j^\lambda\oplus\ger p_j^\lambda$.
5. The odd multiplicity $m_{1,\lambda}$ is even, and $b^\theta$ defines a symplectic form on both $\ger k_1^\lambda$ and $\ger p_1^\lambda$.
The form $b^\theta$ is even, so $b^\theta(\ger g_0,\ger g_1)=0$. For $x\in\ger g^\alpha_j$, $y\in\ger g^\beta_j$, we compute, [for all ]{}$h\in\ger b$, $$\begin{aligned}
(\alpha+\theta\beta)(h)b^\theta(x,y)&=b^\theta([h,x],y)+b^\theta(x,[\theta h,y])\\
&=b^\theta([h,x]+[x,h],y)=0\ .
\end{aligned}$$ Hence, $b^\theta(x,y)=0$ if $\alpha\neq-\theta\beta$. Since $b^\theta$ is non-degenerate and $\ger g/\ger b$ is the sum of root spaces, $b^\theta$ induces a non-degenerate pairing of $\ger g_j^\alpha$ and $\ger g_j^{-\theta\alpha}$. We also know already that $\ger a$ is non-degenerate for $b^\theta$, and (i)-(iii) follow. Statement (iv) is immediate.
We have $$\ger g_{1,\ger a}^\lambda/\ger g_1^\lambda\cong\textstyle\bigoplus_{\alpha\in\Sigma_j(\lambda)\setminus\lambda}\ger g_1^\alpha\ .$$ By (iii) and , this space is even-dimensional. But $\lambda$ is a root if and only if $\lambda=-\theta\lambda$. Then $b^\theta$ defines a symplectic form on $\ger g_1^\lambda$ by (i), and this space is even-dimensional. Thus, $m_{1,\lambda}$ is even, and again by (i), $\ger g_{1,\ger a}^\lambda$ is $b^\theta$-non-degenerate. It is clear that $\ger k_1^\lambda$ and $\ger p_1^{\lambda}$ are $b^\theta$-non-degenerate because $\ger g_{1,\ger a}^\lambda$ and $\ger g_{1,\ger a}^{-\lambda}$ are. Hence, we obtain assertion (v).
Unlike the case of unrestricted roots, there may exist $\lambda\in\Sigma_1$ [such that ]{}$2\lambda\not\in\Sigma$ but $\lambda$ is still anisotropic, *i.e.* $\lambda(A_\lambda)\neq0$. Indeed, consider $\ger g=\ger{osp}(2|2,{\mathbb{C}})$ ($\cong\operatorname{\ger{sl}}(2|1,{\mathbb{C}})$). Then $\ger g_0=\ger o(2,{\mathbb{C}})\oplus\ger{sp}(2,{\mathbb{C}})=\ger{gl}(2,{\mathbb{C}})$ and $\ger g_1$ is the sum of the fundamental representation of $\ger g_0$ and its dual.
Define the involution $\theta$ to be conjugation by the element $\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi\sigma&0\\0&1_2\makeatother$ where $\sigma=\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&1\\1&0\makeatother$. One finds that $\ger k_0=\operatorname{\ger{sl}}(2,{\mathbb{C}})$ and $\ger p_0=\ger a=\ger z(\ger g_0)$ which is one-dimensional and non-degenerate for the supertrace form $b$. On the other hand, $\ger g_1=\ger g_1(\lambda)$ is the sum of the root spaces for certain odd roots $\pm\alpha$, $\pm\theta\alpha$ which restrict to $\pm\lambda$. Clearly, there are no even roots, so $2\lambda$ is not a restricted root. Since $A_\lambda$ generates $\ger a$, it is a $b$-anisotropic vector. We discuss this issue at some length in section 4.2.
We point out that it is also not hard to prove that any such root $\lambda$ occurs in this setup. *I.e.*, given a reductive symmetric superpair $(\ger g,\ger k)$, for any $\lambda\in\Sigma_1$, $2\lambda\not\in\Sigma$, $\lambda(A_\lambda)\neq0$, there exists a $b$-non-degenerate $\theta$-invariant subalgebra $\ger s\cong\ger{osp}(2|2,{\mathbb{C}})$ [such that ]{}$\ger p\cap\ger s_0={\mathbb{C}}A_\lambda=\ger z(\ger s_0)$ (the centre of $\ger s_0$), and $\dim\ger s\cap\ger g_1(\lambda)=4$.
This phenomenon, of course, cannot occur if the symmetric superpair $(\ger g,\ger k)$ is of group type. This reflects the fact that the conditions characterising the invariant algebra may be different in the general case than one might expect from the knowledge of the group case (*i.e.* the theorems of Sergeev and Kac, Gorelik).
The twisted action on the supersymmetric algebra
------------------------------------------------
Let $V=V_0\oplus V_1$ be a finite-dimensional super-vector space over ${\mathbb{C}}$. We define the supersymmetric algebra $S(V)=S(V_0)\otimes\bigwedge(V_1)$. It is ${\mathbb{Z}}$-graded by total degree, as follows: $S^{k,\mathrm{tot}}(V)=\bigoplus_{p+q=k}S^p(V_0)\otimes\bigwedge^q(V_1)$. This grading is not compatible with the ${\mathbb{Z}}_2$-grading, but will of be of use to us nonetheless.
Let $U$ be another finite-dimensional super-vector space, and moreover, let $b:U\times V\to{\mathbb{C}}$ be a bilinear form. Then $b$ extends to a bilinear form $S(U)\times S(V)\to{\mathbb{C}}$: It is defined on linear generators by $$b(x_1\dotsm x_m,y_1\dotsm y_n)=\delta_{mn}\cdot\sum\nolimits_{\sigma\in\ger S_n}\alpha^\sigma_{x_1,\dotsc,x_n}\cdot b(x_{\sigma(1)},y_1)\dotsm b(x_{\sigma(n)},y_n)$$ [for all ]{}$x_1,\dotsc,x_m\in U$, $y_1,\dotsc,y_n\in V$ where $\alpha=\alpha^\sigma_{x_1,\dotsc,x_n}=\pm1$ is determined by the requirement that $\alpha\cdot x_{\sigma(1)}\dotsm x_{\sigma(n)}=x_1\dotsm x_n$ in $S(V)$. If $b$ is even (resp. odd, resp. non-degenerate), then so is its extension. Here, recall that a bilinear form has degree $i$ if $b(V_j,V_k)=0$ whenever $i+j+k\equiv1\ (2)$.
In particular, the natural pairing of $V$ and $V^*$ extends to a non-degenerate even pairing ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{\cdot}\r@m,{\cdot}\r@r\rangle}}$ of $S(V)$ and $S(V^*)$. By this token, $S(V)$ embeds injectively as a subsuperspace in $\widehat S(V)=S(V^*)^*$. Its image coincides with the graded dual $S(V^*)^{*\mathrm{gr}}$ whose elements are the linear forms vanishing on $S^{k,\mathrm{tot}}(V^*)$ for $k\gg1$.
We define a superalgebra homomorphism $\partial:S(V)\to\End0{\widehat S(V^*)}$ by $${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{p}\r@m,{\partial(q)\pi}\r@r\rangle}}={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{pq}\r@m,{\pi}\r@r\rangle}}{\quad\text{{for all }{{}} }}p,q\in S(V)\,,\,\pi\in S(V)^*$$ where $\widehat S(V^*)=S(V)^*$. Clearly, $\partial(q)$ leaves $S(V^*)$ invariant.
If $U$ is an even finite-dimensional vector space over ${\mathbb{C}}$, then we have the well-known isomorphism $S(U^*)\cong{\mathbb{C}}[U]$ as algebras, where ${\mathbb{C}}[U]$ is the set of polynomial mappings $U\to{\mathbb{C}}$. We recall that the isomorphism can be written down as follows.
The pairing ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{\cdot}\r@m,{\cdot}\r@r\rangle}}$ of $S(U)$ and $S(U^*)$ extends to $\widehat S(U)\times S(U^*)$. For any $d\in S(U)$, the exponential $e^d=\sum_{n=0}^\infty\frac{d^n}{n!}$ makes sense as an element of the algebra $\widehat S(U)=\prod_{n=0}^\infty S^n(U)$. Now, define a map $S(U^*)\to{\mathbb{C}}[U]:p\mapsto P$ by $$P(z)={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{e^z}\r@m,{p}\r@r\rangle}}=\textstyle\sum_{n=0}^\infty\frac1{n!}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z^n}\r@m,{p}\r@r\rangle}}=\sum_{n=0}^\infty\frac1{n!}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{1}\r@m,{\partial(z)^np}\r@r\rangle}}\ .$$ Observe $$\tfrac d{dt}P(z_0+tz)\big|_{t=0}=\tfrac d{dt}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{e^{tz}e^{z_0}}\r@m,{p}\r@r\rangle}}\big|_{t=0}={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{ze^{z_0}}\r@m,{p}\r@r\rangle}}\ .$$ Iterating this formula, we obtain ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z_1\dotsm z_n}\r@m,{p}\r@r\rangle}}$ for any $z_j\in U$ as a repeated directional derivative of $P$, and the map is injective. Since it preserves the grading by total degree, it is bijective because of identities of dimension in every degree.
Let $V=V_0\oplus V_1$ be a finite-dimensional super-vector space. We apply the above to define an isomorphism $\phi:S(V^*)\to\Hom[_{S(V_0)}]0{S(V),{\mathbb{C}}[V_0]}$. Here, $S(V_0)$ acts on $S(V)$ by left multiplication, and it acts on ${\mathbb{C}}[V_0]$ by natural extension of the action of $V_0$ by directional derivatives: $$(\partial_zP)(z_0)=\tfrac d{dt}P(z_0+tz)\big|_{t=0}{\quad\text{{for all }{{}} }}P\in{\mathbb{C}}[V_0]\,,\,z,z_0\in V_0\ .$$ The isomorphism $\phi$ is given by the following prescription for $P=\phi(p)$: $$P(d;z)=(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{d}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p}\r@m{}\r@r\rvert}
}}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{e^z}\r@m,{\partial(d)p}\r@r\rangle}}{\quad\text{{for all }{{}} }}p\in S(V^*)\,,\,z\in V_0\,,\,d\in S(V)\ .$$ Here, note that $\widehat S(V_0)\subset\widehat S(V)$ since $S(V^*_0)$ is a direct summand of $S(V^*)$, $S(V^*)=S(V_0^*)\oplus S(V_0^*)\otimes\bigwedge^+(V_1^*)$, where $\bigwedge^+=\bigoplus_{k{\geqslant}1}\bigwedge^k$. Hence, $e^z$ may be considered as an element of $\widehat S(V)$.
The map $\phi$ is an isomorphism as the composition of the isomorphisms $$\begin{aligned}
\Hom[_{S(V_0)}]0{S(V),{\mathbb{C}}[V_0]}&\cong\Hom[_{S(V_0)}]0{S(V_0)\otimes\textstyle\bigwedge V_1,S(V_0^*)}\\
&\cong\textstyle S(V_0^*)\otimes\bigwedge V_1^*\cong S(V^*)\ .
\end{aligned}$$
Let $(\ger g,\ger k)$ be a reductive symmetric superpair of even type, and $\ger a\subset\ger p$ an even Cartan subspace. We apply the isomorphism $\phi$ for $V=\ger p$ to define natural *restriction homomorphisms* $$S(\ger p^*)\to S(\ger p_0^*):p\mapsto\bar p{{\quad\text{{{and}}}\quad}}S(\ger p^*)\to S(\ger a^*):p\mapsto\bar p\ .$$ Here, $\bar p\in S(\ger p_0^*)$ (resp. $\bar p\in S(\ger a^*)$) is defined via its associated polynomial $\bar P\in{\mathbb{C}}[\ger p_0]$ (resp. $\bar P\in{\mathbb{C}}[\ger a]$) where $$\bar P(z)=P(1;z){{\quad\text{{{and}}}\quad}}P=\phi(p)\ .$$ This is a convention we will adhere to in all that follows.
Since $\ger p_0$ is complemented by $\ger p_1$ in $\ger p$, and $\ger a$ is complemented in $\ger p_0$ by $\bigoplus_{\lambda\in\Sigma_0}\ger p_0^\lambda$, we will in the sequel consider $\ger p_0^*\subset\ger p^*$ and $\ger a^*\subset\ger p_0^*$.
Let $K$ be a connected Lie group with Lie algebra $\ger k_0$ [such that ]{}the restricted adjoint representation $\operatorname{\mathrm{ad}}:\ger k_0\to\End0{\ger g}$ lifts to a homomorphism $\operatorname{\mathrm{Ad}}:K\to\operatorname{\mathrm{GL}}(\ger g)$. (For instance, one might take $K$ simply connected.) Then $\ger k$ (resp. $K$) acts on $S(\ger p)$, $S(\ger p^*)$, $\widehat S(\ger p)$, $\widehat S(\ger p^*)$ by suitable extensions of $\operatorname{\mathrm{ad}}$ and $\operatorname{\mathrm{ad}}^*$ (resp. $\operatorname{\mathrm{Ad}}$ and $\operatorname{\mathrm{Ad}}^*$) which we denote by the same symbols. Here, the sign convention for $\operatorname{\mathrm{ad}}^*$ is $${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{y}\r@m,{\operatorname{\mathrm{ad}}^*(x)\eta}\r@r\rangle}}={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{[y,x]}\r@m,{\eta}\r@r\rangle}}=-(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{x}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{y}\r@m{}\r@r\rvert}
}}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{\operatorname{\mathrm{ad}}(x)(y)}\r@m,{\eta}\r@r\rangle}}$$ [for all ]{}$x,y\in\ger g$, $\eta\in\ger g^*$.
Let $z\in\ger p_0$. We have $e^z=\sum_{k=0}^\infty\frac{z^k}{k!}\in\widehat S(\ger p)$, and this element is invertible with inverse $e^{-z}$. Define $$u_z(x)d=\operatorname{\mathrm{ad}}(x)(de^z)e^{-z}{\quad\text{{for all }{{}} }}x\in\ger k\,,\,d\in\widehat S(\ger p)\ .$$ Observe that $$\operatorname{\mathrm{ad}}(x)(e^z)=\textstyle\sum_{n=0}^\infty\frac1{n!}\operatorname{\mathrm{ad}}(x)(z^n)=\sum_{n=1}^\infty\frac n{n!}[x,z]z^{n-1}=[x,z]e^z\ ,$$ because $z$ is even. Hence, $$u_z(x)d=\operatorname{\mathrm{ad}}(x)(de^z)e^{-z}=[x,z]d+\operatorname{\mathrm{ad}}(x)(d)\ .$$ In particular, $u_z(x)$ leaves $S(\ger p)\subset\widehat S(\ger p)$ invariant.
Let $z\in\ger p_0$. Then $u_z$ defines a $\ger k$-module structure on $S(\ger p)$, and [for all ]{}$x\in\ger k$, $k\in K$, we have $$\operatorname{\mathrm{Ad}}(k)\circ u_z(x)=u_{\operatorname{\mathrm{Ad}}(k)(z)}(\operatorname{\mathrm{Ad}}(k)(x))\circ\operatorname{\mathrm{Ad}}(k)\ .$$
We clearly have $$u_z(x)u_z(y)d=(\operatorname{\mathrm{ad}}(x)\operatorname{\mathrm{ad}}(y)(de^z))e^{-z}\ .$$ Now $u_z$ is a $\ger k$-action because $\operatorname{\mathrm{ad}}$ is a homomorphism. Similarly, $$\begin{aligned}
\operatorname{\mathrm{Ad}}(k)(u_z(x)d)&=\operatorname{\mathrm{ad}}(\operatorname{\mathrm{Ad}}(k)(x))(\operatorname{\mathrm{Ad}}(k)(d)e^{\operatorname{\mathrm{Ad}}(k)(z)})e^{-\operatorname{\mathrm{Ad}}(k)(z)}\\
&=u_{\operatorname{\mathrm{Ad}}(k)(z)}(\operatorname{\mathrm{Ad}}(k)(x))\operatorname{\mathrm{Ad}}(k)(d)\ ,
\end{aligned}$$ which manifestly gives the second assertion.
Let $u_z$ also denote the natural extension of $u_z$ to $\Uenv0{\ger k}$. Then we may define an action $\ell$ of $\Uenv0{\ger k}$ on $\Hom[_{S(\ger p_0)}]0{S(\ger p),{\mathbb{C}}[\ger p_0]}$ via $$(\ell_vP)(d;z)=(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{v}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{P}\r@m{}\r@r\rvert}
}}P(u_z(S(v))d;z)$$ [for all ]{}$P\in\Hom[_{S(\ger p_0)}]0{S(\ger p),{\mathbb{C}}[\ger p_0]}$, $v\in\Uenv0{\ger k}$, $d\in S(\ger p)$, $z\in\ger p_0$. Here, we denote by $S:\Uenv0{\ger g}\to\Uenv0{\ger g}$ the unique linear map [such that ]{}$S(1)=1$, $S(x)=-x$ [for all ]{}$x\in\ger g$, and $S(uv)=(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{u}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{v}\r@m{}\r@r\rvert}
}}S(v)S(u)$ [for all ]{}homogeneous $u,v\in\Uenv0{\ger g}$ (*i.e.* the principal anti-automorphism). Compare [@koszul-superaction] for a similar definition in the context of the action of a supergroup on its algebra of superfunctions.
We also define $$(L_kP)(d;z)=P(\operatorname{\mathrm{Ad}}(k^{-1})(d);\operatorname{\mathrm{Ad}}(k^{-1})(z))$$ [for all ]{}$P\in\Hom[_{S(\ger p_0)}]0{S(\ger p),{\mathbb{C}}[\ger p_0]}$, $k\in K$, $d\in S(\ger p)$, $z\in\ger p_0$.
The map $\ell$ (resp. $L$) defines on $\Hom[_{S(\ger p_0)}]0{S(\ger p),{\mathbb{C}}[\ger p_0]}$ the structure of a module over $\ger k$ (resp. $K$) making the isomorphism $\phi$ equivariant for $\ger k$ (resp. $K$).
Let $P=\phi(p)$. Then $$\begin{aligned}
(\ell_xP)(d;z)&=-(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{x}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p}\r@m{}\r@r\rvert}
}}P(u_z(x)d;z)=-(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{d}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p}\r@m{}\r@r\rvert}
}}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{\operatorname{\mathrm{ad}}(x)(e^zd)}\r@m,{p}\r@r\rangle}}\\
&=(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{d}\r@m{}\r@r\rvert}
}({ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{x}\r@m{}\r@r\rvert}
}+{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p}\r@m{}\r@r\rvert}
})}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{e^zd}\r@m,{\operatorname{\mathrm{ad}}^*(x)(p)}\r@r\rangle}}=\phi{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\operatorname{\mathrm{ad}}^*(x)(p)}\r@m{}\r@r)}
}(d;z)\ .
\end{aligned}$$
Similarly, we check that $$\begin{aligned}
(L_kP)(d;z)&=P(\operatorname{\mathrm{Ad}}(k^{-1})(d);\operatorname{\mathrm{Ad}}(k^{-1})(z))\\
&=(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{d}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p}\r@m{}\r@r\rvert}
}}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{e^{\operatorname{\mathrm{Ad}}(k^{-1})(z)}\operatorname{\mathrm{Ad}}(k^{-1})(d)}\r@m,{p}\r@r\rangle}}\\
&=(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{d}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p}\r@m{}\r@r\rvert}
}}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{\operatorname{\mathrm{Ad}}(k^{-1})(e^zd)}\r@m,{p}\r@r\rangle}}=\phi{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\operatorname{\mathrm{Ad}}^*(k)(p)}\r@m{}\r@r)}
}(z;d)\ .
\end{aligned}$$ This proves our assertion.
Chevalley’s restriction theorem
===============================
The map $\gamma_z$
------------------
From now on, let $(\ger g,\ger k)$ be a reductive symmetric superpair of even type, and let $\ger a\subset\ger p_0$ be an even Cartan subspace.
An element $z\in\ger p_0$ is called *oddly regular* whenever the map $\operatorname{\mathrm{ad}}(z):\ger k_1\to\ger p_1$ is surjective. Recall that $z\in\ger p_0$ is called *regular* if $\dim\ger z_{\ger k_0}(z)=\dim\ger z_{\ger k_0}(\ger a)$. We shall call $z$ *super-regular* if it is both regular and oddly regular.
Fix an even Cartan subspace $\ger a$, and let $\Sigma$ be the set of (both odd and even) restricted roots. Let $\Sigma^+\subset\Sigma$ be any subset such that $\Sigma$ is the disjoint union of $\pm\Sigma^+$. Define $\Sigma_j^\pm=\Sigma_j\cap\Sigma^\pm$ for $j\in{\mathbb{Z}}/2{\mathbb{Z}}$. Let $\bar\Sigma_1$ be the set of $\lambda\in\Sigma_1$ [such that ]{}$m\lambda\not\in\Sigma_0$ for $m=1,2$. Denote $\bar\Sigma_1^+=\bar\Sigma_1\cap\Sigma^+$. Note that $\Pi_1\in S(\ger a^*)^W$ where $\Pi_1(h)=\prod_{\lambda\in\Sigma_1}\lambda(h)$, and $W$ is the Weyl group of $\Sigma_0$.
By Chevalley’s restriction theorem, restriction $S(\ger p_0^*)^{\ger k_0}\to S(\ger a^*)^W$ is a bijective map. Let $\Pi_1$ also denote the unique extension to $S(\ger p^*_0)^{\ger k_0}$ of $\Pi_1$.
The space $\ger p_0$ contains non-semi-simple elements, and the definitions we have given above work in this generality. However, it will suffice for our purposes to consider the set of *semi-simple* super-regular elements in $\ger p_0$, by the following reasoning.
First, the set of semi-simple elements in $\ger p_0$ is Zariski dense (a linear endomorphism is semi-simple if and only if its minimal polynomial has only simple zeros). Second, the set of semi-simple elements in $\ger p_0$ equals $\operatorname{\mathrm{Ad}}(K)(\ger a)$ [@helgason-geoman Chapter III, Proposition 4.16]. Thus, given any *semi-simple* $z\in\ger p_0$, $z$ is oddly regular (super-regular) if and only if $\lambda(\operatorname{\mathrm{Ad}}(k)(z))\neq0$ [for all ]{}$\lambda\in\Sigma_1$ ($\lambda\in\Sigma$), and for some (any) $k\in K$ [such that ]{}$\operatorname{\mathrm{Ad}}(k)(z)\in\ger a$. In particular, the set of super-regular elements of $\ger a$ is the complement of a finite union of hyperplanes. Hence, the set of semi-simple super-regular elements of $\ger p_0$ is non-void and therefore Zariski dense; in particular, this holds for the set of semi-simple oddly regular elements.
If $z\in\ger p_0$ is semi-simple, then $\ger k_i=\ger z_{\ger k_i}(z)\oplus[z,\ger p_i]$, and the subspaces $\ger z_{\ger k_i}(z)$ and $[z,\ger p_i]$ are $b$-non-degenerate.
Since $\operatorname{\mathrm{ad}}z$ is a semi-simple endomorphism of $\ger g$ ($\ger g$ is a semi-simple $\ger g_0$-module and $z$ is semi-simple), we have $\ger g_i=\ger z_{\ger g_i}(z)\oplus[z,\ger g_i]$. Taking $\theta$-fixed parts, we deduce $\ger k_i=\ger z_{\ger k_i}(z)\oplus\ger[z,\ger p_i]$. The summands, being $b$-orthogonal, are non-degenerate.
Let $z\in\ger p_0$ be semi-simple and oddly regular. Let $\beta:S(\ger g)\to\Uenv0{\ger g}$ be the supersymmetrisation map. Let $$\Gamma_z:\textstyle\bigwedge(\ger p_1)\otimes S(\ger p_0)\to S(\ger p):q\otimes p\mapsto u_z{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\beta([z,q])}\r@m{}\r@r)}
}p$$ on elementary tensors and extend linearly.
Let $z$ be oddly regular and semi-simple. Then $\Gamma_z$ is bijective, and $\gamma_z=({\varepsilon}\otimes 1)\circ\Gamma_z^{-1}:S(\ger p)\to S(\ger p_0)$ satisfies $$\gamma_{\operatorname{\mathrm{Ad}}(k)(z)}\circ\operatorname{\mathrm{Ad}}(k)=\operatorname{\mathrm{Ad}}(k)\circ\gamma_z{\quad\text{{for all }{{}} }}k\in K\ .$$ Here ${\varepsilon}:\bigwedge(\ger p_1)\to{\mathbb{C}}$ is the unique unital algebra homomorphism.
Moreover, on $S^{m,\mathrm{tot}}(\ger p)$, $\Pi_1(z)^m\gamma_z$ is polynomial in $z$, *i.e.* it extends to an element $\Pi_1(\cdot)^m\gamma_\cdot$ of the space ${\mathbb{C}}[\ger p_0]\otimes\Hom0{S^{m,\mathrm{tot}}(\ger p),S(\ger p_0)}$.
By the assumption on $z$, $\operatorname{\mathrm{ad}}z:\ger p_1\to[z,\ger p_1]$ is bijective. Moreover, $\Gamma_z$ respects the filtrations by total degree, and the degrees of these filtrations are equidimensional by the assumption. Hence, $\Gamma_z$ will be bijective once it is surjective. In degree zero, $\Gamma_z$ is the identity. We proceed to prove the surjectivity in higher degrees by induction.
By assumption, $\operatorname{\mathrm{ad}}z:[z,\ger p_1]\to\ger p_1$ is also bijective (since its kernel is $\ger z_{\ger k_1}(z)\cap[z,\ger p_1]$, which is $0$ by ). Let $y_1,\dotsc,y_m\in\ger p_1$, $y_1',\dotsc,y_n'\in\ger p_0$. Let $x_j\in\ger p_1$ [such that ]{}$[[z,x_j],z]=y_j$. We find $$\Gamma_z(x_1\dotsm x_m\otimes y_1'\dotsm y_n')\equiv y_1\dotsm y_my_1'\dotsm y_n'\quad{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\textstyle\bigoplus\nolimits_{k<m+n}S^{k,\mathrm{tot}}(\ger p)}\r@m{}\r@r)}
}\ ,$$ so the first assertion follows by induction.
As to the covariance property, observe first that we have the identity $\operatorname{\mathrm{Ad}}(k)([z,\ger p_1])=[\operatorname{\mathrm{Ad}}(k)(z),\ger p_1]$. Moreover, $$\begin{aligned}
(\operatorname{\mathrm{Ad}}(k)\circ\gamma_z)(\Gamma_z(v\otimes d))&={\varepsilon}(v)\operatorname{\mathrm{Ad}}(k)(d)={\varepsilon}(\operatorname{\mathrm{Ad}}(k)(v))\operatorname{\mathrm{Ad}}(k)(d)\\
&=\gamma_{\operatorname{\mathrm{Ad}}(k)(z)}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\Gamma_{\operatorname{\mathrm{Ad}}(k)(z)}(\operatorname{\mathrm{Ad}}(k)(v)\otimes\operatorname{\mathrm{Ad}}(k)(d))}\r@m{}\r@r)}
}\\
&=\gamma_{\operatorname{\mathrm{Ad}}(k)(z)}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({u_{\operatorname{\mathrm{Ad}}(k)(z)}(\operatorname{\mathrm{Ad}}(k)(\beta([z,v])))\operatorname{\mathrm{Ad}}(k)(d)}\r@m{}\r@r)}
}\\
&=\gamma_{\operatorname{\mathrm{Ad}}(k)(z)}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\operatorname{\mathrm{Ad}}(k)(u_z(\beta([z,v]))(d))}\r@m{}\r@r)}
}\\
&=(\gamma_{\operatorname{\mathrm{Ad}}(k)(z)}\circ\operatorname{\mathrm{Ad}}(k))(\Gamma_z(v\otimes d))
\end{aligned}$$ [for all ]{}$v\in\bigwedge(\ger p_1)$ and $d\in S(\ger p_0)$, by .
To show that $\Pi_1(z)^m\gamma_z:S^{m,\mathrm{tot}}(\ger p)\to S(\ger p_0)$ is given by the restriction of a polynomial function, we remark that its domain of definition—the set $U$ of semi-simple oddly regular elements in $\ger p_0$—is Zariski dense. We need only prove that $f:U\to\Hom0{\ger p_1,\ger k_1}$, $f(z)=\Pi_1(z)(\operatorname{\mathrm{ad}}z)^{-1}$, is polynomial in $z$, where we consider $(\operatorname{\mathrm{ad}}z)^{-1}:\ger p_1\to[z,\ger p_1]$ as a linear map $\ger p_1\to\ger k_1$.
Thus, let $z\in\ger p_0$ be semi-simple and oddly regular. It is contained in some even Cartan subspace $\ger a$ (say). We have $\ger z_{\ger k_1}(\ger a)=\ger m_1=\ger k_1\cap[z,\ger p_1]^\perp$ by and $(\ger k_1\cap\ger m_1^\perp)\oplus\ger p_1=\bigoplus_{\lambda\in\Sigma_1^+}\ger g_{1,\ger a}^\lambda$. If $x=u+v\in\ger g^\lambda_{1,\ger a}$, and $u\in\ger k_1$, $v\in\ger p_1$, then $[z,u]=\lambda(z)v$. It follows that $\Pi_1(z)(\operatorname{\mathrm{ad}}z)^{-1}$ depends polynomially on $z$, proving our claim.
Let $p\in S(\ger p^*)^{\ger k}$. Then $P(d;z)=P(\gamma_z(d);z)$ [for all ]{}oddly regular and semi-simple $z\in\ger p_0$ and $d\in S(\ger p)$.
Fix an oddly regular $z\in\ger p_0$, and let $x_1,\dotsc,x_n\in\ger p_1$. By , we find for $n>0$ $$P{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\Gamma_z(x_1\dotsm x_n\otimes q);z}\r@m{}\r@r)}
}=(\ell_{\beta([z,x_1\dotsm x_n])}P)(q;z)=0\ .$$ Since $d-\gamma_z(d)\in\Gamma_z(\bigwedge^+(\ger p_1)\otimes S(\ger p_0))$, where $\bigwedge^+(\ger p_1)$ denotes the kernel of ${\varepsilon}:\bigwedge(\ger p_1)\to{\mathbb{C}}$ (*i.e.*, the set of elements without constant term), the assertion follows immediately.
Let $(\ger g,\ger k)$ be a reductive symmetric superpair of even type. The algebra homomorphism $p\mapsto\bar p:I(\ger p^*)=S(\ger p^*)^{\ger k}\to S(\ger p_0^*)$ is injective. In particular, $I(\ger p^*)$ is commutative and purely even.
Let $p\in I(\ger p^*)$. Assume that $\bar p=0$. Let $d\in S(\ger p)$. For all $z\in\ger p_0$ which are oddly regular and semi-simple, $$P(d;z)=P(\gamma_z(d);z)=[\partial_{\gamma_z(d)}\bar P](z)=0\ ,$$ by . It follows that $P(d;-)=0$ on $\ger p_0$, since it is a polynomial. Since $d$ was arbitrary, we have established our contention.
The statement of the Corollary can, of course, be deduced by applying the inverse function theorem for supermanifolds, as in [@sergeev-invpol Proposition 1.1]. Nonetheless, we find it instructive to give the above proof based on the map $\gamma_z$, as it illustrates the approach we will take to determine the image of the restriction map.
Proof of Theorem (A)
--------------------
Let $(\ger g,\ger k)$ be a reductive symmetric superpair of even type, and let $\ger a$ be an even Cartan subspace. We denote by $\ger a'$ the set of super-regular elements of $\ger a$. Let $\mathcal R$ be the algebra of differential operators on $\ger a$ with rational coefficients which are non-singular on $\ger a'$. For any $z\in\ger a'$ and any $D\in\mathcal R$, let $D(z)$ be the local expression of $D$ at $z$. This is defined by the requirement that $D(z)$ be a differential operator with constant coefficients, and $$(Df)(z)=(D(z)f)(z){\quad\text{{for all }{{}} }}z\in\ger a'\ ,$$ and all regular functions $f$.
We associate to $\Sigma\subset\ger a^*$, the restricted root system of $\ger g:\ger a$, the subset $\mathcal R_\Sigma=\bigcup_{\lambda\in\bar\Sigma_1^+}\mathcal R_\lambda\subset\mathcal R$ where $$\mathcal R_\lambda={ \Size{1}{\{}{D\in\mathcal R}{|}{\exists\,d\in S(\ger p_1^\lambda)\colon D(z)=\gamma_z(d)\text{ {for all }}z\in\ger a'}{\}}}\ .$$ *I.e.*, $\mathcal R_\Sigma$ consists of those differential operators which are given as radial parts of operators with constant coefficients on the $\ger p$-projections $\ger p_1^\lambda$ of the restricted root spaces for the $\lambda\in\bar\Sigma_1^+$. For any $D\in\mathcal R$, let the *domain* $\operatorname{\mathrm{dom}}D$ be the set of all $p\in{\mathbb{C}}[\ger a]$ [such that ]{}$Dp\in{\mathbb{C}}[\ger a]$.
As we shall see, the image of the restriction map is the set of $W$-invariant polynomials in the common domain of $\mathcal R_\Sigma$. We will subsequently determine $\mathcal R_\Sigma$ in order to describe this common domain in more explicit terms.
The restriction homomorphism $I(\ger p^*)\to S(\ger a^*)$ from is a bijection onto the subspace $I(\ger a^*)=S(\ger a^*)^W\cap\bigcap_{D\in\mathcal R_\Sigma}\operatorname{\mathrm{dom}}D$.
The *proof* of the Theorem requires a little preparation.
Let $q\in S(\ger p_0^*)^K$, $Q=\phi(q)$, and $z\in\ger p_0$ be super-regular and semi-simple. [For all ]{}$x\in\ger k$, and $w\in S(\ger p)$, we have $$Q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\gamma_z(u_z(x)w);z}\r@m{}\r@r)}
}=0\ .$$
There is no restriction to generality in supposing $z\in\ger a'$, so that $\ger z_{\ger k}(z)=\ger z_{\ger k}(\ger a)=\ger m$ and $\ger z_{\ger k_0}(z)=\ger z_{\ger k_0}(\ger a)=\ger m_0$. We define linear maps $$\gamma_z':S(\ger p_0)\to S(\ger a){{\quad\text{{{and}}}\quad}}\gamma_z'':S(\ger p)\to S(\ger a)$$ by the requirements that $v-\gamma_z'(v)\in u_z(\ger m_0^\perp\cap\ger k_0)(S(\ger p_0))$ [for all ]{}$v\in S(\ger p_0)$ and $w-\gamma_z''(w)\in u_z(\ger m^\perp\cap\ger k)(S(\ger p))$ [for all ]{}$w\in S(\ger p)$. (That such maps exist and are uniquely defined by these properties follows in exactly the same way as for . We remark that $[z,\ger p_i]=\ger k_i\cap\ger m_i^\perp$ by .) Then $$\begin{aligned}
w&-\gamma_z'(\gamma_z(w))=w-\gamma_z(w)+\gamma_z(w)-\gamma_z'(\gamma_z(w))\\
&\in u_z(\ger m_1^\perp\cap\ger k_1)(S(\ger p))+u_z(\ger m_0^\perp\cap\ger k_0)(S(\ger p_0))\subset u_z(\ger m^\perp\cap\ger k)(S(\ger p))
\end{aligned}$$ [for all ]{}$w\in S(\ger p)$, where $\ger m_1=\ger z_{\ger k_1}(\ger a)$. This shows that $\gamma_z''=\gamma_z'\circ\gamma_z$.
Moreover, by the $K$-invariance of $q$, we have $Q(v;z)=Q(\gamma_z'(v);z)$ [for all ]{}$v\in S(\ger p_0)$. We infer $$Q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\gamma_z(u_z(x)w);z}\r@m{}\r@r)}
}=Q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\gamma_z''(u_z(x)w);z}\r@m{}\r@r)}
}=0{\quad\text{{for all }{{}} }}x\in\ger m^\perp\cap\ger k\ ,\ w\in S(\ger p)$$ since $u_z(x)w\in u_z(\ger m^\perp\cap\ger k)(S(\ger p))$ belongs to $\ker\gamma_z''$.
Next, we need to consider the case of $x\in\ger m$. Then $\operatorname{\mathrm{ad}}(x):S(\ger p)\to S(\ger p)$ annihilates the subspace $S(\ger a)$, and moreover, $\operatorname{\mathrm{ad}}(x)(e^z)=0$. From this we find [for all ]{}$y\in\ger m^\perp\cap\ger k$, $d\in S(\ger p)$ $$\begin{aligned}
\operatorname{\mathrm{ad}}(x){ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({u_z(y)(d)}\r@m{}\r@r)}
}&=(\operatorname{\mathrm{ad}}(x)\operatorname{\mathrm{ad}}(y)(de^z))e^{-z}\\
&=(\operatorname{\mathrm{ad}}([x,y])(de^z))e^{-z}+(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{x}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{y}\r@m{}\r@r\rvert}
}}\operatorname{\mathrm{ad}}(y)(\operatorname{\mathrm{ad}}(x)(d)e^z)e^{-z}\\
&=u_z([x,y])d+(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{x}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{y}\r@m{}\r@r\rvert}
}}u_z(y)\operatorname{\mathrm{ad}}(x)(d)\ .
\end{aligned}$$ Since $\ger m$ is a subalgebra and $b$ is $\ger k$-invariant, $\ger m^\perp\cap\ger k$ is $\ger m$-invariant. Hence, the above formula shows that $\ker\gamma_z''=u_z(\ger m^\perp\cap k)(S(\ger p))$ is $\operatorname{\mathrm{ad}}(x)$-invariant.
By the definition of $\gamma_z''$, we find that $$\gamma_z''(\operatorname{\mathrm{ad}}(x)d)=\operatorname{\mathrm{ad}}(x)\gamma_z''(d)=0{\quad\text{{for all }{{}} }}x\in\ger m\,,\,d\in S(\ger p)\ .$$ Reasoning as above, we see that $$Q(\gamma_z(u_z(x)d);z)=Q(\gamma_z(\operatorname{\mathrm{ad}}(x)d);z)=0{\quad\text{{for all }{{}} }}x\in\ger m\,,\,d\in S(\ger p)\ .$$ Since $\ger k=\ger m\oplus(\ger m^\perp\cap\ger k)$, this proves the lemma.
Let $\ger p_0'$ be the set of semi-simple super-regular elements in $\ger p_0$. Recall the polynomial $\Pi_1$, and consider the localisation ${\mathbb{C}}[\ger p_0]_{\Pi_1}$. Let $q\in S(\ger p_0^*)^K$, $Q=\phi(q)$, and define $$P(v;z)=Q(\gamma_z(v);z){\quad\text{{for all }{{}} }}v\in S(\ger p)\ ,\ z\in\ger p_0'\ .$$ By , $P\in\Hom0{S(\ger p),{\mathbb{C}}[\ger p_0]_{\Pi_1}}$. We remark that the $\ger k$-action $\ell$ defined in extends to $\Hom0{S(\ger p),{\mathbb{C}}[\ger p_0]_{\Pi_1}}$, by the same formula.
Retain the above assumptions. Then $P$ is $S(\ger p_0)$-linear and $\ger k$-invariant, *i.e.* $P\in\Hom[_{S(\ger p_0)}]0{S(\ger p),{\mathbb{C}}[\ger p_0]_{\Pi_1}}^{\ger k}$.
By , $P$ is $\ger k$-invariant. It remains to prove that $P$ is $S(\ger p_0)$-linear. To that end, we first establish that $P$ is $K$-equivariant as linear map $S(\ger p)\to{\mathbb{C}}[\ger p_0]_{\Pi_1}$. Since $q$ is $K$-invariant, $$\begin{aligned}
P{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\operatorname{\mathrm{Ad}}(k)(v);\operatorname{\mathrm{Ad}}(k)(z)}\r@m{}\r@r)}
}&=Q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\gamma_{\operatorname{\mathrm{Ad}}(k)(z)}(\operatorname{\mathrm{Ad}}(k)(v));\operatorname{\mathrm{Ad}}(k)(z)}\r@m{}\r@r)}
}\\
&=Q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\operatorname{\mathrm{Ad}}(k)(\gamma_z(v));\operatorname{\mathrm{Ad}}(k)(z)}\r@m{}\r@r)}
}\\
&=Q(\gamma_z(v);z)=P(v;z)\ .
\end{aligned}$$
Next, fix $z\in\ger p_0'$. Then $S(\ger p)=S(\ger p_0)\oplus u_z(\ger z_{\ger k_1}(z)^\perp\cap\ger k_1)(S(\ger p))$ where the second summand equals $\ker\gamma_z$. We may check the $S(\ger p_0)$-linearity on each summand separately.
For $v\in S(\ger p_0)$, we have $P(v;z)=Q(v;z)$, so for any $y\in\ger p_0$ $$[\partial_yP(v;-)](z)=[\partial_yQ(v;-)](z)=Q(yv;z)=P(yv;z)\ .$$
We are reduced to considering $v=u_z(x)v'$ where $x\in\ger z_{\ger k_1}(z)^\perp\cap\ger k_1$ and $v'\in S(\ger p)$. We may assume w.l.o.g. $z\in\ger a$ (since $z$ is semi-simple), so that $\ger z_{\ger k_1}(z)=\ger z_{\ger k_1}(\ger a)=\ger m_1$. By our assumption on $z$, $\ger p_0=\ger a\oplus[\ger k_0,z]$, and we may consider $y$ in each of the two summands separately.
Let $y\in\ger a$. For sufficiently small $t$, we have $z+ty\in\ger a'=\ger a\cap\ger p_0'$, so that $\ger z_{\ger k_1}(z+ty)=\ger m_1=\ger z_{\ger k_1}(z)$. Hence, $\gamma_{z+ty}(u_{z+ty}(x)v')=0$. By the chain rule, $$0=\tfrac d{dt}\gamma_{z+ty}(u_{z+ty}(x)v')\big|_{t=0}=d\gamma_\cdot(v)_z(y)+\gamma_z{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\tfrac d{dt}u_{z+ty}(x)v'\big|_{t=0}}\r@m{}\r@r)}
}\ ,$$ Since $\tfrac d{dt}u_{z+ty}(x)v'\big|_{t=0}=[x,y]v'$, we have $$d\gamma_\cdot(v)_z(y)=-\gamma_z(\tfrac d{dt}u_{z+ty}(x)v'\big|_{t=0})=\gamma_z([y,x]v')\ .$$
Moreover, as operators on $S(\ger p)$, $$[y,u_z(x)]=y[x,z]+y\operatorname{\mathrm{ad}}(x)-[x,z]y-\operatorname{\mathrm{ad}}(x)y=[y,x]\ ,$$ and thus $yv=yu_z(x)v'\equiv[y,x]v'$ modulo $\ker\gamma_z$. We conclude $$d\gamma_\cdot(v)_z(y)=\gamma_z([y,x]v')=\gamma_z(yv)=\gamma_z(yv)-y\gamma_z(v)$$ since $\gamma_z(v)=0$. Hence, $$[\partial_yP(v;-)](z)=Q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({d\gamma_\cdot(v)_z(y)+y\gamma_z(v);z}\r@m{}\r@r)}
}=Q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\gamma_z(yv);z}\r@m{}\r@r)}
}=P(yv;z)\ .$$
Now let $y=[u,z]$ where $u\in\ger k_0$. We may assume that $u\perp\ger z_{\ger k_0}(z)$. Define $k_t=\exp tu$. Then by the $K$-invariance of $P$, $$\begin{aligned}
[\partial_yP(v;-)](z)&=\tfrac d{dt}P{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({v;\operatorname{\mathrm{Ad}}(k_t)(z)}\r@m{}\r@r)}
}\big|_{t=0}
=\tfrac d{dt}P{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\operatorname{\mathrm{Ad}}(k_t^{-1})(v);z}\r@m{}\r@r)}
}\big|_{t=0}\\
&=-P{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\operatorname{\mathrm{ad}}(u)(v);z}\r@m{}\r@r)}
}=P(yv;z)-P{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({u_z(u)v;z}\r@m{}\r@r)}
}=P(yv;z)
\end{aligned}$$ where in the last step, we have used .
The restriction map is injective by and Chevalley’s restriction theorem for $\ger g_0$. By the latter, the image lies in the set of $W$-invariants. Let $\bar p\in S(\ger a^*)$ be the restriction of $p\in I(\ger p^*)$, and $P=\phi(p)$. For any $d\in S(\ger p)$, and $D\in\mathcal R_\Sigma$ given by $D(z)=\gamma_z(d)$, we have by $$(D\bar p)(z)=(\partial_{\gamma_z(d)}\bar P)(z)=P(\gamma_z(d);z)=P(d;z){\quad\text{{for all }{{}} }}z\in\ger a'\ .$$ The result is clearly polynomial in $z$, so $\bar p\in\operatorname{\mathrm{dom}}D$. This shows that the image of the restriction map lies in $I(\ger a^*)$.
Let $r\in I(\ger a^*)$. By Chevalley’s restriction theorem, there exists a unique $q\in I(\ger p_0^*)=S(\ger p_0^*)^K$ [such that ]{}$Q(h)=R(h)$ [for all ]{}$h\in\ger a$.
Next, recall that for $d\in S(\ger p)$ and $z\in\ger p_0'$: $$P(d;z)=Q(\gamma_z(d);z)\ .$$ By , $P\in\Hom[_{S(\ger p_0)}]0{S(\ger p),{\mathbb{C}}[\ger p_0]_{\Pi_1}}^{\ger k}$. Hence, $P$ will define an element $p\in I(\ger p^*)$ by virtue of the isomorphism $\phi$, as soon as it is clear that, as a linear map $S(\ger p)\to{\mathbb{C}}[\ger p_0]_{\Pi_1}$, it takes its values in ${\mathbb{C}}[\ger p_0]$.
We only have to consider $z$ in the Zariski dense set $\ger p_0'$. The function $\Pi_1(z)^k\cdot P(d;z)$ depends polynomially on $z$, where we assume $d\in S^{{\leqslant}k,\mathrm{tot}}(\ger p)$. To prove that $P$ has polynomial values, it will suffice (by the removable singularity theorem and the conjugacy of Cartan subspaces) to prove that $P(d;h)$ is bounded as $h\in\ger a'=\ger a\cap\ger p_0'$ approaches one of the hyperplanes $\lambda^{-1}(0)$ where $\lambda\in\Sigma_1^+$ is arbitrary. Since $r$ is $W$-invariant, $r-r_0$ (where $r_0$ is the constant term of $r$) vanishes on $\lambda^{-1}(0)$ if a multiple of $\lambda$ belongs to $\Sigma_0^+$. Such a multiple could only be $\pm\lambda,\pm2\lambda$. Hence, it will suffice to consider $\lambda\in\bar\Sigma_1^+$. By definition, $2\lambda\not\in\Sigma$.
Consider $P(d;h)$ as a map linear in $d$, and let $N_h=\ker P(-;h)$. Let $d\in S^{{\leqslant}k,\mathrm{tot}}(\ger p)$. Assume that $d=zd'$ where $z$ is defined by $x=y+z$, $y\in\ger k$, $z\in\ger p$, [for some ]{}$x\in\ger g^\mu_{\ger a}$ and $\mu\in\Sigma^+$, $\mu\neq\lambda$. Then, modulo $N_h$, $$d=zd'\equiv zd'+\frac{u_h(y)d'}{\mu(h)}=zd'+\frac{[y,h]d'}{\mu(h)}+\frac{\operatorname{\mathrm{ad}}(y)(d')}{\mu(h)}=\frac{\operatorname{\mathrm{ad}}(y)(d')}{\mu(h)}\ .$$ The root $\mu$ is not proportional to $\lambda$ and the total degree of $\operatorname{\mathrm{ad}}(y)(d')$ is strictly less than that of $d$. By induction, modulo $N_h$, $$d\equiv\frac{\tilde d}{\Pi_{\mu\in\Sigma^+\setminus\lambda}\,\mu(h)^k}$$ [for some ]{}$\tilde d$ which lies in the subalgebra of $S(\ger p)$ generated by $\ger a\oplus\ger p_1^\lambda$, and depends polynomially on $h$ and linearly on $d\in S^{{\leqslant}k,\mathrm{tot}}(\ger p)$.
Hence, the problem of showing that $P(d;h)$ remains bounded as $h$ approaches $\lambda^{-1}(0)$ is reduced to the case of $d\in S(\ger a\oplus\ger p_1^\lambda)$. For $d\in S(\ger p_1^\lambda)$, the polynomiality of $P(d;-)$ immediately follows from the assumption on $r$. If $d=d'd''$ where $d'\in S(\ger a)$ and $d''\in S(\ger p_1^\lambda)$, then $P(d;z)=[\partial(d')P(d'';-)](z)$ since $P$ is $S(\ger p_0)$-linear. But $P(d'';-)\in{\mathbb{C}}[\ger p_0]$ and this space is $S(\ger p_0)$-invariant, so $P(d;-)\in{\mathbb{C}}[\ger p_0]$.
Therefore, there exists $p\in I(\ger p^*)$ [such that ]{}$P=\phi(p)$. By its definition, it is clear that $p$ restricts to $r$, so we have proved the theorem.
Proof of Theorem (B)
--------------------
In order to give a complete description of the image of the restriction map, we need to compute the radial parts $\gamma_h(d)$ for $d\in S(\ger p_1^\lambda)$ and $h\in\ger a'$ explicitly. First, let us choose bases of the spaces $S(\ger p_1^\lambda)$.
Let $\lambda\in\Sigma^+_1$. By (v) we may choose $b^\theta$-symplectic bases $y_i,\tilde y_i\in\ger k_1^\lambda$, $z_i,\tilde z_i\in\ger p_1^\lambda$, $i=1,\dotsc,\frac12m_{1,\lambda}$, $m_{1,\lambda}=\dim\ger g_{1,\ger a}^\lambda$. *I.e.*, $$b(y_i,\tilde y_j)=b(\tilde z_j,z_i)=\delta_{ij}\,,\,b(y_i,y_j)=b(\tilde y_i,\tilde y_j)=b(z_i,z_j)=b(\tilde z_i,\tilde z_j)=0\ .$$ We may impose the conditions $x_i=y_i+z_i,\tilde x_i=\tilde y_i+\tilde z_i\in\ger g_{1,\ger a}^\lambda$, so that $$[h,y_i]=\lambda(h)z_i\,,\,[h,\tilde y_i]=\lambda(h)\tilde z_i\,,\,[h,z_i]=\lambda(h)y_i\,,\,[h,\tilde z_i]=\lambda(h)\tilde y_i$$ [for all ]{}$h\in\ger a$. (Compare (iv).)
Given partitions $I=(i_1<\dotsm<i_k)$, $J=(j_1<\dotsm<j_\ell)$, we define monomials $z_I\tilde z_J=z_{i_1}\dotsm z_{i_k}\tilde z_{j_1}\dotsm\tilde z_{j_\ell}$ in $S(\ger p_1^\lambda)=\bigwedge(\ger p_1^\lambda)$. They form a basis of $S(\ger p_1^\lambda)$.
Fix $\lambda\in\bar\Sigma_1^+$. Let $h\in\ger a$ be oddly regular, $I,J$ be multi-indices where $k={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}$, $\ell={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{J}\r@m{}\r@r\rvert}
}$, and let $m$ be a non-negative integer. Modulo $\ker\gamma_h$, $$z_I\tilde z_JA_\lambda^m\equiv\begin{cases}0&I\neq J\ ,\\
A_\lambda^m & I=J={\varnothing}\ ,\\
(-1)^kz_{I'}\tilde z_{I'}\textstyle\sum_{j=0}^m(-1)^j\tfrac{\lambda(A_\lambda)^j}{\lambda(h)^{j+1}}(m)_jA_\lambda^{m+1-j}&I=J=(i<I')\ ,\end{cases}$$ where $(m)_j$ is the falling factorial $m(m-1)\dotsm(m-j+1)$, and $(m)_0=1$.
For $k=\ell=0$, there is nothing to prove. We assume that $k>0$ or $\ell>0$, and write $I=(i<I')$ if $k>0$, $J=(j<J')$ if $\ell>0$. We claim that modulo $\ker\gamma_h$, $$z_I\tilde z_JA_\lambda^m\equiv\begin{cases}0&k\neq\ell\text{ or }i\neq j\ ,\\
(-1)^kz_{I'}\tilde z_{J'}\textstyle\sum_{n=0}^m(-1)^n\tfrac{\lambda(A_\lambda)^n}{\lambda(h)^{n+1}}(m)_nA_\lambda^{m+1-n}&i=j\ .\end{cases}$$ We argue by induction on $\max(k,\ell)$. There will also be a sub-induction on the integer $m$. First, we assume that $k>0$, and compute $$z_I\tilde z_JA_\lambda^m\equiv z_iz_{I'}\tilde z_JA_\lambda^m+\tfrac1{\lambda(h)}u_h(y_i)(z_{I'}\tilde z_JA_\lambda^m)=\tfrac1{\lambda(h)}\operatorname{\mathrm{ad}}(y_i)(z_{I'}\tilde z_JA_\lambda^m)\ .$$
For any $q$, we have $$b{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({[y_i,z_q],h'}\r@m{}\r@r)}
}=-\lambda(h')b(y_i,y_q)=0{\quad\text{{for all }{{}} }}h'\in\ger a\ ,$$ so $b([y_i,z_q],\ger a)=0$, and $[y_i,z_q]\in\ger p_0$. Hence $[y_i,z_q]\in\ger g_{0,\ger a}^{2\lambda}\oplus\ger g_{0,\ger a}^{-2\lambda}=0$. Similarly, for $i\neq q$, we have $[y_i,\tilde z_q]=0$. Now, assume that $i{\leqslant}J$. Then $$\begin{aligned}
z_I\tilde z_JA_\lambda^m&\equiv(-1)^{k-1}\tfrac1{\lambda(h)}z_{I'}\operatorname{\mathrm{ad}}(y_i)(\tilde z_JA_\lambda^m)\\
&=(-1)^{k-1}\tfrac1{\lambda(h)}[y_i,\tilde z_j]z_{I'}\tilde z_{J'}A_\lambda^m-m\tfrac{\lambda(A_\lambda)}{\lambda(h)}z_I\tilde z_JA_\lambda^{m-1}\tag{$*$}
\end{aligned}$$ since $[y_i,A_\lambda^m]=-m\lambda(A_\lambda)z_iA_\lambda^{m-1}$. As it stands, equation ($*$) only holds for $\ell>0$, but if we take the first summand to be $0$ if $\ell=0$, then it is also true in the latter case.
If $\ell>0$ and $i<J$, then the first summand also vanishes, and arguing by induction on $m$, we find $$z_I\tilde z_JA_\lambda^m\equiv(-1)^mm!\tfrac{\lambda(A_\lambda)^m}{\lambda(h)^m}z_I\tilde z_J=(-1)^{m+k-1}m!\tfrac{\lambda(A_\lambda)^m}{\lambda(h)^{m+1}}[y_i,\tilde z_j]z_{I'}\tilde z_J=0\ .$$ Virtually the same reasoning goes through for $\ell=0$. In particular, whenever $\gamma_h(z_I\tilde z_JA_\lambda^m)\neq0$ and $k>0$, then $i{\leqslant}J$ implies $\ell>0$ and $i=j$.
If $\ell>0$ and $j{\leqslant}I$, then we observe that $z_I\tilde z_J=(-1)^{k\ell}\tilde z_Jz_I$. Formally exchanging the letters $z_s$ and $\tilde z_s$ in the above equations, and reordering all terms in the appropriate fashion, we obtain $$\tag{$**$}
z_I\tilde z_JA_\lambda^m\equiv(-1)^k\tfrac1{\lambda(h)}[\tilde y_j,z_i]z_{I'}\tilde z_{J'}A_\lambda^m-m\tfrac{\lambda(A_\lambda)}{\lambda(h)}z_I\tilde z_JA_\lambda^{m-1}\ ,$$ because $k\ell+\ell-1+(k-1)(\ell-1)=k(2\ell-1)\equiv k\ (2)$. Arguing as above, the right hand side of equation ($**$) is equivalent to $0$ modulo $\ker\gamma_h$ if $k=0$ or $j<I$. Therefore, $\gamma_h(z_I\tilde z_JA_\lambda^m)$ vanishes unless $k,\ell>0$ and $i=j$.
We consider the case of $k,\ell>0$ and $i=j$. Since $[y_i,\tilde z_i]-[\tilde y_i,z_i]=-2A_\lambda$ by standard arguments, we find, by adding equations ($*$) and ($**$), $$z_I\tilde z_JA_\lambda^m\equiv(-1)^k\tfrac1{\lambda(h)}z_{I'}\tilde z_{J'}A_\lambda^{m+1}-m\tfrac{\lambda(A_\lambda)}{\lambda(h)}z_I\tilde z_JA_\lambda^{m-1}\ .$$ We may now apply this formula recursively to the second summand, to conclude $$z_I\tilde z_JA_\lambda^m\equiv(-1)^kz_{I'}\tilde z_{J'}\textstyle\sum_{n=0}^m(-1)^n\tfrac{\lambda(A_\lambda)^n}{\lambda(h)^{n+1}}(m)_nA_\lambda^{m+1-n}\ .$$
By induction on $\max(k,\ell)$, the right hand side belongs to $\ker\gamma_h$ unless $k=\ell$. We have proved our claim, and thus, we arrive at the assertion of the lemma.
Fix $\lambda\in\bar\Sigma_1^+$ and $h\in\ger a'$. Let $I=(i_1<\dotsm<i_k)$ and $1{\leqslant}\ell{\leqslant}k$. Set $I'=(i_{\ell+1}<\dotsm<i_k)$. Let $${\varepsilon}^k_\ell=(-1)^{\sum_{j=k-\ell+1}^kj}=(-1)^{\frac\ell2(2k-\ell+1)}\ .$$ We claim that there are $b_{s\ell}\in{\mathbb{N}}$, $s<\ell$, $b_{01}=1$, such that, modulo $\ker\gamma_h$, $$z_I\tilde z_I\equiv{\varepsilon}^k_\ell z_{I'}\tilde z_{I'}\textstyle\sum_{j=0}^{\ell-1} b_{j\ell}\tfrac{(-\lambda(A_\lambda))^j}{\lambda(h)^{\ell+j}}A_\lambda^{\ell-j}\ .\tag{$***$}$$
The case $\ell=1$ has already been established. To prove the inductive step, let $I''=(i_\ell,\dotsc,i_k)=(i_\ell<I')$, and $J=(i_0<I)$. We compute $$\begin{aligned}
z_J\tilde z_J&\equiv{\varepsilon}_\ell^{k+1}z_{I''}\tilde z_{I''}\textstyle\sum_{j=0}^{\ell-1}b_{j\ell}\tfrac{(-\lambda(A_\lambda))^j}{\lambda(h)^{\ell+j}}A_\lambda^{\ell-j}\\
&\equiv(-1)^{k-\ell+1}{\varepsilon}_\ell^{k+1}z_{I'}\tilde z_{I'}\textstyle\sum_{s=0}^\ell\sum_{j=0}^{\min(s,\ell-1)}(\ell-j)_{s-j}b_{j\ell}\tfrac{(-\lambda(A_\lambda))^s}{\lambda(h)^{\ell+1+s}}A_\lambda^{\ell+1-s}\ ,
\end{aligned}$$ so $$b_{s,\ell+1}=\textstyle\sum_{j=0}^{\min(s,\ell-1)}(\ell-j)_{s-j}b_{j\ell}=\frac1{(\ell-s)!}\sum_{j=0}^{\min(s,\ell-1)}(\ell-j)!b_{j\ell}\ .$$ This proves our claim, where the constants $b_{s\ell}$ obey the recursion relation set out above.
To solve this recursion, we claim that $$b_{s\ell}=\frac{(\ell-1+s)!}{2^s(\ell-1-s)!s!}{\quad\text{{for all }{{}} }}0{\leqslant}s<\ell\ .$$ This is certainly the case for $\ell=1$. By induction, [for all ]{}$0{\leqslant}s{\leqslant}\ell$, $\ell{\geqslant}1$, $$b_{s,\ell+1}=\tfrac1{(\ell-s)!}\textstyle\sum_{j=0}^{\min(s,\ell-1)}(\ell-j)\tfrac{(\ell-1+j)!}{2^jj!}\ .$$ As is easy to show by induction, $\sum_{j=0}^N(\ell-j)\tfrac{(\ell-1+j)!}{2^jj!}=\tfrac{(\ell+N)!}{2^NN!}$. Hence, $$b_{s,\ell+1}=\begin{cases}\frac{(\ell+s)!}{2^s(\ell-s)!s!}&0{\leqslant}s<\ell\\\frac{(2\ell-1)!}{2^{\ell-1}(\ell-1)!}=\frac{(2\ell)!}{2^\ell\ell!} & s=\ell\end{cases}$$ which establishes the claim.
Setting $\ell=k={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}$ in $(***)$, we obtain the following lemma.
Fix $\lambda\in\bar\Sigma_1^+$. Let $h\in\ger a$ be oddly regular, $I$ be a multi-index where $k={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}$. Then $$\gamma_h(z_I\tilde z_I)=(-1)^{\frac{k(k+1)}2}\textstyle\sum_{j=0}^{k-1}\tfrac{(k-1+j)!}{2^j(k-1-j)!j!}\tfrac{(-\lambda(A_\lambda))^j}{\lambda(h)^{k+j}}A_\lambda^{k-j}\ .$$
In passing, note that $b_{k-2,k}=b_{k-1,k}=\frac{(2k-2)!}{2^{k-1}(k-1)!}$. We remark also that $\theta_n(z)=\sum_{j=0}^nb_{j,n+1}z^{n-j}$ are so-called *Bessel polynomials* [@grosswald-besselpol], [@OEIS A001498].
Let $\lambda\in\bar\Sigma_1^+$, $\lambda(A_\lambda)=0$. By , we find [for all ]{}$I$, ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}=k$, that $\gamma_h(z_I\tilde z_I)=(-1)^{\frac12k(k+1)}\lambda(h)^{-k}A_\lambda^k$ ($h\in\ger a'$). Hence, $$\bigcap\nolimits_{D\in\mathcal R_\lambda}\operatorname{\mathrm{dom}}D=\bigcap\nolimits_{k=1}^{\frac12m_{1,\lambda}}\operatorname{\mathrm{dom}}\lambda^{-k}\partial(A_\lambda)^k\ .$$
The situation in the case $\lambda(A_\lambda)\neq0$ is different and requires a more detailed study.
Let $\lambda\in\bar\Sigma_1^+$, $\lambda(A_\lambda)\neq0$. Then ${\mathbb{C}}[\ger a]\cong R[\lambda]$ where $R={\mathbb{C}}[\ker\lambda]$. This isomorphism is equivariant for $S({\mathbb{C}}A_\lambda)$ if we define an action $\partial$ on $R[\lambda]$ by requiring that $\partial(A_\lambda)$ be the unique $R$-derivation for which $\partial(A_\lambda)\lambda=\lambda(A_\lambda)$.
Now, let $R$ be an arbitrary commutative unital ${\mathbb{C}}$-algebra. We define an action $\partial$ of $S({\mathbb{C}}A_\lambda)$ on $R[\lambda,\lambda^{-1}]$ by requiring that $\partial(A_\lambda)$ be the unique $R$-derivation [such that ]{}$\partial(A_\lambda)=\lambda(A_\lambda)$ and $\partial(A_\lambda)\lambda^{-1}=-\lambda(A_\lambda)\lambda^{-2}$. The action $\partial$ is faithful, because $\lambda(A_\lambda)\neq0$.
Let $\mathcal D_\lambda$ be the subalgebra of $\End[_{\mathbb{C}}]0{R[\lambda,\lambda^{-1}]}$ generated by $\partial(S({\mathbb{C}}A_\lambda))$ and ${\mathbb{C}}[\lambda,\lambda^{-1}]$. In particular, we may embed $\mathcal R_\lambda\subset\mathcal D_\lambda$. We consider the action of $D\in\mathcal R_\lambda$, $D(h)=\gamma_h(z_I\tilde z_I)$, ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}=k$, on $p=\sum_{j=0}^Na_j\lambda^j\in R[\lambda]$, $$Dp=(-1)^{\frac{k(k+1)}2}\textstyle\sum_{j=1}^Na_j\lambda(A_\lambda)^k\lambda^{j-2k}\sum_{i=(k-j)_+}^{k-1}(-1)^i(j)_{k-i}b_{ik}\in R[\lambda,\lambda^{-1}]\ .$$ Since $\lambda(A_\lambda)\neq0$, we have $Dp\in R[\lambda]$ if and only if $$a_j\textstyle\sum_{i=(k-j)_+}^{k-1}(-1)^i(j)_{k-i}b_{ik}=0{\quad\text{{for all }{{}} }}j=1,\dotsc,2k-1\ .$$
We need to determine when the number $$\label{eq:ajk-def}
a_{jk}=\sum_{i=(k-j)_+}^{k-1}(-1)^i(j)_{k-i}b_{ik}=\sum_{i=(k-j)_+}^{k-1}{ { \ifx20 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx21 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx22 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx23 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx24 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({-\frac12}\r@m{}\r@r)}
}^i(j)_{k-i}\frac{(k-1+i)!}{(k-1-i)!i!}$$ is non-zero.
Fix $k{\geqslant}1$. For $x\in{\mathbb{R}}$ and $1{\leqslant}j{\leqslant}k$, let $$a_{2k-j,k}(x)=\sum_{i=0}^{k-1}x^i(2k-j)_{k-i}\frac{(k-1+i)!}{(k-1-i)!i!}\ .$$ We claim that $$\label{eq:coeffx}
a_{2k-j,k}(x)=\textstyle\frac{(j-1)!(2k-j)!}{(k-1)!}\sum_{\ell=0}^{j-1}\binom{k-1}\ell\binom{k-1}{j-1-\ell}x^\ell(1+x)^{k-1-\ell}\ .$$ To that end, we rewrite $$a_{2k-j,k}(x)=\frac{(j-1)!(2k-j)!}{(k-1)!}\sum_{i=0}^{k-1}\binom{k-1}i\binom{k+i-1}{j-1}x^i\ .$$ Then, for fixed $x\in{\mathbb{R}}$, we form the generating function $$f(z)=\sum_{j=1}^\infty z^{j-1}\sum_{i=0}^{k-1}\binom{k-1}i\binom{k+i-1}{j-1}x^i\ .$$ It is easy to see $$\begin{aligned}
f(z)&=\sum_{i=0}^{k-1}\binom{k-1}ix^i\sum_{j=1}^{k+i}\binom{k+i-1}{j-1}z^{j-1}\\
&=(1+z)^{2k-2}\sum_{i=0}^{k-1}\binom{k-1}ix^i{ { \ifx20 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx21 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx22 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx23 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx24 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\frac1{1+z}}\r@m{}\r@r)}
}^{k-1-i}\\
&=(1+z)^{k-1}((1+z)x+1)^{k-1}\ .
\end{aligned}$$
On the other hand, we may form the generating function for the right hand side of (\[eq:coeffx\]), $$g(z)=\sum_{j=1}^\infty z^{j-1}\sum_{\ell=0}^{j-1}\binom{k-1}\ell\binom{k-1}{j-1-\ell}x^\ell(1+x)^{k-1-\ell}\ .$$ Then $$\begin{aligned}
g(z)&=\sum_{\ell=0}^{k-1}\binom{k-1}\ell x^\ell(1+x)^{k-1-\ell}\sum_{j=\ell+1}^{k+\ell}\binom{k-1}{j-1-\ell}z^{j-1}\\
&=\sum_{\ell=0}^{k-1}\binom{k-1}\ell(xz)^\ell(1+x)^{k-1-\ell}\sum_{j=0}^{k-1}\binom{k-1}jz^j\\
&=(xz+x+1)^{k-1}(1+z)^{k-1}=f(z)\ .
\end{aligned}$$ Since the generating functions coincide, we have proved (\[eq:coeffx\]).
We notice that for $k{\geqslant}1$ and $j=1,\dotsc,k$, $k-(2k-j)=j-k{\leqslant}0$, so $a_{2k-j,k}=a_{2k-j,k}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({-\tfrac12}\r@m{}\r@r)}
}$ by (\[eq:ajk-def\]). By (\[eq:coeffx\]), we obtain $$a_{2k-j,k}=\frac{(j-1)!(2k-j)!}{2^{k-1}(k-1)!}\sum_{\ell=0}^{j-1}(-1)^\ell\binom{k-1}\ell\binom{k-1}{j-1-\ell}$$ For $j=1$, one gets $$a_{2k-1,k}=\frac{(2k-1)!}{2^{k-1}(k-1)!}\neq0\ .$$ Now, let $j=2n$ where $1{\leqslant}n{\leqslant}\lfloor\tfrac k2\rfloor$. Then $\ell\mapsto(-1)^\ell\binom{k-1}\ell\binom{k-1}{2n-1-\ell}$ is odd under the permutation $\ell\mapsto2n-1-\ell$ of $\{0,\dotsc,2n-1\}$, so $$a_{jk}=0{\quad\text{{for all }{{}} }}j=k,\dotsc,2k-2\ ,\ j\equiv0\ (2)\ .$$
Next, we study the behaviour of $a_{k-j,k}$ for $k{\geqslant}1$ and $j=1,\dotsc,k-1$, by a similar scheme. To that end, write $$\begin{aligned}
a_{k-j,k}&=\sum_{i=j}^{k-1}\frac{(k-j)!(k-1+i)!}{(i-j)!(k-1-i)!i!}{ { \ifx20 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx21 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx22 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx23 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx24 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({-\frac12}\r@m{}\r@r)}
}^i\\
&=\frac{(k-1+j)!(k-j)!}{(k-1)!}\sum_{i=j}^{k-1}\binom{k-1}i\binom{k-1+i}{k-1+j}{ { \ifx20 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx21 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx22 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx23 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx24 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({-\frac12}\r@m{}\r@r)}
}^i\ .
\end{aligned}$$ Observe that we may sum over $i=0,\dotsc,k-1$ since the second binomal coefficient vanishes for $i<j$.
Now, we fix $x\in{\mathbb{R}}$ and define $f(z)=\sum_{j=1}^{k-1}a_{k-j,k}(x)z^{k+j-1}\in{\mathbb{C}}[z]$ where $$a_{k-j,k}(x)=\sum_{i=0}^{k-1}\binom{k-1}i\binom{k-1+i}{k-1+j}x^i\ .$$ We wish to study the coefficients of the polynomial $f$. Observe that the lowest power of $z$ occuring in $f(z)$ is $z^k$. Thus, we compute, modulo ${\mathbb{C}}[z]_{<k}$, $$\begin{aligned}
f(z)&=\sum_{i=0}^{k-1}\binom{k-1}ix^i\sum_{j=1}^i\binom{k-1+i}{k-1+j}z^{k+j-1}\\
&=\sum_{i=0}^{k-1}\binom{k-1}ix^i\sum_{j=k}^{k-1+i}\binom{k-1+i}jz^j\\
&\equiv(1+z)^{k-1}\sum_{i=0}^{k-1}\binom{k-1}i(x(1+z))^i=(1+z)^{k-1}(1+x(1+z))^{k-1}\ .
\end{aligned}$$
For $j=k,\dotsc,2k-2$, $a_{2k-j-1,k}(x)$ is the coefficient of $z^j$ in $f(z)$. Since $$(1+z)^{k-1}(1+x(1+z))^{k-1}=\sum_{j=0}^{2k-2}z^j\sum_{i=0}^j\binom{k-1}{j-i}\binom{k-1}i(1+x)^{k-1-i}x^i\ ,$$ we find, for $j=k,\dotsc,2k-2$, $$\begin{aligned}
a_{2k-j-1,k}(x)&=\sum_{i=0}^j\binom{k-1}{j-i}\binom{k-1}i(1+x)^{k-1-i}x^i\\
&=(1+x)^{k-1}\sum_{i=j-k+1}^{k-1}\binom{k-1}{j-i}\binom{k-1}i{ { \ifx20 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx21 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx22 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx23 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx24 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\frac x{1+x}}\r@m{}\r@r)}
}^i\ .
\end{aligned}$$
In particular, $$a_{2k-j-1,k}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({-\tfrac12}\r@m{}\r@r)}
}=2^{1-k}\sum_{i=j-k+1}^{k-1}(-1)^i\binom{k-1}{j-i}\binom{k-1}i\ .$$ Notice that the function $i\mapsto(-1)^i\binom{k-1}{j-i}\binom{k-1}i$ has parity $j$ with respect to the permutation $i\mapsto j-i$ of $\{j-k+1,\dotsc,k-1\}$. Since $2k-j-1$ is even and only if $j$ is odd, this implies $$a_{jk}=0{\quad\text{{for all }{{}} }}j=2,\dotsc,k-1\ ,\ j\equiv0\ (2)\ .$$
We summarise the above considerations in the following proposition.
Let $R$ be a commutative unital ${\mathbb{C}}$-algebra, and $\lambda\in\bar\Sigma_1^+$ [such that ]{}$\lambda(A_\lambda)\neq0$. Let $m{\geqslant}1$ be an integer, and for $k=1,\dotsc,m$, define $$D_k=(-1)^{\frac{k(k+1)}2}\textstyle\sum_{j=0}^{k-1}\frac{(k-1+j)!}{2^j(k-1-j)!j!}\frac{(-\lambda(A_\lambda))^j}{\lambda^{k+j}}A_\lambda^{k-j}\in \mathcal D_\lambda\ .$$ Let $p=\sum_{j=0}^Na_j\lambda^j\in R[\lambda]$. Then $D_kp\in R[\lambda]$ [for all ]{}$k=1,\dotsc,m$ if and only $a_j=0$ for all $j=1,\dotsc,2m-1$, $j\equiv1\ (2)$.
Let $1{\leqslant}k{\leqslant}m$. We have $a_{2k-1}a_{2k-1,k}=0$ and $a_{2k-1,k}\neq0$, so $a_{2k-1}=0$. Conversely, there are no further conditions, since $a_{km}=0$ for even $k$, $1<k<2m$.
To apply to the determination of the image of the restriction map, let $\lambda\in\bar\Sigma_1^+$, $\lambda(A_\lambda)\neq0$. Note that ${\mathbb{C}}[\ger a]={\mathbb{C}}[\ker\lambda][\lambda]$. Then [for all ]{}$p\in{\mathbb{C}}[\ger a]$, $$p=\textstyle\sum_{j=0}^\infty(j!)^{-1}\partial(A_\lambda)^jp|_{\ker\lambda}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({\frac\lambda{\lambda(A_\lambda)}}\r@m{}\r@r)}
}^j\ .$$ *I.e.*, if we take $R={\mathbb{C}}[\ker\lambda]$, then $p=\sum_ja_j\lambda^j$ where the coefficients are given by $a_j=\frac1{\lambda(A_\lambda)^jj!}\partial(A_\lambda)^jp|_{\ker\lambda}\in R$. Also, $\partial(A_\lambda)^ip|_{\ker\lambda}=0$ [for all ]{}$i=1,\dotsc,j$ if and only if $p\in{\mathbb{C}}\oplus\lambda^{j+1}{\mathbb{C}}[\ger a]$. Together with , we immediately obtain our main result, as follows.
The restriction homomorphism $I(\ger p^*)\to S(\ger a^*)$ is a bijection onto the subspace $I(\ger a^*)=\bigcap_{\lambda\in\bar\Sigma_1^+}S(\ger a^*)^W\cap I_\lambda$ where $$I_\lambda=\textstyle\bigcap_{j=1}^{\frac12m_{1,\lambda}}\operatorname{\mathrm{dom}}\lambda^{-j}\partial(A_\lambda)^j{\quad\text{{if}}\quad}\lambda(A_\lambda)=0$$ and if $\lambda(A_\lambda)\neq0$, then $I_\lambda$ consists of those $p\in{\mathbb{C}}[\ger a]$ [such that ]{} $$\partial(A_\lambda)^kp|_{\ker\lambda}=0{\quad\text{{for all }{odd integers} }} k\ ,\ 1{\leqslant}k{\leqslant}m_{1,\lambda}-1\ .$$
Examples
========
Scope of the theory
-------------------
As remarked in , applies to a symmetric superpair of group type where $\ger k$ is classical and carries a non-degenerate invariant even form. The assumptions are still fulfilled if we add to $\ger k$ an even reductive ideal. Hence, $\ger k$ may be a direct sum of a reductive Lie algebra, and copies of any of the following Lie superalgebras [@kac-liesuperalgs]: $$\begin{gathered}
\ger{gl}(p|q,{\mathbb{C}})\ ,\ \operatorname{\ger{sl}}(p|q,{\mathbb{C}})\ (p\neq q)\ ,\ \operatorname{\ger{sl}}(p|p,{\mathbb{C}})/{\mathbb{C}}\ ,\\
\ger{osp}(p|2q,{\mathbb{C}})\ ,\ D(1,2;\alpha)\ ,\ F(4)\ ,\ G(3)\ .
\end{gathered}$$ As follows from (iv), in this situation one has $\lambda(A_\lambda)=0$ [for all ]{}$\lambda\in\bar\Sigma_1^+$.
If we take $(\ger g,\ger k)$ to be an arbitrary reductive symmetric superpair, then the assumption of *even type* amounts to an additional condition.
As an example, we consider $\ger g=\ger{gl}(p+q|r+s,{\mathbb{C}})$, $p,q,r,s{\geqslant}0$, where $\theta$ is given by conjugation with the diagonal matrix whose diagonal entries are the matrix blocks $1_p$, $-1_q$, $1_r$, $-1_s$. Let $\ger a\subset\ger p_0$ be the maximal Abelian subalgebra of all matrices $$\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&A&0&0\\-A^t&0&0&0\\0&0&0&B\\0&0&-B^t&0\makeatother\in{\mathbb{C}}^{(p+q+r+s)\times(p+q+r+s)}$$ where $A=(D,0)$ or $A=\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fiD\\0\makeatother$ for a diagonal matrix $D\in{\mathbb{C}}^{\min(p,q)\times\min(p,q)}$, and similarly for $B$. Let $x_j$, $j=1,\dotsc,\min(p,q)$, and $y_\ell$, $\ell=1,\dotsc,\min(r,s)$, be the linear forms on $\ger a$ given by the entries of the diagonal blocks of $A,B$.
Consider the $\ger a$-module $\ger g_1$. Then the non-zero weights are $$\pm(x_j\pm y_\ell)\ (2)\ ,\ \pm x_j\ (2{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{r-s}\r@m{}\r@r\rvert}
})\ ,\ \pm y_\ell\ (2{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p-q}\r@m{}\r@r\rvert}
})$$ with multiplicities given in parentheses [@schmittner-zirnbauer]. The sum $U\subset\ger g_1$ of the non-zero weight spaces therefore has dimension $$\begin{aligned}
8\min(p,q)\min(r,s)&+4{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{r-s}\r@m{}\r@r\rvert}
}\min(p,q)+4{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p-q}\r@m{}\r@r\rvert}
}\min(r,s)\\
&=2{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({(p+q)(r+s)-{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p-q}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{r-s}\r@m{}\r@r\rvert}
}}\r@m{}\r@r)}
}\ .
\end{aligned}$$ (The equation follows by applying the formula $2\min(a,b)=a+b-{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{a-b}\r@m{}\r@r\rvert}
}$.)
We have that $U$ is $\theta$-stable, and the action of a generic $h\in\ger a$ induces an automorphism of $U$. Hence, we have $\dim U_{\ger k}=\dim U_{\ger p}=\frac12\dim U$ where $U_{\ger k}$ and $U_{\ger p}$ are the projections of $U$ onto $\ger k_1$ and $\ger p_1$, respectively. It follows that $\dim U_{\ger p}=(p+q)(r+s)-{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p-q}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{r-s}\r@m{}\r@r\rvert}
}$. On the other hand, $$\dim\ger p_1=2(ps+rq)=(p+q)(r+s)-(p-q)(r-s)\ .$$ Hence, $\ger z_{\ger p_1}(\ger a)=0$ if and only if $(p-q)(r-s){\geqslant}0$, and $(\ger g,\ger k)$ is of even type if and only if this condition holds.
We remark that in this case, the set $\bar\Sigma_1^+$ consists of the weights $x_j\pm y_\ell$ (for a suitably chosen positive system). For each $\lambda\in\bar\Sigma_1^+$, one has $\lambda(A_\lambda)=0$.
A similar example arises by restricting the involution from to the subalgebra $\ger g=\ger{osp}(p+q|r+s,{\mathbb{C}})$, where we now assume $r$ and $s$ to be even. We realise $\ger g$ by taking the direct sum of the standard non-degenerate symmetric forms on ${\mathbb{C}}^p\oplus{\mathbb{C}}^q$, and the direct sum of the standard symplectic forms on ${\mathbb{C}}^r\oplus{\mathbb{C}}^s$.
For $k$ even, denote by $J_k\in{\mathbb{C}}^{k\times k}$ the matrix representing the standard symplectic form. Let $\ger a\subset\ger p_0$ be the maximal Abelian subalgebra of all matrices $$\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&A&0&0\\-A^t&0&0&0\\0&0&0&B\\0&0&J_sB^tJ_r&0\makeatother\in{\mathbb{C}}^{(p+q+r+s)\times(p+q+r+s)}$$ where $A=(D,0)$ or $A=\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fiD\\0\makeatother$ for a diagonal matrix $D\in{\mathbb{C}}^{\min(p,q)\times\min(p,q)}$, and $B=(D',0)$ or $B=\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fiD'\\0\makeatother$ for a diagonal matrix $D'\in{\mathbb{C}}^{\frac12\min(r,s)\times\frac12\min(r,s)}$.
By restriction, we obtain the following non-zero $\ger a$-weights in $\ger g_1$, $$\pm(x_j\pm y_\ell)\ (2)\ ,\ \pm x_j\ ({ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{r-s}\r@m{}\r@r\rvert}
})\ ,\ \pm y_\ell\ (2{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p-q}\r@m{}\r@r\rvert}
})\ ,$$ where now $j=1,\dotsc,\min(p,q)\,,\,\ell=1,\dotsc,\tfrac12\min(r,s)$, and the multiplicities are given in parentheses [@schmittner-zirnbauer].
Let $U$ be the sum of all weight spaces for non-zero weights of the $\ger a$-module $\ger g_1$. Then the dimension of $U$ is $$\begin{aligned}
4\min(p,q)\min(r,s)&+2{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{r-s}\r@m{}\r@r\rvert}
}\min(p,q)+2{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p-q}\r@m{}\r@r\rvert}
}\min(r,s)\\
&=(p+q)(r+s)-{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{p-q}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{r-s}\r@m{}\r@r\rvert}
}\ .
\end{aligned}$$ If $U_{\ger p}$ is the projection of $U$ onto $\ger p_1$, then by the same argument as in , $\dim U_{\ger p}=\frac12\dim U$. We have $$\dim\ger p_1=pq+rs=\tfrac12{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({(p+q)(r+s)-(p-q)(r-s)}\r@m{}\r@r)}
}\ ,$$ so, as above, $(\ger g,\ger k)$ is of even type if and only if $(p-q)(r-s){\geqslant}0$. In this case, as in , the set $\bar\Sigma_1^+$ consists of the weights $x_j\pm y_\ell$ (for a suitable choice of positive system), and again we have $\lambda(A_\lambda)=0$ [for all ]{}$\lambda\in\bar\Sigma_1^+$.
An extremal class: $\ger g=C(q+1)=\ger{osp}(2|2q,{\mathbb{C}})$, $\ger k_0=\ger{sp}(2q,{\mathbb{C}})$
-----------------------------------------------------------------------------------------------------
Consider the Lie superalgebra $\ger g=C(q+1)=\ger{osp}(2|2q,{\mathbb{C}})$ where $q{\geqslant}1$ is arbitrary. Let $I=\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&1\\1&0\makeatother\in{\mathbb{C}}^{2\times2}$ and $J=\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&1\\-1&0\makeatother\in{\mathbb{C}}^{2q\times2q}$. If we realise $\ger g$ with respect to the orthosymplectic form $I\oplus J$, it consists of the matrices $$x=\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fia&0&-w^{\prime t}&z^{\prime t}\\0&-a&-w^t&z^t\\ z&z'&A&B\\ w&w'&C&-A^t\makeatother$$ where $a\in{\mathbb{C}}$, $z,z',w,w'\in{\mathbb{C}}^q$, $A,B=B^t,C=C^t\in{\mathbb{C}}^{q\times q}$.
The matrix $g=\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fiI&0\\0&1\makeatother\in{\mathbb{C}}^{(2+2q)\times(2+2q)}$ represents an even automorphism of the super-vector space ${\mathbb{C}}^{2|2q}$, of order $2$. Since $g$ leaves the orthosymplectic form invariant, $\theta(x)=gxg$ defines an involutive automorphism of $\ger g$. Moreover, since $g^2=1$, the supertrace form $b(x,y)=\operatorname{\mathrm{str}}(xy)$ on $\ger g$ is $\theta$-invariant. Hence, $(\ger g,\ger k)$, where $\ger k=\ger g_\theta$, is a reductive symmetric superpair.
We compute $$\theta(x)=\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi-a&0&-w^t&z^t\\0&a&-w^{\prime t}&z^{\prime t}\\z'&z&A&B\\w'&w&C&-A^t\makeatother$$ when $x\in\ger g$ is written as above. Hence, the general elements of $\ger k$ and $\ger p$ are respectively of the form $$x=\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&-w^t&z^t\\0&0&-w^t&z^t\\z&z&A&B\\w&w&C&-A^t\makeatother{{\quad\text{{{and}}}\quad}}x=\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fia&0&w^t&-z^t\\0&-a&-w^t&z^t\\z&-z&0&0\\w&-w&0&0\makeatother\ .$$
It is immediate that the one-dimensional space $\ger a=\ger p_0$ is self-centralising in $\ger p_0$. In particular, any non-zero element of $\ger a$ is $b$-anisotropic (since $\ger p_0$ is non-degenerate). The bracket relation for the general element of $[\ger a,\ger g_1]$ $$\left[\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&a&0&0\\-a&0&0&0\\0&0&0&0\\0&0&0&0\makeatother,\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&-w^{\prime t}&z^{\prime t}\\0&0&-w^t&z^t\\z&z'&0&0\\w&w'&0&0\makeatother\right]=\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&-aw^{\prime t}&az^{\prime t}\\0&0&aw^t&-az^t\\-az&az'&0&0\\-aw&aw'&0&0\makeatother$$ implies in particular that $\ger z_{\ger p_1}(\ger a)=0$. Hence, $\ger a$ is an even Cartan subspace, and $(\ger g,\ger k)$ is of even type.
Also, there are only two restricted roots, $\pm\lambda$, where $\lambda$ maps $x\in\ger a$ (as above) to $a$. Necessarily, $\lambda$ is odd, so $2\lambda\not\in\Sigma=\{\pm\lambda\}$, and $W=W(\Sigma_0)=1$. Since $A_\lambda$ is $b$-anisotropic, we have $\lambda(A_\lambda)\neq0$.
Moreover, we must have $\ger p_1=\ger p_1^\lambda$, and this space has dimension $2q$, so $m_{1,\lambda}=2q$. From , we obtain the following result.
Let $\ger g=\ger{osp}(2|2q,{\mathbb{C}})$, with the involution defined above. The image of the restriction map $S(\ger p^*)^{\ger k}\to S(\ger a^*)={\mathbb{C}}[\lambda]$ is $$I(\ger a^*)={ \Size{1}{\{}{p=\textstyle\sum_ja_j\lambda^j}{|}{a_{2j-1}=0\ \forall j=1,\dotsc,q}{\}}}\ .$$
In particular, the algebra $I(\ger a^*)$ is isomorphic to the commutative unital ${\mathbb{C}}$-algebra defined by the generators $\lambda_2$, $\lambda_{2q+1}$, and the relation $$(\lambda_2)^{2q+1}=(\lambda_{2q+1})^2\ .$$
We only need to prove the presentation of $I(\ger a^*)$. Let $A$ be the unital commutative ${\mathbb{C}}$-algebra defined by the above generators and relations. It is clear that there is a surjective algebra homomorphism from $\phi:A\to I(\ger a^*)$, defined by $\phi(\lambda_n)=\lambda^n$.
Consider on $I(\ger a^*)$ the grading induced by ${\mathbb{C}}[\lambda]$. For any multiindex $\alpha=(\alpha_2,\alpha_{2q+1})$, define $\lambda_\alpha=(\lambda_2)^{\alpha_2}(\lambda_{2q+1})^{\alpha_{2q+1}}$ in the free commutative algebra ${\mathbb{C}}[\lambda_2,\lambda_{2q+1}]$. The latter is graded via ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{\lambda_\alpha}\r@m{}\r@r\rvert}
}={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{\alpha}\r@m{}\r@r\rvert}
}=2\alpha_2+(2q+1)\alpha_{2q+1}$. The relation defining $A$ is homogeneous for this grading, so that $A$ inherits a grading.
By definition, $\phi$ respects the grading, and in fact, it is surjective in each degree of the induced filtration (and hence, in each degree of the grading). The relation of $A$ ensures that the image of $\lambda_\alpha$ in $A$, for any $\alpha$, depends only on ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{\alpha}\r@m{}\r@r\rvert}
}$. Hence, $\dim A_j{\leqslant}1$ [for all ]{}$j$. This proves that $\phi$ is injective.
Under the assumptions of , the algebra $I(\ger a^*)$ defines the singular curve in ${\mathbb{C}}^2$ given by the equation $z^{2q+1}=w^2$.
We substantiate the above by some explicit computations. We have $$\operatorname{\mathrm{str}}\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fia&0&w^t&-z^t\\0&-a&-w^t&z^t\\z&-z&0&0\\w&-w&0&0\makeatother\ifx10 \@smallmattrue \else \@smallmatfalse
\fi \ifx11 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx12 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx13 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx14 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx15 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fia'&0&w'^t&-z'^t\\0&-a'&-w'^t&z'^t\\z'&-z'&0&0\\w'&-w'&0&0\makeatother=2aa'+4(w^tz'-z^tw')$$ for the trace form $b$ on $\ger p=\ger a\oplus\ger p_1^\lambda$. In particular, $$A_\lambda=\tfrac12\left(\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx01 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx02 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx03 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx04 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx05 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi1&0&0&0\\0&-1&0&0\\0&0&0&0\\0&0&0&0\makeatother\right)\ ,\ \lambda(A_\lambda)=\tfrac12\ .$$ Setting $$\begin{gathered}
z_i=\tfrac12\left(\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx01 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx02 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx03 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx04 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx05 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&0&-e_i^t\\0&0&0&e_i^t\\e_i&-e_i&0&0\\0&0&0&0\makeatother\right)\ ,\ \tilde z_i=\tfrac12\left(\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx01 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx02 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx03 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx04 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx05 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&e_i^t&0\\0&0&-e_i^t&0\\0&0&0&0\\e_i&-e_i&0&0\makeatother\right)\ ,\\
y_i=\tfrac12\left(\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx01 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx02 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx03 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx04 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx05 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&0&-e_i^t\\0&0&0&-e_i^t\\-e_i&-e_i&0&0\\0&0&0&0\makeatother\right)\ ,\ \tilde y_i=\tfrac12\left(\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx01 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx02 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx03 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx04 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx05 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&e_i^t&0\\0&0&e_i^t&0\\0&0&0&0\\-e_i&-e_i&0&0\makeatother\right)\ ,
\end{gathered}$$ one verifies the conditions from , namely $$\begin{gathered}
y_i,\tilde y_i\in\ger k_1\,,\,z_i,\tilde z_i\in\ger p_1\,,\,y_i+z_i,\tilde y_i+\tilde z_i\in\ger g_1^\lambda\,,\,b(y_i,\tilde y_j)=b(\tilde z_j,z_i)=\delta_{ij}\,,\\
b(y_i,y_j)=b(\tilde y_i,\tilde y_j)=b(z_i,z_j)=b(\tilde z_i,\tilde z_j)=0\ .
\end{gathered}$$ Then one computes $$\begin{gathered}
[y_i,z_j]=[\tilde y_i,\tilde z_j]=0\ ,\ [y_i,\tilde z_j]=-\delta_{ij}A_\lambda\ ,\ [\tilde y_i,z_j]=\delta_{ij}A_\lambda\ ,\\
[A_\lambda,y_i]=\tfrac12z_i\ ,\ [A_\lambda,z_i]=\tfrac12y_i\ ,\ [A_\lambda,\tilde y_i]=\tfrac12\tilde z_i\ ,\ [A_\lambda,\tilde z_i]=\tfrac12\tilde y_i\ .
\end{gathered}$$
Let $\zeta_i$, $\tilde\zeta_i$, $i=1,\dotsc,q$, be the basis of $\ger p_1^*$, dual to $z_i$, $\tilde z_i$, $i=1,\dotsc,q$, so $${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{\tilde z_j}\r@m,{\zeta_i}\r@r\rangle}}=-{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z_i}\r@m,{\tilde\zeta_j}\r@r\rangle}}=\delta_{ij}\ ,\ { { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z_j}\r@m,{\zeta_i}\r@r\rangle}}={ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{\tilde z_i}\r@m,{\tilde\zeta_j}\r@r\rangle}}=0\ .$$ Then ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z}\r@m,{\zeta_i}\r@r\rangle}}=b(z,z_i)$, ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z}\r@m,{\tilde\zeta_i}\r@r\rangle}}=b(z,\tilde z_i)$, and one has $$\begin{gathered}
\operatorname{\mathrm{ad}}^*(y_i)\zeta_j=\operatorname{\mathrm{ad}}^*(\tilde y_i)\tilde\zeta_j=0\ ,\ -\!\operatorname{\mathrm{ad}}^*(y_i)\tilde\zeta_j=\operatorname{\mathrm{ad}}^*(\tilde y_i)\zeta_j=\delta_{ij}\lambda\ ,\\
\operatorname{\mathrm{ad}}^*(y_i)\lambda=-\tfrac12\zeta_i\ ,\ \operatorname{\mathrm{ad}}^*(\tilde y_i)\lambda=-\tfrac12\tilde\zeta_i\ .
\end{gathered}$$ Also, we observe ${ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z_I\tilde z_Jh^\nu}\r@m,{\zeta_K\tilde\zeta_L\lambda^\mu}\r@r\rangle}}=\delta_{IL}\delta_{JK}\delta_{\nu\mu}(-1)^{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{J}\r@m{}\r@r\rvert}
}}\nu!\lambda(h)^\nu$.
The preimages $p_2,p_{2q+1}$ of the generators $\lambda^2,\lambda^{2q+1}$ in $S(\ger p^*)^{\ger k}$ under the restriction map can be deduced from , because $\ger p=\ger a\oplus\ger p_1^\lambda$. Indeed, let $P=\phi(p_N)$ where $N=2$ or $N=2q+1$. By the formulae from , for $q{\geqslant}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}=k>0$ and $h\in\ger a'$, $$\begin{aligned}
\textstyle\sum_{\nu=0}^\infty\tfrac1{\nu!}{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\langle{z_I\tilde z_Jh^\nu}\r@m,{p_N}\r@r\rangle}}&=P(z_I\tilde z_J;h)=(\partial_{\gamma_h(z_I\tilde z_J)}\lambda^N)(h)\\
&=\delta_{IJ}(-1)^{\tfrac12k(k+1)}2^{-k}a_{Nk}\lambda(h)^{N-2k}
\end{aligned}$$ where $$a_{Nk}=\textstyle\sum_{i=(k-N)_+}^{k-1}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({-\tfrac12}\r@m{}\r@r)}
}^i(N)_{k-i}\frac{(k-1+i)!}{(k-1-i)!i!}\ .$$ Thus, $$p_N=\lambda^N+\textstyle\sum_{k=1}^{\min(N,q)}(-1)^{\tfrac12k(k+3)}2^{-k}a_{Nk}\lambda^{N-2k}\sum_{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}=k}\zeta_I\tilde\zeta_I\ .$$
When $N=2$ and $k{\geqslant}2$, then $a_{Nk}=0$ by and . On the other hand, $a_{21}=2$. Hence, $$p_2=\lambda^2+\textstyle\sum_{i=1}^q\zeta_i\tilde\zeta_i$$ is the super-Laplacian, and $$p_{2q+1}=\lambda^{2q+1}+\textstyle\sum_{k=1}^q(-1)^{\tfrac12k(k+3)}2^{-k}a_{2q+1,k}\lambda^{2(q-k)+1}\sum_{{ { \ifx00 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx01 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx02 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx03 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx04 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l\lvert{I}\r@m{}\r@r\rvert}
}=k}\zeta_I\tilde\zeta_I\ .$$ These elements are clearly subject to the relation $p_2^{2q+1}=p_{2q+1}^2$.
One readily checks $$\operatorname{\mathrm{ad}}^*(y_i)p_2=-\lambda\zeta_i+\zeta_i\lambda=0{{\quad\text{{{and}}}\quad}}\operatorname{\mathrm{ad}}^*(\tilde y_i)p_2=-\lambda\tilde\zeta_i+\lambda\tilde\zeta_i=0\ .$$ In case $q=1$, one has $p_3=\lambda^3+\frac32\lambda\zeta_1\tilde\zeta_1$, and $$\begin{aligned}
\operatorname{\mathrm{ad}}^*(y_1)p_3&=-\tfrac32\lambda^2\zeta_1-\tfrac32\lambda\zeta_1\operatorname{\mathrm{ad}}^*(y_1)\tilde\zeta_1=-\tfrac32\lambda^2\zeta_1+\tfrac32\lambda\zeta_1\lambda=0\ ,\\
\operatorname{\mathrm{ad}}^*(\tilde y_1)p_3&=-\tfrac32\lambda^2\tilde\zeta_1+\tfrac32\lambda\operatorname{\mathrm{ad}}^*(\tilde y_1)(\zeta_1)\tilde\zeta_1=-\tfrac32\lambda^2\tilde\zeta_1+\tfrac32\lambda^2\tilde\zeta_1=0\ .
\end{aligned}$$
To verify the $\ger k_0$-invariance, let $$x=\left(\ifx00 \@smallmattrue \else \@smallmatfalse
\fi \ifx01 \@nonefalse\@parentrue\@brackfalse\@bracefalse\@vlinefalse \else \ifx02 \@nonefalse\@parenfalse\@bracktrue\@bracefalse\@vlinefalse \else \ifx03 \@nonefalse\@parenfalse\@brackfalse\@bracetrue\@vlinefalse \else \ifx04 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinetrue
\else \ifx05 \@nonefalse\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \else \@nonetrue\@parenfalse\@brackfalse\@bracefalse\@vlinefalse \fi \fi \fi \fi \fi \if@smallmat \if@none \begin{smallmatrix} \else \if@paren \bigl(\begin{smallmatrix} \else \if@brack \bigl[\begin{smallmatrix} \else \if@brace \bigl\{\begin{smallmatrix} \else \if@vline \bigl\lvert\begin{smallmatrix} \else \bigl\lVert\begin{smallmatrix} \fi \fi \fi \fi \fi \else \if@none \begin{matrix} \else \if@paren \begin{pmatrix} \else \if@brack \begin{bmatrix} \else \if@brace \begin{Bmatrix} \else \if@vline \begin{vmatrix} \else \begin{Vmatrix} \fi \fi \fi \fi \fi \fi} {\if@smallmat \if@none \end{smallmatrix} \else \if@paren \end{smallmatrix}\bigr) \else \if@brack \end{smallmatrix}\bigr] \else \if@brace \end{smallmatrix}\bigr\} \else \if@vline \end{smallmatrix}\bigr\rvert \else \end{smallmatrix}\bigr\rVert \fi \fi \fi \fi \fi \else \if@none \end{matrix} \else \if@paren \end{pmatrix} \else \if@brack \end{bmatrix} \else \if@brace \end{Bmatrix} \else \if@vline \end{vmatrix} \else \end{Vmatrix} \fi \fi \fi \fi \fi \fi0&0&0&0\\0&0&0&0\\0&0&A&B\\0&0&C&-A^t\makeatother\right)\in\ger k_0=\ger{sp}(2q,{\mathbb{C}})\ .$$ Then $$\operatorname{\mathrm{ad}}^*(x)\zeta_i=\textstyle\sum_{j=1}^q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({A_{ji}\zeta_j+C_{ji}\tilde\zeta_j}\r@m{}\r@r)}
}{{\quad\text{{{and}}}\quad}}\operatorname{\mathrm{ad}}^*(x)\tilde\zeta_i=\sum_{j=1}^q{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({B_{ji}\zeta_j+A_{ji}\tilde\zeta_j}\r@m{}\r@r)}
}\ .$$ This implies $$\operatorname{\mathrm{ad}}^*(x)(\zeta_i\tilde\zeta_i)=\textstyle\sum_{j\neq i}{ { \ifx10 \def\r@l{}\def\r@m{}\def\r@r{} \else \ifx11 \def\r@l{\bigl}\def\r@r{\bigr}\def\r@m{\bigm} \else \ifx12 \def\r@l{\Bigl}\def\r@r{\Bigr}\def\r@m{\Bigm} \else \ifx13 \def\r@l{\biggl}\def\r@r{\biggr}\def\r@m{\biggm} \else
\ifx14 \def\r@l{\Biggl}\def\r@r{\Biggr}\def\r@m{\Biggm} \fi \fi \fi \fi \fi \ifx00 \def\r@m{} \fi \r@l({C_{ji}\tilde\zeta_j\tilde\zeta_i-B_{ji}\zeta_i\zeta_j}\r@m{}\r@r)}
}\ .$$ Since $B=B^t$, $C=C^t$, we deduce $\sum_{i=1}^q\operatorname{\mathrm{ad}}^*(x)(\zeta_i\tilde\zeta_i)=0$. Since $\ger a=\ger z(\ger g_0)$ and thus $\operatorname{\mathrm{ad}}^*(\ger k_0)\lambda=0$, this implies that $p_2$ (for general $q$) and $p_3$ (for $q=1$) are $\ger k$-invariant.
|
---
abstract: 'We propose a class of graded coronagraphic image masks for a high throughput Lyot-type coronagraph that transmits light from an annular region around an extended source and suppresses light, with extremely high ratio, elsewhere. The interior radius of the region is comparable with its exterior radius. The masks are designed using an idea inspired by approach due M.J. Kuchner and W.A. Traub (“band-limited" masks) and approach to optimal apodization by D.Slepian. One potential application of our masks is direct high-resolution imaging of exo-planets with the help of the Solar Gravitational Lens, where apparent radius of the “Einstein ring" image of a planet is of the order of an arc-second and is comparable with the apparent radius of the sun and solar corona.'
---
Igor Loutsenko and Oksana Yermolayeva
[Laboratoire de Physique Mathématique,\
CRM, Université de Montréal\
]{}
------------------------------------------------------------------------------------------------- --
[**Keywords**]{}: [*Coronagraphy, Optimal Band-Limiting, Exo-Planets, Gravitational Lensing*]{}
------------------------------------------------------------------------------------------------- --
Introduction
=============
Although potential applications of our masks are not restricted only to Solar Gravitational Lens (SGL) imaging, we will demonstrate main principles of the mask design in context of SGL imaging, since the latter was our main motivation.
The idea of using the sun as a powerful telescope goes back to Eshleman [@Eshleman]: The gravitational field of the sun acts as a spherical lens and magnifies intensity of electromagnetic radiation from distant objects along a semi-infinite focal line with the nearest point of observations being about $Z_{\rm min}$=550 AU (A good brief, self-contained introduction to the subject and related problems can be found in [@L]. For more details, one can e.g. see [@L1], [@M], [@TA], [@TT] and references therein). For example, an integral intensity of radiation from an Earth-like exo-planet at distance 30 pc can be pre-magnified by the SGL up to six orders of magnitude (see e.g. [@L]). Theoretical angular resolution of the SGL is comparable to that of a telescope with aperture of the order of the sun size. At visible wavelengths this resolution could be as small as $10^{-10}$ arcsec (see e.g. [@TT]).
Recently, properties of the solar gravitational lens attracted attention both due to discovery of numerous exo-planets and the success of the Voyager-1 spacecraft, presently operating at about 140AU. Possibilities of high resolution (up to mega-pixel) imaging of such planets from the focal line of solar gravitational lens are now being discussed.
Without going into much detail, we recall that an observer at distance $Z$ from the sun sees the image of an exo-planet as the “Einstein ring" of the apparent radius $\alpha_{\rm E}$ $$\alpha_{\rm E}(Z)=\alpha_{\rm max}\sqrt{Z_{\rm min}/Z},$$ where $\alpha_{\rm max}=1.75$ arcsec, is the apparent radius of the sun at $Z=Z_{\rm min}=550$ AU. The width (i.e. apparent thickness) of the ring $\delta\alpha$ equals a half of the apparent (angular) diameter of an exo-planet (see Figure \[fig\_SGL\]).
![Left: Geometrical optics image of the sun, Einstein ring image of the planet and two images of the parent star. Right: Ratio of surface brightness of solar corona to surface brightness of the sun center.[]{data-label="fig_SGL"}](ring.png){width="175mm"}
For an Earth-like planet at 30pc from the sun $\delta\alpha\sim 10^{-6}$ arcsec. As a consequence, the Einstein ring width is not resolvable by any realistic telescope. In other words, for any practical purpose the ring can be considered as a circle whose brightness is varying along the circle circumference [^1]. A telescope with radius of aperture $\sim 1$m resolves the circumference of the ring with up to $\sim 10^{2}$ elements. Thus, performing a scan in the observer plane by taking a set of images of the Einstein rings from different $(X,Y)$ - positions, one could then make a tomographic reconstruction of the real image of a planet. Note that amplification of the integral energy flow density by SGL is of order $2\alpha_{\rm E}/\delta\alpha$ (for details see e.g. [@L]).
Apart from formidable technical difficulties of getting to the focal line, there are principal problems of suppression of the diffraction glare from the sun and its corona. Indeed, the apparent radius of the sun at distance $Z$ is $$\alpha_{\rm S}(Z)=\alpha_{\rm max}Z_{\rm min}/Z.$$ The separation between the solar disc and the Einstein ring $\alpha_{\rm E}(Z)-\alpha_{S}(Z)$ is zero at $Z=Z_{\rm min}$ and reaches its maximum at $Z=4Z_{\rm min }\approx 2200$ AU. The maximal separation equals $\alpha_{\rm max}/4\approx 0.44$ arcsec. At this distance the apparent radius of the sun $\alpha_{\rm S}$ equals the separation angle.
The distance $Z=4Z_{\rm min}\approx 2200$ AU is considered as minimal practical distance of observation [@L]: The image of the ring will be superimposed with that of solar corona. The (angular) surface brightness of corona at the Einstein ring (i.e. at $\alpha_{\rm E}=\alpha_{\rm max}/2\approx 0.88$ arcsec) equals the surface brightness of the Earth night sky at full moon (astronomical observations of faint objects are not taken during periods of full moon due to the sky brightness). This surface brightness is about $10^{-8}I_0$, where $I_0$ is the surface brightness of the sun at its center (see Figure \[fig\_SGL\]).
More rigorous argument in favour of the above minimal practical distance is the following: In geometrical optics, the surface brightness is invariant. The surface brightness of an Earth-like planet is about $10^{-5}$ times of that of its host star. Therefore, in geometrical optics the surface brightness of the Einstein ring is $I_{\rm G}\sim 10^{-5}I_0$. Since the width of the ring is not resolved by a telescope, one has to take into account reduction of the surface brightness due to diffraction on the telescope aperture [^2].
Take, for instance, a telescope comparable with the Hubble Space Telescope, i.e. a telescope with the diameter $D\approx 2.5$ m operating at visible wavelengths $\lambda<750$ nm. The characteristic (diffraction) width of the image of Einstein ring in the focal plane of the telescope is of order $\lambda/D$, which is about $10^{-1}$ arcsec. This is about $10^4-10^5$ times bigger than the geometrical optics width of the Einstein ring. Therefore, the surface brightness of the ring in the focal plane of the telescope $I$ will be $10^{-4}-10^{-5}$ times $I_{\rm G}$, i.e. about $10^{-9}-10^{-10}$ times $I_0$. Taking into account that the brightness of corona at $\alpha=\alpha_{\rm E}(4Z_{\rm min})$ is of order $10^{-8}I_0$, we see that it exceeds $I$ by one-two orders of magnitude. In other words, even at $Z=2200$ AU the image of the ring in the focal plane of a realistic space telescope will be one-two orders of magnitude fainter than that of the background corona [^3]. That is why $Z=2200$ AU is considered as a minimal practical distance of observation.
From the above it follows that it is necessary to suppress the diffraction glare of the sun, so that the surface brightness of the glare (in vicinity of the ring image) in the final image plane of the optical system should be about $10^{-10}I_0$. It is important to stress that not only on-axis light, but also off-axis light with incidence angles $\alpha<\alpha_{\rm S}$ should be suppressed, while the light at $\alpha\approx\alpha_{E}$ should be transmitted almost entirely (we recall that $\alpha_{\rm S}=\alpha_{\rm E}/2$ at $Z=4Z_{\rm min}\approx 2200$ AU).
Below we propose a coronagraph that satisfies the above requirements. In addition, it suppresses not only the light from the sun but also the light from most of corona, except the part of corona in a close vicinity of the ring.
In the next section we recall general principles of the Lyot-type coronagraphs as well as those with the “band-limited" masks introduced by M.J. Kuchner and W.A. Traub in [@KT]. Then we introduce a quasi band-limited mask using approach similar to that of optimal apodization by D. Slepian [@S]. Later, we consider problem of suppressing light from the parent star of an exo-planet introducing “one-dimensional" mask based on Slepian’s solution of one-dimensional band-limiting problem. Finally, we introduce the “product" mask which suppresses not only sunlight and the light from a part of corona, but also the light from the parent star. Tolerance to manufacturing errors is discussed in the concluding section of the paper.
Band-Limited Mask
=================
To establish notations, let us first briefly review general principles of the Lyot-type coronagraph (for more detail see e.g. [@SKMBK] and references therein). Stages of propagation of light through the coronagraph are depicted in Figure \[stages\].
We consider a telescope with primary of diameter $D$. Let $\vec{R}$ be two dimensional vector defining the coordinates in the entrance pupil plane. Then it is convenient to introduce dimensionless coordinates $\vec{x}=\vec{R}/D$. Similarly, let $\vec{r}$ be two-dimensional coordinates in the first image (focal) plane of the telescope. We re-scale them in the units of the diffraction characteristic scale $\lambda \mathcal{F}/D$, where $\cal{F}$ is the telescope focal length, introducing dimensionless coordinates $\vec{y}=\vec{r}D/(\lambda \cal{F})$. It is also convenient to re-scale the incidence angle $\vec{\alpha}$ of the plane wave in the units of the characteristic diffraction angle $\lambda/D$, introducing “dimensionless" [^4] incidence $\vec{\beta}=\vec{\alpha}D/\lambda$.
![Schematic representation of principal stages of field propagation through coronagraph.[]{data-label="stages"}](coronagraph.png){width="155mm"}
We consider a non-apodized entrance pupil, so, up to an $\vec{x}$-independent common factor, the dimensionless complex field of the plane wave of the unit amplitude immediately after encountering the primary mirror is $$A_{\vec{\beta}}(\vec{x})=P(x)E_{\vec \beta}(\vec{x}),
\label{A}$$ where $$E_{\vec \beta}(\vec{x})=e^{-2\pi i (\vec{\beta}\cdot\vec{x})}$$ and $P$ is the non-apodized circular aperture function $$P(x)=\left\{\begin{array}{ll}
1, & x<1/2 \\
0, & x>1/2
\end{array} \right. , \quad x:=|\vec{x}|.$$ From the Fraunhofer theory of diffraction it follows that amplitude of the field in the focal plane of the telescope is the Fourier transform of (\[A\]) $$a_{\vec{\beta}}(\vec{y})=\int A_{\vec{\beta}}(\vec{x})e^{2\pi i (\vec{x}\cdot\vec{y})}d^2x$$ and $$a_{\vec{\beta}}(\vec{y})=p\left(|\vec{y}-\vec{\beta}|\right), \quad p(y)=\frac{J_1(\pi y)}{2y},
\label{p}$$ where $J_1$ stands for the Bessel function. In the Lyot-type coronagraph the image is focused on the occulting mask with amplitude transmission factor $m(\vec{y})$, so, immediately after passing the mask the field amplitude becomes $$[ma](\vec{y})= m(\vec{y}) a_{\vec{\beta}}(\vec{y}) .
\label{am}$$ Successive optics in the coronagraph transforms this product to the second pupil plane, where the field is $M*A_{\vec{\beta}}$. Here, $*$ denotes convolution and $M(\vec{x})=\int m(\vec{y})e^{2\pi i (\vec{x}\cdot\vec{y})}d^2y$ is the Fourier transform of $m$. Note that we consider only symmetric graded masks, i.e. the masks with $$m(\vec{y})=m(-\vec{y}), \quad m=|m| .$$ As a consequence, $M$ is real and $M(\vec{x})=M(-\vec{x})$ in our case.
Now, the field passes through the Lyot stop, which can be described by an aperture function $L(x)$. After passing the Lyot stop the field becomes $$F_{\vec{\beta}}(x)=L(\vec{x})[M*A_{\vec{\beta}}](\vec{x}),
\label{L}$$ where $[M*A_{\vec{\beta}}](\vec{x})=\int M(\vec{x}-\vec{x'})A_{\vec{\beta}}(\vec{x'})d^2x'$.
Then, the field passes the final pupil and is focused into the final (i.e. detector’s) image plane where the field amplitude is the Fourier transform of (\[L\]). In other words, the field amplitude in the final image plane equals $$f_{\vec{\beta}}=l*[ma_{\vec{\beta}}],
\label{f}$$ where $l$ is the Fourier transform of $L$. The field $F$ in (\[L\]) is usually referred as the “final field" and we refer its Fourier transform $f$ in (\[f\]) as the “detected field". More precisely, detector registers intensity of image $\mu$ in the final (i.e. detector’s) image plane. This intensity is the Point Spread Function (PSF) of the optical system and equals square of the absolute value of the detected field $$\mu(\vec{\beta},\vec{y})=|f_{\vec{\beta}}(\vec{y})|^2.$$ The surface brightness $\mathcal{I}$ of a final image of a point source with incidence $\beta$ is the PSF divided by the Lyot stop area $$\mathcal{I}(\vec{\beta},\vec{y})=\mu(\vec{\beta},\vec{y})/S, \quad S=\int L(\vec{x})d^2x .
\label{brightness_ps}$$ For an extended incoherent source with surface brightness distribution $I_{\rm s}(\vec{\xi})$, the surface brightness at the detector plane $I$ equals[^5] $$I(\vec{y})=\int I_{\rm s}(\vec{\xi})\mathcal{I}(\vec{\xi},\vec{y})d^2\xi .
\label{brightness}$$ Now, we turn our attention to the band-limited masks. For such masks, the Fourier transform $M$ of the transmission amplitude $m$ has a finite support. In the rest of this and the next section we will consider only circularly symmetric masks $m(\vec{y})=m(y)$ (obviously, $M$ is also circularly symmetric). For such symmetric band limited mask, $M(x)$ is “concentrated" completely within the disc of diameter $\epsilon$ and vanishes elsewhere, i.e. $$M(\vec{x})=0, \quad x>\epsilon/2.
\label{bl}$$ We also consider only circular non-apodized Lyot stops. Let dimensionless diameter of the stop be $\sigma<1$, then $$L(\vec{x})=P(x/\sigma), \quad S=\pi\sigma^2/4.$$ It has been noticed in [@KT] that from (\[L\]) and (\[bl\]) it follows that in the case of Lyot stops, whose dimensionless diameter $\sigma$ does not exceed $1-\epsilon$, the final field equals $$F_{\vec{\beta}}=L[M*A_{\vec{\beta}}]=L[M*E_{\vec{\beta}}]=m(\vec{\beta})LE_{\vec{\beta}} .$$ In other words, the final field of the band-limited coronagraph [^6] coincides with that of the plane wave of amplitude $m(\beta)$ at incidence $\beta$ that passed through a pupil of dimensionless diameter $\sigma\le 1-\epsilon$. Thus the detected field equals $$f_{\vec{\beta}}(y)=m(\beta)l(\vec{y}-\vec{\beta}) ,$$ where for a circular stop of diameter $\sigma$ $$l(y)=\sigma p(\sigma y)=\sigma\frac{J_1(\pi\sigma y)}{2y} .
\label{l}$$ It follows that the intensity in the detector image plane equals $m(\beta)^2$ times the PSF of the stop [^7], i.e $$\mu(\vec{\beta},\vec{y})=m(\beta)^2 l^2\left(|\vec{y}-\vec{\beta}|\right)=m(\beta)^2\mu_{\rm L}(\vec{\beta},\vec{y}) .
\label{psf}$$ Thus, the band-limited coronagraph completely suppresses an on-axis source when $m(0)=0$.
The final throughput of the energy for the incidence $\beta$ equals $$\tau(\beta)=m(\beta)^2\sigma^2 .
\label{tau}$$ Working throughput of the coronagraph equals $\tau(\beta_{\rm W})$, where $\beta_{\rm W}$ is the working incidence. Usually one chooses $m\left(\beta_{\rm W}\right)$ to be close to unity in order to get the maximal useful throughput which is reached when $\sigma=1-\epsilon$.
It is important to note that the band limited coronagraph designed for the wavelength $\lambda$ will work for smaller wavelengths $\lambda'<\lambda$ as well, since the change $\lambda$ to $\lambda'$ is equivalent to the scalings $m(y)\to m(\lambda' y/\lambda)$, $M(x)\to M(\lambda x/\lambda')$ and $\beta\to \lambda \beta/\lambda'$, while $\sigma$ remains invariant. As a consequence, the energy transmission at a given incidence angle $\alpha=\beta \lambda/D$ is the same for $\lambda'<\lambda$.
Quasi Band-Limited Mask
=======================
Our aim is to construct a coronagraph that transmits the light coming from an annular region containing the Einstein ring and suppress it elsewhere. We recall that the band-limited coronagraph completely suppresses light at incidences $\beta$ for which $m(\beta)=0$. However, the band-limited coronagraph that completely suppresses light coming from outside of the annular region is impossible to construct due to the uncertainty principle: A function and its Fourier transform cannot both have finite supports.
Below we propose construction of a graded mask that is “almost" band limited: Quasi Band Limited Mask or QBLM. It is band-limited in the sense that the transmission factor $m(y)$ has a finite support, vanishing outside some annual region of the interior radius $y_1$ and the exterior radius $y_2$ $$m(y)=0, \quad y\not\in y_1<y<y_2,
\label{annular}$$ while its Fourier conjugate $M(x)$ is “concentrated" in the disc of diameter $\epsilon$, “almost" vanishing elsewhere. For instance, in examples that will be presented below, maximum of the “tail" of $M$ outside the disc is about 5 orders of magnitude smaller than maximum of the “main lobe" of $M$ on the disc (see Fig. \[fig\_M\]). In what follows, we refer to functions having the property (\[annular\]) as “annular-limited".
Les us now split $M(x)$ into two parts $M_\epsilon$ and $\delta M$: $$M=M_\epsilon+\delta M ,
\label{M_split}$$ where $M_\epsilon$ is the “main lobe" that vanishes outside the disc of diameter $\epsilon$ $$M_\epsilon(x)=\left\{\begin{array}{ll}
M(x), & x<\epsilon/2 \\
0, & x>\epsilon/2
\end{array}
\right. ,
\label{M_epsilon}$$ and the “tail", that vanishes on the disc $$\delta M =\left\{\begin{array}{ll}
0, & x<\epsilon/2 \\
M(x), & x>\epsilon/2
\end{array}
\right. .
\label{delta_M}$$ Taking a coronagraph with $\sigma\le 1-\epsilon$, from the decomposition (\[M\_split\], \[M\_epsilon\], \[delta\_M\]) and eqs. (\[L\], \[bl\]) we get the final field $$F_{\vec{\beta}}(\vec{x})=m(\vec{\beta})L(x)E_{\vec{\beta}}(\vec{x})-L(x)\Delta_{\vec{\beta}}(\vec{x}) ,$$ where $$\Delta_{\vec{\beta}}(\vec{x})=\delta m(\vec{\beta})E_{\vec{\beta}}(\vec{x})-[\delta M*A_{\vec{\beta}}](\vec{x}) .$$ Here, $\delta m$ is the Fourier transform of the “tail" $\delta M$.
In the detector image plane we have $$f_{\vec{\beta}}(\vec{y})=m(\vec{\beta})l\left(|\vec{y}-\vec{\beta}|\right)-\delta f_{\vec{\beta}}(\vec{y}) ,
\label{qbl}$$ where $$\delta f_{\vec{\beta}}=l*\left[(\delta_{\vec{\beta}}-a_{\vec{\beta}})\delta m\right],
\label{delta_f}$$ and $\delta_{\vec{\beta}}$ stands for the two-dimensional Dirac $\delta$-function: $$\delta_{\vec{\beta}}(\vec{y})=\delta(\vec{y}-\vec{\beta}).$$ We refer to the first term in the RHS of (\[qbl\]) as the “main field" , while the second term $\delta f$ is referred as the “residue field".
Since the transmission amplitude vanishes outside the annulus (\[annular\]), the main field for incidences outside this annular region is completely suppressed and $$f_{\vec{\beta}}(y)=-\delta f_{\vec{\beta}}(y), \quad \mu(\vec{\beta},\vec{y})=\delta f_{\vec{\beta}}(y)^2, \quad \beta \not\in y_1<\beta<y_2 .$$ The residue field is determined by the Fourier transform $\delta m$ of the “tail". In what follows we call $\delta m$ the residue transmission. From (\[M\_split\], \[M\_epsilon\], \[delta\_M\]) it follows that the residue transmission equals $$\delta m =(1-\hat{K}_\epsilon) m ,
\label{mu}$$ where $\hat{K}_\epsilon$ is the following integral operator $$\hat{K}_\epsilon [m](\vec{y})=\int_{y_1<y'<y_2}K_\epsilon\left(\vec{y}-\vec{y'}\right)m(\vec{y'})d^2y, \quad K_\epsilon(\vec{y})=\epsilon\frac{J_1(\pi\epsilon y)}{2y}
\label{K_epsilon}$$ and the integral is taken over the annulus $y_1<y<y_2$.
On the other hand, for working incidences, $m$ is of the order of unity and $m(\beta_{\rm W})\gg \max |\delta m(y)|$. As a consequence, at these incidences the residue field can be neglected and, similarly to the band limited coronagraph (\[psf\]), PSF of the optical system is $m(\beta_{\rm W})^2$ times the PSF of the Lyot Stop.
Optimal Occultation
===================
Now we are going to find the annular-limited $m(y)$, having the “smallest" possible “tail" $\delta M$. The tail is smallest in the sense of the optimal apodization by D.Slepian [@S]: that is, it has the smallest possible energy. In more details, one has to maximize the energy of the main lobe, i.e. the ratio $$\kappa=\frac{\int_{x<\epsilon/2} M(\vec{x})^2d^2x}{\int M(\vec{x})^2d^2x} ,
\label{kappa_x}$$ where the integral in the numerator is taken over the disc of diameter $\epsilon$, while the integral in the denominator is taken over the whole plane. Obviously, $0<\kappa<1$. The above equation can be rewritten in terms of $m$: $$\kappa=\frac{\int d^2y \int d^2y' K_\epsilon\left(\vec{y}-\vec{y'}\right) m(\vec{y})m(\vec{y'})}{\int m(y)^2d^2y} .
\label{kappa_y}$$ It is not difficult to see that optimal $m(y)$ is an eigenfunction of the integral operator $\hat {K}_\epsilon$ corresponding to its maximal eigenvalue $\kappa$: $$\kappa m(\vec{y})=\hat{K}_\epsilon[m](\vec{y}), \quad y_1<y<y_2 .
\label{km}$$ Since we consider radially symmetric $m(\vec{y})=m(y)$, the two-dimensional integral operator (\[K\_epsilon\]) can be reduced to one-dimensional one by integration in polar coordinates in the $y$-plane (see Appendix 1) and $$\kappa m(y)=(\pi\epsilon)^2\int_{y_1}^{y_2}\mathcal{K}(\pi\epsilon y, \pi\epsilon y') m(y') y' dy', \quad y_1<y<y_2 ,
\label{K_reduced}$$ where $$\mathcal{K}(y,y')=\frac{yJ_1(y)J_0(y')-y'J_1(y')J_0(y)}{y^2-{y'}^2}.
\label{Kernel_1d}$$ Our optimization differs from that of the apodization problem considered by D. Slepian [^8] in [@S]. There, an analog of our function $m(y)$ is “disc"-limited, which allows to reduce the apodization problem to solution of an eigenvalue problem for an ordinary differential operator. We do not know if a reduction to some “sparce" operator is possible in the annular-limited case, but the diagonalization (\[K\_reduced\]) can be easily performed numerically [^9] due to symmetry and positive definiteness of the operator.
![Optimal transmission amplitude $m(y)$ for the mask with $y_1=7$, $y_2=28$ and $\epsilon=0.4$. Here $1-\kappa\approx 4\times 10^{-7}$. The working incidence $\beta_{\rm W}=2y_1=14$. The working throughput is 30 percents.[]{data-label="fig_m"}](transmission_amplitude.png){width="175mm"}
![$\log_{10}\frac{M(x)^2}{M(0)^2}$ of the optimal mask from Figure \[fig\_m\] ($y_1=7$, $y_2=28$ and $\epsilon=0.4$). The part of the plot to the left from the vertical dashed line corresponds to the “main lobe" of $M$, while the part on the right corresponds to its “tail" (see eqs. (\[M\_split\], \[M\_epsilon\], \[delta\_M\])).[]{data-label="fig_M"}](log10_M2.png){width="155mm"}
We call the problem of optimization of $m(y)$ for an arbitrary shaped support on the mask the optimal occultation. Similarly to results of the optimal apodization, $\kappa$ is very close to $1$ in our examples of the optimal occultation (see below). There the value of $1-\kappa$ is of order of $10^{-7}$.
To make estimates of coronagraphic suppression, we will need to find the residue transmission $\delta m$. According to (\[annular\], \[mu\], \[km\]) $$\delta m(y)=\left\{\begin{array}{ll}
(1-\kappa)m(y), & y_1<y<y_2 \\
-\hat{K}_\epsilon[m](y), & {\rm otherwise}
\end{array} \right .
\label{delta_m}$$ It is worthy to note that $m(y)$ can be formally continued [@S1964] beyond the annular region by dropping the restriction $y_1<y<y_2$ in equation (\[K\_reduced\]). Then $$\tilde m(y)=\kappa^{-1}\hat{K}_\epsilon[m](y)$$ equals $m(y)$ inside the annular region and continues it outside the region where, according to eq.(\[delta\_m\]), $\delta m=-\kappa \tilde{m}$. Since in all our examples $1-\kappa$ will be extremely small (of the order of $10^{-7}$), we can write that $$\delta m(y)\approx\left\{\begin{array}{ll}
0, & y_1<y<y_2 \\
-\tilde{m}(y), & {\rm otherwise}
\end{array} \right.$$ and at boundaries of the region $\delta m(y_i)\approx -m(y_i), i=1,2$. The residue transmission is an oscillating function with the asymptotics $\delta m(y)\to -M(0)\frac{\sqrt{2\epsilon}}{2\pi y^{3/2}}\sin(\pi\epsilon y-\pi/4)$ for $y\to\infty$.
To get an idea of magnitude of the coronagraphic suppression, one can roughly estimate maximum of PSF for incidences $\beta$ outside the annular region and compare it with maximum of PSF with no mask applied (i.e. with maximum of PSF of the Lyot stop).
From (\[delta\_f\]) it follows that absolute value of the residue field $|\delta f_{\vec{\beta}}|$ is not exceeding the biggest of the two following values $$2|\delta m(\beta)|\max |l|, \quad 2\max \left| l*\left[a_{\vec{\beta}}\delta m\right]\right| .
\label{max_residue}$$ One may assume that these two values are of the same order on average and $$|\delta f_{\vec{\beta}}|<\mathcal{O}(\max |\delta m(y)|)\max |l| .$$ Therefore, for incidences outside the annular region: $$\max \mu(\vec{\beta},\vec{y})= s(\beta) \max \mu_{\rm L}=s(\beta)\left(\frac{\pi \sigma^2}{4}\right)^2, \quad s(\beta)<\mathcal{O}(\max \delta m(y)^2), \quad \beta \not\in y_1<\beta<y_2,
\label{sb}$$ where $\mu_{\rm L}$ stands for PSF of the Lyot stop. In other words, according to our estimate the suppression coefficient $s(\beta)=\max \mu(\vec{\beta},\vec{y})/\max \mu_{\rm L}$ is of order $\max \delta m^2$ or smaller order. This estimate is confirmed by direct numerical simulations of coronagraph which are presented in the next section. In fact, the actual suppression coefficient turns to be about two-three orders of magnitude smaller (i.e. suppression is 2-3 orders stronger) than our very rough estimate $\max \delta m^2$ for incidences that are not in vicinity of boundary of the annular region (see Figures \[fig\_points\], \[fig\_suppression\]). In vicinity of the boundary it is of the same order. A more accurate estimate could use some envelope $\tilde m_{\rm e}(\beta)$ of the oscillating function $\tilde m(\beta)$. The suppression coefficient and square of this envelope are of the same order $s(\beta)=\mathcal{O}\left(\tilde m^2_{\rm e}(\beta)\right)$ (see Figure \[fig\_suppression\]).
Now, to be specific, we consider an example from the introduction section: Take $D=2.5$m telescope at $Z=4Z_{\rm min}$ and $\lambda=750$ nm. For these parameters the magnitude of the incidence corresponding to the Einstein ring (working incidence) $\beta_{\rm W}=\alpha_{\rm E}D/\lambda=14$. Let the apparent boundary of the sun coincide with the interior boundary of the annular region. Then $y_1=\beta_{\rm W}/2=7$. We chose the outer boundary of the region to be $y=y_2=2\beta_{\rm W}=28$.
We choose $\epsilon=0.4$ which corresponds to the maximal throughput $(1-\epsilon)^2=0.36$. The optimal transmission amplitude obtained by numerical solution of (\[K\_reduced\]) is shown on Figure \[fig\_m\]. The difference between the “main lobe" and the “tail" of the Fourier transform of $m$ is demonstrated on Figure \[fig\_M\]. The formal continuation $\tilde m(y)$ of $m$, is shown in separate boxes of Figure \[fig\_m\]. For this solution $1-\kappa\approx 4\times 10^{-7}$. Therefore, outside the annular region, $\delta m\approx-\tilde m$ with the relative precision $4\times 10^{-7}$. The residue transmission reaches maximum by absolute value at $y=y_1$ and $\max \delta m^2\approx 6.4\times 10^{-7}$. Therefore, one can expect the suppression to be of six or more orders of magnitude.
The transmission amplitude at working angle $m(\beta_{\rm W})$ equals $0.925$, so, according to (\[tau\]) the maximal working throughput is $m(\beta_{\rm W})^2(1-\epsilon)^2\approx 0.308$ (i.e. about 30 percents).
Results of Direct Numerical Simulations
=======================================
Direct numerical simulations of a Quasi Band Limited Coronagraph are in agreement with the above estimates. We performed direct simulations of all stages of the field propagation (see eqs. (\[A\]-\[f\]) and Figure \[stages\]) applying the fast Fourier transform at each pupil. Maximal size of the grid was $4096 \times 4096$, but sizes $2048 \times 2048$ and $1024 \times 1024$ give results which differ only by few percents from those of maximal grid. Simulations are run for the Lyot stop of the diameter $\sigma=1-\epsilon=0.6$.
![Bottom: Normalized images of point sources. Top: $\log_{10}$ of PSF in the detector plane (i.e. $\log_{10}\mu$) along the central sections (i.e. along the dashed lines at corresponding bottom images). Panel A: No mask is applied. Panels B to D: optimal mask is applied. Corresponding incidences are: $\beta=0$ (Panel B, on-axis source), $\beta=y_1/2=3.5$ (Panel C) and $\beta=y_1=7$ (Panel D, source at interior boundary of annular region).[]{data-label="fig_points"}](point_sources_comparison.png){width="175mm"}
The suppression is minimal (i.e. suppression coefficients are maximal) in the neighborhood of interior boundary of the annular region. With a good precision the suppression coefficients equal $\delta m(\beta)^2$ in this neighborhood. The above value has its maximum $6.4\times 10^{-7}$ at the boundary and falls rapidly by about two orders of magnitude as $\beta$ decreases: The suppression coefficients are of the order of $10^{-8}-10^{-9}$ at the bigger part of the disc $\beta<y_1$, see Figure \[fig\_suppression\]. The suppression coefficient of maximum of PSF is of the same order as the coefficient of energy suppression $\tau(\beta)/\sigma^2$. The latter coefficient equals the ratio of energy flow passed in the presence of mask to that without mask.
![Thick solid curve: $\log_{10}$ of energy suppression coefficient $\tau(\beta)/\sigma^2$ (the latter is the ratio of energy passed with mask to that passed without mask). Thin solid curve: $\log_{10}$ of suppression of maximum of PSF at given $\beta$ (i.e. $\log_{10} s(\beta)$, see eq. (\[sb\])). Dashed curve: $\log_{10}$ of square of continuation $\tilde{m}(\beta)$ (recall that $\tilde{m}(\beta)^2\approx\delta m(\beta)^2$ for $\beta \not\in y_1<\beta<y_2$).[]{data-label="fig_suppression"}](suppression_circularly_symm.png){width="175mm"}
The detector plane images of point sources at different incidences $\beta\le y_1$ and sections of intensity are shown on Figure \[fig\_points\].
To evaluate the sun glare one has to compute integral (\[brightness\]) of the product of distribution of the surface brightness of the sun with the PSF in the detector plane divided by area of the Lyot stop. We recall that we choose the parameters in such a way that apparent radius of the sun equals $y_1$ in units of characteristic diffraction angles, i.e. edge of the sun disc coincides with interior boundary of the annulus. In our computations we presented the sun as a disc of uniform surface brightness $I_0$, so that the limb darkening is not taken into account. Accounting for limb darkening will give better results [^10] (i.e. the glare brightness will be smaller).
![Left: Detector plane image of the solar disc. Optimal mask is applied. No limb darkening is taken into account (the sun is presented as a disc of uniform surface brightness $I_0$). Center: Radial dependence of the relative surface brightness in the detector plane. Right: Radial dependence without mask applied.[]{data-label="fig_disc"}](disc_uniform.png){width="175mm"}
With the optimal mask applied, the glare brightness at working angle $\beta_{\rm W}=2y_1=14$ (i.e. at position of the Einstein ring) is about $10^{-10}I_0$ (see Figure. \[fig\_disc\]). Therefore, the goal set in the introductory section can be achieved with help of our mask.
Suppressing the Parent Star, One-dimensional Slepian’s Mask, Product Mask
=========================================================================
The circularly symmetric mask can suppress the diffraction glare of the sun to the level of brightness of the detector plane image of an Earth-like exo-planet at about 30pc. However, there remains a problem of suppressing glare from the planet’s parent star. The SGL produces two images of the parent star on the line that passes through the center of the sun (see Figure \[fig\_SGL\]). They appear at opposite sides wrt the center, one inside the circle $\alpha=\alpha_{\rm E}$ (i.e. inside circle $\beta=\beta_{\rm W}$) and the other outside it. Minimal angular distance between images and the circle is about a half of the apparent separation between the planet and its parent star. The size of the star is not resolvable by the telescope, so, for any practical purpose the star can be considered as a point source.
Since the SGL amplifies intensity of radiation from a planet much more stronger than it does from its parent star and apparent diameter of the Einstein ring is relatively big, an acceptable suppression level for observations with SGL is several orders of magnitude weaker than that required for observations without it. Indeed, the ratio of the amplification of radiation from an exo-planet to that from its parent star is of the order of the ratio of planet’s orbit radius to its own radius. This is about $3\times 10^4$. Taking into account that the circumference of the Einstein ring in the detector plane is about $30$ resolution elements, reduction of the flux ratio [^11] by the SGL is $\sim 10^3$. Since, for an Earth-like planet the unreduced flux ratio is about $\sim 10^{10}$, it will be $\sim 10^7$ in presence of the SGL. Also, since the apparent distance between the bigger part of the Einstein ring and images of the parent star is of the order of $10$ diffraction angles, acceptable magnitude of the suppression coefficients can be several orders bigger than $10^{-7}$ (we recall that, according to our definitions, smaller suppression coefficients mean bigger suppression and vice versa).
To solve the problem of simultaneous suppression of the sun and the parent star glare, we first introduce an effectively one-dimensional mask that eliminates light from sources located on the line passing through two images of the parent star. The transmission amplitude of this mask is essentially Slepian’s solution to the one-dimensional optimal apodization problem [@S]. We call the corresponding mask Slepian’s mask. Then we will test the mask that is the product of circularly symmetric mask (introduced in previous sections) and the Slepian’s mask (see Figure \[fig\_masks\]).
![Left: Circularly symmetric mask. Center: One-dimensional “Slepian’s" mask. Right: Product mask. Gray-scale values of image pixels are proportional to the mask transparency $t=m^2$ (also called intensity transmission factor). Black color corresponds to $t=0$, while white color corresponds to $t=1$. The interior/exterior radii of the annular support are $y_1=7$ and $y_2=28$ correspondingly. The half-width of the Slepian’s mask equals $y_1$. For both masks $\epsilon=0.4$. Einstein ring radius $\beta_{\rm W}=2y_1=14$.[]{data-label="fig_masks"}](masks_squared.png){width="135mm"}
Let $y_\perp$ and $y_\parallel$ be coordinates in the $y$-plane, i.e. $\vec{y}=(y_\perp,y_\parallel)$. We consider a mask with the transmission amplitude changing only in one direction $m=m(y_\perp)$. Our aim is to suppress point sources at the line $y_\perp=0$, so that $m(0)=0$. Also $$m(\vec{y})=1-u(y_\perp) ,$$ where $u$ is an even function $u(y_\perp)=u(-y_\perp)$, such that $u(0)=1$ and $u(y_\perp)$ has a finite support $y_1<y_\perp<-y_1$: $$u(y_\perp)=0, \quad |y_\perp|>y_1.$$ In other words, the Slepian’s mask is completely transparent outside the strip $|y_\perp|<y_1$ ($m=1$ for $|y_\perp|>y_1$). The Fourier transform $M$ of $m$ is $$M(\vec{x})=\left[\delta(x_\perp)-U(x_\perp)\right]\delta(x_\parallel) ,$$ where $U(x_\perp)=\int e^{2\pi i x_\perp y_\perp}u(y_\perp)dy_\perp$ is one-dimensional Fourier transform of $u(y_\perp)$ and $\vec{x}=(x_\perp,x_\parallel)$.
![Left: Optimal $u(y)$ for the Slepian’s mask with $y_1=7$ and $\epsilon=0.4$. Here $1-\kappa\approx 5\times 10^{-7}$. Top right: $\log_{10}\frac{U(x)^2}{U(0)^2}$. Bottom right: Formal continuation $\tilde{u}(y)$ of $u(y)$ (We recall that $\tilde{u}(y)\approx\delta m(y)$ when $|y|>y_1$.)[]{data-label="fig_S"}](Slepian.png){width="175mm"}
Similarly to the two-dimensional optimal occultation problem, we split $U(x)$ into the “main lobe" $U_\epsilon(x)$ with support on the interval $-\epsilon/2<x<\epsilon/2$ and the “tail" $\delta U(x)$: $$U=U_\epsilon+\delta U, \quad U_\epsilon(x)=\left\{\begin{array}{ll}
U(x), & |x|<\epsilon/2 \\
0, & |x|>\epsilon/2
\end{array}
\right., \quad
\delta U(x)=\left\{\begin{array}{ll}
0, & |x|<\epsilon/2 \\
U(x), & |x|>\epsilon/2
\end{array}
\right.$$ Energy of the “tail" is minimal when $u$ is an eigenfunction corresponding to the highest eigenvalue $\kappa$ of the one-dimensional integral operator: $$\kappa u(y)=\hat{K}_\epsilon[u](y)=\int_{-y_1}^{y_1}\frac{\sin\pi\epsilon (y-y')}{\pi(y-y')}u(y')dy' .$$ Solution of the above equation for $\epsilon=0.4$ and $y_1=7$ (same values as in the circularly symmetric case) is shown on Figure \[fig\_S\]. Here the deviation of $\kappa$ from unity equals $5\times 10^{-7}$ and is of the same order as that of the example considered in previous section for the circularly symmetric case. The maximal residue of the transmission amplitude $\max|\delta m|\approx u(y_1)$ and is approximately of the same order as in the circularly symmetric example ($u(y_1)\approx 1.5\times 10^{-3}$, compare Figures \[fig\_m\] and \[fig\_S\]). Therefore, one can expect the same level of suppression for a point source at the line $\beta_\perp=0$. And, indeed, direct numerical simulations of a coronagraph with the Slepian’s mask and a circular stop of the diameter $1-\epsilon=0.6$ give the result $\frac{\max \mu}{\max \mu_{\rm L}}\approx 5.25 \times 10^{-10}$ for PSF and the energy suppression coefficient is equal $\approx 2\times 10^{-9}$ when the point source is on the line $\beta_\perp=0$. Obviously, the suppression coefficients do not depend on $\beta_\parallel$ for this one-dimensional mask.
![Center: Detector plane image of the solar disc. Product mask is applied. No limb darkening is taken into account (the sun is presented as a disc of uniform surface brightness $I_0$). Left: $\log_{10}$ of relative surface brightness $I/I_0$ along the circumference of the image of the Einstein ring $y=2y_1=14$. Right: Level contours of $\log_{10}$ of the relative surface brightness.[]{data-label="fig_product_disc"}](disc_product.png){width="175mm"}
Consider now the product mask, i.e. the mask whose amplitude transmission factor is the product of factors of the circularly symmetric and Slepian’s mask (see Figure \[fig\_masks\]). The idea of using the product mask comes from the pure band limited case: A product of two band-limited functions $f(\vec{y})$ and $g(\vec{y})$ is also a band limited function, since the support of the convolution $[F*G](\vec{x})$ is a cartesian sum of supports of $F$ and $G$. In the case of product of the circularly symmetric and one dimensional masks, this is the cartesian sum of a (two-dimensional) disc and a (one-dimensional) segment. However, in the case of the quasi band limited masks, the support of new “main lobe" can be smaller than the cartesian sum of supports of the “main lobes" of $F$ and $G$. Indeed, the definition of support of the “main lobe" as a region where most of the energy is concentrated is somehow arbitrary. Also, the convolution of “main lobes" can be of the same order as the “tail" on a substantial part of a cartesian product. That is why one can try to apply the product mask without changing the size of the Lyot stop.
We have performed direct numerical simulations of the coronagraph with the product mask without changing diameter of the stop $\sigma=1-\epsilon=0.6$, preserving $30$ percent working throughput. The results of simulations do not show substantial degradation in the suppression coefficients: The intensity of glare over the bigger part of the working region is of the same order as in the circularly symmetric case. The results of suppression of the glare from the sun are shown on Figure \[fig\_product\_disc\]: Although, in difference from the circularly symmetric case, the working space is reduced, the glare at the bigger part of the Einstein ring (in total, more than $180$ degrees of the working space along the ring circumference) is of the same order as in the circularly symmetric case.
![$\log_{10}$ of the detector plane PSF for the product mask and different position of point sources along the line $\beta_\perp=0$. Corresponding positions of the sources and the Einstein ring are shown at the rightmost panel ($\beta_\parallel=12$, $\beta_\parallel=14$ and $\beta_\parallel=16$ for the point sources at A, B and C correspondingly).[]{data-label="fig_parent"}](Parent_log10.png){width="175mm"}
![$\log_{10}$ of suppression coefficients of the product mask as functions of $\beta_\parallel$ for $\beta_\perp=0$, $\beta_\perp=0.05$ and $\beta_\perp=0.1$ (from left to right correspondingly). Thick line corresponds to the energy suppression coefficient, while thin line stands for the PSF maximum suppression coefficient.[]{data-label="fig_suppression_product"}](suppression_product.png){width="175mm"}
Finally, the suppression coefficient for the point sources located at the line $\beta_\perp=0$ also do not show a substantial degradation: The level of suppression is more than sufficient for reducing glare from the parent star to an acceptable level. We also performed simulations of cases with off-alignment of the parent stars and the mask, i.e. the cases when $\beta_\perp\not=0$ (see Figure \[fig\_suppression\_product\]). The acceptable deviation from the central line is at least of the order of $0.1$, (i.e. $\sim 1/10$ of the characteristic diffraction angle).
Discussion and Conclusions
==========================
In the present article we proposed graded coronagraph masks that can substantially suppress the diffraction glare from extended sources. In examples considered, the diffraction glare from a source in the form of disc of the apparent diameter of the order of one arc-second and of the uniform surface brightness $I_0$ can be reduced by about $10^8$ times (example of $2.5$m telescope operating at $\lambda=750$nm). Surface brightness of the glare at the apparent distance twice the radius of the disc from the disc center is about $10^{-10}I_0$.
![$\log_{10}$ of suppression coefficients of the circularly symmetric mask designed for wavelength $\lambda$ and operating at $\lambda'=\lambda/2$. Thick curve corresponds to the energy suppression coefficient, while thin line corresponds to the coefficient for maximum of PSF.[]{data-label="fig_suppression_scaled"}](suppression_radial_lambda_scaled.png){width="125mm"}
It is important to note that, similarly to the band-limited masks, the quasi band-limited mask designed for a wavelength $\lambda$ will work for $\lambda'<\lambda$ with the same working throughput and better suppression rates. Figure \[fig\_suppression\_scaled\] illustrates the above statement for our example of the circularly symmetric mask designed for $\lambda=750$nm and operating at $\lambda'=\lambda/2=375$nm (compare with Figure \[fig\_suppression\]).
Examples considered in this paper are centered around the particular application: imaging of exo-planets with the help of the solar gravitation lens (SGL). Such imaging requires not only suppressing sunlight, but also reducing glare from the parent star of an exo-planet. The product mask, introduced in the previous section, can perform both of the above tasks.
Another problem related to the SGL imaging is to suppress light from the solar corona: Our mask reduces significantly the glare from almost all the corona, except immediate neighborhood of the Einstein ring. Appendix 2 presents the related results.
There are two important problems related to effectiveness of our coronagraph in realistic conditions: the one of tolerance to manufacturing errors and a tolerance to wavefront imperfections (wavefront control).
![Comparison of the original mask with dicsretized (degraded) mask. Dashed curves show quantities related to the original mask, while solid curves correspond to degraded mask. Left: transparency $t(y)$ across the annular region $y_1<y<y_2$. Center: Zoom of transparency in neighborhood of interior boundary of annulus. Right: Comparison of the degraded energy suppression coefficient with the original one.[]{data-label="fig_mask_degrade"}](discretized_t.png){width="175mm"}
To test the tolerance of mask to manufacturing errors we divided transparency (also called intensity transmission factor) of the circularly symmetric mask $$t(y)=m(y)^2, \quad 0\le t\le 1,$$ into $10^4$ discrete gray-scale levels with spacing $\delta t=10^{-4}$ between the adjacent levels. Such a discretization leads to about one order of magnitude degradation in the level of suppression which is acceptable for our purposes. The results of the corresponding simulations are shown on Figure \[fig\_mask\_degrade\]. It is worthwhile to mention that our numeric computations were performed on a square spatial grid with spacing $\delta y\approx 0.1$, and, therefore $\delta t$ is, in fact, the minimal spacing between levels. The actual spacing at given $y$ is the biggest of the two values $\delta t$ and $|\nabla t(y)\cdot\delta \vec{y}|$. The second of the above values is $\approx 3\times 10^{-3}$ at points where $t(y)$ is steepest.
Concluding the paper we would like to note that the level of suppression in our examples is comparable with the level of suppression by the WFIRST coronagraph instrument [@WFIRST], where the active wavefront control is used. In our examples, the working angle is much bigger than the minimal working angle of the WFIRST coronagraph instrument, so one can assume that the wavefront control would require less sophisticated technology in our case.
Appendix 1
==========
Below we derive (\[K\_reduced\], \[Kernel\_1d\]): From (\[kappa\_x\], \[kappa\_y\]) it follows that $$\hat{K}_\epsilon[m](\vec{y})=\int_{y_1<y'<y_2}K_\epsilon(\vec{y}-\vec{y'})m(\vec{y'})d^2y'=\int_{y_1<y'<y_2}d^2y'\int_{x<\epsilon/2}d^2xe^{2i\pi \vec{x}\cdot(\vec{y}-\vec{y'})}m(\vec{y'}) .$$ After integrating in polar coordinates in the $x$-plane only, one can express $K_\epsilon$ in terms of the Bessel function as in eq. (\[K\_epsilon\]), but instead we rewrite all variables in polar coordinates as $$\vec{x}=(x\cos\theta,x\sin\theta), \quad \vec{y}=(y\cos\varphi,y\sin\varphi), \quad \vec{y'}=(y'\cos\varphi',y'\sin\varphi') .$$ Then, taking into account that $m$ is circularly symmetric, i.e. $m(\vec{y'})=m(y')$, and integrating first in $\phi'$ and then in $\theta$, we get $$\hat{K}_\epsilon[m](y)=\int_{y_1}^{y_2}\left[(2\pi)^2\int_0^{\epsilon/2}J_0(2\pi xy)J_0(2\pi xy')xdx\right]m(y')y'dy' .$$ Taking integral in the square brackets, we obtain (\[K\_reduced\], \[Kernel\_1d\]).
Appendix 2
==========
With great precision eq. (\[psf\]) holds for incidences corresponding to the light from corona $\beta>y_1$. Taking into account (\[brightness\_ps\]) and (\[brightness\]) we obtain surface brightness of the corona’s glare in the detector plane $$I(\vec{y})=\int I_{\rm c}(\beta)m^2(\beta)\mathcal{I}_{\rm L}(\vec{\beta},\vec{y})d^2\beta ,
\label{brightness_corona}$$ where $I_c(\beta)$ is the distribution of the surface brightness in corona and $\mathcal{I}_{\rm L}(\vec{\beta},\vec{y})=\mu_{\rm L}(\vec{\beta},\vec{y})/S $.
![Dependence of effective distribution of the relative surface brightness of corona on $\beta$, i.e. $I_{\rm eff}(\beta)/I_0$.[]{data-label="fig_Ieff"}](corona_Ieff.png){width="115mm"}
From (\[brightness\_corona\]) it follows that effect of mask is equivalent to reducing distribution of the surface brightness of corona by the factor $m(\beta)^2$. In other words, corona is seen in the detector plane as if there were no mask, but instead the surface brightness distribution of the corona were $I_{\rm eff}(\beta)=m(\beta)^2 I_{\rm c}(\beta)$.
According to [@NK], dependence of the distribution $I_c$ on $\beta$ is $$I_c=\left(\frac{3.67}{\rho^{18}}+\frac{1.939}{\rho^{7.8}}+\frac{0.0551}{\rho^{2.5}}\right)\times 10^{-6}I_0,
\label{Ic}$$ where $$\rho=\beta/\beta_{\rm S}>1$$ is the distance from the center of the sun in units of angular radius of the sun. In our example $\beta_{\rm S}=y_1$. Multiplying (\[Ic\]) and $m(\beta)^2$ from our example of the circularly symmetric mask (corresponding $m(\beta)$ is shown on Figure \[fig\_m\]), we get effective distribution of brightness $I_{\rm eff}$ shown on Figure \[fig\_Ieff\]. Action of the mask results in significant effective reduction of the corona brightness almost everywhere, except close neighbourhood of the Einstein ring. (compare Figure \[fig\_Ieff\] with Figure \[fig\_SGL\])
[101]{}
Eshleman, Von R., [*Gravitational lens of the sun: its potential for observations and communications over interstellar distances*]{}, Science, Vol. 205, no. 4411 (1979) pp. 1133-1135.
Marc J. Kuchner and Wesley A. Traub, [*A coronagraph with a band-limited mask for finding terrestrial planets*]{}, The Astrophysical Journal, 570:900–908, 2002 May 10
Geoffrey A. Landis, [*Mission to the Gravitational Focus of the Sun: A Critical Analysis*]{}, 55th AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2017-1679), https://arxiv.org/ftp/arxiv/papers/1604/1604.06351.pdf
Igor Loutsenko, [*On the role of caustics in solar gravitational lens imaging*]{}, Progress of Theoretical and Experimental Physics, Volume 2018, Issue 12, December 2018, 123A02
Claudio Maccone, [*Deep Space Flight and Communications: Exploiting the Sun as a Gravitational Lens*]{}, Springer Science and Business Media, 2009 - 402 pages
November, L. J., and Koutchmy, S., [*White-Light Coronal Dark Threads and Density Fine Structure*]{}, Astrophysical Journal, 466,, July 1996, pp. 512-529.
S.G. Turyshev, B-G Andersson, [*The 550 AU Mission: A Critical Discussion*]{}, Mon. Not. Roy. Astron. Soc. 341 (2003) 577-582
Slava G. Turyshev, Viktor T. Toth, [*Diffraction of electromagnetic waves in the gravitational field of the Sun*]{}, Phys. Rev. D 96, 024008 (2017)
Anand Sivaramakrishnan1, Christopher D. Koresko, Russell B. Makidon, Thomas Berkefeld, Marc J. Kuchner, [*Ground-based Coronagraphy with High-order Adaptive Optics*]{}, The Astrophysical Journal, Volume 552, Number 1
David Slepian, [*Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-IV: Extensions to Many Dimensions Generalized Prolate Spheroidal Functions*]{}, The Bell System Technical Journal, November 1964, pp 3009-3057
David Slepian, [*Analytic Solution of Two Apodization Problems*]{}, Journal of the Optical Society of America, Vol 55, N 19, Septeber 1965, pp 1110-1115
Rachel Akeson, Lee Armus, Etienne Bachelet, Vanessa Bailey, Lisa Bartusek, Andrea Bellini, …Neil Zimmerman, [*The Wide Field Infrared Survey Telescope: 100 Hubbles for the 2020s*]{}, <https://arxiv.org/abs/1902.05569>
[^1]: The distribution of linear brightness along such a circle is essentially a Radon transform of the image of the planetary disc.
[^2]: For an optical system, the surface brightness is the energy flux per unit solid angle divided by the pupil area.
[^3]: We note that the brightness of corona strongly increases towards the sun disc edge, from $\sim 10^{-8}I_0$ at two solar radii from the center of the sun to $\sim 10^{-5}I_0$ at one solar radius (see Figure \[fig\_SGL\]).
[^4]: In what follows we refer to $\vec{\alpha}$ as “incidence angle" and to $\vec{\beta}$ as “incidence".
[^5]: For a point source (i.e. plane wave) of a dimensionless unit intensity $I_{\rm s}(\vec{\xi})=\delta(\vec{\xi}-\vec{\beta})$.
[^6]: By the “band limited coronagraph" we mean a Lyot-type coronagraph with the band limited mask and dimensionless diameter of the stop not exceeding $1-\epsilon$.
[^7]: We recall that in our case $m$, $l$ and $f$ are all real.
[^8]: General formulation of the problem for arbitrary finite supports has been posed earlier, e.g. in [@S1964]. There general properties of $m$ are given, but solution was presented only for the disc-limited case.
[^9]: Diagonalization of our integral operator takes seconds of CPU time on modern PC, while at the time when works of D.Slepian et al [@S] were published such computation power was unavailable.
[^10]: The surface brightness of the edge of the sun disc is less than half of that of the disc center. Since the boundary regions contribute most to the glare, the surface brightness of the glare obtained with account of the limb darkening will be approximately half of that we obtained for the uniform disc.
[^11]: We define flux ratio as the ratio between the energy flow from the parent star and the energy flow per resolution element from an exo-planet.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.